Commun. Math. Phys. 294, 1–19 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0920-3
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Commun. Math. Phys. 294, 1–19 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0920-3

Communications in

Mathematical Physics

Some Remarks about Semiclassical Trace Invariants and Quantum Normal Forms Victor Guillemin1, , Thierry Paul2 1 Department of Mathematics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA. E-mail: [email protected]

2 CNRS and Département de Mathématiques et Applications, École Normale Supérieure,

45, Rue d’Ulm, F-75730 Paris Cedex 05, France. E-mail: [email protected] Received: 23 January 2009 / Accepted: 15 June 2009 Published online: 11 September 2009 – © Springer-Verlag 2009

Abstract: In this paper we explore the connection between semi-classical and quantum Birkhoff canonical forms (BCF) for Schrödinger operators. In particular we give a “non-symbolic” operator theoretic derivation of the quantum Birkhoff canonical form and provide an explicit recipe for expressing the quantum BCF in terms of the semiclassical BCF. 1. Introduction Let X be a compact manifold and H : L 2 (X ) → L 2 (X ) a self-adjoint first order elliptic pseudodifferential operator with leading symbol H (x, ξ ). From the wave trace eit E k , (1.1) E k ∈Spec(H )

one can read off many properties of the “classical dynamical system” associated with H , i.e. the flow generated by the vector field ξH =

∂H ∂ ∂H ∂ − . ∂ξi ∂ xi ∂ xi ∂ξi

(1.2)

For instance it was observed in the ‘70’s’ by Colin de Verdière, Chazarain and Duistermaat-Guillemin that (1.1) determines the period spectrum of (1.2) and the linear Poincaré map about a non-degenerate periodic trajectory, γ , of (1.2) ([2–4]). More recently it was shown by one of us [5] that (1.1) determines the entire Poincaré map about γ , i.e. determines, up to isomorphism, the classical dynamical system associated with H in a formal neighborhood of γ . The proof of this result involved a microlocal Birkhoff canonical form for H in a formal neighborhood of γ and an algorithm First author supported by NSF grant DMS 890771.

2

V. Guillemin, T. Paul

for computing the wave trace invariants associated with γ from the microlocal Birkhoff canonical form. Subsequently a more compact and elegant algorithm for computing these invariants from the Birkhoff canonical form was discovered by Zelditch [11,12] making the computation of these local trace invariants extremely simple and explicit. In this paper we will discuss some semiclassical analogues of these results. In our set-up H can either be the Schrödinger operator on Rn − 2 + V with V → ∞ as x tends to infinity, or more generally a self-adjoint semiclassical elliptic pseudodifferential operator H (x, Dx ) whose symbol, H (x, ξ ), is proper (as a map from T ∗ X into R). Let E be a regular value of H and γ a non-degenerate periodic trajectory of period Tγ lying on the energy surface H = E 1. Consider the Gutzwiller trace (see [6]) E − Ei ψ , (1.3) where ψ is a C ∞ function whose Fourier transform is compactly supported with support in a small neighborhood of Tγ and is identically one in a still smaller neighborhood. As shown in [8,9] (1.3) has an asymptotic expansion ei

Sγ

+σγ

∞

a k k ,

(1.4)

k=0

and we will show below how to compute the terms of this expansion to all orders in terms of a microlocal Birkhoff canonical form for H in a formal neighborhood of γ by means of a Zelditch-type algorithm 2 . If γ is non-degenerate so are all its iterates γ r . Then, for each of these iterates one gets an expansion of (1.3) similar to (1.4), ei

Sγ r

+σγ r

∞

ak,r k ,

(1.5)

k=0 1 For simplicity we will consider periodic trajectories of elliptic type in this paper, however our results are true for non-degenerate periodic of all types, hyperbolic, mixed elliptic hyperbolic, focus-focus, etc. Unfortunately however the Zelditch algorithm depends upon the type of the trajectory and in dimension n there are roughly as many types of trajectories as there are Cartan subalgebras of Sp(2n) (see for instance [1]) i.e. the number of types can be quite large. 2 For elliptic trajectories non-degeneracy means that the numbers

θ1 , . . . , θn , 2π are linearly independent over the rationals, eiθκ , κ = 1, . . . , n being the eigenvalues of the Poincaré map about γ . The results above are true to order O(r ) providing (κ1 θ1 + · · · + κn θ )n + l2π = 0 for all | κ1 | + · · · + | κn |≤ r , i.e. providing there are no resonances of order ≤ r .

Semiclassical Trace Invariants

3

and for these expansions as well the coefficients ak,r can be computed from the microlocal Birkhoff canonical form theorem for H in a formal neighborhood of γ . Conversely one can show Theorem 1.1. The constants ak,r , κ, r = 0, 1, . . . determine the microlocal Birkhoff canonical form for H in a formal neighborhood of γ (and hence, a fortiori, determine the classical Birkhoff canonical form). One of the main goals of this paper will be to give a proof of this result. Our proof, in Sects. 2, 3 and 6 is, with semiclassical modifications, more or less the same as the proof of the Guillemin-Zelditch results [5,11,12] alluded to above. An alternative proof based on Grushun reductions, flux norms and trace formulas for monodromy operators can be found in [7]. Another main goal of this paper is to develop a purely quantum mechanical approach to the theory of Birkhoff canonical forms in which symbolic expansions get replaced by operator theoretic expansions and estimates involving Hermite functions. This can be seen as a “local” version of the Rayleigh-Schrödinger perturbation formalism where no “” parameter is involved. The virtue of this approach is that the dependence of the normal form is an intrinsic part of the theory, and avoids any additional semiclassical computation. This approach is developed in Sect. 4 and the connection of this to the symbolic approach of Sects. 2–3 is described in Sect. 5. To conclude these prefatory remarks we would like to thank Cyrille Heriveuax for his perusal of the first draft of the our manuscript and we would also like to express our gratitude to the referee for his careful line-by-line reading of the manuscript and his many helpful suggestions. 2. The Classical Birkhoff Canonical Form Theorem Let M be a 2n + 2 dimensional symplectic manifold, H a C ∞ function and ξH =

∂H ∂ ∂H ∂ − ∂ξi ∂ xi ∂ xi ∂ξi

(2.1)

the Hamiltonian vector field associated with H . Let E be a regular value of H and γ a non-degenerate elliptic periodic trajectory of ξ H lying on the energy surface, H = E. Without loss of generality one can assume that the period of γ is 2π . In this section we will review the statement (and give a brief sketch of the proof) of the classical Birkhoff canonical form theorem for the pair (H, γ ). Let (x, ξ, t, τ ) be the standard cotangent coordinates on T ∗ (Rn × S 1 ) and let √ pi = xi2 + ξi2 and qi = arg (xi + −1 yi ). (2.2) Theorem 2.1. There exists a symplectomorphism, ϕ, of a neighborhood of γ in M onto a neighborhood of p = τ = 0 such that ϕ o γ (t) = (0, 0, t, 0) and ϕ ∗ H = H1 ( p, τ ) + H2 (x, ξ, t, τ ), H2 vanishing to infinite order at p = τ = 0. We break the proof of this up into the following five steps.

(2.3)

4

V. Guillemin, T. Paul

Step 1. For small there exists a periodic trajectory, γ , on the energy surface, H = E +, which depends smoothly on and is equal to γ for = 0. The union of these trajectories is a 2 dimensional symplectic submanifold , , of M which is invariant under the flow of ξ H . Using the Weinstein tubular neighborhood theorem one can map a neighborhood of γ symplectically onto a neighborhood of p = τ = 0 in T ∗ (Rn × S 1 ) such that gets mapped onto p = 0 and ϕ o γ (t) = (0, 0, t, 0). Thus we can henceforth assume that M = T ∗ (Rn × S 1 ) and is the set, p = 0. Step 2. We can assume without loss of generality that the restriction of H to is a function of τ alone, i.e. H = E + h(τ ) on . With this normalization, H = E + h(τ ) +

θi (τ ) pi + O( p 2 ),

(2.4)

where h(τ ) = τ + O(τ 2 ) and θi = θi (0), i = 1, . . . , n

(2.5)

are the rotation angles associated with γ . Since γ is non- degenerate, θ1 , . . . , θn , 2π are linearly independent over the rationals. Step 3. Theorem 2.1 can be deduced from the following result (which will also be the main ingredient in our proof of the “microlocal” Birkhoff canonical theorem in the next section). Theorem 2.2. Given a neighborhood, U, of p = τ = 0 and G = G(x, ξ, t, τ ) ∈ C ∞ (U), there exist functions F, G 1 , R ∈ C ∞ (U) with the properties i. G 1 = G 1 ( p, τ ), ii. {H, F} = G + G 1 + R, iii. R vanishes to infinite order on p = τ = 0. Moreover, if G vanishes to order κ on p = τ = 0, one can choose F to have this property as well. Proof of the assertion. Theorem 2.2 ⇒ Theorem 2.1. By induction one can assume that H is of the form, H = H0 ( p, τ ) + G(x, ξ, t, τ ), where G vanishes to order κ on p = τ = 0. We will show that H can be conjugated to a Hamiltonian of the same form with G vanishing to order κ + 1 on p = τ = 0. By Theorem 2.2 there exists an F, G and R such that F vanishes to order κ and R to order ∞ on p = τ = 0, G 1 = G 1 ( p, τ ) and {H, F} = G + G 1 + R. Thus 1 {F, {F, H }} + · · · 2! = H0 ( p, τ ) − G 1 ( p, τ ) + · · · ,

(expξ F )∗ H = H + {F, H } +

the “dots” indicating terms which vanish to order κ + 1 on p = τ = 0. Step 4. Theorem 2.2 follows (by induction on κ) from the following slightly weaker result:

Semiclassical Trace Invariants

5

Lemma 2.3. Given a neighborhood, U, of p = τ = 0 and a function, G ∈ C ∞ (U), which vanishes to order κ on p = τ = 0, there exists functions F, G 1 , R ∈ C ∞ (U) such that i. G 1 = G 1 ( p, τ ), ii. {H, F} = G + G 1 + R, iii. F vanishes to order κ and R to order κ + 1 on p = τ = 0. Step 5. Proof of Lemma 2.3. In proving Lemma 2.3 we can replace H by the Hamiltonian θi pi , H0 = E + τ + since H ( p, q, t, τ ) − H0 ( p, q, t, τ ) vanishes to second order in τ, p. Consider now the identity {H0 , F} = G + G 1 ( p, τ ) + O( p ∞ ). √ √ Introducing the complex coordinates, z = x + −1ξ , and z = x − −1ξ , this can be written as n √ ∂F ∂ ∂ F+ −1 θi z i − zi = G + G 1 + O( p ∞ ). ∂z i ∂z i ∂t i=1

Expanding F, G and G 1 in Fourier-Taylor series about z = z = 0: aµ,ν,m (τ )z µ z ν e2πimt , F= µ=ν

G= G1 =

bµ,ν,m (τ )z µ z ν e2πimt , cµ (τ )z µ z µ ,

µ

one can rewrite this as the system of equations n √ −1 θi (µi − νi ) + 2π m aµ,ν,m (τ ) = bµ,ν,m (τ )

(2.6)

i=1

for µ = ν or µ = ν and m = 0, and − cµ (τ ) = bµ,µ,0 (τ )

(2.7)

for µ = ν and m = 0. By assumption the numbers, θ1 , . . . , θn , 2π, are linearly independent over the rationals, so this system has a unique solution. Moreover, for µ and ν fixed, bµ,ν,m (τ )e2πimt is the (µ, ν) Taylor coefficient of G(z, z, t, τ ) about z = z = 0; so, with µ and ν fixed and j >> 0, | bµ,ν,m (τ ) |≤ Cµ,ν, j m − j

6

V. Guillemin, T. Paul

for all m. Hence, by (2.6), − j−1 | aµ,ν,m (τ ) |≤ Cµ,ν, jm

for all m. Thus aµ,ν (t, τ ) =

aµ,ν,m (τ )e2πimt

is a C ∞ function of t and τ . Now let F(z, z, t, τ ) and G 1 ( p, τ ) be C ∞ functions with Taylor expansion:

aµ,ν (t, τ )z µ z ν

µ=ν

and

cµ (τ )z µ z µ

µ

about z = z = 0. Note, by the way, that if G vanishes to order κ on p = τ = 0, so does F and G; so we have proved Theorem 2.2 (and, a fortiori Lemma 2.3) with H replaced by H0 . 3. The Semiclassical Version of the Birkhoff Canonical Form Theorem Let X be an (n + 1)-dimensional manifold and H : C0∞ (X ) → C ∞ (X ) a semiclassical elliptic pseudo-differential operator with leading symbol, H (x, ξ ), and let γ be a periodic trajectory of the bicharacteristic vector field (2.1). As in Sect. 1 we will assume that γ is elliptic and non-degenerate, with rotation numbers (2.4). Let Pi and Dt be the differential operators on Rn × S 1 associated with the symbols (2.2) and τ , i.e. Pi = −2 ∂x2i + xi2 and Dt = −i∂t . We will prove below the following semiclassical version of Theorem 2.1 ∞ Theorem n 3.1.1 There exists a semiclassical Fourier integral operator Aϕ : C0 (X ) → ∞ C R × S implementing the symplectomorphism (2.3) such that microlocally on a neighborhood, U, of p = τ = 0,

A∗ϕ = A−1 ϕ

(3.1)

Aϕ H A−1 ϕ = H (P1 , . . . , Pn , Dt , ) + H ,

(3.2)

and

the symbol of H vanishing to infinite order on p = τ = 0.

Semiclassical Trace Invariants

7

Proof. Let Bϕ be any Fourier integral operator implementing ϕ and having the property (3.1). Then, by Theorem 2.1, the leading symbol of Bϕ H Bϕ−1 is of the form H0 ( p, τ ) + H0 ( p, q, t, τ ),

(3.3)

H0 ( p, q, t, τ ) being a function which vanishes to infinite order on p = τ = 0. Thus the symbol,H0 , of Bϕ H Bϕ−1 is of the form H0 ( p, τ ) + H0 ( p, q, t, τ ) + H1 ( p, q, t, τ ) + O(2 ).

(3.4)

By Theorem 2.2 there exists a function, F( p, q, t, τ ), with the property {H0 , F} = H1 ( p, q, t, τ ) + H1 ( p, τ ) + H1 ( p, q, t, τ ),

(3.5)

where H1 vanishes to infinite order on p = τ = 0. Let Q be a self-adjoint pseudo-differential operator with leading symbol F and consider the unitary pseudo-differential operator Us = eis Q . Let −1 Hs = Us Bϕ H Us Bϕ = Us Bϕ H Bϕ−1 U−s . Then ∂ Hs = i[Q, Hs ], ∂s

(3.6)

∂ so ∂s Hs is of order −1, and hence the leading symbol of Hs is independent of s. In par∂ Hs is equal, by (3.6) to the leading symbol of i[Q, Hs ] ticular the leading symbol of ∂s which, by (3.5), is: − H1 ( p, q, t, τ ) + H1 ( p, τ ) + H1 ( p, q, t, τ ) .

Thus by (3.4) and (3.5) the symbol of −1 U1 Bϕ H U1 Bϕ = Bϕ H Bϕ−1 +

1 0

∂ Hs ds ∂s

is of the form H0 ( p, τ ) + H1 ( p, τ ) + H0 + H1 + O(2 ),

(3.7)

the term in parentheses being a term which vanishes to infinite order on p = τ = 0. By repeating the argument one can successively replace the terms of order 2 , . . . , r , etc. in (3.7) by expressions of the form r Hr ( p, τ ) + Hr ( p, q, t, τ ) with Hr vanishing to infinite order on p = τ = 0.

8

V. Guillemin, T. Paul

4. A Direct Construction of the Quantum Birkhoff Form In this section we present a “quantum” construction of the quantum Birkhoff normal form which is in a sense algebraically equivalent to the classical one of Sect. 2. To do this we will need to define for operators the equivalent of “a Taylor expansion which vanishes at a given order”. We will first start in the L 2 (Rn × S 1 ) setting, and show at the end of the section the link with Theorem 3.1. Definition 4.1. Let us consider on L 2 (Rn × S 1 , d xdt) the following operators: • ai− = • ai+ = • Dt =

√1 (x i + ∂x ) i 2 √1 (x i − ∂x ) i 2 ∂ −i ∂t

We will say that an operator A on L 2 (Rn × S 1 ) is an “ordered polynomial of order greater than p ∈ N” (OPOG( p)) if there exists P ∈ N such that: i

A=

[2] P

j

αi j (t, )Dt

i= p j=0

i−2 j

bl

(4.1)

l=1

with, ∀l, bl ∈ {a1− , a1+ , . . . , an− , an+ } and αi j ∈ C ∞ (S1 × [0, 1[). i−2 j In (4.1) bl is meant to be the ordered product b1 . . . bi−2 j . l=1

The meaning of this definition is clarified by the following: basis of L 2 (Rn × S 1 ) defined by Hµ (x, t) = −n/4 Lemma√ 4.2. Let Hµ denote √ the iµ t , where the h are the (normalized) Hermite funcn+1 h µ1 (x1 / ) . . . h µn (xn / )e j

tions. Let us define moreover |µ| := µ2 2 . Let A be an OPOG(p). Then: ∀M < +∞ (microlocal “cut-off”), ∃ C = C(A, M) such that, p

||AHµ || L 2 ≤ C|µ| 2 , ∀µ ∈ Nn+1 s.t |µ| ≤ M. Proof. The proof follows immediately from the two well known facts (expressed here in one dimension): a ± Hµ =

(µ ± 1)Hµ±1

and Dt eimt = meimt .

For the rest of this section we will need the following collection of results.

Semiclassical Trace Invariants

9

Proposition 4.3. Let A be a (Weyl) pseudodifferential operator on L 2 (Rn × S 1 ) with symbol of type S1,0 . Then, ∀L ∈ N and ∀M < +∞, there exists an OPOG(1) A L and a constant C = C(A, L , M) such that, ||(A − A L )Hµ || L 2 ≤ C|µ|

L+1 2

), ∀µ ∈ Nn+1 s.t. |µ| ≤ M.

Moreover, if the principal symbol of A is of the form: a0 (x1 , ξ1 , . . . , xn , ξn , t, τ ) = θi (xi2 + ξi2 ) + τ + h.o.t., (or is any function whose symbol vanishes to first order at x = ξ = τ = 0) then A L is an OPOG(2). Proof. Let us take the L th order Taylor expansion of the (total) symbol of A in the variables x, ξ, τ, near the origin. Noticing that a pseudodifferential operator with polynomial symbol in x, ξ, τ, is an OPOG, we just have to estimate the action, on Hµ , of a pseudo-differential operator whose symbol vanishes at the origin to order L in the variable x, ξ, τ . The result is easily obtained for the τ part, as the “t” part of Hµ is an exponential. For the µ part we will prove this result in one dimension, the extension to n dimensions being straightforward. Let us define a coherent state at (q, p) to be a function of the form ψqap (x) := px √ −1/4 a x−q ei , for a in the Schwartz class and ||a|| L 2 = 1. Let us also set ϕq p = ψqap

for a(η) = π −1/2 e−η /2 . It is well known, and easy to check using the generating function of the Hermite polynomials, that

1 t Hµ = − 4 e−i 2 ϕq(t) p(t) dt, 2

S1

where q(t) + i p(t) = eit (q + i p) , q 2 + p 2 = (µ + 21 ). Therefore, for any operator A, ||AHµ || = O(

sup p 2 +q 2 =(µ+ 12 )

1

− 4 ||Aϕq p ||).

(4.2)

Lemma 4.4. let H a pseudodifferential operator whose (total) Weyl symbol vanishes at the origin to order M. Then, if q 2 +p2 = O(1): M ||H ψqap || = O ( p 2 + q 2 ) 2 . Before proving the lemma we observe that the proof of the proposition follows easily from the lemma using (4.2). Proof. An easy computation shows that, if h is the (pseudodifferential) symbol of H , then H ψqap = ψqbp with

√ √ b(η) = h(q + η, p + ν)eiην a(ν)dν, ˆ (4.3) R

where aˆ is the ( independent) Fourier transform of a.

10

V. Guillemin, T. Paul

k k+1 bk 2 2 ), where Developing (4.3) we get that H ψqap = k=K k=M Dk h(q, p)ψq p + O( bk ∈ S and Dk is an homogeneous differential operator of order k. It is easy to conclude, thanks to the hypothesis q 2 +p2 = O(1), that k

2 (q 2 + p 2 )

M−k 2

M

= O((q 2 + p 2 ) 2 ),

M+1 2

M

= O((q 2 + p 2 ) 2 ).

This proposition is crucial for the rest of this section, as it allows us to reduce all computations to the polynomial setting. For example A may have a symbol bounded at infinity (class S(1), an assumption which we will need for the application below of Egorov’s Theorem in the proof of Theorem 4.9), but, with respect to the algebraic equations we will have to solve, one can consider it as a “OPOG” (see Theorem 4.9 below). Lemma 4.5. Let A be a OPOG(1) on L 2 (Rn × S 1 ). Let us suppose that A is a symmetric operator. For P ∈ N (large), let A P := A + (|Dθ |2 + |x|2 + |Dx |2 ) P .

(4.4)

Then A P is an elliptic selfadjoint pseudo-differential operator. Therefore eis family of unitary Fourier integral operators.

AP

is a

Proof. It is enough to observe that A P is, defined on the domain of |Dθ |2 + |x|2 + |Dx |2 , a selfadjoint pseudodifferential operator with symbol of type S1,0 .

Lemma 4.6. Let H0 be the operator H0 =

n

θi ai− ai+ + Dt ,

1

then, if W is an OPOG(r ), so is

[H0 ,W ] i .

d is H0 / e W eis H0 /|s=0 which, since H0 is quadratic, is the same polyProof. [H0i ,W ] = ds nomial as W modulo the substitution ai− → eis ai− , ai+ → e−is ai and shifting of the coefficients in t by s. Therefore the result is immediate.

More generally: Lemma 4.7. For any H and W of type OPOG(m) and OPOG(r ) respectively, an OPOG(m + r − 2).

[H,W ] i

is

The proof is immediate noting that [ai− , a +j ] = δi j and that, for any C ∞ function a(t), [Dt , a] = ia . We can now state the main result of this section: Theorem 4.8. Let H be a (Weyl) pseudo-differential operator on L 2 (Rn × S 1 ) whose principal symbol is of the form: H0 (x, ξ ; t, τ ) =

n 1

θi (xi2 + ξi2 ) + τ + H2 ,

Semiclassical Trace Invariants

11

where H2 vanishes to third order at x = ξ = τ = 0 and θ1 , . . . , θn , 2π are line2 arly independent over the rationals. Let us define, as before, Pi = −2 ∂∂x 2 + xi2 and i

Dt = −i ∂t∂ . Then, ∀M < +∞, there exists a family of unitary operators (U L ) L=3... and constants (C L ) L=3... , and a C ∞ function h( pi , . . . , pn , τ, ) such that: L+1 || U L HU L−1 −h(P1 , . . . , Pn , Dt , ) Hµ || L 2 (Rn ×S 1 ) ≤ C L |µ| 2 ∀µ ∈ Nn+1 s.t |µ| ≤ M.

Proof. The proof of Theorem 4.8 will be a consequence of the following: Theorem 4.9. Let H be as before, and let G be an OPOG(3). Then there exists a function G 1 ( p1 , . . . , pn , τ, ), an OPOG F and an operator R such that: i.

[H,F] i

= G + G 1 + R, L+1

ii. R satisfies: ||R Hµ || = O(|µ| 2 ), ∀µ ∈ Nn+1 , |µ| = O(1) and ∀L ∈ N, iii. if G is an OPOG(κ) so is F, iv. if G is a symmetric operator, so is F and G 1 is real. Let us first prove that Theorem 4.9 implies Theorem 4.8: By induction, as in the “classical” case and thanks to Proposition 4.3, one can assume that H is of the form H = H0 + G, where G is an OPOG(κ). Let us consider the operators ei FP

FP

H e−i

FP

and

FP

H (s) := eis H e−is , where F satisfies Theorem 4.9 and FP is defined by (4.4) for P large enough. Since we are in an iterative perturbative setting, it is easy to check by taking P large FP

F

enough that we can omit the subscript P in H (s) and let e±i stand for e±i in the rest of the computation. We have: [F,H ] F F [F, H ] [F, i ] [F, [F, [F, 10 t0 s0 H (u)dudsdt]/i]/i] ei H e−i = H + + + i i i [F,H ] [F, H ] [F, i ] ˜ + +R = H0 + G + i i ] [F, [F,H i ] ˜ = H0 − G 1 + R + + R. (4.5) i Since we are interested in letting all the operators acting on the Hµ for |µ| = O(1), we can microlocalize near x = ξ = τ = 0 and replace F and H by their microlocalF˜ ized versions F˜ and H˜ . ei is a Fourier integral operator and, byEgorov’s Theorem, H˜ (s) is a family of pseudodifferential operators, and so is 10 t0 s0 H˜ (u)dudsdt. By Proposition 4.3, Lemma 4.7 and Lemma 4.2 we have, since G is an OPOG(κ), || R˜ Hµ || = O(|µ|κ+1 ). [F,H ]

[F, i ] By the same argument, satisfies the same estimate. Developing R˜ by the i Lagrange formula (4.5) to arbitrary order, we get, thanks to Lemma 4.7, R˜ = G˜ + R, where G˜ is an OPOG(κ + 1) and

||R Hµ || = O(|µ|

L+1 2

).

12

Therefore, letting G =

V. Guillemin, T. Paul ] [F, [F,H i ] i

ei

FP

˜ we have: + G,

H e−i

FP

= H0 + G 1 + G + R,

with G an OPOG(κ + 1). By induction Theorem 4.8 follows. Proof of Theorem 4.9. Let us first prove the following Lemma 4.10. Let H0 be as before and let G be an OPOG(r ). Then there exists a OPOG(r ) F and G 1 = G 1 ( p1 , . . . , pn , Dt , ), such that [H0 , F] = G + G1. i

(4.6)

Proof. By Lemma 4.6, if F is an OPOG, it must be an OPOG(r ), since the left-hand side of (4.6) is an OPOG(r ). Let us take the matrix elements of (4.6) relating to the Hµ s. We get: −i.(µ − ν) < µ|F|ν >=< µ|G + G 1 |ν > + < µ|R|ν >, where .(µ − ν) := n1 θi µi + µn+1 and < µ|.|ν >= Hµ , .Hν . We get immediately that G 1 (µ, ) = − < µ|G|µ >. Moreover, let us define F by: < µ|F|ν >:=

< µ|G + G 1 |ν > , −i.(µ − ν)

which exists by the non-resonance condition. To show that F is an OPOG one just j has to decompose G = G l in monomial OPOGs G l = α(t)Dt b1 . . . bm , bi ∈ + + {a1 , a1 , . . . , an , an }. Then, for each ν there is only one µ for which < µ|G +G 1 |ν >= 0 and the difference µ − ν depends obviously only on G l , not on ν. Let us call this difference ρG l . Then F is given by the sum: F=

1 Gl . −i.ρG l

It is easy to check that one can pass from Lemma 4.10 to Theorem 4.9 by induction, writing [H, F + F ] = [H, F] + [H0 , F ] + [H − H0 , F] + [H − H0 , F ].

We will show finally that Theorems 4.8 and 3.1 are equivalent. Once again we can start by considering an Hamiltonian on L 2 (Rn × S 1 ) since any Fourier integral operator Bϕ , as defined in the beginning of the proof of Theorem 3.1, intertwines the original Hamiltonian H : C0∞ (X ) → C ∞ (X ) of Sect. 3 with a pseudodifferential operator on L 2 (Rn × S 1 ) satisfying the hypothesis of Theorem 4.8. W3

W4

Let us remark first of all that if U L = ei ei . . . ei integral operators, then so is U L . Secondly we have

WL

, all ei

Wl

being Fourier

Proposition 4.11. Let A be a pseudodifferential operator of total Weyl symbol a(x, ξ, t, τ, ). Then a vanishes to infinite order at p = τ = 0 if and only if ||AHµ || L 2 (Rn ×S 1 ) = O(|µ|∞ ).

Semiclassical Trace Invariants

13

Proof. The “if” part is exactly Proposition 4.3. For the “only if” part let us observe that, if the total symbol didn’t vanish to infinite order, then it would contain terms of the form αkmnr (t)k (x + iξ )m (x − iξ )n τ r . Let us prove this can’t happen in dimension 1, the extension to dimension n being straightforward. Each term of the form (x + i Dx )m (x − i Dx )n = a m (a + )n gives rise to an operator Am,n such that: Am,n Hµ =

|m+n| 2

∼ |µ| Therefore

(µ + 1) . . . (µ + n)(µ + n − 1) . . . (µ + n − m)Hµ+m−n

m+n 2

Hµ+m−n .

cmn Am,n Hµ = ||

cm,m−l Hµ+l ∼

cmn Am,n Hµ ||2 ∼

cm,m−l |µ|

2m−l 2

Hµ+l . In particular:

|µ|2m−l ,

so || cmn Amn Hµ || = O(|µ|∞ ) implies Cmn = 0. It is easy to check that the same argument is also valid for any ordered product of a’s and a + ’s.

In the next section we will show how the functions H of Theorem 4.8 and h of Theorem 3.1 are related. 5. Link Between the Two Quantum Constructions Consider a symbol (on R2n ) of the form h( p1 , . . . , pn ) ξ 2 +x 2

with pi = i 2 i . There are several ways of quantizing h: one of them consists in associating to h, by the spectral theorem, the operator h(P1 , . . . , Pn ) = h(P), −2 ∂x2 +x 2

i i where Pi = . Another one is the Weyl quantization procedure. 2 In this section we want to compute the Weyl symbol h we of h(P1 , . . . , Pn ) and apply the result to the situation of the preceding sections. By the metaplectic invariance of the Weyl quantization and the fact that h(P1 , . . . , Pn ) commutes with all the Pi ’s, we know that h we has the form

h we ( p1 , . . . , pn ) = h we ( p), that is, is a function of the classical harmonic oscillators pi := ξi2 + xi2 . To see how this h we is related to the h above we note that H is diagonal on the Hermite basis h j . Therefore

x+y xξ d xdξ 1 h(( j + )) =< h j , H h j >= h we ( )2 + ξ 2 ei h j (x)h j (ξ ) n/2 . 2 2 We now claim

14

V. Guillemin, T. Paul

Proposition 5.1. Let h be either in the Schwartz class, or a polynomial function. Let ˆ h(s) = (2π1 )n h( p)e−is. p dp be the Fourier transform of h. Then

2i tan(s/2). p we ˆ h ( p) = h(s)e (s)ds, (5.1) n where tan(s/2). p stands for i tan(si /2) pi and (s) = i=1 (1 − 2i tan(si /2)), and where (5.1) has to be interpreted in the sense of distribution, that is, for each ϕ in the Schwartz class of R,

2i tan(s/2). p ˆ h we ( p)ϕ( p)dp = h(s)e (s)dsϕ( p)dp

2i tan(s/2) ˆ = h(s)(s)ϕˆ ds. Finally, as → 0, h we ∼ h +

∞

cl 2l .

(5.2)

l=1

is.P ds, where eis.P is a zeroth order semiclassical pseudoˆ Proof. Let h(P) = h(s)e differential operator whose Weyl symbol will be computed from its Wick symbol (see n eisi .Pi , it is 5.5 below for the definition). Let us first remark that since eis.P = i=1 enough to prove the theorem in the one-dimensional case. Let ϕxξ be a coherent state at (x, ξ ), that is ξy

1

ϕxξ (y) = (π )− 4 ei e− Let z =

ξ√ +i x , z 2

=

ξ √ +i x 2

and z(t) =

ξ(t)+i √ x(t) . 2

(y−x)2 2

.

A straightforward computation gives

2zz −|z|2 −|z |2 2 ϕxξ , ϕx ξ = e .

(5.3)

Moreover decomposing ϕxξ on the Hermite basis leads to eis P ϕxξ = ei 2 ϕx(s)ξ(s) , s

(5.4)

−2 ∂x2 +x 2

where P = and z(t) = eit z. 2 The Wick symbol of eis P is defined as

σ wi (eis P )(x, ξ ) := ϕxξ , eis P ϕxξ

(5.5)

which, by (5.3) and (5.4), is equal to −is

e

− 1−e

x 2 +ξ 2 2

+i 2s

.

Moreover, using the Weyl quantization formula, it is immediate to see that the Weyl and Wick symbols are related by

where = −

∂2 ∂x2

∂2 + ∂ξ 2 .

σ wi = e−

4

σ we ,

Semiclassical Trace Invariants

15

It is a standard fact that the Wick symbol determines the operator: indeed the function

−2zz +|z|2 +|z |2 2

e (ϕxξ , eis P ϕx ξ ) obviously determines eis P . Moreover it is easily seen to be analytic in z and z . Therefore it is determined by its values on the diagonal z = z i.e., precisely, the Wick symbol of eis P . A straightforward calculation shows that, for (2k+1)π s , k ∈ Z, 2 2 = (1 − 2i tan(s/2))e

(2k+1)π , 2

This shows that, for 2s = σ

we

(e

− 4

e

2i tan(s/2)

x 2 +ξ 2 2

−is

=e

x 2 +ξ 2 2

+i 2s

.

(5.6)

k ∈ Z, we have

)( p) = (1 − 2i tan(s/2))e

is P

− 1−e

2i tan(s/2)

x 2 +ξ 2 2

.

Let us now take ϕ in the Schwartz class of R, and let Bϕ be the operator of (total) Weyl symbol ϕ( x

2 +ξ 2

2

). Let

f (s) := 2π

σ we (eis P )( p)ϕ( p) pdp = Trace[eis P Bϕ ].

Lemma 5.2. f ∈ C ∞ (R). Proof. By metaplectic invariance we know that Bϕ is diagonal on the Hermite basis. Therefore, ∀k ∈ N, (−i)k

1 dk 1 f (s) := Trace[eis P P k Bϕ ] = < h j , Bϕ h j > (( j + ))k eis( j+ 2 ). k ds 2

Since h j is microlocalized on the circle of radius ( j + 21 ) and ϕ is in the Schwartz class, the sum is absolutely convergent for each k.

2i tan(s/2)

x 2 +ξ 2 2

Therefore f (s) = 2π (1 − 2i tan(s/2))e ϕ( p) pdp and (5.6) is valid in the sense of distribution (in the variable p) for all s ∈ R. This expression gives (5.1) immediately for h in the Schwartz class. When h is a polynomial function it is straightforward to check that, since hˆ is a sum of derivatives of the Dirac mass and eis P is a Weyl operator whose symbol is C ∞ with respect to s, the formula also holds in this case. The asymptotic expansion (5.2) is obtained by expanding e

2itg(s/2)

x 2 +ξ 2 2

near eis

x 2 +ξ 2 2

.

Formula (5.1) shows clearly that h we depends only on the 2π periodization of ˆh(s)ei s2 , therefore Corollary 5.3. h we depends only on the values h (k + 21 ) , k ∈ N.

16

V. Guillemin, T. Paul

We mention one application of formula (5.1). Let us suppose first that we have computed the quantum normal form at order K , that is h K ( p) = ck p k := ck p1k1 . . . pnkn , |k|=k1 +···kn ≤K

|k|=k1 +···kn ≤K

and let us define h we K as the Weyl symbol of h K (P). Corollary 5.4. h we K ( p) =

ck

|k|=k1 +···kn ≤K

:=

|k|=k1 +···kn ≤K

ck

2i tan(s/2) p ∂K (s)e |s=0 ∂sk ∂K ∂sk11 . . . ∂ kn sn

(s)e2i

tan(s1 /2) p1 +···+tan(sn /2) pn

|s=0 .

Let us come back now to the comparison between the two constructions of Sects. 2 and 3. Clearly the “θ ” part doesn’t play any role, as the Weyl quantization of any function f (τ ) is exactly f (Dθ ). Therefore we have the following Theorem 5.5. The functions H of Theorem 3.1 and h of Theorem 4.8 are related by the formula

2i tan(s/2). p ˆ Dt , )e H (P1 , . . . , Pn , Dt , ) = h(s, (s)ds, where hˆ is the Fourier transform of h with respect to the variables pi . In particular H − h = O(2 ). Proof. The proof follows immediately from Proposition 5.1, and the unicity of the (quantum) Birkhoff normal form.

6. The Computation of the Semiclassical Birkhoff Canonical Form from the Asymptotics of the Trace Formula In this section we will abandon the quantum approach to Birkhoff canonical forms developed in Sects. 4–5 and revert to the symbolic approach of Sects. 2–3. Using this approach we will prove that the wave trace data coming from the Gutzwiller formula determine the Quantum Birkhoff canonical form constructed in Sect. 3. Our goal will be by “mimicking” (with semiclassical modifications) the proof of this result by Zelditch in [11,12] and in particular avoid the method of “Grushin reduction” used in [7] to equate the trace formula of [6,8,9] with the trace formula for a monodromy operator. Warning. The Aϕ in display 6.3 below is not the family of U L ’s figuring in Theorem 4.8 but is the “symbolic” Aϕ figuring in Theorem 3.1. In particular the estimates in Theorems 4.8 and 4.9 will not play any role in this proof. Let X and H be as in the Introduction. Let γ be a periodic trajectory of the vector field (2.1) of period 2π .

Semiclassical Trace Invariants

17

For l ∈ Z let ψl be a Schwartz function on the real line whose Fourier transform ψˆl is supported in a neighborhood of 2πl containing no other period The semiclassical of (2.1). of the form: trace formula gives an asymptotic expansion for Trace ψl H −E

Trace ψl

H−E

∼

∞

dlm m ,

(6.1)

m=0

where the dl ’s are distributions acting on ψˆl with support concentrated at {2πl}. We will show that the knowledge of the dl s determine the quantum semiclassical Birkhoff form of Sect. 2, and therefore the classical one. Let us first rewrite the l.h.s of (6.1) as

it H −E ˆ Trace ϕψ(t)e dt . (6.2) Since ψˆ is supported near a single period of (2.1) we know from the general theory of Fourier integral operators that one can microlocalize (6.1) near γ . Therefore we can conjugate (6.2) by the semiclassical Fourier integral operator Aϕ of Theorem 3.1. This leads to the computation of

it H −E −1 ˆ Trace Aϕ ψ(t)e dt Aϕ

H (P1 ,...,Pn ,Dt ,)+H −E it ˆ = Tr ψ(t)ρ(P1 , . . . , Pn , Dt )e dt , (6.3) where ρ ∈ C0∞ (Rn+1 ) with ρ = 1 in a neighborhood of p = τ = 0 and Tr stands for the Trace in L 2 (Rn × S 1 ). Let us note that, as is standard in the proof on trace formulas, by the independence condition of the Poincaré angles (see footnote (2)), γ is isolated on its energy shell {H = E}. By standard stationary phase techniques this is enough to show that the contribution of H in (6.3) is of order O(∞ ). Let us write H (P1 , . . . , Pn , Dt , ) as E + Dt + θi Pi + cr,s ()P r Dts . (6.4) r ∈Nn ,s∈Z

We will first prove l (t, θ ) be the function defined by Proposition 6.1. Let gr,s it θ1 +···+θn 2 e ∂ s ∂ r l ˆ gr,s (t, θ ) = −i −i t ψ(t) . t∂θ ∂t i (1 − eitθi )

(6.5)

Let us fix l ∈ Z. Then the knowledge of all the dlm s for m < M in (6.1) determines the following quantities: l cr,s ()gr,s (2πl, θ ) (6.6) |r |+s=m

for all m < M.

18

V. Guillemin, T. Paul

Proof. The r.h.s. of (6.3) can be computed thanks to (6.4) using 1 spectrum Pi = {(µi + ), µi ∈ N}, 2 spectrum Dt = {ν, n ∈ Z}. Thus the r.h.s of (6.3) can be written as

∞ k 1 it ν+θ.(µ+ 12 ) (it) ψˆl (t) ρ (µ + ), ν e 2 k! µ,ν k=0 k 1 r s |µ|+s−1 × cr,s () µ + ν dt, 2 r,s

(6.7)

since the support of ψˆ l contains only one period, and therefore the trace can be microlocalized infinitely close to the periodic trajectory, making the role of H inessential. Using the following remark of S. Zelditch:

1 µ+ 2

r

r

∂ ν = −i t∂θ s

∂ −i ∂t

s e

it ν+θ.(µ+ 12 )

we get, mod(∞ ),

ψˆl (t)

∞ (it)k k=0

k!

r,s

r s k ∂ ∂ it ν+θ.(µ+ 12 ) |µ|+s−1 −i cr,s () −i e dt. t∂θ ∂t µ,ν

(6.8) Since

ν∈Z

eitν

= 2π

l δ(t − 2πl) , and

µ∈Nn

e

itθ. µ+ 21

=

θ1 +···+θn

eit 2 , i 1−eitθi

together

with the fact that ψˆ is supported near 2πl, we get that (6.8) is equal to ⎡ 2π ⎣

∞ (i)k k=0

k!

×

t k ψˆl (t)

|µ|+s−1

r,s θ1 +···+θn

∂ cr,s () −i t∂θ

eit 2 i 1 − eitθi

.

r k ∂ s −i ∂t (6.9)

t=2πl

Rearranging terms in increasing powers of shows that the quantities (6.6) can be computed recursively.

The fact that one can compute the cr,s () from the quantities (6.6) is an easy consequence of the rational independence of the θi s and the Kronecker theorem, and is exactly the same as in [5].

Semiclassical Trace Invariants

19

References 1. Bridges, T.J., Cushman, R.H., MacKay, R.S.: Dynamics near an irrational collision of eigenvalues for symplectic mappings. Lamgford, W.F. (ed.) In: Normal Forms and Homoclinic Chaos. Fields Inst. Commun. 4, Providence, RI: Amer. Math. Soc., 1995, pp. 61–79 2. Chazarain, J.: Formule de Poisson pour les variétés Riemanniennes. Invent. Math. 24, 65–82 (1974) 3. Colin de Verdière, Y.: Spectre du Laplacien et longueurs des géodésiques périodiques. Compos. Math. 27, 83–106 (1973) 4. Duistermaat, J.J., Guillemin, V.: The spectrum of positive elliptic operators and periodic bicharacteristics. Inv. Math. 29, 39–79 (1975) 5. Guillemin, V.: Wave-trace invariants. Duke Math. J. 83, 287–352 (1976) 6. Gutzwiller, M.: Periodic orbits and classical quantization conditions. J. Math. Phys. 12, 343–358 (1971) 7. Iantchenko, A., Sjöstrand, J., Zworski, M.: Birkhoff normal forms in semi-classical inverse problems. Math. Res. Lett. 9, 337–362 (2002) 8. Paul, T., Uribe, A.: Sur la formule semi-classique des traces. C.R. Acad. Sci Paris 313, I, 217–222 (1991) 9. Paul, T., Uribe, A.: The semi-classical trace formula and propagation of wave packets. J. Funct. Anal. 132, 192–249 (1995) 10. Robert, D.: Autour de l’approximation semi-classique. Basel-Boston: Birkhäuser, 1987 11. Zelditch, S.: Wave invariants at elliptic closed geodesics. Geom. Funct. Anal. 7, 145–213 (1997) 12. Zelditch, S.: Wave invariants for non-degenerate closed geodesics. Geom. Funct. Anal. 8, 179–217 (1998) Communicated by B. Simon

Commun. Math. Phys. 294, 21–60 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0899-9

Communications in

Mathematical Physics

On Dilation Symmetries Arising from Scaling Limits Henning Bostelmann1,2, , Claudio D’Antoni2, , Gerardo Morsella2, 1 University of York, Department of Mathematics, Heslington, York YO10 5DD, United Kingdom.

E-mail: [email protected]

2 Università di Roma “Tor Vergata”, Dipartimento di Matematica, Via della Ricerca Scientifica,

I-00133 Roma, Italy. E-mail: [email protected], [email protected] Received: 24 January 2009 / Accepted: 18 March 2009 Published online: 30 August 2009 – © Springer-Verlag 2009

Abstract: Quantum field theories, at short scales, can be approximated by a scaling limit theory. In this approximation, an additional symmetry is gained, namely dilation covariance. To understand the structure of this dilation symmetry, we investigate it in a nonperturbative, model independent context. To that end, it turns out to be necessary to consider non-pure vacuum states in the limit. These can be decomposed into an integral of pure states; we investigate how the symmetries and observables of the theory behave under this decomposition. In particular, we consider several natural conditions of increasing strength that yield restrictions on the decomposed dilation symmetry. 1. Introduction In the analysis of quantum field theories, the notion of scaling limits plays an important role. The physical picture underlying this mathematical concept is as follows: One considers measurements in smaller and smaller space-time regions, at the same time increasing the energy content of the states involved, so that the characteristic action scale remains constant. Passing to the limit of infinitesimal scales, one obtains a new quantum field theory, the scaling limit of the original model. The scaling limit theory can be seen as an approximation of the full theory in the short-distance regime. However, it may differ significantly from the full theory in fundamental aspects, for example regarding its charge structure: In quantum chromodynamics, it is expected that confined charges (color) appear in the limit theory, but are not visible as such in the full theory. The virtues of the scaling limit theory include that it is typically simpler than the original one. In fact, in relevant examples, one expects it to be interactionless (asymptotic freedom). But even where this is not the case, the limit theory should possess Supported in part by the EU network “Noncommutative Geometry” (MRTN-CT-2006-0031962) Supported in part by PRIN-MIUR and GNAMPA-INDAM Supported in part by PRIN-MIUR, GNAMPA-INDAM and the Scuola Normale Superiore

22

H. Bostelmann, C. D’Antoni, G. Morsella

an additional symmetry: It should be dilation covariant, since any finite masses in the original model can be neglected in the limit of large energies. On the mathematical side, a very natural description of scaling limits has been given by Buchholz and Verch [BV95]. This description, formulated in the C∗ algebraic framework of local quantum physics [Haa96], originates directly from the physical notions, and avoids any additional input motivated merely on the technical side, such as a rescaling of coupling constants or mass parameters, or the choice of renormalization factors for quantum fields. This has the advantage of allowing an intrinsic, model-independent description of the short distance properties of the theory at hand. In particular, it has been successfully applied to the analysis of the charge structure of the theory in the scaling limit and to the intrinsic characterization of charge confinement [Buc96b,DMV04]. While the framework of Buchholz and Verch seems rather abstract at first, it has recently been shown that it reproduces the usual picture of multiplicative field renormalization in typical cases [BDM09]. The approach of [BV95] is based on the notion of the scaling algebra A, which consists—roughly speaking—of sequences of observables λ → Aλ at varying scale λ, uniformly bounded in norm, and subject to certain continuity conditions. (We shall recall the precise definition in Sec. 2.1.) The task of passing to the scaling limit is then reduced to finding a suitable state ω0 on the C∗ algebra A that represents the vacuum of the limit theory; it is constructed as a limit of vacuum states at finite scales. The limit theory itself is then obtained by a standard GNS construction with respect to ω0 . It should be easy in this context to describe the additional dilation symmetry that arises in the scaling limit. In fact, the scaling algebra A carries a very natural representation µ → δ µ of the dilation group, which acts by shifting the argument of the functions λ → Aλ : δ µ (A)λ = Aµλ . However, things turn out to be more involved: The limit states ω0 described in [BV95] are not invariant under this group action, and thus one does not obtain a canonical group representation in the limit Hilbert space. In [BDM09], generalized limit states have been introduced, some of which are invariant under dilations, and give rise to a unitary implementation of the dilation group in the limit theory. However, these dilation invariant limit states are never pure; rather they arise as a mixture of states of the Buchholz-Verch type, which are pure in 2+1 or more space-time dimensions. The object of the present paper is to analyze this generalized class of limit states in more detail, in order to describe the structure of the dilation symmetry associated with the dilation invariant ones. In particular we will show that, as briefly mentioned in [BDM09], the decomposition of these states in pure (Buchholz-Verch type) states gives rise to a direct integral decomposition of the limit Hilbert space, which also induces a decomposition of observables and of Poincaré symmetries. It should be noted here that the entire construction is complicated by the fact that uncountably many extremal states are involved in this decomposition, and that the measure space underlying the direct integral is of a very general nature. Because of this, we need to use a notion of direct integral of Hilbert spaces which is more general than the one previously employed in the quantum field theory literature [DS85]. It is also of interest to discuss how the special but physically important class of theories with a unique scaling limit, as defined in [BV95], fits into our generalized framework. It turns out that, up to some technical conditions, uniqueness of the scaling limit in the Buchholz-Verch framework is equivalent to the factorization of our generalized scaling limit into a tensor product of an irreducible scaling limit theory and a commutative part, which is just the image under the scaling limit representation of the center of the

On Dilation Symmetries Arising from Scaling Limits

23

scaling algebra. In particular we show that such factorization holds for a restricted class of theories, those with a convergent scaling limit. This class includes in particular dilation invariant theories and free field models. The technical conditions referred to above consist in a suitable separability requirement of the scaling limit Hilbert space, which is needed in order to be able to employ the full power of direct integrals theory. As a matter of fact, such separability condition is a consequence of a refined version of the Haag-Swieca compactness condition. With these results at hand, it is possible to discuss the structure of the unitarily implemented dilation symmetry in dilation invariant scaling limit states. The outcome is that in general the dilations do not decompose, not even in the factorizing situation. Rather, the dilations intertwine in a suitable sense the different pure limit states that occur in the direct integral decomposition. A complete factorization of the dilation symmetry is however obtained in the convergent scaling limit case. For such theories, therefore, one gets a unitary implementation of the dilation symmetry in the pure limit theory. The remainder of this paper is organized as follows: First, in Sec. 2, we recall the notion of scaling limits in the algebraic approach to quantum field theory, and generalize some fundamental results of [BV95] to our situation. In Sec. 3, we establish the direct integral decomposition mentioned above, including a decomposition of local observables and Poincaré symmetries. Section 4 contains a discussion of unique scaling limits as a special case. We define several conditions that generalize the notion from [BV95], and discuss relations between them. Then, in Sec. 5, we analyze the structure of dilation symmetries in the limit Hilbert space, and their decomposition along the direct integral, on different levels of generality. In Sec. 6 we propose a stronger version of the HaagSwieca compactness condition and we show that it implies the separability property used in the analysis of Sec. 4. Section 7 discusses some simple models as examples, showing in particular that these fulfill all of our conditions proposed in Sec. 4 and 6. We conclude with a brief outlook in Sec. 8. The Appendix reviews the concept of direct integrals of Hilbert spaces, which we need in a more general variant than covered in the standard literature. 2. Definitions and General Results We shall first recall the definition of scaling limits in the algebraic approach to quantum field theory, and prove some fundamental results regarding uniqueness of the limit vacuum state and geometric modular action. 2.1. The setting. We consider quantum field theory on (s + 1) dimensional Minkowski space. For our analysis, we work entirely within the framework of algebraic quantum field theory [Haa96], where observables localized in an open bounded subset O ⊂ Rs+1 of spacetime are described by the selfadjoint elements of a C∗ algebra A(O) in such a way that if O1 ⊂ O2 then A(O1 ) ⊂ A(O2 ). The correspondence A : O → A(O) thus defined is called a net of algebras. The inductive limit C∗ algebra of O → A(O) is denoted again by A and is called the quasilocal algebra. Let us repeat the formal definition of a quantum field theoretical model in this context. Definition 2.1. Let G be a Lie group of point transformations of Minkowski space that includes the translation group. A local net of algebras with symmetry group G is a net of algebras A together with a representation g → αg of G as automorphisms of A, such that

24

H. Bostelmann, C. D’Antoni, G. Morsella

(i) [A1 , A2 ] = 0 if O1 , O2 are two spacelike separated regions, and Ai ∈ A(Oi ); (ii) αg A(O) = A(g.O) for all O, g. We call A a net in a positive energy representation if, in addition, the A(O) are W ∗ algebras acting on a common Hilbert space H, and (iii) there is a strongly continuous unitary representation g → U (g) of G on H such that αg = ad U (g); (iv) the joint spectrum of the generators of translations U (x) lies in the closed forward light cone V¯ + ; (v) there exists a vector Ω ∈ H which is invariant under all U (g) and cyclic for A. We call A a net in the vacuum sector if, in addition, (vi) the vector Ω is unique (up to scalar factors) as an invariant vector for the translation group. Our approach is to start from a local net A in the vacuum sector, with the Poincaré ↑ group P+ as its symmetry group; this net A will be kept fixed in all that follows. Our aim is to describe the short-distance scaling limit of A. Following [BV95], we define B to be the set of bounded functions B : R+ → B(H), λ → B λ . Equipped with pointwise addition, multiplication, and ∗ operation, and with the norm B = supλ B λ , the set B becomes a C∗ algebra. Let G be the group formed by Poincaré transformations and dilations; we will write G g = (µ, x, Λ) with µ ∈ R+ , x ∈ Rs+1 , and Λ a Lorentz matrix. G acts on B via a representation α, given by (α g B)λ = αλµx,Λ (B λµ ) for g = (µ, x, Λ) ∈ G, B ∈ B,

(2.1)

where α is the Poincaré group representation on A. Note the rescaling of translations with the scale parameter λ. We now define new local algebras as subsets of B: A(O) := A ∈ B | Aλ ∈ A(λO) for all λ > 0; g → α g (A) is norm continuous . (2.2) This is a net of local algebras in the sense of Def. 2.1, with the enlarged symmetry group G [BDM09]. We denote by A the associated quasilocal algebra, i.e. the inductive limit of A(O) as O Rs+1 . This A is called the scaling algebra. Note that A has a large center Z(A), consisting of all operators A of the form Aλ = f (λ)1, where f : R+ → C is a bounded uniformly continuous function on R+ as a group under multiplication. We often identify A ∈ Z(A) with the function f without further notice. For a description of the scaling limit, we first consider states on Z(A). Let m be a mean on the bounded uniformly continuous functions1 on R+ , i.e., a positive normalized linear functional on the commutative C∗ algebra Z(A). We say that m is asymptotic if m( f ) = limλ→0 f (λ) whenever the limit on the right-hand side exists; or, equivalently, if m( f ) = 0 whenever f (λ) = 0 for small λ. Asymptotic means are, in this sense, generalizations of the limit λ → 0. Further we consider two important classes of means: (i) m is called multiplicative if m( f g) = m( f )m(g) for all functions f, g. (ii) m is called invariant if m( f µ ) = m( f ) for all functions f and all µ > 0, where f µ = f (µ · ). 1 In contrast to [BDM09], we do not consider means on the bounded functions on R , but rather on the + bounded uniformly continuous functions. While all of them can be extended to the bounded functions, these extensions do not play a role in our current investigation.

On Dilation Symmetries Arising from Scaling Limits

25

It is an important fact that (i) and (ii) are mutually exclusive; there are no multiplicative invariant means in our situation (cf. [Mit66]). We now extend these “generalized limits” of functions to a limit of operator sequences, using a projection technique. Let ω = (Ω| · |Ω) be the vacuum state of A. This state induces a projector (or conditional expectation) in A onto Z(A), which we denote by the same symbol: ω : A → Z(A), (ω(A))λ = ω(Aλ )1.

(2.3)

Using this projector, any mean m defines a state ωm on A by ωm := m ◦ ω. If here m is asymptotic, we call ωm a limit state, and typically denote it by ω0 . These are the states that correspond to scaling limits of the quantum field theory. Since there is a one-to-one correspondence between asymptotic means and limit states, we will usually work with the state ω0 only, and not refer to the mean m explicitly. A limit state ω0 will be called multiplicative2 or invariant if the corresponding mean has this property. Multiplicative limit states correspond to those considered by Buchholz and Verch in [BV95]. Every other limit state arises from these by convex combinations and weak∗ limits; this follows directly from the property of states on the commutative algebra Z(A). Given a limit state ω0 , we can obtain the limit theory via a GNS construction: Let π0 be the GNS representation of A with respect to ω0 , and H0 the representation space, with GNS vector Ω0 . Denoting by G0 the subgroup of G under which ω0 is invariant, we canonically obtain a strongly continuous unitary representation of G0 on H0 by setting U0 (g)π0 (A)Ω0 := π0 (α g (A))Ω0 , g ∈ G0 . The subgroup G0 contains the Poincaré group; and if ω0 is invariant, then G0 = G. The translation part of U0 fulfills the spectrum condition [BDM09]. Setting A0 (O) := π0 (A(O)) , one obtains a local net A0 with symmetry group G0 in a positive energy representation: the limit theory. 2.2. Multiplicity of the vacuum state. If ω0 is a multiplicative limit state, its restriction to Z(A) is pure. It has been shown in [BV95] that in the case s ≥ 2, this property extends to the entire theory: ω0 is a pure vacuum state on A, and π0 is an irreducible representation. On the other hand, if ω0 is not multiplicative, the same must be false, since already ω0 Z(A) is non-pure. However, we shall show that this property of the center is the only “source” of reducibility: namely one has π0 (A) = π0 (Z(A)) . We need some preparations to prove this. In the following, set Z0 := π0 (Z(A)) , and let HZ := clos(Z0 Ω0 ) ⊂ H0 be the representation space of the commutative algebra. Lemma 2.2. Let PZ ∈ B(H0 ) be the orthogonal projector onto HZ. If s ≥ 2, then PZ ∈ π0 (A) , and HZ is the space of all translation-invariant vectors in H0 . Proof. As a consequence of the spectrum condition in the theory A0 , it is known [Ara64] that the translation operators U0 (x) are contained in π0 (A) . Now let U∞ be an ultraweak cluster point of U0 (x) as x goes to spacelike infinity on some fixed sequence within the time-0 plane. (Such cluster points exist by the Alaoglu-Bourbaki theorem.) Then U∞ ∈ π0 (A) ; we will show U∞ = PZ. To that end, we first note that ω0 (A B) = ω0 (ω(A) B) for all A ∈ A, B ∈ Z(A),

(2.4)

2 For clarity, we note that a multiplicative limit state, by this definition, is not a multiplicative functional on A, but is multiplicative only on the center Z(A).

26

H. Bostelmann, C. D’Antoni, G. Morsella

which follows directly from the definition of ω0 . Now we make use of the cluster property of the vacuum at finite scales. As in [BV95, Lemma 4.3], one can obtain the following norm estimate in the algebra A: ω(A α x B) − ω(A)ω(B) ≤ c

rs ˙ + AB ˙ A B |x|s−1

(2.5)

for fixed r > 0, x in the time-0 plane with |x| > 3r , and for A, B chosen from some norm-dense subset of A(Or ), with Or being the standard double cone of radius r around the origin. Here c > 0 is some constant, and the dot denotes the time derivative. This implies that as |x| → ∞, lim ω0 (A α x B) = ω0 (ω(A)ω(B)) x

(2.6)

for these A, B. Now it follows from Eq. (2.4) – with ω(B) in place of B – that (π0 (A)Ω0 |U∞ π0 (B)Ω0 ) = lim ω0 (A∗ α x B) = (π0 (A)Ω0 |π0 (ω(B))Ω0 ). (2.7) x

Continuing this relation from the dense sets chosen, this means U∞ π0 (B)Ω0 = π0 (ω(B))Ω0 for all B ∈ A.

(2.8)

2 = U , and img U This shows that U∞ ∞ ∞ = HZ. Also, again applying Eq. (2.4), one ∗ = U . Thus U is the unique orthogonal projector onto H . For the last obtains U∞ ∞ ∞ Z part, note that translations act trivially on HZ, and that U∞ leaves all translation-invariant vectors unchanged; so HZ is the space of all translation-invariant vectors.

We are now ready to prove the announced result about the commutant of π0 (A). Theorem 2.3. Let s ≥ 2. Let ω0 be a limit state, and let π0 be the corresponding GNS representation. Then π0 (A) = π0 (Z(A)) . Proof. Let B ∈ π0 (A) . By Lemma 2.2, B commutes with PZ; hence BHZ ⊂ HZ, and B HZ ∈ B(HZ) is well-defined. As Z0 ⊂ π0 (A) , we know that [B HZ, C HZ] = 0 for all C ∈ Z0

(2.9)

as an equation in B(HZ). Since Z0 HZ is a maximal abelian algebra in B(HZ) [BR79, Lemma 4.3.15], there exists C ∈ Z0 with B HZ = C HZ. Now for any A ∈ π0 (A) , we can compute B AΩ0 = ABΩ0 = ACΩ0 = C AΩ0 . Since Ω0 is cyclic for π0 , this implies B = C. Thus π0 is trivial.

(A)

(2.10)

⊂ Z0 . The reverse inclusion

It should be noted that the same theorem does not hold in 1+1 space-time dimensions. In this case, it is known even in free field theory [BV98, Sec. 4] that the algebra π0 (A) has a large center, even if π0 (Z(A)) = C1. We can now easily reproduce the known results for multiplicative limit states. In this case, the GNS representation of the abelian algebra Z(A) for the state ω0 must be irreducible; thus Z0 = C1, and dim HZ = 1. The above results imply: Corollary 2.4. Let ω0 be a multiplicative limit state, and let s ≥ 2. Then Ω0 is unique up to a scalar factor as an invariant vector for the translations U0 (x), and the representation π0 is irreducible. A0 is a net in the vacuum sector in the sense of Def. 2.1.

On Dilation Symmetries Arising from Scaling Limits

27

2.3. Wedge algebras and geometric modular action. While we have defined the scaling limit in terms of local algebras for bounded regions, it is also worthwhile to consider algebras associated with unbounded, in particular wedge-shaped regions. This is particularly important in the context of charge analysis for the limit theory [DMV04,DM06]. While we do not enter this topic here, and do not build on it in the following, we wish to discuss briefly how wedge algebras and the condition of geometric modular action fit into our context. Again, this transfers results of [BV95] to our generalized class of limit states. Let W be a wedge region, i.e. W is a Poincaré transform of the right wedge W+ = −W− = {x ∈ R4 | x · e± < 0}, where e± := (±1, 1, 0, 0).

(2.11)

Note that (W + ) = W− . We introduce the one-parameter group (Λt )t∈R of Lorentz boosts leaving W+ invariant, fixed by Λt e± = exp(±t)e± , and acting as the identity on the edge (e± )⊥ of W+ . Let furthermore j be the inversion with respect to the edge of W+ , i.e. je± = −e± and j = 1 on (e± )⊥ . Note that j 2 = 1 and jW+ = W− . For a local net of algebras (resp. for a net of algebras in a positive energy representation) O → A(O), we define the algebra A(W) associated to the wedge W as the C∗ -algebra (resp. W∗ -algebra) generated by the algebras A(O), where O is any double cone whose closure is contained in W (O ⊂⊂ W in symbols). With these definitions, we can adapt the arguments in [BV95, Lemma 6.1], which do not depend on irreducibility of the net. It is then straightforward to verify that, for a net A in a positive energy representation, the vacuum vector Ω is cyclic and separating for all wedge algebras A(W). This allows us to introduce the notion of geometric modular action. Definition 2.5. Let A be a local net in a positive energy representation, and denote by ∆, J the modular objects associated to A(W+ ), Ω. The net A is said to satisfy the condition of geometric modular action if there holds ∆it = U (Λ2π t ),

t ∈ R, ↑

J U (x, Λ)J = U ( j x, jΛj), (x, Λ) ∈ P+ , J A(O)J = A( jO).

(*)

If A satisfies the condition of geometric modular action, then it also satisfies wedge duality, since, according to Tomita-Takesaki theory and equation (*), A(W+ ) = J A(W+ )J = A(W− ).

(2.12)

This also implies that A satisfies essential Haag duality, i.e. that the dual net Ad of A, defined on double cones O as A(W), (2.13) Ad (O) := W ⊃O

is local and such that A(O) ⊂ Ad (O) for each double cone O. From now on, let A be a net in the vacuum sector, and ω0 a scaling limit state, with π0 the corresponding scaling limit representation. It holds that π0 (A(W)) = A0 (W), since clearly π0 (A(W)) =

O⊂⊂W

π0 (A(O))

·

,

(2.14)

28

H. Bostelmann, C. D’Antoni, G. Morsella

and therefore

π0 (A(W)) =

π0 (A(O)) =

O⊂⊂W

A0 (O) = A0 (W) .

(2.15)

O⊂⊂W

Proposition 2.6. Assume that A satisfies the condition of geometric modular action. Then for each limit state ω0 , the corresponding limit theory A0 also satisfies the condition of geometric modular action. Proof. It’s a straightforward adaptation of the proofs of Lemma 6.2 and Proposition 6.3 of [BV95]. The only point which is worth mentioning is the proof that ω0 is a KMS state (at inverse temperature 2π ) for the algebra A(W+ ) with respect to the one-parameter group of automorphisms (α Λt )t∈R , which goes as follows. Let m be the mean which induces ω0 . Then m is a weak∗ limit of convex combinations of multiplicative means, and therefore ω0 is a weak∗ limit of convex combinations of multiplicative limit states. For such states, the arguments in [BV95, Lemma 6.2] show that they are KMS on A(W+ ), and therefore, the set of KMS states at a fixed inverse temperature being convex and weak∗ closed [BR81, Thm. 5.3.30], this holds also for ω0 . 3. Decomposition Theory Our aim is now to decompose an arbitrary limit state ω0 into “simple” limit states of the Buchholz-Verch type, and to obtain corresponding decompositions of the relevant objects in the limit theory. We start by proving an integral decomposition which is a consequence of standard results. Proposition 3.1. Let ω0 be a limit state. There exists a compact Hausdorff space Z, a regular Borel probability measure ν on Z, and for each z ∈ Z a multiplicative limit state ω z , such that ω0 (A) = dν(z) ω z (A) for all A ∈ A. Z

Further, the map Z(A) → C(Z), C → (z → ω z (C)) is surjective. Proof. Let π0 be the GNS representation of A for ω0 . Consider the C∗ algebra π0 (Z(A)). It is well known that this commutative algebra is isomorphic to C(Z) for a compact Hausdorff space Z, with the isomorphism being given by π0 (C) → (z → ρz (π0 (C))), where the ρz are multiplicative functionals. Now by the Riesz representation theorem, the GNS state (Ω0 | · |Ω0 ) on π0 (Z(A)) ∼ = C(Z) is given by a regular Borel measure ν on Z. Explicitly, one has for all C ∈ Z(A), ω0 (C) = (Ω0 |π0 (C)|Ω0 ) = dν(z) ρz ◦ π0 (C). (3.1) Z

It is clear that ν(Z) = 1. In the above expression, mz := ρz ◦ π0 are multiplicative means; they are asymptotic, since π0 (A) = 0 whenever Aλ vanishes for small λ. Thus, setting ω z = ρz ◦ π0 ◦ ω as usual, we obtain multiplicative limit states ω z on A such that ω0 (A) = dν(z) ω z (A) for all A ∈ A. (3.2) Z

As a last point, the map Z(A) → C(Z), C → (z → ω z (C)) = (z → ρz (π0 (C))) is surjective by construction.

On Dilation Symmetries Arising from Scaling Limits

29

We have thus decomposed a general limit state ω0 into multiplicative limit states ω z . In the case s ≥ 2, this will also be a decomposition into pure states; but the above result does not depend on that. Also, we emphasize that our aim is not a decomposition of the von Neumann algebra π0 (A) along its center; rather we work on the C∗ algebraic side only. We would now like to interpret the above decomposition in the sense of decomposing the limit Hilbert space H0 as a direct integral. This is complicated by the fact that our measure spaces (Z, ν) can be of a very general nature, making the limit Hilbert space nonseparable. In fact, if ω0 is an invariant limit state, one finds that all vectors of the form π0 (C)Ω0 are mutually orthogonal if C λ = χ (λ)1, where χ is a character on R+ . Since there are clearly uncountably many characters—just take χ (λ) = λık with k ∈ R—the limit Hilbert space H0 cannot be separable in this case. The theory of direct integrals of Hilbert spaces in the absence of separability assumptions is nonstandard and only partially complete; we give a brief review in Appendix A. Here we note that the notion of a direct integral over Z, with fiber spaces Hz , crucially depends on the specification of a fundamental family Γ ⊂ z∈Z Hz . This Γ is a vector space with certain extra conditions (see Def. A.1) that serves to define which Hilbert space valued functions are considered measurable. Indeed, using the exact notions, we prove: Theorem 3.2. Let ω0 be a limit state, and Z, ν, ω z as in Proposition 3.1. Let πz , Hz , Ωz be the GNS representation objects corresponding to ω z . Then, Hz Γ := {z → πz (A)Ωz | A ∈ A} ⊂ z∈Z

is a fundamental family. With respect to this family, it holds that Γ ∼ H0 = dν(z) Hz , Z

where the isomorphism is given by π0 (A)Ω0 →

Γ

Z

dν(z) πz (A)Ωz , A ∈ A.

Proof. It is clear that Γ is a linear space; and per Prop. 3.1, the function z → πz (A)Ωz 2 = ω z (A∗ A) is integrable for any A ∈ A. Thus Γ is a fundamental family per Definition A.1. The map W : H0 → Hz , π0 (A)Ω0 → (z → πz (A)Ωz ) (3.3) z∈Z

is clearly linear and isometric when A ranges through A; thus W can in fact be extended to a well-defined isometric map from H0 into Γ¯ . It remains to show that W is surjective. In fact, since L ∞ (Z) · Γ is total in the direct integral space, it suffices to show that all vectors of the form Γ dν(z) f (z) πz (A)Ωz , f ∈ L ∞ (Z), A ∈ A, (3.4) Z

can be approximated in norm with vectors of the form W π0 (B)Ω0 , B ∈ A.

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H. Bostelmann, C. D’Antoni, G. Morsella

To that end, let f ∈ L ∞ (Z) and A ∈ A be fixed. We first note that, as a simple consequence of Lusin’s theorem, there exist functions f n ∈ C(Z) such that f n ∞ ≤ f ∞ and limn→∞ f n (z) = f (z) for almost every z ∈ Z. On the other hand, per Proposition 3.1 there exist C n ∈ Z(A) such that ω z (C n ) = f n (z) for all z ∈ Z, which implies πz (C n ) = f n (z)1. Therefore we have lim

n→∞ Z

( f (z)πz (A) − πz (C n A))Ωz 2 dν(z) = 0

by an application of the dominated convergence theorem.

(3.5)

In the following, we will usually not denote the above isomorphism explicitly, but rather identify H0 with its direct integral representation. In this way, the subspace HZ ⊂ H0 is isomorphic to the function space L 2 (Z, ν), where f ∈ L 2 (Z, ν) is identified

Γ with Z dν(z) f (z)Ωz ∈ H0 . The next corollary follows directly from the proof above, since a decomposition of operators needs to be checked on the fundamental family only (Lemma A.2). Corollary 3.3. With respect to the direct integral decomposition in Theorem 3.2, all

Γ operators π0 (A), A ∈ A are decomposable, and one has π0 = Z dν(z)πz . If A ∈ Z(A),

Γ then π0 (A) is diagonal, with π0 (A) = Z dν(z) ω z (A)1. Finally, we remark that Lorentz symmetries U0 (x, Λ) in the limit theory are decomposable. ↑

Proposition 3.4. Let g → Uz (g) be the implementation of P+ on the limit Hilbert space Hz corresponding to ω z . Then, one has U0 (g) =

Γ

Z

↑

dν(z)Uz (g) for all g ∈ P+ .

Proof. Again, it suffices to verify this on vectors from Γ . With W being the isomorphism ↑ introduced in the proof of Theorem 3.2, one obtains for all g ∈ P+ and A ∈ A, W U0 (g)π0 (A)Ω0 = W π0 (α g A)Ω0 = =

Γ

Z

dν(z)Uz (g)πz (A)Ωz .

This proves the proposition.

Γ

Z

dν(z)πz (α g A)Ωz (3.6)

It should be remarked that the same simple structure cannot be expected for dilations, if they exist as a symmetry of the limit. For even if ω0 ◦ α µ = ω0 , the multiplicative limit states ω z cannot be invariant under α µ , not even when restricted to Z(A). Thus, the unitaries U0 (µ) will not commute with π0 (Z(A)), and can therefore not be decomposable. In special situations, there may be a generalized sense in which the dilation unitaries can be decomposed; we will investigate this in more detail in Sec. 5.

On Dilation Symmetries Arising from Scaling Limits

31

4. Unique and Factorizing Scaling Limits The limit theory on the Hilbert space H0 is composed, as discussed in the previous section, of simpler components that live on the “fibre” Hilbert spaces Hz of the direct integral. It is natural to ask whether the theories on these spaces Hz , or more precisely, the nets of algebras Az (O) = πz (A(O)) , are similar or identical in a certain sense. While no models have been explicitly constructed for which the limit theories substantially depend on the choice of a (multiplicative) limit state,3 it does not seem to be excluded that measurable properties, such as the mass spectrum or charge structure of Az , can depend on z. For most applications in physics, however, one expects that the situation is simpler, and that the limit theory does not depend substantially on the choice of ω z . Here it would be much too strict to require that the representations πz are unitarily equivalent. [In fact, for s ≥ 2, the πz are irreducible per Thm. 2.3, and since they do not agree on Z(A), they are even pairwise disjoint.] Rather one can expect that their images, the algebras Az (O), are unique as sets, up to unitaries that identify the different Hilbert spaces Hz ; see Def. 4.1 below. This is the situation of a unique scaling limit in the sense of Buchholz and Verch. In the present section, we want to elaborate how the situation of unique scaling limits, originally formulated for multiplicative limit states, fits into our generalized context. To that end, we will formulate several conditions on the limit theory that roughly correspond to unique limits, and discuss their mutual dependencies. 4.1. Definitions. We shall first motivate and define the conditions to be considered; the proofs of their interrelations are deferred to sections further below. We start by recalling the condition of a unique scaling limit in the sense of [BV95], with some slight modifications. Definition 4.1. The theory A is said to have a unique scaling limit if there exists a local Poincaré covariant net (Au , Hu , Ωu , Uu ) in the vacuum sector such that the following holds. For every multiplicative limit state ω0 , there exists a unitary V : H0 → Hu such ↑ that V Ω0 = Ωu , V U0 (g)V ∗ = Uu (g) for all g ∈ P+ , and V A0 (O)V ∗ ⊂ Au (O) for all open bounded regions O. This includes the aspect of a “unique vacuum structure”. Compared with [BV95], we have somewhat weakened the condition, since we require only inclusion of V A0 (O)V ∗ in Au (O), not equality. This is for the following reason. Supposing that both A and Au fulfill the condition of geometric modular action (Definition 2.5), such that the Haagdualized nets of A0 and Au are well-defined, our condition precisely implies that these dualized nets agree for all multiplicative limit states. Since for many applications, particularly charge analysis [DMV04], the dualized limit nets are seen as the fundamental objects, we think that this is a reasonable generalization of the condition. For a general, not necessarily multiplicative limit state ω0 , we obtain a decomposition ω0 = Z dν(z)ω z into multiplicative states, as discussed in Sec. 3, and thus obtain from Def. 4.1 corresponding unitaries Vz for every z. Due to the very general nature of the measure space Z, and due to a possible arbitrariness in the choice of Vz , particularly if Au possesses inner symmetries, an analysis of ω0 seems impossible in this generality. Rather we will often make use of a regularity condition, which is formulated as follows. 3 See however [Buc96a, Sec. 5] for some ideas to that end.

32

H. Bostelmann, C. D’Antoni, G. Morsella

Definition 4.2. Suppose that the theory A has a unique scaling limit. We say that a limit state ω0 is regular if there is a choice of the unitaries Vz such that for any A ∈ A, the function ϕ A : Z → Hu , z → Vz πz (A)Ωz is Lusin measurable [i.e., is contained in L 2 (Z, ν, Hu )]. We shall later give a sufficient condition for the above regularity, which actually implies that the functions ϕ A can be chosen constant in generic cases. Our concepts so far refer to multiplicative limit states mostly. We will now give a generalization of Def. 4.1 that involves generalized limit states directly, and that seems natural in our context. It is based on the picture that the limit Hilbert space should have a tensor product structure, H0 ∼ = HZ ⊗ Hu , where Hu is the unique representation space associated with multiplicative limit states, and HZ is the representation space of Z(A) under π0 . All objects of the theory—local algebras, Poincaré symmetries, and the vacuum vector—should factorize along this tensor product. We now formulate this in detail. Definition 4.3. The theory A is said to have a factorizing scaling limit if there exists a local Poincaré covariant net (Au , Hu , Ωu , Uu ) in the vacuum sector such that the following holds. For every limit state ω0 , there exists a decomposable unitary V :

Γ,⊕ H0 → L 2 (Z, ν, Hu ), V = Z dν(z)Vz with unitaries Vz : Hz → Hu , such that ↑ V Ω0 = ΩZ ⊗ Ωu , V U0 (g)V ∗ = 1 ⊗ Uu (g) for all g ∈ P+ , and Vz Az (O)Vz∗ ⊂ Au (O) for all open bounded regions O and all z ∈ Z. Here ΩZ ∈ HZ denotes the GNS vector of the commutative algebra. The conditions on local algebras are deliberately chosen quite strict. We require Vz Az (O)Vz∗ ⊂ Au (O) ¯ u (O). This serves to for every z, rather than the weaker condition V A0 (O)V ∗ ⊂ Z0 ⊗A avoid countability problems; see Sec. 4.3 for further discussion. In subsequent sections, we will show that the notion of a unique scaling limit and a factorizing scaling limit are cum grano salis identical, up to the extra regularity condition in Definition 4.2 that we have to assume. We also consider a stronger condition, which is easier to check in models. Our ansatz is to require a sufficiently large subset Aconv ⊂ A such that for each A ∈ Aconv , the function λ → ω(Aλ ) is convergent as λ → 0. Consider the following definition: Definition 4.4. The theory A is said to have a convergent scaling limit if there exists an α-invariant C∗ subalgebra Aconv ⊂ A with the following properties: (i) For each A ∈ Aconv , the function λ → ω(Aλ ) converges as λ → 0. (ii) If ω0 is a multiplicative limit state, then π0 (A(O)∩Aconv ) is weakly dense in A0 (O) for every open bounded region O. It follows directly from (ii) that also π0 (Aconv )Ω0 is dense in H0 . The condition roughly says that “convergent scaling functions” are sufficient for describing the limit theory—considering nonconvergent sequences is only required for technical consistency of our formalism, for describing the image of Z(A), which does not directly relate to quantum theory. This is heuristically expected in many physical models: In usual renormalization approaches in formal perturbation theory, the selection of subsequences or filters to enforce convergence seems not to be widespread, and sequences of pointlike fields can be chosen to converge in matrix elements. We will show that the above condition is sufficient for the scaling limit to be unique, and all limit states to be regular. In fact, we shall see later that also the structure of dilations simplifies.

On Dilation Symmetries Arising from Scaling Limits

33

Unique limit

Factorizing limit

Convergent limit

Regularity condition

Fig. 1. Implications between the conditions on the limit theory. Arrows marked with ∗ are only proven under additional separability assumptions

We also mention that this condition has been employed in [CM08] in order to discuss some functoriality properties of the scaling limit with respect to the formation of subsystems of the observable net. Figure 1 summarizes the different conditions we introduced, and shows the implications we briefly mentioned. We will now go ahead and prove that the individual arrows are indeed correct. However, in order to avoid problems with the direct integral spaces involved, we shall make certain separability assumptions in most cases. Let us comment on these. For multiplicative limit states, it seems a reasonable assumption that the limit Hilbert space H0 is separable. This would follow, from example, from the Haag-Swieca compactness condition; cf. [Buc96a]. For general limit states, in particular if these are invariant, H0 cannot be separable since already HZ ∼ = L 2 (Z, ν) is nonseparable, as discussed in Sec. 3. We can however reasonably assume that H0 fulfills a condition which we call uniform separability; cf. Def. A.3 in the Appendix. This means that a countable

Γ set {χ j } ⊂ H0 = Z dν(z)Hz exists such that {χ j (z)} is dense in every Hz . As we shall see in Sec. 6, uniform separability follows from a sharpened version of the Haag-Swieca compactness condition; and we will show in Sec. 7 that this compactness condition is indeed fulfilled in relevant examples. 4.2. Unique limit ⇒ factorizing limit. In the following, we suppose that A has a unique scaling limit. We fix a regular limit state ω0 , and denote the associated objects Z, ν, H0 , π0 , Ω0 , Hz , πz , Ωz , Vz as before. In order to prove that the scaling limit factorizes, we have to construct a unitary V : H0 → L 2 (Z, ν, Hu ) with appropriate properties. In fact,

Γ,⊕ this V is intuitively given by V = Z dν(z)Vz ; and the key question turns out to be whether this V is surjective. We will prove this only under separability assumptions. Proposition 4.5. Let A have a unique scaling limit; let ω0 be a regular limit state; and suppose that H0 is uniformly separable. Then, Γ,⊕ V : H0 → L 2 (Z, ν, Hu ), V = dν(z)Vz Z

defines a unitary operator. Proof. First, it is clear that if H0 is uniformly separable, then all Hz , and in particular Hu , are separable. Hence L 2 (Z, ν, Hu ) is uniformly separable. Now note that V is well-defined precisely by the regularity condition. Further, writing explicitly V π0 (A)Ω0 = z → Vz πz (A)Ωz , A ∈ A, (4.1)

34

H. Bostelmann, C. D’Antoni, G. Morsella

one has

V π0 (A)Ω0 2 = dν(z) Vz πz (A)Ωz 2 = dν(z) πz (A)Ωz 2 = π0 (A)Ω0 2 , Z

Z

(4.2) so V is isometric. It remains to show that V is surjective. To that end, let P be the orthogonal projector onto img V . Since

⊕ V commutes with all diagonal operators, so does P; thus P is decomposable: P = Z dν(z)P(z). Now compute 0 = (1 − P)V =

Γ,⊕

Z

dν(z)(1 − P(z))Vz .

(4.3)

Using uniform separability of both spaces involved, we obtain that (1 − P(z))Vz = 0 a. e. Since the Vz are surjective onto Hu , this means P(z) = 1 a. e. This implies P = 1, so V is surjective. It is clear that V Ω0 = ΩZ ⊗ Ωu ; and we can also verify from the properties of the Vz with respect to Poincaré symmetries that ↑

V U0 (g)V ∗ = 1 ⊗ Uu (g) for all g ∈ P+ .

(4.4)

Also, by the definition of the unique scaling limit, it must hold that Vz Az (O)Vz∗ ⊂ Au (O) for all z. Summarizing the results of this section, we have shown: Theorem 4.6. Suppose that A has a unique scaling limit, that every limit state ω0 is regular, and that the limit spaces H0 are uniformly separable. Then the scaling limit of A is factorizing. 4.3. Factorizing limit ⇒ unique limit. Now reversing the arrow, we start from a theory with factorizing scaling limit, and want to show that the scaling limit is unique in the sense of Buchholz and Verch, and that the limit states are regular. At first glance, this implication seems to be apparent from the definitions. A detailed investigation however reveals some subtleties, which again lead us to making separability assumptions. Theorem 4.7. Assume that A has a factorizing scaling limit. Then the scaling limit is unique. If the space Hu is separable, all limit states ω0 are regular. Proof. It is clear that the scaling limit is unique by Def. 4.1, specializing the conditions of Def. 4.3 to the case where ω0 is multiplicative, and Z consists of a single point. Now

Γ,⊕ let ω0 be a limit state; we need to show it is regular. Let V = Z dν(z)Vz be the unitary guaranteed by Def. 4.3. By definition, the map z → Vz πz (A)Ωz is measurable for any A ∈ A. But we have to show that each Vz fulfills the conditions of Def. 4.1; in fact, we will have to modify the Vz on a null set. First, we have V Ω0 = ΩZ ⊗ Ωu by assumption. On the other hand, V Ω0 =

⊕ Z dν(z) Vz Ωz , so that Vz Ωz = Ωu for z ∈ Z\NΩ , where NΩ is a null set. Next we consider Poincaré transformations. Starting from Def. 4.3, we know that: ↑

V U0 (g)V ∗ = 1 ⊗ Uu (g) for all g ∈ P+ .

(4.5)

On Dilation Symmetries Arising from Scaling Limits

35

Since U0 (g) factorizes by Prop. 3.4, we can rewrite this equation as

⊕

Z

dν(z) Vz Uz (g)Vz∗

=

⊕

Z

dν(z) Uu (g).

(4.6)

Now if Hu is separable, and thus L 2 (Z, ν, Hu ) uniformly separable, we can conclude that Vz Uz (g)Vz∗ = Uu (g) for all z ∈ Z\Ng , with a null set Ng depending on g. We pick ↑ a countable dense subset Pc of P+ , and consider the null set N := NΩ ∪ (∪g∈Pc Ng ). Our results so far are that Vz Ωz = Ωu , Vz Uz (g)Vz∗ = Uu (g) for all z ∈ Z\N , g ∈ Pc .

(4.7)

↑ Indeed, by continuity of the representations, the same holds for all g ∈ P+ . Now let Vˆz be those unitaries obtained by evaluating Def. 4.3 for the multiplicative limit states ω z . We set Vz for z ∈ Z\N , Wz := ˆ (4.8) Vz for z ∈ N .

Γ,⊕ Then we have V = Z dν(z) Wz , and the Wz fulfill the relations in Eq. (4.7) for all ↑ z ∈ Z and g ∈ P+ . As a last point, Wz Az (O)Wz∗ ⊂ Au (O) holds for every z, since both Vz and Vˆz have this property. Thus ω0 is regular. Let us add some comments on the conditions required for Vz in Def. 4.3, regarding Poincaré transformations and local algebras. We could choose stricter conditions on Vz , requiring that ↑

Vz Uz (g)Vz∗ = Uu (g) for all z ∈ Z and g ∈ P+ .

(4.9)

In this case, the countability problem in the proof above does not occur, and Thm. 4.7 holds without the requirement that Hu is separable. On the other hand, it does not seem reasonable to weaken the conditions on Vz with respect to local algebras, requiring only that ¯ u (O) for all O. V A0 (O)V ∗ ⊂ Z0 ⊗A

(4.10)

(We shall show below that this relation is implied by the chosen conditions on Vz .) For if we require only (4.10), and we wish to apply the techniques used in the proof of Thm. 4.7, it becomes necessary not only to require separability of Hu —which seems reasonable for applications in physics—but also separability of the algebras Au (O). That would however be too strict for our purposes, since the local algebras are expected to be isomorphic to the hyperfinite type III1 factor [BDF87]. We now show that Eq. (4.10) follows from Def. 4.3 as given. Proposition 4.8. Let A have a factorizing scaling limit. With V the unitary of Def. 4.3, ¯ u (O) for any bounded open region O. one has V A0 (O)V ∗ ⊂ Z0 ⊗A

36

H. Bostelmann, C. D’Antoni, G. Morsella

Proof. Let A ∈ A(O), and A ∈ Au (O) . We compute the commutator [1 ⊗ A , V π0 (A)V ∗ ] as a direct integral: ⊕ dν(z) [A , Vz πz (A)Vz∗ ]. (4.11) [1 ⊗ A , V π0 (A)V ∗ ] = Z

Now by our requirements on the Vz , we have Vz πz (A)Vz∗ ∈ Au (O) for all z, hence the commutator under the integral vanishes. Since A ∈ A(O) was arbitrary, this means ¯ u (O). V π0 (A(O))V ∗ ⊂ (1 ⊗ Au (O) ) = Z0 ⊗A By weak closure, this inclusion extends to V A0

(O)V ∗ .

(4.12)

4.4. Convergent limit ⇒ unique limit. We now assume that the theory has a convergent scaling limit, and show that our other conditions follow. The main simplification in the convergent case is as follows: For every A ∈ Aconv , the function λ → ω(Aλ ) converges to a finite limit as λ → 0; so all asymptotic means applied to this function yield the same value. Hence the value of ω0 (A) is the same for all limit states ω0 , multiplicative or not. Theorem 4.9. If the scaling limit of A is convergent, then it is unique. If further a multiplicative limit state exists such that the associated limit space H0 is separable, then all limit states are regular, and H0 is uniformly separable for any limit state. Proof. We pick a fixed multiplicative limit state ωu and denote the corresponding representation objects as Hu , πu , Uu , Ωu . Given any other multiplicative limit state ω0 , we define a map V by V : H0 → Hu , π0 (A)Ω0 → πu (A)Ωu for all A ∈ Aconv .

(4.13)

The convergence property of A ∈ Aconv implies π0 (A)Ω0 2 = ω0 (A∗ A) = ωu (A∗ A) = πu (A)Ωu 2 ,

(4.14)

so the linear map V is both well-defined and isometric. It is also densely defined and surjective by assumption (Def. 4.4). Hence V extends to a unitary. Using the α-invariance of ↑ Aconv , one checks by direct computation that V U0 (g)V ∗ = Uu (g) for all g ∈ P+ . Also, ∗ V Ω0 = Ωu is clear. Further, if A ∈ A(O) ∩ Aconv , it is clear that V π0 (A)V = πu (A). By weak density, this means V A0 (O)V ∗ = Au (O). Thus the scaling limit is unique. Now let ω0 not necessarily be multiplicative. Decomposing it into multiplicative states ω z as in Prop. 3.1, the above construction gives us unitaries Vz : Hz → Hu for every z. In fact, the functions z → Vz πz (A)Ωz = πu (A)Ωu are constant for all A ∈ Aconv , in particular measurable. Now let χ ∈ Hu and B ∈ A. We can choose a sequence (An )n∈N in Aconv such that πu (An )Ωu → χ in norm. Noticing that (Vz πz (B)Ωz |χ ) = lim (Vz πz (B)Ωz |πu (An )Ωu ) = lim ω z (B ∗ An ), n→∞

n→∞

(4.15)

we see that the left-hand side, as a function of z, is the pointwise limit of continuous functions, and hence measurable. Thus z → Vz πz (B)Ωz is weakly measurable. Now if Hu was chosen separable, which is possible by assumption, weak measurability implies Lusin measurability of the function (cf. Appendix). Thus ω0 is regular. Finally, in the separable case, we remark that we can pick a countable subset of Acount ⊂ Aconv such that πu (Acount )Ωu is dense in Hu . Then π0 (Acount )Ω0 becomes a fundamental sequence in H0 , so that this space is uniformly separable.

On Dilation Symmetries Arising from Scaling Limits

37

Of course, it follows as a corollary to the preceding sections that the limit is also factorizing. Let us spell this out more explicitly. Proposition 4.10. Suppose that A has a convergent scaling limit, and that there exists a multiplicative limit state ωu for which the representation space Hu is separable. Let ω0 be

Γ,⊕ any scaling limit state. There exists a unitary V = Z dν(z) Vz : H0 → L 2 (Z, ν, Hu ) such that V π0 (A C)Ω0 = π0 (C)ΩZ ⊗ πu (A)Ωu for all A ∈ Aconv , C ∈ Z(A), and such that the Vz fulfill all requirements of Def. 4.3. Proof. We use notation as in the proof of Thm. 4.9. Let Vz : Hz → Hu be the unitaries constructed there. Then, z → Vz∗ is a measurable family of operators. Namely, for any A ∈ Aconv , we find Vz∗ πu (A)Ωu = πz (A)Ωz

(4.16)

which is in Γ ; hence measurability is checked on the fundamental family (cf. Lemma A.2). So the operator ⊕,Γ ∗ V := dν(z) Vz∗ (4.17) Z

is well-defined. Domain and range of V ∗ are both uniformly separable, see Thm. 4.9.

Γ,⊕ Thus also the adjoint of V ∗ , denoted as V , is decomposable with V = Z dν(z) Vz . It is then clear that V is unitary. Also, we have for A ∈ Aconv and C ∈ Z(A), Γ ⊕ dν(z) πz (C)πz (A)Ωz = dν(z) πz (C)Vz πz (A)Ωz V π0 (A C)Ω0 = V Z Z ⊕ = dν(z) πz (C)πu (A)Ωu = (π0 (C)ΩZ) ⊗ (πu (A)Ωu ). (4.18) Z

As a direct consequence of the discussion following Eq. (4.14), the Vz have all the properties required in Def. 4.3 regarding vacuum vector, symmetries, and local algebras. 5. Dilation Covariance in the Limit Our next aim is to analyze the structure of dilation symmetries in the limit theory. To that end, we consider a scaling limit state ω0 which is invariant under δ µ . As shown in [BDM09, Sec. 2], the associated limit theory is covariant with respect to a strongly continuous unitary representation g ∈ G → U0 (g) of the extended symmetry group G, including both Poincaré symmetries and dilations. Our interest is how the dilation unitaries U0 (µ) relate to decomposition theory in Sec. 3, and how they behave in the more specific situations analyzed in Sec. 4. We will consider three cases of decreasing scope: first, the general situation; second, the factorizing scaling limit; third, the convergent scaling limit. We first consider a general theory as in Sec. 3, and analyze the decomposition of the dilation operators corresponding to the direct integral decomposition of H0 introduced

38

H. Bostelmann, C. D’Antoni, G. Morsella

in Thm. 3.2. To this end, we first note that δ µ leaves Z(A) invariant; thus we have a representation of the dilations UZ(µ) := U0 (µ) HZ on HZ. Identifying HZ with L 2 (Z, ν) as before, the UZ(µ) act on a function space. This action, and its extension to the entire Hilbert space, can be described in more detail. Proposition 5.1. Let ω0 be an invariant limit state. There exist an action of the dilations through homeomorphisms z → µ.z of Z, and unitary operators Uz (µ) : Hz → Hµ.z for µ ∈ R+ , z ∈ Z, such that: (i) the measure ν is invariant under the transformation z → µ.z; (ii) UZ(µ)χ (z) = χ (µ−1 .z) for all χ ∈ L 2 (Z, ν), as an equation in the L 2 sense; (iii) Uz (1) = 1, Uz (µ)∗ = Uµ.z (µ−1 ), Uµ.z (µ )Uz (µ) = Uz (µ µ) for all z ∈ Z, µ, µ ∈ R+ ; ↑ (iv) Uz (µ)Uz (x, Λ) = Uµ.z (µx, Λ)Uz (µ) for all z ∈ Z, (x, Λ) ∈ P+ ;

Γ

Γ −1 (v) U0 (µ)χ = Z dν(z) Uµ−1 .z (µ)χ (µ .z) for all χ ∈ Z dν(z) Hz . Proof. Recalling that Z is the spectrum of the commutative C∗ algebra π0 (Z(A)), we define the homeomorphism z → µ.z as the one induced by the automorphism ad UZ(µ−1 ) of π0 (Z(A)). For C ∈ Z(A), we know that π0 (C)Ω0 ∈ HZ corresponds to the function z → χC (z) = ω z (C), precisely the image of π0 (C) in the Gelfand isomorphism. Applying U0 (µ−1 ) to this vector, one obtains (UZ(µ−1 )χC )(z) = χC (µ.z);

(5.1)

thus (ii) holds for all χ ∈ C(Z). Taking the scalar product of Eq. (5.1) with Ω0 , one sees that Z dν(z)χ (z) = Z dν(z)χ (µ.z) for all µ and χ ∈ C(Z), so (i) follows. Now for general χ ∈ L 2 (Z, ν), statement (ii) follows by density. Expressing the action of z → µ.z on the level of algebras, it is easy to see that ωµ.z ◦ δ µ Z(A) = ω z Z(A).

(5.2)

Since however δ µ commutes with the projector ω : A → Z(A), the same equation holds on all of A. Therefore, the maps Uz (µ) : Hz → Hµ.z given by Uz (µ)πz (A)Ωz := πµ.z (δ µ (A))Ωµ.z

(5.3)

are well-defined and unitary. The properties of Uz (µ) listed in (iii) and (iv) then follow from this definition by easy computations.

Γ Now for (v): As before, we identify H0 with Z dν(z) Hz . Then we have, for all A ∈ A, U0 (µ)π0 (A)Ω0 = π0 (δ µ (A))Ω0 = =

Γ

Z

Γ

Z

dν(z) πz (δ µ (A))Ωz

dν(z) Uµ−1 .z (µ)πµ−1 .z (A)Ωµ−1 .z .

(5.4)

Γ Given now a vector χ ∈ Z dν(z) Hz , we can find a sequence (π0 (An )Ω0 )n∈N converging in norm to χ . Passing to a subsequence, we can also assume that πz (An )Ωz → χ (z)

On Dilation Symmetries Arising from Scaling Limits

39

in norm for almost every z ∈ Z. Hence, using the dominated convergence theorem and (i), we see that Γ Γ lim dν(z) Uµ−1 .z (µ)πµ−1 .z (An )Ωµ−1 .z = dν(z) Uµ−1 .z (µ)χ (µ−1 .z), n→+∞ Z

Z

which gives (v).

Thus dilations act between the fibers of the direct integral decomposition by unitaries Uz (µ), which depend on the fiber. They fulfill the cocycle-type composition rule Uµ.z (µ )Uz (µ) = Uz (µ µ) that one would naively expect; cf. also the theory of equivariant disintegrations for separable C ∗ algebras [Tak02, Ch. X §3]. We shall now further restrict to the situation of a factorizing scaling limit, as in Def. 4.3, in which the fiber spaces Hz are all identified with a unique space Hu . By this identification, we can regard the unitaries Uz (µ) as endomorphisms Uˆ z (µ) of Hu . Our result for these endomorphisms is as follows. Proposition 5.2. Let ω0 be an invariant limit state. Suppose that the scaling limit of A is factorizing, and let V = dν(z)Vz be the unitary of Def. 4.3. Then, the unitary operators Uˆ z (µ) : Hu → Hu , Uˆ z (µ) = Vµ.z Uz (µ)Vz∗ fulfill for any z ∈ Z, µ, µ ∈ R+ the relations Uˆ z (1) = 1, Uˆ z (µ)∗ = Uˆ µ.z (µ−1 ), Uˆ µ.z (µ )Uˆ z (µ) = Uˆ z (µ µ). If H0 is uniformly separable, one has

∗

V U0 (µ)V = (UZ(µ) ⊗ 1)

⊕

Z

dν(z)Uˆ z (µ); ↑

and for every µ > 0, there is a null set N ⊂ Z such that for any (x, Λ) ∈ P+ and any z ∈ Z\N , Uˆ z (µ)Uu (x, Λ) = Uu (µx, Λ)Uˆ z (µ). Proof. It is clear that Uˆ z (µ), defined as above, are unitary, and their composition relations follow from Prop. 5.1 (iii). Now let H0 be uniformly separable. Then, together with V , also V ∗ is decomposable. By a short computation, one finds for any χ ∈ L 2 (Z, ν, Hu ): ⊕ dν(z) Vz Uµ−1 .z (µ)Vµ∗−1 .z χ (µ−1 .z). (5.5) V U0 (µ)V ∗ χ = Z

Now, following Prop. 5.1 (ii), the operator UZ(µ) ⊗ 1 acts on vectors χ via ⊕ (UZ(µ) ⊗ 1)χ = dν(z) χ (µ−1 .z). Z

(5.6)

Together with Eq. (5.5), this entails V U0 (µ)V ∗ χ = (UZ(µ) ⊗ 1)

⊕

Z

dν(z) Vµ.z Uz (µ)Vz∗ χ (z),

(5.7)

40

H. Bostelmann, C. D’Antoni, G. Morsella

of which the second assertion follows. Further, one computes from V U0 (x, Λ)V ∗ = 1 ⊗ Uu (x, Λ) and from Eq. (5.7) that ⊕ ⊕ ˆ dν(z)Uz (µ)Uu (x, Λ) = dν(z)Uu (µx, Λ)Uˆ z (µ). (5.8) Z

Z

Uniform separability implies that the integrands agree except on a null set. This null set may depend on x, Λ. However, we can choose it uniformly on a countable dense set of the group, and hence, by continuity, uniformly for all Poincaré group elements. This shows that the dilation symmetries factorize into a central part, UZ(µ)⊗1, which

⊕ “mixes” the fibers of the direct integral, and a decomposable part, Z dν(z)Uˆ z (µ). The unitaries Uˆ z (µ) will generally depend on z; and like the Uz (µ) before, they do not necessarily fulfill a group relation, but a cocycle equation Uˆ µ.z (µ )Uˆ z (µ) = Uˆ z (µ µ),

(5.9)

as shown above, where µ.z can in general not be replaced with z. However, using the commutation relations with the other parts of the symmetry group, one sees that Uˆ z (µ )Uˆ z (µ)Uˆ z (µ µ)∗ is (a. e.) an inner symmetry of the theory Au . On the other hand, this representation property “up to an inner symmetry” cannot be avoided if such symmetries exist in the theory at all; for they might be multiplied to Vz in a virtually arbitrary fashion at any point z. In this respect, we encounter a similar situation with respect to dilation symmetries as Buchholz and Verch [BV95]. In the present context, however, it seems more transparent how this cocycle arises. Under somewhat stricter assumptions, we can prove a stronger result that avoids the ambiguities discussed above. Let us consider the case of a convergent scaling limit, per Def. 4.4. In this case, we shall see that the Uz (µ) can actually be chosen independent of z, and yield a group representation in the usual sense. Proposition 5.3. Let A have a convergent scaling limit, and let ωu be a multiplicative limit state with separable representation space Hu . Then the Poincaré group representation Uu on Hu extends to a representation of the extended symmetry group G. For any invariant limit state ω0 with associated representation U0 of G, one has V U0 (µ)V ∗ = (U0 (µ) HZ) ⊗ Uu (µ), where V is the unitary introduced in Proposition 4.10. Proof. With ωu , also every ωu ◦ δ µ is a scaling limit state. Thanks to the invariance of Aconv under dilations, we thus have for each A ∈ Aconv , πu (δ µ (A))Ωu 2 = ωu ◦ δ µ (A∗ A) = ωu (A∗ A) = πu (A)Ωu 2 .

(5.10)

This yields the existence of a unitary strongly continuous representation µ → Uu (µ) on Hu such that Uu (µ)πu (A)Ωu = πu (δ µ (A))Ωu ,

A ∈ Aconv .

(5.11)

That also implies Uu (µ)Uu (x, Λ)πu (A)Ωu = πu (α µ,x,Λ (A))Ωu ,

A ∈ Aconv ,

(5.12)

On Dilation Symmetries Arising from Scaling Limits

41

which shows that (µ, Λ, x) → Uu (µ)Uu (Λ, x) is a unitary representation of G on Hu , extending the representation of the Poincaré group. Now if V : H0 → HZ ⊗ Hu is the unitary of Prop. 4.10, a calculation shows that V U0 (µ)V ∗ π0 (C)Ω0 ⊗ πu (A)Ωu = π0 (δ µ (C))Ω0 ⊗ πu (δ µ (A))Ωu , C ∈ Z(A), A ∈ Aconv , (5.13) which entails that V U0 (µ)V ∗ = (U0 (µ) HZ) ⊗ Uu (µ).

Thus, the limit theory is “dilation covariant” in the usual sense, with a unitary acting on Hu . Considering the unitaries 1 ⊗ Uu (g), we actually get a unitary representation in any limit theory, even corresponding to multiplicative states. Only for compatibility with the scaling limit representation π0 it is necessary to consider invariant means, and to take U0 (µ) HZ into account. 6. Phase Space Properties In this section, we wish to investigate how the notion of phase space conditions, specifically the (quite weak) Haag-Swieca compactness condition [HS65], fits into our context, and how it transfers to the limit theory. An important aspect here is that Haag-Swieca compactness of a quantum field theory guarantees that the corresponding Hilbert space is separable; this property transfers to multiplicative limit states in certain circumstances [Buc96a]. We shall give a strengthened version of the compactness condition that guarantees our general limit spaces to be uniformly separable, a property that turned out to be valuable in the previous sections. We need some extra structures to that end. First, we consider “properly rescaled” vector-valued functions χ : R+ → H. Specifically, for A ∈ A, let AΩ denote the function λ → Aλ Ω. We set H = clos{AΩ | A ∈ A},

(6.1)

where the closure is taken in the supremum norm χ = supλ χ λ . Then H is a Banach space, in fact a Banach module over Z(A) in a natural way. Given a limit state, we transfer the limit representation π0 to vector-valued functions. To that end, consider the space C(Γ ) of Γ -continuous vector fields, as defined in the Appendix. We define η0 : H → C(Γ ) on a dense set by η0 (AΩ) := π0 (A)Ω0 .

(6.2)

This is well-defined, since one computes 1/2 π0 (A)Ω0 ∞ = sup πz (A)Ωz = sup ω z (A∗ A) z∈Z

∗

≤ ω(A A)

1/2

z∈Z

= sup Aλ Ω2 λ>0

1/2

= AΩ.

(6.3)

That also shows η0 ≤ 1. Note that η0 fulfills η0 (Cχ ) = π0 (C)η0 (χ ) for all C ∈ Z(A), χ ∈ H,

(6.4)

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H. Bostelmann, C. D’Antoni, G. Morsella

this easily being checked for χ = AΩ. So η0 preserves the module structure in this sense. Further, η0 : H → C(Γ ) clearly has dense range. It is important in our context that H is left invariant under multiplication with suitably rescaled functions of the Hamiltonian. More precisely, we denote these functions as f (H ) for f ∈ S([0, +∞)); they are defined as elements of B by f (H )λ = f (λH ), with norm f (H ) ≤ f ∞ . They act on H by pointwise multiplication. The following lemma generalizes an observation in [Buc96a]. Lemma 6.1. Let f ∈ S([0, +∞)). Then, for each χ ∈ H, we have f (H )χ ∈ H. There exists a test function g ∈ S(R) such that for all A ∈ A,

f (H )AΩ = α g AΩ := dt g(t) α t A Ω. Proof. We continue f to a test function fˆ ∈ S(R), and choose g as the Fourier transform of fˆ. One finds by spectral analysis of H that for any A ∈ A, ∞ fˆ(λE)d P(E)Aλ Ω = dt g(t)eıλH t Aλ Ω = (α g A)λ Ω. (6.5) f (λH )Aλ Ω = 0

This shows that f (H )AΩ has the proposed form, and is an element of H. Since f (λH ) ≤ f ∞ uniformly in λ, we may pass to limits in AΩ and obtain that f (H )χ ∈ H for all χ ∈ H. As a next step towards phase space conditions, let us explain a notion of compact maps adapted to our context. To that end, let E be a Banach space and F a Banach module over the commutative Banach algebra R. We say that a linear map ψ : E → F is of uniform rank 1 if it is of the form ψ = e( · ) f with e : E → R linear and continuous, and f ∈ F. Sums of n such terms are called of uniform rank n.4 We say that ψ is uniformly compact if it is an infinite sum of terms of uniform rank 1, ψ = ∞ j=0 e j ( · ) f j , where the sum converges in the Banach norm. For R = C and F a Hilbert space, these definitions reduce to the usual notions of compact or finite-rank maps. We are now in the position to consider Haag-Swieca compactness. We fix, once and < for all, an element C < ∈ Z(A) with C < ≤ 1, C < λ = 0 for λ > 1, and C λ = 1 for λ < 1/2. For a given β > 0 and any bounded region O, we consider the map Θ (β,O) : A(O) → H,

A → e−β H C < AΩ.

(6.6)

This is indeed well-defined due to Lemma 6.1. Our variant of the Haag-Swieca compactness condition, uniform at small scales, is then as follows. Definition 6.2. A quantum field theory fulfills the uniform Haag-Swieca compactness condition if, for each bounded region O, there is β > 0 such that the map Θ (β,O) is uniformly compact. We note that this property is independent of the choice of C < ; the role of that factor is to ensure that we restrict our attention to the short-distance rather than the long-distance regime. We do not discuss relations of uniform Haag-Swieca compactness with other 4 Note that the “uniform rank” is rather an upper estimate, in the sense that a map of uniform rank n may at the same time be of uniform rank n − 1.

On Dilation Symmetries Arising from Scaling Limits

43

versions of phase space conditions here. Rather, we show in Sec. 7 that the condition is fulfilled in some simple models. We now investigate how the compactness property transfers to the scaling limit. To that end, we consider the corresponding phase space map in the limit theory, (β,O )

Θ0

A → e−β H0 AΩ0 .

: A0 (O) → H0 ,

(6.7)

Its relation to Θ (β,O) is rather direct. (β,O )

Proposition 6.3. For any fixed O and β > 0, one has η0 ◦ Θ (β,O) = Θ0 (β,O ) If Θ (β,O) is uniformly compact, so is Θ0 ◦ π0 .

◦ π0 .

Proof. Given β, we choose a function gβ relating to f β (E) = exp(−β E) per Lemma 6.1. For any A ∈ A(O), we compute η0 Θ (β,O) (A) = η0 (C < α gβ AΩ) = π0 (C < )π0 (α gβ A)Ω0 (β,O )

= α0,gβ π0 (A)Ω0 = Θ0

π0 (A).

(6.8)

(β,O )

◦ π0 as proposed. Now let Θ (β,O) be uniformly comThus η0 ◦ Θ (β,O) = Θ0 pact, Θ (β,O) = j e j ( · ) f j . Then η0 can be exchanged with the infinite sum due to continuity, which yields (β,O ) Θ0 ◦ π0 = η0 (e j ( · ) f j ) = (π0 ◦ e j ( · ))(η0 f j ), (6.9) j (β,O )

using Eq. (6.4). Thus Θ0

j

◦ π0 is uniformly compact.

(β,O )

◦ π0 ⊂ C(Γ ). Since we can The above results show in particular that img Θ0 write (β,O ) Θ0 ◦ π0 (A) = dν(z) Θz(β,O) ◦ πz (A) (6.10) (β,O )

with the obvious definition of Θz , the above proposition establishes a rather strong form of compactness in the limit theory, uniform in z; note that the sum in Eq. (6.9) converges with respect to the supremum norm of C(Γ ). We now come to the main result of the section, showing that compactness in the above form implies uniform separability of the limit Hilbert space. Theorem 6.4. Suppose that the theory A fulfils uniform Haag-Swieca compactness. Then, for any limit state ω0 , the representation space H0 is uniformly separable, where the fundamental sequence can be chosen from C(Γ ). Proof. We choose a sequence of regions Ok such that Ok Rs+1 , and a sequence (βk )k∈N in R+ such that all Θ (βk ,Ok ) are uniformly compact. By Prop. 6.3 above, also (β ,O ) (k) Θ0 k k ◦ π0 are uniformly compact. Explicitly, choose e j : A(Ok ) → C(Z) and f j(k) ∈ C(Γ ) such that

(βk ,Ok )

Θ0

◦ π0 =

j

(k) e(k) j (·) fj .

(6.11)

44

H. Bostelmann, C. D’Antoni, G. Morsella (k)

We will construct a fundamental sequence using the f j . To that end, let A ∈ A(O) for some O. For k large enough, we know that (k) (β ,O ) (k) e−βk H0 π0 (A)Ω0 = Θ0 k k (π0 (A)) = e j (A) f j . (6.12) j

The sum converges in the supremum norm, i.e., uniformly at all points z. Let us choose a fixed z. Then, it is clear that (k) 2 2 (k) e−Hz πz (A)Ωz = e j (A) (z) e−Hz +βk Hz f j (z), (6.13) j (k)

noting that exp(−Hz2 +βk Hz ) is a bounded operator. Observe that (e j (A))(z) are merely numerical factors. Since A and O were arbitrary, and ∪O πz (A(O))Ωz is dense in Hz , this means (k)

e−Hz Hz ⊂ clos span{ e−Hz +βk Hz f j (z)| j, k ∈ N}. 2

2

(6.14)

Now exp(−Hz2 ) is a selfadjoint operator with trivial kernel, thus its image is dense. Hence the exp(−Hz2 + βk Hz ) f j(k) (z) are total in Hz . This holds for all z, thus {exp(−H02 + (k)

βk H0 ) f j | j, k ∈ N} is a fundamental sequence. Applying Lemma 6.1 to f (E) = exp(−E 2 + βk E), we find that the elements of the fundamental sequence lie in C(Γ ). 7. Examples We are now going to investigate the structures discussed in simple models. Particularly, we wish to show that our conditions on “convergent scaling limits” (Def. 4.4) and “uniform Haag-Swieca compactness” (Def. 6.2) can be fulfilled at least in simple situations. To that end, we first consider the situation where the theory A “at finite scales” is equipped with a dilation symmetry. Then, we investigate the real scalar free field as a concrete example. 7.1. Dilation covariant theories. We now consider the case where the net A, which our investigation starts from, is already dilation covariant. One expects that the scaling limit construction reproduces the theory A in this case, and that the dilation symmetry obtained from the scaling algebra coincides with the original one. We shall show that this is indeed the case under a mild phase space condition, and also that this implies the stronger phase space condition in Def. 6.2. This extends a discussion in [BV95, Sec. 5]. Technically, we will assume in the following that A is a local net in the vacuum sector with symmetry group G, which is generated by the Poincaré group and the dilation group. We shall denote the corresponding unitaries as U (µ, x, Λ) = U (µ)U (x, Λ). The mild phase space condition referred to is the Haag-Swieca compactness condition for the original theory: We assume that for each bounded region O in Minkowski space, there exists β > 0 such that the map Θ (β,O) : A(O) → H, A → exp(−β H )AΩ is compact. (This is equivalent to a formulation where the factor exp(−β H ) is replaced with a sharp energy cutoff, as used in [HS65].)

On Dilation Symmetries Arising from Scaling Limits

45

Theorem 7.1. Let A be a dilation covariant net in the vacuum sector which satisfies the Haag-Swieca compactness condition. Then A has a convergent scaling limit. ˆ Proof. For each O, we introduce the C∗ -subalgebra A(O) ⊂ A(O) of those elements A ∈ A(O) for which g → αg (A) is norm continuous. Since the symmetries are impleˆ mented by continuous unitary groups, A(O) is strongly dense in A(O). We then define a C∗ -subalgebra of the scaling algebra A(O), ˆ Aconv (O) := {λ → U (λ)AU (λ)∗ | A ∈ A(O)},

(7.1)

and the α-invariant algebra Aconv ⊂ A is defined as the C∗ -inductive limit of the Aconv (O). It is evident that condition (i) in Def. 4.4 is fulfilled by Aconv , as the functions λ → ω(Aλ ) are constant in the present case. Now let ω0 be a multiplicative limit state. With similar arguments5 as in [BV95, Prop. 5.1], using the Haag-Swieca compactness condition, we can construct a net isomorphism φ from A0 to A, which has the property ˆ that if Aλ = U (λ)AU (λ)∗ with A ∈ A(O), then φ(π0 (A)) = A. From this, and from ˆ the strong density of A(O) in A(O), it follows that π0 (Aconv (O)) is strongly dense in A0 (O). Thus condition (ii) in Def. 4.4 is satisfied as well. Since the isomorphism φ above can be shown to intertwine the respective vacuum states, it is actually the adjoint action of a unitary W : H0 → H. We remark that H, and then also H0 , is separable due to the Haag-Swieca compactness condition. Then as a consequence of Prop. 5.3, A has a factorizing scaling limit and the representation of the symmetry group G0 factorizes too. It is also clear from the proof above and from that of Thm. 4.9, that Au is unitarily equivalent to A through the operator W , taken here for πu in place of π0 . Furthermore, this W also intertwines the dilations in the scaling limit with those of the underlying theory. Corollary 7.2. Under the hypothesis of Thm. 7.1, there holds W Uu (µ)W ∗ = U (µ). ˆ Proof. It is sufficient to verify the relation on vectors of the form AΩ with A ∈ A(O). For such vectors it follows by noting that Aλ = U (λ)AU (λ)∗ is an element of Aconv (O), ˆ and that δ µ (A)λ = U (λ)αµ (A)U (λ)∗ with αµ (A) ∈ A(µO). For showing the consistency of our definitions, we now prove that the Haag-Swieca compactness condition at finite scales, together with dilation covariance, implies our uniform compactness condition of Def. 6.2. Proposition 7.3. If the dilation covariant local net A fulfills the Haag-Swieca compactness condition, then it also fulfills uniform Haag-Swieca compactness. Proof. Let O be fixed, and let β > 0 such that Θ (β,O) is compact; Θ (β,O) =

∞

e j ( · ) f j with e j ∈ A(O)∗ , f j ∈ H.

(7.2)

j=1 5 Since in contrast to [BV95], we here take the A (O) to be W∗ algebras, we need to amend the argument 0 in step (d) of [BV95, Prop. 5.1] slightly: We first construct the isomorphism φ on the C∗ algebra π0 (A(O)), and then continue it to the weak closure; cf. [KR97, Lemma 10.1.10].

46

H. Bostelmann, C. D’Antoni, G. Morsella

Taking the normal part, we can in fact arrange that e j ∈ A(O)∗ . (See [BDF87, Lemma 2.2] for a similar argument.) Now define e j : A(O) → Z(A) by e j (A)λ = e j U (λ)∗ C < (7.3) λ Aλ U (λ) . That the image is indeed in Z(A), i.e., continuous under δ µ , is seen as follows. We compute for λ, µ > 0, e (A)λµ − e (A)λ = e j U (λµ)∗ (C < A)λµ U (λµ) − U (λ)∗ (C < A)λ U (λ) j j ≤ e j δ µ (C < A)−C < A + e j U (µ)∗ · U (µ) −e j C < A. (7.4) Now as µ → 0, the first summand vanishes due to norm continuity of δ µ on A, and the second due to strong continuity of U (µ); both limits are uniform in λ. Thus δ µ acts continuously on e j (A). Further, we define f j ∈ H by f

jλ

= U (λ) f j .

(7.5)

ˆ for suitable O This is indeed an element of H: Namely, given > 0, choose A ∈ A(O) ˆ is as in the proof of Thm. 7.1. Then, Aλ = U (λ)AU (λ)∗ such that AΩ− f j < ; here A defines an element of A, and AΩ − f j ≤ AΩ − f j < . Hence f j is contained in the closure of AΩ. Now we are in the position to show that Θ (β,O) = j e j f j . Let J ∈ N be fixed. It is straightforward to compute that for any A ∈ A(O) and λ > 0, J J Θ (β,O) (A) − e j (A) f j = U (λ) Θ (β,O) (Bλ ) − e j Bλ ) f j , j=1

λ

j=1

where Bλ = U (λ)∗ C < λ Aλ U (λ).

(7.6)

Note here that Bλ ∈ A(O) for any λ. This entails J J (β,O) Θ − e j ( · ) f j ≤ Θ (β,O) − e j ( · ) f j A. j=1

(7.7)

j=1

The right-hand side vanishes as J → ∞, as a consequence of the compactness condition at finite scales. This shows that Θ (β,O) is uniformly compact. 7.2. The scaling limit of a free field. We now show in a simple, concrete example from free field theory that the model has a convergent scaling limit in the sense of Def. 4.4. Specifically, we consider a real scalar free field of mass m > 0, in 2+1 or 3+1 spacetime dimensions. The algebraic scaling limit of this model is the massless real scalar field; this was as already discussed in [BV98], and in parts we rely on the arguments given there. However, we need to consider several aspects that were not handled in that work, in particular continuity aspects of Poincaré and dilation transformations. Also, as mentioned before, in contrast to [BV98] we deal with weakly closed local algebras at fixed scales and in the limit theory.

On Dilation Symmetries Arising from Scaling Limits

47

We start by recalling, for convenience, the necessary notations and definitions from [BV98]. We consider the Weyl algebra W over D(Rs ), s = 2, 3: ı W ( f )W (g) = e− 2 σ ( f,g) W ( f + g), σ ( f, g) = Im dx f (x)g(x). (7.8) ↑

Then, we define a mass dependent automorphic action of P+ on W by (m)

(m)

αx,Λ (W ( f )) = W (τx,Λ f ),

(7.9)

↑

where the action τ (m) of P+ on D(Rs ) is defined by the following formulas. In those, we write f˜(p) = (2π )−s/2 dx f (x)e−ixp for the Fourier transform of f, which we split into f˜ = f˜R + ı f˜I , where f R = Re f and f I = Im f ; also, ωm (p) := m 2 + |p|2 . (τx(m) f )(y) := f (y − x), (m)

˜ ˜ f )∼ R (p) := cos(tωm (p)) f R (p) − ωm (p) sin(tωm (p)) f I (p),

(m)

−1 ˜ f )∼ sin(tωm (p)) f˜R (p), I (p) := cos(tωm (p)) f I (p) + ωm (p)

(τt (τt

(m)

(τΛ f )∼ R (p) := ϕΛ (ωm (p), p),

(7.11)

f

−1 (τΛ(m) f )∼ I (p) := ωm (p) ψΛ (ωm (p), p). f

f

(7.10)

(7.12)

f

Here the functions ϕΛ , ψΛ : Rs+1 → C are defined by 1 1 ˜ f f R (Λ−1 p) + f˜R (ΛT p) + (Λ−1 p)0 f˜I (Λ−1 p) − (ΛT p)0 f˜I (ΛT p) , ϕΛ ( p) := 2 2ı 1 1˜ f −1 T f R (Λ p)− f˜R (Λ p) + (Λ−1 p)0 f˜I (Λ−1 p) + (ΛT p)0 f˜I (ΛT p) , ψΛ ( p) := − 2ı 2 (7.13) where we use the notation Λp = ((Λp)0 , Λp) for Λ ∈ L, p ∈ Rs+1 . One verifies that all the above expressions are even in ωm (p), which, due to the analytic properties of f˜, (m) implies that τx,Λ D(Rs ) ⊂ D(Rs ). We also introduce the action σ of dilations on W by σλ (W ( f )) = W (δλ f ),

(7.14)

(δλ f )(x) := λ−(s+1)/2 ( f R )(λ−1 x) + ıλ−(s−1)/2 ( f I )(λ−1 x).

(7.15)

with

(m)

(λm)

It holds that αλx,Λ ◦ σλ = σλ ◦ αx,Λ . Finally we define the vacuum state of mass m ≥ 0 on W as 1

ω(m) (W ( f )) = e− 2 f m , where f 2m :=

2

2 1 dpωm (p)−1/2 f˜R (p) + ı ωm (p)1/2 f˜I (p) . 2 Rs

(7.16)

(7.17)

48

H. Bostelmann, C. D’Antoni, G. Morsella (m)

There holds clearly ω(m) ◦ αx,Λ = ω(m) , ω(m) ◦ σλ = ω(λm) . Proceeding now along the lines of [BV98], we consider the GNS representation (π (0) , H(0) , Ω (0) ) of W induced by the massless vacuum state ω(0) . For each m ≥ 0, we define a net O → A(m) (O) of von Neumann algebras on H(0) as (m) A(m) (ΛO B + x) := {π (0) αx,Λ (W (g)) : supp g ⊂ B} , (7.18) where O B is any double cone with base the open ball B in the time t = 0 plane. For other open regions we can define the algebras by taking unions, but this will not be relevant for the following discussion. Due to the local normality of the different states ω(m) , m ≥ 0, with respect to each other [EF74], these nets are isomorphic to the nets generated by the free scalar field of mass m on the respective Fock spaces. From now on, we will identify elements of W and of A(m) , and therefore we will drop the indication of the representation π (0) . We denote the (dilation and Poincaré covariant) scaling algebra associated to A(m) by A(m) . The next lemma generalizes the results of [BV98, Lemma 3.2] to the present situation. ↑

Lemma 7.4. Let a > 1 and h D ∈ D((1/a, a)), h P ∈ D(P+ ), f ∈ D(Rs ), and consider the function W : R+ → A(m) given by dµ (m) W λ := d x dΛ h D (µ)h P (x, Λ)αµλx,Λ ◦ σµλ (W ( f )), ↑ µ R + × P+ where dΛ is the left-invariant Haar measure on the Lorentz group and the integral is to be understood in the weak sense. Then: (i) there exists a double cone O such that W ∈ A(m) (O); (ii) there holds in the strong operator topology, dµ (0) −1 d x dΛ h D (µ)h P (x, Λ)αµx,Λ lim σλ (W λ ) = ◦ σµ (W ( f )) =: W0 ; ↑ µ λ→0+ R + × P+ (iii) the span of the operators W0 of the form above, with W ∈ A(m) (O) for fixed O, is strongly dense in A(0) (O). (m)

Proof. Since W is the convolution, with respect to the action (µ, x, Λ) → α µ,x,Λ , of the function h D ⊗ h P with the bounded function λ → σλ (W ( f )), and thanks to the support properties of h D , h P and f , (i) follows. In order to prove (ii), we start by observing that, for each vector χ ∈ H(0) , dµ σλ−1 (W λ ) − W0 χ ≤ d x dΛ |h D (µ)h P (x, Λ)| ↑ µ R + × P+ (µλm) (0) (7.19) × W δµ τx,Λ f − W δµ τx,Λ f χ . Now f → W ( f ) is known to be continuous with respect to · 0 on the initial space and the strong operator topology on the target space [BR81, Prop. 5.2.4]. Since the norm ↑ · 0 is δµ -invariant, it therefore suffices to show that for each fixed (x, Λ) ∈ P+ , (m) (0) lim + τx,Λ f − τx,Λ f 0 = 0; (7.20) m→0

On Dilation Symmetries Arising from Scaling Limits

49

for (ii) then follows from the dominated convergence theorem. In order to show Eq. (7.20), we introduce the following family of functions f (m) (p) of two arguments: F = f : [0, 1] × Rs → C f (m) ( · ) ∈ D(Rs ) for each fixed m ∈ [0, 1]; lim f (m) (p) = f (0) (p) for each fixed p ∈ Rs ;

m→0

f (m) (p)| ≤ g(p) . ∃g ∈ S(Rs ) : ∀m ∈ [0, 1] ∀p ∈ Rs : |

(7.21)

It is clear that for f ∈ F, one has f (m) − f (0) 0 → 0 as m → 0 per dominated convergence. Also, each f ∈ D(Rs ), with trivial dependence on m, falls into F. So it (·) leaves F invariant, where remains to show that the (naturally defined) action of τx,Λ (·)

it suffices to check this for a set of generating subgroups. Indeed, τx,Λ F ⊂ F is clear for spatial translations and rotations. For time translations and boosts, it was already remarked that D(Rs ) is invariant under these at fixed m, and pointwise convergence as m → 0 is clear. Further, from Eqs. (7.11) and (7.13), one sees that f˜ is modified by at most polynomially growing functions, uniform in m ≤ 1, hence uniform S-bounds (m) (m) f as well. (Again, it enters here that all expressions are even in ωm , for hold for τx,Λ which it is needed that f˜ is smooth.) This completes the proof of (ii). Finally, (iii) follows from the observation that as h D and h P converge to delta functions, W0 converges strongly to W ( f ) thanks to the strong continuity of the function (0) (µ, x, Λ) → αµx,Λ ◦ σµ (W ( f )); and of course the span of the Weyl operators with supp f ⊂⊂ O is strongly dense in A(0) (O). Using the above lemma, we can prove the following. Theorem 7.5. The theory of the massive real scalar free field in s = 2, 3 spatial dimensions has a convergent scaling limit. (m)

Proof. Consider the C∗ -subalgebra Aconv (O) of A(m) (O) which is generated by the ele(m) ments W ∈ A(m) (O) defined in the previous lemma, and let Aconv be the corresponding (m) quasi-local algebra. Since α µ,x,Λ (W ) is again an element of the same form, just with (m)

shifted function h D ⊗ h P , the algebra Aconv is α (m) invariant. In order to verify that (m) , we start by observing that, λ → ω(m) (Aλ ) has a limit, as λ → 0, for all A ∈ Aconv thanks to Lemma 7.4 (ii) and to the fact that σλ is unitarily implemented on H(0) , for each such A there exists limλ→0+ σλ−1 (Aλ ) =: A in the strong operator topology. Then (m) (O) there holds the inequality if A ∈ Aconv |ω(m) (Aλ ) − ω(0) (A)| ≤ (ω(m) − ω(0) ) A(0) (λO)A + |ω(0) (σλ−1 (Aλ )) − ω(0) (A)|.

(7.22) Together with the fact that limλ→0+ (ω(m) − ω(0) ) A(0) (λO) = 0 as a consequence of the local normality of ω(m) with respect to ω(0) , this implies that limλ→0+ ω(m) (Aλ ) = (m) (m) ω(0) (A) for all A in some local algebra Aconv (O). This then extends to all of Aconv by density.

50

H. Bostelmann, C. D’Antoni, G. Morsella (m)

It remains to show that for multiplicative limit states, π0 (Aconv ∩ A(m) (O)) is weakly (m) dense in A0 (O) = π0 (A(m) (O)) for any O. To that end, we use similar methods as in Thm. 7.1. With O fixed and U the ultrafilter that underlies the limit state, we define φ : π0 (A(m) (O)) → A(0) (O), π0 (A) → lim σλ−1 (Aλ ),

(7.23)

U

with the limit understood in the weak operator topology. Using methods as in [BV98, Sec. 3], one can show that φ is indeed a well-defined isometric ∗ homomorphism, which further satisfies ω0 = ω(0) ◦ φ on the domain of φ. Hence φ is given by the adjoint action of a partial isometry, and can be continued by weak closure to a ∗ homomorphism (m) φ − : A0(m) (O) → A(0) (O). On the other hand, for W ∈ Aconv (O) as in Lemma 7.4, one finds φ(π0 (W )) = W0 by (ii) of that lemma. However, the double commutant of those W0 is all of A(0) (O), see (iii) of the same lemma. So φ − is in fact an isomorphism; and inverting φ − , one obtains the proposed density. 7.3. Phase space conditions in the free field. Our aim in this section is to prove the uniform compactness condition of Sec. 6 in the case of a real scalar free field, again of mass m ≥ 0, in 3 + 1 or higher dimensions. To that end, we will use a short-distance expansion of local operators, very similar to the method used in the Appendix of [Bos05b], however in a refined formulation. In this section, we will consider a fixed mass m ≥ 0 throughout, and therefore we drop the label (m) from the local algebras, the vacuum state, and the Hilbert space norm for simplicity. We rewrite the Weyl operators of Eq. (7.8) in terms of the familiar free field φ and its time derivative ∂0 φ in the time-0 plane, (7.24) W ( f ) = exp ı φ(Re f ) − ∂0 φ(Im f ) , f ∈ D(Rs ). Also, we need to introduce some multi-index notation. Given n ∈ N0 , we consider multi-indexes ν = (ν1 , . . . , νn ) ∈ ({0, 1} × Ns0 )n ; that is, each ν j has the form ν j = (ν j0 , ν j1 , . . . , ν js ) with ν j0 ∈ {0, 1}, ν jk ∈ N0 for 1 ≤ k ≤ s. These indices will be ν ν used for labeling derivatives in configuration space, ∂ ν j = ∂0 j0 . . . ∂s js . We denote νj! =

s

ν jk ! , ν! =

k=0

n

ν j ! , |ν j | =

j=1

s k=0

ν jk , |ν| =

n

|ν j |.

(7.25)

j=1

Now we can define the following local fields as quadratic forms on a dense domain: φn,ν = :

n

∂ ν j φ: (0).

(7.26)

j=1

These will form a basis in the space of local fields at x = 0. Further, for given r > 0, we choose a test function h ∈ D(Rs ) which is equal to 1 for |x| ≤ r ; then we set ν h ν j (x) = sk=1 xk jk h(x). This is used to define the following functionals on A(Or ).

ı n (−1) j ν j0 (1−ν10 ) Ω [∂0 φ(h ν1 ), [. . . [∂0(1−νn0 ) φ(h νn ), A] . . .]Ω . (7.27) σn,ν (A) = n! ν! One sees that this definition is independent of the choice of h. We can therefore consistently consider σn,ν as a functional on ∪O A(O), though its norm may increase as O grows large. The significance of these functionals becomes clear in the following lemma.

On Dilation Symmetries Arising from Scaling Limits

51

Lemma 7.6. We have for all Weyl operators A = W ( f ) with f ∈ D(Rs ), A=

∞

σn,ν (A) φn,ν

n=0 ν

in the sense of matrix elements between vectors of finite energy and finite particle number. Proof. We indicate only briefly how this combinatorial formula can be obtained; see also [Bos00, Sec. 7.2] and [Bos05b, Appendix]. Using Wick ordering, we rewrite Eq. (7.24) for the Weyl operators as 2 W ( f ) = e− f /2 : exp ı φ(Re f ) − ∂0 φ(Im f ) : = e− f

2 /2

∞ n n ı : φ(Re f ) − ∂0 φ(Im f ) : . n!

(7.28)

n=0

Now, in each factor of the n-fold product :(. . .)n:, we expand both Re f and Im f into a Taylor series in momentum space. Note that this is justified, since those functions have compact support in configuration space, since they are evaluated in scalar products with functions of compact support in momentum space, and since the sum over n is finite in matrix elements. The Taylor expansion in momentum space corresponds to an expansion in derivatives of δ-functions in configuration space, and this is what produces the fields φn,ν localized at 0. We then need to identify the remaining factors with σn,ν (A), which is done using the known commutation relations of W ( f ) with φ and ∂0 φ. Our main task will now be to extend the above formula to more general states and more observables, by showing that the sum converges in a suitable norm. To that end, we need estimates of the fields and functionals involved. Lemma 7.7. Given s ≥ 2, m ≥ 0, and r0 > 0, there exists a constant c such that the following holds for any n, ν: √ e−β H φn,ν Ω ≤ cn (n!)1/2 ν! (β/2 s)−|ν|−n(s−1)/2 for any β > 0, (a) P(E)φn,ν P(E) ≤ cn E |ν|+n(s−1)/2 σn,ν A(Or ) ≤ c (n!) n

−1/2

(ν!)

−1

for any E > 0, provided s ≥ 3, (3r )

|ν|+n(s−1)/2

for any r ≤ r0 .

(b) (c)

Proof. One has n n √ e−β H φn,ν Ω = a ∗ (e−βω p ν j ) Ω ≤ n! e−βω p ν j . j=1

(7.29)

j=1

For the single-particle vectors on the right-hand side, one uses scaling arguments to obtain the estimate √ e−βω p ν j ≤ c1 ν j ! (β/2 s)−|ν j |−(s−1)/2 for β > 0, (7.30) where c1 is a constant (depending on s and m). This implies (a). For (b), we use energy bounds for creation operators a ∗ ( f ), similar to [BP90, Sec. 3.3]. One finds for single 1/2 particle space functions f 1 , . . . , f k in the domain of ωm , P(E)a ∗ (ωm f 1 ) . . . a ∗ (ωm f k ) ≤ E k/2 Q(E) f 1 . . . Q(E) f k , 1/2

1/2

(7.31)

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H. Bostelmann, C. D’Antoni, G. Morsella

with Q(E) being the energy projector for energy E in single particle space. This leads us to P(E)φn,ν P(E) ≤ 2n E n/2

n

−1/2 ν j

ωm

p χ E ,

(7.32)

j=1

ν where p ν j = sk=0 pk jk , and χ E is the characteristic function of ωm (p) ≤ E. For the single-particle functions, one obtains −1/2 ν j

ωm

p χ E ≤ c2 E |ν j |+(s−2)/2

(7.33)

with a constant c2 , which implies (b). Now consider the functional σn,ν . We choose a real-valued test function h 1 ∈ D(R+ ) such that h 1 (x) ≤ 1 for all x, h 1 (x) = 1 on [0, 1], and h 1 (x) = 0 for x ≥ 2. Then, h r (x) = h 1 (|x|/r ) is a valid choice for the test function used in the definition of σn,ν A(Or ), see Eq. (7.27). Expressing the fields φ there in annihilation and creation operators, and writing each commutator as a sum of two terms, we obtain n 4n (1−ν j0 ) σn,ν A(Or ) ≤ √ ωm h r,ν j . n! ν! j=1

(7.34)

Again, we use scaling arguments for the single-particle space functions and obtain for r ≤ r0 , (1−ν j0 )

ωm

h r,ν j ≤ c3 (3r )|ν j |+(s−1)/2

with a constant c3 that depends on r0 . This yields (c).

(7.35)

We now define the “scale-covariant” objects that will allow us to expand the maps Θ (β,O) in a series. They are constructed of the fields φn,ν and the functionals σn,ν by multiplication with appropriate powers of λ. We begin with the quantum fields. Proposition 7.8. For any n, ν and β > 0, the function χ n,ν,β : λ → λ|ν|+n(s−1)/2 e−βλH φn,ν Ω √ defines an element of H, with norm estimate χ n,ν,β ≤ cn (n!)1/2 ν!(β/2 s)−|ν|−n(s−1)/2. Proof. We use techniques from [BDM09], and adopt the notation introduced there. In particular, R denotes the function R λ = (1 + λH )−1 , and we write A() = supλ R λ Aλ R λ , where Aλ may be unbounded quadratic forms. Let n, ν be fixed in the following. We set φ λ = λ|ν|+n(s−1)/2 φn,ν .

(7.36)

From Lemma 7.7, one sees that P(E/λ)φ λ P(E/λ) is bounded uniformly in λ. Hence, applying [BDM09, Lemma 2.6], we obtain φ() < ∞ for sufficiently large . Also, the action g → α g φ of the symmetry group on φ (which extends canonically from bounded operators to quadratic forms) is continuous in some · () : This is clear for

On Dilation Symmetries Arising from Scaling Limits

53

translations by the energy-damping factor; for dilations it is immediate from the definition; and for Lorentz transformations it holds since they act by a finite-dimensional matrix representation on φn,ν . Thus, φ is an element of the space Φ defined in [BDM09, Eq. (2.39)]. Moreover, each φ λ is clearly an element of the Fredenhagen-Hertel field content ΦFH . Thus, [BDM09, Thm. 3.8] provides us with a sequence (An ) in A(O), with O a fixed neighborhood of zero, such that An − φ() → 0 as n → ∞. Now since exp(−β H )R − < ∞, we obtain exp(−β H )(An − φ)Ω → 0. Note here that exp(−β H )An Ω ∈ H by Lemma 6.1. Hence exp(−β H )φΩ = χ n,ν,β lies in H, since this space is closed in norm. The estimate for χ n,ν,β follows directly from Lemma 7.7(a). Next, we rephrase the functionals σn,ν as maps from the scaling algebra A to its center. Lemma 7.9. For any n, ν, the definition (σ n,ν (A))λ = λ−|ν|−n(s−1)/2 σn,ν ((C < A)λ ) yields a linear map σ n,ν : ∪O A(O) → Z(A), with its norm bounded by √ σ n,ν A(Or ) ≤ cn ( n! ν!)−1 (3r )|ν|+n(s−1)/2 for r ≤ r0 ; here r0 , c are the constants of Lemma 7.7. Proof. The norm estimate is a consequence of Lemma 7.7(c), where one notes that (C < A)λ ∈ A(Oλr ) for λ ≤ 1, and (C < A)λ = 0 for λ > 1, so that these operators are always contained in A(Or0 ). It remains to show that σ n,ν (A) ∈ Z(A), i.e., that µ → δ µ (σ n,ν (A)) is continuous. But this follows from continuity of µ → δ µ A and the definition of σ n,ν . We are now in the position to prove that n,ν σ n,ν χ n,ν,β is a norm convergent expansion of the map Θ (β,O) . Theorem 7.10. Let s ≥ 3. For each r > 0, there exists β > 0 such that Θ (β,Or ) =

∞

σ n,ν ( · )χ n,ν,β .

n=0 ν

Proof. We will show below that the series in the statement converges absolutely in the Banach space B(A(Or ), H), i.e. that ∞

σ n,ν A(Or )χ n,ν,β < ∞.

(7.37)

n=0 ν

Once this has been established, the assertion of the theorem is obtained as follows. From Lemma 7.6, we know that ∞ n=0 ν

λ−|ν|−n(s−1)/2 σn,ν (A)(χ n,ν,β )λ = e−λβ H AΩ

(7.38)

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H. Bostelmann, C. D’Antoni, G. Morsella

at each fixed λ, whenever A is a linear combination of Weyl operators, and when evaluated in scalar products with vectors from a dense set. But (7.37) also shows that the left hand side of (7.38) converges in B(A(λOr ), H), and it is therefore strongly continuous for A in norm bounded parts of A(λOr ). Then by Kaplansky’s theorem (7.38) holds for in H. Finally, this entails for all A ∈ A(Or ) that any A ∈ A(λOr ) as an equality (β,Or ) (A) at each fixed λ > 0, i.e. the statement. σ (A) (χ ) = Θ λ n,ν,β λ λ n,ν n,ν We √ now prove Eq. (7.37). Let r0 > 0 be fixed in the following, and r < r0 . Set a := 6 s. From Prop. 7.8 and Lemma 7.9, we obtain the estimate σ n,ν A(Or )χ n,ν,β ≤ c2n (ar/β)|ν|+n(s−1)/2 s n n = c2 (ar/β)(s−1)/2 (ar/β)ν jk . (ar/β)ν j0 j=1

(7.39)

k=1

Factorizing the sum over multi-indexes ν accordingly, we obtain at fixed n, ν

(ar/β)(s−1)/2 n σ n,ν A(Or )χ n,ν,β ≤ 2c2 , (1 − ar/β)s

(7.40)

where we suppose ar/β < 1, and where each sum over ν j0 ∈ {0, 1} has been estimated by introducing a factor of 2. Now if we choose r/β small enough, we can certainly achieve that the expression in Eq. (7.40) is summable over n as a geometric series, and hence the series in Eq. (7.37) converges. This establishes the phase space condition of Def. 6.2 in our context. Corollary 7.11. The theory of a real scalar free field of mass m ≥ 0 in 3 + 1 or more space-time dimensions fulfills the uniform Haag-Swieca compactness condition. While our goal was to show that the maps Θ (β,O) are uniformly compact, it follows from Eq. (7.37) that they are actually uniformly nuclear at all scales, or by a slightly modified argument, even uniformly p-nuclear for any 0 < p ≤ 1. So we can generalize the somewhat stronger Buchholz-Wichmann condition [BW86] to our context. Several other types of phase space conditions can be derived with similar methods as in Thm. 7.10 as well. Particularly, one can show for s ≥ 3 that the sum n,ν σn,ν φn,ν converges in norm under a cutoff in energy E and restriction to a fixed local algebra A(Or ), with estimates uniform in E · r , where this product is small. This implies that Phase Space Condition II of [BDM09], which guarantees a regular behavior of pointlike fields under scaling, is fulfilled for those models. 8. Conclusions In this paper, we have considered short-distance scaling limits in the model independent framework of Buchholz and Verch [BV95]. In order to describe the dilation symmetry that arises in the limit theory, we passed to a generalized class of limit states, which includes states invariant under scaling. The essential results of [BV95] carry over to this generalization, including the structure of Poincaré symmetries and geometric modular action. However, the dilation invariant limit states are not pure. Rather, they can be decomposed into states of the Buchholz-Verch type, which are pure in two or more spatial

On Dilation Symmetries Arising from Scaling Limits

55

dimensions. This decomposition gives rise to a direct integral decomposition of the Hilbert space of the limit theory, under which local observables, Poincaré symmetries, and other relevant objects of the theory can be decomposed—except for dilations. The dilation symmetry has a more intricate structure, intertwining the pure components of the limit state. The situation simplifies if we consider the situation of a “unique limit” in the classification of [BV95]; our condition of a “factorizing limit” turned out to be equivalent modulo technicalities. Under this restriction, the dilation unitaries in the limit are, up to a central part, decomposable operators. The decomposed components do not necessarily fulfill a group relation though, but a somewhat weaker cocycle relation. Only under stronger assumptions (“convergent scaling limits”) we were able to show that the dilation symmetry factorizes into a tensor product of unitary group representations. It is unknown at present which type of models would make the generalized decomposition formulas necessary. In this paper, we have only considered very simple examples, which all turned out to fall into the more restricted class of convergent scaling limits. However, thinking e.g. of infinite tensor products of free fields with increasing masses as suggested in [Buc96a, Sec. 5], it may well be that some models violate the condition of uniqueness of the scaling limit, or even exhibit massive particles in the limit theory. In this case, the direct integral decomposition would be needed to obtain a reasonable description of dilation symmetry in the limit, with the symmetry operators intertwining fibers of the direct integral that correspond to different masses. As a next step in the analysis of dilation symmetries in the limit theory, one would like to investigate further the deviation of the theory at finite scales from the idealized dilation covariant limit theory; so to speak, the next-order term in the approximation λ → 0. This would be interesting e.g. for applications to deep inelastic scattering, which can currently only be treated in formal perturbation theory. It is expected that the dilation symmetry in the limit also contains some information about these next-order terms. Our formalism, however, does at the moment not capture these next-order approximations, and would need to be generalized considerably. Further, it would be interesting to see whether the dilation symmetry we analyzed can be used to obtain restrictions on the type of interaction in the limit theory, possibly leading to criteria for asymptotic freedom. Here we do not refer to restrictions on the form of the Lagrangian, a concept that is not visible in our framework. Rather, we think that dilation symmetries should manifest themselves in the coefficients of the operator product expansion [Bos05a,BDM09,BF08] or in the general structure of local observables [BF77]. In this context, it seems worthwhile to investigate simplified low-dimensional interacting models, such as the 1+1 dimensional massive models with factorizing S matrix that have recently been rigorously constructed in the algebraic framework [Lec08]. By abstract arguments, these models possess a scaling limit theory in our context. Just as in the Schwinger model [Buc96b,BV98], one expects here that even the limit theory for multiplicative states has a nontrivial center. This may be seen as a peculiarity of the 1+1 dimensional situation; we have not specifically dealt with this problem in the present paper. But neglecting these aspects, one would expect that the limit theory corresponds to a massless (and dilation covariant) model with factorizing S matrix, although it would probably not have an interpretation in the usual terms of scattering theory. Such models of “massless scattering” have indeed been considered in the physics literature; see [FS94] for a review. Their mathematical status as quantum field theories, however,

56

H. Bostelmann, C. D’Antoni, G. Morsella

remains largely unclear at this time. Nevertheless, one should be able to treat them with our methods. In fact, these examples may give a hint to the restrictions on interaction that the dilation symmetry implies. At least in a certain class of two-particle S matrices—those which tend to 1 at large momenta—one expects that the limit theory is chiral, i.e., factors into a tensor product of two models living on the left and right light ray, respectively. On the other hand, the theories we consider are always local; and for chiral local models, dilation covariance is the essential property that guarantees conformal covariance [GLW98]. So if those models have a nontrivial scaling limit at all, they underly quite rigid restrictions, since local conformal chiral nets are—at least partially—classifiable in a discrete series [KL04]. The detailed investigation of these aspects of factorizing S matrix models is however the subject of ongoing research, and some surprises are likely to turn up. Acknowledgements. The authors are obliged to Laszlo Zsido and Michael Müger for helpful discussions. They also profited from financial support by the Erwin Schrödinger Institute, Vienna, and from the friendly atmosphere there. HB further wishes to thank the II. Institut für Theoretische Physik, Hamburg, for their hospitality.

A. Direct Integrals of Hilbert Spaces In our investigation, we make use of the concept of direct integrals of Hilbert spaces, H = Z dν(z)Hz , where the integral is defined on some measure space (Z, ν). Due to difficult measure-theoretic problems, the standard literature treats these direct integrals only under separability assumptions on the Hilbert spaces involved; see e.g. [KR97]. These are however not a priori implied in our analysis; and even where we make such assumptions, we need to apply them with care. While we can often reasonably assume the “fiber spaces” Hz to be separable, the measure space Z will, in our applications, be of a very general nature, and even L 2 (Z, ν) is known to be non-separable in some situations. The concept of direct integrals can be generalized to that case. Since however the literature on that topic6 is somewhat scattered and not easily accessible, we give here a brief review for the convenience of the reader, restricted to the case that concerns us. In the following, let Z be a compact topological space and ν a finite regular Borel measure on Z. For each z ∈ Z, we consider a Hilbert space Hz with scalar product · | · "z and associated norm · z . Elements χ ∈ z∈Z Hz will be called vector fields and alternatively denoted as maps, z → χ (z). Direct integrals of this field of Hilbert spaces Hz over Z are not unique, but depend on the choice of a fundamental family. Definition A.1. A fundamental family is a linear subspace Γ ⊂ z∈Z Hz such that for every χ ∈ Γ , the function z → χ (z)2z is ν-integrable. If the same function is always continuous, we say that Γ is a continuous fundamental family. The continuity aspect will be discussed further below, for the moment we focus on measurability. Eachfundamental family Γ uniquely extends to a minimal vector space Γ¯ , with Γ ⊂ Γ¯ ⊂ z∈Z Hz , which has the following properties [Wil70, Corollary 2.3]: 6 In the general case, we largely follow Wils [Wil70], however with some changes in notation. Other, somewhat stronger notions of direct integrals exist, e.g. [God51, Ch. III], [Seg51]; see [Mar69] for a comparison. Under separability assumptions (Definition A.3), all these notions agree, and we are in the case described in [Tak79, Ch. IV.8], [Dix81, Part II Ch. 1].

On Dilation Symmetries Arising from Scaling Limits

(i) (ii) (iii) (iv)

57

z → χ (z)2z is ν-integrable for all χ ∈ Γ¯ . If for χ ∈ z Hz , there exists χˆ ∈ Γ¯ such that χ (z) = χˆ (z) a.e., then χ ∈ Γ¯ . If χ ∈ Γ¯ and f ∈ L ∞ (Z, ν), then f χ ∈ Γ¯ , where ( f χ )(z) := f (z)χ (z). Γ¯ is complete with respect to the seminorm χ = ( Z dν(z)χ (z)2 )1/2 .

Such Γ¯ is called an integrable family. It is obtained from Γ by multiplication with L ∞ functions and closure in · . The elements of Γ¯ are called Γ -measurable functions; they are in fact analogues of square-integrable functions, and the usual measure theoretic results hold for them: Egoroff’s theorem; the dominated convergence theorem; any norm-convergent sequence (χn ) in Γ¯ has a subsequence on which χn (z) converges pointwise a.e.; and if (χn ) is a sequence in Γ¯ that converges pointwise a.e., the limit function χ is in Γ¯ . (Cf. Propositions 1.3 and 1.5 of [Mar69].) After dividing out vectors of zero norm (which we do not denote explicitly), Γ¯ becomes a Hilbert space, which we call the direct integral of the Hz with respect to Γ , and denote it as Γ H= dν(z)Hz , with scalar product (χ |χ) ˆ = dν(z) χ (z)|χ(z)" ˆ z . (A.1) Z

Z

Γ Correspondingly, the elements χ ∈ H are denoted as χ = Z χ (z)dν(z). We also consider bounded operators between such direct integral spaces. Let Hz , Hˆ z be two fields of Hilbert spaces over the same measure space Z, and let Γ, Γˆ be associated fundamental families. We call B ∈ z∈Z B(Hz , Hˆ z ) a measurable field of operators if ess supz B(z) < ∞, and if for every χ ∈ Γ¯ , the vector field z → B(z)χ (z) is Γˆ -measurable, i.e. an element of Γ¯ˆ . In fact it suffices to check the measurability condition on the fundamental family Γ . Lemma A.2. Let B ∈ z∈Z B(Hz , Hˆ z ) such that ess supz B(z) < ∞, and such that z → B(z)χ (z) is Γˆ -measurable for every χ ∈ Γ . Then B is a measurable field of operators. Proof. Evidently, z → B(z)χ (z) is also Γˆ -measurable if χ is taken from L ∞ (Z, ν) · Γ or from its linear span. This span is however dense in H. So let χ ∈ H. There exists a sequence χn ∈ span L ∞ (Z, ν) · Γ such that χn → χ in norm; by the remarks after Def. A.1, we can assume that χn (z) → χ (z) a. e. But then B(z)χn (z) → B(z)χ (z) a. e., due to continuity of each B(z). So B(z)χ (z) is a pointwise a. e. limit of functions in Γ¯ˆ . This implies (z → B(z)χ (z)) ∈ Γ¯ˆ , which was to be shown. A measurable field of operators B now defines a bounded linear operator H → Hˆ

Γ,Γˆ ˆ of this form are which we denote as B = Z dν(z)B(z). Operators in B(H, H) called decomposable. Their decomposition need not be unique however, not even a. e. If here Γ = Γˆ , and if B(z) = f (z)1Hz with a function f ∈ L ∞ (Z , ν), then B is called a diagonalizable operator. We sometimes write this multiplication operator as

Γ M f = Z dν(z) f (z)1. Let A be a C∗ algebra, and let for each z ∈ Z a representation πz of A on Hz be given, such that z → πz (A) is a measurable field of operators for any A ∈ A. Then,

Γ π(A) = Z dν(z)πz (A) defines a new representation π of A on H, which we formally

Γ denote as π = Z dν(z)πz .

58

H. Bostelmann, C. D’Antoni, G. Morsella

In many cases, obtaining useful results regarding decomposable operators requires additional separability assumptions. The following property will usually be general enough for us.

Γ Definition A.3. A fundamental sequence in H = Z dν(z)Hz is a sequence (χ j ) j∈N in Γ¯ such that for every z ∈ Z, the set {χ j (z) | j ∈ N} is total in Hz . If such a fundamental sequence exists for H, we say that H is uniformly separable. This more restrictive situation agrees with the setting of [Tak79,Dix81]; cf. [Mar69, Prop. 1.13]. Note that this property implies that the fiber spaces Hz are separable, but the integral space H does not need to be separable if Z is sufficiently general. Under the above separability assumption, additional desirable properties of decomposable operators hold true.

Γ

Γˆ Theorem A.4. Let H = Z dν(z)Hz and Hˆ = Z dν(z)Hˆ z both be uniformly sepa Γ,Γˆ rable. Then, for each decomposable operator B = Z dν(z)B(z), also B ∗ is decom Γˆ ,Γ posable, with B ∗ = Z dν(z)B(z)∗ . One has B = B ∗ = ess supz B(z).

Γ,Γˆ Decompositions of operators are unique in the following sense: If Z dν(z)B(z) =

Γ,Γˆ ˆ ˆ dν(z) B(z), then B(z) = B(z) for almost every z. Z

For the proof methods, see e.g. [God51, Ch. III Sec. 13]. Note that the theorem is false if the separability assumption is dropped; see Example 7.6 and Remark 7.11 of [Tak79, Ch. IV]. We also obtain an important characterization of decomposable operators. ˆ is decomTheorem A.5. Let H, Hˆ be uniformly separable. An operator B ∈ B(H, H) posable if and only if it commutes with all diagonalizable operators; i.e. M f B = B Mˆ f for all f ∈ L ∞ (Z , ν). ˆ A proof can be found in [Dix81, Ch. II §2 Sec. 5 Thm. 1]. In particular, if H = H, we know that both the decomposable operators and the diagonalizable operators form W∗ algebras, which are their mutual commutants. Note that the “if” part of the theorem is known to be false for sufficiently general direct integrals, violating the separability assumption [Sch90]. We now discuss the case of a continuous fundamental family Γ ; cf. [God51, Ch. III Sec. 2]. In this case, we can consider the space of Γ -continuous functions, denoted C(Γ ), and defined as the closed span of C(Z) · Γ in the supremum norm, χ ∞ = supz∈Z χ (z)z . With this norm, C(Γ ) is a Banach space, in fact a Banach module over the commutative C∗ algebra C(Z). We have C(Γ ) ⊂ Γ¯ in a natural way, and this inclusion is dense, but it is important to note that different norms are used in these two spaces. A simple but particularly important example for direct integrals arises as follows [Tak79, Ch. IV.7]. Let Hu be a fixed Hilbert space, and Z a measure space as above. For each z ∈ Z, set Hz = Hu . Then the set Γ of constant functions Z → Hu is a continuous fundamental family; and the associated integrable family Γ¯ is precisely the space of all square-integrable, Lusin-measurable functions χ : Z → Hu . We denote ⊕ the corresponding direct integral space as L 2 (Z, ν, Hu ) = Z dν(z)Hu (with reference to the “canonical” fundamental family). This space is isomorphic to L 2 (Z, ν) ⊗ Hu ; the canonical isomorphism, which we do not denote explicitly, maps f ⊗ χ to the

On Dilation Symmetries Arising from Scaling Limits

59

function z → f (z)χ . In this way, the algebra of diagonal operators is identified with L ∞ (Z, ν) ⊗ 1. If here Hu is separable, then L 2 (Z, ν, Hu ) is clearly uniformly separable. In this case, a simple criterion identifies the elements of the integral space: A function χ : Z → Hu is Lusin measurable if and only if it is weakly measurable, i.e. if z → χ (z)|η" is measurable for any fixed η ∈ Hu . Also, the algebra of decomposable operators is iso¯ morphic to L ∞ (Z, ν)⊗B(H u ). References [Ara64] [BDF87] [BDM09] [BF77] [BF08] [Bos00] [Bos05a] [Bos05b] [BP90] [BR79] [BR81] [Buc96a] [Buc96b] [BV95] [BV98] [BW86] [CM08] [Dix81] [DM06] [DMV04] [DS85] [EF74] [FS94]

[GLW98] [God51] [Haa96]

Araki, H.: On the algebra of all local observables. Prog. Theor. Phys. 32, 844–854 (1964) Buchholz, D., D’Antoni, C., Fredenhagen, K.: The universal structure of local algebras. Commun. Math. Phys. 111, 123–135 (1987) Bostelmann, H., D’Antoni, C., Morsella, G.: Scaling algebras and pointlike fields. A nonperturbative approach to renormalization. Commun. Math. Phys. 285, 763–798 (2009) Buchholz, D., Fredenhagen, K.: Dilations and interactions. J. Math. Phys. 18, 1107–1111 (1977) Bostelmann, H., Fewster, C.J.: Quantum inequalities from operator product expansions. Commun. Math. Phys. (2009). doi:10.1007/s00220-009-0853-x Bostelmann, H.: Lokale Algebren und Operatorprodukte am Punkt. Thesis, Universität Göttingen, 2000. Available online at http://webdoc.sub.gwdg.de/diss/2000/bostelmann/ Bostelmann, H.: Operator product expansions as a consequence of phase space properties. J. Math. Phys. 46, 082304 (2005) Bostelmann, H.: Phase space properties and the short distance structure in quantum field theory. J. Math. Phys. 46, 052301 (2005) Buchholz, D., Porrmann, M.: How small is the phase space in quantum field theory? Ann Inst. H. Poincaré 52, 237–257 (1990) Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, Volume I. New York: Springer, 1979 Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, Volume II. New York: Springer, 1981 Buchholz, D.: Phase space properties of local observables and structure of scaling limits. Ann. Inst. H. Poincaré 64, 433–459 (1996) Buchholz, D.: Quarks, gluons, colour: facts or fiction? Nucl Phys. B 469, 333–353 (1996) Buchholz, D., Verch, R.: Scaling algebras and renormalization group in algebraic quantum field theory. Rev. Math. Phys. 7, 1195–1239 (1995) Buchholz, D., Verch, R.: Scaling algebras and renormalization group in algebraic quantum field theory. II. Instructive examples. Rev. Math. Phys. 10, 775–800 (1998) Buchholz, D., Wichmann, E.H.: Causal independence and the energy-level density of states in local quantum field theory. Commun. Math. Phys. 106, 321–344 (1986) Conti, R., Morsella, G.: Scaling limit for subsystems and Doplicher-Roberts reconstruction. Ann. H. Poincaré 10, 485–511 (2009) Dixmier, J.: Von Neumann Algebras. Amsterdam: North-Holland, 1981 D’Antoni, C., Morsella, G.: Scaling algebras and superselection sectors: Study of a class of models. Rev. Math. Phys. 18, 565–594 (2006) D’Antoni, C., Morsella, G., Verch, R.: Scaling algebras for charged fields and short-distance analysis for localizable and topological charges. Ann. H. Poincaré 5, 809–870 (2004) Driessler, W., Summers, S.J.: Central decomposition of Poincaré-invariant nets of local field algebras and absence of spontaneous breaking of the Lorentz group. Ann. Inst. H. Poincaré Phys. Theor. 43, 147–166 (1985) Eckmann, J.P., Fröhlich, J.: Unitary equivalence of local algebras in the quasifree representation. Ann. Inst. H. Poincaré Sect. A (N.S.) 20, 201–209 (1974) Fendley, P., Saleur, H.: Massless integrable quantum field theories and massless scattering in 1+1 dimensions. In: Gava, E., Masiero, A., Nariain, K.S., Randjbar-Daemi, S., Shafi, Q. (eds.), Proceedings of the 1993 Summer School on High Energy Physics and Cosmology, Volume 10 of ICTP Series in Theoretical Physics, Singapore: World Scientific, 1994 Guido, D., Longo, R., Wiesbrock, H.W.: Extensions of conformal nets and superselection structures. Commun. Math. Phys. 192, 217–244 (1998) Godement, R.: Sur la théorie des représentations unitaires. Ann. of Math. 53, 68–124 (1951) Haag, R.: Local Quantum Physics. Berlin: Springer, 2nd edition, 1996

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H. Bostelmann, C. D’Antoni, G. Morsella

Haag, R., Swieca, J.A.: When does a quantum field theory describe particles? Commun. Math. Phys. 1, 308–320 (1965) Kawahigashi, Y., Longo, R.: Classification of local conformal nets. Case c < 1. Ann. Math. 160, 493–522 (2004) Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory. Orlando FL: Academic Press, 1997 Lechner, G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008) Maréchal, O.: Champs mesurables d’espaces hilbertiens. Bull. Sci. Math. 93, 113–143 (1969) Mitchell, T.: Fixed points and multiplicative left invariant means. Trans. Amer. Math. Soc. 122, 195–202 (1966) Schaflitzel, R.: The algebra of decomposable operators in direct integrals of not necessarily separable Hilbert spaces. Proc. Amer. Math. Soc. 110, 983–987 (1990) Segal, I.: Decompositions of Operator Algebras I, II, Volume 9 of Mem. Amer. Math. Soc. Providence, RI: Amer. Math. Soc., 1951 Takesaki, M.: Theory of Operator Algebras I. Berlin: Springer, 1979 Takesaki, M.: Theory of Operator Algebras II. Berlin: Springer, 2002 Wils, W.: Direct integrals of Hilbert spaces I. Math. Scand. 26, 73–88 (1970)

Communicated by Y. Kawahigashi

Commun. Math. Phys. 294, 61–72 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0934-x

Communications in

Mathematical Physics

Blowup of Jang’s Equation at Outermost Marginally Trapped Surfaces Jan Metzger Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Potsdam, Germany. E-mail: [email protected] Received: 30 January 2009 / Accepted: 24 July 2009 Published online: 8 October 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract: The aim of this paper is to accurately describe the blowup of Jang’s equation. First, we discuss how to construct solutions that blow up at an outermost MOTS. Second, we exclude the possibility that there are extra blowup surfaces in data sets with non-positive mean curvature. Then we investigate the rate of convergence of the blowup to a cylinder near a strictly stable MOTS and show exponential convergence with an identifiable rate near a strictly stable MOTS. 1. Introduction This paper is concerned with the examination of the relation of Jang’s equation to marginally outer trapped surfaces (MOTS). To set the perspective, we consider Cauchy data (M, g, K ) for the Einstein equations. Such data sets are 3-manifolds M equipped with a Riemannian metric together with a symmetric bilinear form K representing the second fundamental form of the time slice M in space-time. A marginally outer trapped surface is a surface with θ + = H + P = 0, where H is the mean curvature of in M and P = tr K − K (ν, ν) for the normal ν to . In the paper [AM07], inspired by an idea of Schoen [Sch04], we constructed MOTS in the presence of barrier surfaces by inducing a blow-up of Jang’s equation. In this context, Jang’s equation [SY81,Jan78] is an equation of prescribed mean curvature for the graph of a function in M × R. For details we refer to Sect. 2. In this note, we take a slightly different perspective. Consider a data set (M, g, K ) with a non-empty outer boundary ∂ + M and assume that we are given the outermost MOTS in (M, g, K ). Here, outermost means that there is no other MOTS on the outside of . From [AM07] it follows that (M, g, K ) always contains a unique such surface, or does not contain outer trapped surfaces at all, under the assumption that ∂ M is outer untrapped. As stated in Theorem 3.1, we show that there exists a solution f Research on this project started while the author was supported in part by a Feodor-Lynen Fellowship of the Humboldt Foundation.

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to Jang’s equation that actually blows up at , assuming that ∂ M is inner and outer untrapped. By blow-up we mean that outside from the function f is such that graph f is a smooth submanifold of M × R with a cylindrical end converging to × R. There is however a catch, as f may blow up at other surfaces, too. These surfaces are marginally inner trapped. In Theorem 3.4 we show that the other blow-up surfaces can not occur if the data set has non-positive mean curvature. To put the result in perspective note that if the dominant energy condition holds, the graph of f is of non-negative Yamabe type and thus can be equipped with a (singular) metric of zero scalar curvature. This was used by Schoen and Yau in [SY81] to prove the positive mass theorem. Later Bray [Bra01] proposed to use Jang’s equation to relate the Penrose conjecture in its general setting to the Riemannian Penrose inequality [HI01,Bra01] on a manifold constructed from Jang’s graph. One of the main questions in this program is whether or not Jang’s equation can be made to blow up at a specific MOTS. This question was raised in the literature, cf. for example [MÓM04] where this is discussed in the rotationally symmetric case. Here we give the positive answer that blow-up solutions exist at outermost MOTS. The author recently learned that the existence of the blow-up solution is used in [Khu09] to prove a Penrose-like inequality. With the blow-up constructed, we can turn to the asymptotic behavior of the blowup itself. It has been shown in [SY81] that such a blow-up must be asymptotic to a cylinder over the outermost MOTS. In Theorems 4.2 and 4.4 we show that under the assumption of strict stability the convergence rate is exponential with a power directly related to the principal eigenvalue of the MOTS. The general idea is to show the existence of a super-solution with at most logarithmic blow-up of the desired rate. Turning the picture sideways yields exponential decay, when writing the solution as a graph over the cylinder in question. Furthermore, we show that beyond a certain decay rate, the solution must be trivial, thus exhibiting the actual rate. We expect that the knowledge of these asymptotics is tied to the question whether the blow-up solution is unique. Furthermore note that the constant in the Penrose-like inequality in [Khu09] depends on the geometry of the solution. We thus expect that the value on this constant is related to the asymptotic behavior near the blow-up cylinder. Before turning to these results, we introduce some notation in Sect. 2. Section 3 proceeds with the construction of the the blow-up. We will not go into details here, but emphasize the general idea and point to the results needed from the paper [AM07]. In Sect. 4, we perform the calculation of the asymptotics. 2. Preliminaries Let (M, g, K ) be an initial data set for the Einstein equations. That is M is a 3-dimensional manifold, g a Riemannian metric on M and K a symmetric 2-tensor. We do not require any energy condition to hold. Assume that ∂ M is the disjoint union ∂ M = ∂ − M ∪ ∂ + M, where ∂ ± M are smooth surfaces without boundary. We refer to ∂ − M as the inner boundary and endow it with the normal vector ν pointing into M. The outer boundary ∂ + M is endowed with the normal ν pointing out of M. We denote by H [∂ M] the mean curvature of ∂ M with respect to the normal vector field ν, and by P[∂ M] = tr ∂ M K the trace of the tensor K restricted to the 2-dimensional surface ∂ M. Then the inward and outward expansions of ∂ M are defined by θ ± [∂ M] = P[∂ M] ± H [∂ M]. Assume that θ + [∂ − M] = 0, and that θ + [∂ + M] > 0 and θ − [∂ + M] < 0.

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If ⊂ M is a smooth, embedded surface homologous to ∂ + M, then bounds a region together with ∂ + M. In this case, we define θ ± [] as above, where H is computed with respect to the normal vector field pointing into (that is in direction of ∂ + M). is called a marginally outer trapped surface (MOTS), if θ + [] = 0. We say that ∂ M is an outermost MOTS, if there is no other MOTS in M, which is homologous to ∂ + M. In [AM07] it is proved that for any initial data set (M, g, K ) which contains a MOTS, there is also an outermost MOTS surrounding it. Let ⊂ M be a MOTS and consider a normal variation of in M, that is a map ∂ F : × (−ε, ε) → M such that F(·, 0) = id and ∂s F( p, s) = f ν, where f is s=0 a function on and ν is the normal of . Then the change of θ + is given by ∂θ + [F(, s)] = L M f, ∂ds s=0 where L M is a quasi-linear elliptic operator of second order along . It is given by 1 L M f = − f + 2S(∇ f ) + f div S − |χ + |2 − |S|2 + Sc − µ − J (ν) . 2 In this expression ∇, div and denote the gradient, divergence and Laplace-Beltrami operator tangential to . The tangential 1-form S is given by S = K (·, ν)T , χ + is the bilinear form χ + = A + K , where A is the second fundamental form of in M and K is the projection of K to T × T . Furthermore, Sc denotes the scalar curvature of , µ = 21 ( M Sc − |K |2 + (tr K )2 ), and J = M divK − d tr K . For a more detailed investigation of this operator we refer to [AMS05] and [AMS07]. The facts we will need here are that L M has a principal eigenvalue λ, which is real and has a one-dimensional eigenspace which is spanned by a positive function. If λ is non-negative is called stable, and if λ is positive, is called strictly stable. In particular, if is strictly stable as a MOTS, there exists an outward deformation strictly increasing θ + . In M¯ = M × R, we consider Jang’s equation [Jan78,SY81] for the graph of a function f : M → R. Let N := graph f = {(x, z) : z = f (x)}. The mean curvature H[ f ] of N with respect to the downward normal is given by

∇f

H[ f ] = div 1 + |∇ f |2

.

Define K¯ on M¯ by K¯ (x,z) (X, Y ) = K x (π X, π Y ), where π : T M¯ → T M denotes the orthogonal projection onto the horizontal tangent vectors. Let P[ f ] = tr N K¯ . Then Jang’s equation becomes J [ f ] = H[ f ] − P[ f ] = 0.

(2.1)

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Fig. 1. The situation in Theorem 3.1. All of the shaded region belongs to M, whereas f is only defined in 0

3. The Blowup The main result of this section is that we can construct a solution to Jang’s equation which blows up at the outermost MOTS in (M, g, K ) and has zero Dirichlet boundary data at ∂ + M. In fact, we chose the assumptions on the outer boundary ∂ + M so that we can prescribe more general Dirichlet data there. The focus here lies on the blow-up in the interior, so that we do not investigate the optimal conditions for ∂ + M. Theorem 3.1. If (M, g, K ) is an initial data set with ∂ M = ∂ − M ∪ ∂ + M such that ∂ − M is an outermost MOTS, θ + [∂ + M] > 0 and θ − [∂ + M] < 0, then there exists an open set 0 ⊂ M and a function f : 0 → R such that 1. 2. 3. 4. 5. 6.

M\0 does not intersect ∂ M, θ − [∂0 ] = 0 with respect to the normal vector pointing into 0 , J [ f ] = 0, N + = graph f ∩ M × R+ is asymptotic to the cylinder ∂ − M × R+ , N − = graph f ∩ M × R− is asymptotic to the cylinder ∂0 × R− , and f |∂ + M = 0.

For data sets (M, g, K ) which do not contain surfaces with θ − = 0, the above theorem implies the following result. Corollary 3.2. If (M, g, K ) is as in Theorem 3.1, and in addition there are no subsets ⊂ M with θ − [∂] = 0 with respect to the normal pointing out of , then there exists a function f : M → R such that 1. J [ f ] = 0, 2. N = graph f is asymptotic to the cylinder ∂ − M × R+ , 3. f |∂ + M = 0. Remark 3.3. Analogous results hold if (M, g, K ) is asymptotically flat with appropriate decay of g and K instead of having an outer boundary ∂ + M. Then the assertion f |∂ + M = 0 in Theorem 3.1 has to be replaced by f (x) → 0 as x → ∞. The proof of Theorem 3.1 is largely based on the tools developed in [SY81 and AM07]. Thus we will not include all details here, but provide a summary, which facts will have to be used.

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Proof. We will assume that (M, g, K ) is embedded into (M , g , K ) which extends M beyond the boundary ∂ − M such that ∂ − M lies in the interior of M , without further requirements. N where the are the connected components of ∂ M. As ∂ M is Let ∂ − M = ∪i=1 i i an outermost MOTS, each of the i is stable [AM07, Cor. 5.3]. Following the proof of [AM07, Th. 5.1], we deform ∂ − M to a surface s by pushing the components i out of M, into the extension M . To this end, let φi > 0 be the principal eigenfunction of the stability operator of i and extend the vector field X i = −φi νi to a neighborhood of i in M . Flowing i by X i yields a family of surfaces is , s ∈ [0, ε) so that the is form a smooth foliation for small enough ε with is ∈ M \ M. If i is strictly stable then ∂ θ + [s ] = −λφ < 0, ∂s s=0 where λ is the principal eigenvalue of i . Thus, for small enough ε, we have θ + [is ] < 0 for all s ∈ (0, ε). If i has principal eigenvalue λ = 0, then the is satisfy ∂ θ + [s ] = 0. ∂s s=0 In this case it is possible to change the data K on is as follows: K˜ = K − 21 ψ(s)γs ,

(3.1)

where γs is the metric on s and ψ is a smooth function ψ : [0, ε] → R. The operator θ˜ + , which means θ + computed with respect to the data K˜ instead of K , satisfies θ˜ + [is ] = θ + [is ] − ψ(s). It is clear from Eq. (3.1) that ψ can be chosen such that ψ(0) = ψ (0) = 0 and θ˜ + [is ] < 0 for all s ∈ (0, ε) provided ε is small enough. Then K˜ is C 1,1 when extended by K to the rest of M. Replace each original boundary component i of M by a surface iε as constructed above, and replace K with K˜ , such that the following properties are satisfied. Let M˜ denote the manifold with boundary components iε resulting from this procedure. Thus ˜ g , K˜ ) with the following properties: we construct from (M, g, K ) a data set ( M, ˜ 1. M ⊂ M˜ with g | M = g, K˜ | M = K , and ∂ + M = ∂ + M, + − ˜ 2. θ [∂ M] < 0, and 3. the region M˜ \ M is foliated by surfaces s with θ + (s ) < 0. The method developed in Sect. 3.2 in [AM07] now allows the modification of the data ˜ g, ˜ g , K˜ ) to a new data set, which we also denote by ( M, ˜ K˜ ), although K˜ changes ( M, in this step. This data set has the following properties: ˜ 1. M ⊂ M˜ with g | M = g, K˜ | M = K , and ∂ + M = ∂ + M, ˜ < 0, 2. θ + [∂ − M] ˜ > 0, where H is the mean curvature of ∂ − M with respect to the normal 3. H [∂ − M] ˜ pointing out of ∂ − M, 4. the region M˜ \ M is foliated by surfaces s with θ + (s ) < 0.

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By Sect. 3.3 in [AM07] this enables us to solve the boundary value problem ⎧ ⎪ ⎨J [ f τ ] = τ f τ in M˜ f τ = 2τδ on ∂ − M˜ ⎪ ⎩f =0 on ∂ + M˜ τ

(3.2)

where δ is a lower bound for H on ∂ − M. The solvability of this equation follows, provided an estimate for the gradient at the boundary can be found. The barrier construction at ∂ − M˜ was carried out in detail in [AM07], whereas the barrier construction at ∂ + M˜ is standard due to the stronger requirement that θ + [∂ + M] > 0 and θ − [∂ + M] < 0. The solution f τ to Eq. (3.2) satisfies an estimate of the form sup | f τ | + sup |∇ f τ | ≤ M˜

M˜

C , τ

(3.3)

˜ g, where C is a constant depending only on the data ( M, ˜ K˜ ) but not on τ . The gradient estimate implies in particular that there exists an ε > 0 independent of τ such that f τ (x) ≥

δ 4τ

˜ ∀x with dist(x, ∂ − M).

The graphs Nτ have uniformly bounded curvature in M˜ × R away from the boundary. This allows the extraction of a sequence τi → 0 such that the Nτi converge to a manifold N , cf. [AM07, Prop. 3.8], [SY81, Sect. 4]. This convergence determines three open ˜ subsets of M: − := {x ∈ M : f τi (x) → −∞ locally uniformly as i → ∞}, 0 := {x ∈ M : lim sup | f τi (x)| < ∞}, i→∞

+ := {x ∈ M : f τi (x) → ∞ locally uniformly as i → ∞}. ˜ we have that + = ∅ and + contains a From the fact that the f τ blow up near ∂ − M, ˜ As already noted in [SY81] ∂+ \ ∂ M˜ consists of MOTS. As the neighborhood of ∂ − . region M˜ \ M is foliated by surfaces with θ + < 0, we must have that + ⊃ ( M˜ \ M) and hence ∂+ is a MOTS in M. As ∂ − M was assumed to be an outermost MOTS in M, we conclude that the closure of + is M˜ \ M. The barriers near ∂ + M are so that they imply that the f τ are uniformly bounded near + ∂ M. Thus 0 contains a neighborhood of ∂ + M and 0 ⊂ M. The limit manifold N over 0 is a graph satisfying J [ f τ ] = 0, and has the desired asymptotics. We will now discuss a geometric condition to assert that the resulting graph is nonsingular on M, i.e., M = 0 in Theorem 3.1. Theorem 3.4. Let (M, g, K ) be as in Theorem 3.1 with tr K ≤ 0. Then in the assertion of Theorem 3.1 we have that 0 = M, that is f is defined on M and has no other blow-up than near ∂ − M.

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67

Proof. This follows from a simple argument using the maximum principle. Let f τ be a solution to the regularized problem H[ f τ ] − P[ f τ ] − τ f τ = 0

(3.4)

˜ as in the proof of Theorem 3.1. We claim that f τ can not have a negative minimum in M, in the region where the data is unmodified. Assume that x ∈ M is such a minimum. There we have H[ f τ ] ≥ 0, and since graph f is horizontal at x we have that P[ f τ ] = tr K ≤ 0, thus the right-hand side of (3.4) is non-negative, whereas τ f τ is assumed to be negative, a contradiction. Since we know that in the limit τ → 0, the functions f τ must blow-up in the modified region which lies in + , we infer a lower bound for f τ from the above argument. Thus 0 = M as claimed. 4. Asymptotic Behavior Here, we discuss a refinement of [SY81, Cor. 2], which says that N = graph f converges uniformly in C 2 to the cylinder ∂ − M × R for large values of f . A barrier construction allows us to determine the asymptotics of this convergence. Before we present our result, recall the statement of [SY81, Cor. 2]: Theorem 4.1. Let N = graph f be the manifold constructed in the proof of Theorem 3.1 and let be a connected component of ∂ − M. Let U be a neighborhood of with positive distance to ∂ − M \ . Then for all ε > 0 there exists z¯ = z¯ (ε), depending also on the geometry of (M, g, K ), such that N ∩ U × [¯z , ∞) can be written as the graph of a function u over C z¯ := × [¯z , ∞), so that |u( p, z)| + | C z¯ ∇ u( p, z)| + | C z¯ ∇ 2 u( p, z)| < ε for all ( p, z) ∈ C z¯ . Here,

C z¯

∇ denotes covariant differentiation along C z¯ .

If is strictly stable, we can in fact say more about u. Theorem 4.2. Assume the situation of Theorem 4.1. If in addition is strictly stable √ with principal eigenvalue λ > 0, we have that for all δ < λ there exists c = c(δ) depending only on the data (M, g, K ) and δ such that |u( p, z)| + | C z¯ ∇ u( p, z)| + | C z¯ ∇ 2 u( p, z)| ≤ c exp(−δz). Proof. Denote by β > 0 the eigenfunction to the principal eigenvalue λ on normalized such that max β = 1. We denote by ν the normal vector field of pointing into M. Consider the map : × [0, s¯ ] → M : ( p, s) → exp M p (sβν).

(4.1)

Given ε > 0 we can choose s¯ > 0 small enough such that the surfaces s = (, s) with s ∈ [0, s¯ ] form a local foliation near with lapse β such that θ + [s ] ≥ λ(1 − ε)βs.

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Denote the region swiped out by these s by Us¯ . Note that ∂Us¯ = ∪ s¯ and dist(s¯ , ) > 0. We can assume that dist(s¯ , ∂ M) > 0. On Us¯ we consider functions w of the form w = φ(s). For such functions Jang’s operator can be computed as follows: φ + φ φ θ − 1+ P − σ −2 K (ν, ν) + 2 3 , J [w] = βσ βσ β σ where σ 2 = 1+β −2 (φ )2 , and φ denotes the derivative of φ with respect to s, cf. [AM07]. The quantities θ + , K (ν, ν) and P are computed on the respective s . Note that with our normalization σ −2 ≤ β 2 |φ |−2 ≤ |φ |−2 , and if we assume that φ ≥ µ for a large µ = µ(β) we have 1 + φ ≤ 2|φ |−2 . βσ Furthermore, φ |φ | |φ | |φ | β 2 σ 3 = β 2 (1 + β −2 φ 2 )3/2 ≤ β 2 (β −2 φ 2 )3/2 = β |φ |3 . On the other hand, increasing µ = µ(β, ε) if necessary, we have |φ | ≥ 1 − ε, βσ if |φ | ≥ µ. In combination we find that J [w] ≤ −λ(1 − ε)βs +

c1 |φ | + β , |φ |2 |φ |3

(4.2)

with c > 0 depending on ε and the data (M, g, K ), provided |φ | ≥ µ and φ < 0. Choosing φ(s) = a log s with a = (1 − ε)−1 λ−1/2 , we calculate that φ (s) =

a a , φ (s) = − 2 , s s

so that 1 s2 = = (1 − ε)2 λs 2 , |φ |2 a2

φ s = 2 = (1 − ε)2 λs. 3 |φ | a

Thus we can choose s¯ so small that |φ | ≥ µ(β, ε) and the estimate in (4.2) holds. We can then decrease s¯ further, so that s¯ ≤ εβ/(c1 (1−ε)). This choice makes the right-hand side of (4.2) non-positive, that is J [w] ≤ 0. Hence, we obtain a super-solution w with Jτ w ≤ 0 at least where w ≥ 0, that is near .

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As w blows up near the horizon, and the f τ are bounded uniformly in τ on s¯ , we can translate w vertically to w¯ = w + b with a suitable b > 0 so that ¯ s¯ f τ |s¯ ≤ w| for all τ > 0. Then the maximum principle implies that f τ ≤ w¯ for all τ > 0 in Us¯ and consequently the function f constructed in Theorem 3.1 also satisfies f ≤ w. ¯ Near , the graph of w¯ can be written as the graph of a function v¯ over × (¯z , ∞), where v decays exponentially in z. This is due to the fact that by the assumptions on β, the parameter s is comparable to the distance to . By the above construction u ≤ v, where u is the function from Theorem 4.1. Thus we find the claimed estimate for u. Getting the desired estimates for the derivatives of u is then a standard procedure, but as it is a little work to set the stage, we briefly indicate how to proceed. We choose coordinates of a neighborhood × R in a slightly different manner as ¯ : × (−ε, ε) → M be the map above. Let ¯ : × (−ε, ε) × R → M × R : (x, s, z) → expx (sν), z . For a function h on C z¯ we let graph¯ h be the set ¯ graph¯ h = {(x, h(x), z) : (x, z) ∈ × R}. From Theorem 4.1, it is clear that for large enough z¯ the set N ∩ M × [¯z , ∞) can be written as graph¯ h, where h decays exponentially by the above reasoning. We can compute the value of Jang’s operator for h as follows: ¯ ( H¯ − P)[N ] = J h, where J is a quasi-linear elliptic operator of mean curvature type. To be more precise, J h has the form ij

ij

J h = ∂z2 h + γh(x,z) ∇i,2 j h − 2γh(x,z) ∂i (h)K (∂s , ∂ j ) − θ + [h(x,z) ] + Q(h,

C z¯

∇ h,

C z¯

(4.3)

∇ 2 h),

where γs is the metric on s and Q is of the form Q(h,

C z¯

∇ h,

C z¯

∇ 2 h) = h ∗

C z¯

∇h +

C z¯

∇h ∗

C z¯

∇h +

C z¯

∇h ∗

C z¯

∇h ∗

C z¯

∇ 2 h,

where ∗ denotes some contraction with a bounded tensor. Furthermore, the vectors ∂i , i = 1, 2 denote directions tangential to and ∂z the direction along the R-factor in C z¯ . By freezing coefficients, we therefore conclude that h satisfies a linear, uniformly elliptic equation of the form a i j ∂i ∂ j h + b,

C z¯

∇ h − θ + [h(x,z) ] = 0.

By construction we have that |θ + [s ]| ≤ κs for some fixed κ. Thus θ + [h(x,z) ] decays exponentially in z. Now we are in the position to use standard interior estimates for linear elliptic equations to conclude the decay of higher derivatives of h. This decay translates back into the decay of the first and second derivatives of u as the coordinate transformation is smooth and controlled by the geometry of (M, g, K ).

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Remark 4.3. If is not strictly stable, but has positive k th variation, we find that the foliation near satisfies θ + [s ] ≥ κs k . Then a function of the form φ(s) = as − p with large a and p = k−1 2 yields a super-solution. This super-solution can be used to prove that |u| ≤ C z 2/(1−k) as above. We can get even more information about the decay rate. A closer look at Eq. (4.3) yields that the expression for J h on C z¯ can also be written as follows: J h = (∂z2 − L M )h + Q (h,

C z¯

∇ h,

C z¯

∇ 2 h),

since θs+ = s L M 1 + O(s 2 ), ij

γh(x,z) ∇i,2 j h = h + Q 1 (h, ∇h, ∇ 2 h), and ij

γh(x,z) ∂i h K (∂s , ∂ j ) = S(∇h) + Q 2 (h, ∇h), where the differential operators ∇ and are with respect to . Then note that L M h = h L M 1 − h + 2S(∇h). Further investigation of the structure of Q yields that |Q (h,

C z¯

∇ h,

C z¯

∇ 2 h)| ≤ C |h|2 + | C z¯ ∇ h|2 + |h|| C z¯ ∇ 2 h| + | C z¯ ∇ h|2 | C z¯ ∇ 2 h| ,

so that in view of the differential Harnack estimate | C z¯ ∇ h| ≤ c|h| for positive solutions of linear elliptic equations we have that in fact |Q (h,

C z¯

∇ h,

C z¯

∇ 2 h)| ≤ c|h| |h| + | C z¯ ∇ h| + | C z¯ ∇ 2 h| ,

provided |h| ≤ C. By projecting the equation J h = 0 to the one-dimensional eigenspace of L M it is now a somewhat standard ODE argument to show the following result. Theorem 4.4. Under the assumptions of Theorem 4.2 there are no solutions h : × [0, ∞) → R to the equation Jh = 0 with decay |h( p, z)| + | C z¯ ∇ h( p, z)| + | C z¯ ∇ h( p, z)| ≤ C exp(−δz) such that δ >

√ λ and h > 0.

(4.4)

Blowup of Jang’s Equation at Outermost Marginally Trapped Surfaces

71

Proof. Assume that h > 0 is such a solution. We derive a contradiction as follows. Let λ be the principal eigenvalue and φ be the corresponding eigenfunction of L M as before. Let L ∗M be the (formal) adjoint of L M on L 2 () and denote by φ ∗ > 0 its principal eigenfunction, normalized such that φφ ∗ dµ = 1. Then the operator Pu =

φ ∗ u dµ φ

is a projection onto the eigenspace spanned by φ and moreover commutes with L M . We interpret h(z) as a family of functions on , that is h(z)( p) = h( p, z) for p ∈ . Choose α(z) such that Ph(z) = α(z)φ, and β(z) accordingly, β(z)φ = P Q (h(·, z),

C z¯

∇ h(·, z),

C z¯

∇ 2 h(·, p)) .

Then Eq. (4.4) and the fact that P commutes with L M and ∂z imply α (z) − λα(z) = β(z). Using φ ∗ > 0 and h > 0 yields α(z) > 0 and we can furthermore estimate that

φ ∗ |h( p, z)| |h( p, z)| + |∇h( p, z)| + |∇ 2 h( p, z)| dµ ≤ c exp(−δz) φ ∗ |h( p, z)| dµ

|β(z)| ≤ c

≤ c exp(−δz)α(z). Thus, we conclude that on [˜z , 0) the function α > 0 satisfies a differential inequality of the form α (z) − λα ≤ εα, √ where ε > 0 can be chosen arbitrarily small by choosing z˜ large enough. If λ + ε < δ this ODE has no solutions with decay exp(−δz) other than the trivial solution. Thus α ≡ 0 and we arrive at the desired contradiction. Acknowledgement. The author thanks the Mittag-Leffler-Institute, Djursholm, Sweden for hospitality and support during the program Geometry, Analysis, and General Relativity in Fall 2008. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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References [AM07] [AMS05] [AMS07] [Bra01] [HI01] [Jan78] [Khu09] [MÓM04] [Sch04] [SY81]

Andersson, L., Metzger, J.: The area of horizons and the trapped region. Commun. Math. Phys. 290, 941–972 (2009) Andersson, L., Mars, M., Simon, W.: Local existence of dynamical and trapping horizons. Phys. Rev. Lett. 95, 111102 (2005) Andersson, L., Mars, M., Simon, W.: Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes. http://arxiv.org/abs/0704.2889v2[gr-qc], 2007 Bray, H.L.: Proof of the riemannian penrose inequality using the positive mass theorem. J. Diff. Geom. 59(2), 177–267 (2001) Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the riemannian penrose inequality. J. Diff. Geom. 59(3), 353–437 (2001) Jang, P.S.: On the positivity of energy in general relativity. J. Math. Phys. 19, 1152–1155 (1978) Khuri, M.: A penrose-like inequality for general initial data sets. Commun. Math. Phys. 290(2), 779–788 (2009) Malec, E., Murchadha, N.Ó.: The Jang equation, apparent horizons and the Penrose inequality. Class. Quant. Grav. 21(24), 5777–5787 (2004) Schoen, R.: Talk Given at the Miami Waves Conference, January 2004 Schoen, R., Yau, S.-T.: Proof of the positive mass theorem. II. Commun. Math. Phys. 79(2), 231–260 (1981)

Communicated by P. T. Chru´sciel

Commun. Math. Phys. 294, 73–95 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0954-6

Communications in

Mathematical Physics

Incompressible Limits and Propagation of Acoustic Waves in Large Domains with Boundaries Eduard Feireisl Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic. E-mail: [email protected] Received: 3 February 2009 / Accepted: 26 August 2009 Published online: 19 November 2009 – © Springer-Verlag 2009

Abstract: We study the incompressible limit for the full Navier-Stokes-Fourier system on unbounded domains with boundaries, supplemented with the complete slip boundary condition for the velocity field. Using an abstract result of Tosio Kato we show that the energy of acoustic waves decays to zero on any compact subset of the physical space. This in turn implies strong convergence of the velocity field to its limit in the incompressible regime.

1. Introduction Propagation and attenuation of acoustic waves plays an important role in the analysis of fluid flows in the low Mach number regime, in particular in the so-called incompressible limit, when the speed of sound in the material becomes infinite. Recently, several studies have been devoted to a rigorous justification of the low Mach number limit for the complete Navier-Stokes-Fourier system. Alazard [1,2] studied the problem for a quite general class of initial data that give rise to sufficiently regular solutions defined, however, only on a short time interval. In this purely “hyperbolic” approach proposed in the seminal paper by Klainerman and Majda [14], the presence of viscosity in the Navier-Stokes system plays only a marginal role. A different technique, based on the concept of weak solutions, was used by Lions and Masmoudi [22,23], and further developed in a series of papers by Desjardins and Grenier [7], Desjardins et al. [8], Masmoudi [24–26], among others. These results concern the Navier-Stokes system describing a compressible barotropic fluid flow, where the basic framework is provided by the existence theory developed by Lions [21]. The work of E.F. was supported by Grant 201/08/0315 of GA CR ˇ as a part of the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503.

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The same strategy, based on global-in-time weak solutions, was later adapted to the complete Navier-Stokes-Fourier system in [10]. This approach leans on energy-entropy estimates, where the presence of viscosity is indispensable. As frequently observed in practical as well as numerical experiments, the influence of acoustic waves on the fluid motion is negligible in the low Mach number limit (cf. Klein et al. [17]). In terms of the mathematical theory, the acoustic waves are supported by the gradient part of the fluid velocity u, specifically H⊥ [u], where u = H[u] + H⊥ [u] and H is the Helmholtz projection onto the space of solenoidal functions. In the (hypothetical) incompressible limit, the mass density tends to a constant, the speed of sound becomes infinite whereas the gradient component H⊥ “disappears” since the limit velocity field is solenoidal. However, if the initial data are ill prepared, meaning the fluctuations of the fluid density and temperature are of the same order as the Mach number, and if the fluid is contained in a bounded spatial domain with an acoustically hard boundary (see Wilcox [37]), the gradient component of the velocity H⊥ [u] develops fast time oscillations with frequencies inversely proportional to the Mach number (cf. Lions and Masmoudi [22], Schochet [30,31]). Accordingly, the gradient part H⊥ [u] tends to zero only weakly, meaning in the sense of integral averages, with respect to time - a phenomenon that may destabilize certain numerical schemes when applied to the original system. A rather heuristic argument why the acoustic waves can be neglected in the low Mach number flows encountered in the real world applications arising in meteorology, oceanography, or astrophysics, is usually based on the fact that the underlying physical space is unbounded or, more correctly, sufficiently large when compared to the sound speed in the material in question (cf. Klein [15,16]). In accordance with the fundamental observation of Lighthill [20], propagation of acoustic waves in the low Mach number regime may be described by a simple linear wave equation with a source term - Lighthill’s tensor - containing the remaining quantities appearing in the complete Navier-Stokes system. The expected local decay of the acoustic energy then follows immediately from the dispersive estimates. Desjardins and Grenier [7] exploited this idea combined with the non-trivial Strichartz estimates for the acoustic equation in order to show strong (pointwise) convergence of the velocity field in the low Mach number limit for a barotropic fluid flow in the whole physical space R3 . A similar approach was adapted in [11] to the complete Navier-Stokes-Fourier system considered on “large” spatial domains, on which the Strichartz estimates were replaced by global integrability of the local energy established by Burq [4], Smith and Sogge [33]. Note that the concept of the so-called radiation boundary conditions, amply used in numerical analysis, is based on the same physical principle (see Engquist and Majda [9]). In contrast with the simple geometry of the whole space R3 , where the efficient mathematical tools based on Fourier analysis are at hand, any real problem of wave propagation inevitably includes the influence of the boundary representing a wall in the physical space. As is well-known, the Strichartz estimates become much more delicate and usually require severe geometrical restrictions to be imposed on the boundary. For instance, if the fluid domain is exterior to a compact obstacle, the latter must be starshaped or at least non-trapping (see Burq [4], Metcalfe [27], Smith and Sogge [33]), and the references cited therein). In this paper, we propose a simple method that may be used to establish strong (pointwise a.a.) convergence of acoustic waves in the low Mach number limit for fluid flows

Study of Incompressible Limit for Navier – Stokes – Fourier System

75

with ill-prepared data for a sufficiently vast class of physical domains ⊂ R3 . Pursuing the philosophy that any real physical space is always bounded but possibly “large” with respect to the speed of sound in the medium, we consider a family of bounded domains {ε }ε>0 ⊂ R3 such that ε ≈ in a certain sense as ε → 0. More specifically, we suppose that ⊂ R3 is an unbounded domain with a compact boundary ∂,

(1.1)

ε = Br (ε) ∩ ,

(1.2)

and set

where Br (ε) is a ball centered at zero with a radius r (ε), with r (ε) → ∞. Our approach is based on a nowadays classical result of Kato [13] (cf. also Burq et al. [5]) concerning weighted L 2 space-time estimates for a class of abstract operators in a Hilbert space: Theorem 1.1. [ Reed and Simon [29, Theorem XIII.25 and Corollary] ]. Let A be a closed densely defined linear operator and H a self-adjoint densely defined linear operator in a Hilbert space X . For λ ∈ / R, let R H [λ] = (H − λId)−1 denote the resolvent of H . Suppose that =

sup

λ∈ / R, v∈D (A∗ ), v X =1

A ◦ R H [λ] ◦ A∗ [v] X < ∞.

(1.3)

Then π sup w∈X, w X =1 2

∞ −∞

A exp(−it H )[w] 2X dt ≤ 2 .

The paper is organized as follows. In Sect. 2, we introduce a scaled Navier-StokesFourier system and formulate the problem of the incompressible limit for vanishing Mach number. Following the idea of Lighthill [20], we then rewrite the Navier-StokesFourier system as a wave (acoustic) equation, where all “non-hyperbolic” components are considered as a source term (see Sect. 3). In Sect. 4, we collect the necessary uniform bounds, independent of the Mach number, resulting from the total dissipation balance. This step is now well understood, and the results are taken over from [10] without proofs. Section 5 is central and contains the bulk of the analysis of acoustic waves. Assuming the boundary of the limit domain is acoustically hard, meaning the fluid velocity satisfies the complete slip or Navier-like boundary conditions, we show that Theorem 1.1 may be applied in order to deduce a uniform local decay of the acoustic energy. A remarkable feature of the present approach is that we do not need any kind of non-trapping condition to be imposed on the boundary. As a matter of fact, our technique applies whenever the spatial domain admits the limiting absorption principle for the corresponding wave operator (see Vainberg [35, Chap. VIII.2]).

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2. Primitive System and its Incompressible Limit 2.1. Navier-Stokes-Fourier system. We consider a scaled Navier-Stokes-Fourier system in the form: ∂t + divx (u) = 0, 1 ∂t (u) + divx (u ⊗ u) + 2 ∇x p(, ϑ) = divx S(ϑ, ∇x u), ε q(ϑ, ∇x ϑ) = σε , ∂t (s(, ϑ)) + divx (s(, ϑ)u) + divx ϑ supplemented with the total energy balance 2 ε d |u|2 + e(, ϑ) (t, ·) dx = 0, dt ε 2

(2.1) (2.2) (2.3)

(2.4)

where = (t, x) is the density, u = u(t, x) the velocity field, ϑ the temperature, and p = p(, ϑ), e = e(, ϑ), s = s(, ϑ) denote the pressure, the (specific) internal energy, the (specific) entropy, respectively, obeying Gibbs’ relation 1 ϑ Ds(, ϑ) = De(, ϑ) + p(, ϑ)D . (2.5) In addition, the viscous stress tensor S is given by Newton’s rheological law 2 t S(ϑ, ∇x u) = µ(ϑ) ∇x u + ∇x u − Idivx u + η(ϑ)I divx u, 3

(2.6)

while the heat flux q(ϑ, ∇x ϑ) satisfies Fourier’s law q(ϑ, ∇x ϑ) = −κ(ϑ)∇x ϑ.

(2.7)

Finally, in virtue of the Second Law of Thermodynamics, the entropy production rate σε satisfies 1 κ(ϑ) 2 2 ε S : ∇x u + |∇x ϑ| ≥ 0. (2.8) σε ≥ ϑ ϑ Relation (2.8) reflects the fact that the weak solutions considered in the present paper may hypothetically produce “more” entropy than given by the classical formula 1 κ(ϑ) 2 2 (2.9) ε S : ∇x u + |∇x ϑ| . σε = ϑ ϑ Note, however, that (2.8) reduces to (2.9) for any weak solution of problem (2.1–2.8) as soon as this solution is smooth (see [10]). The small parameter ε can be interpreted as the Mach number. When ε → 0, the speed of sound becomes infinite, and, accordingly, the fluid can be considered as incompressible in the asymptotic limit ε → 0. Note that there are different scalings of the primitive system leading to the same mathematical problems, a typical example is provided by a long-time, small-velocity, small-viscosity setting (see Klein et al. [17]).

Study of Incompressible Limit for Navier – Stokes – Fourier System

77

2.2. Energetically insulating boundary conditions. In accordance with the total energy conservation imposed though (2.4), the system is supplemented with conservative boundary conditions u · n|∂ε = 0, [Sn] × n|∂ε = 0, q · n|∂ε = 0.

(2.10) (2.11)

The complete slip boundary condition (2.10) implies, in particular, that the boundary is acoustically hard (cf. Sect. 3 below). Note that the more conventional no-slip boundary condition u|∂ε = 0 results in a fast decay of acoustic waves in a generic class of bounded domains because of creation of a boundary layer effect (see Desjardins et al. [8]). On the other hand, our approach applies if (2.10) is replaced by a more general stipulation of Navier’s type u · n|∂ε = 0, β[u]tan + [S[ϑ, ∇x u]n]tan |∂ε = 0. 2.3. Ill-prepared initial data. The initial state of the system is determined by the following conditions: 1 1 , ϑ(0, ·) = ϑ0,ε = ϑ + εϑ0,ε , (0, ·) = 0,ε = + ε0,ε

where

, ϑ > 0,

(2.12)

ε

1 0,ε dx =

ε

1 ϑ0,ε dx = 0 for all ε > 0,

(2.13)

and 1 1 {0,ε }ε>0 , {ϑ0,ε }ε>0 are bounded in L 2 ∩ L ∞ ().

(2.14)

u(0, ·) = u0,ε ,

(2.15)

{u0,ε }ε>0 is bounded in L 2 ∩ L ∞ (; R3 ).

(2.16)

In addition

where

2.4. Incompressible limit. Let {ε , uε , ϑε }ε>0 be a family of weak solutions to the Navier-Stokes-Fourier system (2.1–2.8) supplemented with the boundary conditions (2.10), (2.11) and the initial condition (2.12), (2.15). The precise meaning of the concept of weak solution will become clear in Sect. 3 in the context of the acoustic equation. Here, we only point out that, in general, the entropy production σε may be interpreted as a non-negative measure satisfying (2.3), (2.8) in the sense of distributions (see also [10, Chap. 2]). Our main goal is to show strong (pointwise a.a.) convergence ⎧ ⎫ ⎨ ε → ⎬ a.a. in (0, T ) × , (2.17) ⎩ ⎭ ϑε → ϑ

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and uε → U a.a. in (0, T ) ×

(2.18)

at least for suitable subsequences. In other words, the convergence imposed on the initial data through (2.12–2.16) “propagates” in time. This is not surprising for ε , ϑε , but far less obvious for the velocity uε . The piece of information contained in (2.17), (2.18) is clearly sufficient to identify the limit problem represented by the incompressible Navier-Stokes system divx U = 0,

(∂t U + divx (U ⊗ U)) + ∇x = divx µ(ϑ) ∇x U + ∇xt U , supplemented with the conventional heat equation c p (∂t + divx (U)) − divx (κ(ϑ)∇x ) = 0, where c p > 0 denotes the specific heat at constant pressure, and stands for the (relative) temperature identified as a weak limit of (ϑε − ϑ)/ε. If, in addition, a driving force is imposed, more sophisticated models as the Oberbeck-Boussinesq approximation may be obtained (see [10, Chap. 5]). As we will see in Sect. 4, the pointwise convergence of the density and the temperature claimed in (2.17) follows easily from the uniform bounds established in Sect. 4 below. On the other hand, the strong convergence of the velocity (2.18) is far more delicate and intimately related to propagation and attenuation of acoustic waves. As a matter of fact, (2.18) is not expected to hold on bounded domains with acoustically hard boundary, where large amplitude rapidly oscillating waves are generated in the limit ε → 0 (see, for instance, Lions and Masmoudi [22], or Schochet [31] ). Accordingly, for (2.18) to hold it is necessary that the target domain be unbounded, more specifically, the two closely related properties must be satisfied: • the point spectrum of the associated wave operator must be empty; • the local acoustic energy decays in time (cf. Morawetz [28], Walker [36]). 3. Lighthill’s Acoustic Equation We begin by introducing a “time lifting” ε of the measure σε through formula ε ; ϕ = σε ; I [ϕ], where we have set ε ; ϕ = σε ; I [ϕ], I [ϕ](t, x) t ϕ(z, x) dz for any ϕ ∈ L 1 (0, T ; C(ε )). =

(3.1)

0

+ It is easy to check that ε can be identified with an abstract function ε ∈ L ∞ weak (0, T ; M (ε )), where

ε (τ ), ϕ = lim σε , ψδ ϕ, δ→0+

Study of Incompressible Limit for Navier – Stokes – Fourier System

with

ψδ (t) =

⎧ 0 ⎪ ⎪ ⎪ ⎨ 1

δ ⎪ ⎪ ⎪ ⎩ 1

79

for t ∈ [0, τ ), (t − τ ), for t ∈ (τ, τ + δ), for t ≥ τ + δ,

in particular, the measure ε is well-defined for any τ ∈ [0, T ), and the mapping τ → ε is non-increasing in the sense of measures. Here the subscript in L ∞ weak stands for “weakly measurable”. Following the idea of Lighthill [20], we rewrite the Navier-Stokes-Fourier system (2.1–2.3) in the form: ε∂t Z ε + divx Vε = εdivx Fε1 , A ε∂t Vε + ω∇x Z ε = ε divx F2ε + ∇x Fε3 + 2 ∇x ε , ε ω

(3.2) (3.3)

supplemented with the homogeneous Neumann boundary conditions Vε · n|∂ε = 0, where

s(ε , ϑε ) − s(, ϑ) ε − A A + ε ε , Vε = ε uε , Zε = + ε ω ε εω

s(ε , ϑε ) − s(, ϑ) A A κ∇x ϑε 1 Fε = ε , uε + ω ε ω εϑε F2ε = Sε − ε uε ⊗ uε ,

and

ε − Fε3 = ω ε2

+ Aε

s(ε , ϑε ) − s(, ϑ) p(ε , ϑε ) − p(, ϑ) − . ε2 ε2

(3.4)

(3.5) (3.6) (3.7)

(3.8)

Here the constants A and ω are chosen to eliminate the first order term in the (formal) asymptotic expansion of (3.8) in terms of the quantities (ε − )/ε, (ϑε − ϑ)/ε , namely A

∂ p(, ϑ) ∂s(, ϑ) ∂ p(, ϑ) ∂s(, ϑ) = , ω+ A = . ∂ϑ ∂ϑ ∂ ∂

(3.9)

In order to guarantee the wave speed ω to be strictly positive, we impose the hypothesis of thermodynamic stability in the form ∂ p(, ϑ) ∂e(, ϑ) > 0, > 0 for all , ϑ > 0 ∂ ∂ϑ

(3.10)

(see Bechtel et al. [3]). Indeed (3.10), together with Gibbs’ relation (2.5), yield ω > 0, in particular, Eqs. (3.2), (3.3) form a hyperbolic system provided the right-hand is considered as given. Relation (3.10) plays a crucial role in the uniform estimates presented in Sect. 4 below.

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System (3.2), (3.3) can be viewed as a variant of Lighthill’s acoustic analogy supplemented with acoustically hard boundary condition (3.4) (cf. Lighthill [19]). We assume that Eqs. (3.2), (3.3) as well as the boundary condition (3.4) are satisfied in a weak sense, more precisely, the integral identity

T

ε

0

[ε Z ε ∂t ϕ + Vε · ∇x ϕ] dx dt =

T 0

ε

εFε1 · ∇x ϕ dx dt

(3.11)

holds for any test function ϕ ∈ Cc∞ ((0, T ) × ε ), and T [εVε · ∂t ϕ + ωZ ε divx ϕ] dx dt 0

=

ε T

A ε ; divx ϕ εF2ε : ∇x ϕ + εFε3 divx ϕ dx dt + εω ε

0

(3.12)

is satisfied for any ϕ ∈ Cc∞ ((0, T ) × ε ; R3 ), ϕ · n|∂ε = 0,

where we have identified ε (τ )ψ dx =< ε (τ ); ψ >, ψ ∈ C(ε ). ε

4. Total Dissipation Balance, Uniform Bounds 4.1. Total dissipation balance. Besides the acoustic equation specified in (3.11), (3.12), any weak solution {ε , uε , ϑε } of the Navier-Stokes-Fourier system (2.1–2.4) satisfies the total dissipation balance

1 1 ε |uε |2 + 2 Hϑ (ε , ϑε )−∂ Hϑ (, ϑ)(ε −)− Hϑ (, ϑ) (τ, ·) dx ε ε 2 ϑ + 2 σε [0, τ ] × ε ε 1 1 0,ε |u0,ε |2 + 2 Hϑ (0,ε , ϑ0,ε )−∂ Hϑ (, ϑ)(0,ε −)− Hϑ (, ϑ) dx = ε ε 2 (4.1)

for a.a. τ ∈ [0, T ],

Study of Incompressible Limit for Navier – Stokes – Fourier System

81

where we have introduced the Helmholtz function Hϑ (, ϑ) = e(, ϑ) − ϑs(, ϑ) (see [10, Chap. 5.2.2]). As a direct consequence of Gibbs’ relation (2.5) and hypothesis of thermodynamic stability (3.10), the function Hϑ enjoys the following remarkable properties: • the function → Hϑ (, ϑ) is strictly convex; • the function ϑ → Hϑ (, ϑ) is decreasing for ϑ < ϑ and increasing for ϑ > ϑ for any fixed > 0. Moreover, we have ⎫

⎧ Hϑ (, ϑ) − ∂ Hϑ (, ϑ)( − ) − Hϑ (, ϑ) ≥ c | − |2 + |ϑ − ϑ|2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ whenever ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ /2 < < 2, ϑ/2 < ϑ < 2ϑ,

(4.2)

and

Hϑ (, ϑ) − ∂ Hϑ (, ϑ)( − ) − Hϑ (, ϑ) ≥ c e(, ϑ) + ϑ|s(, ϑ)| otherwise (4.3) (see [10, Chap. 5, Lemma 5.1]).

4.2. Uniform estimates. Total dissipation balance (4.1), together with the structural properties of the Helmholtz function Hϑ specified in (4.2), (4.3), can be used to deduce uniform bounds on the family of the weak solutions {ε , uε , ϑε }ε>0 . Indeed hypotheses (2.12–2.16) imply that the integral on the right-hand side of (4.1) is bounded, therefore the quantities on the left-hand side are bounded uniformly with respect to ε → 0. Furthermore, in order to exploit (4.3), certain technical assumptions must be imposed on the structural properties of the thermodynamic functions p, e, and s. Motivated by the existence theory developed in [10, Chap. 3], we assume that the pressure p is given through formula a p(, ϑ) = ϑ 5/2 P + ϑ 4 , a > 0, (4.4) ϑ 3/2 3 where P ∈ C 1 [0, ∞), P(0) = 0, P (Z ) > 0 for all Z ≥ 0,

(4.5)

and 0

0,

P(Z ) = p∞ > 0. Z 5/3

(4.6) (4.7)

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E. Feireisl

Moreover, in accordance with Gibbs’ relation (2.5), we suppose that e(, ϑ) =

ϑ4 3 5/2 ϑ P +a , 3/2 2 ϑ

(4.8)

and, finally, the transport coefficients satisfy 0 < c1 (1 + ϑ) ≤ µ(ϑ) ≤ c2 (1 + ϑ), 0 ≤ η(ϑ) ≤ c2 (1 + ϑ) 0 < c1 (1 + ϑ 3 ) ≤ κ(ϑ) ≤ c2 (1 + ϑ 3 )

(4.9) (4.10)

for all ϑ > 0 (see [10, Chaps. 1-3] for the physical background and further discussion concerning the structural hypotheses introduced above). Note that (4.6) is a direct consequence of the hypothesis of thermodynamic stability stated in (3.10). In view of (4.2), (4.3), it is convenient to introduce the essential and residual parts of a function h as h = [h]ess + [h]res , [h]ess = (ε , ϑε )h, [h]res = (1 − (ε , ϑε )) h, where ∈ Cc∞ (0, ∞)2 , 0 ≤ ≤ 1, ≡ 1 in an open neighborhood of the point [, ϑ]. The total dissipation balance established in (4.1), together with the structural properties of the Helmholtz function stated in (4.2), (4.3), and the restrictions imposed through hypotheses (4.4–4.10), give rise to uniform estimates on the quantities appearing in the acoustic equation (3.11), (3.12) independent of ε. Very roughly indeed, we may say that the essential components are bounded in the Lebesgue space L 2 , while the residual parts vanish in the asymptotic limit in the L 1 −norm. We state the resulting list of estimates referring to [10, Chap. 8.2] for the detailed proofs. To begin, we write Z ε = Z ε1 + Z ε2 , with ε − = + ε res ε − 2 Zε = + ε ess

Z ε1

s(ε , ϑε ) − s(, ϑ) A A ε , + ε ω ε εω res s(ε , ϑε ) − s(, ϑ) A ε ω ε ess

(cf. (3.5), where ess sup Z ε1 M(ε ) ≤ εc, ess sup Z ε2 L 2 (ε ) ≤ c. t∈(0,T )

t∈(0,T )

Similarly, Vε = Vε1 + Vε2 ,

(4.11)

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83

where ⎫ ⎧ 1 2 ⎨ ess supt∈(0,T ) Vε L 1 (ε ;R3 ) ≤ εc, ess supt∈(0,T ) Vε L 2 (ε ;R3 ) ≤ c, ⎬ ⎩

⎭

ess supt∈(0,T ) Vε1 L 5/4 (ε ;R3 ) ≤ c.

(4.12)

The constants in (4.11), (4.12) are independent of ε. The driving forces in (3.11), (3.12) admit similar bounds, namely Fε1 = Fε1,1 + Fε1,2 , where

T 0

Fε1,1 2L 1 ( ;R3 ) + Fε1,2 2L 2 ( ;R3 ) dt ≤ c, ε ε F2ε

=

F2,1 ε

(4.13)

+ F2,2 ε ,

where

T 0

2 1,2 2 F1,1 ε L 1 (ε ;R3×3 ) + Fε L 2 (ε ;R3×3 )

dt ≤ c

(4.14)

and, finally, Fε3 +

A ε2 ω

ε = Fε3,1

with ess sup Fε3,1 M(ε ) ≤ c. t∈(0,T )

(4.15)

5. Analysis of Acoustic Waves Having collected all the necessary preliminaries, we are in a position to formulate rigorously the main result of this paper. To simplify the forthcoming analysis, we assume that ⊂ R3 is an unbounded domain with compact boundary. Generalization to a larger class of spatial domains satisfying the so-called limiting absorption principle is straightforward. Theorem 5.1. Let ⊂ R3 be an unbounded domain with a compact boundary of class C 2+ν , ν > 0. Assume that the thermodynamic functions p, e, s satisfy Gibbs’ equation (2.5), together with the structural restrictions (4.4–4.8). In addition, let the transport coefficients µ, η, and κ obey (4.9), (4.10). Let {ε , uε , ϑε }ε>0 be a family of (weak) solutions to the acoustic equation (3.11), (3.12) in (0, T ) × ε satisfying the total dissipation balance (4.1), where • the initial data 0,ε , ϑ0,ε , u0,ε obey (2.12–2.16); • ε = ∩ Br (ε) , where lim εr (ε) = ∞.

ε→0

(5.1)

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Then, passing to a subsequence as the case may be, we have ε → , ϑε → ϑ in L 1 ((0, T ) × K ), and uε → U in L 1 ((0, T ) × K ; R3 )

(5.2)

for any compact K ⊂ . Remark 5.1. The hypotheses of Theorem 5.1 are obviously satisfied provided the trio {ε , uε , ϑε }ε>0 represents a weak solution of Navier-Stokes-Fourier system (2.1–2.11) in (0, T ) × ε in the sense specified in [10, Chap. 3]. The existence of such a solution for a large class of initial data including those specified in (2.12–2.16) was established in [10, Chap. 3, Theorem 3.1]. Remark 5.2. The balls Br (ε) in the definition of ε may be replaced by general bounded domains B˜ ε , namely ε = ∩ B˜ ε , with Br (ε) ⊂ B˜ ε . Hypothesis (5.1) means the distance to ∂ Br (ε) dominates the speed of sound proportional to 1/ε. In particular, the acoustic waves cannot reach the outer boundary ∂ Br (ε) and return to a fixed compact set K ⊂ within the time interval (0, T ). Remark 5.3. The convergence result stated in (5.2) is not optimal with respect to the space variable, where the velocity field enjoys higher regularity, however, the main issue in the proof of Theorem 5.1 is to eliminate fast oscillations of acoustic waves in time. The remaining part of this section is devoted to the proof of Theorem 5.1. Since ε − ϑε − ϑ ϑε − ϑ ε − ε − ϑε − ϑ = = + , + , ε ε ε ε ε ε ess res ess

res

where, by virtue of (4.1), (4.7), and (4.8), ϑ −ϑ ε − ε esssupt∈(0,T ) ≤ c, esssupt∈(0,T ) ε ε ess L 2 (ε )

≤ c,

ess L 2 (ε )

and ε − esssupt∈(0,T ) ε

res L 1 (ε )

ϑ −ϑ ε ≤ εc, esssupt∈(0,T ) ε

≤ εc,

res L 1 (ε )

the proof of Theorem 5.1 reduces to showing the strong convergence of the velocity field stated in (5.2). Moreover, we claim that for (5.2) to hold it is enough to show t → uε (t, ·) · w dx → t → U(t, ·) · w dx in L 1 (0, T ) (5.3)

Study of Incompressible Limit for Navier – Stokes – Fourier System

85

for any fixed w ∈ Cc∞ (K ; R3 ), K ⊂ a given ball. Indeed, by virtue of the dissipation balance (4.1) and Korn’s inequality, we get 0

T

uε 2W 1,2 ( ;R3 ) dt ≤ c ε

(see [10, Chap. 8.2.2] for details), in particular, extending uε outside ε we may infer that uε → U weakly in L 2 (0, T ; W 1,2 (; R3 )). As W 1,2 (; R3 ) is compactly imbedded into L 2 (K ) for any bounded K , it is easy to see that (5.3) yields (5.2). Finally, since [uε ]res → 0 in, say, L 1 ((0, T ) × K ), it is enough to show (5.3) with uε replaced by [uε ]ess , which is equivalent to t → Vε (t, ·) · w dx → t → V(t, ·) · w dx in L 1 (0, T )

(5.4)

for any fixed w ∈ Cc∞ (K ; R3 ), where Vε = ε uε appears in the acoustic equation (3.11), (3.12), and V = U. 5.1. Regularization. To begin, it is useful to observe that we may assume, without loss of generality, that all quantities appearing in system (3.11), (3.12) are smooth. To this end, we deduce from (3.11), (3.12), using the uniform bounds established in (4.11), (4.12), that Z ε ∈ Cweak−(∗) ([0, T ]; M(ε )), Vε ∈ Cweak ([0, T ]; L 5/4 (ε ; R3 )), in particular, the initial values Z ε (0, ·) = Z 0,ε ∈ M(ε ), V(0, ·) = V0,ε = 0,ε u0,ε ∈ L 2 (ε ; R3 ) are well defined. Moreover, in accordance with (4.11), (4.12), 1 2 Z 0,ε = Z 0,ε + Z 0,ε ,

where 1 2 Z 0,ε M() + Z 0,ε L 2 (ε ) + V0,ε L 2 (ε ;R3 ) + ≤ c.

(5.5)

i }δ>0 ⊂ Cc∞ (ε ), For a fixed ε > 0, there exist families of smooth functions {Z 0,ε,δ 3 ∞ i = 1, 2, {V0,ε,δ }δ>0 ⊂ Cc (ε ; R ), such that 1 2 {Z 0,ε,δ }ε,δ>0 is bounded in L 1 (), {Z 0,ε,δ }ε,δ>0 is bounded in L 2 (),

(5.6)

{V0,ε,δ }ε,δ>0 is bounded in L (; R ),

(5.7)

2

3

86

E. Feireisl

and, in addition, 1 1 2 2 Z 0,ε,δ ϕ dx → Z 0,ε ; ϕ, Z 0,ε,δ ϕ dx → Z 0,ε ϕ dx for any ϕ ∈ Cc∞ (ε ), ε V0,ε,δ · ϕ dx → V0,ε · ϕ dx for any ϕ ∈ Cc∞ (ε ; R3 ),

ε

as δ → 0. Similarly, we can find ⎧ 1 ⎫ 1,1 1,2 1,i Fε,δ = Fε,δ + Fε,δ , Fε,δ ∈ Cc∞ ((0, T ) × ε ; R3 ), i = 1, 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ 2,1 2,2 2,i 2 3×3 ∞ Fε,δ = Fε,δ + Fε,δ , Fε,δ ∈ Cc ((0, T ) × ε ; R ), i = 1, 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 3,1 Fε,δ ∈ Cc∞ ((0, T ) × ε ) such that 1 2 Fε,δ → Fε1 in L 2 (0, T ; L 1 (ε ; R3 )), Fε,δ → Fε2 in L 2 (0, T ; L 2 (ε ; R3 )),

F1ε,δ → F1ε

in L (0, T ; L (ε ; R 2

1

3×3

)),

F2ε,δ → F2ε

in L (0, T ; L (ε ; R 2

2

3×3

(5.8) )), (5.9)

and 3,1 sup Fε,δ L 1 (ε ) ≤ c,

t∈[0,T ]

T 0

ε

3,1 Fε,δ ϕ dx dt →

0

T

< Fε3,1 ; ϕ > dt

(5.10)

for any ϕ ∈ Cc∞ ([0, T ] × ε ), as δ → 0. Now, consider the (unique) solution Z ε,δ , Vε,δ of the initial-boundary value problem 1 ε∂t Z ε,δ + divx Vε,δ = εdivx Fε,δ in(0, T ) × ε ,

(5.11)

ε∂t Vε,δ + ω∇x Z ε,δ = in (0, T ) × ε , Vε,δ · n|∂ε = 0, Z ε,δ (0, ·) = Z 0,ε,δ , Vε,δ (0, ·) = V0,ε,δ .

(5.12) (5.13) (5.14)

εdivx F2ε,δ

3 + ε∇x Fε,δ

Keeping ε > 0 fixed and letting δ → 0, we easily check that

Vε − Vε,δ (t, ·) · w dx → 0 as δ → 0 ess sup t∈(0,T )

for any w ∈ Cc∞ (K ; R3 ) as in (5.4). Accordingly, it is enough to show (5.4) with Vε replaced by Vε,δ(ε) for δ(ε) small enough. In what follows, we drop the subscript δ and replace the weak formulation of the acoustic equation (3.11), (3.12) by its classical counterpart (5.11), (5.12), supplemented by (5.13), (5.14). The data appearing in (5.11–5.14) are smooth and satisfy the bounds established in (4.13–4.15) uniformly for ε → 0.

Study of Incompressible Limit for Navier – Stokes – Fourier System

87

5.2. Finite speed of propagation - extension to the set . System (5.11), (5.12) admits a √ finite speed of propagation of order ω/ε. This can be easily seen multiplying Eq. (5.11) by Z ε,δ , taking the scalar product of (5.12) with Vε,δ , and integrating the resulting expression over the set √ ω (t, x) t ∈ [0, τ ], x ∈ ε , |x| < r − t . ε Consequently, by virtue of hypothesis (5.1), we may assume, extending the data in (5.11–5.14) to be zero outside ε , that Z ε,δ , Vε,δ are smooth, compactly supported in the set [0, T ] × , and solve the acoustic equation (5.11–5.14) in (0, T ) × . Thus the resulting problem reads as follows: Show that the family

t →

Vε (t, ·) · w dx

is precompact in L 1 (0, T )

(5.15)

for any w ∈ Cc∞ (K ; R3 ), K ⊂ K ⊂ a bounded ball, provided that ε∂t Z ε + divx Vε = εdivx Fε1 in (0, T ) × , εdivx F2ε

in (0, T ) × , ε∂t Vε + ω∇x Z ε = Vε · n|∂ = 0, Z ε (0, ·) = Z 0,ε , Vε (0, ·) = V0,ε in ,

(5.16) (5.17) (5.18) (5.19)

where 1 2 Z 0,ε = Z 0,ε + Z 0,ε ,

i Z 0,ε ∈ Cc∞ (), i = 1, 2,

V0,ε ∈ Cc∞ (; R3 ), and Fε1 = Fε1,1 + Fε1,2 , Fε1,i ∈ Cc∞ ((0, T ) × ; R3 ), i = 1, 2,

2,2 2,i ∞ 3×3 ), i = 1, 2, F2ε = F2,1 ε + Fε , Fε ∈ C c ((0, T ) × ; R

with 1 2 {Z 0,ε }ε>0 bounded in L 1 (), {Z 0,ε }ε>0 bounded in L 2 (),

(5.20)

{V0,ε }ε>0 bounded in L (; R ), ⎫ ⎧ 1,1 ⎨ {Fε }ε>0 bounded in L 2 (0, T ; L 1 (; R3 )), ⎬

(5.21)

⎭ ⎩ 1,2 {Fε,0 }ε>0 bounded in L 2 (0, T ; L 2 (; R3 )), ⎫ ⎧ 2,1 ⎨ {Fε }ε>0 bounded in L 2 (0, T ; L 1 (; R3×3 )), ⎬

(5.22)

⎭

(5.23)

2

⎩

3

3×3 2 2 {F2,2 )). ε,0 }ε>0 bounded in L (0, T ; L (; R

88

E. Feireisl

5.3. Compactness of the solenoidal part. Consider ψ ∈ W 1,2 ∩W 1,∞ (; R3 ), divx ψ = 0, ψ · n|∂ = 0. Multiplying Eq. (5.17) on ψ and integrating by parts, we obtain d Vε · ψ dx = − F2ε : ∇x ψ dx, Vε (0, ·) · ψ dx = V0,ε · ψ dx, dt in particular the family t → Vε · ψ dx is precompact in C[0, T ].

In other words, introducing the Helmholtz decomposition v = H[v] + H⊥ [v], H⊥ = ∇x , where is the unique solution of the Neumann problem = divx v, ∇x · n|∂ = v · n|∂

(5.24)

such that ∇x ∈ L 2 (; R3 ), ∈ L 6 () whenever v ∈ L 2 (; R3 ), we may infer that t → Vε (t, ·) · H[ψ] dx is precompact in C[0, T ],

(5.25)

provided ψ ∈ Cc∞ (K ; R3 ) is the same as in (5.15). 5.4. Local decay of the gradient component. In light of the previous arguments, it is enough to show (5.15) for the gradient part H⊥ [Vε ], H⊥ [Vε ] = ∇x ε , where ε is uniquely determined through (5.24). Accordingly, problem (5.16–5.19) can be interpreted in terms of ε as follows: ε∂t Z ε + ε = εdivx Fε1 ,

(5.26)

ε∂t ε + ωZ ε = ∇x ε · n|∂ = 0, Z ε (0, ·) = Z 0,ε , ε (0, ·) = 0,ε = −1 N [div x V0,ε ].

(5.27) (5.28) (5.29)

2 ε−1 N [div x div x Fε ],

The symbol N denotes the Laplace operator on the domain endowed with the homogeneous Neumann boundary condition. More precisely, − N is a non-negative self-adjoint operator on the space L 2 () with a domain of definition D(− N ) = v ∈ L 2 () ∇x v ∈ W 1,2 (; R3 ), v ∈ L 2 (), 1,2 (− N )[v]w dx = ∇x v · ∇x w dx for any w ∈ W () = v ∈ W 2,2 () ∇x v · n|∂ = 0 .

Study of Incompressible Limit for Navier – Stokes – Fourier System

89

We denote by {Pλ }λ≥0 the spectral resolution associated to − N - a system of orthogonal projections in L 2 () such that ∞ (− N )[v]w dx = λ d Pλ [v]w dx for any v ∈ D(− N ), w ∈ L 2 ().

0

Using {Pλ }λ≥0 we can define G(− N ) for any (possibly complex valued) Borel function G through formula ∞ G(− N )[v]w dx = G(λ) d Pλ [v]w dx .

0

5.5. Homogeneous equation. To simplify notation, we will assume hereafter that ω = 1. Our goal is to express solutions of problem (5.26–5.29) by means of Duhamel’s formula. To this end, we examine first the associated homogeneous equation ∂t Z + = 0, ∂t + Z = 0 in (0, ∞) × ,

(5.30)

supplemented with the Neumann boundary condition ∇x · n|∂ = 0,

(5.31)

Z (0, ·) = Z 0 , (0, ·) = 0 in .

(5.32)

and the initial conditions √ The (unique) solution of (5.30–5.32) can be written in terms of − N as 1 i 0 + √ [Z 0 ] (t, ·) = exp it − N 2 2 − N 1 i 0 − √ [Z 0 ] , + exp −it − N 2 2 − N √ 1 d i − N Z0 − [0 ] Z (t, ·) = − (t, ·) = exp it − N dt 2 2 √ 1 i − N Z0 + [0 ] . + exp −it − N 2 2

(5.33)

(5.34)

In particular, problem (5.30–5.32) generates a group in the associated energy space (Z , ) ∈ L 2 () × H 1,2 (), where H 1,2 () denotes the homogeneous Sobolev space, H 1,2 () = {v | v ∈ L 6 (), ∇x v ∈ L 2 (; R3 )}. At this stage, we apply Theorem 1.1 taking • X = L√2 (), • H = − N , • A = ϕG(− N ), ϕ ∈ Cc∞ (), G ∈ Cc∞ (0, ∞), for a suitable non-negative function G.

90

E. Feireisl

Since A ◦ R H [λ] ◦ A∗ = ϕG(− N ) √

1 G(− N )ϕ, − N − λ

it is enough to verify hypothesis (1.3) of Theorem 1.1 for the values of the parameter λ belonging to a bounded set Q of the complex plane, namely λ ∈ Q = {z ∈ C | Re[z] ∈ [a, b], 0 < |Im[z]| < d}, where 0 < a < b < ∞, supp[G] ⊂ (a 2 , b2 ). and d > 0. Indeed if λ ∈ / Q, then 1 G(− N ) √ G(− N ) − N − λ is a bounded linear operator, with a norm bounded in terms of the parameters a, b, d. Thus we can rewrite A ◦ R H [λ] ◦ A∗ = ϕ with

M(− N , λ) ϕ, (− N ) − λ2

M(− N , λ) = G(− N )( − N + λ)G(− N )

– a bounded linear operator in L 2 () for λ ∈ Q. At this stage, we recall that the operator − N satisfies the limiting absorption principle, namely 1 sup V ◦ (− ) − µ ◦ V 2 2 ≤ c(α, β, ϕ) < ∞ (5.35) N µ∈C;α0 , {Hε2 }ε>0 bounded in L 2 (0, T ; L 2 (; C)). In accordance with relations (5.40–5.42), formula (5.39) can be recast in the form

t 1 1 N [h 1ε ]+ √ ε (t, ·) = exp ±i [h 2ε ] ± i N [h 3ε ]+ √ [h 4ε ] − N ε − N − N t

t −s exp ±i − N + ε 0

1 N [Hε1 ]+ √

− N

[Hε2 ] ± i

1 N [Hε3 ]+ √

− N

[Hε4 ]

ds,

(5.43) where {h iε }ε>0 is bounded in L 2 (), and {Hεi }ε>0 is bounded in L 2 ((0, T ) × ), i = 1, . . . , 4. (5.44)

Study of Incompressible Limit for Navier – Stokes – Fourier System

5.6.2. Convergence. Taking H (ξ ) =

1 ξ

93

in (5.38) we get

2 ϕG(− N ) exp ±i t − N [ N [h i ]] dt ε ε −∞ L 2 () ∞ 2 =ε dt ≤ εcG h iε L 2 () , i = 1, 3, ϕG(− N ) exp ±it − N [ N [h iε ]] 2 ∞

L ()

−∞

and, for H (ξ ) =

√

ξ,

2 ϕG(− N ) exp ±i t − N ( − N )−1 [h i ] dt ε ε −∞ L 2 () ∞ 2 =ε ϕG(− N ) exp ±it − N ( − N )−1 [h iε ] 2 ∞

L ()

−∞

dt

≤ εcG h iε L 2 () , i = 2, 4. Similarly, 2 ϕG(− N ) exp ±i t − s − N N H i ds dt ε ε 0 0 L 2 () ∞ T 2 ϕG(− N ) exp ±it − N exp i −s − N N H i ds dt ≤εc ε ε −∞ 0 L 2 () 2 T T −s i 2 i exp i H − ds=εc ds, i = 1, 3. ≤εcG Hε 2 N G ε L () ε 2

T

T

L ()

0

0

(5.45) Finally, 0

2 1 i ϕG(− N ) exp ±i t − s − N Hε ds dt √ ε − N 0 L 2 () ∞ T −s ≤ εc ϕG(− N ) exp ±it − N exp i ε − N −∞ 0 2 1 × √ Hεi ds dt 2 − N L () 2 T T −s i 2 i exp i H ≤εcG − ds = εc ds, i = 2, 4. Hε 2 N G ε L () ε 0 0 L 2 () (5.46)

T

T

Thus we conclude that ϕG(− N )[ε ] → 0 in L 2 ((0, T ) × ) as ε → 0 for any G ∈ Cc∞ (0, ∞) and ϕ ∈ Cc∞ ().

(5.47)

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In order to establish (5.4), we take ϕ ∈ Cc∞ () such that ϕ ≡ 1 on the set K containing the support of w and write ∇x ε · w dx = − ε divx w dx = − ϕε divx w dx =− ϕG(− N )[ε ]divx w dx+ ϕ(G(− N )−Id)[ε ]divx w dx,

where, in accordance with (5.47), the first integral on the right-hand side tends to zero for ε → 0 for any fixed G. On the other hand, we can take a family of functions G(λ) 1, in particular, (G(− N ) − Id)[h] → 0 for any fixed h ∈ L 2 (). Consequently, writing ϕ(G(− N ) − Id)[ε ]divx w dx = ϕ(G(− N ) − Id)[ε ]divx w dx ε (G(− N ) − Id)[divx w] dx,

we can deduce (5.4) from (5.43), (5.47) as soon as we observe that 1 N [divx w], √ [divx w] ∈ L 2 (). − N Thus we have proved (5.4), and therefore Theorem 5.1. References 1. Alazard, T.: Low Mach number flows and combustion. SIAM J. Math. Anal. 38(4), 1186–1213 (electronic) (2006) 2. Alazard, T.: Low Mach number limit of the full Navier-Stokes equations. Arch. Rat. Mech. Anal. 180, 1–73 (2006) 3. Bechtel, S.E., Rooney, F.J., Forest, M.G.: Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids. J. Appl. Mech. 72, 299–300 (2005) 4. Burq, N.: Global Strichartz estimates for nontrapping geometries: about an article by H. F. Smith and C. D. Sogge: “Global Strichartz estimates for nontrapping perturbations of the Laplacian”. Comm. Part. Diff. Eqs. 28(9–10), 1675–1683 (2003) 5. Burq, N., Planchon, F., Stalker, J.G., Tahvildar-Zadeh, A.S.: Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J. 53(6), 1665–1680 (2004) 6. Dermejian, Y., Guillot, J.-C.: Théorie spectrale de la propagation des ondes acoustiques dans un milieu stratifié perturbé. J. Diff. Eqs. 62, 357–409 (1986) 7. Desjardins, B., Grenier, E.: Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455(1986), 2271–2279 (1999) 8. Desjardins, B., Grenier, E., Lions, P.-L., Masmoudi, N.: Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 461–471 (1999) 9. Engquist, B., Majda, A.: Radiation boundary conditions for acoustic and elastic wave calculations. Comm. Pure Appl. Math. 32(3), 314–358 (1979) 10. Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Basel: Birkhäuser-Verlag, 2009 11. Feireisl, E., Poul, L.: On compactness of the velicity field in the incompressible limit of the full NavierStokes-Fourier system on large domains. Math. Meth. Appl. Sci. 32, 1269–1286 (2009) 12. Isozaki, H.: Singular limits for the compressible Euler equation in an exterior domain. J. Reine Angew. Math. 381, 1–36 (1987)

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13. Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279, (1965/1966) 14. Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34, 481–524 (1981) 15. Klein, R.: Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. Z. Angw. Math. Mech. 80, 765–777 (2000) 16. Klein, R.: Multiple spatial scales in engineering and atmospheric low Mach number flows. ESAIM: Math. Mod. Numer. Anal. 39, 537–559 (2005) 17. Klein, R., Botta, N., Schneider, T., Munz, C.D., Roller, S., Meister, A., Hoffmann, L., Sonar, T.: Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Engrg. Math. 39, 261–343 (2001) 18. Leis, R.: Initial-boundary Value Problems in Mathematical Physics. Stuttgart: B. G. Teubner, 1986 19. Lighthill, J.: On sound generated aerodynamically I. General theory. Proc. of the Royal Society of London A 211, 564–587 (1952) 20. Lighthill, J.: Waves in Fluids. Cambridge: Cambridge University Press, 1978 21. Lions, P.-L.: Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models. Oxford: Oxford Science Publication, 1998 22. Lions, P.-L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998) 23. Lions, P.-L., Masmoudi, N.: Une approche locale de la limite incompressible. C.R. Acad. Sci. Paris Sér. I Math. 329(5), 387–392 (1999) 24. Masmoudi, N.: Asymptotic problems and compressible and incompressible limits. In: Advances in Mathematical Fluid Mechanics, edited by Málek, J., Neˇcas, J., Rokyta, M., Berlin: Springer-Verlag, 2000, pp. 119–158 25. Masmoudi, N.: Examples of singular limits in hydrodynamics. In: Handbook of Differential Equations, III, Dafermos, C., Feireisl, E., eds., Amsterdam: Elsevier, 2006 26. Masmoudi, N.: Rigorous derivation of the anelastic approximation. J. Math. Pures Appl. 88, 230– 240 (2007) 27. Metcalfe, J.L.: Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle. Trans. Amer. Math. Soc. 356(12), 4839–4855 (electronic) (2004) 28. Morawetz, C.S.: Decay for solutions of the exterior problem for the wave equation. Comm. Pure Appl. Math. 28, 229–264 (1975) 29. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1978 30. Schochet, S.: Fast singular limits of hyperbolic PDE’s. J. Diff. Eqs. 114, 476–512 (1994) 31. Schochet, S.: The mathematical theory of low Mach number flows. M2 AN Math. Model Numer. Anal. 39, 441–458 (2005) 32. Shimizu, S.: The limiting absorption principle. Math. Meth. Appl. Sci. 19, 187–215 (1996) 33. Smith, H.F., Sogge, C.D.: Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Part. Diff. Eqs. 25(11–12), 2171–2183 (2000) 34. Vaigant, V.A.: An example of the nonexistence with respect to time of the global solutions of NavierStokes equations for a compressible viscous barotropic fluid (in Russian). Dokl. Akad. Nauk 339(2), 155– 156 (1994) 35. Va˘ınberg, B.R.: Asimptoticheskie metody v uravneniyakh matematicheskoi fiziki. Moscow: Moskov. Gos. Univ., 1982 36. Walker, H.F.: Some remarks on the local energy decay of solutions of the initial-boundary value problem for the wave equation in unbounded domains. J. Diff. Eqs. 23(3), 459–471 (1977) 37. Wilcox, C.H.: Sound Propagation in Stratified Fluids. Appl. Math. Ser. 50. Berlin: Springer-Verlag, 1984 Communicated by P. Constantin

Commun. Math. Phys. 294, 97–119 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0917-y

Communications in

Mathematical Physics

On q -Deformed gl+1 -Whittaker Function I Anton Gerasimov1,2,3 , Dimitri Lebedev1 , Sergey Oblezin1 1 Institute for Theoretical and Experimental Physics,

117259, Moscow, Russia. E-mail: [email protected]; [email protected]

2 School of Mathematics, Trinity College, Dublin 2, Ireland.

E-mail: [email protected]

3 Hamilton Mathematics Institute, TCD, Dublin 2, Ireland

Received: 4 February 2009 / Accepted: 25 June 2009 Published online: 26 September 2009 – © Springer-Verlag 2009

Abstract: We propose a new explicit form of q-deformed Whittaker functions solving q-deformed gl+1 -Toda chains. In the limit q → 1 the constructed solutions reduce to the classical gl+1 -Whittaker functions of class one in the form proposed by Givental. An important property of the proposed expression for the q-deformed gl+1 -Whittaker function is that it can be represented as a character of C∗ × GL+1 . This provides a q-version of the Shintani-Casselman-Shalika formula for the p-adic Whittaker function. The Shintani-Casselman-Shalika formula is recovered in the limit q → 0 when the q-deformed Whittaker function is reduced to a character of a finite-dimensional representation of gl+1 expressed through the Gelfand-Zetlin basis. Introduction Whittaker functions corresponding to semisimple finite-dimensional Lie algebras arise in various parts of modern mathematics. In particular, these functions appear in representation theory as matrix elements of infinite-dimensional representations, in the theory of quantum integrable systems as a common eigenfunction of Toda chain quantum Hamiltonians, in string theory as generating functions of correlators in Type A topological string theory on flag manifolds, and in number theory in the description of local Archimedean L-factors corresponding to automorphic representations. Although much studied, Whittaker functions seem to have some deep properties that are not yet fully revealed. In this paper we study the q-deformed gl+1 -Whittaker functions. The q-deformed Whittaker functions can be identified with the common eigenfunctions of a set of commuting q-deformed Toda chain Hamiltonians. This q-deformed Toda chain (also known as the relativistic Toda chain [Ru]) was discussed in terms of representation theory of quantum groups in [Se1,Et,Se2] and an integral representation for the q-deformed gl+1 -Whittaker function was constructed in [KLS]. Recently the q-deformed Toda chain attracted a special interest due to its connection with quantum K -theory of flag manifolds

98

A. Gerasimov, D. Lebedev, S. Oblezin

[GiL]. In this paper we pursue another direction. Our principal motivation to study q-deformed Whittaker functions is that in this more general setting, some important hidden properties of classical Whittaker functions become visible. The main result of the paper is given by Theorem 2.1 where a new expression for the q-deformed gl+1 -Whittaker function (for q < 1) is introduced. As a simple corollary of Theorem 2.1, the q-deformed gl+1 -Whittaker function can be represented as a character of C∗ × GL+1 . In the limit q → 1 this leads to a similar representation of the classical gl+1 -Whittaker function. This representation is not easy to perceive looking directly at the classical Whittaker functions. The importance of this representation of a (q-deformed) gl+1 -Whittaker function becomes obvious if we notice that in the limit q → 0 the constructed q-deformed Whittaker function reduces to the p-adic Whittaker function. In this limit, the representation as a character reduces to the well-known Shintani-Casselman-Shalika representation of the p-adic G L +1 -Whittaker function as a character of a finite-dimensional representation of G L +1 [Sh,CS]. Thus, the representation of a (q-deformed) gl+1 -Whittaker function as a character can be considered as a q-version of the Shintani-Casselman-Shalika representation. For instance, the constructed q-deformed Whittaker function vanishes outside a dominant weight cone of gl+1 similarly to the Shintani-Casselman-Shalika p-adic Whittaker function. We expect that the representation of the classical Whittaker function as a character should provide important insights into the arithmetic geometry at an infinite place of Spec(Z). Let us also remark that taking into account the results of [CS] one should expect that in the case of an arbitrary semisimple Lie algebra g, q-deformed g-Whittaker function should be given by a character of C∗ × L G(C), where Lie( L G) = L g is the Langlands dual Lie algebra. It is worth mentioning that the q → 1 limit of the explicit expression of the q-deformed Whittaker function proposed in this paper reduces to the integral representations for classical Whittaker functions introduced by Givental [Gi,GKLO]. We consider this as a sign of an “arithmetic nature” of this integral representation. On the other hand, the explicit solution has an obvious relation with the Gelfand-Zetlin parametrization of finite-dimensional representations of gl+1 (and precisely reproduces to the Gelfand-Zetlin formula for characters of finite-dimensional representations in the limit q → 0). The duality of Gelfand-Zetlin and Givental representations was already noticed in [GLO]. Let us comment on our approach to the derivation of explicit expressions for q-deformed Whittaker functions. It is known [Et] that in a certain limit defining difference equations for Macdonald polynomials are transformed into the eigenfunction equations for the q-deformed Toda chain. This is a simple generalization of the Inozemtsev limit [I], which transforms the Calogero-Sutherland integrable model into the standard Toda chain. The other ingredient we use is a recursive construction of Macdonald polynomials (analogous to the recursive construction for (q-deformed) Toda chain eigenfunctions [KL1,KLS]). We combine these results to obtain a recursive expression for the q-deformed gl+1 -Whittaker functions. The explicit form of the q-deformed Whittaker function implies various interesting interpretations. These include connections with representation theory (via characters of Demazure modules), geometry of quiver varieties, quantum cohomology of flag manifolds and will be discussed elsewhere [GLO2]. Finally, note that eigenfunctions of the q-deformed Toda chain were discussed previously (e.g. [KLS,GKL1,BF and FFJMM]). The relation of these constructions with the one proposed in this paper is an interesting question which deserves further considerations.

On q-Deformed gl+1 -Whittaker Function I

99

The paper is organized as follows. In Sect. 1 we recall a system of mutually commuting Macdonald-Ruijsenaars difference operators and a recursive construction of their common eigenfunctions. In Sect. 2 we derive a recursive expression for solutions of the q-deformed gl+1 -Toda chain. In Sect. 3 various limiting cases elucidating the construction of the q-deformed gl+1 -Whittaker function are discussed. In Sect. 4, details of the proof of Theorem 2.1 are given. 1. Macdonald-Ruijsenaars Difference Operators In this section we recall some relevant facts from the theory of Macdonald polynomials (see e.g. [Mac,Kir,AOS]). Consider symmetric polynomials in variables (x1 , . . . , x+1 ) over the field Q(q, t) of rational functions in q, t. Given a partition = (0 ≤ 1 ≤ 2 ≤ · · · ≤ +1 ), denote by the same symbol the Young diagram containing + 1 rows with k boxes in the k th row, and the upper row having the maximal length +1 . Let m and π be the following two bases in the space of symmetric polynomials indexed by partitions : m =

σ ∈S+1

+1 1 2 xσ(1) xσ(2) · . . . · xσ(+1) ,

π = π1 π2 · . . . · π+1 ,

πn =

+1

xkn ,

k=1

where S+1 is the permutation group. Define a scalar product , q,t on the space of symmetric functions over Q(q, t) as follows: π , π q,t = δ, · z (q, t), where z (q, t) =

nmn m n ! ·

n≥1

N 1 − q k , 1 − t k

m n = |{k| k = n}|.

k=1

In the following we always assume that q, t ∈ R>0 , 0 < q < 1. The following remarkable theorem was proved by Macdonald [Mac]. gl

gl

Theorem 1.1 (Macdonald). There is a unique basis P +1 = P +1 (x; q, t) in the ring of symmetric polynomial function over Q(q, t) such that gl

P +1 = m +

u m ,

p2,2 ,

is a common eigenfunction of commuting Hamiltonians gl

gl

H1 2 = T1 + (1 − q p2,2 − p2,1 +1 )T2 ,

H2 2 = T1 T2 .

Note that the formula (2.5) can be easily rewritten in the recursive form. Corollary 2.1. The following recursive relation holds gl λ +1 ( p +1 )

=

p,i ∈P+1,

( p ) q

λ+1

+1 i=1

p+1,i − i=1 p,i

gl

×Q+1, ( p +1 , p |q)λ ( p ),

(2.6)

where Q+1, ( p +1 , p |q) =

1 i=1

,

( p,i − p+1,i )q ! ( p+1,i+1 − p,i )q !

and ( p ) =

−1

( p,i+1 − p,i )q !,

i=1

where the notations λ = (λ1 , . . . , λ+1 ), λ = (λ1 , . . . , λ ) are used.

(2.7)

On q-Deformed gl+1 -Whittaker Function I

105

Remark 2.1. The solution (2.5) is a q-analog of Givental’s integral representation of gl+1 -Whittaker function [Gi,GKLO]: gl ψλ +1 (x1 , . . . , x+1 )

=

R

gl

i=1

gl Q gl+1 (t +1 , t |λ+1 )

gl

dt,i Q gl+1 (t +1 , t |λ+1 )λ1,...,λ (t ),

= exp ıλ+1

+1

(2.8)

t+1,i

i=1

−

t,i

t −t t,i −t+1,i+1 +1,i ,i , e +e −

i=1

i=1

where λ = (λ1 , . . . , λ+1 ), t k = (tk1 , . . . , tkk ), xi := t+1,i , i = 1, . . . , + 1 and we gl assume that Q gl1 (t11 |λ1 ) = eıλ1 t1,1 . 0

Proposition 2.1. There exists a C∗ × G L +1 (C) module V such that the common eigenfunction (2.5) of the q-deformed Toda chain allows the following representation for p+1,1 ≤ p+1,2 ≤ . . . p+1,+1 : gl+1

λ

( p +1 ) = Tr V q L 0

+1

q λi Hi ,

(2.9)

i=1

where Hi := E i,i , i = 1, . . . , + 1 are Cartan generators of gl+1 = Lie(G L +1 ) and L 0 is a generator of Lie(C∗ ). Proof. It is useful to rewrite (2.5) in the following form: gl+1

λ

( p +1 ) = ( p +1 )−1

q λk+1 (

i

pk+1,i − j pk,i )

pk,i ∈P (+1) k=1

k pk+1, i+1 − pk+1, i × , pk, i − pk+1, i q i=1

( p +1 ) = where

( p+1, j+1 − p+1, j )q !,

j=1

(n)q ! n . = k q (n − k)q ! (k)q !

Now taking into account the identities n n−1 n−1 = + qk , k q k q k−1 q

∞

1 = q kn , n (1 − q ) k=0

106

A. Gerasimov, D. Lebedev, S. Oblezin gl

one obtains an expansion of the function λ +1 ( p +1 ) into the sum of terms q N q m 1 λ1 · · · q m +1 λ+1 with positive integral coefficients K N ,m 1 ,...,m +1 . Let G L 1 ×···×G L 1 ⊂ G L +1 be the diagonal subgroup, then let us define the following C∗ ×G L 1 ×···×G L 1 -module:

V =

VN ,m 1 ,···,m +1 ,

VN ,m 1 ,···,m +1 = C K N ,m 1 ,...,m +1 .

N , m 1 ,...,m +1

Each VN ,m 1 ,···m +1 is acted on by the factor C∗ via multiplication by q N ; the torus (G L 1 )+1 acts on each VN ,m 1 ,···m +1 by multiplication by q m 1 λ1 +...+m +1 λ+1 . Let us note that the q-Toda chain eigenfunction problem (2.2) contains the variables z i := q λi only due to the eigenvalues given by the central functions χr (z) (2.3). Since the initial conditions for the eigenfunction (2.5) can also be expressed through χr (z), therefore we can extend the structure group of the module V from the torus C∗ × (G L 1 )+1 to the whole group C∗ × G L +1 . Thus we obtain the representation (2.9) and complete the proof.

Remark 2.2. There exists a finite-dimensional C∗ × G L +1 (C) module V f such that the following representation holds for p+1,1 ≤ p+1,2 ≤ . . . p+1,+1 : gl+1 ( p ) = ( p ) gl+1 ( p ) = Tr V f q L 0 λ λ +1 +1 +1

+1

q λi Hi .

(2.10)

i=1

The module V entering (2.9) and the module V f entering (2.10) have a structure of modules under the action of (quantum) affine Lie algebras which will be discussed [GLO2]. See however Proposition 3.4 for an explicit description of V f .

3. Various Limits Besides the limit q → 1, which recovers the classical gl+1 -Whittaker function as a solution of the gl+1 -Toda chain, there are other interesting limits elucidating the meaning of the q-deformed Toda chain equations. In a limit q → 0, the q-deformed gl+1 -Whittaker functions are given by the characters of irreducible representations of gl+1 . This will allow us to identify the Whittaker functions with p-adic Whittaker functions according to Shintani-Casselman-Shalika formula [Sh,CS]. There is also another q → 1 limit which clarifies the recursive structure of q-deformed gl+1 -Whittaker functions. 3.1. A limit q → 0. In this subsection we discuss a limit q → 0 of the constructed q-deformed Whittaker function (we restrict the Whittaker function to the domain { p+1,1 ≤ . . . ≤ p+1,+1 }, where it is non-trivial). We will show that in the domain { p+1,1 ≤ . . . ≤ p+1,+1 } the system of equations for common eigenfunctions of q-deformed Toda chain Hamiltonians reduces to the Pieri formulas (a particular case of Littlewood-Richardson rules) for the decomposition of the tensor product of an arbitrary finite-dimensional representation and a fundamental representation of gl+1 .

On q-Deformed gl+1 -Whittaker Function I

107

Let us rewrite the q-deformed Whittaker function (2.5) using the variables z i = q λ+1,i ,

gl+1

( p +1 |z) =

+1

pk,i ∈P (+1)

k=1

(

k

zk

i=1

k−1 pk,i − i=1 pk−1,i )

k−1 ( pk,i+1 − pk,i )q !

×

k=2 i=1 k

,

( pk,i − pk+1,i )q ! ( pk+1,i+1 − pk,i )q !

k=1 i=1

p+1,1 ≤ · · · ≤ p+1,+1 ,

(3.1)

where z = (z 1 , . . . , z +1 ). Proposition 3.1. 1. In the limit q → 0, the eigenfunction (3.1) is given in the domain p+1,1 ≤ · · · ≤ p+1,+1 by gl

χ p +1 (z) := gl+1 ( p +1 |z)|q=0 = +1

+1

(

k

zk

i=1

k−1 pk,i − i=1 pk−1,i )

.

(3.2)

pk,i ∈P +1 k=1

gl

2. The functions χ p +1 (z) satisfy the following set of difference equations: +1

gl+1

χr gl+1

where χr

gl

(z) χ p +1 (z) = +1

Ir

gl

χ p +1+Ir (z), r = 1, . . . , + 1,

(3.3)

+1

(z) are the characters of fundamental representations Vωr = gl χr +1 (z) = z i 1 · · · z ir , r = 1, . . . , + 1,

r

C+1 :

Ir

and Ir = (i 1 < i 2 < · · · < ir ) ⊆ {1, 2, . . . , + 1}. gl 3. The functions χ p +1 (z) can be identified with characters of irreducible finite+1 dimensional representations of G L +1 corresponding to partitions p+1,1 ≤ · · · ≤ p+1,+1 . Proof. The relations (3.2) and (3.3) follow directly from the similar relations for generic q. To prove the last statement note that (3.2) can be identified with the expression for characters of irreducible finite-dimensional representations of G L +1 obtained using the Gelfand-Zetlin bases (see e.g. [ZS]). Let { pi j }, i = 1, . . . , + 1, j = 1, . . . , i be a Gelfand-Zetlin (GZ) pattern P (+1) , that is the integers pi, j should satisfy the conditions pi+1, j ≤ pi, j ≤ pi+1, j+1 . An irreducible finite-dimensional representation can be realized in a vector space with the basis v p parametrized by GZ patterns { pi j } with fixed p+1,i . The action of the Cartan generators on v p is given by s+1 −s +1,+1 z 1E 11 z 2E 22 · · · z +1 v p = z 1s1 z 2s2 −s1 · · · z +1 vp, E

sk =

k i=1

pki .

(3.4)

108

A. Gerasimov, D. Lebedev, S. Oblezin

Thus, we have for the character gl

+1 χ p+1,1 ,..., p+1,+1 (z 1 , . . . , z +1 ) =

pk,i

+1

∈P (+1)

k=1

(

k

zk

i=1

k−1 pk,i − i=1 pk−1,i )

.

(3.5)

Remark 3.1. The second identity in Proposition 3.1 is known as the Pieri formula (see e.g. [FH], Appendix A). Thus, the q-deformed Toda chain equations can be considered as q-deformations of the Pieri formulas. There is a generalization of q-Toda chain equations providing a q-version of a general Littlewood-Richardson rule. The GZ representation of the characters (3.2) has an obvious recursive structure. Namely, one should introduce variables z i = q λi , i = 1, . . . , + 1 and then take the limit q → 0 in (2.6). This leads to the following. Corollary 3.1. Characters satisfy the following recursive relation: gl

+1 χ p+1,1 ,..., p+1,+1 (z 1 , . . . , z +1 ) +1 p − p gl z +1i=1 +1,i i=1 ,i χ p,1 ,..., p, (z 1 , . . . , z ), =

(3.6)

p,i ∈P+1,

where the sum runs over p = ( p,1 , . . . , p, ) satisfying the GZ conditions p+1,i ≤ p,i ≤ p+1,i+1 . Note that these recursive relations can be derived using the classical Cauchy-Littlewood formula C+1,m+1 (x, y) =

+1 m+1

gl 1 gl = χ +1 (x) χ m+1 (y), 1 − xi y j

m ≤ ,

(3.7)

i=1 j=1

gl

gl

where the sum runs over Young diagrams of glm+1 and χ +1 (x) = χ +1 (x1 , . . . , x+1 ) are the characters of the irreducible finite-dimensional representation of G L +1 corresponding to Young diagram . gl

Proposition 3.2. The following integral relations for the characters χ +1 (x) hold: gl χ +1 (x)

gl χ +1 (x)

=

y1 =∞

···

=

y1 =∞

···

ı dyi gl C+1, (x, y −1 )χ (y) (y|0, 0)), (3.8) 2π yi

y+1 =∞ i=1

+1 ı dyi gl C+1,+1 (x, y −1 )χ +1 (y) (y|0, 0), 2π yi y+1 =∞ i=1

⎛ gl+1 ⎝ χ+(+1) k (x) =

+1 j=1

⎞ gl

x kj ⎠ χ +1 (x).

On q-Deformed gl+1 -Whittaker Function I

109

The relations above provide the character of the irreducible finite-dimensional representation of G L +1 corresponding to any Young diagram . Remark 3.2. The relations (3.8) can be obtained from similar relations for Macdonald polynomials in the limit t → 0, q → 0. These recursion relations are analogs of the Mellin-Barnes recursion relations for the classical Whittaker functions (see [KL1,GKL, GLO] for details). According to Shintani-Casselman-Shalika formula, the p-adic Whittaker function corresponding to an algebraic reductive group G is equal to the character of the Langlands dual Lie group L G 0 acting in an irreducible finite-dimensional representation [Sh,CS]. Thus according to Proposition 3.2 we can consider gl+1 -Whittaker functions at q → 0 as an incarnation of p-adic Whittaker functions (this is in complete agreement with the results of [GLO3]). Moreover, taking into account Proposition 2.1 one can consider the main result of this paper as a generalization of the Shintani-CasselmanShalika formula to the q-deformed case which includes a limiting case of classical gl+1 Whittaker functions. This interpretation of classical Whittaker functions evidently deserves further attention. 3.2. The limit q → 1. In this subsection we consider a modified limit q → 1 leading to a very simple degeneration of the q-deformed Toda chain. In this limit the q-deformed Toda chain can be easily solved. Moreover the form of the solution makes the recursive expressions (2.6) for the q-deformed Toda chain solution very natural. Let us redefine the q-deformed Toda chain Hamiltonians and their common eigenfunctions as follows (we assume p+1,1 ≤ · · · ≤ p+1,+1 below): gl+1

Jr

gl+1

= ( p +1 ) Hr

gl+1 ( p |z) = ( p ) · gl+1 ( p |z), +1 +1 +1

( p +1 )−1 ,

( p +1 ) =

( p+1, j+1 − p+1, j )q !,

j=1

where gl+1 ( p +1 |z) is given by (3.1). Explicitly, we have that 1−δi −i , 1 1−δir −i , 1 1−δi −i , 1 gl X i1 2 1 · · · · · Ti1 · · · · · Tir , (3.9) Jr +1 = X ir −1 r −1 · X ir r +1 r Ir

X i = 1 − q p+1,i+1 − p+1, i . Let us now take the limit where we assume ir +1 = + 2 and q → 1: gl+1 ( p |z) = lim gl+1 ( p |z), ψ +1 +1 q→1

gl+1

hr

gl +1

= lim Jr q→1

.

We have limq→1 (1 − q n ) = 0 and, therefore, we obtain from (2.1) that gl+1

hr

= T+2−r · · · · · T+1 ,

where Ti acts on the functions of p+1, j as follows: p +1 ), p+1,k = p+1,k + δk,i , i, k = 1, . . . , + 1. Ti f ( p +1 ) = f ( Now the eigenvalue problem is easily solved.

110

A. Gerasimov, D. Lebedev, S. Oblezin

Proposition 3.3. 1. The function gl+1 ( p |z) ψ +1

p+1, 1 p+1, i+1 − p+1, i gl gl+1 = χ+1+1 (z) χ+1−i (z) ,

r = 1, . . . , + 1, (3.10)

i=1

is an eigenfunction of the family of mutually commuting difference operators gl+1 gl+1

hr gl+1

where χr

ψ

gl+1

( p +1 |z) = χr

gl+1 ( p |z), (z) ψ +1

(z) is the character of the fundamental representation Vωr = gl+1

χr

(z) =

z i1 · · · z ir , z i = q λi ,

(3.11) r

C+1 :

i = 1, . . . , + 1,

Ir

Ir = {i 1 < i 2 < · · · < ir } ⊂ {1, 2, . . . , + 1} and gl+1

hr

= T+2−r · · · · · T+1 , r = 1, . . . , + 1.

2. In the domain p+1,1 ≤ · · · ≤ p+1,+1 , the following recursive relation holds: gl+1 ( p |z) = ψ +1

+1

p,i ∈P+1,

z +1i=1

p+1,i − i=1 p,i

p+1, i+1 − p+1, i gl ( p |z ), ·ψ × p, i − p+1, i

(3.12)

i=1

where z = (z 1 , . . . , z ). Proof. The identity (3.11) follows from the construction. Let us prove that (3.12) follows from (3.11). Using the relation gl+1

χr

gl

gl

(z) = χr (z ) + z +1 · χr −1 (z ),

r = 1, . . . , + 1,

we have

gl

χ+1+1 (z)

p+1, 1 p+1, i+1 − p+1, i gl+1 χ+1−i (z) i=1

=

p+1,1 gl z +1 χ (z )

p+1,i+1

i=1

p,i = p+1,i

gl

z +1 χ−i (z )

p,i − p+1,i p gl +1,i+1 − p+1, i · χ+1−i (z ) · p, i − p+1, i

p+1,i+1 − p,i

On q-Deformed gl+1 -Whittaker Function I

=

p,i ∈P+1,

z +1i

111

p+1,i+1 − i p,i

p+1,i+1 − p+1, i p, i − p+1, i i=1

p, 1 −1 p, i+1 − p, i gl gl · χ (z ) χ−i (z ) i=1

=

p,i ∈P+1,

z +1i

p+1,i+1 − i p,i

p+1,i+1 − p+1, i gl ( p ). ψ λ p, i − p+1, i

(3.13)

i=1

This completes the proof. Remark 3.3. The functions

gl+1 ( p |z), ψ gl+1 ( p +1 |z) = −1 ( p +1 )ψ +1 satisfy the following recursive relations: p − ψ gl+1 ( p +1 |z) = z +1i +1,i i

p,i

p,i ∈P+1,

×

−1 i=1

( p,i

( p,i+1 − p,i )! ψ gl ( p |z ). − p+1,i )! ( p+1,i+1 − p,i )!

(3.14)

This makes the formula (2.6) for the solution of q-deformed Toda chain slightly less mysterious. Proposition 3.4. The following representation holds: gl+1 ( p |z) = Tr V f ψ +1

+1

q λi Hi ,

(3.15)

i=1

where ⊗( p+1,+1 − p+1, )

V f = Vω1

⊗( p+1,2 − p+1,1 )

⊗ · · · ⊗ Vω

⊗p

⊗ Vω+1+1,1 ,

(3.16)

and Vωn = ∧n C are the fundamental representations of G L +1 . Proof. This is an obvious consequence of Proposition 3.3.

The module V f entering (2.10) is isomorphic to (3.16) as the G L +1 -module but has a more refined structure under the action of quantum affine Lie algebras and will be discussed in [GLO2]. 4. Proof of Theorem 2.1 In this section we provide a proof of Theorem 2.1. To derive an explicit expression (2.5) for q-deformed gl+1 -Whittaker function we take as a motivation the limit t → ∞ of recursive relations (1.4) for Macdonald polynomials. Note that the recursion relations in Proposition 1.1 were defined for 0 < t < 1 and thus taking the limit t → ∞ needs some care. In particular we define the analog of the pairing (1.2) from scratch. We start with some useful relations that will be used in the proof of Theorem 2.1.

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4.1. q-deformed Toda chain from Macdonald-Ruijsenaars system. In this subsection we demonstrate that quantum Hamiltonians of the q-deformed Toda chain arise as a limit of Macdonald-Ruijsenaars operators when t → ∞. Let us take t = q −k . Proposition 4.1. The following relations hold: gl+1

Hr

gl

= lim Hr,k+1 k→∞ 1−δi , 1 1−δir −ir −1 , 1 1−δi −i , 1 X i 1 1 · X i 2 2 1 · · · · · X ir Ti1 · · · · · Tir , = Ir

where gl

gl+1

Hr,k+1 = D(x) Hr

(xi q k i ) D(x)−1 ,

D(x) =

+1

−k(+1−i)

xi

,

i=1

and the sum is taken over all subsets Ir = {i 1 < i 2 < · · · < ir } ⊂ {1, 2, . . . , + 1}. We −1 take X i = 1 − xi xi−1 , i = 2, . . . , + 1 , with X 1 = 1 and Ti x j = q δi, j x j Ti . Proof. Make a change of variables xi : xi −→ xi t −i , i = 1, . . . , + 1. Then for any i and any Ir , containing i we have: ⎛

⎞ ⎛ t xi − x j xi − x j t i−1− j xi − xi−1 ⎝ ⎠ −→ ⎝t br,i × i− j x − xj xi − x j t xi t −1 − xi−1 j ∈I / r i j>i ⎞ xi t j+1−i − x j ⎠, × xi t j−i − x j

(4.1)

j i}|. Making a substitution t = q −k and conjugating the where br,i = |{ j ∈ gl+1 Hamiltonians Hr by D(x) =

+1

−ki

xi

,

i=1

ki , leads to the multiplication of each term (4.1) in the sum (1.1) by i∈Ir q i := + 1 − i. Taking into account that for any i and for any subset Ir containing i one has i∈Ir i − br,i = r (r − 1)/2, we obtain in the limit k → ∞, gl+1

Hr

=

1−δi 1 , 1

X i1

1−δi 2 −i 1 , 1

· X i2

1−δir −ir −1 , 1

· · · · · X ir

Ir −1 , i = 2, . . . , + 1 and X 1 = 1 where X i = 1 − xi xi−1

Ti1 · · · · · Tir ,

On q-Deformed gl+1 -Whittaker Function I

113

4.2. Recursive kernel Q +1, (x, y|q) for the q-deformed Whittaker function. In this subsection by taking an appropriate limit of the Cauchy-Littlewood kernel for Macdonald polynomials we derive its analog for q-deformed Whittaker functions and verify the intertwining relations with q-deformed Toda chain Hamiltonians. Let t = q −k , i = + 1 − i. Given the Cauchy-Littlewood kernel C+1, (x, y|q, t) (1.3), define a new kernel by −ki −k Q +1, (x, y|q) = lim (xi y+1−i ) · Rk (q) · C+1, (x, y|q, q ) , (4.2) k→∞

i=1

where Rk (q) =

k

−q a j , (1 − q j )2

j=1

aj =

( + 1) 2

j+

−1 k 3

.

Proposition 4.2. The following explicit expression for Q +1, (x, y|q) defined by (4.2) holds: Q +1, (x, y|q) =

∞ ∞ 1 − (xi yi )−1 q n 1 − xi+1 yi q −1 q n · . 1 − qn 1 − qn i=1 n=1

(4.3)

i=1 n=1

Proof. Making the substitution xi → xi t −i , yi → yi t i in C+1, (x, y|q, t) and taking t = q −k we have C+1, (x, y|q, t) =

∞ 1 − xi yi q n−k 1 − xi+1 yi q n 1 − xi yi q n 1 − xi+1 yi q n+k

n=0 i=1

×

+1 i−2 1 − xi y j q n+(i− j−1)k 1 − x+2−i y+1− j q n+( j−i)k . 1 − xi y j q n+(i− j)k 1 − x+2−i y+1− j q n+( j+1−i)k i=3 j=1

One encounters four types of factors which can be rewritten as k ∞ k 1 − x yq n−k −j k = (1 − x yq ) = (x y) (−q − j ) 1 − (x y)−1 q j , n 1 − x yq

n=0

j=1

j=1

∞ k−1 k 1 − x yq n n −1 j 1 − x yq , = (1 − x yq ) = q 1 − x yq n+k

n=0

n=0

j=1

∞ 2k 2k 1 − x yq n−(m+1)k −j k −j −1 j , = (1 − x yq ) = (x y) (−q ) 1 − (x y) q 1 − x yq n−mk

n=0

j=k+1

∞ n=0

j=k+1

2k 1 − x yq n+mk −1 j 1 − x yq . = q 1 − x yq n+(m+1)k j=k+1

Now it is easy to take the limit k → ∞ and obtain (4.3)

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Let us introduce a set of slightly modified mutually commuting Hamiltonians 1−δi −i , 1 1−δir −i , 1 1−δi −i , 1 rgl (y) = H Yi1 2 1 · · · · · Yir −1 r −1 · Yir r +1 r Ti1 · · · · · Tir , (4.4) Ir

where −1 Yi (y) = 1 − yi yi+1 ,

1 ≤ i < ,

Y = 1.

We assume here Ir = (i 1 < i 2 < · · · < ir ) ⊂ {1, 2, . . . , } and we set ir +1 = + 1. Proposition 4.3. The following intertwining relations hold: gl gl (y) + H gl (y) Q +1, (x, y|q), Hk +1 (x) Q +1, (x, y|q) = H k−1 k gl+1

where k = 1, . . . , + 1. Here Q +1, (x, y|q) , Hk (4.3), (2.1) and (4.4) respectively.

(4.5)

gl (y) are defined by (x) and H k

Proof. Direct calculation similar to the one used in the proof of Proposition 4.1.

Let us introduce a function Q+1, ( p +1 , p |q) on the lattice Z+1 × Z as follows: Q+1, ( p +1 , p |q) = Q +1, (q p+1,i +i−1 , q − p,i −i+1 |q). Corollary 4.1. The following explicit expression for Q+1, ( p +1 , p |q) holds:

Q+1, ( p +1 , p |q) = i=1

( p,i − p+1,i )( p+1,i+1 − p,i )

i=1 ( p,i

− p+1,i )q ! ( p+1,i+1 − p,i )q !

,

where (n) = 1 when n ≥ 0 and (n) = 0 otherwise. Proposition 4.4. For any k = 1, . . . , + 1 the following intertwining relations hold: gl+1

Hk

( p +1 )Q+1, ( p +1 , p |q) gl (− p ) + H gl (− p ) Q+1, ( p , p |q). = H k−1 k +1

Proof. Follows from Proposition 4.3.

(4.6)

4.3. Pairing. Define the pairing: f, gq =

ı dyi (y) f (y −1 )g(y), 2π yi 0

(4.7)

i=1

where (y) =

−1 i=1

∞ ∞

n=1 (1 − q

n)

−1 n n=0 (1 − yi+1 yi q )

,

f (y −1 ) := f (y1−1 , . . . , y−1 ).

(4.8)

The integration domain 0 is such that each yi goes around yi = ∞ and is in the region defined by inequalities |yi+1 yi−1 | ≤ q, i = 1, . . . , .

On q-Deformed gl+1 -Whittaker Function I

115

gl rgl (y) are adjoint with respect to the Proposition 4.5. Hamiltonians Hr (y) and H pairing (4.7): gl gl f, gq , f, Hk gq = H k

k = 1, . . . , .

Proof. Let us adopt the following notations: gl rgl (y) = A Ir (y) TIr , H B Ir (y) TIr , Hr (y) = Ir

Ir

where TIr := Ti1 Ti2 · . . . · Tir . One should prove

dyi (y) f (y −1 ) TIr · TI−1 A Ir (y)TIr g(y) r 0 i=1 2πı yi Ir ⎛ ⎞ −1 T (y)T dyi I r I · TI−1 = (y) ⎝ TI−1 A Ir (y)TIr · r f (y −1 )⎠ g(y). r r 2πı yi (y) 0 i=1

Ir

Let us first prove the following lemma: Lemma 4.1. For any Ir = (i 1 < i 2 < · · · < ir ) ⊂ {1, 2, . . . , } the following relation holds: ∗ B Ir (y) = Ir (y) · TI−1 A (y)T , I I r r r where Ir (y) = ((y))−1 TI−1 (y) TIr , r for all i ∈ Ir . Here for a function f (y) we define f ∗ (y) := f (y −1 ). Proof. By direct calculation one derives 1−δi −i , 1 r k k−1 −1 ∗ −1 yi k −1 1−q , (TIr A Ir (y)TIr ) = yik k=1

and yik 1−δik+1 −ik , 1 1 − r yik +1 ∗Ir (y) = 1−δi −i , 1 , k k−1 k=1 1 − q −1 yi k −1 yik where we set i 0 := 0, ir +1 := + 1. In this way we obtain 1−δi 2 −i 1 , 1

(TI−1 A Ir (y)TIr )∗ · ∗Ir (y) = Yi1 r

1−δir −ir −1 , 1

· · · · · Yir −1

1−δir +1 −ir , 1

· Yir

.

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Using the lemma one arrives at the following identity

dyi (y) f (y −1 ) TIr · TI−1 A Ir (y)TIr g(y) r 2πı y i 0 i=1

=

Ir

Ir

∗ dyi (y) B Ir (y)Tr f (y −1 )g(y), 2πı y i 0 (Ir ) i=1

where the integration domains 0 (Ir ) differ from the original integration domain 0 by multiplying the variables yi , i ∈ Ir by q in the definition of 0 . By the lemma the poles of the integrand are inside the circles |yi+1 yi−1 | < 1, and therefore the contours 0 (Ir ) can be deformed to 0 without encountering poles of the integrand. This proves (4.9) and, therefore, the proposition.

To construct recursive formulas for q-deformed Whittaker functions one should introduce a pairing on functions defined on the lattice {yi = q p,i +i−1 ; i = 1, . . . , ; p,i ∈ Z } with appropriate decay at infinities. Let us define the following analog of (4.7): f, glat = ( p ) f (− p )g( p ), (4.9) p ∈Z

where ( p ) =

−1

( p,i+1 − p,i ) ( p,i+1 − p,i )q !.

(4.10)

i=1

Thus ( p ) provides an extension of the measure ( p ) defined in the dominant domain p ∈ P+1, (see (2.7)) to the lattice Z . Let us note that the variables

yi = q p,i +i−1 for p ∈ P+1, satisfy conditions |yi+1 yi−1 | ≤ q entering the definition of the domain of integration 0 . The following proposition can be easily proved by mimicking the proof of Proposition 4.5.

gl rgl ( p ) are adjoint with respect to the Proposition 4.6. Hamiltonians Hr ( p ) and H pairing (4.9), gl gl f, glat , f, Hk glat = H k

k = 1, . . . , .

(4.11)

4.4. Proof of Theorem 2.1. Now we are ready to prove Theorem 2.1. We use recursion gl over the rank of glk . Set λ11 ( p11 ) = q λ1 p11 and assume that gl

gl

gl

gl

Hr ( p ) · λ1,...,λ ( p ) = χr (q i λi Ei,i ) λ1,...,λ ( p ), gl χr (q i λi Ei,i ) = z i 1 z i 2 · · · · · z ir , z i = q λi .

(4.12)

()

Ir

Here E i, j are the standard generators of gl , Ir() = (i 1 < i 2 < · · · < ir ) ⊂ {1, 2, . . . , } gl and χr (g) are characters of fundamental representations Vr = r C of gl .

On q-Deformed gl+1 -Whittaker Function I

117

gl

Let us define the function λ1+1 ,...,λ+1 ( p +1 ) as follows: gl

λ1+1 ,...,λ+1 ( p +1 ) =

p

∈Z

·q λ+1 (

( p ) Q+1, ( p +1 , p ) +1

p+1,i − i=1 p,i )

i=1

gl

λ1,...,λ ( p ),

(4.13)

where Q+1, ( p +1 , p ) =

( p,i − p+1,i ) ( p+1,i+1 − p,i ) , ( p,i − p+1,i )q ! ( p+1,i+1 − p,i )q ! i=1

and ( p ) =

−1

( p,i+1 − p,i ) ( p,i+1 − p,i )q !.

i=1

One should verify the relations: gl+1

Hr

gl

gl

gl

+1 ( p +1 ) · λ1+1 (q i λi Ei,i ) λ1+1 ,...,λ+1 ( p +1 ) = χr ,...,λ+1 ( p +1 ), gl z i 1 z i 2 · · · · · z ir , z i = q λi , χr +1 (q i λi Ei,i ) =

(4.14)

(+1)

Ir (+1)

where Ir = {i 1 < i 2 < . . . < ir } ⊂ (1, 2, . . . , + 1). gl Applying Hamiltonians Hr +1 ( p +1 ) to (4.13) and using the intertwining relation given in Proposition 4.4, one obtains

gl+1

( p )Q+1, ( p +1 , p ) q λ+1 ( i p+1,i − k p,k ) +1 gl (− p ) + H rgl (− p ) Q+1, ( p , p ) q λ+1 ( i = q λ+1 H r −1 +1

Hr

p+1,i − k p,k )

Now using (4.11), one obtains gl+1

gl

( p +1 )λ1+1 ,...,λ+1 ( p +1 ) gl = ( p ) Hr +1 ( p +1 )Q+1, ( p +1 , p |q)

Hr

p ∈Z

× q λ+1 (

p+1,i − k p,k )

gl

λ1,...,λ ( p ) gl (− p ) + H rgl (− p ) = ( p ) q λ+1 H r −1 p ∈Z

× Q+1, ( p +1 , p ) q λ+1 (

i

i

p+1,i − k p,k )

gl

λ1,...,λ ( p )

.

118

A. Gerasimov, D. Lebedev, S. Oblezin

=

p ∈Z

( p ) Q+1, ( p +1 , p ) q λ+1 (

i

p+1,i − k p,k )

gl gl gl × q λ+1 Hr −1 ( p ) + Hr ( p ) λ1,...,λ ( p ) ⎞ ⎛ ⎟ gl ⎜ = ⎝q λ+1 q λi + q λi ⎠ λ1+1 ,...,λ+1 ( p +1 ) ()

()

Ir −1 i∈Ir −1

⎛ ⎜ =⎝ (+1)

Ir

(+1)

()

⎞

Ir

()

i∈Ir

⎟ gl q λi ⎠ λ1+1 ,...,λ+1 ( p +1 ),

i∈Ir

()

(+1)

= {i 1 < i 2 < · · · < ir } ⊂ where Ir = {i 1 < i 2 < · · · < ir } ⊂ (1, 2, . . . , ) and Ir (1, 2, . . . , + 1). In the last equality above we use the following relation: gl+1

χr

gl

gl

(z) = z +1 χr −1 (z ) + χr (z ),

where z = (z 1 , z 2 , . . . , z ) for z i = q λi . This completes the proof of Theorem 2.1.

Acknowledgements. The research of AG was partly supported by SFI Research Frontier Programme and Marie Curie RTN Forces Universe from EU. The research of SO is partially supported by RF President Grant MK-134.2007.1.

References [AOS] [BF] [CS] [Ch] [CK] [Et] [EK] [I] [FH] [FFJMM] [GKL] [GKL1] [GKLO] [GLO]

Awata, H., Odake, S., Shiraishi, J.: Integral representations of the Macdonald symmetric functions. Commun. Math. Phys. 179, 647–666 (1996) Braverman, A., Finkelberg, M.: Finite-difference quantum Toda lattice via equivariant K-theory. Trans. Groups 10, 363–386 (2005) Casselman, W., Shalika, J.: The unramified principal series of p-adic groups II. The Whittaker Function. Comp. Math. 41, 207–231 (1980) Cherednik, I.V.: Quantum groups as hidden symmetries of classic representation theory. In Differential Geometric Methods in Theoretical Physics (Chester,1988), Teaneck, NJ: World Sci. Publishing, 1989, pp. 47–54 Cheung, P., Kac, V.: Quantum Calculus. Berlin-Heidelberg-New York: Springer, 2001 Etingof, P.: Whittaker functions on quantum groups and q-deformed Toda operators. Amer. Math. Soc. Transl. Ser.2, Vol. 194, Providence, RI: Amer. Math. Soc., 1999, pp. 9–25 Etingof, P.I., Kirillov, A.A. Jr..: Macdonald’s polynomials and representations of quantum groups. Math. Res. Let. 1, 279–296 (1994) Inozemtsev, V.I.: Finite Toda lattice. Commun. Math. Phys. 121, 629–638 (1989) Fulton, W., Harris, J.: Representation Theory. A First Course. Berlin-Heidelberg-New York: Springer, 1991 3 subspaces and quantum Feigin, B., Feigin, E., Jimbo, M., Miwa, T., Mukhin, E.: Principal sl Toda hamiltonians. http://arxiv.org/abs/0707.1635v2[math.QA], 2007 Gerasimov, A., Kharchev, S., Lebedev, D.: Representation Theory and Quantum Inverse Scattering Method: Open Toda Chain and Hyperbolic Sutherland Model, Int. Math. Res. Notes, No.17, 823–854 (2004) Gerasimov, A., Kharchev, S., Lebedev, D.: Representation Theory and Quantum Integrability. Progress in Math. 237, Basel: Birkhäuser, 2005, pp. 133–156 Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a Gauss-Givental representation of quantum Toda chain wave function. Int. Math. Res. Notices, ArticleID 96489, 23 pages, 2006 Gerasimov, A., Lebedev, D., Oblezin, S.: Baxter operator and archimedean Hecke algebra. Commun. Math. Phys. 284(3), 867–896 (2008)

On q-Deformed gl+1 -Whittaker Function I [GLO1] [GLO2] [GLO3] [GK] [Gi] [GiL] [Kir] [KL1] [KLS] [Mac] [Ru] [Se1] [Se2] [Sh] [ZS]

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Gerasimov, A., Lebedev, D., Oblezin, S.: New Integral Representations of Whittaker Functions for Classical Lie Groups. http://arxiv.org/abs/0705.2886v1[math.RT], 2007 Gerasimov, A., Lebedev, D., Oblezin, S.: On q-deformed gl+1 -Whittaker functions II. Commun. Math. Phys., to apper (this issue), 2009, http://arxiv.org/abs/0805.3754v2[math.RT], 2008 Gerasimov, A., Lebedev, D., Oblezin, S.: Baxter q-operators and their arithmetic implications. Lett. Math. Phys. 88(1–3), 3–30 (2009) Givental, A., Kim, B.: Quantum cohomology of flag manifolds and Toda lattices. Commun. Math. Phys. 168, 609–641 (1995) Givental, A.: Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture. In: Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser., 2 Vol. 180, Providence, RI: Amer.Math.Soc., 1997, pp. 103–115 Givental, A., Lee, Y.-P.: Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups. Invent. Math. 151, 193–219 (2003) Kirillov, A. Jr.: Traces of intertwining operators and Macdonald’s polynomials. PhD Thesis, Yale University, May 1995, available at http://arxiv.org/abs/q-alg/9503012v1, 1995 Kharchev, S., Lebedev, D.: Eigenfunctions of G L(n, r ) Toda chain: the Mellin-Barnes representation. JETP Lett. 71, 235–238 (2000) Kharchev, S., Lebedev, D., Semenov-Tian-Shansky, M.: Unitary representations of u q (sl(2, r )), the modular double and the multiparticle q-deformed Toda chains. Commun. Math. Phys. 225, 573–609 (2002) Macdonald, I.G.: A New Class of Symmetric Functions. Publ. I.R.M.A. Strasbourg, 372/S-20, Actes 20e Séminaire Lotharingien de Combinatoire, 131–171 (1988) Ruijsenaars, S.: The relativistic Toda systems. Commun. Math. Phys. 133, 217–247 (1990) Sevostyanov, A.: Regular nilpotent elements and quantum groups. Commun. Math. Phys. 204, 1–16 (1999) Sevostyanov, A.: Quantum deformation of Whittaker modules and Toda lattice. Duke Math. J. 105, 211–238 (2000) Shintani, T.: On an explicit formula for class 1 Whittaker functions on G L n over p-adic fields. Proc. Japan Acad. 52, 180–182 (1976) Zhelobenko, D., Shtern, A.: Representations of Lie Groups. Moscow: Nauka, 1983

Communicated by Y. Kawahigashi

Commun. Math. Phys. 294, 121–143 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0919-9

Communications in

Mathematical Physics

On q -Deformed gl+1 -Whittaker Function II Anton Gerasimov1,2,3 , Dimitri Lebedev1 , Sergey Oblezin1 1 Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia.

E-mail: [email protected]; [email protected]

2 School of Mathematics, Trinity College Dublin, Dublin 2, Ireland.

E-mail: [email protected]

3 Hamilton Mathematical Institute TCD, Dublin 2, Ireland

Received: 4 February 2009 / Accepted: 25 June 2009 Published online: 17 September 2009 – © Springer-Verlag 2009

Abstract: A representation of a specialization of a q-deformed class one lattice gl+1 -Whittaker function in terms of cohomology groups of line bundles on the space QMd (P ) of quasi-maps P1 → P of degree d is proposed. For = 1, this provides an interpretation of the non-specialized q-deformed gl2 -Whittaker function in terms of QMd (P1 ). In particular the (q-version of the) Mellin-Barnes representation of the gl2 -Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of -function as a topological genus in semi-infinite geometry. A relation with the Givental-Lee universal solution (J -function) of q-deformed gl2 -Toda chain is also discussed.

Introduction In our work [GLO1] (which is the first in the series of papers [GLO1,GLO2]) we have proposed an explicit representation of a q-deformed class one lattice gl+1 -Whittaker function defined as a common eigenfunction of a complete set of commuting quantum Hamiltonians of a q-deformed gl+1 -Toda chain. Here “class one” means that the Whittaker function is non-zero only in the dominant domain. The case = 1 was discussed previously in [GLO3] (for related results in this direction see [KLS, GiL,GKL1,BF,FFJMM]). A special feature of the proposed representation is that the gl q-deformed class one gl+1 -Whittaker function z +1 ( p) with z = (z 1 , . . . , z +1 ) and p = ( p1 , . . . , p+1 ) ∈ Z+1 , is given by a character of a C∗ × G L +1 (C)-module V p . The expression in terms of a character can be considered as a q-version of ShintaniCasselman-Shalika representation of class one p-adic Whittaker functions [Sh,CS]. Indeed our representation of a q-deformed gl+1 -Whittaker function reduces, in a certain

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limit, to the Shintani-Casselman-Shalika representation of a p-adic Whittaker function. Note that the representation of a q-deformed Whittaker function as a character is a q-analog of the Givental integral representation [Gi2,GKLO] of the classical gl+1 Whittaker function. The main objective of this paper is to better understand the representation of the q-deformed gl+1 -Whittaker function as a character. Below we consider a specialization of the q-deformed Whittaker function given by the trace over a certain C∗ × G L +1 (C)module Vn,k (in the case = 1 there is actually no specialization). Our main result is presented in Theorem 3.1. We provide a description of the C∗ × G L +1 (C)-module Vn,k as a zero degree cohomology group of a line bundle on an algebraic version LP+ of a semi-infinite cycle LP + in a universal covering LP of the space of loops in P . We define LP+ as an appropriate limit d → ∞ of the space QMd (P ) of degree d quasi-maps of P1 to P [Gi1,CJS]. In particular for = 1 this provides a description of the q-deformed gl2 -Whittaker function in terms of cohomology of line bundles over LP1+ . A universal solution of the q-deformed gl+1 -Toda chain [GiL] was given in terms of cohomology groups of line bundles over QMd (X ), X = G/B for finite d. We demonstrate how our interpretation of the q-deformed gl+1 -Whittaker function is reconciled with the results of [GiL]. Using Theorem 3.1, we interpret a q-version of the Mellin-Barnes integral representation of the specialized q-Whittaker function as an instance of the Riemann-RochHirzebruch theorem in a semi-infinite setting. The corresponding Todd class is expressed in terms of a q-version of the -function. Analogously, the classical -function appears in a description of the fundamental class of semi-infinite homology theory and enters the Mellin-Barnes integral representation of the classical Whittaker function. Let us stress that the C∗ × G L +1 (C)-module Vn,k arising in the description of the q-deformed gl+1 -Whittaker function is not irreducible. It would be natural to look for an interpretation of Vn,k as an irreducible module of a quantum affine Lie group. A relation of the geometry of semi-infinite flags to representation theory of affine Lie algebras was ∞ proposed in [FF]. The semi-infinite flag space is defined as X 2 = G(K)/H (O)N (K), where K = C((t)), O = C[[t]], B = H N is a Borel subgroup of G, N is its unipotent radical and H is the associated Cartan subgroup. The semi-infinite flag spaces are not easy to deal with. An interesting approach to the semi-infinite geometry was proposed by Drinfeld. He introduced a space of quasi-maps QMd (P1 , G/B) that should be con∞ sidered as a finite-dimensional substitute for the semi-infinite flag space X 2 (see e.g. [FM,FFM,Bra]). Thus, taking into account the constructions proposed in this paper one can expect that (q-deformed) gl+1 -Whittaker functions (encoding the Gromov-Witten invariants and their K -theory generalizations) can be expressed in terms of representation theory of affine Lie algebras (see [GiL] for a related conjecture and [FFJMM] for recent progress in this direction). The paper [GLO2] establishes a connection of our results to the representation theory of (quantum) affine Lie groups. The paper is organized as follows. In Sect. 1, explicit solutions of the q-deformed gl+1 -Toda chain (q-versions of Whittaker functions) are recalled. In Sect. 2, we derive integral expressions for the counting of holomorphic sections of line bundles on the space of quasi-maps. In Sect. 3 we derive a representation of the specialized q-Whittaker functions in terms of cohomology of holomorphic line bundles on the space of quasi-maps of P1 to P . We propose an interpretation of the q-Whittaker functions as semi-infinite periods. In Sect. 4 the analogous interpretation of the classical Whittaker functions is discussed. In Sect. 5, we clarify the connection of our interpretation of the q-deformed gl+1 -Whittaker function with the results of [GiL].

On q-Deformed gl+1 -Whittaker Function II

123

1. q-Deformed gl+1 -Whittaker Function In this section we recall a construction [GLO1] of the q-deformed gl+1 -Whittaker funcgl tion z +1 ( p +1 ) defined on the lattice p +1 = ( p+1,1 , . . . , p+1,+1 ) ∈ Z+1 . We will consider only class one Whittaker functions, i.e. Whittaker functions satisfying the condition gl+1

z

( p +1 ) = 0

(1.1)

outside the dominant domain p+1,1 ≥ · · · ≥ p+1,+1 . The q-deformed gl+1 -Whittaker functions are common eigenfunctions of q-deformed gl+1 -Toda chain Hamiltonians: 1−δi −i , 1 1−δir −i , 1 1−δi −i , 1 gl Hr +1 ( p +1 ) = X i1 2 1 · . . . · Ti1 · . . . · Tir , X ir −1 r −1 · X ir r +1 r Ir

(1.2) where the sum is over ordered subsets Ir = {i 1 < i 2 < . . . < ir } ⊂ {1, 2, · · · , + 1} and we assume ir +1 = + 2. In (1.2) we use the following notations Ti f ( p +1 ) = f ( p +1 ), p+1,k = p+1,k + δk,i , i, k = 1, . . . , + 1, X i = 1 − q p+1,i − p+1,i+1 +1 , i = 1, . . . , , and X +1 = 1. We assume q ∈ R, 0 < q < 1. For example, the first nontrivial Hamiltonian has the following form: gl

H1 +1 ( p +1 ) =

(1 − q p+1,i − p+1,i+1 +1 )Ti + T+1 .

(1.3)

i=1

The main result of [GLO1] is a construction of common eigenfunctions of quantum Hamiltonians (1.2): gl gl+1 gl+1 Hr +1 ( p +1 )z 1 ,...,z ( p ) = ( z i ) z 1 ,...,z (1.4) +1 +1 ( p +1 ). +1 Ir i∈Ir

Denote by P (+1) ⊂ Z(+1)/2 a subset of integers pn,i , n = 1, . . . , + 1, i = 1, . . . , n satisfying the Gelfand-Zetlin conditions pk+1,i ≥ pk,i ≥ pk+1,i+1 for k = 1, . . . , . In the following we use the standard notation (n)q ! = (1 − q)...(1 − q n ). gl

+1 Theorem 1.1. Let z 1 ,...,z +1 ( p +1 ) be a function given in the dominant domain p+1,1 ≥ . . . ≥ p+1,+1 by

gl

+1 z 1 ,...,z +1 ( p +1 ) =

+1

zk

i

pk,i − i pk−1,i

pk,i ∈P (+1) k=1 k−1 ( pk,i − pk,i+1 )q !

×

k=2 i=1 k k=1 i=1

( pk+1,i − pk,i )q ! ( pk,i − pk+1,i+1 )q !

,

(1.5)

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A. Gerasimov, D. Lebedev, S. Oblezin gl

+1 and zero otherwise. Then, z 1 ,...,z +1 ( p +1 ) is a common solution of the eigenvalue problem (1.4).

Formula (1.5) can be written also in the recursive form. Corollary 1.1. Let P+1, be a set of p = ( p,1 , . . . , p, ) satisfying the conditions p+1,i ≥ p,i ≥ p+1,i+1 . The following recursive relation holds: gl

+1 z 1 ,...,z +1 ( p +1 ) =

( p ) z +1i

p+1,i − i p,i

p ∈P+1,

gl

Q +1, ( p +1 , p |q)z 1 ,...,z ( p ),

(1.6) where 1

Q +1, ( p +1 , p |q) =

i=1 −1

( p ) =

,

( p+1,i − p,i )q ! ( p,i − p+1,i+1 )q ! (1.7)

( p,i − p,i+1 )q !.

i=1

Remark 1.1. The representation (1.6) is a q-analog of Givental’s integral representation of the classical gl+1 -Whittaker function [Gi2,JK]: gl+1

ψλ

(x1 , . . . , x+1 ) =

R i=1

gl

gl

dt,i Q gl+1 (t +1 , t |λ+1 )ψλ1,...,λ (t ),

(1.8)

gl

Q gl+1 (t +1 , t |λ+1 )

+1 t −t

t,i −t+1,i+1 +1,i ,i = exp ıλ+1 e , t+1,i − t,i − +e i=1

i=1

i=1

where λ = (λ1 , . . . , λ+1 ), t k = (tk1 , . . . , tkk ), xi := t+1,i , i = 1, . . . , + 1 and gl z i = q γi , λi = γi log q and we assume that Q gl1 (t1,1 |λ1 ) = eıλ1 t1,1 . For the represen0 tation theory derivation of this integral representation of gl+1 -Whittaker function see [GKLO]. The representation (1.5) of the q-Whittaker function turns into representation (1.8) of the classical Whittaker function in the appropriate limit. As an example consider g = gl2 . Let p1 := p2,1 ∈ Z, p2 := p2,2 ∈ Z and p := p1,1 ∈ Z. Then the function

gl

2 z 1 ,z 2 ( p1 , p2 ) =

p2 ≤ p≤ p1

gl

2 z 1 ,z 2 ( p1 , p2 ) = 0,

p p +p −p

z1 z2 1 2 , ( p1 − p)q !( p − p2 )q !

p1 < p2 ,

p1 ≥ p2 , (1.9)

On q-Deformed gl+1 -Whittaker Function II

125

is a solution of the system of equations: gl2 gl2 (1 − q p1 − p2 +1 )T1 + T2 z 1 ,z 2 ( p1 , p2 ) = (z 1 + z 2 ) z 1 ,z 2 ( p1 , p2 ), gl

gl

(1.10)

2 2 T1 T2 z 1 ,z 2 ( p1 , p2 ) = z 1 z 2 z 1 ,z 2 ( p1 , p2 ).

Let us consider the following specialization of the q-deformed gl+1 -Whittaker function: gl

gl

+1 +1 z 1 ,...,z +1 (n, k) := z 1 ,...,z +1 (n + k, k, . . . , k).

(1.11)

Let Pn,k be a Gelfand-Zetlin pattern such that ( p+1,1 , . . . , p+1,+1 ) = (n +k, k, . . . , k). Then, the relations p+1,i ≥ p,i ≥ p+1,i+1 for the elements of a Gelfand-Zetlin pattern imply pk,i=1 = 0 and we have that

+1 n+k− p p −p p −k z +1 ,1 z ,1 −1,1 z 1 1,1 gl+1 k ··· zi z 1 ,...,z +1 (n, k) = (n + k − p,1 )q ! ( p,1 − p−1,1 )q ! ( p1,1 − k)q ! i=1 Pn,k

+1 n +1 z +1 z n1 = z ik ··· 1 . (1.12) (n +1 )q ! (n 1 )q ! n +···+n =n i=1

+1

1

gl

+1 Theorem 1.2. z 1 ,...,z +1 (n, k) satisfies following difference equation: +1 gl+1 −1 n gl+1 (1 − z i T ) z 1 ,...,z +1 (n, k) = q z 1 ,...,z +1 (n, k),

(1.13)

i=1

where T · f (n) = f (n + 1). Proof. The proof is based on the explicit expression (1.5). Introduce the generating function gl+1 z 1 ,...,z +1 (t, k)

=

t

n

gl+1 z 1 ,...,z +1 (n, k)

=

n∈Z

+1

z ik , m m=0 (1 − t z i q )

∞

j=1

where we use the identity ∞

xn 1 . = m (n)q ! m=0 (1 − xq )

∞

n=0

gl

gl

+1 +1 Due to the fact that z 1 ,...,z +1 (n, k) = 0 for n < 0, the generating function z 1 ,...,z +1 (t, k) is regular at t = 0. It is easy to check now the following identity:

+1

gl

gl

+1 +1 (1 − t z i ) z 1 ,...,z +1 (t, k) = z 1 ,...,z +1 (qt, k).

j=1

Expanding the latter relation in powers of t, we obtain (1.13) for the coefficients of gl+1 z 1 ,...,z

+1 (t, k).

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Remark 1.2. The difference equation (1.13) for the specialized q-Whittaker function gl+1 z 1 ,...,z +1 (n, k) can be derived directly from the system of equations (1.4) for the nongl

+1 specialized q-deformed Whittaker function z 1 ,...,z +1 ( p1 , p2 , . . . , p+1 ) and the condition

gl

+1 z 1 ,...,z +1 ( p1 , p2 , . . . , p+1 ) = 0

outside the dominant domain p1 ≥ · · · ≥ p+1 . Lemma 1.1. The following integral representation for the specialized q-deformed gl+1 Whittaker functions holds:

+1 +1 dt −n gl +1 t z (n, k) = z ik q (z i t), (1.14) t=0 2πı t i=1

i=1

where q (x) =

∞ n=0

1 . 1 − qnx

Proof. Using the identity ∞ n=0

one obtains, for n ≥ 0, that gl z +1 (n, k)

=

∞ 1 xm = , 1 − xq n (m)q !

+1

m=0

z ik

i=1

=

+1

+1 ∞ dt −n 1 t 1 − z i tq m t=02πı t i=1 m=0

z ik

z 1n 1

n 1 +...+n +1 =n

i=1

(n 1 )q !

· ··· ·

n +1 z +1 . (n +1 )q !

(1.15)

gl

For n < 0, we obviously have that z +1 (n, k) = 0.

The corresponding integral representation for the classical gl2 -Whittaker function is given by the Mellin-Barnes representation for the gl2 -Whittaker function, ıσ +∞ ı ı λ − λ2 λ − λ1 gl2 (λ1 +λ2 )x2 λ(x1 −x2 ) , ψλ1 ,λ2 (x1 , x2 ) = e dλ e ı ı iσ −∞ (1.16) where σ > max{Im λ j , j = 1, . . . , + 1}. Remark 1.3. The expression

+1 gl+1 z 1 ,...,z +1 (n, k) = z ik i=1

gl

+1 z 1 ,...,z +1 (n, k) = 0,

n 1 +···+n +1 =n

n 0, the rescaled cumulative distribution function ρ ε , defined in (1.2), is proved to converge (cf. [14, Th. 2.5]) toward the unique solution of the corresponding initial value problem for nonlinear diffusion equation (−u, u x ), ut = H

(1.6)

is a continuous function and is a Lévy operator of order 1. where the Hamiltonian H It is defined for any function U ∈ Cb2 (R) and for r > 0 by the formula 1 − U (x) = C(1) U (x + z) − U (x) − zU (x)1{|z|≤r } dz (1.7) |z|2 R with a constant C(1) > 0. Finally, in the particular case of c ≡ 0 in (1.3), we have (L , p) = L| p| (cf. [14, Th. 2.6]) which allows us to rewrite Eq. (1.6) in the form H u t + |u x |u = 0.

(1.8)

One can show that the definition of is independent of r > 0, hence, we fix r = 1.

1/2 In fact, for suitably chosen C(1), = 1 = −∂ 2 /∂ x 2 is the pseudodifferential 1 w)(ξ ) = |ξ | operator defined in the Fourier variables by ( w(ξ ) (cf. formulae (2.3) and (2.4) below). In this particular case, Eq. (1.8) is an integrated form of a model studied

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147

by Head [18] for the self-dynamics of a dislocation density represented by u x . Indeed, denoting v = u x we may rewrite Eq. (1.8) as vt + (|v|Hv)x = 0,

(1.9)

where H is the Hilbert transform defined by ) = − i sgn(ξ ) (Hv)(ξ v (ξ ). Let us recall two well known properties of this transform (cf. [30]) 1 v(y) Hv(x) = P.V. dy and 1 v = Hvx . π x − y R

(1.10)

(1.11)

Head [18] called (1.9) the equation of motion of the dislocation continuum and constructed an explicit self-similar solution. Numerical studies of solutions to this equation were performed in [12]. Quasi-geostrophic equations. Let us recall completely different physical motivations which also lead to Eq. (1.9). The 2D quasi-geostrophic equations (QG), modeling the dynamics of the mixture of cold and hot air in a thin layer and the fronts between them, are of the form θt + (u · ∇)θ = 0, u = ∇ ⊥ ψ, θ = −(− )1/2 ψ R2

(1.12)

∇⊥

for x ∈ and t > 0, where = (−∂x2 , ∂x1 ). Here, θ (x, t) represents the air temperature. Pioneering studies concerning a finite time blow up criterion of solutions to (1.12) are due to Constantin et al. [9]. Much earlier, Constantin et al. [8] proposed a one-dimensional version of the Euler equations, namely, ωt + ωHω = 0, and studied formation of singularities of its solutions. Those results motivated the authors of [6,7,10] to consider another simplified model derived from (1.12) in the following way. First, one should write system (1.12) in another equivalent form. From the second and third equation in (1.12) we have the representation u = −∇ ⊥ (− )−1/2 θ = −R ⊥ θ,

(1.13)

where we have used the notation R ⊥ θ = (−R2 θ, R1 θ ), with the Riesz transforms defined by (see e.g. [30]) (x j − y j )θ (y, t) 1 R j (θ )(x, t) = P.V. dy. 2 2π |x − y|3 R Using Eq. (1.13), we find that (1.12) can be transformed into θt − div ((R ⊥ θ )θ ) = 0,

(1.14)

because div (R ⊥ θ ) = 0. To construct the 1D model, the authors of [6,7,10] considered the unknown function θ = θ (x, t) for x ∈ R and t > 0, and replaced the Riesz transform −R ⊥ in (1.14), by the Hilbert transform H (cf. (1.10)–(1.11)). Then Eq. (1.14) is converted into the model equation θt + (θ Hθ )x = 0 for x ∈ R and t > 0.

(1.15)

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P. Biler, G. Karch, R. Monneau

Obviously, for θ ≥ 0, both models (1.9) and (1.15) are identical. However, in the case of Eq. (1.15), it is possible to show that the complex valued function z(x, t) = Hθ (x, t) − iθ (x, t) satisfies the inviscid Burgers equation z t + zz x = 0. This property of solutions has been systematically used in [6,7] to study the existence, the regularity and the blow up in finite time of solutions to Eq. (1.15). We refer the reader to those publications for additional references concerning Eq. (1.15). Below, see Remarks 2.6, 2.10 and 7.7, we explain how our results on Eq. (1.9) and its generalizations contribute to the theory developed for model (1.15). Organization of the paper. In the next section, we state the initial value problem considered in this paper and we formulate our main results. In Sect. 3, we construct explicitly the self-similar solution. In Sect. 4, we recall the necessary material about viscosity solutions, which will be used systematically in the remainder of the paper. In Sect. 5, we prove the uniqueness of the self-similar solution. Under the additional assumption that the solution is confined between its boundary values at infinity, we prove the stability of the self-similar solution, namely Theorem 2.5. In Sect. 6, we prove further decay properties of a solution with compact support. Applying these estimates, we finish the proof of Theorem 2.5 in the general case. In Sect. 7, we introduce an ε-regularized equation, for which we prove both the global existence of a smooth solution and the corresponding gradient estimates. Finally in Sect. 8, we deduce the gradient estimate in the limit case ε = 0, namely Theorem 2.7, using the corresponding estimates for the approximate ε-problem. 2. Main Results Motivated by physics described above, we study the following initial value problem for the nonlinear and nonlocal equation involving u = u(x, t), u t = −|u x | α u on R × (0, +∞), u(x, 0) = u 0 (x) for x ∈ R,

(2.1) (2.2)

where the assumptions on the initial datum u 0 will be made precise later. Here, for

α/2 α ∈ (0, 2), α = −∂ 2 /∂ x 2 is the pseudodifferential operator defined via the Fourier transform α w)(ξ ) = |ξ |α w ( (ξ ).

(2.3)

Recall that the operator α has the Lévy–Khintchine integral representation for every α ∈ (0, 2),

dz − α w(x) = C(α) , (2.4) w(x + z) − w(x) − zw (x)1{|z|≤1} |z|1+α R where C(α) > 0 is a constant. This formula (discussed in, e.g., [13, Th. 1] for functions w in the Schwartz space) allows us to extend the definition of α to functions which are bounded and sufficiently smooth, however, not necessarily decaying at infinity. As we have already explained (cf. Eq. (1.8)), in the particular case α = 1, Eq. (2.1) is a mean field model that has been derived rigorously in [14] as the limit of a system of particles in interactions (cf. (1.1)) with forces V (z) = − 1z . Here, the density u x means the positive density |u x | of dislocations of type of the sign of u x . Moreover, the

Nonlinear Diffusion of Dislocation Density

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occurrence of the absolute value |u x | in the equation allows the vanishing of dislocation particles of the opposite sign. In the present paper, we study the general case α ∈ (0, 2) that could be seen as a mean field model of particles modeled by system (1.1) with repulsive interactions V (z) = − z1α . Here, we would like also to keep in mind that (2.1) is the simplest nonlinear anomalous diffusion model (described by the Lévy operator α ) which degenerates for u x = 0. First note that Eq. (2.1) is invariant under the scaling u λ (x, t) = u(λx, λα+1 t) u(x, t) is a solution to (2.1), then u λ

(2.5) u λ (x, t)

for each λ > 0 which means that if u = = is so. Hence, our first goal is to construct self-similar solutions of Eq. (2.1), i.e. solutions which are invariant under the scaling (2.5). By a standard argument, any self-similar solution should have the following form: x u α (x, t) = α (y) with y = 1/(α+1) , (2.6) t where the self-similar profile α has to satisfy the following equation: − (α + 1)−1 y α (y) = −(α α (y)) α (y) for all y ∈ R.

(2.7)

In our first theorem, we construct solutions to Eq. (2.7). Theorem 2.1 (Existence of self-similar profile). Let α ∈ (0, 2). There exists a nondecreasing function α of the regularity C 1+α/2 at each point and analytic on the interval (−yα , yα ) for some yα > 0, which satisfies

0 on (−∞, −yα ),

α = 1 on (yα , +∞), and (α α )(y) =

y for all y ∈ (−yα , yα ). α+1

Remark 2.2. We can obtain the self-similar solutions corresponding to different boundary values at infinity, simply considering for any γ > 0 and b ∈ R the profiles

γ α γ −1/(α+1) y + b which are also solutions of Eq. (2.7). Remark 2.3. The fact that ∂ y α has compact support reveals a finite velocity propagation of the support of the solution which is typical for solutions the porous medium equation, cf. Remark 2.8 below. At least formally, the function α is the solution of (2.7), and the self-similar function u α given by (2.6) is a solution of Eq. (2.1) with the initial datum being the Heaviside function

0 if x < 0, (2.8) u 0 (x) = H (x) = 1 if x > 0. In order to check that u α given by (2.6) solves (2.1), we introduce a suitable notion of viscosity solutions to the initial value problem (2.1)–(2.2), see Sect. 4. In this setting, we show in Theorem 4.7 the existence and the uniqueness of a solution for any initial condition u 0 in BU C(R), i.e. the space of bounded and uniformly continuous functions on R. Although the initial datum (2.8) is not continuous, we have the following result.

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P. Biler, G. Karch, R. Monneau

Theorem 2.4 (Uniqueness of self-similar solution). Let α ∈ (0, 2). Then the function u α defined in (2.6) with the profile α constructed in Theorem 2.1 is the unique viscosity solution of Eq. (2.1) with the initial datum (2.8). In Theorem 2.4, the uniqueness holds in the sense that if u is another viscosity solution to (2.1), (2.8), then u = u α on (R × [0, +∞))\{(0, 0)}. The self-similar solutions are not only unique, but are also stable in this framework of viscosity solutions, as the following result shows. Theorem 2.5 (Stability of the self-similar solution). Let α ∈ (0, 2). For any initial data u 0 ∈ BU C(R) satisfying lim u 0 (x) = 0 and

x→−∞

lim u 0 (x) = 1,

x→+∞

(2.9)

let us consider the unique viscosity solution u = u(x, t) of (2.1)–(2.2) and, for each λ > 1, its rescaled version u λ = u λ (x, t) given by Eq. (2.5). Then, for any compact set K ⊂ (R × [0, +∞)) \ {(0, 0)}, we have x u λ (x, t) → α 1/(α+1) in L ∞ (K ) as λ → +∞. (2.10) t We stress the fact that Theorem 2.5 contains a result on the long time behaviour of solution because, first, choosing t = 1 in (2.10) and, next, substituting λ = t 1/(α+1) we 1/(α+1) obtain the convergence of u xt , t toward the self-similar profile α (x). On the other hand, convergence (2.10) can be seen as a stability result when we consider initial data which are perturbations of the Heaviside function. This is a nonstandard stability result in the framework of discontinuous viscosity solutions. It shows that the approach by viscosity solutions is a good one in the sense of Hadamard, even if we consider here initial conditions which are perturbations of the Heaviside function. Remark 2.6. In the particular case of α = 1, the nonnegative function U (x, t) = t −1/2 1 (xt −1/2 ), with 1 — the self-similar profile provided by Theorem 2.1, is the compactly supported self-similar solution of (1.15). This function attracts other nonnegative solutions to (1.15) in the sense stated in Theorem 2.5. Finally, we have the following result of independent interest. Theorem 2.7 (Optimal decay estimates). Let α ∈ (0, 1]. For any initial condition u 0 ∈ BU C(R) such that u 0,x ∈ L 1 (R), the unique viscosity solution u of (2.1)–(2.2) satisfies u(·, t) ∞ ≤ u 0 ∞ and u x (·, t) ∞ ≤ u 0,x ∞ for any t > 0. Moreover, for every p ∈ [1, +∞) we have pα+1

u x (·, t) p ≤ C p,α u 0,x 1p(α+1) t

( p−1) − p(α+1)

for any t > 0,

(2.11)

with some constant C p,α > 0 depending only on p and α. The decay given in (2.11) is optimal in the sense that the self-similar solution satisfies (u α )x (·, t) p = ( α ) y (·) p t

( p−1) − p(α+1)

.

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Remark 2.8. The equation satisfied by v = u x , vt = −(|v|α−1 Hv)x

(2.12)

(with the Hilbert transform H defined in (1.10)) can be treated as a nonlocal counterpart of the porous medium equation.

Indeed, for α = 2 and for nonnegative v, Eq. (2.12) reduces to vt = (vvx )x = v 2 /2 x x . As in the case of the porous medium equation (see e.g. [32] and the references therein), estimates (2.11) show a regularizing effect created by the equation, even for the anomalous diffusion: if v0 ∈ L 1 (R) then v ∈ L p (R) for each p > 1. Observealso that Eq. (2.12) has compactly supported self-similar solution 1 1 − α+1 α+1 , where the profile α was constructed in Theorem 2.1. v(x, t) = t

α x/t This function for α = 2 corresponds to the well known Barenblatt–Prattle solution of the porous medium equation. Remark 2.9. As we have already mentioned, see Theorem 4.7 below, the initial value problem (2.1)–(2.2) has the unique global-in-time viscosity solution for any initial datum u 0 ∈ BU C(R). Under the additional assumption u 0,x ∈ L p (R), the corresponding solution satisfies u x (·, t) ∈ L p (R) for all t > 0. Indeed, this is an immediate consequence of the L p -inequalities stated in Remark 7.7 and of a limit argument analogous to that in Step 6 of the proof of Theorem 2.7. Remark 2.10. For any positive, sufficiently regular and vanishing at infinity initial condition v0 ∈ L 2 (R), the corresponding solution v = v(x, t) of (1.15) is global-in-time and analytic, see [6, Th. 2.1]. Since v = u x ≥ 0, this result holds true for solutions of problem (2.1)–(2.2) with α = 1. On the other hand, the nonexistence of global-in-time solutions to the initial value problem for Eq. (1.15) has been always proved assuming that the initial datum is (smooth enough and) negative at some point, see [7, Th. 2.1 and Remark 2.3], [6, Th. 3.1 and 4.8] and [10]. Those arguments cannot be applied to Eq. (2.1) with α = 1 due to the factor |v| (= |u x |) in the nonlinearity and lower regularity of the data. Remark 2.11. For α ∈ (1, 2), we do not know how to define the product |u x | (α u) in the sense of distributions, which is an obstacle for us to prove the result of Theorem 2.7 in this case, see Sect. 7. Note, however, that the inequalities from Theorem 2.7 are valid for α ∈ (1, 2] as well, provided the solution u = u(x, t) is sufficiently regular. 3. Construction of Self-Similar Solutions Proof of Theorem 2.1. The crucial role in the construction of the self-similar profile α is played by the function

2 α/2 for |x| < 1, (3.1) v(x) = K (α) 1 − |x| 0 for |x| ≥ 1, with K (α) = (1/2) [2α (1 + α/2)((1 + α)/2)]−1 . This function (together with its multidimensional counterparts) has an important probabilistic interpretation. Indeed, if {X (t)}t≥0 denotes the symmetric α-stable process in R of order α ∈ (0, 2] and if T = inf{t : |X (t)| > 1} is the first passage time of the process to the exterior of the segment {x : |x| ≤ 1}, Getoor [15] proved that Ex (T ) = v(x), where Ex denotes the expectation under the condition X (0) = x.

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In particular, it was computed in [15, Th. 5.2] using a purely analytical argument (based on definition (2.3) and on properties of the Fourier transform) that α v ∈ L 1 (R) and α v(x) = 1 for |x| < 1.

(3.2)

Now, for the function v, we define the bounded, nondecreasing, C 1+α/2 -function x u(x) = v(y) dy, 0

which obviously satisfies u(x) = M(α) for all x ≥ 1 and u(x) = −M(α) for x ≤ −1 with 1 α/2 π M(α) = K (α) 1 − |y|2 dy =

2 . 0 2α (α + 1) 1+α 2 Then, for any ϕ ∈ Cc∞ (R), we can introduce the following duality:

α u, ϕ = u(y)(α ϕ)(y) dy. R

This defines α u as a distribution, because we can check (using the Lévy–Khintchine formula (2.4)) that there exists a constant C > 0 such that |(α ϕ)(x)| ≤

C ϕ W 2,∞ (R) 1 + |x|1+α

.

If, moreover, supp ϕ ⊂ (−1, 1), it is easy to check using the properties of the function v = v(x) that

∂x (α u), ϕ = − u, α (∂x ϕ) = − u, ∂x (α ϕ) = α (∂x u), ϕ = 1, ϕ, where the last inequality is a consequence of (3.2). From the symmetry of v, we deduce the antisymmetry of u, and then (α u)(−x) = −(α u)(x). Therefore, we get the equality (α u)(x) = x in D (−1, 1), and thus by [21, Cor. 3.1.5], in the classical sense for each y ∈ (−1, 1), too. Finally, we define the nonnegative function γ −1/(α+1) 2M(α) u γ .

α (y) = y + M(α) with γ −1 = α+1 α+1 Now, for yα = γ 1/(α+1) = [2M(α)]−1/(α+1) , we can check easily that α is exactly as stated in Theorem 2.1, which ends the proof. Let us note that we will not use in the sequel the explicit form of the function α , but only its properties listed in Theorem 2.1. Remark 3.1. It has been known since the work of Head and Louat [19] (see also [18]) that

1/2 the function v(x) = K 1 − |x|2 (with a suitably chosen constant K = K (1) > 0) is the solution of the equation (1 v)(x) = 1 on (−1, 1). This result is a consequence of an inversion theorem due to Muskhelishvili, see either [28, p. 251] or [31, Sect. 4.3].

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4. Notion of Viscosity Solutions Here, we consider Eq. (2.1) and its vanishing viscosity approximation, i.e. the following initial value problem for α ∈ (0, 2) and η ≥ 0, u t = ηu x x − |u x | α u on R × (0, +∞), u(x, 0) = u 0 (x) for x ∈ R.

(4.1) (4.2)

In this section, we present the framework of viscosity solutions to problem (4.1)– (4.2). To this end, we recall briefly the necessary material, which can be either found in the literature or is essentially a standard adaptation of those results. We also refer the reader to Crandall et al. [11] for a classical text on viscosity solutions to local (i.e. partial differential) equations. Let us first recall the definition of relaxed lower semi-continuous (lsc, for short) and upper semi-continuous (usc, for short) limits of a family of functions u ε which is locally bounded uniformly with respect to ε, lim sup ∗ u ε (x, t) = lim sup u ε (y, s) and ε→0

ε→0 y→x,s→t

lim inf ∗ u ε (x, t) = lim inf u ε (y, s). ε→0

ε→0 y→x,s→t

If the family consists of a single element, we recognize the usc envelope and the lsc envelope of a locally bounded function u, u ∗ (x, t) = lim sup u(y, s) y→x,s→t

and

u ∗ (x, t) = lim inf u(y, s). y→x,s→t

Now, we recall the definition of a viscosity solution for (4.1)–(4.2). Here, the difficulty is caused by the measure |z|−1−α dz appearing in the Lévy–Khintchine formula (2.4) which is singular at the origin and, consequently, the function has to be at least C 1,1 in space in order that α u(·, t) makes sense (especially for α close to 2). We refer the reader, for instance, to [4,25,29] for the stationary case, and to [23,24] for the evolution equation where this question is discussed in detail. Now, we are in a position to define viscosity solutions. Definition 4.1 (Viscosity solution/subsolution/supersolution). A bounded usc (resp. lsc) function u : R × R+ → R is a viscosity subsolution (resp. supersolution) of Eq. (4.1) on R × (0, +∞) if for any point (x0 , t0 ) with t0 > 0, any τ ∈ (0, t0 ), and any test function φ belonging to C 2 (R × (0, +∞)) ∩ L ∞ (R × (0, +∞)) such that (u − φ) attains a maximum (resp. minimum) at the point (x0 , t0 ) on the cylinder Q τ (x0 , t0 ) := R × (t0 − τ, t0 + τ ), we have ∂t φ(x0 , t0 ) − ηφx x (x0 , t0 ) + |φx (x0 , t0 )| (α φ(·, t0 ))(x0 ) ≤ 0 (resp. ≥ 0), where (α φ(·, t0 ))(x0 ) is given by the Lévy–Khintchine formula (2.4). We say that u is a viscosity subsolution (resp. supersolution) of problem (4.1)–(4.2) on R × [0, +∞), if it satisfies moreover at time t = 0, u(·, 0) ≤ u ∗0 (resp. u(·, 0) ≥ (u 0 )∗ ). A function u : R × R+ → R is a viscosity solution of (4.1) on R × (0, +∞) (resp. R × [0, +∞)) if u ∗ is a viscosity subsolution and u ∗ is a viscosity supersolution of the equation on R × (0, +∞) (resp. R × [0, +∞)). Other equivalent definitions are also natural, see for instance [4].

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Remark 4.2. Any bounded function u ∈ C 1+β (with some β > max{0, α − 1}) which satisfies pointwisely (using the Lévy–Khintchine formula (2.4)) Eq. (4.1) with η = 0, is indeed a viscosity solution. Theorem 4.3 (Comparison principle). Consider a bounded usc subsolution u and a bounded lsc supersolution v of (4.1)–(4.2). If u(x, 0) ≤ u 0 (x) ≤ v(x, 0) for some u 0 ∈ BU C(R), then u ≤ v on R × [0, +∞). Proof. Recall that in [23, Th. 5], the comparison principle is proved for α = 1 and η = 0 under the additional assumption that u 0 ∈ W 1,∞ (R). Looking at the proof of that result, the regularity of the initial data u 0 is only used to show that

sup (u 0 )ε (x) − (u 0 )ε (x) → 0 as ε → 0, (4.3) x∈R

where (u 0 )ε and (u 0 )ε are respectively sup and inf-convolutions. It is easy (and classical) to check that (4.3) is still true for u 0 ∈ BU C(R). The general case can be done either considering a variation of the proof of [23] taking into account the additional Laplace operator, or applying the “maximum principle” from [25], or following, for instance, the lines of [4]. We skip here the detail of this adaptation. This finishes the proof. Theorem 4.4 (Stability). Let {u ε }ε>0 be a sequence of viscosity subsolutions (resp. supersolutions) of Eq. (4.1) which are locally bounded, uniformly in ε. Then u = lim sup∗ u ε (resp. u = lim inf ∗ u ε ) is a subsolution (resp. supersolution) of (4.1) on R × (0, +∞). Proof. A counterpart of Theorem 4.4 is proved in [4, Th.1]. Here, the result for the time dependent problem is again a classical adaptation of that argument, so we skip details. Remark 4.5. One can generalize directly Theorem 4.4 assuming that {u ε }ε>0 are solutions to the sequence of Eq. (4.1) with η = ε. Then, in the limit ε → 0+ , we obtain viscosity subsolutions (resp. supersolutions) of Eq. (2.1). We use this property in the proof of Theorem 2.7. Remark 4.6. In Theorem 4.4, we only claim that the limit u is a supersolution on R × (0, +∞), but not on R × [0, +∞). In other words, we do not claim that u satisfies the initial condition. Without further properties of the initial data u 0 , it may happen that u(·, 0) ≤ u ∗0 is not true. Theorem 4.7 (Existence). Consider u 0 ∈ BU C(R). Then there exists the unique bounded continuous viscosity solution u of (4.1)–(4.2). Proof. Applying the argument of [22] (already adapted from the classical arguments), we can construct a solution by the Perron method, if we are able to construct suitable barriers. Case 1. First, assume that u 0 ∈ W 2,∞ (R). Then the following functions u ± (x, t) = u 0 (x) ± Ct

(4.4)

are barriers for C > 0 large enough (depending on the norm u 0 W 2,∞ (R) ), and we get the existence of solutions by the Perron method.

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Case 2. Let u 0 ∈ BU C(R). For any ε > 0, we can regularize u 0 by a convolution, and get a function u ε0 ∈ W 2,∞ (R) which satisfies, moreover, |u ε0 − u 0 | ≤ ε.

(4.5) u ε0

uε

the solution of (4.1)–(4.2) with the initial condition instead of u 0 . Let us call Then, from the fact that the equation does not see the constants and from the comparison principle Theorem 4.3, we have for any ε, δ > 0, |u ε − u δ | ≤ ε + δ. Therefore, {u ε }ε>0 is the Cauchy sequence which converges in L ∞ (R × [0, +∞)) to some continuous function u (because all the functions u ε are continuous). By the stability result Theorem 4.4, we see that u is a viscosity solution of Eq. (4.1) on R × (0, +∞). To recover the initial boundary condition, we simply remark that u ε (x, 0) = u ε0 (x) satisfies (4.5), and then passing to the limit, we get u(x, 0) = u 0 (x). This shows that u is a viscosity solution of problem (4.1)–(4.2) on R × [0, +∞), and concludes the proof of Theorem 4.7. 5. Uniqueness and Stability of the Self-similar Solution Lemma 5.1 (Comparison with the self-similar solution). Let v be a subsolution (resp. a supersolution) of Eq. (2.1) with the Heaviside initial datum given in (2.8). Then we have v ∗ ≤ (u α )∗ (resp. (u α )∗ ≤ v∗ ). Proof. Using Remark 4.2 and properties of α gathered in Theorem 2.1, it is straightforward to check that the self-similar solution u α (x, t) given in (2.6) is a viscosity solution of Eqs. (4.1)–(4.2) with the initial condition (2.8). Now, we show the inequality (u α )∗ ≤ v∗ . Let v be a viscosity supersolution of (4.1)– (4.2) with the Heaviside initial datum (2.8). Given a > 0 and v a (x, t) = v(a + x, t), we have (u α )∗ (x, 0) ≤ (u 0 )∗ (x) ≤ (u 0 )∗ (a + x) ≤ v a (x, 0). Because of the translation invariance of Eq. (2.1), we see that v a is still a supersolution. Moreover, for any a > 0, we can always find an initial condition u a ∈ BU C(R) such that u α (x, 0) ≤ u a (x) ≤ v a (x, 0). Therefore, applying the comparison principle (Theorem 4.3), we deduce that u α ≤ va . Because this is true for any a > 0, we can take the limit as a → 0 and get (u α )∗ ≤ v∗ . For a subsolution v, we proceed similarly to obtain v ∗ ≤ (u α )∗ . This finishes the proof of Lemma 5.1. Proof of Theorem 2.4. We consider a viscosity solution v of Eq. (2.1) with the Heaviside initial datum (2.8). Using both inequalities of Lemma 5.1, and the fact that (u α )∗ = (u α )∗ on (R × [0, +∞))\ {(0, 0)}, we deduce the equality v = u α on (R × [0, +∞))\ {(0, 0)}, which ends the proof of Theorem 2.4. We will now prove the following weaker version of Theorem 2.5.

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Theorem 5.2 (Convergence for suitable initial data). The convergence in Theorem 2.5 holds true under the following additional assumption lim u 0 (y) = 0 ≤ u 0 (x) ≤ 1 = lim u 0 (y). y→+∞

y→−∞

(5.1)

Proof. Step 1. Limits after rescaling of the solution. Consider a solution u of (2.1)–(2.2) with an initial condition u 0 satisfying (5.1). Recall that for any λ > 0, the rescaled solution is given by u λ (x, t) = u(λx, λα+1 t). Let us define u = lim sup ∗ u λ and u = lim inf ∗ u λ . λ→+∞

λ→+∞

From the stability result Theorem 4.4, we know that u (resp. u) is a subsolution (resp. supersolution) of (2.1) on R × (0, +∞). Step 2. The initial condition. We now want to prove that u(x, 0) = u(x, 0) = H (x) for x ∈ R\ {0},

(5.2)

where H is the Heaviside function. To this end, we remark that u 0 satisfies for some γ > 0 the inequality |u 0 (x)| ≤ γ (note that γ = 1 under assumption (5.1)), and for each ε > 0, there exists M > 0 such that |u 0 (x)| < ε for x ≤ −M. In particular, we get u 0 (x) < ε + γ H (x + M), and then from the comparison principle, we deduce u(x, t) ≤ ε + (u γα )∗ (x + M, t) with u γα (x, t) = γα

x

t 1/(α+1)

and γα (y) = γ α γ −1/(α+1) y .

(5.3)

(5.4)

γ

Here α is the self-similar profile solution of (2.7) with the boundary conditions 0 and γ γ at infinity. Moreover, because u α is continuous off the origin, we can simply drop the star ∗ , when we are interested in points different from the origin. This implies x + Mλ−1 u λ (x, t) ≤ ε + γα , t 1/(α+1) and then u(x, t) ≤ ε + γα

x t 1/(α+1)

.

Therefore, for every x < 0 we have u(x, 0) ≤ ε + γα (−∞) = ε. Because this is true for every ε > 0, we get u(x, 0) ≤ 0 for every x < 0. We get the other inequalities similarly, and finally conclude that (5.2) is valid.

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Step 3. Initial condition at the origin, using assumption (5.1). We now make use of (5.1) to identify the initial values of the limits u and u. We deduce from the comparison principle that 0 ≤ u(x, 0) ≤ u(x, 0) ≤ 1, and then for every x ∈ R we have u(x, 0) ≤ H ∗ (x) and u(x, 0) ≥ H∗ (x). Step 4. Identification of the limits after rescaling. From Lemma 5.1, we obtain u ≤ (u α )∗ = (u α )∗ ≤ u on (R × [0, +∞))\ {(0, 0)}. We have by the construction u ≤ u, hence we infer u = u = u α on (R × [0, +∞))\ {(0, 0)}. Step 5. Conclusion for the convergence. Then for any compact K ⊂ (R × [0, +∞))\ {(0, 0)}, we can easily deduce that sup |u λ (x, t) − u α (x, t)| → 0 as λ → +∞,

(x,t)∈K

which finishes the proof of Theorem 5.2.

6. Further Decay Properties and the End of the Proof of Theorem 2.5 Theorem 6.1 (Decay of a solution with compact support). Let u be the solution to (2.1)– (2.2) with the initial datum u 0 ∈ BU C(R) satisfying for some A > 0, u 0 (x) ≤ 0 for |x| ≥ A,

(6.1)

and u 0 (x) ≤ γ for some γ > 0 and all x ∈ R. Then, there exist on α, but independent of A, γ ) such that

β, β

> 0 (depending

u(x, t) ≤ Ct −β , and u(x, t) ≤ 0 for |x| ≥ C t β

with some constants C = C(α, A, γ ) and C = C (α, A, γ ). First, we need the following Lemma 6.2 (Decay after the first interaction). Consider α and yα defined in Theorem 2.1. Let ν ∈ (1/2, 1) and ξν ∈ (0, yα ) be such that α (ξν ) = ν. Let T > 0 be defined by A = ξν . γ 1/(α+1) T 1/(α+1)

(6.2)

Then, under the assumptions of Theorem 6.1, we have u(x, t) ≤ νγ for all t ≥ T, x ∈ R, and

yα u(x, t) ≤ 0 for all 0 ≤ t ≤ T and |x| ≥ A 1 + ξν

(6.3) .

(6.4)

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γ Proof. Let us denote α (y) = γ α γ −1/(α+1) y . Then we have γ H (x + A) ≥ u 0 (x) for x ∈ R, where γ H (x + A) = lim+ γα t→0

x+A t 1/(α+1)

for x + A = 0.

Now, we apply the comparison principle to deduce that x+A

γα 1/(α+1) ≥ u(x, t) for (x, t) ∈ R × (0, +∞). t Thisargument can be made rigorous by simply replacing the function γ H (x + A) by γ 1/(α+1)

α (x + A + δ)/(tε ) with δ > 0 and some sequence tε → 0+ , and then taking the limit δ → 0+ . Therefore we have x+A ≥ u(x, t) for (x, t) ∈ R × (0, +∞). γ α γ 1/(α+1) t 1/(α+1) From the properties of the support of α , we also deduce that u(x, t) ≤ 0 for x ≤ − A + yα (γ t)1/(α+1) , and then, by symmetry, u(x, t) ≤ 0 for |x| ≥ A + yα (γ t)1/(α+1) . Moreover, it follows from the monotonicity of α that A ≥ u(x, t) γ α (γ t)1/(α+1) for x ≤ 0, and by symmetry we can prove the same property for x ≥ 0. Then for T > 0 defined in (6.2) we easily deduce (6.3) and (6.4). This ends the proof of Lemma (6.2). Proof of Theorem 6.1. We apply recurrently Lemma 6.2. Define A0 = A, γ0 = γ , and yα An , γn+1 = νγn , and An+1 = An 1 + = ξν . ξν (γn Tn )1/(α+1) This gives yα n An = A0 1 + , γn = ν n γ0 , Tn = K µn , ξν with K =

1 γ0

A0 ξν

α+1

, 1 0, x ∈ R, with β=−

log ν > 0. log µ

Similarly, we have u(x, t) ≤ 0 for |x| ≥ An if t ≤ T0 + · · · + Tn−1 = K

µn − 1 . µ−1

In particular, we get for any n ∈ N\ {0},

yα n , if t ≤ K 0 µn u(x, t) ≤ 0 for |x| ≥ A0 1 + ξν

with K 0 = K /µ. This implies

u(x, t) ≤ 0 for |x| ≥ A0 (K 0 )−β t β with β = This ends the proof of Theorem 6.1.

log 1 + log µ

yα ξν

for t ≥ 0,

> 0.

As a corollary, we can now remove assumption (5.1) in Theorem 5.2 and complete the proof of Theorem 2.5. Proof of Theorem 2.5. We simply repeat Step 3 of the proof of Theorem 5.2, but here without assuming (5.1). Then, for any ε > 0 there exists A > 0 such that u 0 (x) ≤ 1 + ε for |x| ≥ A. By Theorem 6.1 applied to the solution u(x, t) − 1 − ε, this implies that there exists a constant C > 0 (depending on ε) such that u(x, t) ≤ 1 + ε + Ct −β . Therefore, for any for λ > 0 the following inequality u λ (x, t) ≤ 1 + ε + Ct −β λ−β

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holds true, which implies that u = lim sup ∗ u λ satisfies λ→+∞

u(x, t) ≤ 1 + ε for (x, t) ∈ R × (0, +∞). Since this is true for any ε > 0, we deduce that u(x, t) ≤ 1 for (x, t) ∈ R × (0, +∞). Let us now define u˜ = min (1, u) . By the construction, ˜ u(x, t) = u(x, t) for (x, t) ∈ R × [0, +∞)\ {(0, 0)}, ˜ 0) ≤ H ∗ (x) for all x ∈ R. Therefore, u˜ is a subsolution of and, by (5.2), we have u(x, (2.1)–(2.2) on R × [0, +∞) with the initial datum being the Heaviside function. Similarly, we can show that u = lim sup ∗ u λ satisfies λ→+∞

u ≥ 0 for (x, t) ∈ R × (0, +∞).

Hence, the function u˜ = max 0, u , which is a supersolution of (2.1)–(2.2) on R × [0, +∞) with the Heaviside initial datum. Finally, the conclusion of the proof is the same as in the proof of Theorem 5.2 where u (resp. u) is replaced by u˜ (resp. u). ˜ This finishes the proof of Theorem 2.5. 7. Approximate Equation and Gradient Estimates In this section, in order to prove our gradient estimates for viscosity solutions stated in Theorem 2.7, we replace Eq. (2.1) by an approximate equation for which smooth solutions do exist. Indeed, with ε > 0, we consider the following initial value problem: u t = εu x x − |u x |α u on R × (0, +∞), u(x, 0) = u 0 (x) for x ∈ R.

(7.1) (7.2)

We have added to Eq. (2.1) an auxiliary viscosity term which is stronger than α u and u x . In the case α ∈ (0, 1], we will see later (in Sect. 8) that it is possible to pass to the limit ε → 0+ in L ∞ (R), which is the required convergence for the framework of viscosity solutions. The difficulty in the case α ∈ (1, 2) comes from the fact that, for the limit equation with ε = 0, we are not able to give a meaning to the product |u x | (α u) in the sense of distributions, while it is possible when α ∈ (0, 1]. Our results on qualitative properties of solutions to the regularized problem (7.1)– (7.2) are stated in the following two theorems. Theorem 7.1 (Approximate equation – existence of solutions). Let α ∈ (0, 1] and ε > 0. Given any initial datum u 0 ∈ C 2 (R) such that u 0,x ∈ L 1 (R) ∩ L ∞ (R), there exists a unique solution u ∈ C(R × [0, +∞)) ∩ C 2,1 (R × (0, +∞)) of (7.1)–(7.2). This solution satisfies u x ∈ C([0, T ], L p (R)) ∩ C((0, T ]; W 1, p (R)) ∩ C 1 ((0, T ], L p (R)) for every p ∈ (1, ∞) and each T > 0.

(7.3)

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Theorem 7.2 (Approximate equation – decay estimates). Under the assumptions of Theorem 7.1, the solution u = u(x, t) of (7.1)–(7.2) satisfies u(·, t) ∞ ≤ u 0 ∞ , u x (·, t) ∞ ≤ u 0,x ∞ ,

(7.4)

and pα+1

u x (·, t) p ≤ C p,α u 0,x 1p(α+1) t

1 − α+1 1− 1p

,

(7.5)

for every p ∈ [1, ∞), all t > 0, and constants C p,α > 0, see (7.20) below), independent of ε > 0, t > 0 and u 0 . Existence theory. First, we construct solutions to the initial value problem for the regularized equation. Proof of Theorem 7.1. Note first that α u = α−1 Hu x ,

(7.6)

where H denotes the Hilbert transform, see (1.10). We recall that the Hilbert transform is bounded on the L p -space for any p ∈ (1, +∞) (see [30, Ch. 2, Th. 1]), i.e. it satisfies for any function v ∈ L p (R) the following inequality Hv p ≤ C p v p

(7.7)

with a constant C p independent of v. For α ∈ (0, 1), the operator α−1 defined analogously as in (2.3) corresponds to the convolution with the Riesz potential α−1 v = Cα | · |−α ∗ v. Hence, by [30, Ch. 5, Th. 1], for any p > 1/α with α ∈ (0, 1] and any function v ∈ L q (R), we have α−1 v p ≤ C p,α v q with

1 1 = + 1 − α. q p

(7.8)

Now, if u = u(x, t) is a solution to (7.1)–(7.2), using identity (7.6), we write the initial value problem for v = u x , vt = εvx x − (|v|α−1 Hv)x on R × (0, +∞), v(·, 0) = v0 = u 0,x ∈ L 1 (R) ∩ L ∞ (R), as well as its equivalent integral formulation t v(t) = G(εt) ∗ v0 − ∂x G(ε(t − τ )) ∗ (|v|α−1 Hv) dτ,

(7.9) (7.10)

(7.11)

0

with the Gauss–Weierstrass kernel G(x, t) = (4π t)−1/2 exp(−x 2 /(4t)). The next step is completely standard and consists in applying the Banach contraction principle to Eq. (7.11) in a ball in the Banach space XT = C([0, T ]; L 1 (R) ∩ L ∞ (R)) endowed with the usual norm v T = supt∈[0,T ] ( v(t) 1 + v(t) ∞ ). Using well known estimates of the heat semigroup and inequalities (7.7)–(7.8) combined with the imbedding L 1 (R) ∩ L ∞ (R) ⊂ L p (R) for each p ∈ [1, ∞], we obtain a solution v = v(x, t)

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to Eq. (7.11) in the space XT provided T > 0 is sufficiently small. We refer the reader to, e.g., [2,5] for examples of such a reasoning. This solution satisfies (7.3) for every p ∈ (1, ∞) and each T > 0, by standard regularity estimates of solutions to parabolic equations. Moreover, following the reasoning in [2], one can show that the solution is regular. Finally, this local-in-time solution can be extended to global-in-time (i.e. for all T > 0) because of the estimates v(t) p ≤ v0 p for every p ∈ [1, ∞] being the immediate consequence of inequalities (7.17), (7.18), and (7.21) below. Gradient estimates. In the proof of the decay estimates of u x , we shall require several properties of the operator α . First, we recall the Nash inequality for the operator α . Lemma 7.3 (Nash inequality). Let α > 0. There exists a constant C N > 0 such that 2(1+α)

w 2

≤ C N α/2 w 22 w 2α 1

(7.12)

for all functions w satisfying w ∈ L 1 (R) and α/2 w ∈ L 2 (R). The proof of inequality (7.12) is given, e.g., in [26, Lemma 2.2]. Our next tool is the, so-called, Stroock–Varopoulos inequality. Lemma 7.4 (Stroock–Varopoulos inequality). Let 0 ≤ α ≤ 2. For every p > 1, we have p 2 α 4( p − 1) α p−2 2 |w| 2 ( w)|w| w dx ≥ dx (7.13) p2 R R for all w ∈ L p (R) such that α w ∈ L p (R). If α w ∈ L 1 (R), we obtain (α w) sgn w dx ≥ 0.

(7.14)

Moreover, if w, α w ∈ L 2 (R), it follows that (α w)w + dx ≥ 0 and (α w)w − dx ≤ 0,

(7.15)

R

R

R

where w + = max{0, w} and w − = max{0, −w}. Inequality (7.13) is well known in the theory of sub-Markovian operators and its statement and the proof is given, e.g., in [27, Th. 2.1, combined with the Beurling–Deny condition (1.7)]. Inequality (7.14), called the (generalized) Kato inequality, is used, e.g., in [13] to construct entropy solutions of conservation laws with a Lévy diffusion. It can be easily deduced from [13, Lemma 1] by an approximation argument. The proof of (7.15) can be found, for example, in [27, Prop. 1.6]. Remark 7.5. Remark that inequality (7.14) appears to be a limit case of (7.13) for p = 1. Inequality (7.15) for w + follows easily from (7.14) by a comparison argument if, for instance, w ∈ Cc∞ (R). Finally, remark that the constant appearing in (7.13) is the same as for the Laplace operator ∂ 2 /∂ x 2 = −2 . Our proof of the decay of v(t) = u x (t) is based on the following Gagliardo–Nirenberg type inequality:

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Lemma 7.6 (Gagliardo–Nirenberg type inequality). Assume that p ∈ (1, ∞) and α > 0 are fixed and arbitrary. For all v ∈ L 1 (R) such that α/2 |v|( p+1)/2 ∈ L 2 (R), the following inequality is valid: 2 v ap ≤ C N α/2 |v|( p+1)/2 v b1 , (7.16) 2

where a=

p( p + α) pα + 1 , b= , p−1 p−1

and C N is the constant from the Nash inequality (7.12). Proof. Without loss of generality, we can assume that v 1 = 0. Substituting w = |v|( p+1)/2 in the Nash inequality (7.12) we obtain 2 ( p+1)(1+α) α( p+1) ≤ C N α/2 |v|( p+1)/2 v ( p+1)/2 . v p+1 2

Next, it suffices to apply two particular cases of the Hölder inequality

v p 1/ p 2 v 1

p2 /( p2 −1) p/( p+1)

≤ v p+1 as well as v ( p+1)/2 ≤ v p

1/( p+1)

v 1

,

and compute carefully all the exponents which appear on both sides of the resulting inequality. Proof of Theorem 7.2. The first inequality in (7.4) is an immediate consequence of the comparison principle from Theorem 4.3, because classical solutions are viscosity solutions, as well. The maximum principle and an argument based on inequalities (7.15) (cf. [26] for more detail) lead to the second inequality in (7.4). We also discuss this inequality in Remark 7.7 below. For the proof of the L 1 -estimate u x (t) 1 ≤ u 0,x 1

(7.17)

(i.e. (7.5) with p = 1 and C p,α = 1), we multiply Eq. (7.9) by sgn v = sgn u x and we integrate with respect to x to obtain d (α−1 Hv)|v| sgn v dx. |v| dx = ε vx x sgn v dx − x dt R R R The first term on the right-hand side is nonpositive by the Kato inequality (i.e. (7.14) with α = 2) hence we skip it. Remark that (formally) α−1 ( Hv)|v| sgn v dx = (α−1 Hv)vx (sgn v)2 + (α−1 Hvx )v dx x R R = (α−1 Hv)v dx = 0. R

x

√ Now, approximating the sign function in a standard way by sgnδ (z) = z/ z 2 + δ, integrating by parts, and passing to the limit δ → 0+ , one can show rigorously that the

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second term on right-hand side of the above inequality is nonpositive. This completes the proof of (7.17) with p = 1. Next, we multiply the equation in (7.9) by |v| p−2 v with p > 1 to get 1 d p p−2 (α−1 Hv)|v| |v| p−2 v dx. |v| dx = ε vx x |v| v dx − x p dt R R R We drop the first term on the right-hand side, because it is nonpositive by (7.13) with α = 2. Integrating by parts and using the elementary identity p − 1 p−1 |v| |v| p−2 v = |v| v , x x p we transform the second quantity on the right-hand side as follows: p−1 α−1 p−2 − ( Hv)|v| |v| v dx = − (α v)|v| p−1 v dx. x p R R Consequently, by the Stroock–Varopoulos inequality (7.13) (with the exponent p replaced by p + 1), we obtain d 4 p( p − 1) α/2 ( p+1)/2 2 p |v| (7.18) v(t) p ≤ − . 2 dt ( p + 1)2 Hence, the interpolation inequality (7.16) combined with (7.17) lead to the following p inequality for v(t) p : 4 p( p − 1) d p ( pα+1)/( p−1) −1 p ( p+α)/( p−1) v(t) p ≤ − C N v0 1 v(t) p . (7.19) 2 dt ( p + 1) Recall now that if a nonnegative (sufficiently smooth function) f = f (t) satisfies, for all t > 0, the inequality f (t) ≤ −K f (t)β with constants K > 0 and β > 1, then f (t) ≤ (K (β − 1)t)−1/(β−1) . Applying this simple result to the differential inequality (7.19), we complete the proof of the L p -decay estimate (7.5) with the constant C p,α

1 4 p(α + 1) − α+1 = C N−1 ( p + 1)2

1− 1p

where C N is the constant from the Nash inequality (7.12).

,

(7.20)

Remark 7.7. Note that, for every fixed α, we have lim p→∞ C p,α = +∞. By this reason, we are not allowed to pass directly to the limit p → +∞ in inequalities (7.5) (as was done in, e.g., [26, Th. 2.3]) in order to obtain a decay estimate of v(t) in the L ∞ -norm. Nevertheless, using (7.19) we immediately deduce the inequality v(·, t) p ≤ v0 (·) p valid for every p ∈ (1, ∞). Hence, passing to the limit p → +∞ we get v(·, t) ∞ ≤ v0 (·) ∞ .

(7.21)

It is natural to expect that, under the assumptions of Theorem 7.2, the quantity v(·, t) ∞ should decay at the rate t −1/(α+1) . For a proof, one might follow an idea from [6, Lemma 4.7] where the decay estimates of vx (·, t) were obtained for solutions of a certain regularization of Eq. (1.15). Here, however, we did not try to go this way, because our main goal was to study decay estimates for the problem (2.1)–(2.2) whose viscosity solutions are not regular enough a priori to handle decay properties of u x x (x, t) = vx (x, t).

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8. Passage to the Limit and the Proof of Theorem 2.7 Now, we are in a position to complete the proof of the gradient estimates (2.11). First, we show that from the sequence {u ε }ε>0 of solutions to the approximate problem (7.1)–(7.2) one can extract, via the Ascoli–Arzelà theorem, a subsequence converging uniformly. Theorem 4.4 on the stability and Remark 4.5 imply that the limit function is a viscosity solution to (2.1)–(2.2). Passing to the limit ε → 0+ in inequalities (7.4) and (7.5) we complete our reasoning. Proof of Theorem 2.7. First, let us suppose that u 0 ∈ C ∞ (R) ∩ W 2,∞ (R) with u 0,x ∈ L 1 (R)∩ L ∞ (R). Denote by u ε = u ε (x, t) the corresponding solution to the approximate problem with ε > 0. Step 1. Modulus of continuity in space. Under this additional assumption, we have u εx (·, t) p ≤ C p t −γ p

(8.1)

pα+1 1 1 − 1p . The Sobolev imbedding theorem with C p = C p,α u 0,x 1p(α+1) and γ p = α+1 implies that there exist some β ∈ (0, 1) and C0 > 0 such that

|u ε (x + h, t) − u ε (x, t)| ≤ |h|β C0 C p t −γ p .

(8.2)

∞ Step 2. Modulus of continuity in time. Let us consider a nonnegative function ϕ ∈ C (R) with supp ϕ ⊂ [−1, 1] such that R ϕ(x) dx = 1, and for any δ > 0 set ϕδ (x) = δ −1 ϕ(δ −1 x). Then, multiplying (7.1) by ϕδ and integrating in space, we get d ε u (x, t)ϕδ (x) dx = ε u ε (·, t), (ϕδ )x x dt R − ϕδ (x) |u εx (x, t)|(H α−1 u εx (x, t)) dx,

R

+ 1/ p

= 1, and then with 1/ p d ε ≤ ε u ε (·, t) ∞ (ϕδ )x x 1 u (x, t)ϕ (x) dx δ dt R

+ ϕδ ∞ u εx (·, t) p H α−1 u εx (·, t) p .

(8.3)

Here, we have used relation (7.6). Combining inequalities (7.7) and (7.8) with estimate (8.1), we get for p > 1/α, H α−1 u εx (·, t) p ≤ C p C p ,α Cq t −γq . Then for any bounded time interval I ⊂ (0, +∞) there exists a constant C I,δ such that for all t ∈ I , we have for any ε ∈ (0, 1], d ε ≤ C I,δ . u (x, t)ϕ (x) dx δ dt R Now, for any t, t + s ∈ I , we get ε u ε (x, t + s)ϕδ (x) dx − u (x, t)ϕδ (x) dx ≤ |s|C I,δ . R

R

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Therefore, the following estimate: |u ε (0, t + s) − u ε (0, t)| ϕδ (x) dx ≤ |s|C I,δ + R

× sup |u ε (x, t + s) − u ε (0, t + s)| + |u ε (x, t) − u ε (0, t)| x∈[−δ,δ]

holds true. Using the Hölder inequality (8.2), we deduce that there exists a constant C I depending on I , but independent of δ and of ε ∈ (0, 1], such that |u ε (0, t + s) − u ε (0, t)| ≤ |s|C I,δ + C I δ β . Since the above inequality is true for any δ, this shows the existence of a modulus of continuity ω I satisfying |u ε (0, t + s) − u ε (0, t)| ≤ ω I (|s|) for any t, t + s ∈ I. By the translation invariance of the problem, this estimate is indeed true for any x ∈ R, i.e. |u ε (x, t + s) − u ε (x, t)| ≤ ω I (|s|) for any t, t + s ∈ I, x ∈ R.

(8.4)

From estimates (8.2) and (8.4), and using the Step 3. Convergence as ε → Ascoli–Arzelà theorem and the Cantor diagonal argument, we deduce that there exists a subsequence (still denoted {u ε }ε ) which converges to a limit u ∈ C(R × (0, +∞)). By the stability result in Theorem 4.4 (see also Remark 4.5), we have that u is a viscosity solution of (2.1) on R × (0, +∞). 0+ .

Step 4. Checking the initial conditions for u 0 smooth. Remark that for u 0 ∈ W 2,∞ we can use the barriers given in (4.4) with some constant C > 0 uniform in ε ∈ (0, 1]. This ensures that u is continuous up to t = 0 and satisfies u(·, 0) = u 0 , so this proves the result under additional assumptions. Step 5. General case. The proof in the case of less regular initial conditions simply follows by an approximation argument as in the proof of Theorem 4.7. Step 6. Gradient estimates. To pass to the limit ε → 0+ in estimates (7.5), we use the inequality h −1 u ε (· + h, t) − u ε (·, t) p ≤ u εx (·, t) p

(8.5)

with fixed h > 0. Hence, by the Fatou lemma combined with the pointwise convergence of u ε toward u, we deduce from (8.5) and (7.5) that pα+1

h −1 u(· + h, t) − u(·, t) p ≤ C p,α u 0,x 1p(α+1) t

1 − α+1 1− 1p

for all h > 0. For every fixed t > 0, the sequence {h −1 (u(·+h, t)−u(·, t))}h>0 is bounded in L p (R) and converges (up to a subsequence) weakly in L p (R) toward u x (·, t) (see, e.g., [30, Ch. V, Prop. 3]). Using the well known property of a weak convergence in Banach spaces we conclude h u x (·, t) p ≤ lim inf + h→0

−1

pα+1 p(α+1)

u(· + h, t) − u(·, t) p ≤ C p,α u 0,x 1

This finishes the proof of Theorem 2.7.

t

1 − α+1 1− 1p

.

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Acknowledgements. The third author is indebted to Cyril Imbert for stimulating and enlightening discussions on the subject of this paper. The authors thank the anonymous referee for calling their attention to the onedimensional counterpart of the quasi-geostrophic equation (1.15). This work was supported by the contract ANR MICA (2006–2009), by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389, and by the Polish Ministry of Science grant N201 022 32/0902.

References 1. Alvarez, O., Hoch, P., Le Bouar, Y., Monneau, R.: Dislocation dynamics: short time existence and uniqueness of the solution. Arch. Rat. Mech. Anal. 181, 449–504 (2006) 2. Amour, L., Ben-Artzi, M.: Global existence and decay for viscous Hamilton-Jacobi equations. Nonlinear Anal. 31, 621–628 (1998) 3. Barles, G., Chasseigne, E., Imbert, C.: On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57, 213–246 (2008) 4. Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. I.H.P., Anal Non-Lin. 25, 567–585 (2008) 5. Ben-Artzi, M., Souplet, Ph., Weissler, F.B.: The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces. J. Math. Pures Appl. 81, 343–378 (2002) 6. Castro, A., Córdoba, D.: Global existence, singularities and ill-posedness for a nonlocal flux. Adv. Math. 219, 1916–1936 (2008) 7. Chae, D., Córdoba, A., Córdoba, D., Fontelos, M.A.: Finite time singularities in a 1D model of the quasi-geostrophic equation. Adv. Math. 194, 203–223 (2005) 8. Constantin, P., Lax, P., Majda, A.: A simple one-dimensional model for the three dimensional vorticity. Comm. Pure Appl. Math. 38, 715–724 (1985) 9. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) 10. Córdoba, A., Córdoba, D., Fontelos, M.A.: Formation of singularities for a transport equation with nonlocal velocity. Ann. Math. 162, 1377–1389 (2005) 11. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.), 27, 1–67 (1992) 12. Deslippe, J., Tedstrom, R., Daw, M.S., Chrzan, D., Neeraj, T., Mills, M.: Dynamics scaling in a simple one-dimensional model of dislocation activity. Phil. Mag. 84, 2445–2454 (2004) 13. Droniou, J., Imbert, C.: Fractal first order partial differential equations. Arch. Rat. Mech. Anal. 182, 299–331 (2006) 14. Forcadel, N., Imbert, C., Monneau, R.: Homogenization of the dislocation dynamics and of some particle systems with two-body interactions. Disc. Contin. Dyn. Syst. Ser. A 23, 785–826 (2009) 15. Getoor, R.K.: First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101, 75–90 (1961) 16. Head, A.K.: Dislocation group dynamics I. Similarity solutions od the n-body problem. Phil. Mag. 26, 43–53 (1972) 17. Head, A.K.: Dislocation group dynamics II. General solutions of the n-body problem. Phil. Mag. 26, 55–63 (1972) 18. Head, A.K.: Dislocation group dynamics III. Similarity solutions of the continuum approximation. Phil. Mag. 26, 65–72 (1972) 19. Head, A.K., Louat, N.: The distribution of dislocations in linear arrays. Austral. J. Phys. 8, 1–7 (1955) 20. Hirth, J.R., Lothe, L.: Theory of Dislocations. Second Ed., Malabar, FL: Krieger, 1992 21. Hörmander,: The Analysis of Linear Partial Differential Operators. Vol. 1, New York: Springer-Verlag, 1990 22. Imbert, C.: A non-local regularization of first order Hamilton-Jacobi equations. J. Differ. Eq. 211, 214–246 (2005) 23. Imbert, C., Monneau, R., Rouy, E.: Homogenization of first order equations, with (u/ε)-periodic Hamiltonians. Part II: application to dislocations dynamics. Comm. Part. Diff. Eq. 33, 479–516 (2008) 24. Jakobsen, E.R., Karlsen, K.H.: Continuous dependence estimates for viscosity solutions of integro-PDEs. J. Differ. Eq. 212, 278–318 (2005) 25. Jakobsen, E.R., Karlsen, K.H.: A maximum principle for semicontinuous functions applicable to integro-partial differential equations. NoDEA Nonlin. Differ. Eqs. Appl. 13, 137–165 (2006) 26. Karch, G., Miao, C., Xu, X.: On the convergence of solutions of fractal Burgers equation toward rarefaction waves. SIAM J. Math. Anal. 39, 1536–1549 (2008)

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27. Liskevich, V.A., Semenov, Yu.A.: Some problems on Markov semigroups. In: Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Math. Top. 11, Berlin: Akademie Verlag, 1996, pp. 163–217 28. Muskhelishvili, N.I.: Singular Integral Equations. Groningen: P. Noordhoff, N. V., 1953 29. Sayah, A.: Équations d’Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I Unicité des solutions de viscosité, II Existence de solutions de viscosité. Comm. Part. Diff. Eq. 16, 1057–1093 (1991) 30. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30, Princeton, NJ: Princeton University Press, 1970 31. Tricomi, F.G.: Integral Equations. New York-London: Interscience Publ., 1957 32. Vázquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type. Oxford Lecture Series in Mathematics and its Applications 33, Oxford: Oxford University Press, 2006 Communicated by P. Constantin

Commun. Math. Phys. 294, 169–197 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0935-9

Communications in

Mathematical Physics

On the Massive Wave Equation on Slowly Rotating Kerr-AdS Spacetimes Gustav Holzegel Department of Mathematics, Fine Hall, Princeton University, Washington Road, Princeton, NJ 08544, United States. E-mail: [email protected] Received: 27 February 2009 / Accepted: 29 July 2009 Published online: 2 October 2009 – © Springer-Verlag 2009

Abstract: The massive wave equation g ψ − α 3 ψ = 0 is studied on a fixed Kerr-anti de Sitter background M, g M,a, . We first prove that in the Schwarzschild case (a = 0), ψ remains uniformly bounded on the black hole exterior provided that α < 49 , i.e. the Breitenlohner-Freedman bound holds. Our proof is based on vectorfield multipliers and commutators: The total flux of the usual energy current arising from the timelike Killing vector field T (which fails to be non-negative pointwise) is shown to be non-negative with the help of a Hardy inequality after integration over a spacelike slice. In addition to T , we construct a vectorfield whose energy identity captures the redshift producing good estimates close to the horizon. The argument is finally generalized to slowly rotating Kerr-AdS backgrounds. This is achieved by replacing the Killing vectorfield T = ∂t with K = ∂t + λ∂φ for an appropriate λ ∼ a, which is also Killing and–in contrast to the asymptotically flat case–everywhere causal on the black hole exterior. The separability properties of the wave equation on Kerr-AdS are not used. As a consequence, the theorem also applies to spacetimes sufficiently close to the Kerr-AdS spacetime, as long as they admit a causal Killing field K which is null on the horizon.

1. Introduction The study of linear wave equations on black hole spacetimes has acquired a prominent role within the subject of general relativity. The main reason is the expectation that understanding the mechanisms responsible for the decay of linear waves on black hole exteriors in a sufficiently robust setting provides important insights for the non-linear black hole stability problem [7]. The mathematical analysis of linear waves in this context was initiated by the pioneering work of Kay and Wald establishing boundedness (up to and including the event horizon) for φ satisfying g φ = 0 on Schwarzschild spacetimes [14,20]. Since then considerable progress has been achieved, especially in the last few years. Most of these

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recent decay and boundedness theorems for linear waves concern black hole spacetimes satisfying the vacuum Einstein equations Rµν −

1 Rgµν + gµν = 0 2

(1)

with = 0 and, motivated by cosmological considerations, > 0. In particular, by now polynomial decay rates have been established for g φ = 0 on Schwarzschild [4,8,16] and more recently, Kerr spacetimes [6,7,19]. In the course of work on the decay problem, a much more robust understanding of boundedness on both Schwarzschild and Kerr spacetimes was also obtained [6,7] allowing one to prove boundedness for a large class of spacetimes, which are not exactly Schwarzschild or Kerr but only assumed to be sufficiently close Many of the above results have been extended to the cosmological case, > 0. Here (much stronger) decay rates have been established for the wave equation on Schwarzschild-de Sitter spacetimes [5,13,17]. For further discussion we refer the reader to the lecture notes [7], which among other things provide an account of previous work on these problems (Sect. 4.4, 5.5, 6.3 ibidem) and a comparison with results that have been obtained in the heuristic tradition (Sect. 4.6). In contrast to the case of a positive cosmological constant, the choice < 0 in (1) has remained relatively unexplored. While this problem certainly deserves mathematical attention in its own right, there is also considerable interest from high energy physics, see [9,18]. In this paper, we study the equation g ψ − α

ψ =0 3

(2)

on a class of spacetimes, which will include slowly rotating Kerr anti-de Sitter spacetimes [3]. These spacetimes generalize the well-known Kerr solution (the latter being the unique two-parameter family of stationary, axisymmetric asymptotically flat black hole solutions to (1) with = 0.). They are axisymmetric, stationary solutions of (1) with < 0, parametrized by their mass M and angular momentum per unit mass a = MJ . Before we comment further on their geometry, let us discuss Eq. (2). The main motivation to include the zeroth order term in (2) is the case α = 2, the conformally invariant case. In pure AdS the Green’s function for (2) with α = 2 is supported purely on the light cone, which makes it a natural analogue of the equation ψ = 0 in asymptotically flat space (and also explains the adjective “massless” which is sometimes used in the physics literature in connection with this choice of α). The case α = 2 also occurs naturally in classical general relativity when studying a Maxwell field or linear gravitational perturbations in AdS [12]. Again for the case of pure AdS, it is well known ([1,2,12]) that (2) is only well-posed for α < 54 , the so-called second Breitenlohner-Freedman bound. While no solutions exist for α ≥ 49 , one has an infinite number of solutions depending on boundary conditions for α in the range 45 ≤ α < 49 , the latter bound being the first Breitenlohner-Freedman bound. For general asymptotically AdS spacetimes, however, the wellposedness of (2) has not yet been established explicitly. In this context it is essential to notice that asymptotically AdS spaces are not globally hyperbolic. To make the dynamics of (2) well-posed suitable boundary conditions will have to be imposed on the timelike boundary of the spacetime. We will address this boundary initial value problem in detail in a separate

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171

paper. For the purpose of the present paper, we will assume that α < 94 and that we are given a solution to (2) which decays suitably near the AdS boundary. This pointwise “radial” decay depends on how close α is to the Breitenlohner Freedman bound and ensures in particular that there is no energy flux through the timelike boundary I. It is precisely the decay suggested by the mode analysis of [2] in pure AdS and expected to hold for all asymptotically AdS spacetimes. The global geometry of the Kerr-AdS background spacetimes is quite different from their asymptotically flat counterparts. Most notably perhaps, null-infinity is now timelike, entailing the non-globally hyperbolic nature of the spacetime mentioned above. With the boundary conditions imposed there will be no radiation flux through infinity and the only possible decay mechanism is provided by an energy flux through the horizon. Another geometric feature, which we will exploit to a great extent in the present paper, is the existence of an everywhere causal Killing vectorfield on the black hole exterior of slowly rotating Kerr-AdS spacetimes.1 This vectorfield was used previously by Hawking and Reall [10] to obtain a positive conserved energy for any matter fields satisfying the dominant energy condition. The scalar field (2) in the subcase α ≤ 0 provides an example. In particular, the argument of [10] excluded a negative energy flux through the horizon and hence superradiance as a mechanism of instability, at least if 2 with r |a| − 3 < rhoz hoz being the location of the event horizon.

For 0 < α < 49 , however, the energy momentum tensor associated with ψ does not satisfy the dominant energy condition and the energy current associated with the causal Killing field fails to be positive pointwise. In particular, as noted for instance in [15], the positivity argument of [10] breaks down for 0 < α < 94 , including the most interesting case α = 2. In this paper we present a simple resolution of this problem: Using a Hardy inequality we show that the energy flux arising from the Killing field is still positive in an integrated sense. With the existence of the everywhere causal Killing field and its associated globally positive energy, superradiance is eliminated as an obstacle to stability.2 This makes the problem much easier to deal with than the asymptotically flat case, where such a vectorfield is not available. In particular, we can avoid the intricate bootstrap and harmonic analysis techniques of [6], which were necessary to deal with superradiant phenomena and trapping (see also [19]).3 It is well-known that the notion of positive energy outlined above is not sufficient to prevent the scalar field from blowing up on the horizon in evolution. This issue was first addressed and resolved for Schwarzschild in the celebrated work of Kay and Wald [14,20] exploiting the special symmetry properties of the background spacetime. With the recent work of Dafermos and Rodnianski, in particular their mathematical understanding of the celebrated redshift, there is now a much more stable argument available, which does not hinge on the discrete symmetries of Schwarzschild and is in fact applicable to any black hole event horizon with positive surface gravity [7]. This geometric understanding of the role of the event horizon for the boundedness and decay mechanism 1 This is crucially different from the asymptotically flat case, where any linear combination of the two available Killing fields K = ∂t + λ∂φ (for some constant λ) is somewhere spacelike. 2 As expected perhaps, the restriction on a (which can be computed explicitly) becomes tighter compared to the case α ≤ 0 of Hawking and Reall. 3 Superradiance is induced by the existence of an ergosphere [i.e. an effect which is not present in Schwarzschild] arising from the fact that the energy density associated with the Killing field ∂t can be negative inside the ergosphere. This allows for a negative energy flux through the horizon and hence an amplification of the amplitude for backscattered waves. See [21] for a nice discussion and also [7] for a detailed mathematical treatment.

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was first developed in [8] and plays a crucial role in the recent proof of boundedness [6] and polynomial decay [7] of scalar waves on slowly rotating Kerr spacetimes. The boundedness statement of [6] holds for a much more general class of spacetimes nearby Kerr, while the decay statement of [7] requires the background to be exactly Kerr. For our considerations at the horizon we will adapt the ideas developed by Dafermos and Rodnianski. A vectorfield Y is constructed whose energy identity, if coupled to the timelike Killing vectorfield K in an appropriate way, provides control over Sobolev norms whose weights do not degenerate at the horizon. An additional complication compared to the asymptotically-flat massless case is that due to the zeroth order term in the wave equation (2), a zeroth order flux-term of the redshift vectorfield on the horizon has the wrong sign. However, this term can, after a computation, be absorbed by the “good” positive terms at our disposal. With Y and K at hand, boundedness is shown adapting the argument of [6], taking care of the different weights in r which appear due to the asymptotically hyperbolic nature of the background space.4 It should be emphasized that the proof does not require the construction of a globally positive spacetime integral arising from a virial vectorfield but that it suffices to use the timelike Killing field and the redshift vectorfield alone. The fact that the more elementary statement of boundedness can be obtained from the use of these two vectorfields alone in the non-superradiant regime was observed by Dafermos and Rodnianski in [6] in the asymptotically-flat case. Previously, even the understanding of boundedness was intriniscially tied to the understanding of a globally positive spacetime term and hence decay, see [8] and also Sect. 3.4 of [7]. Finally, here is an outline of the paper. We introduce the AdS-Schwarzschild and Kerr backgrounds equipped with regular coordinate systems on the black hole exterior (defining in particular the spacelike slices we are going to work with) in Sect. 2. In the following section the class of solutions we wish to consider is defined and the notion of vectorfield multipliers discussed. For reasons of presentation we first state and prove the boundedness theorem for Schwarzschild-AdS in Sect. 4, before we turn to the generalization to Kerr-AdS in Sect. 5. The paper concludes with some final remarks and future directions. 2. The Black Hole Backgrounds 2.1. Schwarzschild-AdS. In the familiar (t, r ) coordinates, the Schwarzschild AdS metric reads −1 2M r 2 2M r 2 g =− 1− + 2 dt 2 + 1 − + 2 dr 2 + r 2 dω2 , r l r l

(3)

where = − l32 is the cosmological constant. This coordinate system is not well-behaved at the zeros of 2M r 2 + 2 . 1−µ= 1− r l

(4)

4 Since angular momentum operators do not commute with the wave operator for a = 0, we have to commute the wave equation with T and the redshift vectorfield Y to obtain L 2 control of certain derivatives,

which leads to control over all derivatives using elliptic estimates on the asymptotically hyperbolic spacelike slices .

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

173

Let us define ⎛

p = ⎝ Ml 2 +

⎛

⎞1 3 6 l 2 4 ⎠ M l + 27

and q = ⎝ Ml 2 −

⎞1 3 6 l 2 4 ⎠ M l + . 27

(5)

2

Clearly, p > 0, q < 0 with pq = − l3 . The expression (4) has a single real root at rhoz = p + q > 0,

(6)

the location of the black hole event horizon. Note that for l → ∞ we have rhoz → 2M. In general we have the estimate 2 − 3 pq (r − rhoz ) r 2 + rrhoz + rhoz r 3 + l 2 r − 2Ml 2 = 1−µ = 2 rl 2 3 rl 2 3 2 2 r − rhoz (r − rhoz ) r + rrhoz + rhoz + l = ≥ , (7) rl 2 rl 2 which will be useful later. Here is a coordinate system which is well behaved everywhere on the black hole exterior and the horizon: It arises from the coordinate transformation r t = t + r (r ) − l arctan , (8) l where r (r ) is a solution of the differential equation 1 dr

= 2M dr 1− r +

r2 l2

=

1 1−µ

and r (3M) = 0.

(9)

The variable r is often called the tortoise coordinate. In the new (t , r ) coordinates the metric reads 2

r 1 + 2M 2 4M r + l2

g = − (1 − µ) dt + dr + dr 2 + r 2 dω2 , dt 2 2 r2 r 1 + rl 2 1 + l2

(10)

which is clearly regular on the horizon. For notational convenience let us agree on the shorthand notation k± = 1 ±

2M r 2 + 2 r l

and

k0 = 1 +

r2 . l2

(11)

Slices τ of constant t will play a prominent role for stating energy identities in the paper. Their normal 2

r 1 + 2M 2M r + l2 ∂r ∂ − − ∇t = 2 t 2 2 r 1 + rl 2 1 + rl 2

(12)

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G. Holzegel

is everywhere timelike: k+

g ∇t , ∇t = g t t = − 2 . k0

(13)

We denote the unit-normal by

√ k+ ∇t

2M n = − √ ∂t − √ ∂r . =

k0 −g (∇t , ∇t ) r k+

(14)

We also define the (Killing) vectorfields i (i = 1, 2, 3) to be a basis of generators of the Lie-algebra of S O (3) corresponding to the spherical symmetry of the Schwarzschild / to denote the gradient of the metric induced on the metric. Moreover, we will write ∇ S O (3)-orbits. Finally, a Penrose diagram of (the black hole exterior of) Schwarzschild-AdS spacetime with two slices of constant t is depicted below.

H+ τ I

0

2.2. Kerr-AdS. The Kerr-AdS metric in Boyer Lindquist coordinates reads 2

θ r 2 + a 2 − − a 2 sin2 θ 2 2 sin2 θ d φ˜ 2 dr + dθ + g=

−

θ 2

θ r 2 + a 2 − −

− θ a 2 sin2 θ 2 ˜ − − a sin2 θ d φdt dt −2 with the identifications = r 2 + a 2 cos2 θ, r2

± = r 2 + a 2 1 + 2 ± 2Mr, l a2 cos2 θ, l2 a2 = 1− 2. l

θ = 1 −

(15)

(16) (17) (18) (19)

2

Once again, let k0 = 1 + rl 2 . A coordinate system which is regular on the horizon is obtained by the transformations t = t + A (r )

and

φ = φ˜ + B (r ) ,

(20)

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

175

where 2Mr dA = dr

− 1 +

r2 l2

a dB = . dr

−

and

(21)

The new metric coefficients become , gt t = gtt ,

θ = gφ˜ φ˜ , gt φ = gt φ˜ , 1 2 2

, = − a sin θ k + + 0 2 l2 r2 1 + l2

gθθ =

(22)

gφφ

(23)

grr

a sin2 θ [ + 2Mr ] , k0 a2 1 2Mr − 2 sin2 θ . = k0 l

(24)

gφr = −

(25)

gt r

(26)

Note that the angular momentum term in the last expression grows faster in r than the mass term.5 For a = 0 the metric reduces to the Schwarzschild-AdS metric in t , r coordinates, cf. (10). The inverse components are

gt

t

=−

+ + a 2 sin2 θ −k0 + k02 θ

2Mr l2

,

θ 2 a

, g φφ = , , gt φ = k 0 l 2 θ

θ sin2 θ

− a 2Mr

, g φr = , gt r = . = k0

(27)

g θθ =

(28)

grr

(29)

The unit normal of a constant t slice is n =

−g t

t

gt r gt φ

∂t − ∂ − ∂ , g(n , n ) = −1, r

φ −g t t −g t t

(30)

with the determinant of the metric induced on constant t slices being √

det h =

det gt =const

= sin θ

t

t . −g

(31)

5 Hence the Kerr-AdS metric is not uniformly close to the Schwarzschild metric in these coordinates! It follows that for the statement that “Schwarzschild-AdS is close to Kerr-AdS” one has to use both a regular coordinate patch for r ≤ R and a Boyer Lindquist patch for r > R (for some R away from the horizon). For the asymptotically-flat case this is not necessary.

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3. The Dynamics 3.1. The class of solutions. Having defined a proper coordinate system on the black hole backgrounds in the previous section we can turn to the wave equation we would like to study (recall = − l32 ): g ψ +

α ψ = 0, l2

(32)

for α ∈ R. The choice α = 2 in (32) corresponds to the conformally invariant case, cf. the discussion in the Introduction. Generally we will assume α < 49 , the upper Breitenlohner-Freedman bound. Before we state a theorem on the global dynamics of the above scalar field, let us address the issue of well-posedness of (32). It turns out that this has not yet been proven explicitly for asymptotically anti de Sitter spacetimes, nor has it for the special cases of Schwarzschild and Kerr-AdS. For pure AdS (M = a = 0), however, it can be deduced from [1]. The peculiarity of this problem arises from the fact that due to the timelike nature of I asymptotically AdS spacetimes are not globally hyperbolic. Hence in general “appropriate” boundary conditions have to be imposed on I to make the dynamics well-posed. We will formulate and prove the precise well-posedness statement for asymptotically AdS spacetimes in a separate paper. For the purpose of this paper, we content ourselves with considering a class of ψ defined as follows: Definition 3.1. Fix α < 49 , a Kerr-AdS spacetime M, g M,a, and a constant t slice 0 in D = J + (I) ∩ J − (I), as well as an integer k ≥ 0. We say that the function ψ is k if in J + a solution of class Cdec ( 0 ) ∩ J − (I) , • ψ is C k , • ψ satisfies (32), √ • for all δ < 21 9 − 4α 3

lim |r 2 +n+δ ∂rn ψ| = 0

r →∞

holds for n = 0, 1, . . . , k.

(33)

k there is no In particular, the decay (33) ensures that for a solution of class Cdec energy flux (cf. Sect. 4.1) through the AdS boundary. The decay (33) is precisely the one expected from the AdS case [1] and also strongly suggested from an asymptotic expansion in r of Eq. (32) (performed in [2] for pure AdS). The existence of a large class of solutions with the properties of Definition 3.1, which arise from appropriate initial data prescribed on 0 , would follow from a general well-posedness statement phrased in terms of weighted Sobolev norms (cf. also the Appendix). We emphasize again that uniqueness is only expected for α < 45 .

3.2. Vectorfield multipliers and commutators. We will obtain estimates for the field ψ via vectorfield multipliers (and eventually commutators). Since the general technique is well known and reviewed in detail in [7] we will only give a brief summary. The starting point is the energy momentum tensor of the above scalar field α 1 Tµν = ∂µ ψ∂ν ψ − gµν ∂β ψ∂ β ψ − 2 ψ 2 . 2 l

(34)

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

177

It satisfies ∇ µ Tµν = 0,

(35)

provided that (32) holds. Consider a vectorfield X on spacetime. We define its associated currents JµX = Tµν X ν , K = Tµν X

(X ) µν

π

(36) ,

(37)

where (X ) π µν = 21 (∇ µ X ν + ∇ ν X µ ) is the deformation tensor of the vectorfield X . One has (using (35)) the identity (38) ∇ µ JµX = ∇ µ Tµν X ν = K X . Note that π vanishes if X is Killing. Integrating (38) over regions of spacetime relates boundary and volume terms via Stokes’ theorem. In particular, for background Killing vectorfields we obtain conservation laws. In writing out the aforementioned integral identities we will sometimes not spell out the measure explicitly (e.g. Eq. (72)), it being implicit that the measure is the one induced on the slices (cf. Eq. (31)) or the spacetime measure respectively. Besides using vectorfields as multipliers we will also use them as commutators. If X is a vectorfield and ψ satisfies (2) then X (ψ) satisfies (cf. the Appendix of [7]) α g X (ψ) + 2 X (ψ) = −2(X ) π γβ ∇γ ∇β ψ − 2 2 ∇ γ (X ) πγ µ − ∇µ (X ) πγγ ∇ µ ψ. l Note that if X is Killing the right-hand side vanishes. In general one may apply multipliers to the commuted equation to derive estimates for higher order derivatives. 4. Boundedness in the Schwarzschild Case Here is our boundedness theorem for Schwarzschild-anti de Sitter:

Theorem 4.1. Fix a Schwarzschild-anti de Sitter spacetime M, g M>0, and 0 = τ0 a slice of constant t = τ0 in D = J + (I) ∩ J − (I).6 Let α < 49 and ψ n+1 with k ψ ∈ C n+1−k for k = 0, ..., n, where n ≥ 0 is be a solution to (32) of class Cdec i dec an integer. If n 2 2 1 k 2 k k 2 /

r 2 dr dω < ∞, ∂ ∂ ψ + r ψ + | ∇ ψ| (39) t r 2 r 0 k=0

then

2 2 1 k 2 k k 2 /

r 2 dr dω ∂ ∂ ψ + r ψ + | ∇ ψ| t r 2 r k=0 τ n 2 2 1 k 2 k k 2 2 / ψ| r dr dω ∂t ψ + r ∂r ψ + |∇ ≤C 2 0 r

n

(40)

k=0

for a constant C which just depends on M, l and α. Here τ denotes any constant t

slice to the future of 0 and restricted to r ≥ rhoz . 6 Note that such slices satisfy in particular H− ∩ = ∅. 0

178

G. Holzegel

By Sobolev embedding on S 2 we immediately obtain Corollary 4.1. The pointwise bound 2 2 2 1 k / k ψ|2 r 2 dr dω + r 2 ∂r k ψ + |∇ k=0 0 r 2 ∂t ψ |ψ| ≤ C (41) 3 r2 holds in the exterior J − I + ∩ J + ( 0 ) for a constant C just depending on the initial data, M, l and α. The remainder of this section is spent proving the above theorem. In the following we denote by B and b constants which just depend on the fixed parameters M, l and α. We also define R (τ1 , τ2 ) = ∪τ1 ≤τ ≤τ2 τ

(42)

to be the region enclosed by the slices τ1 and τ2 , a piece of I, and the horizon piece (43) H (τ1 , τ2 ) = H ∩ J + τ1 ∩ J − τ2 . Compare the figure above. 4.1. Positivity of energy. The first step is to obtain a positive energy arising from the Killing vectorfield T = ∂t . For this we apply the vectorfield identity (38) in the region R t1 , t2 . For the energy flux through a slice we obtain ∞

T (∂t , n ) g¯ dr dω E t = 1 = 2

rhoz

∞

rhoz

S2

S2

(∂t ψ)

2

k+ k02

α 2 2 2 / − 2ψ + (∂r ψ) (1 − µ) + |∇ψ| r dr dω. (44) l 2

The flux through the horizon is E H[t1 ,t2 ] =

H(t1 ,t2 )

(∂t ψ)2 r 2 dt dω,

(45)

hence in particular non-negative. Finally, the flux through I, the AdS boundary, vanishes because of the boundary conditions imposed. Combining these facts we obtain the energy identity E t2 = E t1 + E H[t1 ,t2 ] , (46) stating in particular that E (t ) is non-increasing. Next we show that the energy flux through the slices is positive: Lemma 4.1. We have 1 E t ≥ 2

∞

S2

rhoz

/ +|∇ψ|

(∂ ψ) t

2

r 2 dr dω.

2

k+ k02

4 + 1 − α (∂r ψ)2 (1 − µ) 9 (47)

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

179

Proof.

∞

rhoz

ψ r dr = 2 2

3 r 3 − rhoz

3

∞ ψ 2 rhoz

2 − 3

∞

rhoz

3 dr. ψψr r 3 − rhoz

(48)

The boundary term vanishes because of the decay of ψ at infinity and we obtain the Hardy inequality 3 2 ∞ 3 r − rhoz 4 ∞ 4 ∞ 2 2 2 ψ r dr ≤ dr ≤ (∂r ψ) (∂r ψ)2 r 2 l 2 (1 − µ) dr, 9 rhoz r2 9 rhoz rhoz (49) 3 ≤ r 3 in the last step. Hence where we used (7) and that r 3 − rhoz ∞ ∞ α 4 2 2 ψ r dr ≤ α (∂r ψ)2 r 2 (1 − µ) dr. l 2 rhoz 9 rhoz

Inserting this into (44) yields the result.

(50)

The lemma clearly reveals the relevance of the Breitenlohner-Freedman bound in ensuring a positive energy. We emphasize that one has positivity of energy only in an integrated sense! Lemma 4.1 also answers a question posed in [15] on whether one can construct a positive energy for the range 0 < α < 49 . Finally, we note that a similar Hardy inequality was used previously in the Appendix of [1], where the massive wave equation is studied on pure AdS. Having established positivity of the energy we only have to deal with the fact that the control over the (∂r ψ)2 term degenerates at the horizon. This is achieved using a so-called “redshift vectorfield”. See [8] for its first appearance in the context of asymptotically flat Schwarzschild black holes and the recent [7] for a version applicable to all non-extremal black holes. An additional difficulty in the present context lies in the fact that the energy momentum tensor does not satisfy the dominant energy condition. 4.2. The redshift. Define γ 1−µ k+ γ Y = ∂t + − + β ∂r . +β 2k0 2k0 2 2

(51)

Here γ ≥ 0 and β ≥ 0 are both functions of r only, which we will define below. We compute the current 2 3 k2

k2

ψ) (∂ t µ JµY n = T (Y, n ) = k+ 0 + β k+2 0 γ 2 2 2k04

(∂t ψ) (∂r ψ) − k+ γ + β k+ k− 2k0 2k β k− (∂r ψ)2 γ 0 k+ k0 + √ + 2k0 2 2 k+ +

+

/ 2− |∇ψ| 2

α 2 ψ l2

γ β k+ . √ + 2 k+ 2

(52)

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G. Holzegel

Next consider the current associated with N = T + eY for some small constant e (depending on α): µ JµN n

3 k2

k2 (∂t ψ)2 3 0 2 0 = T (N , n ) = k+ k0 + eγ k+ + eβ k+ 2 2 2k04

(∂t ψ) (∂r ψ) − k+ eγ + eβ k+ k− + 2k0 2k k02 γ β k− (∂r ψ)2 0 k+ k0 + e √ + √ k− + e 2k0 2 2 k+ k+ / 2 − α2 ψ 2 k0 |∇ψ| γ β l k+ . + √ +e √ +e 2 2 k+ 2 k+

(53)

The bulk term of the vectorfield N reads 1 N αβ K = πY Tαβ e 2M 2r k+ (∂t ψ)2 2 γ − γ + k = + k β + 2 − µ) − γ (β (1 ) ,r − ,r + r2 l2 r 2k02

2M 2r 8M (∂t ψ) (∂r ψ) −2 γ − γ +2β + k k k − + (β (1−µ)−γ ) ,r − ,r − + 2k0 r2 l2 r2 2 γ,r k− k− k− M r 2 + (∂r ψ) γ − + β,r − + (β (1 − µ) − γ ) r 2 l2 2 2 r k− β,r γ,r β 2M 2r 2 / − − + |∇ψ| + 2 + 2 2 r2 l 2 γ,r α 2 1 γ k− β,r β 2M 2r βk− − + + . (54) + 2ψ + 2 − l r r 2 2 r2 l 2 KY =

Let us define the functions γ and β. Set γ = ξ (r ) (1 + (1 − µ)) and β = 1δ (1 − µ) ξ (r ). Here ξ is a smooth positive function equal to 1 in r ≤ r0 and identically zero for r ≥ r1 . The quantity δ is a small parameter. (Note that β,r is positive in r ≤ r0 ). Let us regard ξ as being fixed. We choose r0 and δ such that the following conditions hold in r ≤ r0 : 2M 2r k+ γ + 2 − γ,r k− + k+2 β,r + 2 (β (1 − µ) − γ ) ≥ b, r2 l r 2 γ,r k− k− k− M r − + + β,r − γ (β (1 − µ) − γ ) ≥ b, r 2 l2 2 2 r γ,r β 2M 2r β,r k− + − ≥ b, + − 2 2 r2 l2 2

(55)

(56)

(57)

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

181

and

2M 2r 8M (∂t ψ) (∂r ψ) −2 γ + 2 − γ,r k− + 2β,r k− k+ − 2 (β (1 − µ) − γ ) 2k0 r2 l r 2 1 (∂t ψ) 2M 2r k+ 2 ≤ γ + 2 − γ,r k− + k+ β,r + 2 (β (1 − µ) − γ ) 2 r2 l r 2k02 2 γ,r k− k− k− M r 2 + (∂r ψ) γ − + β,r − + (58) (β (1 − µ) − γ ) . r 2 l2 2 2 r

It is easily seen all these inequalities can be achieved by inserting the expressions for γ and β, then choosing δ sufficiently small and finally r0 sufficiently close to rhoz to exploit factors of k− = (1 − µ). With γ and β being determined choose e so small that eγ ≤

1 2k0

and, in case that 0 < α < 49 , also 9 1 − 49 α 1 e sup (γ + βk+ ) < 2k0 8α r

(59)

(60)

hold for all r . 4.2.1. The N bulk term. Let us denote by K0N the expression for K N with the zeroth order term removed. Lemma 4.2. The quantities δ, r0 and r1 can be chosen such that / 2 K0N ≥ b (∂t ψ)2 + (∂r ψ)2 + |∇ψ|

(61)

pointwise in r ≤ r0 . On the other hand, / 2 − K0N ≤ B (∂t ψ)2 + (∂r ψ)2 + |∇ψ|

(62)

pointwise in r0 ≤ r ≤ r1 . Proof. The second inequality is immediate since we are away from the horizon. The first is a consequence of the inequalities (55)-(58). 4.2.2. The N boundary terms. Let us investigate what the current J N actually controls. We can write √ √ √ k+ k+ eγ k+ (∂t ψ)2 (∂r ψ)2 k0 N µ − eγ 2 + Jµ n = T (N , n ) = √ k− + 2 k0 2 4 k+ 2k0 √ 2 γ k+ ∂r ψ 2 k− ∂t ψ β +e k+ 2 − k+ ∂t ψ + √ ∂r ψ +e 4 k0 2 4 k0 k+ α 2 2 / − l2 ψ |∇ψ| k0 γ β (63) + k+ √ +e √ +e 2 2 k+ 2 k+

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G. Holzegel

with the term in the penultimate line being manifestly non-negative. From (59) one easily obtains a lower bound for the square bracket multiplying the (∂t ψ)2 -term. Inequality (59) in turn allows us to control the bad (for 0 < α < 49 !) zeroth order term by borrowing from the r -derivative term using the Hardy inequality (50). Namely, if 0 < α < 49 , then: e α 2 ψ 1+ (γ + βk+ ) r 2 dr 2 l 2k 0 rhoz 4 ∞ α 9 1 − 9α ≤ 1+ ψ 2 r 2 dr 2 8α rhoz l ∞ 4 1 1+ α ≤ (∂r ψ)2 (1 − µ) r 2 dr. 2 9 rhoz

∞

(64)

Hence finally

∞

√

k+ 2 r dr dω k02 rhoz √ ∞ k+ (∂t ψ)2 ≥ 2 2 2k 0 rhoz S √ 4 γ k+ 2 (∂r ψ)2 1 1 − α (1 − µ) + e r dr dω + 2 2 9 4 ∞ / 2 2 |∇ψ| r dr dω, + 2 rhoz S 2 S2

µ JµN n

(65)

(66)

and we have established the following Lemma 4.3. ≥b

∞

rhoz

S2

µ JµN n

=

∞

S2

rhoz

µ JµN n

√

k+ 2 r dr dω k0

(∂t ψ) / 2 r 2 dr dω. + (∂r ψ)2 r 2 + |∇ψ| r2 2

(67)

Note that the degeneration of the (∂r ψ)2 -term which occurred in the T -energy has disappeared.

4.3. The boundedness. Proposition 4.1. There exist constants B and b such that for any τ2 ≥ τ1 ≥ 0, τ2 µ N µ Jµ n H+ + b dτ JµN n − H(τ1 ,τ2 ) τ τ12 τ µ ≤ KN + B dτ JµT n . R(τ1 ,τ2 )

τ1

τ

(68)

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

183

Proof. We first note that all zeroth order terms can be absorbed by the last term in (68) using inequality (50). Hence it suffices to prove the inequality for the first order terms. We have τ2 M α 2 N µ 2 2 /

eγ |∇ψ| − 2 ψ + (∂t ψ) r 2 dt dω. Jµ n H+ = 2r k0 l H(τ1 ,τ2 ) τ1 S2 (69) For α ≤ 0 the term has a good sign while for 0 < α < 49 , τ2 r1 Mr α 2 N µ

− Jµ n H+ ≤ dt dω dr ∂r −eγ ψ 2k0 l 2 H(τ1 ,τ2 ) τ1 S2 rhoz Bα 2 η (∂r ψ)2 + ψ dt dr dω ≤e η l2 R(τ1 ,τ2 )∩{r ≤r1 }

(70)

for any η > 0 using that γ is supported for r < r1 only. So we only need a little bit of the ∂r term which we can borrow from the good bulk term K N in r ≤ r0 (by Lemma 4.2) and from the T -energy term in the remaining region. We have τ2 Bα µ −K N + eη (∂r ψ)2 + e 2 ψ 2 +b dτ JµN n η l R(τ1 ,τ2 )∩{r ≤r1 } τ1 τ τ2 µ T ≤ B˜ dτ Jµ n (71) τ1

τ

for a small constant b and large constants B and B˜ as a consequence of Lemma 4.2. Combining this inequality with (70) yields (68). Remark. We have discarded good terms (the t and angular derivative) on the horizon in the proof of the proposition as they are not needed for the following argument. Later, when we start commuting the equation with the redshift vectorfield we will need to keep those positive terms in order to estimate certain errorterms (cf. Sec. 5.3). Using the previous proposition we can prove boundedness as follows. The N identity reads µ µ µ JµN n + JµN n H+ + KN = JµN n 0 . (72) H+

τ

R(0,τ )

An application of Proposition 4.1 yields the inequality τ2 τ2 µ µ JµN n + b dτ JµN n ≤ B dτ τ2

τ1

τ

τ1

τ

µ

τ

JµT n +

τ1

µ

JµN n τ . 1

(73)

Using (46) (in particular the fact that the T -energy from initial data) is non-increasing µ µ and setting f (τ ) = τ JµN n as well as D = 0 JµT n we arrive at τ2 f (τ2 ) + b f (τ ) dτ ≤ B D (τ2 − τ1 ) + f (τ1 ) (74) τ1

for any τ2 ≥ τ1 ≥ 0 from which boundedness of f (τ ) follows from a pigeonhole argument, cf. [7].

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G. Holzegel

Due to the spherical symmetry of the background we can commute Eq. (32) with angular momentum operators and obtain boundedness of µ JµN k ψ n (75)

for any integer k assuming such a bound on the data. Theorem 4.1 follows. 4.4. Pointwise bounds. For completeness we give the proof of Corollary 4.1. We have ∞ ∞

1 |∂r ψ t , r, ω |dr ≤ √ 3 |∂r ψ (t , r, ω) |2 r 4 dr , (76) |ψ t , r, ω | ≤ r r 3r 2 and by Sobolev embedding on S 2 ,

∞

|∂r ψ t , r, ω |2 r 4 dr ≤ C˜

r

∞

r

≤ C˜

2

2 |∂r k (ψ) t , r, ω |2 r 4 dr dω S 2

S 2 k=0

k=0

µ JµN k ψ n ,

(77)

from which the statement of the corollary follows.

4.5. Higher order quantities. Clearly, via commuting with T and angular momentum operators, we obtain control over certain higher order energies as well. Using just this collection of vectorfields however, we will still not be able to estimate the derivative transverse to the horizon. This problem was resolved for the asymptotically-flat Kerr case in [6,7], roughly speaking by commuting with the redshift vectorfield as well. We will adapt their argument to the present asymptotically hyperbolic case in Sect. 5.3, when we deal with the Kerr-AdS metric. There the argument is unavoidable even to obtain a pointwise bound on ψ since one can no longer (trivially) commute with angular momentum operators! 5. The Kerr-AdS Case To generalize the argument to include the case of Kerr-AdS we have to circumvent several difficulties. First of all, the timelike Killing field T = ∂t is no longer timelike everywhere on the black hole exterior due to the presence of an ergoregion close to the horizon. Hence the T -identity alone will not produce positive boundary terms. The resolution is to consider the Killing field K = T + λ

(78)

for an appropriate constant λ and = ∂φ being the Killing field corresponding to the axisymmetry of the Kerr-AdS metric. K is everywhere timelike on the black hole exte2 . This rior (it is null on the horizon, coinciding with the generators) as long as |a|l < rhoz

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

185

is a consequence of the asymptotically AdS nature of the background7 : Note that the analogous vectorfield in asymptotically flat space would turn spacelike near infinity! These properties of K were first exploited by Hawking and Reall [10] to obtain a pointwise positive energy for fields whose energy momentum tensor satisfies the dominant energy condition, see the discussion in the Introduction. Not surprisingly, we will show below that for sufficiently small a the energy identity associated to K produces boundary terms with manifestly non-negative first order terms. The zeroth order term however still has the wrong sign for 0 < α < 94 (as in Schwarzschild-AdS), a consequence of the dominant energy condition being violated in this range. We resolve this issue by generalizing the Hardy inequality (50) to the Kerr-AdS case allowing us to again control the zeroth order term from the derivative term in the K energy. Because we have to “borrow” from the r -derivative term for the Hardy inequality, we have to impose stronger 2 . restrictions on the smallness of a than just |a|l < rhoz Since the argument close to the horizon involving the vectorfield Y is stable in itself (it applies to any black hole event horizon with positive surface gravity by Theorem 7.1 of [7]) our previous argument using the vectorfield N = K + eY immediately yields boundedness of the L 2 -norm of all derivatives. In particular, no further restrictions on the size of a arise. The only remaining difficulty is that we can no longer commute with angular momentum operators to obtain pointwise bounds because the spherical symmetry of Schwarzschild has been broken. The way around this is to commute (32) with K and Y and to derive elliptic estimates on the asymptotically hyperbolic slices . This is carried out in Sect. 5.3.

5.1. The vectorfield K : Positive energy. We first compute the currents associated with the Killing vectors T = ∂t and = ∂φ ,

− 1 1 α 1

−g t t (∂t ψ)2 + ψ2 (∂r ψ)2 − 2 2 2 −g t t

l 2 −g t t

2 1 φφ θθ 2 rφ g + ψ + g ψ) + 2g ψ) ∂ ψ (79) ∂ (∂ (∂ φ θ r φ

2 −g t t

T (∂t , n ) =

and t r t φ 2 g g

T ∂φ , n = −g t t . (80) ∂φ ψ (∂t φ) + t t ∂φ ψ (∂r φ) + t t ∂φ ψ g g We observe that we cannot control the (∂r ψ) ∂φ ψ term in the T -energy, because the (∂r ψ)2 term from which we have to borrow degenerates on the horizon. However, consider the vectorfield K = T + λ

with λ =

7 More precisely, the fact that g ∼ g ∼ g 2 φφ ∼ r . t t t φ

a + a2

2 rhoz

(81)

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G. Holzegel

and its current on constant t slices α

2 −g t t · T (K , n ) = − 2 ψ 2 + g θθ (∂θ ψ)2 l 2a

∂φ ψ (∂t ψ) + −g t t (∂t ψ)2 − g t t 2 2 rhoz + a 2 2a

+ g φφ − g t φ 2 ψ ∂ φ rhoz + a 2

− a

+ . (∂r ψ)2 + 2 (∂r ψ) ∂φ ψ gr φ − g t r 2 rhoz + a 2

(82)

We choose a so small that 1 φφ 2a 2 2a 2 2

g − gt φ 2 = ≥ 0. − 2 + a2 k l 2 4 rhoz rhoz + a 2 4 θ sin2 θ 0 θ

(83)

This is possible because both terms decay like r12 in r . In the asymptotically flat case

the g t φ -term will eventually dominate because k0 = 1 in the asymptotically flat case 2 (k0 = 1 + rl 2 in AdS!). We also choose a small enough so that 2 1 t t 2a 1 φφ

ψ) + −g g ψ ψ ≥ 0. ∂ ∂ (∂t ψ)2 − g t t 2 (∂ φ t φ 2 8 rhoz + a 2

(84)

This is easily achieved since again all terms decay like r12 at infinity and the cross-term has a factor of a. Of the first order terms it remains to control the r φ cross-term. For this we note (cf. Lemma 5.1) a a 2 2 k = + a − 2Mr r 0 hoz 2 + a2 2 + a2 rhoz k0 rhoz a ( − − k0 (rhoz + r ) (r − rhoz )) = 2 k0 rhoz + a 2 l2 a (r − rhoz ) r− . (85) = k0 l2 rhoz The reason for the above choice of K is now obvious: The (∂r ψ) ∂φ ψ -term has acquired a weight which degenerates on the horizon. Hence we can borrow from the (∂r ψ)2 term whose weight also degenerates. Moreover, the decay of (85) in r is strong, which will be exploited soon. Before we continue we obtain rather precise control over how the quantity − deteriorates on the horizon: gr φ − g t

r

Lemma 5.1. We have a 2l 2 1 3 2 2 2 2

− = 2 (r − rhoz ) r + r rhoz + r rhoz + l + a − l rhoz

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187

and 3 2 3 + 3a 2 (r − r r − rhoz hoz )

− − = f, r 2 + a2 l 2

(86)

where 1 a 2l 2 3 2 (r − rhoz ) r 3 l 2 − 4a 2 + r 2 rhoz f = 2 − + a r hoz rhoz r + a2 l 2 2 a 4 l 2 4 2 2 + r rhoz . − 8a 4 + a 2 rhoz + l 2 + rhoz rhoz + 3a 2 − rhoz

(87)

Proof. Direct computation. Corollary 5.1. For sufficiently small a we have f ≥ 0. Proof. All coefficients of the polynomial in the square bracket of (87) are positive for 2 is seen to be sufficient). small enough a (the condition |a| < 21 l in addition to |a|l < rhoz Corollary 5.2. For sufficiently small a we have 16 g φφ

2 a (r − rhoz ) l2 r− < f. k0 l2 rhoz

(88)

Proof. Direct computation. Of course we recover our previous result (7) for a = 0. Note also that f grows slower (like ∼ r 2 ) than − (which grows like ∼ r 4 ). Going back to (85) we have the estimate rφ

2 (∂r φ) ∂φ ψ g − g t r

a 2 + a2 rhoz 2 2 8 a (r − rhoz ) l2 1 r− ≤ (∂r ψ)2 + g φφ ∂φ ψ φφ 2 g k0 l rhoz 8 ≤

2 f (∂r φ)2 1 φφ + g ∂φ ψ 2 8

(89)

for sufficiently small a using Corollary 5.2. It is here where the good decay of (85) is exploited, allowing us to borrow from f (and not the full − ) only. Now that positivity has been obtained for all first order terms we can turn to the zeroth order term and generalize the Hardy inequality (50). We note α l2

dθ dφ S2

α ≤ 2 l

S2

∞

2

ψ t , r, θ, φ rhoz ∞ dθ dφ sin θ dr r 2 + a 2 ψ 2 dr sin θ

rhoz

(90)

188

G. Holzegel

and estimate as previously, 2 ∞ 3 3 + 3a 2 (r − r r − rhoz α ∞ 2 4 hoz ) 2 2 dr r + a ψ ≤ α dr (∂r ψ)2 l 2 rhoz 9 rhoz r 2 + a2 l 2 ∞ 4 = α dr ( − − f ) (∂r ψ)2 (91) 9 rhoz using Lemma 5.1. We have obtained the Hardy inequality ∞ 2 α

ψ t , r, θ, φ dθ dφ dr sin θ 2 2 l S rhoz ∞ 4 sin θ ≤ α dθ dφ dr ( − − f ) (∂r ψ)2 . 9 S2 rhoz

(92)

Inserting the estimates (83), (84), (89) and (92) into (82) we finally have Proposition 5.1. For sufficiently small a the estimate ∞ µ µ

K Jµ (ψ) n = dθ dφ dr sin θ −g t t JµK n 2 S rhoz ∞ 1

− ≥b dθ dφ dr sin θ 2 (∂t ψ)2 + 2 (∂r ψ)2 r r rhoz S2 2 θ + ∂φ ψ + (∂θ ψ)2 2

θ sin θ

(93)

holds for a constant b which just depends on M, l and α < 49 . How large is a allowed to be? Note that in the case α ≥ 0 our estimates reduce to the simple geometric condition that K is everywhere timelike outside the black hole. 2 , the This in turn translates into a single inequality for a, thereby recovering |a|l < rhoz 9 result of [10]. If on the other hand 0 < α < 4 , the additional restriction of Corollary 5.2 has to be imposed. The reason is that the − (∂r φ)2 -term in the K -identity has to control both the mixed term (89) and the zeroth order term (92). We used a coefficient f (independent of α) to control the former and ( − − f ) to control the latter establishing that a uniform a can be chosen for α < 49 . Finally we compute the flux term on the horizon, where the normal is proportional to K : 2 a T (K , K ) = ∂φ ψ > 0. (94) ∂t ψ + 2 rhoz + a 2 H H 5.2. The vectorfield N. Now that we again have a positive, non-increasing K -energy we can invoke the identical argument as in the Schwarzschild case close to the horizon, involving the vectorfields Y and N = K + eY . This will eventually produce a bound ∞ 2 1 dθ dφ dr sin θ 2 (∂t ψ)2 + r 2 (∂r ψ)2 + ∂φ ψ 2 r

θ sin θ S2 rhoz

θ µ µ 2 N ≤B + Jµ (ψ) n ≤ B JµN (ψ) n 0 , (95) (∂θ ψ) 0

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

189

where the “bad” − -weight from the r -derivative term has now disappeared. Note that no further smallness restrictions on a arise as long as the horizon has positive surface gravity.

5.3. Higher order energies and pointwise bounds. We finally address the issue of pointwise bounds. Unfortunately, we can no longer exploit the commutation with the angular derivatives because of the broken spherical symmetry. However, we can certainly commute with T (or K respectively) to obtain integral bounds for certain higher order m the inequality energies. Given a solution of class Cdec

µ JµN K k ψ n ≤ B

0

µ JµN K k ψ n

(96)

for any non-negative integer k < m is an automatic consequence of the fact that K commutes with the wave operator. We write the wave equation as √ 1 √ ∂i g i j g∂ j ψ g = −g t

t

√ 1 α

(∂t ∂t ψ) − 2g t i ∂t ∂i ψ − √ ∂r g t r g (∂t ψ) − 2 ψ, g l

(97)

where i = 1, 2, 3 (or r, θ, φ respectively). Using both the decay assumed for ψ at infinity and that we control the right-hand side by (96), we obtain for any r0 > rhoz the elliptic estimate 1

i j kl

sin θ h h ∇ ψ) ∇ ∇ ψ −g t t dr dθ dφ (∇ i k j l

−g t t ∩{r ≥r0 >rhoz } µ µ

sin θ −g t t dr dθ dφ (98) JµN (ψ) n + JµN (K ψ) n ≤ C (r0 ) ∩{r ≥rhoz } away from the horizon, where h i j is the inverse of the induced metric on . Note that the argument breaks down at the horizon because of the degenerating weight of grr (whereas h rr is well-behaved there).8 To obtain good estimates close to the horizon we again adapt the ideas of [6,7]. Their resolution is to commute the equation with a version of the redshift vector field 1 1 Yˆ = ∂t − ∂r 2k0 2

(99)

which yields the equation 2 α 4r Yˆ ψ = + − Yˆ Yˆ (ψ) − Yˆ T (ψ) + λ1 (∂r ψ) + λ2 (∂t ψ) + 2 Yˆ (ψ), l

(100)

8 Actually, the estimate (98) remains true if one inserts an additional weight of r n for any n < √ max 2, 9 − 4α into the integrand on the left-hand side. This improvement is outlined in the Appendix where stronger weighted Sobolev norms are derived.

190

G. Holzegel

where λ1 =

1 1

∼ , 2 − r 0

λ2 = −

(101)

1 a 2 + 5r 2 ∼ 2, 2 l k0 r

(102)

and a prime denotes taking a derivative with respect to r . What will be crucial is that the coefficient of the first term on the right-hand side of (100) has a (good) sign close to the horizon. There is a geometric reason for this which was observed in [7]: Computing the surface gravity κ defined by (cf. [21]) ∂b K a K a = −2κ K b (103) r =rhoz

r =rhoz

we obtain κ=

− (rhoz ) 2 2 rhoz + a2

(104)

for the Kerr-AdS black hole under consideration. This means that on the horizon the coefficient of the Yˆ Yˆ (ψ)-term on the right-hand side of (100) is proportional to the surface gravity of the black hole, which is positive for a non-degenerate horizon! Remarkably, this observation is not bound to the special properties of the Kerr metric but a general fact about black hole event horizons with positive surface gravity. For details the reader may consult [7]. Let us investigate why this sign allows us to derive good estimates close to the horizon. The first step is to apply the identity (38) with the vectorfield multiplier N to Eq. (100): µ µ N ˆ N ˆ Jµ Y ψ n + Jµ Y ψ n H + K N Yˆ ψ τ H+ (0,τ ) R(0,τ )∩{r ≤r0 } µ =− K N Yˆ ψ + JµN Yˆ ψ n 0 R(0,τ )∩{r0 ≤r ≤r1 } 0 (105) + E N Yˆ ψ + E N Yˆ ψ , R(0,τ )∩{rhoz ≤r ≤r0 }

R(0,τ )∩{r >r0 }

where 2 2

− N − eYˆ Yˆ ψ Yˆ Yˆ (ψ) E N Yˆ ψ = −2e − Yˆ Yˆ (ψ) − 4r ˆ ˆ + N Y ψ Y T ψ − λ1 (∂r ψ) N Yˆ ψ − λ2 (∂t ψ) N Yˆ ψ α (106) − 2 N Yˆ ψ (Y (ψ)). l Let us start with the horizon term in (105). As previously (cf. (69)) this flux has a good sign except for the lowest order term when 0 < α < 49 .

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

191

This latter term is estimated as previously (cf. (70)) borrowing a bit from the good K N term: eγ M α ˆ 2 Yψ 2 H+ (0,τ ) 2r k0 l 1 τ N ˆ ≤ K Y ψ + B dt JµN (ψ) n µ ≤ K N Yˆ ψ + B Dτ, (107) 0 τ with D=

0

µ µ JµN (ψ) n 0 + JµN (K ψ) n 0 .

(108)

We turn to the E N -terms in (105). Note that N − eYˆ = K on the horizon. Hence in r ≤ r0 (where weights in r don’t matter) we have −

2 − N − eYˆ Yˆ ψ Yˆ Yˆ (ψ) ≤ B K + (1 − µ) Yˆ Yˆ ψ Yˆ Yˆ (ψ) 2 2 1ˆ ≤ Y K ψ + Yˆ Yˆ (ψ) (109)

by choosing r0 sufficiently close to the horizon to exploit the (1 − µ)-term as a smallness factor. We are going to choose small enough to borrow from the good first term in (106). Similarly, 2 2 2 4r ˆ ˆ N Y ψ Y T ψ ≤ Yˆ Yˆ (ψ) + B Yˆ K ψ + B|λ| · ∂φ Yˆ (ψ) . (110) Using several integrations by parts and the elliptic estimate (119) we derive the following estimate for the last term: 2 B|λ| ∂φ Yˆ (ψ) ≤ K N Yˆ ψ + B D (τ + 1) R(0,τ ) R(0,τ )∩{r ≤r0 } 1 1 1 µ µ µ + JµN Yˆ ψ )n H+ + JµN Yˆ ψ n + JµN Yˆ ψ n . (111) 8 H(0,τ ) 8 τ 8 0 Let us denote the three boundary-terms in the last line collectively by P1 . The other terms of E N involve lower order terms and are estimated via Cauchy’s inequality, putting a small weight on the Yˆ Yˆ -term so that one can borrow again from the good term. Hence finally R(0,τ )∩{r ≤r0 }

E N Yˆ ψ ≤

R(0,τ )∩{r ≤r0 }

K N Yˆ ψ

2 1 1 ˆ N Y K ψ + K (ψ) + B D (τ + 1) + P1 + R(0,τ )∩{r ≤r0 } ≤ K N Yˆ ψ + B D (τ + 1) + P1 .

R(0,τ )∩{r ≤r0 }

(112)

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G. Holzegel

The penultimate term in the second line deals in particular with the first order terms (which have the wrong sign if α > 0) arising in the K N terms. For r ≥ r0 we only have to be careful with the weights in r : E N Yˆ ψ

E N Yˆ ψ ≤ dt . (113)

−g t t R(0,τ )∩{r ≥r0 } τ

In the Ad S case g t t ∼ r12 , so in comparison with the asymptotically flat case (where this quantity approaches 1 at infinity) one loses a power of r . However, the decay of N E Yˆ ψ in r is easily seen to be strong enough to have

E N Yˆ ψ

E N Yˆ ψ ≤ dt

−g t t R(0,τ )∩{r ≥r0 } τ 0 τ µ JµN (ψ) n + JµN (K ψ) ≤ B Dτ, ≤ dt

τ

τ

0

(114)

using the elliptic estimate (98) above. For K N we have the analogous estimates to Lemma 4.2. In fact the lowest (first) order terms of the wrong sign can now be simply controlled τ by adding 0 dt τ JµN (ψ) n µ ≤ B Dτ to the right-hand side – a Hardy inequality is no longer necessary. Putting these estimates together the identity (105) turns into the estimate τ µ µ N ˆ Jµ Y ψ n + b dτ JµN Yˆ ψ n τ τ 0 µ ≤ B D (τ + 1) + 2 JµN Yˆ ψ n 0 . (115) 0

This is the analogue of (74) and hence µ N ˆ Jµ Y ψ n ≤ B D + τ

0

JµN

µ ˆ Y ψ n 0

(116)

follows as previously. We have finally obtained control over the rr derivative at the horizon. We now show how to control the second derivatives on the topological twospheres close to the horizon defined by constant (t , r ). For this we employ a second elliptic estimate. Write the wave equation as 1 1

∂θ (sin θ θ ∂θ ψ) + g φφ − 2λg t φ + λ2 g t t ∂φ2 ψ sin θ

= −g t t ∂t (K ψ) − 2g t r ∂r (K ψ) + g t t λ − 2g t φ ∂φ (K ψ) − grr ∂r ∂r ψ + ξ (r − rhoz ) ∂r ∂φ ψ + lower order terms,

(117)

where ξ is some bounded function. Note also the degenerating weight of grr on the horizon. From (117) we derive µ µ µ / 2 ψ|2 ≤ B JµN (ψ) n + JµN (K ψ) n + JµN Yˆ ψ n (118) |∇ Sr

Sr

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193

on a sphere of radius r . Close to the horizon we obtain µ / 2 ψ|2 ≤ B JµN (ψ) n + JµN (K ψ) + · JµN Yˆ ψ , (119) |∇ τ ∩{r ≤r0 }

τ ∩{r ≤r0 }

with arising from choosing r0 small enough to exploit the degenerating weights in (117). Together with the previous elliptic estimate finally 1 µ µ i j kl N N

h h (∇i ∇k ψ) ∇ j ∇l ψ + J (K ψ) n + J (ψ) n

−g t t µ µ µ JµN (ψ) n + JµN (K ψ) n + JµN Yˆ ψ n ≤B τ µ µ µ ≤B JµN (ψ) n + JµN (K ψ) n 0 + JµN Yˆ ψ n 0 . (120) 0

To write the estimate in a more geometric form we observe 1 µ ij

h ∇i ψ∇ j ψ ≤ B JµN (ψ) n

−g t t and, from (92),

1

−g t t

The factor of √ r2

ψ2 ≤ B

∞

S2

rhoz

∼ r arises because

1

−g t t

r 2 ψ 2 dr dω ≤ B √

(121)

µ

JµN (ψ) n .

(122)

h ∼ r for the hyperbolic metric (instead of

in the asymptotically flat case). In the other direction we have µ JµN Yˆ ψ n 1 µ µ N N i j kl ≤B h h (∇i ∇k ψ) ∇ j ∇l ψ , (123) J (K ψ) n + J (ψ) n +

−g t t

which is seen by direct computation taking care of the weights of r . Hence the inequality (120) becomes µ µ ψ J N (K ψ) n + J N (ψ) n + 2 Hw ( ) µ µ ≤ B ψ 2 J N (K ψ) n 0 + J N (ψ) n 0 (124) + Hw ( 0 )

with Hw2 denoting the n ψ

0

√ 1 -weighted −g t t

H⊥1

:=

≤ B ψ

Hw2 ( )

1

H 2 norm of . Finally, we define 2 r 2 ∇ · n ψ + (n ψ)2

−g t t µ µ J N (K ψ) n + J N (ψ) n +

and state the boundedness theorem for the massive wave equation on Kerr-AdS:

(125)

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G. Holzegel

Theorem 5.1. Fix a cosmological constant = − l32 , a mass M > 0 and some α < 49 . There exists an amax > 0, depending only on , M and α, such that the following statement is true for all a with |a| < amax . Let 0 = τ0 bea slice of constant t = τ0 in D = J + (I) ∩ J − (I) of the Kerr-anti 2 . If de Sitter spacetime M, g M,a, . Let ψ be a solution to (32) of class Cdec µ µ JµK (ψ) n 0 + JµK (K ψ) n 0 < ∞, (126) 0

0

then ψ

Hw2 ( )

+ n ψ

≤ C ψ˜

Hw2 ( 0 )

H⊥1 ( )

+

1 m=0

+ n ψ˜

H⊥1 ( 0 )

µ JµK K m ψ n +

1 m=0 0

µ JµK K m ψ n 0

(127)

holds on J + ( 0 ) ∩ D for a uniform constant C depending only on the parameters M, a, l and α. Here denotes any constant t -slice to the future of 0 . We remark that in contrast to the statement in the asymptotically flat case [7] we need the K boundary term in this estimate. This is because the weighted L 2 norm of the second time-derivative that one obtains from the current J K is stronger than what can be derived from the wave equation in combination with the boundedness of |ψ| Hw2 ( ) and |n ψ| H 1 ( ) alone. ⊥

r2

As mentioned previously, if α ≤ 0 then amax = hoz l since all that was needed in the proof is that the vectorfield K is timelike on the exterior (cf. [10]). In general, the restriction on amax depends on α and becomes tighter as α approaches the Breitenlohner Freedman bound, α → 49 . However, there is a uniform lower bound on amax (i.e. amax ≥ auni f or m > 0 for any α < 49 ) which can be worked out explicitly from the estimates of Sect. 5.1. The Sobolev embedding theorem for asymptotically hyperbolic space yields Corollary 5.3. We have the pointwise bound 1 |ψ| ≤ C ψ˜ 2 + n ψ˜ 1 + Hw ( 0 )

H⊥ ( 0 )

m=0 0

µ JµK K m ψ n 0

(128)

on J + ( 0 ) ∩ D. To prove the corollary we rely on the following general Sobolev embedding theorem (cf. Theorem 3.4 in [11]) Theorem 5.2. Let (N , h) be a smooth complete Riemannian 3-manifold with Ricci curvature bounded from below and positive injectivity radius and u ∈ H 2 (N ) a function on N . Then 2 2 sup |u| ≤ B |∇ j u|2 dv (h). (129) N

j=0

N

Remark. Our τ is only complete with respect to the asymptotically hyperbolic end, but it is straightforward to incorporate the boundary at r = rhoz > 0.

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6. Final Comments As mentioned in the abstract of the paper, the result does not make use of the separability properties of (2) with respect to the Kerr background. In fact it does not make use of the axisymmetry either! All that was needed was the causal Killing vectorfield K on the black hole exterior. In view of this fact, Theorem 5.1 can be stated in the following generalized setting: Fix the Killing vectorfield K = T + λ of a slowly rotating Kerr-AdS background as in Sect. 5. Perturb the metric such that it stays C 1 -close to the Kerr-AdS metric and such that K remains both Killing and null on the horizon.9 Then Theorem 5.1 remains true for such spacetimes. The main motivation for generalizations of this type are non-linear situations, in which the metric is not known explicitly a-priori but is itself dynamical. In view of this one should use as less quantitative assumptions on the metric as possible to obtain bounds on the fields. Compare [7] for a more detailed discussion. As a further generalization one may assume only an approximate causal Killing field, i.e. a vectorfield whose deformation tensor decays sufficiently fast in t. Treating the latter as a decaying error term in the estimates one can prove boundedness of (2) for all spacetimes approaching a spacetime that is C 1 -close in the sense above10 to a slowly rotating Kerr-AdS solution. The question whether ψ decays in time and if so, at what rate remains open. Acknowledgements. I would like to thank Mihalis Dafermos and Igor Rodnianski for stimulating discussions and useful comments. I am also grateful to an anonymous referee for his careful reading of the manuscript and many insightful remarks and suggestions.

A. Radial Decay In this Appendix we outline how – assuming the boundary conditions (33) – one can establish boundedness of appropriate higher weighted Sobolev-norms. The idea is the 3 , let T k (ψ), denote the application of k times following. Given a solution ψ of class Cdec the vectorfield T to ψ. We know that the T -energy associated with T k (ψ) is conserved for k = 0, 1, 2. If we now revisit the wave equation to do elliptic estimates, this will introduce natural weighted Sobolev norms whose r -weights depend on 0 < α < 49 . These optimized norms are expected to play a crucial role for the local existence theorem and are hence presented here. Recall from (97) that (for Kerr-AdS) one may write the wave equation in the form √ α 1 √ ∂i g i j g∂ j ψ + 2 ψ g l √ 1

= −g t t (∂t ∂t ψ) − 2g t i ∂t ∂i ψ − √ ∂r g t r g (∂t ψ). (130) g 2

The weighted of the right-hand side decays very fast in view of the boundedness L norm µ µ of JµK T k ψ n + JµK (ψ) n . In particular, denoting the right-hand side by f we have ∞ µ µ (131) dr dωr 4 f 2 < B JµK (T ψ) n + JµK (ψ) n . rhoz

S2

9 The C 1 regularity is necessary because the surface gravity, whose positivity was essential for the argument, is C 1 in the metric. 10 I.e. in particular admitting a timelike Killing field K on the black hole exterior which becomes null on the horizon.

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Let now χ = χ (r ) be a cut-off function which is equal to 1 for r > R for √ some fixed 2n R > rhoz and is zero close to the horizon. Set σ = n+3 for some n < 9 − 4α and multiply (130) by √ 1 α √ n ij g gχ ∂ ψ − χ σ ψ gr ∂ √ i j g l2 α √ n √ = χ f + (1 − σ )) 2 ψ gr + χ,r gr j ∂ j φ gr n . l Integrating over the slice , we observe (in view of (131)) that we can bound the right hand side as long as n ≤ 2 (provided we can borrow an of the term r 2+n ψ 2 dr dω from the left). The radial derivatives11 on the left-hand side can after several integrations by parts be estimated by ∞ dr dωr 6+n (∂r ∂r ψ)2 S2 R ∞ + dr dω (16 − 4 (5 + n) − α + ασ ) r 4+n (∂r ψ)2 S2 R ∞ (132) 1−σ 2+n 2 dr dωα −σ α + n (n + 3) r ψ 2 S2 R µ µ K K ≤B Jµ (T ψ) n + Jµ (ψ) n .

The boundary terms arising in the computations all vanish: on the left because of the √ cut-off function, and on the right in view of (33) (the restriction n < 9 − 4α is imposed in order to make these boundary terms vanish). Note also that the error-terms introduced by the derivatives of the cut-off function are well in the interior (bounded r ) and hence unproblematic. The coefficient of the zeroth order term in (132)√is positive if n(n+3) 2n n+3 = σ < 2α+n(n+3) , which an easy calculation reveals to be true for n < 9 − 4α. (In particular we have room to borrow the aforementioned of this term for the right-hand side.) The second term on the left of (132) is negative. How large can we allow n to be to still absorb this term by the first term using a Hardy inequality? As one easily checks, the Hardy condition (4 + 4n + α (1 − σ ))

4 (n + 5)2

0 for all ξ ∈ (0, ∞). Then the following hold for all N -chains for both the continuous-time process P t and its associated discrete-time process p n : (a) There is at most one invariant probability measure (which is therefore ergodic). (b) This measure has a density with respect to Lebesgue measure. The conditions above can be weakened as follows: There is a function φ(N ) = O( N12 ) such that (a) and (b) hold for the N -chain if L(ξ ) and R(ξ ) are strictly positive on open sets I L and I R , and there exist ξ ∈ I L ∪ I R and ξ ∈ I L ∩ I R such that ξ = φ(N )ξ . 1 . The form of our assumption An example of φ(N ) that is sufficient is φ(N ) = (18N )2 involving φ(N ) is quite possibly an artifact of our proof, but it is essential for L(ξ ) and R(ξ ) to have some spread. Our results imply, in particular, that the processes we consider have no singular invariant probability measures.

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We do not prove the existence of invariant measures in this paper. For existence, the two issues are: (1) tightness, which requires that one controls the dynamics of super-fast and super-slow particles. (2) Discontinuities of transition probabilities p X as a function of X due to same-site double collisions. Techniques for treating (1) in nonequilibrium situations are generally lacking, and (2) is likely to involve very technical arguments. We have elected to leave these problems for future work. Our next result gives an explicit formula for the equilibrium distribution in terms of bath distributions. In general, no such explicit expressions exist for nonequilibrium steady states, and our models appear to be no exception. Theorem 2. Let L(ξ ) = R(ξ ) = ρ(ξ ) be a probability density on (0, ∞) satisfying (i) 1 √ ρ(ξ )dξ < ∞ and (ii) ρ(ξ ) = O(ξ −κ ) for some κ > 3/2 as ξ → ∞. Then the ξ following measure is invariant for the (continuous-time) process defined in Section 1.1: N N 1 dξ1 · · · dξ N · ρ(ξi ) × dη0 · · · dη N · √ ρ(ηi ) × λ N +1 × ω N +1 , ηi i=1

where λ N +1 = Lebesgue measure on configurations in {(σ1 , · · · , σ N +1 )}.

i=0

[0, 1] N +1

and ω N +1 assigns equal weight to the

The following corollary suggests some examples to which our results apply: Corollary 1. Any finite chain with bath injections L(ξ ) = R(ξ ) given by either (a) an exponential distribution or (b) a Gaussian truncated at 0 and normalized has a unique (hence ergodic) equilibrium distribution, and it is given by the expression in Theorem 2. 3. Density and Ergodicity: Outline of Proof For definiteness, we will work with the discrete-time Markov chain on M+ . By Lemma 1.1, the assertions in Theorem 1 for p n imply the corresponding assertions for P t . Proving uniqueness of invariant measures or ergodicity requires, roughly speaking, that we be able to steer a trajectory from one location of the phase space to another. If the transition probabilities have densities on open sets, then one needs only to do so in an approximate way. Our model, unfortunately, has highly degenerate transition probabilities. One must, therefore, tackle hand in hand the problems of (i) acquisition of densities for p nX and (ii) steering of trajectories. The purpose of this section is to outline how we plan to do this. Section 3.1 introduces definitions and ideas that will be used. In Section 3.2, we formulate three propositions to which the proof of Theorem 1 will be reduced. 3.1. Basic ingredients of the proof. (A) Acquiring densities. Given a finite Borel measure ν on M+ , we let ν = ν⊥ + νac denote the decomposition of ν into a singular and an absolutely continuous part with respect to Lebesgue measure, and say ν has an absolutely continuous component when νac = 0. We say X ∈ M+ eventually acquires a density if for some n > 0, p nX has an absolutely continuous component, i.e. ( p nX )ac = 0.

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Lemma 3.1. If every X ∈ M+ acquires a density eventually, then every invariant probability measure of p n has a density. Proof. For any measure ν, if ν is absolutely continuous with respect to Lebesgue, then so is νn := p nX dν(X ). This is because under the dynamics, Lebesgue measure is carried to a measure equivalent to Lebesgue, followed, at certain steps, by a diffusion in one or two directions corresponding to bath injections. Thus (νn )ac (M+ ) ≥ νac (M+ ) for all n ≥ 1. The hypothesis of this lemma implies that this inequality is strict for some n unless ν⊥ = 0. Now let µ be an invariant probability measure for p n . Since µn = µ, it follows that µ⊥ = 0.

Recall that I L = {L > 0} and I R = {R > 0}. We call a finite or infinite sequence of points X 0 , X 1 , . . . in M+ a sample path if such a sequence can, in principle, occur. In particular, if X n−1 is followed by a collision with the right bath, then the ξ N -coordinate of X n must lie in I R , and similarly with the left bath. Let X = (X 0 , X 1 , . . . , X n ) be a sample path obtained by injecting into the system the energy sequence ε = (ε1 , . . . , εm ) in the order shown. (One does not specify whether the injection is from the left or the right; it is forced by the sequence.) We write

X 0 (ε) = X n . Suppose X has no multiple collisions. It is easy to see that there is a small neighborhood E of ε in Rm such that for all ε ∈ E, (i) ε is a feasible sequence of injections, i.e., if the sequence dictates that εj be injected from the left (right) bath, then εj ∈ I L (resp. I R ), and (ii) the sample path X = (X 0 , X 1 , . . . , X n ) produced by injecting ε has the exact same sequence of collisions as X . In particular, it also has no multiple collisions. We may therefore extend = X 0 to a mapping from E to M+ with (ε ) = X n . As such, is clearly continuous. Lemma 3.2. : E → M+ is continuously differentiable. In the proof below, it will be useful to adopt the viewpoint expressed in Section 1.3, i.e. to track the movements of the injected energies through time. Notice that for a sample path with no multiple collisions, there is no ambiguity whatsoever about the trajectory of an injected energy. ∂ xi i ∂ξi Proof. We verify that ∂η δε j , δε j and δε j exist and are continuous on E. For ηi and ξi , it is easy: either ε j is carried by the particle in question, or it is not. If it is, then the partial derivative is = 1; if not, then it is = 0. For xi , consider first the case where ηi carries the injection εk , k = j. We let p be the total number of times ε j crosses the interval [i, i + 1] (in either direction) before X n . ∂ xi = 0. Suppose not. Then perturbing (only) ε j to ε j + δε, the time If p = 0, then δε j

gained per crossing (with a sign) will be ∂ xi √ = lim p εk · δε→0 δε j

√1 εj

√1 εj

−√

−√

1 . ε j +δε

1 ε j +δε

δε

The case where ε j is carried by ηi is left to the reader.

Consequently

=

p 21 − 32 ε ε . 2 k j

In the setting above, if D (ε) is onto as a linear map, then by the implicit function theorem, there is an open set E ⊂ E with ε ∈ E such that carries Lebesgue measure on E to a measure in M+ with a strictly positive density on a neighborhood of (ε). This implies in particular ( p nX 0 )ac > 0. We summarize as follows:

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Definition 3.1. We say X ∈ M+ has full rank2 if there is a sequence of injections ε = (ε1 , · · · , εm ) leading to a sample path X = (X 0 , X 1 , . . . , X n ) with X 0 = X such that (i) X has no multiple collisions, and (ii) D X 0 (ε) is onto. Corollary 3.1. If X ∈ M+ has full rank and X is as above, then p nX has strictly positive densities on an open set containing X n . Having full rank is obviously an open condition, meaning if X has full rank, then so does Y for all Y sufficiently close to X . (B) Ergodic components. One way to force two points to be in the same ergodic component (in the sense to be made precise) is to show that they have “overlapping futures”. This motivates the following relation: Definition 3.2. For X, Y ∈ M+ , we write X ∼ Y if there exist a positive Lebesgue n measure set A = A(X, Y ) ⊂ M+ and m, n ∈ Z+ such that ( p m X )ac |A and ( pY )ac |A have strictly positive densities. Here is how this condition will be used: Suppose µ and ν are ergodic measures, and there is a positive µ-measure set Aµ and a positive ν-measure set Aν such that X ∼ Y for all X ∈ Aµ and Y ∈ Aν . Then the ergodic theorem tells us that µ = ν since they have the same ergodic averages along positive measure sets of sample paths. As an immediate corollary of the ideas in Part (A), we obtain Corollary 3.2. If X ∈ M+ has full rank, then there is a neighborhood N of X such that Y ∼ Z for all Y, Z ∈ N . (C) Constant energy configurations. For fixed e ∈ (0, ∞), let Qe = {X ∈ M+ : ηi = ξi = e for all i}. If we start from X ∈ Qe and inject only energies having value e, then the resulting sample path(s) will remain in the set Qe . (Obviously, this is feasible only if e ∈ I L ∩ I R .) We call these constant-energy sample paths, and say X ∈ Qe has no multiple collisions if that is true of its constant-energy sample path. Needless to say, constant-energy sample paths occur with probability zero. However, they have very simple dynamics, and perturbations are relatively easy to control. Our plan is to exploit these facts by driving all sample paths to some Qe and to work from there. 3.2. Intermediate propositions. We claim that Theorem 1 follows readily from the following two propositions: Proposition 3.1. Every X ∈ M+ eventually acquires a density. Proposition 3.2. X ∼ Y for a.e. X, Y ∈ M+ with respect to Lebesgue measure. We remark that to rule out the presence of singular invariant measures, we need Proposition 3.1 to hold for every X , not just almost every X . 2 This property has the flavor of Hörmander’s condition for hypoellipticity for SDEs – in a setting that is largely deterministic and has discontinuities.

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Proof of Theorem 1 assuming Propositions 3.1 and 3.2. Proposition 3.1 and Lemma 3.1 together imply that every invariant probability measure has a density, proving part (b). To prove part (a), let µ and ν be ergodic measures. Since they have densities, Proposition 3.2 implies, as noted in Paragraph (B), that µ = ν.

Next we identify three (more concrete) conditions that will imply Propositions 3.1 and 3.2. The following shorthand is convenient: For X, Y ∈ M+ , we write X ⇒ Y if given any neighborhood N of Y , there exists n such that p nX (N ) > 0, X Y if given any neighborhood N of Y , there is a neighborhood N of X such that X ⇒ Y for all X ∈ N . Proposition 3.3. Given X ∈ M+ and e ∈ I L ∩ I R , there exists Z = Z (X ) ∈ Qe such that X ⇒ Z . Proposition 3.4. Z Z for all Z , Z ∈ Qe , any e ∈ I L ∩ I R . Proposition 3.5. Every Z ∈ Qe with no multiple collisions has full rank. Proof of Propositions 3.1–3.2 assuming Propositions 3.3–3.5. To prove Proposition 3.1, we concatenate Propositions 3.3 and 3.4 to show that for every X ∈ M+ , X ⇒ Z for some Z = Z (X ) ∈ Qe with no multiple collisions. The assertion is proved if there is a neighborhood N Z of Z such that every Z ∈ N Z eventually acquires a density. This follows from Proposition 3.5, Corollary 3.1 and the fact that full rank is an open condition. To prove Proposition 3.2, it is necessary to produce a common Z = Z (X, Y ) ∈ Qe with no multiple collisions such that X, Y ⇒ Z . To this end, we first apply Proposition 3.3 to produce Z (X ) ∈ Qe with X ⇒ Z (X ) and Z (Y ) ∈ Qe with Y ⇒ Z (Y ). We then fix an (arbitrary) Z ∈ Qe with no multiple collisions. Since Z (X ), Z (Y ) Z (Proposition 3.4), we have X, Y ⇒ Z as before. Proposition 3.5 and Corollary 3.2 then give the desired result.

Proving Theorem 1 has thus been reduced to proving Propositions 3.3–3.5. 4. Proofs of Propositions 3.3–3.5 In Sections 4.1–4.2, we give an algorithm for driving sample paths from given initial conditions to constant-energy surfaces. This can be viewed as changing the energies in a configuration. In Sections 4.3–4.4, we focus on changing the relative positions of the moving particles. 4.1. Sample paths from X to Qe (“typical” initial conditions). The hypothesis of Theorem 1 guarantees the existence of some ξ ∈ I L ∩ I R . We may assume, without loss of generality, that ξ = 1. Given X ∈ M+ , we seek sample paths that lead to the constant energy surface Q1 . Consider first the following specific question: By injecting only particles with energy 1, will we eventually “flush out” all the energies in X , replacing them with new particles having energy 1? To study this question, we propose to suppress some information, to focus on the following evolution of arrays of energies: To each X ∈ M+ , we associate the array E(X ) = (η0 , ξ1 , η1 , ξ2 , . . . , ξ N , η N ), where ηi and ξi are the energies of X and they are arranged in the order shown. Then corresponding to each sample path X = (X 0 , X 1 , . . .)

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is the sequence of moves E(X 0 ) → E(X 1 ) → E(X 2 ) → · · ·. For example, collision between site i and the particle on its right corresponds to swapping the (2i + 1)st entry in the array with the (2i)th ; the rightmost particle exiting the system and an energy of value ξ R entering corresponds to replacing the (2N + 1)st entry by ξ R , and so on. For as long as there are no same-site double collisions, we can trace the movements of energies in (E(X 0 ), E(X 1 ), · · · ) as discussed in Section 1.3. We define the exit time of the j th entry in E(X 0 ) to be the number of moves before this energy exits the system, and define T (X ) to be the last exit time of all the elements in E(X 0 ). A priori, T (X ) ≤ ∞. Lemma 4.1. Given X 0 ∈ M+ , suppose that injecting all 1s gives rise to a sample path X with no same-site double collisions. Then T (X ) < ∞. At each step, we will refer to the original energies of X 0 that remain as the “old energies”, and the ones that are injected as “new energies”. In particular, all new energies have value 1 (some old energies may also have value 1). The lemma asserts that in finite time, the number of old energies remaining will decrease to 0. Proof. For each n, we let E(X n ) = (η0 , ξ1 , η1 , ξ2 , · · · , ξ N , η N ), and say X n is in state (·, i) if the leftmost old energy is ξi , (+, i) if the leftmost old energy is ηi and σi = +, (−, i) if the leftmost old energy is ηi and σi = −. The terminology above is not intended to suggest that we are working with a reduced system. No such system is defined; nevertheless it makes sense to discuss which transitions among these states are permissible in the underlying Markov dynamics. Notice first that independent of the state of a system, it will change eventually. This is because all energies are nonzero, so a collision involving the leftmost old energy is guaranteed to occur at some point. We list below all the transitions between states that are feasible, skipping over (many) steps in the Markov chain that do not involve the leftmost old energy: (1) Suppose the system is in state (−, i). If i = 0, then the only possible transition is (−, i) → (·, i). If i = 0, then the leftmost old energy exits the system, and the new state is determined by the next leftmost old energy (if one remains). (2) Suppose the system is in state (+, i). If i = N , then T (X ) is reached as this last remaining old energy exits the system. If not, we claim the only two possibilities are (+, i) → (·, i + 1) and (+, i) → (−, i). The first case corresponds to the energy originally at site i + 1 being new, the second case old. (3) Finally, consider the case where the system is in state (·, i). If the next collision is with the particle from the left, then (·, i) → (−, i − 1). If it is with the particle from the right, then there are two possibilities corresponding to the approaching energy being new or old, namely (·, i) → (+, i) or (·, i) → (·, i). Notice that the system cannot remain in state (·, i) forever, since the particle from the left will arrive sooner or later, causing the state to change. For this reason, let us agree not to count (·, i) → (·, i) as a transition. To summarize, the only possible transitions are (−, i) → (·, i), (·, i) → (−, i − 1), (+, i), (+, i) → (·, i + 1), (−, i) except where the transition leads to an old energy exiting the system. When that happens, either there is no old energy left, or the system can start again in any state. The following observation is crucial:

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Sublemma 4.1. The transition sequence (−, i) → (·, i) → (+, i) is forbidden. We first complete the proof of Lemma 4.1 assuming the result in this sublemma: Observe first that if one starts from (+, i), either the evolution is (+, i) → (·, i + 1) → (+, i + 1) → (·, i + 2) → · · · leading to an exit at the right, or a state of the form (−, j) is reached; a similar assertion holds if one starts from (·, i). On the other hand, starting from (−, i), the only possible sequence permitted by the sublemma is (−, i) → (·, i) → (−, i − 1) → (·, i − 1) → · · · leading to an exit at the left. This proves that in finite time (meaning a finite number of steps with respect to the Markov chain p n ), the number of old energies remaining will decrease by one.

Proof of Sublemma. Suppose the transition (−, i) → (·, i) takes place at t = 0. Since all the energies to the left of site i are new and therefore have speed one, a particle from the left is guaranteed to reach site i at time t = t0 < 2, since the previous collision with the site from the left took place strictly before t = 0 (no same-site double collisions). We will argue that when this particle arrives, it will find an old energy at site i, that in fact the state of the system has not changed between t = 0 and t0 . Thus the next transition has to be (·, i) → (−, i − 1). Here are the events that may transpire between t = 0 and t0 on the segment [i, i + 1]: Observe that the transition at t = 0 is necessarily between an old and a new energy (otherwise the state at t = 0− could not be (−, i)). This will result in a new energy leaving site i for site i + 1 at t = 0+ , and arriving at t = 1. The only way (·, i) → (+, i) can happen is for a new energy to move leftward on [i, i + 1], and to arrive at site i before time t0 . This cannot happen, since new energies travel at unit speed. (Notice that the old energy at site i may change between t = 1 and t0 , but the state of the system does not.)

Partial proof of Proposition 3.3. We prove the result for X = X 0 under the additional assumption that injecting 1s gives rise to a sample path with no same-site double collisions. By Lemma 4.1, there is a sample path X = (X 0 , . . . , X n ) with X n ∈ Q1 . Let N be a neighborhood of X n . If X has no multiple collisions, then all nearby sample paths will end in N as discussed in Section 3.1. If X has multiple (but not same-site) collisions,3 then injecting a slightly perturbed energy sequence may – is likely to, in fact – desynchronize the simultaneous jumps. If, for example, two collisions at sites i and i occur simultaneously at step j, then for perturbed injected energies, X j may be replaced by X j and X j corresponding to two collisions that happen in quick succession. This aside, the situation is similar to that with no multiple collisions, and we still have p kX (N ) > 0 but possibly for some k > n.

4.2. Sample paths from X to Qe (“exceptional” cases). As noted in Section 3.2, to rule out the existence of singular invariant measures, it is necessary to show that every X ∈ M+ eventually acquires a density. Our strategy is to inject energy 1s into the system and argue that this produces a sample path X that leads to a point Z ∈ Q1 . To prove X ⇒ Z , however, requires more than that: it requires that a positive measure set of sample paths starting from X follow X . This is not a problem if X has no same-site double collisions, for in the absence of such collisions, the dynamics are essentially continuous (as explained above). Yet it is unavoidable that for some X ∈ M+ , injecting 1s will lead 3 These points are discontinuities only for the (discrete-time) Markov chain, affecting the number of steps. They are not discontinuities at all for the continuous-time jump process.

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to same-site double collisions. It is an exceptional situation, but one that we must deal with if we are to follow the same route of proof. We now address the potential problems. Recall that at each same-site double collision, there are two ways to continue the sample path. Partitioning nearby sample paths according to itineraries. Let X = (X 0 , X 1 , . . . , X n ) be a sample path obtained by injecting a sequence of m energies all of which are 1s and making a specific choice at each same-site double collision. As in Section 3.1(A), we consider energies ε = (ε1 , . . . , εm ) near 1 = (1, . . . , 1), and ask the following question: Which injection sequence ε will give rise to a sample path X (ε) that has the same collision sequence as X ? To focus on the issues at hand, we will ignore multiple collisions that do not involve same-site double collisions, for they are harmless as explained earlier. The computation in Lemma 3.2 motivates the following coordinate change, which is not essential but simplifies the notation: Let ψ(ε) = √1ε − 1, and let (ε1 , . . . , εm ) = (ψ(ε1 ), . . . , ψ(εm )). Then maps a neighborhood of 1 diffeomorphically onto a neighborhood of 0 = (0, . . . , 0). We assume that X (ε) has the same itinerary as X through step i − 1, and that at step i, X has a double collision at site j. To determine whether the left or right particle will arrive first at site j for X (ε) , we use X as the point of reference, and let t j (ε) and t j−1 (ε) be the times gained (with a sign) by the particles approaching site j from the right and left respectively. Then t j (ε) = p1 ψ(ε1 ) + · · · + pm ψ(εm ) and t j−1 (ε) = q1 ψ(ε1 ) + · · · + qm ψ(εm ), where pk and qk are the numbers of times – up to and including the approach to site j – the injected energy εk has passed through the intervals [ j, j + 1] and [ j − 1, j] respectively. Thus for X (ε) , the right particle arrives first if and only if t j (ε) > t j−1 (ε). Thus given X = (X 0 , X 1 , . . . , X n ) as above, there is a decreasing sequence of subsets Rm = V0 ⊃ V1 ⊃ V2 ⊃ · · · ⊃ Vn defined as follows: Vi = Vi−1 except where a same-site double collision occurs; when a same-site double collision occurs, say at site j, we let Vi = Vi−1 ∩ H , where H = {ε ∈ Rm : t j (ε) ≥ t j−1 (ε)} or {t j (ε) ≤ t j−1 (ε)} depending on whether we have chosen to let the right or left particle arrive first in X . If there are multiple same-site double collisions, then we intersect with the half-spaces corresponding to all of them. The sequence of Vi so obtained will have the property that all ε close enough to 1 with (ε) ∈ Vn give sample paths that have the same collision sequence as X . In particular, if Vn is nontrivial, meaning it has interior, then the set of ε for which X (ε) shadows X will have positive measure. That is to say, X 0 ⇒ X n . In general, there is no guarantee that Vn is nontrivial. Supposing Vn for a sample path is nontrivial, we say the choices made at step n + 1 are viable if they lead to a nontrivial Vn+1 . Observe that inductively, a viable choice can be made at each step, since one of the two half-spaces must intersect nontrivially Vn from the previous step. We first treat a special situation involving the same-site collision of three energies all of which are 1s. The setting is as above. Sublemma 4.2. Suppose Vn is nontrivial for a sample path X , and at step n + 1, the energy εk , which is = 1, is involved in a same-site double collision with two other energies both of which are 1s. Then the choice to have εk arrive first is viable if εk has the following history:

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(i) its movement up to the collision above is monotonically from left to right; (ii) all of its previous same-site double collisions are with two other energy 1s. Proof. By (i), we have that in the collision at step n + 1, pk = 0 and qk = 1. Choosing to have εk arrive first corresponds to choosing the half-space {(εk ) ≥ G(ε)}, where G(ε) is a linear combination of (εi ), i = k. By (ii), all the other half-spaces in the definition of Vn either do not involve εk or are of the same type as the one above.

Completing the proof of Proposition 3.3. Given X 0 ∈ M+ , we now inject 1s and make a choice at every same-site double collision with the aim of obtaining a sample path X 0 , X 1 , . . . , X n with X n ∈ Q1 and X 0 ⇒ X n . Our plan is to follow the scheme in Section 4.1 to reach Q1 , and to make viable choices along the way. The proof of Lemma 4.1 is based on the following observation: the leftmost old energy either moves right monotonically until it exits, or it turns around and starts to go left, and by Sublemma 4.1, once that happens it must move left monotonically until it exits. Revisiting the arguments, we see that with the exception of Sublemma 4.1, all statements in the proof of Lemma 4.1 hold for any sample path, with whatever choices are made at double collisions. Moreover, the only way Sublemma 4.1 can fail is that at t = 0, a double collision occurs at site i, we choose to have the left particle arrive first, and this is followed by another double collision at t = 2 at which time we choose to have the right particle arrive first. It is only through these “two bad decisions” that the leftmost old energy can turn around and head right again. The choices at same-site double collisions are arbitrary provided the following two conditions are met: (i) The choice must be viable. (ii) In the setting of Sublemma 4.2, we choose to have εk arrive first. Following these rules, it suffices to produce X n ∈ Q1 for some n; the nontriviality of Vn follows from (i). Suppose, to derive a contradiction, that no X n ever reaches Q1 , i.e., a set of old energies is trapped in the chain forever. Let i 0 be the leftmost site that it visits infinitely often. That means from some step n 0 on, the leftmost energy never ventures to the left of site i 0 , but it returns to site i 0 infinitely many times, each time repeating the “two bad decisions” scenario above. The situation to the left of site i 0 is as follows: From time n 0 on, all the energies strictly to the left of this site have speed one. Thus by rule (ii), after a new energy enters from the left bath, it will march monotonically to the right until at least site i 0 , whether or not it is involved in any same-site double collisions. To complete the proof, consider a moment after time n 0 when the leftmost old energy arrives at site i 0 . From the discussion above we know it has to be involved in a samesite double collision, and that the energy approaching from the left is new. By (ii), we choose to have the new energy arrive first, making the first “bad decision”. But then in the collision at the same site 2 units of time later, again we choose to have the energy from the left arrive first, causing the old energy to move to site i 0 − 1 and contradicting the definition of site i 0 .

4.3. Moving about on constant energy surfaces. This subsection focuses on getting from Z ∈ Qe to Z ∈ Qe for e ∈ I L ∩ I R . In general, this cannot be accomplished by considering sample paths that lie on Qe alone, since such sample paths are periodic as we will see momentarily. Instead we will make controlled excursions from Qe aimed at returning to specific target points. As before, we assume 1 ∈ I L ∩ I R and work with Q1 .

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Circular tracks between sites. It is often useful to represent the pair (xi , σi ) by a single coordinate z i and to view the particle as making laps in a circle of length 2. More precisely, if we represent this circle as [0, 2]/ ∼ with end points identified, then z i = xi for σi = + and z i = 2 − xi when σi = −. A continuous-time sample path Z (t), t ≥ 0, for the process described in Section 1 then gives rise to a curve Z ∗ (t) = (z 0 (t), z 1 (t), . . . , z N (t)) ∈ T N +1 . (We write T N +1 even though each factor has length 2.) For each i, z i (t) goes around the circle in uniform motion, changing speed only at z i = 0 and 1. Accordingly, we denote by ∂(T N +1 ) the set of points such that z i = 0 or z i = 1 for (at least) one i. Asterisks will be used to signify the use of circular-track notation: A discrete-time sample path Z = (Z 0 , Z 1 , · · · ) in the notation of previous sections corresponds to Z ∗ = (Z 0∗ , Z 1∗ , · · · ) with Z i∗ ∈ ∂(T N +1 ). Returns of Z to M+0 = M+ ∩ {x0 = 0} translates into returns of Z ∗ to ∂(T N +1 ) ∩ {z 0 = 0}. We now identify two features of constant energy configurations that make them easy to work with, beginning with the simplest case: Lemma 4.2. Suppose we start from Z ∈ Q1 with no multiple collisions. (a) By injecting all 1s, the sample path returns to Z in 2N + 2 steps. (b) If all the injected energies are 1s except possibly for one, which we call e, and if we assume the resulting sample path has no same-site double collisions, then the energy e moves monotonically through the system until it exits, after which the system rejoins the periodic sample path in (a). Proof. (a) This is especially easy to see in continuous time and in circular-track notation: each z i (t) is periodic with period 2. Thus the continuous-time sample path Z (t) is periodic with period 2, and in every two units of time, there are exactly 2N +2 collisions. (b) Suppose the energy e has just arrived at site j from the left. Then the next collision at site j is with the particle from the right, because all cycles take 2 units of time to complete and the right one has a (strict) headstart. This proves the monotonic movement of the energy e. After it leaves the system, the relative positions between z i and z j remain unchanged for all i, j, since the passage of the energy e through the chain leads to identical time gains for z i and z j .

Another nice property of constant energy sample paths is that the process is continuous along these paths. We make precise this idea: Definition 4.1. Given a sample path X = (X 0 , X 1 , . . . , X n ), we say the process is continuous at X if the following holds: Let ε = (ε1 , . . . , εm ) be the sequence in injections. Then given any neighborhood N of X n , there are neighborhoods N0 of X 0 and U of ε such that for all X 0 ∈ N0 and ε ∈ U, all the sample paths generated follow X into N . (We do not require that it reaches N in exactly n steps; it is likely to take more than n steps if X has multiple collisions.) Lemma 4.3. Let X = (X 0 , . . . , X n ) be a sample path. (a) If X either has no same-site double collisions or every such collision involves three identical energies, then the process is continuous at X . (b) If the process is continuous at X , then X 0 X n .

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The ideas behind part (a) are the same as those discussed in Sections 4.1 and 4.2 and will not be repeated here. Part (b) is immediate, for “” requires less than continuity. The following lemma is the analog of Lemma 4.2(b) without the assumption of “no multiple collisions”: Lemma 4.4. Given any Z ∈ Q1 , let all the energies injected be 1s except possibly for one which has value e. Then (i) there is a sample path in which the energy e moves through the system monotonically; (ii) except for a discrete set of values e, it will do so without being involved in any same-site double collisions. Proof. The monotonic motion of e is clear if it is not involved in any same-site double collisions. If it is, choose to have it arrive first every time. This proves (i). To prove (ii), let J be a finite interval of possible energies to be injected at t = 0, and suppose e is on its way from site 0 to site 1. In circular-tracks notation, z 1 is periodic with period 2, and except for a finite subset of e ∈ J , z 0 = 1 at exactly the same time that z 1 = 0. We avoid this “bad set” of e, ensuring that the special energy will not be in a double collision at site 1. Now this energy leaves site 1 exactly when z 1 = 0, and at such a moment, the configuration to the right of site 1 is identical (independent of the value of e in use). The same argument as before says that to avoid a double collision at site 2, another finite subset of J may have to be removed.

Injecting a single energy different than 1 will not produce the desired positional variations. We now show that injecting a second energy – both appropriately chosen and injected at appropriate times – will do the trick. The hypothesis of Theorem 1 guarantees, after scaling to put 1 ∈ I L ∩ I R , that there exists e ∈ I L ∪ I R with e = o(1/N 2 ). (Lemma 4.5 below is the only part of the proof in which this hypothesis is used.) Let us assume for definiteness that e ∈ I L , and consider initial conditions Z ∈ M+0 = M+ ∩ {x0 = 0}. Lemma 4.5. There exists a0 > 0 for which the following hold: Given a with |a| < a0 , Z 0 ∈ Q1 ∩ M+0 and k ∈ {1, . . . , N }, there is a sample path Z = (Z 0 , . . . , Z n ) with Z n ∈ Q1 ∩ M+0 such that if Z 0∗ = (0, z 1 , . . . , z N ), then Z n∗ = (0, z 1 , . . . , z k−1 , z k − a, z k+1 − a, . . . , z N − a). Moreover, the sample path Z may be chosen so that it has no same-site double collisions except where all the energies involved are 1s. Proof. First we focus on constructing a sample path that leads to the desired Z n without attempting to avoid double collisions. In the construction to follow, all the injections except for two will be 1s. At t = 0, we inject energy e = o(1/N 2 ) from the left bath, letting it move monotonically from left to right. When it is about halfway between sites k and k + 1, we inject from the left an energy e = e(a) ≈ 1 (the relation between e and a will be clarified later). The energy e also moves monotonically from left to right until it reaches the k th site, which takes O(N ) units of time. Since the slow particle takes longer than O(N ) units of time to reach site k + 1, the e-energy waiting at site k is met by a particle from the left. So it turns around and moves monotonically left, eventually exiting at the left end. As for the slow energy, it eventually reaches site k + 1, and continues its way monotonically to the right until it exits the chain. Let Z n be the first return to M+0 after the exit of e (the e-energy exits much earlier). By definition, Z n ∈ Q1 . It remains to investigate its z i -coordinates. Reasoning as in

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the proof of Lemma 4.2, we see that the e -energy does not cause a shift in the relative positions of the z i because its effects on all of the z i are identical. The e-energy, on the other hand, has made one full lap (first half on its way to site k and second half on its way back) on the segments [ j, j + 1] for j = 0, 1, . . . , k − 1 but not for j ≥ k. Let 1 a = 2(1 − e− 2 ) be the (signed) time gain per lap for a particle with energy e over a particle with energy 1. Then the position of z 0 relative to z j for j < k is maintained while relative to z j for j ≥ k, z 0 is ahead by distance a, leading to the coordinates of Z n as claimed. We now see that the bound imposed on a is that e(a) must be in I L ∩ I R . This completes the proof of the lemma except for the claim regarding double collisions. From Lemma 4.4(b), we know that for e outside of a discrete set of values, the particle carrying this special energy will not be involved in any same-site double collisions. It remains to arrange for the energy e to avoid such collisions. Here the situation is different: since a is given, e is fixed, but notice that increasing e is tantamount to delaying the injection of the energy e. Thus an argument similar to that in Lemma 4.4(b) can be used to show that by avoiding a further discrete set of values for e (depending on Z 0 , e and k), the energy e will not be involved in same-site double collisions. This completes the proof.

Remark. The following two facts make this result very useful: (a) The dynamical events described in Lemma 4.5 require no pre-conditions on Z 0 other than Z 0 ∈ Q1 ∩ M+0 . (b) The range of admissible a is independent of Z 0 . Suppose a > 0, and for some j > k, the j th moving particle in Z 0 is such that x j < a and σ j = +. Then it will follow that σ j = − in Z n , since it is distance a behind its original position. This is very natural when one thinks about it in terms of circular tracks, but it is a jump in the phase space topology in Section 1.1.

4.4. Proofs of Propositions 3.4 and 3.5. Proof of Proposition 3.4. Given Z , Z ∈ Q1 , we will construct a sample path Z = (Z 0 , Z 1 , · · · , Z n ) with Z 0 = Z , Z n = Z and Z 0 Z n . Since all constant energy sample paths pass through M+0 , we may assume Z , Z ∈ M+0 . Let Z ∗ = (0, z 1 , . . . , z N ) and (Z )∗ = (0, z 1 , . . . , z N ). Our plan is to apply Lemma 4.5 as many times as needed to nudge each z i toward z i one i at a time beginning with i = 1. Write z 1 − z 1 = ja for some j ∈ Z+ and a small enough for Lemma 4.5. Applying Lemma 4.5 j times with k = 1, we produce Z n 1 ∈ Q1 ∩ M+0 with Z n∗1 = (0, z 1 , z 2 + (z 1 − z 1 ), . . . , z n + (z 1 − z 1 )). We then apply Lemma 4.5 to Z n 1 with k = 2 repeatedly to change it to Z n 2 with Z n∗2 = (0, z 1 , z 2 , z 3 + (z 2 − z 2 − (z 1 − z 1 )), . . . , z n + (z 2 − z 2 − (z 1 − z 1 ))), and so on until Z n = Z n N is reached. Notice that the sample path Z so obtained has the property that it has no same-site double collisions other than those that involve three identical energies (all of which 1s). Lemma 4.3 says that the dynamics are continuous at such sample paths. Hence Z Z .

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Proof of Proposition 3.5. Given Z ∈ Q1 with no multiple collisions, our aim here is to produce a sample path Z 0 = Z , Z 1 , . . . , Z n , also free of multiple collisions, by injecting a sequence of energies ε = (ε1 , . . . , εm ) so that in the notation of Section 3.1, the map

has the property that D (ε) is onto. We first give the algorithm, with explanations to follow: (1) We inject 1s until n 0 , the first time Z n 0 ∈ M+0 . (2) At step n 0 , we inject into the system an energy e = o(1/N 2 ) chosen to avoid multiple collisions, followed by all 1s, until the first time the sample path returns to Q1 ∩ M+0 ; call this step n 1 . (3) By Lemma 4.1, T (Z n 1 ) < ∞. Let n be the smallest integer ≥ n 1 + T (Z n 1 ) for which Z n ∈ M+0 . We now prove that D (ε) has full rank: Let δηi denote an infinitesimal displacement in the ηi variable at Z n . To see that δηi is in the range of D (ε), we trace this energy backwards (in the sense of Section 4.1) to locate its point of origin. (3) above ensures that it was injected at step j for some n 1 < j ≤ n. Varying ε j , therefore, leads directly to variations in ηi . The same argument applies to δξi . Next we consider displacements in xi . Assume for definiteness that e enters the system from the left. We fix k ∈ {1, . . . , N }, and let ε j , j = j (k), be one of the energies injected from the left when e is about halfway between sites k and k + 1. By Lemma 4.5, varying ε j leads to a variation of the form δk = δxk + . . . + δx N in Z n 1 . By Lemma 4.2, this displacement is retained between Z n 1 and Z n since only 1s are injected. The vectors {δk , k = 1, 2, . . . , N } span the subspace corresponding to positional variations. The proof of Proposition 3.5 is complete.

5. Equilibrium Distributions 5.1. Systems with a single site. This subsection treats exclusively the case N = 1. As we will see, nearly all of the ideas in a complete proof of Theorem 2 (for general N ) show up already in this very simple situation. Following the notation in Section 1.1, we consider phase variables X = (η0 , ξ, η1 ; x0 , x1 ; σ0 , σ1 ) ∈ (0, ∞)3 × [0, 1]2 × {−, +}2 ; the phase space has 4 components corresponding to the 4 elements of {−, +}2 . Define ∞ 1 ρ(η) ρ(η) , where Z = τ (η) = √ √ dη . Z η η 0 Theorem 2 asserts the invariance of the probability measure µ = τ (η0 )ρ(ξ )τ (η1 )I[0,1]2 (x0 , x1 ) dη0 dξ dη1 d x0 d x1 × w4 , where I(·) is the indicator function and w4 gives equal weight to the 4 points. Let µt = PXt dµ(X ) be the distribution at time t with initial distribution µ. To prove Theorem 2, we need to show µt = µ for all t > 0, equivalently for all small enough t > 0. We say a system undergoes a simple change in configuration between times 0 and t if during this period at most one collision occurs and this collision involves only a single

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moving particle. We will show in a sense to be made precise that for any system (independent of size), in short enough time intervals it suffices to consider simple changes in configuration. This observation simplifies the proof for N = 1. It is crucial in the analysis of N -chains; without it, the complexity of the situation gets out of hand quickly. In stochastic processes, analogous ideas are provided by independence and exponential clocks. The dynamics being largely deterministic here, we will have to argue it “by hand”, deducing it from tail assumptions on ρ. In what follows, |ν| denotes the total variation norm of a finite signed measure ν, and µ = ν + O(t) means |µ − ν| = O(t). For notational simplicity, we assume ρ(η) = O(η−2 ) as η → ∞. Proof of Theorem 2 for the case N = 1. Step 1. Reduction to simple changes in configuration. We will prove |µt − µ| = O(t 1+δ ) for some δ > 0 as t → 0. To see that this implies the invariance of µ, fix s > 0. By a repeated application of the estimate above with step size ns , we obtain µs = µ + n · O(( ns )1+δ ), which tends to µ as n → ∞. In what follows, we focus on the component M (+,+) = {(σ0 , σ1 ) = (+, +)}; other components are analyzed similarly. Instead of comparing µ and µt directly on all of M (+,+) , we will compare µ with some µˆ t ≈ µt (to be defined) on a large subset t ⊂ M (+,+) (to be defined). (A) We consider in the place of µt the measure µˆ t = PXt d µ(X ˆ ), where µˆ is the restriction of µ to the set {η0 , ξ, η1 < t −2 }. Notice that if all particles in a system have energy < t −2 , then their speeds are < t −1 , so no moving particle colliding with a site or a bath can set off a second collision to take place within t units of time. We claim that confusing µt with µˆ t leads to an error of a size we can tolerate, since ∞ ∞ ρ(ξ )dξ = O(t 2 ) and τ (η)dη = O(t 3 ). t −2

t −2

(B) Next, we permit at most one√moving particle to have a collision. For a particle at x ∈ (0, 1) with√σ = + and speed η, a collision occurred in the previous t units of time if and only if t η > x. Let √ √ t = {X ∈ M (+,+) : (i) η0 , ξ, η1 < t −2 and (ii) t η0 < x0 or t η1 < x1 }. 2 √ We claim that µ(t ) = 1− O(t 1+δ ). This is because if t η > x, then either (a) η > t − 3 , ∞ 1 2 or (b) x < t · t − 3 = t 3 . Now (a) occurs with probability − 2 τ (η)dη = O(t), and (b) t 3 2 2 4 √ √ occurs with probability t 3 . Thus µ{t η0 > x0 and t η1 > x1 } = [O(t 3 )]2 = O(t 3 ). The implications of (A) and (B) above are as follows: Suppose we show µˆ t |t = µ|t + O(t 1+δ ) for some δ with 0 < δ ≤

1 3.

(∗)

Then by (B), we have

µ = µ · It + O(t 1+δ ) = µˆ t · It + O(t 1+δ ). Now we know from (A) that |µˆ t | = 1 − O(t 1+δ ). This together with the equality above implies µˆ t (tc ) = O(t 1+δ ), so that µˆ t · It = µˆ t + O(t 1+δ ). Thus µ = µˆ t + O(t 1+δ ) = µt + O(t 1+δ ) , the second equality following from (A). This is what we seek.

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Step 2. Analysis of simple changes in configuration. Let ϕ and ϕˆt be the densities of µ and µˆ t respectively, and fix X¯ = (η¯ 0 , ξ¯ , η¯ 1 ; x¯0 , x¯1 ; +, +) ∈ t . We seek to compare ϕ( X¯ ) and ϕˆt ( X¯ ), remembering that to obtain µˆ t , one considers only initial conditions −2 in which √ all energies are < √t . Case 1: t η¯ 0 < √ x¯0 and t η¯ 1√< x¯1 . The only way to reach X¯ in time t is to start from (η¯ 0 , ξ¯ , η¯ 1 ;√x¯0 − t η¯ 0 , x¯1 −√t η¯ 1 ; +, +). No√collision occurs, and ϕ( X¯ ) = ϕˆt ( X¯ ). Case 2: t η¯ 0 < x¯0 and t η¯ 1 > x¯1 . Let s η¯ 1 √ = x¯1 . We claim that the only way to reach X¯ in time t is to start from ( η ¯ , ξ, η ; x ¯ − t η¯ 0 , x1 ; +, −), where ξ = η¯ 1 , η1 = ξ¯ 0 1 0 √ and x1 = (t − s) η1 . Starting here, one reaches X¯ after a single exchange of energy between the right particle and the site. To compute the densities at X¯ , we let ε > 0 be a verysmall number. Initial conditions with x1 ∈ ((t − s) ξ¯ − , (t − s) ξ¯ + ) and √ / ξ¯ = ε/ η¯ 1 will reach the target interval (x¯1 − ε, x¯1 + ε) at time t. Since ρ(η¯ 1 )τ (ξ¯ ) = ρ(η¯ 1 )

1 ρ(ξ¯ ) 1 ρ(ξ¯ ) = ρ(η¯ 1 ) √ ε = ρ(ξ¯ )τ (η¯ 1 )ε , Z ξ¯ Z η¯ 1

¯ we conclude ϕˆt ( X¯ ). √ that ϕ( X ) = √ √ Case 3: t η¯ 0 > x¯0 and t η¯ 1 < x¯1 . Here we let s be such that s η¯ 0 = x¯0 . The only √ way to reach X¯ is for the left particle to start at x0 = (t − s) η0 < 1 if η0 is its initial energy, go left, reach the left bath at time t − s, and have the newly emitted energy η¯ 0 reach x¯0 at time t. To compare densities, again√fix a small target interval (√ x¯0 − ε, x¯0√ + ε). √ This forces x0 ∈ ((t − s) η0 − (η0 ), (t − s) η0 + (η0 )) with (η0 )/ η0 = ε/ η¯ 0 . Thus −2 √ t η0 1 ϕˆt ( X¯ ) = ρ(ξ¯ )τ (η¯ 1 ) · τ (η0 ) √ dη0 · ρ(η¯ 0 ) 4 η¯ 0 0 t −2 1 ρ(η¯ 0 ) 1 ρ(η0 )dη0 = ϕ( X¯ ) · (1 − O(t 2 )). = ρ(ξ¯ )τ (η¯ 1 ) · √ 4 Z η¯ 0 0 √ √ Since t η¯ 0 > x¯0 and t η¯ 1 > x¯1 is not permitted for X¯ ∈ t , we have exhausted all viable cases, completing the proof of (∗).

5.2. Proof of Theorem 2. We follow closely Section 5.1, adapting the ideas there to chains with N sites. We begin with the analogous set of reductions. As in Section 5.1, we seek to show µt = µ + O(t 1+δ ) for some δ > 0, focusing on a fixed component of the phase space M (σ¯ ) = {(σ0 , · · · , σ N ) = σ¯ }. Instead of µt , we consider µˆ t = PXt d µ(X ˆ ) where µˆ is the restriction of µ to {ηi , ξi < t −2 , all i}. It follows from the estimates in Section 5.1 that (for fixed N ) this introduces an error within the tolerable range: ∞ ∞ |µt − µˆ t | < N ρ + (N + 1) τ = O(t 2 ). t −2

(σ¯ )

We define t = t

t −2

to be

{X ∈ M (σ¯ ) : (i) ξi , ηi < t −2 for all i; √ √ (ii) t ηi < xi if σi = +, t ηi < 1 − xi if σi = − for all except at most one i} ,

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√ the idea being that when the above relation between t, ηi and xi holds, no collision occurs in the t units of time prior to arrival in that configuration. The same argument as 4 before shows that µ(t ) = 1 − O(t 3 ): the set on which this relation is violated by two 4 of the moving particles has measure O(t 3 ); the set on which it is violated by more than two particles is smaller. As in Section 5.1, we seek to compare ϕˆt and ϕ on the set t . As before, all changes in configuration involved are simple: there are 3 cases corresponding to no collisions, a collision with a site, and a collision with a bath. The analysis is identical to that in Section 5.1.

Part II. Numerical Results We report here some results on nonequilibrium steady states (NESS) obtained by studying numerically two families of models: the first has exponential bath injections and the second, which we call the “two-energy model”, is chosen for the transparency of the role played by the integrable dynamics.

6. Exponential Baths We consider in this section processes defined in Section 1.1 for which the bath injections have exponential distributions, i.e., L(ξ ) = β L e−β L ξ and R(ξ ) = β R e−β R ξ , where, as usual, β L = TL−1 and β R = TR−1 are to be thought of as inverse temperatures. We are interested in steady states of systems that are in contact with two baths at unequal temperatures, i.e., where TL = TR . It follows from Theorem 1 that all invariant distributions that arise from these bath injections are ergodic. In Section 6.1, we demonstrate that mean energy profiles are well defined, both for finite N and as N → ∞. In Section 6.2, we focus on specific points along the chain, and investigate marginals of the NESS on (very) short segments. In the simulations shown, mean bath temperatures are TL = 1 and TR = 10, and chains of various lengths up to N = 1600 are used. 6.1. Macroscopic energy profiles. For a chain with N sites, we let E[ξi ] denote the mean energy at site i, “mean” being taken with respect to the unique steady state distribution. Let f N : [0, 1] → R be the function which linearly interpolates f N ( Ni+1 ) = E[ξi ], i = 1, · · · , N . If as N increases, f N converges (pointwise) to a function f on [0, 1], we will call f the site-energy profile of this model. Similarly, we let f N be the function that interpolates f N ( Ni+1 +2 ) = E[ηi ], i = 0, 1, . . . , N , and call f = lim N f N the gap-energy profile. Profiles of gap energies conditioned on σi being + or − are denoted f N+ and f N− . Figure 1 shows plots of f N , f N± , and f N . Finite-chain profiles are found to vary little as N goes from 100 to 1600, so we assume the limit profile will not be too different. Convergence time to steady state is slow, and increases with N as expected. (See numerical details in the caption.) 6.2. Local equilibrium properties. This subsection is about local properties of NESS, by which we refer to marginals of the form µˆ x,,N , where x ∈ (0, 1), N is the length of

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5

4

Mean gap energy

Mean site energy

8

6

4

2

3

2

1

0

0 0.2

0.4

0.6

0.8

x (a) Site energies, N = 1600

0

0.2

0.4

0.6

0.8

1

x (b) Gap energies, N =200

Fig. 1. Mean energy profiles for the exponential baths model. The bath temperatures are TL = 1 and T R = 10. Panel (a) shows the mean site energies f N for N = 1600. In (b), we have superimposed the gap energy profiles f N+ , f N− , and f N for N = 200. Numerical details: We impose a constraint that injected energies are ≥ 0.01 but do not impose a high energy cutoff. We simulate the system for 10k events (an “event” being one collision anywhere in the chain), increasing k until the computed profile stabilizes. This occurred after 1013 events in (a), 4 × 1011 events in (b)

the chain, N , and µˆ x,,N is the marginal distribution of the NESS on the -chain centered at site [x N ]. For fixed (unequal) bath distributions, since the energy gradient on the -chain tends to zero as N → ∞, one might expect the µˆ x,,N to resemble equilibrium distributions, i.e., invariant measures on chains with equal bath injections. Theorem 2 gives an explicit formula for the equilibrium distribution µρ,N of the N -chain when the bath distributions are L = R = ρ. This result is valid for very general ρ. Notice that when specialized to the case ρ(ξ ) = βe−βξ , the density of µρ,N has the √ familiar form Z1 e−β H . More precisely, let us use (xi , vi ), vi = σi ηi , as coordinates in the gaps instead of (ηi , xi , σi ). Then −(N +1) −β i vi2 e i d xi dvi . dµρ,N = β N e−β i ξi i dξi × Z β

(1)

Returning to the situation of unequal exponential baths, one way to define local thermodynamic equilibrium (LTE) is to require that for every x ∈ (0, 1) and ∈ Z+ , as N → ∞, the marginals µˆ x,,N tend to a probability measure having the form in (1) with N replaced by and β = β(x) for some β(x) > 0. For a more physical notion of LTE, one sometimes considers only -chains for 1 N . A. Single-site and single-gap marginals. We consider marginal distributions of the NESS at single sites and single gaps at [x N ], where x ∈ (0, 1) is fixed and N is varied. To show that the system does not tend to LTE (by any definition), it suffices to show that these marginals are not Gibbsian. This is because if marginals on -chains tended to µˆ x,,N , projecting onto the single site or gap at [x N ] one would again obtain a distribution of the form in (1). Figure 2 shows two examples of single-site marginal densities at x = 0.25 and 0.6 for a chain of length N = 1600. The plots are log-linear, so that exponential functions

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Fig. 2. Site energy distributions for the exponential baths model in log-linear scales. The parameters are TL = 1, T R = 10, N = 1600. Empirical data are plotted in open squares; the solid curve represents the mixture of left and right bath distributions that best fits the data (in the total variation norm)

would appear as straight lines. Without a doubt both of the distributions shown are very far from exponential. These distributions are compared to mixtures of the bath distributions, meaning distributions of the form a L(ξ ) + (1 − a)R(ξ ), a = a(x) ∈ (0, 1). In Fig. 2, open squares represent empirical data (from simulations), while the solid curves are given by the formulas for mixtures with a chosen in each of the two cases to fit the data best. Given the numerical cutoffs, etc., we think the fits between empirical data and the mixtures curves are excellent. Plots (not shown) at other locations evenly spaced along the chain show the same phenomenon, with the “knee” moving steadily to the left as x increases. Likewise, single-gap marginals are found to be mixtures: At gaps adjacent to [x N ] 2 for fixed x, instead of Z β−1 e−βv for some β = β(x), we find numerically the marginals to be of the form a·

e−β L v Z βL

2

+ (1 − a) ·

e−β R v Z βR

2

for some a = a(x), in clear violation of the prescription of Gibbs for Hamiltonian systems in equilibrium. We explain how we arrived at the idea of “mixtures”: First, the energies in a chain should reflect those injected, and if there is a nontrivial discrepancy between β L and β R , it is difficult to imagine having an abundance of the energies in the “middle” to constitute all the exponential distributions along the chain. We are also influenced by the following stochastic model, which can be thought of as a “zeroth-order” approximation to our Hamiltonian systems.

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The random swaps model. This is a stochastic model defined by N random variables ξ1 , . . . , ξ N , to be thought of as energies, located at sites 1, . . . , N . As usual there are two baths, situated at sites 0 and N +1. At bonds between sites, i and i +1 for i ∈ {0, 1, . . . , N } are exponential clocks which ring independently at rate 1. When a clock goes off, energies between the 2 sites are swapped. That is to say, for i = 0, N , when the clock between sites i and i + 1 rings, the values of ξi and ξi+1 are interchanged. Swapping energy with the left bath means that ξ1 is replaced by an energy drawn randomly from the left bath distribution L(ξ ), and similarly for ξ N and the right bath, which has distribution R(ξ ). The energies emitted by each bath are assumed to be i.i.d. and independent of emissions by the other bath. A model along similar lines was studied in [10]. Proposition 6.1. In the random swaps model, for each i ∈ {1, . . . , N }, the marginal distribution ρi of the unique NESS is given by

i N +1−i L+ R ρi = N +1 N +1 for arbitrary bath distributions L and R. Sketch of proof. First, we distinguish only between whether an energy is “from the left” or “from the right”. In this reduced system, it is easy to see that there is a unique steady state, the single-site marginals of which are, by definition, mixtures of these two kinds of energies. Since the trajectory of each energy, from the time it enters the system to when it leaves, is that of a simple unbiased random walk, the weights in the formula for ρi follow from basic recursive relations. Finally, each of the energies in this reduced system can be assigned independently any one of the allowed values, so that the marginals are really mixtures of L and R.

Since Proposition 6.1 holds for arbitrary bath distributions L and R, it is natural to ask if a similar result holds for the Hamiltonian chain with non-exponential baths. We investigated this question and found the answer to be negative. For example, when L and R are uniform distributions, marginal distributions are far from mixtures of uniform distributions, see Fig. 3. When the densities of L and R have shapes closer to exponential distributions, single-site marginals are closer to mixtures, but noticeable differences were seen in each of the half-dozen or so cases tested. The numerical results above raise the following questions: (A) In the case of exponential baths, are single-site marginals as N → ∞ genuinely mixtures of the two bath distributions, or are they simply close to mixtures? (B) If the mixtures result here is exact, are exponential baths the only distributions that have this property, and if so, what are the underlying reasons? Independent of the answer to (A), we believe we have shown very definitively that local marginal distributions do not have the form in (1). Hence the concept of LTE does not apply to this class of Hamiltonian chains. B. Vanishing of spatial correlations. We investigate next if, as N → ∞, µˆ x,,N → µρ, where ρ is the mixture found in Paragraph A. Since spatial correlations are expected to be largest between adjacent sites and gaps, we verify only that (i) marginal distributions on two adjacent sites are product measures; (ii) marginal distributions on two adjacent gaps are product measures, and the directions of travel are independent;

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for the exponential-baths model. The 4 curves in each panel show P ξ[N x]+1 ∈ I | ξ[N x] ∈ [2k, 2(k + 1)] for k = 0, 1, 2, · · · , 14 and I = [0, 2], [2, 4], [4, 6], and [6, 8] (top to bottom). In both panels, x = 0.6. Bath parameters are TL = 1, T R = 10

(iii) marginal distributions on a√site and its adjacent gap are product measures, with gap density = const·ρ(η)/ η, where site density = ρ. A sample of simulation results in support of (i) is shown in Fig. 4. In the horizontal axis are values of ξi for i = [0.6N ], and plotted are conditional distributions of ξi+1 given the various values of ξi . Here we group the values of ξi+1 into intervals (0, 2), (2, 4), (4, 6), and (6, 8), the 4 graphs representing the conditional probabilities of ξi+1 being in each one of these intervals given the value of ξi in the horizontal axis. The two plots, for N = 100 and 1600 respectively, show weak dependence of ξi and ξi+1 in the first and close to zero dependence for the longer chain.

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Table 1. Joint distributions of gap energies conditioned on directions of travel. Here, N = 1600 and i = [0.6 N ]. The numbers are raw probabilities of the specified events

A small sample of data is shown in Table 1 to illustrate the combinations of joint densities that need to be checked to verify (ii). Here we focus on two adjacent gaps at x = 0.6, and distinguish only between “high” and “low” energies, referring to energies η > 3 as “high” and η < 3 as “low”. The fractions of time in all configurations of high-low and (σi , σi+1 ), referring to the directions of travel in these gaps, are tabulated. Finally, we have computed energy distributions for adjacent sites and gaps at various sites along the chain. The data confirm property (iii) above. Remark. The “local product structure” finding above may be valid beyond the case of exponential baths. Our simulations show that it holds both for the uniform distribution models in Fig. 3 and for the “two-energy models” discussed in the next section. Since the latter are rather “extreme” from many points of view, it gives reason to conjecture that this phenomenon – which is very natural – may hold widely. 7. Two-Energy Models The bath distributions in this section are sharply peaked Gaussians truncated at 0. In the simulations shown, these Gaussians have means 1 and 5, and their standard deviations are chosen to be 0.2, so that the probability of being within ±0.5 of the means is roughly 99%. Even though these systems are small perturbations of integrable models, we know from Theorem 1 that all of their invariant measures are ergodic. 7.1. Macroscopic energy profiles and local distributions. Figure 5(a) shows site-energy profiles f N for a range of N . Unlike the exponential case, these profiles vary substantially with N : For N = 10 (not shown), the profile is essentially flat; as N increases, it acquires a gradient and appears to be stabilizing, but even at N = 3200, the profile is still moving a little. From these profiles, one suspects – correctly – that for (very) small N , most energies move monotonically along the chain, entering from one end and exiting at the other. As N increases, some of the energies turn around, some doing so a number of times before exiting, creating a gradient in the profile. Figure 5(b) shows profiles for gap energies with specified directions, showing that for the chain with N = 200, f N+ and f N− are substantially different, with more energy

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Fig. 5. Mean energy profiles for two-energy model. Panel (a) shows mean site energies; the curves are N =40, 200, and 3200, in order of increasing values at x = 1. Panel (b) shows mean gap energies f N− , f N , and f N+ (top to bottom) for N = 200. The rules used to determine when a steady state is reached are as Fig. 1. See text for the bath distributions

1s moving to the right and 5s moving to the left. This is to be contrasted with Fig. 1(b), which shows that for exponential baths, this left–right discrepancy has by and large vanished by N = 200. Indeed in the present model, f N+ and f N− remain quite far apart in some of the gaps even at N = 3200, leaving open the question whether or not the distributions conditioned on σi = + and − will eventually equalize. A follow-up investigation is discussed in Section 7.2. With regard to local distributions, computations on site-to-site, gap-to-gap, and siteto-gap correlations similar to those in Section 6 were carried out. Independence was found to be achieved already at relatively small N . Figure 6 captures the two main points: The two curves representing P(ξ[0.6N ]+1 = 1|ξ[0.6N ] = k), k = 1 and 5, are virtually on top of each other for N ≥ 400, illustrating the rapid vanishing of correlations between adjacent sites, while both curves continue to have a slightly negative slope at N = 3200, illustrating the slow convergence of mean site energies. 7.2. Ballistic transport vs. diffusion: a phenomenological explanation. The purpose of this subsection is to examine more closely the way in which energy is transported from one end of the chain to the other under near-integrable dynamics. Since the vast majority of the energies in this system are very close to 1 or 5, let us assume there are only two kinds of energies in the system. For definiteness, we adopt the view in Section 1.3, and focus on the movement of 1s on the right side of the chain. We will demonstrate that modulo certain time changes, the statistics generated by the movements of these energies resemble those of a particle undergoing a 1-D diffusion with a spatially-varying diffusion coefficient. We begin by introducing an especially simple model which will be used for comparison purposes: Model A. Consider a Markov chain with state space {0, 1, · · · , N + 1} defined as follows: Starting from 1, one performs a simple, unbiased random walk until either 0 or N + 1 is reached, then returns to 1 to start over again. For 1 ≤ i ≤ N − 1, let n (i) and rn (i) denote the number of left and right crossings respectively between sites i and i + 1

Prob( ξ[Nx]+1 | ξ[Nx] )

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N Fig. 6. Nearest-neighbor conditional probabilities in the two-energy model. This plot shows P(ξ[N x]+1 < 3 | ξ[N x] < 3) (open squares) and P(ξ[N x]+1 < 3 | ξ[N x] > 3) (solid discs) as functions of N . The location is x = 0.6; N ranges from 100 to 3200

in the first n steps, and let E N ,i = limn→∞ E N [n (i)]/E N [rn (i)]. Here are some easy facts: (i) For all N and i, 0 < E N ,i < 1. (ii) For each N , E N ,i decreases as i increases. (iii) For each x ∈ (0, 1), E N ,[x N ] increases to 1 as N → ∞. These facts follow from the following observations: Fix i and N , and focus only on left and right crossings between sites i and i + 1. Then a left crossing is necessarily followed by a right crossing, while a right crossing can be followed by either. The probability that a right crossing is followed by another right crossing is equal to the probability of starting from site i + 1 and reaching site N + 1 before site i in the random walk. The numbers E N ,i can be explicitly estimated if one so desires. To bring this simple model closer to our Hamiltonian chains, we first identify some relevant features of the chains. The following idea is used in the rigorous part of this paper (Lemmas 4.2 and 4.4): Consider a scenario in which an energy 1 is in the midst of many energy 5s, i.e. in some segment of the chain, ξ j = η j = 5 for all j except for a ˆ where η ˆ = 1. Then the energy 1 will move monotonically in some direction single j, j until the pattern is disrupted by the approach of another (oncoming) energy 1. After such a “collision” a variety of things can happen; the resulting motion of the energies depends on the details of the interaction. Since the energy profiles have nonzero gradient (see Fig. 5), 1s are more sparse on the right side of the chain. Thus the closer to the right end, the larger the “mean free distance” between “collisions” of energy 1s. With regard to the ratio of left to right crossings (and not the actual number of crossings per unit time), the observations above suggest the following modification of Model A: Model B. Let λ : (0, 1) → (0, ∞) be a function that monotonically increases from 1 to ∞ on some subinterval (x∗ , 1) ⊂ (0, 1). We consider a process similar to that in Model A but with transition rules modified as follows: when at site i = [x N ], go X sites to the

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Fig. 7. Statistics on lengths of runs. (a) Mean lengths of runs in different intervals [a, b]. Note that runs that reach the boundary of [a, b] are artificially cut short by our numerical procedure (see “numerical details” below); we do not consider averages based on too many (> 10%) such runs meaningful and have omitted those averages. (*) marks averages for which 5 − 10% of the runs were cut short. (b) Log-linear plot of the histogram of run-lengths in the interval [0.6, 0.8], for N = 3200. (c) The ratio P(ηi ≈ 1, σ = −)/P(ηi ≈ 1, σ = +) as a function of x = i/N . The curves are, from bottom to top, N =100, 400, 1600, 3200. Numerical details for (a) and (b): For a given interval [a, b], we begin with a suitably prepared system and wait until an energy ≈ 1 hits the site at x = (a + b)/2. The energy is tracked until it hits either x = a or x = b; the lengths of runs executed by this energy are recorded. This process is repeated (after a delay to ensure that the energies are not too correlated). Each data point in Table (a) based on 104 – 106 runs collected this way

left/right with probability 21 / 21 , where X is an exponential random variable with mean λ(x), and return as before to 1 to start over once 0 or N + 1 is reached. First we confirm numerically that in terms of the statistics generated, Model B gives a good approximation of movements of energy 1s on the right side of the real chain. (Obviously, we do not claim that the two models are equivalent.) We study the lengths of runs of randomly picked 1s in specified segments of the chain, a run being defined to be a consecutive sequence of moves in the same direction. Some mean length-of-runs are tabulated in Fig. 7(a). The numbers in each row increase as one moves to the right, a trend consistent with the environment becoming increasingly dominated by energy 5s. As N increases, these means stabilize, as one would expect them to when the local marginals tend to a distribution of the form µρ, . Histograms of lengths of runs within specified intervals are plotted and found to have roughly exponential distributions for large N ; one such plot is shown in Fig. 7(b). Returning to Models A and B, observe that they are qualitatively similar if we imagine that the sites in Model B are “packed closer and closer” as x → 1. More precisely, the ratio of left to right crossings at site [x N ] in Model B should resemble that at [y N ] in Model A for some y > x, with (1 − y)/(1 − x) → 0 as x → 1. Properties (i-iii) in

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Model A therefore pass to Model B. Assuming they pass from there to our Hamiltonian chain, one would conclude that for our chain: – left-right traffic will equalize everywhere as N → ∞; – this equalization occurs more slowly than at corresponding sites in Model A, the discrepancy increasing as x → 1. Simulations to compute the ratios of left/right crossings were performed. The results, shown in Fig. 7(c), are very much consistent with the predictions above: for each N , the x → Prob(σ[N x] = −)/Prob(σ[N x] = +) curves are decreasing, and at each x ∈ (0, 1), this ratio increases with N and appears to head toward 1. Remarks on connection with theory. We have used the fact, proven rigorously in Section 4.3, that in regions of the chain occupied by significantly more energies of one kind than the other, the ones with low density will, on average, have relatively long runs. The same is true, in fact, for the energies with high density. Our rigorous work leaves untreated the situation where the two energies occur in roughly comparable proportions. For these regions, we have seen via simulations that the mean lengths of runs tend still to be greater than 2, the value for simple unbiased random walks. For example, when TL = 1 and TR = 5, the region with the most even mix of 1s and 5s is [0.2, 0.4], and there the mean run-length for 1s is > 5 (see Fig. 7(a)). Thus the phenomena discussed in this subsection appear to be valid for both energies on a good part of the chain.

Summary and Conclusion We considered a Hamiltonian chain with many conserved quantities and studied its nonequilibrium steady states when the two ends of the chain are put in contact with unequal heat baths. Our main findings are: Rigorous results. First, under mild restrictions on the bath distributions L(ξ ) and R(ξ ), we proved ergodicity of the invariant measure (assuming existence). Second, we identified a class of equilibrium measures {µρ,N }, where µρ,N is the unique invariant probability on the N -chain with L = R = ρ; the measures µρ,N are product measures. Numerical results. Simulations were performed for chains with two kinds of bath distributions: exponential distributions and sharply peaked Gaussians, the latter giving rise to what is called the “two-energy model”. 1. We demonstrated numerically that (a) NESS exist for finite N and mean energy profiles converge as N → ∞; (b) as N → ∞ local marginals at x ∈ (0, 1) tend to measures of the form µρ, for some distribution ρ = ρ(x). 2. For exponential bath distributions, the limits of local marginals are definitively not Gibbs measures, i.e., this chain violates the concept of LTE. The marginals appear to be weighted averages of Gibbs measures. 3. For the 2-energy model, the paths traced out by energies resemble the sample paths of a random walk with a bias in favor of continuing in the same direction, with this bias increasing as x → 0 or 1. We conclude that the resulting transport behavior is more normal, i.e., more diffusive, than one might have expected given the integrable character of the dynamics.

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References 1. Bernardin, C., Olla, S.: Fourier’s law for a microscopic model of heat conduction. J. Stat. Phys. 118, 271–289 (2005) 2. Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory for stationary non-equilibrium states. J. Statist. Phys. 107, 635–675 (2002) 3. Bonetto, F., Lebowitz, J., Rey-Bellet, L.: Fourier law: a challenge to theorists. In: Mathematical Physics 2000, edited by Fokas, A., Grigoryan, A., Kibble, T., Zegarlinski, B., London: Imp. Coll. Press, 2000 4. Bricmont, J., Kupiainen, A.: Fourier’s law from closure equations. Phys. Rev. Lett. 98, 214301 (2007) 5. Bricmont, J., Kupiainen, A.: Towards a derivation of Fourier’s law for coupled anharmonic oscillators. Commun. Math. Phys. 274, 555–626 (2007) 6. Collet, P., Eckmann, J.-P.: A model of heat conduction. Preprint, http://arxiv.org/abs/0804.3025v1[mathph], 2008 7. Collet, P., Eckmann, J.-P., Mejía-Monasterio, C.: Superdiffusive heat transport in a class of deterministic one-dimensional many-particle Lorentz gases. Preprint, http://arxiv.org/abs/0810.4461v1[cond-mat. stat.-mech], 2008 8. Derrida, B.: Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. J. Stat. Mech. (2007) P07023, doi:10.1088/1742-5468/2007/07/p02023, July 2007 9. Derrida, B., Lebowitz, J.L., Speer, E.R.: Large deviation of the density profile in the steady state of the open symmetric simple exclusion process. J. Stat. Phys. 107, 599–634 (2002) 10. Dhar, A., Dhar, D.: Absence of local thermal equilibrium in two models of heat conduction. Phys. Rev. Lett. 82, 480–483 (1999) 11. de Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. Amsterdam: North-Holland, 1962 12. Eckmann, J.-P., Hairer, M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212, 105–164 (2000) 13. Eckmann, J.-P., Jacquet, P.: Controllability for chains of dynamical scatterers. Nonlinearity 20, 1601– 1617 (2007) 14. Eckmann, J.-P., Mejía-Monasterio, C., Zabey, E.: Memory effects in nonequilibrium transport for deterministic Hamiltonian systems. to appear in J. Stat. Phys., 2006 15. Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Commun. Math. Phys. 201, 657–697 (1999) 16. Eckmann, J.-P., Young, L.-S.: Nonequilibrium energy profiles for a class of 1-D models. Commun. Math. Phys. 262, 237–267 (2006) 17. Gaspard, P., Gilbert, T.: Heat conduction and Fourier’s law in a class of many particle dispersing billiards. New J. Phys. 10, 103004 (2008) 18. Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Berlin: Springer-Verlag, 1999 19. Kipnis, C., Marchioro, C., Presutti, E.: Heat flow in an exactly solvable model. J. Stat. Phys. 27, 65–74 (1982) 20. Larralde, H., Leyvraz, F., Mejía-Monasterio, C.: Transport properties of a modified Lorentz gas. J. Stat. Phys. 113, 197–231 (2003) 21. Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377(1), 1–80 (2003) 22. Li, B., Casati, G., Wang, J., Prosen, T.: Fourier Law in the alternate-mass hard-core potential chain. Phys. Rev. Lett. 92, 254301 (2004) 23. Lin, K.K., Young, L.-S.: Correlations in nonequilibrium steady states of random-halves models. J. Stat. Phys. 128, 607–639 (2007) 24. Olla, S., Varadhan, S.R.S., Yau, H.T.: Hydrodynamical limit for a Hamiltonian system with weak noise. Commun. Math. Phys. 155, 523–560 (1993) 25. Rateitschak, K., Klages, R., Nicolis, G.: Thermostating by deterministic scattering: the periodic Lorentz gas. J. Stat. Phys. 99, 1339–1364 (2000) 26. Ravishankar, K., Young, L.-S.: Local thermodynamic equilibrium for some stochastic models of Hamiltonian origin. J Stat. Phys. 128, 641–665 (2007) 27. Rey-Bellet, L., Thomas, L.E.: Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Commun. Math. Phys. 225, 305–329 (2002) 28. Rieder, Z., Lebowitz, J.L., Lieb, E.: Properties of a harmonic crystal in a stationary nonequilibrium state. J. Math. Phys. 8, 1073 (1967) 29. Spohn, H.: Long range correlations for stochastic lattice gases in a nonequilibrium steady state. J. Phys. A 16, 4275–4291 (1983) Communicated by A. Kupiainen

Commun. Math. Phys. 294, 229–249 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0869-2

Communications in

Mathematical Physics

Ground State Energy of Large Atoms in a Self-Generated Magnetic Field László Erd˝os1, , Jan Philip Solovej2, 1 Institute of Mathematics, University of Munich, Theresienstr. 39,

D-80333 Munich, Germany. E-mail: [email protected]

2 Department of Mathematics, University of Copenhagen, Universitetsparken 5,

DK-2100 Copenhagen, Denmark. E-mail: [email protected] Received: 11 March 2009 / Accepted: 16 April 2009 Published online: 3 July 2009 – © Springer-Verlag 2009

Abstract: We consider a large atom with nuclear charge Z described by non-relativistic quantum mechanics with classical or quantized electromagnetic field. We prove that the absolute ground state energy, allowing for minimizing over all possible self-generated electromagnetic fields, is given by the non-magnetic Thomas-Fermi theory to leading order in the simultaneous Z → ∞, α → 0 limit if Z α 2 ≤ κ for some universal κ, where α is the fine structure constant. 1. Introduction The ground state energy of non-relativistic atoms and molecules with large nuclear charge Z can be described by Thomas-Fermi theory to leading order in the Z → ∞ limit [L,LS]. Magnetic fields in this context were taken into account only as an external field, either a homogeneous one [LSY1,LSY2] or an inhomogeneous one [ES] but subject to certain regularity conditions. Self-generated magnetic fields, obtained from Maxwell’s equation are not known to satisfy these conditions. In this paper we extend the validity of Thomas-Fermi theory by allowing a self-generated magnetic field that interacts with the electrons. This means we look for the absolute ground state of the system, after minimizing for both the electron wave function and for the magnetic field and we show that the additional magnetic field does not change the leading order Thomas-Fermi energy. Apart from finite energy, no other assumption is assumed on the magnetic field. The nonrelativistic model of an atom in three spatial dimensions with nuclear charge Z ≥ 1 and with N electrons in a classical magnetic field is given by the Hamiltonian N 1 1 Z cl H N ,Z (A) = T j (A) − + + B2 (1.1) |x j | |xi − x j | 8π α 2 R3 j=1

i< j

Partially supported by SFB-TR12 of the German Science Foundation. Work partially supported by the Danish Natural Science Research Council and by a Mercator Guest Professorship from the German Science Foundation.

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acting on the space of antisymmetric functions 1N H with a single particle Hilbert space H = L 2 (R3 ) ⊗ C2 . The coordinates of the N electrons are denoted by x = (x1 , x2 , . . . , x N ). The vector potential A : R3 → R3 generates the magnetic field B = ∇ × A and it can be chosen divergence-free, ∇ · A = 0. The last term in (1.1) is the energy of the magnetic field. The kinetic energy of an electron is given by the Pauli operator T (A) = [σ · (p + A)]2 = (p + A)2 + σ · B,

p = −i∇x .

Here σ is the vector of Pauli matrices. We use the convention that for any one-body operator T , the subscript in T j indicates that the operator acts on the j th variable, i.e. T j (A) = [σ j · (−i∇x j + A(x j ))]2 . The term −Z |x j |−1 describes the attraction of the j th electron to the nucleus located at the origin and the term |xi − x j |−1 is the electrostatic repulsion between the i th and j th electron. Our units are 2 (2me2 )−1 for the length, 2me4 −2 for the energy and 2mec−1 for the magnetic vector potential, where m is the electron mass, e is the electron charge and is the Planck constant. In these units, the only physical parameter that appears in (1.1) is the dimensionless fine structure constant α = e2 (c)−1 . We will assume that Z α 2 ≤ κ with some sufficiently small universal constant κ ≤ 1 and we will investigate the simultaneous limit Z → ∞, α → 0. Note that the field energy is added to the total energy of the system and by the condition ∇ · A = 0 we have B2 = |∇ ⊗ A|2 , (1.2) R3

R3

where ∇ ⊗ A denotes the 3 × 3 matrix of all derivatives ∂i A j and |∇ ⊗ A|2 = 3 2 i, j=1 |∂i A j | . We will always assume that the vector potential belongs to the space of divergence free H 1 -vector fields A := {A ∈ H 1 (R3 , R3 ), ∇ · A = 0}. In the analogous nonrelativistic model of quantum electrodynamics, the electromagnetic vector potential is quantized. In the Coulomb gauge it is given by A (x) = A(x) = A− (x) + A− (x)∗ with A− (x) =

α 1/2 2π

R3

g(k) aλ (k)eλ (k)eik·x dk. √ |k| λ=±

Here g(k) is a cutoff function, satisfying |g(k)| ≤ 1 and supp g ⊂ {k ∈ R3 : |k| ≤ } with a constant < ∞ (ultraviolet cutoff). The field operators A(x) depend on the cutoff function g(k) whose precise form is unimportant; the only relevant parameter is . For each k, the two polarization vectors e− (k), e+ (k) ∈ R3 are chosen such that together with the direction of propagation k/|k| they are orthonormal. The operators aλ (k), aλ (k)∗ are annihilation and creation operators acting on the bosonic Fock space F over L 2 (R3 ) and satisfying the canonical commutation relations [aλ (k), aλ (k )] = [aλ (k)∗ , aλ (k )∗ ] = 0, [aλ (k), aλ (k )∗ ] = δλλ δ(k − k ).

Energy of Atoms in a Self-Generated Field

The field energy is given by Hf = α The total Hamiltonian is qed H N ,Z

=

N j=1

−1

231

R3

|k|

aλ (k)∗ aλ (k)dk.

λ=±

1 Z T j (A ) − + + Hf , |x j | |xi − x j |

(1.3)

i< j

and it acts on ( 1N H) ⊗ F. The stability of atoms in a classical magnetic field [F,FLL,LL,LLS] implies that the operator (1.1) is bounded from below uniformly in A, if Z α 2 is small enough. It is known [LY,ES2] that stability fails if Z α 2 is too large. The analogous stability result for quantized field [BFG] states that (1.3) is bounded from below if Z α 2 is small. In particular, we can for each fixed A define the operators in (1.1) and (1.3) as the Friedrichs extensions of these operators defined on smooth functions with compact support. The ground state energy of the operator with a classical field is given by N

cl cl ∞ 3 2 E N ,Z (A) = inf , H N ,Z (A) : ∈ C0 (R ) ⊗ C , = 1 , 1

and after minimizing in A we set cl E cl N ,Z = inf E N ,Z (A). A∈A

We note that it is sufficient to minimize over all A ∈ A0 , where A0 = Hc1 (R3 , R3 ) denotes the space of compactly supported H 1 vector fields. It is easy to see that the Euler-Lagrange equations for the above minimizations in and A correspond to the stationary version of the coupled Maxwell-Pauli system, i.e., the eigenvalue problem 2 H Ncl,Z (A) = E cl N ,Z together with the Maxwell equation ∇ × B = 4π α J , where J is the current of the wave function . It is for this reason that it is natural to refer to B as a self-generated magnetic field in this context. In the case of the quantized field, we define N

qed qed ∞ 3 2 C0 (R ) ⊗ C ⊗ F, = 1 . E N ,Z = inf , H N ,Z : ∈ 1 qed

The stability results of [F,FLL,LL,LLS,BFG] imply that E cl N ,Z > −∞ and E N ,Z > −∞ 2 if Z α is small enough. Finally, we define the ground state energy with no magnetic field as cl E nf N ,Z := E N ,Z (A = 0).

In all three cases we define E #Z := inf E #N ,Z , N ∈N

# ∈ {cl, qed, nf},

for the absolute (grand canonical) ground state energy. The main result of this paper states that the magnetic field does not change the leading term of the absolute ground state energy of a large atom in the Z → ∞ limit. In particular, Thomas-Fermi theory is correct to leading order even with including self-generated magnetic field.

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Theorem 1.1. There exists a positive constant κ such that if Z α 2 ≤ κ, then 7

1

cl nf 3 − 63 . E nf Z ≥ EZ ≥ EZ − C Z

(1.4)

For the quantized case, if we additionally assume 1

7

1

≤ κ 4 Z 12 −γ α − 4 with some 0 ≤ γ ≤

1 63 ,

(1.5)

then qed

2 E nf Z + C Z α ≥ E Z

7

3 −γ . ≥ E nf Z − CZ

(1.6)

We note that Z α2 Z 7/3 if κ −1/4 Z 11/12 in the Z → ∞ limit. Remark 1. The leading term asymptotics of the non-magnetic problem is given by the 7 2 3 Thomas-Fermi theory and E nf Z = −cTF Z + O(Z ) as Z → ∞, where cTF = 3.678 · 2

(3π 2 ) 3 is the Thomas-Fermi constant. The leading order asymptotics was established in [LS] (see also [L]). The correction to order Z 2 is known as the Scott correction and was established in [H,SW1] and for molecules in [IS] (see also [SW2,SW3,SS]). The next term in the expansion of order Z 5/3 was rigorously established for atoms in [FS]. Remark 2. The exponents in the error terms are far from being optimal. They can be improved by strengthening our general semiclassical result Lemma 1.3 for special Coulomb like potentials using multi-scale analysis. Remark 3. For simplicity, we state and prove our results for atoms, but the same proofs work for molecules as well; if the number of nuclei K is fixed, each has a charge Z , and assume that the nuclei centers {R1 , . . . , R K } are at least at distance Z −1/3 away, i.e. |Ri − R j | ≥ cZ −1/3 , i = j. Remark 4. Theorem 1.1 holds for the magnetic Schrödinger operator as well, i.e. if we replace the Pauli operator T (A) = [σ · (p + A)]2 by T (A) = (p + A)2 everywhere. The proof is a trivial modification of the Pauli case. The argument is in fact even easier; instead of the magnetic Lieb-Thirring inequality for the Pauli operator one uses the usual Lieb-Thirring inequality that holds for the magnetic Schrödinger operator uniformly in the magnetic field. We leave the details to the reader. Note that although the condition Z α 2 ≤ κ is not needed in order to ensure stability in the Schrödinger case, we still need it in the statement in Theorem 1.1. In the Schrödinger case this condition is not optimal. The upper bound in (1.4) is trivial by using a non-magnetic trial state. The upper bound in (1.6) is obtained by a trial state that is the tensor product of a non-magnetic electronic trial function with the vacuum | of F. The field energy H f and all terms that are linear in A give zero expectation value in the vacuum. The only effect of the quantized field is in the nonlinear term A2 . A simple calculation shows that |A2 | ≤ Cα2 . The main task is to prove the lower bounds. Using the results from [BFG], the result for the quantized field (1.6) will directly follow from an analogous result for a slightly modified Hamiltonian with a classical field. Let N 1 1 Z + + Ti (A)− |∇ ⊗ A|2 H N ,Z (A) = H N ,Z ,α (A) = |xi | |xi − x j | 8π α 2 |x|≤3r i=1

i< j

(1.7)

Energy of Atoms in a Self-Generated Field

233

with some r = D Z −1/3 with D ≥ 1.

(1.8)

Note that instead of the local field energy, the total local H 1 -norm of A is added in (1.7). By (1.2), we have H Ncl,Z (A) ≥ H N ,Z (A)

(1.9)

for any A ∈ A. We define the ground state energy of the modified Hamiltonian (1.7), N

∞ 3 2 E N ,Z (A) := inf , H N ,Z (A) : ∈ C0 (R ) ⊗ C , = 1 , 1

and set E N ,Z := inf E N ,Z (A), A∈A

E Z := inf E N ,Z , N

where the infimum for A ∈ A can again be restricted to compactly supported vector potentials A ∈ A0 . For the modified classical Hamiltonian we have the following theorem: Theorem 1.2. Let Z α 2 ≤ κ and assume that r = D Z −1/3 with 1 ≤ D ≤ Z 1/63 . Then nf 7/3 −1 E nf D . Z ≥ EZ ≥ EZ − C Z

(1.10)

Taking into account (1.9), Theorem 1.2 immediately implies the lower bound in (1.4). The proof of the lower bound in (1.6) follows from Theorem 1.2 adapting an argument in [BFG] that we will review in Sect. 6 for completeness. One of the key ingredients of the proof of Theorem 1.2 is the following semiclassical statement that is of interest in itself. The first version is formulated under general conditions but without an effective error term. In our proof we actually use the second version that has a quantitative error term. Theorem 1.3. Let Th (A) = [σ · (hp + A)]2 or Th (A) = (hp + A)2 , h ≤ 1, and V ≥ 0. 1) If V ∈ L 5/2 (R3 ) ∩ L 4 (R3 ), then

Tr [Th (A) − V ]−+ h −2 B2 ≥ Tr −h 2 − V +o(h −3 ) as h → 0. (1.11) R3

−

2) Assume that V ∞ ≤ K with some 1 ≤ K ≤ Ch −2 and consider the operators with Dirichlet boundary condition on ⊂ R3 . Let B R√denote the ball of radius R about the origin and let √h := + B√h denote the h-neighborhood of the set . We set |√h | for the Lebesgue measure of √h . Then B2 Tr [(Th (A) − V ) ]− + h −2 3 R

≥ Tr (−h 2 − V ) −

1/2

1/2 −3 5/2 √ 3/2 3/2 1 + hK . (1.12) −Ch K | h | h K

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L. Erd˝os, J. P. Solovej

Remark. Despite the electrons being confined to , their motion generates a magnetic field in the whole R3 , so the magnetic field energy in (1.12) is given by integration over R3 . We use the convention that letters C, c denote positive universal constants whose values may change from line to line. 2. Reduction to the Main Lemmas Proof of Theorem 1.2. We focus on the lower bound, the upper bound is trivial. We start with two localizations, one on scale r ≥ Z −1/3 and the other one on scale d ≤ Z −1/3 . The first one is designed to address the difficulty that the H 1 -norm of A is available only locally around the nucleus. This step would not be needed for the direct proof of (1.4). The second localization removes the “Coulomb tooth”, i.e. the Coulomb singularity near the nucleus. In this section we reduce the proof of the lower bound in Theorem 1.2 to two lemmas. Lemma 2.1 will show that the Coulomb tooth is indeed negligible. Lemma 2.2 shows that the magnetic field cannot substantially lower the energy for the problem without the Coulomb tooth. In the proof of Lemma 2.2 we will use Theorem 1.3. Recall that B R denotes the ball of radius R about the origin. We construct a pair of smooth cutoff functions satisfying the following conditions: θ02 + θ12 ≡ 1, supp θ1 ⊂ B2d , θ1 ≡ 1 on Bd , |∇θ0 |, |∇θ1 | ≤ Cd −1 . We will choose d = δ Z −1/3

(2.13)

with some δ ≤ 1, in particular d ≤ r . We split the Hamiltonian as H N ,Z (A) = H N0 ,Z (A) + H N1 ,Z (A) with H N0 ,Z (A)

N

Z 2 2 θ0 T (A) − − |∇θ0 | + |∇θ1 | θ0 = |x| i i=1 1 1 + + |∇ ⊗ A|2 , |xi − x j | 16π α 2 B3r

(2.14)

i< j

H N1 ,Z (A) =

N

Z θ1 T (A) − − |∇θ0 |2 + |∇θ1 |2 θ1 |x| i i=1 1 + |∇ ⊗ A|2 , 16π α 2 B3r

where we used the IMS localization formula that is valid for the Pauli operator as well as for the Schrödinger operator. In Sect. 3 we deal with H N1 ,Z , to prove that it is negligible:

Energy of Atoms in a Self-Generated Field

235

Lemma 2.1. There is a positive universal constant κ such that for any Z , α with Z α 2 ≤ κ we have inf inf H N1 ,Z (A) ≥ −C Z 7/3 δ 1/2 − Z 2/3 δ −2 N A∈A0

if C Z −2/3 ≤ δ ≤ D with a sufficiently large constant C. Starting Sect. 4 we will treat H N0 ,Z (A) and we prove the following: Lemma 2.2. There is a positive universal constant κ such that for any Z , α with Z α 2 ≤ κ we have

inf inf H N0 ,Z (A) ≥ −cTF Z 7/3 − C Z 7/3 Z −1/30 + D −1 (2.15) N

A

with a sufficiently large constant C if Z −1/6 ≤ δ ≤ 1 and D ≤ Z 1/24 δ 13/16 . The main ingredient in the proof is Theorem 1.3 that will be proven in Sect. 5. The proof of the lower bound in Theorem 1.2 then follows from Lemmas 2.1 and 2.2 after choosing δ = Z −2/63 . 3. Estimating the Coulomb Tooth Proof of the Lemma 2.1. Let χ 0 be a smooth cutoff function supported on B3r such that |∇ χ0 | ≤ Cr −1 and χ 0 ≡ 1 on B2r . Let A := |B3r |−1 B3r A. We define A0 := (A − A ) χ0 ,

B0 := ∇ × A0 ,

(3.16)

0 ∇ ⊗ A + (A − A ) ⊗ ∇ χ0 . Clearly then ∇ ⊗ A0 = χ 2 2 2 2 −2 B0 ≤ |∇ ⊗ A0 | ≤ 2 χ 0 |∇ ⊗ A| + Cr (A − A )2 R3 R3 R3 B3r ≤ C1 |∇ ⊗ A|2 (3.17) B3r

for some universal constant C1 , where in the last step we used the Poincaré inequality. Let ϕ be a real phase such that ∇ϕ = A . Since χ 0 ≡ 1 on the support of θ1 by D ≥ δ, we have θ1 T (A)θ1 = θ1 e−iϕ T (A − A )eiϕ θ1 = θ1 e−iϕ T (A0 )eiϕ θ1 . After these localizations, we have H N1 ,Z (A) ≥

N

θ1 e−iϕ (T (A0 ) − W (x)) eiϕ θ1 + j

j=1

with

W (x) =

Z + Cd −2 1(|x| ≤ 2d). |x|

1 2C1 α 2

B20

(3.18)

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L. Erd˝os, J. P. Solovej

Now we use the “running energy scale” argument in [LLS]. ∞ N

θ1 e−iϕ [T (A0 ) − W ] eiϕ θ1 ≥ − N−e (T (A0 ) − W )de j=1

j

µ

≥−

0 µ

≥− 0

N−e (T (A0 ) − W )de −

0

∞

µ ∞

N0

N−e (T (A0 ) − W )de −

µ

µ

N0

e

T (A0 ) − W + e de

e2 e T (A0 ) − W + µ µ

de,

(3.19)

where N−e (A) denotes the number of eigenvalues of a self-adjoint operator A below −e. In the first term we use the bound T (A0 ) ≥ (p + A0 )2 − |B0 | and the CLR bound: µ µ 3/2 N−e (T (A0 ) − W )de ≤ C de (W + |B0 | − e)+ R3 0 0 µ 3/2 ≤C de (W − e/2)+ 3 0 µ R 3/2 +C de (|B0 | − e2 /2µ)+ 3 R 0 5/2 ≤C W + Cµ1/2 B20 R3 R3 5/2 1/2 −2 1/2 = C Z d + Cd + Cµ B20 . (3.20) R3

In the second term of (3.19) we use 1 e 2eZ 2 (p + A0 ) − 1(|x| ≤ 2d) T (A0 ) − W ≥ µ 2 µ|x| 1 Ce + (p + A0 )2 − |B0 | − 1(|x| ≤ 2d), 2 µd 2 and that (p + A0 )2 − i.e. T (A0 ) −

2eZ 4eZ 2eZ 2 , 1(|x| ≤ 2d) ≥ (p + A0 )2 − ≥− µ|x| µ|x| µ

(3.21)

2 1 e eZ Ce W ≥ (p + A0 )2 − 2 − |B0 | − 1(|x| ≤ 2d). µ 2 µ µd 2

We choose µ = 4Z 2 , then using Ce/µd 2 ≤ e2 /4µ for µ ≤ e (i.e. C ≤ (δ Z 2/3 )2 ), we get ∞ ∞ 1 e2 e e2 2 de ≤ (p + A0 ) − |B0 | + de N0 T (A0 ) − W + N0 µ µ 2 4µ µ µ µ 3/2 ≤C de (|B0 | − e2 /4µ)+ 3 0 R 1/2 ≤ Cµ B20 . (3.22) R3

Energy of Atoms in a Self-Generated Field

237

Note that if Z α 2 ≤ κ with some sufficiently small universal constant κ, then the magnetic energy terms in (3.20) and (3.22) can be controlled by the corresponding term in (3.18). Combining the estimates (3.18), (3.19), (3.20) and (3.22) we obtain H N1 ,Z (A) ≥ −C Z 5/2 d 1/2 − Cd −2 and Lemma 2.1 follows.

(3.23)

4. Removing the Magnetic Field Proof of Lemma 2.2. We start with two preparations. In Sect. 4.1 we give an upper bound for the number of electrons N in the truncated model described by H N0 ,Z (A). In Sect. 4.2 we then reduce the problem to a one-body semiclassical statement on boxes. The semiclassical problem will be investigated in Sect. 5 and this will complete the proof of Lemma 2.2. 4.1. Upper bound on the number of electrons N . Let N

E 0N ,Z (A) := inf , H N0 ,Z (A) : ∈ C0∞ (R3 ) ⊗ C2 , = 1 1

be the ground state energy of the truncated Hamiltonian H N0 ,Z (A) defined in (2.14). The following lemma shows that we can assume N ≤ C Z when taking the infimum over N in (2.15). The proof is a slight modification of the proof of the Ruskai-Sigal theorem as presented in [CFKS]. We note that the original proof was given for the non-magnetic case and it can be trivially extended to the Schrödinger operator with a magnetic field but not to the Pauli operator. This is because a key element of the proof, the standard lower bound on the hydrogen atom, − − Z /|x| ≥ −Z 2 /4, is valid if − = p2 replaced by (p + A)2 but there is no lower bound for the ground state energy of the hydrogen atom with the Pauli kinetic energy that is independent of the magnetic field. However, for the truncated Coulomb potential the trivial lower bound can be used. Lemma 4.1. There exist universal constants c and C such that for any fixed A ∈ A0 and Z we have E 0N ,Z (A) = E 0N −1,Z (A) whenever N ≥ C Z and Z −1/6 ≤ δ ≤ c. In particular inf inf E 0N ,Z (A) = inf N A∈A0

inf E 0N ,Z (A)

N ≤C Z A∈A0

(4.24)

if Z −1/6 ≤ δ ≤ c. Proof. We mostly follow the proof of Theorem 3.15 in [CFKS] and we will indicate only the necessary changes. For any x = (x1 , x2 , . . . x N ) ∈ R3N we define x∞ (x) := max{|xi |, : i = 1, 2, . . . , N }, A0 := {x : |x j | < ∀ j = 1, 2, . . . N }, Ai := x : |xi | ≥ (1 − ζ )x∞ (x), x∞ (x) > 2

238

L. Erd˝os, J. P. Solovej

for some fixed positive and ζ < 1/2 to be chosen later. According to Lemma 3.16 in [CFKS], there is partition of unity {Ji }i=0,1,...N , with i Ji2 ≡ 1, supp Ji ⊂ Ai such that the gradient estimates L(x) =

N

|∇ Ji (x)|2 ≤

C N 1/2 2

|∇ Ji (x)|2 ≤

C N 1/2 x∞ (x)

i=0

L(x) =

N i=0

if x ∈ A0 , if x ∈ A j , j ≥ 1

hold with a suitable universal constant C. Moreover, J0 is symmetric in all variables, while Ji , i ≥ 1, is symmetric in all variables except xi . We subtract the local field energy that is an irrelevant constant, i.e. define 1 0 H N := H N ,Z (A) − |∇ ⊗ A|2 16π α 2 B3r and E N = inf Spec H N . We will show that E N = E N −1 for N ≥ C Z . By removing one electron to infinity, clearly E N ≤ E N −1 ≤ 0; here we used the fact that A is compactly supported. By the IMS localization H N = J0 (H N − L)J0 +

N

Ji (H N − L)Ji .

(4.25)

i=1

In the first term we use that on the support of θ0 we have −Z |x|−1 ≥ −Z d −1 . Hence N (N − 1) C N 1/2 −1 −2 J0 . (4.26) − J0 (H N − L)J0 ≥ J0 −C Z N d − C N d + 4 2 Choosing = 8d we see that J0 (H N − L)J0 ≥ 0 if N ≥ C Z with a constant C if δ ≥ C Z −2/3 . To estimate the terms Ji (H N − L)Ji for i = 0, we define H N(i)−1

N

Z 1 2 2 θ0 T (A) − − |∇θ0 | + |∇θ1 | . θ0 + := |x| |x − xj| k j j=1 k< j j =i

k, j =i

On the support of Ji we have |xi | ≥ /4 = 2d, so ∇θ0 and ∇θ1 vanish. Then we can estimate N −1 C N 1/2 Z (i) + − Ji Ji (H N − L)Ji ≥ Ji H N −1 − |xi | 2x∞ (x) x∞ (x) C N 1/2 Z 1/3 1 N −1 (1 − ζ ) − Z − Ji ≥ Ji E N −1 + |xi | 2 δ ≥ Ji E N −1 Ji (4.27) if N ≥ C Z and N is large. Thus we conclude from (4.25), (4.26) and (4.27) that E N ≥ E N −1 if N ≥ C Z .

Energy of Atoms in a Self-Generated Field

239

4.2. Reduction to a one-body problem. We start by presenting an abstract lemma whose proof is given in Appendix A. Lemma 4.2. Let h be a one-particle operator on H = L 2 (R3 ) and let W be a twoparticle operator defined on H ∧ H. We assume that the domains of h and W include the C0∞ functions. Let θ ∈ C ∞ (R3 ) with compact support := supp θ . Then ⎧ ⎡ ⎫ ⎤ N N ⎨ ⎬ inf , ⎣ θi hi θi + θi θ j Wi j θ j θi ⎦ : ∈ C0∞ (R N ), = 1 ⎩ ⎭ 1 i=1 1≤i< j≤N ⎧ ⎛ ⎫ ⎞ n n ⎨ ⎬ ≥ inf inf , ⎝ hi + Wi j ⎠ : ∈ C0∞ (), = 1 , ⎩ ⎭ n≤N i=1

1

1≤i< j≤n

(4.28) where hi denotes the operator h acting on the component of the tensor product, and similar convention is used for the two-particle operators. The same result holds with obvious changes if H = L 2 (R3 ) ⊗ C2 . i th

To continue the proof of Lemma 2.2, we first localize H N0 ,Z (A) onto a ball Br of radius r = D Z −1/3 (see (1.8)) and we also localize the magnetic field as in Sect. 3. We introduce smooth cutoff functions χ0 and χ1 with χ02 + χ12 ≡ 1, supp χ0 ⊂ B2r , χ0 ≡ 1 on Br , |∇χ0 |, |∇χ1 | ≤ Cr −1 . We get H N0 ,Z (A)

≥

N

θ0 χ0 e

i=1

1 + 2 Cα

−iϕ0

1 Z Ti (A0 ) − eiϕ0 χ0 θ0 + |xi | |xi − x j | i< j

R3

B20 − C N d −2 − C N Zr −1

(4.29)

using that the new localization error |∇χ1 |2 + |∇χ0 |2 ≤ Cr −2 ≤ Cd −2 and that −Z /|x| ≥ −Zr −1 on the support of χ1 . We also used (3.17). Let Ad,r = {x : d ≤ |x| ≤ r } ⊂ R3 . Using (4.24), the positivity of the Coulomb repulsion |xi − x j |−1 > 0 and Lemma 4.2 with θ := θ0 χ0 we obtain ⎧ ⎪ ⎨

⎡

inf inf H N0 ,Z (A) ≥ inf inf inf , ⎣ N A∈A0 N ≤C Z A0 ∈A0 ⎪ ⎩

+

⎫ ⎪ ⎬

⎤ 1 Z ⎦ T (A0 ) − + |x| i |x − x j | i=1 i< j i Ad,r

N

1 B2 − C Z 5/3 δ −2 − C Z 7/3 D −1 , Cα 2 R3 0 ⎪ ⎭

(4.30)

where the infimum is over all antisymmetric wave functions ∈ 1N C0∞ (Ad,r ) ⊗ C2 with 2 = 1. The notation [H ] Q indicates the N -particle operator H with Dirichlet boundary condition on the domain Q N ⊂ R3N . We define f (x)g(y) 1 dxdy. D( f, g) := 2 R3 ×R3 |x − y|

240

L. Erd˝os, J. P. Solovej

Lemma 4.3. There is a universal constant C0 > 0 such that for any ∈ 1N C0∞ (R3 )⊗ C2 with 2 = 1, for any nonnegative function : R3 → R with D(, ) < ∞, for any A ∈ A0 , and for any ε > 0 we have

⎡

⎤

1 ⎦ + C0 Ti (A) + B2 , ⎣ε 3 |xi − x j | R i=1 i< j N

∗ |xi |−1 − Cε−1 N . ≥ −D(, ) + , N

(4.31)

i=1

Proof. By the Lieb-Oxford inequality [LO] and by the positivity of the quadratic form D(·, ·), 1 4/3 ≥ D( , ) − C , 3 |xi − x j | R i< j N 4/3 ≥ −D(, ) + , ( ∗ |xi |−1 ) − C ,

(4.32)

R3

i=1

where (x) is the one-particle density of . The error term is controlled by the following kinetic energy inequality for the Pauli operator

% ,

N

&

T (A)i ≥ c

i=1

5/3

R3

4/3

min{ , γ } − γ

B2

(4.33)

R3

with some positive universal constant c and for any γ > 0. For the proof of (4.33) use the magnetic Lieb-Thirring inequality

%

& N , [T (A) − U ]i ≥ −C i=1

U R3

5/2

− Cγ

−3

U −γ 4

R3

R3

B2 .

With the choice U = β min{ , γ } we can ensure that 21 U ≥ CU 5/2 +Cγ −3 U 4 if β is sufficiently small (independent of γ ) and this proves (4.33). Thus 4/3 5/3 4/3 ≤ γ −1 min{ , γ } + γ R3 R3 R3 % N & −1 ≤ (cγ ) T (A)i + c−1 B2 + γ N (4.34) , 2/3

1/3

i=1

R3

so choosing γ = Cε−1 with a sufficiently large constant C, we obtain (4.31).

Energy of Atoms in a Self-Generated Field

241

Using Lemma 4.3 we can continue the estimate (4.30) (writing A instead of A0 in the infimum) as inf inf H N0 ,Z (A) N

A

⎧ % & N ⎨ [T (A) + W ]i ≥ (1 − ε) inf inf inf , N ≤C Z A∈A0 ⎩ i=1

Ad,r

1 + Cα 2

B2 R3

−D(, ) − Cε−1 Z − C Z 5/3 δ −2 − C Z 7/3 D −1 with W (x) :=

⎫ ⎬ ⎭

(4.35)

Z 1 − + ∗ |x|−1 1−ε |x|

and assuming that α ≤ α0 with some small universal α0 . We now perform a rescaling: x = Z −1/3 X , p = Z 1/3 P and A(X ) = Z −2/3 A(Z −1/3 X ),

B(X ) = ∇ × A(X ) = Z −1 B(Z −1/3 X ).

A) := [(hP + A) · σ ]2 , we obtain that the kinetic energy Introducing h = Z −1/3 and Th ( changes as A) · σ ]2 = Z 4/3 Th ( A) [(p + A) · σ ]2 = Z 4/3 [(Z −1/3 P + and the field energy changes as 2 B (x)dx = Z R3

R3

B2 (X )dX.

The new potential energy is (X ) = Z −4/3 W (Z −1/3 X ) = W

1 1 − + ∗ |X |−1 , 1−ε |X |

where (X ) = Z −2 (Z −1/3 X ) and D( , ) = Z −7/3 D(, ). After rescaling, we get from (4.35), inf inf H N0 ,Z (A) N A∈A0

⎧ % & N ⎨ 4/3 ]i ≥ (1 − ε)Z inf inf inf , [Th ( A) + W N ≤C Z A∈A0 ⎩ i=1

+ Aδ,D

−Z 7/3 D( , ) − Cε−1 Z − C Z 5/3 δ −2 − C Z 7/3 D −1 ,

h −2 C Z α2

B2

R3

⎫ ⎬ ⎭

(4.36)

where Aδ,D = {X : δ ≤ |X | ≤ D} and inf denotes the infimum over all normalized antisymmetric functions. Using (1.12) from Theorem 1.3 and the fact that Z α 2 ≤ κ, we get

) Aδ,D − C Z 13/6 D 3 δ −13/4 inf inf H N0 ,Z (A) ≥ (1 − ε)Z 4/3 Tr (−h 2 + W N

A

−

−Z 7/3 D( , ) − Cε−1 Z − C Z 5/3 δ −2 − C Z 7/3 D −1 , (4.37)

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assuming δ ≥ Z −2/9 . By a standard semiclassical result for Coulomb-like potentials (see e.g. the result in Sect. V.2 of [L]):

2 2 −3 −5/2 − Ch −3+1/10 , ≥ Tr −h + W ≥ −Csc h Tr (−h + W ) Aδ,D W −

−

R3

(4.38) 1 where Csc = 2/(15π 2 ) is the Weyl constant in semiclassics. The 10 exponent in the error term is far from being optimal; the methods developed to prove the Scott correction can yield an exponent up to one (see Remark 1 after Theorem 1.1). Taking the optimal to be the Thomas-Fermi density for Z = 1 = TF (see, e.g. Sect. II of [L]) and defining the Thomas-Fermi constant as

5/2 1 −1 , − + TF ∗ |X | |X | R3 −

cTF := D(TF , TF ) + C SC we get ' inf inf N

A

H N0 ,Z (A)

≥ (1 − ε)

−3/2

Z

7/3

−D( , ) − Csc

5/2 ( 1 −1 − + ∗ |X | |X | R3 −

−C Z 7/3−1/30 − C Z 13/6 D 3 δ −13/4 − Cε−1 Z − C Z 5/3 δ −2 − C Z 7/3 D −1 ≥ −(1 − ε)−3/2 cTF Z 7/3 − C Z 7/3−1/30 − Cε−1 Z − C Z 5/3 δ −2 − C Z 7/3 D −1

(4.39) ≥ −cTF Z 7/3 − C Z 7/3 Z −1/30 + D −1 , where we optimized for ε and we used that D ≤ Z 1/24 δ 13/16 and Z −1/6 ≤ δ ≤ 1. This completes the proof of Lemma 2.2. 5. Semiclassics: Proof of Theorem 1.3 We present the Schrödinger and Pauli cases in parallel. We prove the statement with Dirichlet boundary conditions (1.12) in detail and in Sect. 5.4 we comment on the necessary changes for the proof of the (1.11). The potential V is defined only on , but we extend it to be zero on R3 \; we continue to denote by V its extension. 5.1. Localization onto boxes. We choose a length L with h ≤ L ≤ 13 h 1/2 . Let L = + B L be the L-neighborhood of . Let Q k = {y ∈ R3 : y − k∞ < L/2} with k ∈ (LZ)3 ∩ L denote a non-overlapping covering of with boxes of size L. In this sectionthe index k will always run over the set (LZ)3 ∩ L . Let ξk be a partition of unity, k ξk2 ≡ 1, subordinated to the collection of boxes Q k , such that supp ξk ⊂ (2Q)k ,

|∇ξk | ≤ C L −1 ,

where (2Q)k denotes the cube of side-length 2L with center k. Let ξk be a cutoff funck := (3Q)k and tion such that ξk ≡ 1 on (2Q)k (i.e. on the support of ξk ), supp ξk ⊂ Q |∇ ξk | ≤ C L −1 .

Energy of Atoms in a Self-Generated Field

k |−1 Let A k = | Q Bk := ∇ × Ak , then

k Q

243

A. Similarly to (3.16), we define Ak := (A − A k ) ξk and

R3

B2k ≤ C

k Q

|∇ ⊗ A|2

(5.40)

as in (3.17). From the IMS localization with ψk satisfying h∇ψk = Ak we have ∗ −2 2 −2 Tr [[Th (A) − V ] ]− + h B = inf Tr (γ [Th (A) − V ]) + h |∇ ⊗ A|2 R3

γ

≥ inf γ

R3

∗

Ek (γ )

k∈(LZ)3 ∩

(5.41)

L

with

Ek (γ ) := Tr γ ξk e−iψk [Th (A − A k ) − V ]eiψk ξk − γ |h∇ξk |2 + c0 h −2

k Q

|∇ ⊗ A|2

with some universal constant c0 . Here inf ∗γ denotes infimum over all density matrices 0 ≤ γ ≤ 1 that are supported on , i.e. they are operators on L 2 () ⊗ C2 . We also used R3 B2 = R3 |∇ ⊗ A|2 and we reallocated the second integral. We introduce the notation −2 Fk := c0 h |∇ ⊗ A|2 . k Q

5.2. A priori bound on the local field energy. In case of the Pauli operator, for each fixed k we apply the magnetic Lieb-Thirring inequality [LLS] together with (5.40) and box Q with the bound V ∞ ≤ K to obtain that for any density matrix γ ,

Ek (γ ) ≥ Tr [Th (Ak ) − V − Ch 2 L −2 ] Qk + Fk − ≥ −Ch −3 [V + Ch 2 L −2 ]5/2 k Q

−C

k Q

[V + Ch 2 L −2 ]4

1/4

h −2

3/4

k Q

B2k

+ Fk

c 0 −2 h ≥ −C h −3 K 5/2 L 3 + h 2 L −2 + K 4 L 3 + h 8 L −5 − |∇ ⊗ A|2 + Fk 2 k Q 1 ≥ −Ch −3 K 5/2 L 3 + Fk , (5.42) 2 using h ≤ L and 1 ≤ K ≤ Ch −2 . In the Schrödinger case we use the usual LiebThirring inequality [LT] that holds with a magnetic field as well. The estimate (5.42) is then valid even without the third term in the second line. Let S ⊂ (LZ)3 ∩ L denote the set of those k indices such that Fk ≤ Ch −3 K 5/2 L 3

(5.43)

holds with some large constant C. In particular Ek (γ ) ≥ 0, for all k ∈ S and for any γ .

(5.44)

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5.3. Improved bound. We use the Schwarz inequality in the form Th (A − A k ) ≥ −(1 − εk )h 2 − Cεk−1 (A − A k )2 , with some 0 < εk < 13 . We have for any γ supported on that

Ek (γ ) ≥ Tr 1 ξk [−(1 − 2εk )h 2 − V − Ch 2 L −2 ]ξk 1 −

−1 2 2 +Tr 1 Qk [−εk h − Cεk (A − A k ) ]1 Qk + Fk .

(5.45)

We will show at the end of the section that

Tr 1 ξk [−(1 − 2εk )h 2 − V − Ch 2 L −2 ]ξk 1 −

2 −3 5/2 k |. ≥ Tr 1 ξk (−h − V )ξk 1 − Ch K (εk + h 2 L −2 )| Q

(5.46)

−

−

Using (5.44) and (5.46), ∗ ∗ inf Ek (γ ) ≥ inf Ek (γ ) γ

γ

k

≥

k

≥

k∈S

k

Tr 1 ξk [−h 2 − V ]ξk 1 + Dk −

k∈S

inf Tr ξk γk ξk 1 [−h 2 − V ]1 + Dk γk

≥ Tr (−h 2 − V ) + Dk −

k∈S

(5.47)

k∈S

with

k |(εk + h 2 L −2 ) + Fk . Dk := Tr [−εk h 2 − Cεk−1 (A − A k )2 ] Qk − Ch −3 K 5/2 | Q −

(5.48) In the last step in (5.47) we used that for any collection of density matrices γk , the density matrix k 1 ξk γk ξk 1 is admissible in the variational principle

Tr (−h 2 − V ) = inf Tr γ −h 2 − V : 0 ≤ γ ≤ 1, supp γ ⊂ . −

(5.49) We estimate Dk for k ∈ S as follows: −4 −3 k |(εk + h 2 L −2 ) + Fk Dk ≥ −Cεk h (A − A k )5 − Ch −3 K 5/2 | Q k Q

≥ Fk − Cεk−4 h 2 L 1/2 Fk

5/2

k |(εk + h 2 L −2 ). − Ch −3 K 5/2 | Q

(5.50)

In the first step we used the Lieb-Thirring inequality, in the second step the Hölder and Sobolev inequalities in the form 5/2 5 1/2 2 (A − A k ) ≤ C L |∇ ⊗ A| . k Q

k Q

Energy of Atoms in a Self-Generated Field

245

We choose εk = h L −1/2 K −1/2 Fk , 1/2

and using the a priori bound (5.43), we see that εk ≤ Ch −1/2 L K 3/4 . Thus, assuming L ≤ ch 1/2 K −3/4

(5.51)

with a sufficiently small constant c, we get εk ≤ 1/3. With this choice of εk , and recalling k | = 9|Q k | = 9L 3 , we have |Q Dk ≥ Fk − Ch −2 L 5/2 K 2 Fk − Ch −3 K 5/2 L 3 h 2 L −2

≥ −Ch −3 L 3 K 5/2 h −1 L 2 K 3/2 + h 2 L −2 . 1/2

(5.52)

If we choose L = h 3/4 K −3/8 , then

1/2 Dk ≥ −Ch −3 L 3 K 5/2 h K 3/2 . This choice is allowed by (5.51) if K ≤ ch −2/3 . If ch −2/3 ≤ K ≤ h −2 , then we choose L = ch 1/2 K −3/4 and we get from (5.52), Dk ≥ −Ch −3 L 3 K 5/2 (1 + h K 3/2 ). Combining these two inequalities, we get that

1/2

1/2 Dk ≥ −Ch −3 L 3 K 5/2 h K 3/2 1 + h K 3/2

(5.53)

always holds. Summing up (5.53) for all k and using that L 3 ≤ C|3L | ≤ C|√h | k∈(LZ)3 ∩ L

(recall that √h is a and (5.53), ∗

inf γ

√

k

h-neighborhood of and 3L ≤ h 1/2 ), we obtain from (5.47)

Ek (γ ) ≥ Tr (−h 2 − V )

−

1/2

1/2 1 + h K 3/2 , −Ch −3 K 5/2 |√h | h K 3/2

(5.54)

and this proves (1.12). Finally, we prove (5.46). Let γ be a trial density matrix for the left hand side of (5.46). We can assume that

0 ≥ Tr γ 1 ξk [−(1 − 2εk )h 2 − V − Ch 2 L −2 ]ξk 1 .

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L. Erd˝os, J. P. Solovej

Then

1 1 0 ≥ Tr γ 1 ξk [− h 2 + K ]ξk 1 + Tr γ 1 ξk [− h 2 −V −Ch 2 L −2 − K ]ξk 1 6 6 1 2 ≥ Tr γ 1 ξk [− h + K ]ξk 1 − Ch −3 [V + K + Ch 2 L −2 ]5/2 , (5.55) 6 k Q

where we used the Lieb-Thirring inequality. Thus, using |V | ≤ K , h ≤ L and K ≥ 1, we have 1 2 k |. Tr γ 1 ξk [− h + K ]ξk 1 ≤ Ch −3 K 5/2 | Q 6 Therefore

Tr γ 1 ξk [−(1 − 2εk )h 2 − V − Ch 2 L −2 ]ξk 1

k |. ≥ Tr γ 1 ξk (−h 2 − V )ξk 1 − Ch −3 K 5/2 (εk + h 2 L −2 )| Q

Now (5.46) follows by variational principle.

(5.56)

5.4. Reduction of (1.11) to (1.12). We approximate V ∈ L 5/2 ∩ L 4 by a bounded , V ∞ ≤ K , that is supported on a ball B R/2 and V ≤ V . By choosing K potential V and R sufficiently large, we can make V − V 5/2 +V − V 4 arbitrarily small. We choose a cutoff function χ R that is supported on B R , χ R ≡ 1 on B R/2 and |∇χ R | ≤ C R −1 and R2 ≡ 1. let χ R satisfy χ R2 + χ Borrowing a small part of the kinetic energy, by IMS localization we have ]χ R Th (A) − V ≥ (1 − ε)χ R [Th (A) − V ) − |∇χ R |2 − |∇ +εTh (A) − (V − (1 − ε)V χ R |2 .

(5.57)

Using the magnetic Lieb-Thirring inequality [LLS] to estimate the second term, we get ) B R ]− Tr [Th (A) − V ]− ≥ (1 − ε)Tr [(Th (A) − V −3/2 −3 5/2 −Cε h |U | − C

1 |U | − h −2 3 2 R

R3

4

B2 (5.58) R3

with ) + |∇χ R |2 + |∇ U := (V − (1 − ε)V χ R |2 . For 2the first term in (5.58) we use (1.12) (and that it holds even with a 1/2 in front of B ) and the fact that

) B R Tr (−h 2 − V ≥ Tr −h 2 − V −

−

≤ V . The second and the third terms in (5.58) can be made arbiby monotonicity, V trarily small compared with h −3 for any fixed ε if R and K are sufficiently large and h is small. Finally, choosing ε sufficiently small, we proved (1.11).

Energy of Atoms in a Self-Generated Field

247

6. Proof of the Quantized Field Case For the proof of the lower bound in (1.6), we follow the argument of [BFG] to reduce the problem to the classical bound (1.10). We set |g(k)|2 |k| aλ (k)∗ aλ (k)dk Hg = α −1 R3

λ=±

to be the cutoff field energy, then H f ≥ Hg and only the modes appearing in Hg interact with the electron. By Lemma 3 of [BFG], for any real function f ∈ L 1 (R3 ) ∩ L ∞ (R3 ) we have 1 f (x)|∇ ⊗ A(x)|2 dx ≤ α 2 f ∞ Hg + Cα4 f 1 . 8π R3 Applying it with f being the characteristic function of the ball B3r with r = D Z −1/3 (the radius of the ball is here chosen differently from [BFG]) and using Z α 2 ≤ κ we get 2 Zα Z H f ≥ Hg ≥ Hg ≥ |∇ ⊗ A(x)|2 dx − Cκ −1 α4 D 3 . κ 8π κ B3r Setting α = (κ/Z )1/2 , i.e. Z α 2 = κ, we have for κ sufficiently small, −1 4 3 E N ,Z ≥ E N ,Z , α − Cκ α D , qed

where E N ,Z , α is the ground state energy of the Hamiltonian (1.7) with fine structure constant α . Applying (1.10) to this Hamiltonian, we get qed

EZ

7

−1 3 ≥ E nf − Cκ −1 α4 D 3 Z − CZ D

1

whenever 1 ≤ D ≤ Z 63 . Writing D = Z γ and applying the upper bound (1.5) on , we obtain the lower bound in (1.6). A. Proof of Lemma 4.2 Let the function χ (x) ∈ C ∞ (R3 ) be defined such that θ 2 (x)+χ 2 (x) ≡ 1. For any subset α ⊂ {1, 2, . . . , N } we denote by xα the collection of variables {xi : i ∈ α} and define ) ) α = α (xα ) := θ (xi ), α = α (xα ) := χ (xi ). i∈α

i∈α

We set the notation α c = {1, 2, . . . , N } \ α for the complement of the set α and set n := {1, 2, . . . , n}. Let |α| denote the cardinality of the set α. For an arbitrary function ∈ 1N C0∞ (R3 ), = 1, and for 0 ≤ n ≤ N we define * + , n := n Tr n c n c | |n c n , where Tr n c denotes taking the partial trace for the xn+1 , xn+2 , . . . , x N variables. Define .N the fermionic Fock space as F = n=0 Hn with Hn := n H and we define a density matrix N N n |α| on F. := = n α⊂{1,2,...,N }

n=0

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We first prove that ≤ I on F. It is sufficient to show that ≤ I on the n-particle sectors for each n. Let n ≤ N , choose ∈ Hn , and compute , |α| = dxα dxα (xα ) n (xα , xα )(xα ) α⊂{1,2,...N } |α|=n

=

α⊂{1,2,...N } |α|=n

dxα dxα dyα c (xα )α (xα )α c (yα c ) (xα , yα c )

α

× (xα , yα c )α c (yα c )α (xα )(xα ) ≤ dxα dxα dyα c 2α (xα )2α c (yα c )| (xα , yα c )|2 |(xα )|2 α

=

22

dx| (x)|2

α

= 22 ,

2α (xα )2α c (xα c )

/N

(A.59)

using Schwarz inequality and that 1 ≡ j=1 [θ 2 (x j )+χ 2 (x j )] = α 2α (xα )2α c (xα c ). Second, for a fixed n ≤ N , we compute % & % & N N n n 0 N n Tr F Tr F hi = hi n n=0 i=1 n=0 i=1 = dxα dxα c (xα , xα c )α (xα )α c (xα c )hi (α (xα )α c (xα c ) (xα , xα c )) α

=

N

i∈α

i=1 α : i∈α

=

N

dx (xα , xα c )2α\{i} (xα\{i} )2α c (xα c )θ (xi )hi (θ (xi ) (xα , xα c ))

, θi hi θi ,

(A.60)

i=1

where the trace on the left/hand side is computed on F. In the last step we used that for 2 2 , where the summation any fixed i, we have 1 ≡ j=i [θ 2 (x j ) + χ 2 (x j )] = 1 α 1 α 1 αc c is over all 1 α ⊂ {1, 2, . . . , N }\{i} and 1 α = {1, 2, . . . , N }\{i}\α. A similar calculation for the two-body potential shows that ⎡ ⎤ N 0 2 3 Tr F ⎣ , θi θ j Wi j θ j θi . Wi j ⎦ = n=0 1≤i< j≤n

1≤i< j≤N

Thus, by the variational principle, ⎛ N , ⎝ θi hi θi + i=1

⎞

n=0

i=1

θi θ j Wi j θ j θi ⎠

1≤i< j≤N

⎛ N n 0 ⎝ ≥ inf Tr F ⎣ hi +

⎡

⎞⎤ Wi j ⎠⎦.

1≤i< j≤n

Since is a density matrix supported on , we obtain (4.28).

Energy of Atoms in a Self-Generated Field

249

References [BFG] [CFKS] [ES]

[ES2] [F] [FLL] [FS] [H] [IS] [L] [LL] [LLS] [LO] [LS] [LSY1] [LSY2] [LT]

[LY] [SW1] [SW2] [SW3] [SS]

Bugliaro, L., Fröhlich, J., Graf, G.M.: Stability of quantum electrodynamics with nonrelativistic matter. Phys. Rev. Lett. 77, 3494–3497 (1996) Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Berlin-Heidelberg-New York: Springer-Verlag, 1987 Erd˝os, L., Solovej, J.P.: Semiclassical eigenvalue estimates for the pauli operator with strong non-homogeneous magnetic fields. II. leading order asymptotic estimates. Commun. Math. Phys. 188, 599–656 (1997) Erd˝os, L., Solovej, J.P.: The kernel of dirac operators on S3 and R3 . Rev. Math. Phys. 13, 1247– 1280 (2001) Fefferman, C.: Stability of coulomb systems in a magnetic field. Proc. Nat. Acad. Sci. USA 92, 5006–5007 (1995) Fröhlich, J., Lieb, E.H., Loss, M.: Stability of coulomb systems with magnetic fields. I. the oneelectron atom. Commun. Math. Phys. 104, 251–270 (1986) Fefferman, C., Seco, L.A.: On the energy of a large atom. Bull. AMS 23(2), 525–530 (1990) Hughes, W.: An atomic energy bound that gives scott’s correction. Adv. Math. 79, 213–270 (1990) Ivrii, V.I., Sigal, I.M.: Asymptotics of the ground state energies of large coulomb systems. Ann. of Math. (2) 138, 243–335 (1993) Lieb, E.H.: Thomas-fermi and related theories of atoms and molecules. Rev. Mod. Phys. 65(4), 603–641 (1981) Lieb, E.H., Loss, M.: Stability of coulomb systems with magnetic fields. II. the many-electron atom and the one-electron molecule. Commun. Math. Phys. 104, 271–282 (1986) Lieb, E.H., Loss, M., Solovej, J.P.: Stability of matter in magnetic fields. Phys. Rev. Lett. 75, 985–989 (1995) Lieb, E.H., Oxford, S.: Improved lower bound on the indirect coulomb energy. Int. J. Quant. Chem. 19, 427–439 (1981) Lieb, E.H., Simon, B.: The thomas-fermi theory of atoms. molecules and solids. Adv. Math. 23, 22–116 (1977) Lieb, E.H., Solovej, J.P., Yngvason, J.: Asymptotics of heavy atoms in high magnetic fields: I. lowest landau band region. Commun. Pure Appl. Math. 47, 513–591 (1994) Lieb, E.H., Solovej, J.P., Yngvason, J.: Asymptotics of heavy atoms in high magnetic fields: II. semiclassical regions. Commun. Math. Phys. 161, 77–124 (1994) Lieb, E.H., Thirring, W.: A bound on the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann. E. H. Lieb, B. Simon, A. Wightman, eds., Princeton, NJ: Princeton University Press, 1976, pp. 269–303 Loss, M., Yau, H.T.: Stabilty of coulomb systems with magnetic fields. III. zero energy bound states of the pauli operator. Commun. Math. Phys. 104, 283–290 (1986) Siedentop, H., Weikard, R.: On the leading energy correction for the statistical model of an atom: interacting case. Commun. Math. Phys. 112, 471–490 (1987) Siedentop, H., Weikard, R.: On the leading correction of the thomas-fermi model: lower bound. Invent. Math. 97, 159–193 (1990) Siedentop, H., Weikard, R.: A new phase space localization technique with application to the sum of negative eigenvalues of schrödinger operators. Ann. Sci. École Norm. Sup. (4) 24(2), 215–225 (1991) Solovej, J.P., Spitzer, W.L.: A new coherent states approach to semiclassics which gives scott’s correction. Commun. Math. Phys. 241, 383–420 (2003)

Communicated by I. M. Sigal

Commun. Math. Phys. 294, 251–272 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0903-4

Communications in

Mathematical Physics

The Hermitian Laplace Operator on Nearly Kähler Manifolds Andrei Moroianu1 , Uwe Semmelmann2 1 CMLS, École Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau, France.

E-mail: [email protected]

2 Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany.

E-mail: [email protected] Received: 23 March 2009 / Accepted: 11 May 2009 Published online: 9 August 2009 – © Springer-Verlag 2009

Abstract: The moduli space N K of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1, 1) forms (cf. Moroianu et al. in Pacific J Math 235:57–72, 2008). Using the Hermitian Laplace operator and some representation theory, we compute the space N K on all 6-dimensional homogeneous nearly Kähler manifolds. It turns out that the nearly Kähler structure is rigid except for the flag manifold F(1, 2) = SU3 /T 2 , which carries an 8-dimensional moduli space of infinitesimal nearly Kähler deformations, modeled on the Lie algebra su3 of the isometry group. 1. Introduction Nearly Kähler manifolds were introduced in the 70’s by A. Gray [8] in the context of weak holonomy. More recently, 6-dimensional nearly Kähler manifolds turned out to be related to a multitude of topics among which we mention: Spin manifolds with Killing spinors (Grunewald), SU3 -structures, geometries with torsion (Cleyton, Swann), stable forms (Hitchin), or super-symmetric models in theoretical physics (Friedrich, Ivanov). Up to now, the only sources of compact examples are the naturally reductive 3-symmetric spaces, classified by Gray and Wolf [13], and the twistor spaces over positive quaternion-Kähler manifolds, equipped with the non-integrable almost complex structure. Based on previous work by R. Cleyton and A. Swann [6], P.-A. Nagy has shown in 2002 that every simply connected nearly Kähler manifold is a Riemannian product of factors which are either of one of these two types, or 6-dimensional [12]. Moreover, J.-B. Butruille has shown [5] that every homogeneous 6-dimensional nearly Kähler manifold is a 3-symmetric space G/K , more precisely isometric with S 6 = G 2 /SU3 , S 3 × S 3 = SU2 × SU2 × SU2 /SU2 , CP3 = SO5 /U2 × S 1 or F(1, 2) = SU3 /T 2 , all endowed with the metric defined by the Killing form of G. This work was supported by the French-German cooperation project Procope no. 17825PG.

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A method of finding new examples is to take some homogeneous nearly Kähler manifold and try to deform its structure. In [10] we have studied the deformation problem for 6-dimensional nearly Kähler manifolds (M 6 , g) and proved that if M is compact, and has normalized scalar curvature scalg = 30, then the space N K of infinitesimal deformations of the nearly Kähler structure is isomorphic to the eigenspace for the eigenvalue 12 of the restriction of the Laplace operator g to the space of co-closed (1,1) primitive (1, 1)-forms 0 M. It is thus natural to investigate the Laplace operator on the known 3-symmetric examples (besides the sphere S 6 , whose space of nearly Kähler structures is well-understood, and isomorphic to SO7 /G 2 ∼ = RP7 , see [7] or [5, Prop. 7.2]). Recall that the spectrum of the Laplace operator on symmetric spaces can be computed in terms of Casimir eigenvalues using the Peter-Weyl formalism. It turns out that a similar method can be applied ¯ (called the Hermitian in order to compute the spectrum of a modified Laplace operator Laplace operator) on 3-symmetric spaces. This operator is SU3 -equivariant and coincides with the usual Laplace operator on co-closed primitive (1, 1)-forms. The space of infinitesimal nearly Kähler deformations is thus identified with the space of co-closed ¯ = 12α}. Our main result is that the forms in 0(1,1) (12) := {α ∈ C ∞ (0(1,1) M) | α nearly Kähler structure is rigid on S 3 × S 3 and CP3 , and that the space of infinitesimal nearly Kähler deformations of the flag manifold F(1, 2) is eight-dimensional. The paper is organized as follows. After some preliminaries on nearly Kähler man(1,1) ifolds, we give two general procedures for constructing elements in 0 (12) out of Killing vector fields or eigenfunctions of the Laplace operator for the eigenvalue 12 (Corollary 4.5 and Proposition 4.11). We show that these elements can not be co-closed, thus obtaining an upper bound for the dimension of the space of infinitesimal nearly Kähler deformations (Proposition 4.12). We then compute this upper bound explicitly on the 3-symmetric examples and find that it vanishes for S 3 × S 3 and CP3 , which therefore have no infinitesimal nearly Kähler deformation. This upper bound is equal to 8 on the flag manifold F(1, 2) = SU3 /T 2 and in the last section we construct an explicit isomorphism between the Lie algebra of the isometry group su3 and the space of infinitesimal nearly Kähler deformations on F(1, 2). In addition, our explicit computations (in Sect. 5) of the spectrum of the Hermitian Laplace operator on the 3-symmetric spaces, together with the results in [11] show that every infinitesimal Einstein deformation on a 3-symmetric space is automatically an infinitesimal nearly Kähler deformation. 2. Preliminaries on Nearly Kähler Manifolds An almost Hermitian manifold (M 2m , g, J ) is called nearly Kähler if (∇ X J )(X ) = 0,

∀ X ∈ T M,

(1)

where ∇ denotes the Levi-Civita connection of g. The canonical Hermitian connection ¯ defined by ∇, ∇¯ X Y := ∇ X Y − 21 J (∇ X J )Y,

∀ X ∈ T M, ∀ Y ∈ C ∞ (M)

(2)

¯ = 0 and ∇¯ J = 0) with torsion T¯X Y = −J (∇ X J )Y . is a Um connection on M (i.e. ∇g A fundamental observation, which goes back to Gray, is the fact that ∇¯ T¯ = 0 on every nearly Kähler manifold (see [2]).

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We denote the Kähler form of M by ω := g(J., .). The tensor + := ∇ω is totally skew-symmetric and of type (3, 0) + (0, 3) by (1). From now on we assume that the dimension of M is 2m = 6 and that the nearly Kähler structure is strict, i.e. (M, g, J ) is not Kähler. It is well-known that M is Einstein in this case. We will always normalize the scalar curvature of M to scal = 30, in which case we also have | + |2 = 4 point¯ wise. The form + can be seen as the real part of a ∇-parallel complex volume form + + i − on M, where − = ∗ + is the Hodge dual of + . Thus M carries a SU3 ¯ Notice that Hitchin has shown structure whose minimal connection (cf. [6]) is exactly ∇. + − that a SU3 structure (ω, , ) is nearly Kähler if and only if the following exterior system holds: dω = 3 + (3) d − = −2ω ∧ ω. Let A ∈ 1 M ⊗ EndM denote the tensor A X := J (∇ X J ) = − +J X , where Y+ denotes the endomorphism associated to Y + via the metric. Since for every unit vector X , A X defines a complex structure on the 4-dimensional space X ⊥ ∩ (J X )⊥ , we easily get in a local orthonormal basis {ei } the formulas |A X |2 = 2|X |2 , Aei Aei (X ) = −4X,

∀X ∈ T M, ∀X ∈ T M,

(4) (5)

where here and henceforth, we use Einstein’s summation convention on repeating subscripts. The following algebraic relations are satisfied for every SU3 structure (ω, + ) on T M (notice that we identify vectors and 1-forms via the metric): A X ei ∧ ei + X − + (X ) ∧ + (J X + ) ∧ ω

= −2X ∧ ω, ∀X ∈ T M, + = −J X , ∀X ∈ T M, = X ∧ ω2 , ∀X ∈ T M, = X ∧ +, ∀X ∈ T M.

(6) (7) (8) (9)

The Hodge operator satisfies ∗2 = (−1) p on p M and moreover ∗ (X ∧ + ) = J X + , ∗(φ ∧ ω) = −φ,

∀φ ∈

∗(J X ∧ ω ) = −2X, 2

∀X ∈ T M,

(10)

(1,1) 0 M,

(11)

∀X ∈ T M.

(12)

From now on we assume that (M, g) is compact 6-dimensional not isometric to the round sphere (S 6 , can). It is well-known that every Killing vector field ξ on M is an automorphism of the whole nearly Kähler structure (see [10]). In particular, L ξ ω = 0,

L ξ + = 0,

L ξ − = 0.

(13)

¯ Then the formula (cf. [1]) Let now R and R¯ denote the curvature tensors of ∇ and ∇. RW X Y Z = R¯ W X Y Z − 41 g(Y, W )g(X, Z ) + 41 g(X, Y )g(Z , W ) + 43 g(Y, J W )g(J X, Z ) − 43 g(Y, J X )g(J W, Z ) − 21 g(X, J W )g(J Y, Z ) may be rewritten as R X Y = − X ∧ Y + R CY XY

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and R¯ X Y = − 43 (X ∧ Y + J X ∧ J Y − 23 ω(X, Y )J ) + R CY XY , where R CY X Y is a curvature tensor of Calabi-Yau type. We will recall the definition of the curvature endomorphism q(R) (cf. [10]). Let E M be the vector bundle associated to the bundle of orthonormal frames via a representation π : SO(n) → Aut(E). The Levi-Civita connection of M induces a connection on E M, whose curvature satisfies R XE YM = π∗ (R X Y ) = π∗ (R(X ∧ Y )), where we denote with π∗ the differential of π and identify the Lie algebra of S O(n), i.e. the skew-symmetric endomorphisms, with 2 . In order to keep notations as simple as possible, we introduce the notation π∗ (A) = A∗ . The curvature endomorphism q(R) ∈ End(E M) is defined as q(R) = 21 (ei ∧ e j )∗ R(ei ∧ e j )∗

(14)

for any local orthonormal frame {ei }. In particular, q(R) = Ric on T M. By the same formula we may define for any curvature tensor S, or more generally any endomorphism S of 2 T M, a bundle morphism q(S). In any point q : R → q(R) defines an equivariant map from the space of algebraic curvature tensors to the space of endomorphisms of E. Since a Calabi-Yau algebraic curvature tensor has vanishing Ricci curvature, q(R CY ) = 0 holds on T M. Let R 0X Y be defined by R 0X Y = X ∧ Y + J X ∧ J Y − 23 ω(X, Y )J . Then a direct calculation gives (ei ∧ e j )∗ (ei ∧ e j )∗ + 21 (ei ∧ e j )∗ (J ei ∧ J e j )∗ − 23 ω∗ ω∗ . q(R 0 ) = 21 We apply this formula on T M. The first summand is exactly the SO(n)-Casimir, which acts as −5id. The third summand is easily seen to be 23 id, whereas the second summand acts as −id (cf. [11]). Altogether we obtain q(R 0 ) = − 16 3 id, which gives the following ¯ acting on T M: expression for q( R) ¯ T M = 4 id T M . q( R)|

(15)

3. The Hermitian Laplace Operator In the next two sections (M 6 , g, J ) will be a compact nearly Kähler manifold with scalar curvature normalized to scalg = 30. We denote as usual by the Laplace operator = d ∗ d + dd ∗ = ∇ ∗ ∇ + q(R) on differential forms. We introduce the Hermitian Laplace operator ¯ ¯ = ∇¯ ∗ ∇¯ + q( R),

(16)

which can be defined on any associated bundle E M. In [11] we have computed the ¯ on a primitive (1, 1)-form φ: difference of the operators and ¯ ( − )φ = (J d ∗ φ) + .

(17)

¯ coincide on co-closed primitive (1, 1)-forms. We now compute In particular, and ¯ on 1-forms. Using the calculation in [11] (or directly from (15)) we the difference −

The Hermitian Laplace Operator on Nearly Kähler Manifolds

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¯ = id on T M. It remains to compute the operator P = ∇ ∗ ∇ − ∇¯ ∗ ∇¯ have q(R) − q( R) on T M. A direct calculation using (5) gives for every 1-form θ , P(θ ) = − 41 Aei Aei θ − Aei ∇¯ ei θ = θ − Aei ∇¯ ei θ = θ + 21 Aei Aei θ − Aei ∇ei θ = −θ − Aei ∇ei θ. In order to compute the last term, we introduce the metric adjoint α : 2 M → T M of the bundle homomorphism X ∈ T M → X + ∈ 2 M. It is easy to check that α(X + ) = 2X (cf. [10]). Keeping in mind that A is totally skew-symmetric, we compute for an arbitrary vector X ∈ T M,

Aei (∇ei θ ), X = A X ei , ∇ei θ = A X , ei ∧ ∇ei θ = A X , dθ = − +J X , dθ = − J X, α(dθ ) = J α(dθ ), X , whence Aei (∇ei θ ) = J α(dθ ). Summarizing our calculations we have proved the following Proposition 3.1. Let (M 6 , g, J ) be a nearly Kähler manifold with scalar curvature normalized to scalg = 30. Then for any 1-form θ it holds that ¯ = −J α(dθ ). ( − )θ The next result is a formula for the commutator of J and α ◦ d on 1-forms. Lemma 3.2. For all 1-forms θ , the following formula holds: α(dθ ) = 4J θ + J α(d J θ ). Proof. Differentiating the identity θ ∧ + = J θ ∧ − gives dθ ∧ + = d J θ ∧ − +2J θ ∧ ω2 . With respect to the SU3 -invariant decomposition 2 M = (1,1) M ⊕ (2,0)+(0,2) M, we can write dθ = (dθ )(1,1) + 21 α(dθ ) + and d J θ = (d J θ )(1,1) + 21 α(d J θ ) + . Since the wedge product of forms of type (1, 1) and (3, 0) vanishes we derive the equation + + 1 2 (α(dθ ) ) ∧

= 21 (α(d J θ ) + ) ∧ − + 2J θ ∧ ω2 .

Using (8) and (9) we obtain 2 1 2 α(dθ ) ∧ ω

= 21 J α(d J θ ) ∧ ω2 + 2J θ ∧ ω2 .

Taking the Hodge dual of this equation and using (12) gives J α(dθ ) = −α(d J θ ) − 4θ, which proves the lemma. Finally we note two interesting consequences of Proposition 3.1 and Lemma 3.2. Corollary 3.3. For any closed 1-form θ it holds that ¯ = 0, ( − )θ

¯ θ = 4J θ. ( − )J

¯ coincide on θ . Proof. For a closed 1-form θ Lemma 3.1 directly implies that and For the second equation we use Proposition 3.1 together with Lemma 3.2 to conclude ¯ θ = −J α(d J θ ) = 4J θ − α(dθ ) = 4J θ, ( − )J since θ is closed. This completes the proof of the corollary.

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¯ 4. Special -Eigenforms on Nearly Kähler Manifolds In this section we assume moreover that (M, g) is not isometric to the standard sphere ¯ (S 6 , can). In the first part of this section we will show how to construct -eigenforms on M starting from Killing vector fields. Let ξ be a non-trivial Killing vector field on (M, g), which in particular implies d ∗ ξ = 0 and ξ = 2Ric(ξ ) = 10ξ . As an immediate consequence of the Cartan formula and (13) we obtain d J ξ = L ξ ω − ξ dω = −3ξ ψ +

(18)

so by (4), the square norm of d J ξ (as a 2-form) is |d J ξ |2 = 18|ξ |2 .

(19)

In [9] we showed already that the vector field J ξ is co-closed if ξ is a Killing vector field and has unit length. However it turns out that this also holds more generally. Proposition 4.1. Let ξ be a Killing vector field on M. Then d ∗ J ξ = 0. Proof. Let dv denote the volume form of (M, g). We start with computing the L 2 -norm of d ∗ J ξ . ∗ 2 ∗ ∗ d J ξ L 2 =

d J ξ, d J ξ dv = [ J ξ, J ξ − d ∗ d J ξ, J ξ ]dv

M

[ ∇ ∗ ∇ J ξ, J ξ + 5|J ξ |2 − |d J ξ |2 ]dv

=

M

M

=

[|∇ J ξ |2 + 5|ξ |2 − |d J ξ |2 ]dv =

M

[|∇ J ξ |2 − 13|ξ |2 ]dv. M

Here we used the well-known Bochner formula for 1-forms, i.e. θ = ∇ ∗ ∇θ + Ric(θ ), with Ric(θ ) = 5θ in our case. Next we consider the decomposition of ∇ J ξ into its symmetric and skew-symmetric parts 2∇ J ξ = d J ξ + L J ξ g, which together with (19) leads to |∇ J ξ |2 = 41 (|d J ξ |2 + |L J ξ g|2 ) = 9|ξ |2 + 14 |L J ξ g|2 .

(20)

(Recall that the endomorphism square norm of a 2-form is twice its square norm as a form.) In order to compute the last norm, we express L J ξ g as follows: L J ξ g(X, Y ) = g(∇ X J ξ, Y ) + g(X, ∇Y J ξ ) = g(J ∇ X ξ, Y ) + g(X, J ∇Y ξ ) + + (X, ξ, Y ) + + (Y, ξ, X ) = −g(∇ X ξ, J Y ) − g(J X, ∇Y ξ ) = −dξ (1,1) (X, J Y ), whence L J ξ g2L 2 = 2dξ (1,1) 2L 2 .

(21)

On the other hand, as an application of Lemma 3.2 together with Eq. (18) we get α(dξ ) = 4J ξ + J α(d J ξ ) = −2J ξ , so dξ (2,0) = −J ξ + .

(22)

The Hermitian Laplace Operator on Nearly Kähler Manifolds

257

Moreover, ξ = 10ξ since ξ is a Killing vector field, which yields dξ (1,1) 2L 2 = dξ 2L 2 − dξ (2,0) 2L 2 = 10ξ 2L 2 − 2ξ 2L 2 = 8ξ 2L 2 . This last equation, together with (20) and (21) gives ∇ J ξ 2L 2 = 13ξ 2L 2 . Substituting this into the first equation proves that d ∗ J ξ has vanishing L 2 -norm and thus that J ξ is co-closed. Proposition 4.2. Let ξ be a Killing vector field on M. Then ξ = 10ξ,

and

J ξ = 18J ξ.

In particular, J ξ can never be a Killing vector field. Proof. The first equation holds for every Killing vector field on an Einstein manifold with Ric = 5id. From (18) we know d J ξ = −3ξ + . Hence the second assertion follows from: (10) (12) d ∗ d J ξ = − ∗ d ∗ d J ξ = −3 ∗ d(J ξ ∧ + ) = 9 ∗ (ξ ∧ ω2 ) = 18J ξ. Since the differential d commutes with the Laplace operator , every Killing vector field ξ defines two -eigenforms of degree 2: d J ξ = 18d J ξ

and

dξ = 10dξ.

As a direct consequence of Proposition 4.2, together with formulas (18), (22), and Proposition 3.1 we get: Corollary 4.3. Every Killing vector field on M satisfies ¯ = 12ξ, ξ

¯ ξ = 12J ξ. J

¯ -eigenform. By (22) we have Our next goal is to show that the (1, 1)-part of dξ is a dξ = φ − J ξ + ,

(23)

for some (1, 1)-form φ. Using Proposition 4.1, we can write in a local orthonormal basis {ei }:

dξ, ω = 21 dξ, ei ∧ J ei = ∇ei ξ, J ei = d ∗ J ξ = 0, thus showing that φ is primitive. The differential of φ can be computed from the Cartan formula: (23) (7) dφ = d(J ξ + + dξ ) = −d(ξ − ) (13) = −L ξ − + ξ d − = −2ξ ω2 = −4J ξ ∧ ω. From here we obtain ∗dφ = −4 ∗ (J ξ ∧ ω) = 4ξ ∧ ω,

(24)

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whence (23) d ∗ dφ = 4dξ ∧ ω − 12ξ ∧ + = 4φ ∧ ω − 4(J ξ + ) ∧ ω − 12ξ ∧ + (9) = 4φ ∧ ω − 16ξ ∧ + . Using (10) and (11), we thus get d ∗ dφ = − ∗ d ∗ dφ = 4φ + 16J ξ + . On the other hand, (11) (24) (12) d ∗ φ = − ∗ d ∗ φ = ∗d(φ ∧ ω) = X (−4J ξ ∧ ω2 + 3φ ∧ + ) = 8ξ and finally dd ∗ φ = 8dξ = 8φ − 8J ξ + . The calculations above thus prove the following proposition Proposition 4.4. Let (M 6 , g, J ) be a compact nearly Kähler manifold with scalar curvature scalg = 30, not isometric to the standard sphere. Let ξ be a Killing vector field on (1,1) M and let φ be the (1, 1)-part of dξ . Then φ is primitive, i.e. φ = (dξ )0 . Moreover ∗ + d φ = 8ξ and φ = 12φ + 8J ξ . Corollary 4.5. The primitive (1, 1)-form ϕ satisfies ¯ = 12φ. φ Proof. From (17) and the proposition above we get ¯ = φ − ( − )φ ¯ φ = 12φ + 8J ξ + − (J d ∗ φ) + = 12φ. In the second part of this section we will present another way of obtaining primi¯ tive -eigenforms of type (1, 1), starting from eigenfunctions of the Laplace operator. Let f be such an eigenfunction, i.e. f = λ f . We consider the primitive (1, 1)-form (1,1) η := (d J d f )0 . Lemma 4.6. The form η is explicitly given by η = d J d f + 2d f + +

λ 3

f ω.

Proof. According to the decomposition of 2 M into irreducible SU3 -summands, we can write d J d f = η + γ + + hω for some vector field γ and function h. From Lemma 3.2 we get 2γ = α(d J d f ) = −4d f . In order to compute h, we write (12) 6h dv = hω ∧ ω2 = d J d f ∧ ω2 = d(J d f ∧ ω2 ) = 2d ∗ d f = 2λ f dv.

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We will now compute the Laplacian of the three summands of η separately. First, we ¯ f = λd f . Since ¯ commutes with J , we have d f = λd f and Corollary 3.3 yields d ¯ also have J d f = λJ d f and from the second equation in Corollary 3.3 we obtain ¯ d f + ( − )J ¯ d f = (λ + 4)J d f. J d f = J Hence, d J d f is a -eigenform for the eigenvalue λ + 4. Lemma 4.7. The co-differential of the (1, 1)-form η is given by d ∗ η = 2λ 3 − 4 J d f. Proof. Notice that d ∗ ( f ω) = −d f ω and that d ∗ J d f = − ∗ d ∗ J d f = − 21 ∗ d(d f ∧ ω2 ) = 0, since dω2 = 0. Using this we obtain d ∗ η = J d f + 2d ∗ (d f + ) − λ3 d f ω = (λ + 4)J d f − 2 ∗ d(d f ∧ − ) − λ3 J d f (12) = (λ + 4 − λ3 )J d f − 4 ∗ (d f ∧ ω2 ) = ( 2λ 3 − 4)J d f. In order to compute of the second summand of η we need three additional formulas Lemma 4.8. ¯ ) + . ¯ + ) = (X (X ¯ Since + is ∇-parallel ¯ ¯ = ∇¯ ∗ ∇¯ + q( R). Proof. Recall that we immediately obtain ¯ + ) = −∇¯ ei ∇¯ ei (X + ) = −(∇¯ ei ∇¯ ei X ) + . ∇¯ ∗ ∇(X The map A → A∗ + is a SU3 -equivariant map from 2 to 3 . But since 3 does not (1,1) contain the representation 0 as an irreducible summand, it follows that A∗ + = 0 for any skew-symmetric endomorphism A corresponding to some primitive (1, 1)-form. Hence we conclude ¯ ¯ ) + , ¯ i )∗ (X + ) = (ωi∗ R(ω ¯ i )∗ X ) + = (q( R)X q( R)(X + ) = ωi∗ R(ω where, since the holonomy of ∇¯ is included in SU3 , the sum goes over some ortho(1,1) ¯ + ) = normal basis {ωi } of 0 M. Combining these two formulas we obtain (X ¯ ) + . (X Lemma 4.9. ¯ ( − )(d f + ) = 6(d f + ) −

4λ 3

f ω − 2η.

Proof. From Proposition 3.4 in [11] we have ¯ ¯ ¯ ( − )(d f + ) = (∇ ∗ ∇ − ∇¯ ∗ ∇)(d f + ) f + ) + (q(R) − q( R))(d ¯ = (∇ ∗ ∇ − ∇¯ ∗ ∇)(d f + ) + 4d f + . The first part of the right hand side reads ¯ (∇ ∗ ∇ − ∇¯ ∗ ∇)(d f + ) = − 41 Aei ∗ Aei ∗ d f + − Aei ∗ ∇¯ ei (d f + ).

(25)

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From (5) we get Aei ∗ Aei ∗ d f + = Aei ∗ (Aei ek ∧ + (d f, ek , ·)) = Aei Aei ek ∧ + (d f, ek , ·) + Aei ek ∧ Aei + (d f, ek , ·) = −4ek ∧ ek d+f + Aei ek ∧ Aei e j + (d f, ek , e j ) = −8d+f , where we used the vanishing of the expression E = Aei ek ∧ Aei e j + (d f, ek , e j ): E = A J ei ek ∧ A J ei e j + (d f, ek , e j ) = Aei J ek ∧ Aei J e j + (d f, ek , e j ) = Aei ek ∧ Aei e j + (d f, J ek , J e j ) = −E. It remains to compute the second term in (25). We notice that by Schur’s Lemma, every SU3 -equivariant map from the space of symmetric tensors Sym2 M to T M vanishes, so in particular (since ∇d f is symmetric), one has Aei ∇ei d f = 0. We then compute Aei ∗ ∇¯ ei d+f = Aei ∗ ((∇¯ ei d f ) + ) = (Aei ∇¯ ei d f ) + + (∇¯ ei d f )Aei ∗ + (6) = (Aei ∇ei d f ) + − 21 (Aei Aei d f ) + − 2(∇¯ ei d f )(ei ∧ ω) = 2d+f + 2d ∗ d f ω + Aei d f, ei ω + 2ei ∧ J ∇¯ ei d f = 2d+f +2λ f ω + 2ei ∧ ∇¯ ei J d f = 2d+f +2λ f ω + 2d J d f − ei ∧ Aei J d f = 2d+f + 2λ f ω + 2d J d f + 2 A J d f = 4d+f + 2λ f ω + 2d J d f. Plugging back what we obtained into (25) yields ¯ f + ) = −(2d+f + 2λ f ω + 2d J d f ), (∇ ∗ ∇ − ∇¯ ∗ ∇)(d which together with Lemma 4.6 and the first equation prove the desired formula.

Lemma 4.10. f ω = (λ + 12) f ω − 2(d f + ). Proof. Since d ∗ ( f ω) = −d f ω = −J d f we have dd ∗ ( f ω) = −d J d f . For the second summand of ( f ω) we first compute d( f ω) = d f ∧ ω + 3 f + . Since d ∗ + = 1 ∗ ∗ + + ∗ + + 3 d dω = 4ω, we get d f = −d f + f d = −d f + 4 f ω. Moreover d ∗ (d f ∧ ω) = − ∗ d(J d f ∧ ω) = − ∗ (d J d f ∧ ω − 3J d f ∧ + ) = − ∗ ([η − 2d f + −

λ 3

f ω] ∧ ω) + 3 ∗ (J d f ∧ + )

= η + 2 ∗ ((d f + ) ∧ ω) + = η + 2d f + + Recalling that η = d J d f + 2d f + +

2λ 3

λ 3

2λ 3

f ω − 3d f +

f ω − 3d f + .

f ω, we obtain

f ω = −d J d f − 3d f + + 12 f ω + η − d f + +

2λ 3

f ω = (λ + 12) f ω − 2d f + .

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261

Applying these three lemmas we conclude ¯ f + ) + ( − )(d ¯ f + ) = (λ + 6)(d f + ) − (d f + ) = (d

4λ 3

f ω − 2η,

and thus η = (λ + 4)d J d f + (2λ + 12)(d f + ) − + = λη + 4 − 2λ 3 (d f ).

8λ 3

f ω − 4η + λ3 (λ + 12) f ω −

2λ + 3 (d f )

¯ on Finally we have once again to apply the formula for the difference of and primitive (1, 1)-forms. We obtain ¯ = η − J d ∗ η + = η + 2λ − 4 (d f + ) = λη. η 3 Summarizing our calculations we obtain the following result. Proposition 4.11. Let f be an -eigenfunction with f = λ f Then the primitive (1, 1)form η := (d J d f )0(1,1) satisfies ¯ = λη and d ∗ η = 2λ − 4 J d f. η 3 ¯ ¯ = Let 0 (12) ⊂ C ∞ (M) be the -eigenspace for the eigenvalue 12 (notice that (1,1) ¯ on functions) and let 0 (12) denote the space of primitive (1, 1)-eigenforms of corresponding to the eigenvalue 12. Summarizing Corollary 4.5 and Proposition 4.11, we have constructed a linear mapping (1,1)

: i(M) → 0

(1,1)

(12),

(ξ ) := dξ0 (1,1)

from the space of Killing vector fields into 0 : 0 (12) → 0(1,1) (12),

(12) and a linear mapping

( f ) := (d J d f )0(1,1) .

Let moreover N K ⊂ 0(1,1) (12) denote the space of nearly Kähler deformations, which (1,1) by [10] is just the space of co-closed forms in 0 (12). Proposition 4.12. The linear mappings and defined above are injective and the sum Im() + Im() + N K ⊂ 0(1,1) (12) is a direct sum. In particular, dim(N K) ≤ dim(0(1,1) (12)) − dim(i(M)) − dim(0 (12)).

(26)

Proof. It is enough to show that if ξ ∈ i(M), f ∈ 0 (12) and α ∈ N K satisfy (1,1)

dξ0

(1,1)

+ (d J d f )0

+ α = 0,

(27)

then ξ = 0 and f = 0. We apply d ∗ to (27). Using Propositions 4.4 and 4.11 to express the co-differentials of the first two terms we get 8ξ + 8J d f = 0.

(28)

Since J ξ is co-closed (Proposition 4.1), formula (28) implies 0 = d ∗ J ξ = d ∗ d f = 12 f , i.e. f = 0. Plugging back into (28) yields ξ = 0 too.

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5. The Homogeneous Laplace Operator on Reductive Homogeneous Spaces 5.1. The Peter-Weyl formalism. Let M = G/K be a homogeneous space with compact Lie groups K ⊂ G and let π : K → Aut(E) be a representation of K . We denote by E M := G ×π E the associated vector bundle over M. The Peter-Weyl theorem and the Frobenius reciprocity yield the following isomorphism of G-representations: Vγ ⊗ Hom K (Vγ , E), (29) L 2 (E M) ∼ = γ ∈Gˆ

where Gˆ is the set of (non-isomorphic) irreducible G-representations. If not otherwise stated we will consider only complex representations. Recall that the space of smooth sections C ∞ (E M) can be identified with the space C ∞ (G; E) K of K -invariant E-valued functions, i.e. functions f : G → E with f (gk) = π(k)−1 f (g). This space is made into a G-representation by the left-regular representation , defined by ((g) f )(a) = f (g −1 a). Let v ∈ Vγ and A ∈ Hom K (Vγ , E), then the invariant E-valued function corresponding to v ⊗ A is defined by g → A(g −1 v). In particular, each summand in the Hilbert space direct sum (29) is a subset of C ∞ (E M) ⊂ L 2 (E M). Let g be the Lie algebra of G. We denote by B the Killing form of g, B(X, Y ) := tr(ad X ◦ adY ). The Killing form is non-degenerate and negative definite if G is compact and semi-simple, which will be the case in all examples below. If π : G → Aut(E) is a G-representation, the Casimir operator of (G, π ) acts on E by the formula CasπG = (π∗ X i )2 , (30) where {X i } is a (−B)-orthonormal basis of g and π∗ : g → End(E) denotes the differential of the representation π . Remark 5.1. Notice that the Casimir operator is divided by k if one uses the scalar product −k B instead of −B. If G is simple, the adjoint representation ad on the complexification gC is irreducible, so, by Schur’s Lemma, its Casimir operator acts as a scalar. Taking the trace in (30) for G = −1. π = ad yields the useful formula Casad Let Vγ be an irreducible G-representation of highest weight γ . By Freudenthal’s formula the Casimir operator acts on Vγ by scalar multiplication with ρ2 − ρ + γ 2 , where ρ denotes the half-sum of the positive roots and · is the norm induced by −B on the dual of the Lie algebra of the maximal torus of G. Notice that these scalars are always non-positive. Indeed ρ2 − ρ + γ 2 = − γ , γ + 2ρ B and γ , ρ ≥ 0, since γ is a dominant weight, i.e. it is in the the closure of the fixed Weyl chamber, whereas ρ is the half-sum of positive weights and thus by definition has a non-negative scalar product with γ . 5.2. The homogeneous Laplace operator. We denote by ∇¯ the canonical homogeneous connection on M = G/K . It coincides with the Levi-Civita connection only in the case that G/K is a symmetric space. A crucial observation is that the canonical homogeneous connection coincides with the canonical Hermitian connection on naturally reduc¯ ∈ tive 3-symmetric spaces (see below). We define the curvature endomorphism q( R) ¯ ¯ ¯ π = ∇¯ ∗ ∇+q( End(E M) as in (14) and introduce as in (16) the second order operator R) acting on sections of the associated bundle E M := G ×π E.

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Lemma 5.2. Let G be a compact semi-simple Lie group, K ⊂ G a compact subgroup, and let M = G/K be the naturally reductive homogeneous space equipped with the Riemannian metric induced by −B. For every K -representation π on E, let E M := ¯ acts G ×π E be the associated vector bundle over M. Then the endomorphism q( R) ¯ = −CasπK . Moreover the differential operator ¯ acts on the fibre-wise on E M as q( R) space of sections of E M, considered as G-representation via the left-regular represen¯ = −CasG . tation, as Proof. Consider the Ad(K )-invariant decomposition g = k ⊕ p. For any vector X ∈ g we write X = X k + X p, with X k ∈ k and X p ∈ p. The canonical homogeneous connection is the left-invariant connection in the principal K -fibre bundle G → G/K corresponding to the projection X → X k. It follows that one can do for the canonical homogeneous connection on G/K the same identifications as for the Levi Civita connection on Riemannian symmetric spaces. In particular, the covariant derivative of a section φ ∈ (E M) with respect to ˆ of the corresponding function φˆ ∈ some X ∈ p translates into the derivative X (φ) ∞ K ˆ = C (G; E) , which is minus the differential of the left-regular representation X (φ) ∗ ¯ ¯ ˆ −∗ (X )φ. Hence, if {eµ } is an orthonormal basis in p, the rough Laplacian ∇ ∇ trans¯ it remains ¯ = ∇¯ ∗ ∇¯ +q( R) lates into the sum −∗ (eµ )∗ (eµ ) = (−CasG +CasK ). Since K K ¯ = −Cas = −Casπ in order to complete the proof of the lemma. to show that q( R) We claim that the differential i ∗ : k → so(p) ∼ = 2 p of the isotropy representation i : K → SO(p) is given by i ∗ (A) = − 21 eµ ∧ [A, eµ ] for any A ∈ k. Indeed ( 21 eµ ∧ [A, eµ ])∗ X = − 21 B(eµ , X )[A, eµ ] + 21 B([A, eµ ], X )eµ = −[A, X ]. Next we recall that for X, Y ∈ p the curvature R¯ X,Y of the canonical connection acts by −π∗ ([X, Y ]k) on every associated vector bundle E M, defined by the representation π . Hence the curvature operator R¯ can be written for any X, Y ∈ p as ¯ ∧ Y ) = 1 eµ ∧ R¯ X,Y eµ = − 1 eµ ∧ [[X, Y ]k, eµ ] = i ∗ ([X, Y ]k). R(X 2 2 Let PSO(p) = G ×i SO(p) be the bundle of orthonormal frames of M = G/K . Then any SO(p)-representation π˜ defines a K -representation by π = π˜ ◦ i. Moreover any vector bundle E M associated to PSO(p) via π˜ can be written as a vector bundle associated via π to the K -principle bundle G → G/K , i.e. E M = PSO(p) ×π˜ E = G ×π E. ¯ we have Let { f α } be an orthonormal basis of k. Then by the definition of q( R) ¯ µ ∧ eν )) = 1 π˜ ∗ (eµ ∧ eν ) π∗ ([eµ , eν ]k) ¯ = 1 π˜ ∗ (eµ ∧ eν ) π˜ ∗ ( R(e q( R) 2 2 = −21 B([eµ , eν ], f α )π˜ ∗ (eµ ∧eν ) π∗ ( f α ) = − 21 B(eν , [ f α , eµ ])π˜ ∗ (eµ ∧ eν ) π∗ ( f α ) = 21 π˜ ∗ (eµ ∧ [ f α , eµ ]) π∗ ( f α ) = −π∗ ( f α ) π∗ ( f α ) = −CasπK . ¯ ∈ End(E M) acts fibre-wise as −CasπK . Let Z ∈ k and We have shown that q( R) f ∈ C ∞ (G; E) K , then the K -invariance of f implies π∗ (Z ) f = −Z ( f ) = ∗ (Z ) f and also CasπK = CasK , which concludes the proof of the lemma.

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¯ on sections of E M is the set It follows from this lemma that the spectrum of of numbers λγ = ρ + γ 2 − ρ2 , where γ is the highest weight of an irreducible G-representation Vγ such that Hom K (Vγ , E) = 0, i.e. such that the decomposition of Vγ , considered as a K -representation, contains components of the K -representation E. 5.3. Nearly Kähler deformations and Laplace eigenvalues. Let (M, g, J ) be a compact simply connected 6-dimensional nearly Kähler manifold not isometric to the round sphere, with scalar curvature normalized to scalg = 30. Recall the following result from [10]: Theorem 5.3. The Laplace operator coincides with the Hermitian Laplace operator ¯ on co-closed primitive (1, 1)-forms. The space N K of infinitesimal deformations of the nearly Kähler structure of M is isomorphic to the eigenspace for the eigenvalue 12 ¯ to the space of co-closed primitive (1, 1)-forms on M. of the restriction of (or ) Assume from now on that M is a 6-dimensional naturally reductive 3-symmetric space G/K in the list of Gray and Wolf, i.e. SU2 × SU2 × SU2 /SU2 , SO5 /U2 or SU3 /T 2 . As was noticed before, the canonical homogeneous and the canonical Hermitian connection coincide, since for the later it can be shown that its torsion and its curvature are parallel, a property, which by the Ambrose-Singer-Theorem characterizes the canonical homogeneous connection (cf. [5]). In order to determine the space N K on M we thus ¯ need to apply the previous calculations to compute the -eigenspace for the eigenvalue 12 on primitive (1, 1)-forms and decide which of these eigenforms are co-closed. According to Lemma 5.2 and the decomposition (29) we have to carry out three 1,1 steps: first to determine the K -representation 1,1 0 p defining the bundle 0 T M, then to compute the Casimir eigenvalues with the Freudenthal formula, which gives all pos¯ sible -eigenvalues and finally to check whether the G-representation Vγ realizing the eigenvalue 12 satisfies Hom K (Vγ , 1,1 0 p) = {0} and thus really appears as eigenspace. Before going on, we make the following useful observation Lemma 5.4. Let (G/K , g) be a 6-dimensional homogeneous strict nearly Kähler manifold of scalar curvature scalg = 30. Then the homogeneous metric g is induced from 1 B, where B is the Killing form of G. − 12 Proof. Let G/K be a 6-dimensional homogeneous strict nearly Kähler manifold. Then the metric is induced from a multiple of the Killing form, i.e. G/K is a normal homogeneous space with Ad(K )-invariant decomposition g = k ⊕ p. The scalar curvature of the metric h induced by −B may be computed as (cf. [3]) scalh =

3 2

− 3CasλK ,

where λ : K → so(p) is the isotropy representation. From Lemma 5.2 we know that ¯ which on the tangent bundle was computed in Lemma 15 as q( R) ¯ = CasλK = −q( R), 2scalh 3 2 5 15 id. Hence we obtain the equation scalh = 2 + 5 scalh and it follows scalh = 2 , i.e. 1 the metric g corresponding to − 12 B has scalar curvature scalg = 30. ¯ 5.4. The -spectrum on S 3 × S 3 . Let K = SU2 with Lie algebra k = su2 and G = K × K × K with Lie algebra g = k ⊕ k ⊕ k. We consider the 6-dimensional manifold

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M = G/K , where K is diagonally embedded. The tangent space at o = eK can be identified with p = {(X, Y, Z ) ∈ k ⊕ k ⊕ k | X + Y + Z = 0}. 1 Let B be the Killing form of k and define B0 = − 12 B. Then it follows from Lemma 5.4 that the invariant scalar product

B0 ((X, Y, Z ), (X, Y, Z )) = B0 (X, X ) + B0 (Y, Y ) + B0 (Z , Z ) defines a normal metric, which is the homogeneous nearly Kähler metric g of scalar curvature scalg = 30. The canonical almost complex structure on the 3-symmetric space M, corresponding to the 3rd order G-automorphism σ , with σ (k1 , k2 , k3 ) = (k2 , k3 , k1 ), is defined as J (X, Y, Z ) =

√2 (Z , 3

X, Y ) +

√1 (X, Y, 3

Z ).

The (1, 0)-subspace p1,0 of pC defined by J is isomorphic to the complexified adjoint 2 representation of SU2 on suC 2 . Let E = C denote the standard representation of SU2 ∼ ∼ ¯ (notice that E = E because every SU2 = Sp1 representation is quaternionic). (1,1)

Lemma 5.5. The SU2 -representation defining the bundle 0 ducible summands Sym4 E and Sym2 E.

T M splits into the irre-

Proof. The defining SU2 -representation of (1,1) T M is p1,0 ⊗ p0,1 ∼ = Sym2 E ⊗ 4 E ⊕ Sym 2 E ⊕ Sym 0 E from the Clebsch-Gordan formula. Since we Sym2 E ∼ Sym = are interested in primitive (1, 1)-forms, we still have to delete the trivial summand Sym0 E ∼ = C. Since G = SU2 × SU2 × SU2 , every irreducible G-representation is isomorphic to one of the representations Va,b,c = Syma E ⊗Symb E ⊗Symc E. The Casimir operator of the SU2 -representation Symk E (with respect to B) is − 18 k(k + 2) and the Casimir operator ¯ of G is the sum of the three SU2 -Casimir operators. Hence all possible -eigenvalues with respect to the metric B0 are of the form 3 2 (a(a

+ 2) + b(b + 2) + c(c + 2))

(31)

for non-negative integers a, b, c. It is easy to check that the eigenvalue 12 is obtained only for (a, b, c) equal to (2, 0, 0), (0, 2, 0) or (0, 0, 2). The restrictions to SU2 (diagonally embedded in G) of the three corresponding G-representations are all equal to the SU2 -representation Sym2 E, thus dim HomSU2 (V2,0,0 , 0(1,1) p) = 1, and similarly ¯ on primitive (1, 1)-forms for for the two other summands. Hence the eigenspace of the eigenvalue 12 is isomorphic to V2,0,0 ⊕ V0,2,0 ⊕ V0,0,2 and its dimension, i.e. the multiplicity of the eigenvalue 12, is equal to 9. Since the isometry group of the nearly Kähler manifold M = SU2 ×SU2 ×SU2 /SU2 has dimension 9, the inequality (26) yields (1,1)

dim(N K) ≤ dim(0

(12)) − dim(i(M)) − dim(0 (12)) = − dim(0 (12)) ≤ 0.

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We thus have obtained the following Theorem 5.6. The homogeneous nearly Kähler structure on S 3 × S 3 does not admit any infinitesimal nearly Kähler deformations. Finally we remark that there are also no infinitesimal Einstein deformations either. In [11] we showed that the space of infinitesimal Einstein deformations of a nearly Kähler metric g, with normalized scalar curvature scalg = 30, is isomorphic to the direct sum ¯ of -eigenspaces of primitive co-closed (1, 1)-forms for the eigenvalues 2, 6 and 12. It ¯ is clear from (31) that neither 2 nor 6 can be realized as -eigenvalues. Corollary 5.7. The homogeneous nearly Kähler metric on S 3 × S 3 does not admit any infinitesimal Einstein deformations. ¯ 5.5. The -spectrum on CP 3 . In this section we consider the complex projective space CP 3 = SO5 /U2 , where U2 is embedded by U2 ⊂ SO4 ⊂ SO5 . Let G = SO5 with Lie algebra g and K = U2 with Lie algebra k. We denote the Killing form of G with B. Then we have the B-orthogonal decomposition g = k ⊕ p, where p can be identified with the tangent space in o = eK . The space p splits as p = m ⊕ n, where m resp. n can be identified with the horizontal resp. vertical tangent space at o of the twistor space 1 fibration SO5 /U2 → SO5 /SO4 = S 4 . We know from Lemma 5.4 that B0 = − 12 B defines the homogeneous nearly Kähler metric g of scalar curvature scalg = 30. Let {ε1 , ε2 } denote the canonical basis of R2 . Then the positive roots of SO5 are α1 = ε1 , α2 = ε2 , α3 = ε1 + ε2 , α4 = ε1 − ε2 , with ρ = 23 ε1 + 21 ε2 . Let gα ⊂ gC be the root space corresponding to the root α. Then mC = gα1 ⊕ g−α1 ⊕ gα2 ⊕ g−α2 ,

nC = gα3 ⊕ g−α3 .

The invariant almost complex structure J may be defined by specifying the (1, 0)subspace p1,0 of pC : p1,0 = {X − i J X | X ∈ p} = gα1 ⊕ gα2 ⊕ g−α3 . It follows that J is not integrable, since the restricted root system {α1 , α2 , −α3 } is not closed under addition (cf. [4]). We note that replacing −α3 by α3 yields an integrable almost complex structure. This corresponds to the well-known fact that on the twistor space the non integrable almost complex structure J is transformed into the integrable one by replacing J with −J on the vertical tangent space. Let Ck denote the U1 -representation on C defined by (z, v) → z k v, for v ∈ C and z ∈ U1 ∼ = C∗ . Then, since U2 = (SU2 × U1 )/Z2 , any irreducible U2 -representation is of the form E a,b = Syma E ⊗ Cb , with a ∈ N, b ∈ Z and a ≡ b mod 2. As usual let E = C2 denote the standard representation of SU2 . With this notation we obtain the following decomposition of p1,0 considered as a U2 -representation: p1,0 ∼ = E 0,−2 ⊕ E 1,1

E 0,−2 ∼ = g−α3 and E 1,1 ∼ = gα1 ⊕ gα2 . (32) Since p0,1 is obtained from p1,0 by conjugation we have p0,1 ∼ = E 0,2 ⊕ E 1,−1 . The definwith

ing U2 -representation of (1,1) T M is p1,0 ⊗ p0,1 , which obviously decomposes into 5 irreducible summands, among which, two are isomorphic to the trivial representation E 0,0 . Considering only primitive (1, 1)-forms we still have to delete one of the trivial summands and obtain

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Lemma 5.8. The U2 -representation defining the bundle 0 decomposition into irreducible summands: (1,1)

0

T M has the following

p = E 0,0 ⊕ E 1,3 ⊕ E 1,−3 ⊕ E 2,0 .

Let Va,b be an irreducible SO5 -representation of highest weight γ = (a, b) with a, b ∈ N and a ≥ b ≥ 0, e.g. V1,0 = 1 and V1,1 = 2 . The scalar product induced by the Killing form B on the dual t∗ ∼ = R2 of the maximal torus of SO5 is − 16 times the Euclidean scalar product. By the Freudenthal formula we thus get CasVa,b = γ , γ + 2ρ B = − 16 (a(a + 3) + b(b + 1)).

(33)

G Notice that we have V1,1 = soC 5 and Cas V1,1 = −1, which is consistent with Casad = −1. ¯ It follows (cf. Remark 5.1) that all possible -eigenvalues with respect to the metric induced by B0 are of the form 2(a(a + 3) + b(b + 1)). The eigenvalue 12 is realized if and only if (a, b) = (1, 1). We still have to decide whether the SO5 -representation V1,1 actually appears in the decomposition (29) of L 2 (1,1 0 T M). However this follows from

Lemma 5.9. The SO5 -representation V1,1 restricted to U2 ⊂ SO5 has the following decomposition as U2 -representation: V1,1 ∼ = (E 0,0 ⊕ E 2,0 ) ⊕ (E 0,−2 ⊕ E 1,1 ⊕ E 0,2 ⊕ E 1,−1 ) and in particular C dim HomU2 (V1,1 , 1,1 0 p )=2

and

dim HomU2 (V1,1 , C) = 1.

Proof. We know already that V1,1 = soC 5 is the complexified adjoint representation and C ⊕(p1,0 ⊕p0,1 ). The decomposition of the last two summands is contained = u that soC 2 5 in (32). Hence it remains to make explicit the adjoint representation of U2 on uC 2 . It is clear that its restriction to U1 acts trivially, whereas its restriction to SU2 decomposes C ∼ into C ⊕ suC 2 , i.e. u2 = E 0,0 ⊕ E 2,0 . ¯ on primitive (1, 1)-forms for the eigenvalue 12 is thus isomorThe eigenspace of phic to the sum of two copies of V1,1 , i.e. the eigenvalue 12 has multiplicity 2 · 10 = 20. It is now easy to calculate the smallest eigenvalue and the corresponding eigenspace ¯ which coincides of the Laplace operator on non-constant functions. We do this for , (1,1) with on functions. Then we have to replace 0 p in the calculations above with the trivial representation C and to look for SO5 -representations Va,b containing the zero weight. It follows from Lemma 5.9 and (33) that the -eigenspace on functions 0 (12) is isomorphic to V1,1 and is thus 10-dimensional. Since the dimension of the isometry group of the nearly Kähler manifold SO(5)/U2 is 10, the inequality (26) shows that (1,1)

dim(N K) ≤ dim(0

(12)) − dim(i(M)) − dim(0 (12)) = 20 − 10 − 10 = 0,

so there are no infinitesimal nearly Kähler deformations in this case either. Finally, we remark like before that there are also no other infinitesimal Einstein defor¯ on mations, since by (33), the eigenvalues 2 and 6 do not occur in the spectrum of (1,1) 0 M. Summarizing, we have obtained the following: Theorem 5.10. The homogeneous nearly Kähler structure on CP3 = SO5 /U2 does not admit any infinitesimal nearly Kähler or Einstein deformations.

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¯ 5.6. The -spectrum on the flag manifold F(1, 2). In this section we consider the flag manifold M = SU3 /T 2 , where T 2 ⊂ SU3 is the maximal torus. Let g = su3 and let k = t, the Lie algebra of T 2 . We have the decomposition g=k⊕p

and

p = m ⊕ n.

Denoting by E i j , Si j the “real and imaginary” part of the projection of the vector X i j ∈ gl3 (equal to 1 on i th row and j th column and 0 elsewhere) onto su3 : E i j = X i j − X ji

Si j = i(X i j + X ji ),

the subspaces m and n are explicitly given by m = span{E 12 , S12 , E 13 , S13 } = span{e1 , e2 , e3 , e4 }, n = span{E 23 , S23 } = span{e5 , e6 }. The dual of the Lie algebra t of the maximal torus T 2 can be identified with t∗ ∼ = {(λ1 , λ2 , λ3 ) ∈ R3 | λ1 + λ2 + λ3 = 0}. If {εi } denotes the canonical basis in R3 then the set of positive roots is given as φ + = {αi j = εi − ε j | 1 ≤ i < j ≤ 3} and the half-sum of the positive roots is ρ = ε1 − ε3 . 1 B defines the Let B denote the Killing form of SU3 . By Lemma 5.4, B0 = − 12 homogeneous nearly Kähler metric g of scalar curvature scalg = 30. The almost complex structure J is explicitly defined on p by J (e1 ) = e2 ,

J (e3 ) = −e4 ,

J (e5 ) = e6 .

Alternatively we may define the (1, 0)-subspace of pC : p1,0 = gα12 ⊕ gα31 ⊕ gα23 = span{X 12 , X 31 , X 23 }, where gα is the root space for α. It follows that J is not integrable, since the restricted root system {α12 , α31 , α23 } is not closed under addition (cf. [4]). ¯ Let E = C3 be the standard representation of SU3 with conjugate representation E. Any irreducible representations of SU3 is isomorphic to one of the representations ¯ 0, Vk,l := (Symk E ⊗ Syml E) where the right-hand side denotes the kernel of the contraction map ¯ Symk E ⊗ Syml E¯ → Symk−1 E ⊗ Syml−1 E, ¯ The weights of Symk E are i.e. Vk,l is the Cartan summand in Symk E ⊗ Syml E. aε1 + bε2 + cε3 ,

with a, b, c ≥ 0, a + b + c = k.

If v1 , v2 , v3 are the weight vectors of E, then these weights correspond to the weight vectors v1a · v2b · v3c in Symk E. Since the weights of Syml E¯ are just minus the weights of Syml E, we see that the weights of Vk,l are (a − a )ε1 + (b − b )ε2 + (c − c )ε3 , a, b, c, a , b , c ≥ 0, a + b + c = k, a + b + c = l.

(34)

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From the given definition of the almost complex structure J it is clear that the T 2 -representation on p1,0 splits in three one-dimensional T 2 -representations with the weights α12 , α31 , α23 . Since the weights of a tensor product representation are the sums of weights of each factor and since ε1 + ε2 + ε3 = 0 on the Lie algebra of the maximal torus of SU3 , we immediately obtain Corollary 5.11. The weights of the T 2 -representation on 1,1 p ∼ = p1,0 ⊗ p0,1 are ±3ε1 , ±3ε2 , ±3ε3 , and 0. It remains to compute the Casimir operator of the irreducible SU3 -representations Vk,l . The highest weight of Vk,l is γ = kε1 − lε3 and ρ = ε1 − ε3 , thus CasVk,l = γ , γ + 2ρ B = − 16 (k(k + 2) + l(l + 2)).

(35)

Here we use again the Freudenthal formula and the fact that the Killing form B induces − 16 times the Euclidean scalar product on t∗ ⊂ R3 (easy calculation). Notice that we G have V1,1 = suC 3 and CasV1,1 = −1, which is consistent with Casad = −1 as in the previous cases. ¯ It follows that all possible -eigenvalues (with respect to the metric B0 ) are of the form 2(k(k + 2) + l(l + 2)). Obviously the eigenvalue 12 can only be obtained for k = l = 1. Moreover, the restriction of the SU3 -representation V1,1 contains the zero weight space. In fact, from (34), the zero weight appears in Vk,l if and only if there exist a, b, c, a , b , c ≥ 0, a + b + c = k, a + b + c = l such that (a − a )ε1 + (b − b )ε2 + (1,1) (c − c )ε3 = 0, which is equivalent to k = l. We see that dim Hom T 2 (V1,1 , 0 p) = 2 · 2 = 4. ¯ on primitive (1, 1)-forms for the eigenvalue 12 is isomorHence the eigenspace of phic to the sum of four copies of V1,1 , i.e. the eigenvalue 12 has multiplicity 4 · 8 = 32. Computing the smallest eigenvalue and the corresponding eigenspace of the Laplace operator on non-constant functions we find V0,0 for the eigenvalue 0 and V1,1 for the eigenvalue 12. All other possible representations give a larger eigenvalue. Hence, the -eigenspace on functions 0 (12) is isomorphic to two copies of V1,1 , i.e. the eigenvalue 12 has multiplicity 8 · 2 = 16. Since the dimension of the isometry group of the nearly Kähler manifold SU3 /T 2 is 8, we obtain from (26) (1,1)

dim(N K) ≤ dim(0

(12)) − dim(i(M)) − dim(0 (12)) = 8.

(36)

In the next section we will show by an explicit construction that actually the equality holds, so the flag manifold has an 8-dimensional space of infinitesimal nearly Kähler deformations. Before describing this construction we note that there are no infinitesimal Einstein deformations other than the nearly Kähler deformations. It follows from (35) that the ¯ on (1,1) M. The eigenvalue 6 could eigenvalue 2 does not occur in the spectrum of 0 be realized on the SU3 -representations V = V1,0 or V = V0,1 . However it is easy to (1,1) check that Hom T 2 (V, 0 p) = {0}. Corollary 5.12. Every infinitesimal Einstein deformation of the homogeneous nearly Kähler metric on F(1, 2) = SU3 /T 2 is an infinitesimal nearly Kähler deformation.

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6. The Infinitesimal Nearly Kähler Deformations on SU3 / T 2 In this section we describe by explicit computation the space of infinitesimal nearly Kähler deformations of the flag manifold F(1, 2) = SU3 /T 2 . The Lie algebra u3 is spanned by {h 1 , h 2 , h 3 , e1 , . . . , e6 }, where h 1 = i E 11 , h 2 = i E 22 , h 3 = i E 33 , e1 = E 12 − E 21 , e3 = E 13 − E 31 , e5 = E 23 − E 32 , e2 = i(E 12 + E 21 ), e4 = i(E 13 + E 31 ), e6 = i(E 23 + E 32 ). We consider the bi-invariant metric g on SU3 induced by −B/12, where B denotes the Killing form of su3 . It is easy to check that |ei |2 =1 and |h i − h j |2 = 1 with respect √ to g. We extend this metric to U3 in the obvious way which makes the frame {ei , 2h j } orthonormal. This defines a metric, also denoted by g, on the manifold M = F(1, 2). From now on we identify vectors and 1-forms using this metric and use the notation ei j = ei ∧ e j , etc. An easy explicit commutator calculation yields the exterior derivative of the leftinvariant 1-forms ei on U3 : de1 de2 de3 de4 de5 de6

= = = = = =

−2e2 ∧ (h 1 − h 2 ) + e35 + e46 , 2e1 ∧ (h 1 − h 2 ) + e45 − e36 , 2e4 ∧ (h 3 − h 1 ) − e15 + e26 , −2e3 ∧ (h 3 − h 1 ) − e25 − e16 , −2e6 ∧ (h 2 − h 3 ) + e13 + e24 , 2e5 ∧ (h 2 − h 3 ) + e14 − e23 .

(37)

Let J denote the almost complex structure on M = F(1, 2) whose Kähler form is ω = e12 − e34 + e56 . (It is easy to check that ω, which a priori is a left-invariant 2-form on U3 , projects to M because L h i ω = 0.) J induces an orientation on M with volume form −e123456 . Let + + i − denote the associated complex volume form on M defined by the adT 3 -invariant form (e2 + i J e2 ) ∧ (e4 + i J e4 ) ∧ (e6 + i J e6 ). Explicitly, + = e136 + e246 + e235 − e145 ,

− = e236 − e146 − e135 − e245 .

Using (37) we readily obtain d(e12 ) = −d(e34 ) = d(e56 ) = + ,

(38)

so dω = 3 + ,

and

d − = −2ω2 .

The pair (g, J ) thus defines a nearly Kähler structure on M (a fact which we already knew). We fix now an element ξ ∈ su3 ⊂ u3 , and denote by X the right-invariant vector field on U3 defined by ξ . Consider the functions xi = g(X, ei ),

vi = g(X, h i ).

(39)

The functions vi are projectable to M and clearly v1 + v2 + v3 = 0. Let us introduce the vector fields on U3 , a1 = x6 e5 − x5 e6 ,

a2 = x3 e4 − x4 e3 ,

a3 = x2 e1 − x1 e2 .

The Hermitian Laplace Operator on Nearly Kähler Manifolds

271

One can check that they project to M. Of course, one has J a1 = x5 e5 + x6 e6 ,

J a2 = x3 e3 + x4 e4 ,

J a3 = x1 e1 + x2 e2 .

The commutator relations in SU3 yield dv1 = a2 − a3 ,

dv2 = a3 − a1 ,

dv3 = a1 − a2 .

(40)

Using (37) and some straightforward computations we obtain d(J a1 ) = (−a1 + a2 + a3 ) + + 4(v2 − v3 )e56 , d(J a2 ) = (a1 − a2 + a3 ) + + 4(v1 − v3 )e34 , d(J a3 ) = (a1 + a2 − a3 ) + + 4(v1 − v2 )e12 .

(41)

We claim that the 2-form ϕ = v1 e56 − v2 e34 + v3 e12

(42)

on M is of type (1,1), primitive, co-closed, and satisfies ϕ = 12ϕ. The first two assertions are obvious (recall that v1 + v2 + v3 = 0). In order to prove that ϕ is co-closed, it is enough to prove that dϕ ∧ ω = 0. Using (38) and (40) we compute: dϕ ∧ ω = [(a2 − a3 ) ∧ e56 − (a3 − a1 ) ∧ e34 + (a1 − a2 ) ∧ e12 ] ∧ (e12 − e34 + e56 ) = (a1 − a2 ) ∧ e1256 − (a3 − a2 ) ∧ e1234 + (a1 − a2 ) ∧ e3456 = 0. Finally, using (41), we get ϕ = d ∗ dϕ = − ∗ d ∗ [(a2 − a3 ) ∧ e56 − (a3 − a1 ) ∧ e34 + (a1 − a2 ) ∧ e12 ] = −∗ d[J a2 ∧ e12 + J a3 ∧ e34 + J a3 ∧ e56 − J a1 ∧ e12 − J a1 ∧ e34 − J a2 ∧ e56 ] = − ∗ [d(J a2 ) ∧ (e12 − e56 ) + d(J a3 ) ∧ (e34 + e56 ) − d(J a1 ) ∧ (e12 + e34 )] = − ∗ [(a1 + a2 + a3 ) + ∧ (e12 − e56 + e34 + e56 − e12 − e34 ) −2 (a2 + ) ∧ (e12 − e56 )−2(a3 + ) ∧ (e34 +e56 )+2(a1 + ) ∧ (e12 + e34 ) + 4(v1 − v3 )e34 ∧ (e12 − e56 ) + 4(v1 − v2 )e12 ∧ (e34 + e56 ) − 4(v2 − v3 )e56 ∧ (e12 + e34 )] = − ∗ [4(2v1 − v2 − v3 )e1234 + 4(v1 + v3 − 2v2 )e1256 + 4(2v3 − v1 − v2 )e3456 ] = − ∗ [12v1 e1234 − 12v2 e1256 + 12v3 e3456 ] = 12ϕ. Taking into account the inequality (36), we deduce at once the following Corollary 6.1. The space of infinitesimal nearly Kähler deformations of the nearly Kähler structure on F(1, 2) is isomorphic to the Lie algebra of SU3 . More precisely, every right-invariant vector field X on SU3 defines an element ϕ ∈ N K via the formulas (39) and (42). Acknowledgements. We are grateful to Gregor Weingart for helpful discussions and in particular for suggesting the statement of Lemma 5.4.

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A. Moroianu, U. Semmelmann

References 1. Baum, H., Friedrich, Th., Grunewald, R., Kath, I.: Twistor and Killing Spinors on Riemannian Manifolds. Stuttgart–Leipzig: Teubner–Verlag, 1991 2. Belgun, F., Moroianu, A.: Nearly Kähler 6-manifolds with reduced holonomy. Ann. Global Anal. Geom. 19, 307–319 (2001) 3. Besse, A.: Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10. Berlin: SpringerVerlag, 1987 4. Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces I. 80, 458–538 (1958) 5. Butruille, J.-B.: Classification des variétés approximativement kähleriennes homogènes. Ann. Global Anal. Geom. 27, 201–225 (2005) 6. Cleyton, R., Swann, A.: Einstein metrics via intrinsic or parallel torsion. Math. Z. 247, 513–528 (2004) 7. Friedrich, Th.: Nearly Kähler and nearly parallel G 2 -structures on spheres. Arch. Math. (Brno) 42, 241– 243 (2006) 8. Gray, A.: The structure of nearly Kähler manifolds. Math. Ann. 223, 233–248 (1976) 9. Moroianu, A., Nagy, P.-A., Semmelmann, U.: Unit Killing Vector Fields on Nearly Kähler Manifolds. Internat. J. Math. 16, 281–301 (2005) 10. Moroianu, A., Nagy, P.-A., Semmelmann, U.: Deformations of Nearly Kähler Structures. Pacific J. Math. 235, 57–72 (2008) 11. Moroianu, A., Semmelmann, U.: Infinitesimal Einstein Deformations of Nearly Kähler Metrics. to appear in Trans. Amer. Math. Soc., 2009 12. Nagy, P.-A.: Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 3, 481–504 (2002) 13. Wolf, J., Gray, A.: Homogeneous spaces defined by Lie group automorphisms I, II. J. Differ. Geom. 2, 77–114, 115–159 (1968) Communicated by A. Connes

Commun. Math. Phys. 294, 273–301 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0933-y

Communications in

Mathematical Physics

Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I. Manoussos G. Grillakis 1 , Matei Machedon 1 , Dionisios Margetis 1,2,3 1 Department of Mathematics, University of Maryland, College Park,

MD 20742, USA. E-mail: [email protected]

2 Institute for Physical Science and Technology, University of Maryland,

College Park, MD 20742, USA

3 Center for Scientific Computation and Mathematical Modeling, University of Maryland,

College Park, MD 20742, USA Received: 31 March 2009 / Accepted: 30 July 2009 Published online: 2 October 2009 – © Springer-Verlag 2009

Abstract: Inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schrödinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential v(x) = χ (x)|x|−1 , where is sufficiently small and χ ∈ C0∞ even, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (Part II) of this paper. 1. Introduction An advance in physics in 1995 was the first experimental observation of atoms with integer spin (Bosons) occupying a macroscopic quantum state (condensate) in a dilute gas at very low temperatures [1,4]. This phenomenon of Bose-Einstein condensation has been observed in many similar experiments since. These observations have rekindled interest in the quantum theory of large Boson systems. For recent reviews, see e.g. [23,29]. A system of N interacting Bosons at zero temperature is described by a symmetric wave function satisfying the N -body Schrödinger equation. For large N , this description is impractical. It is thus desirable to replace the many-body evolution by effective (in an appropriate sense) partial differential equations for wave functions in much lower space dimensions. This approach has led to “mean-field” approximations in which the single particle wave function for the condensate satisfies nonlinear Schrödinger equations (in 3 + 1 dimensions). Under this approximation, the N -body wave function is viewed simply as a tensor product of one-particle states. For early related works, see the papers by Gross [15,16], Pitaevskii [28] and Wu [34,35]. In particular, Wu [34,35] introduced a second-order approximation for the Boson many-body wave function in terms of the pair-excitation function, a suitable kernel that describes the scattering of atom pairs

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M. G. Grillakis, M. Machedon, D. Margetis

from the condensate to other states. Wu’s formulation forms a nontrivial extension of works by Lee, Huang and Yang [21] for the periodic Boson system. Approximations carried out for pair excitations [21,34,35] make use of quantized fields in the Fock space. (The Fock space formalism and Wu’s formulation are reviewed in Sects. 1.1 and 1.3, respectively.) Connecting mean-field approaches to the actual many-particle Hamiltonian evolution raises fundamental questions. One question is the rigorous derivation and interpretation of the mean field limit. Elgart, Erd˝os, Schlein and Yau [6–11] showed rigorously how mean-field limits for Bosons can be extracted in the limit N → ∞ by using Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchies for reduced density matrices. Another issue concerns the convergence of the microscopic evolution towards the mean field dynamics. Recently, Rodnianski and Schlein [31] provided estimates for the rate of convergence in the case with Hartree dynamics by invoking the formalism of Fock space. In this paper, inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schrödinger equation describing an improved approximation for the evolution of the Boson system. This approximation offers a second-order correction to the usual tensor product (mean field limit) for the many-body wave function. Our equation yields a corresponding new estimate in Fock space, which complements nicely the previous estimate [31]. The static version of the many-body problem is not studied here. The energy spectrum was addressed by Dyson [5] and by Lee, Huang and Yang [21]. A mathematical proof of the Bose-Einstein condensation for the time-independent case was provided recently by Lieb, Seiringer, Solovej and Yngvanson [22–25]. 1.1. Fock space formalism. Next, we review the Fock space F over L 2 (R3 ), following Rodnianski and Schlein [31]. The elements of F are vectors of the form ψ = (ψ0 , ψ1 (x1 ), ψ2 (x1 , x2 ), . . .), where ψ0 ∈ C and ψn ∈ L 2s (R3n ) are symmetric in x1 , . . . , xn . The Hilbert space structure of F is given by (φ, ψ) = n φn ψn d x. For f ∈ L 2 (R3 ) the (unbounded, closed, densely defined) creation operator a ∗ ( f ) : F → F and annihilation operator a( f¯) : F → F are defined by n ∗ 1 a ( f )ψn−1 (x1 , x2 , . . . , xn ) = √ f (x j )ψn−1 (x1 , . . . , x j−1 , x j+1 , . . . xn ), n j=1 √ a( f )ψn+1 (x1 , x2 , . . . , xn ) = n + 1 ψ(n+1) (x, x1 , . . . , xn ) f (x) d x.

The operator valued distributions ax∗ and ax defined by ∗ f (x)ax∗ d x, a (f) = a( f ) = f (x) ax d x. These distributions satisfy the canonical commutation relations [ax , a ∗y ] = δ(x − y), [ax , a y ] = [ax∗ , a ∗y ] = 0.

(1)

Second-Order Corrections for Weakly Interacting Bosons I

275

Let N be a fixed integer (the total number of particles), and v(x) be an even potential. Consider the Fock space Hamiltonian H N : F → F defined by 1 HN = ax∗ ∆ax d x + v(x − y)ax∗ a ∗y ax a y d x d y 2N 1 (2) =: H0 + V. N This H N is a diagonal operator which acts on each ψn in correspondence to the Hamiltonian H N ,n =

n

∆x j +

j=1

n 1 v(xi − x j ). 2N i, j=1

In the particular case n = N , this is the mean field Hamiltonian. Except for the Introduction, this paper deals only with the Fock space Hamiltonian. The reader is alerted that “PDE” Hamiltonians such as H N ,n will always have two subscripts. The sign of v will not play a role in our analysis. However, the reader is alerted that due to our sign convention, v ≤ 0 is the “good” sign. The time evolution in the coordinate space for Bose-Einstein condensation deals with the function eitHn,n ψ0

(3)

for tensor product initial data, i.e., if ψ0 (x1 , x2 , . . . , xn ) = φ0 (x1 )φ0 (x2 ) . . . φ0 (xn ), where φ0 L 2 (R3 ) = 1. This approach has been highly successful, even for very singular potentials, in the work of Elgart, Erd˝os, Schlein and Yau [6–11]. In this context, the convergence of evolution to the appropriate mean field limit (tensor product) as N → ∞ is established at the level of marginal density matrices γi(N ) in the trace norm topology. The density matrices are defined as (N ) γi (t, x1 , . . . , xi ; x1 , . . . xi ) = ψ(t, x1 , . . . , x N )ψ(t, x1 , . . . , x N ) d xi+1 · · · d x N .

1.2. Coherent states. There are alternative approaches, due to Hepp [17], Ginibre and Velo [13], and, most recently, Rodnianski and Schlein [31] which can treat Coulomb potentials v. These approaches rely on studying the Fock space evolution eit HN ψ 0 , where the initial data ψ 0 is a coherent state, ψ 0 = (c0 , c1 φ0 (x1 ), c2 φ0 (x1 )φ0 (x2 ), · · · ); see (4) below. The evolution (3) can then be extracted as a “Fourier coefficient” from the Fock space evolution, see [31]. Under the assumption that v is a Coulomb potential, this approach leads to strong L 2 -convergence, still at the level of the density matrices (N ) γi , as we will briefly explain below. To clarify the issues involved, let us consider the one-particle wave function φ(t, x) (to be determined later as the solution of a Hartree equation), satisfying the initial condition φ(0, x) = φ0 (x). Define the skew-Hermitian unbounded operator A(φ) = a(φ) − a ∗ (φ)

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and the vacuum state Ω = (1, 0, 0, . . .) ∈ F. Accordingly, consider the operator √

W (φ) = e−

N A(φ)

,

which is the Weyl operator used by Rodnianski and Schlein [31]. The coherent state for the initial data φ0 is √

ψ 0 = W (φ0 )Ω = e− =e

−N φ2 /2

N A(φ0 )

1, . . . ,

Ω n 1/2

N n!

φ0 (x1 ) . . . φ0 (xn ), · · ·

.

(4)

Hence, the top candidate approximation for eitHN ψ 0 reads √

ψ tensor (t) = e−

N A(φ(t,·))

Ω.

(5)

Rodnianski and Schlein [31] showed that this approximation works (under suitable assumptions on v), in the sense that √ √ 1 itHN e ψ 0 , a ∗y ax eit HN ψ 0 − e− N A(φ(t,·)) Ω, a ∗y ax e− N A(φ(t,·)) Ω Tr N eCt ) N → ∞; = O( N the symbol Tr here stands for the trace norm in x ∈ R3 and y ∈ R3 . The first term in the last relation, including N1 , is essentially the density matrix γ1(N ) (t, x, y). For the precise statement of the problem and details of the proof, see Theorem 3.1 of Rodnianski and Schlein [31]. Our goal here is to find an explicit approximation for the evolution in the Fock space. For this purpose, we adopt an idea germane to Wu’s second-order approximation for the N -body wave function in Fock space [34,35].

1.3. Wu’s approach. We first comment on the case with periodic boundary conditions, when the condensate is the zero-momentum state. For this setting, Lee, Huang and Yang [21] studied systematically the scattering of atoms from the condensate to states of opposite momenta. By diagonalizing an approximation for the Hamiltonian in Fock space, these authors derived a formula for the N -particle wave function that deviates from the usual tensor product, as it expresses excitation of particles from zero monentum to pairs of opposite momenta. For non-periodic settings, Wu [34,35] invokes the splitting ax = a0 (t)φ(t, x) + ax,1 (t), where a0 corresponds to the condensate, [a0 , a0∗ ] = 1, and ax,1 corresponds to ∗ ]. Wu applies the following states orthogonal to the condensate, [a0 , ax,1 ] = 0 = [a0 , ax,1 ansatz for the N -body wave function in Fock space: N (t) eP [K 0 ] ψ N0 (t),

(6)

where ψ N0 (t) describes the tensor product, N (t) is a normalization factor, and P[K 0 ] is an operator that averages out in space the excitation of particles from the condensate φ to

Second-Order Corrections for Weakly Interacting Bosons I

277

other states with the effective kernel (pair excitation function) K 0 . An explicit formula for P[K 0 ] is ∗ P[K 0 ] = [2N0 (t)]−1 ax,1 a ∗y,1 K 0 (t, x, y) a0 (t)2 , (7) where N0 is the expectation value of particle number at the condensate. This K 0 is not a-priori known (in contrast to the case of the classical Boltzmann gas) but is determined by means consistent with the many-body dynamics. In the periodic case, (6) reduces to the many-body wave function of Lee, Huang and Yang [21]. Wu derives a coupled system of dispersive hyperbolic partial differential equations for (φ, K 0 ) via an approximation for the N -body Hamiltonian that is consistent with ansatz (6). A feature of this system is the spatially nonlocal couplings induced by K 0 . Observable quantities such that the depletion of the condensate can be computed directly from solutions of this PDE system. This system has been solved only in a limited number of cases [26,27,35].

1.4. Scope and outline. Our objective in this work is to find an explicit approximation for the evolution eitHN ψ 0 in the Fock space norm, where ψ 0 is the coherent state (4). This would imply an approximation for the evolution eitHN ,N ψ0 in L 2 (R3N ) as N → ∞. To the best of our knowledge, no such approximation is available in the mathematics or physics literature. In particular, the tensor product type approximation (5) for φ satisfying a Hartree equation, as in [31], is not known to be such a Fock space approximation (nor do we expect it to be). To accomplish our goal, we propose to modify (5) in two ways. One minor correction is the multiplication by an oscillatory term. A second correction is a composition with a second-order “Weyl operator”. Both corrections are inspired by the work of Wu [34,35]; see also [26,27]. However, our set-up and derived equation are essentially different from these works. We proceed to describe the second order correction. Let k(t, x, y) = k(t, y, x) be a function (or kernel) to be determined later, with k(0, x, y) = 0. The minimum regularity expected of k is k ∈ L 2 (d x d y) for a.e. t. We define the operator 1 (8) B= k(t, x, y)ax a y − k(t, x, y)ax∗ a ∗y d x d y. 2 Notice that B is skew-Hermitian, i.e., iB is self-adjoint. The operator e B could be defined by the spectral theorem; see [30]. However, we prefer the more direct approach of defining it first on the dense subset of vectors with finitely many non-zero components, where it can be defined by a convergent Taylor series if k L 2 (d xd y) is sufficiently small. Indeed, B restricted to the subspace of vectors with all entries past the first N identically zero has norm ≤ C N k L 2 . Then e B is extended to F as a unitary operator.

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Now we have described all ingredients needed to state our results and derivations. The remainder of the paper is organized as follows. In Section 2 we state our main result and outline its proof. In Sect. 3 we study implications of the Hartree equation satisfied by the one-particle wave function φ(t, x). In Sect. 4 we develop bookkeeping tools of Lie algebra for computing requisite operators containing B. In Sect. 5 we study the evolution equation for a matrix K that involves the kernel k. In Sect. 6 we develop an argument for the existence of solution to the equation for the kernel k. In Sect. 7 we find conditions under which terms involved in the error term e B V e−B are bounded. In Sect. 8 we study similarly the error term e B [A, V ]e−B . In Sect. 9 we show that we can control traces needed in derivations. 2. Statement of Main Result and Outline of Proof In this section we state our strategy for general potentials satisfying certain properties. Later in the paper we show that all assumptions of the related theorem are satisfied , : sufficiently small, and χ ∈ C0∞ : even. locally in time for v(x) = χ (x) |x| Theorem 1. Suppose that v is an even potential. Let φ be a smooth solution of the Hartree equation i

∂φ + ∆φ + (v ∗ |φ|2 )φ = 0 ∂t

(9)

with initial conditions φ0 , and assume the three conditions listed below: 1. Assume that we have k(t, x, y) ∈ L 2 (d xd y) for a.e. t, where k is symmetric, and solves (iut + ug T + gu − (1 + p)m) = (i pt + [g, p] + um)(1 + p)−1 u,

(10)

where all products in (10) are interpreted as spatial compositions of kernels, “1” is the identity operator, and u(t, x, y) := sh(k) := k +

1 kkk + · · · , 3!

1 kk + · · · , (11) 2! g(t, x, y) := −∆x δ(x − y) − v(x − y)φ(t, x)φ(t, y) − (v ∗ |φ|2 )(t, x)δ(x − y),

δ(x − y) + p(t, x, y) := ch(k) := δ(x − y) + m(t, x, y) := v(x − y)φ(t, x)φ(t, y). 2. Also, assume that the functions

f (t) := e B [A, V ]e−B ΩF and g(t) := e B V e−B ΩF are locally integrable (V is defined in (2)).

Second-Order Corrections for Weakly Interacting Bosons I

3. Finally, assume that

279

d(t, x, x) d x is locally integrable in time, where d(t, x, y) = ish(k)t + sh(k)g T + gsh(k) sh(k) − (ich(k)t + [g, ch(k)]) ch(k) −sh(k)mch(k) − ch(k)msh(k).

Then, there exist real functions χ0 , χ1 such that √

e−

N A(t) −B(t) −i

t

≤

0

e

e t

f (s)ds + √ N

t

0

0 (N χ0 (s)+χ1 (s))ds

Ω − eitHN ψ 0 F

g(s)ds . N

(12)

Recall that we defined (see Sect. 1) √

ψ 0 = e−

N A(0)

Ω an arbitrary coherent state (initial data),

A(t) = a(φ(t, ·)) − a ∗ (φ(t, ·)), 1 B(t) = k(t, x, y)ax a y − k(t, x, y)ax∗ a ∗y d x d y. 2 A few remarks on Theorem 1 are in order. Remark 1. Written explicitly, the left-hand side of (10) equals ∂ iut + ug T + gu − (1 + p)m = i − ∆x − ∆ y u(t, x, y) ∂t −φ(t, x) v(x − z)φ(t, z)u(t, z, y) dz − φ(t, y) u(t, x, z)v(z − y)φ(t, z) dz −(v ∗ |φ|2 )(t, x)u(t, x, y) − (v ∗ |φ|2 )(t, y)u(t, x, y) −v(x − y)φ(t, x)φ(t, y) −φ(t, y) (1 + p)(t, x, z)v(z − y)φ(t, z) dz. The main term in the right-hand side equals ∂ i pt + [g, p] + um = i p(t, x, y) + −∆x + ∆ y p(t, x, y) ∂t −φ(t, x) v(x − z)φ(t, z) p(t, z, y) dz +φ(t, y) p(t, x, z)v(z − y)φ(t, z) dz −(v ∗ |φ|2 )(t, x) p(t, x, y) + (v ∗ |φ|2 )(t, y) p(t, x, y) + u(t, x, z)v(z − y)φ(t, z)φ(t, x) dz. Remark 2. The algebra, as well as the local analysis presented in this paper do not depend on the sign of v. However, the global in time analysis of our equations would require v to be non-positive.

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Remark 3. Our √techniques would allow us to consider more general initial data of the form ψ 0 = e− N A(0) e−B(0) Ω. For convenience, we only consider the case of tensor products (B(0) = 0) in this paper. √

Proof. Since e ei

NA

and e B are unitary, the left-hand side of (12) equals

t

0 (N χ0 (s)+χ1 (s))ds

e B(t) e

√

√ N A(t) itH N − N A(0)

e

e

Ω − ΩF .

Define Ψ (t) = e B(t) e

√

√ N A(t) it H − N A(0)

e

e

Ω.

In Corollary 1 of Sect. 5 we show that our equations for φ, k insure that 1 ∂ Ψ = LΨ, i ∂t where L = L − N χ0 − χ1 for some L: Hermitian, i.e. L = L ∗ , where L commutes with functions of time, χ0 , χ1 are real functions of time, and, most importantly (see Corollary 1 of Sect. 5 and the remark following it), LΩF ≤ N −1/2 e B [A, V ]e−B ΩF + N −1 e B V e−B ΩF .

(13)

We apply energy estimates to

t 1 ∂

− L (ei 0 (N χ0 (s)+χ1 (s))ds Ψ − Ω) = LΩ. i ∂t

Explicitly, ∂ i t (N χ0 (s)+χ1 )ds (e 0 Ψ − Ω)2F ∂t t ∂ i t (N χ0 (s)+χ1 )ds i 0 (N χ0 (s)+χ1 )ds 0 = 2 (e Ψ − Ω), e Ψ −Ω ∂t t t ∂ i 0 (N χ0 (s)+χ1 )ds i 0 (N χ0 (s)+χ1 )ds

− i L (e Ψ − Ω), e Ψ −Ω = 2 ∂t t = 2 i LΩ, ei 0 (N χ0 (s)+χ1 )ds Ψ − Ω t ≤ 2 N−1/2 e B [A, V ]e−B ΩF + N−1 e B V e−B ΩF (ei 0 (N χ0 (s)+χ1 )ds Ψ −Ω)F . Thus t ∂ (ei 0 (N χ0 (s)+χ1 )ds Ψ − Ω) ≤ N −1/2 e B [A, V ]e−B ΩF + N −1 e B V e−B ΩF ∂t

and (12) holds. This concludes the proof.

Second-Order Corrections for Weakly Interacting Bosons I

281

3. The Hartree Equation In this section we see how far we can go by using only the Hartree equation for the one-particle wave function φ. Lemma 1. The following commutation relations hold (where the t dependence is suppressed, A denotes A(φ) and V is defined by formula (2)): [A, V ] = v(x − y) φ(y)ax∗ ax a y + φ(y)ax∗ a ∗y ax d x d y, [A, [A, V ]] = v(x − y) φ(y)φ(x)ax a y + φ(y)φ(x)ax∗ a ∗y + 2φ(y)φ(x)ax∗ a y d x d y, +2 v ∗ |φ 2 | (x)ax∗ ax d x, A, A, [A, V ] = 6 v ∗ |φ 2 | (x) φ(x)ax∗ + φ(x)ax d x,

A, A, [ A, [A, V ]]

= 12 v ∗ |φ 2 | (x)|φ(x)|2 d x.

(14)

Proof. This is an elementary calculation and is left to the interested reader. √

√

Now, we consider Ψ1 (t) = e N A(t) eit H e− N A(0) Ω for which we have the basic calculation in the spirit of Hepp [17], Ginibre-Velo [13], and Rodnianski-Schlein [31]; see Eq. (3.7) in [31]. Proposition 1. If φ satisfies the Hartree equation i

∂φ + ∆φ + (v ∗ |φ|2 )φ = 0 ∂t

while Ψ1 (t) = e

√

√ N A(t) it H − N A(0)

e

e

Ω,

then Ψ1 (t) satisfies 1 1 ∂ Ψ1 (t) = H0 + [A, [A, V ]] i ∂t 2 N +N −1/2 [A, V ] + N −1 V − v(x − y)|φ(t, x)|2 |φ(t, y)|2 d x d y Ψ1 (t). 2 Proof. Recall the formulas ∂ C(t) −C(t) 1 ˙ + ··· ˙ + 1 C, [C, C] e = C˙ + [C, C] e ∂t 2! 3! and eC H e−C = H + [C, H ] +

1 [C, [C, H ]] + · · · 2!

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Applying these relations to C =

√

N A we get

1 ∂ ψ1 (t) = L 1 ψ1 , i ∂t

(15)

where

√ √ 1 ∂ √ N A(t) −√ N A(t) e e + e N A(t) H e− N A(t) i ∂t 1 N ˙ + H + N 1/2 [A, H0 ] N 1/2 A˙ + [A, A] = i 2 N +N −1/2 [A, V ] + [A, [A, H0 ]] 2 N 1 N 1/2 A, A, [A, V ] + A, A, [ A, [A, V ]] . × [A, [A, V ]] + 2 3! 4! √ Eliminating the terms with a weight of N , or setting 1 1 ˙ A, A, [A, V ] = 0, (16) A + [A, H0 ] + i 3! L1 =

is exactly equivalent to the Hartree equation (9). By taking an additional bracket with A in (16), we have 1 1 ˙ [A, A] + [A, [A, H0 ]] + A, A, [ A, [A, V ]] = 0, i 3! and thus simplify (15) to 1 ∂ 1 ψ1 (t) = H0 + [A, [A, V ]] i ∂t 2 +N This concludes the proof.

−1/2

[A, V ] + N

−1

1 A, A, [A, [A, V ]] ψ1 . V−N 4!

on the right-hand side are the main ones. The next two terms are two terms The first 1 1 and O N . The last term equals O √ N

−

N 2

v(x − y)|φ(t, x)|2 |φ(t, y)|2 d x d y := −N χ0 .

Notice that L 1 (Ω) is not small because of the presence of ax∗ a ∗y in [A, [A, V ]]. In order to eliminate these terms, we introduce B (see (8)) and take ψ = e B ψ1 . Accordingly, we compute 1 ∂ ψ = Lψ, i ∂t

Second-Order Corrections for Weakly Interacting Bosons I

where 1 L= i

283

∂ B −B e e + e B L 1 e−B ∂t

= L Q + N −1/2 e B [A, V ]e−B + N −1 e B V e−B − N χ0 , and LQ

1 = i

∂ B −B 1 B H0 + [A, [A, V ]] e−B e e +e ∂t 2

(17)

contains all quadratics in the operators a, a ∗ . Equation (10) for k turns out to be equivalent to the requirement that L has no terms of the form a ∗ a ∗ . Terms of the form aa ∗ will occur, and will be converted to a ∗ a at the expense of χ1 . In other words, we require that L Q have no terms of the form a ∗ a ∗ . For a similar argument (but for a different set-up), see Wu [35]. 4. The Lie Algebra of “Symplectic Matrices” In this section we describe the bookkeeping tools needed to compute L Q of (17) in closed form. The results of this section are essentially standard, but they are included here for the sake of completeness. We start with the remark that [a( f 1 ) + a ∗ (g1 ), a( f 2 ) + a ∗ (g2 )] = f 1 g2 − f 2 g1 f = − f 1 g1 J 2 , (18) g2 where

0 −δ(x − y) J= . δ(x − y) 0

This observation explains why we have to invoke symplectic linear algebra. We thus consider the infinite-dimensional Lie algebra sp of “matrices” of the form d k S(d, k, l) = l −d T for symmetric kernels k = k(t, x, y) and l = l(t, x, y), and arbitrary kernel d(t, x, y). (The dependence on t will be suppressed when not needed.) This situation is analogous to the Lie algebra of the finite-dimensional complex symplectic group, with x, y playing the role of i and j. We also consider the Lie algebra Quad of quadratics of the form ∗ d k 1 −a y ∗ ax ax Q(d, k, l) := l −d T ay 2 ∗ ∗ ax a y + a y ax 1 = − d(x, y) dx dy + k(x, y)ax a y d x d y 2 2 1 − (19) l(x, y)ax∗ a ∗y d x d y 2

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(k, l and d as before). Furthermore, we agree to identify operators which differ (formally) by a scalar operator. Thus, d(x, y)ax a ∗y is considered equivalent to d(x, y)a ∗y ax . We recall the following result related to the metaplectic representation (see, e.g. [12]). Theorem 2. Let S = S(d, k, l), Q = Q(d, k, l) related as above. Let f , g be functions (or distributions). Denote f := f (x)ax + g(x)ax∗ d x. (ax , ax∗ ) g We have the following commutation relation: f f ∗ ∗ = (ax , ax )S , Q, (ax , ax ) g g where products are interpreted as compositions. We also have f f Q ∗ −Q ∗ S , e e (ax , ax ) = (ax , ax )e g g

(20)

(21)

provided that e Q makes sense as a unitary operator (Q: skew-Hermitian). Proof. The commutation relation (20) can be easily checked directly, but we point out that it follows from (18). In fact, using (18), for any rank one quadratic we have a( f 1 ) + a ∗ (g1 ) a( f 2 ) + a ∗ (g2 ) , a( f ) + a ∗ (g) f1 f2 f ∗ f 1 g1 + f 2 g2 J = − ax ax . g g2 g1 Thus, for any R we have a f ax ax∗ R ∗y , a( f ) + a ∗ (g) = − ax ax∗ R + R T J . ay g Now specialize to R = 21 S J , S ∈ sp, and use S T = J S J to complete the proof. The second part, Eq. (21), follows from the identity e Q Ce−Q = C + [Q, C] +

1 [Q, [Q, C]] + · · · , 2!

or, in the language of adjoint representations, Ad(e Q )(C) = ead(Q) (C), which is applied to C = a( f ) + a ∗ (g).

A closely related result is provided by the following theorem. Theorem 3. 1. The linear map I : sp → Quad defined by S(d, k, l) → Q(d, k, l) is a Lie algebra isomorphism. 2. Moreover, if S = S(t), Q = Q(t) and I(S(t)) = Q(t) is skew-Hermitian, so that e Q is well defined, we have ∂ Q −Q ∂ S −S = e . e e e (22) I ∂t ∂t

Second-Order Corrections for Weakly Interacting Bosons I

285

3. Also, if R ∈ sp, we have I e S Re−S = e Q I(R)e−Q .

(23)

Remark 4. In the finite-dimensional case, this is (closely related to) the “infinitesimal metaplectic representation”; see p. 186 in [12] . In the infinite dimensional case, we must be careful, as some of our operators are not of trace class. For instance, ax ax∗ does not make sense. Proof. First, we point out that (21) implies (23), at least in the case where R is the “rank one” matrix f hi . R= g Notice that (21) can also be written as ax −Q S T ax f g eQ f g e . = e ax∗ ax∗ In conclusion, we find ∗ −a y −Q ∗ e ax ax R e ay f a Q ∗ h i a a J a ∗y e−Q =e x x g y f a Q −Q Q ∗ a h i a e e J a ∗y e−Q =e x x g y f J S J ay h i Je = ax ax∗ e S a ∗y g ∗ −a y , = ax ax∗ e S Re−S ay Q

since S T = J S J if S ∈ sp, and J e J S J = e−S J . We now give a direct proof that (19) preserves Lie brackets. Denote the quadratic 1 ∗ ∗ ∗ ∗ ∗ building blocks by Q x y = ax a y , Q x y = ax a y , N x y = 2 ax a y + a y ax . One can verify the following commutation relations, which will be also needed below: Q x y , Q ∗zw = δ(x − z)N yw + δ(x − w)N yz + δ(y − z)N xw + δ(y − w)N x z , (24) (25) Q x y , Nzw = δ(x − w)Q yz + δ(y − w)Q x z , ∗ ∗ (26) N x y , Q zw = δ(x − z)Q yw + δ(x − w)Q yz , N x y , Nzw = δ(x − w)Nzy − δ(y − z)N xw . (27) Using (24) we compute 1 1 k(x, y)ax a y d xd y, − l(x, y)ax∗ a ∗y d xd y = − (kl)(x, y)N x y d x d y, 2 2

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which corresponds to the relation 0 0 k , l 0 0

0 0

=

kl 0 . 0 −lk

The other three cases are similar. To prove (22), expand both the left-hand side and the right-hand side as ∂ S −S I e e ∂t 1 ˙ ˙ = I S + [S, S] + · · · 2 1 ˙ + ··· = Q˙ + [Q, Q] 2 ∂ Q −Q e . = e ∂t The proof of (23) is along the same lines.

Remark 5. Note on rigor: All the Lie algebra results that we have used are standard in the finite-dimensional case. In our applications, S will be K , where K is a matrix of the form (29), see below, and Q will be B = I(K ). The unbounded operator B is skew-Hermitian and e B ψ is defined by a convergent Taylor series if ψ ∈ F has only finitely many non-zero components, provided k(t, ·, ·) L 2 (d x d y) is small. We then extend e B to all F as a unitary operator. The norm k(t, ·, ·) L 2 (d x d y) iterates under compositions; thus, the kernel e K is well defined by its convergent Taylor expansion. In the expression e B Pe−B = P + [B, P] + · · ·

(28)

for P, a first- or second-order polynomial in a, a ∗ , we point out that the right-hand side stays a polynomial of the same degree, and converges when applied to a Fock space vector with finitely many non-zero components. For our application, we need to know if (28) is true when applied to Ω. The same comment applies to the series ∂ B −B 1 ˙ + ··· . e e = B˙ + [B, B] ∂t 2 5. Equation for Kernel k Now apply the isomorphism of the previous section to the operator B = I(K ) for

0 k(t, x, y) . k(t, x, y) 0

K =

(29)

This agrees to the letter with the isomorphism (19). The next two isomorphisms, (30) and (31), require special treatment because aa ∗ terms mirroring the a ∗ a terms are missing

Second-Order Corrections for Weakly Interacting Bosons I

287

in (2), (14). However, the discrepancy only happens on the diagonal. Once the relevant terms are commuted with B, they fit the pattern exactly. It isn’t quite true that −(∆δ)(x − y) 0 H0 = I 0 (∆δ)(x − y) −∆ 0 =I (30) 0 ∆ since, strictly speaking, ∗ ax ∆ax + ax ∆ax∗ −(∆δ)(x − y) 0 = I dx 0 (∆δ)(x − y) 2 is undefined. However, one can compute directly that [∆x ax , a ∗y ] = (∆δ)(x − y). Using that, we compute 1 [B, H0 ] = (∆x + ∆ y )k(x, y)ax a y + (∆x + ∆ y )k(x, y)ax∗ a ∗y d x d y. 2 This commutator is in agreement with (29), (30), and the result can be represented in accordance with (19), namely 0k −(∆δ)(x − y) 0 [B, H0 ] = I , . 0 (∆δ)(x − y) k0 We also have e B H0 e−B − H0 0 −(∆δ)(x − y) 0 K −(∆δ)(x − y) −K e =I e − , 0 (∆δ)(x − y) 0 (∆δ)(x − y) since e B H0 e−B − H0 = [B, H0 ] + 21 [B, [B, H0 ]] + · · ·. The same comment applies to the diagonal part of 1 v12φ 1 φ 2 −v12 φ 1 φ2 − v ∗ |φ|2 δ12 , (31) [A, [A, V ]] = I −v12 φ1 φ2 v12 φ1 φ 2 + v ∗ |φ|2 δ12 2 where v12 φ1 φ2 is an abbreviation for the product v(x − y)φ(x)φ(y), etc. Formula (31) isn’t quite true either, but becomes true after commuting with B. To apply our isomorphism, we quarantine the “bad” terms in (30) and the diagonal part of (31). Define g 0 0 m , G= and M= −m 0 0 −g T where g = −∆δ12 − v12 φ 1 φ2 − (v ∗ |φ|2 )δ12 , m = v12 φ 1 φ 2 , and split H0 +

1 [A, [A, V ]] = HG + I(M), 2

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where

HG = H0 + v(x − y)φ(y)φ(x)ax∗ a y d x d y + v ∗ |φ 2 | (x)ax∗ ax d x.

(32)

By the above discussion we have [B, HG ] = I([K , G]) and B −B K [e , HG ]e = I([e , G]e−K ). Write

1 ∂ B −B 1 e e + e B H0 + [A, [A, V ]] e−B i ∂t 2 1 ∂ B −B e e + HG + [e B , HG ]e−B + e B I(M)e−B = i ∂t 1 ∂ K −K e e = HG + I + [e K , G]e−K + e K Me−K i ∂t = HG + I(M1 + M2 + M3 ).

LQ =

(33)

Notice that if K is given by (29), then ch(k) sh(k) K e = , sh(k) ch(k) where 1 1 ch(k) = I + kk + kkkk + · · · , (34) 2 4! and similarly for sh(k). Products are interpreted, of course, as compositions of operators. We compute 1 ch(k)t sh(k)t ch(k) −sh(k) M1 = −sh(k) ch(k) i sh(k)t ch(k)t 1 ch(k)t ch(k) − sh(k)t sh(k) −ch(k)t sh(k) + sh(k)t ch(k) = ∗ ∗ i T [ch(k), g] −sh(k)g − gsh(k) [e K , G] = ∗ ∗ and M2 = [e K , G]e−K =

[ch, g] ch + (shg T + gsh)sh −[ch, g]sh − (shg T + gsh)ch , ∗ ∗

where sh is an abbreviation for sh(k), etc, and −shm ch − chmsh shmsh + chmch . M3 = e K Me−K = ∗ ∗ Now define M = M1 + M2 + M3 . We have proved the following theorem.

Second-Order Corrections for Weakly Interacting Bosons I

289

Theorem 4. Recall the isomorphism (19) of Theorem 3. 1. If L Q is given by (17), then

L Q = H0 + v(x − y)φ(y)φ(x)ax∗ a y d x d y + v ∗ |φ 2 | (x)ax∗ ax d x + I (M) .

(35)

2. The coefficient of ax a y in I (M) is −M12 or (ish(k)t + sh(k)g T + gsh(k))ch(k) − (ich(k)t − [ch(k), g])sh(k) −sh(k)msh(k) − ch(k)mch(k). 3. The coefficient of ax∗ a ∗y equals minus the complex conjugate of the coefficient of ax a y . 4. The coefficient of −

ax a ∗y + a ∗y ax 2

is M11 , or d(t, x, y) = ish(k)t + sh(k)g T + gsh(k) sh(k) − (ich(k)t + [g, ch(k)]) ch(k) −sh(k)mch(k) − ch(k)msh(k).

(36)

Corollary 1. If φ and k satisfy (9) and (10) of Theorem 1, then the coefficients of ax a y and ax∗ a ∗y drop out and L Q becomes ∗ v ∗ |φ 2 | (x)ax∗ ax d x L Q = H0 + v(x − y)φ(t, y)φ(t, x)ax a y d x d y + ax a ∗y + a ∗y ax − d(t, x, y) d x d y, 2 where d is given by (36) and the full operator reads ∗ v ∗ |φ 2 | (x)ax∗ ax d x L = H0 + v(x − y)φ(y)φ(t, x)ax a y d xd y + − d(t, x, y)a ∗y ax d x + N −1/2 e B [A, V ]e−B + N −1 e B V e−B − N χ0 − χ1 := L − N χ 0 − χ1 , and 1 χ0 = 2

v(x − y)|φ(t, x)|2 |φ(t, y)|2 d x d y, 1 χ1 (t) = − d(t, x, x)d x. 2 Remark 6. Notice that

LΩ = N −1/2 e B [A, V ]e−B + N −1 e B V e−B Ω,

and therefore we can derive the bound LΩ ≤ N −1/2 e B [A, V ]e−B Ω + N −1 e B V e−B Ω.

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M. G. Grillakis, M. Machedon, D. Margetis

Also, L is (formally) self-adjoint by construction. The kernel d(t, x, y), being the sum of the (1,1) entry of the self-adjoint matrices 1i ∂t∂ e K e−K , [e K , G]e−K = e K Ge−K − G and the visibly self-adjoint term −sh(k)mch(k) − ch(k)msh(k), is self-adjoint; thus, it has a real trace. Hence, L is also self-adjoint. In the remainder of this paper, we check that the hypotheses of our main theorem are satisfied, locally in time, for the potential v(x) = χ (x) |x| . 6. Solutions to Equation 10 0 0 Theorem 5. Let 0 be sufficiently small and assume that v(x) = |x| , or v(x) = χ (x) |x| for χ ∈ C0∞ (R3 ). Assume that φ is a smooth solution to the Hartree equation (16), φ L 2 (d x) = 1. Then there exists k ∈ L ∞ ([0, 1])L 2 (d xd y) solving (10) with initial conditions k(0, x, y) = 0 for 0 ≤ t ≤ 1. The solution k satisfies the following additional properties:

1.

2.

3.

∂ i − ∆x − ∆ y k L ∞ [0,1]L 2 (d xd y) ≤ C, ∂t ∂ i − ∆x − ∆ y sh(k) L ∞ [0,1]L 2 (d xd y) ≤ C, ∂t ∂ i − ∆x + ∆ y p L ∞ [0,1]L 2 (d xd y) ≤ C. ∂t

4. The kernel k agrees on [0, 1] with a kernel k for which k

1 1

X 2,2+

≤ C;

see (38) for the definition of the space X s,δ and, of course, 21 + denotes a fixed number slightly bigger than 21 . Proof. We first establish some notation. Let S denote the Schrödinger operator S=i

∂ − ∆x − ∆ y ∂t

and let T be the transport operator T =i

∂ − ∆x + ∆ y . ∂t

Let : L 2 (d xd y) → L 2 (d xd y) denote schematically any linear operator of operator norm ≤ C0 , where C is a “universal constant”. In practice, will be (composition with)

Second-Order Corrections for Weakly Interacting Bosons I

291

a kernel of the type φ(t, x)φ(t, y)v(x − y), or multiplication by v ∗ |φ|2 . Also, recall the inhomogeneous term m(t, x, y) = v(x − y)φ(t, x)φ(t, y). Then, Eq. (10), written explicitly, becomes Sk = m + S(k − u) + (u) + ( p) + (T p + ( p) + (u))(1 + p)−1 u.

(37)

Note that ch(k)2 − sh(k)sh(k) = 1; thus, 1 + p = ch(k) ≥ 1 as an operator and (1 + p)−1 is bounded from L 2 to L 2 . We plan to iterate in the norm N (k) = k L ∞ [0,1]L 2 (d xd y) + Sk L ∞ [0,1]L 2 (d xd y) . Notice that m L 2 (d xd y) ≤ C0 . Now solve Sk0 = m with initial conditions k0 (0, ·, ·) = 0, where N (k0 ) ≤ C0 . Define u 0 , p0 corresponding to k0 . For the next iterate, solve Sk1 = m + S(k0 − u 0 ) + (u 0 ) + ( p0 ) + (T p0 + ( p0 ) + (u 0 ))(1 + p0 )−1 u 0 ; the non-linear terms satisfy S(u 0 − k0 ) L ∞ [0,1]L 2 (d xd y) 1 (Sk0 )k 0 k0 − k0 (Sk0 )k0 + k0 k 0 Sk0 + · · · L ∞ [0,1]L 2 (d xd y) = 3! = O(N (k0 )3 ). Also, recalling that p0 = ch(k0 ) − 1, we have 1 T ( p0 ) L ∞ [0,1]L 2 (d xd y) = (Sk0 )k 0 − k0 (Sk0 ) + · · · L ∞ [0,1]L 2 (d xd y) 2 = O N (k0 )2 . Thus, N (k1 ) ≤ C0 + C02 . Continuing this way, we obtain a fixed point solution in this space which satisfies the first three requirements of Theorem 5. N a N a In fact, we can apply the same argument to ∂t∂ D k, since ∂t∂ D m ∈ L ∞ [0, 1] 1 2 L (d x d y) for 0 ≤ a < 2 . However, we cannot repeat the argument for D 1/2 k. We would like to have S D 1/2 k L ∞ [0,1]L 2 (d x d y) finite. Unfortunately, this misses “logarithmically” because of the singularity of v. Fortunately, we can use the well-known X s,δ spaces (see [2,18,20]) to show that |S|s D 1/2 u L 2 (dt)L 2 (d x d y) is finite locally in time for (all) 1 > s > 21 . This assertion will be sufficient for our purposes. Recall the definition of X s,δ : δ u L 2 (dτ dξ ) := u X s,δ . (38) |ξ |s |τ − |ξ |2 | + 1 Going back to (37), we write S(k) = m + F,

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where we define the expression F(k) := S(k − u) − (u) + pm + (T ( p) + ( p) + um) (1 + p)−1 u. The idea is to localize in time on the right-hand side: S( k) = χ (t) (m + F), k = k on [0, 1]. where χ ∈ C0∞ (R), χ = 1 on [0, 1]. Then, N a D k L 2 [0,1]L 2 (d x d y) ≤ C for As we already pointed out, we can estimate S ∂t∂ 1

0 ≤ a < 2 . We can further localize k in time to insure that these relations hold globally in time. By using the triangle inequality |τ −|ξ |2 |+|τ | ≥ |ξ |2 , we immediately conclude that 1+ 3 2 |ξ | 2 − |τ − |ξ |2 | + 1 kχ L 2 (dτ dξ ) ≤ C.

7. Error Term e B V e−B The goal of this section is to list explicitly all terms in e B V e−B and to find conditions under which these terms are bounded. Recall that V is defined by V = v(x0 − y0 )Q ∗x0 y0 Q x0 y0 d x0 dy0 . For simplicity, shb(k) denotes either sh(k) or sh(k), and chb(k) denotes either ch(k) or ch(k). Let x0 = y0 ; we obtain e B Q ∗x0 y0 Q x0 y0 e−B = e B Q ∗x0 y0 e−B e B Q x0 y0 e−B . According to the isomorphism (19), we have Q ∗x0 y0 = I

0 −2δ(x − x0 )δ(y − y0 )

0 , 0

where the operator e

B

Q ∗x0 y0 e−B

ch(k) −sh(k) ch(k) sh(k) 0 0 =I −2δ(x − x0 )δ(y − y0 ) 0 −sh(k) ch(k) sh(k) ch(k)

is a linear combination of the terms chb(k)(x, x0 )chb(k)(y0 , y)Q ∗x y d x d y, shb(k)(x, x0 )chb(k)(y0 , y)N x y d x d y, shb(k)(x, x0 )shb(k)(y0 , y)Q x y d x d y.

(39)

Second-Order Corrections for Weakly Interacting Bosons I

A similar calculation shows that e B Q x0 y0 e−B is a linear combination of chb(k)(x, x0 )chb(k)(y0 , y)Q x y d x d y, shb(k)(x, x0 )chb(k)(y0 , y)N x y d x d y, shb(k)(x, x0 )shb(k)(y0 , y)Q ∗x y d x d y.

293

(40)

Thus, e B Q ∗x0 y0 Q x0 y0 e−B is a linear combination of the nine possible terms obtained by combining the above. Now we list all terms in e B V e−B Ω. Terms in e B V e−B ending in Q x y are automatically discarded because they contribute nothing when applied to Ω. The remaining six terms are listed below. chb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 )chb(k)(y0 , y2 )

v(x0 − y0 )Q ∗x1 y1 N x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 ,

(41)

chb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 )shb(k)(y0 , y2 )

v(x0 − y0 )Q ∗x1 y1 Q ∗x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 ,

(42)

shb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 )chb(k)(y0 , y2 )

v(x0 − y0 )N x1 y1 N x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 ,

(43)

shb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 )shb(k)(y0 , y2 )

v(x0 − y0 )N x1 y1 Q ∗x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 ,

(44)

shb(k)(x1 , x0 )shb(k)(y0 , y1 )shb(k)(x2 , x0 )chb(k)(y0 , y2 )

v(x0 − y0 )Q x1 y1 N x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 ,

(45)

shb(k)(x1 , x0 )shb(k)(y0 , y1 )shb(k)(x2 , x0 )shb(k)(y0 , y2 ) v(x0 − y0 )Q x1 y1 Q ∗x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 .

(46)

To compute the above six terms,recall (24) through (27) as well as (1). In general, N x y Ω = 1/2δ(x − y)Ω, while f (x, y)Q ∗x y d xd yΩ = (0, 0, f (x, y), 0, . . .) up to symmetrization and normalization. The resulting contributions (neglecting symmetrization and normalization) follow. From (41): ψ(x1 , y1 ) = chb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 ) ×chb(k)(y0 , x2 )v(x0 − y0 )d x2 d x0 dy0 . From (42): ψ(x1 , y1 , x2 , y2 ) =

(47)

chb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 ) ×shb(k)(y0 , y2 )v(x0 − y0 )d x0 dy0 .

(48)

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From (43): ψ=

shb(k)(x1 , x0 )chb(k)(y0 , x1 )shb(k)(x2 , x0 ) ×chb(k)(y0 , x2 )v(x0 − y0 )d x1 d x2 d x0 dy0 .

(49)

From (44), with the N and Q ∗ reversed, we get ψ(x2 , y2 ) =

shb(k)(x1 , x0 )chb(k)(y0 , x1 )shb(k)(x2 , x0 ) ×shb(k)(y0 , y2 )v(x0 − y0 )d x1 d x0 dy0 ,

(50)

as well as the contribution from [N , Q ∗ ], i.e. ψ(y1 , y2 ) =

shb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x1 , x0 ) ×shb(k)(y0 , y2 )v(x0 − y0 )d x1 d x0 dy0 .

(51)

The contribution of (45) is zero, and, finally, the contribution of (46), using (24), consists of four numbers, which can be represented by the two formulas ψ=

shb(k)(x1 , x0 )shb(k)(y0 , x1 )shb(k)(x2 , x0 )

(52)

×shb(k)(y0 , x2 )v(x0 − y0 )d x1 d x2 d x0 dy0 and |shb(k)|2 (x1 , x0 )|shb(k)|2 (y0 , y1 )v(x0 − y0 )d x1 dy1 d x0 dy0 .

ψ=

(53)

We can now state the following proposition. Proposition 2. The state e B V e−B Ω has entries on the zeroth, second and fourth slot of a Fock space vector of the form given above. In addition, if ∂ i − ∆x − ∆ y sh(k) L 1 [0,T ]L 2 (d xd y) ≤ C1 , ∂t ∂ i − ∆x + ∆ y p L 1 [0,T ]L 2 (d xd y) ≤ C2 , ∂t and v(x) =

1 |x| ,

1 or v(x) = χ (x) |x| , then

T 0

e B V e−B Ω2F dt ≤ C,

where C only depends on C1 and C2 .

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Proof. This follows by writing ch(k) = δ(x − y)+ p and applying Cauchy-Schwartz and local smoothing estimates as in the work of Sjölin [32], Vega [33]; see also Constantin and Saut [3]. In fact, we need the following slight generalization (see Lemma 2 below): If ∂ i − ∆x1 − ∆x2 ± ∆x3 · · · ± ∆xn f (t, x1 , . . . xn ) L 1 [0,T ]L 2 (dtd x) ≤ C, ∂t with initial conditions 0, then

f (t, x1 , x2 , . . .) L 2 [0,T ]L 2 (d xd y) ≤ C. |x1 − x2 |

(54)

We will check a typical term, (48). This amounts to proving the following three terms are in L 2 . 1. ψ pp (t, x1 , y1 , x2 , y2 ) = p(t, x1 , x0 ) p(t, y0 , y1 )shb(k)(t, x2 , x0 )shb(k)(t, y0 , y2 )v(x0 − y0 ) d x0 dy0 . We use Cauchy-Schwartz in x0 , y0 to get

T

|ψ pp |2 dt d x1 d x2 dy1 dy2 0 ≤ sup | p(t, x1 , x0 ) p(t, y0 , y1 )|2 d x1 d x0 dy1 dy0 t

×

T

|shb(k)(t, x2 , x0 )shb(k)(t, y0 , y2 )v(x0 − y0 )|2 dt d x2 d x0 dy2 dy0 ≤ C.

0

The first term is estimated by energy, and the second one is an application of (54) with f = shb(k)shb(k). Notice that, because of the absolute value, we can choose either sh(k) or sh(k) to insure that the Laplacians in x0 , y0 have the same signs. 2. ψ pδ (t, x1 , y1 , x2 , y2 ) = p(t, x1 , x0 )shb(k)(t, x2 , x0 )shb(k)(t, y1 , y2 )v(x0 − y1 ) d x0 . Here, we use Cauchy-Schwartz in x0 to estimate, in a similar fashion,

T

|ψ pδ |2 dt d x1 d x2 dy1 dy2 ≤ sup | p(t, x1 , x0 )|2 d x1 d x0

0

t

×

T 0

|shb(k)(t, x2 , x0 )shb(k)(t, y1 , y2 )v(x0 − y1 )|2 dt d x2 d x0 dy2 dy0 ≤ C.

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3. ψδδ (x1 , y1 , x2 , y2 ) = shb(k)(t, x2 , x1 )shb(k)(t, y1 , y2 )v(x1 − y1 ), which is just a direct application of (54). All other terms are similar.

We have to sketch the proof of the local smoothing estimate that we used above. Lemma 2. If f : R3n+1 → C satisfies ∂ i − ∆x1 − ∆x2 ± ∆x3 · · · ± ∆xn f (t, x1 , . . . xn ) L 1 [0,T ]L 2 (d xd y) ≤ C ∂t with initial conditions f (0, · · · ) = 0, then

f (t, x1 , x2 , . . .) L 2 [0,T ]L 2 (d x) ≤ C. |x1 − x2 |

Proof. We follow the general outline of Sjölin, [32]. Using Duhamel’s principle, it suffices to assume that ∂ i − ∆x1 − ∆x2 ± ∆x3 · · · ± ∆xn f (t, x1 , · · · xn ) = 0 (55) ∂t with initial conditions f (0, · · · ) = f 0 ∈ L 2 . Furthermore, after the change of variables x1 → x1√+ x2 , x2 → x2√−x1 , it suffices to prove that 2

2

f (t, x1 , x2 , . . .) L 2 [0,T ]L 2 (d x) ≤ C, |x1 |

where f satisfies the same equation (55). Changing notation, denote x = (x2 , x3 , . . .) and let < ξ >2 be the relevant expression ±|ξ2 |2 ± |ξ3 |2 . . .. Write 2 2 f (t, x1 , x) = eit (|ξ1 | + ) ei x1 ·ξ1 +i x·ξ f 0 (ξ1 , ξ ) dξ1 dξ. Thus, we obtain | f (t, x1 , x)|2 dtd x1 d x |x1 |2 i x ·(ξ −η )+i x·(ξ −η) 2 2 2 2 e 1 1 1 = eit (|ξ1 | −|η1 | + − ) |x1 |2 × f (ξ , ξ ) f 0 (η1 , η)dξ1 dξ dη1 dηdt d x d x1 0 1 1 = c δ(|ξ1 |2 − |η1 |2 ) f 0 (η1 , ξ )dξ1 dη1 dξ f 0 (ξ1 , ξ ) |ξ1 − η1 | ≤ | f 0 (ξ1 , ξ )|2 d x1 dξ, because one can easily check that sup δ(|ξ1 |2 − |η1 |2 ) ξ1

1 dη1 ≤ C. |ξ1 − η1 |

1 is bounded from L 2 (dη1 ) to L 2 (dξ1 ). Thus, the kernel δ(|ξ1 |2 − |η1 |2 ) |ξ1 −η 1|

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8. Error Terms e B [ A, V ]e−B We proceed to check the operator e B [A, V ]e−B . The calculations of this section are similar to those of the preceding section with the notable exception of (60)–(63). Recall the calculations of Lemma 1 and write B −B e [A, V ]e = v(x − y) φ(y)e B ax∗ e−B e B ax a y e−B (56) +φ(y)e B ax∗ a ∗y e−B e B ax e−B d x d y. Now fix x0 . We start with the term (56). According to Theorem 2, we have B ∗ −B e ax0 e = sh(k)(x, x0 )ax + ch(k)(x, x0 )ax∗ d x, while e B ax0 a y0 e−B has been computed in (40). The relevant terms are and shb(k)(x, x0 )chb(k)(y0 , y)N x y d x d y shb(k)(x, x0 )shb(k)(y0 , y)Q ∗x y d x d y. Combining these two terms, there are three non-zero terms (which will act on Ω): 1.

v(x0 − y0 )φ(y0 )shb(k)(x1 , x0 )shb(k)(x2 , x0 ) ×shb(k)(y0 , y2 )ax1 Q ∗x2 y2 Ωd x1 d x2 dy2 d x0 dy0 .

(57)

This term contributes terms of the form ψ(t, y2 ) = v(x0 − y0 )φ(t, y0 )(shb(k)(t, x1 , x0 ))2 shb(k)(t, y0 , y2 )d x1 d x0 dy0 (58) as well as the term ψ(t, x2 ) =

v(x0 − y0 )φ(t, y0 )shb(k)(t, x1 , x0 )shb(k)(t, x2 , x0 ) ×shb(k)(t, y0 , x1 )d x1 d x0 dy0 ,

(59)

which we know how to estimate. The second contribution is: 2.

v(x0 − y0 )φ(y0 )chb(k)(x1 , x0 )shb(k)(x2 , x0 )Ω ×chb(k)(y0 , y2 )ax∗1 N x2 y2 d x1 d x2 dy2 d x0 dy0 .

(60)

ax∗1

with ax2 , we find that (60) contributes ψ(t, y2 ) = v(x0 − y0 )φ(t, y0 )chb(k)(t, x1 , x0 )shb(k)(t, x1 , x0 )

Commuting

×chb(k)(t, y0 , y2 )d x1 d x0 dy0 .

(61)

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We expand chb(k)(t, x1 , x0 ) = δ(x1 − x0 ) + p(k)(t, x1 − x0 ). The contributions of p are similar to previous terms, but δ(x1 − x0 ) presents a new type of term, which will be addressed in Lemma 3. These contributions are ψδp (t, y2 ) = v(x1 − y0 )φ(t, y0 )shb(k)(t, x1 , x1 )p(k)(t, y0 , y2 ) d x1 dy0 (62) and

ψδδ (t, y2 ) = φ(t, y2 )

v(x1 − y2 )shb(k)(t, x1 , x1 )d x1 .

(63)

The last contribution of (56) is: 3.

v(x0 − y0 )φ(y0 )chb(k)(x1 , x0 )shb(k)(x2 , x0 )shb(k)(y0 , y2 )ax∗1 Q ∗x2 y2 Ω ×d x1 d x2 dy2 d x0 dy0 ∼ ψ(x1 , x2 , y2 ),

where

ψ(t, x1 , x2 , y2 ) = v(x0 − y0 )φ(t, y0 ) ×chb(k)(t, x1 , x0 )shb(k)(t, x2 , x0 )shb(k)(t, y0 , y2 )d x0 dy0 ,

modulo normalization and symmetrization. This term, as well as all the terms in (57), are similar to previous ones and are omitted. We can now state the following proposition: Proposition 3. The state e B [A, V ]e−B Ω has entries in the first and third slot of a Fock space vector of the form given above. In addition, if ∂ i − ∆x − ∆ y sh(k) L 1 [0,T ]L 2 (d xd y) ≤ C1 , ∂t ∂ i − ∆x + ∆ y p L 1 [0,T ]L 2 (d xd y) ≤ C1 ∂t and shb(k)(t, x, x) L 2 ([0,T ]L 2 (d x)) ≤ C3 , and v(x) =

χ (x) |x|

(64)

for χ a C0∞ cut-off function, then

T 0

e B [A, V ]e−B Ω2F ≤ C,

where C only depends on C1 , C2 , C3 . Proof. The proof is similar to that of Proposition 2, the only exception being the terms (62), (63). It is only for the purpose of handling these terms that the Coulomb potential has to be truncated, since the convolution of the Coulomb potential with the L 2 function shb(k)(x, x) does not make sense. If v is truncated to be in L 1 (d x), then we estimate the convolution in L 2 (d x), and take φ ∈ L ∞ (dydt).

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To apply this proposition, we need the following lemma. 1 1

Lemma 3. Let u ∈ X 2 , 2 + . Then, u(t, x, x) L 2 (dt d x) ≤ Cu

1 1

X 2,2+

.

Proof. As it is well known, it suffices to prove the result for u satisfying ∂ i − ∆x − ∆ y u(t, x, y) = 0 ∂t 1

with initial conditions u(0, x, y) = u 0 (x, y) ∈ H 2 . This can be proved as a “Morawetz estimate”, see [14], or as a space-time estimate as in [19]. Following the second approach, the space-time Fourier transform of u (evaluated at 2ξ rather than ξ for neatness) is u 0 (ξ − η, ξ + η)dη

u (τ, 2ξ ) = c δ(τ − |ξ − η|2 − |ξ + η|2 ) δ(τ − |ξ − η|2 − |ξ + η|2 ) F(ξ − η, ξ + η)dη, =c (|ξ − η| + |ξ + η|)1/2 where F(ξ − η, ξ + η) = (|ξ − η| + |ξ + η|)1/2 u 0 (ξ − η, ξ + η). By Plancherel’s theorem, it suffices to show that u L 2 (dτ dξ ) ≤ CF L 2 (dξ dη) . This, in turn, follows from the pointwise estimate (Cauchy-Schwartz with measures) δ(τ − |ξ − η|2 − |ξ + η|2 ) dη | u (τ, 2ξ )|2 ≤ c |ξ − η| + |ξ + η| × δ(τ − |ξ − η|2 − |ξ + η|2 )|F(ξ − η, ξ + η)|2 dη, and the remark that

δ(τ − |ξ − η|2 − |ξ + η|2 ) dη ≤ C. |ξ − η| + |ξ + η|

9. The Trace

d(t, x, x)d x

This section addresses the control of traces involved in derivations. Recall that d(t, x, y) = ish(k)t + sh(k)g T + gsh(k) sh(k) − (ich(k)t + [g, ch(k)]) ch(k) −sh(k)mch(k) − ch(k)msh(k). Notice that if k1 (x, y) ∈ L 2 (d x d y) and k2 (x, y) ∈ L 2 (d x d y) then |k1 k2 |(x, x)d x ≤ |k1 (x, y)||k2 (y, x)|d y d x ≤ k1 L 2 k2 L 2 .

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Recall from Theorem 5 that if v(x) = |x| or v(x) = χ (x) |x| then ish(k)t + sh(k)g T + gsh(k), ich(k)t + [g, ch(k)] and sh(k) are in L ∞ ([0, 1])L 2 (d xd y). This allows us to control all traces except the contribution of δ(x − y) to the second term. But, in fact, we have ich(k)t + [g, ch(k)] = ikt − ∆x k − ∆ y k k − k ikt − ∆x k − ∆ y k + · · · ,

which has bounded trace, uniformly in [0, 1]. Acknowledgements. The first two authors thank William Goldman and John Millson for discussions related to the Lie algebra of the symplectic group, and Sergiu Klainerman for the interest shown for this work. The third author is grateful to Tai Tsun Wu for useful discussions on the physics of the Boson system. The third author’s research was partially supported by the NSF-MRSEC grant DMR-0520471 at the University of Maryland, and by the Maryland NanoCenter.

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23. Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvanson, J.: The Mathematics of the Bose Gas and its Condensation. Birkhaüser Verlag, Basel, 2005 24. Lieb, E.H., Seiringer, R., Yngvanson, J.: Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev. A 61, 043602 (2006) 25. Lieb, E.H., Seiringer, R., Yngvason, J.: A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Commun. Math. Phys. 224, 17–31 (2001) 26. Margetis, D.: Studies in Classical Electromagnetic Radiation and Bose-Einstein Condensation. Ph.D. thesis, Harvard University, 1999 27. Margetis, D.: Solvable model for pair excitation in trapped Boson gas at zero temperature. J. Phys. A: Math. Theor. 41, 235004 (2008); Corrigendum. J. Phys. A: Math. Theor. 41, 459801 (2008) 28. Pitaevskii, L.P.: Vortex lines in an imperfect Bose gas. Soviet Phys. JETP 13, 451–454 (1961) 29. Pitaevskii, L.P., Stringari, S.: Bose-Einstein Condensation. Oxford: Oxford University Press, 2003 30. Riesz, F., Nagy, B.: Functional analysis. New York: Frederick Ungar Publishing, 1955 31. Rodnianski, I., Schlein, B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291(2), 31–61 (2009) 32. Sjölin, P.: Regularity of solutions to the Schrödinger equation. Duke Math. J. 55, 699–715 (1987) 33. Vega, L.: Schrödinger equations: Pointwise convergence to the initial data. Proc. AMS 102, 874–878 (1988) 34. Wu, T.T.: Some nonequilibrium properties of a Bose system of hard spheres at extremely low temperatures. J. Math. Phys. 2, 105–123 (1961) 35. Wu, T.T.: Bose-Einstein condensation in an external potential at zero temperature: General theory. Phys. Rev. A 58, 1465–1474 (1998) Communicated by H.-T. Yau

Commun. Math. Phys. 294, 303–342 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0952-8

Communications in

Mathematical Physics

Entropic Bounds on Semiclassical Measures for Quantized One-Dimensional Maps Boris Gutkin Fachbereich Physik, Universität Duisburg-Essen, Lotharstrasse 1, 47048 Duisburg, Germany. E-mail: [email protected] Received: 15 April 2008 / Accepted: 16 September 2009 Published online: 24 November 2009 – © Springer-Verlag 2009

Abstract: Quantum ergodicity asserts that almost all infinite sequences of eigenstates of quantized ergodic Hamiltonian systems are equidistributed in phase space. This, however, does not prohibit existence of exceptional sequences which might converge to different (non-Liouville) classical invariant measures. It has been recently shown by N. Anantharaman and S. Nonnenmacher in [20,21] (with H. Koch) that for Anosov geodesic flows the metric entropy of any semiclassical measure µ must satisfy a certain bound. This remarkable result seems to be optimal for manifolds of constant negative curvature, but not in the general case, where it might become even trivial if the (negative) curvature of the Riemannian manifold varies a lot. It has been conjectured by the same authors, that in fact, a stronger bound (valid in the general case) should hold. In the present work we consider such entropic bounds using the model of quantized piecewise linear one-dimensional maps. For a certain class of maps with non-uniform expansion rates we prove the Anantharaman-Nonnenmacher conjecture. Furthermore, for these maps we are able to construct some explicit sequences of eigenstates which saturate the bound. This demonstrates that the conjectured bound is actually optimal in that case. 1. Introduction The theory of quantum chaos deals with quantum systems whose classical limit is chaotic. It is assumed in general that chaotic dynamics induce certain characteristic patterns. For instance, the Random Matrix conjecture predicts that statistical distribution of high-lying eigenvalues in a chaotic system is the same as in certain ensembles of random matrices and depends only on symmetries of the system [1]. In the same spirit, it is believed that eigenstates of chaotic systems are delocalized over the entire available part of the phase space [2,3] which is totally different from the case of quasi-integrable systems, where eigenstates are known to concentrate near KAM tori [4]. The rigorous implementation of that idea is known as Quantum Ergodicity Theorem. It was first proven by

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B. Gutkin

A. I. Schnirelman for Laplacians on surfaces of negative curvature [5] and later generalized [6,7] and extended to other systems e.g., ergodic billiards [8,9], quantized maps [10] and general Hamiltonians [11]. Very generally, the Quantum Ergodicity Theorem states that for a classically ergodic system “almost all” eigenstates ψk become uniformly distributed over the phase space in the semiclassical limit k → ∞. To give a more precise meaning of this statement it is convenient to use the notion of measure. Given some Hamiltonian system, let ψk , k = 1, 2, . . . be a sequence of normalized eigenstates of the corresponding quantum Hamiltonian operator. With any ψk one can associate the distribution µk , such that µk ( f ) = ψk Opk ( f ), ψk ,

f ∈ Cc∞ (X ),

where f (q, p) is a classical observable on the phase space X of the system and Opk ( f ) is the corresponding quantum observable. Here Opk ( · ) is a quantization procedure set at the scale k which in turn is fixed by the state ψk under consideration. As k → ∞, k → 0 and we are looking for possible semiclassical limits of the measure µk . Although the exact form of µk depends on the quantization procedure Opk ( · ) (e.g., Weyl, AntiWick quantization, etc.), a weak limit of µk as k → ∞, in the distributional topology does not depend on the choice of the quantization. Any such limit µ (called semiclassical measure) is a probability measure which is, in addition, invariant under the classical flow. The Quantum Ergodicity theorem asserts that for “almost all” sequences of eigenstates the corresponding semiclassical measure is actually the Liouville measure. Since the Quantum Ergodicity theorem does not exclude the possibility that exceptional sequences of eigenstates produce non-Liouville classically invariant measures, it makes sense to ask whether such measures might actually appear. In the context of Anosov geodesic flows on surfaces of negative curvature it was conjectured [12] that a typical system possesses Quantum Unique Ergodicity property, meaning that all sequences of eigenstates converge to the Liouville measure. However, so far there have been only a limited number of rigorous results supporting this conjecture. One of the most significant results in that direction was obtained by E. Lindenstrauss in [13], where he proved that all Hecke eigenstates of the Laplacian on some compact arithmetic surfaces are equidistributed. If (as widely believed) all the Laplacian eigenstates are non-degenerate, this result would amount to the proof of Quantum Unique Ergodicity for the arithmetic case. On the other hand, it is known that exceptional sequences actually do appear in some quantum systems. For instance, as has been recently shown by A. Hassell [14], Quantum Unique Ergodicity fails for “almost all” stadia billiards. Another example is provided by quantized hyperbolic symplectomorphisms of two-dimensional torus (known as “cat maps” in physics literature) [15,16]. Here the semiclassical measures induced by exceptional sequences of eigenstates can be, for instance, composed of two ergodic components: µ = aµL + (1 − a)µD ,

1 ≥ a ≥ 1/2,

(1)

where the first part µL is the Liouville measure equidistributed over the entire phase space and the second part µD is the Dirac peak concentrated on a single unstable periodic orbit. Similar sequences of eigenstates have been also constructed for the “Walsh quantization” of the baker’s map [17]. For quantized hyperbolic symplectomorphisms of higher-dimensional tori there exists a different type of semiclassical measures which are Lebesgue measures on some invariant co-isotropic subspaces of the torus [18]. As we know that non-Liouville semiclassical measures do appear (at least) in some systems, it would be of great interest to understand which kind might exist in the general

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case. Quite recently, it has been proven by N. Anantharaman and S. Nonnenmacher [19– 21] (with H. Koch) that for the Laplacian on a compact Riemannian manifold X with Anosov geodesic flow on the unit cotangent bundle S ∗ X the metric (Kolmogorov-Sinai) entropy HKS (µ) of any semiclassical measure µ on S ∗ X must satisfy a certain bound. Particularly, in the two-dimensional case the following result holds [21]: 1 HKS (µ) ≥ | log J u (x)|dµ − λmax , (2) 2 S∗ X where J u (x) is the unstable Jacobian of the flow at the point x ∈ S ∗ X and λmax is the maximum expansion rate of the flow. If the maximum expansion rate is close to its average value, this remarkable bound gives valuable information on µ itself. In particular, for surfaces with a constant negative curvature it implies that the fraction of any semiclassical measure concentrated on periodic orbits cannot exceed “half” of the total measure of the phase space. On the other hand, if the expansion rate varies a lot, the above bound might provide little information, as the right hand side of (2) can even become negative. Thus, it is natural to expect that (2) is not an optimal result, and a stronger bound might exist in a general case. Such a bound has been conjectured in [17,19]. The conjecture states that for an Anosov canonical map (resp. Hamiltonian flow) on a compact symplectic manifold (resp. on a compact energy shell) any semiclassical measure must satisfy: 1 HKS (µ) ≥ (3) | log J u (x)|dµ. 2 Assuming that the bound is true, it provides certain restriction on the class of possible semiclassical measures in the general case. Suppose, for instance, that a semiclassical measure takes the form (1). (Note that for Anosov geodesic flows we don’t know whether such semiclassical measures actually exist.) Then the bound (3) implies that the LiouD ville part should be always present and its proportion satisfy a ≥ λavλ+λ , where λav is D the average Lyapunov exponent (with respect to the Liouville measure) and λD is the Lyapunov exponent for the periodic orbit where µD is localized. 2. Model and Statement of the Main Results The central purpose of this paper is to provide support for the conjectured bound (3) using the model of quantized one-dimensional piecewise linear maps. A procedure for quantization of one-dimensional linear maps was originally introduced in [22] in order to generate families of quantum graphs with some special properties. Being much simpler on the technical level, these models still exhibit characteristic properties of typical quantum chaotic Hamiltonian systems. Most importantly, it turns out that the quantum evolution here follows the classical evolution till the (Ehrenfest) time which grows logarithmically with the dimension N of the “quantum” Hilbert space.1 Note that in such a model N is always finite and N −1 plays the role of the Planck’s constant. As will be shown in the body of the paper, the construction of quantized one-dimensional maps is also closely related to the Walsh quantized baker’s maps in [17]. We consider a class of piecewise linear maps T : [0, 1) → [0, 1) =: I which preserve the Lebesgue measure on I . More specifically, let {I j , j = 1, . . . l} be a partition of 1 As we deal in the present paper with a discrete time evolution, the term “time” stands here and after for the number of iterations of either classical or quantum maps.

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the unit interval I = ∪lj=1 I j into l subintervals I j = β− (I j ), β+ (I j ) , j = 1, . . . l, where β− (I j ) and β+ (I j ) denote the left and right endpoints of I j respectively. At each subinterval I j , T is then defined as a simple linear map T : I j → I : T (x) = j (x − β− (I j )),

j = 1/|I j |,

for x ∈ I j , j = 1, . . . l,

(4)

with |I j | standing for the length of I j . Furthermore, we will assume that the slopes j are positive integers satisfying l

−1 j = 1,

j > 1, j = 1, . . . l.

j=1

These conditions guarantee that the map is both Lebesgue-measure preserving and chaotic. Note that each such map T is uniquely determined by the ordered set of its slopes = {1 , . . . l }, so the notation T will be sometimes used to define the corresponding map. We will now briefly describe the procedure introduced by P. Pako´nski et al [22] for quantization of such maps. Let M = {E i , i = 1, . . . N } be a partition of I¯ = [0, 1] into N intervals E i = [(i − 1)/N , i/N ], i = 1, . . . N of equal lengths. For each interval E i we will denote by β+ (E i ) (β− (E i )) the right (resp. left) endpoint of E i and by N β (E ) the set of all endpoints of the partition M. Obviously both M β(M) = ∪i=1 ± i and β(M) are uniquely determined by the size N of the partition. In what follows we will consider an increasingly refined sequence of the above partitions Mk whose sizes Nk , k = 1, . . . ∞ grow exponentially. Conditions 1. Given a map T of the form (4) we impose the following conditions on the sequence of Mk : • Each partition Mk is a refinement of the previous one. That means for each k ≥ 1, Nk+1 /Nk is an integer number greater than one. • The set of the endpoints of the initial partition M1 must include all singular points of T , i.e., β(M1 ) ⊇ β(Ii ) for all i = 1, . . . l. For a sequence of partitions Mk , k = 1, . . . ∞ satisfying Conditions 1 consider the corresponding sequence of the transfer (Frobenius-Perron) operators given by Nk × Nk doubly stochastic matrices Bk , whose elements read as: −1 |E i ∩ T −1 E j | Ei if E i ∩ T −1 E j = ∅ Bk (i, j) = = (5) 0 otherwise, |E i | where Ei is the slope of T at the partition element E i . We will call a piecewise linear map T quantizable if there exists a sequence of partitions Mk , k = 1, . . . ∞ such that for each matrix Bk one can find a unitary matrix Uk of the same dimension satisfying Bk ( j, i) = |Uk (i, j)|2

(6)

for each matrix element ( j, i); j, i ∈ {1, . . . Nk }.2 For quantizable maps the matrices Uk are regarded as “quantizations” of Bk and play the role of quantum evolution operators acting on the Nk -dimensional Hilbert space Hk C Nk equipped with the Nk ¯ ψ(i)ϕ(i) for vectors ψ = (ψ(1), . . . ψ(Nk )), standard scalar product: ψ, ϕ = i=1 ϕ = (ϕ(1), . . . ϕ(Nk )). As an example, consider the following linear map (see Fig. 1): 2 Note that our definition for U matrix corresponds to the adjoint of the corresponding quantum evolution in [22,23].

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1

307

1

1

1/2

0

1

0

1/2

1

0

1/2

1

0

1

Fig. 1. Linear maps with uniform (left) and non-uniform slopes (middle) which allow “tensorial” quantization. On the right is shown the asymmetric Baker map corresponding to the linear map in the middle

T (x) = 2x

mod 1,

x ∈ I.

(7)

Here for the sequence of partitions Mk of the unit interval into Nk = 2k equal pieces, the matrix elements Bk (i, j) of the classical transfer operators take the values 1/2 if j = 2i, j = 2i − 1, j + Nk = 2i, j + Nk = 2i − 1 and 0 otherwise: ⎞ ⎛ 1 1 0 0 1 1 1 1 ⎜0 0 1 1⎟ ··· . , B4 = ⎝ B2 = ⎠, 2 1 1 2 1 1 0 0 0 0 1 1 It is worth mentioning that the structure of Bk , actually, resembles the structure of the map T (rotated clockwise by π/2). It is also easy to see that the map (7) is quantizable. By a permutation of rows Bk can be brought into the block diagonal form such that every block is 2×2 matrix B2 whose elements are all 1/2. Thus the question of the quantization of T reduces to finding a unitary 2 × 2 matrix U satisfying |U(l, m)|2 = 1/2 for all its elements. An appropriate choice for U is provided, for instance, by the two-dimensional matrix F2 , where F p is the p-dimensional discrete Fourier transform: 2πi 1 (l − 1)(m − 1) , l, m = 1, . . . p. (8) F p (l, m) = √ exp − p p This construction can be straightforwardly generalized to all other maps with uniform slopes. For a general piecewise linear map it becomes, however, a non-trivial problem to determine whether the corresponding doubly stochastic matrices Bk allow the representation (6) in terms of unitary matrices Uk (see [22,24] and references there). Nevertheless, we show in the Appendix to the paper that the class of quantizable piecewise linear maps is quite wide and contains many maps with non-uniform slopes. Notice that the above quantization of one-dimensional piecewise linear maps is just a formal procedure for generation of unitary matrices Uk . To turn it to a “meaningful” quantization one needs, in addition, to make a connection between classical observables on the unit interval and the corresponding quantum observables on the Hilbert space Hk . Such a quantization procedure has been introduced in [23]. With a classical observable f ∈ L 2 [0, 1] one associates the sequence of the quantum observables Opk ( f ), defined by the diagonal matrices of the dimension Nk whose components Opk ( f ) j, j equal the average value of f at j’s element of the partition Mk . The key observation making the above quantization interesting is the existence of the semiclassical correspondence (Egorov property) between evolutions of classical and quantum observables. Precisely, for a Lipschitz continuous observable f (x) one has [23]:

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Uk∗ Opk ( f )Uk − Opk ( f ◦ T ) = O

1 Nk

.

(9)

Recall that the size Nk of the partition plays here the role of the inverse Planck constant −1 k and the semiclassical limit corresponds to k → ∞. Equipped with the above quantization procedure we can define now the sequence of the semiclassical measures associated with the eigenstates of Uk . For any normalized eigenstate ψk = (ψ(1), . . . ψ(Nk )) ∈ Hk , Uk ψk = eiθk ψk , let µk be the associated probability measure, whose density is given by the piecewise constant function ρk (x) = Nk |ψ(i)|2 , for x ∈ E i , i = 1, . . . Nk . By this definition, the quantum average of any observable f ∈ L 2 [0, 1] can be written as: ψk Opk ( f )ψk = f (x) ρk (x)d x ≡ f (x) dµk . (10) I

I

We will be concerned with the possible weak limits of µk as k → ∞ and call any such limiting measure µ a semiclassical measure. Speaking informally µ characterizes the possible sets of the localization on I for the eigenstates of quantized maps. (Alternatively (see [23]), one can think that µ determines the limiting eigenstates distribution on the sequence of quantum graphs provided by Uk ’s.) An immediate consequence of the Egorov property is that any semiclassical measure µ must be invariant under the map T . Indeed, since ψk is an eigenstate of Uk : 1 , (11) f (x) dµk (x) = ψk Uk∗ Opk ( f )Uk ψk = f (T (x)) dµk (x) + O N k I I and the invariance of µ follows immediately after taking the limit k → ∞. As there exist many classical measures preserved by T , the invariance alone does not determine all possible outcomes for the semiclassical measures. Similarly to the case of Hamiltonian systems, using the Egorov property one can show by standard methods (see e.g., [26]) that almost any sequence of the eigenstates gives rise to the Lebesgue measure in the semiclassical limit (this was proved in [23] by a somewhat different method). Theorem 1 (Quantum Ergodicity [23, Thm. 2]). Let T be a quantizable piecewise linear map of the form (4) and let Uk , k = 1, . . . ∞ be a sequence of its quantizations (i) with eigenstates ψk , i = 1, . . . Nk . Then for each k there exists a subsequence of Nk (i N )

eigenstates: k := {ψk(i1 ) , . . . ψk k } such that limk→∞ Nk /Nk and for any sequence of eigenstates ψk j ∈ k j , j = 1, . . . ∞ and any Lipschitz continuous function f one has: lim ψk j Opk j ( f )ψk j = f (x) d x. (12) j→∞

I

In the present paper we go beyond the Quantum Ergodicity and ask about the possible exceptional semiclassical measures. In what follows we will restrict our treatment to a special subclass of the piecewise linear maps (4): Definition 1. Let I = ∪li=1 I j be a partition of I into a number of subintervals. We define T p as a piecewise linear map (4) whose precise form at each subinterval is given by: T p (x) = j x

mod 1,

for x ∈ I j , j = 1, 2 . . . l.

(13)

Furthermore, the slopes j = p n j , j = 1, 2 . . . l are given by positive powers n j ∈ N of an integer p > 1, satisfying condition: lj=1 p −n j = 1.

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Using the method developed in [17,20,21] we can easily prove for these maps an analog of the bound (2): Theorem 2 Let Uk , k = 1, . . . ∞ be a sequence of quantizations of T p and let µ be a semiclassical measure induced by a subsequence ψk , k = 1, . . . ∞ of the eigenstates of Uk ’s. Then, the following bound holds for the metric entropy of µ:

1 1 log (x) dµ(x) − log max = µ(I j ) log j − log max , 2 2 l

HKS (T p , µ) ≥ I

j=1

(14) where max := max1≤ j≤l j and µ(I j ) are the measures of the intervals I j . As the right hand side of (14) can in principle be negative, one might suspect that this bound is not optimal for the maps T p with non-uniform slopes. The main result of the present paper is a proof of a stronger bound on the metric entropy of semiclassical measures. Namely, in the body of the paper we show that the maps T p allow a special type of “tensorial” quantizations. For the maps T p quantized in that way we prove the precise analog of the Anantharaman-Nonnenmacher conjecture (3). Theorem 3 Let T p be a map as in Definition 1 and let Uk , k = 1, . . . ∞ be a sequence of “tensorial” quantization of T p . Then for any sequence of eigenstates ψk of Uk , k = 1, . . . ∞ the corresponding semiclassical measure µ satisfies: 1 µ(I j ) log j . 2 l

HKS (T p , µ) ≥

(15)

j=1

Furthermore, for certain “tensorial” quantizations of T p it becomes possible to construct explicit subsequences of eigenstates of Uk . Using these eigenstates we obtain a set of semiclassical measures which can be subsequently analyzed to test (15). It turns out that some of these semiclassical measures, in fact, saturate the bound implying that the result is sharp. Remark 1. In what follows we will distinguish the class of “tensorial” quantizations (which will be introduced in the next section) from general quantizations of one dimensional maps which only need to satisfy the condition (6). Some of the results in the paper are specific for “tensorial” quantizations and it will be always explicitly mentioned that we deal with such class of quantization whenever it is relevant. Let us also notice that the results of the present paper can in fact be proven for a more general class of one dimensional linear maps. For instance, the bound (14) can be proven for all quantizable one dimensional maps, while the main result (15) can be straightforwardly extended to all maps (with general quantizations) whose slopes are given by integer powers of an integer number [33]. However, the restriction of the discussion to the class of maps T p provided by Definition 1 and “tensorial” quantizations, allows to represent the main ideas and results in a particularly transparent way, reducing the technical details to the minimum.

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The paper is organized as follows. In Sect. 3 we introduce a special class of “tensorial” quantizations for the maps T p and demonstrate their connection with the Walsh quantized Baker maps. In Sect. 4 we review the construction in [23] for quantization of observables and prove the Egorov property up to the Ehrenfest time. In Sect. 5 we connect metric entropy of semiclassical measures with a certain type of quantum entropy functions. Based on the method of [20] we then prove Theorem 2 in Sect. 6 using the Entropic Uncertainty Principle. In the next two sections we deal with “tensorial” quantizations of T p . Section 7 is devoted to the proof of Theorem 3. In Sect. 8 we explicitly construct a certain class of semiclassical measures for the maps T p and test the bound (15). The concluding remarks are presented in Sect. 9. 3. “Tensorial” Quantizations of One-Dimensional Piecewise Linear Maps We will consider now in detail the quantization procedure for the piecewise linear maps T p given by Definition 1. As we show below, these maps allow a special type of “tensorial” quantizations mimicking the action of the corresponding classical shift map. This quantization procedure is basically restricted to the class of maps T p . For the sake of completeness, we also consider in Appendix A a different approach suitable for quantization of a more general class of maps (4). Maps with uniform slopes. We will first consider the piecewise linear maps T¯ p := T{ p,... p} with the uniform slope j ≡ p ∈ N i.e, the maps: T¯ p (x) = px mod 1,

x ∈ I.

(16)

(Here and after we will use the bar symbol to distinguish the above uniform maps from non-uniform ones.) For any point x ∈ I it will be convenient to use a p-base numeral system: x = 0.x1 x2 x3 . . ., xi ∈ {0, . . . , p − 1} to represent x. Obviously, each point is then encoded by an infinite sequence (not necessarily unique) of symbols x1 , x2 , x3 . . .. With such representation for the points of I the action of T¯ p becomes equivalent to the shift map: T¯ p : x1 x2 x3 x4 · · · → x2 x3 x4 x5 . . . .

(17)

In the following we will use the symbol x = x1 x2 x3 . . . xm for both finite and infinite sequences with the notation |x| := m reserved for the length of the sequence. So for x with |x| = ∞ the symbol x will stand for the corresponding point x = 0.x in the interval I . For a sequence x, with finite |x| = m we will use notation x to denote the corresponding cylinder set. A point x belongs to x if the first m digits of x after the point coincide with x1 , x2 , . . . xm . For any map T¯ p , there exists a sequence of natural Markov partitions Mk into Nk = p k cylinder sets of the length k: {E x = x, |x| = k}. The corresponding transfer operator is then given by the matrix Bk , whose elements: −1 , i = 1, . . . k − 1 if xi = xi+1 p (18) Bk (x, x ) = 0 otherwise, give the transition probabilities for reaching E x , x = x1 x2 . . . xk starting from E x , x = x1 x2 . . . xk after one step of the classical evolution. These matrices can be now

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“quantized” as follows. Let H C p , be a vector space of the dimension p with a scalar product ·, · and an orthonormal basis {| j, j ∈ {0 . . . p − 1}}. Take U to be a unitary transformation on H such that in the basis above: |Ui, j |2 = 1/ p,

Ui, j := i|U| j.

(19)

(One possible choice for the matrix Ui, j is provided by the p-dimensional discrete Fourier transform.) With each partition Mk we now associate Nk -dimensional Hilbert space: Hk = H ⊗ H ⊗ · · · ⊗ H . k

Using the orthonormal basis in Hk given by the vectors: |x := |x1 ⊗ |x2 ⊗ · · · ⊗ |xk ,

x = x1 . . . xk , xi ∈ {0 . . . p − 1},

one defines the unitary transformation U¯ k as: U¯ k |x = |x2 ⊗ |x3 ⊗ · · · ⊗ |xk ⊗ U|x1 ,

(20)

and the corresponding adjoint: U¯ k∗ |x = U∗ |xk ⊗ |x1 ⊗ |x2 ⊗ · · · ⊗ |xk−1 .

(21)

The action of U¯ k basically mimics the action of the shift map. From this and property (19) of the U matrix it follows immediately that U¯ k satisfies (6) and therefore, indeed, a quantization of Bk . Note that if U∗ is given by the discrete Fourier transform F p , the matrix U¯ k coincides with the evolution operator of the Walsh-quantized Baker map in [17]. In that case U¯ k4 = 1 and the spectrum of U¯ k is highly degenerate. Note also that the U matrix in the definition (20) of U¯ k should not necessarily be constant. More general construction is obtained if one takes U in the form U(x) = exp(iφ(x))U (x2 , x3 . . . xk ), where φ(x) is a real function of x and U (x2 , x3 . . . xk ) is a unitary matrix depending on x2 , x3 . . . xk and satisfying (19). Maps with non-uniform slopes. Let us consider now maps T p given by Definition 1 such that not all n j are equal. For a given p we will use exactly the same representation x = x1 x2 x3 x4 . . ., xi ∈ {0, 1 . . . p − 1} for the point x = 0.x, and the same set of the partitions Mk , k ≥ n max := max j∈{1,...l} n j as for the map T¯ p with the uniform expansion rate. The action of T p in that representation is given by the shift map, where the size of the shift depends on the point itself: T p : x1 x2 x3 x4 · · · → xn j +1 xn j +2 xn j +3 . . . ,

if 0.x ∈ I j , j = 1, 2 . . . l.

(22)

The corresponding classical evolution matrix for the partition Mk is then given by −n p i if x ⊆ Ii and xj = xn i + j , j = 1, . . . k − n i (23) Bk (x, x ) = 0 otherwise.

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Remark 2. It should be emphasized that for the uniform map T¯ p the above p-base encoding (for the points on the interval I ) provides a standard symbolic representation of the dynamical system defined with respect to the initial Markov partition lj=1 I j . That means the sequence x = x1 x2 x3 . . .; xi ∈ {0, . . . p − 1} describes the future history of the point 0.x regarding that partition under the action of T¯ p . On the other hand, for maps T p with non-uniform slopes the above representation possesses no such a dynamical significance. In particular, T p does not act as a simple shift map on the sequences x. Later we will also use the dynamical symbolic representation for the points on I which is defined with respect to the initial Markov partition lj=1 I j . In that case the points on I are represented by sequences of symbols ε = ε0 ε1 ε2 . . .; εi ∈ {1, . . . l} which encode their future histories under the action of T p . In that representation T p acts as the standard shift map on ε. To distinguish between two different representation systems we will always use x and ε letters to denote the corresponding symbols. Note that with this notation ε0 ε1 . . . εn and x1 x2 . . . xn refer, in general, to different types of sets. While the first set is a cylinder defined with respect to the action of the map T p , the second set is a cylinder for the corresponding uniform map T¯ p (and its own Markov partition). Now it is not difficult to “quantize” the matrices (23) using exactly the same Hilbert space as in the uniform case. For each state |x, x = x1 . . . xk such that x ⊆ I j , define the action of Uk on |x by: Uk |x = |xn j +1 ⊗ · · · ⊗ |xk ⊗ Un j |xn j ⊗ Un j −1 |xn j −1 ⊗ · · · ⊗ U1 |x1 , (24) where all the matrices Ui , i = 1, . . . n j satisfy (19). Note that the unitarity of Uk requires the flip in the order of the shifted p-bits on the right hand side of (24). It follows straightforwardly from the definition that Uk is indeed unitary and fulfills (6), thereby it provides a “quantization” of Bk . As for the maps with uniform slopes, the matrices Ui , i = 1, . . . n max should not necessarily be constant and can, for instance, depend on xn max +1 , xn max +2 . . . xk , as well. Example. As an example of the above quantization construction consider the map T2 = T{2,4,4} (see Fig. 1) which will be a principle model for us in what follows. Explicitly, for x = x1 x2 x3 . . ., xi ∈ {0, 1} the action of T2 on x = 0.x is given by 2x mod 1 if 0 ≤ x < 1/2 T2 (x) = (25) 4x mod 1 if 1/2 ≤ x < 1. For the vector space Hk = H ⊗ · · · ⊗ H (k times), H C2 the corresponding quantum evolution acts on |x ∈ Hk as: |x2 ⊗ |x3 ⊗ · · · ⊗ |xk ⊗ U1 |x1 if x1 = 0 (26) Uk |x = |x3 ⊗ · · · ⊗ U2 |x2 ⊗ U1 |x1 if x1 = 1, where U1,2 are constant matrices whose elements satisfy: |U1,2 (i, j)| =

√

1/2.

Connection with Walsh quantized Baker maps. Although one-dimensional maps are not Hamiltonian systems, there exists a close connection between their quantizations and Walsh quantized Baker maps. This is clear for the maps with uniform expansion rates, since in that case Uk defined by Eq. (20) with U∗ = F p , gives precisely the evolution operator of the Walsh quantized standard Baker map. Let us show now that in a similar way some natural quantizations of non-uniformly expanding maps provide

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the time evolution operator for à la Walsh quantized asymmetric Baker maps. For the sake of concreteness consider the map (25). The corresponding asymmetric Baker map T2 : (q, p) → (q, p), q, p ∈ [0, 1) (see Fig. 1) is then defined by: if q ∈ [0, 1/2) (T2 (q), p + 21 ) (27) T2 (q, p) = (T2 (q), p + 41 4q) if q ∈ [1/2, 1). Using the binary representation q = 0.x1 x2 . . ., p = 0.x−1 x−2 . . ., xi ∈ {0, 1} for the space and momentum coordinates of the phase space one can assign to the point x = (q, p) the bi-sequence x = . . . x−2 x−1 · x1 x2 . . .. In this representation the action of the map T2 on the point x takes a simple form . . . x−2 x−1 x1 · x2 x3 . . . if x1 = 0 T2 (. . . x−2 x−1 · x1 x2 . . . ) = (28) . . . x−1 x2 x1 · x3 x4 . . . if x1 = 1. To quantize such a map one can apply the same procedure as in [17]. With each cylinder (rectangle) set x− · x+ m , x− = x−(k−m) . . . x−2 x−1 , x+ = x1 x2 . . . xm , m ∈ {0, . . . k} in the phase space one associates the following coherent state in Hk : |x− · x+ m := |x1 ⊗ |x2 ⊗ · · · ⊗ |xm ⊗ F2∗ |x−(k−m) ⊗ · · · ⊗ F2∗ |x−2 ⊗ F2∗ |x−1 . Note that each coherent state |x− · x+ m is strictly localized in the rectangle x− · x+ m . One then uses these states in order to quantize observables by means of the Anti-Wick quantization procedure (see [17] for details). It is a simple observation that the operator Uk defined by Eq. (26) with U1 = U2 = F2∗ acts on the coherent states in accordance with the action of the classical map (28): Uk |x− · x+ m = |T2 (x− · x+ )m− ,

Uk∗ |x− · x+ m = |T2−1 (x− · x+ )m+ , m ∈ {1, . . . k − 2},

where equals either 1 or 2. Thereby, the operator Uk plays the role of the quantum evolution operator for à la Walsh quantized map T2 . In an analogous way the quantization (24) of the map T p with Ui = F p∗ , i = 1, . . . n max supplies the quantum evolution operator for à la Walsh quantized asymmetric Baker map T p whose expansion rates are given by the powers of p. As both quantizations provide the same set of eigenfunctions, one can utilize the results of the present paper in order to deduce the bound (3) for semiclassical measures of à la Walsh quantized T p . 4. Quantization of Observables We recall now the procedure for the quantization of observables introduced in [23]. Let Mk be the partition of the unit interval I¯ into Nk intervals {E i = [(i − 1)Nk−1 , i Nk−1 ], i = 1, . . . Nk } and let Hk C Nk denote the corresponding Hilbert space. For each function f ∈ L 2 ( I¯) the corresponding quantum observable Op( f ) is given by the matrix, whose elements are Op( f )i, j := δi, j Nk f (x) d x, i, j = 1, . . . Nk . (29) Ei

Let Ic be the circle corresponding to I = [0, 1), where the endpoints 0 and 1 are identified. It will be assumed that Ic is equipped with the standard Euclidean metric

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coming from R. In particular, the distance d(x, y) between two points x, y ∈ Ic is defined by d(x, y) := min{|x − y|, |x − y − 1|}. Below we will deal with a class of observables f ∈ Lip(Ic ) which are Lipschitz continuous on Ic . Recall that the space Lip(Ic ) is equipped with the Lipschitz norm: f Lip = sup | f (x)| + sup x∈I

x= y∈I

| f (x) − f (y)| d(x, y)

(30)

and f ∈ Lip(Ic ) iff f Lip is finite. The definition (29) is strongly motivated by the existence of the correspondence between classical and quantum evolutions of observables (Egorov property). In the context of quantized one-dimensional maps the Egorov property was proved in [23, Thm. 3] for Lipschitz continuous observables undergoing one step evolution. The following theorem is a straightforward extension of that result to larger times. Theorem 4 Let U = Uk be a quantum evolution operator (satisfying (6)) for a quantizable one-dimensional map T of the form (4) and let f be a Lipschitz continuous function on Ic , then U −n Op( f )U n − Op( f ◦ T n ) ≤ D(T ) f Lip

nmax , Nk

(31)

where D(T ) is a constant independent of n and Nk . Proof. For n = 1 the following bound was proved in [23]: U −1 Op( f )U − Op( f ◦ T ) ≤ f Lip

D(T ) . Nk

(32)

From this one immediately gets for n iterations: U −n Op( f )U n − Op( f ◦ T n ) n U −i Op( f ◦ T n−i )U i − U 1−i Op( f ◦ T n−i+1 )U i−1 ≤ i=1 n D(T ) n ≤ f ◦ T i−1 Lip ≤ D(T ) f Lip max , Nk Nk

(33)

i=1

where we used the fact that f ◦ T i ∈ Lip(Ic ) and f ◦ T i Lip ≤ imax f Lip . The inequality (31) implies that the quantum evolution follows the classical one up to the time: n E := log Nk / log max which plays the role of the Ehrenfest time for the model. (Here and after y denotes the largest integer smaller than y.) It is worth noticing that for a certain class of observables the Egorov property turns out to be exact. Let x1 , x2 be two points on the lattice β(Mk ), then with an interval X = [x1 , x2 ] ⊂ I we can associate the projection operator PX := Op(χ X ), where χ X is the characteristic function on the set X . For such operators one has the following result.

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

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Proposition 1. Let X ⊂ I be an interval (or union of intervals) such that all the endpoints β(X ) and β(T −1 X ) belong to β(Mk ), then U −1 PX U = PT −1 X . Proof. Written in matrix form the left side of (34) is given by (U j,l )∗ U j,m , (U ∗ PX U )l,m =

(34)

(35)

{ j|E j ⊆X }

where E j denotes j’s element of the partition Mk . Observe that when E j ⊆ X , the elements (U j,l )∗ = 0, (U j,m = 0) only if T (El ) ⊆ X (resp. T (E m ) ⊆ X ). On the other hand, if the last condition holds, one can extend the summation in (35) to all values of j. By the unitarity of U it gives the right side of (34). For the class of maps T p the proposition above implies the exact correspondence between classical and quantum evolutions of some projection operators up to the times of order nE . Corollary 1. Let T p , be a map of the form (13). Denote U a general quantization of T p (satisfying (6)) acting on the vector space Hk of the dimension Nk = p k . For a cylinder x of the length |x| ≤ k the evolution of the corresponding projection operator Px is given by U −n Px U n = PT p−n x

for all n + |x|/n max ≤ n E ,

(36)

where max = p n max is the maximum slope of T p . 5. Metric Entropy of Semiclassical Measures Let Uk : Hk → Hk , k = 1, · · · ∞ be a sequence of unitary quantizations of a quantizable map T of the form (4). For a given sequence of eigenstates: ψk ∈ Hk , Uk ψk = eiθk ψk , the corresponding measures µk , k = 1, · · · ∞ are defined by Eq. (10) through the Riesz representation theorem. We will be concerned with the possible outcome for semiclassical T -invariant measures µ = limk→∞ µk . Following the approach of [17,19,20] we will consider the metric entropy HKS (T, µ) of µ. Below we recall some basic properties of classical entropies and connect them to a certain type of quantum entropies. s Let π = i=1 Ii be a certain partition of I into s intervals. Given a measure µ on I the entropy function of µ with respect to the partition π is defined by h π (µ) := −

s

µ(Ii ) log(µ(Ii )).

i=1

More generally, one can consider the pressure function: pπ,v (µ) := −

s

µ(Ii ) log(vi2 µ(Ii )),

i=1

where the weights v = {vi |i = 1, . . . s} are given by a set of real numbers fixed for a given partition. Obviously, if all vi equal one, then pπ,v is just the entropy defined

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s above. An important feature of h π (µ) is its subadditivity property. If π = i=1 Ii and s τ = i=1 Ji are two partitions, then for the partition π ∨ τ consisting of the elements Ii ∩ J j and a measure µ one has: h π ∨τ (µ) ≤ h π (µ) + h τ (µ).

(37)

Now consider dynamically generated refinements of π . Define ε = ε0 ε1 . . . εn−1 , and let it be a sequence of the elements εi ∈ {1, . . . s} of the length |ε| = n. For any n ≥ 1 set partition: π (n) = |ε|=n ε of I is the collection of the sets: ε := T −(n−1) Iεn−1 ∩ T −(n−2) Iεn−2 ∩ . . . Iε0 . Each cylinder ε has a simple meaning as the set of the points with the same “ε-future” up to n iteration. One is interested in the entropies of T -invariant measures µ with respect to the partitions π (n) : h n (µ) := h π (n) (µ) = − µ(ε) log(µ(ε)). |ε|=n

If µ is T -invariant, it follows (see e.g., [25]) by the subadditivity (37) that: h n+m (µ) ≤ h n (µ) + h m (µ).

(38)

For the entropy function this implies the existence of the limit: 1 h n (µ). n→∞ n

Hπ (T, µ) = lim

(39)

The metric (Kolmogorov-Sinai) entropy is then defined as the supremum over all finite measurable initial partitions π : HKS (T, µ) = sup Hπ (T, µ). π

In the quantum mechanical framework one needs to define a quantum entropy (preslimit. Note that a measure of sure) reproducing h n (µ) (resp. pn,v (µ)) in the semiclassical each set Ii can be written as the average µ(Ii ) = χIi (x) dµ over the classical observable χIi (x) which is the characteristic function of the set Ii . The quantum observable corresponding to χIi is then simply the projection operatorPi := PIi = Op(χIi ) on the set Ii . Now we need to “quantize” the refined partitions |ε|=n ε. The most straightforward approach would be considering quantization of the observables χε . A different scheme was suggested in [20]. Instead of taking classically refined observables χε and then quantizing them, one considers a natural quantum dynamical refinement of the initial quantum partition. We will say that a sequence of operators πˆ = {πˆ i , i = 1 . . . s} defines a quantum partition of Hk if they resolve the unity operator: 1Hk =

s

πˆ i∗ πˆ i .

i=1

For a quantum partition πˆ the entropy (resp. pressure) of a state ψ ∈ Hk is given by hˆ πˆ (ψ) := −

s i=1

πˆ i ψ2 log(πˆ i ψ2 ),

pˆ π,v ˆ (ψ) := −

s i=1

πˆ i ψ2 log(πˆ i ψ2 vi2 ).

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

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Now with each set ε of π (n) one associates the operator defined by: Pεi ( p) = U − p Pεi U p .

Pε := Pεn−1 (n − 1) . . . Pε1 (1)Pε0 (0),

(40)

As follows immediately from the definition of Pε , the sets of the operators πˆ (n) = {Pε , |ε| = n}, πˆ ∗(n) = {Pε∗ , |ε| = n} define quantum partitions of 1Hk . Note that Pε∗ and Pε differ only by the order of the components Pεi (i) and both πˆ (n) , πˆ ∗(n) correspond to the same classical partition π (n) . For an eigenfunction ψk ∈ Hk of the operator Uk let hˆ πˆ (n) (ψk ), hˆ πˆ ∗(n) (ψk ) be the corresponding entropies. After introducing the weight functions: µˆ ∗k (ε) := Pε∗ ψk 2

µˆ k (ε) := Pε ψk 2 ,

(41)

for the elements ε of the corresponding classical partition π (n) , the quantum entropies of ψk can be written with a slight abuse of notation (in principle, µˆ k , µˆ ∗k are not measures but merely positive weight functions defined only on the elements of the partitions) as the classical entropy function of µˆ k , µˆ ∗k : hˆ πˆ (n) (ψk ) = h n (µˆ k ),

h n (µˆ k ) = −

hˆ πˆ ∗(n) (ψk ) = h n (µˆ ∗k ),

h n (µˆ ∗k ) = −

µˆ k (ε) log µˆ k (ε),

|ε|=n

|ε|=n

µˆ ∗k (ε) log µˆ ∗k (ε).

(42)

Note that the weight function µˆ k satisfies an important compatibility condition: µˆ k (ε0 . . . εn−1 ) =

µˆ k (ε0 . . . εn ).

εn ∈{1...s}

As Pεi (i), i ≤ n = |ε| approximately commute with each other for a finite n by virtue of the Egorov property, the same property also holds (up to semiclassically small errors) for µˆ ∗k . Furthermore, for a finite n the Egorov property guarantees that both µˆ k (ε) and µˆ ∗k (ε) equal (up to semiclassically small errors) the measure µk (ε) induced by the eigenstate ψk . Hence in the semiclassical limit: lim h n (µˆ k ) = lim h n (µˆ ∗k ) = h n (µ),

k→∞

k→∞

(43)

where µ = limk→∞ µk is the corresponding semiclassical measure. To extract from h n (µ) the metric entropy HKS (T, µ) of the measure µ it is necessary to apply the classical limit (39). In complete analogy, the quantum pressures of ψk : pˆ πˆ (n) ,v (ψk ) = pn,v (µˆ k ),

pn,v (µˆ k ) = −

pˆ πˆ ∗(n) ,v (ψk ) = pn,v (µˆ ∗k ),

pn,v (µˆ ∗k ) = −

|ε|=n

|ε|=n

µˆ k (ε) log µˆ k (ε)vε2 , µˆ ∗k (ε) log µˆ ∗k (ε)vε2 (44)

converge in the limit k → ∞ to the classical pressure pn,v (µ) of µ.

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6. Bound on Metric Entropy The main purpose of this section is to prove the bound (14) on the possible values of HKS (T p , µ). In what follows we will closely follow the approach developed in [20,21] for Anosov geodesic flows. The main technical tool is a variant of the entropic uncertainty relation first proposed in [27,28] and later generalized and proved in [29]. Here we will make use of a particular case of the statement appearing in [20,21]. s , τˆ = Theorem 5 (Entropic Uncertainty Principle [20, Thm. 6.5]). Let πˆ = {πˆ i }i=1 s , be two partitions of the unity operator 1 on a complex Hilbert space (H, ., .) {τˆi }i=1 H s , w = {w }s be the families of the associated weights. For any and let v = {vi }i=1 i i=1 normalized ψ ∈ H and any isometry U on H the corresponding pressures satisfy:

pˆ π,v ˆ ∗j ). ˆ (ψ) + pˆ τˆ ,w (Uψ) ≥ −2 log(sup v j wk τˆk U π

(45)

j,k

Let T p be a map satisfying Definition 1 and let Uk be its quantization satisfying (6) which acts on the Hilbert space Hk . In what follows we will use Theorem 5 for the Hilbert space Hk , quantum partitions πˆ = {Pε∗ , |ε| = n}, τˆ = {Pε , |ε| = n}, defined by (40) as “quantizations” of n-times refinement π (n) of the classical partition π = li=1 Ii . For each ε-element of the partitions τˆ , πˆ the corresponding weights are then defined by n−1 1/2 vε = wε = i=0 εi , ε = ε0 . . . εn−1 , where εi is the expansion rate of the map T p at the interval Iεi , εi = 1, . . . l. Finally, the isometry U will be the unitary transformation (Uk )n and the normalized state ψ will be an eigenstate ψk of Uk . With such a choice the left side of (45) reads: µˆ k (ε) log(µˆ k (ε)vε2 ) + µˆ ∗k (ε) log(µˆ ∗k (ε)vε2 ). pn,v (µˆ k ) + pn,v (µˆ ∗k ) = − |ε|=n

Thus, in order to bound pn,v (µk ) from below we need an estimation on the right hand side of (45). This amounts to the control over the elements: Pε U n Pε = U Pεn−1 U Pεn−2 . . . U Pε0 U Pεn−1 . . . U Pε0 ,

where U = Uk . The following proposition gives the required estimation. Proposition 2. Let π = li=1 Ii be the classical partition of I and let πˆ = {Pi := Op(χ Ii ), i = 1 . . . l} be the corresponding quantum partition. Then for any sequence ε = ε0 . . . εn−1 , n > 0, the product Pε = U Pεn−1 U Pεn−2 . . . U Pε0 , satisfies the bound: 1/2

Pε ≤ Nk

n−1

−1/2 . εi

(46)

i=0

Proof. It is easy to understand the source of (46). Since the structure of U basically mimics the structure of the corresponding classical map, each time when U Pε j acts on a −1/2

vector from the Hilbert space, it “decreases” its components by the factor ε j . More precisely, for any v ∈ Hk , the absolute value of j’s component of the vector v = U Pεi v satisfies the bound |v j | ≤ (−1/2 ) εi

max

m=1,...Nk

|vm |.

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

319

Applying this inequality n times one gets for the components of the vector v (n) = Pε v: n−1 −1/2 (n) |v j | ≤ εi max |vm |, j = 1, . . . Nk . m=1,...Nk

i=0

From this the desired estimation follows immediately, since |v j | ≤ v for all j and 1/2 v (n) ≤ Nk max |v (n) j |. The entropic uncertainty principle together with Proposition 2 give now the bound on the pressure of ψk : 1/2 , (47) pn,v (µˆ k ) + pn,v (µˆ ∗k ) ≥ −2 log Nk which can be also written as h n (µˆ k ) + h n (µˆ ∗k ) ≥

1/2 . µˆ k (ε) + µˆ ∗k (ε) log vε2 − 2 log Nk

(48)

|ε|=n

n 1/2 Note that such a bound becomes nontrivial only for times n when vε = i=1 εi is 1/2 comparable with Nk . In other words, n should be of the same order as the Ehrenfest time n E . For shorter times (48) would only imply that h n (µˆ k ) + h n (µˆ ∗k ) > C0 , where C0 < 0 (which is completely redundant as h n is a positive function). It is now tempting to use the inequality (48) for n = n E to get a bound on the metric entropy. Recall, however, that in such a case the relevant partition used to define h n E is of the quantum size Nk−1 . On the other hand, the correct order of the semiclassical and classical limits in the definition of HKS (T, µ) requires a bound on the entropy function for partitions of a finite (classical) size, independent of k. Thus in order to extract useful information from (47,48) it is necessary to connect the pressure pn E ,v (µˆ k ) for the quantum time n E with the pressure pn,v (µˆ k ) for an arbitrary classical time n (independent of k). To this end it has been suggested in [17] to make use of the subadditivity of the metric entropy. More specifically, for a classical invariant measure µ the subadditivity of the entropy function implies: pn+m,v (µ) ≤ pn,v (µ) + pm,v (µ), ⇒ pm,v (µ) ≤ qpn,v (µ) + pr,v (µ), m = qn + r.

(49)

The subadditivity property (49) cannot be utilized straightforwardly, as the weights µˆ ∗k , µˆ k are not T -invariant, in general. However, as follows from the following lemma, by virtue of the Egorov property the measures µk (ε) of cylinders ε turn out to be invariant for sufficiently short times. Lemma 1. Let ε be a cylinder of the size |ε| = m ≤ n E , and let µk (ε) be its measure as defined by Eq. (10). Then µk (ε) = µˆ ∗k (ε) = µˆ k (ε), where µˆ ∗k (ε), µˆ k (ε) as in Eq. (41). Furthermore, for any integer 0 ≤ n such that n + m ≤ n E the measures remain invariant under the action of T p−n : µk (ε) = µk (T p−n ε).

(50)

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B. Gutkin

Proof. Note that any set ε of the size |ε| = m ≤ n E can be written in the x-representation (see Remark 2) as x1 x2 . . . xm , where m ≤ k. The proof of the lemma then straightforwardly follows from Corollary 1 and the fact that all Px commute with each other for |x| ≤ k. From this lemma the desired connection between the pressures for partitions of classical and quantum sizes immediately follows. Proposition 3. Let Uk , k = 1, . . . ∞ be a sequence of unitary quantizations of a map T p (satisfying Definition 1), and let {ψk } be a sequence of their eigenstates with the corresponding measures µk , as defined by eq. (10). Set the pressure for each µk to be pn,v (µk ) := − µk (ε) log µk (ε)vε2 . |ε|=n

Then for n E = qn + r , q, n, r ∈ N, 0 ≤ r < n we have: pn E ,v (µk ) ≤ qpn,v (µk ) + pr,v (µk ).

(51)

Proof. Straightforwardly follows from the subadditivity of h n and (50). Equipped with the above proposition we can prove now the bound (14) on the metric entropy for maps T p . Theorem 6 Let Uk , k = 1, . . . ∞ be a sequence of unitary quantizations of a map T p , and let {ψk } be a sequence of their eigenstates. Assuming that the corresponding limiting invariant measure µ = limk→∞ µk exists, µ must satisfy the following bound: 1 HKS (T p , µ) ≥ µ(I j ) log j − log max . (52) 2 j

Proof. From the bound (48) and Proposition 3 it follows that the pressure for the partition of an arbitrary fixed size 0 < n < n E satisfies the inequality: pn,v (µk ) 1 pr,v (µk ) r pn,v (µk ) ≥ − log max − − . n 2 nE n nE

(53)

Because r , pr,v are bounded for a fixed n, the last two terms in the right hand side of (53) vanish when k → ∞ and one gets: pn,v (µ) 1 ≥ − log max . n 2 To complete the proof it remains to notice that n µ(ε) log εi , pn,v (µ) = h n (µ) − |ε|=n

and

µ(ε) log

|ε|=n

since µ is an invariant measure.

n i=1

i=1

εi

(54)

=

j

µ(I j ) log j ,

(55)

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

321

7. Proof of the Anantharaman-Nonnenmacher Conjecture for T p Maps As we have shown in the previous section, the method of N. Anantharaman and S. Nonnenmacher can be employed for the proof of the bound (14). However, exactly as for Anosov geodesics flows, such an approach does not allow to prove a stronger result (15). Very roughly, the reason for this can be explained in the following way. For a generic map the entropy function h n (µk ) is a “non-homogeneous” quantity which contains contributions from the cylinders ε with different “expansion rates” ε . The domain of validity for subadditivity of the entropy function is determined by an entry (cylinder) with the largest expansion rate and thus, restricted to the times n ≤ n E . On the other max , hand, the bound (47) becomes informative for times n ≥ n, ¯ where n¯ = n E log 2 log l log = i=1 µ(Ii ) log i . When the expansion rate is highly non-uniform one is unable to match long “quantum” times n > n¯ with short “classical” times n < n E , see Fig. 2. This results in the bound (14) which is, in general, weaker than (15) for maps with non-uniform slopes. Below we formulate a certain modification of the original strategy to overcome the problem.

7.1. General idea. Speaking informally, the basic idea here is to “homogenize” the original system, making it uniformly expanding first and only then apply the method used in the previous section. More specifically, we consider the class of maps T = T p , given by Definition 1. In what follows we adopt the tower construction widely used in the theory of dynamical systems (see e.g., [30]). As we show in the next subsection, T can be regarded as the first return map for a certain uniformly expanding dynamical system. Namely, the action of T on I turns out to be equivalent to the action of the : so-called tower map T I → I on a subset (“zero level”) of the tower phase space I. By a standard construction for first return maps, any invariant measure µ for T induces . The corresponding metric entropies HKS (T , a measure µ on I invariant under T µ), HKS (T, µ) are then related to each other by Abramov’s formula and the entropic bound (15) turns out to be equivalent to: 1 , µ) ≥ log p. HKS (T 2

(56)

Thus, in order to prove the conjecture of S. Nonnenmacher and N. Anantharaman for maps T p one needs to show (56) for the measure µ. It turns out that a pure classical construction above can be “lifted” to the quantum level. Recall that µ is a semiclassical measure generated by eigenstates of a sequence {Uk } of unitary quantizations of T . The crucial observation is that µ is actually a semik } of quantizations of T . In Subsect. 7.3 we show classical measure for a sequence {U that for each sequence {ψk } of the eigenstates of {Uk } generating in the semiclassical k } generating the limit the measure µ there exists a sequence { k } of eigenstates of {U measure µ. This is schematically depicted by the following diagram: µψk k→∞

µ = µ ◦ T −1

Quantum

+3 µ k k→∞

Classical3+ −1 µ= µ◦T

(57)

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B. Gutkin

n

classical

quantum

nE classical

I1

n

I0 [0]

quantum

[1]

nE Fig. 2. On the left down (up) the case when (14) provides non-trivial (resp. trivial) bound on the metric entropy HKS (T, µ) is shown. On the right is depicted the tower for the map T{2,4,4}

is a map with a uniform expansion rate one can apply the method used in Since T the previous section in order to prove (56). From this the metric bound (15) follows immediately. As we would like to keep the exposition and notation below as simple as possible, we will first consider in detail the map T2 = T{2,4,4} defined in (25). The results can then be straightforwardly extended to all other maps T p . 7.2. Classical towers. In what follows we construct the tower dynamical system corresponding to the map T := T{2,4,4} as defined by Eq. (25). To this end let us double the original phase space and consider the set I := I × {0, 1}. We will refer to the sets I0 := I × {0} = {(x, 0), x ∈ I }, I1 := I × {1} = {(x, 1), x ∈ I } as zero and first levels of the tower I = I0 ∪ I1 respectively. Let D0 := [0, 1/2), D1 := [1/2, 1). The tower : map T I → I is then defined by: ¯ (x, η) = (T (x), 0) if η = 0, x ∈ D0 or η = 1 and any x, (58) T (T¯ (x), 1) if η = 0, x ∈ D1 where T¯ := T{2,2} is the uniformly expanding map corresponding to T . Consider now I on the set I on the first return map T I0 . It is straightforward to see that the action of T 0 0 I0 ∼ = I coincides with the action of T on I . In other words, T can be regarded as the first return map for the lowest level of the tower (see Fig. 2). I ) one can construct (using Given an invariant measure µ for T (equivalently for T 0 a standard procedure, see e.g., [25,31]) the probability measure µ which is invariant . Precisely, for a set A ⊆ I one defines the measures of the sets under the tower map T (A × {0}), (A × {1}) by µ(A × {0}) = −1 µ(A),

µ(A × {1}) = −1 µ(T¯ −1 A ∩ [1/2, 1]),

with the normalization constant = 1+µ([1/2, 1]). If A = x is a cylinder set (defined with respect to the action of T¯ ) this can be rewritten as: µ(x × {0}) = −1 µ(x),

µ(x × {1}) = −1 µ(1x).

(59)

it makes sense to consider the corresponding metric entropy Since µ is invariant under T , , HKS (T µ). An important observation is that HKS (T µ) is related to HKS (T, µ). As T is −1 the first return map for I0 , and µ( I0 ) = , by Abramov’s formula (see e.g., [25]): , µ). HKS (T, µ) = HKS (T

(60)

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

323

Having an invariant measure µ on I it is possible in turn to construct a measure µ¯ on I which is invariant under the homogeneous map T¯ . Let π I : I → I be a natural projection on the tower: π I (x, η) = x, for all x ∈ I , η ∈ {0, 1}. As = T¯ ◦ π I , πI ◦ T it follows immediately that the measure µ¯ := µ ◦ π ∗I

(61)

is invariant under T¯ . Furthermore, the metric entropy of µ¯ turns out to be equal to the metric entropy of µ: , HKS (T¯ , µ) ¯ = HKS (T µ). (62) This equality can be deduced, from a version of the Abramov-Rokhlin relative entropy formula in [32]. For the sake of completeness we give a simple proof of (62) in Appendix B. The above construction allows a straightforward extension to the case of an arbitrary map T p satisfying Definition 1. The tower phase space here is defined as lt = n max (n max = max j=1,...l {n j }) copies of I : I = I × {0, 1, . . . , lt − 1} ∼ =

l t −1

Ij,

(63)

j=0

p : I → I where the set I j = I × { j} stands for j’s level of the tower. The tower map T is then defined with the help of the uniformly expanding map T¯ p given by Eq. (16). For each level η ∈ {0, 1, . . . , lt − 1} define the set: Ij. Dη := { j|n j =η+1}

p is given by: Then the action of the map T ¯ if x ∈ T¯ η Dη p (x, η) = (T p (x), 0) T (T¯ p (x), η + 1) if x ∈ T¯ η Dη .

(64)

p a point x ∈ Dη × {0} climbs η Such a definition implies that under the action of T steps upstairs in the tower phase space, then it “jumps” downstairs to zero level and the process is repeated. It is now straightforward to see that the map T p coincides with the first return map p for zero level of T I0 of the tower. As a result, starting from an invariant measure µ p . For a set µ for the tower map T for T p one can easily construct the invariant measure A × {η} ⊆ I , with A ⊆ I and level η ∈ {0, . . . lt − 1} the corresponding measure is given by µ(A × {η}) = −1

{k|n k ≥η}

µ(T¯ p−η (A) ∩ Ik ),

=

l

n j µ(I j ),

(65)

j=1

where is the average return time to zero level of the tower. Precisely as for the map T{2,4,4} , one can also construct the measure µ¯ invariant under the action of T¯ p . The corresponding metric entropies are then related by: p , HKS (T p , µ) = HKS (T¯ p , µ) ¯ = HKS (T µ). (66)

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7.3. Quantum towers. We are going now to consider the quantum analog of the above tower construction. Construction. Let U = Uk be a tensorial quantization of the map T = T{2,4,4} , acting on the Hilbert space Hk of the dimension 2k = dim(Hk ). We will assume that U is of the form (26). In that case U allows an obvious decomposition: U = U¯ P0 + V¯ U¯ P1 ,

V¯ = σ U¯ .

(67)

Here U¯ : Hk → Hk , U¯ : Hk → Hk stand for tensorial quantization (20) of the uniformly expanding map T¯ = T{2,2} with U given by U1 and U2 respectively, and σ stands for the unitary transformation reversing the order of the last two symbols in |x1 ⊗ · · · ⊗ |xk−1 ⊗ |xk ∈ Hk : σ |x1 ⊗ · · · ⊗ |xk−1 ⊗ |xk = |x1 ⊗ · · · ⊗ |xk ⊗ |xk−1 . In addition to P0 , P1 it will be also convenient to use the projection operator: P1 = U¯ P1 U¯ ∗ .

(68)

Explicitly its action on the basis states of Hk is given by: P1 |x1 ⊗ · · · ⊗ |xk−1 ⊗ |xk = |x1 ⊗ · · · ⊗ |xk−1 ⊗ U1 P1 U1∗ |xk , where P1 |i = δi,1 |i, i = 0, 1. It worth to notice that P1 commutes with V¯ : V¯ P1 = P1 V¯ .

(69)

=H 0 ⊕ H 1 , dim(H) = 2k + 2k−1 with We define now the “tower” Hilbert space H 0 := Hk , and H 1 := U¯ P1 Hk ≡ P Hk , H 1

(70)

0 , H 1 with the corresponding to zero and the first levels of the tower. As we identify H 0 (resp. φ ∈ H 1 ) can also be regarded space (resp. the subspace of) Hk , any vector φ ∈ H as a vector from Hk . By this identification, operations defined on Hk can be “lifted” to 0 , H 1 , as well. In particular, we can define the scalar product on H the Hilbert spaces H using the scalar product on Hk . Namely for = (φ0 , φ1 ) ∈ H, = (φ0 , φ1 ) ∈ H, 0 and φ1 , φ ∈ H 1 : with φ0 , φ0 ∈ H 1 ( , ) = φ0 , φ0 + φ1 , φ1 . can be easily constructed from an orthonormal basis in Hk . An orthonormal basis in H A convenient choice is provided by the vectors: E(x,0) := (|x1 ⊗ · · · ⊗ |xk−1 ⊗ |xk , 0), x = x1 . . . xk−1 xk , E(x,1) := (0, |x1 ⊗ · · · ⊗ |xk−1 ⊗ |1 ),

x = x1 . . . xk−2 xk−1 ,

(71)

where |0 := U1 |0, |1 := U1 |1 and xi , i = 1, . . . k (resp. i = 1, . . . k − 1) run over all possible sequences of {0, 1}.

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

325

In what follows we will consider a one-parameter family of tower evolution operators →H defined in the following way. For any = (φ0 , φ1 ) ∈ H, with φ0 ∈ H 0 θ : H U and φ1 ∈ H1 : θ := (V¯ P φ1 + U¯ P0 φ0 , eiθ U¯ P1 φ0 ). U 1

(72)

→H is given by ∗ : H Correspondingly, the adjoint operation U θ θ∗ = (e−iθ P1 U¯ ∗ φ1 + P0 U¯ ∗ φ0 , P V¯ ∗ φ0 ). U 1

(73)

and U ∗ ∈ H θ is a unitary θ , U Main properties. It is straightforward to see that U θ operation on H: ∗ be as above, then θ , U Proposition 4. Let U θ θ U θ∗ U θ∗ = U θ = 1. U ∗ U Proof. In the block matrix representation the product U θ θ takes the form P0 + P1 P0 U¯ ∗ V¯ P1 P0 U¯ ∗ e−iθ P1 U¯ ∗ U¯ P0 V¯ P1 = . P1 V¯ ∗ 0 P V¯ ∗ U¯ P0 P eiθ U¯ P1 0 1

1

(74) Using Eqs. (68, 69) it is now straightforward to see that the off-diagonal terms in the right side of (74) vanish. θ . Specifically the short Below we demonstrate that the Egorov property holds for U time evolution of projection operators is prescribed by the classical evolution of the corresponding tower map. Proposition 5. Let x ⊆ I be a cylinder of the length m = |x| < k − 1, then: θ = (P0 P ¯ −1 ⊕ P P ¯ −1 ), θ∗ (Px ⊕ 0) U U T x 1 T x ∗ Uθ (0 ⊕ P1 Px ) Uθ = (P1 PT¯ −1 x ⊕ 0). Proof. In the matrix representation the left side of (75) reads: ¯ P0 U¯ ∗ e−iθ P1 U¯ ∗ U P0 V¯ P1 Px 0 . P1 V¯ ∗ 0 0 0 eiθ U¯ P1 0

(75) (76)

(77)

By Eqs. (68, 69) the off diagonal terms of the above product are zeros, and the diagonal part: P0 U¯ ∗ Px U¯ P0 0 0 P1 V¯ ∗ Px V¯ P1 coincides with the right side of Eq. (75), as follows from Corollary 1 and the equality: σ Px σ = Px . Equation (76) is then proved analogously.

326

B. Gutkin

Let X be a subset of I and let PX be the corresponding projection operator. The projection operator corresponding to the subset X × {0} ∪ X × {1} of the tower, is then defined as: X = PX ⊕ P PX . P 1 The following corollary follows immediately from Proposition 5. Corollary 2. For a given cylinder x ⊂ I of the length |x| = m and an integer n such ¯ −n be the projection operator corresponding to the subset that n + m < k − 1, let P T x −n −n ¯ ¯ T x × {0} ∪ T x × {1} of the tower. Then: ¯ −n U θ∗ P U T x θ = PT¯ −n−1 x .

(78)

Eigenfunctions and semiclassical measures. Given an eigenfunction ψ of the original θ = eiθ evolution operator U , U ψ = eiθ ψ we can construct the eigenfunction , U of the tower evolution operator. Precisely, −1 = ψ, U¯ P1 ψ ψ 2 ,

ψ = 1 + ψ, P1 ψ,

(79)

θ . Indeed, this is so, since by (68), is the normalized eigenstate of the operator U 1 1 θ = (U¯ P0 + U¯ 1 P U¯ P1 )ψ, eiθ U¯ P1 ψ − 2 = U ψ, eiθ U¯ P1 ψ − 2 , U ψ ψ 1 and ψ is the eigenstate of U . It is also easy to verify that ( , )) = 1. For any sequence of quantizations {Uk } of T and their eigenstates {ψk }, let us conθk } of quantizations of the tower map T determined sider the corresponding sequence {U by Eq. (72). (Note that these quantizations depend on the eigenvalues eiθk of ψk ’s.) θk } can be constructed applying Eq. (79). As a result, Then the eigenstates { k } of {U a sequence of semiclassical measures µk on I induces the sequence of semiclassical measures µk on I . For a cylinder x ⊂ I , |x| ≤ k the measures µk of the tower sets x × {0}, x × {1} are defined as: µk (x × {0}) = ( k , (Px ⊕ 0) k) ,

µk (x × {1}) = ( k , (0 ⊕ P1 Px ) k) .

By Eq. (79) these measures are related to the measure µk of the set x: µk (x × {0}) = k−1 µk (x),

µk (x × {1}) = k−1 µk (1x),

(80)

where we set k = ψk . Note that after taking the limit k → ∞ in (80) one obtains Eqs. (59), where µ = limk→∞ µk is precisely the measure of the classical tower obtained from the semiclassical measure µ = limk→∞ µk by the procedure from the previous section. Also, defining the measure µ¯ k on I by x k) = −1 µk (x) + µk (1x) , µ¯ k (x) := ( k , P (81) k one reveals in the semiclassical limit the measure µ¯ = limk→∞ µ¯ k related to µ by Eq. (61).

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

327

Extension to other maps T p . The above construction of quantum towers can be straightforwardly extended to other maps T p . Let U be a quantization of T p acting on the Hilbert space Hk , as defined by Eq. (24). Note that U can be cast in the form: U=

l t −1

U¯ (0) = 1, U¯ (η) := U¯ η U¯ η−1 . . . U¯ 1 for η > 0,

ση U¯ (η+1) PDη ,

η=0

where U¯ i , i = 1, . . . lt are quantizations of the uniformly expanding map T¯ p , σ j is the operator which reverse the order of the last j elements in the products |x1 ⊗· · ·⊗|xk−1 , lt is the height of the corresponding classical tower and PDη is the projection operator corresponding to the set Dη as defined in Sect. 7.2. Mimicking the construction of the classical towers one defines then the tower Hilbert space as the direct sum: = H

lt −1

η , H

η=0

η ∼ H = Pη + P¯η Hk ,

with the projection operators Pη , P¯η defined by Pη := U¯ (η) PDη (U¯ (η) )∗ ,

P¯η := U¯ (η) P¯Dη (U¯ (η) )∗ ,

P¯Dη :=

PD j .

j>η

θ whose action on the states = φ0 , φ1 , . . . Now take the tower evolution operator U is given by: φlt −1 ∈ H ⎛

l t −1

θ = ⎝ U

η=0

⎞ ση U¯ η+1 Pη φη , eiθ U¯ 1 P¯0 φ0 , eiθ U¯ 2 P¯1 φ1 , . . . , eiθ U¯ lt P¯lt −1 φlt −1 ⎠ . (82)

θ follow from It can be easily checked that the unitarity and the Egorov property for U the same properties for the operator U . Finally, if U ψ = eiθ ψ then = ψ, U¯ (1) P¯D0 ψ, U¯ (2) P¯D1 ψ, . . . , U¯ (lt −1) P¯Dlt −2 ψ / ψ

(83)

θ with the eigenvalue eiθ and the normalization constant given by is the eigenstate of U ψ = 1 +

l t −2 j=0

ψ, P¯D j ψ =

l

ψ, PI j ψn j .

j=1

This shows that any semiclassical measure µ = limk→∞ µk on I induced by a sequence of eigenstates {ψk } of Uk , generates through Eq. (83) the semiclassical measures: µ= µk on I and µ¯ = limk→∞ µk ◦ π ∗I on I which are invariant under the action limk→∞ p and T¯ p , respectively. Note also that the corresponding metric entropies are related of T to each other by Eq. (66).

328

B. Gutkin

7.4. Proof of Theorem 3. Let us now prove the bound (15) for the map T{2,4,4} . Theorem 7 Let {Uk }∞ k=1 be a sequence of tensorial quantizations of T = T{2,4,4} . For ∞ a sequence {ψk }k=1 of eigenstates Uk ψk = eiθk ψk let µ = limk→∞ µk be the corresponding semiclassical measure, then HKS (T, µ) ≥

µ(0) + 2µ(1) log 2. 2

(84)

Proof. To prove (84) it is possible, in principle, to follow precisely the scheme described , µ) for the corresponding in the beginning of the section i.e., to prove the bound on HKS (T semiclassical measure µ on the tower and then deduce the bound (84) using Abramov’s formula. From the technical point of view, however, it turns out to be easier to prove an equivalent bound for the metric entropy HKS (T¯ , µ) ¯ of the measure µ. ¯ Let { k }∞ be the sequence of the tower eigenstates corresponding to the sequence k=1 ˆ of ψk ’s, and let h n ( k ) ≡ h n (µ¯ k ) be the entropy function for the corresponding measures µ¯ k : x k 2 log( P x k 2 ). (85) h n (µ¯ k ) = − µ¯ k (x) log µ¯ k (x) = − P |x|=n

|x|=n

Then the metric entropy HKS (T¯ , µ) ¯ is obtained after first applying the semiclassical limit: h n (µ) ¯ = lim h n (µ¯ k ),

(86)

1 HKS (T¯ , µ) ¯ = lim h n (µ). ¯ n→∞ n

(87)

k→∞

and then the classical limit:

¯ corresponds to the “probing” of towers with parAs can be seen from Eq. (85), HKS (T¯ , µ) x ). This explains titions made of “vertical rectangles” (represented by the projections P , µ). ¯ and HKS (T ˜ To prove the source of the equality between metric entropies HKS (T¯ , µ) the bound on HKS (T¯ , µ) ¯ we will make use of the same scheme as in [17]. The first step is to get the bound on the entropy function, when n is of the same order as k. This is provided by the following proposition. Proposition 6. Let h n (µ¯ k ) be as in (85) and set n = k − 1, then k−1 − 1 log 2. h k−1 (µ¯ k ) ≥ 2

(88)

Proof. We will use the Uncertainty Entropic principle (Theorem 5) for the partitions: y , |y| = k − 1}, weights: vy = wy ≡ 1 and the isometry operation π = τ = {P k −1 θk ) . Since k is an eigenstate of U θk it follows immediately from (45): U = (U h k−1 ( k ) ≥ − log(

sup

|y|=|y |=k−1

y (U θk )k−1 P y ). P

(89)

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

329

y (U θk )k−1 P y . To Thus one needs to estimate the norm of the operator C(y , y) = P this end let us calculate the matrix elements of C(y , y): ( E(x ,i ) , C(y , y)E(x,i)) , in the basis of orthogonal states (71) with the parameters: i, i ∈ {0, 1}, |x| = k, (|x | = k) if i = 0 (resp. i = 0) and |x| = k − 1, (|x | = k − 1) if i = 1 (resp. i = 1). The action of the projection operator on the basis states is given by ⎛ y E(x,i) = E(x,i) ⎝ P

k −1

⎞ δxm ,ym ⎠ .

(90)

m=1

Hence for each pair of y, y there exist at most two values of x and two values of x such that the matrix elements ( E(x ,i ) , C(y , y)E(x,i)) are not zeros. From that follows: C(y , y) ≤ 2

max

(x,i),(x ,i )

=2

max

|((E(x ,i ) , C(y , y)E(x,i)) |

(x,i),(x ,i )

θk )k−1 E(x,i)) |. |((E(x ,i ) , (U

(91)

θk )k−1 in the basis of Therefore, it remains to estimate the elements of the operator (U θk on {E(x,i) } up to times k closely {E(x,i) }. To this end, let us notice that the action of U on the sets x × {i} of connected to the action of the corresponding tower map T I. k −1 Specifically, let E = (Uθk ) E(x,i) . Then, as follows from Eq. (72), depending on x, i the state E takes the values (e, 0) or (0, e), where e is of the form: e=

ei Qθk |xk ⊗ Ui1 |xi1 ⊗ Ui2 |xi2 ⊗ · · · ⊗ Uik−1 |xik−1 if i = 0, ei Qθk Ui1 |xi1 ⊗ |1 ⊗ Ui2 |xi2 ⊗ · · · ⊗ Uik−1 |xik−1 if i = 1.

(92)

Here Uim is either U1 or U2 , xi1 , xi2 . . . xik−1 is some permutation of the original sequence √ x1 , x2 . . . xk−1 and Q is an integer number. Since |x j , U1,2 xi | = 1/ 2 for any pair xi , x j ∈ {0, 1}, θk ) |((E(x ,i ) , (U

k−1

E(x,i)) | = |((E(x ,i ) , E))| ≤ 2

− k−1 2

.

Together with (89) and (91) this gives the proof of the proposition. The second necessary step is to connect values h k−1 (µ¯ k ) of the entropy at quantum times of order k to its values h n (µ¯ k ) at short fixed classical times n. Proposition 7. Let h n (µ¯ k ) be as in (85), and let k − 1 = qn + r , r < n, where n < k − 1 is a fixed (classical) time and q, r are integers, then 1 n log 2 1 h n (µ¯ k ) ≥ h k−1 (µ¯ k ) − . n k−1 k−1

(93)

330

B. Gutkin

Proof. The prove of (93) is analogous to the proof of Proposition 3. One makes use of the fact that the measure µ¯ k is invariant under the transformation T¯ j for “short” times. From the definition of µ¯ k and Eq. (78) it follows that for any cylinder x of length |x| = m: (94) µ¯ k (x) = µ¯ k (T¯ −n x), for m + n ≤ k − 1. Let n, q, r be as in the conditions of the proposition. Then the subadditivity property of the entropy function implies: h k−1 (µ¯ k ) ≤ qh n (µ¯ k ) + h r (µ¯ k ).

(95)

Since |h r (µ¯ k )| is bounded from above by n log 2, one immediately obtains the inequality (93). End of the proof of Theorem 7: The final step is to combine Propositions 6 and 7: 1 log 2 (n + 1) log 2 h n (µ¯ k ) ≥ − , for all n < k. n 2 k−1 Taking in (96) first limit k → ∞ and then n → ∞ gives: log 2 HKS (T¯ , µ) , ¯ ≥ 2 which by (62, 60) implies the bound: log 2 HKS (T, µ) ≥ . 2 Since = µ(0) + 2µ(1), this gives the bound (84).

(96)

(97)

Sketch of proof of Theorem 3. All the ingredients of the above construction can be straightforwardly extended from the map T{2,4,4} to a general map T p satisfying Definition 1. Specificaly, given a sequence of eigenstates Uk ψk = eiθk ψk , k = 1, . . . ∞, one k }∞ of eigenstates for quantizations {U θk }∞ of k ∈ H first constructs the sequence { k=1 k=1 the tower map T p . These sets of eigenstates induce then two sequences of related measures (defined as in Eq. (81)): {µk }∞ ¯ k }∞ k=1 , {µ k=1 on I . Assuming that in the semiclassical limit µk ’s converge to an invariant measure µ of T p , the sequence of the measures µ¯ k must converge to the measure µ¯ which remains invariant under the action of the corresponding uniformly expanding map T¯ p . Repeating the same steps as in the proof of Proposition 6 it is strightforward to get the metric bound on the measures µ¯ k : k + 1 − lt h k+1−lt (µ¯ k ) ≥ (98) + 1 − lt log p. 2 Furtheremore, the subadditivity property of h n together with Eq. (98) then imply that the limiting measure µ¯ = limk→∞ µ¯ k must satisfy the bound: log p . HKS (T¯ p , µ) ¯ ≥ 2 Since the metric entropies HKS (T¯ p , µ), ¯ HKS (T p , µ) are connected to each other by Eq. (66) one immediately gets log p HKS (T p , µ) ≥ , (99) 2 where is as in Eq. (65). Finally, it remains to check that the right side of (99) coincides with the right side of (15).

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

331

8. Explicit Sequences of “Non-ergodic” Eigenstates Below we construct some explicit sequences of eigenstates for maps T¯ p , T p quantized as in Sect. 3. Having such sequences we can calculate the induced semiclassical measures and test the bound (15) for the corresponding metric entropies. 8.1. Maps with uniform slopes. Let us first consider the map T¯ p whose quantization is given by Eq. (20). Note that if U∗ is given by the discrete Fourier transform F p , the evolution operator U¯ k and the corresponding eigenstates are precisely the same as for the Walsh-quantized baker’s map treated in [17]. As we show below, the same construction can be carried out for a general U. Similarly to the case of the Walsh-quantized baker’s map, the resulting semiclassical measures on I turn out to be Bernoulli measures. Let w ∈ H be an eigenstate of U; then it is easy to see that (w)

ψk

(w)

= w ⊗ w ⊗ · · · ⊗ w,

ψk

k

∈ Hk = H ⊗ H ⊗ · · · ⊗ H

(100)

k

is the eigenstate of U¯ k . It is now straightforward to compute the semiclassical measure p−1 (w) µw corresponding to the sequence ψk , k = 1, . . . ∞. Assuming that w = i=0 wi |i, where {|i, i = 0, . . . p−1} is an orthonormal basis in H, the µw -measure of the cylinder set x, x = x1 . . . xm , xi ∈ {0, . . . p − 1} is given by: (w)

(w)

µw (x) = lim ψk Px ψk = k→∞

m

|wxi |2 .

(101)

i=1

As this is a simple product measure (i.e., a measure µ which factorizes: µ(x) = m i=1 µ(xi ) with respect to the corresponding Markov partition), one gets for the metric entropy: HKS (T¯ p , µw ) = −

p−1

|wi |2 log(|wi |2 ).

(102)

i=0

Now let us show that for some quantizations of T¯ p there also exist semiclassical measures which are linear combinations of Bernoulli measures constructed in the previous example. Assume that U is such that there exists an eigenstate w(0) ∈ H of Ud : Ud w (0) = eiθ w (0) for some integer d > 0, while w (0) is not an eigenstate of Ui for any integer i < d. (This, for instance, is possible if Ud = 1 for some d > 1, as in the case: U = F p∗ ). Taking then such normalized state w (0) one can form the corresponding sequence w( j) := e−iθ j/d U j w (0) , j = 0, . . . d − 1 of cyclically related states: e−iθ/d Uw ( j) = w ( j+1 mod d) , with |w ( j) |w (i) | < 1 if i = j. Define w0 := w (0) ⊗ w (1) · · · ⊗ w (d−1) and let w := {w0 , w1 , . . . wd−1 } be the sequence of the states obtained from w0 by the cyclic permutation of its components: w j := w ( j

mod d)

⊗ w (1+ j

mod d)

⊗ · · · ⊗ w (d−1+ j

mod d)

,

j = 0, . . . d − 1.

For each k satisfying: k mod d = 1 one looks for normalized eigenstates of U¯ k in the form (w)

ψk

=

d−1 j=0

(k)

Cj

w j ⊗ w j ⊗ · · · ⊗ w j ⊗w j , (k−1)/d

(w)

ψk

∈ Hk .

(103)

332

B. Gutkin (k)

Equation (103) then defines an eigenstate of U¯ k if all C j are equal to some constant C (k) (w)

fixed by the normalisation condition ψk = 1. (Note that the eigenstates provided by Eq. (100) could be seen as a particular case of (103) when d = 1.) The corresponding semiclassical measure is then given by (w)

(w)

µw (x) = lim ψk Px ψk k→∞ = lim

k→∞

d−1

|C (k) |2 w j ⊗ · · · ⊗ w j Px wi ⊗ · · · ⊗ wi .

i, j=0

Since |w ( j) |w (i) | < 1 if i = j, only the diagonal terms survive the limit k → ∞ and one has: d−1 1 ( j) µw (x), µw (x) = d

( j) µw (x)

j=0

=

m

|wx( kj+k−1

| ,

mod d) 2

(104)

k=1

( j)

where wi is i’s component of the vector w( j) in the basis {|i, i = 0, . . . p − 1}. Although µw is not a simple product measure, it is still possible to calculate the metric entropy by noticing that the product measures entering in (104) are cyclically related to ( j+1 mod d) ( j) each other: µw = T¯ p∗ ◦ µw and remain invariant under the action of (T¯ p )d = T¯ pd . Since the metric entropy is an affine function of the measure, one has: 1 HKS (T¯ p , µw ) = HKS (T¯ pd , µw ) d d−1 d−1 p−1 1 1 ( j) 2 ( j) ( j) = 2 HKS (T¯ pd , µw ) = − |wi | log(|wi |2 ), d d j=0

j=0 i=0

(105) ( j)

where we have used the fact that each µw is a simple product measure and all ( j) HKS (T¯ pd , µw ) are equal to each other. Furthermore, by a simple application of the Uncertainty Entropic Principle one has for all j = 0, . . . d − 1, ( j) ˆ ( j) ) + h(Uw ˆ h(w ) ≥ −2 log(max |Uk,m |) = log p, k,m

ˆ ( j) ) = h(w

p−1

( j)

( j)

|wi |2 log(|wi |2 ).

i=0

As e−iθ j/d Uw ( j) = w ( j+1 mod d) , it follows immediately by (105) that HKS (T¯ p , µw ) ≥ 1 2 log p which is precisely the bound (14) (equivalent to (15) in that case). It is worth mentioning that for U = F p∗ and d = 1 there exist vectors w such that the measures µw saturate this bound [17]. ( j)

Remark 3. Note that if all wi = 0 the measures above are supported on the whole I . As has been shown in [17] in the case when U = F p∗ , it is also possible to construct an entirely different class of exceptional sequences of eigenstates, where parts of the

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

333

corresponding semiclassical measures are localized on some periodic trajectories. This construction uses the fact that U¯ kn = 1 for a “short” time n, meaning the spectrum of U¯ k becomes highly degenerate. Since no such degeneracies are expected for quantized maps with non-uniform slopes, it seems that this type of semiclassical measures can be constructed only for the maps T¯ p . We refer the reader to [15, 17] for the details of the construction.

8.2. Maps with non-uniform slopes. For the maps T p we will look for sequences of eigenstates exactly in the same form (103), as in the uniform case. As we will show, for certain quantizations of T p one can, indeed, construct such sequences by choosing the (k) constants Ci in an appropriate way. Rather than consider a general case, we will provide below several concrete examples of such a construction for the map T2 (see Eq. (25)) whose quantization is given by (26) with some choice for U1 , U2 . In order to calculate the metric entropy of a semiclassical measure µ we need, in general, a knowledge of the measures µ(ε0 . . . εn−1 ), where ε0 . . . εn−1 , εi ∈ {1, 2, 3} are cylinder sets defined with respect to the Markov partition j=1,2,3 I j . On the other hand, because of the eigenstates, structure, it turns out to be easier to calculate the measures µ(x1 . . . xn ) of the “binary” cylinders, where xi ∈ {0, 1} (as defined in Sect. 3). The transition between two representations is straightforward. Given some cylinder ε0 . . . εn−1 in ε-representation its binary representation can be obtained by switching every element εi , i = 0, . . . n − 1 to 0 if εi = 1, to 10 if εi = 2 and to 11 if εi = 3 respectively. (For instance the cylinder 123 in the binary representation becomes 01011.) Thus, in order to obtain the measures of the sets ε0 . . . εn−1 , it is convenient first to transform them into the binary form using the above procedure and then find their measures as in the previous subsection. Example 1. Below we construct a semiclassical measure which is totally concentrated on a part of I , where T2 has a uniform slope. Let U1 = U2 = U be a two-by-two √ matrix satisfying U2 = 1, |U(i, j)| = 1/ 2, e.g., the discrete Fourier transform. Let U|1 =: |e+ . Since U|e+ = |1 it can be easily seen that for even k, (1)

ψk = |1 ⊗ |e+ ⊗ · · · ⊗ |1 ⊗ |e+

(106)

k

(1)

is an eigenstate of Uk . For the sequence of states ψk the induced semiclassical measure µ(1) has entire support on the Cantor set: {x = 0.x1 x2 x3 · · · ∈ I, x2i+1 = 1}. Correspondingly, µ(1) (ε0 . . . εn−1 ) = 1/2n if εi ∈ {2, 3} for all i, and µ(1) (ε0 . . . εn−1 ) = 0 otherwise. The metric entropy of µ(1) can be easily calculated and it is given by: HKS (T2 , µ(1) ) = log 2. This saturates the bound (15) coinciding in that case with (14). Let us mention that since µ(1) is supported on the interval [1/2, 1] (where the slope of T2 is 4), it is also an eigenmeasure of T¯4 map, with the same metric entropy as for T2 . Example 2. Let us show now that for some quantisations of T2 one can construct sequences of eigenstates precisely in the form (100). As a result, the corresponding semiclassical measure µ(2) is given by Eq. (101). Although it remains invariant under the action of both T2 and T¯2 , the metric entropies of µ(2) with respect to T2 and T¯2 turn out to be different.

334

B. Gutkin

Consider the following quantization of T2 . Let U be an arbitrary unitary matrix whose √ elements have modules 1/ 2 and let w be one of its eigenvectors with the eigenvalue eiγ . We now fix U1 , U2 by the conditions U2 = e−iγ U, U1 = U. The state (2)

ψk = w ⊗ w ⊗ · · · ⊗ w, k

is then an eigenstate of Uk . (Note that this example allows a straightforward general(2) ization to all maps T p .) Denote µw the corresponding semiclassical measure. Unlike (2) the previous example, in general, µw is supported over all I . For a given state w = w0 |0 + w1 |1 the measures of the sets ε0 , ε0 = {1, 2, 3} are: ⎧ ⎨ p for ε0 = 1 (2) (2) pq for ε0 = 2 µ(2) w (ε0 ) = lim ψk Pε0 ψk = ⎩ q 2 for ε = 3, k→∞ 0 (2)

where we introduced notation |w0 |2 = p, |w1 |2 = q. Since µw is a simple product measure the corresponding metric entropy is given by: (2) µ(2) HKS (T2 , µ(2) w )=− w (ε0 ) log µw (ε0 ) ε0 ={1,2,3}

= −( p log p + pq log( pq) + q 2 log q 2 ). This should be compared to the bound given by the right side of (15). For the measure (2) µw this bound is equal to 21 (1 + q) log 2. Recall that w is an eigenvector of a unitary matrix whose entries have the same This restricts the possible values of q, √ modules. √ p = 1 − q to the interval [(2 − 2)/4, (2 + 2)/4]. As can be easily checked, for all values of q, p in this interval the strict inequality (15) holds. It is worth to mention (2) that µw is, in fact, an eigenmeasure of the map T¯2 , as well. The corresponding metric entropy HKS (T¯2 , µ(2) w ), however, is given by Eq. (102) and it is obviously different from (2) the expression for HKS (T2 , µw ). Example 3. Unlike two previous examples, here we construct semiclassical measures which somewhat differ from the semiclassical measures obtained for the maps with uniform slopes. Most importantly, we demonstrate that for a certain choice of parameters the bound (15) is actually saturated. To construct such semiclassical measures we are looking for normalized eigenstates of Uk in the form of two state products: (3)

(k)

(1) (2) (1) (2) ⊗ w (2) ⊗ w (1) ⊗ w ψk = C1 w ⊗ · · · ⊗ w ⊗ w

k

(2) + C2(k) w

⊗w

(1)

⊗w

(2)

(1) (2) (1) ⊗w ⊗ · · · ⊗ w ⊗ w .

k

Take U2 = U1 = U,

1 U= √ 2

1 eiα , e−iα −1

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

335

HKS 0.7 0.6 0.5 0.4

0

0.5

1

1.5

Re z 2

Fig. 3. Metric entropy (green, solid line) and the corresponding bound (blue, dashed line) (15) for the semiclassical measure from Example 3 as functions of Re(z) when Im(z) = 0 (k)

and for a given z ∈ C set C1

√ (k) = zC2 , c = 1 + |z 2 − 1|2 ,

√ w (1) = c−1/2 |0 + e−iα (z 2 − 1)|1 ,

√ w (2) = c−1/2 z|0 + e−iα ( 2 − z)|1 .

With such a choice (U)2 = 1 and Uw (1) = w (2) , Uw (2) = w (1) . It is easy to check that ψk(3) is an eigenstate of Uk for any z ∈ C. The resulting semiclassical measure (k)

(3)

µz = (C1 )2 µ1z + (C2 )2 µ2z , Ci = limk→∞ |Ci |, i = 1, 2 is a linear combination of two simple product measures µ1z , µ2z = µ1z −1 defined in Eq. (104). Note that this measure has the same structure as in the case of maps with uniform slopes. However, here the con(2) (2) (1) stants C1 , C2 are not equal, in general. Denote p1 = |w0 |2 , p2 = |w1 |2 , q1 = |w0 |2 , (1) 2 (i) (i) q2 = |w1 | , where w0 , w1 are the first and second component of the vectors w(i) , i = 1, 2. As will be shown below, the metric entropy of µ(3) z can be explicitly calculated and it is given by HKS (T2 , µ(3) z )=−

pk log pk + qk log qk , 2 k=1,2

where = 2(µ([10]) + µ([11])) + µ([0]) = 1 + (C1 )2 p1 + (C2 )2 p2 , |z| . C1 = |z|C2 = # |z|2 + 1 The plot in Fig. 3 shows the metric entropy versus the bound of (15) as functions of the real part of z when Im(z) = 0.

2

log 2 given by right side

Some special cases: 1) |z| = 1. In this case p1 = p2 , q1 = q2 and the resulting measures are of a simple product type. Furthermore, both w(1) and w (2) are the eigenvectors of the same unitary matrix whose elements have equal modulus. Thus one actually, (2) gets the measures of the same type as for one vector product states ψk in Example 2. 2) z = 0, z = ∞. In that case either C2 or C1 vanishes and we get the states considered

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√ √ in Example 1. 3) z = 2 (or z −1 = 2). In such a case p1 = q1 = 1/2, p2 = 0, q2 = 1 √ ) = 2 log 2 saturates the bound (15). and the metric entropy HKS (T2 , µ(3) 3 2 Calculation of metric entropy. Using the same approach as in the last example, one can, in principle, construct d-state product eigenstates of the type (103) for other maps T p . For such a construction it is necessary to define the quantum evolution operator Uk by Eq. (24) with Ui = U, i = 1, . . . n max , where Ud = 1 and the vectors w ( j) , j = 0 . . . d − 1 are cyclically related to each other by U (precisely as in Sect. 8.1). (k) Assume that by an appropriate choice of constants Ci one can obtain a sequence of (w) eigenstates ψk of Uk ’s in the form (103). The corresponding semiclassical measures ( j)

µw then have the form of a linear combination of the components µw from Eq. (104). Let us show now how the metric entropy of µw can be computed. First, note that ψk(w) being an eigenstate of Uk , is in addition, an eigenstate for the operator (U¯ k )d , where U¯ k is the quantization (20) of the map T¯ p with the uniform slope p. Since (U¯ k )d is also a quantization of the map T¯ pd , the semiclassical measure µw and all its components ( j) µw turn out to be invariant under T¯ pd . Furthermore, as has been shown in Sect. 7.2, using measure µw one can construct the corresponding measure µ¯ w invariant under the action of T¯ p . An important observation is that this measure has the same structure as µw . Namely, it is a linear combination of the simple product measures given by Eq. (104): µ¯ w =

d−1 j=0

( j)

α j µw ,

d−1

α j = 1, α j ∈ R.

j=0

It is now easy to calculate HKS (T¯ p , µ¯ w ) using the affineness of the metric entropy and ( j) the fact that HKS (T¯ pd , µw ) is equal to HKS (T¯ pd , µw ) for each j: d−1 1 1 1 ( j) HKS (T¯ p , µ¯ w ) = HKS (T¯ pd , µ¯ w ) = α j HKS (T¯ pd , µw ) = HKS (T¯ pd , µw ). d d d j=0

(107) Using now connection (66) between the metric entropies of µw , µ¯ w and expression (105) for HKS (T¯ pd , µw ) one obtains: d−1 p−1 ( j) 2 ( j) ¯ HKS (T p , µw ) = HKS (T pd , µw ) = − |wi | log(|wi |2 ). d d

(108)

j=0 i=0

Note that as the right side of (15) amounts to log2 p the proof of the AnantharamanNonnenmacher conjecture for the measure µw amounts to the proof of the bound log p d , HKS (T¯ pd , µw ) ≥ 2

(109)

for the uniformly expanding map. This bound has been shown already in Sect. 8.1. Note also that the bound (15) saturates if (109) saturates for the corresponding uniformly expanding map T¯ pd .

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Finally, it is instructive to see how the formula (108) can be understood in an intuitive way. Recall that each point ζ ∈ I can be encoded by sequences ε(ζ ) = ε0 ε1 . . ., εi ∈ {1, . . . l} in accordance with the dynamical (forward) “history” of ζ regarding the action of T p . Furthermore, this sequence generates the set of cylinders: G n (ζ ) := {ε0 . . . εn−1 , n = 1, 2, . . . } corresponding to the “partial histories” of the point evolution. Assuming that µw is ergodic, the Shannon-McMillan-Breiman theorem asserts that for almost every (with respect to µw ) point ζ ∈ I the metric entropy of T p is given by: 1 log µw (G n (ζ )). (110) n Speaking informally, this means that on average the measures of cylinders G n = ε0 . . . εn−1 decay exponentially as functions of n: µw (G n ) ∼ exp(−n HKS (T p , µw )) with the exponent given by HKS (T p , µw ). Analogously, one can argue that HKS (T¯ pd , µw ) determines the measures of G n = x1 . . . xm n in the x-representation: µw (G n ) ∼ exp(− mdn HKS (T¯ pd , µw )). On the other hand, by Birkhoff’s ergodic theorem the lengths of G n in both representations can be easily connected. Namely, for almost every ζ ∈ I (with respect to µw ), HKS (T p , µw ) = − lim

n→∞

lim m n /n =

n→∞

l

n i µw (Ii ) = ,

i=1

where m n , n is the length of G n (ζ ) in the x and ε-representation, respectively. Comparing the asymptotics of µw (G n ) in both representations gives Eq. (108). 9. Conclusions and Outlook In the current paper we have proved the Anantharaman-Nonnenmacher conjecture for the “tensorial” quantizations of the one-dimensional piecewise linear maps T p . It should be emphasized that we deal here with the “tensorial” quantizations mostly for the sake of convenience, as these quantizations allow very explicit treatment. Actually the current method with minimal adjustments can be used to prove the result for general quantizations of the maps T p . On the other hand, the present approach is clearly restricted to the class of maps whose slopes are all powers of an integer p > 1, since only such maps can be represented as first return maps for towers with uniform expansion rates. To prove the conjecture for a more general class of maps (or other chaotic systems) a more flexible tower construction is needed. We believe that such a modified tower construction is in fact possible and can be utilized to prove the result for all quantizable piecewise linear (chaotic) maps T , [33]. Note also that in the recent preprint [34] of G. Riviere a related approach has been used to prove the Anantharaman-Nonnenmacher conjecture for geodesics flows on two-dimensional Riemannian manifolds of non-positive curvature. The present application demonstrates that quantized one-dimensional maps (or similarly à la Walsh quantized asymmetric Baker maps) can be useful as toy models for understanding general features of quantum chaotic systems. On the technical level these models are much simpler than generic chaotic Hamiltonian systems, but still exhibit their most important features. A quite rare opportunity (for chaotic systems) to construct explicit sequences of eigenstates make them potentially useful as test systems. Another possibility is to use one dimensional maps as models for scattering systems. By opening a “gap” in the unit interval one can produce quantized one-dimensional maps with

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an “absorption” (in complete analogy with the open Walsh-Baker maps introduced in [35]). Finally, since we know already that various exceptional semiclassical measures appear for the “tensorial” quantizations of the maps T p it would be of interest to identify an opposite class of quantizations for which quantum unique ergodicity holds i.e, no exceptional sequences of eigenstates are present. Acknowledgements. I would like to thank Andreas Knauf and Christoph Schumacher for numerous fruitful discussions. I am particularly grateful to Stephane Nonnenmacher for reading the preliminary version of the manuscript and very helpful comments. Most of the present work was accomplished during my pleasant stay in Erlangen-Nuremberg University. I am grateful to all my colleagues at the Mathematical Department for the hospitality extended to me. The financial support of the Minerva Foundation and SFB/TR12 of the Deutsche Forschungsgemainschaft is acknowledged.

Appendix A: Quantization of General Maps We have shown in Sect. 3 that the maps T p can be quantized by means of the “tensorial” quantization procedure. Here we discuss how a more general class of maps T , = {1 , . . . l } can be quantized. Let i1 , . . . i , > 1 be the maximal set of different slopes in , i.e., in = im for n = m. Assuming that each slope ik has a multiplicity m k ≥ 1, the Lebesgue measure preservation condition mk = 1, ik

(111)

k=1

imposes certain restrictions on the values of ik , m k . In particular, it is clear that the set ik , k = 1, . . . must have a greatest common divisor p larger than one. This means ¯ k, ik = p

¯ k ∈ N for k ∈ {1, . . . }.

¯ i ’s are relatively prime. Then it follows immediately from (111) Assume now that all ¯ k , m¯ k ∈ N, k ∈ {1, . . . l}, where lk=1 m¯ k = p. that m k ’s are of the form m k = m¯ k We are going now to show that the maps T whose slopes satisfy the above conditions are quantizable. ¯ i , p ∈ N, ¯ i+1 ≥ ¯i Proposition 8. Let T be a map (4) with the slopes i = p ¯ i ’s are relatively prime integers, then T is having multiplicities m i , i ∈ {1, . . . l}. If “quantizable”. Proof. As the first step notice that T can be represented as a composition of the uniformly expanding map T¯ p and the “block diagonal” map TBD , whose slopes are uniform at each block. Lemma 2. Let T be a map as defined above, then T = T¯ p ◦ TBD , where T¯ p (x) = px mod 1 and mj for x ∈ [bi , bi+1 ], bi = . TBD (x) = (i x mod 1) / p + bi , j j

Proof. Straightforward calculation.

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

1

1

339

1

= 0

1/2

1

0

1/2

1

0

1/2

1

Fig. 4. A “generic” map (112) and its decomposition into the uniformly expanding and the “block diagonal” parts

The parameters entering into the definition of TBD have the following simple meaning. m The points bi , bi+1 mark the position of i’s block which is the square of the size jj . Inside of each such block the map TBD acts as a piecewise linear map with the uniform ¯ i. expansion rate Example. To illustrate the above lemma consider as an example the map with the slopes 6 and 4: 6x mod 1 if x ∈ [0, 1/2) T (x) = (112) 4x mod 1 if x ∈ [1/2, 1). As shown in Fig. 4, it can be decomposed into the uniformly expanding map T¯2 = 2x mod 1 and the “block diagonal” map: TBD (x) =

(6x mod 1)/2 if x ∈ [0, 1/2) (4x mod 1)/2 + 1/2 if x ∈ [1/2, 1).

Let us now define a set of partitions Mk of I by setting their sizes Nk . Take N0 = ¯ i , then Nk = (N0 )k for k = 1, . . . ∞. It is clear that these partitions satisfy p li=1 Conditions 1. For each partition Mk denote by B¯ k , BkBD the corresponding evolution operators for the map T¯ p and TBD respectively. Note that both B¯ k and BkBD are quantizable i.e., one can find unitary matrices U¯ k , UkBD satisfying (6). Indeed, this is completely obvious for B¯ k as T¯ p has the uniform slope. Since BkBD is of the block diagonal form, the corresponding quantum evolution UkBD can be defined as the block diagonal matrix of the same structure where each block is quantized with the help of the discrete Fourier transform. Given matrices B¯ k , BkBD , and the quantizations U¯ k , UkBD one can easily construct the transfer operator for the composition map T = T¯ p ◦ TBD and the corresponding quantization. Lemma 3. Let T , Mk be the map and partition as above and let Bk be the corresponding evolution operator, then Bk = BkBD B¯ k and the matrix Uk = U¯ k UkBD satisfies (6). Proof. Straightforward check. From this the proof of the theorem follows immediately.

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It worth mentioning that using the above quantization procedure it is also possible to obtain the tensorial quantization of the maps T p . For instance, the map T{2,4,4} can be decomposed into the uniformly expanding T¯2 and the map TBD consisting of two “blocks” with the slope two and one, respectively. Correspondingly, the tensorial quantization (67) ¯ BD , where U¯ , U BD = P0 + U¯ ∗ U¯ 1 U¯ P1 of T{2,4,4} can be cast in the form: U = UU give the “uniform” and the “block-diagonal” components.

Appendix B: Proof of Eq. (62) Let T , T¯ be as in Sect. 7.2 and T˜ be the corresponding tower map given by (58). From the Markov partition of I : {ε0 , ε0 ∈ {1, 2}} one can easily construct the Markov partition of I˜: {ε0 × {η}, ε0 ∈ {1, 2} and η ∈ {0, 1}}. The corresponding n-times refined (with respect to T˜ ) partition is given then by the set of cylinders: {˜ε, ε˜ = ε˜ 0 . . . ε˜ n−1 }, where ε˜ i = (εi , ηi ), εi ∈ {1, 2} and ηi ∈ {0, 1}}. The metric entropy HKS (T˜ , µ) ˜ is determined by the corresponding limit of the entropy function: ˜ =− h n (µ)

µ(˜ ˜ ε) log µ(˜ ˜ ε).

(113)

|˜ε |=n

For a cylinder ˜ε let ε = π I ˜ε be the corresponding cylinder in I containing exactly the same sequence of ε as in ε˜ . Note that the time evolution of any point ζ˜ ∈ I˜ is completely determined by the sequence ε and the initial level η0 . Therefore, for a given ε there are precisely two non-empty cylinders ˜ε, ˜ε such that π I ˜ε = π I ˜ε = ε. Furthermore, µ(˜ ˜ ε) = −1 µ(ε), µ(˜ ˜ ε ) = −1 µ(1ε) and h n (µ) ˜ can be rewritten as: h n (µ) ˜ = − −1

µ(ε) log µ(ε) −1 + µ(1ε) log µ(1ε) −1 .

|ε|=n

On the other hand, the entropy of the measure µ¯ is given by h n (µ) ¯ = −

−1

µ(ε) ¯ + µ(1ε) ¯ . µ(ε) ¯ + µ(1ε) ¯ log

|ε|=n

It remains to see that two limits limn→∞ h n (µ)/n, ˜ limn→∞ h n (µ)/n ¯ coincide. By the convexity of the entropy function ¯ ≥ h n (µ) ˜ + log 2. h n (µ)

(114)

Since, log(x + y) ≥ log x one also has: ¯ ≤ h n (µ). ˜ h n (µ) From (114, 115) the claim immediately follows.

(115)

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References 1. Bohigas, O.: Random matrix theory and chaotic dynamics. In: Giannoni, M.J., Voros, A., Zinn-Justin, J., eds., Chaos et physique quantique, (École d’été des Houches, Session LII, 1989), Amsterdam: North Holland, 1991 2. Berry, M.V.: Regular and irregular semiclassical wave functions. J. Phys. A 10, 2083–2091 (1977) 3. Voros, A.: Semiclassical ergodicity of quantum eigenstates in the Wigner representation. In: Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Casati, G., Ford, J., eds., Proceedings of the Volta Memorial Conference, Como, Italy, 1977, Lecture Notes in Phys. 93, Berlin: Springer, 1979, pp. 326–333 4. Lazutkin, V.F.: Semiclassical asymptotics of eigenfunctions. In: Partial Differential Equations V, Berlin: Springer, 1999 5. Schnirelman, A.I.: Ergodic properties of eigenfunctions. Usp. Mat. Nauk 29, no. 6 (180), 181–182 (1974) 6. Zelditch, S.: Uniform distribution of the eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987) 7. Colinde Verdière, Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102, 497–502 (1985) 8. Gérard, P., Leichtnam, É.: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71(2), 559–607 (1993) 9. Zworski, M., Zelditch, S.: Ergodicity of eigenfunctions for ergodic billiards. Commun. Math. Phys. 175, 673–682 (1996) 10. Bouzouina, A., De Bièvre, S.: Equipartition of the eigenfunctions of quantized ergodic maps on the torus. Commun. Math. Phys. 178, 83–105 (1996) 11. Helffer, B., Martinez, A., Robert, D.: Ergodicité et limite semi-classique. Commun. Math. Phys. 109, 313–326 (1987) 12. Rudnick, Z., Sarnak, P.: The behavior of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161, 195–213 (1994) 13. Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. Math. 163, 165–219 (2006) 14. Hassell, A.: Ergodic billiards that are not quantum unique ergodic, with an appendix by A. Hassell, L. Hillairet. Preprint (2008) http://arxiv.org/abs/0807.0666v3[math,AP], 2008, to appear in Ann. of Math 15. Faure, F., Nonnenmacher, S., De Bièvre, S.: Scarred eigenstates for quantum cat maps of minimal periods. Commun. Math. Phys. 239, 449–492 (2003) 16. Faure, F., Nonnenmacher, S.: On the maximal scarring for quantum cat map eigenstates. Commun. Math. Phys. 245, 201–214 (2004) 17. Anantharaman, N., Nonnenmacher, S.: Entropy of semiclassical measures of the Walsh-quantized baker’s map. Ann. H. Poincaré 8, 37–74 (2007) 18. Kelmer, D.: Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus. Preprint (2005), http://arxiv.org/abs/math-ph/0510079v5, 2007, to appear in Ann. of Math. 19. Anantharaman, N.: Entropy and the localization of eigenfunctions. Ann. of Math. 168(2), 435–475 (2008) 20. Anantharaman, N., Nonnenmacher, S.: Half–delocalization of eigenfunctions of the Laplacian on an Anosov manifold. Ann. de l’Inst. Fourier 57(7), 2465–2523 (2007) 21. Anantharaman, N., Nonnenmacher, S., Koch, H.: Entropy of eigenfunctions. http://arXiv.org/abs/0704. 1564v1[math-ph], 2007 ˙ 22. Pako´nski, P., Zyczkowski, K., Ku´s, M.: Classical 1D maps, quantum graphs and ensembles of unitary matrices. J. Phys. A 34(43), 9303–9317 (2001) 23. Berkolaiko, G., Keating, J.K., Smilansky, U.: Quantum Ergodicity for Graphs Related to Interval Maps. Commun. Math. Phys. 273, 137–159 (2007) 24. Zyczkowski, K., Ku´s, M., Słomczy´nski, W., Sommers, H.-J.: Random unistochastic matrices. J. Phys. A 36(12), 3425–3450 (2003) 25. Keller, G.: Equilibrium States in Ergodic Theory. London Mathematical Society Student Texts 42 Cambridge: Cambridge University Press, 1998 26. De Bièvre, S.: Quantum chaos: a brief first visit. In: Second Summer School in Analysis and Mathematical Physics (Cuernavaca, 2000), Vol. 289 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2001, pp. 161–218 27. Deutsch, D.: Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631–633 (1983) 28. Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987) 29. Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103–1106 (1988) 30. Baladi, V.: Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics, Vol 16, Singapore: World Scientific, 2000

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Commun. Math. Phys. 294, 343–352 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0967-1

Communications in

Mathematical Physics

The Vanishing of Two-Point Functions for Three-Loop Superstring Scattering Amplitudes Samuel Grushevsky1 , Riccardo Salvati Manni2 1 Mathematics Department, Princeton University, Fine Hall, Washington Road,

Princeton, NJ 08544, USA. E-mail: [email protected]

2 Dipartimento di Matematica, Università “La Sapienza”,

Piazzale A. Moro 2, Roma, I 00185, Italy. E-mail: [email protected] Received: 16 June 2008 / Accepted: 18 December 2008 Published online: 12 December 2009 – © Springer-Verlag 2009

Abstract: In this paper we show that the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen [2] vanishes. Our proof uses the reformulation of the ansatz given in [8], theta functions, and specifically the theory of the 00 linear system, introduced by van Geemen and van der Geer [6], on Jacobians. At the two-loop level, where the amplitudes were computed by D’Hoker and Phong [11–14,17,18], we give a new proof of the vanishing of the two-point function (which was proven by them). We also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera [3,8,25]. 1. Introduction An investigation of the problem of computing the superstring measure explicitly for arbitrary genus of the worldsheet was begun by the work of Green and Schwarz [7], who gave an explicit formula in genus 1 using operator methods. D’Hoker and Phong in a series of papers [11–14] introduced a gauge-fixing procedure and computed from first principles the genus 2 superstring measure, verifying that it satisfied the physical constraints, e.g. the vanishing of the 1,2,3-point functions. They also proposed in [15,16] to search for an ansatz for the superstring measure in arbitrary genus as the product of the bosonic measure and a modular form. The ansatz for three-loop measure in this form was then proposed by Cacciatori, Dalla Piazza, and van Geemen in [2]. The genus g ≤ 3 ansatze were reformulated in terms of syzygetic subspaces by the first author in [8], where an ansatz for general genus was proposed, under the assumption on holomorphicity of certain 2r -roots. Cacciatori, Dalla Piazza, and van Geemen in [3] give the genus 4 ansatz in terms of quadrics in the theta constants. The second author in [25] showed that the proposed ansatz is holomorphic in genus 5. Dalla Piazza and van Geemen in [4] proved the uniqueness of the Research is supported in part by National Science Foundation under the grant DMS-05-55867.

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modular form in genus 3 satisfying the factorization constraints. Morozov in [23] surveyed this work and gave an alternative proof that factorization constraints are satisfied for the ansatz; in [24] he has also investigated the 1,2,3-point functions of the proposed ansatz, proving under certain non-trivial mathematical assumption that they vanish on the hyperelliptic locus. In this paper we use the techniques of theta functions, and especially the 00 sublinear system of the linear system |2| introduced by van Geemen and van der Geer [6] to prove the vanishing of the 2-point function in genus 3. We also obtain a new proof of the vanishing of the 2-point function in genus 2. 2. Notations and Definitions We denote by Ag the moduli space of complex principally polarized abelian varieties (ppav for short) of dimension g, and by Hg the Siegel upper half-space of symmetric complex matrices with positive-definite imaginary part, called period matrices. The space Hg is the universal cover of Ag , with the deck group Sp(2g, Z), so that we have Ag = Hg / Sp(2g, Z) for a certain action of the symplectic group. A function f : Hg → C is called a (scalar) modular form of weight k with respect to a subgroup ⊂ Sp(2g, Z) if f (γ ◦ τ ) = det(Cτ + D)k f (τ )

∀γ ∈ , ∀τ ∈ Hg ,

where C and D are the lower blocks if we write γ as four g × g blocks. For a period matrix τ ∈ Hg the principal polarization τ on the abelian variety Aτ := Cg /(Zg + τ Zg ) is the divisor of the theta function exp(πi(n t τ n + 2n t z)). θ (τ, z) := n∈Zg

Notice that for fixed τ theta is a function of z ∈ Cg , and its automorphy properties under the lattice Zg + τ Zg define the bundle τ . Given a point of order two on Aτ , which can be uniquely represented as τ ε+δ 2 for g ε, δ ∈ Z2 (where Z2 = {0, 1} is the additive group), the associated theta function with characteristic is ε θ (τ, z) := exp(πi((n + ε)t τ (n + ε) + 2(n + ε)t (z + δ)). δ n∈Zg

ε is odd or even depending on whether the scalar product δ ε · δ ∈ Z2 is equal to 1 or 0, respectively. The theta function with characteristic is the generator of the space of sections of the bundle τ + τ ε+δ 2 (where we have implicitly identified the principally polarized abelian variety with its dual, and think of points as bundles of degree 0). Thus the square of any theta function with characteristic is a section of 2τ , and the basis for the space of sections of this bundle is given by theta functions of the second order ε (2τ, 2z) [ε](τ, z) := θ 0 As a function of z, θ

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g

for all ε ∈ Z2 . Riemann’s addition formula is an explicit expression of the squares of theta functions with characteristics in this basis: ε θ (τ, z)2 = (−1)δ·σ [σ ](τ, 0)[σ + ε](τ, z). (1) δ g σ ∈Z2

Theta constants are restrictions of theta functions to z = 0; thus all theta constants with odd characteristics vanish identically in τ , while theta constants with even characteristics and all theta constants of the second order do not vanish identically. All theta constants with characteristics are modular forms of weight one half with respect to a certain normal subgroup of finite index (4, 8) ⊂ Sp(2g, Z), while all theta constants of the second order are modular forms of weight one half with respect to a bigger normal subgroup (2, 4) ⊃ (4, 8). Theta constants with characteristics are not algebraically independent, and satisfy a host of algebraic identities, some of which follow from Riemann’s addition formula. However, the theta constants of the second order are algebraically independent for g = 1, 2, and the only relation among them in genus 3 is of degree 16, and has been known classically. It is discussed in detail in [6] — here we give the explicit formula for easy reference. Indeed, a special case of Riemann’s quartic addition theorem in genus 3 is the following identity for theta constants (where we suppress the argument τ ): θ

000 000 000 000 θ θ θ 110 010 100 000 001 001 001 000 000 000 000 001 θ θ . =θ θ θ θ +θ θ 100 000 111 011 101 001 110 010

If we denote the three terms in this relation by ri , so that the relation is r1 = r2 + r3 , then multiplying the 4 “conjugate” relations r1 = ±r2 ± r3 yields the identity F := r14 + r24 + r34 − 2r12 r22 − 2r22 r32 − 2r32 r12 = 0.

(2)

Notice that F is a polynomial of degree 8 in the squares of theta constants with characteristics, and thus by applying Riemann’s addition formula (1) F can be rewritten as a polynomial of degree 16 in theta constants of the second order. We refer to [1,19] for details on theta functions and modular forms, and the current knowledge about the ideal of relations among theta constants of the second order for g > 3 (which is not known completely even for g = 4). 3. The Linear System 00 In this section we review the definition and some facts about the linear system 00 ⊂ |2| introduced and studied in [6]. We refer to that paper for details, as well as to [5,9,20] for a review and results on the importance of the linear system 00 for the Schottky problem of characterizing Jacobians. The linear system 00 ⊂ |2| is defined to consist of all sections vanishing to order at least four at the origin. Since all sections of 2 are even, this is equivalent to the value and all the second derivatives ∂zi ∂z j vanishing at zero. These conditions turn out be

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independent, when (Aτ , ) is an indecomposable ppav (i.e. not isomorphic to a product of lower-dimensional ppavs). In this case the matrix ⎛ ⎞ ](τ,0) ](τ,0) [ε1 ](τ, 0) ∂[ε∂τ111 . . . ∂[ε∂τ1gg ⎜ ⎟ .. .. .. .. ⎜ ⎟ . . . . ⎝ ⎠ ∂[ε2g ](τ,0) ∂[ε2g ](τ,0) g [ε2 ](τ, 0) ... ∂τ11 ∂τgg has rank

g(g+1) 2

+ 1, cf. [26] and thus ([6], Prop. 1.1)

dim 00 = dim |2| − 1 −

1 = 2g − 1 −

1≤i≤ j≤g

g(g + 1) . 2

(3)

Thus the linear system 00 is zero for g ≤ 2, has dimension 1 for g = 3, and higher dimension for all other genera. The above description leads to a simple construction of a basis for the space 00 . Proposition 1. Let τ0 be an irreducible point of Hg (i.e. corresponding to indecomposg able ppav). Denote N := 1 + g(g+1) 2 , and choose ε1 , . . . , ε N ∈ Z2 such that the modular form ⎞ ⎛ ](τ,0) ](τ,0) . . . ∂[ε∂τ1gg [ε1 ](τ, 0) ∂[ε∂τ111 ⎟ ⎜ .. .. .. ⎟ gε1 ,...,ε N (τ ) := det ⎜ . . . ⎠ ⎝ ∂[ε N ](τ,0) ∂[ε N ](τ,0) ... [ε N ](τ, 0) ∂τ11 ∂τgg does not vanish at τ0 . Then the sections ⎛ [ε1 ](τ0 , z) [ε1 ](τ0 , 0) ⎜ .. .. ⎜ ⎜ . . f ε (τ0 , z) := det ⎜ ⎜[ε N ](τ0 , z) [ε N ](τ0 , 0) ⎝ [ε](τ0 , z) [ε](τ0 , 0)

∂[ε1 ](τ0 ,0) ∂τ11

.. .

∂[ε N ](τ0 ,0) ∂τ0 τ11 ∂[ε](τ0 ,0) ∂τ11

... .. . ... ...

∂[ε1 ](τ0 ,0) ∂τgg

⎞

⎟ ⎟ ⎟ , ∂[ε N ](τ0 ,0) ⎟ ⎟ ∂τgg ⎠ ∂[ε](τ0 ,0) ∂τgg

g

for ε ∈ Z2 \ {ε1 , . . . , ε N } form a basis of 00 ⊂ |2τ0 |. Proof. The proof is a simple linear algebra argument that we recall for completeness. First note that f ε (τ0 , z) belongs to 00 , as the determinant and all the second z-derivatives (equal to the first τ -derivatives by the heat equation) vanish for z = 0, as two of the columns of the matrix become identical. It thus remains to show that the functions f ε for various ε are linearly independent. Indeed, recall that theta functions of the second order form a basis of sections of 2, and now note that the basis element [ε](τ0 , z) enters only the expression of f ε (τ0 , z), and that with non-zero coefficient gε1 ,...,ε N (τ0 ). Remark 2. It can be shown that on the open sets {gε1 ,...,ε N (τ ) = 0} the coefficients of the basis vectors are in fact modular forms of weight g + 1 + N /2, see [10]. In particular when g = 3 we have a global expression of the unique section f (τ, z) of the space 00 . Remark 3. Observe that if the period matrix τ is decomposable, then the dimension of 00 increases; however, a basis can still be constructed by using the same method.

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There exists another method for constructing elements of 00 — it is described in [6], and is as follows. Suppose I is an algebraic relation among theta constants of the second order (in genus g). This is to say, suppose I ∈ C[x0...0 , . . . , x1...1 ] is a polynomial in 2g variables such that for any τ ∈ Hg we have I ([ε](τ )) = 0. Then the function f I (z) :=

∂I ([0 . . . 0](τ, 0), . . . , [1 . . . 1](τ, 0)) [ε](τ, z) g ∂xε

ε∈Z2

lies in 00 ⊂ |2τ |. Indeed, since I vanishes identically on Hg , by Euler’s formula we have f I (0) = 0. Moreover, by the heat equation 2πi(1 + δ j,k )

∂ I ∂[ε](τ ) ∂2 fI ∂ I ([0 . . . 0], . . . [1 . . . 1]) |z=0 = = , ∂z j ∂z k ∂x ∂τ ∂τ jk ε jk g ε∈Z2

which is zero since I vanishes identically on Hg , and thus its derivative in any direction is also zero. In [6], Prop. 1.2 it is shown that as I ranges over the ideal of relations among theta constants, the functions f I generate the linear system 00 . Since for g ≥ 4 the ideal of algebraic relations among theta constants of the second order is not completely known, for g ≥ 4 this method does not yield a complete description of the basis of 00 . However, the geometry of these relations is intriguing, and this method produces elements of 00 with coefficients algebraic in theta constants, rather than involving their derivatives as well. 4. The Proposed Ansatz for the Superstring Measure An ansatz for the 3-loop superstring measure was proposed in [2]. The reformulation of this ansatz in terms of products of theta constants with characteristics in a syzygetic subspace given in [8] is as follows. For any i = 0 . . . g define (g) Gi

ε (τ ) := δ

2g V ⊂Z2 ; dim V =i

α β

4−i ε+α θ (τ )2 . δ+β

(4)

∈V

Noticethat since any i-dimensional linear subspace contains zero, all products will conε . Since all odd theta constants vanish identically, it is enough to sum over the tain θ δ even cosets of syzygetic i-dimensional subspaces containing [ε, δ], see [8,25]. 2g To simplify notations, we write m := [ε, δ] ∈ Z2 for characteristics and similarly ε . Then the proposed ansatz for the superstring measure is the product write θm := θ δ of the bosonic measure (which is a form on Mg ) and, for any even characteristic m, the expression (g) m :=

g i(i−1) (g) (−1)i 2 2 G i [m] i=0

(5)

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which is a modular form of weight 8 with respect to a subgroup of Sp(2g, Z) conjugate to (1, 2). In particular for genus 3 we have (3)

(3)

(3)

(3)

(3) m := G 0 [m] − G 1 [m] + 2G 2 [m] − 8G 3 [m].

(3) In [25] it is shown that the sum m m is a non-zero multiple of the modular form F given by (2), and thus vanishes identically on H3 . (g) (g) From definition (4) of the summands G i [m] of the measure m it follows that (g) G i [m] is a polynomial in the squares of theta constants with characteristics for i ≤ 3, divisible by θm2 (τ ). Since this is the only kind of summands appearing in the definition (g) of m for g ≤ 3, by applying Riemann’s addition formula (1) we get (g)

(g)

Proposition 4. For g ≤ 3 the modular form m defined by (5) and the ratio m /θm2 (τ, 0) are both polynomials in theta constants of the second order, of degrees 16 and 14, respectively. 5. The Vanishing of the 2-Point Function We recall (see [12] for explicit formulas) that the vanishing of the cosmological constant

(g) reduces to the identity m m = 0 (proven for the proposed ansatz for g ≤ 4 in [25]), and this also implies the vanishing of the 1-point function, while as shown in [18] the vanishing of the two-point function is equivalent to the vanishing of 2 (g) m Sm (a, b) m

for any points a, b on the Riemann surface (thought of as embedded into its Jacobian), where Sm is the Szeg˝o kernel Sm (a, b) :=

θm (a − b) , θm (0)E(a, b)

with E being the prime form on the Riemann surface. Since the prime form does not depend on m, it is a common factor in all summands above, and thus does not matter for the vanishing of the 2-point function, so the vanishing of the 2-point function is equivalent to the vanishing of X 2 (a, b) :=

(g) m (τ ) θ (τ, a − b)2 , 2 (τ, 0) m θ m m

where τ is the period matrix of the Jacobian J ac(C) of a Riemann surface C, and a, b ∈ C ⊂ J ac(C) are arbitrary. We will now relate the vanishing of the 2-point function and the 00 linear system. Set m (τ ) θ (τ, z)2 (6) X 2 (τ, z) := 2 (τ, 0) m θ m m and note that for τ fixed this function is a section of |2τ |. The vanishing of the 2-point function is then equivalent to X 2 (z) vanishing along the surface C − C ⊂ J ac(C). By Proposition 2.1 in [6] and the subsequent remark, a section of |2| vanishes along the surface C − C if and only if it lies in 00 . We thus get

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Theorem 5. The 2-point function for the proposed superstring measure ansatz vanishes (for genus g) if and only if for the period matrix τ of any Jacobian of a Riemann surface of genus g the section X 2 (τ, z) of 2τ defined above lies in 00 . Since for g = 1, 2 the linear system 00 is zero, the vanishing of the two-point function is equivalent to X 2 (τ, z) vanishing identically in z and τ for g ≤ 2. If we write out X 2 (τ, z) as a linear combination of the basis for sections of 2τ given by theta functions of the second order X 2 (τ, z) = cε (τ )[ε](τ, z), (7) then X 2 vanishes identically if and only if each cε (τ ) vanishes identically. This allows us to recover the result of D’Hoker and Phong in genus 2. Proposition 6. The 2-point function for the proposed superstring ansatz vanishes identically for g ≤ 2. Proof. Let us apply Riemann’s addition formula (1) to the definition (6) of the two-point function to rewrite it in terms of theta functions of the second order (notice that the only term depending on z is θm (τ, z)2 , and we apply the addition formula to it as well). Notice that by Proposition 4 the coefficients cε in (7) obtained in this way are explicit polynomials of degree 15 in theta constants of the second order, and since there are no algebraic relations among theta constants of the second order for g ≤ 2, one needs to verify that all polynomials cε are zero. This can be done on a computer (we used Maple). Note that the computation can be made easier by noting that since X 2 (τ, z) has a transformation formula with respect to the entire symplectic group, (i.e. it is a Jacobi form) and the coefficients cε (τ ) are permuted under the action of a suitable subgroup Sp(2g, Z) that acts monomially on the theta constants of the second order, it is enough to check that just one of cε is the zero polynomial. In the case of g = 3, recall from (3) that the space 00 is one-dimensional, and by the results of [6] we know that it is generated by F2 :=

∂F ([000](τ, 0), . . . , [111](τ, 0)) [ε](τ, z), ∂xε 3

ε∈Z2

where we recall that F, given by (2), is the only polynomial relation of degree 16 among the 8 theta constants of the second order for g = 3. Proposition 7. For any τ ∈ H3 the sections F2 and X 2 of 2τ are proportional; more precisely F2 = − 14 5 X 2. Proof. We have explicit expressions for F2 and X 2 as linear combinations of the basis of the sections of 2 given by the second order theta functions. Thus what we need to verify is that the coefficient in 5F2 + 14X 2 of any [ε](z) is equal to zero. This coefficient is a polynomial of degree 15 in theta constants of the second order and can be verified to be zero using Maple (since the only relation among theta constants of the second order is of degree 16, a polynomial in theta constants of the second order of degree 15 vanishes identically only if it is zero). Notice that by modularity it is again enough to verify that the coefficient of [000](z) in 5F2 + 14X 2 is equal to zero.

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Theorem 8. The 2-point function for the proposed ansatz for the 3-loop superstring measure vanishes identically. Proof. By the above proposition we see that for any τ ∈ H3 the function X 2 , being a constant multiple of F2 , lies in the linear system 00 ⊂ |2τ |. By Theorem 5 this is equivalent to the identical vanishing of the two-point function. Remark 9. We note that the global section F2 is proportional also to the global section f (τ, z). Really we have that F2 (τ, z) = c(τ ) f (τ, z) for any irreducible τ ∈ Hg . Moreover c(τ ) results to be a modular function with respect to Sp(6, Z) that is regular on the set of irreducible point, so it is regular everywhere (modular form) and hence it is a non-zero constant. This identity produces eight nontrivial identities expressing each jacobian determinant gε1 ,...,ε7 (τ ) as a polynomial of degree 15 in the theta constants [σ ](τ, 0). 6. Conclusion There are two generalizations that it is natural to try to prove. First, one could ask whether the vanishing of the 2-point function can be obtained for the proposed in [8] ansatz in higher genera. For genus 4 the ansatz is also given in [3] and is manifestly holomorphic in either formulation. The holomorphicity of the ansatz in genus 5 was proven in [25], and thus it is natural to ask whether the 2-point function vanishes for g ≤ 5. By Theorem 5 we know that this is equivalent to X 2 lying in the linear system 00 . However, already for genus 4 the geometry of the situation is much more complicated: instead of just one relation F in genus 3 the ideal of relations among theta constants of the second order in genus 4 is unknown, and an explicit basis for 00 is unknown for g = 4. Moreover, it could be that here the fact that we are working on the moduli space of curves M4 rather than A4 plays a role — the geometry of 00 depends on this, see [20]. Second, one could try to prove the vanishing of the 3-point function. As shown in [18] for genus 2, this is equivalent to proving that the sum (g) m Sm (a, b)Sm (b, c)Sm (c, a) m

vanishes. Using the explicit formula for the Szeg˝o kernel and canceling the m-independent factor, this is equivalent to the function X 3 (a, b, c) :=

m (τ ) θm (a − b)θm (b − c)θm (c − a) θm3 (τ, 0) m

vanishing identically for a, b, c ∈ C. However, in this case we do not know a natural function on J ac(C)×n of which X 3 is a restriction, and there is no analog of the theory of the 00 for more points. It seems that the identity among the third order theta functions obtained by Krichever in his proof of the trisecant conjecture ([21], formula (1.18)) may potentially be useful in relating the 3-point and 2-point functions, but so far we have not been able to find an explicit way to do this.

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Added in proof. The recent paper of Matone and Volpato [22], written after the current manuscript was submitted for publication, uses other identities relating theta functions and abelian differentials to show that the expression X 3 above in fact does not vanish identically in genus 3. Note, however, that as shown by D’Hoker and Phong in their series of papers, the full expression for N -point function involves many more terms resulting from gauge-fixing, and thus the result of [22] suggests that these terms have a non-zero contribution starting from the 3-point function in genus 3. Acknowledgements. We are grateful to Eric D’Hoker and Duong Phong for introducing us to questions about the superstring scattering amplitudes and explanations regarding the conjectured properties of N -point functions. The computations for this paper were done using Maplesoft’s Maple© software. We would like to thank the referee.

References 1. Birkenhake, Ch., Lange, H.: Complex abelian varieties: second, augmented edition. Grundlehren der mathematischen Wissenschaften 302, Berlin: Springer-Verlag, 2004 2. Cacciatori, S.L., Dalla Piazza, F., van Geemen, B.: Modular forms and three loop superstring amplitudes. Nucl. Phys. B. 800, 565–590 (2008) 3. Cacciatori, S.L., Dalla Piazza, F., van Geemen, B.: Genus four superstring measures. Lett. Math. Phys. 85, 185–193 (2008) 4. Dalla Piazza, F., van Geemen, B.: Siegel modular forms and finite symplectic groups. http://arXiv.org/ abs/0804.3769v2[math.AG], 2008 5. van Geemen, B.: The Schottky problem and second order theta functions. In: Taller de variedades abelianas y funciones theta, Sociedad Matemática Mexicana, Aportaciones Matemáticas, Investigación 13, 41–84 (1998) 6. van Geemen, B., van der Geer, G.: Kummer varieties and the moduli spaces of abelian varieties. Amer. J. of Math. 108, 615–642 (1986) 7. Green, M.B., Schwarz, J.H.: Supersymmetrical string theories. Phys. Lett. B 109, 444–448 (1982) 8. Grushevsky, S.: Superstring scattering amplitudes in higher genus. Commun. Math. Phys. 287, 749–767 (2009) 9. Grushevsky, S.: A special case of the 00 conjecture. http://arXiv.org/abs/0804.0525v2[math.AG], 2009 10. Grushevsky, S., Salvati Manni, R.: Two generalizations of Jacobi’s derivative formula. Math. Res. Lett. 12(5-6), 921–932 (2005) 11. D’Hoker, E., Phong, D.H.: Two-loop superstrings I, main formulas. Phys. Lett. B 529, 241–255 (2002) 12. D’Hoker, E., Phong, D.H.: Two-loop superstrings II, the chiral measure on moduli space. Nucl. Phys. B 636, 3–60 (2002) 13. D’Hoker, E., Phong, D.H.: Two-loop superstrings III, slice independence and absence of ambiguities. Nucl. Phys. B 636, 61–79 (2002) 14. D’Hoker, E., Phong, D.H.: Two-loop superstrings IV, The cosmological constant and modular forms. Nucl. Phys. B 639, 129–181 (2002) 15. D’Hoker, E., Phong, D.H.: Asyzygies, modular forms, and the superstring measure I. Nucl. Phys. B 710, 58–82 (2005) 16. D’Hoker, E., Phong, D.H.: Asyzygies, modular forms, and the superstring measure. II. Nucl. Phys. B 710, 83–116 (2005) 17. D’Hoker, E., Phong, D.H.: Two-loop superstrings V, Gauge slice independence of the N-point function. Nucl. Phys. B 715, 91–119 (2005) 18. D’Hoker, E., Phong, D.H.: Two-loop superstrings VI, non-renormalization theorems and the 4-point function. Nucl. Phys. B 715, 3–90 (2005) 19. Igusa, J.-I.: Theta functions. Die Grundlehren der mathematischen Wissenschaften, Band 194. New YorkHeidelberg: Springer-Verlag, 1972 20. Izadi, E.: The geometric structure of A4 , the structure of the Prym map, double solids and 00 -divisors. J. Reine Angew. Math. 462, 93–158 (1995) 21. Krichever, I.: Characterizing Jacobians via trisecants of the Kummer Variety. Ann. of Math., to appear, http://arXiv.org/abs/math/0605625v4[math.AG], 2008 22. Matone, M., Volpato, R.: Superstring measure and non-renormalization of the three-point amplitude. Nucl. Phys. B 806, 735–747 (2009)

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23. Morozov, A.: NSR superstring measures revisited. JHEP 0805, 086 (2008) 24. Morozov, A.: NSR measures on hyperelliptic locus and non-renormalization of 1,2,3-point functions. Phys. Lett. B 664, 116–122 (2008) 25. Salvati Manni, R.: Remarks on Superstring amplitudes in higher genus. Nucl. Phys. B 801, 163–173 (2008) 26. Sasaki, R.: Modular forms vanishing at the reducible points of the Siegel upper-half space. J. Reine Angew. Math. 345, 111–121 (1983) Communicated by N.A. Nekrasov

Commun. Math. Phys. 294, 353–388 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0941-y

Communications in

Mathematical Physics

Escape Rates and Physically Relevant Measures for Billiards with Small Holes Mark Demers1, , Paul Wright2, , Lai-Sang Young3, 1 Department of Mathematics and Computer Science, Fairfield University,

Fairfield, USA. E-mail: [email protected]

2 Department of Mathematics, University of Maryland,

College Park, USA. E-mail: [email protected]

3 Courant Institute of Mathematical Sciences, New York University,

New York, USA. E-mail: [email protected] Received: 4 November 2008 / Accepted: 3 April 2009 Published online: 25 November 2009 – © Springer-Verlag 2009

Abstract: We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map. This paper is about leaky dynamical systems, or dynamical systems with holes. Consider a dynamical system defined by a map or a flow on a phase space M, and let H ⊂ M be a hole through which orbits escape, that is to say, once an orbit enters H , we stop considering it from that point on. Starting from an initial probability distribution µ0 on M, mass will leak out of the system as it evolves. Let µn denote the distribution remaining at time n. The most basic question one can ask about a leaky system is its rate of escape, i.e. whether µn (M) ∼ ϑ n for some ϑ. Another important question concerns the nature of the remaining distribution. One way to formulate that is to normalize µn , and to inquire about properties of µn /µn (M) as n tends to infinity. Such limiting distributions, when they exist, are not invariant; they are conditionally invariant, meaning they are invariant up to a normalization. Comparisons of systems with small holes with the corresponding closed systems, i.e. systems for which the holes have been plugged, are also natural. These are some of the questions we will address in this paper. We do not consider these questions in the abstract, however; for a review paper in this direction, see [DY]. Our context here is that of billiard systems with small holes.

This research is partially supported by NSF grant DMS-0801139. This research is partially supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. This research is partially supported by a grant from the NSF.

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Specifically, we carry out our analysis for the collision map of a 2-dimensional periodic Lorentz gas, and expect our results to be extendable to other dispersing billiards. Our holes are “physical” holes, in the sense that they are derived from holes in the physical domain of the system, i.e., the billiard table: we consider both convex holes away from the scatterers and holes that live on the boundaries of the scatterers. The holes considered in this paper are very small, but their placements are immaterial. For these leaky systems, we prove that there is a common rate of escape and a common limiting distribution for a large class of natural initial distributions including those with densities with respect to Liouville measure. These conditionally invariant measures, therefore, can be viewed as characteristic of the leaky systems in question, in a way that is analogous to physical measures or SRB measures for closed systems. We show, in fact, that as hole size tends to zero, these measures tend to the natural invariant measure of the corresponding closed billiard system. Our proof involves constructing a Markov tower extension with a special property over the billiard map, the new requirement being that it respects the hole. Let us backtrack a little for readers not already familiar with these ideas: In much the same way that Markov partitions have proved to be very useful in the study of Anosov and Axiom A diffeomorphisms, it was shown, beginning with [Y] and continued in a number of other papers, that many systems with sufficiently strong hyperbolic properties (but which are not necessarily uniformly hyperbolic) admit countable Markov extensions. Roughly speaking, these extensions behave like countable state Markov chains “with nonlinearity”; they have considerably simpler structures than the original dynamical system. The idea behind this work is that escape dynamics are much simpler in a Markov setting when the hole corresponds to a collection of “states”; this is what we mean by the Markov extension “respecting the hole.” All this is not for free, however. We pay a price with a somewhat elaborate construction of the tower, and again when we pass the information back to the billiard system, in exchange for having a Markov structure to work with in the treatment of the hole. There are advantages to this route of proof: First, once a Markov extension is constructed for a system, it can be used many times over for entirely different purposes. For the billiard maps studied here, these extensions were constructed in [Y]; our main task is to adapt them to holes. Second, once results on escape dynamics are established on towers, they apply to all Markov extensions. Here, the desired results are already known in a special case, namely expanding towers [BDM]; we need to extend them to the general, hyperbolic setting. What we propose here is a unified, generic approach for dealing with holes in dynamical systems, one that can, in principle, be carried out for all systems that admit Markov towers. Such systems include logistic maps, rank one attractors including the Hénon family, piecewise hyperbolic maps and other dispersing billiards in 2 or more dimensions. Conditionally invariant measures were first introduced in probabilistic settings, namely countable state Markov chains and topological Markov chains, beginning with [V] and more recently in [FKMP and CMS3]. In this setting, such measures are called quasi-stationary distributions and the existence of a Yaglom limit corresponds to the limit µn /µn (M), which we use here to identify a physical conditionally invariant measure for the leaky system. The first works to study deterministic systems with holes took advantage of finite Markov partitions. These include: Expanding maps on Rn with holes which are elements of a finite Markov partition [PY,CMS1,CMS2]; Smale horseshoes [C1,C2]; Anosov diffeomorphisms [CM1,CM2,CMT1,CMT2]; billiards with convex scatterers satisfying a

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non-eclipsing condition [LM,R] and large parameter logistic maps whose critical point maps out of the interval [HY]. In the latter two, the holes are chosen in such a way that the surviving dynamics are uniformly expanding or hyperbolic with Markov partitions. First results which drop Markov requirements on the map include piecewise expanding maps of the interval [BaK,CV,LiM,D1,BDM]; Misiurewicz [D2] and Collet-Eckmann [BDM] maps with generic holes; and piecewise uniformly hyperbolic maps [DL]. The tower construction is used in the one-dimensional studies [D1,D2,BDM]. Typically a restriction on the size of the hole is introduced in order to control the dynamics when a finite Markov partition is absent. General conditions ensuring the existence of conditionally invariant measures are first given in [CMM]. The physical relevance of such measures, however, is unclear without further qualifications. As noted in [DY], under very weak assumptions on the dynamical system, many such measures exist: for any prescribed rate of escape, one can construct infinitely many conditionally invariant densities. This is the reason for the emphasis placed in this paper on the limit µn /µn (M), which identifies a unique, physically relevant conditionally invariant measure. This paper is organized as follows: Our results are formulated in Sect. 1. In Sects. 2 and 3, the geometry of billiard maps and holes are looked at carefully as we modify previous constructions to give a generalized horseshoe that respects the hole. Out of this horseshoe, a Markov tower extension is constructed and results on escape dynamics on it proved; this is carried out in Sects. 4 and 5. These results are passed back to the billiard system in Sect. 6, where the remaining theorems are also proved. 1. Formulation of Results 1.1. Basic definitions. We consider a closed dynamical system defined by a self-map f of a manifold M, and let H ⊂ M be a hole through which orbits escape, i.e., we stop considering an orbit once it enters H . In this paper we are primarily concerned with holes that are open subsets of the phase space; they are not too large and generally not f -invariant. We will refer to the triplet ( f, M, H ) as a leaky system. First we introduce some notation. Let M˚ = M\H . At least to begin with, let us make ˚ : M˚ ∩ f −1 M˚ → M, ˚ and a formal distinction between f and f˚ = f |( M˚ ∩ f −1 M) n ˚ Let η be a probability measure on M. ˚ We define f˚∗ η to write f˚n = f n |( i=0 f −i M). ˚ If η be the measure on M˚ defined by ( f˚∗ η)(A) = η( f˚−1 A) for each Borel set A ⊂ M. (n) n n ˚ ˚ ˚ is an initial distribution on M, then η := f ∗ η/| f ∗ η| is the normalized distribution of points remaining in M˚ after n units of time. Given an initial distribution η, the most basic question is the rate at which mass is leaked out of the system. We define the escape rate starting from η to be − log ϑ(η), where n 1 −i ˚ log ϑ(η) = lim log η assuming such a limit exists. f M n→∞ n i=0

Another basic object is the limiting distribution η(∞) defined to be η(∞) = limn→∞ η(n) if this weak limit exists. Of particular interest is when there is a number ϑ∗ and a probability measure µ∗ with the property that for all η in a large class of natural initial distributions (such as those having densities with respect to Lebesgue measure), we have ϑ(η) = ϑ∗ and η(∞) = µ∗ . In such a situation, µ∗ can be thought of as a physical measure for the leaky system ( f, M, H ), in analogy with the idea of physical measures for closed systems.

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A Borel probability measure η on M is said to be conditionally invariant if it satisfies f˚∗ η = ϑη for some ϑ ∈ (0, 1]. Clearly, the escape rate of a conditionally invariant measure η is well defined and is equal to − log ϑ. Most leaky dynamical systems admit many conditionally invariant measures; see [DY]. In particular, limiting distributions, when they exist, are often conditionally invariant; they are among the more important conditionally invariant measures from an observational point of view. Finally, when a physical measure η for a leaky system ( f, M, H ) has absolutely continuous conditional measures on the unstable manifolds of the underlying closed system ( f, M), we will call it an SRB measure for the leaky system, in analogy with the idea of SRB measures for closed systems. 1.2. Setting of present work. The underlying closed dynamical system here is the billiard map associated with a 2-dimensional periodic Lorentz gas. Let {i : i = 1, · · · , d} be pairwise disjoint C 3 simply-connected curves on T2 with strictly positive curvature, and consider the billiard flow on the “table” X = T2 \ i {interiori }. We assume the “finite horizon” condition, which imposes an upper bound on the number of consecutive tangential collisions with ∪i . The phase space of the unit-speed billiard flow is M = (X × S1 )/ ∼ with suitable identifications at the boundary. Let M = ∪i i × [− π2 , π2 ] ⊂ M be the cross-section to the billiard flow corresponding to collision with the scatterers, and let f : M → M be the Poincaré map. The coordinates on M are denoted by (r, ϕ), where r ∈ ∪i is parametrized by arc length and ϕ is the angle a unit tangent vector at r makes with the normal pointing into the domain X . We denote by ν the invariant probability measure induced on M by Liouville measure on M, i.e., dν = c cos ϕdr dϕ, where c is the normalizing constant. We consider the following two types of holes: Holes of Type I. In the table X , a hole σ of this type is an open interval in the boundary of a scatterer. When q0 ∈ ∪i , we refer to {q0 } as an infinitesimal hole, and let h (q0 ) denote the collection of all open intervals σ ⊂ ∪i in the h-neighborhood of q0 . A hole σ in X of this type corresponds to a set Hσ ⊂ M of the form (a, b) × [− π2 , π2 ]. Holes of Type II. A hole σ of this type is an open convex subset of X away from ∪i i and bounded by a C 3 simple closed curve with strictly positive curvature. As above, we regard {q0 } ⊂ X \ ∪i as an infinitesimal hole, and use h (q0 ) to denote the set of all σ in the h-neighborhood of q0 . In this case, σ ⊂ X does not correspond directly to a set in M. Rather, σ corresponds directly to a set in M, the phase space for the billiard flow, and we must make a choice as to which set in the cross section M will represent the hole for the billiard map. There is a well defined set Bσ ⊂ M consisting of all (r, ϕ) whose trajectories under the billiard flow on M will enter σ × S1 before reaching M again. Thus Hσ = f (Bσ ) is a natural candidate for the hole in M representing σ , and will be taken as such in this work. However, it would also have been possible to take Bσ as the representative set. The geometry of Bσ and Hσ in phase space will be discussed in detail in Sect. 3.1. Also, we note that the requirement that ∂σ be a C 3 simple closed curve with strictly positive curvature can be considerably relaxed. It is even possible to allow some holes σ that are not convex. See the remark at the end of Sect. 3.1. 1.3. Statement of results. Let G = G(Hσ ) denote the set of finite Borel measures η on M that are absolutely continuous with respect to ν with dη/dν being (i) Lipschitz on ∞ f −i M. ˚ Notice that each connected component of M and (ii) strictly positive on ∩i=0 measures on M with Lipschitz dη/dν correspond to measures on M having a Lipschitz density with respect to Liouville measure.

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Standing hypotheses for Theorems 1–3. We assume (1) f : M → M is the billiard map defined in Sect. 1.2, (2) {q0 } is an infinitesimal hole of either Type I or Type II, and (3) σ ∈ h (q0 ), where h > 0 is assumed to be sufficiently small. Theorem 1 (Common escape rate). All initial distributions η ∈ G have a common escape rate − log ϑ∗ for some ϑ∗ < 1; more precisely, for all η ∈ G, ϑ(η) is well defined and is equal to ϑ∗ . Theorem 2 (Common limiting distribution). (a) For all η ∈ G, the normalized surviving distributions f˚∗n η/| f˚∗n η| converge weakly to a common conditionally invariant distribution µ∗ with ϑ(µ∗ ) = ϑ∗ . (b) In fact, for all η ∈ G, there is a constant c(η) > 0 s.t. ϑ∗−n f˚∗n η converges weakly to c(η)µ∗ . Thus from an observational point of view, − log ϑ∗ is the escape rate and µ∗ the physical measure for the leaky system ( f, M, Hσ ). Theorem 3 (Geometry of limiting distribution). (a) µ∗ is singular with respect to ν; (b) µ∗ has strictly positive conditional densities on local unstable manifolds. The precise meaning of the statement in part (b) of Theorem 3 is that there are countably many “patches” (Vi , µi ), i = 1, 2, . . ., where for each i, (i) Vi ⊂ M is the union of a continuous family of unstable curves {γ u }; (ii) µi is a measure on Vi whose conditional measures on {γ u } have strictly positive densities with respect to the Riemannian measures on γ u ; (iii) µi ≤ µ∗ for each i, and i µi ≥ µ∗ . This justifies viewing µ∗ as the SRB measure for the leaky system ( f, M, Hσ ). Our final result can be interpreted as a kind of stability for the natural invariant measure ν of the billiard map without holes. Theorem 4 (Small-hole limit). We assume (1) and (2) in the Standing Hypotheses above. Let σh ∈ h (q0 ), h > 0, be an arbitrary family of holes, and let − log ϑ∗ (σh ) and µ∗ (σh ) be the escape rate and physical measure for the leaky system ( f, M, Hσh ). Then ϑ∗ (σh ) → 1 and µ∗ (σh ) → ν as h → 0. Some straightforward generalizations: Our proofs continue to hold under the more general conditions below, but we have elected not to discuss them (or to include them formally in the statement of our theorems) because keeping track of an increased number of objects will necessitate more cumbersome notation. 1. Holes. Our results apply to more general classes of holes than those described above. For example, we could fix a finite number of infinitesimal holes {q0 }, . . . , {qk } and consider σ = ∪i σi with σi ∈ h (qi ). In fact, we may take more than one σi in each h (qi ) for as long as the total number of holes is uniformly bounded. See Sect. 3.4 for further generalizations on the types of holes allowed. 2. Initial distributions. Theorems 1 and 2 (and consequently Theorems 3 and 4) remain true with G replaced by a broader class of measures. For example, we use only the Lipschitz property of dη/dν along unstable leaves, and it is sufficient for dη/dν to be strictly positive on large enough open sets (see Remark 6.3). Moreover, dη/dν need not be bounded provided it blows up sufficiently slowly near the singularity set for f .

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Finally, we remark that Theorem 2(b) continues to hold without requiring that dη/dν be strictly positive anywhere, except that now c(η) might be 0. 2. Relevant Dynamical Structures Our plan is to show that the billiard maps described in Sect. 1.2 admit certain structures called “generalized horseshoes” which can be arranged to “respect the holes.” The main results are summarized in Proposition 2.2 in Sect. 2.2 and proved in Sect. 3. 2.1. Generalized horseshoes. We begin by recalling the idea of a horseshoe with infinitely many branches and variable return times introduced in [Y] for general dynamical systems without holes. These objects will be referred to in this paper as “generalized horseshoes”. Following the notation in Sect. 1.1 of [Y], we consider a smooth or piecewise smooth invertible map f : M → M, and let µ and µγ denote respectively the Riemannian measure on M and on γ where γ ⊂ M is a submanifold. We say the pair ( , R) defines a generalized horseshoe if (P1)–(P5) below hold (see [Y] for precise formulation): (P1) is a compact subset of M with a hyperbolic product structure, i.e., = (∪ u ) ∩ (∪ s ), where s and u are continuous families of local stable and unstable manifolds, and µγ {γ ∩ } > 0 for every γ ∈ u . (P2) R : → Z+ is a return time function to . Modulo a set of µ-measure zero, is the disjoint union of s-subsets j , j = 1, 2, . . . , with the property that for each j, R| j = R j ∈ Z+ and f R j ( j ) is a u-subset of . There is a notion of separation time s0 (·, ·), depending only on the unstable coordinate, defined for pairs of points in , and there are numbers C > 0 and α < 1 such that the following hold for all x, y ∈ : (P3) For y ∈ γ s (x), d( f n x, f n y) ≤ Cα n for all n ≥ 0. (P4) For y ∈ γ u (x) and 0 ≤ k ≤ n < s0 (x, y), (a) d( f n x, f n y) ≤ Cα s0 (x,y)−n ; n det D f u ( f i x) ≤ Cα s0 (x,y)−n . (b) log i=k det D f u ( f i y)

∞ det D f ( f x) ≤ Cα n for all n ≥ 0. (P5) (a) For y ∈ γ s (x), log i=n det D f u ( f i y) u (b) For γ , γ ∈ , if : γ ∩ → γ ∩ is defined by (x) = γ s (x) ∩ γ , then u

is absolutely continuous and

i

d(−1 ∗ µγ ) (x) dµγ

∞ det D f ( f x) . = i=0 det D f u ( f i x) u

i

The meanings of the last three conditions are as follows: Orbits that have not “separated” are related by local hyperbolic estimates; they also have comparable derivatives. Specifically, (P3) and (P4)(a) are (nonuniform) hyperbolic conditions on orbits starting from . (P4)(b) and (P5) treat more refined properties such as distortion and absolute continuity of s , conditions that are known to hold for C 1+ε hyperbolic systems. We say the generalized horseshoe ( , R) has exponential return times if there exist C0 > 0 and θ0 > 0 such that for all γ ∈ u , µγ {R > n} ≤ C0 θ0n for all n ≥ 0. The setting described above is that of [Y]; it does not involve holes. In this setting, we now identify a set H ⊂ M (to be regarded later as the hole) and introduce a few relevant terminologies. Let ( , R) be a generalized horseshoe for f with ⊂ (M \ H ). We say ( , R) respects H if for every i and every with 0 ≤ ≤ Ri , f ( i ) either does not intersect H or is completely contained in H .

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The following definitions of “mixing” are motivated by Markov-chain considerations: Let s ⊂ be an s-subset. We say s makes a full return to at time n if there are numR +···+Ri j ( s ) ⊂ i j+1 bers i 0 , i 1 , . . . , i k with n = Ri0 + · · · + Rik such that s ⊂ i0 , f i0 n s for j < k, and f ( ) is a u-subset of . (i) We say the horseshoe ( , R) is mixing if there exists N such that for every n ≥ N , some s-subset s (n) makes a full return at time n. (ii) If ( , R) respects H , then when we treat H as a hole, we say the surviving dynamics are mixing if in addition to the condition in (i), we require that f s (n) ∩ H = ∅ for all with 0 ≤ ≤ n. This is equivalent to requiring that s (n) makes a full return to at time n under the dynamics of f˚, where f˚ is the map defined in Sect. 1.1. We note that the mixing of f in the usual sense of ergodic theory does not imply that any generalized horseshoe constructed is necessarily mixing in the sense of the last paragraph, nor does mixing of the horseshoe imply that of its surviving dynamics. 2.2. Main Proposition for billiards with holes. With these general ideas out of the way, we now return to the setting of the present paper. From here on, f : M → M is the billiard map of the 2-D Lorentz gas as in Sect. 1.2. The following result lies at the heart of the approach taken in this paper: Proposition 2.1 (Theorem 6(a) of [Y]). The map f admits a generalized horseshoe with exponential return times. A few more definitions are needed before we are equipped to state our main proposition: We call Q ⊂ M a rectangular region if ∂ Q = ∂ u Q ∪ ∂ s Q, where ∂ u Q consists of two unstable curves and ∂ s Q two stable curves. We let Q( ) denote the smallest rectangular region containing , and define µu ( ) := inf γ ∈ u µγ ( ∩ γ ). Finally, for a generalized horseshoe ( , R) respecting a hole H , we define n( , R; H ) = sup{n ∈ Z+ : no point in falls into H in the first n iterates}. In the rest of this paper, C and α will be the constants in (P3)–(P5) for the closed system f . All notation is as in Sect. 1.2. Proposition 2.2. Given an infinitesimal hole {q0 } of Type I or II, there exist C0 , κ > 0, θ0 ∈ (0, 1), and a rectangular region Q such that for all small enough h we have the following: (a) For each σ ∈ h (q0 ), (i) f admits a generalized horseshoe ( (σ ) , R (σ ) ) respecting Hσ ; (ii) both ( (σ ) , R (σ ) ) and the corresponding surviving dynamics are mixing. (b) All σ ∈ h (q0 ) have the following uniform properties: (i) Q( (σ ) ) ≈ Q 1 , and µu ( (σ ) ) ≥ κ; (ii) µγ {R (σ ) > n} < C0 θ0n for all n ≥ 0; (iii) (P3)–(P5) hold with the constants C and α. ¯ → ∞ as h → 0. Moreover, if n(h) ¯ = inf σ ∈ h (q0 ) n( , R; Hσ ), then n(h) Clarification: 1. Here and in Sect. 3, there is a set, namely Hσ , that is identified to be “the hole,” and a horseshoe is constructed to respect it. Notice that the construction is continued after a set enters Hσ . For reasons to become clear in Sect. 6, we cannot simply disregard those parts of the phase space that lie in the forward images of Hσ . 1 By Q( (σ ) ) ≈ Q, we only wish to convey that both rectangular regions are located in roughly the same region of the phase space, M, and not anything technical in the sense of convergence.

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2. Proposition 2.2 treats only small h, i.e. small holes. The smallness of the holes and the uniformness of the estimates in part (b) are needed for the spectral arguments in Sect. 4 to apply. Without any restriction on h, all the conclusions of Proposition 2.2 remain true except for the following: (a)(ii), where for large holes the surviving dynamics need not be mixing, (b)(i), and (b)(ii), where C0 and θ0 may be σ -dependent. The assertions for large h will be evident from our proofs; no separate arguments will be provided. A proof of Proposition 2.2 will require that we repeat the construction in the proof of Proposition 2.1 – and along the way, to carry out a treatment of holes and related issues. We believe it is more illuminating conceptually (and more efficient in terms of journal pages) to focus on what is new rather than to provide a proof written from scratch. We will, therefore, proceed as follows: The rest of this section contains a review of all the arguments used in the proof of Proposition 2.1, with technical estimates omitted and specific references given in their place. A proof of Proposition 2.2 is given in Sect. 3. There we go through the same arguments point by point, explain where modifications are needed and treat new issues that arise. For readers willing to skip more technical aspects of the analysis not related to holes, we expect that they will get a clear idea of the proof from this paper alone. For readers who wish to see all detail, we ask that they read this proof alongside the papers referenced. 2.3. Outline of construction in [Y]. In this subsection, the setting and notation are both identical to that in Sect. 8 of [Y]. Referring the reader to [Y] for detail, we identify below 7 main ideas that form the crux of the proof of Proposition 2.1. We will point out the use of billiard properties and other geometric facts that may potentially be impacted by the presence of holes. Holes are not discussed explicitly, however, until Sect. 3. Notation and conventions. In [Y], S0 and ∂ M were used interchangeably. Here we use exclusively ∂ M. Clearly, f −1 ∂ M is the discontinuity set of f . (i) u- and s-curves. Invariant cones C u and C s are fixed at each point, and curves all of whose tangent vectors are in C u (resp. C s ) are called u-curves (resp. s-curves). (ii) The p-metric. Euclidean distance on M is denoted by d(·, ·). Unless declared otherwise, distances and derivatives along u- and s-curves are measured with respect to a semi-metric called the p-metric defined by cos ϕdr . These two metrics are 1 related by cp(x, y) ≤ d(x, y) ≤ p(x, y) 2 . By Wδu (x), we refer to the piece of local unstable curve of p-length 2δ centered at x. (P3)–(P5) in Sect. 2.1 hold with respect to the p-metric. See Sect. 8.3 in [Y] for details. (iii) Derivative bounds. With respect to the p-metric, there is a number λ > 1 so that all vectors in C u are expanded by ≥ λ and all vectors in C s contracted by ≤ λ−1 . Furthermore, derivatives at x along u-curves are ∼ d(x, ∂ M)−1 . For purposes of distortion control, homogeneity strips of the form 1 1 π π , k ≥ k0 , Ik = (r, ϕ) : − 2 < ϕ < − 2 k 2 (k + 1)2 are used, with {I−k } defined similarly in a neighborhood of ϕ = − π2 . For convenience, we will refer to M \ (∪|k|≥k0 Ik ) as one of the “Ik ”. Important Geometric Facts (†). The following facts are used many times in the proof: (a) the discontinuity set f −1 ∂ M is the union of a finite number of compact piecewise smooth decreasing curves, each of which stretches from {ϕ = π/2} to {ϕ = −π/2};

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(b) u-curves are uniformly transversal (with angles bounded away from zero) to ∂ M and to f −1 ∂ M. 1. Local stable and unstable manifolds. Only homogeneous local stable and unstable curves are considered. Homogeneity for Wδu , for example, means that for all n ≥ 0, f −n Wδu lies in no more than 3 contiguous Ik . Let δ1 > 0 be a small number to be 1

chosen. We let λ1 = λ 4 , δ = δ14 , and define

for all n ≥ 0}, Bλ+1 ,δ1 = {x ∈ M : d( f n x, ∂ M ∪ f −1 (∂ M)) ≥ δ1 λ−n 1 Bλ−1 ,δ1 = {x ∈ M : d( f −n x, ∂ M ∪ f (∂ M)) ≥ δ1 λ−n for all n ≥ 0}. 1

We require d( f n x, f −1 (∂ M)) ≥ δ1 λ−n 1 to ensure the existence of a local unstable curve through x, while the requirement on d( f n x, ∂ M) is to ensure its homogeneity.2 Similar s (x) is well defined reasons apply to stable curves. Observe that (i) for all x ∈ Bλ+1 ,δ1 , W10δ and homogeneous (this is straightforward since δ δ homogeneous Wloc on both sides of the curves in s . The set , which is defined to be (∪ u ) ∩ (∪ s ), clearly has a hyperbolic product structure. (P5)(b) is standard. This together with the choice of x1 guarantees µγ {γ ∩ } > 0 for γ ∈ u , completing the proof of (P1). A natural definition of separation time for x, y ∈ γ u is as follows: Let [x, y] be the subsegment of γ u connecting x and y. Then f n x and f n y are “not yet separated,” i.e. s0 (x, y) ≥ n, if for all i ≤ n, f i [x, y] is connected and is contained in at most 3 contiguous Ik . With this definition of s0 (·, ·), (P3)–(P5)(a) are checked using previously known billiard estimates. 3. The return map f R : → . We point out that there is some flexibility in choosing the return map f R : Certain conditions have to be met when a return takes place, but when these conditions are met, we are not obligated to call it a return; in particular, R is not necessarily the first time an s-subrectangle of Q u-crosses Q, where Q = Q( ). ˜ n = n \ {R ≤ n}. On ˜ n is a partition P˜ n whose We first define f R on ∞ . Let elements are segments representing distinct trajectories. The rules are different before and after a certain time R1 , a lower bound for which is determined by λ1 , δ1 and the derivative of f .4 2 In fact, provided δ is chosen sufficiently small, one can verify that d( f n x, f −1 (∂ M)) ≥ δ λ−n implies 1 1 1 −(n+1) for all n ≥ 0. This fact, which was not used in [Y], will be used in item 2 that d( f n+1 x, ∂ M) ≥ δ1 λ1

below to simplify our presentation. 3 Later we will impose one further technical condition on the choice of x . See the very end of Sect. 2.4. 1 4 In [Y], properties of R are used in 4 places: (I)(i) in Sect. 3.2, Sublemma 3 in Sect. 7.3, the paragraph 1 following (**) in Sect. 8.4, and a requirement in Sect. 8.3 that stable manifolds pushed forward more than R1 times are sufficiently contracted.

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(a) For n < R1 , P˜ n is constructed from the results of the previous step5 as follows: Let ω ∈ P˜ n−1 , and let ω be a component of ω ∩ n . Inserting cut-points only where necessary, we divide ω into subsegments ωi with the property that f n (ωi ) is homogeneous. These are the elements of P˜ n . No point returns before time R1 . (b) For n ≥ R1 , we proceed as in (a) to obtain ωi . If f n (ωi ) u-crosses the middle of Q with ≥ 1.5δ sticking out on each side, then we declare that R = n on ωi ∩ f −n , and the elements of P˜ n |ωi ∩˜ n are the connected components of ωi \ f −n . Otherwise put ωi ∈ P˜ n as before. This defines R on a subset of ∞ (which we do not know yet has full measure); the defis -curves. nition is extended to the associated s-subset of by making R constant on Wloc −n The s-subsets associated with ωi ∩ f in (b) above are the j in (P2). It remains to check that f R ( j ) is in fact a u-subset of . This is called the “matching of Cantor sets” in [Y] and is a consequence of the fact that ∞ is dynamically defined and that R1 is chosen sufficiently large. It remains to prove that p{R ≥ n} decays exponentially with n. Paragraphs 4, 5 and 6 contain the 3 main ingredients of the proof, with the final count given in 7. 4. Growth of u-curves to “long” segments. This is probably the single most important point, so we include a few more details. We first give the main idea before adapting it to the form it is used. Let ε0 > 0 be a number the significance of which we will explain later. Here we think of a u-curve whose p-length exceeds ε0 > 0 as “long”. Consider a u-curve ω. We introduce a stopping time T on ω as follows. For n = 1, 2, . . ., we divide f n ω into homogeneous segments representing distinguishable trajectories. For x ∈ ω, let T (x) = inf{n > 0 : the segment of f n ω containing f n x has p−length > ε0 }. Lemma 2.3. There exist D1 > 0 and θ1 < 1 such that for any u-curve ω, p(ω \ {T ≤ n}) < D1 θ1n

for all n ≥ 1.

This lemma relies on the following important geometric property of the class of billiards in question. This choice of ε0 > 0 is closely connected to this property: n f −i (∂ M) passing through or (*) ([BSC1], Lemma 8.4) The number of curves in ∪i=1 ending in any one point in M is ≤ K 0 n, where K 0 is a constant depending only on the “table” X . 1 Let α0 := 2 ∞ k=k0 k 2 , where {Ik , |k| ≥ k0 } are the homogeneity strips, and assume 1

that λ−1 + α0 < 1. Choose m large enough that θ1 := (K 0 m + 1) m (λ−1 + α0 ) < 1. u -curve of p-length ≤ ε We may then fix ε0 < δ to be small enough that every Wloc 0 −i has the property that it intersects ≤ K 0 m smooth segments of ∪m 1 f (∂ M), so that the u -curve has ≤ (K m + 1) connected components. f m -image of such a Wloc 0 The proof of Lemma 2.3, which follows [BSC2], goes as follows: Consider a large n, which we may assume is a multiple of m. (Once Lemma 2.3 is proved for multiples of m, the estimate can be extended to intermediate values by enlarging the constant D1 .) 5 In [Y], it was sufficient to allow returns to at times that were multiples of a large fixed integer m. Not only is this not necessary (see Paragraph 4), here it is essential that we avoid such periodic behavior to ensure mixing. Thus we take m = 1 when choosing return times in Paragraph 3. This is the only substantial departure we make from the construction in [Y].

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We label distinguishable trajectories by their Ik -itineraries. Notice that because f i ω is the union of a number of (disconnected) u-curves, it is possible for many distinguishable trajectories to have the same Ik -itinerary. Specifically, by (*), each trajectory of length jm, j ∈ Z+ , gives birth to at most (K 0 m + 1) trajectories of length ( j + 1)m with the same Ik -itinerary. To estimate p(ω \ {T ≤ n}), we assume the worst case scenario, in which the f n -images of subsegments of ω corresponding to all distinguishable trajectories have length ≤ ε0 . We then sum over all possible itineraries using bounds on D f along u-curves in Ik . We now adapt Lemma 2.3 to the form in which it will be used. Let ω = f k ω for some ω ∈ P˜ k in the construction in Paragraph 3. As we continue to evolve ω, f n ω is not just chopped up by the discontinuity set, bits of it that go near f −1 (∂ M) will be lost by intersecting with f k+n k+n , and we need to estimate p(ωn \ {T ≤ n}), where ωn := ω ∩ f k (k+n ) takes into consideration these intersections and T is redefined accordingly. A priori this may require a larger bound than that given in Lemma 2.3: it is conceivable that there are segments that will grow to length ε0 without losing these “bits” but which do not now reach this reference length. We claim that all such segments have been counted, because (i) the deletion procedure does not create new connected components; it merely trims the ends of segments adjacent to cut-points; and (ii) the combinatorics in Lemma 2.1 count all possible itineraries (and not just those that lead to “short” segments). This yields the desired estimate on p(ωn \ {T ≤ n}), which is Sublemma 2 in Sect. 8.4 of [Y]. 5. Growth of “gaps” of . Let ω be the subsegment of some γ u ∈ u connecting the two s-boundaries of Q. We think of this as a return in the construction outlined in Paragraph 3, with the connected components ω of ωc = ω \ being f k -images of elements of P˜ k . We define a stopping time T on ωc by considering one ω at a time and defining on it the stopping time in Paragraph 4. Lemma 2.4. There exist D2 > 0 and θ2 < 1 independent of ω such that p(ωnc \ {T ≤ n}) < D2 θ2n

for all n ≥ 1.

The idea of the proof is as follows. We may identify ω with (see Paragraph 2), so that the collection of ω is precisely the collection of gaps in . We say ω is of generation q if this is the first time a part of ω is removed in the construction of ∞ . There are two separate estimates: (I ) := p(ω ); (I I ) := p(ωn \{T ≤ n}). q>εn gen(ω )=q

q≤εn gen(ω )=q

(I) has exponentially small p-measure: this follows from a comparison of the growth rate of D f along u-curves versus the rate at which these curves get cut (see Paragraph 4). (II) is bounded above by

q≤εn gen(ω )=q

C p(ω ) n−q−1 · D1 θ1 . p( f q−1 ω )

This is obtained by applying the modified version of Lemma 2.3 to f q−1 ω . A lower bound on p( f q−1 ω ) can be estimated as these curves have not been cut by f −1 (∂ M) (though they may have been shortened to maintain homogeneity), reducing the estimate to q gen(ω )=q p(ω ), which is ≤ p(ω).

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6. Return of “long” segments. This concerns the evolution of unstable curves after they have grown “long”, where “long” has the same meaning as in Paragraph 4. The following geometric fact from [BSC2] is used: u -curve ω with p(ω) > ε and (**) Given ε0 > 0, ∃n 0 s.t. for every homogeneous Wloc 0 q every q ≥ n 0 , f ω contains a homogeneous segment which u-crosses the middle half of Q with > 2δ sticking out from each side.

We choose ε0 > 0 as explained in Paragraph 4 above, and apply (**) with q = n 0 to the segments that arise in Paragraphs 4 and 5 when the stopping time T is reached. For example, ω here may be equal to f n ω , where ω is a subsegment of the ω in the last paragraph of Paragraph 4 with T |ω = n. We claim that a fixed fraction of such a segment will make a return within n 0 iterates. To guarantee that, two other facts need to be established: (i) The small bits deleted by intersecting with f n+k n+k before the return still leave a segment which u-crosses the middle half of Q with > 1.5δ sticking out from each side; this is easily checked. (ii) For q ≤ n 0 , ( f q ) is uniformly bounded on f −q -images of homogeneous segments that u-cross Q. This is true because a segment contained in Ik for too large a k cannot grow to length δ in n 0 iterates. 7. Tail estimate of return time. We now prove p{R ≥ n} ≤ C0 θ0n for some θ0 < 1. On , introduce a sequence of stopping times T1 < T2 < · · · as follows: A stopping time T of the type in Paragraph 4 or 5 is initiated on a segment as soon as Tk is reached, and Tk+1 is set equal to Tk + T . In this process, we stop considering points that are lost to deletions or have returned to . The desired bound follows immediately from the following two estimates: (i) There exists ε > 0, D3 ≥ 1, and θ3 < 1 such that p(T[ε n] > n) < D3 θ3n . (ii) There exists ε1 > 0 such that if Tk |ω = n, then p(ω∩{R > n +n 0 }) ≤ (1−ε1 ) p(ω), where n 0 is as in (**) in Paragraph 6. (ii) is explained in Paragraph 6. To prove (i), we let p = [ε n], decompose into sets of the form A(k1 , . . . , k p ) = {x ∈ : T1 (x), . . . , T p (x) are defined with Ti = ki }, apply Lemmas 2.1 and 2.2 to each set and recombine the results. The argument here is combinatorial, and does not use further geometric information about the system. 2.4. Sketch of proof of (**) following [BSC2]. Property (**) is a weaker version of Theorem 3.13 in [BSC2]. We refer the reader to [BSC2] for detail, but include an outline of its proof because a modified version of the argument will be needed in the proof of Proposition 2.2. We omit the proof of the following elementary fact, which relies on the geometry of the discontinuity set including Property (*): Sublemma A. Given any u-curve γ , through µγ -a.e. x ∈ γ passes a homogeneous s (x) for some δ(x) > 0. The analogous statement holds for s-curves. Wδ(x) u -curve as required in (**), the problem is reduced Instead of considering every Wloc to a finite number of “mixing boxes” U1 , U2 , . . . , Uk with the following properties:

(i) U j is a hyperbolic product set defined by (homogeneous) families u (U j ) and s (U j ); located in the middle third of U j is an s-subset U˜ j with ν(U˜ j ) > 0; (ii) ∪ u (U j ) fills up nearly 100% of the measure of Q(U j ); and u -curve ω with p(ω) > ε passes through the middle third of one of the (iii) every Wloc 0 Q(U j ) in the manner shown in Fig. 1 (left).

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Fig. 1. Left: A mixing box U j . Right: The target box U0

That (i) and (ii) can be arranged follows from Sublemma A. That a finite number of U j suffices for (iii) follows from a compactness argument. Next we choose a suitable subset U˜ 0 ⊂ to be used in the mixing. To do that, first pick a hyperbolic product set U0 related to Q( ) as shown in Fig. 1 (right). We require that it meet Q( ) in a set of positive measure, that it sticks out of Q( ) in the u-direction by more than 2δ, and that the curves in u (U0 ) fill up nearly 100% of Q(U0 ). Let 0 > 0 be a small number, and let U˜ 0 ⊂ U0 consist of those density points of U0 ∩ Q( ) with the additional property that if a homogeneous stable curve γ s with p(γ s ) < 0 meets such a point, then p(γ s ∩ U0 )/ p(γ s ) ≈ 1. For 0 small enough, ν(U˜ 0 ) > 0 because the u -curves is absolutely continuous. foliation into Wloc By the mixing property of ( f, ν), there exists n 0 such that for all q ≥ n 0 , ν( f q (U˜ j ) ∩ U˜ 0 ) > 0 for every U˜ j . We may assume also that n 0 is so large that for q ≥ n 0 , if x ∈ U˜ j is such that f q x ∈ U˜ 0 , then p( f q (γ s (x))) < 0 , where γ s (x) is the stable curve in s (U˜ j ) passing through x. Let q ≥ n 0 and j be fixed, and let x ∈ U˜ j be as above. From the high density of unstable curves in both U j and U0 , we are guaranteed that there are two elements γ1u , γ2u ∈ u (U j ) sandwiching the middle third of Q(U j ) such that for each i, a subsegment of γiu containing γ s (x) ∩ γiu is mapped under f q onto some γˆiu ∈ u (U0 ). Let Q ∗ = Q ∗ (q, j) be the u-subrectangle of Q(U0 ) with ∂ u Q ∗ = γˆ1u ∪ γˆ2u . Sublemma B. f −q | Q ∗ is continuous, equivalently, Q ∗ ∩ (∪0 f i (∂ M)) = ∅. q

Sublemma B is an immediate consequence of the geometry of the discontinuity set: q By the choice of x1 in item 2 of Sect. 2.3, Q ∗ ∩∂ M = ∅. Suppose Q ∗ ∩(∪1 f i (∂ M)) = ∅. q i Since ∪1 f (∂ M) is the union of finitely many piecewise smooth (increasing) u-curves each connected component of which stretches from {ϕ = −π/2} to {ϕ = π/2}, and q these curves cannot touch ∂ u Q ∗ , a piecewise smooth segment from ∪1 f i (∂ M) that ∗ s ∗ enters Q through one component of ∂ Q must exit through the other. In particular, it must cross f q γ s (x), which is a contradiction. u -curve with p(ω) > ε . We pick U so that ω passes To prove (**), let ω be a Wloc 0 j through the middle third of U j as in (iii) above. Sublemma B then guarantees that f q (ω∩ f −q Q ∗ ) connects the two components of ∂ s Q ∗ . This completes the proof of (**), except that we have not yet verified that f q (ω ∩ f −q Q ∗ ) is homogeneous. To finish this last point, we modify the above argument as follows: First, we define a u curve γ to be strictly homogeneous if for all n ≥ 0, f −n γ is contained inside one Wloc s curves is defined analogously. The homogeneity strip Ik (n). Strict homogeneity for Wloc conclusions of Sublemma A remain valid if, in its statement, the word “homogeneous” is replaced by “strictly homogeneous.” Thus the mixing boxes U1 , . . . , Uk can be chosen so that their defining families are comprised entirely of strictly homogeneous local manifolds. Furthermore, if x1 is also chosen as a density point of points with sufficiently long strictly homogeneous unstable curves, u (U0 ) can be chosen to be comprised entirely u -curves. Having done this, an argument very similar to of strictly homogeneous Wloc

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the proof of Sublemma B shows that f −i Q ∗ ∩ (∪k ∂ Ik ) = ∅ for 0 ≤ i ≤ q, and this completes the proof of (**). 3. Horseshoes Respecting Holes for Billiard Maps 3.1. Geometry of holes in phase space. We summarize here some relevant geometric properties and explain how we plan to incorporate holes into our horseshoe construction. Holes of Type I. Recall from Sect. 1.2 that for q0 ∈ ∪i and σ ∈ h (q0 ), Hσ ⊂ M is a rectangle of the form (a, b) × [− π2 , π2 ]. We define ∂ Hσ := {a, b} × [− π2 , π2 ], i.e. ∂ Hσ is the boundary of Hσ viewed as a subset of M. It will also be convenient to let H0 ⊂ M denote the vertical line {q0 } × [− π2 , π2 ]. To construct a horseshoe respecting Hσ , it is necessary to view two nearby points as having separated when they lie on opposite sides of ∂ Hσ or on opposite sides of Hσ in M \ Hσ . Thus it is convenient to view f −1 (∂ Hσ ) as part of the discontinuity set of f . For simplicity, consider first the case where q0 does not lie on a line in the table X tangent to more than one scatterer. Then f −1 (∂ Hσ ) is a finite union of pairs of roughly parallel, smooth s-curves. (Recall that s-curves are negatively sloped, with slopes uniformly bounded away from 0 and −∞.) Each of the curves comprising f −1 (∂ Hσ ) begins and ends in ∂ M ∪ f −1 (∂ M), that is to say, the geometric properties of f −1 (∂ Hσ ) ∪ f −1 (∂ M) are similar to those of f −1 (∂ M). Likewise, f (∂ Hσ ) is a finite union of pairs of (increasing) u-curves that begin and end in ∂ M ∪ f (∂ M), and it will be convenient to regard that as part of the discontinuity set of f −1 . Let Nε (·) denote the ε-neighborhood of a set. We will need the following lemma. Lemma 3.1. For each ε > 0 there is an h > 0 such that for each σ ∈ h , Hσ ⊂ Nε (H0 ), f Hσ ⊂ Nε ( f H0 ), and f −1 Hσ ⊂ Nε ( f −1 H0 ). As f is discontinuous, Lemma 3.1 is not immediate. However, it can be easily verified, and we leave the proof to the reader. Points q0 that lie on lines in X with multiple tangencies to scatterers lead to slightly more complicated geometries, and special care is needed when defining what is meant by f H0 and f −1 H0 . For example, consider the case where q0 ∈ 3 lies on a line that

Fig. 2. An infinitesmal hole aligned with multiple tangencies. Left: q0 lies on a line segment in the billiard table X that is tangent to two scatterers. Right: Induced singularity curves in the subset 1 × [−π/2, π/2] of the phase space M

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is tangent to 1 and 2 , but which is not tangent to any other scatterer including 3 . Suppose further that r1 ∈ 1 , r2 ∈ 2 are the points of tangency, that r2 is closer to q0 than r1 is, that no other scatterer touches the line segment [q0 , r1 ], and that 1 and 2 both lie on the same side of [q0 , r1 ]; see Fig. 2 (left). Let σ be a small hole of Type I with q0 ∈ σ . Then in 2 × [−π/2, π/2], f −1 (∂ Hσ ) appears as described above. However, 2 “obstructs” the view of σ from 1 , and so in 1 × [−π/2, π/2], f −1 (Hσ ) is a small triangular region whose three sides are composed of a segment from 1 × {π/2}, a segment from f −1 (2 × {π/2}), and a single segment from f −1 (∂ Hσ ). See Fig. 2 (right). As a consequence, when we write f −1 H0 , we include in this set not just (r2 , π/2), but also f −1 (r2 , π/2) = (r1 , π/2). This is necessary in order for Lemma 3.1 to continue to hold. Aside from such minor modifications, the case of multiple tangencies is no different than when they are not present, and we leave further details to the reader. Holes of Type II. For simplicity, consider first the case where q0 does not lie on a line in the “table” X tangent to more than one scatterer. Recall from Sect. 1.2 that “the hole” Hσ here is taken to be f (Bσ ), where Bσ consists of points in M which enter σ ×S1 under the billiard flow before returning to the section M. As with holes of Type I, we define ∂ Hσ to be the boundary of Hσ viewed as a subset of M. The set Bσ as a subset of M has similar geometric properties as f −1 Hσ for Type I holes, i.e., f −1 (∂ Hσ )\(∂ M ∪ f −1 (∂ M)) consists of pairs of negatively sloped curves ending in ∂ M ∪ f −1 (∂ M). The slopes of these curves are uniformly bounded (independent of σ ) away from −∞ and 0. For the reasons discussed, it will be convenient to view this set as part of the discontinuity set of f . The infinitesimal hole H0 ⊂ M is defined in the natural way, and the analog of Lemma 3.1 can be verified. We will say more about the geometry of Hσ in Sect. 3.3. Points q0 that lie on multiple tangencies lead to slightly more complicated geometries, and special care is needed when defining what is meant by the sets f −1 H0 , H0 , and f H0 as in the case of Type I holes. Further generalizations on holes of Type II: In addition to the generalizations discussed in Sect. 1.3, sufficient conditions on the holes allowed in h for Prop. 2.2 to remain true are the following, as can be seen from our proofs: (1) There exist N and L for which the following hold for all sufficiently small h: (a) f −1 (∂ Hσ ), ∂ Hσ , and f (∂ Hσ ) each consist of no more than N smooth curves, all of which have length no greater than L. (b) For each σ ∈ h , f −1 (∂ Hσ )\(∂ M ∪ f −1 (∂ M)) consists of piecewise smooth, negatively sloped curves (with slopes uniformly bounded away from −∞ and 0), and the end points of these curves must lie on ∂ M ∪ f −1 (∂ M). (2) The analog of Lemma 3.1 holds. Thus it would be permissible to allow a convex hole σ to be in h that did not have a C 3 simple closed curve with strictly positive curvature as its boundary. For example, conditions (a) and (b) above hold if ∂σ is a piecewise C 3 simple closed curve which consists of finitely many smooth segments that are either strictly positively curved or flat. As another generalization, consider the case when any line segment in the table X with its endpoints on two scatterers that passes through the convex hull of σ also intersects σ . Then it is no loss of generality to replace σ by its convex hull. Using this, one can often verify that the set Hσ that arises satisfies properties (a) and (b) above, even if σ is not itself convex. See Fig. 3. In Sect. 3.2, the discussion is for holes of Type I with a single interval deleted. The proof follows mutatis mutandis for holes of Type II, with the necessary minor modifications discussed in Sect. 3.3.

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Fig. 3. Examples of Type II holes that are permissible

3.2. Proof of Proposition 2.2 (for holes of Type I). The idea of the proof is as follows. First we construct a horseshoe ( (0) , R (0) ) with the desired properties for the infinitesimal hole {q0 }. Then we construct ( (σ ) , R (σ ) ) for all σ ∈ h (q0 ), and show that with (σ ) sufficiently close to (0) in a sense to be made precise, ( (σ ) , R (σ ) ) will inherit the desired properties with essentially the same bounds. To ensure that (σ ) can be taken “close enough” to (0) , we decrease the size of the hole, i.e., we let h → 0. Now the constructions of ( (0) , R (0) ) and ( (σ ) , R (σ ) ) are essentially identical. To avoid repeating ourselves more than needed, we will carry out the two constructions simultaneously. It is useful to keep in mind, however, that logically, the case of the infinitesimal hole is treated first, and some of the information so obtained is used to guide the arguments for positive-size holes. As explained in Sect. 3.1, to ensure that the horseshoe respects the hole, it is convenient to include f −1 (∂ Hσ ) as part of the discontinuity set for f . Since Hσ will be viewed as a perturbation of H0 , we include f −1 (H0 ) in this set as well. The following convention will be adopted when we consider a system with hole Hσ : (a) Suppose for definiteness q0 ∈ 1 . The new phase space Mσ is obtained from M by cutting 1 × [− π2 , π2 ] along the lines comprising H0 ∪ ∂ Hσ , splitting it into three connected components. (b) As a consequence, the new discontinuity set of f is f −1 (∂ Mσ ), and the new discontinuity set of f −1 is f (∂ Mσ ). We use the notation “σ = 0” for the infinitesimal hole, so that M0 is obtained from M by cutting along H0 . Notice immediately that this changes the definitions of stable and unstable curves, in the sense that if γ was a stable curve for the system without holes, then γ continues to be a stable curve if and only if (i) γ ∩ ∂ Mσ = ∅, and (ii) f n (γ ) ∩ f −1 ∂ Mσ = ∅ for all n ≥ 0; a similar characterization holds for unstable curves. All objects constructed below will be σ -dependent, but we will suppress mention of σ except where it is necessary. Observe also that the Important Geometric Facts (†) in Sect. 2.3 with f −1 ∂ Mσ instead of f −1 ∂ M as the new discontinuity set remains valid. We now follow sequentially the 7 points outlined in Sect. 2.3 and discuss the modifications needed. These modifications, along with two additional points (8 and 9) form a complete proof of Proposition 2.2. We believe we have prepared ourselves adequately in Sects. 2.3 and 2.4 so that the discussion to follow can be understood on its own, but encourage readers who wish to see proofs complete with all technical detail to read the rest of this section alongside the relevant parts of [Y and BSC2]. The notation within each item below is as in Sect. 2.3.

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369 (σ )±

1. The relationships λ = λ41 and δ = δ14 are as before, and the sets Bλ1 ,δ1 are defined in a manner similar to that in Sect. 2.3. For example, Bλ(σ1 ,δ)+1 = {x ∈ Mσ : d(x, ∂ Mσ ) ≥ δ1 and d( f n x, f −1 ∂ Mσ ) ≥ δ1 λ−n 1 ∀n ≥ 0}. As in Sect. 2.3, the condition on d( f n x, f −1 ∂ Mσ ) is to ensure the existence of stable curves, and the necessity for x to be away from ∂ Mσ is obvious (cf. footnote in item 1 of Sect. 2.3). Properties (i) and (ii) continue to hold for each σ given the geometry of the new discontinuity set. With regard to the choice of δ1 , we let δ1 be as in [Y], (0)+ (0)− and shrink it if necessary to ensure that Bλ1 ,2δ1 ∩ Bλ1 ,2δ1 has positive ν-measure away from f −1 (∂ M0 ) ∪ ∂ M0 ∪ f (∂ M0 ). This is where the sets (σ ) will be located (see Paragraph 2). (σ )± (0)± The following lemma relates Bλ1 ,δ1 and Bλ1 ,δ1 : (σ )±

(0)±

Lemma 3.2. (i) For all σ ∈ h , we have Bλ1 ,δ1 ⊂ Bλ1 ,δ1 . (ii) As h → 0, sup ν(Bλ(0)+ \ Bλ(σ1 ,δ)+1 ) → 0, 1 ,δ1

sup ν(Bλ(0)− \ Bλ(σ1 ,δ)−1 ) → 0. 1 ,δ1

σ ∈ h

σ ∈ h

Proof. (i) follows immediately from ∂ Mσ ⊃ ∂ M0 . As for (ii), let ε > 0 be given. Recall that Nα (·) denotes the α-neighborhood of a set. By Lemma 3.1 we may choose h small enough that for all σ ∈ h , ∂ Hσ ∈ Nε (H0 ) and f −1 (∂ Hσ ) ∈ Nε ( f −1 H0 ). Then if (0)+ (σ )+ x ∈ Bλ1 ,δ1 \ Bλ1 ,δ1 , either x ∈ Nδ1 (∂ Hσ ) \ Nδ1 (H0 ), or x ∈ ∪n≥0 f −n (Nδ1 λ−n ( f −1 ∂ Hσ ) \ Nδ1 λ−n ( f −1 H0 )). 1

1

We estimate the ν-measure of the right side separately for ∪n≥n ε and ∪n 0 is determined by properties of ( (0) , R (0) ) (requirements will appear below, and

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in items 6 and 9). Suffice it to say here that however small δ2 may be, Lemma 3.2 guarantees that this can be done by shrinking h. Once x1(σ ) is chosen, we set = Wδu (x1(σ ) ) and n = {y ∈ : d( f i y, f −1 (∂ Mσ )) ≥ δ1 λ−i 1 for 0 ≤ i ≤ n}. Then the sets ∞ , s , u and (σ ) are constructed as before. (σ ) (0) That Q( (σ ) ) ≈ Q( (0) ) follows immediately from the proximity of x1 to x1 . u (σ ) u (0) Since δ is fixed, µ ( ) ≈ µ ( ) > 0 can be arranged by taking δ2 sufficiently small and using Lemma 3.2 with h sufficiently small. This proves Proposition 2.2(b)(i). With the separation time happening sooner due to the enlarged discontinuity set, (P3)– (P5) remain true with the same C and α for the closed system; in other words, Proposition 2.2(b)(iii) requires no further work. 3. To arrange for mixing properties (not done in [Y]), we will need to delay the return times to by forbidding returns before time R2 for some R2 ≥ R1 determined by ( (0) , R (0) ); see Lemma 3.4. This aside, the construction of f R is as before. The matching of Cantor sets argument should be looked at again since the Cantor sets are different, but the proof goes through as before because the sets are dynamically defined. Notice that for ω ∈ P˜ n , f i ω is either entirely in the hole or outside of the hole, as is i f ( s ), where s is the s-subset of associated with ω, for 0 ≤ i ≤ n; this is a direct consequence of our taking the boundary of the hole into consideration in our definition of the discontinuity set. Together with the fact that is away from ∂ Mσ , it ensures that the generalized horseshoe we are constructing respects the hole. 4. This is where one of the more substantial modifications occurs: Lemma 2.3, which is based largely on the competition between expansion along u-curves and the rate at which they are cut, is clearly affected by the additional cutting due to our enlarged discontinuity set. The condition (*) in Sect. 2.3 must now be replaced by Lemma 3.3. There exists K 1 such that for any m ∈ Z+ , there exists ε0 > 0 with the property that for any u-curve with p(ω) < ε0 , f m (ω) has ≤ (K 1 m 2 + 4) connected components with respect to the enlarged discontinuity set. Proof. Let m ∈ Z+ be given. As in Sect. 2.3, choose ε0 > 0 small enough such that if ω is a u-curve with p(ω) < ε0 , f m (ω) has ≤ (K 0 m + 1) connected components with respect to the original discontinuity set f −1 S0 . Let ω j be the f −m -image of one of these connected components. This means that for 0 ≤ k ≤ m, f k (ω j ) is, in reality, a connected u-curve even though it may not be connected with respect to our enlarged discontinuity set. Since f k ω j is an (increasing) u-curve, it can meet the three vertical lines making up (∂ Hσ ) ∪ H0 in no more than three points. (As the slopes dϕ/dr of u-curves are never less than the curvature of i at r , connected u-curves cannot wrap around the cylinder i × [−π/2, π/2] and meet (∂ Hσ ) ∪ H0 more than once.) Hence the cardinality −k ((∂ H ) ∪ H )} is ≤ 3(m + 1), and as (∂ H ) ∪ H is the additional of {ω j ∩ m σ 0 σ 0 k=0 f set added to ∂ M to create ∂ Mσ , it follows that f m ω has ≤ (K 0 m + 1) · (3(m + 1) + 1) connected components with respect to the enlarged discontinuity set. Using Lemma 3.3, one adapts easily the proof of Lemma 2.3 to the present setup with 1 θ1 = (K 1 m 2 + 4) m (λ−1 + α0 ), where m is chosen large enough so that this number is < 1. The constant D1 depends only on the properties of D f and is unchanged. Hence Lemma 2.3 is valid with D1 and θ1 modified but independent of σ . As in Sect. 2.3, these estimates can then be adapted to estimate p(ωn \{T ≤ n}).

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5. Lemma 2.4 remains valid with modified constants which are independent of σ . Returning to the sketch of the proof provided in Sect. 2.3, we see that both sets of estimates boil down to the geometry of the new discontinuity set and the rates of growth versus cutting, which has been taken care of for the enlarged discontinuity set in Paragraph 4 above. 6. We need to show that there exist n 1 and ε1 > 0 independent of σ such that for every homogeneous u-curve with p-length > ε0 , a fraction ≥ ε1 of ω returns within the next n 1 steps. Before we enlarged the discontinuity set, this property followed from property (**) in Sect. 2.3. We replace (**) here with the following: Lemma 3.4. Given ε0 > 0, provided h and δ2 are sufficiently small, there exists n 1 u -curve ω such that the following holds for each σ ∈ h : for every homogeneous Wloc with p(ω) > ε0 and each q ∈ {n 1 , n 1 + 1}, f q ω contains a homogeneous segment that u-crosses the middle half of Q( (σ ) ) with greater than 2δ sticking out from each side. Once Lemma 3.4 is proved, the fact that a fraction ε1 (independent of σ ) has the desired properties follows from derivative estimates as in Sect. 2.3 and our uniform lower bound on µu ( (σ ) ). The reason we want q to take two consecutive values in the statement of Lemma 3.4 has to do with the mixing property in item 9 below. Proof. Fix ε0 > 0. We first prove the following for the case σ = 0: u -curve ω with p(ω) > (**)’ For σ = 0, there exists n 1 such that any homogeneous Wloc q ε0 and every q ≥ n 1 , f ω contains a homogeneous segment that u-crosses the middle fourth of Q( (0) ) with greater than 4δ sticking out from each side.

The proof of (**)’ is completely analogous to the proof of (**) outlined in Sect. 2.4. Sublemmas A and B continue to hold due to the similar geometry of the discontinuity set. Notice that unlike (**)’, the assertion in Lemma 3.4 is only for q = n 1 and n 1 + 1, so that its proof involves only a finite number of mixing boxes U j and a finite number of iterates. This will be important in the perturbative argument to follow. Consider now σ = 0, and consider a homogeneous unstable curve ω with p(ω) > ε0 . First, ω continues to be an unstable curve with respect to the discontinuity set f −1 ∂ M0 , so by the proof of (**)’, for q ∈ {n 1 , n 1 + 1} and every j, there is a rectangular region Q ∗ = Q ∗ (q, j) such that (i) Q ∗ u-crosses the middle fourth of Q( (0) ) with > 4δ sticking out, (ii) f −q Q ∗ is an s-subrectangle in the middle third of Q(U j ), and (iii) for i = 0, 1, · · · , q, f −i Q ∗ stays clear of f −1 ∂ M0 by some amount. Lemma 3.1 ensures that for h small enough, (iii) continues to hold with f −1 ∂ M0 replaced by f −1 ∂ Mσ . Finally, provided δ2 is small enough, (i) holds for Q( (σ ) ) with > 2δ sticking out on each side. 7. Once steps 4, 5 and 6 have been completed, the argument here is unchanged (as it is largely combinatorial), guaranteeing constants C0 and θ0 independent of σ with p{R ≥ n} ≤ C0 θ0n . This completes the proof of Proposition 2.2(a)(i) and (b)(ii). We have reached the end of the 7 steps outlined in Sect. 2.3. Two items remain: 8. That n(h) ¯ → ∞ as h → 0 is easy: Orbits from (σ ) start away from H0 and cannot −1 approach f H0 faster than a fixed rate. Thus using Lemma 3.1, we can arrange for orbits starting from (σ ) to stay out of Hσ for as long as we wish by taking h small. 9. The mixing of ( (σ ) , R (σ ) ) follows from Lemma 3.5. There exists R2 ≥ R1 (independent of σ ) such that for small enough h, the construction in Step 3 can be modified to give the following:

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(i) no returns are allowed before time R2 , and (ii) at both times R2 and R2 + 1, there are s-subsets of (σ ) making full returns. Proof. Again we first consider the case σ = 0. Here R2 is chosen as follows: Without allowing any returns, let R1 be the smallest time greater than or equal to R1 such that there exists ω ∈ P˜ R1 with p( f R1 ω) > ε0 > 0. With ε0 chosen as before, we take n 1 from Lemma 3.4 and set R2 = R1 + n 1 . Using Lemma 3.4, we find two subsegments ω and ω ⊂ ω such that f R2 ω and f R2 +1 ω are both homogeneous segments that u-cross the middle half of Q( (0) ) with greater than 2δ sticking out from each side. We may suppose that ω and ω are disjoint since f has no fixed points. They give rise to two s-subsets of (0) with the properties in (ii). From time R2 on, returns to (0) are allowed as before. When σ = 0, we follow the same procedure as above to ensure the mixing of ( (σ ) , R (σ ) ). The only concern is that R1 = R1 (σ ) (and hence also R2 = R1 + n 1 ) might not be independent of σ . This is not a problem as the construction above involves only a finite number of steps: With h and δ2 sufficiently small, the elements of P˜ n(σ ) can (0) be defined in such a way that they are in a one-to-one correspondence with those of P˜ n for n ≤ R1 (0). Finally, mixing of the surviving dynamics is ensured by choosing h small enough that n(h) ¯ > R2 + 1. This ensures that the s-subsets s that make full returns at times R2 and R2 + 1 cannot fall into the hole prior to returning. The proof of Proposition 2.2 for holes of Type I is now complete. 3.3. Modifications needed for holes of Type II. The proof for Type II holes is very similar to that for Type I holes. There are, however, some differences due to the more complicated geometry of ∂ Hσ . In the discussion below, we assume the infinitesimal hole {q0 } does not lie on any segment in the table tangent to more than one scatterer. The general situation is left to the reader. From the discussion of the geometry of Type II holes in Sect. 3.1, we see that the Important Geometric Facts (†) in Sect. 2.3 continue to hold with Mσ in the place of M, except that u-curves need not be transversal to the ∂ Hσ ∪ H0 part of ∂ Mσ . Potential problems that may arise are discussed below. The discontinuity set of f , i.e. f −1 ∂ Mσ , has the same geometric properties as before. We now go through the 9 points in Sect. 3.2. No modifications are needed in items 1–3. As expected, item 4 is where the most substantial modifications occur: Modifications in Item 4. Lemma 3.3 is still true as stated, but the geometry is different. In the discussion below related to this lemma, the discontinuity set refers to f −1 ∂ M, not the enlarged discontinuity set f −1 ∂ Mσ , and unstable curves are defined accordingly. For Type I holes, the proof relies on the fact that any (increasing) connected u-curve ω meets ∂ Hσ ∪ H0 , which is the union of three vertical lines, in at most three points. Lemma 3.6. Any unstable curve ω meets ∂ Hσ ∪ H0 in at most three points. Even though Lemma 3.3 is stated for u-curves, we need it only for unstable curves (and the argument here is slightly simpler for unstable curves). Proof. Let us distinguish between two different types of curves that comprise ∂ Hσ : Primary segments, which are the forward images of curves in ∂ Bσ \ f −1 (∂ M),

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Fig. 4. Representative examples of the geometry of Type II holes. Top set: On the left is a configuration on the billiard table X , while on the right is the resulting configuration in the subset 1 × [− π2 , π2 ] of phase space. In this subset, H0 consists of a single primary segment whose endpoints lie on f (2 × { π2 }) and 1 × { π2 }. For σ = 0, ∂ Hσ contains two primary segments and a single secondary segment that lies on f (2 × { π2 }). (Recall that by convention ∂ Hσ does not include subsegments of ∂ M.) Bottom set: The analogous situation when the view of 1 from q0 is obstructed by two scatterers, instead of just one. Observe that now ∂ Hσ contains two secondary segments in 1 × [− π2 , π2 ]. The situation when the view of 1 from q0 is unobstructed by other scatterers is simple and is left to the reader

and secondary segments, which are subsegments of f (∂ M). For examples, see Fig. 4. In general, when q0 does not lie on a line segment with multiple tangencies to the scatterers, secondary segments are absent in H0 , while each component of H0 gives rise to two primary segments in ∂ Hσ for σ = 0. To prove the lemma, observe first that H0 can have no more than one component in any connected component of M \ f (∂ M). Second, ω must also be entirely contained inside one connected component of M \ f (∂ M). This is because unstable curves for f cannot cross the discontinuity set of f −1 . As a consequence, ω also cannot cross any secondary segment as secondary segments of ∂ Hσ are contained in f (∂ M). It remains to show that ω can meet each primary segment in at most one point. Although primary segments are increasing, their tangent vectors lie outside of unstable cones (except at ∂ M where the unstable cone is degenerate). This is because the curves in ∂ Bσ \ f −1 ∂ M are decreasing, while the unstable cones are defined to be the forward images of {0 ≤ dϕ dr ≤ ∞} under D f . Hence primary segments have greater slopes than ω.

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As pointed out in Sect. 2.3, item 4, Lemma 3.3 must be modified to account for the deletions that arise from intersections with forward images of n , and one might be concerned about the absence of uniform estimates on transversality in (†) between ∂ Hσ and unstable curves. This, in fact, is not a problem, because such deletions occur only in neighborhoods of f −1 ∂ Mσ , which are decreasing curves and hence uniformly transversal to u-curves. This completes the modifications associated with item 4. No modifications are required for items 5, 7, 8 and 9. Modifications in Item 6. In the proof of (**)’, the argument needs to be modified, again q due to the difference in geometry: In order to prove that Q ∗ ∩ (∪0 f i (∂ M0 )) = ∅ (Subq i lemma B), in the case of Type I holes we use that ∪1 f (∂ M0 ) is the union of finitely many piecewise smooth increasing curves that stretch from {ϕ = − π2 } to {ϕ = π2 }. For Type II holes, this is not true. However, it can be arranged that Sublemma B will continue to hold as we now explain: First,

q q q−1 ∪1 f i (∂ M0 ) ⊂ (∪1 f i (∂ M)) ∪ (∪0 f i (H0 )) ∪ f q (H0 ). If we write the right side as A ∪ f q (H0 ), then A has the desired geometry, i.e. it is the union of finitely many piecewise smooth increasing curves that stretch from {ϕ = − π2 } to {ϕ = π2 }. Thus the same argument as before shows that this set is disjoint from Q ∗ . One way to ensure that Q ∗ ∩ f q (H0 ) = ∅ is to choose the mixing boxes U j disjoint from H0 , which can easily be arranged given the geometry of primary segments discussed above. This completes the proof of Proposition 2.2 for Type II holes. 4. Escape Dynamics on Markov Towers In this section and the next, we lift the problems from the billiard systems in question to their Markov tower extensions, and solve the problems there. In Sect. 4, we review relevant works and formulate results on towers. Proofs are given in Sect. 5. 4.1. From generalized horseshoes to Markov towers (review). It is shown in [Y] that given a map f : M → M with a generalized horseshoe ( , R) as defined in Sect. 2.1, one can associate a Markov extension F : → which focuses on the return dynamics to (and suppresses details between returns). We first recall some facts about this very general construction, taking the opportunity to introduce some notation. Let = {(x, n) ∈ × N : n < R(x)}, and define F : → as follows: For < R(x) − 1, we let F(x, ) = (x, + 1), and define F(x, R(x) − 1) = ( f R(x) (x), 0). Equivalently, one can view as the disjoint union ∪≥0 , where , the th level of the tower, is a copy of {x ∈ : R(x) > }. This is the representation we will use. There is a natural projection π : → M such that π ◦ F = f ◦ π . In general, π is not one-to-one, but for each ≥ 0, it maps bijectively onto f ( ∩ {R ≥ }). In the construction of ( , R), one usually introduces an increasing sequence of partitions of into s-subsets representing distinguishable itineraries in the first n steps. ˜ .) These partitions (In Sects. 2.3 and 3.2, these partitions were given by P˜ of

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375

induce a partition {, j } of which is finite on each level and and is a (countable) Markov partition for F. We define a separation time s(x, y) ≤ s0 (x, y) by inf{n > 0 : F n x, F n y lie in different , j }. We borrow the following language from ( , R) for use on : For each , j, recall that s (π(, j )) and u (π(, j )) are the stable and unstable families defining the hyperbolic product set π(, j ). We will say γ˜ ⊂ , j is an unstable leaf of , j if π(γ˜ ) = γ ∩ π(, j ) for some γ ∈ u (π(, j )), and use u (, j ) to denote the set of all such γ˜ . Let u () = ∪, j u (, j ) be the set of all unstable leaves of . Stable leaves of , j and the families s (, j ) and s () are defined similarly. Associated with F : → , which we may think of as a “hyperbolic tower”, is its quotient “expanding tower” obtained by collapsing stable leaves to points. Topologically, = /∼, where for x, y ∈ , x ∼ y if and only if y ∈ γ (x) for some γ ∈ s (). Let π : → be the projection defined by ∼, and let F : → be the induced map on satisfying F ◦ π = π ◦ F. We will use the notation = π ( ), , j = π(, j ), and so on. It is shown in [Y] that there is a well defined differential structure on preserved by F. Recall that µγ is the Riemannian measure on γ , and for γ , γ ∈ u ( ), γ ,γ : γ ∩ → γ ∩ is the holonomy map obtained by sliding along stable curves, i.e. γ ,γ (x) = γ s (x) ∩ γ . We introduce the following notation: For x ∈ i ∩ γ , let γ be such that f Ri (γ ) ⊂ γ . Then J u ( f R )(x) = Jm γ ,m γ ( f Ri |(γ ∩ i ))(x) is the Jacobian of f R with respect to the measures m γ and m γ . Lemma 1 of [Y], which we recall below, is key to the differential structure on . Lemma 4.1. There is a function u : → R such that for each γ ∈ u ( ), if m γ is the measure whose density with respect to µγ is eu Iγ ∩ , then we have the following: (1) For all γ , γ ∈ u ( ), (γ ,γ )∗ m γ = m γ . (2) J u ( f R )(x) = J u ( f R )(y) for all y ∈ γ s (x). (3) ∃C1 > 0 (depending on C and α) such that for each i and all x, y ∈ i ∩ γ , u R J ( f )(x) s( f R x, f R y)/2 . J u ( f R )(y) − 1 ≤ C1 α

(1)

1

The properties of u include |u| ≤ C and |u(x) − u(y)| ≤ 4Cα 2 s(x,y) on each γ . (1) and (2) together imply that there is a natural measure m on with respect to which the Jacobian of F, J F, is well defined: First, identify 0 with γ ∩ for any γ ∈ u ( ), and let m|0 be the measure that corresponds to m γ . (1) says that m so R

defined is independent of γ , and (2) says that with respect to m, J F (x) = J u ( f R )(y) for any y ∈ γ s (x). We then extend m to ∪>0 in such a way that J F ≡ 1 on all of −1 \ F ( 0 ). In the rest of Sect. 4.1 we will assume m{R > n} < C0 θ0n for some C0 ≥ 1 and θ0 < 1.6 One of the reasons for passing from the hyperbolic tower to the expanding tower is that the spectral properties of the transfer√operator associated with the latter can be leveraged. We fix β with 1 > β > max{θ0 , α}, and define a symbolic metric on by √ dβ (x, y) = β s(x,y) . Since β > α, Lemma 4.1(3) implies that J F is log-Lipshitz with 6 Our default rule is to use the same symbol for corresponding objects for f, F and F when no ambiguity can arise given context. Thus R is the name of the return time function on , 0 and 0 .

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respect to this metric. A natural function space on is B = {ρ ∈ L 1 ( , m) : ρ < ∞}, where ρ = ρ∞ + ρLip and ρ∞ = sup sup |ρ(x)|β ,

ρLip = sup Lip(ρ| , j )β .

, j x∈ , j

, j

Lip(·) above is with respect to the symbolic metric dβ . The weights β provide the needed contraction from one level to the next, and β > θ0 is needed to maintain exponential tail estimates. 4.2. Towers with Markov holes. Now consider a leaky system ( f, M, H ) as defined in Sect. 2.1, and suppose ( , R) is a generalized horseshoe respecting the hole H . Let F : → be the associated tower map with π : → M, and let H˜ = π −1 (H ). Then (F, , H˜ ) is a leaky system in itself. With the horseshoe respecting H , we have that H˜ is the union of a collection of , j , usually an infinite number of them; we refer to holes of this type as “Markov holes”. The notation H := H˜ ∩ will be used. Projecting and letting H = π ( H˜ ), we obtain the quotient leaky system (F, , H ). Let us say (F, , H˜ ) and (F, , H ) are mixing if the surviving dynamics of the horseshoe that gives rise to these towers are mixing; see Sect. 2.1. ˚ = \ H˜ , we introduce the notation Letting n ˚ = {x ∈ : F i x ∈ n = ∩i=0 F −1 / H˜ for 0 ≤ i ≤ n},

˚ = 0 . Corresponding objects for (F, , H ) are denoted by n . so that in particular 4.2.1. What is known: Spectral properties of expanding towers. Expanding towers (that are not necessarily quotients of hyperbolic towers) with Markov holes were studied in [D1 and BDM]. The following theorem summarizes several results proved in [BDM, Proposition 2.4, Corollary 2.5], under some conditions on the tower that are easily satisfied here. We refer the reader to [BDM] for detail, and state their results in our context of (F, , H ). Let B˚ = {ρ ∈ L 1 ( 0 , m) : ρ < ∞}, where ρ is as above, and let L denote the ˚ i.e., for ρ ∈ B˚ and x ∈ 0 , transfer operator associated with F| 1 defined on B, Lρ(x) = ρ(y)(J F(y))−1 . y∈ 0 ∩F

−1

x

Theorem 4.2 [BDM]. Let (F, , H ) be such that (i) (F, ) has exponential return times and (ii) (F, , H ) is mixing. Assume the following condition on hole size: ≥1

β −(−1) m(H )

β, and it has a unique eigenfunction h ∗ ∈ B˚ with h ∗ dm = 1. In addition, ˚ there exist constants D > 0 and τ < 1 such that for all ρ ∈ B,

Escape Rates and Physically Relevant Measures for Billiards with Small Holes n

ϑ∗−n L ρ − d(ρ)h ∗ ≤ Dρτ n ,

where d(ρ) = lim λ−n n→∞

(2) The eigenvalue ϑ∗ satisfies ϑ∗ > 1 −

1+C1 m( 0 )

≥1 β

377

n

ρ dm < ∞.

−(−1) m(H ).

The spectral property of L as described in Theorem 4.2(1) implies that all ρ except for those in a codimension 1 subspace have d(ρ) = 0. Given the pivotal role played by the base 0 of the tower , one would guess that for a density ρ, if ρ > 0 on 0 , then d(ρ) = 0. A slightly more general condition is given in Corollary 4.3 below. We call , j a surviving element of the tower if some part of , j returns to 0 before entering H . Corollary 4.3 [BDM]. Let ρ ∈ B˚ be a nonnegative function that is > 0 on a surviving , j . Then d(ρ) > 0. 4.2.2. What is desired: Results for hyperbolic towers. Here we formulate a set of results for the hyperbolic tower that connect the results in Sect. 4.2.1 to the stated theorems for billiards. Let G˜ be the class of measures η on with the following properties: (i) η has absolutely continuous conditional measures on unstable leaves; and (ii) π ∗ η = ρdm for some ρ ∈ B˚ with d(ρ) > 0. Let ( (σ ) , R (σ ) ) be a generalized horseshoe with the properties in Proposition 2.2, and let (F, ) be its associated tower. Let n(, H˜ ) := sup{ : H = ∅}, i.e., n(, H˜ ) = n( (σ ) , R (σ ) ; Hσ ) as defined in Sect. 2.2. Theorem 4.4. Assume that n(, H˜ ) is large enough that

β −(−1) m( )

0 such that

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(i) the conditional densities ργ of µ˜ ∗ | with respect to µγ on unstable leaves satisfy C2−1 ϑ∗− ≤ ργ ≤ C2 ϑ∗− ; (ii) µ˜ ∗ (∪>L ) ≤ Kβ −L θ0L ; and (iii) ϑ∗ → 1 as n(, H˜ ) → ∞. 5. Proofs of Theorems on the Tower The following notational abbreviations are used only in this section: – We will sometimes drop the ˜ used to distinguish between objects on M and corresponding objects on ; there can be no ambiguity as long as we restrict ourselves to the towers. ˚ Specifically, F∗n η is to be interpreted as F˚∗n η, and – We will at times drop the ˚ in F. n F ∗ η is to be interpreted the same way. We focus on the stable direction, since that is what lies between Theorem 4.2 and The˚ that are Lipschitz in the stable orem 4.4. The following is a class of test functions on s s s direction. For γ ∈ () and x, y ∈ γ , we denote by d s (x, y) the distance between π(x) and π(y) according to the p-metric, so that d s (F n x, F n y) ≤ λ−n d s (x, y) for some λ > 1 (see Sect. 2.3). Let Fb be the set of bounded, measurable functions on ˚ For ϕ ∈ Fb , we define |ϕ|sLip to be the Lipshitz constant of ϕ restricted to stable . leaves, i.e. |ϕ|sLip =

sup

sup

˚ x,y∈γ s γ s ∈ s ()

ϕ(x) − ϕ(y) , ds (x, y)

˚ = {ϕ ∈ Fb : |ϕ|sLip < ∞}. and let Lips () 5.1. Proof of Theorem 4.4. A. Escape rates. Theorem 4.4(a) follows easily from Theorem 4.2 as (F, , H ) and (F, , H ) have the same escape rate. In more detail, let η ∈ G˜ and notice that since H is a union of , j , we have, for each n, η(n ) = η( n ), ˜ dη = ρ ∈ B˚ with d(ρ) > 0. Theorem 4.2(1) then where η = π ∗ η. By definition of G, dm n implies that ϑ∗−n L ρ converges to d(ρ)h ∗ . Since the convergence is in the · -norm, n we may integrate with respect to m. Noting that L ρ dm = n ρ dm = η( n ), we have lim ϑ −n η(n ) n→∞ ∗

= lim ϑ∗−n η( n ) = d(ρ). n→∞

(4)

Thus − log ϑ∗ , where ϑ∗ is the eigenvalue in Theorem 4.2, is the common escape rate ˜ of (F, , H ) for initial distributions in G. B. Uniqueness of limiting distributions. We first prove uniqueness postponing the proof of existence of limiting distributions. ˜ we define a measure ηs on u (), i.e. a measure transverse to unstaGiven η ∈ G, ble leaves, as follows: Set ηs ( u (, j )) = 0 if η(, j ) = 0. If η(, j ) = 0, then ηs | u (, j ) is the factor measure of η|, j normalized, and {ρdm γ , γ ∈ u (, j )} is the disintegration of η into measures on unstable leaves. We will use the convention that ηs (, j ) = 1, and ρ|γ is the density with respect to m γ , so that ρ|γ dηs (γ ) = ρ, where dπ ∗ η = ρdm.

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˜ Suppose for i = 1, 2, there exists µi∗ such that Lemma 5.1. Let η1 and η2 ∈ G. lim ϑ −n F∗n ηi n→∞ ∗

= d(ρ i )µi∗ ,

where ρ i is the density of π ∗ ηi Then µ1∗ = µ2∗ . The crux of the argument for Lemma 5.1 is contained in ˚ Lemma 5.2. Let η1 and η2 be as above, and assume ρ 1 = ρ 2 . Then for all ϕ ∈ Lips (), −n n n ϑ∗ |F∗ η1 (ϕ) − F∗ η2 (ϕ)| → 0 exponentially fast as n → ∞. Proof. For i = 1, 2, let ηis and ρi be the (normalized) factor measure and (unnormalized) densities on γ ∈ u () of ηi as described above. We consider functions which are constant along stable leaves to be defined on both ˚ and 0 and do not distinguish between the two versions of such functions. For each , j , let γˆ ∈ u (, j ) be a representative leaf. Then |F∗n η1 (ϕ) − F∗n η2 (ϕ)| ≤

, j

γˆ ∩n, j

dm γˆ

γs

ρ1 ϕ ◦ F n dη1s −

γs

ρ2 ϕ ◦ F n dη2s . (5)

Next fix x ∈ γˆ ∩ n and estimate the integrals on γ s (x). Define ϕ n = Then, n s n s s ρ1 ϕ ◦ F dη1 − s ρ2 ϕ ◦ F dη2 γ γ n s n s ≤ ρ1 (ϕ ◦ F − ϕ n )dη1 + ρ2 (ϕ ◦ F − ϕ n )dη2 γs γs + ϕ n ρ1 dη1s − ϕ n ρ2 dη2s . γs

γs

ϕ ◦ F n dη1s .

γs

Since ϕ n is constant on γ s and ρ 1 = ρ 2 , the third term above is 0. For the first two terms, we note that for each y ∈ γ s (x), |ϕ n (y) − ϕ ◦ F n (y)| ≤ |ϕ|sLip λ−n . Thus ϑ∗−n |F∗n µ1 (ϕ) − F∗n µ2 (ϕ)| ≤ ϑ∗−n = n

n, j

2ρ 1 dm |ϕ|sLip λ−n

, j −n n 2ϑ∗ |L ρ 1 |1 |ϕ|sLip λ−n ,

which proves the lemma since ϑ∗−n |L ρ 1 | → d(ρ 1 ) by Theorem 4.2.

(6)

Remark 5.3. We have used in the proof above a property of the billiard maps, namely d s (F n x, F n y) ≤ λ−n d s (x, y). For general towers, one has only the contraction guaranteed by (P3) which is nonuniform. It is not hard to see that the lemma holds in the n more general case with the exponential rate given by max{α 2 , β −n θ0n } in the place of λ−n ; we leave the proof to the interested reader.

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Proof of Lemma 5.1. Let µ∗ = h ∗ m be the conditionally invariant measure given by Theorem 4.2. For i = 1, 2, we have, on the one hand, n lim ϑ −n F ∗ ηi n→∞ ∗

= d(ρ i )µ∗ ,

which follows from Theorem 4.2, and on the other, lim ϑ −n π ∗ F∗n ηi n→∞ ∗

= d(ρ i )π ∗ µi∗ , n

which follows from the hypothesis of the lemma. Since π ∗ F∗n ηi = F ∗ π ∗ ηi for each n ≥ 0, we have π ∗ µ1∗ = µ∗ = π ∗ µ2∗ . Thus ϑ∗−n |F∗n µ1∗ − F∗n µ2∗ | → 0 as n → ∞ by Lemma 5.2. But ϑ∗−n F∗n µi∗ = µi∗ since µi∗ is conditionally invariant. Hence µ1∗ = µ2∗ . ˚ C. Convergence to conditionally invariant measure. For a probability measure η on , n n n −n n ˜ |F∗ η| = η( ) = π ∗ η( ). So for η ∈ G, (4) implies limn→∞ ϑ∗ |F∗ η| = d(ρ) > 0, where ρ is the density of η = π ∗ η. More than that is true: Lemma 5.4. ϑ∗−n F∗n η/d(ρ) converges weakly to a conditionally invariant probability measure µ∗ as n → ∞. This is half of Theorem 4.4(b). Once we have this, it will follow immediately that F∗n η ϑ∗−n F∗n η = lim = µ∗ , n→∞ |F∗n η| n→∞ ϑ∗−n |F n η| ∗ lim

(7)

which is the other half. We will use the following algorithm to “lift” measures from to : Fix a measure µs on u () with µs ( u (, j )) = 1. Given η on with density ρ, we define π −1 ∗ η to η decomposes be the measure on with the property that restricted to each , j , π −1 ∗ into the factor measure µs and leaf measures {ρdm γ }, where ρ|π −1 (x) ≡ ρ(x). Notice that π ∗ π −1 ∗ η = η. ˚ and show that ϑ∗−n F∗n η(ϕ) Proof of Lemma 5.4. Our first step is to fix ϕ ∈ Lips () s is a Cauchy sequence. For a fixed µ as above, let ϕ n (x) = γ s (x) ϕ ◦ F˚ n dµs . Define ˜ η has density ρ ∈ B˚ with d(ρ) > 0. Then by definition of π −1 η = π ∗ η. Since η ∈ G, ∗ , ˚n (π −1 dµs (γ ) ϕ ◦ F˚ n ρ dm γ ∗ π ∗ η)(ϕ ◦ F ) = =

, j

u (, j )

, j

, j

γu

ρ ϕ n dm = π ∗ η(ϕ n ).

(8)

For n, k1 , k2 ≥ 0, write |ϑ∗−n−k1 F∗n+k1 η(ϕ) − ϑ∗−n−k2 F∗n+k2 η(ϕ)

k1 ≤ ϑ∗−n−k1 |F∗n+k1 η(ϕ) − F∗n π −1 ∗ π ∗ F∗ η(ϕ)|

k1 −n−k2 n −1 F∗ π ∗ π ∗ F∗k2 η(ϕ)| +|ϑ∗−n−k1 F∗n π −1 ∗ π ∗ F∗ η(ϕ) − ϑ∗ k2 n+k2 +ϑ∗−n−k2 |F∗n π −1 η(ϕ)|. ∗ π ∗ F∗ η(ϕ)η(ϕ) − F∗

(9)

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The first and third terms of (9) are estimated using Lemma 5.2 since π ∗ (ϑ∗−ki F∗ki η) = ki π ∗ (ϑ∗−ki π −1 ∗ π ∗ F∗ η) for i = 1, 2. Thus by Lemma 5.2, ki s n ϑ∗−n−ki |F∗n+ki η(ϕ) − F∗n π −1 ∗ π ∗ F∗ η(ϕ)| ≤ C d(ρ)(|ϕ|Lip + |ϕ|∞ )ζ

for some C > 0 and ζ < 1. We now fix n and estimate the second term of (9). Due to (8), for any k ≥ 0 we have k −n−k −1 ϑ∗−n−k F∗n π −1 π ∗ π ∗ F∗k η(ϕ ◦ F n · 1˚ n ) = ϑ∗−n−k π ∗ F∗k η(ϕ n · 1 n ) ∗ π ∗ F∗ η(ϕ) = ϑ∗ k k ϕ n · L ρ dm. = ϑ∗−n−k F ∗ η(ϕ n · 1 n ) = ϑ∗−n−k n

˜ we estimate Recalling that ρ ∈ B˚ and d(ρ) > 0 since η ∈ G, k1 −n−k2 n −1 |ϑ∗−n−k1 F∗n π −1 F∗ π ∗ π ∗ F∗k2 η(ϕ)| ∗ π ∗ F∗ η(ϕ) − ϑ∗ −k1 k1 ≤ |ϕ|∞ ϑ∗−n ϑ∗ L ρ − d(ρ)h ∗ dm n −k2 k2 −n +|ϕ|∞ ϑ∗ ϑ∗ L ρ − d(ρ)h ∗ dm.

(10)

n

Both terms of ϑ∗−n

n

(10) are small: By Theorem 4.2(1), −k k −n −k k dm L ρ − d(ρ)h ≤ ϑ L ρ − d(ρ)h ϑ∗ ∗ ∗ ∗ ϑ∗ n

≤ ϑ∗−n |L 1β |1 Dρτ k

n

1β dm

˚ ϑ −n |Ln 1β |1 converges ˚ . Since 1β ∈ B, for k = ki , where 1β (x) = β − for x ∈ ∗ to d(1β ) as n → ∞. Thus (10) can be made arbitrarily small by choosing k1 and k2 sufficiently large. We have shown that ϑ∗−n F∗n η(ϕ)/d(ρ) is a Cauchy sequence and therefore converges to a number Q(ϕ). The functional Q(ϕ) := limn→∞ ϑ∗−n F∗n η(ϕ)/d(ρ) is clearly linear in ϕ, positive and satisfies Q(1) = 1. Also |Q(ϕ)| ≤ |ϕ|∞ Q(1) so that Q extends to a ˚ the set of bounded functions which are continuous bounded linear functional on Cb0 (), ˚ on each , j . By the Riesz representation theorem, there exists a unique Borel probability measure ˚ [H, Sect. 56]. Also, µ∗ satisfying µ∗ (ϕ) = Q(ϕ) for each ϕ ∈ Cb0 () ˚ = lim ϑ∗−n F∗n η(ϕ ◦ F) ˚ = ϑ∗ lim ϑ∗−n−1 F∗n+1 η(ϕ) = ϑ∗ d(ρ)µ∗ (ϕ), d(ρ)µ∗ (ϕ ◦ F) n→∞

n→∞

so that µ∗ is a conditionally invariant measure for F˚ with escape rate − log ϑ∗ .

This completes the proof of parts (a) and (b) of Theorem 4.4. To prove part (c), we must show that µ∗ has absolutely continuous conditional measures on unstable leaves. Proof of a stronger version of this fact is contained in the proof of Proposition 4.6(i) below. Notice that from the proof of Lemma 5.1, we have π ∗ µ∗ = µ∗ so that the density of ˜ π ∗ µ∗ is precisely h ∗ . Since h ∗ ∈ B˚ and d(h ∗ ) = 1 > 0, we conclude µ∗ ∈ G.

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5.2. Proof of Proposition 4.6. Consider the set of holes h (q0 ) for fixed q0 where h is small enough as required in Proposition 2.2. For σ ∈ h (q0 ), let (σ ) be the tower with ) holes induced by the generalized horseshoe, and let µ(σ ∗ be the conditionally invariant measure given by Theorem 4.4. Proof of (i). We drop the superscript (σ ) in what follows and point out that the constants we use are uniform for all σ ∈ h (q0 ) and h sufficiently small. Choose γ0 ∈ u (0 ) and let η0 be the measure supported on γ0 with uniform den˜ It is immediate that π ∗ η0 has density sity with respect to µγ . We claim that η0 ∈ G. −u ˚ ρ = e |γ0 with respect to m, which is in B by Lemma 4.1. To see that d(ρ) > 0, notice that the mixing assumption on (F, ) implies that 0 is necessarily a surviving partition element. Since ρ > 0 on 0 , Corollary 4.3 implies that d(ρ) > 0. (n) By Theorem 4.4(b), η(n) := F∗n η0 /|F∗n η0 | converges to µ∗ . Let ργ denote the density of η(n) with respect to µγ on γ ∈ u (). Notice that inverse branches of F˚ n on γ are well defined. For any x1 , x2 ∈ γ , treating one branch at a time and summing over all branches, we obtain that −1 (n) ˚n Jµγ F˚ n (y2 ) ργ (x1 ) y1 ∈ F˚ −n x1 (Jµγ F (y1 )) = ≤ sup ≤ eC (n) n (y ))−1 n (y ) ˚ ˚ F F (J J −n ργ (x2 ) ˚ −n µ 2 µ 1 ˚ y ∈F x γ γ y2 ∈ F

x2

1

1

by Property (P4)(b), where Jµγ F˚ n is the Jacobian of F˚ n with respect to µγ . Since by Proposition 2.2 the constant C is independent of σ , x and n, we have e−C ≤

(n)

supx∈γ ργ (x) inf x∈γ ργ(n) (x)

≤ eC .

(11)

This estimate plus the minimum length κ of µu ( ) given by Proposition 2.2 yields the desired uniform upper and lower bounds on the conditional densities of η(n) with respect to µγ (and hence to m γ ) on 0 . The uniformity of these bounds in n implies that they pass to the conditional densities of µ∗ in the limit as n → ∞. Since µ∗ is conditionally invariant, µ∗ |˚ = ϑ∗−1 µ∗ | F˚ −1 ˚ . The required bounds on the densities extend easily to ˚ for > 0. Proof of (ii). We decompose µ∗ into a normalized factor measure µs∗ on u ( ) and densities ργ with respect to m γ on γ ∈ u ( ). Then (σ ) µ∗ (∪≥L ) = dµs∗ ργ dm γ ≤ C2 ϑ∗− m( ) ≤ C2 C0 θ0 β − . (σ )

u ≥L ( )

γ

≥L

≥L

Here we have used Proposition 4.6(i) to estimate ργ , Proposition 2.2 and Lemma 4.1 for the uniformity of C0 and θ0 , and the fact that ϑ∗ > β. The sum can be made arbitrarily small since β > θ0 . Proof of (iii). Notice that n(, H˜ ) ≥ n(h) ¯ by definition of n(h) ¯ in Sect. 2.2. From Theorem 4.2(2), we know that the escape rate − log ϑ∗ satisfies 1 + C1 −1 1 + C1 −1 ϑ∗ > 1 − β m(H ∩ ) > 1 − β C0 θ0 . κ κ ≥1

By Proposition 2.2, n(h) ¯ → ∞ as h → 0, so that ϑ∗ → 1.

≥n(h) ¯

Escape Rates and Physically Relevant Measures for Billiards with Small Holes

383

6. Proofs of Theorems for Billiards In the Proofs of Theorems 1–3, we fix a hole σ that is acceptable with respect to Proposition 2.2 and for which n( (σ ) , R (σ ) , Hσ ) is large enough to meet the condition in Theorem 4.4. We suppress mention of σ , and let (F, , H˜ ) be the tower constructed n ˚ M ∞ = ∩n≥0 M n . from ( , R, H ). Define M n = ∩i=0 f −n M, 6.1. Proof of Theorems 1 and 2. The first order of business is to show that each η ∈ G ˚ with can be lifted to a measure η˜ ∈ G˜ in such a way that the escape dynamics on initial distribution η˜ reflect those on M˚ with initial distribution η. Recall that the natural invariant probability measure for the closed billiard system f : M → M is denoted by ν. In [Y, Sect. 2], it is shown that there is a unique invariant probability measure ν˜ for the tower map F : → with absolutely continuous conditional measures on unstable leaves, and this measure has the property π∗ ν˜ = ν. Given η ∈ G, we define η˜ on as follows: By definition, every η ∈ G is absolutely continuous with respect to ν. Let ˜ ˜ where ψ˜ = ψ ◦ π . This ψ = dη dν . We take η˜ to be the measure given by d η˜ = ψd ν, implies in particular that π∗ η˜ = η. ˜ Lemma 6.1. If η ∈ G, then η˜ ∈ G. As before, let Fb denote the set of bounded functions on . For ϕ ∈ Fb and γ ∈ u (), we let Lipu (ϕ|γ ) be the Lipschitz constant of ϕ|γ with respect to the dβ -metric (notice that dβ , the symbolic metric defined on , can be thought of as a metric on unstable leaves). Let |ϕ|uLip =

sup

γ ∈ u ()

Lipu (ϕ|γ ),

and Lipu () = {ϕ ∈ Fb : |ϕ|uLip < ∞}. The first step toward proving Lemma 6.1 is ˜ uLip ≤ Lemma 6.2. Let ϕ : M → R be Lipschitz. Then ϕ˜ := ϕ ◦ π ∈ Lipu () with |ϕ| CLip(ϕ). Proof. Recall that for x, y ∈ M lying in a piece of local unstable manifold, we have d(x, y) ≤ p(x, y)1/2 , where p(·, ·) is the p-metric (see Sect. 2.3). Now for γ ∈ u () and x, y ∈ γ , we have 1

|ϕ(x) ˜ − ϕ(y)| ˜ = |ϕ(π x) − ϕ(π y)| ≤ Lip(ϕ)d(π x, π y) ≤ Lip(ϕ) p(π x, π y) 2 . 1

By (P4)(a), p(π x, π y) 2 ≤ Cα s(π x,π y)/2 , which is ≤ Cdβ (x, y) since s ≤ s0 and √ β ≥ α. ˚ Let ψ = Proof of Lemma 6.1. (i) First we show π ∗ η˜ = ρ m with ρ ∈ B. s u disintegrating η˜ into η˜ and {ργ dm γ , γ ∈ ()}, we obtain ργ := ψ˜ ·

Then

d ν˜ dµγ · . dµγ dm γ

Now ψ˜ is bounded by assumption and is ∈ Lipu () by Lemma 6.2, Lipu ()

dη dν .

dµγ dm γ

d ν˜ dµγ

is bounded

([Y], Sect. 2), as is (Lemma 4.1). Thus we conclude that ργ ∈ and is ∈ u Lip () and is bounded. Recall that ρ(x) = γ s (x) ργ d η˜ s . It follows immediately that |ρ|∞ ≤ supγ |ργ |∞ and Lip(ρ) ≤ supγ Lipu (ργ ).

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(ii) It remains to show d(ρ) > 0. By definition of G, ψ > 0 on M ∞ , the set of points −u are ˚ so ψ˜ > 0 on ∞ . The fact that d ν/dµ which never escape from M, ˜ γ and e ∞ strictly positive implies that ρ > 0 on ; hence it is > 0 on a surviving cylinder set, k i.e. a set E k such that F maps E k onto a surviving , j before any part of it enters the k n hole. By Corollary 4.3, d(L ρ) > 0. Since n g dm = L g dm for each n ≥ 0 and k

g ∈ L 1 (m), we have d(L ρ) = ϑ∗k d(ρ) so that d(ρ) > 0 as well.

Proof of Theorems 1 and 2. Given η ∈ G, let η˜ be as defined earlier. Then η˜ ∈ G˜ by Lemma 6.1. For ϕ ∈ C 0 (M), let ϕ˜ = ϕ ◦ π . Then ϕ˜ ∈ Cb0 () and for n ≥ 0 we have, f˚∗n η(ϕ) = η(ϕ ◦ f n · 1 M n ) = η( ˜ ϕ˜ ◦ F n · 1n ) = F˚∗n η( ˜ ϕ). ˜

(12)

˚ = F˚∗n η( ˚ = η( Setting ϕ ≡ 1 in (12), we have η(M n ) = f˚∗n η( M) ˜ ) ˜ n ) for n > 0, so lim

n→∞

1 1 log η(M n ) = lim log η( ˜ n ) = log ϑ∗ n→∞ n n

by Theorem 4.4(a). This proves Theorem 1. Let µ∗ = π∗ µ˜ ∗ , where µ˜ ∗ is given by Theorem 4.4. Then µ∗ (ϕ) = µ˜ ∗ (ϕ), ˜ and f˚∗ µ∗ (ϕ) = F˚∗ µ˜ ∗ (ϕ) ˜ = ϑ∗ µ˜ ∗ (ϕ) ˜ = ϑ∗ µ∗ (ϕ), proving µ∗ is conditionally invariant. Using (12) again, the fact that the normalizations are equal, and Theorem 4.4(b), we obtain lim

n→∞

f˚∗n η(ϕ) = lim n→∞ ˚ f˚∗n η( M)

F˚∗n η( ˜ ϕ) ˜ ˜ = µ∗ (ϕ). = µ˜ ∗ (ϕ) n ˚ ˚ F∗ η()

Thus f˚∗n η/η(M n ) → µ∗ weakly. Finally, lim ϑ −n f˚∗n η(ϕ) n→∞ ∗

= lim ϑ∗−n F˚∗n η( ˜ ϕ) ˜ = d(ρ) · µ˜ ∗ (ϕ) ˜ = d(ρ) · µ∗ (ϕ), n→∞

˜ This completes the proof of Theorem 2. where d(ρ) > 0 since η˜ ∈ G.

Remark 6.3. In the proof of Lemma 6.1, step (i) holds for any η that has Lipschitz densities on unstable leaves. Thus for this class of measures, Theorem 2(b) holds (with c(η) possibly equal to zero). It is also clear from step (ii) that to show d(ρ) > 0, it suffices to assume ψ > 0 on M ∞ ∩ , or on M ∞ ∩π(, j ), where π (, j ) is any surviving element. 6.2. Proof of Theorem 3. Let µ∗ = π∗ µ˜ ∗ be as above. (a) Since f˚∗ µ∗ = ϑ∗ µ∗ , it follows that µ∗ is supported on M \ ∪n≥0 f n (H ), where H = Hσ . This set has Lebesgue measure zero since by the ergodicity of f , ∪n≥0 f n (H ) has full Lebesgue measure. Thus µ∗ is singular with respect to Lebesgue measure. (b) First, we argue that µ∗ has absolutely continuous conditional measures on unstable leaves (without claiming that the densities are strictly positive). This is true because for each , j, µ˜ ∗ |, j has absolutely continuous conditional measures on γ ∈ u (, j ), and π |, j , which is one-to-one, identifies each γ with a positive Lebesgue measure subset of a local unstable manifold of f .

Escape Rates and Physically Relevant Measures for Billiards with Small Holes

385

The rest of the proof is concerned with showing that the conditional densities of µ∗ are strictly positive. To do that, it is not productive to view µ∗ as π∗ µ˜ ∗ . Instead, we will view µ∗ as the weak limit of ν (n) := f˚∗n ν/| f˚∗n ν| as n → ∞, where ν is the natural invariant measure for f . This convergence of ν (n) is guaranteed by Theorem 2. We will prove that µ∗ has the properties immediately following the statement of Theorem 3 in Sect. 1.3. Step 1: Our first patch is built on V = ∪{γ u : γ u ∈ u ( )}, where u = u ( ) is the defining family of unstable curves for . To understand the geometric properties of ν (n) |V , observe that in backward time, each γ u ∈ u either falls into the hole completely or stays out completely. This is because f (∂ H ) is regarded as part of the discontinuity set for f −1 when we constructed the horseshoe (see Sect. 3.2). Thus there is a decreasing sequence of sets Un = ∪{γ u ∈ u : f −i γ u ∩ H = ∅ for all 0 ≤ i ≤ n} ⊂ V consisting of whole γ u -curves. Assuming ν(Un ) > 0 for now, we have ν (n) |V = cn ν|Un for some constant cn > 0 as ν is f -invariant. Let ζ be a limit point of ν (n) |V , i.e., ζ = limn k ν (n k ) |V . Assuming ζ (V ) > 0, lower bounds for conditional probability densities of ν (n k ) |V , equivalently those of ν|Un , are passed to ζ , and these bounds are strictly positive. To see that ζ (V ) > 0, recall that ν = π∗ ν˜ for some ν˜ on the tower , so that ν (n) = π∗ ν˜ (n) , where ν˜ (n) = F˚∗n ν˜ /| F˚∗n ν˜ |. Since π(0 ) ⊂ V , we have ζ = lim ν (n k ) |V ≥ lim π∗ (˜ν (n k ) |0 ) = π∗ (µ˜ ∗ |0 ). nk

nk

We have written an inequality (as opposed to equality) above because parts of for ≥ 1 may get mapped into V as well. Clearly, µ˜ ∗ (0 ) > 0, thereby ensuring ζ (V ) > 0, hence ν(Un ) > 0 and the strictly positive conditional densities property above. This together with ζ ≤ µ∗ |V (equality is not claimed because it is possible for part of ν (n k ) from outside of V to leak into V in the limit) proves that (V, ζ ) is an acceptable patch. Step 2: Next we use (V, ζ ) to build patches (V, j , ζ, j ) corresponding to partition elements , j of the tower with > 0 and µ˜ ∗ (, j ) > 0. From Sects. 3 and 4, we know that π(, j ) is a hyperbolic product set, and π(, j ) = f ( s ) for some s-subset s ⊂ . Moreover, f i ( s ) ∩ H = ∅ for all 0 < i ≤ . Thus we may assume V, j = ∪{γ u : γ u ∈ u (π(, j ))} ⊂ f (V ). Let ζ, j = ϑ∗− ( f ∗ ζ )|V, j . Then ζ, j has strictly positive conditional densities on unstable curves because ζ does, and ζ, j ≤ µ∗ |V, j as µ∗ satisfies f˚∗ µ∗ = ϑ∗ µ∗ . Finally, since ζ, j ≥ π∗ (µ˜ ∗ |, j ) for each , j, it follows that , j ζ, j ≥ µ∗ , completing the proof of Theorem 3. 6.3. Proof of Theorem 4. Suppose h n is a sequence of numbers tending to 0, σh n ∈ h n (q0 ) is a sequence of holes in the billiard table, and Hn = Hσh n the corresponding holes in M. For each n, let ϑn be the escape rate and µn the physical measure for the leaky system ( f, M, Hn ) given by Theorem 2. By Proposition 4.6(iii), we have ϑn → 1 as n → ∞. To prove µn → ν, we will assume, having passed to a subsequence, that µn converges weakly to some µ∞ , and show that (i) µ∞ is f -invariant, and (ii) it has absolutely continuous conditional measures on unstable leaves. These two properties together uniquely characterize ν. The following notation will be used: (n) is the generalized horseshoe respecting the hole Hn , (n) = ∪ (n) is the corresponding tower, Fn : (n) → n is the tower

386

M. Demers, P. Wright, L.-S. Young

map, πn : (n) → M is the projection, and µ˜ n is the conditionally invariant measure on (n) that projects to µn . (i) Proof of f -invariance: Let S = ∂ M ∪ f −1 ∂ M. Lemma 6.4. µ∞ (S) = 0 Proof. Let δ1 and λ1 be as in Sect. 2.3, and let Nε (S) denote the ε-neighborhood of S. We claim that there exist constants C3 , ς > 0 such that for ε < δ1 and for all n, µn (Nε (S)) ≤ C3 ες . By the construction of = (n), any n, d( f ( ), S) ≥ δ1 λ− 1 . Thus f ( ) ∩ Nε (S) = ∅ for all ≤ − log(ε/δ1 )/ log λ1 . Hence µn (Nε (S)) ≤ µ˜ n ( (n)), >− log(ε/δ1 )/ log λ1

which by Proposition 4.6(ii) is ≤ K (β −1 θ0 )− log(ε/δ1 )/ log λ1 , proving the claim above with C3 = K /δ1 and ς = log(βθ0−1 )/ log λ1 . Since C3 and ς are independent of n, these bounds pass to µ∞ , implying µ∞ (S) = 0. Having established that f is well defined µ∞ -a.e., we now verify that µ∞ is f -invariant: Let ϕ : M → R be a continuous function. Then (ϕ ◦ f )dµ∞ = lim (ϕ ◦ f )dµn = lim ϕ d( f ∗ µn ), n→∞

and

n→∞

ϕ d( f ∗ µn ) =

M\Hn

ϕ d( f ∗ µn ) +

Hn

ϕ d( f ∗ µn ).

(13)

Since( f ∗ µn )| M\Hn = f˚∗ µn = ϑn µn , the first integral on the right side of (13) is equal to ϑn ϕ dµn , while the absolute value of the second is bounded by (1 − ϑn )|ϕ|∞ . Since ϑn → 1 as n → ∞, the right side of (13) tends to ϕ dµ∞ . (ii) Absolutely continuous conditional measures on unstable leaves: Since the measures µ˜ n do not live on the same space for different n, a first task here is to find common domains in M on which (πn )∗ µ˜ n can be compared. In the constructions to follow, the discontinuity set refers to the real discontinuity set of f , not the ones that include boundaries of holes (as was done in Sect. 3). We choose a rectangular region Qˇ slightly larger than Q in Proposition 2.2, large enough that Qˇ ⊃ (n) for all n, and let ˇ u denote the set of all homogeneous unstable ˇ Let Vˇ = ∪{γ u ∈ ˇ u }. Then (n) ⊂ Vˇ curves connecting the two components of ∂ s Q. u u ˇ ˇ for all n, for γ ∩ Q ∈ for every γ ∈ ( (n)) (defined using the enlarged discontinuity set). Now for all n, (πn )∗ (µ˜ n |0 (n) ) is a sequence of measures on Vˇ with absolutely continuous conditional measures on the elements of ˇ u . Moreover, the conditional densities are uniformly bounded from above with a bound independent of n (Proposition 4.6(i)). Let µ∞,0 be a limit point of (πn )∗ (µ˜ n |0 (n) ). Assuming µ∞,0 (Vˇ ) > 0, these density bounds are inherited by µ∞,0 . To show µ∞,0 (Vˇ ) > 0, we will argue there exists b > 0 such that µ˜ n (0 (n)) > b for all n, and that is true because the µ˜ n are probability measures, there is a uniform lower bound on µ˜ n (∪ 0, we define Qˇ to be the finite union of s-subrectangles of Qˇ retained in steps in the construction of when f has no holes, i.e. roughly speaking, Qˇ consists of points that stay away from S = ∂ M ∪ f −1 ∂ M by a distance ≥ δ1 λ−i 1 at step i. Let − ˚ ˇ ˇ ˇ ˇ V = V ∩ Q . Then πn ( Fn (n)) ⊂ V for all n. In fact, for each j, πn ( F˚n− , j (n)) is contained in a connected component of Qˇ . The argument for µ∞,0 can now be repeated to conclude the existence of a limit point of (πn ◦ F˚n− )∗ (µ˜ n | (n) ) with absolutely continuous conditional measures on unstable leaves. Pushing all measures forward by f ∗ , this gives a limit point µ∞, of (πn )∗ (µ˜ n | (n) ) as n → ∞ with the same property. To proceed systematically, we perform a Cantor diagonal argument, choosing a single subsequence n k with the property that for each ≥ 0, (πn k )∗ (µ˜ n k | (n k ) ) converges to a measure µ∞, on f Vˇ . Finally, to conclude µ∞ = µ∞, , we need a tightness condition as the towers are noncompact. This is given by Proposition 4.6(ii). The proof of Theorem 4 is now complete. Acknowledgements. The second author (P.W.) would like to thank The Courant Institute of Mathematical Sciences, New York University, where he was affiliated when this project began. The authors thank MSRI, Berkeley, and ESI, Vienna, where part of this work was carried out.

References [BaK] [BDM] [BSC1] [BSC2] [B] [C1] [C2] [CM1] [CM2] [CM3] [CMT1] [CMT2] [CMM] [CMS1] [CMS2] [CMS3] [CV]

Baladi, V., Keller, G.: Zeta functions and transfer operators for piecewise monotonic transformations. Commun. Math. Phys. 127, 459–477 (1990) Bruin, H., Demers, M., Melbourne, I.: Convergence properties and an equilibrium principle for certain dynamical systems with holes. To appear in Ergod. Th. and Dynam. Sys. Bunimovich, L.A., Sina˘ı, Ya.G., Chernov, N.I.: Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv. 45(3), 105–152 (1990) Bunimovich, L.A., Sina˘ı, Ya.G., Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46(4), 47–106 (1991) Buzzi, J.: Markov extensions for multidimensional dynamical systems. Israel J. of Math. 112, 357–380 (1999) Cenvoca, N.N.: A natural invariant measure on smale’s horseshoe. Soviet Math. Dokl. 23, 87–91 (1981) Cenvoca, N.N.: Statistical properties of smooth Smale horseshoes. In: Mathematical Problems of Statistical Mechanics and Dynamics, R.L. Dobrushin, ed. Dordrecht: Reidel, 1986, pp. 199–256 Chernov, N., Markarian, R.: Ergodic properties of anosov maps with rectangular holes. Bol. Soc. Bras. Mat. 28, 271–314 (1997) Chernov, N., Markarian, R.: Anosov maps with rectangular holes. Nonergodic Cases. Bol. Soc. Bras. Mat. 28, 315–342 (1997) Chernov, N., Markarian, R.: Chaotic Billiards. Number 127 in Mathematical Surveys and Monographs, Providence, RI: Amer. Math. Soc., 2006 Chernov, N., Markarian, R., Troubetzkoy, S.: Conditionally invariant measures for anosov maps with small holes. Ergod. Th. and Dynam. Sys. 18, 1049–1073 (1998) Chernov, N., Markarian, R., Troubetzkoy, S.: Invariant measures for anosov maps with small holes. Ergod. Th. and Dynam. Sys. 20, 1007–1044 (2000) Collet, P., Martínez, S., Maume-Deschamps, V.: On the existence of conditionally invariant probability measures in dynamical systems. Nonlinearity 13, 1263–1274 (2000) Collet, P., Martínez, S., Schmitt, B.: The Yorke-Pianigiani measure and the asymptotic law on the limit cantor set of expanding systems. Nonlinearity 7, 1437–1443 (1994) Collet, P., Martínez, S., Schmitt, B.: Quasi-stationary distribution and Gibbs measure of expanding systems. In: Instabilities and Nonequilibrium Structures. V. E. Tirapegui, W. Zeller, eds. Dordrecht: Kluwer, 1996, pp. 205–219 Collet, P., Martínez, S., Schmitt, B.: The Pianigiani-Yorke measure for topological Markov chains. Israel J. Math. 97, 61–70 (1997) Chernov, N., van dem Bedem, H.: Expanding maps of an interval with holes. Ergod. Th. and Dynam. Sys. 22, 637–654 (2002)

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M. Demers, P. Wright, L.-S. Young

Demers, M.: Markov extensions for dynamical systems with holes: an application to expanding maps of the interval. Israel J. of Math. 146, 189–221 (2005) Demers, M.: Markov extensions and conditionally invariant measures for certain logistic maps with small holes. Ergod. Th. and Dynam. Sys. 25(4), 1139–1171 (2005) Demers, M., Liverani, C.: Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360(9), 4777–4814 (2008) Demers, M., Young, L.-S.: Escape rates and conditionally invariant measures. Nonlinearity 19, 377–397 (2006) Ferrari, P.A., Kesten, H., Martínez, S., Picco, P.: Existence of quasi-stationary distributions. A Renewal Dynamical Approach. Annals of Prob. 23(2), 501–521 (1995) Halmos, P.R.: Measure Theory. University Series in Higher Mathematics, Princeton, NJ: D. Van Nostrand Co., Inc., 1950, 304 p. Homburg, A., Young, T.: Intermittency in families of unimodal maps. Ergod. Th. and Dynam. Sys. 22(1), 203–225 (2002) Katok, A., Strelcyn, J.M.: Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. Volume 1222, Springer Lecture Notes in Math., Berlin-Heidelberg-NewYork: Springer, 1986 Liverani, C., Maume-Deschamps, V.: Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set. Annales de l’Institut Henri Poincaré Probability and Statistics 39, 385–412 (2003) Lopes, A., Markarian, R.: Open billiards: cantor sets, invariant and conditionally invariant probabilities. SIAM J. Appl. Math. 56, 651–680 (1996) Pianigiani, G., Yorke, J.: Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252, 351–366 (1979) Richardson, P.A., Jr.: Natural Measures on the Unstable and Invariant Manifolds of Open Billiard Dynamical Systems. Doctoral Dissertation, Department of Mathematics, University of North Texas, 1999 Sina˘ı, Ya.G.: Dynamical systems with elastic collisions. Ergodic Properties of Dispersing Billiards. Usp. Mat. Nauk 25(2), 141–192 (1970) Vere-Jones, D.: Geometric ergodicity in denumerable Markov chains. Quart. J. Math. 13, 7–28 (1962) Wojtkowski, M.: Invariant families of cones and Lyapunov exponents. Ergod. Th. Dynam. Sys. 5(1), 145–161 (1985) Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Annals of Math. 147(3), 585–650 (1998)

Communicated by G. Gallavotti

Commun. Math. Phys. 294, 389–410 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0938-6

Communications in

Mathematical Physics

Argyres-Seiberg Duality and the Higgs Branch Davide Gaiotto1 , Andrew Neitzke2 , Yuji Tachikawa1 1 School of Natural Sciences, Institute for Advanced Study, Princeton,

New Jersey 08540, USA. E-mail: [email protected]

2 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

Received: 10 November 2008 / Accepted: 20 August 2009 Published online: 1 December 2009 – © Springer-Verlag 2009

Abstract: We demonstrate the agreement between the Higgs branches of two N = 2 theories proposed by Argyres and Seiberg to be S-dual, namely the SU(3) gauge theory with six quarks, and the SU(2) gauge theory with one pair of quarks coupled to the superconformal theory with E 6 flavor symmetry. In mathematical terms, we demonstrate the equivalence between a hyperkähler quotient of a linear space and another hyperkähler quotient involving the minimal nilpotent orbit of E 6 , modulo the identification of the twistor lines. Contents 1. 2. 3. 4.

5.

6. 7.

Introduction . . . . . . . . . . . . . . . 1.1 Argyres-Seiberg duality . . . . . . . 1.2 Higgs branch . . . . . . . . . . . . Rudiments of Hyperkähler Cones . . . . Geometry of the Minimal Nilpotent Orbit SU(3) Side . . . . . . . . . . . . . . . . 4.1 Poisson brackets . . . . . . . . . . . 4.2 Conjugation . . . . . . . . . . . . . 4.3 Constraints . . . . . . . . . . . . . Exceptional Side . . . . . . . . . . . . . 5.1 Poisson brackets . . . . . . . . . . . 5.2 Conjugation . . . . . . . . . . . . . 5.3 Constraints . . . . . . . . . . . . . 5.4 Gauge invariant operators . . . . . . Comparison . . . . . . . . . . . . . . . 6.1 Identification of operators . . . . . . 6.2 Constraints . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . .

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A. Conventions . . . . . . . . . . . . . B. Twistor Spaces of Hyperkähler Cones C. Comparison of the Kähler Potential . C.1 Exceptional side . . . . . . . . . C.2 SU(3) side . . . . . . . . . . . . C.3 Comparison . . . . . . . . . . . D. Mathematical Summary . . . . . . .

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404 405 406 406 407 407 408

1. Introduction 1.1. Argyres-Seiberg duality. In a remarkable paper [1], a new type of strong-weak duality of four-dimensional N = 2 theories was introduced. Consider an N = 2 supersymmetric SU(3) gauge theory with six quarks in the fundamental representation. This theory has vanishing one-loop beta function, and the gauge coupling constant τ=

θ 8πi + 2 π g

(1.1)

is exactly marginal. Argyres and Seiberg carried out a detailed study of the behavior of the Seiberg-Witten curve close to the point τ → 1, where the theory is infinitely strongly-coupled, and were led to conjecture a dual description involving an SU(2) group with gauge coupling τ =

1 . 1−τ

(1.2)

To understand the matter content of the dual theory, one first needs to recall the interacting superconformal field theory (SCFT) with flavor symmetry E 6 first described by [2]. This theory has one-dimensional Coulomb branch parametrized by u whose scaling dimension is 3, and is realized as the low-energy limit of the worldvolume theory on a D3-brane probing the transverse geometry of an F-theory 7-brane with E 6 gauge group. The gauge group on the 7-brane then manifests as a flavor symmetry from the point of view of the D3-brane. We denote this theory by SCFT[E 6 ] following [1]. Now, the theory Argyres and Seiberg proposed as the dual of the SU(3) gauge theory with six quarks consists of the SU(2) gauge bosons, coupled to one hypermultiplet in the doublet representation, and also to a subgroup SU(2) ⊂ E 6 of SCFT[E 6 ]. The SU(2) subgroup is chosen so that the raising operator of SU(2) maps to the raising operator for the highest root of E 6 . In the following, we refer to two sides of the duality as the SU(3) side and the exceptional side, respectively. Argyres and Seiberg provided a few compelling pieces of evidence for this duality. First, the flavor symmetry agrees. On the SU(3) side, there is a U(6) = U(1) × SU(6) symmetry which rotates the six quarks. On the exceptional side, there is an SO(2) symmetry which rotates a pair of quarks in the doublet representation, which can be identified with the U(1) part of U(6). Then, the flavor symmetry of the SCFT with E 6 is broken down to the maximal subgroup commuting with SU(2) ⊂ E 6 , which is SU(6). Second, the scaling dimensions of Coulomb-branch operators agree. Indeed, on the SU(3) side one has tr φ 2 and tr φ 3 , where φ is the adjoint chiral multiplet of SU(3). The dimensions are thus 2 and 3. On the exceptional side, one has tr ϕ 2 (where ϕ is the adjoint chiral multiplet of SU(2)), which has dimension 2, and the Coulomb-branch operator u of SCFT[E 6 ], which has dimension 3.

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391

Third, Argyres and Seiberg studied in detail the deformation of the Seiberg-Witten curve under the SU(6) mass deformation, and found remarkable agreement. Fourth, they computed the current algebra central charge of the SU(6) flavor symmetry on the SU(3) side, which agreed with the central charge of the E 6 symmetry on the exceptional side, inferred from the fact that the beta function of the SU(2) gauge group coupling is zero. This is as it should be, because SU(6) arises as a subgroup of E 6 on the exceptional side. This provided a prediction of the current central charge of SCFT[E 6 ] for the first time, which was later reproduced holographically by [3]. There are generalizations to similar duality pairs involving SCFTs with flavor symmetries other than E 6 [1,4]. Our aim in this note is to present further convincing evidence for this duality, by showing that the Higgs branches of the two sides of the duality are equivalent as hyperkähler cones. Mathematically speaking, we will show the agreement of their twistor spaces as complex varieties with real structure, but we have not been able to prove that they share the same family of twistor lines. Instead we give numerical evidence that their Kähler potentials agree in Appendix C. 1.2. Higgs branch. On the SU(3) side, let us denote the squark fields by Q ia ,

Q˜ ia ,

(1.3)

where i = 1, . . . , 6 are the flavor indices and a = 1, 2, 3 the color indices. The Higgs branch is the locus where the F-term and the D-term both vanish, divided by the action of the gauge group SU(3). As is well known, this space can also be obtained by setting F = 0 without setting D = 0, and dividing by the complexified gauge group SL(3, C). Thus the Higgs branch is parametrized by gauge invariant composite operators M i j = Q ia Q˜ aj ,

j

B i jk = abc Q ia Q b Q kc ,

B˜ i jk = abc Q˜ ia Q˜ bj Q˜ ck

(1.4)

which satisfy various constraints, e.g. B [i jk M l] m = 0

(1.5)

to which we will come back later. The fields Q ia , Q˜ ia have 36 complex components, while the F-term condition imposes 8 complex constraints. The quotient by SL(3, C) reduces the complex dimension further by 8, so the Higgs branch has complex dimension 2 × 3 × 6 − 8 − 8 = 20.

(1.6)

Our problem is to understand how this structure of the Higgs branch is realized on the exceptional side. Firstly, we have one hypermultiplet in the doublet representation, which we denote as vα , v˜α in N = 1 superfield notation. Here α = 1, 2 is the doublet index. We also have the Higgs branch of SCFT[E 6 ], the structure of which is known through the F-theoretic construction of the SCFT. Recall that this theory is the worldvolume theory on one D3-brane probing a F-theory 7-brane of type E 6 . Say the D3-brane extends along the directions 0123, and the 7-brane along the directions 01234567. The onedimensional Coulomb branch of this theory is identified with the transverse directions 89 to the 7-brane. The theory becomes superconformal when the D3-brane hits the 7-brane, at which point the Higgs branch emanates. This is identified as the process where a D3-brane is absorbed into the worldvolume of the 7-brane as an E 6 instanton

392

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along the directions 4567. The real dimension of the N -instanton moduli space of E 6 is 4h E 6 N with the dual Coxeter number h E 6 = 12. The center-of-mass motion of the instanton corresponds to a decoupled free hypermultiplet, and thus the genuine moduli space is the so-called ‘centered’ one-instanton moduli space without the center-of-mass motion, which has complex dimension 22. The SU(2) gauge group couples to the quark fields vα , v˜α , and this instanton moduli space. Imposing the F-term condition and dividing by the complexified gauge group, we find the complexified dimension of the Higgs branch as 2 × 2 + 22 − 3 − 3 = 20,

(1.7)

which correctly reproduces the dimension of the Higgs branch on the SU(3) side. We would like to perform more detailed checks, and for that purpose one needs to have a concrete description of the instanton moduli. It is well known that the ADHM description is available for classical gauge groups, but how can we proceed for exceptional groups? Luckily, there is another description of the 1-instanton moduli spaces, applicable to any group G, which identifies the centered 1-instanton moduli space with the minimal nilpotent orbit of G [5]. Let us now define the minimal nilpotent orbit. G C acts on the complexified Lie algebra gC , which has the Cartan generators H i and the raising/lowering operators E ±ρ for roots ρ. G C also acts on the dual vector space g∗C of gC via the coadjoint action,1 and the minimal nilpotent orbit Omin (G) of G is the orbit of (E θ )∗ , where θ denotes the highest root: Omin (G) = G C · (E θ )∗ .

(1.8)

The minimal nilpotent orbit is known to have polynomial defining equations. Moreover, they can be chosen to be quadratic, transforming covariantly under G C . These relations are known under the name of the Joseph ideal [6]. The simplest example is the case G = SU(2). In this case gC is three-dimensional; denote its three coordinates by a, b and c, which transform as a triplet of SU(2). The minimal nilpotent orbit is then given by a 2 + b2 + c2 = 0

(1.9)

which describes the space C2 /Z2 , and as is well-known, the centered one-instanton moduli space of SU(2) is exactly this orbifold. Let us come back to the case of E 6 . We fix an SU(2) subalgebra generated by E ±θ . The maximal commuting subalgebra is then SU(6). The E 6 algebra can be decomposed under the subgroup SU(2) × SU(6) into Xi j,

Yα[i jk] ,

Z αβ ,

(1.10)

where i, j, k = 1, . . . , 6 are the SU(6) indices, α, β = 1, 2 those for SU(2). Here X ij i jk

and Z αβ are adjoints of SU(6) and SU(2) respectively, and Yα transforms as the threeindex anti-symmetric tensor of SU(6) times the doublet of SU(2). The minimal nilpotent orbit is then given by the simultaneous zero locus of quadratic equations in X , Y and Z which we describe in detail later. 1 One can of course identify g∗ and g using the Killing form, but it is more mathematically natural to C C use the coadjoint representation here.

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393

For now let us see what are the gauge-invariant coordinates of the Higgs branch of the exceptional side. The SU(2) gauge group is identified to the SU(2) ⊂ E 6 just chosen above, i.e. the SU(2) gauge bosons couple to the current of this SU(2) subgroup of the E 6 symmetry. We also have the quarks vα and v˜α in addition to the fields X , Y and Z , and we need to make SU(2)-invariant combinations of them. Moreover, we need to impose the F-term equation, which is Z αβ + v(α v˜β) = 0

(1.11)

as we argue later. Thus, any appearance of Z inside a composite operator can be eliminated in favor of v and v. ˜ Therefore we have the following natural gauge-invariant composites, from which all gauge-invariant operators can be generated as will be shown in Sec. 5.4: (v v), ˜

Xi j,

(Y i jk v),

(Yi jk v). ˜

(1.12)

Here we defined (uw) ≡ u α wβ αβ

(1.13) i jk

for two doublets u α and wα , and Yi jk,α is defined by lowering the indices of Yα by the epsilon tensor, see Appendix A. This suggests the following identifications between the operators on the two sides of the duality: tr M ↔ (v v), ˜ B

i jk

↔ (Y

i jk

v),

Mˆ i j ↔ X i j , B˜ i jk ↔ (Yi jk v), ˜

(1.14) (1.15)

where Mˆ ij is the traceless part of M ij . The identifications preserve the dimensions of the operators if we assign dimensions 2 to the fields X , Y and Z . The SU(6) transformation nicely agrees. The U(1) part of the flavor symmetry can be matched if one assigns charge ˜ and charge ±3 to v, v. ±1 to Q, Q, ˜ This factor of 3 was predicted in the original paper [1] from a totally different point of view, by demanding that the two-point function of two U(1) currents should agree under the duality. Let us quickly recall how it was derived there. The form of the twopoint function of the U(1) current jµ is strongly constrained by the conservation and the conformal symmetry, and we have jµ (x) jν (0) ∝ k

x 2 gµν − 2xµ xν + ··· . x8

(1.16)

k is called the central charge, and · · · stands for less singular terms. Let us normalize k such that one hypermultiplet of charge q contributes q 2 to k. Assign Q, Q˜ the charge ±1, and let the charge of v, v˜ be ±q. Then k calculated from the SU(3) side is 6 × 3 = 18, while k determined from the exceptional side is 2q 2 . Equating these, Argyres and Seiberg concluded that the charge of v, v˜ should be q = ±3. The agreement is already impressive at this stage, but we would like to see how the constraints are mapped. We would also like to study how the hyperkähler structures agree, because so far we considered the Higgs branch only as a complex manifold. For that purpose we need to recall more about the hyperkähler cone.

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The structure of the rest of the paper is as follows: We discuss in Sec. 2 what data are mathematically necessary to show the equivalence of the Higgs branches. Section 3 is devoted to the description of the minimal nilpotent orbit, i.e. the 1-instanton moduli space, as a hyperkähler space. Sections 4 and 5 will be spent in calculating the necessary data on the SU(3) side and the exceptional side, respectively. Then they are compared in Sec. 6 which shows remarkable agreement. We conclude in Sec. 7. We have four Appendices: Appendix A collects our conventions, Appendix B gathers the machinery of twistor spaces required to show the equivalence of hyperkähler cones, and Appendix C compares the Kähler potentials of the duality pair. Appendix D is a summary for mathematicians. 2. Rudiments of Hyperkähler Cones Here we collect the basics of the hyperkähler cones in a physics language. Mathematically precise formulation can be found in [7,8]. The Higgs branch M of an N = 2 gauge theory is a hyperkähler manifold, i.e. one has three complex structures J 1,2,3 satisfying J 1 J 2 = J 3 , compatible with the metric g, and the associated two-forms ω1,2,3 are all closed. We choose a particular N = 1 supersymmetry subgroup of the N = 2 supersymmetry group, which distinguishes one of the complex structures, say J ≡ J 3 . M is then thought of as a Kähler manifold with the Kähler form ω = ω3 . = ω1 + iω2 is a closed (2, 0)-form on M which then defines a holomorphic symplectic structure on M. Physically this means that the N = 1 chiral ring, i.e. the ring of holomorphic functions on M, has a natural holomorphic Poisson bracket [ f 1 , f 2 ] = ( −1 )i j ∂i f 1 ∂ j f 2

(2.1)

for two holomorphic functions f 1,2 on M. Second, we are dealing with the Higgs branch of an N = 2 superconformal theory, which has the dilation and the SU(2) R symmetry built in the symmetry algebra. The dilation makes M into a cone with the metric 2 2 dsM = dr 2 + r 2 dsbase ,

(2.2)

and SU(2) R symmetry acts on the base of the cone as an isometry, rotating the three complex structures as a triplet. These two conditions make M into a hyperkähler cone. K = r 2 is a Kähler potential with respect to any of the complex structures J 1,2,3 , and is called the hyperkähler potential in the mathematical literature. The dilatation assigns the scaling dimensions, or equivalently the weights, to the chiral operators on M. Let us consider an element of SU(2) R which acts on the three complex structures as (J 1 , J 2 , J 3 ) → (J 1 , −J 2 , −J 3 ). This element defines an anti-holomorphic involution σ : M → M because it reverses J ≡ J 3 . This induces an operation σ ∗ on holomorphic functions on M via (σ ∗ ( f ))(x) ≡ f (σ (x)). σ ∗ maps holomorphic functions to anti-holomorphic functions, but is a linear operation, not a conjugate-linear operation. We call this operation the conjugation. As will be detailed in Appendix B, the space M as a complex manifold, with the Poisson brackets, the scaling weights and the conjugation, almost suffices to reconstruct the hyperkähler metric on M. Therefore, our main task in checking the agreement of the Higgs branches of the duality pair is to identify them as complex manifolds, and to show that the extra data defined on them also coincide. In order to complete the proof we need to show that the families of the twistor lines coincide, which we have not been

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able to do. Instead we will give numerical support by calculating the Kähler potential directly on both sides in Appendix C. The Higgs branches M that we treat here are gauge theory moduli spaces. They can be described by the hyperkähler quotient construction [9], which we now review. Let us start with an N = 2 gauge theory with the gauge group G, whose hypermultiplets take value in the hyperkähler manifold X . The action of G on X preserves three Kähler structures, and thus there are three moment maps µas (s = 1, 2, 3; a = 1, . . . , dim G) which satisfy dµas = ιξ a ωs ,

(2.3)

where ξ a is the Killing vector associated to the a th generator of G. The Higgs branch of the gauge theory, in the absence of any non-zero Fayet-Iliopoulos parameter, is then given by M ≡ X ///G ≡ {x ∈ X µas (x) = 0}/G. (2.4) With one complex structure J = J 3 chosen, it is convenient to call D a = µa3 ,

F a = µa1 + iµa2 .

(2.5)

M = {x ∈ X F a = 0}/G C .

(2.6)

Then, as a complex manifold,

It is instructive to note that F a is exactly the Hamiltonian which generates the G action on the chiral ring of X , under the Poisson bracket associated to = ω1 + iω2 . The conjugation σ ∗ and the Poisson bracket [·, ·] on the quotient M are given by the restriction of the corresponding operations on X . It is instructive to see why the Poisson bracket of the quotient is well-defined: two G-invariant holomorphic functions f 1,2 on X lead to the same function on M if and only if f 1 = f 2 + u a F a with holomorphic functions u a . Then we have, for a G-invariant holomorphic function h, [ f 1 , h] − [ f 2 , h] = [u a F a , h] = [u a , h]F a + u a [F a , h]

(2.7)

on X . The first term in the right hand side is zero on M because we set F a = 0, while the second term is zero because h is G-invariant. Therefore [ f 1 , h] and [ f 2 , h] determine the same holomorphic function on M. The Kähler potential of M is similarly the restriction of that of X to the zero locus of the moment maps in our situation, as discussed in Sec. 2B of [9]. To illustrate the procedure, let us consider an N = 1 supersymmetric U(1) gauge theory coupled to chiral fields i of charge qi whose Lagrangian is 4 ∗ 2qi V L= d θ i e i + ξ V , (2.8) i

where ξ is the Fayet-Iliopoulos parameter. The moduli space can be determined by taking the gauge coupling to be formally infinite, i.e. treating the linear superfield V as an auxiliary field. Then V is determined via its equation of motion qi i∗ e2qi V i + ξ = 0, (2.9) i

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i.e. i = eqi V i solve the usual D-term equation. The Kähler potential of the moduli space is then given by plugging the solution to (2.9) into (2.8). It can be generalized to any gauge group, and the result agrees with the mathematical formula given in Sec. 3.1 of [10]. This shows that the Kähler potential is given just by the restriction of the original one if ξ = 0. This analysis does not incorporate quantum corrections, but it is wellknown that for N = 2 theories the quantum effect does not modify the hyperkähler structure, see Sec. 3 of [11]. 3. Geometry of the Minimal Nilpotent Orbit Here we gather the relevant information on the hyperkähler geometry of the minimal nilpotent orbit of any simple group G, which coincides with the centered moduli space of single instantons with gauge group G [5,8]. We hope this section might be useful for anyone who wants to deal with the one-instanton moduli space. In the following G stands for a compact simple Lie group, gR its Lie algebra. We let G C and gC be complexifications of G and gR respectively. The existence of a uniform description of the one-instanton moduli space applicable to any G might be understood as follows: we can construct a one-instanton configuration easily by taking a BPST instanton of SU(2) and regard it as an instanton of G via a group embedding SU(2) ⊂ G. It is known that any one-instanton of G arises in this manner [12]. The one-instanton moduli space is then parameterized by the position, the size, and the gauge orientation of the BPST instanton inside G. This description realizes the one-instanton moduli space as a cone over a homogeneous manifold G/H , where H is the maximal subgroup of G which commutes with the SU(2) used in the embedding. It is however not directly suitable for the analysis of its complex structure. For that purpose we use another realization of the one-instanton moduli space as the minimal nilpotent orbit Omin of G [5]. Let us define Omin . First we decompose gC into the Cartan generators H i and the raising/lowering operators E ±ρ for roots ρ. The minimal nilpotent orbit Omin (G) is then the orbit of (E θ )∗ in g∗C , where θ denotes the highest root: Omin (G) = G C · (E θ )∗ ⊂ g∗C .

(3.1)

We will write Omin without explicitly writing G for the sake of simplicity when there is no confusion. We think of elements of gC as holomorphic functions on Omin , i.e. we have holomorphic functions2 Xa (a = 1, . . . , dim G ) on Omin . The defining equations of Omin are a set of quadratic equations which we call the Joseph relations [6].3 These relations can be studied using a theorem of Kostant [13]: Let V (α) denote the representation space of a semisimple group G with the highest weight α, and let v ∈ V (α)∗ be a vector in the highest weight space. The orbit G C · v is then an affine algebraic variety whose defining ideal I is generated by its degree-two part I2 . Furthermore, I2 is given by the relation Sym2 V (α) = V (2α) ⊕ I2 ,

(3.2)

∗ 2 More mathematically, one has a natural holomorphic g∗ -valued function X : O min → gC given by C the embedding. Then every element t ∈ gC gives a holomorphic function (X, t) on Omin via x ∈ Omin → (X(x), t). Our Xa is (X, T a ) for a generator T a of gC . We take a real basis of gC , so in fact T a ∈ gR ⊂ gC . 3 Strictly speaking, the Joseph ideal is a two-sided ideal in the universal enveloping algebra of g , and C what we use below is its associated ideal in the polynomial algebra.

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where we identify Sym2 V (α) as the space of degree-two polynomials on V (α)∗ . The minimal nilpotent orbit is exactly of this form where V (α) is the adjoint representation, i.e. Omin = {X ∈ g∗C (X ⊗ X)|I2 = 0}. (3.3) For practice, let us apply this to the case G = SU(2). There, V (α) is the triplet representation, so by (3.2) I2 is the singlet representation. Therefore, if we parameterize su(2) by (a, b, c), the minimal nilpotent orbit is given by the equation a 2 + b2 + c2 = 0,

(3.4)

which is C2 /Z2 as it should be. Now that we have given Omin as a complex manifold, let us describe its hyperkähler structure. The main fact we use is that G acts isometrically on Omin , preserving the hyperkähler structure. There is a triplet of moment maps µas for this action where a = 1, . . . , dim G and s = 1, 2, 3. The functions Xa are the holomorphic moment maps of the G action, i.e. Xa = µa1 + iµa2 . It follows that their Poisson bracket is [Xa , Xb ] = f ab c Xc ,

(3.5)

where f ab c are the structure constants of G. Phrased differently, the holomorphic symplectic structure underlying the hyperkähler structure of the nilpotent orbit is the standard Kirilov–Kostant–Souriau symplectic form on the coadjoint orbit [5,8]. The conjugation is given by the SU(2) R action, which sends (µ1 , µ2 , µ3 ) to (µ1 , −µ2 , −µ3 ). Therefore σ ∗ (Xa ) = (Xa )∗ .

(3.6)

The scaling dimension of X is fixed to be two, as it should be for the F-term in an N = 2 supersymmetric theory. Let us next describe a Kähler potential for Omin , which was determined in [14]. The derivation boils down to the following: G acts on Omin with cohomogeneity one, and by averaging over this action we can consider K to be G-invariant; so K is a √ function of tr XX∗ . K should be of scaling dimension two, so that K is proportional to tr XX∗ up to a constant. The constant factor can be fixed by considering a particular element on Omin . For this purpose we again turn to the minimal nilpotent orbit of SU(2), which is C2 /Z2 . The normalization of the Kähler potential of the minimal nilpotent orbit of a general group can then be determined because it contains the minimal nilpotent orbit of SU(2) as a subspace. We parameterize C2 by (u, u) ˜ and divide by the multiplication by −1. We define our conventions for the holomorphic Poisson bracket and the Kähler potential of a flat H as follows: K = |u|2 + |u| ˜ 2,

[u, u] ˜ = 1.

(3.7)

Now, C2 /Z2 is parametrized by Z 11 = u 2 /2,

Z 12 = Z 21 = u u/2, ˜

Z 22 = u˜ 2 /2

(3.8)

which satisfy Z 11 Z 22 = Z 12 2 .

(3.9)

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The Kähler potential is now K = 2 |Z 11 |2 + |Z 22 |2 + 2|Z 12 |2 = 2 Z αβ Z¯ αβ .

(3.10)

Then, the moment map associated to the generator J3 of non-R SU(2) acting on C2 /Z2 can be explicitly calculated, with the result F = Z 12 ,

D=

2 (|Z 11 |2 − |Z 22 |2 ). K

(3.11)

Now that the preparation is done, we move on to the calculation of the Higgs branch on both sides of the duality. 4. SU(3) Side The theory has six quarks in the fundamental representation, Q ia ,

Q˜ ia ,

(4.1)

where a = 1, . . . , 6 and i = 1, 2, 3. As is well known, any SU(3)-invariant polynomial constructed out of these fields is a polynomial in the operators [11]: M i j = Q ia Q˜ aj ,

j

B i jk = abc Q ia Q b Q kc ,

B˜ i jk = abc Q˜ ia Q˜ bj Q˜ ck .

(4.2)

In the following we study the Poisson brackets, the action of the conjugation, and the constraints in turn. 4.1. Poisson brackets. The Poisson bracket of the basic fields is given by [Q ia , Q˜ bj ] = δ i j δ b a .

(4.3)

[M ij , Q ak ] = −δ kj Q ia ,

(4.4)

Then we have, for example,

i.e. M ij is the generator of U(6). We define tr M to be the trace of M i j , and 1 Mˆ ij = M ij − δ ij tr M 6

(4.5)

is its traceless part. Mˆ i j is the SU(6) generator and tr M the U(1) generator. We define the U(1) charge q of an operator O to be given by [tr M, O] = −qO.

(4.6)

The most complicated bracket is [B i jk , B˜ lmn ] = 18M [i [l M j m δ k] n] 1 = 18 Mˆ [i [l Mˆ j m δ k] n] + 6(tr M) Mˆ [i [l δ j m δ k] n] + (tr M)2 δ [i [l δ j m δ k] n] . 2 (4.7)

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4.2. Conjugation. We choose the involution on the elementary fields to be σ ∗ (Q ia ) = ( Q˜ ia )∗ ,

σ ∗ ( Q˜ ia ) = −(Q ia )∗ .

(4.8)

Then the transformation of the composites are σ ∗ (M ij ) = −(M j i )∗ ,

σ ∗ (tr M) = −(tr M)∗ ,

(4.9)

σ ∗ (B i jk ) = ( B˜ i jk )∗ ,

σ ∗ ( B˜ i jk ) = −(B i jk )∗ .

(4.10)

4.3. Constraints. The constraints were studied in [11]. Those which come before imposing the F-term constraint are B i jk B˜ lmn = 6M [i l M j m M k] n , B i j[k B lmn] = 0, B˜ i j[k B˜ lmn] = 0, M [i j B klm] = 0, M i [ j B˜ klm] = 0.

(4.11) (4.12) (4.13)

The F-term constraint 1 ˜ =0 Q ia Q˜ ib − δab (Q Q) 3

(4.14)

Mˆ ij B jkl = 16 (tr M)B ikl ,

(4.15)

Mˆ ij B˜ ikl = 16 (tr M) B˜ jkl ,

(4.16)

Mˆ ij M j k

(4.17)

further imposes

=

1 6 (tr

M)Mki .

We will find it convenient later to have constraints in terms of irreducible representations (irreps) of SU(6). We use the Dynkin labels to distinguish the irreps in the following. The M B = 0 relations (4.13), (4.15), (4.16) give Mˆ {i l B [ jk]}l = 0, Mˆ l {i B[ jk]}l = 0,

Mˆ {i l B˜ [ jk]}l = 0, Mˆ l {i B˜ [ jk]}l = 0.

(4.18) (4.19)

Here we defined the projector from a tensor with the structure Ai[ jk] to the irrep (1, 1, 0, 0, 0) by A{i[ jk]} ≡ Ai[ jk] − A[i[ jk]] . We also have 1 Mˆ [i l B jk]l = (tr M)B i jk , 6

1 Mˆ l [i B˜ jk]l = (tr M) B˜ i jk . 6

(4.20)

The M M = 0 relation (4.17) gives 1 j Mˆ ij Mˆ k = δki Mˆ nm Mˆ mn , 6 1 j i Mˆ j Mˆ i = (tr M)2 . 6

(4.21) (4.22)

The B B = 0 relation (4.12) gives B ikl B jkl = 0,

B˜ ikl B˜ jkl = 0.

(4.23)

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Finally, the decomposition of the B B˜ = M M M relation gives, using (4.21) and (4.22) repeatedly, 2 (tr M)3 , 9 2 = (tr M)2 Mˆ ij , 9 2 = (tr M) Mˆ [i [k Mˆ j] l] 0,1,0,1,0 , 3 j = 6 Mˆ [i l Mˆ m Mˆ k] n 0,0,2,0,0 .

B i jk B˜ i jk = B ikl B˜ jkl adj B i jm B˜ klm 0,1,0,1,0 B i jk B˜ lmn

0,0,2,0,0

(4.24) (4.25) (4.26) (4.27)

5. Exceptional Side 5.1. Poisson brackets. We have chiral fields Xa which transform in the adjoint of E 6 , and satisfy the quadratic Joseph identities. We decompose Xa under the subgroup SU(2) × SU(6) ⊂ E 6 . It gives X ij , Yα[i jk] ,

Z αβ ,

(5.1) i jk

where X ij and Z αβ are the adjoints of SU(6) and SU(2) respectively, and Yα is in the doublet of SU(2) and in the representation (0, 0, 1, 0, 0), i.e. the three-index antisymmetric tensor, of SU(6). The Poisson brackets of the fields X , Y and Z are exactly the Lie brackets as explained above, which we take to be

[Z αβ , Z γ δ ] =

[X ij , X lk ] = δli X kj − δ kj X li ,

(5.2)

1 (αγ Z βδ + βγ Z αδ + αδ Z βγ + βδ Z αγ ) 2

(5.3)

and 1 [X ij , Yαklm ] = −3δ [k j Yαlm]i + δ ij Yαklm , 2 i jk [Z αβ , Yγi jk ] = Y(α β)γ ,

(5.4) (5.5)

and finally [Yαi jk , Yβlmn ] = i jklmn Z αβ −

3 αβ (X [i p jk]lmnp + X [l p mn]i jkp ). 2

(5.6)

The final commutation relation can also be written as [Yαi jk , Ylmn β ] = −18X [i [l δ j m δ k] n] − 6Z αβ δ [i [l δ j m δ k] n] .

(5.7)

As we explained above, X , Y and Z are the holomorphic moment maps of the E 6 action. Therefore the contribution from Omin to the F-term constraint for the SU(2) gauge group is given just by Z αβ . We take the bracket of v and v˜ to be [vα , v˜β ] = αβ .

(5.8)

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401

Then we have [v(α v˜β) , vγ ] = v(α β)γ ,

(5.9)

and [(v v), ˜ vα ] = vα ,

[(v v), ˜ v˜α ] = −v˜α .

(5.10)

Recall that we define (uw) ≡ u α wβ αβ for two doublets u α and wα . It is straightforward to check that v(α v˜β) is the moment map of the SU(2) action on v and v. ˜ Thus the F-term condition is v(α v˜β) + Z αβ = 0.

(5.11)

5.2. Conjugation. We take the conjugation on the variables v, v˜ to be σ ∗ (vα ) = (v˜β )∗ αβ ,

σ ∗ (v˜α ) = (vβ )∗ αβ .

(5.12)

i jk

In terms of our variables (X ij , Yα , Z αβ ), the conjugation acts as follows: σ ∗ (X ij ) = −(X j i )∗ , σ ∗ (Yαi jk ) = (Yi jk β )∗ αβ , ∗

(5.13)

σ ∗ (Yi jk α ) = −(Yβ )∗ αβ , i jk

∗

σ (Z αβ ) = (Z γ δ ) αγ βδ .

(5.14) (5.15)

5.3. Constraints. As explained in Sec. 3, the Joseph relations are given by (X ⊗ X)|I2 = 0,

(5.16)

Sym2 V (adj) = V (2adj) ⊕ I2 .

(5.17)

where I2 is given by the relation

Here, V (adj) is the adjoint representation of E 6 whose Dynkin label is adj = 0 0 01 0 0 . We then have (5.18) I2 = V 1 0 00 0 1 ⊕ V 0 0 00 0 0 . The representations which appear in I2 , decomposed under SU(2) × SU(6), are summarized in Table 1. The table reads as follows: e.g. for relation 4, the fourth column tells us there is one Joseph identity transforming as a doublet in SU(2) and as (0, 0, 1, 0, 0) under SU(6), but the fifth column says one can construct two objects in i jk this representation from bilinears in X ij , Yα and Z αβ . This means the identity has the form 0 = Yαi jk Z βγ αβ + c4 X [i l Yγjk]l ,

(5.19)

where c4 needs to be fixed, which can be done e.g. by explicitly evaluating the right hand side on a few elements on the nilpotent orbit. Elements on the nilpotent orbit can be readily generated, because one knows that the point X ij = 0,

Yαi jk = 0,

Z 11 = 1,

Z 12 = Z 22 = 0

(5.20)

402

D. Gaiotto, A. Neitzke, Y. Tachikawa Table 1. Decomposition of I2 in terms of SU(2) × SU(6) ⊂ E 6

1. 2. 3. 4. 5. 6. 7.

SU(2)

SU(6)

in I2

in Sym2 V (adj)

3 2 2 2 1 1 1

(1,0,0,0,1) (1,1,0,0,0) (0,0,0,1,1) (0,0,1,0,0) (0,1,0,1,0) (1,0,0,0,1) (0,0,0,0,0)

1 1 1 1 1 1 2

2 1 1 2 2 1 3

is on the nilpotent orbit by definition. Then the rest of the points can be generated by the coadjoint action of E 6 , which can be obtained by exponentiating the structure constants. Carrying out this program, we obtain the following full set of Joseph identities: 1. 2. 3.

1 ikl 0 = X ij Z αβ + Y(α Y jklβ) , 4 0 = X l {i Y[ jk]}lα , 0=

4.

0=

5.

0=

6. 7. 7’.

X l Yα[ jk]}l , Yαi jk Z βγ αβ

(5.21) (5.22)

{i

(5.23) [i

+ X l Yγjk]l , (Yαi jm Yklmβ αβ − 4X [i [k X j] l] ) 0,1,0,1,0 ,

1 0 = X ki X k j − δ ij X k l X l k , 6 0 = Yαi jk Yi jkβ αβ + 24Z αβ Z γ δ αγ βδ , 0=

X ij X j i

+ 3Z αβ Z γ δ

αγ βδ

.

(5.24) (5.25) (5.26) (5.27) (5.28)

5.4. Gauge invariant operators. Let us enumerate the generators of the SU(2)-invariant i jk operators constructed out of vα , v˜α , and X ij , Yα , Z αβ , using the F-term equation (5.11) and the Joseph identities (5.21) ∼ (5.28). Suppose we have a monomial constructed from those fields. We first replace every appearance of Z αβ by −v(α v˜β) . All the SU(2) indices are contracted by epsilon tensors of SU(2). Therefore the monomial is a product of X ij , (v v), ˜ (Y i jk v), (Y i jk v) ˜ and (Y i jk Y lmn ). The last of these can be eliminated using the Joseph identities. Indeed, the combination of the relations (5.25), (5.27) and (5.28) gives a Joseph identity of the form Yαi jk Ylmn β αβ = 18X [i [l X j m δ k] n] − 3Z αβ Z γ δ αγ βδ δ [i [l δ j m δ k] n] .

(5.29)

We conclude that any SU(2)-invariant polynomial is a polynomial in ˜ (Y i jk v), and (Y i jk v). ˜ X ij , (v v),

(5.30)

6. Comparison 6.1. Identification of operators. Let us now proceed to the comparison of the structures we studied in Sec. 4 and in Sec. 5. We first make the following identification: Mˆ ij = X ij ,

tr M = −3(v v). ˜

(6.1)

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403

These are the moment maps of the flavor symmetries SU(6) and U(1), so the identification is fixed including the coefficients, and then the Poisson brackets involving either Mˆ ˜ also agrees with that or tr M automatically agree. The conjugation acting on X ij , (v v) i ˆ on M j and tr M. We then set B i jk = c(Y i jk v),

B˜ i jk = c(Y ˜ i jk v). ˜

(6.2)

One has σ ((Y i jk v)) = (Yi jk v) ˜ ∗.

(6.3)

To be consistent with (4.10), we need to have c˜ = c∗ .

(6.4)

Let us then calculate the Poisson bracket of (Y i jk v) and (Ylmn v) ˜ using (5.29). We have 9 ˜ 2 . (6.5) ˜ = −18X [i [l X j m δ k] n] + 18(v v)X ˜ [i [l X j m δ k] n] − (v v) [(Y i jk v), (Ylmn v)] 2 Comparing with the bracket [B i jk , B˜ lmn ] calculated in (4.7), we find they indeed agree if cc˜ = −1. Thus we conclude c = c˜ = i, i.e. B i jk = i(Y i jk v),

B˜ i jk = i(Yi jk v). ˜

(6.6)

6.2. Constraints. Now, let us check using the Joseph relations that the constraints on the SU(3) side, listed in Eqs. (4.18) ∼ (4.27), can be correctly reproduced on the exceptional side. • • • • • •

(4.18): Contract v or v˜ to the relation 2, (5.22). (4.19): Contract v or v˜ to the relation 3, (5.23). (4.20): Contract v or v˜ to the relation 4, (5.24). (4.21): This is exactly the relation 6, (5.26). (4.22): This is exactly the relation 7’, (5.28). (4.23): Contract vα vβ or v˜α v˜β to the relation 1 (5.21).

As for the relation of the type B B˜ = M M M, • • • •

(4.24): The singlet part. Contract vα v˜β to the relation 7 (5.27). (4.25): The adjoint part. Contract vα v˜β to the relation 1 (5.21). (4.26): The (0, 1, 0, 1, 0) part. Contract vα v˜β to the relation 5 (5.25). (4.27):This is the (0, 0, 2, 0, 0) part and is slightly trickier, but it follows from a cubic Joseph identity 0 = αγ βδ Z αβ Yγi jk Ylmn,δ 0,0,2,0,0 −6X [i l X j m X k] n 0,0,2,0,0 (6.7) upon replacing Z αβ with v(α v˜β) . This cubic Joseph identity itself can be derived from the quadratic Joseph identities, as it should be. First, we use the relation 4 (5.24) to show αγ βδ Z αβ Yγi jk Ylmn δ 0,0,2,0,0 ∝ X [i p Yαjk] p Ylmn β αβ . (6.8)

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Now, the antisymmetric product of two Y ’s contain both the singlet and the (0, 1, 0, 1, 0) part. One sees the singlet drops out inside the projector to the (0, 0, 2, 0, 0) part, so we have ∝ X [i p (Y jk] p Ylmn ) 0,1,0,1,0 0,0,2,0,0 . (6.9) Then we use the relation 5 (5.25) to transform this to ∝ X li X j m X k n 0,0,2,0,0 .

(6.10)

The proportionality constant can be fixed, e.g. by evaluating on a few points on the orbit. This concludes the comparison of the constraints. 7. Conclusions In the previous three sections, we determined the Higgs branches both on the SU(3) side and on the exceptional side. We demonstrated that their defining equations agree, and furthermore exhibited that the Poisson bracket and the conjugation are the same on both sides. As was stated in Sec. 2 and will be detailed in Appendix B, these are (almost) sufficient to conclude that they are the same as hyperkähler manifolds. To remove any remaining doubts, we compare the Kähler potentials of the two sides in Appendix C. Again, they show remarkable agreement with one another. Thus we definitely showed the agreement of the Higgs branches of the new S-duality pair proposed by Argyres and Seiberg in [1], which provides a convincing check of their conjecture. In this paper we only dealt with the example involving E 6 , but there are more examples of similar dualities in [1 and 4]. It would be interesting to carry out the same analysis of the Higgs branches to those examples. A pressing issue is to understand the Argyres-Seiberg duality more fully. For example, it would be nicer to have an embedding of this duality in string/M-theory. We hope to revisit these problems in the future. Acknowledgements. The authors thank Alfred D. Shapere for collaboration at an early stage of the project. They would also like to thank S. Cherkis, C. R. LeBrun, H. Nakajima for helpful discussions. They also relied heavily on the softwares LiE4 and Mathematica. DG is supported in part by the DOE grant DE-FG0290ER40542 and in part by the Roger Dashen membership in the Institute for Advanced Study. AN was supported in part by the Martin A. and Helen Chooljian Membership at the Institute for Advanced Study, and in part by the NSF under grant numbers PHY-0503584 and PHY-0804450. YT is supported in part by the NSF grant PHY-0503584, and in part by the Marvin L. Goldberger membership in the Institute for Advanced Study.

A. Conventions Greek indices α, β are for the doublets of SU(2), a, b, c, . . . for the triplets of SU(3) and i, j, k, . . . for the sextets of SU(6). We define (uw) ≡ u α wβ αβ for two doublets u α and wα , 4 It can be downloaded from http://www-math.univ-poitiers.fr/~maavl/LiE/.

(A.1)

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405

We use the following sign conventions for the epsilon tensors of SU(2), SU(3) and SU(6): αβ = −αβ , abc = abc , i jklmn = i jklmn .

(A.2)

We normalize the antisymmetrizer [abc...] and the symmetrizer (abc...) so that they are projectors, i.e. T i jk = T [i jk]

(A.3)

for the antisymmetric tensors T i jk , etc. We raise and lower three antisymmetrized indices of SU(6) via the following rule: T i jk =

1 i jklmn Tlmn , 6

Tlmn =

1 i jk T i jklmn . 6

(A.4)

Our convention for the placement of the indices of the complex conjugate is e.g. Z¯ αβ ≡ (Z αβ )∗ ,

(A.5)

i.e. the complex conjugation is always accompanied by the exchange of subscripts and superscripts, as is suitable for the action of SU groups. We take the Kähler potential of a flat C parameterized by z with the standard metric to be K = |z|2 .

(A.6)

B. Twistor Spaces of Hyperkähler Cones Recall that a hyperkähler manifold M admits a continuous family of complex structures Jζ , parameterized by ζ ∈ CP1 . The full information in the hyperkähler metric is captured by this family of complex structures and their Poisson brackets. It can be encoded into purely holomorphic data on a complex manifold Z, the twistor space of M, as we now review. Topologically Z = M × CP1 . Its complex structure can be specified by specifying which functions on Z are holomorphic: they are f (x, ζ ) which are holomorphic in ζ for fixed x ∈ M, and also holomorphic in x with respect to complex structure Jζ for fixed ζ . Hence we may view Z as a holomorphic fiber bundle over CP1 , where the fiber over ζ is just a copy of M, equipped with complex structure Jζ . The Poisson brackets on the holomorphic functions in each fiber glue together globally to give a bracket operation on Z. This bracket operation is globally twisted by the line bundle O(−2): i.e. given local holomorphic functions f 1 , f 2 we get a local section { f 1 , f 2 } of O(−2), and more generally if f 1 , f 2 are sections of O(d1 ), O(d2 ) then { f 1 , f 2 } is a section of O(d1 + d2 − 2). Finally there is an involution σ on Z, simply defined by (x, ζ ) → (x, −1/ζ¯ ). This is an antiholomorphic involution, since the complex structure Jζ is opposite to J−1/ζ¯ . As a complex manifold Z is a fibration over CP1 , and (x, ζ ) with x fixed gives a holomorphic section of this fibration, which is invariant under σ . The normal bundle to this section is isomorphic to the line bundle O(1)⊕n , where n is the complex dimension of M. Conversely, a holomorphic section of Z which is invariant under σ and whose normal bundle is isomorphic to O(1)⊕n is called a twistor line. Therefore, the points on M give rise to a n-dimensional family of twistor lines on Z.

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It was shown in [9] that given Z, together with its Poisson brackets and antiholomorphic involution, one can canonically reconstruct a hyperkähler metric on the space of twistor lines. Therefore, to check that our two hyperkähler cones are the same is essentially to check that their twistor spaces Z are the same. Now, the twistor space of a hyperkähler cone can be constructed from the data we described in Sec. 2, i.e. the Poisson bracket, the dilatation and the conjugation on M. We pick one complex structure induced from the hyperkähler structure, and regard M as a complex manifold. We then form Z as a complex manifold as Z = ((C2 \ (0, 0)) × M)/C× ,

(B.1)

where C× acts on the first factor by multiplication, and on the second factor as the natural

complexification of the action of the dilatation. Then the Poisson bracket on M naturally induces one on Z. We define σ on Z to send (z, w, x) ∈ C2 ×M to (−w, ¯ z¯ , σ (x)). Then it is straightforward to check that this Z is the twistor space of M, using the SU(2) R action on M rotating three complex structures. There is a subtle problem remaining, however. Namely, the theorem in [9] asserts that there is a component of the space of the twistor lines of Z which agrees metrically with the original hyperkähler manifold M, but does not exclude the possibility that the space of twistor lines has many components, each of which is a hyperkähler manifold with the same complex structure but with a different metric. Mathematicians the authors consulted know no concrete example where this latter possibility is realized, so the authors think it quite unlikely that our two hyperkähler manifolds are the same as holomorphic symplectic manifolds but not as hyperkähler manifolds. To dispel this last possibility, in the next appendix we directly compare the Kähler potential of our two hyperkähler manifolds. C. Comparison of the Kähler Potential In this Appendix, we describe the method to calculate and compare the Kähler potential of the Higgs branches on the two sides of the duality. C.1. Exceptional side. The invariant norm of E 6 in our notation is 1 Z αβ Z¯ αβ + Yαi jk Y¯iαjk + X ij X¯ j i . 6 Therefore the correctly normalized Kähler potential is

1 i jk K E 6 = 2 Z αβ Z¯ αβ + Yα Y¯iαjk + X ij X¯ j i , 6

(C.1)

(C.2)

and the D-term for the SU(2) ⊂ E 6 is 2 1 i jk ¯ γ i jk ¯ γ (E 6 ) γδ γδ ¯ ¯ Z αγ Z δβ + Z βγ Z δα + (Yα Yi jk γβ + Yβ Yi jk γ α ) . Dαβ = K E6 12 (C.3) We also have quarks vα , v˜α which have (|vα |2 + |v˜α |2 ) K v,v˜ = α

(C.4)

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407

and (v,v) ˜

Dαβ

=

1 (vα v¯ γ γβ + vβ v¯ γ γ α + v˜α v¯˜ γ γβ + v˜β v¯˜ γ γ α ). 2

(C.5)

The Kähler potential of the exceptional side is thus given by K v,v˜ + K E 6

(C.6)

restricted to the locus v(α v˜β) + Z αβ = 0,

(v,v) ˜

Dαβ

(E )

+ Dαβ 6 = 0

(C.7)

expressed as a function of M ij , B i jk , B˜ i jk and their complex conjugates. C.2. SU(3) side. We start from the Kähler potential |Q ia |2 + | Q˜ a |2 . K =

(C.8)

i

i,a

Using the analysis in [11], the Kähler potential on the quotient was determined in [15] as

ν2 (C.9) K =2 m i2 + , 4 i=1,2,3

where (m 21 , m 22 , m 23 , 0, 0, 0) are the eigenvalues of M ij M¯ j k , and ν is defined by 3ν =

|Q ia |2 − | Q˜ ia |2 ,

(C.10)

i,a

i.e. 1/3 of the U(1) D-term. In terms of gauge invariants we have ⎛ ⎞ 2 ν ν ⎝ m2 + + ⎠ = 16 B i jk B¯ i jk , i 4 2 i=1,2,3 ⎛ ⎞ 2 ν ν ⎝ m2 + − ⎠ = 16 B˜ i jk B¯˜ i jk . i 4 2

(C.11)

(C.12)

i=1,2,3

C.3. Comparison. Now, the Kähler potentials of the two sides, (C.6) and (C.9) should ˜ but we have not been able to check that analytically. agree as functions of M, B and B, Instead, one can check it numerically on as many points on the quotient as computer time allows. The algorithm is as follows: i jk

1. Generate a point X = (X ij , Yα , Z αβ ) on the nilpotent orbit of E 6 , by applying an i jk

element of the group E 6 to the point (Z 11 , Z 12 , Z 22 ) = (1, 0, 0), X ij = Yα

= 0.

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2. Find vα , v˜α which satisfy v(α v˜β) + Z αβ = 0.

(C.13)

˜ This is more or less unique up to C× action on v, v. 3. Apply SL(2, C) action to (v, v, ˜ X) to find the solution of the D-term equation, (v,v) ˜

Dαβ

(E )

+ Dαβ 6 = 0.

(C.14)

This is equivalent to the minimization of ˜ + K E6 (g(X)), K v,v˜ (g(v), g(v))

(C.15)

where g is an SL(2, C) action. 4. Form M, B, B˜ from v, v˜ and X thus obtained, and calculate ν and m i . At this point, two checks of the sanity of the calculation are possible. One is to see that three eigenvalues of M M¯ are zero. Another is to see that ν determined from (C.11), (C.12) is equal to ν= |vα |2 − |v˜α |2 . (C.16) α

The latter fact follows from the identification of ν as 1/3 of the U(1) moment map on the quotient. 5. Evaluate the Kähler potential of the SU(3) side using (C.9) and compare it to that of the exceptional side (C.6). We implemented the algorithm above in Mathematica, and found that the value of the Kähler potential at any points agrees on both sides of the duality to arbitrary accuracy.5 An analytic proof of the agreement of the Kähler potential will be welcomed. D. Mathematical Summary Let us summarize briefly in the language of mathematics what was done in this paper. Let M(m, n) be M(m, n) = Hom(V, W ) ⊕ Hom(W, V )

where V = Cm , W = Cn

(D.1)

which is a flat hyperkähler space of quaternionic dimension mn. It has a natural triholomorphic action of U(m) × U(n) induced from its action on V and W . Let N (m, n) be the flat hyperkähler space N (m, n) = Rm ⊗R Hn

(D.2)

of quaternionic dimension mn, which has a natural triholomorphic action of SO(m) × Sp(n). One then defines a hyperkähler quotient A by A1 = M(6, 3)///SU(3). 5 We thank H. Elvang for improvement of the accuracy in the calculation.

(D.3)

Argyres-Seiberg Duality and the Higgs Branch

409

We consider another hyperkähler quotient A2 = (N (2, 1) × Omin (E 6 ))///Sp(1)

(D.4)

where Omin (G) is the minimal nilpotent orbit of the group G, and the Sp(1) action on Omin (E 6 ) is given by considering the maximally compact subgroup Sp(1)×SU(6) ⊂ E 6 . One sees easily that A1,2 are both of quaternionic dimension 10, both carry a natural triholomorphic action of SU(6) × U(1). Our claim is that A1 = A2 as hyperkähler cones. We demonstrated that A1 and A2 match as holomorphic symplectic varieties by explicitly showing that their defining equations and the holomorphic symplectic forms are the same. We also found that the twistor spaces of A1 and A2 are the same as complex manifolds with antiholomorphic involution, but could not show that A1 and A2 correspond to the same family of twistor lines. Instead we directly compared the Kähler potentials of A1 and A2 . Again we could not rigorously prove the equivalence, but we performed numerical calculations of the Kähler potential which convinced us that they agree. The equivalence of A1,2 was suggested by the analysis of a new type of S-duality in four-dimensional N = 2 supersymmetric gauge theories in [1]. In [1,4], more examples of the same type of duality were described, of which we record two more here. Now consider B1 = N (12, 2)///Sp(2)

(D.5)

B2 = Omin (E 7 )///Sp(1).

(D.6)

and

Here Sp(1) acts on Omin (E 7 ) through the maximal subgroup Sp(1)×SO(12) ⊂ E 7 . The quaternionic dimension of B1,2 is 14, and both have triholomorphic actions of SO(12). We believe B1 = B2 as hyperkähler cones. For an example which involves Omin (E 8 ), consider C1 = (Z ⊕ N (11, 3))///Sp(3).

(D.7)

Here Z is a pseudoreal irreducible representation of Sp(3) of quaternionic dimension 7, which arises as ∧3C X = Z ⊕ X,

(D.8)

where X = C6 is the defining representation of Sp(3). Let us take another hyperkähler quotient C2 = Omin (E 8 )///SO(5),

(D.9)

where SO(5) acts via embedding SO(5) × SO(11) ⊂ SO(16) ⊂ E 8 .

(D.10)

It is easy to check that C1,2 are both of quaternionic dimension 19, and SO(11) acts triholomorphically on both C1 and C2 . We predict that C1 = C2 as hyperkähler cones.

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References 1. Argyres, P.C., Seiberg, N.: S-duality in N = 2 supersymmetric gauge theories. JHEP 0712, 088 (2007) 2. Minahan, J.A., Nemeschansky, D.: An N = 2 superconformal fixed point with E 6 global symmetry. Nucl. Phys. B 482, 142 (1996) 3. Aharony, O., Tachikawa, Y.: A holographic computation of the central charges of d = 4, N = 2 SCFTs. JHEP 0801, 037 (2008) 4. Argyres, P.C., Wittig, J.R.: Infinite coupling duals of N = 2 gauge theories and new rank 1 superconformal field theories. JHEP 0801, 074 (2008) 5. Kronheimer, P.B.: Instantons and the geometry of the nilpotent variety. J. Diff. Geom. 32, 473 (1990) 6. Joseph, A.: The minimal orbit in a simple Lie algebra and its associated maximal ideal. Ann. Sci. École Norm. Sup. Ser. 4 9, 1 (1976) 7. Swann, A.: Hyperkähler and quaternionic Käher geometry. Math. Ann. 289, 421 (1991) 8. Brylinski, R.: Instantons and Kähler geometry of nilpotent orbits. In: Representation Theories and Algebraic Geometry. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Dordrecht: Kluwer, 1998, pp. 85–125 9. Hitchin, N.J., Karlhede, A., Lindström, U., Roˇcek, M.: Hyperkähler metrics and supersymmetry. Commun. Math. Phys. 108, 535 (1987) 10. Biquard, O., Gauduchon, P.: Hyper-Kähler metrics on cotangent bundles of Hermitian symmetric spaces. In: Lecture Notes in Pure and Appl. Math. 184, Newyork: Dekker, 1997, pp. 287–298 11. Argyres, P.C., Plesser, M.R., Seiberg, N.: The Moduli Space of N = 2 SUSY QCD and Duality in N = 1 SUSY QCD. Nucl. Phys. B 471, 159 (1996) 12. Vainshtein, A.I., Zakharov, V.I., Novikov, V.A., Shifman, M.A.: ABC of instantons. Sov. Phys. Usp. 25, 195 (1982) [Usp. Fiz. Nauk 136, 553 (1982)] 13. Garfinkle, D.: A new construction of the Joseph ideal. MIT thesis, 1982. Available on-line at the service ‘MIT Theses in DSpace.’ http://dspace.mit.edu/handle/1721.1/15620, 1982 (see chap. III) 14. Kobak, P., Swann, A.: The hyperkähler geometry associated to Wolf spaces. Boll. Unione Mat. Ital. Serie 8, Sez. B Artic. Ric. Mat. 4, 587 (2001) 15. Antoniadis, I., Pioline, B.: Higgs branch, hyperkähler quotient and duality in SUSY N = 2 Yang-Mills theories. Int. J. Mod. Phys. A 12, 4907 (1997) Communicated by A. Kapustin

Commun. Math. Phys. 294, 411–437 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0955-5

Communications in

Mathematical Physics

Isometric Immersions and Compensated Compactness Gui-Qiang Chen1 , Marshall Slemrod2 , Dehua Wang3 1 Department of Mathematics, Northwestern University, Evanston, IL 60208, USA.

E-mail: [email protected]

2 Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA.

E-mail: [email protected]

3 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA.

E-mail: [email protected] Received: 27 December 2008 / Accepted: 10 September 2009 Published online: 27 November 2009 – © Springer-Verlag 2009

Abstract: A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold M2 which can be realized as isometric immersions into R3 . This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in R3 . The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in R3 . As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a C 1,1 isometric immersion of the two-dimensional manifold in R3 satisfying our prescribed initial conditions. To achieve this, we introduce a vanishing viscosity method depending on the features of initial value problems for isometric immersions and present a technique to make the a priori estimates including the L ∞ control and H −1 –compactness for the viscous approximate solutions. This yields the weak convergence of the vanishing viscosity approximate solutions and the weak continuity of the Gauss-Codazzi system for the approximate solutions, hence the existence of an isometric immersion of the manifold into R3 satisfying our initial conditions. The theory is applied to a specific example of the metric associated with the catenoid. 1. Introduction A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold M2 which can be realized as isometric immersions into R3 (cf. Yau [40]; also see [21,34,36]). Important results have been achieved

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for the embedding of surfaces with positive Gauss curvature which can be formulated as an elliptic boundary value problem (cf. [21]). For the case of surfaces of negative Gauss curvature where the underlying partial differential equations are hyperbolic, the complimentary problem would be an initial or initial-boundary value problem. Hong in [23] first proved that complete negatively curved surfaces can be isometrically immersed in R3 if the Gauss curvature decays at a certain rate in the time-like direction. In fact, a crucial lemma in Hong [23] (also see Lemma 10.2.9 in [21]) shows that, for such a decay rate of the negative Gauss curvature, there exists a unique global smooth, small solution forward in time for prescribed smooth, small initial data. Our main theorem, Theorem 5.1(i), indicates that in fact we can solve the corresponding problem for a class of large non-smooth initial data. Possible implication of our approach may be in existence theorems for equilibrium configurations of a catenoidal shell as detailed in Vaziri-Mahedevan [39]. When the Gauss curvature changes sign, the immersion problem then becomes an initial-boundary value problem of mixed elliptic-hyperbolic type, which is still under investigation. The purpose of this paper is to introduce a general approach, which combines a fluid dynamic formulation of balance laws with a compensated compactness framework, to deal with the isometric immersion problem in R3 (even when the Gauss curvature changes sign). In Sect. 2, we formulate the isometric immersion problem for two-dimensional Riemannian manifolds in R3 via solvability of the Gauss-Codazzi system. In Sect. 3, we introduce a fluid dynamic formulation of balance laws for the Gauss-Codazzi system for isometric immersions. Then, in Sect. 4, we provide a compensated compactness framework and present one of our main observations that this framework is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in R3 . A generalization of this approach to higher dimensional immersions has been given in Chen-Slemrod-Wang [8]. In Sect. 5, as a first application of this approach, we focus on the isometric immersion problem of two-dimensional Riemannian manifolds with strictly negative Gauss curvature. Since the local existence of smooth solutions follows from the standard hyperbolic theory, we are concerned here with the global existence of solutions of the initial value problem with large initial data. The metrics gi j we study have special structures and forms usually associated with the catenoid of revolution when g11 = g22 = cosh(x) and g12 = 0. For these cases, while Hong’s theorem [23] applies to obtain the existence of a solution for small smooth initial data, our result yields a large-data existence theorem for a C 1,1 isometric immersion. To achieve this, we introduce a vanishing viscosity method depending on the features of the initial value problem for isometric immersions and present a technique to make the a priori estimates including the L ∞ control and H −1 –compactness for the viscous approximate solutions. This yields the weak convergence of the vanishing viscosity approximate solutions and the weak continuity of the Gauss-Codazzi system for the approximate solutions, hence the existence of a C 1,1 –isometric immersion of the manifold into R3 with prescribed initial conditions. We remark in passing that, for the fundamental ideas and early applications of compensated compactness, see the classical papers by Tartar [38] and Murat [32]. For applications to the theory of hyperbolic conservation laws, see for example [4,10,13,18,37]. In particular, the compensated compactness approach has been applied in [3,6,11,12, 25,26] to the one-dimensional Euler equations for unsteady isentropic flow, allowing for cavitation, in Morawetz [29,30] and Chen-Slemrod-Wang [7] for two-dimensional

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413

steady transonic flow away from stagnation points, and in Chen-Dafermos-SlemrodWang [5] for subsonic-sonic flows. 2. The Isometric Immersion Problem for Two-Dimensional Riemannian Manifolds in R3 In this section, we formulate the isometric immersion problem for two-dimensional Riemannian manifolds in R3 via solvability of the Gauss-Codazzi system. Let ⊂ R2 be an open set. Consider a map r : → R3 so that, for (x, y) ∈ , the two vectors {∂x r, ∂y r} in R3 span the tangent plane at r(x, y) of the surface r() ⊂ R3 . Then n=

∂x r × ∂y r |∂x r × ∂y r|

is the unit normal of the surface r() ⊂ R3 . The metric on the surface in R3 is ds 2 = dr · dr

(2.1)

ds 2 = (∂x r · ∂x r) (dx)2 + 2(∂x r · ∂y r) dxdy + (∂y r · ∂y r) (dy)2 .

(2.2)

or, in local (x, y)–coordinates,

Let gi j , i, j = 1, 2, be the given metric of a two-dimensional Riemannian manifold M parameterized on . The first fundamental form I for M on is I := g11 (dx)2 + 2g12 dxdy + g22 (dy)2 .

(2.3)

Then the isometric immersion problem is to seek a map r : → R3 such that dr · dr = I, that is, ∂x r · ∂x r = g11 , ∂x r · ∂y r = g12 , ∂y r · ∂y r = g22 ,

(2.4)

so that {∂x r, ∂y r} in R3 are linearly independent. The equations in (2.4) are three nonlinear partial differential equations for the three components of r(x, y). The corresponding second fundamental form is II := −dn · dr = h 11 (dx)2 + 2h 12 dxdy + h 22 (dy)2 ,

(2.5)

and (h i j )1≤i, j≤2 is the orthogonality of n to the tangent plane. Since n · dr = 0, then d(n · dr) = 0 implies −II + n · d 2 r = 0, i.e.,

2 II = (n · ∂x2 r) (dx)2 + 2(n · ∂xy r) dxdy + (n · ∂y2 r) (dy)2 .

The fundamental theorem of surface theory (cf. [14,21]) indicates that there exists a surface in R3 whose first and second fundamental forms are I and II if the smooth coefficients (gi j ) and (h i j ) of the two given quadratic forms I and II with (gi j ) > 0 satisfy the Gauss-Codazzi system. It is indicated in Mardare [28] (Theorem 9; also see [27]) that this theorem holds even when (h i j ) is only in L ∞ for given (gi j ) in C 1,1 , for

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which the immersion surface is C 1,1 . This shows that, for the realization of a two-dimensional Riemannian manifold in R3 with given metric (gi j ) > 0, it suffices to solve for (h i j ) ∈ L ∞ determined by the Gauss-Codazzi system to recover r a posteriori. The simplest way to write the Gauss-Codazzi system (cf. [14,21]) is as (2)

(2)

(2)

∂x M − ∂y L = 22 L − 212 M + 11 N , (1)

(1)

(1)

(2.6)

∂x N − ∂y M = −22 L + 212 M − 11 N , with L N − M 2 = κ.

(2.7)

Here h 11 L=√ , |g|

h 12 M=√ , |g|

h 22 N=√ , |g|

2 , κ(x, y) is the Gauss curvature that is determined by the |g| = det (gi j ) = g11 g22 − g12 relation: R1212 (m) (m) (n) (m) (n) (m) κ(x, y) = , Ri jkl = glm ∂k i j − ∂ j ik + i j nk − ik n j , |g|

Ri jkl is the curvature tensor and depends on (gi j ) and its first and second derivatives, and (k)

i j =

1 kl g ∂ j gil + ∂i g jl − ∂l gi j 2

is the Christoffel symbol and depends on the first derivatives of (gi j ), where the summation convention is used, (g kl ) denotes the inverse of (gi j ), and (∂1 , ∂2 ) = (∂x , ∂y ). Therefore, given a positive definite metric (gi j ) ∈ C 1,1 , the Gauss-Codazzi system gives us three equations for the three unknowns (L , M, N ) determining the second fundamental form II . Note that, although (gi j ) is positive definite, R1212 may change sign and so does the Gauss curvature κ. Thus, as we will discuss in Sect. 3, the Gauss-Codazzi system (2.6)–(2.7) generically is of mixed hyperbolic-elliptic type, as in transonic flow (cf. [2,7,9,31]). In §3–4, we introduce a general approach to deal with the isometric immersion problem involving nonlinear partial differential equations of mixed hyperbolic-elliptic type by combining a fluid dynamic formulation of balance laws in §3 with a compensated compactness framework in §4. As an example of direct applications of this approach, in §5 we show how this approach can be applied to establish an isometric immersion of a two-dimensional Riemannian manifold with negative Gauss curvature in R3 . 3. Fluid Dynamic Formulation for the Gauss-Codazzi System From the viewpoint of geometry, the constraint condition (2.7) is a Monge-Ampère equation and the equations in (2.6) are integrability relations. However, our goal here is to put the problem into a fluid dynamic formulation so that the isometric immersion problem may be solved via the approaches that have shown to be useful in fluid dynamics

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for solving nonlinear systems of balance laws. To achieve this, we formulate the isometric immersion problem via solvability of the Gauss-Codazzi system (2.6)–(2.7), that is, solving first for h i j , i, j = 1, 2, via (2.6) with constraint (2.7) and then recovering r a posteriori. To do this, we set L = ρv 2 + p,

M = −ρuv,

N = ρu 2 + p,

and set q 2 = u 2 + v 2 as usual. Then the Codazzi equations in (2.6) become the familiar balance laws of momentum: (2)

(2)

(2)

(1)

(1)

(1)

∂x (ρuv) + ∂y (ρv 2 + p) = −(ρv 2 + p)22 − 2ρuv12 − (ρu 2 + p)11 ,

(3.1)

∂x (ρu 2 + p) + ∂y (ρuv) = −(ρv 2 + p)22 − 2ρuv12 − (ρu 2 + p)11 , and the Gauss equation (2.7) becomes ρpq 2 + p 2 = κ.

(3.2)

From this, we can see that, if the Gauss curvature κ is allowed to be both positive and negative, the “pressure” p cannot be restricted to be positive. Our simple choice for p is the Chaplygin-type gas: 1 p=− . ρ Then, from (3.2), we find −q 2 +

1 = κ, ρ2

and hence we have the “Bernoulli” relation: 1 ρ= . 2 q +κ

(3.3)

This yields p = − q 2 + κ,

(3.4)

and the formulas for u 2 and v 2 : u 2 = p( p − M),

v 2 = p( p − L),

M 2 = (N − p)(L − p).

The last relation for M 2 gives the relation for p in terms of (L , M, N ), and then the first two give the relations for (u, v) in terms of (L , M, N ). We rewrite (3.1) as ∂x (ρuv) + ∂y (ρv 2 + p) = R1 , ∂x (ρu 2 + p) + ∂y (ρuv) = R2 , where R1 and R2 denote the right-hand sides of (3.1).

(3.5)

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We now find the corresponding “geometric rotationality–continuity equations”. Multiplying the first equation of (3.5) by v and the second by u, and setting ∂x v − ∂y u = −σ, we see v R1 1 div(ρu, ρv) − ∂y κ = + σ u, ρ 2 ρ R2 1 u div(ρu, ρv) − ∂x κ = − σ v, ρ 2 ρ and hence 1ρ ∂y κ + 2v 1ρ div(ρu, ρv) = ∂x κ + 2u

div(ρu, ρv) =

R1 ρuσ + , v v R2 ρvσ − . u u

Thus, the right hand sides of (3.6) are equal, which gives a formula for σ : 1 1 1 σ = v − u . ρ∂ ρ∂ κ + R κ + R x 2 y 1 ρq 2 2 2

(3.6)

(3.7)

If we substitute this formula for σ into (3.6), we can write down our “rotationality-continuity equations” as 1 1 1 ∂x v − ∂y u = u − v , (3.8) κ + R κ + R ρ∂ ρ∂ y 1 x 2 ρq 2 2 2 1 ρu 1 ρv v u ∂x (ρu) + ∂y (ρv) = ∂x κ + ∂y κ + 2 R 1 + 2 R 2 . (3.9) 2 2 2q 2q q q In summary, the Gauss-Codazzi system (2.6)–(2.7), the momentum equations (3.1)– (3.4), and the rotationality-continuity equations (3.3) and (3.8)–(3.9) are all formally equivalent. However, for weak solutions, we know from our experience with gas dynamics that this equivalence breaks down. In Chen-Dafermos-Slemrod-Wang [5], the decision was made (as is standard in gas dynamics) to solve the rotationality-continuity equations and view the momentum equations as “entropy” equalities which may become inequalities for weak solutions. In geometry, this situation is just the reverse. It is the Gauss-Codazzi system that must be solved exactly and hence the rotationality-continuity equations will become “entropy” inequalities for weak solutions. The above issue becomes apparent when we set up “viscous” regularization that preserves the “divergence” form of the equations, which will be introduced in §5.3. This is crucial since we need to solve (3.8)–(3.9) exactly, as we have noted. To continue further our analogy, let us define the “sound” speed: c2 = p (ρ),

(3.10)

which in our case gives c2 =

1 . ρ2

(3.11)

Isometric Immersions and Compensated Compactness

417

Since our “Bernoulli” relation is (3.3), we see c2 = q 2 + κ.

(3.12)

Hence, under this formulation, (i) when κ > 0, the “flow” is subsonic, i.e., q < c, and system (3.1)–(3.2) is elliptic; (ii)

Communications in

Mathematical Physics

Some Remarks about Semiclassical Trace Invariants and Quantum Normal Forms Victor Guillemin1, , Thierry Paul2 1 Department of Mathematics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA. E-mail: [email protected]

2 CNRS and Département de Mathématiques et Applications, École Normale Supérieure,

45, Rue d’Ulm, F-75730 Paris Cedex 05, France. E-mail: [email protected] Received: 23 January 2009 / Accepted: 15 June 2009 Published online: 11 September 2009 – © Springer-Verlag 2009

Abstract: In this paper we explore the connection between semi-classical and quantum Birkhoff canonical forms (BCF) for Schrödinger operators. In particular we give a “non-symbolic” operator theoretic derivation of the quantum Birkhoff canonical form and provide an explicit recipe for expressing the quantum BCF in terms of the semiclassical BCF. 1. Introduction Let X be a compact manifold and H : L 2 (X ) → L 2 (X ) a self-adjoint first order elliptic pseudodifferential operator with leading symbol H (x, ξ ). From the wave trace eit E k , (1.1) E k ∈Spec(H )

one can read off many properties of the “classical dynamical system” associated with H , i.e. the flow generated by the vector field ξH =

∂H ∂ ∂H ∂ − . ∂ξi ∂ xi ∂ xi ∂ξi

(1.2)

For instance it was observed in the ‘70’s’ by Colin de Verdière, Chazarain and Duistermaat-Guillemin that (1.1) determines the period spectrum of (1.2) and the linear Poincaré map about a non-degenerate periodic trajectory, γ , of (1.2) ([2–4]). More recently it was shown by one of us [5] that (1.1) determines the entire Poincaré map about γ , i.e. determines, up to isomorphism, the classical dynamical system associated with H in a formal neighborhood of γ . The proof of this result involved a microlocal Birkhoff canonical form for H in a formal neighborhood of γ and an algorithm First author supported by NSF grant DMS 890771.

2

V. Guillemin, T. Paul

for computing the wave trace invariants associated with γ from the microlocal Birkhoff canonical form. Subsequently a more compact and elegant algorithm for computing these invariants from the Birkhoff canonical form was discovered by Zelditch [11,12] making the computation of these local trace invariants extremely simple and explicit. In this paper we will discuss some semiclassical analogues of these results. In our set-up H can either be the Schrödinger operator on Rn − 2 + V with V → ∞ as x tends to infinity, or more generally a self-adjoint semiclassical elliptic pseudodifferential operator H (x, Dx ) whose symbol, H (x, ξ ), is proper (as a map from T ∗ X into R). Let E be a regular value of H and γ a non-degenerate periodic trajectory of period Tγ lying on the energy surface H = E 1. Consider the Gutzwiller trace (see [6]) E − Ei ψ , (1.3) where ψ is a C ∞ function whose Fourier transform is compactly supported with support in a small neighborhood of Tγ and is identically one in a still smaller neighborhood. As shown in [8,9] (1.3) has an asymptotic expansion ei

Sγ

+σγ

∞

a k k ,

(1.4)

k=0

and we will show below how to compute the terms of this expansion to all orders in terms of a microlocal Birkhoff canonical form for H in a formal neighborhood of γ by means of a Zelditch-type algorithm 2 . If γ is non-degenerate so are all its iterates γ r . Then, for each of these iterates one gets an expansion of (1.3) similar to (1.4), ei

Sγ r

+σγ r

∞

ak,r k ,

(1.5)

k=0 1 For simplicity we will consider periodic trajectories of elliptic type in this paper, however our results are true for non-degenerate periodic of all types, hyperbolic, mixed elliptic hyperbolic, focus-focus, etc. Unfortunately however the Zelditch algorithm depends upon the type of the trajectory and in dimension n there are roughly as many types of trajectories as there are Cartan subalgebras of Sp(2n) (see for instance [1]) i.e. the number of types can be quite large. 2 For elliptic trajectories non-degeneracy means that the numbers

θ1 , . . . , θn , 2π are linearly independent over the rationals, eiθκ , κ = 1, . . . , n being the eigenvalues of the Poincaré map about γ . The results above are true to order O(r ) providing (κ1 θ1 + · · · + κn θ )n + l2π = 0 for all | κ1 | + · · · + | κn |≤ r , i.e. providing there are no resonances of order ≤ r .

Semiclassical Trace Invariants

3

and for these expansions as well the coefficients ak,r can be computed from the microlocal Birkhoff canonical form theorem for H in a formal neighborhood of γ . Conversely one can show Theorem 1.1. The constants ak,r , κ, r = 0, 1, . . . determine the microlocal Birkhoff canonical form for H in a formal neighborhood of γ (and hence, a fortiori, determine the classical Birkhoff canonical form). One of the main goals of this paper will be to give a proof of this result. Our proof, in Sects. 2, 3 and 6 is, with semiclassical modifications, more or less the same as the proof of the Guillemin-Zelditch results [5,11,12] alluded to above. An alternative proof based on Grushun reductions, flux norms and trace formulas for monodromy operators can be found in [7]. Another main goal of this paper is to develop a purely quantum mechanical approach to the theory of Birkhoff canonical forms in which symbolic expansions get replaced by operator theoretic expansions and estimates involving Hermite functions. This can be seen as a “local” version of the Rayleigh-Schrödinger perturbation formalism where no “” parameter is involved. The virtue of this approach is that the dependence of the normal form is an intrinsic part of the theory, and avoids any additional semiclassical computation. This approach is developed in Sect. 4 and the connection of this to the symbolic approach of Sects. 2–3 is described in Sect. 5. To conclude these prefatory remarks we would like to thank Cyrille Heriveuax for his perusal of the first draft of the our manuscript and we would also like to express our gratitude to the referee for his careful line-by-line reading of the manuscript and his many helpful suggestions. 2. The Classical Birkhoff Canonical Form Theorem Let M be a 2n + 2 dimensional symplectic manifold, H a C ∞ function and ξH =

∂H ∂ ∂H ∂ − ∂ξi ∂ xi ∂ xi ∂ξi

(2.1)

the Hamiltonian vector field associated with H . Let E be a regular value of H and γ a non-degenerate elliptic periodic trajectory of ξ H lying on the energy surface, H = E. Without loss of generality one can assume that the period of γ is 2π . In this section we will review the statement (and give a brief sketch of the proof) of the classical Birkhoff canonical form theorem for the pair (H, γ ). Let (x, ξ, t, τ ) be the standard cotangent coordinates on T ∗ (Rn × S 1 ) and let √ pi = xi2 + ξi2 and qi = arg (xi + −1 yi ). (2.2) Theorem 2.1. There exists a symplectomorphism, ϕ, of a neighborhood of γ in M onto a neighborhood of p = τ = 0 such that ϕ o γ (t) = (0, 0, t, 0) and ϕ ∗ H = H1 ( p, τ ) + H2 (x, ξ, t, τ ), H2 vanishing to infinite order at p = τ = 0. We break the proof of this up into the following five steps.

(2.3)

4

V. Guillemin, T. Paul

Step 1. For small there exists a periodic trajectory, γ , on the energy surface, H = E +, which depends smoothly on and is equal to γ for = 0. The union of these trajectories is a 2 dimensional symplectic submanifold , , of M which is invariant under the flow of ξ H . Using the Weinstein tubular neighborhood theorem one can map a neighborhood of γ symplectically onto a neighborhood of p = τ = 0 in T ∗ (Rn × S 1 ) such that gets mapped onto p = 0 and ϕ o γ (t) = (0, 0, t, 0). Thus we can henceforth assume that M = T ∗ (Rn × S 1 ) and is the set, p = 0. Step 2. We can assume without loss of generality that the restriction of H to is a function of τ alone, i.e. H = E + h(τ ) on . With this normalization, H = E + h(τ ) +

θi (τ ) pi + O( p 2 ),

(2.4)

where h(τ ) = τ + O(τ 2 ) and θi = θi (0), i = 1, . . . , n

(2.5)

are the rotation angles associated with γ . Since γ is non- degenerate, θ1 , . . . , θn , 2π are linearly independent over the rationals. Step 3. Theorem 2.1 can be deduced from the following result (which will also be the main ingredient in our proof of the “microlocal” Birkhoff canonical theorem in the next section). Theorem 2.2. Given a neighborhood, U, of p = τ = 0 and G = G(x, ξ, t, τ ) ∈ C ∞ (U), there exist functions F, G 1 , R ∈ C ∞ (U) with the properties i. G 1 = G 1 ( p, τ ), ii. {H, F} = G + G 1 + R, iii. R vanishes to infinite order on p = τ = 0. Moreover, if G vanishes to order κ on p = τ = 0, one can choose F to have this property as well. Proof of the assertion. Theorem 2.2 ⇒ Theorem 2.1. By induction one can assume that H is of the form, H = H0 ( p, τ ) + G(x, ξ, t, τ ), where G vanishes to order κ on p = τ = 0. We will show that H can be conjugated to a Hamiltonian of the same form with G vanishing to order κ + 1 on p = τ = 0. By Theorem 2.2 there exists an F, G and R such that F vanishes to order κ and R to order ∞ on p = τ = 0, G 1 = G 1 ( p, τ ) and {H, F} = G + G 1 + R. Thus 1 {F, {F, H }} + · · · 2! = H0 ( p, τ ) − G 1 ( p, τ ) + · · · ,

(expξ F )∗ H = H + {F, H } +

the “dots” indicating terms which vanish to order κ + 1 on p = τ = 0. Step 4. Theorem 2.2 follows (by induction on κ) from the following slightly weaker result:

Semiclassical Trace Invariants

5

Lemma 2.3. Given a neighborhood, U, of p = τ = 0 and a function, G ∈ C ∞ (U), which vanishes to order κ on p = τ = 0, there exists functions F, G 1 , R ∈ C ∞ (U) such that i. G 1 = G 1 ( p, τ ), ii. {H, F} = G + G 1 + R, iii. F vanishes to order κ and R to order κ + 1 on p = τ = 0. Step 5. Proof of Lemma 2.3. In proving Lemma 2.3 we can replace H by the Hamiltonian θi pi , H0 = E + τ + since H ( p, q, t, τ ) − H0 ( p, q, t, τ ) vanishes to second order in τ, p. Consider now the identity {H0 , F} = G + G 1 ( p, τ ) + O( p ∞ ). √ √ Introducing the complex coordinates, z = x + −1ξ , and z = x − −1ξ , this can be written as n √ ∂F ∂ ∂ F+ −1 θi z i − zi = G + G 1 + O( p ∞ ). ∂z i ∂z i ∂t i=1

Expanding F, G and G 1 in Fourier-Taylor series about z = z = 0: aµ,ν,m (τ )z µ z ν e2πimt , F= µ=ν

G= G1 =

bµ,ν,m (τ )z µ z ν e2πimt , cµ (τ )z µ z µ ,

µ

one can rewrite this as the system of equations n √ −1 θi (µi − νi ) + 2π m aµ,ν,m (τ ) = bµ,ν,m (τ )

(2.6)

i=1

for µ = ν or µ = ν and m = 0, and − cµ (τ ) = bµ,µ,0 (τ )

(2.7)

for µ = ν and m = 0. By assumption the numbers, θ1 , . . . , θn , 2π, are linearly independent over the rationals, so this system has a unique solution. Moreover, for µ and ν fixed, bµ,ν,m (τ )e2πimt is the (µ, ν) Taylor coefficient of G(z, z, t, τ ) about z = z = 0; so, with µ and ν fixed and j >> 0, | bµ,ν,m (τ ) |≤ Cµ,ν, j m − j

6

V. Guillemin, T. Paul

for all m. Hence, by (2.6), − j−1 | aµ,ν,m (τ ) |≤ Cµ,ν, jm

for all m. Thus aµ,ν (t, τ ) =

aµ,ν,m (τ )e2πimt

is a C ∞ function of t and τ . Now let F(z, z, t, τ ) and G 1 ( p, τ ) be C ∞ functions with Taylor expansion:

aµ,ν (t, τ )z µ z ν

µ=ν

and

cµ (τ )z µ z µ

µ

about z = z = 0. Note, by the way, that if G vanishes to order κ on p = τ = 0, so does F and G; so we have proved Theorem 2.2 (and, a fortiori Lemma 2.3) with H replaced by H0 . 3. The Semiclassical Version of the Birkhoff Canonical Form Theorem Let X be an (n + 1)-dimensional manifold and H : C0∞ (X ) → C ∞ (X ) a semiclassical elliptic pseudo-differential operator with leading symbol, H (x, ξ ), and let γ be a periodic trajectory of the bicharacteristic vector field (2.1). As in Sect. 1 we will assume that γ is elliptic and non-degenerate, with rotation numbers (2.4). Let Pi and Dt be the differential operators on Rn × S 1 associated with the symbols (2.2) and τ , i.e. Pi = −2 ∂x2i + xi2 and Dt = −i∂t . We will prove below the following semiclassical version of Theorem 2.1 ∞ Theorem n 3.1.1 There exists a semiclassical Fourier integral operator Aϕ : C0 (X ) → ∞ C R × S implementing the symplectomorphism (2.3) such that microlocally on a neighborhood, U, of p = τ = 0,

A∗ϕ = A−1 ϕ

(3.1)

Aϕ H A−1 ϕ = H (P1 , . . . , Pn , Dt , ) + H ,

(3.2)

and

the symbol of H vanishing to infinite order on p = τ = 0.

Semiclassical Trace Invariants

7

Proof. Let Bϕ be any Fourier integral operator implementing ϕ and having the property (3.1). Then, by Theorem 2.1, the leading symbol of Bϕ H Bϕ−1 is of the form H0 ( p, τ ) + H0 ( p, q, t, τ ),

(3.3)

H0 ( p, q, t, τ ) being a function which vanishes to infinite order on p = τ = 0. Thus the symbol,H0 , of Bϕ H Bϕ−1 is of the form H0 ( p, τ ) + H0 ( p, q, t, τ ) + H1 ( p, q, t, τ ) + O(2 ).

(3.4)

By Theorem 2.2 there exists a function, F( p, q, t, τ ), with the property {H0 , F} = H1 ( p, q, t, τ ) + H1 ( p, τ ) + H1 ( p, q, t, τ ),

(3.5)

where H1 vanishes to infinite order on p = τ = 0. Let Q be a self-adjoint pseudo-differential operator with leading symbol F and consider the unitary pseudo-differential operator Us = eis Q . Let −1 Hs = Us Bϕ H Us Bϕ = Us Bϕ H Bϕ−1 U−s . Then ∂ Hs = i[Q, Hs ], ∂s

(3.6)

∂ so ∂s Hs is of order −1, and hence the leading symbol of Hs is independent of s. In par∂ Hs is equal, by (3.6) to the leading symbol of i[Q, Hs ] ticular the leading symbol of ∂s which, by (3.5), is: − H1 ( p, q, t, τ ) + H1 ( p, τ ) + H1 ( p, q, t, τ ) .

Thus by (3.4) and (3.5) the symbol of −1 U1 Bϕ H U1 Bϕ = Bϕ H Bϕ−1 +

1 0

∂ Hs ds ∂s

is of the form H0 ( p, τ ) + H1 ( p, τ ) + H0 + H1 + O(2 ),

(3.7)

the term in parentheses being a term which vanishes to infinite order on p = τ = 0. By repeating the argument one can successively replace the terms of order 2 , . . . , r , etc. in (3.7) by expressions of the form r Hr ( p, τ ) + Hr ( p, q, t, τ ) with Hr vanishing to infinite order on p = τ = 0.

8

V. Guillemin, T. Paul

4. A Direct Construction of the Quantum Birkhoff Form In this section we present a “quantum” construction of the quantum Birkhoff normal form which is in a sense algebraically equivalent to the classical one of Sect. 2. To do this we will need to define for operators the equivalent of “a Taylor expansion which vanishes at a given order”. We will first start in the L 2 (Rn × S 1 ) setting, and show at the end of the section the link with Theorem 3.1. Definition 4.1. Let us consider on L 2 (Rn × S 1 , d xdt) the following operators: • ai− = • ai+ = • Dt =

√1 (x i + ∂x ) i 2 √1 (x i − ∂x ) i 2 ∂ −i ∂t

We will say that an operator A on L 2 (Rn × S 1 ) is an “ordered polynomial of order greater than p ∈ N” (OPOG( p)) if there exists P ∈ N such that: i

A=

[2] P

j

αi j (t, )Dt

i= p j=0

i−2 j

bl

(4.1)

l=1

with, ∀l, bl ∈ {a1− , a1+ , . . . , an− , an+ } and αi j ∈ C ∞ (S1 × [0, 1[). i−2 j In (4.1) bl is meant to be the ordered product b1 . . . bi−2 j . l=1

The meaning of this definition is clarified by the following: basis of L 2 (Rn × S 1 ) defined by Hµ (x, t) = −n/4 Lemma√ 4.2. Let Hµ denote √ the iµ t , where the h are the (normalized) Hermite funcn+1 h µ1 (x1 / ) . . . h µn (xn / )e j

tions. Let us define moreover |µ| := µ2 2 . Let A be an OPOG(p). Then: ∀M < +∞ (microlocal “cut-off”), ∃ C = C(A, M) such that, p

||AHµ || L 2 ≤ C|µ| 2 , ∀µ ∈ Nn+1 s.t |µ| ≤ M. Proof. The proof follows immediately from the two well known facts (expressed here in one dimension): a ± Hµ =

(µ ± 1)Hµ±1

and Dt eimt = meimt .

For the rest of this section we will need the following collection of results.

Semiclassical Trace Invariants

9

Proposition 4.3. Let A be a (Weyl) pseudodifferential operator on L 2 (Rn × S 1 ) with symbol of type S1,0 . Then, ∀L ∈ N and ∀M < +∞, there exists an OPOG(1) A L and a constant C = C(A, L , M) such that, ||(A − A L )Hµ || L 2 ≤ C|µ|

L+1 2

), ∀µ ∈ Nn+1 s.t. |µ| ≤ M.

Moreover, if the principal symbol of A is of the form: a0 (x1 , ξ1 , . . . , xn , ξn , t, τ ) = θi (xi2 + ξi2 ) + τ + h.o.t., (or is any function whose symbol vanishes to first order at x = ξ = τ = 0) then A L is an OPOG(2). Proof. Let us take the L th order Taylor expansion of the (total) symbol of A in the variables x, ξ, τ, near the origin. Noticing that a pseudodifferential operator with polynomial symbol in x, ξ, τ, is an OPOG, we just have to estimate the action, on Hµ , of a pseudo-differential operator whose symbol vanishes at the origin to order L in the variable x, ξ, τ . The result is easily obtained for the τ part, as the “t” part of Hµ is an exponential. For the µ part we will prove this result in one dimension, the extension to n dimensions being straightforward. Let us define a coherent state at (q, p) to be a function of the form ψqap (x) := px √ −1/4 a x−q ei , for a in the Schwartz class and ||a|| L 2 = 1. Let us also set ϕq p = ψqap

for a(η) = π −1/2 e−η /2 . It is well known, and easy to check using the generating function of the Hermite polynomials, that

1 t Hµ = − 4 e−i 2 ϕq(t) p(t) dt, 2

S1

where q(t) + i p(t) = eit (q + i p) , q 2 + p 2 = (µ + 21 ). Therefore, for any operator A, ||AHµ || = O(

sup p 2 +q 2 =(µ+ 12 )

1

− 4 ||Aϕq p ||).

(4.2)

Lemma 4.4. let H a pseudodifferential operator whose (total) Weyl symbol vanishes at the origin to order M. Then, if q 2 +p2 = O(1): M ||H ψqap || = O ( p 2 + q 2 ) 2 . Before proving the lemma we observe that the proof of the proposition follows easily from the lemma using (4.2). Proof. An easy computation shows that, if h is the (pseudodifferential) symbol of H , then H ψqap = ψqbp with

√ √ b(η) = h(q + η, p + ν)eiην a(ν)dν, ˆ (4.3) R

where aˆ is the ( independent) Fourier transform of a.

10

V. Guillemin, T. Paul

k k+1 bk 2 2 ), where Developing (4.3) we get that H ψqap = k=K k=M Dk h(q, p)ψq p + O( bk ∈ S and Dk is an homogeneous differential operator of order k. It is easy to conclude, thanks to the hypothesis q 2 +p2 = O(1), that k

2 (q 2 + p 2 )

M−k 2

M

= O((q 2 + p 2 ) 2 ),

M+1 2

M

= O((q 2 + p 2 ) 2 ).

This proposition is crucial for the rest of this section, as it allows us to reduce all computations to the polynomial setting. For example A may have a symbol bounded at infinity (class S(1), an assumption which we will need for the application below of Egorov’s Theorem in the proof of Theorem 4.9), but, with respect to the algebraic equations we will have to solve, one can consider it as a “OPOG” (see Theorem 4.9 below). Lemma 4.5. Let A be a OPOG(1) on L 2 (Rn × S 1 ). Let us suppose that A is a symmetric operator. For P ∈ N (large), let A P := A + (|Dθ |2 + |x|2 + |Dx |2 ) P .

(4.4)

Then A P is an elliptic selfadjoint pseudo-differential operator. Therefore eis family of unitary Fourier integral operators.

AP

is a

Proof. It is enough to observe that A P is, defined on the domain of |Dθ |2 + |x|2 + |Dx |2 , a selfadjoint pseudodifferential operator with symbol of type S1,0 .

Lemma 4.6. Let H0 be the operator H0 =

n

θi ai− ai+ + Dt ,

1

then, if W is an OPOG(r ), so is

[H0 ,W ] i .

d is H0 / e W eis H0 /|s=0 which, since H0 is quadratic, is the same polyProof. [H0i ,W ] = ds nomial as W modulo the substitution ai− → eis ai− , ai+ → e−is ai and shifting of the coefficients in t by s. Therefore the result is immediate.

More generally: Lemma 4.7. For any H and W of type OPOG(m) and OPOG(r ) respectively, an OPOG(m + r − 2).

[H,W ] i

is

The proof is immediate noting that [ai− , a +j ] = δi j and that, for any C ∞ function a(t), [Dt , a] = ia . We can now state the main result of this section: Theorem 4.8. Let H be a (Weyl) pseudo-differential operator on L 2 (Rn × S 1 ) whose principal symbol is of the form: H0 (x, ξ ; t, τ ) =

n 1

θi (xi2 + ξi2 ) + τ + H2 ,

Semiclassical Trace Invariants

11

where H2 vanishes to third order at x = ξ = τ = 0 and θ1 , . . . , θn , 2π are line2 arly independent over the rationals. Let us define, as before, Pi = −2 ∂∂x 2 + xi2 and i

Dt = −i ∂t∂ . Then, ∀M < +∞, there exists a family of unitary operators (U L ) L=3... and constants (C L ) L=3... , and a C ∞ function h( pi , . . . , pn , τ, ) such that: L+1 || U L HU L−1 −h(P1 , . . . , Pn , Dt , ) Hµ || L 2 (Rn ×S 1 ) ≤ C L |µ| 2 ∀µ ∈ Nn+1 s.t |µ| ≤ M.

Proof. The proof of Theorem 4.8 will be a consequence of the following: Theorem 4.9. Let H be as before, and let G be an OPOG(3). Then there exists a function G 1 ( p1 , . . . , pn , τ, ), an OPOG F and an operator R such that: i.

[H,F] i

= G + G 1 + R, L+1

ii. R satisfies: ||R Hµ || = O(|µ| 2 ), ∀µ ∈ Nn+1 , |µ| = O(1) and ∀L ∈ N, iii. if G is an OPOG(κ) so is F, iv. if G is a symmetric operator, so is F and G 1 is real. Let us first prove that Theorem 4.9 implies Theorem 4.8: By induction, as in the “classical” case and thanks to Proposition 4.3, one can assume that H is of the form H = H0 + G, where G is an OPOG(κ). Let us consider the operators ei FP

FP

H e−i

FP

and

FP

H (s) := eis H e−is , where F satisfies Theorem 4.9 and FP is defined by (4.4) for P large enough. Since we are in an iterative perturbative setting, it is easy to check by taking P large FP

F

enough that we can omit the subscript P in H (s) and let e±i stand for e±i in the rest of the computation. We have: [F,H ] F F [F, H ] [F, i ] [F, [F, [F, 10 t0 s0 H (u)dudsdt]/i]/i] ei H e−i = H + + + i i i [F,H ] [F, H ] [F, i ] ˜ + +R = H0 + G + i i ] [F, [F,H i ] ˜ = H0 − G 1 + R + + R. (4.5) i Since we are interested in letting all the operators acting on the Hµ for |µ| = O(1), we can microlocalize near x = ξ = τ = 0 and replace F and H by their microlocalF˜ ized versions F˜ and H˜ . ei is a Fourier integral operator and, byEgorov’s Theorem, H˜ (s) is a family of pseudodifferential operators, and so is 10 t0 s0 H˜ (u)dudsdt. By Proposition 4.3, Lemma 4.7 and Lemma 4.2 we have, since G is an OPOG(κ), || R˜ Hµ || = O(|µ|κ+1 ). [F,H ]

[F, i ] By the same argument, satisfies the same estimate. Developing R˜ by the i Lagrange formula (4.5) to arbitrary order, we get, thanks to Lemma 4.7, R˜ = G˜ + R, where G˜ is an OPOG(κ + 1) and

||R Hµ || = O(|µ|

L+1 2

).

12

Therefore, letting G =

V. Guillemin, T. Paul ] [F, [F,H i ] i

ei

FP

˜ we have: + G,

H e−i

FP

= H0 + G 1 + G + R,

with G an OPOG(κ + 1). By induction Theorem 4.8 follows. Proof of Theorem 4.9. Let us first prove the following Lemma 4.10. Let H0 be as before and let G be an OPOG(r ). Then there exists a OPOG(r ) F and G 1 = G 1 ( p1 , . . . , pn , Dt , ), such that [H0 , F] = G + G1. i

(4.6)

Proof. By Lemma 4.6, if F is an OPOG, it must be an OPOG(r ), since the left-hand side of (4.6) is an OPOG(r ). Let us take the matrix elements of (4.6) relating to the Hµ s. We get: −i.(µ − ν) < µ|F|ν >=< µ|G + G 1 |ν > + < µ|R|ν >, where .(µ − ν) := n1 θi µi + µn+1 and < µ|.|ν >= Hµ , .Hν . We get immediately that G 1 (µ, ) = − < µ|G|µ >. Moreover, let us define F by: < µ|F|ν >:=

< µ|G + G 1 |ν > , −i.(µ − ν)

which exists by the non-resonance condition. To show that F is an OPOG one just j has to decompose G = G l in monomial OPOGs G l = α(t)Dt b1 . . . bm , bi ∈ + + {a1 , a1 , . . . , an , an }. Then, for each ν there is only one µ for which < µ|G +G 1 |ν >= 0 and the difference µ − ν depends obviously only on G l , not on ν. Let us call this difference ρG l . Then F is given by the sum: F=

1 Gl . −i.ρG l

It is easy to check that one can pass from Lemma 4.10 to Theorem 4.9 by induction, writing [H, F + F ] = [H, F] + [H0 , F ] + [H − H0 , F] + [H − H0 , F ].

We will show finally that Theorems 4.8 and 3.1 are equivalent. Once again we can start by considering an Hamiltonian on L 2 (Rn × S 1 ) since any Fourier integral operator Bϕ , as defined in the beginning of the proof of Theorem 3.1, intertwines the original Hamiltonian H : C0∞ (X ) → C ∞ (X ) of Sect. 3 with a pseudodifferential operator on L 2 (Rn × S 1 ) satisfying the hypothesis of Theorem 4.8. W3

W4

Let us remark first of all that if U L = ei ei . . . ei integral operators, then so is U L . Secondly we have

WL

, all ei

Wl

being Fourier

Proposition 4.11. Let A be a pseudodifferential operator of total Weyl symbol a(x, ξ, t, τ, ). Then a vanishes to infinite order at p = τ = 0 if and only if ||AHµ || L 2 (Rn ×S 1 ) = O(|µ|∞ ).

Semiclassical Trace Invariants

13

Proof. The “if” part is exactly Proposition 4.3. For the “only if” part let us observe that, if the total symbol didn’t vanish to infinite order, then it would contain terms of the form αkmnr (t)k (x + iξ )m (x − iξ )n τ r . Let us prove this can’t happen in dimension 1, the extension to dimension n being straightforward. Each term of the form (x + i Dx )m (x − i Dx )n = a m (a + )n gives rise to an operator Am,n such that: Am,n Hµ =

|m+n| 2

∼ |µ| Therefore

(µ + 1) . . . (µ + n)(µ + n − 1) . . . (µ + n − m)Hµ+m−n

m+n 2

Hµ+m−n .

cmn Am,n Hµ = ||

cm,m−l Hµ+l ∼

cmn Am,n Hµ ||2 ∼

cm,m−l |µ|

2m−l 2

Hµ+l . In particular:

|µ|2m−l ,

so || cmn Amn Hµ || = O(|µ|∞ ) implies Cmn = 0. It is easy to check that the same argument is also valid for any ordered product of a’s and a + ’s.

In the next section we will show how the functions H of Theorem 4.8 and h of Theorem 3.1 are related. 5. Link Between the Two Quantum Constructions Consider a symbol (on R2n ) of the form h( p1 , . . . , pn ) ξ 2 +x 2

with pi = i 2 i . There are several ways of quantizing h: one of them consists in associating to h, by the spectral theorem, the operator h(P1 , . . . , Pn ) = h(P), −2 ∂x2 +x 2

i i where Pi = . Another one is the Weyl quantization procedure. 2 In this section we want to compute the Weyl symbol h we of h(P1 , . . . , Pn ) and apply the result to the situation of the preceding sections. By the metaplectic invariance of the Weyl quantization and the fact that h(P1 , . . . , Pn ) commutes with all the Pi ’s, we know that h we has the form

h we ( p1 , . . . , pn ) = h we ( p), that is, is a function of the classical harmonic oscillators pi := ξi2 + xi2 . To see how this h we is related to the h above we note that H is diagonal on the Hermite basis h j . Therefore

x+y xξ d xdξ 1 h(( j + )) =< h j , H h j >= h we ( )2 + ξ 2 ei h j (x)h j (ξ ) n/2 . 2 2 We now claim

14

V. Guillemin, T. Paul

Proposition 5.1. Let h be either in the Schwartz class, or a polynomial function. Let ˆ h(s) = (2π1 )n h( p)e−is. p dp be the Fourier transform of h. Then

2i tan(s/2). p we ˆ h ( p) = h(s)e (s)ds, (5.1) n where tan(s/2). p stands for i tan(si /2) pi and (s) = i=1 (1 − 2i tan(si /2)), and where (5.1) has to be interpreted in the sense of distribution, that is, for each ϕ in the Schwartz class of R,

2i tan(s/2). p ˆ h we ( p)ϕ( p)dp = h(s)e (s)dsϕ( p)dp

2i tan(s/2) ˆ = h(s)(s)ϕˆ ds. Finally, as → 0, h we ∼ h +

∞

cl 2l .

(5.2)

l=1

is.P ds, where eis.P is a zeroth order semiclassical pseudoˆ Proof. Let h(P) = h(s)e differential operator whose Weyl symbol will be computed from its Wick symbol (see n eisi .Pi , it is 5.5 below for the definition). Let us first remark that since eis.P = i=1 enough to prove the theorem in the one-dimensional case. Let ϕxξ be a coherent state at (x, ξ ), that is ξy

1

ϕxξ (y) = (π )− 4 ei e− Let z =

ξ√ +i x , z 2

=

ξ √ +i x 2

and z(t) =

ξ(t)+i √ x(t) . 2

(y−x)2 2

.

A straightforward computation gives

2zz −|z|2 −|z |2 2 ϕxξ , ϕx ξ = e .

(5.3)

Moreover decomposing ϕxξ on the Hermite basis leads to eis P ϕxξ = ei 2 ϕx(s)ξ(s) , s

(5.4)

−2 ∂x2 +x 2

where P = and z(t) = eit z. 2 The Wick symbol of eis P is defined as

σ wi (eis P )(x, ξ ) := ϕxξ , eis P ϕxξ

(5.5)

which, by (5.3) and (5.4), is equal to −is

e

− 1−e

x 2 +ξ 2 2

+i 2s

.

Moreover, using the Weyl quantization formula, it is immediate to see that the Weyl and Wick symbols are related by

where = −

∂2 ∂x2

∂2 + ∂ξ 2 .

σ wi = e−

4

σ we ,

Semiclassical Trace Invariants

15

It is a standard fact that the Wick symbol determines the operator: indeed the function

−2zz +|z|2 +|z |2 2

e (ϕxξ , eis P ϕx ξ ) obviously determines eis P . Moreover it is easily seen to be analytic in z and z . Therefore it is determined by its values on the diagonal z = z i.e., precisely, the Wick symbol of eis P . A straightforward calculation shows that, for (2k+1)π s , k ∈ Z, 2 2 = (1 − 2i tan(s/2))e

(2k+1)π , 2

This shows that, for 2s = σ

we

(e

− 4

e

2i tan(s/2)

x 2 +ξ 2 2

−is

=e

x 2 +ξ 2 2

+i 2s

.

(5.6)

k ∈ Z, we have

)( p) = (1 − 2i tan(s/2))e

is P

− 1−e

2i tan(s/2)

x 2 +ξ 2 2

.

Let us now take ϕ in the Schwartz class of R, and let Bϕ be the operator of (total) Weyl symbol ϕ( x

2 +ξ 2

2

). Let

f (s) := 2π

σ we (eis P )( p)ϕ( p) pdp = Trace[eis P Bϕ ].

Lemma 5.2. f ∈ C ∞ (R). Proof. By metaplectic invariance we know that Bϕ is diagonal on the Hermite basis. Therefore, ∀k ∈ N, (−i)k

1 dk 1 f (s) := Trace[eis P P k Bϕ ] = < h j , Bϕ h j > (( j + ))k eis( j+ 2 ). k ds 2

Since h j is microlocalized on the circle of radius ( j + 21 ) and ϕ is in the Schwartz class, the sum is absolutely convergent for each k.

2i tan(s/2)

x 2 +ξ 2 2

Therefore f (s) = 2π (1 − 2i tan(s/2))e ϕ( p) pdp and (5.6) is valid in the sense of distribution (in the variable p) for all s ∈ R. This expression gives (5.1) immediately for h in the Schwartz class. When h is a polynomial function it is straightforward to check that, since hˆ is a sum of derivatives of the Dirac mass and eis P is a Weyl operator whose symbol is C ∞ with respect to s, the formula also holds in this case. The asymptotic expansion (5.2) is obtained by expanding e

2itg(s/2)

x 2 +ξ 2 2

near eis

x 2 +ξ 2 2

.

Formula (5.1) shows clearly that h we depends only on the 2π periodization of ˆh(s)ei s2 , therefore Corollary 5.3. h we depends only on the values h (k + 21 ) , k ∈ N.

16

V. Guillemin, T. Paul

We mention one application of formula (5.1). Let us suppose first that we have computed the quantum normal form at order K , that is h K ( p) = ck p k := ck p1k1 . . . pnkn , |k|=k1 +···kn ≤K

|k|=k1 +···kn ≤K

and let us define h we K as the Weyl symbol of h K (P). Corollary 5.4. h we K ( p) =

ck

|k|=k1 +···kn ≤K

:=

|k|=k1 +···kn ≤K

ck

2i tan(s/2) p ∂K (s)e |s=0 ∂sk ∂K ∂sk11 . . . ∂ kn sn

(s)e2i

tan(s1 /2) p1 +···+tan(sn /2) pn

|s=0 .

Let us come back now to the comparison between the two constructions of Sects. 2 and 3. Clearly the “θ ” part doesn’t play any role, as the Weyl quantization of any function f (τ ) is exactly f (Dθ ). Therefore we have the following Theorem 5.5. The functions H of Theorem 3.1 and h of Theorem 4.8 are related by the formula

2i tan(s/2). p ˆ Dt , )e H (P1 , . . . , Pn , Dt , ) = h(s, (s)ds, where hˆ is the Fourier transform of h with respect to the variables pi . In particular H − h = O(2 ). Proof. The proof follows immediately from Proposition 5.1, and the unicity of the (quantum) Birkhoff normal form.

6. The Computation of the Semiclassical Birkhoff Canonical Form from the Asymptotics of the Trace Formula In this section we will abandon the quantum approach to Birkhoff canonical forms developed in Sects. 4–5 and revert to the symbolic approach of Sects. 2–3. Using this approach we will prove that the wave trace data coming from the Gutzwiller formula determine the Quantum Birkhoff canonical form constructed in Sect. 3. Our goal will be by “mimicking” (with semiclassical modifications) the proof of this result by Zelditch in [11,12] and in particular avoid the method of “Grushin reduction” used in [7] to equate the trace formula of [6,8,9] with the trace formula for a monodromy operator. Warning. The Aϕ in display 6.3 below is not the family of U L ’s figuring in Theorem 4.8 but is the “symbolic” Aϕ figuring in Theorem 3.1. In particular the estimates in Theorems 4.8 and 4.9 will not play any role in this proof. Let X and H be as in the Introduction. Let γ be a periodic trajectory of the vector field (2.1) of period 2π .

Semiclassical Trace Invariants

17

For l ∈ Z let ψl be a Schwartz function on the real line whose Fourier transform ψˆl is supported in a neighborhood of 2πl containing no other period The semiclassical of (2.1). of the form: trace formula gives an asymptotic expansion for Trace ψl H −E

Trace ψl

H−E

∼

∞

dlm m ,

(6.1)

m=0

where the dl ’s are distributions acting on ψˆl with support concentrated at {2πl}. We will show that the knowledge of the dl s determine the quantum semiclassical Birkhoff form of Sect. 2, and therefore the classical one. Let us first rewrite the l.h.s of (6.1) as

it H −E ˆ Trace ϕψ(t)e dt . (6.2) Since ψˆ is supported near a single period of (2.1) we know from the general theory of Fourier integral operators that one can microlocalize (6.1) near γ . Therefore we can conjugate (6.2) by the semiclassical Fourier integral operator Aϕ of Theorem 3.1. This leads to the computation of

it H −E −1 ˆ Trace Aϕ ψ(t)e dt Aϕ

H (P1 ,...,Pn ,Dt ,)+H −E it ˆ = Tr ψ(t)ρ(P1 , . . . , Pn , Dt )e dt , (6.3) where ρ ∈ C0∞ (Rn+1 ) with ρ = 1 in a neighborhood of p = τ = 0 and Tr stands for the Trace in L 2 (Rn × S 1 ). Let us note that, as is standard in the proof on trace formulas, by the independence condition of the Poincaré angles (see footnote (2)), γ is isolated on its energy shell {H = E}. By standard stationary phase techniques this is enough to show that the contribution of H in (6.3) is of order O(∞ ). Let us write H (P1 , . . . , Pn , Dt , ) as E + Dt + θi Pi + cr,s ()P r Dts . (6.4) r ∈Nn ,s∈Z

We will first prove l (t, θ ) be the function defined by Proposition 6.1. Let gr,s it θ1 +···+θn 2 e ∂ s ∂ r l ˆ gr,s (t, θ ) = −i −i t ψ(t) . t∂θ ∂t i (1 − eitθi )

(6.5)

Let us fix l ∈ Z. Then the knowledge of all the dlm s for m < M in (6.1) determines the following quantities: l cr,s ()gr,s (2πl, θ ) (6.6) |r |+s=m

for all m < M.

18

V. Guillemin, T. Paul

Proof. The r.h.s. of (6.3) can be computed thanks to (6.4) using 1 spectrum Pi = {(µi + ), µi ∈ N}, 2 spectrum Dt = {ν, n ∈ Z}. Thus the r.h.s of (6.3) can be written as

∞ k 1 it ν+θ.(µ+ 12 ) (it) ψˆl (t) ρ (µ + ), ν e 2 k! µ,ν k=0 k 1 r s |µ|+s−1 × cr,s () µ + ν dt, 2 r,s

(6.7)

since the support of ψˆ l contains only one period, and therefore the trace can be microlocalized infinitely close to the periodic trajectory, making the role of H inessential. Using the following remark of S. Zelditch:

1 µ+ 2

r

r

∂ ν = −i t∂θ s

∂ −i ∂t

s e

it ν+θ.(µ+ 12 )

we get, mod(∞ ),

ψˆl (t)

∞ (it)k k=0

k!

r,s

r s k ∂ ∂ it ν+θ.(µ+ 12 ) |µ|+s−1 −i cr,s () −i e dt. t∂θ ∂t µ,ν

(6.8) Since

ν∈Z

eitν

= 2π

l δ(t − 2πl) , and

µ∈Nn

e

itθ. µ+ 21

=

θ1 +···+θn

eit 2 , i 1−eitθi

together

with the fact that ψˆ is supported near 2πl, we get that (6.8) is equal to ⎡ 2π ⎣

∞ (i)k k=0

k!

×

t k ψˆl (t)

|µ|+s−1

r,s θ1 +···+θn

∂ cr,s () −i t∂θ

eit 2 i 1 − eitθi

.

r k ∂ s −i ∂t (6.9)

t=2πl

Rearranging terms in increasing powers of shows that the quantities (6.6) can be computed recursively.

The fact that one can compute the cr,s () from the quantities (6.6) is an easy consequence of the rational independence of the θi s and the Kronecker theorem, and is exactly the same as in [5].

Semiclassical Trace Invariants

19

References 1. Bridges, T.J., Cushman, R.H., MacKay, R.S.: Dynamics near an irrational collision of eigenvalues for symplectic mappings. Lamgford, W.F. (ed.) In: Normal Forms and Homoclinic Chaos. Fields Inst. Commun. 4, Providence, RI: Amer. Math. Soc., 1995, pp. 61–79 2. Chazarain, J.: Formule de Poisson pour les variétés Riemanniennes. Invent. Math. 24, 65–82 (1974) 3. Colin de Verdière, Y.: Spectre du Laplacien et longueurs des géodésiques périodiques. Compos. Math. 27, 83–106 (1973) 4. Duistermaat, J.J., Guillemin, V.: The spectrum of positive elliptic operators and periodic bicharacteristics. Inv. Math. 29, 39–79 (1975) 5. Guillemin, V.: Wave-trace invariants. Duke Math. J. 83, 287–352 (1976) 6. Gutzwiller, M.: Periodic orbits and classical quantization conditions. J. Math. Phys. 12, 343–358 (1971) 7. Iantchenko, A., Sjöstrand, J., Zworski, M.: Birkhoff normal forms in semi-classical inverse problems. Math. Res. Lett. 9, 337–362 (2002) 8. Paul, T., Uribe, A.: Sur la formule semi-classique des traces. C.R. Acad. Sci Paris 313, I, 217–222 (1991) 9. Paul, T., Uribe, A.: The semi-classical trace formula and propagation of wave packets. J. Funct. Anal. 132, 192–249 (1995) 10. Robert, D.: Autour de l’approximation semi-classique. Basel-Boston: Birkhäuser, 1987 11. Zelditch, S.: Wave invariants at elliptic closed geodesics. Geom. Funct. Anal. 7, 145–213 (1997) 12. Zelditch, S.: Wave invariants for non-degenerate closed geodesics. Geom. Funct. Anal. 8, 179–217 (1998) Communicated by B. Simon

Commun. Math. Phys. 294, 21–60 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0899-9

Communications in

Mathematical Physics

On Dilation Symmetries Arising from Scaling Limits Henning Bostelmann1,2, , Claudio D’Antoni2, , Gerardo Morsella2, 1 University of York, Department of Mathematics, Heslington, York YO10 5DD, United Kingdom.

E-mail: [email protected]

2 Università di Roma “Tor Vergata”, Dipartimento di Matematica, Via della Ricerca Scientifica,

I-00133 Roma, Italy. E-mail: [email protected], [email protected] Received: 24 January 2009 / Accepted: 18 March 2009 Published online: 30 August 2009 – © Springer-Verlag 2009

Abstract: Quantum field theories, at short scales, can be approximated by a scaling limit theory. In this approximation, an additional symmetry is gained, namely dilation covariance. To understand the structure of this dilation symmetry, we investigate it in a nonperturbative, model independent context. To that end, it turns out to be necessary to consider non-pure vacuum states in the limit. These can be decomposed into an integral of pure states; we investigate how the symmetries and observables of the theory behave under this decomposition. In particular, we consider several natural conditions of increasing strength that yield restrictions on the decomposed dilation symmetry. 1. Introduction In the analysis of quantum field theories, the notion of scaling limits plays an important role. The physical picture underlying this mathematical concept is as follows: One considers measurements in smaller and smaller space-time regions, at the same time increasing the energy content of the states involved, so that the characteristic action scale remains constant. Passing to the limit of infinitesimal scales, one obtains a new quantum field theory, the scaling limit of the original model. The scaling limit theory can be seen as an approximation of the full theory in the short-distance regime. However, it may differ significantly from the full theory in fundamental aspects, for example regarding its charge structure: In quantum chromodynamics, it is expected that confined charges (color) appear in the limit theory, but are not visible as such in the full theory. The virtues of the scaling limit theory include that it is typically simpler than the original one. In fact, in relevant examples, one expects it to be interactionless (asymptotic freedom). But even where this is not the case, the limit theory should possess Supported in part by the EU network “Noncommutative Geometry” (MRTN-CT-2006-0031962) Supported in part by PRIN-MIUR and GNAMPA-INDAM Supported in part by PRIN-MIUR, GNAMPA-INDAM and the Scuola Normale Superiore

22

H. Bostelmann, C. D’Antoni, G. Morsella

an additional symmetry: It should be dilation covariant, since any finite masses in the original model can be neglected in the limit of large energies. On the mathematical side, a very natural description of scaling limits has been given by Buchholz and Verch [BV95]. This description, formulated in the C∗ algebraic framework of local quantum physics [Haa96], originates directly from the physical notions, and avoids any additional input motivated merely on the technical side, such as a rescaling of coupling constants or mass parameters, or the choice of renormalization factors for quantum fields. This has the advantage of allowing an intrinsic, model-independent description of the short distance properties of the theory at hand. In particular, it has been successfully applied to the analysis of the charge structure of the theory in the scaling limit and to the intrinsic characterization of charge confinement [Buc96b,DMV04]. While the framework of Buchholz and Verch seems rather abstract at first, it has recently been shown that it reproduces the usual picture of multiplicative field renormalization in typical cases [BDM09]. The approach of [BV95] is based on the notion of the scaling algebra A, which consists—roughly speaking—of sequences of observables λ → Aλ at varying scale λ, uniformly bounded in norm, and subject to certain continuity conditions. (We shall recall the precise definition in Sec. 2.1.) The task of passing to the scaling limit is then reduced to finding a suitable state ω0 on the C∗ algebra A that represents the vacuum of the limit theory; it is constructed as a limit of vacuum states at finite scales. The limit theory itself is then obtained by a standard GNS construction with respect to ω0 . It should be easy in this context to describe the additional dilation symmetry that arises in the scaling limit. In fact, the scaling algebra A carries a very natural representation µ → δ µ of the dilation group, which acts by shifting the argument of the functions λ → Aλ : δ µ (A)λ = Aµλ . However, things turn out to be more involved: The limit states ω0 described in [BV95] are not invariant under this group action, and thus one does not obtain a canonical group representation in the limit Hilbert space. In [BDM09], generalized limit states have been introduced, some of which are invariant under dilations, and give rise to a unitary implementation of the dilation group in the limit theory. However, these dilation invariant limit states are never pure; rather they arise as a mixture of states of the Buchholz-Verch type, which are pure in 2+1 or more space-time dimensions. The object of the present paper is to analyze this generalized class of limit states in more detail, in order to describe the structure of the dilation symmetry associated with the dilation invariant ones. In particular we will show that, as briefly mentioned in [BDM09], the decomposition of these states in pure (Buchholz-Verch type) states gives rise to a direct integral decomposition of the limit Hilbert space, which also induces a decomposition of observables and of Poincaré symmetries. It should be noted here that the entire construction is complicated by the fact that uncountably many extremal states are involved in this decomposition, and that the measure space underlying the direct integral is of a very general nature. Because of this, we need to use a notion of direct integral of Hilbert spaces which is more general than the one previously employed in the quantum field theory literature [DS85]. It is also of interest to discuss how the special but physically important class of theories with a unique scaling limit, as defined in [BV95], fits into our generalized framework. It turns out that, up to some technical conditions, uniqueness of the scaling limit in the Buchholz-Verch framework is equivalent to the factorization of our generalized scaling limit into a tensor product of an irreducible scaling limit theory and a commutative part, which is just the image under the scaling limit representation of the center of the

On Dilation Symmetries Arising from Scaling Limits

23

scaling algebra. In particular we show that such factorization holds for a restricted class of theories, those with a convergent scaling limit. This class includes in particular dilation invariant theories and free field models. The technical conditions referred to above consist in a suitable separability requirement of the scaling limit Hilbert space, which is needed in order to be able to employ the full power of direct integrals theory. As a matter of fact, such separability condition is a consequence of a refined version of the Haag-Swieca compactness condition. With these results at hand, it is possible to discuss the structure of the unitarily implemented dilation symmetry in dilation invariant scaling limit states. The outcome is that in general the dilations do not decompose, not even in the factorizing situation. Rather, the dilations intertwine in a suitable sense the different pure limit states that occur in the direct integral decomposition. A complete factorization of the dilation symmetry is however obtained in the convergent scaling limit case. For such theories, therefore, one gets a unitary implementation of the dilation symmetry in the pure limit theory. The remainder of this paper is organized as follows: First, in Sec. 2, we recall the notion of scaling limits in the algebraic approach to quantum field theory, and generalize some fundamental results of [BV95] to our situation. In Sec. 3, we establish the direct integral decomposition mentioned above, including a decomposition of local observables and Poincaré symmetries. Section 4 contains a discussion of unique scaling limits as a special case. We define several conditions that generalize the notion from [BV95], and discuss relations between them. Then, in Sec. 5, we analyze the structure of dilation symmetries in the limit Hilbert space, and their decomposition along the direct integral, on different levels of generality. In Sec. 6 we propose a stronger version of the HaagSwieca compactness condition and we show that it implies the separability property used in the analysis of Sec. 4. Section 7 discusses some simple models as examples, showing in particular that these fulfill all of our conditions proposed in Sec. 4 and 6. We conclude with a brief outlook in Sec. 8. The Appendix reviews the concept of direct integrals of Hilbert spaces, which we need in a more general variant than covered in the standard literature. 2. Definitions and General Results We shall first recall the definition of scaling limits in the algebraic approach to quantum field theory, and prove some fundamental results regarding uniqueness of the limit vacuum state and geometric modular action. 2.1. The setting. We consider quantum field theory on (s + 1) dimensional Minkowski space. For our analysis, we work entirely within the framework of algebraic quantum field theory [Haa96], where observables localized in an open bounded subset O ⊂ Rs+1 of spacetime are described by the selfadjoint elements of a C∗ algebra A(O) in such a way that if O1 ⊂ O2 then A(O1 ) ⊂ A(O2 ). The correspondence A : O → A(O) thus defined is called a net of algebras. The inductive limit C∗ algebra of O → A(O) is denoted again by A and is called the quasilocal algebra. Let us repeat the formal definition of a quantum field theoretical model in this context. Definition 2.1. Let G be a Lie group of point transformations of Minkowski space that includes the translation group. A local net of algebras with symmetry group G is a net of algebras A together with a representation g → αg of G as automorphisms of A, such that

24

H. Bostelmann, C. D’Antoni, G. Morsella

(i) [A1 , A2 ] = 0 if O1 , O2 are two spacelike separated regions, and Ai ∈ A(Oi ); (ii) αg A(O) = A(g.O) for all O, g. We call A a net in a positive energy representation if, in addition, the A(O) are W ∗ algebras acting on a common Hilbert space H, and (iii) there is a strongly continuous unitary representation g → U (g) of G on H such that αg = ad U (g); (iv) the joint spectrum of the generators of translations U (x) lies in the closed forward light cone V¯ + ; (v) there exists a vector Ω ∈ H which is invariant under all U (g) and cyclic for A. We call A a net in the vacuum sector if, in addition, (vi) the vector Ω is unique (up to scalar factors) as an invariant vector for the translation group. Our approach is to start from a local net A in the vacuum sector, with the Poincaré ↑ group P+ as its symmetry group; this net A will be kept fixed in all that follows. Our aim is to describe the short-distance scaling limit of A. Following [BV95], we define B to be the set of bounded functions B : R+ → B(H), λ → B λ . Equipped with pointwise addition, multiplication, and ∗ operation, and with the norm B = supλ B λ , the set B becomes a C∗ algebra. Let G be the group formed by Poincaré transformations and dilations; we will write G g = (µ, x, Λ) with µ ∈ R+ , x ∈ Rs+1 , and Λ a Lorentz matrix. G acts on B via a representation α, given by (α g B)λ = αλµx,Λ (B λµ ) for g = (µ, x, Λ) ∈ G, B ∈ B,

(2.1)

where α is the Poincaré group representation on A. Note the rescaling of translations with the scale parameter λ. We now define new local algebras as subsets of B: A(O) := A ∈ B | Aλ ∈ A(λO) for all λ > 0; g → α g (A) is norm continuous . (2.2) This is a net of local algebras in the sense of Def. 2.1, with the enlarged symmetry group G [BDM09]. We denote by A the associated quasilocal algebra, i.e. the inductive limit of A(O) as O Rs+1 . This A is called the scaling algebra. Note that A has a large center Z(A), consisting of all operators A of the form Aλ = f (λ)1, where f : R+ → C is a bounded uniformly continuous function on R+ as a group under multiplication. We often identify A ∈ Z(A) with the function f without further notice. For a description of the scaling limit, we first consider states on Z(A). Let m be a mean on the bounded uniformly continuous functions1 on R+ , i.e., a positive normalized linear functional on the commutative C∗ algebra Z(A). We say that m is asymptotic if m( f ) = limλ→0 f (λ) whenever the limit on the right-hand side exists; or, equivalently, if m( f ) = 0 whenever f (λ) = 0 for small λ. Asymptotic means are, in this sense, generalizations of the limit λ → 0. Further we consider two important classes of means: (i) m is called multiplicative if m( f g) = m( f )m(g) for all functions f, g. (ii) m is called invariant if m( f µ ) = m( f ) for all functions f and all µ > 0, where f µ = f (µ · ). 1 In contrast to [BDM09], we do not consider means on the bounded functions on R , but rather on the + bounded uniformly continuous functions. While all of them can be extended to the bounded functions, these extensions do not play a role in our current investigation.

On Dilation Symmetries Arising from Scaling Limits

25

It is an important fact that (i) and (ii) are mutually exclusive; there are no multiplicative invariant means in our situation (cf. [Mit66]). We now extend these “generalized limits” of functions to a limit of operator sequences, using a projection technique. Let ω = (Ω| · |Ω) be the vacuum state of A. This state induces a projector (or conditional expectation) in A onto Z(A), which we denote by the same symbol: ω : A → Z(A), (ω(A))λ = ω(Aλ )1.

(2.3)

Using this projector, any mean m defines a state ωm on A by ωm := m ◦ ω. If here m is asymptotic, we call ωm a limit state, and typically denote it by ω0 . These are the states that correspond to scaling limits of the quantum field theory. Since there is a one-to-one correspondence between asymptotic means and limit states, we will usually work with the state ω0 only, and not refer to the mean m explicitly. A limit state ω0 will be called multiplicative2 or invariant if the corresponding mean has this property. Multiplicative limit states correspond to those considered by Buchholz and Verch in [BV95]. Every other limit state arises from these by convex combinations and weak∗ limits; this follows directly from the property of states on the commutative algebra Z(A). Given a limit state ω0 , we can obtain the limit theory via a GNS construction: Let π0 be the GNS representation of A with respect to ω0 , and H0 the representation space, with GNS vector Ω0 . Denoting by G0 the subgroup of G under which ω0 is invariant, we canonically obtain a strongly continuous unitary representation of G0 on H0 by setting U0 (g)π0 (A)Ω0 := π0 (α g (A))Ω0 , g ∈ G0 . The subgroup G0 contains the Poincaré group; and if ω0 is invariant, then G0 = G. The translation part of U0 fulfills the spectrum condition [BDM09]. Setting A0 (O) := π0 (A(O)) , one obtains a local net A0 with symmetry group G0 in a positive energy representation: the limit theory. 2.2. Multiplicity of the vacuum state. If ω0 is a multiplicative limit state, its restriction to Z(A) is pure. It has been shown in [BV95] that in the case s ≥ 2, this property extends to the entire theory: ω0 is a pure vacuum state on A, and π0 is an irreducible representation. On the other hand, if ω0 is not multiplicative, the same must be false, since already ω0 Z(A) is non-pure. However, we shall show that this property of the center is the only “source” of reducibility: namely one has π0 (A) = π0 (Z(A)) . We need some preparations to prove this. In the following, set Z0 := π0 (Z(A)) , and let HZ := clos(Z0 Ω0 ) ⊂ H0 be the representation space of the commutative algebra. Lemma 2.2. Let PZ ∈ B(H0 ) be the orthogonal projector onto HZ. If s ≥ 2, then PZ ∈ π0 (A) , and HZ is the space of all translation-invariant vectors in H0 . Proof. As a consequence of the spectrum condition in the theory A0 , it is known [Ara64] that the translation operators U0 (x) are contained in π0 (A) . Now let U∞ be an ultraweak cluster point of U0 (x) as x goes to spacelike infinity on some fixed sequence within the time-0 plane. (Such cluster points exist by the Alaoglu-Bourbaki theorem.) Then U∞ ∈ π0 (A) ; we will show U∞ = PZ. To that end, we first note that ω0 (A B) = ω0 (ω(A) B) for all A ∈ A, B ∈ Z(A),

(2.4)

2 For clarity, we note that a multiplicative limit state, by this definition, is not a multiplicative functional on A, but is multiplicative only on the center Z(A).

26

H. Bostelmann, C. D’Antoni, G. Morsella

which follows directly from the definition of ω0 . Now we make use of the cluster property of the vacuum at finite scales. As in [BV95, Lemma 4.3], one can obtain the following norm estimate in the algebra A: ω(A α x B) − ω(A)ω(B) ≤ c

rs ˙ + AB ˙ A B |x|s−1

(2.5)

for fixed r > 0, x in the time-0 plane with |x| > 3r , and for A, B chosen from some norm-dense subset of A(Or ), with Or being the standard double cone of radius r around the origin. Here c > 0 is some constant, and the dot denotes the time derivative. This implies that as |x| → ∞, lim ω0 (A α x B) = ω0 (ω(A)ω(B)) x

(2.6)

for these A, B. Now it follows from Eq. (2.4) – with ω(B) in place of B – that (π0 (A)Ω0 |U∞ π0 (B)Ω0 ) = lim ω0 (A∗ α x B) = (π0 (A)Ω0 |π0 (ω(B))Ω0 ). (2.7) x

Continuing this relation from the dense sets chosen, this means U∞ π0 (B)Ω0 = π0 (ω(B))Ω0 for all B ∈ A.

(2.8)

2 = U , and img U This shows that U∞ ∞ ∞ = HZ. Also, again applying Eq. (2.4), one ∗ = U . Thus U is the unique orthogonal projector onto H . For the last obtains U∞ ∞ ∞ Z part, note that translations act trivially on HZ, and that U∞ leaves all translation-invariant vectors unchanged; so HZ is the space of all translation-invariant vectors.

We are now ready to prove the announced result about the commutant of π0 (A). Theorem 2.3. Let s ≥ 2. Let ω0 be a limit state, and let π0 be the corresponding GNS representation. Then π0 (A) = π0 (Z(A)) . Proof. Let B ∈ π0 (A) . By Lemma 2.2, B commutes with PZ; hence BHZ ⊂ HZ, and B HZ ∈ B(HZ) is well-defined. As Z0 ⊂ π0 (A) , we know that [B HZ, C HZ] = 0 for all C ∈ Z0

(2.9)

as an equation in B(HZ). Since Z0 HZ is a maximal abelian algebra in B(HZ) [BR79, Lemma 4.3.15], there exists C ∈ Z0 with B HZ = C HZ. Now for any A ∈ π0 (A) , we can compute B AΩ0 = ABΩ0 = ACΩ0 = C AΩ0 . Since Ω0 is cyclic for π0 , this implies B = C. Thus π0 is trivial.

(A)

(2.10)

⊂ Z0 . The reverse inclusion

It should be noted that the same theorem does not hold in 1+1 space-time dimensions. In this case, it is known even in free field theory [BV98, Sec. 4] that the algebra π0 (A) has a large center, even if π0 (Z(A)) = C1. We can now easily reproduce the known results for multiplicative limit states. In this case, the GNS representation of the abelian algebra Z(A) for the state ω0 must be irreducible; thus Z0 = C1, and dim HZ = 1. The above results imply: Corollary 2.4. Let ω0 be a multiplicative limit state, and let s ≥ 2. Then Ω0 is unique up to a scalar factor as an invariant vector for the translations U0 (x), and the representation π0 is irreducible. A0 is a net in the vacuum sector in the sense of Def. 2.1.

On Dilation Symmetries Arising from Scaling Limits

27

2.3. Wedge algebras and geometric modular action. While we have defined the scaling limit in terms of local algebras for bounded regions, it is also worthwhile to consider algebras associated with unbounded, in particular wedge-shaped regions. This is particularly important in the context of charge analysis for the limit theory [DMV04,DM06]. While we do not enter this topic here, and do not build on it in the following, we wish to discuss briefly how wedge algebras and the condition of geometric modular action fit into our context. Again, this transfers results of [BV95] to our generalized class of limit states. Let W be a wedge region, i.e. W is a Poincaré transform of the right wedge W+ = −W− = {x ∈ R4 | x · e± < 0}, where e± := (±1, 1, 0, 0).

(2.11)

Note that (W + ) = W− . We introduce the one-parameter group (Λt )t∈R of Lorentz boosts leaving W+ invariant, fixed by Λt e± = exp(±t)e± , and acting as the identity on the edge (e± )⊥ of W+ . Let furthermore j be the inversion with respect to the edge of W+ , i.e. je± = −e± and j = 1 on (e± )⊥ . Note that j 2 = 1 and jW+ = W− . For a local net of algebras (resp. for a net of algebras in a positive energy representation) O → A(O), we define the algebra A(W) associated to the wedge W as the C∗ -algebra (resp. W∗ -algebra) generated by the algebras A(O), where O is any double cone whose closure is contained in W (O ⊂⊂ W in symbols). With these definitions, we can adapt the arguments in [BV95, Lemma 6.1], which do not depend on irreducibility of the net. It is then straightforward to verify that, for a net A in a positive energy representation, the vacuum vector Ω is cyclic and separating for all wedge algebras A(W). This allows us to introduce the notion of geometric modular action. Definition 2.5. Let A be a local net in a positive energy representation, and denote by ∆, J the modular objects associated to A(W+ ), Ω. The net A is said to satisfy the condition of geometric modular action if there holds ∆it = U (Λ2π t ),

t ∈ R, ↑

J U (x, Λ)J = U ( j x, jΛj), (x, Λ) ∈ P+ , J A(O)J = A( jO).

(*)

If A satisfies the condition of geometric modular action, then it also satisfies wedge duality, since, according to Tomita-Takesaki theory and equation (*), A(W+ ) = J A(W+ )J = A(W− ).

(2.12)

This also implies that A satisfies essential Haag duality, i.e. that the dual net Ad of A, defined on double cones O as A(W), (2.13) Ad (O) := W ⊃O

is local and such that A(O) ⊂ Ad (O) for each double cone O. From now on, let A be a net in the vacuum sector, and ω0 a scaling limit state, with π0 the corresponding scaling limit representation. It holds that π0 (A(W)) = A0 (W), since clearly π0 (A(W)) =

O⊂⊂W

π0 (A(O))

·

,

(2.14)

28

H. Bostelmann, C. D’Antoni, G. Morsella

and therefore

π0 (A(W)) =

π0 (A(O)) =

O⊂⊂W

A0 (O) = A0 (W) .

(2.15)

O⊂⊂W

Proposition 2.6. Assume that A satisfies the condition of geometric modular action. Then for each limit state ω0 , the corresponding limit theory A0 also satisfies the condition of geometric modular action. Proof. It’s a straightforward adaptation of the proofs of Lemma 6.2 and Proposition 6.3 of [BV95]. The only point which is worth mentioning is the proof that ω0 is a KMS state (at inverse temperature 2π ) for the algebra A(W+ ) with respect to the one-parameter group of automorphisms (α Λt )t∈R , which goes as follows. Let m be the mean which induces ω0 . Then m is a weak∗ limit of convex combinations of multiplicative means, and therefore ω0 is a weak∗ limit of convex combinations of multiplicative limit states. For such states, the arguments in [BV95, Lemma 6.2] show that they are KMS on A(W+ ), and therefore, the set of KMS states at a fixed inverse temperature being convex and weak∗ closed [BR81, Thm. 5.3.30], this holds also for ω0 . 3. Decomposition Theory Our aim is now to decompose an arbitrary limit state ω0 into “simple” limit states of the Buchholz-Verch type, and to obtain corresponding decompositions of the relevant objects in the limit theory. We start by proving an integral decomposition which is a consequence of standard results. Proposition 3.1. Let ω0 be a limit state. There exists a compact Hausdorff space Z, a regular Borel probability measure ν on Z, and for each z ∈ Z a multiplicative limit state ω z , such that ω0 (A) = dν(z) ω z (A) for all A ∈ A. Z

Further, the map Z(A) → C(Z), C → (z → ω z (C)) is surjective. Proof. Let π0 be the GNS representation of A for ω0 . Consider the C∗ algebra π0 (Z(A)). It is well known that this commutative algebra is isomorphic to C(Z) for a compact Hausdorff space Z, with the isomorphism being given by π0 (C) → (z → ρz (π0 (C))), where the ρz are multiplicative functionals. Now by the Riesz representation theorem, the GNS state (Ω0 | · |Ω0 ) on π0 (Z(A)) ∼ = C(Z) is given by a regular Borel measure ν on Z. Explicitly, one has for all C ∈ Z(A), ω0 (C) = (Ω0 |π0 (C)|Ω0 ) = dν(z) ρz ◦ π0 (C). (3.1) Z

It is clear that ν(Z) = 1. In the above expression, mz := ρz ◦ π0 are multiplicative means; they are asymptotic, since π0 (A) = 0 whenever Aλ vanishes for small λ. Thus, setting ω z = ρz ◦ π0 ◦ ω as usual, we obtain multiplicative limit states ω z on A such that ω0 (A) = dν(z) ω z (A) for all A ∈ A. (3.2) Z

As a last point, the map Z(A) → C(Z), C → (z → ω z (C)) = (z → ρz (π0 (C))) is surjective by construction.

On Dilation Symmetries Arising from Scaling Limits

29

We have thus decomposed a general limit state ω0 into multiplicative limit states ω z . In the case s ≥ 2, this will also be a decomposition into pure states; but the above result does not depend on that. Also, we emphasize that our aim is not a decomposition of the von Neumann algebra π0 (A) along its center; rather we work on the C∗ algebraic side only. We would now like to interpret the above decomposition in the sense of decomposing the limit Hilbert space H0 as a direct integral. This is complicated by the fact that our measure spaces (Z, ν) can be of a very general nature, making the limit Hilbert space nonseparable. In fact, if ω0 is an invariant limit state, one finds that all vectors of the form π0 (C)Ω0 are mutually orthogonal if C λ = χ (λ)1, where χ is a character on R+ . Since there are clearly uncountably many characters—just take χ (λ) = λık with k ∈ R—the limit Hilbert space H0 cannot be separable in this case. The theory of direct integrals of Hilbert spaces in the absence of separability assumptions is nonstandard and only partially complete; we give a brief review in Appendix A. Here we note that the notion of a direct integral over Z, with fiber spaces Hz , crucially depends on the specification of a fundamental family Γ ⊂ z∈Z Hz . This Γ is a vector space with certain extra conditions (see Def. A.1) that serves to define which Hilbert space valued functions are considered measurable. Indeed, using the exact notions, we prove: Theorem 3.2. Let ω0 be a limit state, and Z, ν, ω z as in Proposition 3.1. Let πz , Hz , Ωz be the GNS representation objects corresponding to ω z . Then, Hz Γ := {z → πz (A)Ωz | A ∈ A} ⊂ z∈Z

is a fundamental family. With respect to this family, it holds that Γ ∼ H0 = dν(z) Hz , Z

where the isomorphism is given by π0 (A)Ω0 →

Γ

Z

dν(z) πz (A)Ωz , A ∈ A.

Proof. It is clear that Γ is a linear space; and per Prop. 3.1, the function z → πz (A)Ωz 2 = ω z (A∗ A) is integrable for any A ∈ A. Thus Γ is a fundamental family per Definition A.1. The map W : H0 → Hz , π0 (A)Ω0 → (z → πz (A)Ωz ) (3.3) z∈Z

is clearly linear and isometric when A ranges through A; thus W can in fact be extended to a well-defined isometric map from H0 into Γ¯ . It remains to show that W is surjective. In fact, since L ∞ (Z) · Γ is total in the direct integral space, it suffices to show that all vectors of the form Γ dν(z) f (z) πz (A)Ωz , f ∈ L ∞ (Z), A ∈ A, (3.4) Z

can be approximated in norm with vectors of the form W π0 (B)Ω0 , B ∈ A.

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H. Bostelmann, C. D’Antoni, G. Morsella

To that end, let f ∈ L ∞ (Z) and A ∈ A be fixed. We first note that, as a simple consequence of Lusin’s theorem, there exist functions f n ∈ C(Z) such that f n ∞ ≤ f ∞ and limn→∞ f n (z) = f (z) for almost every z ∈ Z. On the other hand, per Proposition 3.1 there exist C n ∈ Z(A) such that ω z (C n ) = f n (z) for all z ∈ Z, which implies πz (C n ) = f n (z)1. Therefore we have lim

n→∞ Z

( f (z)πz (A) − πz (C n A))Ωz 2 dν(z) = 0

by an application of the dominated convergence theorem.

(3.5)

In the following, we will usually not denote the above isomorphism explicitly, but rather identify H0 with its direct integral representation. In this way, the subspace HZ ⊂ H0 is isomorphic to the function space L 2 (Z, ν), where f ∈ L 2 (Z, ν) is identified

Γ with Z dν(z) f (z)Ωz ∈ H0 . The next corollary follows directly from the proof above, since a decomposition of operators needs to be checked on the fundamental family only (Lemma A.2). Corollary 3.3. With respect to the direct integral decomposition in Theorem 3.2, all

Γ operators π0 (A), A ∈ A are decomposable, and one has π0 = Z dν(z)πz . If A ∈ Z(A),

Γ then π0 (A) is diagonal, with π0 (A) = Z dν(z) ω z (A)1. Finally, we remark that Lorentz symmetries U0 (x, Λ) in the limit theory are decomposable. ↑

Proposition 3.4. Let g → Uz (g) be the implementation of P+ on the limit Hilbert space Hz corresponding to ω z . Then, one has U0 (g) =

Γ

Z

↑

dν(z)Uz (g) for all g ∈ P+ .

Proof. Again, it suffices to verify this on vectors from Γ . With W being the isomorphism ↑ introduced in the proof of Theorem 3.2, one obtains for all g ∈ P+ and A ∈ A, W U0 (g)π0 (A)Ω0 = W π0 (α g A)Ω0 = =

Γ

Z

dν(z)Uz (g)πz (A)Ωz .

This proves the proposition.

Γ

Z

dν(z)πz (α g A)Ωz (3.6)

It should be remarked that the same simple structure cannot be expected for dilations, if they exist as a symmetry of the limit. For even if ω0 ◦ α µ = ω0 , the multiplicative limit states ω z cannot be invariant under α µ , not even when restricted to Z(A). Thus, the unitaries U0 (µ) will not commute with π0 (Z(A)), and can therefore not be decomposable. In special situations, there may be a generalized sense in which the dilation unitaries can be decomposed; we will investigate this in more detail in Sec. 5.

On Dilation Symmetries Arising from Scaling Limits

31

4. Unique and Factorizing Scaling Limits The limit theory on the Hilbert space H0 is composed, as discussed in the previous section, of simpler components that live on the “fibre” Hilbert spaces Hz of the direct integral. It is natural to ask whether the theories on these spaces Hz , or more precisely, the nets of algebras Az (O) = πz (A(O)) , are similar or identical in a certain sense. While no models have been explicitly constructed for which the limit theories substantially depend on the choice of a (multiplicative) limit state,3 it does not seem to be excluded that measurable properties, such as the mass spectrum or charge structure of Az , can depend on z. For most applications in physics, however, one expects that the situation is simpler, and that the limit theory does not depend substantially on the choice of ω z . Here it would be much too strict to require that the representations πz are unitarily equivalent. [In fact, for s ≥ 2, the πz are irreducible per Thm. 2.3, and since they do not agree on Z(A), they are even pairwise disjoint.] Rather one can expect that their images, the algebras Az (O), are unique as sets, up to unitaries that identify the different Hilbert spaces Hz ; see Def. 4.1 below. This is the situation of a unique scaling limit in the sense of Buchholz and Verch. In the present section, we want to elaborate how the situation of unique scaling limits, originally formulated for multiplicative limit states, fits into our generalized context. To that end, we will formulate several conditions on the limit theory that roughly correspond to unique limits, and discuss their mutual dependencies. 4.1. Definitions. We shall first motivate and define the conditions to be considered; the proofs of their interrelations are deferred to sections further below. We start by recalling the condition of a unique scaling limit in the sense of [BV95], with some slight modifications. Definition 4.1. The theory A is said to have a unique scaling limit if there exists a local Poincaré covariant net (Au , Hu , Ωu , Uu ) in the vacuum sector such that the following holds. For every multiplicative limit state ω0 , there exists a unitary V : H0 → Hu such ↑ that V Ω0 = Ωu , V U0 (g)V ∗ = Uu (g) for all g ∈ P+ , and V A0 (O)V ∗ ⊂ Au (O) for all open bounded regions O. This includes the aspect of a “unique vacuum structure”. Compared with [BV95], we have somewhat weakened the condition, since we require only inclusion of V A0 (O)V ∗ in Au (O), not equality. This is for the following reason. Supposing that both A and Au fulfill the condition of geometric modular action (Definition 2.5), such that the Haagdualized nets of A0 and Au are well-defined, our condition precisely implies that these dualized nets agree for all multiplicative limit states. Since for many applications, particularly charge analysis [DMV04], the dualized limit nets are seen as the fundamental objects, we think that this is a reasonable generalization of the condition. For a general, not necessarily multiplicative limit state ω0 , we obtain a decomposition ω0 = Z dν(z)ω z into multiplicative states, as discussed in Sec. 3, and thus obtain from Def. 4.1 corresponding unitaries Vz for every z. Due to the very general nature of the measure space Z, and due to a possible arbitrariness in the choice of Vz , particularly if Au possesses inner symmetries, an analysis of ω0 seems impossible in this generality. Rather we will often make use of a regularity condition, which is formulated as follows. 3 See however [Buc96a, Sec. 5] for some ideas to that end.

32

H. Bostelmann, C. D’Antoni, G. Morsella

Definition 4.2. Suppose that the theory A has a unique scaling limit. We say that a limit state ω0 is regular if there is a choice of the unitaries Vz such that for any A ∈ A, the function ϕ A : Z → Hu , z → Vz πz (A)Ωz is Lusin measurable [i.e., is contained in L 2 (Z, ν, Hu )]. We shall later give a sufficient condition for the above regularity, which actually implies that the functions ϕ A can be chosen constant in generic cases. Our concepts so far refer to multiplicative limit states mostly. We will now give a generalization of Def. 4.1 that involves generalized limit states directly, and that seems natural in our context. It is based on the picture that the limit Hilbert space should have a tensor product structure, H0 ∼ = HZ ⊗ Hu , where Hu is the unique representation space associated with multiplicative limit states, and HZ is the representation space of Z(A) under π0 . All objects of the theory—local algebras, Poincaré symmetries, and the vacuum vector—should factorize along this tensor product. We now formulate this in detail. Definition 4.3. The theory A is said to have a factorizing scaling limit if there exists a local Poincaré covariant net (Au , Hu , Ωu , Uu ) in the vacuum sector such that the following holds. For every limit state ω0 , there exists a decomposable unitary V :

Γ,⊕ H0 → L 2 (Z, ν, Hu ), V = Z dν(z)Vz with unitaries Vz : Hz → Hu , such that ↑ V Ω0 = ΩZ ⊗ Ωu , V U0 (g)V ∗ = 1 ⊗ Uu (g) for all g ∈ P+ , and Vz Az (O)Vz∗ ⊂ Au (O) for all open bounded regions O and all z ∈ Z. Here ΩZ ∈ HZ denotes the GNS vector of the commutative algebra. The conditions on local algebras are deliberately chosen quite strict. We require Vz Az (O)Vz∗ ⊂ Au (O) ¯ u (O). This serves to for every z, rather than the weaker condition V A0 (O)V ∗ ⊂ Z0 ⊗A avoid countability problems; see Sec. 4.3 for further discussion. In subsequent sections, we will show that the notion of a unique scaling limit and a factorizing scaling limit are cum grano salis identical, up to the extra regularity condition in Definition 4.2 that we have to assume. We also consider a stronger condition, which is easier to check in models. Our ansatz is to require a sufficiently large subset Aconv ⊂ A such that for each A ∈ Aconv , the function λ → ω(Aλ ) is convergent as λ → 0. Consider the following definition: Definition 4.4. The theory A is said to have a convergent scaling limit if there exists an α-invariant C∗ subalgebra Aconv ⊂ A with the following properties: (i) For each A ∈ Aconv , the function λ → ω(Aλ ) converges as λ → 0. (ii) If ω0 is a multiplicative limit state, then π0 (A(O)∩Aconv ) is weakly dense in A0 (O) for every open bounded region O. It follows directly from (ii) that also π0 (Aconv )Ω0 is dense in H0 . The condition roughly says that “convergent scaling functions” are sufficient for describing the limit theory—considering nonconvergent sequences is only required for technical consistency of our formalism, for describing the image of Z(A), which does not directly relate to quantum theory. This is heuristically expected in many physical models: In usual renormalization approaches in formal perturbation theory, the selection of subsequences or filters to enforce convergence seems not to be widespread, and sequences of pointlike fields can be chosen to converge in matrix elements. We will show that the above condition is sufficient for the scaling limit to be unique, and all limit states to be regular. In fact, we shall see later that also the structure of dilations simplifies.

On Dilation Symmetries Arising from Scaling Limits

33

Unique limit

Factorizing limit

Convergent limit

Regularity condition

Fig. 1. Implications between the conditions on the limit theory. Arrows marked with ∗ are only proven under additional separability assumptions

We also mention that this condition has been employed in [CM08] in order to discuss some functoriality properties of the scaling limit with respect to the formation of subsystems of the observable net. Figure 1 summarizes the different conditions we introduced, and shows the implications we briefly mentioned. We will now go ahead and prove that the individual arrows are indeed correct. However, in order to avoid problems with the direct integral spaces involved, we shall make certain separability assumptions in most cases. Let us comment on these. For multiplicative limit states, it seems a reasonable assumption that the limit Hilbert space H0 is separable. This would follow, from example, from the Haag-Swieca compactness condition; cf. [Buc96a]. For general limit states, in particular if these are invariant, H0 cannot be separable since already HZ ∼ = L 2 (Z, ν) is nonseparable, as discussed in Sec. 3. We can however reasonably assume that H0 fulfills a condition which we call uniform separability; cf. Def. A.3 in the Appendix. This means that a countable

Γ set {χ j } ⊂ H0 = Z dν(z)Hz exists such that {χ j (z)} is dense in every Hz . As we shall see in Sec. 6, uniform separability follows from a sharpened version of the Haag-Swieca compactness condition; and we will show in Sec. 7 that this compactness condition is indeed fulfilled in relevant examples. 4.2. Unique limit ⇒ factorizing limit. In the following, we suppose that A has a unique scaling limit. We fix a regular limit state ω0 , and denote the associated objects Z, ν, H0 , π0 , Ω0 , Hz , πz , Ωz , Vz as before. In order to prove that the scaling limit factorizes, we have to construct a unitary V : H0 → L 2 (Z, ν, Hu ) with appropriate properties. In fact,

Γ,⊕ this V is intuitively given by V = Z dν(z)Vz ; and the key question turns out to be whether this V is surjective. We will prove this only under separability assumptions. Proposition 4.5. Let A have a unique scaling limit; let ω0 be a regular limit state; and suppose that H0 is uniformly separable. Then, Γ,⊕ V : H0 → L 2 (Z, ν, Hu ), V = dν(z)Vz Z

defines a unitary operator. Proof. First, it is clear that if H0 is uniformly separable, then all Hz , and in particular Hu , are separable. Hence L 2 (Z, ν, Hu ) is uniformly separable. Now note that V is well-defined precisely by the regularity condition. Further, writing explicitly V π0 (A)Ω0 = z → Vz πz (A)Ωz , A ∈ A, (4.1)

34

H. Bostelmann, C. D’Antoni, G. Morsella

one has

V π0 (A)Ω0 2 = dν(z) Vz πz (A)Ωz 2 = dν(z) πz (A)Ωz 2 = π0 (A)Ω0 2 , Z

Z

(4.2) so V is isometric. It remains to show that V is surjective. To that end, let P be the orthogonal projector onto img V . Since

⊕ V commutes with all diagonal operators, so does P; thus P is decomposable: P = Z dν(z)P(z). Now compute 0 = (1 − P)V =

Γ,⊕

Z

dν(z)(1 − P(z))Vz .

(4.3)

Using uniform separability of both spaces involved, we obtain that (1 − P(z))Vz = 0 a. e. Since the Vz are surjective onto Hu , this means P(z) = 1 a. e. This implies P = 1, so V is surjective. It is clear that V Ω0 = ΩZ ⊗ Ωu ; and we can also verify from the properties of the Vz with respect to Poincaré symmetries that ↑

V U0 (g)V ∗ = 1 ⊗ Uu (g) for all g ∈ P+ .

(4.4)

Also, by the definition of the unique scaling limit, it must hold that Vz Az (O)Vz∗ ⊂ Au (O) for all z. Summarizing the results of this section, we have shown: Theorem 4.6. Suppose that A has a unique scaling limit, that every limit state ω0 is regular, and that the limit spaces H0 are uniformly separable. Then the scaling limit of A is factorizing. 4.3. Factorizing limit ⇒ unique limit. Now reversing the arrow, we start from a theory with factorizing scaling limit, and want to show that the scaling limit is unique in the sense of Buchholz and Verch, and that the limit states are regular. At first glance, this implication seems to be apparent from the definitions. A detailed investigation however reveals some subtleties, which again lead us to making separability assumptions. Theorem 4.7. Assume that A has a factorizing scaling limit. Then the scaling limit is unique. If the space Hu is separable, all limit states ω0 are regular. Proof. It is clear that the scaling limit is unique by Def. 4.1, specializing the conditions of Def. 4.3 to the case where ω0 is multiplicative, and Z consists of a single point. Now

Γ,⊕ let ω0 be a limit state; we need to show it is regular. Let V = Z dν(z)Vz be the unitary guaranteed by Def. 4.3. By definition, the map z → Vz πz (A)Ωz is measurable for any A ∈ A. But we have to show that each Vz fulfills the conditions of Def. 4.1; in fact, we will have to modify the Vz on a null set. First, we have V Ω0 = ΩZ ⊗ Ωu by assumption. On the other hand, V Ω0 =

⊕ Z dν(z) Vz Ωz , so that Vz Ωz = Ωu for z ∈ Z\NΩ , where NΩ is a null set. Next we consider Poincaré transformations. Starting from Def. 4.3, we know that: ↑

V U0 (g)V ∗ = 1 ⊗ Uu (g) for all g ∈ P+ .

(4.5)

On Dilation Symmetries Arising from Scaling Limits

35

Since U0 (g) factorizes by Prop. 3.4, we can rewrite this equation as

⊕

Z

dν(z) Vz Uz (g)Vz∗

=

⊕

Z

dν(z) Uu (g).

(4.6)

Now if Hu is separable, and thus L 2 (Z, ν, Hu ) uniformly separable, we can conclude that Vz Uz (g)Vz∗ = Uu (g) for all z ∈ Z\Ng , with a null set Ng depending on g. We pick ↑ a countable dense subset Pc of P+ , and consider the null set N := NΩ ∪ (∪g∈Pc Ng ). Our results so far are that Vz Ωz = Ωu , Vz Uz (g)Vz∗ = Uu (g) for all z ∈ Z\N , g ∈ Pc .

(4.7)

↑ Indeed, by continuity of the representations, the same holds for all g ∈ P+ . Now let Vˆz be those unitaries obtained by evaluating Def. 4.3 for the multiplicative limit states ω z . We set Vz for z ∈ Z\N , Wz := ˆ (4.8) Vz for z ∈ N .

Γ,⊕ Then we have V = Z dν(z) Wz , and the Wz fulfill the relations in Eq. (4.7) for all ↑ z ∈ Z and g ∈ P+ . As a last point, Wz Az (O)Wz∗ ⊂ Au (O) holds for every z, since both Vz and Vˆz have this property. Thus ω0 is regular. Let us add some comments on the conditions required for Vz in Def. 4.3, regarding Poincaré transformations and local algebras. We could choose stricter conditions on Vz , requiring that ↑

Vz Uz (g)Vz∗ = Uu (g) for all z ∈ Z and g ∈ P+ .

(4.9)

In this case, the countability problem in the proof above does not occur, and Thm. 4.7 holds without the requirement that Hu is separable. On the other hand, it does not seem reasonable to weaken the conditions on Vz with respect to local algebras, requiring only that ¯ u (O) for all O. V A0 (O)V ∗ ⊂ Z0 ⊗A

(4.10)

(We shall show below that this relation is implied by the chosen conditions on Vz .) For if we require only (4.10), and we wish to apply the techniques used in the proof of Thm. 4.7, it becomes necessary not only to require separability of Hu —which seems reasonable for applications in physics—but also separability of the algebras Au (O). That would however be too strict for our purposes, since the local algebras are expected to be isomorphic to the hyperfinite type III1 factor [BDF87]. We now show that Eq. (4.10) follows from Def. 4.3 as given. Proposition 4.8. Let A have a factorizing scaling limit. With V the unitary of Def. 4.3, ¯ u (O) for any bounded open region O. one has V A0 (O)V ∗ ⊂ Z0 ⊗A

36

H. Bostelmann, C. D’Antoni, G. Morsella

Proof. Let A ∈ A(O), and A ∈ Au (O) . We compute the commutator [1 ⊗ A , V π0 (A)V ∗ ] as a direct integral: ⊕ dν(z) [A , Vz πz (A)Vz∗ ]. (4.11) [1 ⊗ A , V π0 (A)V ∗ ] = Z

Now by our requirements on the Vz , we have Vz πz (A)Vz∗ ∈ Au (O) for all z, hence the commutator under the integral vanishes. Since A ∈ A(O) was arbitrary, this means ¯ u (O). V π0 (A(O))V ∗ ⊂ (1 ⊗ Au (O) ) = Z0 ⊗A By weak closure, this inclusion extends to V A0

(O)V ∗ .

(4.12)

4.4. Convergent limit ⇒ unique limit. We now assume that the theory has a convergent scaling limit, and show that our other conditions follow. The main simplification in the convergent case is as follows: For every A ∈ Aconv , the function λ → ω(Aλ ) converges to a finite limit as λ → 0; so all asymptotic means applied to this function yield the same value. Hence the value of ω0 (A) is the same for all limit states ω0 , multiplicative or not. Theorem 4.9. If the scaling limit of A is convergent, then it is unique. If further a multiplicative limit state exists such that the associated limit space H0 is separable, then all limit states are regular, and H0 is uniformly separable for any limit state. Proof. We pick a fixed multiplicative limit state ωu and denote the corresponding representation objects as Hu , πu , Uu , Ωu . Given any other multiplicative limit state ω0 , we define a map V by V : H0 → Hu , π0 (A)Ω0 → πu (A)Ωu for all A ∈ Aconv .

(4.13)

The convergence property of A ∈ Aconv implies π0 (A)Ω0 2 = ω0 (A∗ A) = ωu (A∗ A) = πu (A)Ωu 2 ,

(4.14)

so the linear map V is both well-defined and isometric. It is also densely defined and surjective by assumption (Def. 4.4). Hence V extends to a unitary. Using the α-invariance of ↑ Aconv , one checks by direct computation that V U0 (g)V ∗ = Uu (g) for all g ∈ P+ . Also, ∗ V Ω0 = Ωu is clear. Further, if A ∈ A(O) ∩ Aconv , it is clear that V π0 (A)V = πu (A). By weak density, this means V A0 (O)V ∗ = Au (O). Thus the scaling limit is unique. Now let ω0 not necessarily be multiplicative. Decomposing it into multiplicative states ω z as in Prop. 3.1, the above construction gives us unitaries Vz : Hz → Hu for every z. In fact, the functions z → Vz πz (A)Ωz = πu (A)Ωu are constant for all A ∈ Aconv , in particular measurable. Now let χ ∈ Hu and B ∈ A. We can choose a sequence (An )n∈N in Aconv such that πu (An )Ωu → χ in norm. Noticing that (Vz πz (B)Ωz |χ ) = lim (Vz πz (B)Ωz |πu (An )Ωu ) = lim ω z (B ∗ An ), n→∞

n→∞

(4.15)

we see that the left-hand side, as a function of z, is the pointwise limit of continuous functions, and hence measurable. Thus z → Vz πz (B)Ωz is weakly measurable. Now if Hu was chosen separable, which is possible by assumption, weak measurability implies Lusin measurability of the function (cf. Appendix). Thus ω0 is regular. Finally, in the separable case, we remark that we can pick a countable subset of Acount ⊂ Aconv such that πu (Acount )Ωu is dense in Hu . Then π0 (Acount )Ω0 becomes a fundamental sequence in H0 , so that this space is uniformly separable.

On Dilation Symmetries Arising from Scaling Limits

37

Of course, it follows as a corollary to the preceding sections that the limit is also factorizing. Let us spell this out more explicitly. Proposition 4.10. Suppose that A has a convergent scaling limit, and that there exists a multiplicative limit state ωu for which the representation space Hu is separable. Let ω0 be

Γ,⊕ any scaling limit state. There exists a unitary V = Z dν(z) Vz : H0 → L 2 (Z, ν, Hu ) such that V π0 (A C)Ω0 = π0 (C)ΩZ ⊗ πu (A)Ωu for all A ∈ Aconv , C ∈ Z(A), and such that the Vz fulfill all requirements of Def. 4.3. Proof. We use notation as in the proof of Thm. 4.9. Let Vz : Hz → Hu be the unitaries constructed there. Then, z → Vz∗ is a measurable family of operators. Namely, for any A ∈ Aconv , we find Vz∗ πu (A)Ωu = πz (A)Ωz

(4.16)

which is in Γ ; hence measurability is checked on the fundamental family (cf. Lemma A.2). So the operator ⊕,Γ ∗ V := dν(z) Vz∗ (4.17) Z

is well-defined. Domain and range of V ∗ are both uniformly separable, see Thm. 4.9.

Γ,⊕ Thus also the adjoint of V ∗ , denoted as V , is decomposable with V = Z dν(z) Vz . It is then clear that V is unitary. Also, we have for A ∈ Aconv and C ∈ Z(A), Γ ⊕ dν(z) πz (C)πz (A)Ωz = dν(z) πz (C)Vz πz (A)Ωz V π0 (A C)Ω0 = V Z Z ⊕ = dν(z) πz (C)πu (A)Ωu = (π0 (C)ΩZ) ⊗ (πu (A)Ωu ). (4.18) Z

As a direct consequence of the discussion following Eq. (4.14), the Vz have all the properties required in Def. 4.3 regarding vacuum vector, symmetries, and local algebras. 5. Dilation Covariance in the Limit Our next aim is to analyze the structure of dilation symmetries in the limit theory. To that end, we consider a scaling limit state ω0 which is invariant under δ µ . As shown in [BDM09, Sec. 2], the associated limit theory is covariant with respect to a strongly continuous unitary representation g ∈ G → U0 (g) of the extended symmetry group G, including both Poincaré symmetries and dilations. Our interest is how the dilation unitaries U0 (µ) relate to decomposition theory in Sec. 3, and how they behave in the more specific situations analyzed in Sec. 4. We will consider three cases of decreasing scope: first, the general situation; second, the factorizing scaling limit; third, the convergent scaling limit. We first consider a general theory as in Sec. 3, and analyze the decomposition of the dilation operators corresponding to the direct integral decomposition of H0 introduced

38

H. Bostelmann, C. D’Antoni, G. Morsella

in Thm. 3.2. To this end, we first note that δ µ leaves Z(A) invariant; thus we have a representation of the dilations UZ(µ) := U0 (µ) HZ on HZ. Identifying HZ with L 2 (Z, ν) as before, the UZ(µ) act on a function space. This action, and its extension to the entire Hilbert space, can be described in more detail. Proposition 5.1. Let ω0 be an invariant limit state. There exist an action of the dilations through homeomorphisms z → µ.z of Z, and unitary operators Uz (µ) : Hz → Hµ.z for µ ∈ R+ , z ∈ Z, such that: (i) the measure ν is invariant under the transformation z → µ.z; (ii) UZ(µ)χ (z) = χ (µ−1 .z) for all χ ∈ L 2 (Z, ν), as an equation in the L 2 sense; (iii) Uz (1) = 1, Uz (µ)∗ = Uµ.z (µ−1 ), Uµ.z (µ )Uz (µ) = Uz (µ µ) for all z ∈ Z, µ, µ ∈ R+ ; ↑ (iv) Uz (µ)Uz (x, Λ) = Uµ.z (µx, Λ)Uz (µ) for all z ∈ Z, (x, Λ) ∈ P+ ;

Γ

Γ −1 (v) U0 (µ)χ = Z dν(z) Uµ−1 .z (µ)χ (µ .z) for all χ ∈ Z dν(z) Hz . Proof. Recalling that Z is the spectrum of the commutative C∗ algebra π0 (Z(A)), we define the homeomorphism z → µ.z as the one induced by the automorphism ad UZ(µ−1 ) of π0 (Z(A)). For C ∈ Z(A), we know that π0 (C)Ω0 ∈ HZ corresponds to the function z → χC (z) = ω z (C), precisely the image of π0 (C) in the Gelfand isomorphism. Applying U0 (µ−1 ) to this vector, one obtains (UZ(µ−1 )χC )(z) = χC (µ.z);

(5.1)

thus (ii) holds for all χ ∈ C(Z). Taking the scalar product of Eq. (5.1) with Ω0 , one sees that Z dν(z)χ (z) = Z dν(z)χ (µ.z) for all µ and χ ∈ C(Z), so (i) follows. Now for general χ ∈ L 2 (Z, ν), statement (ii) follows by density. Expressing the action of z → µ.z on the level of algebras, it is easy to see that ωµ.z ◦ δ µ Z(A) = ω z Z(A).

(5.2)

Since however δ µ commutes with the projector ω : A → Z(A), the same equation holds on all of A. Therefore, the maps Uz (µ) : Hz → Hµ.z given by Uz (µ)πz (A)Ωz := πµ.z (δ µ (A))Ωµ.z

(5.3)

are well-defined and unitary. The properties of Uz (µ) listed in (iii) and (iv) then follow from this definition by easy computations.

Γ Now for (v): As before, we identify H0 with Z dν(z) Hz . Then we have, for all A ∈ A, U0 (µ)π0 (A)Ω0 = π0 (δ µ (A))Ω0 = =

Γ

Z

Γ

Z

dν(z) πz (δ µ (A))Ωz

dν(z) Uµ−1 .z (µ)πµ−1 .z (A)Ωµ−1 .z .

(5.4)

Γ Given now a vector χ ∈ Z dν(z) Hz , we can find a sequence (π0 (An )Ω0 )n∈N converging in norm to χ . Passing to a subsequence, we can also assume that πz (An )Ωz → χ (z)

On Dilation Symmetries Arising from Scaling Limits

39

in norm for almost every z ∈ Z. Hence, using the dominated convergence theorem and (i), we see that Γ Γ lim dν(z) Uµ−1 .z (µ)πµ−1 .z (An )Ωµ−1 .z = dν(z) Uµ−1 .z (µ)χ (µ−1 .z), n→+∞ Z

Z

which gives (v).

Thus dilations act between the fibers of the direct integral decomposition by unitaries Uz (µ), which depend on the fiber. They fulfill the cocycle-type composition rule Uµ.z (µ )Uz (µ) = Uz (µ µ) that one would naively expect; cf. also the theory of equivariant disintegrations for separable C ∗ algebras [Tak02, Ch. X §3]. We shall now further restrict to the situation of a factorizing scaling limit, as in Def. 4.3, in which the fiber spaces Hz are all identified with a unique space Hu . By this identification, we can regard the unitaries Uz (µ) as endomorphisms Uˆ z (µ) of Hu . Our result for these endomorphisms is as follows. Proposition 5.2. Let ω0 be an invariant limit state. Suppose that the scaling limit of A is factorizing, and let V = dν(z)Vz be the unitary of Def. 4.3. Then, the unitary operators Uˆ z (µ) : Hu → Hu , Uˆ z (µ) = Vµ.z Uz (µ)Vz∗ fulfill for any z ∈ Z, µ, µ ∈ R+ the relations Uˆ z (1) = 1, Uˆ z (µ)∗ = Uˆ µ.z (µ−1 ), Uˆ µ.z (µ )Uˆ z (µ) = Uˆ z (µ µ). If H0 is uniformly separable, one has

∗

V U0 (µ)V = (UZ(µ) ⊗ 1)

⊕

Z

dν(z)Uˆ z (µ); ↑

and for every µ > 0, there is a null set N ⊂ Z such that for any (x, Λ) ∈ P+ and any z ∈ Z\N , Uˆ z (µ)Uu (x, Λ) = Uu (µx, Λ)Uˆ z (µ). Proof. It is clear that Uˆ z (µ), defined as above, are unitary, and their composition relations follow from Prop. 5.1 (iii). Now let H0 be uniformly separable. Then, together with V , also V ∗ is decomposable. By a short computation, one finds for any χ ∈ L 2 (Z, ν, Hu ): ⊕ dν(z) Vz Uµ−1 .z (µ)Vµ∗−1 .z χ (µ−1 .z). (5.5) V U0 (µ)V ∗ χ = Z

Now, following Prop. 5.1 (ii), the operator UZ(µ) ⊗ 1 acts on vectors χ via ⊕ (UZ(µ) ⊗ 1)χ = dν(z) χ (µ−1 .z). Z

(5.6)

Together with Eq. (5.5), this entails V U0 (µ)V ∗ χ = (UZ(µ) ⊗ 1)

⊕

Z

dν(z) Vµ.z Uz (µ)Vz∗ χ (z),

(5.7)

40

H. Bostelmann, C. D’Antoni, G. Morsella

of which the second assertion follows. Further, one computes from V U0 (x, Λ)V ∗ = 1 ⊗ Uu (x, Λ) and from Eq. (5.7) that ⊕ ⊕ ˆ dν(z)Uz (µ)Uu (x, Λ) = dν(z)Uu (µx, Λ)Uˆ z (µ). (5.8) Z

Z

Uniform separability implies that the integrands agree except on a null set. This null set may depend on x, Λ. However, we can choose it uniformly on a countable dense set of the group, and hence, by continuity, uniformly for all Poincaré group elements. This shows that the dilation symmetries factorize into a central part, UZ(µ)⊗1, which

⊕ “mixes” the fibers of the direct integral, and a decomposable part, Z dν(z)Uˆ z (µ). The unitaries Uˆ z (µ) will generally depend on z; and like the Uz (µ) before, they do not necessarily fulfill a group relation, but a cocycle equation Uˆ µ.z (µ )Uˆ z (µ) = Uˆ z (µ µ),

(5.9)

as shown above, where µ.z can in general not be replaced with z. However, using the commutation relations with the other parts of the symmetry group, one sees that Uˆ z (µ )Uˆ z (µ)Uˆ z (µ µ)∗ is (a. e.) an inner symmetry of the theory Au . On the other hand, this representation property “up to an inner symmetry” cannot be avoided if such symmetries exist in the theory at all; for they might be multiplied to Vz in a virtually arbitrary fashion at any point z. In this respect, we encounter a similar situation with respect to dilation symmetries as Buchholz and Verch [BV95]. In the present context, however, it seems more transparent how this cocycle arises. Under somewhat stricter assumptions, we can prove a stronger result that avoids the ambiguities discussed above. Let us consider the case of a convergent scaling limit, per Def. 4.4. In this case, we shall see that the Uz (µ) can actually be chosen independent of z, and yield a group representation in the usual sense. Proposition 5.3. Let A have a convergent scaling limit, and let ωu be a multiplicative limit state with separable representation space Hu . Then the Poincaré group representation Uu on Hu extends to a representation of the extended symmetry group G. For any invariant limit state ω0 with associated representation U0 of G, one has V U0 (µ)V ∗ = (U0 (µ) HZ) ⊗ Uu (µ), where V is the unitary introduced in Proposition 4.10. Proof. With ωu , also every ωu ◦ δ µ is a scaling limit state. Thanks to the invariance of Aconv under dilations, we thus have for each A ∈ Aconv , πu (δ µ (A))Ωu 2 = ωu ◦ δ µ (A∗ A) = ωu (A∗ A) = πu (A)Ωu 2 .

(5.10)

This yields the existence of a unitary strongly continuous representation µ → Uu (µ) on Hu such that Uu (µ)πu (A)Ωu = πu (δ µ (A))Ωu ,

A ∈ Aconv .

(5.11)

That also implies Uu (µ)Uu (x, Λ)πu (A)Ωu = πu (α µ,x,Λ (A))Ωu ,

A ∈ Aconv ,

(5.12)

On Dilation Symmetries Arising from Scaling Limits

41

which shows that (µ, Λ, x) → Uu (µ)Uu (Λ, x) is a unitary representation of G on Hu , extending the representation of the Poincaré group. Now if V : H0 → HZ ⊗ Hu is the unitary of Prop. 4.10, a calculation shows that V U0 (µ)V ∗ π0 (C)Ω0 ⊗ πu (A)Ωu = π0 (δ µ (C))Ω0 ⊗ πu (δ µ (A))Ωu , C ∈ Z(A), A ∈ Aconv , (5.13) which entails that V U0 (µ)V ∗ = (U0 (µ) HZ) ⊗ Uu (µ).

Thus, the limit theory is “dilation covariant” in the usual sense, with a unitary acting on Hu . Considering the unitaries 1 ⊗ Uu (g), we actually get a unitary representation in any limit theory, even corresponding to multiplicative states. Only for compatibility with the scaling limit representation π0 it is necessary to consider invariant means, and to take U0 (µ) HZ into account. 6. Phase Space Properties In this section, we wish to investigate how the notion of phase space conditions, specifically the (quite weak) Haag-Swieca compactness condition [HS65], fits into our context, and how it transfers to the limit theory. An important aspect here is that Haag-Swieca compactness of a quantum field theory guarantees that the corresponding Hilbert space is separable; this property transfers to multiplicative limit states in certain circumstances [Buc96a]. We shall give a strengthened version of the compactness condition that guarantees our general limit spaces to be uniformly separable, a property that turned out to be valuable in the previous sections. We need some extra structures to that end. First, we consider “properly rescaled” vector-valued functions χ : R+ → H. Specifically, for A ∈ A, let AΩ denote the function λ → Aλ Ω. We set H = clos{AΩ | A ∈ A},

(6.1)

where the closure is taken in the supremum norm χ = supλ χ λ . Then H is a Banach space, in fact a Banach module over Z(A) in a natural way. Given a limit state, we transfer the limit representation π0 to vector-valued functions. To that end, consider the space C(Γ ) of Γ -continuous vector fields, as defined in the Appendix. We define η0 : H → C(Γ ) on a dense set by η0 (AΩ) := π0 (A)Ω0 .

(6.2)

This is well-defined, since one computes 1/2 π0 (A)Ω0 ∞ = sup πz (A)Ωz = sup ω z (A∗ A) z∈Z

∗

≤ ω(A A)

1/2

z∈Z

= sup Aλ Ω2 λ>0

1/2

= AΩ.

(6.3)

That also shows η0 ≤ 1. Note that η0 fulfills η0 (Cχ ) = π0 (C)η0 (χ ) for all C ∈ Z(A), χ ∈ H,

(6.4)

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H. Bostelmann, C. D’Antoni, G. Morsella

this easily being checked for χ = AΩ. So η0 preserves the module structure in this sense. Further, η0 : H → C(Γ ) clearly has dense range. It is important in our context that H is left invariant under multiplication with suitably rescaled functions of the Hamiltonian. More precisely, we denote these functions as f (H ) for f ∈ S([0, +∞)); they are defined as elements of B by f (H )λ = f (λH ), with norm f (H ) ≤ f ∞ . They act on H by pointwise multiplication. The following lemma generalizes an observation in [Buc96a]. Lemma 6.1. Let f ∈ S([0, +∞)). Then, for each χ ∈ H, we have f (H )χ ∈ H. There exists a test function g ∈ S(R) such that for all A ∈ A,

f (H )AΩ = α g AΩ := dt g(t) α t A Ω. Proof. We continue f to a test function fˆ ∈ S(R), and choose g as the Fourier transform of fˆ. One finds by spectral analysis of H that for any A ∈ A, ∞ fˆ(λE)d P(E)Aλ Ω = dt g(t)eıλH t Aλ Ω = (α g A)λ Ω. (6.5) f (λH )Aλ Ω = 0

This shows that f (H )AΩ has the proposed form, and is an element of H. Since f (λH ) ≤ f ∞ uniformly in λ, we may pass to limits in AΩ and obtain that f (H )χ ∈ H for all χ ∈ H. As a next step towards phase space conditions, let us explain a notion of compact maps adapted to our context. To that end, let E be a Banach space and F a Banach module over the commutative Banach algebra R. We say that a linear map ψ : E → F is of uniform rank 1 if it is of the form ψ = e( · ) f with e : E → R linear and continuous, and f ∈ F. Sums of n such terms are called of uniform rank n.4 We say that ψ is uniformly compact if it is an infinite sum of terms of uniform rank 1, ψ = ∞ j=0 e j ( · ) f j , where the sum converges in the Banach norm. For R = C and F a Hilbert space, these definitions reduce to the usual notions of compact or finite-rank maps. We are now in the position to consider Haag-Swieca compactness. We fix, once and < for all, an element C < ∈ Z(A) with C < ≤ 1, C < λ = 0 for λ > 1, and C λ = 1 for λ < 1/2. For a given β > 0 and any bounded region O, we consider the map Θ (β,O) : A(O) → H,

A → e−β H C < AΩ.

(6.6)

This is indeed well-defined due to Lemma 6.1. Our variant of the Haag-Swieca compactness condition, uniform at small scales, is then as follows. Definition 6.2. A quantum field theory fulfills the uniform Haag-Swieca compactness condition if, for each bounded region O, there is β > 0 such that the map Θ (β,O) is uniformly compact. We note that this property is independent of the choice of C < ; the role of that factor is to ensure that we restrict our attention to the short-distance rather than the long-distance regime. We do not discuss relations of uniform Haag-Swieca compactness with other 4 Note that the “uniform rank” is rather an upper estimate, in the sense that a map of uniform rank n may at the same time be of uniform rank n − 1.

On Dilation Symmetries Arising from Scaling Limits

43

versions of phase space conditions here. Rather, we show in Sec. 7 that the condition is fulfilled in some simple models. We now investigate how the compactness property transfers to the scaling limit. To that end, we consider the corresponding phase space map in the limit theory, (β,O )

Θ0

A → e−β H0 AΩ0 .

: A0 (O) → H0 ,

(6.7)

Its relation to Θ (β,O) is rather direct. (β,O )

Proposition 6.3. For any fixed O and β > 0, one has η0 ◦ Θ (β,O) = Θ0 (β,O ) If Θ (β,O) is uniformly compact, so is Θ0 ◦ π0 .

◦ π0 .

Proof. Given β, we choose a function gβ relating to f β (E) = exp(−β E) per Lemma 6.1. For any A ∈ A(O), we compute η0 Θ (β,O) (A) = η0 (C < α gβ AΩ) = π0 (C < )π0 (α gβ A)Ω0 (β,O )

= α0,gβ π0 (A)Ω0 = Θ0

π0 (A).

(6.8)

(β,O )

◦ π0 as proposed. Now let Θ (β,O) be uniformly comThus η0 ◦ Θ (β,O) = Θ0 pact, Θ (β,O) = j e j ( · ) f j . Then η0 can be exchanged with the infinite sum due to continuity, which yields (β,O ) Θ0 ◦ π0 = η0 (e j ( · ) f j ) = (π0 ◦ e j ( · ))(η0 f j ), (6.9) j (β,O )

using Eq. (6.4). Thus Θ0

j

◦ π0 is uniformly compact.

(β,O )

◦ π0 ⊂ C(Γ ). Since we can The above results show in particular that img Θ0 write (β,O ) Θ0 ◦ π0 (A) = dν(z) Θz(β,O) ◦ πz (A) (6.10) (β,O )

with the obvious definition of Θz , the above proposition establishes a rather strong form of compactness in the limit theory, uniform in z; note that the sum in Eq. (6.9) converges with respect to the supremum norm of C(Γ ). We now come to the main result of the section, showing that compactness in the above form implies uniform separability of the limit Hilbert space. Theorem 6.4. Suppose that the theory A fulfils uniform Haag-Swieca compactness. Then, for any limit state ω0 , the representation space H0 is uniformly separable, where the fundamental sequence can be chosen from C(Γ ). Proof. We choose a sequence of regions Ok such that Ok Rs+1 , and a sequence (βk )k∈N in R+ such that all Θ (βk ,Ok ) are uniformly compact. By Prop. 6.3 above, also (β ,O ) (k) Θ0 k k ◦ π0 are uniformly compact. Explicitly, choose e j : A(Ok ) → C(Z) and f j(k) ∈ C(Γ ) such that

(βk ,Ok )

Θ0

◦ π0 =

j

(k) e(k) j (·) fj .

(6.11)

44

H. Bostelmann, C. D’Antoni, G. Morsella (k)

We will construct a fundamental sequence using the f j . To that end, let A ∈ A(O) for some O. For k large enough, we know that (k) (β ,O ) (k) e−βk H0 π0 (A)Ω0 = Θ0 k k (π0 (A)) = e j (A) f j . (6.12) j

The sum converges in the supremum norm, i.e., uniformly at all points z. Let us choose a fixed z. Then, it is clear that (k) 2 2 (k) e−Hz πz (A)Ωz = e j (A) (z) e−Hz +βk Hz f j (z), (6.13) j (k)

noting that exp(−Hz2 +βk Hz ) is a bounded operator. Observe that (e j (A))(z) are merely numerical factors. Since A and O were arbitrary, and ∪O πz (A(O))Ωz is dense in Hz , this means (k)

e−Hz Hz ⊂ clos span{ e−Hz +βk Hz f j (z)| j, k ∈ N}. 2

2

(6.14)

Now exp(−Hz2 ) is a selfadjoint operator with trivial kernel, thus its image is dense. Hence the exp(−Hz2 + βk Hz ) f j(k) (z) are total in Hz . This holds for all z, thus {exp(−H02 + (k)

βk H0 ) f j | j, k ∈ N} is a fundamental sequence. Applying Lemma 6.1 to f (E) = exp(−E 2 + βk E), we find that the elements of the fundamental sequence lie in C(Γ ). 7. Examples We are now going to investigate the structures discussed in simple models. Particularly, we wish to show that our conditions on “convergent scaling limits” (Def. 4.4) and “uniform Haag-Swieca compactness” (Def. 6.2) can be fulfilled at least in simple situations. To that end, we first consider the situation where the theory A “at finite scales” is equipped with a dilation symmetry. Then, we investigate the real scalar free field as a concrete example. 7.1. Dilation covariant theories. We now consider the case where the net A, which our investigation starts from, is already dilation covariant. One expects that the scaling limit construction reproduces the theory A in this case, and that the dilation symmetry obtained from the scaling algebra coincides with the original one. We shall show that this is indeed the case under a mild phase space condition, and also that this implies the stronger phase space condition in Def. 6.2. This extends a discussion in [BV95, Sec. 5]. Technically, we will assume in the following that A is a local net in the vacuum sector with symmetry group G, which is generated by the Poincaré group and the dilation group. We shall denote the corresponding unitaries as U (µ, x, Λ) = U (µ)U (x, Λ). The mild phase space condition referred to is the Haag-Swieca compactness condition for the original theory: We assume that for each bounded region O in Minkowski space, there exists β > 0 such that the map Θ (β,O) : A(O) → H, A → exp(−β H )AΩ is compact. (This is equivalent to a formulation where the factor exp(−β H ) is replaced with a sharp energy cutoff, as used in [HS65].)

On Dilation Symmetries Arising from Scaling Limits

45

Theorem 7.1. Let A be a dilation covariant net in the vacuum sector which satisfies the Haag-Swieca compactness condition. Then A has a convergent scaling limit. ˆ Proof. For each O, we introduce the C∗ -subalgebra A(O) ⊂ A(O) of those elements A ∈ A(O) for which g → αg (A) is norm continuous. Since the symmetries are impleˆ mented by continuous unitary groups, A(O) is strongly dense in A(O). We then define a C∗ -subalgebra of the scaling algebra A(O), ˆ Aconv (O) := {λ → U (λ)AU (λ)∗ | A ∈ A(O)},

(7.1)

and the α-invariant algebra Aconv ⊂ A is defined as the C∗ -inductive limit of the Aconv (O). It is evident that condition (i) in Def. 4.4 is fulfilled by Aconv , as the functions λ → ω(Aλ ) are constant in the present case. Now let ω0 be a multiplicative limit state. With similar arguments5 as in [BV95, Prop. 5.1], using the Haag-Swieca compactness condition, we can construct a net isomorphism φ from A0 to A, which has the property ˆ that if Aλ = U (λ)AU (λ)∗ with A ∈ A(O), then φ(π0 (A)) = A. From this, and from ˆ the strong density of A(O) in A(O), it follows that π0 (Aconv (O)) is strongly dense in A0 (O). Thus condition (ii) in Def. 4.4 is satisfied as well. Since the isomorphism φ above can be shown to intertwine the respective vacuum states, it is actually the adjoint action of a unitary W : H0 → H. We remark that H, and then also H0 , is separable due to the Haag-Swieca compactness condition. Then as a consequence of Prop. 5.3, A has a factorizing scaling limit and the representation of the symmetry group G0 factorizes too. It is also clear from the proof above and from that of Thm. 4.9, that Au is unitarily equivalent to A through the operator W , taken here for πu in place of π0 . Furthermore, this W also intertwines the dilations in the scaling limit with those of the underlying theory. Corollary 7.2. Under the hypothesis of Thm. 7.1, there holds W Uu (µ)W ∗ = U (µ). ˆ Proof. It is sufficient to verify the relation on vectors of the form AΩ with A ∈ A(O). For such vectors it follows by noting that Aλ = U (λ)AU (λ)∗ is an element of Aconv (O), ˆ and that δ µ (A)λ = U (λ)αµ (A)U (λ)∗ with αµ (A) ∈ A(µO). For showing the consistency of our definitions, we now prove that the Haag-Swieca compactness condition at finite scales, together with dilation covariance, implies our uniform compactness condition of Def. 6.2. Proposition 7.3. If the dilation covariant local net A fulfills the Haag-Swieca compactness condition, then it also fulfills uniform Haag-Swieca compactness. Proof. Let O be fixed, and let β > 0 such that Θ (β,O) is compact; Θ (β,O) =

∞

e j ( · ) f j with e j ∈ A(O)∗ , f j ∈ H.

(7.2)

j=1 5 Since in contrast to [BV95], we here take the A (O) to be W∗ algebras, we need to amend the argument 0 in step (d) of [BV95, Prop. 5.1] slightly: We first construct the isomorphism φ on the C∗ algebra π0 (A(O)), and then continue it to the weak closure; cf. [KR97, Lemma 10.1.10].

46

H. Bostelmann, C. D’Antoni, G. Morsella

Taking the normal part, we can in fact arrange that e j ∈ A(O)∗ . (See [BDF87, Lemma 2.2] for a similar argument.) Now define e j : A(O) → Z(A) by e j (A)λ = e j U (λ)∗ C < (7.3) λ Aλ U (λ) . That the image is indeed in Z(A), i.e., continuous under δ µ , is seen as follows. We compute for λ, µ > 0, e (A)λµ − e (A)λ = e j U (λµ)∗ (C < A)λµ U (λµ) − U (λ)∗ (C < A)λ U (λ) j j ≤ e j δ µ (C < A)−C < A + e j U (µ)∗ · U (µ) −e j C < A. (7.4) Now as µ → 0, the first summand vanishes due to norm continuity of δ µ on A, and the second due to strong continuity of U (µ); both limits are uniform in λ. Thus δ µ acts continuously on e j (A). Further, we define f j ∈ H by f

jλ

= U (λ) f j .

(7.5)

ˆ for suitable O This is indeed an element of H: Namely, given > 0, choose A ∈ A(O) ˆ is as in the proof of Thm. 7.1. Then, Aλ = U (λ)AU (λ)∗ such that AΩ− f j < ; here A defines an element of A, and AΩ − f j ≤ AΩ − f j < . Hence f j is contained in the closure of AΩ. Now we are in the position to show that Θ (β,O) = j e j f j . Let J ∈ N be fixed. It is straightforward to compute that for any A ∈ A(O) and λ > 0, J J Θ (β,O) (A) − e j (A) f j = U (λ) Θ (β,O) (Bλ ) − e j Bλ ) f j , j=1

λ

j=1

where Bλ = U (λ)∗ C < λ Aλ U (λ).

(7.6)

Note here that Bλ ∈ A(O) for any λ. This entails J J (β,O) Θ − e j ( · ) f j ≤ Θ (β,O) − e j ( · ) f j A. j=1

(7.7)

j=1

The right-hand side vanishes as J → ∞, as a consequence of the compactness condition at finite scales. This shows that Θ (β,O) is uniformly compact. 7.2. The scaling limit of a free field. We now show in a simple, concrete example from free field theory that the model has a convergent scaling limit in the sense of Def. 4.4. Specifically, we consider a real scalar free field of mass m > 0, in 2+1 or 3+1 spacetime dimensions. The algebraic scaling limit of this model is the massless real scalar field; this was as already discussed in [BV98], and in parts we rely on the arguments given there. However, we need to consider several aspects that were not handled in that work, in particular continuity aspects of Poincaré and dilation transformations. Also, as mentioned before, in contrast to [BV98] we deal with weakly closed local algebras at fixed scales and in the limit theory.

On Dilation Symmetries Arising from Scaling Limits

47

We start by recalling, for convenience, the necessary notations and definitions from [BV98]. We consider the Weyl algebra W over D(Rs ), s = 2, 3: ı W ( f )W (g) = e− 2 σ ( f,g) W ( f + g), σ ( f, g) = Im dx f (x)g(x). (7.8) ↑

Then, we define a mass dependent automorphic action of P+ on W by (m)

(m)

αx,Λ (W ( f )) = W (τx,Λ f ),

(7.9)

↑

where the action τ (m) of P+ on D(Rs ) is defined by the following formulas. In those, we write f˜(p) = (2π )−s/2 dx f (x)e−ixp for the Fourier transform of f, which we split into f˜ = f˜R + ı f˜I , where f R = Re f and f I = Im f ; also, ωm (p) := m 2 + |p|2 . (τx(m) f )(y) := f (y − x), (m)

˜ ˜ f )∼ R (p) := cos(tωm (p)) f R (p) − ωm (p) sin(tωm (p)) f I (p),

(m)

−1 ˜ f )∼ sin(tωm (p)) f˜R (p), I (p) := cos(tωm (p)) f I (p) + ωm (p)

(τt (τt

(m)

(τΛ f )∼ R (p) := ϕΛ (ωm (p), p),

(7.11)

f

−1 (τΛ(m) f )∼ I (p) := ωm (p) ψΛ (ωm (p), p). f

f

(7.10)

(7.12)

f

Here the functions ϕΛ , ψΛ : Rs+1 → C are defined by 1 1 ˜ f f R (Λ−1 p) + f˜R (ΛT p) + (Λ−1 p)0 f˜I (Λ−1 p) − (ΛT p)0 f˜I (ΛT p) , ϕΛ ( p) := 2 2ı 1 1˜ f −1 T f R (Λ p)− f˜R (Λ p) + (Λ−1 p)0 f˜I (Λ−1 p) + (ΛT p)0 f˜I (ΛT p) , ψΛ ( p) := − 2ı 2 (7.13) where we use the notation Λp = ((Λp)0 , Λp) for Λ ∈ L, p ∈ Rs+1 . One verifies that all the above expressions are even in ωm (p), which, due to the analytic properties of f˜, (m) implies that τx,Λ D(Rs ) ⊂ D(Rs ). We also introduce the action σ of dilations on W by σλ (W ( f )) = W (δλ f ),

(7.14)

(δλ f )(x) := λ−(s+1)/2 ( f R )(λ−1 x) + ıλ−(s−1)/2 ( f I )(λ−1 x).

(7.15)

with

(m)

(λm)

It holds that αλx,Λ ◦ σλ = σλ ◦ αx,Λ . Finally we define the vacuum state of mass m ≥ 0 on W as 1

ω(m) (W ( f )) = e− 2 f m , where f 2m :=

2

2 1 dpωm (p)−1/2 f˜R (p) + ı ωm (p)1/2 f˜I (p) . 2 Rs

(7.16)

(7.17)

48

H. Bostelmann, C. D’Antoni, G. Morsella (m)

There holds clearly ω(m) ◦ αx,Λ = ω(m) , ω(m) ◦ σλ = ω(λm) . Proceeding now along the lines of [BV98], we consider the GNS representation (π (0) , H(0) , Ω (0) ) of W induced by the massless vacuum state ω(0) . For each m ≥ 0, we define a net O → A(m) (O) of von Neumann algebras on H(0) as (m) A(m) (ΛO B + x) := {π (0) αx,Λ (W (g)) : supp g ⊂ B} , (7.18) where O B is any double cone with base the open ball B in the time t = 0 plane. For other open regions we can define the algebras by taking unions, but this will not be relevant for the following discussion. Due to the local normality of the different states ω(m) , m ≥ 0, with respect to each other [EF74], these nets are isomorphic to the nets generated by the free scalar field of mass m on the respective Fock spaces. From now on, we will identify elements of W and of A(m) , and therefore we will drop the indication of the representation π (0) . We denote the (dilation and Poincaré covariant) scaling algebra associated to A(m) by A(m) . The next lemma generalizes the results of [BV98, Lemma 3.2] to the present situation. ↑

Lemma 7.4. Let a > 1 and h D ∈ D((1/a, a)), h P ∈ D(P+ ), f ∈ D(Rs ), and consider the function W : R+ → A(m) given by dµ (m) W λ := d x dΛ h D (µ)h P (x, Λ)αµλx,Λ ◦ σµλ (W ( f )), ↑ µ R + × P+ where dΛ is the left-invariant Haar measure on the Lorentz group and the integral is to be understood in the weak sense. Then: (i) there exists a double cone O such that W ∈ A(m) (O); (ii) there holds in the strong operator topology, dµ (0) −1 d x dΛ h D (µ)h P (x, Λ)αµx,Λ lim σλ (W λ ) = ◦ σµ (W ( f )) =: W0 ; ↑ µ λ→0+ R + × P+ (iii) the span of the operators W0 of the form above, with W ∈ A(m) (O) for fixed O, is strongly dense in A(0) (O). (m)

Proof. Since W is the convolution, with respect to the action (µ, x, Λ) → α µ,x,Λ , of the function h D ⊗ h P with the bounded function λ → σλ (W ( f )), and thanks to the support properties of h D , h P and f , (i) follows. In order to prove (ii), we start by observing that, for each vector χ ∈ H(0) , dµ σλ−1 (W λ ) − W0 χ ≤ d x dΛ |h D (µ)h P (x, Λ)| ↑ µ R + × P+ (µλm) (0) (7.19) × W δµ τx,Λ f − W δµ τx,Λ f χ . Now f → W ( f ) is known to be continuous with respect to · 0 on the initial space and the strong operator topology on the target space [BR81, Prop. 5.2.4]. Since the norm ↑ · 0 is δµ -invariant, it therefore suffices to show that for each fixed (x, Λ) ∈ P+ , (m) (0) lim + τx,Λ f − τx,Λ f 0 = 0; (7.20) m→0

On Dilation Symmetries Arising from Scaling Limits

49

for (ii) then follows from the dominated convergence theorem. In order to show Eq. (7.20), we introduce the following family of functions f (m) (p) of two arguments: F = f : [0, 1] × Rs → C f (m) ( · ) ∈ D(Rs ) for each fixed m ∈ [0, 1]; lim f (m) (p) = f (0) (p) for each fixed p ∈ Rs ;

m→0

f (m) (p)| ≤ g(p) . ∃g ∈ S(Rs ) : ∀m ∈ [0, 1] ∀p ∈ Rs : |

(7.21)

It is clear that for f ∈ F, one has f (m) − f (0) 0 → 0 as m → 0 per dominated convergence. Also, each f ∈ D(Rs ), with trivial dependence on m, falls into F. So it (·) leaves F invariant, where remains to show that the (naturally defined) action of τx,Λ (·)

it suffices to check this for a set of generating subgroups. Indeed, τx,Λ F ⊂ F is clear for spatial translations and rotations. For time translations and boosts, it was already remarked that D(Rs ) is invariant under these at fixed m, and pointwise convergence as m → 0 is clear. Further, from Eqs. (7.11) and (7.13), one sees that f˜ is modified by at most polynomially growing functions, uniform in m ≤ 1, hence uniform S-bounds (m) (m) f as well. (Again, it enters here that all expressions are even in ωm , for hold for τx,Λ which it is needed that f˜ is smooth.) This completes the proof of (ii). Finally, (iii) follows from the observation that as h D and h P converge to delta functions, W0 converges strongly to W ( f ) thanks to the strong continuity of the function (0) (µ, x, Λ) → αµx,Λ ◦ σµ (W ( f )); and of course the span of the Weyl operators with supp f ⊂⊂ O is strongly dense in A(0) (O). Using the above lemma, we can prove the following. Theorem 7.5. The theory of the massive real scalar free field in s = 2, 3 spatial dimensions has a convergent scaling limit. (m)

Proof. Consider the C∗ -subalgebra Aconv (O) of A(m) (O) which is generated by the ele(m) ments W ∈ A(m) (O) defined in the previous lemma, and let Aconv be the corresponding (m) quasi-local algebra. Since α µ,x,Λ (W ) is again an element of the same form, just with (m)

shifted function h D ⊗ h P , the algebra Aconv is α (m) invariant. In order to verify that (m) , we start by observing that, λ → ω(m) (Aλ ) has a limit, as λ → 0, for all A ∈ Aconv thanks to Lemma 7.4 (ii) and to the fact that σλ is unitarily implemented on H(0) , for each such A there exists limλ→0+ σλ−1 (Aλ ) =: A in the strong operator topology. Then (m) (O) there holds the inequality if A ∈ Aconv |ω(m) (Aλ ) − ω(0) (A)| ≤ (ω(m) − ω(0) ) A(0) (λO)A + |ω(0) (σλ−1 (Aλ )) − ω(0) (A)|.

(7.22) Together with the fact that limλ→0+ (ω(m) − ω(0) ) A(0) (λO) = 0 as a consequence of the local normality of ω(m) with respect to ω(0) , this implies that limλ→0+ ω(m) (Aλ ) = (m) (m) ω(0) (A) for all A in some local algebra Aconv (O). This then extends to all of Aconv by density.

50

H. Bostelmann, C. D’Antoni, G. Morsella (m)

It remains to show that for multiplicative limit states, π0 (Aconv ∩ A(m) (O)) is weakly (m) dense in A0 (O) = π0 (A(m) (O)) for any O. To that end, we use similar methods as in Thm. 7.1. With O fixed and U the ultrafilter that underlies the limit state, we define φ : π0 (A(m) (O)) → A(0) (O), π0 (A) → lim σλ−1 (Aλ ),

(7.23)

U

with the limit understood in the weak operator topology. Using methods as in [BV98, Sec. 3], one can show that φ is indeed a well-defined isometric ∗ homomorphism, which further satisfies ω0 = ω(0) ◦ φ on the domain of φ. Hence φ is given by the adjoint action of a partial isometry, and can be continued by weak closure to a ∗ homomorphism (m) φ − : A0(m) (O) → A(0) (O). On the other hand, for W ∈ Aconv (O) as in Lemma 7.4, one finds φ(π0 (W )) = W0 by (ii) of that lemma. However, the double commutant of those W0 is all of A(0) (O), see (iii) of the same lemma. So φ − is in fact an isomorphism; and inverting φ − , one obtains the proposed density. 7.3. Phase space conditions in the free field. Our aim in this section is to prove the uniform compactness condition of Sec. 6 in the case of a real scalar free field, again of mass m ≥ 0, in 3 + 1 or higher dimensions. To that end, we will use a short-distance expansion of local operators, very similar to the method used in the Appendix of [Bos05b], however in a refined formulation. In this section, we will consider a fixed mass m ≥ 0 throughout, and therefore we drop the label (m) from the local algebras, the vacuum state, and the Hilbert space norm for simplicity. We rewrite the Weyl operators of Eq. (7.8) in terms of the familiar free field φ and its time derivative ∂0 φ in the time-0 plane, (7.24) W ( f ) = exp ı φ(Re f ) − ∂0 φ(Im f ) , f ∈ D(Rs ). Also, we need to introduce some multi-index notation. Given n ∈ N0 , we consider multi-indexes ν = (ν1 , . . . , νn ) ∈ ({0, 1} × Ns0 )n ; that is, each ν j has the form ν j = (ν j0 , ν j1 , . . . , ν js ) with ν j0 ∈ {0, 1}, ν jk ∈ N0 for 1 ≤ k ≤ s. These indices will be ν ν used for labeling derivatives in configuration space, ∂ ν j = ∂0 j0 . . . ∂s js . We denote νj! =

s

ν jk ! , ν! =

k=0

n

ν j ! , |ν j | =

j=1

s k=0

ν jk , |ν| =

n

|ν j |.

(7.25)

j=1

Now we can define the following local fields as quadratic forms on a dense domain: φn,ν = :

n

∂ ν j φ: (0).

(7.26)

j=1

These will form a basis in the space of local fields at x = 0. Further, for given r > 0, we choose a test function h ∈ D(Rs ) which is equal to 1 for |x| ≤ r ; then we set ν h ν j (x) = sk=1 xk jk h(x). This is used to define the following functionals on A(Or ).

ı n (−1) j ν j0 (1−ν10 ) Ω [∂0 φ(h ν1 ), [. . . [∂0(1−νn0 ) φ(h νn ), A] . . .]Ω . (7.27) σn,ν (A) = n! ν! One sees that this definition is independent of the choice of h. We can therefore consistently consider σn,ν as a functional on ∪O A(O), though its norm may increase as O grows large. The significance of these functionals becomes clear in the following lemma.

On Dilation Symmetries Arising from Scaling Limits

51

Lemma 7.6. We have for all Weyl operators A = W ( f ) with f ∈ D(Rs ), A=

∞

σn,ν (A) φn,ν

n=0 ν

in the sense of matrix elements between vectors of finite energy and finite particle number. Proof. We indicate only briefly how this combinatorial formula can be obtained; see also [Bos00, Sec. 7.2] and [Bos05b, Appendix]. Using Wick ordering, we rewrite Eq. (7.24) for the Weyl operators as 2 W ( f ) = e− f /2 : exp ı φ(Re f ) − ∂0 φ(Im f ) : = e− f

2 /2

∞ n n ı : φ(Re f ) − ∂0 φ(Im f ) : . n!

(7.28)

n=0

Now, in each factor of the n-fold product :(. . .)n:, we expand both Re f and Im f into a Taylor series in momentum space. Note that this is justified, since those functions have compact support in configuration space, since they are evaluated in scalar products with functions of compact support in momentum space, and since the sum over n is finite in matrix elements. The Taylor expansion in momentum space corresponds to an expansion in derivatives of δ-functions in configuration space, and this is what produces the fields φn,ν localized at 0. We then need to identify the remaining factors with σn,ν (A), which is done using the known commutation relations of W ( f ) with φ and ∂0 φ. Our main task will now be to extend the above formula to more general states and more observables, by showing that the sum converges in a suitable norm. To that end, we need estimates of the fields and functionals involved. Lemma 7.7. Given s ≥ 2, m ≥ 0, and r0 > 0, there exists a constant c such that the following holds for any n, ν: √ e−β H φn,ν Ω ≤ cn (n!)1/2 ν! (β/2 s)−|ν|−n(s−1)/2 for any β > 0, (a) P(E)φn,ν P(E) ≤ cn E |ν|+n(s−1)/2 σn,ν A(Or ) ≤ c (n!) n

−1/2

(ν!)

−1

for any E > 0, provided s ≥ 3, (3r )

|ν|+n(s−1)/2

for any r ≤ r0 .

(b) (c)

Proof. One has n n √ e−β H φn,ν Ω = a ∗ (e−βω p ν j ) Ω ≤ n! e−βω p ν j . j=1

(7.29)

j=1

For the single-particle vectors on the right-hand side, one uses scaling arguments to obtain the estimate √ e−βω p ν j ≤ c1 ν j ! (β/2 s)−|ν j |−(s−1)/2 for β > 0, (7.30) where c1 is a constant (depending on s and m). This implies (a). For (b), we use energy bounds for creation operators a ∗ ( f ), similar to [BP90, Sec. 3.3]. One finds for single 1/2 particle space functions f 1 , . . . , f k in the domain of ωm , P(E)a ∗ (ωm f 1 ) . . . a ∗ (ωm f k ) ≤ E k/2 Q(E) f 1 . . . Q(E) f k , 1/2

1/2

(7.31)

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H. Bostelmann, C. D’Antoni, G. Morsella

with Q(E) being the energy projector for energy E in single particle space. This leads us to P(E)φn,ν P(E) ≤ 2n E n/2

n

−1/2 ν j

ωm

p χ E ,

(7.32)

j=1

ν where p ν j = sk=0 pk jk , and χ E is the characteristic function of ωm (p) ≤ E. For the single-particle functions, one obtains −1/2 ν j

ωm

p χ E ≤ c2 E |ν j |+(s−2)/2

(7.33)

with a constant c2 , which implies (b). Now consider the functional σn,ν . We choose a real-valued test function h 1 ∈ D(R+ ) such that h 1 (x) ≤ 1 for all x, h 1 (x) = 1 on [0, 1], and h 1 (x) = 0 for x ≥ 2. Then, h r (x) = h 1 (|x|/r ) is a valid choice for the test function used in the definition of σn,ν A(Or ), see Eq. (7.27). Expressing the fields φ there in annihilation and creation operators, and writing each commutator as a sum of two terms, we obtain n 4n (1−ν j0 ) σn,ν A(Or ) ≤ √ ωm h r,ν j . n! ν! j=1

(7.34)

Again, we use scaling arguments for the single-particle space functions and obtain for r ≤ r0 , (1−ν j0 )

ωm

h r,ν j ≤ c3 (3r )|ν j |+(s−1)/2

with a constant c3 that depends on r0 . This yields (c).

(7.35)

We now define the “scale-covariant” objects that will allow us to expand the maps Θ (β,O) in a series. They are constructed of the fields φn,ν and the functionals σn,ν by multiplication with appropriate powers of λ. We begin with the quantum fields. Proposition 7.8. For any n, ν and β > 0, the function χ n,ν,β : λ → λ|ν|+n(s−1)/2 e−βλH φn,ν Ω √ defines an element of H, with norm estimate χ n,ν,β ≤ cn (n!)1/2 ν!(β/2 s)−|ν|−n(s−1)/2. Proof. We use techniques from [BDM09], and adopt the notation introduced there. In particular, R denotes the function R λ = (1 + λH )−1 , and we write A() = supλ R λ Aλ R λ , where Aλ may be unbounded quadratic forms. Let n, ν be fixed in the following. We set φ λ = λ|ν|+n(s−1)/2 φn,ν .

(7.36)

From Lemma 7.7, one sees that P(E/λ)φ λ P(E/λ) is bounded uniformly in λ. Hence, applying [BDM09, Lemma 2.6], we obtain φ() < ∞ for sufficiently large . Also, the action g → α g φ of the symmetry group on φ (which extends canonically from bounded operators to quadratic forms) is continuous in some · () : This is clear for

On Dilation Symmetries Arising from Scaling Limits

53

translations by the energy-damping factor; for dilations it is immediate from the definition; and for Lorentz transformations it holds since they act by a finite-dimensional matrix representation on φn,ν . Thus, φ is an element of the space Φ defined in [BDM09, Eq. (2.39)]. Moreover, each φ λ is clearly an element of the Fredenhagen-Hertel field content ΦFH . Thus, [BDM09, Thm. 3.8] provides us with a sequence (An ) in A(O), with O a fixed neighborhood of zero, such that An − φ() → 0 as n → ∞. Now since exp(−β H )R − < ∞, we obtain exp(−β H )(An − φ)Ω → 0. Note here that exp(−β H )An Ω ∈ H by Lemma 6.1. Hence exp(−β H )φΩ = χ n,ν,β lies in H, since this space is closed in norm. The estimate for χ n,ν,β follows directly from Lemma 7.7(a). Next, we rephrase the functionals σn,ν as maps from the scaling algebra A to its center. Lemma 7.9. For any n, ν, the definition (σ n,ν (A))λ = λ−|ν|−n(s−1)/2 σn,ν ((C < A)λ ) yields a linear map σ n,ν : ∪O A(O) → Z(A), with its norm bounded by √ σ n,ν A(Or ) ≤ cn ( n! ν!)−1 (3r )|ν|+n(s−1)/2 for r ≤ r0 ; here r0 , c are the constants of Lemma 7.7. Proof. The norm estimate is a consequence of Lemma 7.7(c), where one notes that (C < A)λ ∈ A(Oλr ) for λ ≤ 1, and (C < A)λ = 0 for λ > 1, so that these operators are always contained in A(Or0 ). It remains to show that σ n,ν (A) ∈ Z(A), i.e., that µ → δ µ (σ n,ν (A)) is continuous. But this follows from continuity of µ → δ µ A and the definition of σ n,ν . We are now in the position to prove that n,ν σ n,ν χ n,ν,β is a norm convergent expansion of the map Θ (β,O) . Theorem 7.10. Let s ≥ 3. For each r > 0, there exists β > 0 such that Θ (β,Or ) =

∞

σ n,ν ( · )χ n,ν,β .

n=0 ν

Proof. We will show below that the series in the statement converges absolutely in the Banach space B(A(Or ), H), i.e. that ∞

σ n,ν A(Or )χ n,ν,β < ∞.

(7.37)

n=0 ν

Once this has been established, the assertion of the theorem is obtained as follows. From Lemma 7.6, we know that ∞ n=0 ν

λ−|ν|−n(s−1)/2 σn,ν (A)(χ n,ν,β )λ = e−λβ H AΩ

(7.38)

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H. Bostelmann, C. D’Antoni, G. Morsella

at each fixed λ, whenever A is a linear combination of Weyl operators, and when evaluated in scalar products with vectors from a dense set. But (7.37) also shows that the left hand side of (7.38) converges in B(A(λOr ), H), and it is therefore strongly continuous for A in norm bounded parts of A(λOr ). Then by Kaplansky’s theorem (7.38) holds for in H. Finally, this entails for all A ∈ A(Or ) that any A ∈ A(λOr ) as an equality (β,Or ) (A) at each fixed λ > 0, i.e. the statement. σ (A) (χ ) = Θ λ n,ν,β λ λ n,ν n,ν We √ now prove Eq. (7.37). Let r0 > 0 be fixed in the following, and r < r0 . Set a := 6 s. From Prop. 7.8 and Lemma 7.9, we obtain the estimate σ n,ν A(Or )χ n,ν,β ≤ c2n (ar/β)|ν|+n(s−1)/2 s n n = c2 (ar/β)(s−1)/2 (ar/β)ν jk . (ar/β)ν j0 j=1

(7.39)

k=1

Factorizing the sum over multi-indexes ν accordingly, we obtain at fixed n, ν

(ar/β)(s−1)/2 n σ n,ν A(Or )χ n,ν,β ≤ 2c2 , (1 − ar/β)s

(7.40)

where we suppose ar/β < 1, and where each sum over ν j0 ∈ {0, 1} has been estimated by introducing a factor of 2. Now if we choose r/β small enough, we can certainly achieve that the expression in Eq. (7.40) is summable over n as a geometric series, and hence the series in Eq. (7.37) converges. This establishes the phase space condition of Def. 6.2 in our context. Corollary 7.11. The theory of a real scalar free field of mass m ≥ 0 in 3 + 1 or more space-time dimensions fulfills the uniform Haag-Swieca compactness condition. While our goal was to show that the maps Θ (β,O) are uniformly compact, it follows from Eq. (7.37) that they are actually uniformly nuclear at all scales, or by a slightly modified argument, even uniformly p-nuclear for any 0 < p ≤ 1. So we can generalize the somewhat stronger Buchholz-Wichmann condition [BW86] to our context. Several other types of phase space conditions can be derived with similar methods as in Thm. 7.10 as well. Particularly, one can show for s ≥ 3 that the sum n,ν σn,ν φn,ν converges in norm under a cutoff in energy E and restriction to a fixed local algebra A(Or ), with estimates uniform in E · r , where this product is small. This implies that Phase Space Condition II of [BDM09], which guarantees a regular behavior of pointlike fields under scaling, is fulfilled for those models. 8. Conclusions In this paper, we have considered short-distance scaling limits in the model independent framework of Buchholz and Verch [BV95]. In order to describe the dilation symmetry that arises in the limit theory, we passed to a generalized class of limit states, which includes states invariant under scaling. The essential results of [BV95] carry over to this generalization, including the structure of Poincaré symmetries and geometric modular action. However, the dilation invariant limit states are not pure. Rather, they can be decomposed into states of the Buchholz-Verch type, which are pure in two or more spatial

On Dilation Symmetries Arising from Scaling Limits

55

dimensions. This decomposition gives rise to a direct integral decomposition of the Hilbert space of the limit theory, under which local observables, Poincaré symmetries, and other relevant objects of the theory can be decomposed—except for dilations. The dilation symmetry has a more intricate structure, intertwining the pure components of the limit state. The situation simplifies if we consider the situation of a “unique limit” in the classification of [BV95]; our condition of a “factorizing limit” turned out to be equivalent modulo technicalities. Under this restriction, the dilation unitaries in the limit are, up to a central part, decomposable operators. The decomposed components do not necessarily fulfill a group relation though, but a somewhat weaker cocycle relation. Only under stronger assumptions (“convergent scaling limits”) we were able to show that the dilation symmetry factorizes into a tensor product of unitary group representations. It is unknown at present which type of models would make the generalized decomposition formulas necessary. In this paper, we have only considered very simple examples, which all turned out to fall into the more restricted class of convergent scaling limits. However, thinking e.g. of infinite tensor products of free fields with increasing masses as suggested in [Buc96a, Sec. 5], it may well be that some models violate the condition of uniqueness of the scaling limit, or even exhibit massive particles in the limit theory. In this case, the direct integral decomposition would be needed to obtain a reasonable description of dilation symmetry in the limit, with the symmetry operators intertwining fibers of the direct integral that correspond to different masses. As a next step in the analysis of dilation symmetries in the limit theory, one would like to investigate further the deviation of the theory at finite scales from the idealized dilation covariant limit theory; so to speak, the next-order term in the approximation λ → 0. This would be interesting e.g. for applications to deep inelastic scattering, which can currently only be treated in formal perturbation theory. It is expected that the dilation symmetry in the limit also contains some information about these next-order terms. Our formalism, however, does at the moment not capture these next-order approximations, and would need to be generalized considerably. Further, it would be interesting to see whether the dilation symmetry we analyzed can be used to obtain restrictions on the type of interaction in the limit theory, possibly leading to criteria for asymptotic freedom. Here we do not refer to restrictions on the form of the Lagrangian, a concept that is not visible in our framework. Rather, we think that dilation symmetries should manifest themselves in the coefficients of the operator product expansion [Bos05a,BDM09,BF08] or in the general structure of local observables [BF77]. In this context, it seems worthwhile to investigate simplified low-dimensional interacting models, such as the 1+1 dimensional massive models with factorizing S matrix that have recently been rigorously constructed in the algebraic framework [Lec08]. By abstract arguments, these models possess a scaling limit theory in our context. Just as in the Schwinger model [Buc96b,BV98], one expects here that even the limit theory for multiplicative states has a nontrivial center. This may be seen as a peculiarity of the 1+1 dimensional situation; we have not specifically dealt with this problem in the present paper. But neglecting these aspects, one would expect that the limit theory corresponds to a massless (and dilation covariant) model with factorizing S matrix, although it would probably not have an interpretation in the usual terms of scattering theory. Such models of “massless scattering” have indeed been considered in the physics literature; see [FS94] for a review. Their mathematical status as quantum field theories, however,

56

H. Bostelmann, C. D’Antoni, G. Morsella

remains largely unclear at this time. Nevertheless, one should be able to treat them with our methods. In fact, these examples may give a hint to the restrictions on interaction that the dilation symmetry implies. At least in a certain class of two-particle S matrices—those which tend to 1 at large momenta—one expects that the limit theory is chiral, i.e., factors into a tensor product of two models living on the left and right light ray, respectively. On the other hand, the theories we consider are always local; and for chiral local models, dilation covariance is the essential property that guarantees conformal covariance [GLW98]. So if those models have a nontrivial scaling limit at all, they underly quite rigid restrictions, since local conformal chiral nets are—at least partially—classifiable in a discrete series [KL04]. The detailed investigation of these aspects of factorizing S matrix models is however the subject of ongoing research, and some surprises are likely to turn up. Acknowledgements. The authors are obliged to Laszlo Zsido and Michael Müger for helpful discussions. They also profited from financial support by the Erwin Schrödinger Institute, Vienna, and from the friendly atmosphere there. HB further wishes to thank the II. Institut für Theoretische Physik, Hamburg, for their hospitality.

A. Direct Integrals of Hilbert Spaces In our investigation, we make use of the concept of direct integrals of Hilbert spaces, H = Z dν(z)Hz , where the integral is defined on some measure space (Z, ν). Due to difficult measure-theoretic problems, the standard literature treats these direct integrals only under separability assumptions on the Hilbert spaces involved; see e.g. [KR97]. These are however not a priori implied in our analysis; and even where we make such assumptions, we need to apply them with care. While we can often reasonably assume the “fiber spaces” Hz to be separable, the measure space Z will, in our applications, be of a very general nature, and even L 2 (Z, ν) is known to be non-separable in some situations. The concept of direct integrals can be generalized to that case. Since however the literature on that topic6 is somewhat scattered and not easily accessible, we give here a brief review for the convenience of the reader, restricted to the case that concerns us. In the following, let Z be a compact topological space and ν a finite regular Borel measure on Z. For each z ∈ Z, we consider a Hilbert space Hz with scalar product · | · "z and associated norm · z . Elements χ ∈ z∈Z Hz will be called vector fields and alternatively denoted as maps, z → χ (z). Direct integrals of this field of Hilbert spaces Hz over Z are not unique, but depend on the choice of a fundamental family. Definition A.1. A fundamental family is a linear subspace Γ ⊂ z∈Z Hz such that for every χ ∈ Γ , the function z → χ (z)2z is ν-integrable. If the same function is always continuous, we say that Γ is a continuous fundamental family. The continuity aspect will be discussed further below, for the moment we focus on measurability. Eachfundamental family Γ uniquely extends to a minimal vector space Γ¯ , with Γ ⊂ Γ¯ ⊂ z∈Z Hz , which has the following properties [Wil70, Corollary 2.3]: 6 In the general case, we largely follow Wils [Wil70], however with some changes in notation. Other, somewhat stronger notions of direct integrals exist, e.g. [God51, Ch. III], [Seg51]; see [Mar69] for a comparison. Under separability assumptions (Definition A.3), all these notions agree, and we are in the case described in [Tak79, Ch. IV.8], [Dix81, Part II Ch. 1].

On Dilation Symmetries Arising from Scaling Limits

(i) (ii) (iii) (iv)

57

z → χ (z)2z is ν-integrable for all χ ∈ Γ¯ . If for χ ∈ z Hz , there exists χˆ ∈ Γ¯ such that χ (z) = χˆ (z) a.e., then χ ∈ Γ¯ . If χ ∈ Γ¯ and f ∈ L ∞ (Z, ν), then f χ ∈ Γ¯ , where ( f χ )(z) := f (z)χ (z). Γ¯ is complete with respect to the seminorm χ = ( Z dν(z)χ (z)2 )1/2 .

Such Γ¯ is called an integrable family. It is obtained from Γ by multiplication with L ∞ functions and closure in · . The elements of Γ¯ are called Γ -measurable functions; they are in fact analogues of square-integrable functions, and the usual measure theoretic results hold for them: Egoroff’s theorem; the dominated convergence theorem; any norm-convergent sequence (χn ) in Γ¯ has a subsequence on which χn (z) converges pointwise a.e.; and if (χn ) is a sequence in Γ¯ that converges pointwise a.e., the limit function χ is in Γ¯ . (Cf. Propositions 1.3 and 1.5 of [Mar69].) After dividing out vectors of zero norm (which we do not denote explicitly), Γ¯ becomes a Hilbert space, which we call the direct integral of the Hz with respect to Γ , and denote it as Γ H= dν(z)Hz , with scalar product (χ |χ) ˆ = dν(z) χ (z)|χ(z)" ˆ z . (A.1) Z

Z

Γ Correspondingly, the elements χ ∈ H are denoted as χ = Z χ (z)dν(z). We also consider bounded operators between such direct integral spaces. Let Hz , Hˆ z be two fields of Hilbert spaces over the same measure space Z, and let Γ, Γˆ be associated fundamental families. We call B ∈ z∈Z B(Hz , Hˆ z ) a measurable field of operators if ess supz B(z) < ∞, and if for every χ ∈ Γ¯ , the vector field z → B(z)χ (z) is Γˆ -measurable, i.e. an element of Γ¯ˆ . In fact it suffices to check the measurability condition on the fundamental family Γ . Lemma A.2. Let B ∈ z∈Z B(Hz , Hˆ z ) such that ess supz B(z) < ∞, and such that z → B(z)χ (z) is Γˆ -measurable for every χ ∈ Γ . Then B is a measurable field of operators. Proof. Evidently, z → B(z)χ (z) is also Γˆ -measurable if χ is taken from L ∞ (Z, ν) · Γ or from its linear span. This span is however dense in H. So let χ ∈ H. There exists a sequence χn ∈ span L ∞ (Z, ν) · Γ such that χn → χ in norm; by the remarks after Def. A.1, we can assume that χn (z) → χ (z) a. e. But then B(z)χn (z) → B(z)χ (z) a. e., due to continuity of each B(z). So B(z)χ (z) is a pointwise a. e. limit of functions in Γ¯ˆ . This implies (z → B(z)χ (z)) ∈ Γ¯ˆ , which was to be shown. A measurable field of operators B now defines a bounded linear operator H → Hˆ

Γ,Γˆ ˆ of this form are which we denote as B = Z dν(z)B(z). Operators in B(H, H) called decomposable. Their decomposition need not be unique however, not even a. e. If here Γ = Γˆ , and if B(z) = f (z)1Hz with a function f ∈ L ∞ (Z , ν), then B is called a diagonalizable operator. We sometimes write this multiplication operator as

Γ M f = Z dν(z) f (z)1. Let A be a C∗ algebra, and let for each z ∈ Z a representation πz of A on Hz be given, such that z → πz (A) is a measurable field of operators for any A ∈ A. Then,

Γ π(A) = Z dν(z)πz (A) defines a new representation π of A on H, which we formally

Γ denote as π = Z dν(z)πz .

58

H. Bostelmann, C. D’Antoni, G. Morsella

In many cases, obtaining useful results regarding decomposable operators requires additional separability assumptions. The following property will usually be general enough for us.

Γ Definition A.3. A fundamental sequence in H = Z dν(z)Hz is a sequence (χ j ) j∈N in Γ¯ such that for every z ∈ Z, the set {χ j (z) | j ∈ N} is total in Hz . If such a fundamental sequence exists for H, we say that H is uniformly separable. This more restrictive situation agrees with the setting of [Tak79,Dix81]; cf. [Mar69, Prop. 1.13]. Note that this property implies that the fiber spaces Hz are separable, but the integral space H does not need to be separable if Z is sufficiently general. Under the above separability assumption, additional desirable properties of decomposable operators hold true.

Γ

Γˆ Theorem A.4. Let H = Z dν(z)Hz and Hˆ = Z dν(z)Hˆ z both be uniformly sepa Γ,Γˆ rable. Then, for each decomposable operator B = Z dν(z)B(z), also B ∗ is decom Γˆ ,Γ posable, with B ∗ = Z dν(z)B(z)∗ . One has B = B ∗ = ess supz B(z).

Γ,Γˆ Decompositions of operators are unique in the following sense: If Z dν(z)B(z) =

Γ,Γˆ ˆ ˆ dν(z) B(z), then B(z) = B(z) for almost every z. Z

For the proof methods, see e.g. [God51, Ch. III Sec. 13]. Note that the theorem is false if the separability assumption is dropped; see Example 7.6 and Remark 7.11 of [Tak79, Ch. IV]. We also obtain an important characterization of decomposable operators. ˆ is decomTheorem A.5. Let H, Hˆ be uniformly separable. An operator B ∈ B(H, H) posable if and only if it commutes with all diagonalizable operators; i.e. M f B = B Mˆ f for all f ∈ L ∞ (Z , ν). ˆ A proof can be found in [Dix81, Ch. II §2 Sec. 5 Thm. 1]. In particular, if H = H, we know that both the decomposable operators and the diagonalizable operators form W∗ algebras, which are their mutual commutants. Note that the “if” part of the theorem is known to be false for sufficiently general direct integrals, violating the separability assumption [Sch90]. We now discuss the case of a continuous fundamental family Γ ; cf. [God51, Ch. III Sec. 2]. In this case, we can consider the space of Γ -continuous functions, denoted C(Γ ), and defined as the closed span of C(Z) · Γ in the supremum norm, χ ∞ = supz∈Z χ (z)z . With this norm, C(Γ ) is a Banach space, in fact a Banach module over the commutative C∗ algebra C(Z). We have C(Γ ) ⊂ Γ¯ in a natural way, and this inclusion is dense, but it is important to note that different norms are used in these two spaces. A simple but particularly important example for direct integrals arises as follows [Tak79, Ch. IV.7]. Let Hu be a fixed Hilbert space, and Z a measure space as above. For each z ∈ Z, set Hz = Hu . Then the set Γ of constant functions Z → Hu is a continuous fundamental family; and the associated integrable family Γ¯ is precisely the space of all square-integrable, Lusin-measurable functions χ : Z → Hu . We denote ⊕ the corresponding direct integral space as L 2 (Z, ν, Hu ) = Z dν(z)Hu (with reference to the “canonical” fundamental family). This space is isomorphic to L 2 (Z, ν) ⊗ Hu ; the canonical isomorphism, which we do not denote explicitly, maps f ⊗ χ to the

On Dilation Symmetries Arising from Scaling Limits

59

function z → f (z)χ . In this way, the algebra of diagonal operators is identified with L ∞ (Z, ν) ⊗ 1. If here Hu is separable, then L 2 (Z, ν, Hu ) is clearly uniformly separable. In this case, a simple criterion identifies the elements of the integral space: A function χ : Z → Hu is Lusin measurable if and only if it is weakly measurable, i.e. if z → χ (z)|η" is measurable for any fixed η ∈ Hu . Also, the algebra of decomposable operators is iso¯ morphic to L ∞ (Z, ν)⊗B(H u ). References [Ara64] [BDF87] [BDM09] [BF77] [BF08] [Bos00] [Bos05a] [Bos05b] [BP90] [BR79] [BR81] [Buc96a] [Buc96b] [BV95] [BV98] [BW86] [CM08] [Dix81] [DM06] [DMV04] [DS85] [EF74] [FS94]

[GLW98] [God51] [Haa96]

Araki, H.: On the algebra of all local observables. Prog. Theor. Phys. 32, 844–854 (1964) Buchholz, D., D’Antoni, C., Fredenhagen, K.: The universal structure of local algebras. Commun. Math. Phys. 111, 123–135 (1987) Bostelmann, H., D’Antoni, C., Morsella, G.: Scaling algebras and pointlike fields. A nonperturbative approach to renormalization. Commun. Math. Phys. 285, 763–798 (2009) Buchholz, D., Fredenhagen, K.: Dilations and interactions. J. Math. Phys. 18, 1107–1111 (1977) Bostelmann, H., Fewster, C.J.: Quantum inequalities from operator product expansions. Commun. Math. Phys. (2009). doi:10.1007/s00220-009-0853-x Bostelmann, H.: Lokale Algebren und Operatorprodukte am Punkt. Thesis, Universität Göttingen, 2000. Available online at http://webdoc.sub.gwdg.de/diss/2000/bostelmann/ Bostelmann, H.: Operator product expansions as a consequence of phase space properties. J. Math. Phys. 46, 082304 (2005) Bostelmann, H.: Phase space properties and the short distance structure in quantum field theory. J. Math. Phys. 46, 052301 (2005) Buchholz, D., Porrmann, M.: How small is the phase space in quantum field theory? Ann Inst. H. Poincaré 52, 237–257 (1990) Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, Volume I. New York: Springer, 1979 Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, Volume II. New York: Springer, 1981 Buchholz, D.: Phase space properties of local observables and structure of scaling limits. Ann. Inst. H. Poincaré 64, 433–459 (1996) Buchholz, D.: Quarks, gluons, colour: facts or fiction? Nucl Phys. B 469, 333–353 (1996) Buchholz, D., Verch, R.: Scaling algebras and renormalization group in algebraic quantum field theory. Rev. Math. Phys. 7, 1195–1239 (1995) Buchholz, D., Verch, R.: Scaling algebras and renormalization group in algebraic quantum field theory. II. Instructive examples. Rev. Math. Phys. 10, 775–800 (1998) Buchholz, D., Wichmann, E.H.: Causal independence and the energy-level density of states in local quantum field theory. Commun. Math. Phys. 106, 321–344 (1986) Conti, R., Morsella, G.: Scaling limit for subsystems and Doplicher-Roberts reconstruction. Ann. H. Poincaré 10, 485–511 (2009) Dixmier, J.: Von Neumann Algebras. Amsterdam: North-Holland, 1981 D’Antoni, C., Morsella, G.: Scaling algebras and superselection sectors: Study of a class of models. Rev. Math. Phys. 18, 565–594 (2006) D’Antoni, C., Morsella, G., Verch, R.: Scaling algebras for charged fields and short-distance analysis for localizable and topological charges. Ann. H. Poincaré 5, 809–870 (2004) Driessler, W., Summers, S.J.: Central decomposition of Poincaré-invariant nets of local field algebras and absence of spontaneous breaking of the Lorentz group. Ann. Inst. H. Poincaré Phys. Theor. 43, 147–166 (1985) Eckmann, J.P., Fröhlich, J.: Unitary equivalence of local algebras in the quasifree representation. Ann. Inst. H. Poincaré Sect. A (N.S.) 20, 201–209 (1974) Fendley, P., Saleur, H.: Massless integrable quantum field theories and massless scattering in 1+1 dimensions. In: Gava, E., Masiero, A., Nariain, K.S., Randjbar-Daemi, S., Shafi, Q. (eds.), Proceedings of the 1993 Summer School on High Energy Physics and Cosmology, Volume 10 of ICTP Series in Theoretical Physics, Singapore: World Scientific, 1994 Guido, D., Longo, R., Wiesbrock, H.W.: Extensions of conformal nets and superselection structures. Commun. Math. Phys. 192, 217–244 (1998) Godement, R.: Sur la théorie des représentations unitaires. Ann. of Math. 53, 68–124 (1951) Haag, R.: Local Quantum Physics. Berlin: Springer, 2nd edition, 1996

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H. Bostelmann, C. D’Antoni, G. Morsella

Haag, R., Swieca, J.A.: When does a quantum field theory describe particles? Commun. Math. Phys. 1, 308–320 (1965) Kawahigashi, Y., Longo, R.: Classification of local conformal nets. Case c < 1. Ann. Math. 160, 493–522 (2004) Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory. Orlando FL: Academic Press, 1997 Lechner, G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008) Maréchal, O.: Champs mesurables d’espaces hilbertiens. Bull. Sci. Math. 93, 113–143 (1969) Mitchell, T.: Fixed points and multiplicative left invariant means. Trans. Amer. Math. Soc. 122, 195–202 (1966) Schaflitzel, R.: The algebra of decomposable operators in direct integrals of not necessarily separable Hilbert spaces. Proc. Amer. Math. Soc. 110, 983–987 (1990) Segal, I.: Decompositions of Operator Algebras I, II, Volume 9 of Mem. Amer. Math. Soc. Providence, RI: Amer. Math. Soc., 1951 Takesaki, M.: Theory of Operator Algebras I. Berlin: Springer, 1979 Takesaki, M.: Theory of Operator Algebras II. Berlin: Springer, 2002 Wils, W.: Direct integrals of Hilbert spaces I. Math. Scand. 26, 73–88 (1970)

Communicated by Y. Kawahigashi

Commun. Math. Phys. 294, 61–72 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0934-x

Communications in

Mathematical Physics

Blowup of Jang’s Equation at Outermost Marginally Trapped Surfaces Jan Metzger Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Potsdam, Germany. E-mail: [email protected] Received: 30 January 2009 / Accepted: 24 July 2009 Published online: 8 October 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract: The aim of this paper is to accurately describe the blowup of Jang’s equation. First, we discuss how to construct solutions that blow up at an outermost MOTS. Second, we exclude the possibility that there are extra blowup surfaces in data sets with non-positive mean curvature. Then we investigate the rate of convergence of the blowup to a cylinder near a strictly stable MOTS and show exponential convergence with an identifiable rate near a strictly stable MOTS. 1. Introduction This paper is concerned with the examination of the relation of Jang’s equation to marginally outer trapped surfaces (MOTS). To set the perspective, we consider Cauchy data (M, g, K ) for the Einstein equations. Such data sets are 3-manifolds M equipped with a Riemannian metric together with a symmetric bilinear form K representing the second fundamental form of the time slice M in space-time. A marginally outer trapped surface is a surface with θ + = H + P = 0, where H is the mean curvature of in M and P = tr K − K (ν, ν) for the normal ν to . In the paper [AM07], inspired by an idea of Schoen [Sch04], we constructed MOTS in the presence of barrier surfaces by inducing a blow-up of Jang’s equation. In this context, Jang’s equation [SY81,Jan78] is an equation of prescribed mean curvature for the graph of a function in M × R. For details we refer to Sect. 2. In this note, we take a slightly different perspective. Consider a data set (M, g, K ) with a non-empty outer boundary ∂ + M and assume that we are given the outermost MOTS in (M, g, K ). Here, outermost means that there is no other MOTS on the outside of . From [AM07] it follows that (M, g, K ) always contains a unique such surface, or does not contain outer trapped surfaces at all, under the assumption that ∂ M is outer untrapped. As stated in Theorem 3.1, we show that there exists a solution f Research on this project started while the author was supported in part by a Feodor-Lynen Fellowship of the Humboldt Foundation.

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to Jang’s equation that actually blows up at , assuming that ∂ M is inner and outer untrapped. By blow-up we mean that outside from the function f is such that graph f is a smooth submanifold of M × R with a cylindrical end converging to × R. There is however a catch, as f may blow up at other surfaces, too. These surfaces are marginally inner trapped. In Theorem 3.4 we show that the other blow-up surfaces can not occur if the data set has non-positive mean curvature. To put the result in perspective note that if the dominant energy condition holds, the graph of f is of non-negative Yamabe type and thus can be equipped with a (singular) metric of zero scalar curvature. This was used by Schoen and Yau in [SY81] to prove the positive mass theorem. Later Bray [Bra01] proposed to use Jang’s equation to relate the Penrose conjecture in its general setting to the Riemannian Penrose inequality [HI01,Bra01] on a manifold constructed from Jang’s graph. One of the main questions in this program is whether or not Jang’s equation can be made to blow up at a specific MOTS. This question was raised in the literature, cf. for example [MÓM04] where this is discussed in the rotationally symmetric case. Here we give the positive answer that blow-up solutions exist at outermost MOTS. The author recently learned that the existence of the blow-up solution is used in [Khu09] to prove a Penrose-like inequality. With the blow-up constructed, we can turn to the asymptotic behavior of the blowup itself. It has been shown in [SY81] that such a blow-up must be asymptotic to a cylinder over the outermost MOTS. In Theorems 4.2 and 4.4 we show that under the assumption of strict stability the convergence rate is exponential with a power directly related to the principal eigenvalue of the MOTS. The general idea is to show the existence of a super-solution with at most logarithmic blow-up of the desired rate. Turning the picture sideways yields exponential decay, when writing the solution as a graph over the cylinder in question. Furthermore, we show that beyond a certain decay rate, the solution must be trivial, thus exhibiting the actual rate. We expect that the knowledge of these asymptotics is tied to the question whether the blow-up solution is unique. Furthermore note that the constant in the Penrose-like inequality in [Khu09] depends on the geometry of the solution. We thus expect that the value on this constant is related to the asymptotic behavior near the blow-up cylinder. Before turning to these results, we introduce some notation in Sect. 2. Section 3 proceeds with the construction of the the blow-up. We will not go into details here, but emphasize the general idea and point to the results needed from the paper [AM07]. In Sect. 4, we perform the calculation of the asymptotics. 2. Preliminaries Let (M, g, K ) be an initial data set for the Einstein equations. That is M is a 3-dimensional manifold, g a Riemannian metric on M and K a symmetric 2-tensor. We do not require any energy condition to hold. Assume that ∂ M is the disjoint union ∂ M = ∂ − M ∪ ∂ + M, where ∂ ± M are smooth surfaces without boundary. We refer to ∂ − M as the inner boundary and endow it with the normal vector ν pointing into M. The outer boundary ∂ + M is endowed with the normal ν pointing out of M. We denote by H [∂ M] the mean curvature of ∂ M with respect to the normal vector field ν, and by P[∂ M] = tr ∂ M K the trace of the tensor K restricted to the 2-dimensional surface ∂ M. Then the inward and outward expansions of ∂ M are defined by θ ± [∂ M] = P[∂ M] ± H [∂ M]. Assume that θ + [∂ − M] = 0, and that θ + [∂ + M] > 0 and θ − [∂ + M] < 0.

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If ⊂ M is a smooth, embedded surface homologous to ∂ + M, then bounds a region together with ∂ + M. In this case, we define θ ± [] as above, where H is computed with respect to the normal vector field pointing into (that is in direction of ∂ + M). is called a marginally outer trapped surface (MOTS), if θ + [] = 0. We say that ∂ M is an outermost MOTS, if there is no other MOTS in M, which is homologous to ∂ + M. In [AM07] it is proved that for any initial data set (M, g, K ) which contains a MOTS, there is also an outermost MOTS surrounding it. Let ⊂ M be a MOTS and consider a normal variation of in M, that is a map ∂ F : × (−ε, ε) → M such that F(·, 0) = id and ∂s F( p, s) = f ν, where f is s=0 a function on and ν is the normal of . Then the change of θ + is given by ∂θ + [F(, s)] = L M f, ∂ds s=0 where L M is a quasi-linear elliptic operator of second order along . It is given by 1 L M f = − f + 2S(∇ f ) + f div S − |χ + |2 − |S|2 + Sc − µ − J (ν) . 2 In this expression ∇, div and denote the gradient, divergence and Laplace-Beltrami operator tangential to . The tangential 1-form S is given by S = K (·, ν)T , χ + is the bilinear form χ + = A + K , where A is the second fundamental form of in M and K is the projection of K to T × T . Furthermore, Sc denotes the scalar curvature of , µ = 21 ( M Sc − |K |2 + (tr K )2 ), and J = M divK − d tr K . For a more detailed investigation of this operator we refer to [AMS05] and [AMS07]. The facts we will need here are that L M has a principal eigenvalue λ, which is real and has a one-dimensional eigenspace which is spanned by a positive function. If λ is non-negative is called stable, and if λ is positive, is called strictly stable. In particular, if is strictly stable as a MOTS, there exists an outward deformation strictly increasing θ + . In M¯ = M × R, we consider Jang’s equation [Jan78,SY81] for the graph of a function f : M → R. Let N := graph f = {(x, z) : z = f (x)}. The mean curvature H[ f ] of N with respect to the downward normal is given by

∇f

H[ f ] = div 1 + |∇ f |2

.

Define K¯ on M¯ by K¯ (x,z) (X, Y ) = K x (π X, π Y ), where π : T M¯ → T M denotes the orthogonal projection onto the horizontal tangent vectors. Let P[ f ] = tr N K¯ . Then Jang’s equation becomes J [ f ] = H[ f ] − P[ f ] = 0.

(2.1)

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Fig. 1. The situation in Theorem 3.1. All of the shaded region belongs to M, whereas f is only defined in 0

3. The Blowup The main result of this section is that we can construct a solution to Jang’s equation which blows up at the outermost MOTS in (M, g, K ) and has zero Dirichlet boundary data at ∂ + M. In fact, we chose the assumptions on the outer boundary ∂ + M so that we can prescribe more general Dirichlet data there. The focus here lies on the blow-up in the interior, so that we do not investigate the optimal conditions for ∂ + M. Theorem 3.1. If (M, g, K ) is an initial data set with ∂ M = ∂ − M ∪ ∂ + M such that ∂ − M is an outermost MOTS, θ + [∂ + M] > 0 and θ − [∂ + M] < 0, then there exists an open set 0 ⊂ M and a function f : 0 → R such that 1. 2. 3. 4. 5. 6.

M\0 does not intersect ∂ M, θ − [∂0 ] = 0 with respect to the normal vector pointing into 0 , J [ f ] = 0, N + = graph f ∩ M × R+ is asymptotic to the cylinder ∂ − M × R+ , N − = graph f ∩ M × R− is asymptotic to the cylinder ∂0 × R− , and f |∂ + M = 0.

For data sets (M, g, K ) which do not contain surfaces with θ − = 0, the above theorem implies the following result. Corollary 3.2. If (M, g, K ) is as in Theorem 3.1, and in addition there are no subsets ⊂ M with θ − [∂] = 0 with respect to the normal pointing out of , then there exists a function f : M → R such that 1. J [ f ] = 0, 2. N = graph f is asymptotic to the cylinder ∂ − M × R+ , 3. f |∂ + M = 0. Remark 3.3. Analogous results hold if (M, g, K ) is asymptotically flat with appropriate decay of g and K instead of having an outer boundary ∂ + M. Then the assertion f |∂ + M = 0 in Theorem 3.1 has to be replaced by f (x) → 0 as x → ∞. The proof of Theorem 3.1 is largely based on the tools developed in [SY81 and AM07]. Thus we will not include all details here, but provide a summary, which facts will have to be used.

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Proof. We will assume that (M, g, K ) is embedded into (M , g , K ) which extends M beyond the boundary ∂ − M such that ∂ − M lies in the interior of M , without further requirements. N where the are the connected components of ∂ M. As ∂ M is Let ∂ − M = ∪i=1 i i an outermost MOTS, each of the i is stable [AM07, Cor. 5.3]. Following the proof of [AM07, Th. 5.1], we deform ∂ − M to a surface s by pushing the components i out of M, into the extension M . To this end, let φi > 0 be the principal eigenfunction of the stability operator of i and extend the vector field X i = −φi νi to a neighborhood of i in M . Flowing i by X i yields a family of surfaces is , s ∈ [0, ε) so that the is form a smooth foliation for small enough ε with is ∈ M \ M. If i is strictly stable then ∂ θ + [s ] = −λφ < 0, ∂s s=0 where λ is the principal eigenvalue of i . Thus, for small enough ε, we have θ + [is ] < 0 for all s ∈ (0, ε). If i has principal eigenvalue λ = 0, then the is satisfy ∂ θ + [s ] = 0. ∂s s=0 In this case it is possible to change the data K on is as follows: K˜ = K − 21 ψ(s)γs ,

(3.1)

where γs is the metric on s and ψ is a smooth function ψ : [0, ε] → R. The operator θ˜ + , which means θ + computed with respect to the data K˜ instead of K , satisfies θ˜ + [is ] = θ + [is ] − ψ(s). It is clear from Eq. (3.1) that ψ can be chosen such that ψ(0) = ψ (0) = 0 and θ˜ + [is ] < 0 for all s ∈ (0, ε) provided ε is small enough. Then K˜ is C 1,1 when extended by K to the rest of M. Replace each original boundary component i of M by a surface iε as constructed above, and replace K with K˜ , such that the following properties are satisfied. Let M˜ denote the manifold with boundary components iε resulting from this procedure. Thus ˜ g , K˜ ) with the following properties: we construct from (M, g, K ) a data set ( M, ˜ 1. M ⊂ M˜ with g | M = g, K˜ | M = K , and ∂ + M = ∂ + M, + − ˜ 2. θ [∂ M] < 0, and 3. the region M˜ \ M is foliated by surfaces s with θ + (s ) < 0. The method developed in Sect. 3.2 in [AM07] now allows the modification of the data ˜ g, ˜ g , K˜ ) to a new data set, which we also denote by ( M, ˜ K˜ ), although K˜ changes ( M, in this step. This data set has the following properties: ˜ 1. M ⊂ M˜ with g | M = g, K˜ | M = K , and ∂ + M = ∂ + M, ˜ < 0, 2. θ + [∂ − M] ˜ > 0, where H is the mean curvature of ∂ − M with respect to the normal 3. H [∂ − M] ˜ pointing out of ∂ − M, 4. the region M˜ \ M is foliated by surfaces s with θ + (s ) < 0.

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By Sect. 3.3 in [AM07] this enables us to solve the boundary value problem ⎧ ⎪ ⎨J [ f τ ] = τ f τ in M˜ f τ = 2τδ on ∂ − M˜ ⎪ ⎩f =0 on ∂ + M˜ τ

(3.2)

where δ is a lower bound for H on ∂ − M. The solvability of this equation follows, provided an estimate for the gradient at the boundary can be found. The barrier construction at ∂ − M˜ was carried out in detail in [AM07], whereas the barrier construction at ∂ + M˜ is standard due to the stronger requirement that θ + [∂ + M] > 0 and θ − [∂ + M] < 0. The solution f τ to Eq. (3.2) satisfies an estimate of the form sup | f τ | + sup |∇ f τ | ≤ M˜

M˜

C , τ

(3.3)

˜ g, where C is a constant depending only on the data ( M, ˜ K˜ ) but not on τ . The gradient estimate implies in particular that there exists an ε > 0 independent of τ such that f τ (x) ≥

δ 4τ

˜ ∀x with dist(x, ∂ − M).

The graphs Nτ have uniformly bounded curvature in M˜ × R away from the boundary. This allows the extraction of a sequence τi → 0 such that the Nτi converge to a manifold N , cf. [AM07, Prop. 3.8], [SY81, Sect. 4]. This convergence determines three open ˜ subsets of M: − := {x ∈ M : f τi (x) → −∞ locally uniformly as i → ∞}, 0 := {x ∈ M : lim sup | f τi (x)| < ∞}, i→∞

+ := {x ∈ M : f τi (x) → ∞ locally uniformly as i → ∞}. ˜ we have that + = ∅ and + contains a From the fact that the f τ blow up near ∂ − M, ˜ As already noted in [SY81] ∂+ \ ∂ M˜ consists of MOTS. As the neighborhood of ∂ − . region M˜ \ M is foliated by surfaces with θ + < 0, we must have that + ⊃ ( M˜ \ M) and hence ∂+ is a MOTS in M. As ∂ − M was assumed to be an outermost MOTS in M, we conclude that the closure of + is M˜ \ M. The barriers near ∂ + M are so that they imply that the f τ are uniformly bounded near + ∂ M. Thus 0 contains a neighborhood of ∂ + M and 0 ⊂ M. The limit manifold N over 0 is a graph satisfying J [ f τ ] = 0, and has the desired asymptotics. We will now discuss a geometric condition to assert that the resulting graph is nonsingular on M, i.e., M = 0 in Theorem 3.1. Theorem 3.4. Let (M, g, K ) be as in Theorem 3.1 with tr K ≤ 0. Then in the assertion of Theorem 3.1 we have that 0 = M, that is f is defined on M and has no other blow-up than near ∂ − M.

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67

Proof. This follows from a simple argument using the maximum principle. Let f τ be a solution to the regularized problem H[ f τ ] − P[ f τ ] − τ f τ = 0

(3.4)

˜ as in the proof of Theorem 3.1. We claim that f τ can not have a negative minimum in M, in the region where the data is unmodified. Assume that x ∈ M is such a minimum. There we have H[ f τ ] ≥ 0, and since graph f is horizontal at x we have that P[ f τ ] = tr K ≤ 0, thus the right-hand side of (3.4) is non-negative, whereas τ f τ is assumed to be negative, a contradiction. Since we know that in the limit τ → 0, the functions f τ must blow-up in the modified region which lies in + , we infer a lower bound for f τ from the above argument. Thus 0 = M as claimed. 4. Asymptotic Behavior Here, we discuss a refinement of [SY81, Cor. 2], which says that N = graph f converges uniformly in C 2 to the cylinder ∂ − M × R for large values of f . A barrier construction allows us to determine the asymptotics of this convergence. Before we present our result, recall the statement of [SY81, Cor. 2]: Theorem 4.1. Let N = graph f be the manifold constructed in the proof of Theorem 3.1 and let be a connected component of ∂ − M. Let U be a neighborhood of with positive distance to ∂ − M \ . Then for all ε > 0 there exists z¯ = z¯ (ε), depending also on the geometry of (M, g, K ), such that N ∩ U × [¯z , ∞) can be written as the graph of a function u over C z¯ := × [¯z , ∞), so that |u( p, z)| + | C z¯ ∇ u( p, z)| + | C z¯ ∇ 2 u( p, z)| < ε for all ( p, z) ∈ C z¯ . Here,

C z¯

∇ denotes covariant differentiation along C z¯ .

If is strictly stable, we can in fact say more about u. Theorem 4.2. Assume the situation of Theorem 4.1. If in addition is strictly stable √ with principal eigenvalue λ > 0, we have that for all δ < λ there exists c = c(δ) depending only on the data (M, g, K ) and δ such that |u( p, z)| + | C z¯ ∇ u( p, z)| + | C z¯ ∇ 2 u( p, z)| ≤ c exp(−δz). Proof. Denote by β > 0 the eigenfunction to the principal eigenvalue λ on normalized such that max β = 1. We denote by ν the normal vector field of pointing into M. Consider the map : × [0, s¯ ] → M : ( p, s) → exp M p (sβν).

(4.1)

Given ε > 0 we can choose s¯ > 0 small enough such that the surfaces s = (, s) with s ∈ [0, s¯ ] form a local foliation near with lapse β such that θ + [s ] ≥ λ(1 − ε)βs.

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Denote the region swiped out by these s by Us¯ . Note that ∂Us¯ = ∪ s¯ and dist(s¯ , ) > 0. We can assume that dist(s¯ , ∂ M) > 0. On Us¯ we consider functions w of the form w = φ(s). For such functions Jang’s operator can be computed as follows: φ + φ φ θ − 1+ P − σ −2 K (ν, ν) + 2 3 , J [w] = βσ βσ β σ where σ 2 = 1+β −2 (φ )2 , and φ denotes the derivative of φ with respect to s, cf. [AM07]. The quantities θ + , K (ν, ν) and P are computed on the respective s . Note that with our normalization σ −2 ≤ β 2 |φ |−2 ≤ |φ |−2 , and if we assume that φ ≥ µ for a large µ = µ(β) we have 1 + φ ≤ 2|φ |−2 . βσ Furthermore, φ |φ | |φ | |φ | β 2 σ 3 = β 2 (1 + β −2 φ 2 )3/2 ≤ β 2 (β −2 φ 2 )3/2 = β |φ |3 . On the other hand, increasing µ = µ(β, ε) if necessary, we have |φ | ≥ 1 − ε, βσ if |φ | ≥ µ. In combination we find that J [w] ≤ −λ(1 − ε)βs +

c1 |φ | + β , |φ |2 |φ |3

(4.2)

with c > 0 depending on ε and the data (M, g, K ), provided |φ | ≥ µ and φ < 0. Choosing φ(s) = a log s with a = (1 − ε)−1 λ−1/2 , we calculate that φ (s) =

a a , φ (s) = − 2 , s s

so that 1 s2 = = (1 − ε)2 λs 2 , |φ |2 a2

φ s = 2 = (1 − ε)2 λs. 3 |φ | a

Thus we can choose s¯ so small that |φ | ≥ µ(β, ε) and the estimate in (4.2) holds. We can then decrease s¯ further, so that s¯ ≤ εβ/(c1 (1−ε)). This choice makes the right-hand side of (4.2) non-positive, that is J [w] ≤ 0. Hence, we obtain a super-solution w with Jτ w ≤ 0 at least where w ≥ 0, that is near .

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As w blows up near the horizon, and the f τ are bounded uniformly in τ on s¯ , we can translate w vertically to w¯ = w + b with a suitable b > 0 so that ¯ s¯ f τ |s¯ ≤ w| for all τ > 0. Then the maximum principle implies that f τ ≤ w¯ for all τ > 0 in Us¯ and consequently the function f constructed in Theorem 3.1 also satisfies f ≤ w. ¯ Near , the graph of w¯ can be written as the graph of a function v¯ over × (¯z , ∞), where v decays exponentially in z. This is due to the fact that by the assumptions on β, the parameter s is comparable to the distance to . By the above construction u ≤ v, where u is the function from Theorem 4.1. Thus we find the claimed estimate for u. Getting the desired estimates for the derivatives of u is then a standard procedure, but as it is a little work to set the stage, we briefly indicate how to proceed. We choose coordinates of a neighborhood × R in a slightly different manner as ¯ : × (−ε, ε) → M be the map above. Let ¯ : × (−ε, ε) × R → M × R : (x, s, z) → expx (sν), z . For a function h on C z¯ we let graph¯ h be the set ¯ graph¯ h = {(x, h(x), z) : (x, z) ∈ × R}. From Theorem 4.1, it is clear that for large enough z¯ the set N ∩ M × [¯z , ∞) can be written as graph¯ h, where h decays exponentially by the above reasoning. We can compute the value of Jang’s operator for h as follows: ¯ ( H¯ − P)[N ] = J h, where J is a quasi-linear elliptic operator of mean curvature type. To be more precise, J h has the form ij

ij

J h = ∂z2 h + γh(x,z) ∇i,2 j h − 2γh(x,z) ∂i (h)K (∂s , ∂ j ) − θ + [h(x,z) ] + Q(h,

C z¯

∇ h,

C z¯

(4.3)

∇ 2 h),

where γs is the metric on s and Q is of the form Q(h,

C z¯

∇ h,

C z¯

∇ 2 h) = h ∗

C z¯

∇h +

C z¯

∇h ∗

C z¯

∇h +

C z¯

∇h ∗

C z¯

∇h ∗

C z¯

∇ 2 h,

where ∗ denotes some contraction with a bounded tensor. Furthermore, the vectors ∂i , i = 1, 2 denote directions tangential to and ∂z the direction along the R-factor in C z¯ . By freezing coefficients, we therefore conclude that h satisfies a linear, uniformly elliptic equation of the form a i j ∂i ∂ j h + b,

C z¯

∇ h − θ + [h(x,z) ] = 0.

By construction we have that |θ + [s ]| ≤ κs for some fixed κ. Thus θ + [h(x,z) ] decays exponentially in z. Now we are in the position to use standard interior estimates for linear elliptic equations to conclude the decay of higher derivatives of h. This decay translates back into the decay of the first and second derivatives of u as the coordinate transformation is smooth and controlled by the geometry of (M, g, K ).

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Remark 4.3. If is not strictly stable, but has positive k th variation, we find that the foliation near satisfies θ + [s ] ≥ κs k . Then a function of the form φ(s) = as − p with large a and p = k−1 2 yields a super-solution. This super-solution can be used to prove that |u| ≤ C z 2/(1−k) as above. We can get even more information about the decay rate. A closer look at Eq. (4.3) yields that the expression for J h on C z¯ can also be written as follows: J h = (∂z2 − L M )h + Q (h,

C z¯

∇ h,

C z¯

∇ 2 h),

since θs+ = s L M 1 + O(s 2 ), ij

γh(x,z) ∇i,2 j h = h + Q 1 (h, ∇h, ∇ 2 h), and ij

γh(x,z) ∂i h K (∂s , ∂ j ) = S(∇h) + Q 2 (h, ∇h), where the differential operators ∇ and are with respect to . Then note that L M h = h L M 1 − h + 2S(∇h). Further investigation of the structure of Q yields that |Q (h,

C z¯

∇ h,

C z¯

∇ 2 h)| ≤ C |h|2 + | C z¯ ∇ h|2 + |h|| C z¯ ∇ 2 h| + | C z¯ ∇ h|2 | C z¯ ∇ 2 h| ,

so that in view of the differential Harnack estimate | C z¯ ∇ h| ≤ c|h| for positive solutions of linear elliptic equations we have that in fact |Q (h,

C z¯

∇ h,

C z¯

∇ 2 h)| ≤ c|h| |h| + | C z¯ ∇ h| + | C z¯ ∇ 2 h| ,

provided |h| ≤ C. By projecting the equation J h = 0 to the one-dimensional eigenspace of L M it is now a somewhat standard ODE argument to show the following result. Theorem 4.4. Under the assumptions of Theorem 4.2 there are no solutions h : × [0, ∞) → R to the equation Jh = 0 with decay |h( p, z)| + | C z¯ ∇ h( p, z)| + | C z¯ ∇ h( p, z)| ≤ C exp(−δz) such that δ >

√ λ and h > 0.

(4.4)

Blowup of Jang’s Equation at Outermost Marginally Trapped Surfaces

71

Proof. Assume that h > 0 is such a solution. We derive a contradiction as follows. Let λ be the principal eigenvalue and φ be the corresponding eigenfunction of L M as before. Let L ∗M be the (formal) adjoint of L M on L 2 () and denote by φ ∗ > 0 its principal eigenfunction, normalized such that φφ ∗ dµ = 1. Then the operator Pu =

φ ∗ u dµ φ

is a projection onto the eigenspace spanned by φ and moreover commutes with L M . We interpret h(z) as a family of functions on , that is h(z)( p) = h( p, z) for p ∈ . Choose α(z) such that Ph(z) = α(z)φ, and β(z) accordingly, β(z)φ = P Q (h(·, z),

C z¯

∇ h(·, z),

C z¯

∇ 2 h(·, p)) .

Then Eq. (4.4) and the fact that P commutes with L M and ∂z imply α (z) − λα(z) = β(z). Using φ ∗ > 0 and h > 0 yields α(z) > 0 and we can furthermore estimate that

φ ∗ |h( p, z)| |h( p, z)| + |∇h( p, z)| + |∇ 2 h( p, z)| dµ ≤ c exp(−δz) φ ∗ |h( p, z)| dµ

|β(z)| ≤ c

≤ c exp(−δz)α(z). Thus, we conclude that on [˜z , 0) the function α > 0 satisfies a differential inequality of the form α (z) − λα ≤ εα, √ where ε > 0 can be chosen arbitrarily small by choosing z˜ large enough. If λ + ε < δ this ODE has no solutions with decay exp(−δz) other than the trivial solution. Thus α ≡ 0 and we arrive at the desired contradiction. Acknowledgement. The author thanks the Mittag-Leffler-Institute, Djursholm, Sweden for hospitality and support during the program Geometry, Analysis, and General Relativity in Fall 2008. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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References [AM07] [AMS05] [AMS07] [Bra01] [HI01] [Jan78] [Khu09] [MÓM04] [Sch04] [SY81]

Andersson, L., Metzger, J.: The area of horizons and the trapped region. Commun. Math. Phys. 290, 941–972 (2009) Andersson, L., Mars, M., Simon, W.: Local existence of dynamical and trapping horizons. Phys. Rev. Lett. 95, 111102 (2005) Andersson, L., Mars, M., Simon, W.: Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes. http://arxiv.org/abs/0704.2889v2[gr-qc], 2007 Bray, H.L.: Proof of the riemannian penrose inequality using the positive mass theorem. J. Diff. Geom. 59(2), 177–267 (2001) Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the riemannian penrose inequality. J. Diff. Geom. 59(3), 353–437 (2001) Jang, P.S.: On the positivity of energy in general relativity. J. Math. Phys. 19, 1152–1155 (1978) Khuri, M.: A penrose-like inequality for general initial data sets. Commun. Math. Phys. 290(2), 779–788 (2009) Malec, E., Murchadha, N.Ó.: The Jang equation, apparent horizons and the Penrose inequality. Class. Quant. Grav. 21(24), 5777–5787 (2004) Schoen, R.: Talk Given at the Miami Waves Conference, January 2004 Schoen, R., Yau, S.-T.: Proof of the positive mass theorem. II. Commun. Math. Phys. 79(2), 231–260 (1981)

Communicated by P. T. Chru´sciel

Commun. Math. Phys. 294, 73–95 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0954-6

Communications in

Mathematical Physics

Incompressible Limits and Propagation of Acoustic Waves in Large Domains with Boundaries Eduard Feireisl Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic. E-mail: [email protected] Received: 3 February 2009 / Accepted: 26 August 2009 Published online: 19 November 2009 – © Springer-Verlag 2009

Abstract: We study the incompressible limit for the full Navier-Stokes-Fourier system on unbounded domains with boundaries, supplemented with the complete slip boundary condition for the velocity field. Using an abstract result of Tosio Kato we show that the energy of acoustic waves decays to zero on any compact subset of the physical space. This in turn implies strong convergence of the velocity field to its limit in the incompressible regime.

1. Introduction Propagation and attenuation of acoustic waves plays an important role in the analysis of fluid flows in the low Mach number regime, in particular in the so-called incompressible limit, when the speed of sound in the material becomes infinite. Recently, several studies have been devoted to a rigorous justification of the low Mach number limit for the complete Navier-Stokes-Fourier system. Alazard [1,2] studied the problem for a quite general class of initial data that give rise to sufficiently regular solutions defined, however, only on a short time interval. In this purely “hyperbolic” approach proposed in the seminal paper by Klainerman and Majda [14], the presence of viscosity in the Navier-Stokes system plays only a marginal role. A different technique, based on the concept of weak solutions, was used by Lions and Masmoudi [22,23], and further developed in a series of papers by Desjardins and Grenier [7], Desjardins et al. [8], Masmoudi [24–26], among others. These results concern the Navier-Stokes system describing a compressible barotropic fluid flow, where the basic framework is provided by the existence theory developed by Lions [21]. The work of E.F. was supported by Grant 201/08/0315 of GA CR ˇ as a part of the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503.

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The same strategy, based on global-in-time weak solutions, was later adapted to the complete Navier-Stokes-Fourier system in [10]. This approach leans on energy-entropy estimates, where the presence of viscosity is indispensable. As frequently observed in practical as well as numerical experiments, the influence of acoustic waves on the fluid motion is negligible in the low Mach number limit (cf. Klein et al. [17]). In terms of the mathematical theory, the acoustic waves are supported by the gradient part of the fluid velocity u, specifically H⊥ [u], where u = H[u] + H⊥ [u] and H is the Helmholtz projection onto the space of solenoidal functions. In the (hypothetical) incompressible limit, the mass density tends to a constant, the speed of sound becomes infinite whereas the gradient component H⊥ “disappears” since the limit velocity field is solenoidal. However, if the initial data are ill prepared, meaning the fluctuations of the fluid density and temperature are of the same order as the Mach number, and if the fluid is contained in a bounded spatial domain with an acoustically hard boundary (see Wilcox [37]), the gradient component of the velocity H⊥ [u] develops fast time oscillations with frequencies inversely proportional to the Mach number (cf. Lions and Masmoudi [22], Schochet [30,31]). Accordingly, the gradient part H⊥ [u] tends to zero only weakly, meaning in the sense of integral averages, with respect to time - a phenomenon that may destabilize certain numerical schemes when applied to the original system. A rather heuristic argument why the acoustic waves can be neglected in the low Mach number flows encountered in the real world applications arising in meteorology, oceanography, or astrophysics, is usually based on the fact that the underlying physical space is unbounded or, more correctly, sufficiently large when compared to the sound speed in the material in question (cf. Klein [15,16]). In accordance with the fundamental observation of Lighthill [20], propagation of acoustic waves in the low Mach number regime may be described by a simple linear wave equation with a source term - Lighthill’s tensor - containing the remaining quantities appearing in the complete Navier-Stokes system. The expected local decay of the acoustic energy then follows immediately from the dispersive estimates. Desjardins and Grenier [7] exploited this idea combined with the non-trivial Strichartz estimates for the acoustic equation in order to show strong (pointwise) convergence of the velocity field in the low Mach number limit for a barotropic fluid flow in the whole physical space R3 . A similar approach was adapted in [11] to the complete Navier-Stokes-Fourier system considered on “large” spatial domains, on which the Strichartz estimates were replaced by global integrability of the local energy established by Burq [4], Smith and Sogge [33]. Note that the concept of the so-called radiation boundary conditions, amply used in numerical analysis, is based on the same physical principle (see Engquist and Majda [9]). In contrast with the simple geometry of the whole space R3 , where the efficient mathematical tools based on Fourier analysis are at hand, any real problem of wave propagation inevitably includes the influence of the boundary representing a wall in the physical space. As is well-known, the Strichartz estimates become much more delicate and usually require severe geometrical restrictions to be imposed on the boundary. For instance, if the fluid domain is exterior to a compact obstacle, the latter must be starshaped or at least non-trapping (see Burq [4], Metcalfe [27], Smith and Sogge [33]), and the references cited therein). In this paper, we propose a simple method that may be used to establish strong (pointwise a.a.) convergence of acoustic waves in the low Mach number limit for fluid flows

Study of Incompressible Limit for Navier – Stokes – Fourier System

75

with ill-prepared data for a sufficiently vast class of physical domains ⊂ R3 . Pursuing the philosophy that any real physical space is always bounded but possibly “large” with respect to the speed of sound in the medium, we consider a family of bounded domains {ε }ε>0 ⊂ R3 such that ε ≈ in a certain sense as ε → 0. More specifically, we suppose that ⊂ R3 is an unbounded domain with a compact boundary ∂,

(1.1)

ε = Br (ε) ∩ ,

(1.2)

and set

where Br (ε) is a ball centered at zero with a radius r (ε), with r (ε) → ∞. Our approach is based on a nowadays classical result of Kato [13] (cf. also Burq et al. [5]) concerning weighted L 2 space-time estimates for a class of abstract operators in a Hilbert space: Theorem 1.1. [ Reed and Simon [29, Theorem XIII.25 and Corollary] ]. Let A be a closed densely defined linear operator and H a self-adjoint densely defined linear operator in a Hilbert space X . For λ ∈ / R, let R H [λ] = (H − λId)−1 denote the resolvent of H . Suppose that =

sup

λ∈ / R, v∈D (A∗ ), v X =1

A ◦ R H [λ] ◦ A∗ [v] X < ∞.

(1.3)

Then π sup w∈X, w X =1 2

∞ −∞

A exp(−it H )[w] 2X dt ≤ 2 .

The paper is organized as follows. In Sect. 2, we introduce a scaled Navier-StokesFourier system and formulate the problem of the incompressible limit for vanishing Mach number. Following the idea of Lighthill [20], we then rewrite the Navier-StokesFourier system as a wave (acoustic) equation, where all “non-hyperbolic” components are considered as a source term (see Sect. 3). In Sect. 4, we collect the necessary uniform bounds, independent of the Mach number, resulting from the total dissipation balance. This step is now well understood, and the results are taken over from [10] without proofs. Section 5 is central and contains the bulk of the analysis of acoustic waves. Assuming the boundary of the limit domain is acoustically hard, meaning the fluid velocity satisfies the complete slip or Navier-like boundary conditions, we show that Theorem 1.1 may be applied in order to deduce a uniform local decay of the acoustic energy. A remarkable feature of the present approach is that we do not need any kind of non-trapping condition to be imposed on the boundary. As a matter of fact, our technique applies whenever the spatial domain admits the limiting absorption principle for the corresponding wave operator (see Vainberg [35, Chap. VIII.2]).

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2. Primitive System and its Incompressible Limit 2.1. Navier-Stokes-Fourier system. We consider a scaled Navier-Stokes-Fourier system in the form: ∂t + divx (u) = 0, 1 ∂t (u) + divx (u ⊗ u) + 2 ∇x p(, ϑ) = divx S(ϑ, ∇x u), ε q(ϑ, ∇x ϑ) = σε , ∂t (s(, ϑ)) + divx (s(, ϑ)u) + divx ϑ supplemented with the total energy balance 2 ε d |u|2 + e(, ϑ) (t, ·) dx = 0, dt ε 2

(2.1) (2.2) (2.3)

(2.4)

where = (t, x) is the density, u = u(t, x) the velocity field, ϑ the temperature, and p = p(, ϑ), e = e(, ϑ), s = s(, ϑ) denote the pressure, the (specific) internal energy, the (specific) entropy, respectively, obeying Gibbs’ relation 1 ϑ Ds(, ϑ) = De(, ϑ) + p(, ϑ)D . (2.5) In addition, the viscous stress tensor S is given by Newton’s rheological law 2 t S(ϑ, ∇x u) = µ(ϑ) ∇x u + ∇x u − Idivx u + η(ϑ)I divx u, 3

(2.6)

while the heat flux q(ϑ, ∇x ϑ) satisfies Fourier’s law q(ϑ, ∇x ϑ) = −κ(ϑ)∇x ϑ.

(2.7)

Finally, in virtue of the Second Law of Thermodynamics, the entropy production rate σε satisfies 1 κ(ϑ) 2 2 ε S : ∇x u + |∇x ϑ| ≥ 0. (2.8) σε ≥ ϑ ϑ Relation (2.8) reflects the fact that the weak solutions considered in the present paper may hypothetically produce “more” entropy than given by the classical formula 1 κ(ϑ) 2 2 (2.9) ε S : ∇x u + |∇x ϑ| . σε = ϑ ϑ Note, however, that (2.8) reduces to (2.9) for any weak solution of problem (2.1–2.8) as soon as this solution is smooth (see [10]). The small parameter ε can be interpreted as the Mach number. When ε → 0, the speed of sound becomes infinite, and, accordingly, the fluid can be considered as incompressible in the asymptotic limit ε → 0. Note that there are different scalings of the primitive system leading to the same mathematical problems, a typical example is provided by a long-time, small-velocity, small-viscosity setting (see Klein et al. [17]).

Study of Incompressible Limit for Navier – Stokes – Fourier System

77

2.2. Energetically insulating boundary conditions. In accordance with the total energy conservation imposed though (2.4), the system is supplemented with conservative boundary conditions u · n|∂ε = 0, [Sn] × n|∂ε = 0, q · n|∂ε = 0.

(2.10) (2.11)

The complete slip boundary condition (2.10) implies, in particular, that the boundary is acoustically hard (cf. Sect. 3 below). Note that the more conventional no-slip boundary condition u|∂ε = 0 results in a fast decay of acoustic waves in a generic class of bounded domains because of creation of a boundary layer effect (see Desjardins et al. [8]). On the other hand, our approach applies if (2.10) is replaced by a more general stipulation of Navier’s type u · n|∂ε = 0, β[u]tan + [S[ϑ, ∇x u]n]tan |∂ε = 0. 2.3. Ill-prepared initial data. The initial state of the system is determined by the following conditions: 1 1 , ϑ(0, ·) = ϑ0,ε = ϑ + εϑ0,ε , (0, ·) = 0,ε = + ε0,ε

where

, ϑ > 0,

(2.12)

ε

1 0,ε dx =

ε

1 ϑ0,ε dx = 0 for all ε > 0,

(2.13)

and 1 1 {0,ε }ε>0 , {ϑ0,ε }ε>0 are bounded in L 2 ∩ L ∞ ().

(2.14)

u(0, ·) = u0,ε ,

(2.15)

{u0,ε }ε>0 is bounded in L 2 ∩ L ∞ (; R3 ).

(2.16)

In addition

where

2.4. Incompressible limit. Let {ε , uε , ϑε }ε>0 be a family of weak solutions to the Navier-Stokes-Fourier system (2.1–2.8) supplemented with the boundary conditions (2.10), (2.11) and the initial condition (2.12), (2.15). The precise meaning of the concept of weak solution will become clear in Sect. 3 in the context of the acoustic equation. Here, we only point out that, in general, the entropy production σε may be interpreted as a non-negative measure satisfying (2.3), (2.8) in the sense of distributions (see also [10, Chap. 2]). Our main goal is to show strong (pointwise a.a.) convergence ⎧ ⎫ ⎨ ε → ⎬ a.a. in (0, T ) × , (2.17) ⎩ ⎭ ϑε → ϑ

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and uε → U a.a. in (0, T ) ×

(2.18)

at least for suitable subsequences. In other words, the convergence imposed on the initial data through (2.12–2.16) “propagates” in time. This is not surprising for ε , ϑε , but far less obvious for the velocity uε . The piece of information contained in (2.17), (2.18) is clearly sufficient to identify the limit problem represented by the incompressible Navier-Stokes system divx U = 0,

(∂t U + divx (U ⊗ U)) + ∇x = divx µ(ϑ) ∇x U + ∇xt U , supplemented with the conventional heat equation c p (∂t + divx (U)) − divx (κ(ϑ)∇x ) = 0, where c p > 0 denotes the specific heat at constant pressure, and stands for the (relative) temperature identified as a weak limit of (ϑε − ϑ)/ε. If, in addition, a driving force is imposed, more sophisticated models as the Oberbeck-Boussinesq approximation may be obtained (see [10, Chap. 5]). As we will see in Sect. 4, the pointwise convergence of the density and the temperature claimed in (2.17) follows easily from the uniform bounds established in Sect. 4 below. On the other hand, the strong convergence of the velocity (2.18) is far more delicate and intimately related to propagation and attenuation of acoustic waves. As a matter of fact, (2.18) is not expected to hold on bounded domains with acoustically hard boundary, where large amplitude rapidly oscillating waves are generated in the limit ε → 0 (see, for instance, Lions and Masmoudi [22], or Schochet [31] ). Accordingly, for (2.18) to hold it is necessary that the target domain be unbounded, more specifically, the two closely related properties must be satisfied: • the point spectrum of the associated wave operator must be empty; • the local acoustic energy decays in time (cf. Morawetz [28], Walker [36]). 3. Lighthill’s Acoustic Equation We begin by introducing a “time lifting” ε of the measure σε through formula ε ; ϕ = σε ; I [ϕ], where we have set ε ; ϕ = σε ; I [ϕ], I [ϕ](t, x) t ϕ(z, x) dz for any ϕ ∈ L 1 (0, T ; C(ε )). =

(3.1)

0

+ It is easy to check that ε can be identified with an abstract function ε ∈ L ∞ weak (0, T ; M (ε )), where

ε (τ ), ϕ = lim σε , ψδ ϕ, δ→0+

Study of Incompressible Limit for Navier – Stokes – Fourier System

with

ψδ (t) =

⎧ 0 ⎪ ⎪ ⎪ ⎨ 1

δ ⎪ ⎪ ⎪ ⎩ 1

79

for t ∈ [0, τ ), (t − τ ), for t ∈ (τ, τ + δ), for t ≥ τ + δ,

in particular, the measure ε is well-defined for any τ ∈ [0, T ), and the mapping τ → ε is non-increasing in the sense of measures. Here the subscript in L ∞ weak stands for “weakly measurable”. Following the idea of Lighthill [20], we rewrite the Navier-Stokes-Fourier system (2.1–2.3) in the form: ε∂t Z ε + divx Vε = εdivx Fε1 , A ε∂t Vε + ω∇x Z ε = ε divx F2ε + ∇x Fε3 + 2 ∇x ε , ε ω

(3.2) (3.3)

supplemented with the homogeneous Neumann boundary conditions Vε · n|∂ε = 0, where

s(ε , ϑε ) − s(, ϑ) ε − A A + ε ε , Vε = ε uε , Zε = + ε ω ε εω

s(ε , ϑε ) − s(, ϑ) A A κ∇x ϑε 1 Fε = ε , uε + ω ε ω εϑε F2ε = Sε − ε uε ⊗ uε ,

and

ε − Fε3 = ω ε2

+ Aε

s(ε , ϑε ) − s(, ϑ) p(ε , ϑε ) − p(, ϑ) − . ε2 ε2

(3.4)

(3.5) (3.6) (3.7)

(3.8)

Here the constants A and ω are chosen to eliminate the first order term in the (formal) asymptotic expansion of (3.8) in terms of the quantities (ε − )/ε, (ϑε − ϑ)/ε , namely A

∂ p(, ϑ) ∂s(, ϑ) ∂ p(, ϑ) ∂s(, ϑ) = , ω+ A = . ∂ϑ ∂ϑ ∂ ∂

(3.9)

In order to guarantee the wave speed ω to be strictly positive, we impose the hypothesis of thermodynamic stability in the form ∂ p(, ϑ) ∂e(, ϑ) > 0, > 0 for all , ϑ > 0 ∂ ∂ϑ

(3.10)

(see Bechtel et al. [3]). Indeed (3.10), together with Gibbs’ relation (2.5), yield ω > 0, in particular, Eqs. (3.2), (3.3) form a hyperbolic system provided the right-hand is considered as given. Relation (3.10) plays a crucial role in the uniform estimates presented in Sect. 4 below.

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System (3.2), (3.3) can be viewed as a variant of Lighthill’s acoustic analogy supplemented with acoustically hard boundary condition (3.4) (cf. Lighthill [19]). We assume that Eqs. (3.2), (3.3) as well as the boundary condition (3.4) are satisfied in a weak sense, more precisely, the integral identity

T

ε

0

[ε Z ε ∂t ϕ + Vε · ∇x ϕ] dx dt =

T 0

ε

εFε1 · ∇x ϕ dx dt

(3.11)

holds for any test function ϕ ∈ Cc∞ ((0, T ) × ε ), and T [εVε · ∂t ϕ + ωZ ε divx ϕ] dx dt 0

=

ε T

A ε ; divx ϕ εF2ε : ∇x ϕ + εFε3 divx ϕ dx dt + εω ε

0

(3.12)

is satisfied for any ϕ ∈ Cc∞ ((0, T ) × ε ; R3 ), ϕ · n|∂ε = 0,

where we have identified ε (τ )ψ dx =< ε (τ ); ψ >, ψ ∈ C(ε ). ε

4. Total Dissipation Balance, Uniform Bounds 4.1. Total dissipation balance. Besides the acoustic equation specified in (3.11), (3.12), any weak solution {ε , uε , ϑε } of the Navier-Stokes-Fourier system (2.1–2.4) satisfies the total dissipation balance

1 1 ε |uε |2 + 2 Hϑ (ε , ϑε )−∂ Hϑ (, ϑ)(ε −)− Hϑ (, ϑ) (τ, ·) dx ε ε 2 ϑ + 2 σε [0, τ ] × ε ε 1 1 0,ε |u0,ε |2 + 2 Hϑ (0,ε , ϑ0,ε )−∂ Hϑ (, ϑ)(0,ε −)− Hϑ (, ϑ) dx = ε ε 2 (4.1)

for a.a. τ ∈ [0, T ],

Study of Incompressible Limit for Navier – Stokes – Fourier System

81

where we have introduced the Helmholtz function Hϑ (, ϑ) = e(, ϑ) − ϑs(, ϑ) (see [10, Chap. 5.2.2]). As a direct consequence of Gibbs’ relation (2.5) and hypothesis of thermodynamic stability (3.10), the function Hϑ enjoys the following remarkable properties: • the function → Hϑ (, ϑ) is strictly convex; • the function ϑ → Hϑ (, ϑ) is decreasing for ϑ < ϑ and increasing for ϑ > ϑ for any fixed > 0. Moreover, we have ⎫

⎧ Hϑ (, ϑ) − ∂ Hϑ (, ϑ)( − ) − Hϑ (, ϑ) ≥ c | − |2 + |ϑ − ϑ|2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ whenever ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ /2 < < 2, ϑ/2 < ϑ < 2ϑ,

(4.2)

and

Hϑ (, ϑ) − ∂ Hϑ (, ϑ)( − ) − Hϑ (, ϑ) ≥ c e(, ϑ) + ϑ|s(, ϑ)| otherwise (4.3) (see [10, Chap. 5, Lemma 5.1]).

4.2. Uniform estimates. Total dissipation balance (4.1), together with the structural properties of the Helmholtz function Hϑ specified in (4.2), (4.3), can be used to deduce uniform bounds on the family of the weak solutions {ε , uε , ϑε }ε>0 . Indeed hypotheses (2.12–2.16) imply that the integral on the right-hand side of (4.1) is bounded, therefore the quantities on the left-hand side are bounded uniformly with respect to ε → 0. Furthermore, in order to exploit (4.3), certain technical assumptions must be imposed on the structural properties of the thermodynamic functions p, e, and s. Motivated by the existence theory developed in [10, Chap. 3], we assume that the pressure p is given through formula a p(, ϑ) = ϑ 5/2 P + ϑ 4 , a > 0, (4.4) ϑ 3/2 3 where P ∈ C 1 [0, ∞), P(0) = 0, P (Z ) > 0 for all Z ≥ 0,

(4.5)

and 0

0,

P(Z ) = p∞ > 0. Z 5/3

(4.6) (4.7)

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E. Feireisl

Moreover, in accordance with Gibbs’ relation (2.5), we suppose that e(, ϑ) =

ϑ4 3 5/2 ϑ P +a , 3/2 2 ϑ

(4.8)

and, finally, the transport coefficients satisfy 0 < c1 (1 + ϑ) ≤ µ(ϑ) ≤ c2 (1 + ϑ), 0 ≤ η(ϑ) ≤ c2 (1 + ϑ) 0 < c1 (1 + ϑ 3 ) ≤ κ(ϑ) ≤ c2 (1 + ϑ 3 )

(4.9) (4.10)

for all ϑ > 0 (see [10, Chaps. 1-3] for the physical background and further discussion concerning the structural hypotheses introduced above). Note that (4.6) is a direct consequence of the hypothesis of thermodynamic stability stated in (3.10). In view of (4.2), (4.3), it is convenient to introduce the essential and residual parts of a function h as h = [h]ess + [h]res , [h]ess = (ε , ϑε )h, [h]res = (1 − (ε , ϑε )) h, where ∈ Cc∞ (0, ∞)2 , 0 ≤ ≤ 1, ≡ 1 in an open neighborhood of the point [, ϑ]. The total dissipation balance established in (4.1), together with the structural properties of the Helmholtz function stated in (4.2), (4.3), and the restrictions imposed through hypotheses (4.4–4.10), give rise to uniform estimates on the quantities appearing in the acoustic equation (3.11), (3.12) independent of ε. Very roughly indeed, we may say that the essential components are bounded in the Lebesgue space L 2 , while the residual parts vanish in the asymptotic limit in the L 1 −norm. We state the resulting list of estimates referring to [10, Chap. 8.2] for the detailed proofs. To begin, we write Z ε = Z ε1 + Z ε2 , with ε − = + ε res ε − 2 Zε = + ε ess

Z ε1

s(ε , ϑε ) − s(, ϑ) A A ε , + ε ω ε εω res s(ε , ϑε ) − s(, ϑ) A ε ω ε ess

(cf. (3.5), where ess sup Z ε1 M(ε ) ≤ εc, ess sup Z ε2 L 2 (ε ) ≤ c. t∈(0,T )

t∈(0,T )

Similarly, Vε = Vε1 + Vε2 ,

(4.11)

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83

where ⎫ ⎧ 1 2 ⎨ ess supt∈(0,T ) Vε L 1 (ε ;R3 ) ≤ εc, ess supt∈(0,T ) Vε L 2 (ε ;R3 ) ≤ c, ⎬ ⎩

⎭

ess supt∈(0,T ) Vε1 L 5/4 (ε ;R3 ) ≤ c.

(4.12)

The constants in (4.11), (4.12) are independent of ε. The driving forces in (3.11), (3.12) admit similar bounds, namely Fε1 = Fε1,1 + Fε1,2 , where

T 0

Fε1,1 2L 1 ( ;R3 ) + Fε1,2 2L 2 ( ;R3 ) dt ≤ c, ε ε F2ε

=

F2,1 ε

(4.13)

+ F2,2 ε ,

where

T 0

2 1,2 2 F1,1 ε L 1 (ε ;R3×3 ) + Fε L 2 (ε ;R3×3 )

dt ≤ c

(4.14)

and, finally, Fε3 +

A ε2 ω

ε = Fε3,1

with ess sup Fε3,1 M(ε ) ≤ c. t∈(0,T )

(4.15)

5. Analysis of Acoustic Waves Having collected all the necessary preliminaries, we are in a position to formulate rigorously the main result of this paper. To simplify the forthcoming analysis, we assume that ⊂ R3 is an unbounded domain with compact boundary. Generalization to a larger class of spatial domains satisfying the so-called limiting absorption principle is straightforward. Theorem 5.1. Let ⊂ R3 be an unbounded domain with a compact boundary of class C 2+ν , ν > 0. Assume that the thermodynamic functions p, e, s satisfy Gibbs’ equation (2.5), together with the structural restrictions (4.4–4.8). In addition, let the transport coefficients µ, η, and κ obey (4.9), (4.10). Let {ε , uε , ϑε }ε>0 be a family of (weak) solutions to the acoustic equation (3.11), (3.12) in (0, T ) × ε satisfying the total dissipation balance (4.1), where • the initial data 0,ε , ϑ0,ε , u0,ε obey (2.12–2.16); • ε = ∩ Br (ε) , where lim εr (ε) = ∞.

ε→0

(5.1)

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Then, passing to a subsequence as the case may be, we have ε → , ϑε → ϑ in L 1 ((0, T ) × K ), and uε → U in L 1 ((0, T ) × K ; R3 )

(5.2)

for any compact K ⊂ . Remark 5.1. The hypotheses of Theorem 5.1 are obviously satisfied provided the trio {ε , uε , ϑε }ε>0 represents a weak solution of Navier-Stokes-Fourier system (2.1–2.11) in (0, T ) × ε in the sense specified in [10, Chap. 3]. The existence of such a solution for a large class of initial data including those specified in (2.12–2.16) was established in [10, Chap. 3, Theorem 3.1]. Remark 5.2. The balls Br (ε) in the definition of ε may be replaced by general bounded domains B˜ ε , namely ε = ∩ B˜ ε , with Br (ε) ⊂ B˜ ε . Hypothesis (5.1) means the distance to ∂ Br (ε) dominates the speed of sound proportional to 1/ε. In particular, the acoustic waves cannot reach the outer boundary ∂ Br (ε) and return to a fixed compact set K ⊂ within the time interval (0, T ). Remark 5.3. The convergence result stated in (5.2) is not optimal with respect to the space variable, where the velocity field enjoys higher regularity, however, the main issue in the proof of Theorem 5.1 is to eliminate fast oscillations of acoustic waves in time. The remaining part of this section is devoted to the proof of Theorem 5.1. Since ε − ϑε − ϑ ϑε − ϑ ε − ε − ϑε − ϑ = = + , + , ε ε ε ε ε ε ess res ess

res

where, by virtue of (4.1), (4.7), and (4.8), ϑ −ϑ ε − ε esssupt∈(0,T ) ≤ c, esssupt∈(0,T ) ε ε ess L 2 (ε )

≤ c,

ess L 2 (ε )

and ε − esssupt∈(0,T ) ε

res L 1 (ε )

ϑ −ϑ ε ≤ εc, esssupt∈(0,T ) ε

≤ εc,

res L 1 (ε )

the proof of Theorem 5.1 reduces to showing the strong convergence of the velocity field stated in (5.2). Moreover, we claim that for (5.2) to hold it is enough to show t → uε (t, ·) · w dx → t → U(t, ·) · w dx in L 1 (0, T ) (5.3)

Study of Incompressible Limit for Navier – Stokes – Fourier System

85

for any fixed w ∈ Cc∞ (K ; R3 ), K ⊂ a given ball. Indeed, by virtue of the dissipation balance (4.1) and Korn’s inequality, we get 0

T

uε 2W 1,2 ( ;R3 ) dt ≤ c ε

(see [10, Chap. 8.2.2] for details), in particular, extending uε outside ε we may infer that uε → U weakly in L 2 (0, T ; W 1,2 (; R3 )). As W 1,2 (; R3 ) is compactly imbedded into L 2 (K ) for any bounded K , it is easy to see that (5.3) yields (5.2). Finally, since [uε ]res → 0 in, say, L 1 ((0, T ) × K ), it is enough to show (5.3) with uε replaced by [uε ]ess , which is equivalent to t → Vε (t, ·) · w dx → t → V(t, ·) · w dx in L 1 (0, T )

(5.4)

for any fixed w ∈ Cc∞ (K ; R3 ), where Vε = ε uε appears in the acoustic equation (3.11), (3.12), and V = U. 5.1. Regularization. To begin, it is useful to observe that we may assume, without loss of generality, that all quantities appearing in system (3.11), (3.12) are smooth. To this end, we deduce from (3.11), (3.12), using the uniform bounds established in (4.11), (4.12), that Z ε ∈ Cweak−(∗) ([0, T ]; M(ε )), Vε ∈ Cweak ([0, T ]; L 5/4 (ε ; R3 )), in particular, the initial values Z ε (0, ·) = Z 0,ε ∈ M(ε ), V(0, ·) = V0,ε = 0,ε u0,ε ∈ L 2 (ε ; R3 ) are well defined. Moreover, in accordance with (4.11), (4.12), 1 2 Z 0,ε = Z 0,ε + Z 0,ε ,

where 1 2 Z 0,ε M() + Z 0,ε L 2 (ε ) + V0,ε L 2 (ε ;R3 ) + ≤ c.

(5.5)

i }δ>0 ⊂ Cc∞ (ε ), For a fixed ε > 0, there exist families of smooth functions {Z 0,ε,δ 3 ∞ i = 1, 2, {V0,ε,δ }δ>0 ⊂ Cc (ε ; R ), such that 1 2 {Z 0,ε,δ }ε,δ>0 is bounded in L 1 (), {Z 0,ε,δ }ε,δ>0 is bounded in L 2 (),

(5.6)

{V0,ε,δ }ε,δ>0 is bounded in L (; R ),

(5.7)

2

3

86

E. Feireisl

and, in addition, 1 1 2 2 Z 0,ε,δ ϕ dx → Z 0,ε ; ϕ, Z 0,ε,δ ϕ dx → Z 0,ε ϕ dx for any ϕ ∈ Cc∞ (ε ), ε V0,ε,δ · ϕ dx → V0,ε · ϕ dx for any ϕ ∈ Cc∞ (ε ; R3 ),

ε

as δ → 0. Similarly, we can find ⎧ 1 ⎫ 1,1 1,2 1,i Fε,δ = Fε,δ + Fε,δ , Fε,δ ∈ Cc∞ ((0, T ) × ε ; R3 ), i = 1, 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ 2,1 2,2 2,i 2 3×3 ∞ Fε,δ = Fε,δ + Fε,δ , Fε,δ ∈ Cc ((0, T ) × ε ; R ), i = 1, 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 3,1 Fε,δ ∈ Cc∞ ((0, T ) × ε ) such that 1 2 Fε,δ → Fε1 in L 2 (0, T ; L 1 (ε ; R3 )), Fε,δ → Fε2 in L 2 (0, T ; L 2 (ε ; R3 )),

F1ε,δ → F1ε

in L (0, T ; L (ε ; R 2

1

3×3

)),

F2ε,δ → F2ε

in L (0, T ; L (ε ; R 2

2

3×3

(5.8) )), (5.9)

and 3,1 sup Fε,δ L 1 (ε ) ≤ c,

t∈[0,T ]

T 0

ε

3,1 Fε,δ ϕ dx dt →

0

T

< Fε3,1 ; ϕ > dt

(5.10)

for any ϕ ∈ Cc∞ ([0, T ] × ε ), as δ → 0. Now, consider the (unique) solution Z ε,δ , Vε,δ of the initial-boundary value problem 1 ε∂t Z ε,δ + divx Vε,δ = εdivx Fε,δ in(0, T ) × ε ,

(5.11)

ε∂t Vε,δ + ω∇x Z ε,δ = in (0, T ) × ε , Vε,δ · n|∂ε = 0, Z ε,δ (0, ·) = Z 0,ε,δ , Vε,δ (0, ·) = V0,ε,δ .

(5.12) (5.13) (5.14)

εdivx F2ε,δ

3 + ε∇x Fε,δ

Keeping ε > 0 fixed and letting δ → 0, we easily check that

Vε − Vε,δ (t, ·) · w dx → 0 as δ → 0 ess sup t∈(0,T )

for any w ∈ Cc∞ (K ; R3 ) as in (5.4). Accordingly, it is enough to show (5.4) with Vε replaced by Vε,δ(ε) for δ(ε) small enough. In what follows, we drop the subscript δ and replace the weak formulation of the acoustic equation (3.11), (3.12) by its classical counterpart (5.11), (5.12), supplemented by (5.13), (5.14). The data appearing in (5.11–5.14) are smooth and satisfy the bounds established in (4.13–4.15) uniformly for ε → 0.

Study of Incompressible Limit for Navier – Stokes – Fourier System

87

5.2. Finite speed of propagation - extension to the set . System (5.11), (5.12) admits a √ finite speed of propagation of order ω/ε. This can be easily seen multiplying Eq. (5.11) by Z ε,δ , taking the scalar product of (5.12) with Vε,δ , and integrating the resulting expression over the set √ ω (t, x) t ∈ [0, τ ], x ∈ ε , |x| < r − t . ε Consequently, by virtue of hypothesis (5.1), we may assume, extending the data in (5.11–5.14) to be zero outside ε , that Z ε,δ , Vε,δ are smooth, compactly supported in the set [0, T ] × , and solve the acoustic equation (5.11–5.14) in (0, T ) × . Thus the resulting problem reads as follows: Show that the family

t →

Vε (t, ·) · w dx

is precompact in L 1 (0, T )

(5.15)

for any w ∈ Cc∞ (K ; R3 ), K ⊂ K ⊂ a bounded ball, provided that ε∂t Z ε + divx Vε = εdivx Fε1 in (0, T ) × , εdivx F2ε

in (0, T ) × , ε∂t Vε + ω∇x Z ε = Vε · n|∂ = 0, Z ε (0, ·) = Z 0,ε , Vε (0, ·) = V0,ε in ,

(5.16) (5.17) (5.18) (5.19)

where 1 2 Z 0,ε = Z 0,ε + Z 0,ε ,

i Z 0,ε ∈ Cc∞ (), i = 1, 2,

V0,ε ∈ Cc∞ (; R3 ), and Fε1 = Fε1,1 + Fε1,2 , Fε1,i ∈ Cc∞ ((0, T ) × ; R3 ), i = 1, 2,

2,2 2,i ∞ 3×3 ), i = 1, 2, F2ε = F2,1 ε + Fε , Fε ∈ C c ((0, T ) × ; R

with 1 2 {Z 0,ε }ε>0 bounded in L 1 (), {Z 0,ε }ε>0 bounded in L 2 (),

(5.20)

{V0,ε }ε>0 bounded in L (; R ), ⎫ ⎧ 1,1 ⎨ {Fε }ε>0 bounded in L 2 (0, T ; L 1 (; R3 )), ⎬

(5.21)

⎭ ⎩ 1,2 {Fε,0 }ε>0 bounded in L 2 (0, T ; L 2 (; R3 )), ⎫ ⎧ 2,1 ⎨ {Fε }ε>0 bounded in L 2 (0, T ; L 1 (; R3×3 )), ⎬

(5.22)

⎭

(5.23)

2

⎩

3

3×3 2 2 {F2,2 )). ε,0 }ε>0 bounded in L (0, T ; L (; R

88

E. Feireisl

5.3. Compactness of the solenoidal part. Consider ψ ∈ W 1,2 ∩W 1,∞ (; R3 ), divx ψ = 0, ψ · n|∂ = 0. Multiplying Eq. (5.17) on ψ and integrating by parts, we obtain d Vε · ψ dx = − F2ε : ∇x ψ dx, Vε (0, ·) · ψ dx = V0,ε · ψ dx, dt in particular the family t → Vε · ψ dx is precompact in C[0, T ].

In other words, introducing the Helmholtz decomposition v = H[v] + H⊥ [v], H⊥ = ∇x , where is the unique solution of the Neumann problem = divx v, ∇x · n|∂ = v · n|∂

(5.24)

such that ∇x ∈ L 2 (; R3 ), ∈ L 6 () whenever v ∈ L 2 (; R3 ), we may infer that t → Vε (t, ·) · H[ψ] dx is precompact in C[0, T ],

(5.25)

provided ψ ∈ Cc∞ (K ; R3 ) is the same as in (5.15). 5.4. Local decay of the gradient component. In light of the previous arguments, it is enough to show (5.15) for the gradient part H⊥ [Vε ], H⊥ [Vε ] = ∇x ε , where ε is uniquely determined through (5.24). Accordingly, problem (5.16–5.19) can be interpreted in terms of ε as follows: ε∂t Z ε + ε = εdivx Fε1 ,

(5.26)

ε∂t ε + ωZ ε = ∇x ε · n|∂ = 0, Z ε (0, ·) = Z 0,ε , ε (0, ·) = 0,ε = −1 N [div x V0,ε ].

(5.27) (5.28) (5.29)

2 ε−1 N [div x div x Fε ],

The symbol N denotes the Laplace operator on the domain endowed with the homogeneous Neumann boundary condition. More precisely, − N is a non-negative self-adjoint operator on the space L 2 () with a domain of definition D(− N ) = v ∈ L 2 () ∇x v ∈ W 1,2 (; R3 ), v ∈ L 2 (), 1,2 (− N )[v]w dx = ∇x v · ∇x w dx for any w ∈ W () = v ∈ W 2,2 () ∇x v · n|∂ = 0 .

Study of Incompressible Limit for Navier – Stokes – Fourier System

89

We denote by {Pλ }λ≥0 the spectral resolution associated to − N - a system of orthogonal projections in L 2 () such that ∞ (− N )[v]w dx = λ d Pλ [v]w dx for any v ∈ D(− N ), w ∈ L 2 ().

0

Using {Pλ }λ≥0 we can define G(− N ) for any (possibly complex valued) Borel function G through formula ∞ G(− N )[v]w dx = G(λ) d Pλ [v]w dx .

0

5.5. Homogeneous equation. To simplify notation, we will assume hereafter that ω = 1. Our goal is to express solutions of problem (5.26–5.29) by means of Duhamel’s formula. To this end, we examine first the associated homogeneous equation ∂t Z + = 0, ∂t + Z = 0 in (0, ∞) × ,

(5.30)

supplemented with the Neumann boundary condition ∇x · n|∂ = 0,

(5.31)

Z (0, ·) = Z 0 , (0, ·) = 0 in .

(5.32)

and the initial conditions √ The (unique) solution of (5.30–5.32) can be written in terms of − N as 1 i 0 + √ [Z 0 ] (t, ·) = exp it − N 2 2 − N 1 i 0 − √ [Z 0 ] , + exp −it − N 2 2 − N √ 1 d i − N Z0 − [0 ] Z (t, ·) = − (t, ·) = exp it − N dt 2 2 √ 1 i − N Z0 + [0 ] . + exp −it − N 2 2

(5.33)

(5.34)

In particular, problem (5.30–5.32) generates a group in the associated energy space (Z , ) ∈ L 2 () × H 1,2 (), where H 1,2 () denotes the homogeneous Sobolev space, H 1,2 () = {v | v ∈ L 6 (), ∇x v ∈ L 2 (; R3 )}. At this stage, we apply Theorem 1.1 taking • X = L√2 (), • H = − N , • A = ϕG(− N ), ϕ ∈ Cc∞ (), G ∈ Cc∞ (0, ∞), for a suitable non-negative function G.

90

E. Feireisl

Since A ◦ R H [λ] ◦ A∗ = ϕG(− N ) √

1 G(− N )ϕ, − N − λ

it is enough to verify hypothesis (1.3) of Theorem 1.1 for the values of the parameter λ belonging to a bounded set Q of the complex plane, namely λ ∈ Q = {z ∈ C | Re[z] ∈ [a, b], 0 < |Im[z]| < d}, where 0 < a < b < ∞, supp[G] ⊂ (a 2 , b2 ). and d > 0. Indeed if λ ∈ / Q, then 1 G(− N ) √ G(− N ) − N − λ is a bounded linear operator, with a norm bounded in terms of the parameters a, b, d. Thus we can rewrite A ◦ R H [λ] ◦ A∗ = ϕ with

M(− N , λ) ϕ, (− N ) − λ2

M(− N , λ) = G(− N )( − N + λ)G(− N )

– a bounded linear operator in L 2 () for λ ∈ Q. At this stage, we recall that the operator − N satisfies the limiting absorption principle, namely 1 sup V ◦ (− ) − µ ◦ V 2 2 ≤ c(α, β, ϕ) < ∞ (5.35) N µ∈C;α0 , {Hε2 }ε>0 bounded in L 2 (0, T ; L 2 (; C)). In accordance with relations (5.40–5.42), formula (5.39) can be recast in the form

t 1 1 N [h 1ε ]+ √ ε (t, ·) = exp ±i [h 2ε ] ± i N [h 3ε ]+ √ [h 4ε ] − N ε − N − N t

t −s exp ±i − N + ε 0

1 N [Hε1 ]+ √

− N

[Hε2 ] ± i

1 N [Hε3 ]+ √

− N

[Hε4 ]

ds,

(5.43) where {h iε }ε>0 is bounded in L 2 (), and {Hεi }ε>0 is bounded in L 2 ((0, T ) × ), i = 1, . . . , 4. (5.44)

Study of Incompressible Limit for Navier – Stokes – Fourier System

5.6.2. Convergence. Taking H (ξ ) =

1 ξ

93

in (5.38) we get

2 ϕG(− N ) exp ±i t − N [ N [h i ]] dt ε ε −∞ L 2 () ∞ 2 =ε dt ≤ εcG h iε L 2 () , i = 1, 3, ϕG(− N ) exp ±it − N [ N [h iε ]] 2 ∞

L ()

−∞

and, for H (ξ ) =

√

ξ,

2 ϕG(− N ) exp ±i t − N ( − N )−1 [h i ] dt ε ε −∞ L 2 () ∞ 2 =ε ϕG(− N ) exp ±it − N ( − N )−1 [h iε ] 2 ∞

L ()

−∞

dt

≤ εcG h iε L 2 () , i = 2, 4. Similarly, 2 ϕG(− N ) exp ±i t − s − N N H i ds dt ε ε 0 0 L 2 () ∞ T 2 ϕG(− N ) exp ±it − N exp i −s − N N H i ds dt ≤εc ε ε −∞ 0 L 2 () 2 T T −s i 2 i exp i H − ds=εc ds, i = 1, 3. ≤εcG Hε 2 N G ε L () ε 2

T

T

L ()

0

0

(5.45) Finally, 0

2 1 i ϕG(− N ) exp ±i t − s − N Hε ds dt √ ε − N 0 L 2 () ∞ T −s ≤ εc ϕG(− N ) exp ±it − N exp i ε − N −∞ 0 2 1 × √ Hεi ds dt 2 − N L () 2 T T −s i 2 i exp i H ≤εcG − ds = εc ds, i = 2, 4. Hε 2 N G ε L () ε 0 0 L 2 () (5.46)

T

T

Thus we conclude that ϕG(− N )[ε ] → 0 in L 2 ((0, T ) × ) as ε → 0 for any G ∈ Cc∞ (0, ∞) and ϕ ∈ Cc∞ ().

(5.47)

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In order to establish (5.4), we take ϕ ∈ Cc∞ () such that ϕ ≡ 1 on the set K containing the support of w and write ∇x ε · w dx = − ε divx w dx = − ϕε divx w dx =− ϕG(− N )[ε ]divx w dx+ ϕ(G(− N )−Id)[ε ]divx w dx,

where, in accordance with (5.47), the first integral on the right-hand side tends to zero for ε → 0 for any fixed G. On the other hand, we can take a family of functions G(λ) 1, in particular, (G(− N ) − Id)[h] → 0 for any fixed h ∈ L 2 (). Consequently, writing ϕ(G(− N ) − Id)[ε ]divx w dx = ϕ(G(− N ) − Id)[ε ]divx w dx ε (G(− N ) − Id)[divx w] dx,

we can deduce (5.4) from (5.43), (5.47) as soon as we observe that 1 N [divx w], √ [divx w] ∈ L 2 (). − N Thus we have proved (5.4), and therefore Theorem 5.1. References 1. Alazard, T.: Low Mach number flows and combustion. SIAM J. Math. Anal. 38(4), 1186–1213 (electronic) (2006) 2. Alazard, T.: Low Mach number limit of the full Navier-Stokes equations. Arch. Rat. Mech. Anal. 180, 1–73 (2006) 3. Bechtel, S.E., Rooney, F.J., Forest, M.G.: Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids. J. Appl. Mech. 72, 299–300 (2005) 4. Burq, N.: Global Strichartz estimates for nontrapping geometries: about an article by H. F. Smith and C. D. Sogge: “Global Strichartz estimates for nontrapping perturbations of the Laplacian”. Comm. Part. Diff. Eqs. 28(9–10), 1675–1683 (2003) 5. Burq, N., Planchon, F., Stalker, J.G., Tahvildar-Zadeh, A.S.: Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J. 53(6), 1665–1680 (2004) 6. Dermejian, Y., Guillot, J.-C.: Théorie spectrale de la propagation des ondes acoustiques dans un milieu stratifié perturbé. J. Diff. Eqs. 62, 357–409 (1986) 7. Desjardins, B., Grenier, E.: Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455(1986), 2271–2279 (1999) 8. Desjardins, B., Grenier, E., Lions, P.-L., Masmoudi, N.: Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 461–471 (1999) 9. Engquist, B., Majda, A.: Radiation boundary conditions for acoustic and elastic wave calculations. Comm. Pure Appl. Math. 32(3), 314–358 (1979) 10. Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Basel: Birkhäuser-Verlag, 2009 11. Feireisl, E., Poul, L.: On compactness of the velicity field in the incompressible limit of the full NavierStokes-Fourier system on large domains. Math. Meth. Appl. Sci. 32, 1269–1286 (2009) 12. Isozaki, H.: Singular limits for the compressible Euler equation in an exterior domain. J. Reine Angew. Math. 381, 1–36 (1987)

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13. Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279, (1965/1966) 14. Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34, 481–524 (1981) 15. Klein, R.: Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. Z. Angw. Math. Mech. 80, 765–777 (2000) 16. Klein, R.: Multiple spatial scales in engineering and atmospheric low Mach number flows. ESAIM: Math. Mod. Numer. Anal. 39, 537–559 (2005) 17. Klein, R., Botta, N., Schneider, T., Munz, C.D., Roller, S., Meister, A., Hoffmann, L., Sonar, T.: Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Engrg. Math. 39, 261–343 (2001) 18. Leis, R.: Initial-boundary Value Problems in Mathematical Physics. Stuttgart: B. G. Teubner, 1986 19. Lighthill, J.: On sound generated aerodynamically I. General theory. Proc. of the Royal Society of London A 211, 564–587 (1952) 20. Lighthill, J.: Waves in Fluids. Cambridge: Cambridge University Press, 1978 21. Lions, P.-L.: Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models. Oxford: Oxford Science Publication, 1998 22. Lions, P.-L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998) 23. Lions, P.-L., Masmoudi, N.: Une approche locale de la limite incompressible. C.R. Acad. Sci. Paris Sér. I Math. 329(5), 387–392 (1999) 24. Masmoudi, N.: Asymptotic problems and compressible and incompressible limits. In: Advances in Mathematical Fluid Mechanics, edited by Málek, J., Neˇcas, J., Rokyta, M., Berlin: Springer-Verlag, 2000, pp. 119–158 25. Masmoudi, N.: Examples of singular limits in hydrodynamics. In: Handbook of Differential Equations, III, Dafermos, C., Feireisl, E., eds., Amsterdam: Elsevier, 2006 26. Masmoudi, N.: Rigorous derivation of the anelastic approximation. J. Math. Pures Appl. 88, 230– 240 (2007) 27. Metcalfe, J.L.: Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle. Trans. Amer. Math. Soc. 356(12), 4839–4855 (electronic) (2004) 28. Morawetz, C.S.: Decay for solutions of the exterior problem for the wave equation. Comm. Pure Appl. Math. 28, 229–264 (1975) 29. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1978 30. Schochet, S.: Fast singular limits of hyperbolic PDE’s. J. Diff. Eqs. 114, 476–512 (1994) 31. Schochet, S.: The mathematical theory of low Mach number flows. M2 AN Math. Model Numer. Anal. 39, 441–458 (2005) 32. Shimizu, S.: The limiting absorption principle. Math. Meth. Appl. Sci. 19, 187–215 (1996) 33. Smith, H.F., Sogge, C.D.: Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Part. Diff. Eqs. 25(11–12), 2171–2183 (2000) 34. Vaigant, V.A.: An example of the nonexistence with respect to time of the global solutions of NavierStokes equations for a compressible viscous barotropic fluid (in Russian). Dokl. Akad. Nauk 339(2), 155– 156 (1994) 35. Va˘ınberg, B.R.: Asimptoticheskie metody v uravneniyakh matematicheskoi fiziki. Moscow: Moskov. Gos. Univ., 1982 36. Walker, H.F.: Some remarks on the local energy decay of solutions of the initial-boundary value problem for the wave equation in unbounded domains. J. Diff. Eqs. 23(3), 459–471 (1977) 37. Wilcox, C.H.: Sound Propagation in Stratified Fluids. Appl. Math. Ser. 50. Berlin: Springer-Verlag, 1984 Communicated by P. Constantin

Commun. Math. Phys. 294, 97–119 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0917-y

Communications in

Mathematical Physics

On q -Deformed gl+1 -Whittaker Function I Anton Gerasimov1,2,3 , Dimitri Lebedev1 , Sergey Oblezin1 1 Institute for Theoretical and Experimental Physics,

117259, Moscow, Russia. E-mail: [email protected]; [email protected]

2 School of Mathematics, Trinity College, Dublin 2, Ireland.

E-mail: [email protected]

3 Hamilton Mathematics Institute, TCD, Dublin 2, Ireland

Received: 4 February 2009 / Accepted: 25 June 2009 Published online: 26 September 2009 – © Springer-Verlag 2009

Abstract: We propose a new explicit form of q-deformed Whittaker functions solving q-deformed gl+1 -Toda chains. In the limit q → 1 the constructed solutions reduce to the classical gl+1 -Whittaker functions of class one in the form proposed by Givental. An important property of the proposed expression for the q-deformed gl+1 -Whittaker function is that it can be represented as a character of C∗ × GL+1 . This provides a q-version of the Shintani-Casselman-Shalika formula for the p-adic Whittaker function. The Shintani-Casselman-Shalika formula is recovered in the limit q → 0 when the q-deformed Whittaker function is reduced to a character of a finite-dimensional representation of gl+1 expressed through the Gelfand-Zetlin basis. Introduction Whittaker functions corresponding to semisimple finite-dimensional Lie algebras arise in various parts of modern mathematics. In particular, these functions appear in representation theory as matrix elements of infinite-dimensional representations, in the theory of quantum integrable systems as a common eigenfunction of Toda chain quantum Hamiltonians, in string theory as generating functions of correlators in Type A topological string theory on flag manifolds, and in number theory in the description of local Archimedean L-factors corresponding to automorphic representations. Although much studied, Whittaker functions seem to have some deep properties that are not yet fully revealed. In this paper we study the q-deformed gl+1 -Whittaker functions. The q-deformed Whittaker functions can be identified with the common eigenfunctions of a set of commuting q-deformed Toda chain Hamiltonians. This q-deformed Toda chain (also known as the relativistic Toda chain [Ru]) was discussed in terms of representation theory of quantum groups in [Se1,Et,Se2] and an integral representation for the q-deformed gl+1 -Whittaker function was constructed in [KLS]. Recently the q-deformed Toda chain attracted a special interest due to its connection with quantum K -theory of flag manifolds

98

A. Gerasimov, D. Lebedev, S. Oblezin

[GiL]. In this paper we pursue another direction. Our principal motivation to study q-deformed Whittaker functions is that in this more general setting, some important hidden properties of classical Whittaker functions become visible. The main result of the paper is given by Theorem 2.1 where a new expression for the q-deformed gl+1 -Whittaker function (for q < 1) is introduced. As a simple corollary of Theorem 2.1, the q-deformed gl+1 -Whittaker function can be represented as a character of C∗ × GL+1 . In the limit q → 1 this leads to a similar representation of the classical gl+1 -Whittaker function. This representation is not easy to perceive looking directly at the classical Whittaker functions. The importance of this representation of a (q-deformed) gl+1 -Whittaker function becomes obvious if we notice that in the limit q → 0 the constructed q-deformed Whittaker function reduces to the p-adic Whittaker function. In this limit, the representation as a character reduces to the well-known Shintani-Casselman-Shalika representation of the p-adic G L +1 -Whittaker function as a character of a finite-dimensional representation of G L +1 [Sh,CS]. Thus, the representation of a (q-deformed) gl+1 -Whittaker function as a character can be considered as a q-version of the Shintani-Casselman-Shalika representation. For instance, the constructed q-deformed Whittaker function vanishes outside a dominant weight cone of gl+1 similarly to the Shintani-Casselman-Shalika p-adic Whittaker function. We expect that the representation of the classical Whittaker function as a character should provide important insights into the arithmetic geometry at an infinite place of Spec(Z). Let us also remark that taking into account the results of [CS] one should expect that in the case of an arbitrary semisimple Lie algebra g, q-deformed g-Whittaker function should be given by a character of C∗ × L G(C), where Lie( L G) = L g is the Langlands dual Lie algebra. It is worth mentioning that the q → 1 limit of the explicit expression of the q-deformed Whittaker function proposed in this paper reduces to the integral representations for classical Whittaker functions introduced by Givental [Gi,GKLO]. We consider this as a sign of an “arithmetic nature” of this integral representation. On the other hand, the explicit solution has an obvious relation with the Gelfand-Zetlin parametrization of finite-dimensional representations of gl+1 (and precisely reproduces to the Gelfand-Zetlin formula for characters of finite-dimensional representations in the limit q → 0). The duality of Gelfand-Zetlin and Givental representations was already noticed in [GLO]. Let us comment on our approach to the derivation of explicit expressions for q-deformed Whittaker functions. It is known [Et] that in a certain limit defining difference equations for Macdonald polynomials are transformed into the eigenfunction equations for the q-deformed Toda chain. This is a simple generalization of the Inozemtsev limit [I], which transforms the Calogero-Sutherland integrable model into the standard Toda chain. The other ingredient we use is a recursive construction of Macdonald polynomials (analogous to the recursive construction for (q-deformed) Toda chain eigenfunctions [KL1,KLS]). We combine these results to obtain a recursive expression for the q-deformed gl+1 -Whittaker functions. The explicit form of the q-deformed Whittaker function implies various interesting interpretations. These include connections with representation theory (via characters of Demazure modules), geometry of quiver varieties, quantum cohomology of flag manifolds and will be discussed elsewhere [GLO2]. Finally, note that eigenfunctions of the q-deformed Toda chain were discussed previously (e.g. [KLS,GKL1,BF and FFJMM]). The relation of these constructions with the one proposed in this paper is an interesting question which deserves further considerations.

On q-Deformed gl+1 -Whittaker Function I

99

The paper is organized as follows. In Sect. 1 we recall a system of mutually commuting Macdonald-Ruijsenaars difference operators and a recursive construction of their common eigenfunctions. In Sect. 2 we derive a recursive expression for solutions of the q-deformed gl+1 -Toda chain. In Sect. 3 various limiting cases elucidating the construction of the q-deformed gl+1 -Whittaker function are discussed. In Sect. 4, details of the proof of Theorem 2.1 are given. 1. Macdonald-Ruijsenaars Difference Operators In this section we recall some relevant facts from the theory of Macdonald polynomials (see e.g. [Mac,Kir,AOS]). Consider symmetric polynomials in variables (x1 , . . . , x+1 ) over the field Q(q, t) of rational functions in q, t. Given a partition = (0 ≤ 1 ≤ 2 ≤ · · · ≤ +1 ), denote by the same symbol the Young diagram containing + 1 rows with k boxes in the k th row, and the upper row having the maximal length +1 . Let m and π be the following two bases in the space of symmetric polynomials indexed by partitions : m =

σ ∈S+1

+1 1 2 xσ(1) xσ(2) · . . . · xσ(+1) ,

π = π1 π2 · . . . · π+1 ,

πn =

+1

xkn ,

k=1

where S+1 is the permutation group. Define a scalar product , q,t on the space of symmetric functions over Q(q, t) as follows: π , π q,t = δ, · z (q, t), where z (q, t) =

nmn m n ! ·

n≥1

N 1 − q k , 1 − t k

m n = |{k| k = n}|.

k=1

In the following we always assume that q, t ∈ R>0 , 0 < q < 1. The following remarkable theorem was proved by Macdonald [Mac]. gl

gl

Theorem 1.1 (Macdonald). There is a unique basis P +1 = P +1 (x; q, t) in the ring of symmetric polynomial function over Q(q, t) such that gl

P +1 = m +

u m ,

p2,2 ,

is a common eigenfunction of commuting Hamiltonians gl

gl

H1 2 = T1 + (1 − q p2,2 − p2,1 +1 )T2 ,

H2 2 = T1 T2 .

Note that the formula (2.5) can be easily rewritten in the recursive form. Corollary 2.1. The following recursive relation holds gl λ +1 ( p +1 )

=

p,i ∈P+1,

( p ) q

λ+1

+1 i=1

p+1,i − i=1 p,i

gl

×Q+1, ( p +1 , p |q)λ ( p ),

(2.6)

where Q+1, ( p +1 , p |q) =

1 i=1

,

( p,i − p+1,i )q ! ( p+1,i+1 − p,i )q !

and ( p ) =

−1

( p,i+1 − p,i )q !,

i=1

where the notations λ = (λ1 , . . . , λ+1 ), λ = (λ1 , . . . , λ ) are used.

(2.7)

On q-Deformed gl+1 -Whittaker Function I

105

Remark 2.1. The solution (2.5) is a q-analog of Givental’s integral representation of gl+1 -Whittaker function [Gi,GKLO]: gl ψλ +1 (x1 , . . . , x+1 )

=

R

gl

i=1

gl Q gl+1 (t +1 , t |λ+1 )

gl

dt,i Q gl+1 (t +1 , t |λ+1 )λ1,...,λ (t ),

= exp ıλ+1

+1

(2.8)

t+1,i

i=1

−

t,i

t −t t,i −t+1,i+1 +1,i ,i , e +e −

i=1

i=1

where λ = (λ1 , . . . , λ+1 ), t k = (tk1 , . . . , tkk ), xi := t+1,i , i = 1, . . . , + 1 and we gl assume that Q gl1 (t11 |λ1 ) = eıλ1 t1,1 . 0

Proposition 2.1. There exists a C∗ × G L +1 (C) module V such that the common eigenfunction (2.5) of the q-deformed Toda chain allows the following representation for p+1,1 ≤ p+1,2 ≤ . . . p+1,+1 : gl+1

λ

( p +1 ) = Tr V q L 0

+1

q λi Hi ,

(2.9)

i=1

where Hi := E i,i , i = 1, . . . , + 1 are Cartan generators of gl+1 = Lie(G L +1 ) and L 0 is a generator of Lie(C∗ ). Proof. It is useful to rewrite (2.5) in the following form: gl+1

λ

( p +1 ) = ( p +1 )−1

q λk+1 (

i

pk+1,i − j pk,i )

pk,i ∈P (+1) k=1

k pk+1, i+1 − pk+1, i × , pk, i − pk+1, i q i=1

( p +1 ) = where

( p+1, j+1 − p+1, j )q !,

j=1

(n)q ! n . = k q (n − k)q ! (k)q !

Now taking into account the identities n n−1 n−1 = + qk , k q k q k−1 q

∞

1 = q kn , n (1 − q ) k=0

106

A. Gerasimov, D. Lebedev, S. Oblezin gl

one obtains an expansion of the function λ +1 ( p +1 ) into the sum of terms q N q m 1 λ1 · · · q m +1 λ+1 with positive integral coefficients K N ,m 1 ,...,m +1 . Let G L 1 ×···×G L 1 ⊂ G L +1 be the diagonal subgroup, then let us define the following C∗ ×G L 1 ×···×G L 1 -module:

V =

VN ,m 1 ,···,m +1 ,

VN ,m 1 ,···,m +1 = C K N ,m 1 ,...,m +1 .

N , m 1 ,...,m +1

Each VN ,m 1 ,···m +1 is acted on by the factor C∗ via multiplication by q N ; the torus (G L 1 )+1 acts on each VN ,m 1 ,···m +1 by multiplication by q m 1 λ1 +...+m +1 λ+1 . Let us note that the q-Toda chain eigenfunction problem (2.2) contains the variables z i := q λi only due to the eigenvalues given by the central functions χr (z) (2.3). Since the initial conditions for the eigenfunction (2.5) can also be expressed through χr (z), therefore we can extend the structure group of the module V from the torus C∗ × (G L 1 )+1 to the whole group C∗ × G L +1 . Thus we obtain the representation (2.9) and complete the proof.

Remark 2.2. There exists a finite-dimensional C∗ × G L +1 (C) module V f such that the following representation holds for p+1,1 ≤ p+1,2 ≤ . . . p+1,+1 : gl+1 ( p ) = ( p ) gl+1 ( p ) = Tr V f q L 0 λ λ +1 +1 +1

+1

q λi Hi .

(2.10)

i=1

The module V entering (2.9) and the module V f entering (2.10) have a structure of modules under the action of (quantum) affine Lie algebras which will be discussed [GLO2]. See however Proposition 3.4 for an explicit description of V f .

3. Various Limits Besides the limit q → 1, which recovers the classical gl+1 -Whittaker function as a solution of the gl+1 -Toda chain, there are other interesting limits elucidating the meaning of the q-deformed Toda chain equations. In a limit q → 0, the q-deformed gl+1 -Whittaker functions are given by the characters of irreducible representations of gl+1 . This will allow us to identify the Whittaker functions with p-adic Whittaker functions according to Shintani-Casselman-Shalika formula [Sh,CS]. There is also another q → 1 limit which clarifies the recursive structure of q-deformed gl+1 -Whittaker functions. 3.1. A limit q → 0. In this subsection we discuss a limit q → 0 of the constructed q-deformed Whittaker function (we restrict the Whittaker function to the domain { p+1,1 ≤ . . . ≤ p+1,+1 }, where it is non-trivial). We will show that in the domain { p+1,1 ≤ . . . ≤ p+1,+1 } the system of equations for common eigenfunctions of q-deformed Toda chain Hamiltonians reduces to the Pieri formulas (a particular case of Littlewood-Richardson rules) for the decomposition of the tensor product of an arbitrary finite-dimensional representation and a fundamental representation of gl+1 .

On q-Deformed gl+1 -Whittaker Function I

107

Let us rewrite the q-deformed Whittaker function (2.5) using the variables z i = q λ+1,i ,

gl+1

( p +1 |z) =

+1

pk,i ∈P (+1)

k=1

(

k

zk

i=1

k−1 pk,i − i=1 pk−1,i )

k−1 ( pk,i+1 − pk,i )q !

×

k=2 i=1 k

,

( pk,i − pk+1,i )q ! ( pk+1,i+1 − pk,i )q !

k=1 i=1

p+1,1 ≤ · · · ≤ p+1,+1 ,

(3.1)

where z = (z 1 , . . . , z +1 ). Proposition 3.1. 1. In the limit q → 0, the eigenfunction (3.1) is given in the domain p+1,1 ≤ · · · ≤ p+1,+1 by gl

χ p +1 (z) := gl+1 ( p +1 |z)|q=0 = +1

+1

(

k

zk

i=1

k−1 pk,i − i=1 pk−1,i )

.

(3.2)

pk,i ∈P +1 k=1

gl

2. The functions χ p +1 (z) satisfy the following set of difference equations: +1

gl+1

χr gl+1

where χr

gl

(z) χ p +1 (z) = +1

Ir

gl

χ p +1+Ir (z), r = 1, . . . , + 1,

(3.3)

+1

(z) are the characters of fundamental representations Vωr = gl χr +1 (z) = z i 1 · · · z ir , r = 1, . . . , + 1,

r

C+1 :

Ir

and Ir = (i 1 < i 2 < · · · < ir ) ⊆ {1, 2, . . . , + 1}. gl 3. The functions χ p +1 (z) can be identified with characters of irreducible finite+1 dimensional representations of G L +1 corresponding to partitions p+1,1 ≤ · · · ≤ p+1,+1 . Proof. The relations (3.2) and (3.3) follow directly from the similar relations for generic q. To prove the last statement note that (3.2) can be identified with the expression for characters of irreducible finite-dimensional representations of G L +1 obtained using the Gelfand-Zetlin bases (see e.g. [ZS]). Let { pi j }, i = 1, . . . , + 1, j = 1, . . . , i be a Gelfand-Zetlin (GZ) pattern P (+1) , that is the integers pi, j should satisfy the conditions pi+1, j ≤ pi, j ≤ pi+1, j+1 . An irreducible finite-dimensional representation can be realized in a vector space with the basis v p parametrized by GZ patterns { pi j } with fixed p+1,i . The action of the Cartan generators on v p is given by s+1 −s +1,+1 z 1E 11 z 2E 22 · · · z +1 v p = z 1s1 z 2s2 −s1 · · · z +1 vp, E

sk =

k i=1

pki .

(3.4)

108

A. Gerasimov, D. Lebedev, S. Oblezin

Thus, we have for the character gl

+1 χ p+1,1 ,..., p+1,+1 (z 1 , . . . , z +1 ) =

pk,i

+1

∈P (+1)

k=1

(

k

zk

i=1

k−1 pk,i − i=1 pk−1,i )

.

(3.5)

Remark 3.1. The second identity in Proposition 3.1 is known as the Pieri formula (see e.g. [FH], Appendix A). Thus, the q-deformed Toda chain equations can be considered as q-deformations of the Pieri formulas. There is a generalization of q-Toda chain equations providing a q-version of a general Littlewood-Richardson rule. The GZ representation of the characters (3.2) has an obvious recursive structure. Namely, one should introduce variables z i = q λi , i = 1, . . . , + 1 and then take the limit q → 0 in (2.6). This leads to the following. Corollary 3.1. Characters satisfy the following recursive relation: gl

+1 χ p+1,1 ,..., p+1,+1 (z 1 , . . . , z +1 ) +1 p − p gl z +1i=1 +1,i i=1 ,i χ p,1 ,..., p, (z 1 , . . . , z ), =

(3.6)

p,i ∈P+1,

where the sum runs over p = ( p,1 , . . . , p, ) satisfying the GZ conditions p+1,i ≤ p,i ≤ p+1,i+1 . Note that these recursive relations can be derived using the classical Cauchy-Littlewood formula C+1,m+1 (x, y) =

+1 m+1

gl 1 gl = χ +1 (x) χ m+1 (y), 1 − xi y j

m ≤ ,

(3.7)

i=1 j=1

gl

gl

where the sum runs over Young diagrams of glm+1 and χ +1 (x) = χ +1 (x1 , . . . , x+1 ) are the characters of the irreducible finite-dimensional representation of G L +1 corresponding to Young diagram . gl

Proposition 3.2. The following integral relations for the characters χ +1 (x) hold: gl χ +1 (x)

gl χ +1 (x)

=

y1 =∞

···

=

y1 =∞

···

ı dyi gl C+1, (x, y −1 )χ (y) (y|0, 0)), (3.8) 2π yi

y+1 =∞ i=1

+1 ı dyi gl C+1,+1 (x, y −1 )χ +1 (y) (y|0, 0), 2π yi y+1 =∞ i=1

⎛ gl+1 ⎝ χ+(+1) k (x) =

+1 j=1

⎞ gl

x kj ⎠ χ +1 (x).

On q-Deformed gl+1 -Whittaker Function I

109

The relations above provide the character of the irreducible finite-dimensional representation of G L +1 corresponding to any Young diagram . Remark 3.2. The relations (3.8) can be obtained from similar relations for Macdonald polynomials in the limit t → 0, q → 0. These recursion relations are analogs of the Mellin-Barnes recursion relations for the classical Whittaker functions (see [KL1,GKL, GLO] for details). According to Shintani-Casselman-Shalika formula, the p-adic Whittaker function corresponding to an algebraic reductive group G is equal to the character of the Langlands dual Lie group L G 0 acting in an irreducible finite-dimensional representation [Sh,CS]. Thus according to Proposition 3.2 we can consider gl+1 -Whittaker functions at q → 0 as an incarnation of p-adic Whittaker functions (this is in complete agreement with the results of [GLO3]). Moreover, taking into account Proposition 2.1 one can consider the main result of this paper as a generalization of the Shintani-CasselmanShalika formula to the q-deformed case which includes a limiting case of classical gl+1 Whittaker functions. This interpretation of classical Whittaker functions evidently deserves further attention. 3.2. The limit q → 1. In this subsection we consider a modified limit q → 1 leading to a very simple degeneration of the q-deformed Toda chain. In this limit the q-deformed Toda chain can be easily solved. Moreover the form of the solution makes the recursive expressions (2.6) for the q-deformed Toda chain solution very natural. Let us redefine the q-deformed Toda chain Hamiltonians and their common eigenfunctions as follows (we assume p+1,1 ≤ · · · ≤ p+1,+1 below): gl+1

Jr

gl+1

= ( p +1 ) Hr

gl+1 ( p |z) = ( p ) · gl+1 ( p |z), +1 +1 +1

( p +1 )−1 ,

( p +1 ) =

( p+1, j+1 − p+1, j )q !,

j=1

where gl+1 ( p +1 |z) is given by (3.1). Explicitly, we have that 1−δi −i , 1 1−δir −i , 1 1−δi −i , 1 gl X i1 2 1 · · · · · Ti1 · · · · · Tir , (3.9) Jr +1 = X ir −1 r −1 · X ir r +1 r Ir

X i = 1 − q p+1,i+1 − p+1, i . Let us now take the limit where we assume ir +1 = + 2 and q → 1: gl+1 ( p |z) = lim gl+1 ( p |z), ψ +1 +1 q→1

gl+1

hr

gl +1

= lim Jr q→1

.

We have limq→1 (1 − q n ) = 0 and, therefore, we obtain from (2.1) that gl+1

hr

= T+2−r · · · · · T+1 ,

where Ti acts on the functions of p+1, j as follows: p +1 ), p+1,k = p+1,k + δk,i , i, k = 1, . . . , + 1. Ti f ( p +1 ) = f ( Now the eigenvalue problem is easily solved.

110

A. Gerasimov, D. Lebedev, S. Oblezin

Proposition 3.3. 1. The function gl+1 ( p |z) ψ +1

p+1, 1 p+1, i+1 − p+1, i gl gl+1 = χ+1+1 (z) χ+1−i (z) ,

r = 1, . . . , + 1, (3.10)

i=1

is an eigenfunction of the family of mutually commuting difference operators gl+1 gl+1

hr gl+1

where χr

ψ

gl+1

( p +1 |z) = χr

gl+1 ( p |z), (z) ψ +1

(z) is the character of the fundamental representation Vωr = gl+1

χr

(z) =

z i1 · · · z ir , z i = q λi ,

(3.11) r

C+1 :

i = 1, . . . , + 1,

Ir

Ir = {i 1 < i 2 < · · · < ir } ⊂ {1, 2, . . . , + 1} and gl+1

hr

= T+2−r · · · · · T+1 , r = 1, . . . , + 1.

2. In the domain p+1,1 ≤ · · · ≤ p+1,+1 , the following recursive relation holds: gl+1 ( p |z) = ψ +1

+1

p,i ∈P+1,

z +1i=1

p+1,i − i=1 p,i

p+1, i+1 − p+1, i gl ( p |z ), ·ψ × p, i − p+1, i

(3.12)

i=1

where z = (z 1 , . . . , z ). Proof. The identity (3.11) follows from the construction. Let us prove that (3.12) follows from (3.11). Using the relation gl+1

χr

gl

gl

(z) = χr (z ) + z +1 · χr −1 (z ),

r = 1, . . . , + 1,

we have

gl

χ+1+1 (z)

p+1, 1 p+1, i+1 − p+1, i gl+1 χ+1−i (z) i=1

=

p+1,1 gl z +1 χ (z )

p+1,i+1

i=1

p,i = p+1,i

gl

z +1 χ−i (z )

p,i − p+1,i p gl +1,i+1 − p+1, i · χ+1−i (z ) · p, i − p+1, i

p+1,i+1 − p,i

On q-Deformed gl+1 -Whittaker Function I

=

p,i ∈P+1,

z +1i

111

p+1,i+1 − i p,i

p+1,i+1 − p+1, i p, i − p+1, i i=1

p, 1 −1 p, i+1 − p, i gl gl · χ (z ) χ−i (z ) i=1

=

p,i ∈P+1,

z +1i

p+1,i+1 − i p,i

p+1,i+1 − p+1, i gl ( p ). ψ λ p, i − p+1, i

(3.13)

i=1

This completes the proof. Remark 3.3. The functions

gl+1 ( p |z), ψ gl+1 ( p +1 |z) = −1 ( p +1 )ψ +1 satisfy the following recursive relations: p − ψ gl+1 ( p +1 |z) = z +1i +1,i i

p,i

p,i ∈P+1,

×

−1 i=1

( p,i

( p,i+1 − p,i )! ψ gl ( p |z ). − p+1,i )! ( p+1,i+1 − p,i )!

(3.14)

This makes the formula (2.6) for the solution of q-deformed Toda chain slightly less mysterious. Proposition 3.4. The following representation holds: gl+1 ( p |z) = Tr V f ψ +1

+1

q λi Hi ,

(3.15)

i=1

where ⊗( p+1,+1 − p+1, )

V f = Vω1

⊗( p+1,2 − p+1,1 )

⊗ · · · ⊗ Vω

⊗p

⊗ Vω+1+1,1 ,

(3.16)

and Vωn = ∧n C are the fundamental representations of G L +1 . Proof. This is an obvious consequence of Proposition 3.3.

The module V f entering (2.10) is isomorphic to (3.16) as the G L +1 -module but has a more refined structure under the action of quantum affine Lie algebras and will be discussed in [GLO2]. 4. Proof of Theorem 2.1 In this section we provide a proof of Theorem 2.1. To derive an explicit expression (2.5) for q-deformed gl+1 -Whittaker function we take as a motivation the limit t → ∞ of recursive relations (1.4) for Macdonald polynomials. Note that the recursion relations in Proposition 1.1 were defined for 0 < t < 1 and thus taking the limit t → ∞ needs some care. In particular we define the analog of the pairing (1.2) from scratch. We start with some useful relations that will be used in the proof of Theorem 2.1.

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4.1. q-deformed Toda chain from Macdonald-Ruijsenaars system. In this subsection we demonstrate that quantum Hamiltonians of the q-deformed Toda chain arise as a limit of Macdonald-Ruijsenaars operators when t → ∞. Let us take t = q −k . Proposition 4.1. The following relations hold: gl+1

Hr

gl

= lim Hr,k+1 k→∞ 1−δi , 1 1−δir −ir −1 , 1 1−δi −i , 1 X i 1 1 · X i 2 2 1 · · · · · X ir Ti1 · · · · · Tir , = Ir

where gl

gl+1

Hr,k+1 = D(x) Hr

(xi q k i ) D(x)−1 ,

D(x) =

+1

−k(+1−i)

xi

,

i=1

and the sum is taken over all subsets Ir = {i 1 < i 2 < · · · < ir } ⊂ {1, 2, . . . , + 1}. We −1 take X i = 1 − xi xi−1 , i = 2, . . . , + 1 , with X 1 = 1 and Ti x j = q δi, j x j Ti . Proof. Make a change of variables xi : xi −→ xi t −i , i = 1, . . . , + 1. Then for any i and any Ir , containing i we have: ⎛

⎞ ⎛ t xi − x j xi − x j t i−1− j xi − xi−1 ⎝ ⎠ −→ ⎝t br,i × i− j x − xj xi − x j t xi t −1 − xi−1 j ∈I / r i j>i ⎞ xi t j+1−i − x j ⎠, × xi t j−i − x j

(4.1)

j i}|. Making a substitution t = q −k and conjugating the where br,i = |{ j ∈ gl+1 Hamiltonians Hr by D(x) =

+1

−ki

xi

,

i=1

ki , leads to the multiplication of each term (4.1) in the sum (1.1) by i∈Ir q i := + 1 − i. Taking into account that for any i and for any subset Ir containing i one has i∈Ir i − br,i = r (r − 1)/2, we obtain in the limit k → ∞, gl+1

Hr

=

1−δi 1 , 1

X i1

1−δi 2 −i 1 , 1

· X i2

1−δir −ir −1 , 1

· · · · · X ir

Ir −1 , i = 2, . . . , + 1 and X 1 = 1 where X i = 1 − xi xi−1

Ti1 · · · · · Tir ,

On q-Deformed gl+1 -Whittaker Function I

113

4.2. Recursive kernel Q +1, (x, y|q) for the q-deformed Whittaker function. In this subsection by taking an appropriate limit of the Cauchy-Littlewood kernel for Macdonald polynomials we derive its analog for q-deformed Whittaker functions and verify the intertwining relations with q-deformed Toda chain Hamiltonians. Let t = q −k , i = + 1 − i. Given the Cauchy-Littlewood kernel C+1, (x, y|q, t) (1.3), define a new kernel by −ki −k Q +1, (x, y|q) = lim (xi y+1−i ) · Rk (q) · C+1, (x, y|q, q ) , (4.2) k→∞

i=1

where Rk (q) =

k

−q a j , (1 − q j )2

j=1

aj =

( + 1) 2

j+

−1 k 3

.

Proposition 4.2. The following explicit expression for Q +1, (x, y|q) defined by (4.2) holds: Q +1, (x, y|q) =

∞ ∞ 1 − (xi yi )−1 q n 1 − xi+1 yi q −1 q n · . 1 − qn 1 − qn i=1 n=1

(4.3)

i=1 n=1

Proof. Making the substitution xi → xi t −i , yi → yi t i in C+1, (x, y|q, t) and taking t = q −k we have C+1, (x, y|q, t) =

∞ 1 − xi yi q n−k 1 − xi+1 yi q n 1 − xi yi q n 1 − xi+1 yi q n+k

n=0 i=1

×

+1 i−2 1 − xi y j q n+(i− j−1)k 1 − x+2−i y+1− j q n+( j−i)k . 1 − xi y j q n+(i− j)k 1 − x+2−i y+1− j q n+( j+1−i)k i=3 j=1

One encounters four types of factors which can be rewritten as k ∞ k 1 − x yq n−k −j k = (1 − x yq ) = (x y) (−q − j ) 1 − (x y)−1 q j , n 1 − x yq

n=0

j=1

j=1

∞ k−1 k 1 − x yq n n −1 j 1 − x yq , = (1 − x yq ) = q 1 − x yq n+k

n=0

n=0

j=1

∞ 2k 2k 1 − x yq n−(m+1)k −j k −j −1 j , = (1 − x yq ) = (x y) (−q ) 1 − (x y) q 1 − x yq n−mk

n=0

j=k+1

∞ n=0

j=k+1

2k 1 − x yq n+mk −1 j 1 − x yq . = q 1 − x yq n+(m+1)k j=k+1

Now it is easy to take the limit k → ∞ and obtain (4.3)

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Let us introduce a set of slightly modified mutually commuting Hamiltonians 1−δi −i , 1 1−δir −i , 1 1−δi −i , 1 rgl (y) = H Yi1 2 1 · · · · · Yir −1 r −1 · Yir r +1 r Ti1 · · · · · Tir , (4.4) Ir

where −1 Yi (y) = 1 − yi yi+1 ,

1 ≤ i < ,

Y = 1.

We assume here Ir = (i 1 < i 2 < · · · < ir ) ⊂ {1, 2, . . . , } and we set ir +1 = + 1. Proposition 4.3. The following intertwining relations hold: gl gl (y) + H gl (y) Q +1, (x, y|q), Hk +1 (x) Q +1, (x, y|q) = H k−1 k gl+1

where k = 1, . . . , + 1. Here Q +1, (x, y|q) , Hk (4.3), (2.1) and (4.4) respectively.

(4.5)

gl (y) are defined by (x) and H k

Proof. Direct calculation similar to the one used in the proof of Proposition 4.1.

Let us introduce a function Q+1, ( p +1 , p |q) on the lattice Z+1 × Z as follows: Q+1, ( p +1 , p |q) = Q +1, (q p+1,i +i−1 , q − p,i −i+1 |q). Corollary 4.1. The following explicit expression for Q+1, ( p +1 , p |q) holds:

Q+1, ( p +1 , p |q) = i=1

( p,i − p+1,i )( p+1,i+1 − p,i )

i=1 ( p,i

− p+1,i )q ! ( p+1,i+1 − p,i )q !

,

where (n) = 1 when n ≥ 0 and (n) = 0 otherwise. Proposition 4.4. For any k = 1, . . . , + 1 the following intertwining relations hold: gl+1

Hk

( p +1 )Q+1, ( p +1 , p |q) gl (− p ) + H gl (− p ) Q+1, ( p , p |q). = H k−1 k +1

Proof. Follows from Proposition 4.3.

(4.6)

4.3. Pairing. Define the pairing: f, gq =

ı dyi (y) f (y −1 )g(y), 2π yi 0

(4.7)

i=1

where (y) =

−1 i=1

∞ ∞

n=1 (1 − q

n)

−1 n n=0 (1 − yi+1 yi q )

,

f (y −1 ) := f (y1−1 , . . . , y−1 ).

(4.8)

The integration domain 0 is such that each yi goes around yi = ∞ and is in the region defined by inequalities |yi+1 yi−1 | ≤ q, i = 1, . . . , .

On q-Deformed gl+1 -Whittaker Function I

115

gl rgl (y) are adjoint with respect to the Proposition 4.5. Hamiltonians Hr (y) and H pairing (4.7): gl gl f, gq , f, Hk gq = H k

k = 1, . . . , .

Proof. Let us adopt the following notations: gl rgl (y) = A Ir (y) TIr , H B Ir (y) TIr , Hr (y) = Ir

Ir

where TIr := Ti1 Ti2 · . . . · Tir . One should prove

dyi (y) f (y −1 ) TIr · TI−1 A Ir (y)TIr g(y) r 0 i=1 2πı yi Ir ⎛ ⎞ −1 T (y)T dyi I r I · TI−1 = (y) ⎝ TI−1 A Ir (y)TIr · r f (y −1 )⎠ g(y). r r 2πı yi (y) 0 i=1

Ir

Let us first prove the following lemma: Lemma 4.1. For any Ir = (i 1 < i 2 < · · · < ir ) ⊂ {1, 2, . . . , } the following relation holds: ∗ B Ir (y) = Ir (y) · TI−1 A (y)T , I I r r r where Ir (y) = ((y))−1 TI−1 (y) TIr , r for all i ∈ Ir . Here for a function f (y) we define f ∗ (y) := f (y −1 ). Proof. By direct calculation one derives 1−δi −i , 1 r k k−1 −1 ∗ −1 yi k −1 1−q , (TIr A Ir (y)TIr ) = yik k=1

and yik 1−δik+1 −ik , 1 1 − r yik +1 ∗Ir (y) = 1−δi −i , 1 , k k−1 k=1 1 − q −1 yi k −1 yik where we set i 0 := 0, ir +1 := + 1. In this way we obtain 1−δi 2 −i 1 , 1

(TI−1 A Ir (y)TIr )∗ · ∗Ir (y) = Yi1 r

1−δir −ir −1 , 1

· · · · · Yir −1

1−δir +1 −ir , 1

· Yir

.

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Using the lemma one arrives at the following identity

dyi (y) f (y −1 ) TIr · TI−1 A Ir (y)TIr g(y) r 2πı y i 0 i=1

=

Ir

Ir

∗ dyi (y) B Ir (y)Tr f (y −1 )g(y), 2πı y i 0 (Ir ) i=1

where the integration domains 0 (Ir ) differ from the original integration domain 0 by multiplying the variables yi , i ∈ Ir by q in the definition of 0 . By the lemma the poles of the integrand are inside the circles |yi+1 yi−1 | < 1, and therefore the contours 0 (Ir ) can be deformed to 0 without encountering poles of the integrand. This proves (4.9) and, therefore, the proposition.

To construct recursive formulas for q-deformed Whittaker functions one should introduce a pairing on functions defined on the lattice {yi = q p,i +i−1 ; i = 1, . . . , ; p,i ∈ Z } with appropriate decay at infinities. Let us define the following analog of (4.7): f, glat = ( p ) f (− p )g( p ), (4.9) p ∈Z

where ( p ) =

−1

( p,i+1 − p,i ) ( p,i+1 − p,i )q !.

(4.10)

i=1

Thus ( p ) provides an extension of the measure ( p ) defined in the dominant domain p ∈ P+1, (see (2.7)) to the lattice Z . Let us note that the variables

yi = q p,i +i−1 for p ∈ P+1, satisfy conditions |yi+1 yi−1 | ≤ q entering the definition of the domain of integration 0 . The following proposition can be easily proved by mimicking the proof of Proposition 4.5.

gl rgl ( p ) are adjoint with respect to the Proposition 4.6. Hamiltonians Hr ( p ) and H pairing (4.9), gl gl f, glat , f, Hk glat = H k

k = 1, . . . , .

(4.11)

4.4. Proof of Theorem 2.1. Now we are ready to prove Theorem 2.1. We use recursion gl over the rank of glk . Set λ11 ( p11 ) = q λ1 p11 and assume that gl

gl

gl

gl

Hr ( p ) · λ1,...,λ ( p ) = χr (q i λi Ei,i ) λ1,...,λ ( p ), gl χr (q i λi Ei,i ) = z i 1 z i 2 · · · · · z ir , z i = q λi .

(4.12)

()

Ir

Here E i, j are the standard generators of gl , Ir() = (i 1 < i 2 < · · · < ir ) ⊂ {1, 2, . . . , } gl and χr (g) are characters of fundamental representations Vr = r C of gl .

On q-Deformed gl+1 -Whittaker Function I

117

gl

Let us define the function λ1+1 ,...,λ+1 ( p +1 ) as follows: gl

λ1+1 ,...,λ+1 ( p +1 ) =

p

∈Z

·q λ+1 (

( p ) Q+1, ( p +1 , p ) +1

p+1,i − i=1 p,i )

i=1

gl

λ1,...,λ ( p ),

(4.13)

where Q+1, ( p +1 , p ) =

( p,i − p+1,i ) ( p+1,i+1 − p,i ) , ( p,i − p+1,i )q ! ( p+1,i+1 − p,i )q ! i=1

and ( p ) =

−1

( p,i+1 − p,i ) ( p,i+1 − p,i )q !.

i=1

One should verify the relations: gl+1

Hr

gl

gl

gl

+1 ( p +1 ) · λ1+1 (q i λi Ei,i ) λ1+1 ,...,λ+1 ( p +1 ) = χr ,...,λ+1 ( p +1 ), gl z i 1 z i 2 · · · · · z ir , z i = q λi , χr +1 (q i λi Ei,i ) =

(4.14)

(+1)

Ir (+1)

where Ir = {i 1 < i 2 < . . . < ir } ⊂ (1, 2, . . . , + 1). gl Applying Hamiltonians Hr +1 ( p +1 ) to (4.13) and using the intertwining relation given in Proposition 4.4, one obtains

gl+1

( p )Q+1, ( p +1 , p ) q λ+1 ( i p+1,i − k p,k ) +1 gl (− p ) + H rgl (− p ) Q+1, ( p , p ) q λ+1 ( i = q λ+1 H r −1 +1

Hr

p+1,i − k p,k )

Now using (4.11), one obtains gl+1

gl

( p +1 )λ1+1 ,...,λ+1 ( p +1 ) gl = ( p ) Hr +1 ( p +1 )Q+1, ( p +1 , p |q)

Hr

p ∈Z

× q λ+1 (

p+1,i − k p,k )

gl

λ1,...,λ ( p ) gl (− p ) + H rgl (− p ) = ( p ) q λ+1 H r −1 p ∈Z

× Q+1, ( p +1 , p ) q λ+1 (

i

i

p+1,i − k p,k )

gl

λ1,...,λ ( p )

.

118

A. Gerasimov, D. Lebedev, S. Oblezin

=

p ∈Z

( p ) Q+1, ( p +1 , p ) q λ+1 (

i

p+1,i − k p,k )

gl gl gl × q λ+1 Hr −1 ( p ) + Hr ( p ) λ1,...,λ ( p ) ⎞ ⎛ ⎟ gl ⎜ = ⎝q λ+1 q λi + q λi ⎠ λ1+1 ,...,λ+1 ( p +1 ) ()

()

Ir −1 i∈Ir −1

⎛ ⎜ =⎝ (+1)

Ir

(+1)

()

⎞

Ir

()

i∈Ir

⎟ gl q λi ⎠ λ1+1 ,...,λ+1 ( p +1 ),

i∈Ir

()

(+1)

= {i 1 < i 2 < · · · < ir } ⊂ where Ir = {i 1 < i 2 < · · · < ir } ⊂ (1, 2, . . . , ) and Ir (1, 2, . . . , + 1). In the last equality above we use the following relation: gl+1

χr

gl

gl

(z) = z +1 χr −1 (z ) + χr (z ),

where z = (z 1 , z 2 , . . . , z ) for z i = q λi . This completes the proof of Theorem 2.1.

Acknowledgements. The research of AG was partly supported by SFI Research Frontier Programme and Marie Curie RTN Forces Universe from EU. The research of SO is partially supported by RF President Grant MK-134.2007.1.

References [AOS] [BF] [CS] [Ch] [CK] [Et] [EK] [I] [FH] [FFJMM] [GKL] [GKL1] [GKLO] [GLO]

Awata, H., Odake, S., Shiraishi, J.: Integral representations of the Macdonald symmetric functions. Commun. Math. Phys. 179, 647–666 (1996) Braverman, A., Finkelberg, M.: Finite-difference quantum Toda lattice via equivariant K-theory. Trans. Groups 10, 363–386 (2005) Casselman, W., Shalika, J.: The unramified principal series of p-adic groups II. The Whittaker Function. Comp. Math. 41, 207–231 (1980) Cherednik, I.V.: Quantum groups as hidden symmetries of classic representation theory. In Differential Geometric Methods in Theoretical Physics (Chester,1988), Teaneck, NJ: World Sci. Publishing, 1989, pp. 47–54 Cheung, P., Kac, V.: Quantum Calculus. Berlin-Heidelberg-New York: Springer, 2001 Etingof, P.: Whittaker functions on quantum groups and q-deformed Toda operators. Amer. Math. Soc. Transl. Ser.2, Vol. 194, Providence, RI: Amer. Math. Soc., 1999, pp. 9–25 Etingof, P.I., Kirillov, A.A. Jr..: Macdonald’s polynomials and representations of quantum groups. Math. Res. Let. 1, 279–296 (1994) Inozemtsev, V.I.: Finite Toda lattice. Commun. Math. Phys. 121, 629–638 (1989) Fulton, W., Harris, J.: Representation Theory. A First Course. Berlin-Heidelberg-New York: Springer, 1991 3 subspaces and quantum Feigin, B., Feigin, E., Jimbo, M., Miwa, T., Mukhin, E.: Principal sl Toda hamiltonians. http://arxiv.org/abs/0707.1635v2[math.QA], 2007 Gerasimov, A., Kharchev, S., Lebedev, D.: Representation Theory and Quantum Inverse Scattering Method: Open Toda Chain and Hyperbolic Sutherland Model, Int. Math. Res. Notes, No.17, 823–854 (2004) Gerasimov, A., Kharchev, S., Lebedev, D.: Representation Theory and Quantum Integrability. Progress in Math. 237, Basel: Birkhäuser, 2005, pp. 133–156 Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin, S.: On a Gauss-Givental representation of quantum Toda chain wave function. Int. Math. Res. Notices, ArticleID 96489, 23 pages, 2006 Gerasimov, A., Lebedev, D., Oblezin, S.: Baxter operator and archimedean Hecke algebra. Commun. Math. Phys. 284(3), 867–896 (2008)

On q-Deformed gl+1 -Whittaker Function I [GLO1] [GLO2] [GLO3] [GK] [Gi] [GiL] [Kir] [KL1] [KLS] [Mac] [Ru] [Se1] [Se2] [Sh] [ZS]

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Gerasimov, A., Lebedev, D., Oblezin, S.: New Integral Representations of Whittaker Functions for Classical Lie Groups. http://arxiv.org/abs/0705.2886v1[math.RT], 2007 Gerasimov, A., Lebedev, D., Oblezin, S.: On q-deformed gl+1 -Whittaker functions II. Commun. Math. Phys., to apper (this issue), 2009, http://arxiv.org/abs/0805.3754v2[math.RT], 2008 Gerasimov, A., Lebedev, D., Oblezin, S.: Baxter q-operators and their arithmetic implications. Lett. Math. Phys. 88(1–3), 3–30 (2009) Givental, A., Kim, B.: Quantum cohomology of flag manifolds and Toda lattices. Commun. Math. Phys. 168, 609–641 (1995) Givental, A.: Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture. In: Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser., 2 Vol. 180, Providence, RI: Amer.Math.Soc., 1997, pp. 103–115 Givental, A., Lee, Y.-P.: Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups. Invent. Math. 151, 193–219 (2003) Kirillov, A. Jr.: Traces of intertwining operators and Macdonald’s polynomials. PhD Thesis, Yale University, May 1995, available at http://arxiv.org/abs/q-alg/9503012v1, 1995 Kharchev, S., Lebedev, D.: Eigenfunctions of G L(n, r ) Toda chain: the Mellin-Barnes representation. JETP Lett. 71, 235–238 (2000) Kharchev, S., Lebedev, D., Semenov-Tian-Shansky, M.: Unitary representations of u q (sl(2, r )), the modular double and the multiparticle q-deformed Toda chains. Commun. Math. Phys. 225, 573–609 (2002) Macdonald, I.G.: A New Class of Symmetric Functions. Publ. I.R.M.A. Strasbourg, 372/S-20, Actes 20e Séminaire Lotharingien de Combinatoire, 131–171 (1988) Ruijsenaars, S.: The relativistic Toda systems. Commun. Math. Phys. 133, 217–247 (1990) Sevostyanov, A.: Regular nilpotent elements and quantum groups. Commun. Math. Phys. 204, 1–16 (1999) Sevostyanov, A.: Quantum deformation of Whittaker modules and Toda lattice. Duke Math. J. 105, 211–238 (2000) Shintani, T.: On an explicit formula for class 1 Whittaker functions on G L n over p-adic fields. Proc. Japan Acad. 52, 180–182 (1976) Zhelobenko, D., Shtern, A.: Representations of Lie Groups. Moscow: Nauka, 1983

Communicated by Y. Kawahigashi

Commun. Math. Phys. 294, 121–143 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0919-9

Communications in

Mathematical Physics

On q -Deformed gl+1 -Whittaker Function II Anton Gerasimov1,2,3 , Dimitri Lebedev1 , Sergey Oblezin1 1 Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia.

E-mail: [email protected]; [email protected]

2 School of Mathematics, Trinity College Dublin, Dublin 2, Ireland.

E-mail: [email protected]

3 Hamilton Mathematical Institute TCD, Dublin 2, Ireland

Received: 4 February 2009 / Accepted: 25 June 2009 Published online: 17 September 2009 – © Springer-Verlag 2009

Abstract: A representation of a specialization of a q-deformed class one lattice gl+1 -Whittaker function in terms of cohomology groups of line bundles on the space QMd (P ) of quasi-maps P1 → P of degree d is proposed. For = 1, this provides an interpretation of the non-specialized q-deformed gl2 -Whittaker function in terms of QMd (P1 ). In particular the (q-version of the) Mellin-Barnes representation of the gl2 -Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of -function as a topological genus in semi-infinite geometry. A relation with the Givental-Lee universal solution (J -function) of q-deformed gl2 -Toda chain is also discussed.

Introduction In our work [GLO1] (which is the first in the series of papers [GLO1,GLO2]) we have proposed an explicit representation of a q-deformed class one lattice gl+1 -Whittaker function defined as a common eigenfunction of a complete set of commuting quantum Hamiltonians of a q-deformed gl+1 -Toda chain. Here “class one” means that the Whittaker function is non-zero only in the dominant domain. The case = 1 was discussed previously in [GLO3] (for related results in this direction see [KLS, GiL,GKL1,BF,FFJMM]). A special feature of the proposed representation is that the gl q-deformed class one gl+1 -Whittaker function z +1 ( p) with z = (z 1 , . . . , z +1 ) and p = ( p1 , . . . , p+1 ) ∈ Z+1 , is given by a character of a C∗ × G L +1 (C)-module V p . The expression in terms of a character can be considered as a q-version of ShintaniCasselman-Shalika representation of class one p-adic Whittaker functions [Sh,CS]. Indeed our representation of a q-deformed gl+1 -Whittaker function reduces, in a certain

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limit, to the Shintani-Casselman-Shalika representation of a p-adic Whittaker function. Note that the representation of a q-deformed Whittaker function as a character is a q-analog of the Givental integral representation [Gi2,GKLO] of the classical gl+1 Whittaker function. The main objective of this paper is to better understand the representation of the q-deformed gl+1 -Whittaker function as a character. Below we consider a specialization of the q-deformed Whittaker function given by the trace over a certain C∗ × G L +1 (C)module Vn,k (in the case = 1 there is actually no specialization). Our main result is presented in Theorem 3.1. We provide a description of the C∗ × G L +1 (C)-module Vn,k as a zero degree cohomology group of a line bundle on an algebraic version LP+ of a semi-infinite cycle LP + in a universal covering LP of the space of loops in P . We define LP+ as an appropriate limit d → ∞ of the space QMd (P ) of degree d quasi-maps of P1 to P [Gi1,CJS]. In particular for = 1 this provides a description of the q-deformed gl2 -Whittaker function in terms of cohomology of line bundles over LP1+ . A universal solution of the q-deformed gl+1 -Toda chain [GiL] was given in terms of cohomology groups of line bundles over QMd (X ), X = G/B for finite d. We demonstrate how our interpretation of the q-deformed gl+1 -Whittaker function is reconciled with the results of [GiL]. Using Theorem 3.1, we interpret a q-version of the Mellin-Barnes integral representation of the specialized q-Whittaker function as an instance of the Riemann-RochHirzebruch theorem in a semi-infinite setting. The corresponding Todd class is expressed in terms of a q-version of the -function. Analogously, the classical -function appears in a description of the fundamental class of semi-infinite homology theory and enters the Mellin-Barnes integral representation of the classical Whittaker function. Let us stress that the C∗ × G L +1 (C)-module Vn,k arising in the description of the q-deformed gl+1 -Whittaker function is not irreducible. It would be natural to look for an interpretation of Vn,k as an irreducible module of a quantum affine Lie group. A relation of the geometry of semi-infinite flags to representation theory of affine Lie algebras was ∞ proposed in [FF]. The semi-infinite flag space is defined as X 2 = G(K)/H (O)N (K), where K = C((t)), O = C[[t]], B = H N is a Borel subgroup of G, N is its unipotent radical and H is the associated Cartan subgroup. The semi-infinite flag spaces are not easy to deal with. An interesting approach to the semi-infinite geometry was proposed by Drinfeld. He introduced a space of quasi-maps QMd (P1 , G/B) that should be con∞ sidered as a finite-dimensional substitute for the semi-infinite flag space X 2 (see e.g. [FM,FFM,Bra]). Thus, taking into account the constructions proposed in this paper one can expect that (q-deformed) gl+1 -Whittaker functions (encoding the Gromov-Witten invariants and their K -theory generalizations) can be expressed in terms of representation theory of affine Lie algebras (see [GiL] for a related conjecture and [FFJMM] for recent progress in this direction). The paper [GLO2] establishes a connection of our results to the representation theory of (quantum) affine Lie groups. The paper is organized as follows. In Sect. 1, explicit solutions of the q-deformed gl+1 -Toda chain (q-versions of Whittaker functions) are recalled. In Sect. 2, we derive integral expressions for the counting of holomorphic sections of line bundles on the space of quasi-maps. In Sect. 3 we derive a representation of the specialized q-Whittaker functions in terms of cohomology of holomorphic line bundles on the space of quasi-maps of P1 to P . We propose an interpretation of the q-Whittaker functions as semi-infinite periods. In Sect. 4 the analogous interpretation of the classical Whittaker functions is discussed. In Sect. 5, we clarify the connection of our interpretation of the q-deformed gl+1 -Whittaker function with the results of [GiL].

On q-Deformed gl+1 -Whittaker Function II

123

1. q-Deformed gl+1 -Whittaker Function In this section we recall a construction [GLO1] of the q-deformed gl+1 -Whittaker funcgl tion z +1 ( p +1 ) defined on the lattice p +1 = ( p+1,1 , . . . , p+1,+1 ) ∈ Z+1 . We will consider only class one Whittaker functions, i.e. Whittaker functions satisfying the condition gl+1

z

( p +1 ) = 0

(1.1)

outside the dominant domain p+1,1 ≥ · · · ≥ p+1,+1 . The q-deformed gl+1 -Whittaker functions are common eigenfunctions of q-deformed gl+1 -Toda chain Hamiltonians: 1−δi −i , 1 1−δir −i , 1 1−δi −i , 1 gl Hr +1 ( p +1 ) = X i1 2 1 · . . . · Ti1 · . . . · Tir , X ir −1 r −1 · X ir r +1 r Ir

(1.2) where the sum is over ordered subsets Ir = {i 1 < i 2 < . . . < ir } ⊂ {1, 2, · · · , + 1} and we assume ir +1 = + 2. In (1.2) we use the following notations Ti f ( p +1 ) = f ( p +1 ), p+1,k = p+1,k + δk,i , i, k = 1, . . . , + 1, X i = 1 − q p+1,i − p+1,i+1 +1 , i = 1, . . . , , and X +1 = 1. We assume q ∈ R, 0 < q < 1. For example, the first nontrivial Hamiltonian has the following form: gl

H1 +1 ( p +1 ) =

(1 − q p+1,i − p+1,i+1 +1 )Ti + T+1 .

(1.3)

i=1

The main result of [GLO1] is a construction of common eigenfunctions of quantum Hamiltonians (1.2): gl gl+1 gl+1 Hr +1 ( p +1 )z 1 ,...,z ( p ) = ( z i ) z 1 ,...,z (1.4) +1 +1 ( p +1 ). +1 Ir i∈Ir

Denote by P (+1) ⊂ Z(+1)/2 a subset of integers pn,i , n = 1, . . . , + 1, i = 1, . . . , n satisfying the Gelfand-Zetlin conditions pk+1,i ≥ pk,i ≥ pk+1,i+1 for k = 1, . . . , . In the following we use the standard notation (n)q ! = (1 − q)...(1 − q n ). gl

+1 Theorem 1.1. Let z 1 ,...,z +1 ( p +1 ) be a function given in the dominant domain p+1,1 ≥ . . . ≥ p+1,+1 by

gl

+1 z 1 ,...,z +1 ( p +1 ) =

+1

zk

i

pk,i − i pk−1,i

pk,i ∈P (+1) k=1 k−1 ( pk,i − pk,i+1 )q !

×

k=2 i=1 k k=1 i=1

( pk+1,i − pk,i )q ! ( pk,i − pk+1,i+1 )q !

,

(1.5)

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A. Gerasimov, D. Lebedev, S. Oblezin gl

+1 and zero otherwise. Then, z 1 ,...,z +1 ( p +1 ) is a common solution of the eigenvalue problem (1.4).

Formula (1.5) can be written also in the recursive form. Corollary 1.1. Let P+1, be a set of p = ( p,1 , . . . , p, ) satisfying the conditions p+1,i ≥ p,i ≥ p+1,i+1 . The following recursive relation holds: gl

+1 z 1 ,...,z +1 ( p +1 ) =

( p ) z +1i

p+1,i − i p,i

p ∈P+1,

gl

Q +1, ( p +1 , p |q)z 1 ,...,z ( p ),

(1.6) where 1

Q +1, ( p +1 , p |q) =

i=1 −1

( p ) =

,

( p+1,i − p,i )q ! ( p,i − p+1,i+1 )q ! (1.7)

( p,i − p,i+1 )q !.

i=1

Remark 1.1. The representation (1.6) is a q-analog of Givental’s integral representation of the classical gl+1 -Whittaker function [Gi2,JK]: gl+1

ψλ

(x1 , . . . , x+1 ) =

R i=1

gl

gl

dt,i Q gl+1 (t +1 , t |λ+1 )ψλ1,...,λ (t ),

(1.8)

gl

Q gl+1 (t +1 , t |λ+1 )

+1 t −t

t,i −t+1,i+1 +1,i ,i = exp ıλ+1 e , t+1,i − t,i − +e i=1

i=1

i=1

where λ = (λ1 , . . . , λ+1 ), t k = (tk1 , . . . , tkk ), xi := t+1,i , i = 1, . . . , + 1 and gl z i = q γi , λi = γi log q and we assume that Q gl1 (t1,1 |λ1 ) = eıλ1 t1,1 . For the represen0 tation theory derivation of this integral representation of gl+1 -Whittaker function see [GKLO]. The representation (1.5) of the q-Whittaker function turns into representation (1.8) of the classical Whittaker function in the appropriate limit. As an example consider g = gl2 . Let p1 := p2,1 ∈ Z, p2 := p2,2 ∈ Z and p := p1,1 ∈ Z. Then the function

gl

2 z 1 ,z 2 ( p1 , p2 ) =

p2 ≤ p≤ p1

gl

2 z 1 ,z 2 ( p1 , p2 ) = 0,

p p +p −p

z1 z2 1 2 , ( p1 − p)q !( p − p2 )q !

p1 < p2 ,

p1 ≥ p2 , (1.9)

On q-Deformed gl+1 -Whittaker Function II

125

is a solution of the system of equations: gl2 gl2 (1 − q p1 − p2 +1 )T1 + T2 z 1 ,z 2 ( p1 , p2 ) = (z 1 + z 2 ) z 1 ,z 2 ( p1 , p2 ), gl

gl

(1.10)

2 2 T1 T2 z 1 ,z 2 ( p1 , p2 ) = z 1 z 2 z 1 ,z 2 ( p1 , p2 ).

Let us consider the following specialization of the q-deformed gl+1 -Whittaker function: gl

gl

+1 +1 z 1 ,...,z +1 (n, k) := z 1 ,...,z +1 (n + k, k, . . . , k).

(1.11)

Let Pn,k be a Gelfand-Zetlin pattern such that ( p+1,1 , . . . , p+1,+1 ) = (n +k, k, . . . , k). Then, the relations p+1,i ≥ p,i ≥ p+1,i+1 for the elements of a Gelfand-Zetlin pattern imply pk,i=1 = 0 and we have that

+1 n+k− p p −p p −k z +1 ,1 z ,1 −1,1 z 1 1,1 gl+1 k ··· zi z 1 ,...,z +1 (n, k) = (n + k − p,1 )q ! ( p,1 − p−1,1 )q ! ( p1,1 − k)q ! i=1 Pn,k

+1 n +1 z +1 z n1 = z ik ··· 1 . (1.12) (n +1 )q ! (n 1 )q ! n +···+n =n i=1

+1

1

gl

+1 Theorem 1.2. z 1 ,...,z +1 (n, k) satisfies following difference equation: +1 gl+1 −1 n gl+1 (1 − z i T ) z 1 ,...,z +1 (n, k) = q z 1 ,...,z +1 (n, k),

(1.13)

i=1

where T · f (n) = f (n + 1). Proof. The proof is based on the explicit expression (1.5). Introduce the generating function gl+1 z 1 ,...,z +1 (t, k)

=

t

n

gl+1 z 1 ,...,z +1 (n, k)

=

n∈Z

+1

z ik , m m=0 (1 − t z i q )

∞

j=1

where we use the identity ∞

xn 1 . = m (n)q ! m=0 (1 − xq )

∞

n=0

gl

gl

+1 +1 Due to the fact that z 1 ,...,z +1 (n, k) = 0 for n < 0, the generating function z 1 ,...,z +1 (t, k) is regular at t = 0. It is easy to check now the following identity:

+1

gl

gl

+1 +1 (1 − t z i ) z 1 ,...,z +1 (t, k) = z 1 ,...,z +1 (qt, k).

j=1

Expanding the latter relation in powers of t, we obtain (1.13) for the coefficients of gl+1 z 1 ,...,z

+1 (t, k).

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Remark 1.2. The difference equation (1.13) for the specialized q-Whittaker function gl+1 z 1 ,...,z +1 (n, k) can be derived directly from the system of equations (1.4) for the nongl

+1 specialized q-deformed Whittaker function z 1 ,...,z +1 ( p1 , p2 , . . . , p+1 ) and the condition

gl

+1 z 1 ,...,z +1 ( p1 , p2 , . . . , p+1 ) = 0

outside the dominant domain p1 ≥ · · · ≥ p+1 . Lemma 1.1. The following integral representation for the specialized q-deformed gl+1 Whittaker functions holds:

+1 +1 dt −n gl +1 t z (n, k) = z ik q (z i t), (1.14) t=0 2πı t i=1

i=1

where q (x) =

∞ n=0

1 . 1 − qnx

Proof. Using the identity ∞ n=0

one obtains, for n ≥ 0, that gl z +1 (n, k)

=

∞ 1 xm = , 1 − xq n (m)q !

+1

m=0

z ik

i=1

=

+1

+1 ∞ dt −n 1 t 1 − z i tq m t=02πı t i=1 m=0

z ik

z 1n 1

n 1 +...+n +1 =n

i=1

(n 1 )q !

· ··· ·

n +1 z +1 . (n +1 )q !

(1.15)

gl

For n < 0, we obviously have that z +1 (n, k) = 0.

The corresponding integral representation for the classical gl2 -Whittaker function is given by the Mellin-Barnes representation for the gl2 -Whittaker function, ıσ +∞ ı ı λ − λ2 λ − λ1 gl2 (λ1 +λ2 )x2 λ(x1 −x2 ) , ψλ1 ,λ2 (x1 , x2 ) = e dλ e ı ı iσ −∞ (1.16) where σ > max{Im λ j , j = 1, . . . , + 1}. Remark 1.3. The expression

+1 gl+1 z 1 ,...,z +1 (n, k) = z ik i=1

gl

+1 z 1 ,...,z +1 (n, k) = 0,

n 1 +···+n +1 =n

n 0, the rescaled cumulative distribution function ρ ε , defined in (1.2), is proved to converge (cf. [14, Th. 2.5]) toward the unique solution of the corresponding initial value problem for nonlinear diffusion equation (−u, u x ), ut = H

(1.6)

is a continuous function and is a Lévy operator of order 1. where the Hamiltonian H It is defined for any function U ∈ Cb2 (R) and for r > 0 by the formula 1 − U (x) = C(1) U (x + z) − U (x) − zU (x)1{|z|≤r } dz (1.7) |z|2 R with a constant C(1) > 0. Finally, in the particular case of c ≡ 0 in (1.3), we have (L , p) = L| p| (cf. [14, Th. 2.6]) which allows us to rewrite Eq. (1.6) in the form H u t + |u x |u = 0.

(1.8)

One can show that the definition of is independent of r > 0, hence, we fix r = 1.

1/2 In fact, for suitably chosen C(1), = 1 = −∂ 2 /∂ x 2 is the pseudodifferential 1 w)(ξ ) = |ξ | operator defined in the Fourier variables by ( w(ξ ) (cf. formulae (2.3) and (2.4) below). In this particular case, Eq. (1.8) is an integrated form of a model studied

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147

by Head [18] for the self-dynamics of a dislocation density represented by u x . Indeed, denoting v = u x we may rewrite Eq. (1.8) as vt + (|v|Hv)x = 0,

(1.9)

where H is the Hilbert transform defined by ) = − i sgn(ξ ) (Hv)(ξ v (ξ ). Let us recall two well known properties of this transform (cf. [30]) 1 v(y) Hv(x) = P.V. dy and 1 v = Hvx . π x − y R

(1.10)

(1.11)

Head [18] called (1.9) the equation of motion of the dislocation continuum and constructed an explicit self-similar solution. Numerical studies of solutions to this equation were performed in [12]. Quasi-geostrophic equations. Let us recall completely different physical motivations which also lead to Eq. (1.9). The 2D quasi-geostrophic equations (QG), modeling the dynamics of the mixture of cold and hot air in a thin layer and the fronts between them, are of the form θt + (u · ∇)θ = 0, u = ∇ ⊥ ψ, θ = −(− )1/2 ψ R2

(1.12)

∇⊥

for x ∈ and t > 0, where = (−∂x2 , ∂x1 ). Here, θ (x, t) represents the air temperature. Pioneering studies concerning a finite time blow up criterion of solutions to (1.12) are due to Constantin et al. [9]. Much earlier, Constantin et al. [8] proposed a one-dimensional version of the Euler equations, namely, ωt + ωHω = 0, and studied formation of singularities of its solutions. Those results motivated the authors of [6,7,10] to consider another simplified model derived from (1.12) in the following way. First, one should write system (1.12) in another equivalent form. From the second and third equation in (1.12) we have the representation u = −∇ ⊥ (− )−1/2 θ = −R ⊥ θ,

(1.13)

where we have used the notation R ⊥ θ = (−R2 θ, R1 θ ), with the Riesz transforms defined by (see e.g. [30]) (x j − y j )θ (y, t) 1 R j (θ )(x, t) = P.V. dy. 2 2π |x − y|3 R Using Eq. (1.13), we find that (1.12) can be transformed into θt − div ((R ⊥ θ )θ ) = 0,

(1.14)

because div (R ⊥ θ ) = 0. To construct the 1D model, the authors of [6,7,10] considered the unknown function θ = θ (x, t) for x ∈ R and t > 0, and replaced the Riesz transform −R ⊥ in (1.14), by the Hilbert transform H (cf. (1.10)–(1.11)). Then Eq. (1.14) is converted into the model equation θt + (θ Hθ )x = 0 for x ∈ R and t > 0.

(1.15)

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P. Biler, G. Karch, R. Monneau

Obviously, for θ ≥ 0, both models (1.9) and (1.15) are identical. However, in the case of Eq. (1.15), it is possible to show that the complex valued function z(x, t) = Hθ (x, t) − iθ (x, t) satisfies the inviscid Burgers equation z t + zz x = 0. This property of solutions has been systematically used in [6,7] to study the existence, the regularity and the blow up in finite time of solutions to Eq. (1.15). We refer the reader to those publications for additional references concerning Eq. (1.15). Below, see Remarks 2.6, 2.10 and 7.7, we explain how our results on Eq. (1.9) and its generalizations contribute to the theory developed for model (1.15). Organization of the paper. In the next section, we state the initial value problem considered in this paper and we formulate our main results. In Sect. 3, we construct explicitly the self-similar solution. In Sect. 4, we recall the necessary material about viscosity solutions, which will be used systematically in the remainder of the paper. In Sect. 5, we prove the uniqueness of the self-similar solution. Under the additional assumption that the solution is confined between its boundary values at infinity, we prove the stability of the self-similar solution, namely Theorem 2.5. In Sect. 6, we prove further decay properties of a solution with compact support. Applying these estimates, we finish the proof of Theorem 2.5 in the general case. In Sect. 7, we introduce an ε-regularized equation, for which we prove both the global existence of a smooth solution and the corresponding gradient estimates. Finally in Sect. 8, we deduce the gradient estimate in the limit case ε = 0, namely Theorem 2.7, using the corresponding estimates for the approximate ε-problem. 2. Main Results Motivated by physics described above, we study the following initial value problem for the nonlinear and nonlocal equation involving u = u(x, t), u t = −|u x | α u on R × (0, +∞), u(x, 0) = u 0 (x) for x ∈ R,

(2.1) (2.2)

where the assumptions on the initial datum u 0 will be made precise later. Here, for

α/2 α ∈ (0, 2), α = −∂ 2 /∂ x 2 is the pseudodifferential operator defined via the Fourier transform α w)(ξ ) = |ξ |α w ( (ξ ).

(2.3)

Recall that the operator α has the Lévy–Khintchine integral representation for every α ∈ (0, 2),

dz − α w(x) = C(α) , (2.4) w(x + z) − w(x) − zw (x)1{|z|≤1} |z|1+α R where C(α) > 0 is a constant. This formula (discussed in, e.g., [13, Th. 1] for functions w in the Schwartz space) allows us to extend the definition of α to functions which are bounded and sufficiently smooth, however, not necessarily decaying at infinity. As we have already explained (cf. Eq. (1.8)), in the particular case α = 1, Eq. (2.1) is a mean field model that has been derived rigorously in [14] as the limit of a system of particles in interactions (cf. (1.1)) with forces V (z) = − 1z . Here, the density u x means the positive density |u x | of dislocations of type of the sign of u x . Moreover, the

Nonlinear Diffusion of Dislocation Density

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occurrence of the absolute value |u x | in the equation allows the vanishing of dislocation particles of the opposite sign. In the present paper, we study the general case α ∈ (0, 2) that could be seen as a mean field model of particles modeled by system (1.1) with repulsive interactions V (z) = − z1α . Here, we would like also to keep in mind that (2.1) is the simplest nonlinear anomalous diffusion model (described by the Lévy operator α ) which degenerates for u x = 0. First note that Eq. (2.1) is invariant under the scaling u λ (x, t) = u(λx, λα+1 t) u(x, t) is a solution to (2.1), then u λ

(2.5) u λ (x, t)

for each λ > 0 which means that if u = = is so. Hence, our first goal is to construct self-similar solutions of Eq. (2.1), i.e. solutions which are invariant under the scaling (2.5). By a standard argument, any self-similar solution should have the following form: x u α (x, t) = α (y) with y = 1/(α+1) , (2.6) t where the self-similar profile α has to satisfy the following equation: − (α + 1)−1 y α (y) = −(α α (y)) α (y) for all y ∈ R.

(2.7)

In our first theorem, we construct solutions to Eq. (2.7). Theorem 2.1 (Existence of self-similar profile). Let α ∈ (0, 2). There exists a nondecreasing function α of the regularity C 1+α/2 at each point and analytic on the interval (−yα , yα ) for some yα > 0, which satisfies

0 on (−∞, −yα ),

α = 1 on (yα , +∞), and (α α )(y) =

y for all y ∈ (−yα , yα ). α+1

Remark 2.2. We can obtain the self-similar solutions corresponding to different boundary values at infinity, simply considering for any γ > 0 and b ∈ R the profiles

γ α γ −1/(α+1) y + b which are also solutions of Eq. (2.7). Remark 2.3. The fact that ∂ y α has compact support reveals a finite velocity propagation of the support of the solution which is typical for solutions the porous medium equation, cf. Remark 2.8 below. At least formally, the function α is the solution of (2.7), and the self-similar function u α given by (2.6) is a solution of Eq. (2.1) with the initial datum being the Heaviside function

0 if x < 0, (2.8) u 0 (x) = H (x) = 1 if x > 0. In order to check that u α given by (2.6) solves (2.1), we introduce a suitable notion of viscosity solutions to the initial value problem (2.1)–(2.2), see Sect. 4. In this setting, we show in Theorem 4.7 the existence and the uniqueness of a solution for any initial condition u 0 in BU C(R), i.e. the space of bounded and uniformly continuous functions on R. Although the initial datum (2.8) is not continuous, we have the following result.

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P. Biler, G. Karch, R. Monneau

Theorem 2.4 (Uniqueness of self-similar solution). Let α ∈ (0, 2). Then the function u α defined in (2.6) with the profile α constructed in Theorem 2.1 is the unique viscosity solution of Eq. (2.1) with the initial datum (2.8). In Theorem 2.4, the uniqueness holds in the sense that if u is another viscosity solution to (2.1), (2.8), then u = u α on (R × [0, +∞))\{(0, 0)}. The self-similar solutions are not only unique, but are also stable in this framework of viscosity solutions, as the following result shows. Theorem 2.5 (Stability of the self-similar solution). Let α ∈ (0, 2). For any initial data u 0 ∈ BU C(R) satisfying lim u 0 (x) = 0 and

x→−∞

lim u 0 (x) = 1,

x→+∞

(2.9)

let us consider the unique viscosity solution u = u(x, t) of (2.1)–(2.2) and, for each λ > 1, its rescaled version u λ = u λ (x, t) given by Eq. (2.5). Then, for any compact set K ⊂ (R × [0, +∞)) \ {(0, 0)}, we have x u λ (x, t) → α 1/(α+1) in L ∞ (K ) as λ → +∞. (2.10) t We stress the fact that Theorem 2.5 contains a result on the long time behaviour of solution because, first, choosing t = 1 in (2.10) and, next, substituting λ = t 1/(α+1) we 1/(α+1) obtain the convergence of u xt , t toward the self-similar profile α (x). On the other hand, convergence (2.10) can be seen as a stability result when we consider initial data which are perturbations of the Heaviside function. This is a nonstandard stability result in the framework of discontinuous viscosity solutions. It shows that the approach by viscosity solutions is a good one in the sense of Hadamard, even if we consider here initial conditions which are perturbations of the Heaviside function. Remark 2.6. In the particular case of α = 1, the nonnegative function U (x, t) = t −1/2 1 (xt −1/2 ), with 1 — the self-similar profile provided by Theorem 2.1, is the compactly supported self-similar solution of (1.15). This function attracts other nonnegative solutions to (1.15) in the sense stated in Theorem 2.5. Finally, we have the following result of independent interest. Theorem 2.7 (Optimal decay estimates). Let α ∈ (0, 1]. For any initial condition u 0 ∈ BU C(R) such that u 0,x ∈ L 1 (R), the unique viscosity solution u of (2.1)–(2.2) satisfies u(·, t) ∞ ≤ u 0 ∞ and u x (·, t) ∞ ≤ u 0,x ∞ for any t > 0. Moreover, for every p ∈ [1, +∞) we have pα+1

u x (·, t) p ≤ C p,α u 0,x 1p(α+1) t

( p−1) − p(α+1)

for any t > 0,

(2.11)

with some constant C p,α > 0 depending only on p and α. The decay given in (2.11) is optimal in the sense that the self-similar solution satisfies (u α )x (·, t) p = ( α ) y (·) p t

( p−1) − p(α+1)

.

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Remark 2.8. The equation satisfied by v = u x , vt = −(|v|α−1 Hv)x

(2.12)

(with the Hilbert transform H defined in (1.10)) can be treated as a nonlocal counterpart of the porous medium equation.

Indeed, for α = 2 and for nonnegative v, Eq. (2.12) reduces to vt = (vvx )x = v 2 /2 x x . As in the case of the porous medium equation (see e.g. [32] and the references therein), estimates (2.11) show a regularizing effect created by the equation, even for the anomalous diffusion: if v0 ∈ L 1 (R) then v ∈ L p (R) for each p > 1. Observealso that Eq. (2.12) has compactly supported self-similar solution 1 1 − α+1 α+1 , where the profile α was constructed in Theorem 2.1. v(x, t) = t

α x/t This function for α = 2 corresponds to the well known Barenblatt–Prattle solution of the porous medium equation. Remark 2.9. As we have already mentioned, see Theorem 4.7 below, the initial value problem (2.1)–(2.2) has the unique global-in-time viscosity solution for any initial datum u 0 ∈ BU C(R). Under the additional assumption u 0,x ∈ L p (R), the corresponding solution satisfies u x (·, t) ∈ L p (R) for all t > 0. Indeed, this is an immediate consequence of the L p -inequalities stated in Remark 7.7 and of a limit argument analogous to that in Step 6 of the proof of Theorem 2.7. Remark 2.10. For any positive, sufficiently regular and vanishing at infinity initial condition v0 ∈ L 2 (R), the corresponding solution v = v(x, t) of (1.15) is global-in-time and analytic, see [6, Th. 2.1]. Since v = u x ≥ 0, this result holds true for solutions of problem (2.1)–(2.2) with α = 1. On the other hand, the nonexistence of global-in-time solutions to the initial value problem for Eq. (1.15) has been always proved assuming that the initial datum is (smooth enough and) negative at some point, see [7, Th. 2.1 and Remark 2.3], [6, Th. 3.1 and 4.8] and [10]. Those arguments cannot be applied to Eq. (2.1) with α = 1 due to the factor |v| (= |u x |) in the nonlinearity and lower regularity of the data. Remark 2.11. For α ∈ (1, 2), we do not know how to define the product |u x | (α u) in the sense of distributions, which is an obstacle for us to prove the result of Theorem 2.7 in this case, see Sect. 7. Note, however, that the inequalities from Theorem 2.7 are valid for α ∈ (1, 2] as well, provided the solution u = u(x, t) is sufficiently regular. 3. Construction of Self-Similar Solutions Proof of Theorem 2.1. The crucial role in the construction of the self-similar profile α is played by the function

2 α/2 for |x| < 1, (3.1) v(x) = K (α) 1 − |x| 0 for |x| ≥ 1, with K (α) = (1/2) [2α (1 + α/2)((1 + α)/2)]−1 . This function (together with its multidimensional counterparts) has an important probabilistic interpretation. Indeed, if {X (t)}t≥0 denotes the symmetric α-stable process in R of order α ∈ (0, 2] and if T = inf{t : |X (t)| > 1} is the first passage time of the process to the exterior of the segment {x : |x| ≤ 1}, Getoor [15] proved that Ex (T ) = v(x), where Ex denotes the expectation under the condition X (0) = x.

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In particular, it was computed in [15, Th. 5.2] using a purely analytical argument (based on definition (2.3) and on properties of the Fourier transform) that α v ∈ L 1 (R) and α v(x) = 1 for |x| < 1.

(3.2)

Now, for the function v, we define the bounded, nondecreasing, C 1+α/2 -function x u(x) = v(y) dy, 0

which obviously satisfies u(x) = M(α) for all x ≥ 1 and u(x) = −M(α) for x ≤ −1 with 1 α/2 π M(α) = K (α) 1 − |y|2 dy =

2 . 0 2α (α + 1) 1+α 2 Then, for any ϕ ∈ Cc∞ (R), we can introduce the following duality:

α u, ϕ = u(y)(α ϕ)(y) dy. R

This defines α u as a distribution, because we can check (using the Lévy–Khintchine formula (2.4)) that there exists a constant C > 0 such that |(α ϕ)(x)| ≤

C ϕ W 2,∞ (R) 1 + |x|1+α

.

If, moreover, supp ϕ ⊂ (−1, 1), it is easy to check using the properties of the function v = v(x) that

∂x (α u), ϕ = − u, α (∂x ϕ) = − u, ∂x (α ϕ) = α (∂x u), ϕ = 1, ϕ, where the last inequality is a consequence of (3.2). From the symmetry of v, we deduce the antisymmetry of u, and then (α u)(−x) = −(α u)(x). Therefore, we get the equality (α u)(x) = x in D (−1, 1), and thus by [21, Cor. 3.1.5], in the classical sense for each y ∈ (−1, 1), too. Finally, we define the nonnegative function γ −1/(α+1) 2M(α) u γ .

α (y) = y + M(α) with γ −1 = α+1 α+1 Now, for yα = γ 1/(α+1) = [2M(α)]−1/(α+1) , we can check easily that α is exactly as stated in Theorem 2.1, which ends the proof. Let us note that we will not use in the sequel the explicit form of the function α , but only its properties listed in Theorem 2.1. Remark 3.1. It has been known since the work of Head and Louat [19] (see also [18]) that

1/2 the function v(x) = K 1 − |x|2 (with a suitably chosen constant K = K (1) > 0) is the solution of the equation (1 v)(x) = 1 on (−1, 1). This result is a consequence of an inversion theorem due to Muskhelishvili, see either [28, p. 251] or [31, Sect. 4.3].

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4. Notion of Viscosity Solutions Here, we consider Eq. (2.1) and its vanishing viscosity approximation, i.e. the following initial value problem for α ∈ (0, 2) and η ≥ 0, u t = ηu x x − |u x | α u on R × (0, +∞), u(x, 0) = u 0 (x) for x ∈ R.

(4.1) (4.2)

In this section, we present the framework of viscosity solutions to problem (4.1)– (4.2). To this end, we recall briefly the necessary material, which can be either found in the literature or is essentially a standard adaptation of those results. We also refer the reader to Crandall et al. [11] for a classical text on viscosity solutions to local (i.e. partial differential) equations. Let us first recall the definition of relaxed lower semi-continuous (lsc, for short) and upper semi-continuous (usc, for short) limits of a family of functions u ε which is locally bounded uniformly with respect to ε, lim sup ∗ u ε (x, t) = lim sup u ε (y, s) and ε→0

ε→0 y→x,s→t

lim inf ∗ u ε (x, t) = lim inf u ε (y, s). ε→0

ε→0 y→x,s→t

If the family consists of a single element, we recognize the usc envelope and the lsc envelope of a locally bounded function u, u ∗ (x, t) = lim sup u(y, s) y→x,s→t

and

u ∗ (x, t) = lim inf u(y, s). y→x,s→t

Now, we recall the definition of a viscosity solution for (4.1)–(4.2). Here, the difficulty is caused by the measure |z|−1−α dz appearing in the Lévy–Khintchine formula (2.4) which is singular at the origin and, consequently, the function has to be at least C 1,1 in space in order that α u(·, t) makes sense (especially for α close to 2). We refer the reader, for instance, to [4,25,29] for the stationary case, and to [23,24] for the evolution equation where this question is discussed in detail. Now, we are in a position to define viscosity solutions. Definition 4.1 (Viscosity solution/subsolution/supersolution). A bounded usc (resp. lsc) function u : R × R+ → R is a viscosity subsolution (resp. supersolution) of Eq. (4.1) on R × (0, +∞) if for any point (x0 , t0 ) with t0 > 0, any τ ∈ (0, t0 ), and any test function φ belonging to C 2 (R × (0, +∞)) ∩ L ∞ (R × (0, +∞)) such that (u − φ) attains a maximum (resp. minimum) at the point (x0 , t0 ) on the cylinder Q τ (x0 , t0 ) := R × (t0 − τ, t0 + τ ), we have ∂t φ(x0 , t0 ) − ηφx x (x0 , t0 ) + |φx (x0 , t0 )| (α φ(·, t0 ))(x0 ) ≤ 0 (resp. ≥ 0), where (α φ(·, t0 ))(x0 ) is given by the Lévy–Khintchine formula (2.4). We say that u is a viscosity subsolution (resp. supersolution) of problem (4.1)–(4.2) on R × [0, +∞), if it satisfies moreover at time t = 0, u(·, 0) ≤ u ∗0 (resp. u(·, 0) ≥ (u 0 )∗ ). A function u : R × R+ → R is a viscosity solution of (4.1) on R × (0, +∞) (resp. R × [0, +∞)) if u ∗ is a viscosity subsolution and u ∗ is a viscosity supersolution of the equation on R × (0, +∞) (resp. R × [0, +∞)). Other equivalent definitions are also natural, see for instance [4].

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Remark 4.2. Any bounded function u ∈ C 1+β (with some β > max{0, α − 1}) which satisfies pointwisely (using the Lévy–Khintchine formula (2.4)) Eq. (4.1) with η = 0, is indeed a viscosity solution. Theorem 4.3 (Comparison principle). Consider a bounded usc subsolution u and a bounded lsc supersolution v of (4.1)–(4.2). If u(x, 0) ≤ u 0 (x) ≤ v(x, 0) for some u 0 ∈ BU C(R), then u ≤ v on R × [0, +∞). Proof. Recall that in [23, Th. 5], the comparison principle is proved for α = 1 and η = 0 under the additional assumption that u 0 ∈ W 1,∞ (R). Looking at the proof of that result, the regularity of the initial data u 0 is only used to show that

sup (u 0 )ε (x) − (u 0 )ε (x) → 0 as ε → 0, (4.3) x∈R

where (u 0 )ε and (u 0 )ε are respectively sup and inf-convolutions. It is easy (and classical) to check that (4.3) is still true for u 0 ∈ BU C(R). The general case can be done either considering a variation of the proof of [23] taking into account the additional Laplace operator, or applying the “maximum principle” from [25], or following, for instance, the lines of [4]. We skip here the detail of this adaptation. This finishes the proof. Theorem 4.4 (Stability). Let {u ε }ε>0 be a sequence of viscosity subsolutions (resp. supersolutions) of Eq. (4.1) which are locally bounded, uniformly in ε. Then u = lim sup∗ u ε (resp. u = lim inf ∗ u ε ) is a subsolution (resp. supersolution) of (4.1) on R × (0, +∞). Proof. A counterpart of Theorem 4.4 is proved in [4, Th.1]. Here, the result for the time dependent problem is again a classical adaptation of that argument, so we skip details. Remark 4.5. One can generalize directly Theorem 4.4 assuming that {u ε }ε>0 are solutions to the sequence of Eq. (4.1) with η = ε. Then, in the limit ε → 0+ , we obtain viscosity subsolutions (resp. supersolutions) of Eq. (2.1). We use this property in the proof of Theorem 2.7. Remark 4.6. In Theorem 4.4, we only claim that the limit u is a supersolution on R × (0, +∞), but not on R × [0, +∞). In other words, we do not claim that u satisfies the initial condition. Without further properties of the initial data u 0 , it may happen that u(·, 0) ≤ u ∗0 is not true. Theorem 4.7 (Existence). Consider u 0 ∈ BU C(R). Then there exists the unique bounded continuous viscosity solution u of (4.1)–(4.2). Proof. Applying the argument of [22] (already adapted from the classical arguments), we can construct a solution by the Perron method, if we are able to construct suitable barriers. Case 1. First, assume that u 0 ∈ W 2,∞ (R). Then the following functions u ± (x, t) = u 0 (x) ± Ct

(4.4)

are barriers for C > 0 large enough (depending on the norm u 0 W 2,∞ (R) ), and we get the existence of solutions by the Perron method.

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Case 2. Let u 0 ∈ BU C(R). For any ε > 0, we can regularize u 0 by a convolution, and get a function u ε0 ∈ W 2,∞ (R) which satisfies, moreover, |u ε0 − u 0 | ≤ ε.

(4.5) u ε0

uε

the solution of (4.1)–(4.2) with the initial condition instead of u 0 . Let us call Then, from the fact that the equation does not see the constants and from the comparison principle Theorem 4.3, we have for any ε, δ > 0, |u ε − u δ | ≤ ε + δ. Therefore, {u ε }ε>0 is the Cauchy sequence which converges in L ∞ (R × [0, +∞)) to some continuous function u (because all the functions u ε are continuous). By the stability result Theorem 4.4, we see that u is a viscosity solution of Eq. (4.1) on R × (0, +∞). To recover the initial boundary condition, we simply remark that u ε (x, 0) = u ε0 (x) satisfies (4.5), and then passing to the limit, we get u(x, 0) = u 0 (x). This shows that u is a viscosity solution of problem (4.1)–(4.2) on R × [0, +∞), and concludes the proof of Theorem 4.7. 5. Uniqueness and Stability of the Self-similar Solution Lemma 5.1 (Comparison with the self-similar solution). Let v be a subsolution (resp. a supersolution) of Eq. (2.1) with the Heaviside initial datum given in (2.8). Then we have v ∗ ≤ (u α )∗ (resp. (u α )∗ ≤ v∗ ). Proof. Using Remark 4.2 and properties of α gathered in Theorem 2.1, it is straightforward to check that the self-similar solution u α (x, t) given in (2.6) is a viscosity solution of Eqs. (4.1)–(4.2) with the initial condition (2.8). Now, we show the inequality (u α )∗ ≤ v∗ . Let v be a viscosity supersolution of (4.1)– (4.2) with the Heaviside initial datum (2.8). Given a > 0 and v a (x, t) = v(a + x, t), we have (u α )∗ (x, 0) ≤ (u 0 )∗ (x) ≤ (u 0 )∗ (a + x) ≤ v a (x, 0). Because of the translation invariance of Eq. (2.1), we see that v a is still a supersolution. Moreover, for any a > 0, we can always find an initial condition u a ∈ BU C(R) such that u α (x, 0) ≤ u a (x) ≤ v a (x, 0). Therefore, applying the comparison principle (Theorem 4.3), we deduce that u α ≤ va . Because this is true for any a > 0, we can take the limit as a → 0 and get (u α )∗ ≤ v∗ . For a subsolution v, we proceed similarly to obtain v ∗ ≤ (u α )∗ . This finishes the proof of Lemma 5.1. Proof of Theorem 2.4. We consider a viscosity solution v of Eq. (2.1) with the Heaviside initial datum (2.8). Using both inequalities of Lemma 5.1, and the fact that (u α )∗ = (u α )∗ on (R × [0, +∞))\ {(0, 0)}, we deduce the equality v = u α on (R × [0, +∞))\ {(0, 0)}, which ends the proof of Theorem 2.4. We will now prove the following weaker version of Theorem 2.5.

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Theorem 5.2 (Convergence for suitable initial data). The convergence in Theorem 2.5 holds true under the following additional assumption lim u 0 (y) = 0 ≤ u 0 (x) ≤ 1 = lim u 0 (y). y→+∞

y→−∞

(5.1)

Proof. Step 1. Limits after rescaling of the solution. Consider a solution u of (2.1)–(2.2) with an initial condition u 0 satisfying (5.1). Recall that for any λ > 0, the rescaled solution is given by u λ (x, t) = u(λx, λα+1 t). Let us define u = lim sup ∗ u λ and u = lim inf ∗ u λ . λ→+∞

λ→+∞

From the stability result Theorem 4.4, we know that u (resp. u) is a subsolution (resp. supersolution) of (2.1) on R × (0, +∞). Step 2. The initial condition. We now want to prove that u(x, 0) = u(x, 0) = H (x) for x ∈ R\ {0},

(5.2)

where H is the Heaviside function. To this end, we remark that u 0 satisfies for some γ > 0 the inequality |u 0 (x)| ≤ γ (note that γ = 1 under assumption (5.1)), and for each ε > 0, there exists M > 0 such that |u 0 (x)| < ε for x ≤ −M. In particular, we get u 0 (x) < ε + γ H (x + M), and then from the comparison principle, we deduce u(x, t) ≤ ε + (u γα )∗ (x + M, t) with u γα (x, t) = γα

x

t 1/(α+1)

and γα (y) = γ α γ −1/(α+1) y .

(5.3)

(5.4)

γ

Here α is the self-similar profile solution of (2.7) with the boundary conditions 0 and γ γ at infinity. Moreover, because u α is continuous off the origin, we can simply drop the star ∗ , when we are interested in points different from the origin. This implies x + Mλ−1 u λ (x, t) ≤ ε + γα , t 1/(α+1) and then u(x, t) ≤ ε + γα

x t 1/(α+1)

.

Therefore, for every x < 0 we have u(x, 0) ≤ ε + γα (−∞) = ε. Because this is true for every ε > 0, we get u(x, 0) ≤ 0 for every x < 0. We get the other inequalities similarly, and finally conclude that (5.2) is valid.

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Step 3. Initial condition at the origin, using assumption (5.1). We now make use of (5.1) to identify the initial values of the limits u and u. We deduce from the comparison principle that 0 ≤ u(x, 0) ≤ u(x, 0) ≤ 1, and then for every x ∈ R we have u(x, 0) ≤ H ∗ (x) and u(x, 0) ≥ H∗ (x). Step 4. Identification of the limits after rescaling. From Lemma 5.1, we obtain u ≤ (u α )∗ = (u α )∗ ≤ u on (R × [0, +∞))\ {(0, 0)}. We have by the construction u ≤ u, hence we infer u = u = u α on (R × [0, +∞))\ {(0, 0)}. Step 5. Conclusion for the convergence. Then for any compact K ⊂ (R × [0, +∞))\ {(0, 0)}, we can easily deduce that sup |u λ (x, t) − u α (x, t)| → 0 as λ → +∞,

(x,t)∈K

which finishes the proof of Theorem 5.2.

6. Further Decay Properties and the End of the Proof of Theorem 2.5 Theorem 6.1 (Decay of a solution with compact support). Let u be the solution to (2.1)– (2.2) with the initial datum u 0 ∈ BU C(R) satisfying for some A > 0, u 0 (x) ≤ 0 for |x| ≥ A,

(6.1)

and u 0 (x) ≤ γ for some γ > 0 and all x ∈ R. Then, there exist on α, but independent of A, γ ) such that

β, β

> 0 (depending

u(x, t) ≤ Ct −β , and u(x, t) ≤ 0 for |x| ≥ C t β

with some constants C = C(α, A, γ ) and C = C (α, A, γ ). First, we need the following Lemma 6.2 (Decay after the first interaction). Consider α and yα defined in Theorem 2.1. Let ν ∈ (1/2, 1) and ξν ∈ (0, yα ) be such that α (ξν ) = ν. Let T > 0 be defined by A = ξν . γ 1/(α+1) T 1/(α+1)

(6.2)

Then, under the assumptions of Theorem 6.1, we have u(x, t) ≤ νγ for all t ≥ T, x ∈ R, and

yα u(x, t) ≤ 0 for all 0 ≤ t ≤ T and |x| ≥ A 1 + ξν

(6.3) .

(6.4)

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γ Proof. Let us denote α (y) = γ α γ −1/(α+1) y . Then we have γ H (x + A) ≥ u 0 (x) for x ∈ R, where γ H (x + A) = lim+ γα t→0

x+A t 1/(α+1)

for x + A = 0.

Now, we apply the comparison principle to deduce that x+A

γα 1/(α+1) ≥ u(x, t) for (x, t) ∈ R × (0, +∞). t Thisargument can be made rigorous by simply replacing the function γ H (x + A) by γ 1/(α+1)

α (x + A + δ)/(tε ) with δ > 0 and some sequence tε → 0+ , and then taking the limit δ → 0+ . Therefore we have x+A ≥ u(x, t) for (x, t) ∈ R × (0, +∞). γ α γ 1/(α+1) t 1/(α+1) From the properties of the support of α , we also deduce that u(x, t) ≤ 0 for x ≤ − A + yα (γ t)1/(α+1) , and then, by symmetry, u(x, t) ≤ 0 for |x| ≥ A + yα (γ t)1/(α+1) . Moreover, it follows from the monotonicity of α that A ≥ u(x, t) γ α (γ t)1/(α+1) for x ≤ 0, and by symmetry we can prove the same property for x ≥ 0. Then for T > 0 defined in (6.2) we easily deduce (6.3) and (6.4). This ends the proof of Lemma (6.2). Proof of Theorem 6.1. We apply recurrently Lemma 6.2. Define A0 = A, γ0 = γ , and yα An , γn+1 = νγn , and An+1 = An 1 + = ξν . ξν (γn Tn )1/(α+1) This gives yα n An = A0 1 + , γn = ν n γ0 , Tn = K µn , ξν with K =

1 γ0

A0 ξν

α+1

, 1 0, x ∈ R, with β=−

log ν > 0. log µ

Similarly, we have u(x, t) ≤ 0 for |x| ≥ An if t ≤ T0 + · · · + Tn−1 = K

µn − 1 . µ−1

In particular, we get for any n ∈ N\ {0},

yα n , if t ≤ K 0 µn u(x, t) ≤ 0 for |x| ≥ A0 1 + ξν

with K 0 = K /µ. This implies

u(x, t) ≤ 0 for |x| ≥ A0 (K 0 )−β t β with β = This ends the proof of Theorem 6.1.

log 1 + log µ

yα ξν

for t ≥ 0,

> 0.

As a corollary, we can now remove assumption (5.1) in Theorem 5.2 and complete the proof of Theorem 2.5. Proof of Theorem 2.5. We simply repeat Step 3 of the proof of Theorem 5.2, but here without assuming (5.1). Then, for any ε > 0 there exists A > 0 such that u 0 (x) ≤ 1 + ε for |x| ≥ A. By Theorem 6.1 applied to the solution u(x, t) − 1 − ε, this implies that there exists a constant C > 0 (depending on ε) such that u(x, t) ≤ 1 + ε + Ct −β . Therefore, for any for λ > 0 the following inequality u λ (x, t) ≤ 1 + ε + Ct −β λ−β

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holds true, which implies that u = lim sup ∗ u λ satisfies λ→+∞

u(x, t) ≤ 1 + ε for (x, t) ∈ R × (0, +∞). Since this is true for any ε > 0, we deduce that u(x, t) ≤ 1 for (x, t) ∈ R × (0, +∞). Let us now define u˜ = min (1, u) . By the construction, ˜ u(x, t) = u(x, t) for (x, t) ∈ R × [0, +∞)\ {(0, 0)}, ˜ 0) ≤ H ∗ (x) for all x ∈ R. Therefore, u˜ is a subsolution of and, by (5.2), we have u(x, (2.1)–(2.2) on R × [0, +∞) with the initial datum being the Heaviside function. Similarly, we can show that u = lim sup ∗ u λ satisfies λ→+∞

u ≥ 0 for (x, t) ∈ R × (0, +∞).

Hence, the function u˜ = max 0, u , which is a supersolution of (2.1)–(2.2) on R × [0, +∞) with the Heaviside initial datum. Finally, the conclusion of the proof is the same as in the proof of Theorem 5.2 where u (resp. u) is replaced by u˜ (resp. u). ˜ This finishes the proof of Theorem 2.5. 7. Approximate Equation and Gradient Estimates In this section, in order to prove our gradient estimates for viscosity solutions stated in Theorem 2.7, we replace Eq. (2.1) by an approximate equation for which smooth solutions do exist. Indeed, with ε > 0, we consider the following initial value problem: u t = εu x x − |u x |α u on R × (0, +∞), u(x, 0) = u 0 (x) for x ∈ R.

(7.1) (7.2)

We have added to Eq. (2.1) an auxiliary viscosity term which is stronger than α u and u x . In the case α ∈ (0, 1], we will see later (in Sect. 8) that it is possible to pass to the limit ε → 0+ in L ∞ (R), which is the required convergence for the framework of viscosity solutions. The difficulty in the case α ∈ (1, 2) comes from the fact that, for the limit equation with ε = 0, we are not able to give a meaning to the product |u x | (α u) in the sense of distributions, while it is possible when α ∈ (0, 1]. Our results on qualitative properties of solutions to the regularized problem (7.1)– (7.2) are stated in the following two theorems. Theorem 7.1 (Approximate equation – existence of solutions). Let α ∈ (0, 1] and ε > 0. Given any initial datum u 0 ∈ C 2 (R) such that u 0,x ∈ L 1 (R) ∩ L ∞ (R), there exists a unique solution u ∈ C(R × [0, +∞)) ∩ C 2,1 (R × (0, +∞)) of (7.1)–(7.2). This solution satisfies u x ∈ C([0, T ], L p (R)) ∩ C((0, T ]; W 1, p (R)) ∩ C 1 ((0, T ], L p (R)) for every p ∈ (1, ∞) and each T > 0.

(7.3)

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Theorem 7.2 (Approximate equation – decay estimates). Under the assumptions of Theorem 7.1, the solution u = u(x, t) of (7.1)–(7.2) satisfies u(·, t) ∞ ≤ u 0 ∞ , u x (·, t) ∞ ≤ u 0,x ∞ ,

(7.4)

and pα+1

u x (·, t) p ≤ C p,α u 0,x 1p(α+1) t

1 − α+1 1− 1p

,

(7.5)

for every p ∈ [1, ∞), all t > 0, and constants C p,α > 0, see (7.20) below), independent of ε > 0, t > 0 and u 0 . Existence theory. First, we construct solutions to the initial value problem for the regularized equation. Proof of Theorem 7.1. Note first that α u = α−1 Hu x ,

(7.6)

where H denotes the Hilbert transform, see (1.10). We recall that the Hilbert transform is bounded on the L p -space for any p ∈ (1, +∞) (see [30, Ch. 2, Th. 1]), i.e. it satisfies for any function v ∈ L p (R) the following inequality Hv p ≤ C p v p

(7.7)

with a constant C p independent of v. For α ∈ (0, 1), the operator α−1 defined analogously as in (2.3) corresponds to the convolution with the Riesz potential α−1 v = Cα | · |−α ∗ v. Hence, by [30, Ch. 5, Th. 1], for any p > 1/α with α ∈ (0, 1] and any function v ∈ L q (R), we have α−1 v p ≤ C p,α v q with

1 1 = + 1 − α. q p

(7.8)

Now, if u = u(x, t) is a solution to (7.1)–(7.2), using identity (7.6), we write the initial value problem for v = u x , vt = εvx x − (|v|α−1 Hv)x on R × (0, +∞), v(·, 0) = v0 = u 0,x ∈ L 1 (R) ∩ L ∞ (R), as well as its equivalent integral formulation t v(t) = G(εt) ∗ v0 − ∂x G(ε(t − τ )) ∗ (|v|α−1 Hv) dτ,

(7.9) (7.10)

(7.11)

0

with the Gauss–Weierstrass kernel G(x, t) = (4π t)−1/2 exp(−x 2 /(4t)). The next step is completely standard and consists in applying the Banach contraction principle to Eq. (7.11) in a ball in the Banach space XT = C([0, T ]; L 1 (R) ∩ L ∞ (R)) endowed with the usual norm v T = supt∈[0,T ] ( v(t) 1 + v(t) ∞ ). Using well known estimates of the heat semigroup and inequalities (7.7)–(7.8) combined with the imbedding L 1 (R) ∩ L ∞ (R) ⊂ L p (R) for each p ∈ [1, ∞], we obtain a solution v = v(x, t)

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to Eq. (7.11) in the space XT provided T > 0 is sufficiently small. We refer the reader to, e.g., [2,5] for examples of such a reasoning. This solution satisfies (7.3) for every p ∈ (1, ∞) and each T > 0, by standard regularity estimates of solutions to parabolic equations. Moreover, following the reasoning in [2], one can show that the solution is regular. Finally, this local-in-time solution can be extended to global-in-time (i.e. for all T > 0) because of the estimates v(t) p ≤ v0 p for every p ∈ [1, ∞] being the immediate consequence of inequalities (7.17), (7.18), and (7.21) below. Gradient estimates. In the proof of the decay estimates of u x , we shall require several properties of the operator α . First, we recall the Nash inequality for the operator α . Lemma 7.3 (Nash inequality). Let α > 0. There exists a constant C N > 0 such that 2(1+α)

w 2

≤ C N α/2 w 22 w 2α 1

(7.12)

for all functions w satisfying w ∈ L 1 (R) and α/2 w ∈ L 2 (R). The proof of inequality (7.12) is given, e.g., in [26, Lemma 2.2]. Our next tool is the, so-called, Stroock–Varopoulos inequality. Lemma 7.4 (Stroock–Varopoulos inequality). Let 0 ≤ α ≤ 2. For every p > 1, we have p 2 α 4( p − 1) α p−2 2 |w| 2 ( w)|w| w dx ≥ dx (7.13) p2 R R for all w ∈ L p (R) such that α w ∈ L p (R). If α w ∈ L 1 (R), we obtain (α w) sgn w dx ≥ 0.

(7.14)

Moreover, if w, α w ∈ L 2 (R), it follows that (α w)w + dx ≥ 0 and (α w)w − dx ≤ 0,

(7.15)

R

R

R

where w + = max{0, w} and w − = max{0, −w}. Inequality (7.13) is well known in the theory of sub-Markovian operators and its statement and the proof is given, e.g., in [27, Th. 2.1, combined with the Beurling–Deny condition (1.7)]. Inequality (7.14), called the (generalized) Kato inequality, is used, e.g., in [13] to construct entropy solutions of conservation laws with a Lévy diffusion. It can be easily deduced from [13, Lemma 1] by an approximation argument. The proof of (7.15) can be found, for example, in [27, Prop. 1.6]. Remark 7.5. Remark that inequality (7.14) appears to be a limit case of (7.13) for p = 1. Inequality (7.15) for w + follows easily from (7.14) by a comparison argument if, for instance, w ∈ Cc∞ (R). Finally, remark that the constant appearing in (7.13) is the same as for the Laplace operator ∂ 2 /∂ x 2 = −2 . Our proof of the decay of v(t) = u x (t) is based on the following Gagliardo–Nirenberg type inequality:

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Lemma 7.6 (Gagliardo–Nirenberg type inequality). Assume that p ∈ (1, ∞) and α > 0 are fixed and arbitrary. For all v ∈ L 1 (R) such that α/2 |v|( p+1)/2 ∈ L 2 (R), the following inequality is valid: 2 v ap ≤ C N α/2 |v|( p+1)/2 v b1 , (7.16) 2

where a=

p( p + α) pα + 1 , b= , p−1 p−1

and C N is the constant from the Nash inequality (7.12). Proof. Without loss of generality, we can assume that v 1 = 0. Substituting w = |v|( p+1)/2 in the Nash inequality (7.12) we obtain 2 ( p+1)(1+α) α( p+1) ≤ C N α/2 |v|( p+1)/2 v ( p+1)/2 . v p+1 2

Next, it suffices to apply two particular cases of the Hölder inequality

v p 1/ p 2 v 1

p2 /( p2 −1) p/( p+1)

≤ v p+1 as well as v ( p+1)/2 ≤ v p

1/( p+1)

v 1

,

and compute carefully all the exponents which appear on both sides of the resulting inequality. Proof of Theorem 7.2. The first inequality in (7.4) is an immediate consequence of the comparison principle from Theorem 4.3, because classical solutions are viscosity solutions, as well. The maximum principle and an argument based on inequalities (7.15) (cf. [26] for more detail) lead to the second inequality in (7.4). We also discuss this inequality in Remark 7.7 below. For the proof of the L 1 -estimate u x (t) 1 ≤ u 0,x 1

(7.17)

(i.e. (7.5) with p = 1 and C p,α = 1), we multiply Eq. (7.9) by sgn v = sgn u x and we integrate with respect to x to obtain d (α−1 Hv)|v| sgn v dx. |v| dx = ε vx x sgn v dx − x dt R R R The first term on the right-hand side is nonpositive by the Kato inequality (i.e. (7.14) with α = 2) hence we skip it. Remark that (formally) α−1 ( Hv)|v| sgn v dx = (α−1 Hv)vx (sgn v)2 + (α−1 Hvx )v dx x R R = (α−1 Hv)v dx = 0. R

x

√ Now, approximating the sign function in a standard way by sgnδ (z) = z/ z 2 + δ, integrating by parts, and passing to the limit δ → 0+ , one can show rigorously that the

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second term on right-hand side of the above inequality is nonpositive. This completes the proof of (7.17) with p = 1. Next, we multiply the equation in (7.9) by |v| p−2 v with p > 1 to get 1 d p p−2 (α−1 Hv)|v| |v| p−2 v dx. |v| dx = ε vx x |v| v dx − x p dt R R R We drop the first term on the right-hand side, because it is nonpositive by (7.13) with α = 2. Integrating by parts and using the elementary identity p − 1 p−1 |v| |v| p−2 v = |v| v , x x p we transform the second quantity on the right-hand side as follows: p−1 α−1 p−2 − ( Hv)|v| |v| v dx = − (α v)|v| p−1 v dx. x p R R Consequently, by the Stroock–Varopoulos inequality (7.13) (with the exponent p replaced by p + 1), we obtain d 4 p( p − 1) α/2 ( p+1)/2 2 p |v| (7.18) v(t) p ≤ − . 2 dt ( p + 1)2 Hence, the interpolation inequality (7.16) combined with (7.17) lead to the following p inequality for v(t) p : 4 p( p − 1) d p ( pα+1)/( p−1) −1 p ( p+α)/( p−1) v(t) p ≤ − C N v0 1 v(t) p . (7.19) 2 dt ( p + 1) Recall now that if a nonnegative (sufficiently smooth function) f = f (t) satisfies, for all t > 0, the inequality f (t) ≤ −K f (t)β with constants K > 0 and β > 1, then f (t) ≤ (K (β − 1)t)−1/(β−1) . Applying this simple result to the differential inequality (7.19), we complete the proof of the L p -decay estimate (7.5) with the constant C p,α

1 4 p(α + 1) − α+1 = C N−1 ( p + 1)2

1− 1p

where C N is the constant from the Nash inequality (7.12).

,

(7.20)

Remark 7.7. Note that, for every fixed α, we have lim p→∞ C p,α = +∞. By this reason, we are not allowed to pass directly to the limit p → +∞ in inequalities (7.5) (as was done in, e.g., [26, Th. 2.3]) in order to obtain a decay estimate of v(t) in the L ∞ -norm. Nevertheless, using (7.19) we immediately deduce the inequality v(·, t) p ≤ v0 (·) p valid for every p ∈ (1, ∞). Hence, passing to the limit p → +∞ we get v(·, t) ∞ ≤ v0 (·) ∞ .

(7.21)

It is natural to expect that, under the assumptions of Theorem 7.2, the quantity v(·, t) ∞ should decay at the rate t −1/(α+1) . For a proof, one might follow an idea from [6, Lemma 4.7] where the decay estimates of vx (·, t) were obtained for solutions of a certain regularization of Eq. (1.15). Here, however, we did not try to go this way, because our main goal was to study decay estimates for the problem (2.1)–(2.2) whose viscosity solutions are not regular enough a priori to handle decay properties of u x x (x, t) = vx (x, t).

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8. Passage to the Limit and the Proof of Theorem 2.7 Now, we are in a position to complete the proof of the gradient estimates (2.11). First, we show that from the sequence {u ε }ε>0 of solutions to the approximate problem (7.1)–(7.2) one can extract, via the Ascoli–Arzelà theorem, a subsequence converging uniformly. Theorem 4.4 on the stability and Remark 4.5 imply that the limit function is a viscosity solution to (2.1)–(2.2). Passing to the limit ε → 0+ in inequalities (7.4) and (7.5) we complete our reasoning. Proof of Theorem 2.7. First, let us suppose that u 0 ∈ C ∞ (R) ∩ W 2,∞ (R) with u 0,x ∈ L 1 (R)∩ L ∞ (R). Denote by u ε = u ε (x, t) the corresponding solution to the approximate problem with ε > 0. Step 1. Modulus of continuity in space. Under this additional assumption, we have u εx (·, t) p ≤ C p t −γ p

(8.1)

pα+1 1 1 − 1p . The Sobolev imbedding theorem with C p = C p,α u 0,x 1p(α+1) and γ p = α+1 implies that there exist some β ∈ (0, 1) and C0 > 0 such that

|u ε (x + h, t) − u ε (x, t)| ≤ |h|β C0 C p t −γ p .

(8.2)

∞ Step 2. Modulus of continuity in time. Let us consider a nonnegative function ϕ ∈ C (R) with supp ϕ ⊂ [−1, 1] such that R ϕ(x) dx = 1, and for any δ > 0 set ϕδ (x) = δ −1 ϕ(δ −1 x). Then, multiplying (7.1) by ϕδ and integrating in space, we get d ε u (x, t)ϕδ (x) dx = ε u ε (·, t), (ϕδ )x x dt R − ϕδ (x) |u εx (x, t)|(H α−1 u εx (x, t)) dx,

R

+ 1/ p

= 1, and then with 1/ p d ε ≤ ε u ε (·, t) ∞ (ϕδ )x x 1 u (x, t)ϕ (x) dx δ dt R

+ ϕδ ∞ u εx (·, t) p H α−1 u εx (·, t) p .

(8.3)

Here, we have used relation (7.6). Combining inequalities (7.7) and (7.8) with estimate (8.1), we get for p > 1/α, H α−1 u εx (·, t) p ≤ C p C p ,α Cq t −γq . Then for any bounded time interval I ⊂ (0, +∞) there exists a constant C I,δ such that for all t ∈ I , we have for any ε ∈ (0, 1], d ε ≤ C I,δ . u (x, t)ϕ (x) dx δ dt R Now, for any t, t + s ∈ I , we get ε u ε (x, t + s)ϕδ (x) dx − u (x, t)ϕδ (x) dx ≤ |s|C I,δ . R

R

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Therefore, the following estimate: |u ε (0, t + s) − u ε (0, t)| ϕδ (x) dx ≤ |s|C I,δ + R

× sup |u ε (x, t + s) − u ε (0, t + s)| + |u ε (x, t) − u ε (0, t)| x∈[−δ,δ]

holds true. Using the Hölder inequality (8.2), we deduce that there exists a constant C I depending on I , but independent of δ and of ε ∈ (0, 1], such that |u ε (0, t + s) − u ε (0, t)| ≤ |s|C I,δ + C I δ β . Since the above inequality is true for any δ, this shows the existence of a modulus of continuity ω I satisfying |u ε (0, t + s) − u ε (0, t)| ≤ ω I (|s|) for any t, t + s ∈ I. By the translation invariance of the problem, this estimate is indeed true for any x ∈ R, i.e. |u ε (x, t + s) − u ε (x, t)| ≤ ω I (|s|) for any t, t + s ∈ I, x ∈ R.

(8.4)

From estimates (8.2) and (8.4), and using the Step 3. Convergence as ε → Ascoli–Arzelà theorem and the Cantor diagonal argument, we deduce that there exists a subsequence (still denoted {u ε }ε ) which converges to a limit u ∈ C(R × (0, +∞)). By the stability result in Theorem 4.4 (see also Remark 4.5), we have that u is a viscosity solution of (2.1) on R × (0, +∞). 0+ .

Step 4. Checking the initial conditions for u 0 smooth. Remark that for u 0 ∈ W 2,∞ we can use the barriers given in (4.4) with some constant C > 0 uniform in ε ∈ (0, 1]. This ensures that u is continuous up to t = 0 and satisfies u(·, 0) = u 0 , so this proves the result under additional assumptions. Step 5. General case. The proof in the case of less regular initial conditions simply follows by an approximation argument as in the proof of Theorem 4.7. Step 6. Gradient estimates. To pass to the limit ε → 0+ in estimates (7.5), we use the inequality h −1 u ε (· + h, t) − u ε (·, t) p ≤ u εx (·, t) p

(8.5)

with fixed h > 0. Hence, by the Fatou lemma combined with the pointwise convergence of u ε toward u, we deduce from (8.5) and (7.5) that pα+1

h −1 u(· + h, t) − u(·, t) p ≤ C p,α u 0,x 1p(α+1) t

1 − α+1 1− 1p

for all h > 0. For every fixed t > 0, the sequence {h −1 (u(·+h, t)−u(·, t))}h>0 is bounded in L p (R) and converges (up to a subsequence) weakly in L p (R) toward u x (·, t) (see, e.g., [30, Ch. V, Prop. 3]). Using the well known property of a weak convergence in Banach spaces we conclude h u x (·, t) p ≤ lim inf + h→0

−1

pα+1 p(α+1)

u(· + h, t) − u(·, t) p ≤ C p,α u 0,x 1

This finishes the proof of Theorem 2.7.

t

1 − α+1 1− 1p

.

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Acknowledgements. The third author is indebted to Cyril Imbert for stimulating and enlightening discussions on the subject of this paper. The authors thank the anonymous referee for calling their attention to the onedimensional counterpart of the quasi-geostrophic equation (1.15). This work was supported by the contract ANR MICA (2006–2009), by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389, and by the Polish Ministry of Science grant N201 022 32/0902.

References 1. Alvarez, O., Hoch, P., Le Bouar, Y., Monneau, R.: Dislocation dynamics: short time existence and uniqueness of the solution. Arch. Rat. Mech. Anal. 181, 449–504 (2006) 2. Amour, L., Ben-Artzi, M.: Global existence and decay for viscous Hamilton-Jacobi equations. Nonlinear Anal. 31, 621–628 (1998) 3. Barles, G., Chasseigne, E., Imbert, C.: On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57, 213–246 (2008) 4. Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. I.H.P., Anal Non-Lin. 25, 567–585 (2008) 5. Ben-Artzi, M., Souplet, Ph., Weissler, F.B.: The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces. J. Math. Pures Appl. 81, 343–378 (2002) 6. Castro, A., Córdoba, D.: Global existence, singularities and ill-posedness for a nonlocal flux. Adv. Math. 219, 1916–1936 (2008) 7. Chae, D., Córdoba, A., Córdoba, D., Fontelos, M.A.: Finite time singularities in a 1D model of the quasi-geostrophic equation. Adv. Math. 194, 203–223 (2005) 8. Constantin, P., Lax, P., Majda, A.: A simple one-dimensional model for the three dimensional vorticity. Comm. Pure Appl. Math. 38, 715–724 (1985) 9. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) 10. Córdoba, A., Córdoba, D., Fontelos, M.A.: Formation of singularities for a transport equation with nonlocal velocity. Ann. Math. 162, 1377–1389 (2005) 11. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.), 27, 1–67 (1992) 12. Deslippe, J., Tedstrom, R., Daw, M.S., Chrzan, D., Neeraj, T., Mills, M.: Dynamics scaling in a simple one-dimensional model of dislocation activity. Phil. Mag. 84, 2445–2454 (2004) 13. Droniou, J., Imbert, C.: Fractal first order partial differential equations. Arch. Rat. Mech. Anal. 182, 299–331 (2006) 14. Forcadel, N., Imbert, C., Monneau, R.: Homogenization of the dislocation dynamics and of some particle systems with two-body interactions. Disc. Contin. Dyn. Syst. Ser. A 23, 785–826 (2009) 15. Getoor, R.K.: First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101, 75–90 (1961) 16. Head, A.K.: Dislocation group dynamics I. Similarity solutions od the n-body problem. Phil. Mag. 26, 43–53 (1972) 17. Head, A.K.: Dislocation group dynamics II. General solutions of the n-body problem. Phil. Mag. 26, 55–63 (1972) 18. Head, A.K.: Dislocation group dynamics III. Similarity solutions of the continuum approximation. Phil. Mag. 26, 65–72 (1972) 19. Head, A.K., Louat, N.: The distribution of dislocations in linear arrays. Austral. J. Phys. 8, 1–7 (1955) 20. Hirth, J.R., Lothe, L.: Theory of Dislocations. Second Ed., Malabar, FL: Krieger, 1992 21. Hörmander,: The Analysis of Linear Partial Differential Operators. Vol. 1, New York: Springer-Verlag, 1990 22. Imbert, C.: A non-local regularization of first order Hamilton-Jacobi equations. J. Differ. Eq. 211, 214–246 (2005) 23. Imbert, C., Monneau, R., Rouy, E.: Homogenization of first order equations, with (u/ε)-periodic Hamiltonians. Part II: application to dislocations dynamics. Comm. Part. Diff. Eq. 33, 479–516 (2008) 24. Jakobsen, E.R., Karlsen, K.H.: Continuous dependence estimates for viscosity solutions of integro-PDEs. J. Differ. Eq. 212, 278–318 (2005) 25. Jakobsen, E.R., Karlsen, K.H.: A maximum principle for semicontinuous functions applicable to integro-partial differential equations. NoDEA Nonlin. Differ. Eqs. Appl. 13, 137–165 (2006) 26. Karch, G., Miao, C., Xu, X.: On the convergence of solutions of fractal Burgers equation toward rarefaction waves. SIAM J. Math. Anal. 39, 1536–1549 (2008)

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27. Liskevich, V.A., Semenov, Yu.A.: Some problems on Markov semigroups. In: Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Math. Top. 11, Berlin: Akademie Verlag, 1996, pp. 163–217 28. Muskhelishvili, N.I.: Singular Integral Equations. Groningen: P. Noordhoff, N. V., 1953 29. Sayah, A.: Équations d’Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I Unicité des solutions de viscosité, II Existence de solutions de viscosité. Comm. Part. Diff. Eq. 16, 1057–1093 (1991) 30. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30, Princeton, NJ: Princeton University Press, 1970 31. Tricomi, F.G.: Integral Equations. New York-London: Interscience Publ., 1957 32. Vázquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type. Oxford Lecture Series in Mathematics and its Applications 33, Oxford: Oxford University Press, 2006 Communicated by P. Constantin

Commun. Math. Phys. 294, 169–197 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0935-9

Communications in

Mathematical Physics

On the Massive Wave Equation on Slowly Rotating Kerr-AdS Spacetimes Gustav Holzegel Department of Mathematics, Fine Hall, Princeton University, Washington Road, Princeton, NJ 08544, United States. E-mail: [email protected] Received: 27 February 2009 / Accepted: 29 July 2009 Published online: 2 October 2009 – © Springer-Verlag 2009

Abstract: The massive wave equation g ψ − α 3 ψ = 0 is studied on a fixed Kerr-anti de Sitter background M, g M,a, . We first prove that in the Schwarzschild case (a = 0), ψ remains uniformly bounded on the black hole exterior provided that α < 49 , i.e. the Breitenlohner-Freedman bound holds. Our proof is based on vectorfield multipliers and commutators: The total flux of the usual energy current arising from the timelike Killing vector field T (which fails to be non-negative pointwise) is shown to be non-negative with the help of a Hardy inequality after integration over a spacelike slice. In addition to T , we construct a vectorfield whose energy identity captures the redshift producing good estimates close to the horizon. The argument is finally generalized to slowly rotating Kerr-AdS backgrounds. This is achieved by replacing the Killing vectorfield T = ∂t with K = ∂t + λ∂φ for an appropriate λ ∼ a, which is also Killing and–in contrast to the asymptotically flat case–everywhere causal on the black hole exterior. The separability properties of the wave equation on Kerr-AdS are not used. As a consequence, the theorem also applies to spacetimes sufficiently close to the Kerr-AdS spacetime, as long as they admit a causal Killing field K which is null on the horizon.

1. Introduction The study of linear wave equations on black hole spacetimes has acquired a prominent role within the subject of general relativity. The main reason is the expectation that understanding the mechanisms responsible for the decay of linear waves on black hole exteriors in a sufficiently robust setting provides important insights for the non-linear black hole stability problem [7]. The mathematical analysis of linear waves in this context was initiated by the pioneering work of Kay and Wald establishing boundedness (up to and including the event horizon) for φ satisfying g φ = 0 on Schwarzschild spacetimes [14,20]. Since then considerable progress has been achieved, especially in the last few years. Most of these

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recent decay and boundedness theorems for linear waves concern black hole spacetimes satisfying the vacuum Einstein equations Rµν −

1 Rgµν + gµν = 0 2

(1)

with = 0 and, motivated by cosmological considerations, > 0. In particular, by now polynomial decay rates have been established for g φ = 0 on Schwarzschild [4,8,16] and more recently, Kerr spacetimes [6,7,19]. In the course of work on the decay problem, a much more robust understanding of boundedness on both Schwarzschild and Kerr spacetimes was also obtained [6,7] allowing one to prove boundedness for a large class of spacetimes, which are not exactly Schwarzschild or Kerr but only assumed to be sufficiently close Many of the above results have been extended to the cosmological case, > 0. Here (much stronger) decay rates have been established for the wave equation on Schwarzschild-de Sitter spacetimes [5,13,17]. For further discussion we refer the reader to the lecture notes [7], which among other things provide an account of previous work on these problems (Sect. 4.4, 5.5, 6.3 ibidem) and a comparison with results that have been obtained in the heuristic tradition (Sect. 4.6). In contrast to the case of a positive cosmological constant, the choice < 0 in (1) has remained relatively unexplored. While this problem certainly deserves mathematical attention in its own right, there is also considerable interest from high energy physics, see [9,18]. In this paper, we study the equation g ψ − α

ψ =0 3

(2)

on a class of spacetimes, which will include slowly rotating Kerr anti-de Sitter spacetimes [3]. These spacetimes generalize the well-known Kerr solution (the latter being the unique two-parameter family of stationary, axisymmetric asymptotically flat black hole solutions to (1) with = 0.). They are axisymmetric, stationary solutions of (1) with < 0, parametrized by their mass M and angular momentum per unit mass a = MJ . Before we comment further on their geometry, let us discuss Eq. (2). The main motivation to include the zeroth order term in (2) is the case α = 2, the conformally invariant case. In pure AdS the Green’s function for (2) with α = 2 is supported purely on the light cone, which makes it a natural analogue of the equation ψ = 0 in asymptotically flat space (and also explains the adjective “massless” which is sometimes used in the physics literature in connection with this choice of α). The case α = 2 also occurs naturally in classical general relativity when studying a Maxwell field or linear gravitational perturbations in AdS [12]. Again for the case of pure AdS, it is well known ([1,2,12]) that (2) is only well-posed for α < 54 , the so-called second Breitenlohner-Freedman bound. While no solutions exist for α ≥ 49 , one has an infinite number of solutions depending on boundary conditions for α in the range 45 ≤ α < 49 , the latter bound being the first Breitenlohner-Freedman bound. For general asymptotically AdS spacetimes, however, the wellposedness of (2) has not yet been established explicitly. In this context it is essential to notice that asymptotically AdS spaces are not globally hyperbolic. To make the dynamics of (2) well-posed suitable boundary conditions will have to be imposed on the timelike boundary of the spacetime. We will address this boundary initial value problem in detail in a separate

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171

paper. For the purpose of the present paper, we will assume that α < 94 and that we are given a solution to (2) which decays suitably near the AdS boundary. This pointwise “radial” decay depends on how close α is to the Breitenlohner Freedman bound and ensures in particular that there is no energy flux through the timelike boundary I. It is precisely the decay suggested by the mode analysis of [2] in pure AdS and expected to hold for all asymptotically AdS spacetimes. The global geometry of the Kerr-AdS background spacetimes is quite different from their asymptotically flat counterparts. Most notably perhaps, null-infinity is now timelike, entailing the non-globally hyperbolic nature of the spacetime mentioned above. With the boundary conditions imposed there will be no radiation flux through infinity and the only possible decay mechanism is provided by an energy flux through the horizon. Another geometric feature, which we will exploit to a great extent in the present paper, is the existence of an everywhere causal Killing vectorfield on the black hole exterior of slowly rotating Kerr-AdS spacetimes.1 This vectorfield was used previously by Hawking and Reall [10] to obtain a positive conserved energy for any matter fields satisfying the dominant energy condition. The scalar field (2) in the subcase α ≤ 0 provides an example. In particular, the argument of [10] excluded a negative energy flux through the horizon and hence superradiance as a mechanism of instability, at least if 2 with r |a| − 3 < rhoz hoz being the location of the event horizon.

For 0 < α < 49 , however, the energy momentum tensor associated with ψ does not satisfy the dominant energy condition and the energy current associated with the causal Killing field fails to be positive pointwise. In particular, as noted for instance in [15], the positivity argument of [10] breaks down for 0 < α < 94 , including the most interesting case α = 2. In this paper we present a simple resolution of this problem: Using a Hardy inequality we show that the energy flux arising from the Killing field is still positive in an integrated sense. With the existence of the everywhere causal Killing field and its associated globally positive energy, superradiance is eliminated as an obstacle to stability.2 This makes the problem much easier to deal with than the asymptotically flat case, where such a vectorfield is not available. In particular, we can avoid the intricate bootstrap and harmonic analysis techniques of [6], which were necessary to deal with superradiant phenomena and trapping (see also [19]).3 It is well-known that the notion of positive energy outlined above is not sufficient to prevent the scalar field from blowing up on the horizon in evolution. This issue was first addressed and resolved for Schwarzschild in the celebrated work of Kay and Wald [14,20] exploiting the special symmetry properties of the background spacetime. With the recent work of Dafermos and Rodnianski, in particular their mathematical understanding of the celebrated redshift, there is now a much more stable argument available, which does not hinge on the discrete symmetries of Schwarzschild and is in fact applicable to any black hole event horizon with positive surface gravity [7]. This geometric understanding of the role of the event horizon for the boundedness and decay mechanism 1 This is crucially different from the asymptotically flat case, where any linear combination of the two available Killing fields K = ∂t + λ∂φ (for some constant λ) is somewhere spacelike. 2 As expected perhaps, the restriction on a (which can be computed explicitly) becomes tighter compared to the case α ≤ 0 of Hawking and Reall. 3 Superradiance is induced by the existence of an ergosphere [i.e. an effect which is not present in Schwarzschild] arising from the fact that the energy density associated with the Killing field ∂t can be negative inside the ergosphere. This allows for a negative energy flux through the horizon and hence an amplification of the amplitude for backscattered waves. See [21] for a nice discussion and also [7] for a detailed mathematical treatment.

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was first developed in [8] and plays a crucial role in the recent proof of boundedness [6] and polynomial decay [7] of scalar waves on slowly rotating Kerr spacetimes. The boundedness statement of [6] holds for a much more general class of spacetimes nearby Kerr, while the decay statement of [7] requires the background to be exactly Kerr. For our considerations at the horizon we will adapt the ideas developed by Dafermos and Rodnianski. A vectorfield Y is constructed whose energy identity, if coupled to the timelike Killing vectorfield K in an appropriate way, provides control over Sobolev norms whose weights do not degenerate at the horizon. An additional complication compared to the asymptotically-flat massless case is that due to the zeroth order term in the wave equation (2), a zeroth order flux-term of the redshift vectorfield on the horizon has the wrong sign. However, this term can, after a computation, be absorbed by the “good” positive terms at our disposal. With Y and K at hand, boundedness is shown adapting the argument of [6], taking care of the different weights in r which appear due to the asymptotically hyperbolic nature of the background space.4 It should be emphasized that the proof does not require the construction of a globally positive spacetime integral arising from a virial vectorfield but that it suffices to use the timelike Killing field and the redshift vectorfield alone. The fact that the more elementary statement of boundedness can be obtained from the use of these two vectorfields alone in the non-superradiant regime was observed by Dafermos and Rodnianski in [6] in the asymptotically-flat case. Previously, even the understanding of boundedness was intriniscially tied to the understanding of a globally positive spacetime term and hence decay, see [8] and also Sect. 3.4 of [7]. Finally, here is an outline of the paper. We introduce the AdS-Schwarzschild and Kerr backgrounds equipped with regular coordinate systems on the black hole exterior (defining in particular the spacelike slices we are going to work with) in Sect. 2. In the following section the class of solutions we wish to consider is defined and the notion of vectorfield multipliers discussed. For reasons of presentation we first state and prove the boundedness theorem for Schwarzschild-AdS in Sect. 4, before we turn to the generalization to Kerr-AdS in Sect. 5. The paper concludes with some final remarks and future directions. 2. The Black Hole Backgrounds 2.1. Schwarzschild-AdS. In the familiar (t, r ) coordinates, the Schwarzschild AdS metric reads −1 2M r 2 2M r 2 g =− 1− + 2 dt 2 + 1 − + 2 dr 2 + r 2 dω2 , r l r l

(3)

where = − l32 is the cosmological constant. This coordinate system is not well-behaved at the zeros of 2M r 2 + 2 . 1−µ= 1− r l

(4)

4 Since angular momentum operators do not commute with the wave operator for a = 0, we have to commute the wave equation with T and the redshift vectorfield Y to obtain L 2 control of certain derivatives,

which leads to control over all derivatives using elliptic estimates on the asymptotically hyperbolic spacelike slices .

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

173

Let us define ⎛

p = ⎝ Ml 2 +

⎛

⎞1 3 6 l 2 4 ⎠ M l + 27

and q = ⎝ Ml 2 −

⎞1 3 6 l 2 4 ⎠ M l + . 27

(5)

2

Clearly, p > 0, q < 0 with pq = − l3 . The expression (4) has a single real root at rhoz = p + q > 0,

(6)

the location of the black hole event horizon. Note that for l → ∞ we have rhoz → 2M. In general we have the estimate 2 − 3 pq (r − rhoz ) r 2 + rrhoz + rhoz r 3 + l 2 r − 2Ml 2 = 1−µ = 2 rl 2 3 rl 2 3 2 2 r − rhoz (r − rhoz ) r + rrhoz + rhoz + l = ≥ , (7) rl 2 rl 2 which will be useful later. Here is a coordinate system which is well behaved everywhere on the black hole exterior and the horizon: It arises from the coordinate transformation r t = t + r (r ) − l arctan , (8) l where r (r ) is a solution of the differential equation 1 dr

= 2M dr 1− r +

r2 l2

=

1 1−µ

and r (3M) = 0.

(9)

The variable r is often called the tortoise coordinate. In the new (t , r ) coordinates the metric reads 2

r 1 + 2M 2 4M r + l2

g = − (1 − µ) dt + dr + dr 2 + r 2 dω2 , dt 2 2 r2 r 1 + rl 2 1 + l2

(10)

which is clearly regular on the horizon. For notational convenience let us agree on the shorthand notation k± = 1 ±

2M r 2 + 2 r l

and

k0 = 1 +

r2 . l2

(11)

Slices τ of constant t will play a prominent role for stating energy identities in the paper. Their normal 2

r 1 + 2M 2M r + l2 ∂r ∂ − − ∇t = 2 t 2 2 r 1 + rl 2 1 + rl 2

(12)

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G. Holzegel

is everywhere timelike: k+

g ∇t , ∇t = g t t = − 2 . k0

(13)

We denote the unit-normal by

√ k+ ∇t

2M n = − √ ∂t − √ ∂r . =

k0 −g (∇t , ∇t ) r k+

(14)

We also define the (Killing) vectorfields i (i = 1, 2, 3) to be a basis of generators of the Lie-algebra of S O (3) corresponding to the spherical symmetry of the Schwarzschild / to denote the gradient of the metric induced on the metric. Moreover, we will write ∇ S O (3)-orbits. Finally, a Penrose diagram of (the black hole exterior of) Schwarzschild-AdS spacetime with two slices of constant t is depicted below.

H+ τ I

0

2.2. Kerr-AdS. The Kerr-AdS metric in Boyer Lindquist coordinates reads 2

θ r 2 + a 2 − − a 2 sin2 θ 2 2 sin2 θ d φ˜ 2 dr + dθ + g=

−

θ 2

θ r 2 + a 2 − −

− θ a 2 sin2 θ 2 ˜ − − a sin2 θ d φdt dt −2 with the identifications = r 2 + a 2 cos2 θ, r2

± = r 2 + a 2 1 + 2 ± 2Mr, l a2 cos2 θ, l2 a2 = 1− 2. l

θ = 1 −

(15)

(16) (17) (18) (19)

2

Once again, let k0 = 1 + rl 2 . A coordinate system which is regular on the horizon is obtained by the transformations t = t + A (r )

and

φ = φ˜ + B (r ) ,

(20)

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

175

where 2Mr dA = dr

− 1 +

r2 l2

a dB = . dr

−

and

(21)

The new metric coefficients become , gt t = gtt ,

θ = gφ˜ φ˜ , gt φ = gt φ˜ , 1 2 2

, = − a sin θ k + + 0 2 l2 r2 1 + l2

gθθ =

(22)

gφφ

(23)

grr

a sin2 θ [ + 2Mr ] , k0 a2 1 2Mr − 2 sin2 θ . = k0 l

(24)

gφr = −

(25)

gt r

(26)

Note that the angular momentum term in the last expression grows faster in r than the mass term.5 For a = 0 the metric reduces to the Schwarzschild-AdS metric in t , r coordinates, cf. (10). The inverse components are

gt

t

=−

+ + a 2 sin2 θ −k0 + k02 θ

2Mr l2

,

θ 2 a

, g φφ = , , gt φ = k 0 l 2 θ

θ sin2 θ

− a 2Mr

, g φr = , gt r = . = k0

(27)

g θθ =

(28)

grr

(29)

The unit normal of a constant t slice is n =

−g t

t

gt r gt φ

∂t − ∂ − ∂ , g(n , n ) = −1, r

φ −g t t −g t t

(30)

with the determinant of the metric induced on constant t slices being √

det h =

det gt =const

= sin θ

t

t . −g

(31)

5 Hence the Kerr-AdS metric is not uniformly close to the Schwarzschild metric in these coordinates! It follows that for the statement that “Schwarzschild-AdS is close to Kerr-AdS” one has to use both a regular coordinate patch for r ≤ R and a Boyer Lindquist patch for r > R (for some R away from the horizon). For the asymptotically-flat case this is not necessary.

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3. The Dynamics 3.1. The class of solutions. Having defined a proper coordinate system on the black hole backgrounds in the previous section we can turn to the wave equation we would like to study (recall = − l32 ): g ψ +

α ψ = 0, l2

(32)

for α ∈ R. The choice α = 2 in (32) corresponds to the conformally invariant case, cf. the discussion in the Introduction. Generally we will assume α < 49 , the upper Breitenlohner-Freedman bound. Before we state a theorem on the global dynamics of the above scalar field, let us address the issue of well-posedness of (32). It turns out that this has not yet been proven explicitly for asymptotically anti de Sitter spacetimes, nor has it for the special cases of Schwarzschild and Kerr-AdS. For pure AdS (M = a = 0), however, it can be deduced from [1]. The peculiarity of this problem arises from the fact that due to the timelike nature of I asymptotically AdS spacetimes are not globally hyperbolic. Hence in general “appropriate” boundary conditions have to be imposed on I to make the dynamics well-posed. We will formulate and prove the precise well-posedness statement for asymptotically AdS spacetimes in a separate paper. For the purpose of this paper, we content ourselves with considering a class of ψ defined as follows: Definition 3.1. Fix α < 49 , a Kerr-AdS spacetime M, g M,a, and a constant t slice 0 in D = J + (I) ∩ J − (I), as well as an integer k ≥ 0. We say that the function ψ is k if in J + a solution of class Cdec ( 0 ) ∩ J − (I) , • ψ is C k , • ψ satisfies (32), √ • for all δ < 21 9 − 4α 3

lim |r 2 +n+δ ∂rn ψ| = 0

r →∞

holds for n = 0, 1, . . . , k.

(33)

k there is no In particular, the decay (33) ensures that for a solution of class Cdec energy flux (cf. Sect. 4.1) through the AdS boundary. The decay (33) is precisely the one expected from the AdS case [1] and also strongly suggested from an asymptotic expansion in r of Eq. (32) (performed in [2] for pure AdS). The existence of a large class of solutions with the properties of Definition 3.1, which arise from appropriate initial data prescribed on 0 , would follow from a general well-posedness statement phrased in terms of weighted Sobolev norms (cf. also the Appendix). We emphasize again that uniqueness is only expected for α < 45 .

3.2. Vectorfield multipliers and commutators. We will obtain estimates for the field ψ via vectorfield multipliers (and eventually commutators). Since the general technique is well known and reviewed in detail in [7] we will only give a brief summary. The starting point is the energy momentum tensor of the above scalar field α 1 Tµν = ∂µ ψ∂ν ψ − gµν ∂β ψ∂ β ψ − 2 ψ 2 . 2 l

(34)

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

177

It satisfies ∇ µ Tµν = 0,

(35)

provided that (32) holds. Consider a vectorfield X on spacetime. We define its associated currents JµX = Tµν X ν , K = Tµν X

(X ) µν

π

(36) ,

(37)

where (X ) π µν = 21 (∇ µ X ν + ∇ ν X µ ) is the deformation tensor of the vectorfield X . One has (using (35)) the identity (38) ∇ µ JµX = ∇ µ Tµν X ν = K X . Note that π vanishes if X is Killing. Integrating (38) over regions of spacetime relates boundary and volume terms via Stokes’ theorem. In particular, for background Killing vectorfields we obtain conservation laws. In writing out the aforementioned integral identities we will sometimes not spell out the measure explicitly (e.g. Eq. (72)), it being implicit that the measure is the one induced on the slices (cf. Eq. (31)) or the spacetime measure respectively. Besides using vectorfields as multipliers we will also use them as commutators. If X is a vectorfield and ψ satisfies (2) then X (ψ) satisfies (cf. the Appendix of [7]) α g X (ψ) + 2 X (ψ) = −2(X ) π γβ ∇γ ∇β ψ − 2 2 ∇ γ (X ) πγ µ − ∇µ (X ) πγγ ∇ µ ψ. l Note that if X is Killing the right-hand side vanishes. In general one may apply multipliers to the commuted equation to derive estimates for higher order derivatives. 4. Boundedness in the Schwarzschild Case Here is our boundedness theorem for Schwarzschild-anti de Sitter:

Theorem 4.1. Fix a Schwarzschild-anti de Sitter spacetime M, g M>0, and 0 = τ0 a slice of constant t = τ0 in D = J + (I) ∩ J − (I).6 Let α < 49 and ψ n+1 with k ψ ∈ C n+1−k for k = 0, ..., n, where n ≥ 0 is be a solution to (32) of class Cdec i dec an integer. If n 2 2 1 k 2 k k 2 /

r 2 dr dω < ∞, ∂ ∂ ψ + r ψ + | ∇ ψ| (39) t r 2 r 0 k=0

then

2 2 1 k 2 k k 2 /

r 2 dr dω ∂ ∂ ψ + r ψ + | ∇ ψ| t r 2 r k=0 τ n 2 2 1 k 2 k k 2 2 / ψ| r dr dω ∂t ψ + r ∂r ψ + |∇ ≤C 2 0 r

n

(40)

k=0

for a constant C which just depends on M, l and α. Here τ denotes any constant t

slice to the future of 0 and restricted to r ≥ rhoz . 6 Note that such slices satisfy in particular H− ∩ = ∅. 0

178

G. Holzegel

By Sobolev embedding on S 2 we immediately obtain Corollary 4.1. The pointwise bound 2 2 2 1 k / k ψ|2 r 2 dr dω + r 2 ∂r k ψ + |∇ k=0 0 r 2 ∂t ψ |ψ| ≤ C (41) 3 r2 holds in the exterior J − I + ∩ J + ( 0 ) for a constant C just depending on the initial data, M, l and α. The remainder of this section is spent proving the above theorem. In the following we denote by B and b constants which just depend on the fixed parameters M, l and α. We also define R (τ1 , τ2 ) = ∪τ1 ≤τ ≤τ2 τ

(42)

to be the region enclosed by the slices τ1 and τ2 , a piece of I, and the horizon piece (43) H (τ1 , τ2 ) = H ∩ J + τ1 ∩ J − τ2 . Compare the figure above. 4.1. Positivity of energy. The first step is to obtain a positive energy arising from the Killing vectorfield T = ∂t . For this we apply the vectorfield identity (38) in the region R t1 , t2 . For the energy flux through a slice we obtain ∞

T (∂t , n ) g¯ dr dω E t = 1 = 2

rhoz

∞

rhoz

S2

S2

(∂t ψ)

2

k+ k02

α 2 2 2 / − 2ψ + (∂r ψ) (1 − µ) + |∇ψ| r dr dω. (44) l 2

The flux through the horizon is E H[t1 ,t2 ] =

H(t1 ,t2 )

(∂t ψ)2 r 2 dt dω,

(45)

hence in particular non-negative. Finally, the flux through I, the AdS boundary, vanishes because of the boundary conditions imposed. Combining these facts we obtain the energy identity E t2 = E t1 + E H[t1 ,t2 ] , (46) stating in particular that E (t ) is non-increasing. Next we show that the energy flux through the slices is positive: Lemma 4.1. We have 1 E t ≥ 2

∞

S2

rhoz

/ +|∇ψ|

(∂ ψ) t

2

r 2 dr dω.

2

k+ k02

4 + 1 − α (∂r ψ)2 (1 − µ) 9 (47)

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

179

Proof.

∞

rhoz

ψ r dr = 2 2

3 r 3 − rhoz

3

∞ ψ 2 rhoz

2 − 3

∞

rhoz

3 dr. ψψr r 3 − rhoz

(48)

The boundary term vanishes because of the decay of ψ at infinity and we obtain the Hardy inequality 3 2 ∞ 3 r − rhoz 4 ∞ 4 ∞ 2 2 2 ψ r dr ≤ dr ≤ (∂r ψ) (∂r ψ)2 r 2 l 2 (1 − µ) dr, 9 rhoz r2 9 rhoz rhoz (49) 3 ≤ r 3 in the last step. Hence where we used (7) and that r 3 − rhoz ∞ ∞ α 4 2 2 ψ r dr ≤ α (∂r ψ)2 r 2 (1 − µ) dr. l 2 rhoz 9 rhoz

Inserting this into (44) yields the result.

(50)

The lemma clearly reveals the relevance of the Breitenlohner-Freedman bound in ensuring a positive energy. We emphasize that one has positivity of energy only in an integrated sense! Lemma 4.1 also answers a question posed in [15] on whether one can construct a positive energy for the range 0 < α < 49 . Finally, we note that a similar Hardy inequality was used previously in the Appendix of [1], where the massive wave equation is studied on pure AdS. Having established positivity of the energy we only have to deal with the fact that the control over the (∂r ψ)2 term degenerates at the horizon. This is achieved using a so-called “redshift vectorfield”. See [8] for its first appearance in the context of asymptotically flat Schwarzschild black holes and the recent [7] for a version applicable to all non-extremal black holes. An additional difficulty in the present context lies in the fact that the energy momentum tensor does not satisfy the dominant energy condition. 4.2. The redshift. Define γ 1−µ k+ γ Y = ∂t + − + β ∂r . +β 2k0 2k0 2 2

(51)

Here γ ≥ 0 and β ≥ 0 are both functions of r only, which we will define below. We compute the current 2 3 k2

k2

ψ) (∂ t µ JµY n = T (Y, n ) = k+ 0 + β k+2 0 γ 2 2 2k04

(∂t ψ) (∂r ψ) − k+ γ + β k+ k− 2k0 2k β k− (∂r ψ)2 γ 0 k+ k0 + √ + 2k0 2 2 k+ +

+

/ 2− |∇ψ| 2

α 2 ψ l2

γ β k+ . √ + 2 k+ 2

(52)

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G. Holzegel

Next consider the current associated with N = T + eY for some small constant e (depending on α): µ JµN n

3 k2

k2 (∂t ψ)2 3 0 2 0 = T (N , n ) = k+ k0 + eγ k+ + eβ k+ 2 2 2k04

(∂t ψ) (∂r ψ) − k+ eγ + eβ k+ k− + 2k0 2k k02 γ β k− (∂r ψ)2 0 k+ k0 + e √ + √ k− + e 2k0 2 2 k+ k+ / 2 − α2 ψ 2 k0 |∇ψ| γ β l k+ . + √ +e √ +e 2 2 k+ 2 k+

(53)

The bulk term of the vectorfield N reads 1 N αβ K = πY Tαβ e 2M 2r k+ (∂t ψ)2 2 γ − γ + k = + k β + 2 − µ) − γ (β (1 ) ,r − ,r + r2 l2 r 2k02

2M 2r 8M (∂t ψ) (∂r ψ) −2 γ − γ +2β + k k k − + (β (1−µ)−γ ) ,r − ,r − + 2k0 r2 l2 r2 2 γ,r k− k− k− M r 2 + (∂r ψ) γ − + β,r − + (β (1 − µ) − γ ) r 2 l2 2 2 r k− β,r γ,r β 2M 2r 2 / − − + |∇ψ| + 2 + 2 2 r2 l 2 γ,r α 2 1 γ k− β,r β 2M 2r βk− − + + . (54) + 2ψ + 2 − l r r 2 2 r2 l 2 KY =

Let us define the functions γ and β. Set γ = ξ (r ) (1 + (1 − µ)) and β = 1δ (1 − µ) ξ (r ). Here ξ is a smooth positive function equal to 1 in r ≤ r0 and identically zero for r ≥ r1 . The quantity δ is a small parameter. (Note that β,r is positive in r ≤ r0 ). Let us regard ξ as being fixed. We choose r0 and δ such that the following conditions hold in r ≤ r0 : 2M 2r k+ γ + 2 − γ,r k− + k+2 β,r + 2 (β (1 − µ) − γ ) ≥ b, r2 l r 2 γ,r k− k− k− M r − + + β,r − γ (β (1 − µ) − γ ) ≥ b, r 2 l2 2 2 r γ,r β 2M 2r β,r k− + − ≥ b, + − 2 2 r2 l2 2

(55)

(56)

(57)

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

181

and

2M 2r 8M (∂t ψ) (∂r ψ) −2 γ + 2 − γ,r k− + 2β,r k− k+ − 2 (β (1 − µ) − γ ) 2k0 r2 l r 2 1 (∂t ψ) 2M 2r k+ 2 ≤ γ + 2 − γ,r k− + k+ β,r + 2 (β (1 − µ) − γ ) 2 r2 l r 2k02 2 γ,r k− k− k− M r 2 + (∂r ψ) γ − + β,r − + (58) (β (1 − µ) − γ ) . r 2 l2 2 2 r

It is easily seen all these inequalities can be achieved by inserting the expressions for γ and β, then choosing δ sufficiently small and finally r0 sufficiently close to rhoz to exploit factors of k− = (1 − µ). With γ and β being determined choose e so small that eγ ≤

1 2k0

and, in case that 0 < α < 49 , also 9 1 − 49 α 1 e sup (γ + βk+ ) < 2k0 8α r

(59)

(60)

hold for all r . 4.2.1. The N bulk term. Let us denote by K0N the expression for K N with the zeroth order term removed. Lemma 4.2. The quantities δ, r0 and r1 can be chosen such that / 2 K0N ≥ b (∂t ψ)2 + (∂r ψ)2 + |∇ψ|

(61)

pointwise in r ≤ r0 . On the other hand, / 2 − K0N ≤ B (∂t ψ)2 + (∂r ψ)2 + |∇ψ|

(62)

pointwise in r0 ≤ r ≤ r1 . Proof. The second inequality is immediate since we are away from the horizon. The first is a consequence of the inequalities (55)-(58). 4.2.2. The N boundary terms. Let us investigate what the current J N actually controls. We can write √ √ √ k+ k+ eγ k+ (∂t ψ)2 (∂r ψ)2 k0 N µ − eγ 2 + Jµ n = T (N , n ) = √ k− + 2 k0 2 4 k+ 2k0 √ 2 γ k+ ∂r ψ 2 k− ∂t ψ β +e k+ 2 − k+ ∂t ψ + √ ∂r ψ +e 4 k0 2 4 k0 k+ α 2 2 / − l2 ψ |∇ψ| k0 γ β (63) + k+ √ +e √ +e 2 2 k+ 2 k+

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G. Holzegel

with the term in the penultimate line being manifestly non-negative. From (59) one easily obtains a lower bound for the square bracket multiplying the (∂t ψ)2 -term. Inequality (59) in turn allows us to control the bad (for 0 < α < 49 !) zeroth order term by borrowing from the r -derivative term using the Hardy inequality (50). Namely, if 0 < α < 49 , then: e α 2 ψ 1+ (γ + βk+ ) r 2 dr 2 l 2k 0 rhoz 4 ∞ α 9 1 − 9α ≤ 1+ ψ 2 r 2 dr 2 8α rhoz l ∞ 4 1 1+ α ≤ (∂r ψ)2 (1 − µ) r 2 dr. 2 9 rhoz

∞

(64)

Hence finally

∞

√

k+ 2 r dr dω k02 rhoz √ ∞ k+ (∂t ψ)2 ≥ 2 2 2k 0 rhoz S √ 4 γ k+ 2 (∂r ψ)2 1 1 − α (1 − µ) + e r dr dω + 2 2 9 4 ∞ / 2 2 |∇ψ| r dr dω, + 2 rhoz S 2 S2

µ JµN n

(65)

(66)

and we have established the following Lemma 4.3. ≥b

∞

rhoz

S2

µ JµN n

=

∞

S2

rhoz

µ JµN n

√

k+ 2 r dr dω k0

(∂t ψ) / 2 r 2 dr dω. + (∂r ψ)2 r 2 + |∇ψ| r2 2

(67)

Note that the degeneration of the (∂r ψ)2 -term which occurred in the T -energy has disappeared.

4.3. The boundedness. Proposition 4.1. There exist constants B and b such that for any τ2 ≥ τ1 ≥ 0, τ2 µ N µ Jµ n H+ + b dτ JµN n − H(τ1 ,τ2 ) τ τ12 τ µ ≤ KN + B dτ JµT n . R(τ1 ,τ2 )

τ1

τ

(68)

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

183

Proof. We first note that all zeroth order terms can be absorbed by the last term in (68) using inequality (50). Hence it suffices to prove the inequality for the first order terms. We have τ2 M α 2 N µ 2 2 /

eγ |∇ψ| − 2 ψ + (∂t ψ) r 2 dt dω. Jµ n H+ = 2r k0 l H(τ1 ,τ2 ) τ1 S2 (69) For α ≤ 0 the term has a good sign while for 0 < α < 49 , τ2 r1 Mr α 2 N µ

− Jµ n H+ ≤ dt dω dr ∂r −eγ ψ 2k0 l 2 H(τ1 ,τ2 ) τ1 S2 rhoz Bα 2 η (∂r ψ)2 + ψ dt dr dω ≤e η l2 R(τ1 ,τ2 )∩{r ≤r1 }

(70)

for any η > 0 using that γ is supported for r < r1 only. So we only need a little bit of the ∂r term which we can borrow from the good bulk term K N in r ≤ r0 (by Lemma 4.2) and from the T -energy term in the remaining region. We have τ2 Bα µ −K N + eη (∂r ψ)2 + e 2 ψ 2 +b dτ JµN n η l R(τ1 ,τ2 )∩{r ≤r1 } τ1 τ τ2 µ T ≤ B˜ dτ Jµ n (71) τ1

τ

for a small constant b and large constants B and B˜ as a consequence of Lemma 4.2. Combining this inequality with (70) yields (68). Remark. We have discarded good terms (the t and angular derivative) on the horizon in the proof of the proposition as they are not needed for the following argument. Later, when we start commuting the equation with the redshift vectorfield we will need to keep those positive terms in order to estimate certain errorterms (cf. Sec. 5.3). Using the previous proposition we can prove boundedness as follows. The N identity reads µ µ µ JµN n + JµN n H+ + KN = JµN n 0 . (72) H+

τ

R(0,τ )

An application of Proposition 4.1 yields the inequality τ2 τ2 µ µ JµN n + b dτ JµN n ≤ B dτ τ2

τ1

τ

τ1

τ

µ

τ

JµT n +

τ1

µ

JµN n τ . 1

(73)

Using (46) (in particular the fact that the T -energy from initial data) is non-increasing µ µ and setting f (τ ) = τ JµN n as well as D = 0 JµT n we arrive at τ2 f (τ2 ) + b f (τ ) dτ ≤ B D (τ2 − τ1 ) + f (τ1 ) (74) τ1

for any τ2 ≥ τ1 ≥ 0 from which boundedness of f (τ ) follows from a pigeonhole argument, cf. [7].

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G. Holzegel

Due to the spherical symmetry of the background we can commute Eq. (32) with angular momentum operators and obtain boundedness of µ JµN k ψ n (75)

for any integer k assuming such a bound on the data. Theorem 4.1 follows. 4.4. Pointwise bounds. For completeness we give the proof of Corollary 4.1. We have ∞ ∞

1 |∂r ψ t , r, ω |dr ≤ √ 3 |∂r ψ (t , r, ω) |2 r 4 dr , (76) |ψ t , r, ω | ≤ r r 3r 2 and by Sobolev embedding on S 2 ,

∞

|∂r ψ t , r, ω |2 r 4 dr ≤ C˜

r

∞

r

≤ C˜

2

2 |∂r k (ψ) t , r, ω |2 r 4 dr dω S 2

S 2 k=0

k=0

µ JµN k ψ n ,

(77)

from which the statement of the corollary follows.

4.5. Higher order quantities. Clearly, via commuting with T and angular momentum operators, we obtain control over certain higher order energies as well. Using just this collection of vectorfields however, we will still not be able to estimate the derivative transverse to the horizon. This problem was resolved for the asymptotically-flat Kerr case in [6,7], roughly speaking by commuting with the redshift vectorfield as well. We will adapt their argument to the present asymptotically hyperbolic case in Sect. 5.3, when we deal with the Kerr-AdS metric. There the argument is unavoidable even to obtain a pointwise bound on ψ since one can no longer (trivially) commute with angular momentum operators! 5. The Kerr-AdS Case To generalize the argument to include the case of Kerr-AdS we have to circumvent several difficulties. First of all, the timelike Killing field T = ∂t is no longer timelike everywhere on the black hole exterior due to the presence of an ergoregion close to the horizon. Hence the T -identity alone will not produce positive boundary terms. The resolution is to consider the Killing field K = T + λ

(78)

for an appropriate constant λ and = ∂φ being the Killing field corresponding to the axisymmetry of the Kerr-AdS metric. K is everywhere timelike on the black hole exte2 . This rior (it is null on the horizon, coinciding with the generators) as long as |a|l < rhoz

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

185

is a consequence of the asymptotically AdS nature of the background7 : Note that the analogous vectorfield in asymptotically flat space would turn spacelike near infinity! These properties of K were first exploited by Hawking and Reall [10] to obtain a pointwise positive energy for fields whose energy momentum tensor satisfies the dominant energy condition, see the discussion in the Introduction. Not surprisingly, we will show below that for sufficiently small a the energy identity associated to K produces boundary terms with manifestly non-negative first order terms. The zeroth order term however still has the wrong sign for 0 < α < 94 (as in Schwarzschild-AdS), a consequence of the dominant energy condition being violated in this range. We resolve this issue by generalizing the Hardy inequality (50) to the Kerr-AdS case allowing us to again control the zeroth order term from the derivative term in the K energy. Because we have to “borrow” from the r -derivative term for the Hardy inequality, we have to impose stronger 2 . restrictions on the smallness of a than just |a|l < rhoz Since the argument close to the horizon involving the vectorfield Y is stable in itself (it applies to any black hole event horizon with positive surface gravity by Theorem 7.1 of [7]) our previous argument using the vectorfield N = K + eY immediately yields boundedness of the L 2 -norm of all derivatives. In particular, no further restrictions on the size of a arise. The only remaining difficulty is that we can no longer commute with angular momentum operators to obtain pointwise bounds because the spherical symmetry of Schwarzschild has been broken. The way around this is to commute (32) with K and Y and to derive elliptic estimates on the asymptotically hyperbolic slices . This is carried out in Sect. 5.3.

5.1. The vectorfield K : Positive energy. We first compute the currents associated with the Killing vectors T = ∂t and = ∂φ ,

− 1 1 α 1

−g t t (∂t ψ)2 + ψ2 (∂r ψ)2 − 2 2 2 −g t t

l 2 −g t t

2 1 φφ θθ 2 rφ g + ψ + g ψ) + 2g ψ) ∂ ψ (79) ∂ (∂ (∂ φ θ r φ

2 −g t t

T (∂t , n ) =

and t r t φ 2 g g

T ∂φ , n = −g t t . (80) ∂φ ψ (∂t φ) + t t ∂φ ψ (∂r φ) + t t ∂φ ψ g g We observe that we cannot control the (∂r ψ) ∂φ ψ term in the T -energy, because the (∂r ψ)2 term from which we have to borrow degenerates on the horizon. However, consider the vectorfield K = T + λ

with λ =

7 More precisely, the fact that g ∼ g ∼ g 2 φφ ∼ r . t t t φ

a + a2

2 rhoz

(81)

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G. Holzegel

and its current on constant t slices α

2 −g t t · T (K , n ) = − 2 ψ 2 + g θθ (∂θ ψ)2 l 2a

∂φ ψ (∂t ψ) + −g t t (∂t ψ)2 − g t t 2 2 rhoz + a 2 2a

+ g φφ − g t φ 2 ψ ∂ φ rhoz + a 2

− a

+ . (∂r ψ)2 + 2 (∂r ψ) ∂φ ψ gr φ − g t r 2 rhoz + a 2

(82)

We choose a so small that 1 φφ 2a 2 2a 2 2

g − gt φ 2 = ≥ 0. − 2 + a2 k l 2 4 rhoz rhoz + a 2 4 θ sin2 θ 0 θ

(83)

This is possible because both terms decay like r12 in r . In the asymptotically flat case

the g t φ -term will eventually dominate because k0 = 1 in the asymptotically flat case 2 (k0 = 1 + rl 2 in AdS!). We also choose a small enough so that 2 1 t t 2a 1 φφ

ψ) + −g g ψ ψ ≥ 0. ∂ ∂ (∂t ψ)2 − g t t 2 (∂ φ t φ 2 8 rhoz + a 2

(84)

This is easily achieved since again all terms decay like r12 at infinity and the cross-term has a factor of a. Of the first order terms it remains to control the r φ cross-term. For this we note (cf. Lemma 5.1) a a 2 2 k = + a − 2Mr r 0 hoz 2 + a2 2 + a2 rhoz k0 rhoz a ( − − k0 (rhoz + r ) (r − rhoz )) = 2 k0 rhoz + a 2 l2 a (r − rhoz ) r− . (85) = k0 l2 rhoz The reason for the above choice of K is now obvious: The (∂r ψ) ∂φ ψ -term has acquired a weight which degenerates on the horizon. Hence we can borrow from the (∂r ψ)2 term whose weight also degenerates. Moreover, the decay of (85) in r is strong, which will be exploited soon. Before we continue we obtain rather precise control over how the quantity − deteriorates on the horizon: gr φ − g t

r

Lemma 5.1. We have a 2l 2 1 3 2 2 2 2

− = 2 (r − rhoz ) r + r rhoz + r rhoz + l + a − l rhoz

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187

and 3 2 3 + 3a 2 (r − r r − rhoz hoz )

− − = f, r 2 + a2 l 2

(86)

where 1 a 2l 2 3 2 (r − rhoz ) r 3 l 2 − 4a 2 + r 2 rhoz f = 2 − + a r hoz rhoz r + a2 l 2 2 a 4 l 2 4 2 2 + r rhoz . − 8a 4 + a 2 rhoz + l 2 + rhoz rhoz + 3a 2 − rhoz

(87)

Proof. Direct computation. Corollary 5.1. For sufficiently small a we have f ≥ 0. Proof. All coefficients of the polynomial in the square bracket of (87) are positive for 2 is seen to be sufficient). small enough a (the condition |a| < 21 l in addition to |a|l < rhoz Corollary 5.2. For sufficiently small a we have 16 g φφ

2 a (r − rhoz ) l2 r− < f. k0 l2 rhoz

(88)

Proof. Direct computation. Of course we recover our previous result (7) for a = 0. Note also that f grows slower (like ∼ r 2 ) than − (which grows like ∼ r 4 ). Going back to (85) we have the estimate rφ

2 (∂r φ) ∂φ ψ g − g t r

a 2 + a2 rhoz 2 2 8 a (r − rhoz ) l2 1 r− ≤ (∂r ψ)2 + g φφ ∂φ ψ φφ 2 g k0 l rhoz 8 ≤

2 f (∂r φ)2 1 φφ + g ∂φ ψ 2 8

(89)

for sufficiently small a using Corollary 5.2. It is here where the good decay of (85) is exploited, allowing us to borrow from f (and not the full − ) only. Now that positivity has been obtained for all first order terms we can turn to the zeroth order term and generalize the Hardy inequality (50). We note α l2

dθ dφ S2

α ≤ 2 l

S2

∞

2

ψ t , r, θ, φ rhoz ∞ dθ dφ sin θ dr r 2 + a 2 ψ 2 dr sin θ

rhoz

(90)

188

G. Holzegel

and estimate as previously, 2 ∞ 3 3 + 3a 2 (r − r r − rhoz α ∞ 2 4 hoz ) 2 2 dr r + a ψ ≤ α dr (∂r ψ)2 l 2 rhoz 9 rhoz r 2 + a2 l 2 ∞ 4 = α dr ( − − f ) (∂r ψ)2 (91) 9 rhoz using Lemma 5.1. We have obtained the Hardy inequality ∞ 2 α

ψ t , r, θ, φ dθ dφ dr sin θ 2 2 l S rhoz ∞ 4 sin θ ≤ α dθ dφ dr ( − − f ) (∂r ψ)2 . 9 S2 rhoz

(92)

Inserting the estimates (83), (84), (89) and (92) into (82) we finally have Proposition 5.1. For sufficiently small a the estimate ∞ µ µ

K Jµ (ψ) n = dθ dφ dr sin θ −g t t JµK n 2 S rhoz ∞ 1

− ≥b dθ dφ dr sin θ 2 (∂t ψ)2 + 2 (∂r ψ)2 r r rhoz S2 2 θ + ∂φ ψ + (∂θ ψ)2 2

θ sin θ

(93)

holds for a constant b which just depends on M, l and α < 49 . How large is a allowed to be? Note that in the case α ≥ 0 our estimates reduce to the simple geometric condition that K is everywhere timelike outside the black hole. 2 , the This in turn translates into a single inequality for a, thereby recovering |a|l < rhoz 9 result of [10]. If on the other hand 0 < α < 4 , the additional restriction of Corollary 5.2 has to be imposed. The reason is that the − (∂r φ)2 -term in the K -identity has to control both the mixed term (89) and the zeroth order term (92). We used a coefficient f (independent of α) to control the former and ( − − f ) to control the latter establishing that a uniform a can be chosen for α < 49 . Finally we compute the flux term on the horizon, where the normal is proportional to K : 2 a T (K , K ) = ∂φ ψ > 0. (94) ∂t ψ + 2 rhoz + a 2 H H 5.2. The vectorfield N. Now that we again have a positive, non-increasing K -energy we can invoke the identical argument as in the Schwarzschild case close to the horizon, involving the vectorfields Y and N = K + eY . This will eventually produce a bound ∞ 2 1 dθ dφ dr sin θ 2 (∂t ψ)2 + r 2 (∂r ψ)2 + ∂φ ψ 2 r

θ sin θ S2 rhoz

θ µ µ 2 N ≤B + Jµ (ψ) n ≤ B JµN (ψ) n 0 , (95) (∂θ ψ) 0

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

189

where the “bad” − -weight from the r -derivative term has now disappeared. Note that no further smallness restrictions on a arise as long as the horizon has positive surface gravity.

5.3. Higher order energies and pointwise bounds. We finally address the issue of pointwise bounds. Unfortunately, we can no longer exploit the commutation with the angular derivatives because of the broken spherical symmetry. However, we can certainly commute with T (or K respectively) to obtain integral bounds for certain higher order m the inequality energies. Given a solution of class Cdec

µ JµN K k ψ n ≤ B

0

µ JµN K k ψ n

(96)

for any non-negative integer k < m is an automatic consequence of the fact that K commutes with the wave operator. We write the wave equation as √ 1 √ ∂i g i j g∂ j ψ g = −g t

t

√ 1 α

(∂t ∂t ψ) − 2g t i ∂t ∂i ψ − √ ∂r g t r g (∂t ψ) − 2 ψ, g l

(97)

where i = 1, 2, 3 (or r, θ, φ respectively). Using both the decay assumed for ψ at infinity and that we control the right-hand side by (96), we obtain for any r0 > rhoz the elliptic estimate 1

i j kl

sin θ h h ∇ ψ) ∇ ∇ ψ −g t t dr dθ dφ (∇ i k j l

−g t t ∩{r ≥r0 >rhoz } µ µ

sin θ −g t t dr dθ dφ (98) JµN (ψ) n + JµN (K ψ) n ≤ C (r0 ) ∩{r ≥rhoz } away from the horizon, where h i j is the inverse of the induced metric on . Note that the argument breaks down at the horizon because of the degenerating weight of grr (whereas h rr is well-behaved there).8 To obtain good estimates close to the horizon we again adapt the ideas of [6,7]. Their resolution is to commute the equation with a version of the redshift vector field 1 1 Yˆ = ∂t − ∂r 2k0 2

(99)

which yields the equation 2 α 4r Yˆ ψ = + − Yˆ Yˆ (ψ) − Yˆ T (ψ) + λ1 (∂r ψ) + λ2 (∂t ψ) + 2 Yˆ (ψ), l

(100)

8 Actually, the estimate (98) remains true if one inserts an additional weight of r n for any n < √ max 2, 9 − 4α into the integrand on the left-hand side. This improvement is outlined in the Appendix where stronger weighted Sobolev norms are derived.

190

G. Holzegel

where λ1 =

1 1

∼ , 2 − r 0

λ2 = −

(101)

1 a 2 + 5r 2 ∼ 2, 2 l k0 r

(102)

and a prime denotes taking a derivative with respect to r . What will be crucial is that the coefficient of the first term on the right-hand side of (100) has a (good) sign close to the horizon. There is a geometric reason for this which was observed in [7]: Computing the surface gravity κ defined by (cf. [21]) ∂b K a K a = −2κ K b (103) r =rhoz

r =rhoz

we obtain κ=

− (rhoz ) 2 2 rhoz + a2

(104)

for the Kerr-AdS black hole under consideration. This means that on the horizon the coefficient of the Yˆ Yˆ (ψ)-term on the right-hand side of (100) is proportional to the surface gravity of the black hole, which is positive for a non-degenerate horizon! Remarkably, this observation is not bound to the special properties of the Kerr metric but a general fact about black hole event horizons with positive surface gravity. For details the reader may consult [7]. Let us investigate why this sign allows us to derive good estimates close to the horizon. The first step is to apply the identity (38) with the vectorfield multiplier N to Eq. (100): µ µ N ˆ N ˆ Jµ Y ψ n + Jµ Y ψ n H + K N Yˆ ψ τ H+ (0,τ ) R(0,τ )∩{r ≤r0 } µ =− K N Yˆ ψ + JµN Yˆ ψ n 0 R(0,τ )∩{r0 ≤r ≤r1 } 0 (105) + E N Yˆ ψ + E N Yˆ ψ , R(0,τ )∩{rhoz ≤r ≤r0 }

R(0,τ )∩{r >r0 }

where 2 2

− N − eYˆ Yˆ ψ Yˆ Yˆ (ψ) E N Yˆ ψ = −2e − Yˆ Yˆ (ψ) − 4r ˆ ˆ + N Y ψ Y T ψ − λ1 (∂r ψ) N Yˆ ψ − λ2 (∂t ψ) N Yˆ ψ α (106) − 2 N Yˆ ψ (Y (ψ)). l Let us start with the horizon term in (105). As previously (cf. (69)) this flux has a good sign except for the lowest order term when 0 < α < 49 .

Massive Wave Equation on Slowly Rotating Kerr AdS Spacetimes

191

This latter term is estimated as previously (cf. (70)) borrowing a bit from the good K N term: eγ M α ˆ 2 Yψ 2 H+ (0,τ ) 2r k0 l 1 τ N ˆ ≤ K Y ψ + B dt JµN (ψ) n µ ≤ K N Yˆ ψ + B Dτ, (107) 0 τ with D=

0

µ µ JµN (ψ) n 0 + JµN (K ψ) n 0 .

(108)

We turn to the E N -terms in (105). Note that N − eYˆ = K on the horizon. Hence in r ≤ r0 (where weights in r don’t matter) we have −

2 − N − eYˆ Yˆ ψ Yˆ Yˆ (ψ) ≤ B K + (1 − µ) Yˆ Yˆ ψ Yˆ Yˆ (ψ) 2 2 1ˆ ≤ Y K ψ + Yˆ Yˆ (ψ) (109)

by choosing r0 sufficiently close to the horizon to exploit the (1 − µ)-term as a smallness factor. We are going to choose small enough to borrow from the good first term in (106). Similarly, 2 2 2 4r ˆ ˆ N Y ψ Y T ψ ≤ Yˆ Yˆ (ψ) + B Yˆ K ψ + B|λ| · ∂φ Yˆ (ψ) . (110) Using several integrations by parts and the elliptic estimate (119) we derive the following estimate for the last term: 2 B|λ| ∂φ Yˆ (ψ) ≤ K N Yˆ ψ + B D (τ + 1) R(0,τ ) R(0,τ )∩{r ≤r0 } 1 1 1 µ µ µ + JµN Yˆ ψ )n H+ + JµN Yˆ ψ n + JµN Yˆ ψ n . (111) 8 H(0,τ ) 8 τ 8 0 Let us denote the three boundary-terms in the last line collectively by P1 . The other terms of E N involve lower order terms and are estimated via Cauchy’s inequality, putting a small weight on the Yˆ Yˆ -term so that one can borrow again from the good term. Hence finally R(0,τ )∩{r ≤r0 }

E N Yˆ ψ ≤

R(0,τ )∩{r ≤r0 }

K N Yˆ ψ

2 1 1 ˆ N Y K ψ + K (ψ) + B D (τ + 1) + P1 + R(0,τ )∩{r ≤r0 } ≤ K N Yˆ ψ + B D (τ + 1) + P1 .

R(0,τ )∩{r ≤r0 }

(112)

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G. Holzegel

The penultimate term in the second line deals in particular with the first order terms (which have the wrong sign if α > 0) arising in the K N terms. For r ≥ r0 we only have to be careful with the weights in r : E N Yˆ ψ

E N Yˆ ψ ≤ dt . (113)

−g t t R(0,τ )∩{r ≥r0 } τ

In the Ad S case g t t ∼ r12 , so in comparison with the asymptotically flat case (where this quantity approaches 1 at infinity) one loses a power of r . However, the decay of N E Yˆ ψ in r is easily seen to be strong enough to have

E N Yˆ ψ

E N Yˆ ψ ≤ dt

−g t t R(0,τ )∩{r ≥r0 } τ 0 τ µ JµN (ψ) n + JµN (K ψ) ≤ B Dτ, ≤ dt

τ

τ

0

(114)

using the elliptic estimate (98) above. For K N we have the analogous estimates to Lemma 4.2. In fact the lowest (first) order terms of the wrong sign can now be simply controlled τ by adding 0 dt τ JµN (ψ) n µ ≤ B Dτ to the right-hand side – a Hardy inequality is no longer necessary. Putting these estimates together the identity (105) turns into the estimate τ µ µ N ˆ Jµ Y ψ n + b dτ JµN Yˆ ψ n τ τ 0 µ ≤ B D (τ + 1) + 2 JµN Yˆ ψ n 0 . (115) 0

This is the analogue of (74) and hence µ N ˆ Jµ Y ψ n ≤ B D + τ

0

JµN

µ ˆ Y ψ n 0

(116)

follows as previously. We have finally obtained control over the rr derivative at the horizon. We now show how to control the second derivatives on the topological twospheres close to the horizon defined by constant (t , r ). For this we employ a second elliptic estimate. Write the wave equation as 1 1

∂θ (sin θ θ ∂θ ψ) + g φφ − 2λg t φ + λ2 g t t ∂φ2 ψ sin θ

= −g t t ∂t (K ψ) − 2g t r ∂r (K ψ) + g t t λ − 2g t φ ∂φ (K ψ) − grr ∂r ∂r ψ + ξ (r − rhoz ) ∂r ∂φ ψ + lower order terms,

(117)

where ξ is some bounded function. Note also the degenerating weight of grr on the horizon. From (117) we derive µ µ µ / 2 ψ|2 ≤ B JµN (ψ) n + JµN (K ψ) n + JµN Yˆ ψ n (118) |∇ Sr

Sr

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193

on a sphere of radius r . Close to the horizon we obtain µ / 2 ψ|2 ≤ B JµN (ψ) n + JµN (K ψ) + · JµN Yˆ ψ , (119) |∇ τ ∩{r ≤r0 }

τ ∩{r ≤r0 }

with arising from choosing r0 small enough to exploit the degenerating weights in (117). Together with the previous elliptic estimate finally 1 µ µ i j kl N N

h h (∇i ∇k ψ) ∇ j ∇l ψ + J (K ψ) n + J (ψ) n

−g t t µ µ µ JµN (ψ) n + JµN (K ψ) n + JµN Yˆ ψ n ≤B τ µ µ µ ≤B JµN (ψ) n + JµN (K ψ) n 0 + JµN Yˆ ψ n 0 . (120) 0

To write the estimate in a more geometric form we observe 1 µ ij

h ∇i ψ∇ j ψ ≤ B JµN (ψ) n

−g t t and, from (92),

1

−g t t

The factor of √ r2

ψ2 ≤ B

∞

S2

rhoz

∼ r arises because

1

−g t t

r 2 ψ 2 dr dω ≤ B √

(121)

µ

JµN (ψ) n .

(122)

h ∼ r for the hyperbolic metric (instead of

in the asymptotically flat case). In the other direction we have µ JµN Yˆ ψ n 1 µ µ N N i j kl ≤B h h (∇i ∇k ψ) ∇ j ∇l ψ , (123) J (K ψ) n + J (ψ) n +

−g t t

which is seen by direct computation taking care of the weights of r . Hence the inequality (120) becomes µ µ ψ J N (K ψ) n + J N (ψ) n + 2 Hw ( ) µ µ ≤ B ψ 2 J N (K ψ) n 0 + J N (ψ) n 0 (124) + Hw ( 0 )

with Hw2 denoting the n ψ

0

√ 1 -weighted −g t t

H⊥1

:=

≤ B ψ

Hw2 ( )

1

H 2 norm of . Finally, we define 2 r 2 ∇ · n ψ + (n ψ)2

−g t t µ µ J N (K ψ) n + J N (ψ) n +

and state the boundedness theorem for the massive wave equation on Kerr-AdS:

(125)

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G. Holzegel

Theorem 5.1. Fix a cosmological constant = − l32 , a mass M > 0 and some α < 49 . There exists an amax > 0, depending only on , M and α, such that the following statement is true for all a with |a| < amax . Let 0 = τ0 bea slice of constant t = τ0 in D = J + (I) ∩ J − (I) of the Kerr-anti 2 . If de Sitter spacetime M, g M,a, . Let ψ be a solution to (32) of class Cdec µ µ JµK (ψ) n 0 + JµK (K ψ) n 0 < ∞, (126) 0

0

then ψ

Hw2 ( )

+ n ψ

≤ C ψ˜

Hw2 ( 0 )

H⊥1 ( )

+

1 m=0

+ n ψ˜

H⊥1 ( 0 )

µ JµK K m ψ n +

1 m=0 0

µ JµK K m ψ n 0

(127)

holds on J + ( 0 ) ∩ D for a uniform constant C depending only on the parameters M, a, l and α. Here denotes any constant t -slice to the future of 0 . We remark that in contrast to the statement in the asymptotically flat case [7] we need the K boundary term in this estimate. This is because the weighted L 2 norm of the second time-derivative that one obtains from the current J K is stronger than what can be derived from the wave equation in combination with the boundedness of |ψ| Hw2 ( ) and |n ψ| H 1 ( ) alone. ⊥

r2

As mentioned previously, if α ≤ 0 then amax = hoz l since all that was needed in the proof is that the vectorfield K is timelike on the exterior (cf. [10]). In general, the restriction on amax depends on α and becomes tighter as α approaches the Breitenlohner Freedman bound, α → 49 . However, there is a uniform lower bound on amax (i.e. amax ≥ auni f or m > 0 for any α < 49 ) which can be worked out explicitly from the estimates of Sect. 5.1. The Sobolev embedding theorem for asymptotically hyperbolic space yields Corollary 5.3. We have the pointwise bound 1 |ψ| ≤ C ψ˜ 2 + n ψ˜ 1 + Hw ( 0 )

H⊥ ( 0 )

m=0 0

µ JµK K m ψ n 0

(128)

on J + ( 0 ) ∩ D. To prove the corollary we rely on the following general Sobolev embedding theorem (cf. Theorem 3.4 in [11]) Theorem 5.2. Let (N , h) be a smooth complete Riemannian 3-manifold with Ricci curvature bounded from below and positive injectivity radius and u ∈ H 2 (N ) a function on N . Then 2 2 sup |u| ≤ B |∇ j u|2 dv (h). (129) N

j=0

N

Remark. Our τ is only complete with respect to the asymptotically hyperbolic end, but it is straightforward to incorporate the boundary at r = rhoz > 0.

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6. Final Comments As mentioned in the abstract of the paper, the result does not make use of the separability properties of (2) with respect to the Kerr background. In fact it does not make use of the axisymmetry either! All that was needed was the causal Killing vectorfield K on the black hole exterior. In view of this fact, Theorem 5.1 can be stated in the following generalized setting: Fix the Killing vectorfield K = T + λ of a slowly rotating Kerr-AdS background as in Sect. 5. Perturb the metric such that it stays C 1 -close to the Kerr-AdS metric and such that K remains both Killing and null on the horizon.9 Then Theorem 5.1 remains true for such spacetimes. The main motivation for generalizations of this type are non-linear situations, in which the metric is not known explicitly a-priori but is itself dynamical. In view of this one should use as less quantitative assumptions on the metric as possible to obtain bounds on the fields. Compare [7] for a more detailed discussion. As a further generalization one may assume only an approximate causal Killing field, i.e. a vectorfield whose deformation tensor decays sufficiently fast in t. Treating the latter as a decaying error term in the estimates one can prove boundedness of (2) for all spacetimes approaching a spacetime that is C 1 -close in the sense above10 to a slowly rotating Kerr-AdS solution. The question whether ψ decays in time and if so, at what rate remains open. Acknowledgements. I would like to thank Mihalis Dafermos and Igor Rodnianski for stimulating discussions and useful comments. I am also grateful to an anonymous referee for his careful reading of the manuscript and many insightful remarks and suggestions.

A. Radial Decay In this Appendix we outline how – assuming the boundary conditions (33) – one can establish boundedness of appropriate higher weighted Sobolev-norms. The idea is the 3 , let T k (ψ), denote the application of k times following. Given a solution ψ of class Cdec the vectorfield T to ψ. We know that the T -energy associated with T k (ψ) is conserved for k = 0, 1, 2. If we now revisit the wave equation to do elliptic estimates, this will introduce natural weighted Sobolev norms whose r -weights depend on 0 < α < 49 . These optimized norms are expected to play a crucial role for the local existence theorem and are hence presented here. Recall from (97) that (for Kerr-AdS) one may write the wave equation in the form √ α 1 √ ∂i g i j g∂ j ψ + 2 ψ g l √ 1

= −g t t (∂t ∂t ψ) − 2g t i ∂t ∂i ψ − √ ∂r g t r g (∂t ψ). (130) g 2

The weighted of the right-hand side decays very fast in view of the boundedness L norm µ µ of JµK T k ψ n + JµK (ψ) n . In particular, denoting the right-hand side by f we have ∞ µ µ (131) dr dωr 4 f 2 < B JµK (T ψ) n + JµK (ψ) n . rhoz

S2

9 The C 1 regularity is necessary because the surface gravity, whose positivity was essential for the argument, is C 1 in the metric. 10 I.e. in particular admitting a timelike Killing field K on the black hole exterior which becomes null on the horizon.

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Let now χ = χ (r ) be a cut-off function which is equal to 1 for r > R for √ some fixed 2n R > rhoz and is zero close to the horizon. Set σ = n+3 for some n < 9 − 4α and multiply (130) by √ 1 α √ n ij g gχ ∂ ψ − χ σ ψ gr ∂ √ i j g l2 α √ n √ = χ f + (1 − σ )) 2 ψ gr + χ,r gr j ∂ j φ gr n . l Integrating over the slice , we observe (in view of (131)) that we can bound the right hand side as long as n ≤ 2 (provided we can borrow an of the term r 2+n ψ 2 dr dω from the left). The radial derivatives11 on the left-hand side can after several integrations by parts be estimated by ∞ dr dωr 6+n (∂r ∂r ψ)2 S2 R ∞ + dr dω (16 − 4 (5 + n) − α + ασ ) r 4+n (∂r ψ)2 S2 R ∞ (132) 1−σ 2+n 2 dr dωα −σ α + n (n + 3) r ψ 2 S2 R µ µ K K ≤B Jµ (T ψ) n + Jµ (ψ) n .

The boundary terms arising in the computations all vanish: on the left because of the √ cut-off function, and on the right in view of (33) (the restriction n < 9 − 4α is imposed in order to make these boundary terms vanish). Note also that the error-terms introduced by the derivatives of the cut-off function are well in the interior (bounded r ) and hence unproblematic. The coefficient of the zeroth order term in (132)√is positive if n(n+3) 2n n+3 = σ < 2α+n(n+3) , which an easy calculation reveals to be true for n < 9 − 4α. (In particular we have room to borrow the aforementioned of this term for the right-hand side.) The second term on the left of (132) is negative. How large can we allow n to be to still absorb this term by the first term using a Hardy inequality? As one easily checks, the Hardy condition (4 + 4n + α (1 − σ ))

4 (n + 5)2

0 for all ξ ∈ (0, ∞). Then the following hold for all N -chains for both the continuous-time process P t and its associated discrete-time process p n : (a) There is at most one invariant probability measure (which is therefore ergodic). (b) This measure has a density with respect to Lebesgue measure. The conditions above can be weakened as follows: There is a function φ(N ) = O( N12 ) such that (a) and (b) hold for the N -chain if L(ξ ) and R(ξ ) are strictly positive on open sets I L and I R , and there exist ξ ∈ I L ∪ I R and ξ ∈ I L ∩ I R such that ξ = φ(N )ξ . 1 . The form of our assumption An example of φ(N ) that is sufficient is φ(N ) = (18N )2 involving φ(N ) is quite possibly an artifact of our proof, but it is essential for L(ξ ) and R(ξ ) to have some spread. Our results imply, in particular, that the processes we consider have no singular invariant probability measures.

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We do not prove the existence of invariant measures in this paper. For existence, the two issues are: (1) tightness, which requires that one controls the dynamics of super-fast and super-slow particles. (2) Discontinuities of transition probabilities p X as a function of X due to same-site double collisions. Techniques for treating (1) in nonequilibrium situations are generally lacking, and (2) is likely to involve very technical arguments. We have elected to leave these problems for future work. Our next result gives an explicit formula for the equilibrium distribution in terms of bath distributions. In general, no such explicit expressions exist for nonequilibrium steady states, and our models appear to be no exception. Theorem 2. Let L(ξ ) = R(ξ ) = ρ(ξ ) be a probability density on (0, ∞) satisfying (i) 1 √ ρ(ξ )dξ < ∞ and (ii) ρ(ξ ) = O(ξ −κ ) for some κ > 3/2 as ξ → ∞. Then the ξ following measure is invariant for the (continuous-time) process defined in Section 1.1: N N 1 dξ1 · · · dξ N · ρ(ξi ) × dη0 · · · dη N · √ ρ(ηi ) × λ N +1 × ω N +1 , ηi i=1

where λ N +1 = Lebesgue measure on configurations in {(σ1 , · · · , σ N +1 )}.

i=0

[0, 1] N +1

and ω N +1 assigns equal weight to the

The following corollary suggests some examples to which our results apply: Corollary 1. Any finite chain with bath injections L(ξ ) = R(ξ ) given by either (a) an exponential distribution or (b) a Gaussian truncated at 0 and normalized has a unique (hence ergodic) equilibrium distribution, and it is given by the expression in Theorem 2. 3. Density and Ergodicity: Outline of Proof For definiteness, we will work with the discrete-time Markov chain on M+ . By Lemma 1.1, the assertions in Theorem 1 for p n imply the corresponding assertions for P t . Proving uniqueness of invariant measures or ergodicity requires, roughly speaking, that we be able to steer a trajectory from one location of the phase space to another. If the transition probabilities have densities on open sets, then one needs only to do so in an approximate way. Our model, unfortunately, has highly degenerate transition probabilities. One must, therefore, tackle hand in hand the problems of (i) acquisition of densities for p nX and (ii) steering of trajectories. The purpose of this section is to outline how we plan to do this. Section 3.1 introduces definitions and ideas that will be used. In Section 3.2, we formulate three propositions to which the proof of Theorem 1 will be reduced. 3.1. Basic ingredients of the proof. (A) Acquiring densities. Given a finite Borel measure ν on M+ , we let ν = ν⊥ + νac denote the decomposition of ν into a singular and an absolutely continuous part with respect to Lebesgue measure, and say ν has an absolutely continuous component when νac = 0. We say X ∈ M+ eventually acquires a density if for some n > 0, p nX has an absolutely continuous component, i.e. ( p nX )ac = 0.

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Lemma 3.1. If every X ∈ M+ acquires a density eventually, then every invariant probability measure of p n has a density. Proof. For any measure ν, if ν is absolutely continuous with respect to Lebesgue, then so is νn := p nX dν(X ). This is because under the dynamics, Lebesgue measure is carried to a measure equivalent to Lebesgue, followed, at certain steps, by a diffusion in one or two directions corresponding to bath injections. Thus (νn )ac (M+ ) ≥ νac (M+ ) for all n ≥ 1. The hypothesis of this lemma implies that this inequality is strict for some n unless ν⊥ = 0. Now let µ be an invariant probability measure for p n . Since µn = µ, it follows that µ⊥ = 0.

Recall that I L = {L > 0} and I R = {R > 0}. We call a finite or infinite sequence of points X 0 , X 1 , . . . in M+ a sample path if such a sequence can, in principle, occur. In particular, if X n−1 is followed by a collision with the right bath, then the ξ N -coordinate of X n must lie in I R , and similarly with the left bath. Let X = (X 0 , X 1 , . . . , X n ) be a sample path obtained by injecting into the system the energy sequence ε = (ε1 , . . . , εm ) in the order shown. (One does not specify whether the injection is from the left or the right; it is forced by the sequence.) We write

X 0 (ε) = X n . Suppose X has no multiple collisions. It is easy to see that there is a small neighborhood E of ε in Rm such that for all ε ∈ E, (i) ε is a feasible sequence of injections, i.e., if the sequence dictates that εj be injected from the left (right) bath, then εj ∈ I L (resp. I R ), and (ii) the sample path X = (X 0 , X 1 , . . . , X n ) produced by injecting ε has the exact same sequence of collisions as X . In particular, it also has no multiple collisions. We may therefore extend = X 0 to a mapping from E to M+ with (ε ) = X n . As such, is clearly continuous. Lemma 3.2. : E → M+ is continuously differentiable. In the proof below, it will be useful to adopt the viewpoint expressed in Section 1.3, i.e. to track the movements of the injected energies through time. Notice that for a sample path with no multiple collisions, there is no ambiguity whatsoever about the trajectory of an injected energy. ∂ xi i ∂ξi Proof. We verify that ∂η δε j , δε j and δε j exist and are continuous on E. For ηi and ξi , it is easy: either ε j is carried by the particle in question, or it is not. If it is, then the partial derivative is = 1; if not, then it is = 0. For xi , consider first the case where ηi carries the injection εk , k = j. We let p be the total number of times ε j crosses the interval [i, i + 1] (in either direction) before X n . ∂ xi = 0. Suppose not. Then perturbing (only) ε j to ε j + δε, the time If p = 0, then δε j

gained per crossing (with a sign) will be ∂ xi √ = lim p εk · δε→0 δε j

√1 εj

√1 εj

−√

−√

1 . ε j +δε

1 ε j +δε

δε

The case where ε j is carried by ηi is left to the reader.

Consequently

=

p 21 − 32 ε ε . 2 k j

In the setting above, if D (ε) is onto as a linear map, then by the implicit function theorem, there is an open set E ⊂ E with ε ∈ E such that carries Lebesgue measure on E to a measure in M+ with a strictly positive density on a neighborhood of (ε). This implies in particular ( p nX 0 )ac > 0. We summarize as follows:

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Definition 3.1. We say X ∈ M+ has full rank2 if there is a sequence of injections ε = (ε1 , · · · , εm ) leading to a sample path X = (X 0 , X 1 , . . . , X n ) with X 0 = X such that (i) X has no multiple collisions, and (ii) D X 0 (ε) is onto. Corollary 3.1. If X ∈ M+ has full rank and X is as above, then p nX has strictly positive densities on an open set containing X n . Having full rank is obviously an open condition, meaning if X has full rank, then so does Y for all Y sufficiently close to X . (B) Ergodic components. One way to force two points to be in the same ergodic component (in the sense to be made precise) is to show that they have “overlapping futures”. This motivates the following relation: Definition 3.2. For X, Y ∈ M+ , we write X ∼ Y if there exist a positive Lebesgue n measure set A = A(X, Y ) ⊂ M+ and m, n ∈ Z+ such that ( p m X )ac |A and ( pY )ac |A have strictly positive densities. Here is how this condition will be used: Suppose µ and ν are ergodic measures, and there is a positive µ-measure set Aµ and a positive ν-measure set Aν such that X ∼ Y for all X ∈ Aµ and Y ∈ Aν . Then the ergodic theorem tells us that µ = ν since they have the same ergodic averages along positive measure sets of sample paths. As an immediate corollary of the ideas in Part (A), we obtain Corollary 3.2. If X ∈ M+ has full rank, then there is a neighborhood N of X such that Y ∼ Z for all Y, Z ∈ N . (C) Constant energy configurations. For fixed e ∈ (0, ∞), let Qe = {X ∈ M+ : ηi = ξi = e for all i}. If we start from X ∈ Qe and inject only energies having value e, then the resulting sample path(s) will remain in the set Qe . (Obviously, this is feasible only if e ∈ I L ∩ I R .) We call these constant-energy sample paths, and say X ∈ Qe has no multiple collisions if that is true of its constant-energy sample path. Needless to say, constant-energy sample paths occur with probability zero. However, they have very simple dynamics, and perturbations are relatively easy to control. Our plan is to exploit these facts by driving all sample paths to some Qe and to work from there. 3.2. Intermediate propositions. We claim that Theorem 1 follows readily from the following two propositions: Proposition 3.1. Every X ∈ M+ eventually acquires a density. Proposition 3.2. X ∼ Y for a.e. X, Y ∈ M+ with respect to Lebesgue measure. We remark that to rule out the presence of singular invariant measures, we need Proposition 3.1 to hold for every X , not just almost every X . 2 This property has the flavor of Hörmander’s condition for hypoellipticity for SDEs – in a setting that is largely deterministic and has discontinuities.

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Proof of Theorem 1 assuming Propositions 3.1 and 3.2. Proposition 3.1 and Lemma 3.1 together imply that every invariant probability measure has a density, proving part (b). To prove part (a), let µ and ν be ergodic measures. Since they have densities, Proposition 3.2 implies, as noted in Paragraph (B), that µ = ν.

Next we identify three (more concrete) conditions that will imply Propositions 3.1 and 3.2. The following shorthand is convenient: For X, Y ∈ M+ , we write X ⇒ Y if given any neighborhood N of Y , there exists n such that p nX (N ) > 0, X Y if given any neighborhood N of Y , there is a neighborhood N of X such that X ⇒ Y for all X ∈ N . Proposition 3.3. Given X ∈ M+ and e ∈ I L ∩ I R , there exists Z = Z (X ) ∈ Qe such that X ⇒ Z . Proposition 3.4. Z Z for all Z , Z ∈ Qe , any e ∈ I L ∩ I R . Proposition 3.5. Every Z ∈ Qe with no multiple collisions has full rank. Proof of Propositions 3.1–3.2 assuming Propositions 3.3–3.5. To prove Proposition 3.1, we concatenate Propositions 3.3 and 3.4 to show that for every X ∈ M+ , X ⇒ Z for some Z = Z (X ) ∈ Qe with no multiple collisions. The assertion is proved if there is a neighborhood N Z of Z such that every Z ∈ N Z eventually acquires a density. This follows from Proposition 3.5, Corollary 3.1 and the fact that full rank is an open condition. To prove Proposition 3.2, it is necessary to produce a common Z = Z (X, Y ) ∈ Qe with no multiple collisions such that X, Y ⇒ Z . To this end, we first apply Proposition 3.3 to produce Z (X ) ∈ Qe with X ⇒ Z (X ) and Z (Y ) ∈ Qe with Y ⇒ Z (Y ). We then fix an (arbitrary) Z ∈ Qe with no multiple collisions. Since Z (X ), Z (Y ) Z (Proposition 3.4), we have X, Y ⇒ Z as before. Proposition 3.5 and Corollary 3.2 then give the desired result.

Proving Theorem 1 has thus been reduced to proving Propositions 3.3–3.5. 4. Proofs of Propositions 3.3–3.5 In Sections 4.1–4.2, we give an algorithm for driving sample paths from given initial conditions to constant-energy surfaces. This can be viewed as changing the energies in a configuration. In Sections 4.3–4.4, we focus on changing the relative positions of the moving particles. 4.1. Sample paths from X to Qe (“typical” initial conditions). The hypothesis of Theorem 1 guarantees the existence of some ξ ∈ I L ∩ I R . We may assume, without loss of generality, that ξ = 1. Given X ∈ M+ , we seek sample paths that lead to the constant energy surface Q1 . Consider first the following specific question: By injecting only particles with energy 1, will we eventually “flush out” all the energies in X , replacing them with new particles having energy 1? To study this question, we propose to suppress some information, to focus on the following evolution of arrays of energies: To each X ∈ M+ , we associate the array E(X ) = (η0 , ξ1 , η1 , ξ2 , . . . , ξ N , η N ), where ηi and ξi are the energies of X and they are arranged in the order shown. Then corresponding to each sample path X = (X 0 , X 1 , . . .)

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is the sequence of moves E(X 0 ) → E(X 1 ) → E(X 2 ) → · · ·. For example, collision between site i and the particle on its right corresponds to swapping the (2i + 1)st entry in the array with the (2i)th ; the rightmost particle exiting the system and an energy of value ξ R entering corresponds to replacing the (2N + 1)st entry by ξ R , and so on. For as long as there are no same-site double collisions, we can trace the movements of energies in (E(X 0 ), E(X 1 ), · · · ) as discussed in Section 1.3. We define the exit time of the j th entry in E(X 0 ) to be the number of moves before this energy exits the system, and define T (X ) to be the last exit time of all the elements in E(X 0 ). A priori, T (X ) ≤ ∞. Lemma 4.1. Given X 0 ∈ M+ , suppose that injecting all 1s gives rise to a sample path X with no same-site double collisions. Then T (X ) < ∞. At each step, we will refer to the original energies of X 0 that remain as the “old energies”, and the ones that are injected as “new energies”. In particular, all new energies have value 1 (some old energies may also have value 1). The lemma asserts that in finite time, the number of old energies remaining will decrease to 0. Proof. For each n, we let E(X n ) = (η0 , ξ1 , η1 , ξ2 , · · · , ξ N , η N ), and say X n is in state (·, i) if the leftmost old energy is ξi , (+, i) if the leftmost old energy is ηi and σi = +, (−, i) if the leftmost old energy is ηi and σi = −. The terminology above is not intended to suggest that we are working with a reduced system. No such system is defined; nevertheless it makes sense to discuss which transitions among these states are permissible in the underlying Markov dynamics. Notice first that independent of the state of a system, it will change eventually. This is because all energies are nonzero, so a collision involving the leftmost old energy is guaranteed to occur at some point. We list below all the transitions between states that are feasible, skipping over (many) steps in the Markov chain that do not involve the leftmost old energy: (1) Suppose the system is in state (−, i). If i = 0, then the only possible transition is (−, i) → (·, i). If i = 0, then the leftmost old energy exits the system, and the new state is determined by the next leftmost old energy (if one remains). (2) Suppose the system is in state (+, i). If i = N , then T (X ) is reached as this last remaining old energy exits the system. If not, we claim the only two possibilities are (+, i) → (·, i + 1) and (+, i) → (−, i). The first case corresponds to the energy originally at site i + 1 being new, the second case old. (3) Finally, consider the case where the system is in state (·, i). If the next collision is with the particle from the left, then (·, i) → (−, i − 1). If it is with the particle from the right, then there are two possibilities corresponding to the approaching energy being new or old, namely (·, i) → (+, i) or (·, i) → (·, i). Notice that the system cannot remain in state (·, i) forever, since the particle from the left will arrive sooner or later, causing the state to change. For this reason, let us agree not to count (·, i) → (·, i) as a transition. To summarize, the only possible transitions are (−, i) → (·, i), (·, i) → (−, i − 1), (+, i), (+, i) → (·, i + 1), (−, i) except where the transition leads to an old energy exiting the system. When that happens, either there is no old energy left, or the system can start again in any state. The following observation is crucial:

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Sublemma 4.1. The transition sequence (−, i) → (·, i) → (+, i) is forbidden. We first complete the proof of Lemma 4.1 assuming the result in this sublemma: Observe first that if one starts from (+, i), either the evolution is (+, i) → (·, i + 1) → (+, i + 1) → (·, i + 2) → · · · leading to an exit at the right, or a state of the form (−, j) is reached; a similar assertion holds if one starts from (·, i). On the other hand, starting from (−, i), the only possible sequence permitted by the sublemma is (−, i) → (·, i) → (−, i − 1) → (·, i − 1) → · · · leading to an exit at the left. This proves that in finite time (meaning a finite number of steps with respect to the Markov chain p n ), the number of old energies remaining will decrease by one.

Proof of Sublemma. Suppose the transition (−, i) → (·, i) takes place at t = 0. Since all the energies to the left of site i are new and therefore have speed one, a particle from the left is guaranteed to reach site i at time t = t0 < 2, since the previous collision with the site from the left took place strictly before t = 0 (no same-site double collisions). We will argue that when this particle arrives, it will find an old energy at site i, that in fact the state of the system has not changed between t = 0 and t0 . Thus the next transition has to be (·, i) → (−, i − 1). Here are the events that may transpire between t = 0 and t0 on the segment [i, i + 1]: Observe that the transition at t = 0 is necessarily between an old and a new energy (otherwise the state at t = 0− could not be (−, i)). This will result in a new energy leaving site i for site i + 1 at t = 0+ , and arriving at t = 1. The only way (·, i) → (+, i) can happen is for a new energy to move leftward on [i, i + 1], and to arrive at site i before time t0 . This cannot happen, since new energies travel at unit speed. (Notice that the old energy at site i may change between t = 1 and t0 , but the state of the system does not.)

Partial proof of Proposition 3.3. We prove the result for X = X 0 under the additional assumption that injecting 1s gives rise to a sample path with no same-site double collisions. By Lemma 4.1, there is a sample path X = (X 0 , . . . , X n ) with X n ∈ Q1 . Let N be a neighborhood of X n . If X has no multiple collisions, then all nearby sample paths will end in N as discussed in Section 3.1. If X has multiple (but not same-site) collisions,3 then injecting a slightly perturbed energy sequence may – is likely to, in fact – desynchronize the simultaneous jumps. If, for example, two collisions at sites i and i occur simultaneously at step j, then for perturbed injected energies, X j may be replaced by X j and X j corresponding to two collisions that happen in quick succession. This aside, the situation is similar to that with no multiple collisions, and we still have p kX (N ) > 0 but possibly for some k > n.

4.2. Sample paths from X to Qe (“exceptional” cases). As noted in Section 3.2, to rule out the existence of singular invariant measures, it is necessary to show that every X ∈ M+ eventually acquires a density. Our strategy is to inject energy 1s into the system and argue that this produces a sample path X that leads to a point Z ∈ Q1 . To prove X ⇒ Z , however, requires more than that: it requires that a positive measure set of sample paths starting from X follow X . This is not a problem if X has no same-site double collisions, for in the absence of such collisions, the dynamics are essentially continuous (as explained above). Yet it is unavoidable that for some X ∈ M+ , injecting 1s will lead 3 These points are discontinuities only for the (discrete-time) Markov chain, affecting the number of steps. They are not discontinuities at all for the continuous-time jump process.

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to same-site double collisions. It is an exceptional situation, but one that we must deal with if we are to follow the same route of proof. We now address the potential problems. Recall that at each same-site double collision, there are two ways to continue the sample path. Partitioning nearby sample paths according to itineraries. Let X = (X 0 , X 1 , . . . , X n ) be a sample path obtained by injecting a sequence of m energies all of which are 1s and making a specific choice at each same-site double collision. As in Section 3.1(A), we consider energies ε = (ε1 , . . . , εm ) near 1 = (1, . . . , 1), and ask the following question: Which injection sequence ε will give rise to a sample path X (ε) that has the same collision sequence as X ? To focus on the issues at hand, we will ignore multiple collisions that do not involve same-site double collisions, for they are harmless as explained earlier. The computation in Lemma 3.2 motivates the following coordinate change, which is not essential but simplifies the notation: Let ψ(ε) = √1ε − 1, and let (ε1 , . . . , εm ) = (ψ(ε1 ), . . . , ψ(εm )). Then maps a neighborhood of 1 diffeomorphically onto a neighborhood of 0 = (0, . . . , 0). We assume that X (ε) has the same itinerary as X through step i − 1, and that at step i, X has a double collision at site j. To determine whether the left or right particle will arrive first at site j for X (ε) , we use X as the point of reference, and let t j (ε) and t j−1 (ε) be the times gained (with a sign) by the particles approaching site j from the right and left respectively. Then t j (ε) = p1 ψ(ε1 ) + · · · + pm ψ(εm ) and t j−1 (ε) = q1 ψ(ε1 ) + · · · + qm ψ(εm ), where pk and qk are the numbers of times – up to and including the approach to site j – the injected energy εk has passed through the intervals [ j, j + 1] and [ j − 1, j] respectively. Thus for X (ε) , the right particle arrives first if and only if t j (ε) > t j−1 (ε). Thus given X = (X 0 , X 1 , . . . , X n ) as above, there is a decreasing sequence of subsets Rm = V0 ⊃ V1 ⊃ V2 ⊃ · · · ⊃ Vn defined as follows: Vi = Vi−1 except where a same-site double collision occurs; when a same-site double collision occurs, say at site j, we let Vi = Vi−1 ∩ H , where H = {ε ∈ Rm : t j (ε) ≥ t j−1 (ε)} or {t j (ε) ≤ t j−1 (ε)} depending on whether we have chosen to let the right or left particle arrive first in X . If there are multiple same-site double collisions, then we intersect with the half-spaces corresponding to all of them. The sequence of Vi so obtained will have the property that all ε close enough to 1 with (ε) ∈ Vn give sample paths that have the same collision sequence as X . In particular, if Vn is nontrivial, meaning it has interior, then the set of ε for which X (ε) shadows X will have positive measure. That is to say, X 0 ⇒ X n . In general, there is no guarantee that Vn is nontrivial. Supposing Vn for a sample path is nontrivial, we say the choices made at step n + 1 are viable if they lead to a nontrivial Vn+1 . Observe that inductively, a viable choice can be made at each step, since one of the two half-spaces must intersect nontrivially Vn from the previous step. We first treat a special situation involving the same-site collision of three energies all of which are 1s. The setting is as above. Sublemma 4.2. Suppose Vn is nontrivial for a sample path X , and at step n + 1, the energy εk , which is = 1, is involved in a same-site double collision with two other energies both of which are 1s. Then the choice to have εk arrive first is viable if εk has the following history:

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(i) its movement up to the collision above is monotonically from left to right; (ii) all of its previous same-site double collisions are with two other energy 1s. Proof. By (i), we have that in the collision at step n + 1, pk = 0 and qk = 1. Choosing to have εk arrive first corresponds to choosing the half-space {(εk ) ≥ G(ε)}, where G(ε) is a linear combination of (εi ), i = k. By (ii), all the other half-spaces in the definition of Vn either do not involve εk or are of the same type as the one above.

Completing the proof of Proposition 3.3. Given X 0 ∈ M+ , we now inject 1s and make a choice at every same-site double collision with the aim of obtaining a sample path X 0 , X 1 , . . . , X n with X n ∈ Q1 and X 0 ⇒ X n . Our plan is to follow the scheme in Section 4.1 to reach Q1 , and to make viable choices along the way. The proof of Lemma 4.1 is based on the following observation: the leftmost old energy either moves right monotonically until it exits, or it turns around and starts to go left, and by Sublemma 4.1, once that happens it must move left monotonically until it exits. Revisiting the arguments, we see that with the exception of Sublemma 4.1, all statements in the proof of Lemma 4.1 hold for any sample path, with whatever choices are made at double collisions. Moreover, the only way Sublemma 4.1 can fail is that at t = 0, a double collision occurs at site i, we choose to have the left particle arrive first, and this is followed by another double collision at t = 2 at which time we choose to have the right particle arrive first. It is only through these “two bad decisions” that the leftmost old energy can turn around and head right again. The choices at same-site double collisions are arbitrary provided the following two conditions are met: (i) The choice must be viable. (ii) In the setting of Sublemma 4.2, we choose to have εk arrive first. Following these rules, it suffices to produce X n ∈ Q1 for some n; the nontriviality of Vn follows from (i). Suppose, to derive a contradiction, that no X n ever reaches Q1 , i.e., a set of old energies is trapped in the chain forever. Let i 0 be the leftmost site that it visits infinitely often. That means from some step n 0 on, the leftmost energy never ventures to the left of site i 0 , but it returns to site i 0 infinitely many times, each time repeating the “two bad decisions” scenario above. The situation to the left of site i 0 is as follows: From time n 0 on, all the energies strictly to the left of this site have speed one. Thus by rule (ii), after a new energy enters from the left bath, it will march monotonically to the right until at least site i 0 , whether or not it is involved in any same-site double collisions. To complete the proof, consider a moment after time n 0 when the leftmost old energy arrives at site i 0 . From the discussion above we know it has to be involved in a samesite double collision, and that the energy approaching from the left is new. By (ii), we choose to have the new energy arrive first, making the first “bad decision”. But then in the collision at the same site 2 units of time later, again we choose to have the energy from the left arrive first, causing the old energy to move to site i 0 − 1 and contradicting the definition of site i 0 .

4.3. Moving about on constant energy surfaces. This subsection focuses on getting from Z ∈ Qe to Z ∈ Qe for e ∈ I L ∩ I R . In general, this cannot be accomplished by considering sample paths that lie on Qe alone, since such sample paths are periodic as we will see momentarily. Instead we will make controlled excursions from Qe aimed at returning to specific target points. As before, we assume 1 ∈ I L ∩ I R and work with Q1 .

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Circular tracks between sites. It is often useful to represent the pair (xi , σi ) by a single coordinate z i and to view the particle as making laps in a circle of length 2. More precisely, if we represent this circle as [0, 2]/ ∼ with end points identified, then z i = xi for σi = + and z i = 2 − xi when σi = −. A continuous-time sample path Z (t), t ≥ 0, for the process described in Section 1 then gives rise to a curve Z ∗ (t) = (z 0 (t), z 1 (t), . . . , z N (t)) ∈ T N +1 . (We write T N +1 even though each factor has length 2.) For each i, z i (t) goes around the circle in uniform motion, changing speed only at z i = 0 and 1. Accordingly, we denote by ∂(T N +1 ) the set of points such that z i = 0 or z i = 1 for (at least) one i. Asterisks will be used to signify the use of circular-track notation: A discrete-time sample path Z = (Z 0 , Z 1 , · · · ) in the notation of previous sections corresponds to Z ∗ = (Z 0∗ , Z 1∗ , · · · ) with Z i∗ ∈ ∂(T N +1 ). Returns of Z to M+0 = M+ ∩ {x0 = 0} translates into returns of Z ∗ to ∂(T N +1 ) ∩ {z 0 = 0}. We now identify two features of constant energy configurations that make them easy to work with, beginning with the simplest case: Lemma 4.2. Suppose we start from Z ∈ Q1 with no multiple collisions. (a) By injecting all 1s, the sample path returns to Z in 2N + 2 steps. (b) If all the injected energies are 1s except possibly for one, which we call e, and if we assume the resulting sample path has no same-site double collisions, then the energy e moves monotonically through the system until it exits, after which the system rejoins the periodic sample path in (a). Proof. (a) This is especially easy to see in continuous time and in circular-track notation: each z i (t) is periodic with period 2. Thus the continuous-time sample path Z (t) is periodic with period 2, and in every two units of time, there are exactly 2N +2 collisions. (b) Suppose the energy e has just arrived at site j from the left. Then the next collision at site j is with the particle from the right, because all cycles take 2 units of time to complete and the right one has a (strict) headstart. This proves the monotonic movement of the energy e. After it leaves the system, the relative positions between z i and z j remain unchanged for all i, j, since the passage of the energy e through the chain leads to identical time gains for z i and z j .

Another nice property of constant energy sample paths is that the process is continuous along these paths. We make precise this idea: Definition 4.1. Given a sample path X = (X 0 , X 1 , . . . , X n ), we say the process is continuous at X if the following holds: Let ε = (ε1 , . . . , εm ) be the sequence in injections. Then given any neighborhood N of X n , there are neighborhoods N0 of X 0 and U of ε such that for all X 0 ∈ N0 and ε ∈ U, all the sample paths generated follow X into N . (We do not require that it reaches N in exactly n steps; it is likely to take more than n steps if X has multiple collisions.) Lemma 4.3. Let X = (X 0 , . . . , X n ) be a sample path. (a) If X either has no same-site double collisions or every such collision involves three identical energies, then the process is continuous at X . (b) If the process is continuous at X , then X 0 X n .

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The ideas behind part (a) are the same as those discussed in Sections 4.1 and 4.2 and will not be repeated here. Part (b) is immediate, for “” requires less than continuity. The following lemma is the analog of Lemma 4.2(b) without the assumption of “no multiple collisions”: Lemma 4.4. Given any Z ∈ Q1 , let all the energies injected be 1s except possibly for one which has value e. Then (i) there is a sample path in which the energy e moves through the system monotonically; (ii) except for a discrete set of values e, it will do so without being involved in any same-site double collisions. Proof. The monotonic motion of e is clear if it is not involved in any same-site double collisions. If it is, choose to have it arrive first every time. This proves (i). To prove (ii), let J be a finite interval of possible energies to be injected at t = 0, and suppose e is on its way from site 0 to site 1. In circular-tracks notation, z 1 is periodic with period 2, and except for a finite subset of e ∈ J , z 0 = 1 at exactly the same time that z 1 = 0. We avoid this “bad set” of e, ensuring that the special energy will not be in a double collision at site 1. Now this energy leaves site 1 exactly when z 1 = 0, and at such a moment, the configuration to the right of site 1 is identical (independent of the value of e in use). The same argument as before says that to avoid a double collision at site 2, another finite subset of J may have to be removed.

Injecting a single energy different than 1 will not produce the desired positional variations. We now show that injecting a second energy – both appropriately chosen and injected at appropriate times – will do the trick. The hypothesis of Theorem 1 guarantees, after scaling to put 1 ∈ I L ∩ I R , that there exists e ∈ I L ∪ I R with e = o(1/N 2 ). (Lemma 4.5 below is the only part of the proof in which this hypothesis is used.) Let us assume for definiteness that e ∈ I L , and consider initial conditions Z ∈ M+0 = M+ ∩ {x0 = 0}. Lemma 4.5. There exists a0 > 0 for which the following hold: Given a with |a| < a0 , Z 0 ∈ Q1 ∩ M+0 and k ∈ {1, . . . , N }, there is a sample path Z = (Z 0 , . . . , Z n ) with Z n ∈ Q1 ∩ M+0 such that if Z 0∗ = (0, z 1 , . . . , z N ), then Z n∗ = (0, z 1 , . . . , z k−1 , z k − a, z k+1 − a, . . . , z N − a). Moreover, the sample path Z may be chosen so that it has no same-site double collisions except where all the energies involved are 1s. Proof. First we focus on constructing a sample path that leads to the desired Z n without attempting to avoid double collisions. In the construction to follow, all the injections except for two will be 1s. At t = 0, we inject energy e = o(1/N 2 ) from the left bath, letting it move monotonically from left to right. When it is about halfway between sites k and k + 1, we inject from the left an energy e = e(a) ≈ 1 (the relation between e and a will be clarified later). The energy e also moves monotonically from left to right until it reaches the k th site, which takes O(N ) units of time. Since the slow particle takes longer than O(N ) units of time to reach site k + 1, the e-energy waiting at site k is met by a particle from the left. So it turns around and moves monotonically left, eventually exiting at the left end. As for the slow energy, it eventually reaches site k + 1, and continues its way monotonically to the right until it exits the chain. Let Z n be the first return to M+0 after the exit of e (the e-energy exits much earlier). By definition, Z n ∈ Q1 . It remains to investigate its z i -coordinates. Reasoning as in

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the proof of Lemma 4.2, we see that the e -energy does not cause a shift in the relative positions of the z i because its effects on all of the z i are identical. The e-energy, on the other hand, has made one full lap (first half on its way to site k and second half on its way back) on the segments [ j, j + 1] for j = 0, 1, . . . , k − 1 but not for j ≥ k. Let 1 a = 2(1 − e− 2 ) be the (signed) time gain per lap for a particle with energy e over a particle with energy 1. Then the position of z 0 relative to z j for j < k is maintained while relative to z j for j ≥ k, z 0 is ahead by distance a, leading to the coordinates of Z n as claimed. We now see that the bound imposed on a is that e(a) must be in I L ∩ I R . This completes the proof of the lemma except for the claim regarding double collisions. From Lemma 4.4(b), we know that for e outside of a discrete set of values, the particle carrying this special energy will not be involved in any same-site double collisions. It remains to arrange for the energy e to avoid such collisions. Here the situation is different: since a is given, e is fixed, but notice that increasing e is tantamount to delaying the injection of the energy e. Thus an argument similar to that in Lemma 4.4(b) can be used to show that by avoiding a further discrete set of values for e (depending on Z 0 , e and k), the energy e will not be involved in same-site double collisions. This completes the proof.

Remark. The following two facts make this result very useful: (a) The dynamical events described in Lemma 4.5 require no pre-conditions on Z 0 other than Z 0 ∈ Q1 ∩ M+0 . (b) The range of admissible a is independent of Z 0 . Suppose a > 0, and for some j > k, the j th moving particle in Z 0 is such that x j < a and σ j = +. Then it will follow that σ j = − in Z n , since it is distance a behind its original position. This is very natural when one thinks about it in terms of circular tracks, but it is a jump in the phase space topology in Section 1.1.

4.4. Proofs of Propositions 3.4 and 3.5. Proof of Proposition 3.4. Given Z , Z ∈ Q1 , we will construct a sample path Z = (Z 0 , Z 1 , · · · , Z n ) with Z 0 = Z , Z n = Z and Z 0 Z n . Since all constant energy sample paths pass through M+0 , we may assume Z , Z ∈ M+0 . Let Z ∗ = (0, z 1 , . . . , z N ) and (Z )∗ = (0, z 1 , . . . , z N ). Our plan is to apply Lemma 4.5 as many times as needed to nudge each z i toward z i one i at a time beginning with i = 1. Write z 1 − z 1 = ja for some j ∈ Z+ and a small enough for Lemma 4.5. Applying Lemma 4.5 j times with k = 1, we produce Z n 1 ∈ Q1 ∩ M+0 with Z n∗1 = (0, z 1 , z 2 + (z 1 − z 1 ), . . . , z n + (z 1 − z 1 )). We then apply Lemma 4.5 to Z n 1 with k = 2 repeatedly to change it to Z n 2 with Z n∗2 = (0, z 1 , z 2 , z 3 + (z 2 − z 2 − (z 1 − z 1 )), . . . , z n + (z 2 − z 2 − (z 1 − z 1 ))), and so on until Z n = Z n N is reached. Notice that the sample path Z so obtained has the property that it has no same-site double collisions other than those that involve three identical energies (all of which 1s). Lemma 4.3 says that the dynamics are continuous at such sample paths. Hence Z Z .

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Proof of Proposition 3.5. Given Z ∈ Q1 with no multiple collisions, our aim here is to produce a sample path Z 0 = Z , Z 1 , . . . , Z n , also free of multiple collisions, by injecting a sequence of energies ε = (ε1 , . . . , εm ) so that in the notation of Section 3.1, the map

has the property that D (ε) is onto. We first give the algorithm, with explanations to follow: (1) We inject 1s until n 0 , the first time Z n 0 ∈ M+0 . (2) At step n 0 , we inject into the system an energy e = o(1/N 2 ) chosen to avoid multiple collisions, followed by all 1s, until the first time the sample path returns to Q1 ∩ M+0 ; call this step n 1 . (3) By Lemma 4.1, T (Z n 1 ) < ∞. Let n be the smallest integer ≥ n 1 + T (Z n 1 ) for which Z n ∈ M+0 . We now prove that D (ε) has full rank: Let δηi denote an infinitesimal displacement in the ηi variable at Z n . To see that δηi is in the range of D (ε), we trace this energy backwards (in the sense of Section 4.1) to locate its point of origin. (3) above ensures that it was injected at step j for some n 1 < j ≤ n. Varying ε j , therefore, leads directly to variations in ηi . The same argument applies to δξi . Next we consider displacements in xi . Assume for definiteness that e enters the system from the left. We fix k ∈ {1, . . . , N }, and let ε j , j = j (k), be one of the energies injected from the left when e is about halfway between sites k and k + 1. By Lemma 4.5, varying ε j leads to a variation of the form δk = δxk + . . . + δx N in Z n 1 . By Lemma 4.2, this displacement is retained between Z n 1 and Z n since only 1s are injected. The vectors {δk , k = 1, 2, . . . , N } span the subspace corresponding to positional variations. The proof of Proposition 3.5 is complete.

5. Equilibrium Distributions 5.1. Systems with a single site. This subsection treats exclusively the case N = 1. As we will see, nearly all of the ideas in a complete proof of Theorem 2 (for general N ) show up already in this very simple situation. Following the notation in Section 1.1, we consider phase variables X = (η0 , ξ, η1 ; x0 , x1 ; σ0 , σ1 ) ∈ (0, ∞)3 × [0, 1]2 × {−, +}2 ; the phase space has 4 components corresponding to the 4 elements of {−, +}2 . Define ∞ 1 ρ(η) ρ(η) , where Z = τ (η) = √ √ dη . Z η η 0 Theorem 2 asserts the invariance of the probability measure µ = τ (η0 )ρ(ξ )τ (η1 )I[0,1]2 (x0 , x1 ) dη0 dξ dη1 d x0 d x1 × w4 , where I(·) is the indicator function and w4 gives equal weight to the 4 points. Let µt = PXt dµ(X ) be the distribution at time t with initial distribution µ. To prove Theorem 2, we need to show µt = µ for all t > 0, equivalently for all small enough t > 0. We say a system undergoes a simple change in configuration between times 0 and t if during this period at most one collision occurs and this collision involves only a single

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moving particle. We will show in a sense to be made precise that for any system (independent of size), in short enough time intervals it suffices to consider simple changes in configuration. This observation simplifies the proof for N = 1. It is crucial in the analysis of N -chains; without it, the complexity of the situation gets out of hand quickly. In stochastic processes, analogous ideas are provided by independence and exponential clocks. The dynamics being largely deterministic here, we will have to argue it “by hand”, deducing it from tail assumptions on ρ. In what follows, |ν| denotes the total variation norm of a finite signed measure ν, and µ = ν + O(t) means |µ − ν| = O(t). For notational simplicity, we assume ρ(η) = O(η−2 ) as η → ∞. Proof of Theorem 2 for the case N = 1. Step 1. Reduction to simple changes in configuration. We will prove |µt − µ| = O(t 1+δ ) for some δ > 0 as t → 0. To see that this implies the invariance of µ, fix s > 0. By a repeated application of the estimate above with step size ns , we obtain µs = µ + n · O(( ns )1+δ ), which tends to µ as n → ∞. In what follows, we focus on the component M (+,+) = {(σ0 , σ1 ) = (+, +)}; other components are analyzed similarly. Instead of comparing µ and µt directly on all of M (+,+) , we will compare µ with some µˆ t ≈ µt (to be defined) on a large subset t ⊂ M (+,+) (to be defined). (A) We consider in the place of µt the measure µˆ t = PXt d µ(X ˆ ), where µˆ is the restriction of µ to the set {η0 , ξ, η1 < t −2 }. Notice that if all particles in a system have energy < t −2 , then their speeds are < t −1 , so no moving particle colliding with a site or a bath can set off a second collision to take place within t units of time. We claim that confusing µt with µˆ t leads to an error of a size we can tolerate, since ∞ ∞ ρ(ξ )dξ = O(t 2 ) and τ (η)dη = O(t 3 ). t −2

t −2

(B) Next, we permit at most one√moving particle to have a collision. For a particle at x ∈ (0, 1) with√σ = + and speed η, a collision occurred in the previous t units of time if and only if t η > x. Let √ √ t = {X ∈ M (+,+) : (i) η0 , ξ, η1 < t −2 and (ii) t η0 < x0 or t η1 < x1 }. 2 √ We claim that µ(t ) = 1− O(t 1+δ ). This is because if t η > x, then either (a) η > t − 3 , ∞ 1 2 or (b) x < t · t − 3 = t 3 . Now (a) occurs with probability − 2 τ (η)dη = O(t), and (b) t 3 2 2 4 √ √ occurs with probability t 3 . Thus µ{t η0 > x0 and t η1 > x1 } = [O(t 3 )]2 = O(t 3 ). The implications of (A) and (B) above are as follows: Suppose we show µˆ t |t = µ|t + O(t 1+δ ) for some δ with 0 < δ ≤

1 3.

(∗)

Then by (B), we have

µ = µ · It + O(t 1+δ ) = µˆ t · It + O(t 1+δ ). Now we know from (A) that |µˆ t | = 1 − O(t 1+δ ). This together with the equality above implies µˆ t (tc ) = O(t 1+δ ), so that µˆ t · It = µˆ t + O(t 1+δ ). Thus µ = µˆ t + O(t 1+δ ) = µt + O(t 1+δ ) , the second equality following from (A). This is what we seek.

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Step 2. Analysis of simple changes in configuration. Let ϕ and ϕˆt be the densities of µ and µˆ t respectively, and fix X¯ = (η¯ 0 , ξ¯ , η¯ 1 ; x¯0 , x¯1 ; +, +) ∈ t . We seek to compare ϕ( X¯ ) and ϕˆt ( X¯ ), remembering that to obtain µˆ t , one considers only initial conditions −2 in which √ all energies are < √t . Case 1: t η¯ 0 < √ x¯0 and t η¯ 1√< x¯1 . The only way to reach X¯ in time t is to start from (η¯ 0 , ξ¯ , η¯ 1 ;√x¯0 − t η¯ 0 , x¯1 −√t η¯ 1 ; +, +). No√collision occurs, and ϕ( X¯ ) = ϕˆt ( X¯ ). Case 2: t η¯ 0 < x¯0 and t η¯ 1 > x¯1 . Let s η¯ 1 √ = x¯1 . We claim that the only way to reach X¯ in time t is to start from ( η ¯ , ξ, η ; x ¯ − t η¯ 0 , x1 ; +, −), where ξ = η¯ 1 , η1 = ξ¯ 0 1 0 √ and x1 = (t − s) η1 . Starting here, one reaches X¯ after a single exchange of energy between the right particle and the site. To compute the densities at X¯ , we let ε > 0 be a verysmall number. Initial conditions with x1 ∈ ((t − s) ξ¯ − , (t − s) ξ¯ + ) and √ / ξ¯ = ε/ η¯ 1 will reach the target interval (x¯1 − ε, x¯1 + ε) at time t. Since ρ(η¯ 1 )τ (ξ¯ ) = ρ(η¯ 1 )

1 ρ(ξ¯ ) 1 ρ(ξ¯ ) = ρ(η¯ 1 ) √ ε = ρ(ξ¯ )τ (η¯ 1 )ε , Z ξ¯ Z η¯ 1

¯ we conclude ϕˆt ( X¯ ). √ that ϕ( X ) = √ √ Case 3: t η¯ 0 > x¯0 and t η¯ 1 < x¯1 . Here we let s be such that s η¯ 0 = x¯0 . The only √ way to reach X¯ is for the left particle to start at x0 = (t − s) η0 < 1 if η0 is its initial energy, go left, reach the left bath at time t − s, and have the newly emitted energy η¯ 0 reach x¯0 at time t. To compare densities, again√fix a small target interval (√ x¯0 − ε, x¯0√ + ε). √ This forces x0 ∈ ((t − s) η0 − (η0 ), (t − s) η0 + (η0 )) with (η0 )/ η0 = ε/ η¯ 0 . Thus −2 √ t η0 1 ϕˆt ( X¯ ) = ρ(ξ¯ )τ (η¯ 1 ) · τ (η0 ) √ dη0 · ρ(η¯ 0 ) 4 η¯ 0 0 t −2 1 ρ(η¯ 0 ) 1 ρ(η0 )dη0 = ϕ( X¯ ) · (1 − O(t 2 )). = ρ(ξ¯ )τ (η¯ 1 ) · √ 4 Z η¯ 0 0 √ √ Since t η¯ 0 > x¯0 and t η¯ 1 > x¯1 is not permitted for X¯ ∈ t , we have exhausted all viable cases, completing the proof of (∗).

5.2. Proof of Theorem 2. We follow closely Section 5.1, adapting the ideas there to chains with N sites. We begin with the analogous set of reductions. As in Section 5.1, we seek to show µt = µ + O(t 1+δ ) for some δ > 0, focusing on a fixed component of the phase space M (σ¯ ) = {(σ0 , · · · , σ N ) = σ¯ }. Instead of µt , we consider µˆ t = PXt d µ(X ˆ ) where µˆ is the restriction of µ to {ηi , ξi < t −2 , all i}. It follows from the estimates in Section 5.1 that (for fixed N ) this introduces an error within the tolerable range: ∞ ∞ |µt − µˆ t | < N ρ + (N + 1) τ = O(t 2 ). t −2

(σ¯ )

We define t = t

t −2

to be

{X ∈ M (σ¯ ) : (i) ξi , ηi < t −2 for all i; √ √ (ii) t ηi < xi if σi = +, t ηi < 1 − xi if σi = − for all except at most one i} ,

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√ the idea being that when the above relation between t, ηi and xi holds, no collision occurs in the t units of time prior to arrival in that configuration. The same argument as 4 before shows that µ(t ) = 1 − O(t 3 ): the set on which this relation is violated by two 4 of the moving particles has measure O(t 3 ); the set on which it is violated by more than two particles is smaller. As in Section 5.1, we seek to compare ϕˆt and ϕ on the set t . As before, all changes in configuration involved are simple: there are 3 cases corresponding to no collisions, a collision with a site, and a collision with a bath. The analysis is identical to that in Section 5.1.

Part II. Numerical Results We report here some results on nonequilibrium steady states (NESS) obtained by studying numerically two families of models: the first has exponential bath injections and the second, which we call the “two-energy model”, is chosen for the transparency of the role played by the integrable dynamics.

6. Exponential Baths We consider in this section processes defined in Section 1.1 for which the bath injections have exponential distributions, i.e., L(ξ ) = β L e−β L ξ and R(ξ ) = β R e−β R ξ , where, as usual, β L = TL−1 and β R = TR−1 are to be thought of as inverse temperatures. We are interested in steady states of systems that are in contact with two baths at unequal temperatures, i.e., where TL = TR . It follows from Theorem 1 that all invariant distributions that arise from these bath injections are ergodic. In Section 6.1, we demonstrate that mean energy profiles are well defined, both for finite N and as N → ∞. In Section 6.2, we focus on specific points along the chain, and investigate marginals of the NESS on (very) short segments. In the simulations shown, mean bath temperatures are TL = 1 and TR = 10, and chains of various lengths up to N = 1600 are used. 6.1. Macroscopic energy profiles. For a chain with N sites, we let E[ξi ] denote the mean energy at site i, “mean” being taken with respect to the unique steady state distribution. Let f N : [0, 1] → R be the function which linearly interpolates f N ( Ni+1 ) = E[ξi ], i = 1, · · · , N . If as N increases, f N converges (pointwise) to a function f on [0, 1], we will call f the site-energy profile of this model. Similarly, we let f N be the function that interpolates f N ( Ni+1 +2 ) = E[ηi ], i = 0, 1, . . . , N , and call f = lim N f N the gap-energy profile. Profiles of gap energies conditioned on σi being + or − are denoted f N+ and f N− . Figure 1 shows plots of f N , f N± , and f N . Finite-chain profiles are found to vary little as N goes from 100 to 1600, so we assume the limit profile will not be too different. Convergence time to steady state is slow, and increases with N as expected. (See numerical details in the caption.) 6.2. Local equilibrium properties. This subsection is about local properties of NESS, by which we refer to marginals of the form µˆ x,,N , where x ∈ (0, 1), N is the length of

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5

4

Mean gap energy

Mean site energy

8

6

4

2

3

2

1

0

0 0.2

0.4

0.6

0.8

x (a) Site energies, N = 1600

0

0.2

0.4

0.6

0.8

1

x (b) Gap energies, N =200

Fig. 1. Mean energy profiles for the exponential baths model. The bath temperatures are TL = 1 and T R = 10. Panel (a) shows the mean site energies f N for N = 1600. In (b), we have superimposed the gap energy profiles f N+ , f N− , and f N for N = 200. Numerical details: We impose a constraint that injected energies are ≥ 0.01 but do not impose a high energy cutoff. We simulate the system for 10k events (an “event” being one collision anywhere in the chain), increasing k until the computed profile stabilizes. This occurred after 1013 events in (a), 4 × 1011 events in (b)

the chain, N , and µˆ x,,N is the marginal distribution of the NESS on the -chain centered at site [x N ]. For fixed (unequal) bath distributions, since the energy gradient on the -chain tends to zero as N → ∞, one might expect the µˆ x,,N to resemble equilibrium distributions, i.e., invariant measures on chains with equal bath injections. Theorem 2 gives an explicit formula for the equilibrium distribution µρ,N of the N -chain when the bath distributions are L = R = ρ. This result is valid for very general ρ. Notice that when specialized to the case ρ(ξ ) = βe−βξ , the density of µρ,N has the √ familiar form Z1 e−β H . More precisely, let us use (xi , vi ), vi = σi ηi , as coordinates in the gaps instead of (ηi , xi , σi ). Then −(N +1) −β i vi2 e i d xi dvi . dµρ,N = β N e−β i ξi i dξi × Z β

(1)

Returning to the situation of unequal exponential baths, one way to define local thermodynamic equilibrium (LTE) is to require that for every x ∈ (0, 1) and ∈ Z+ , as N → ∞, the marginals µˆ x,,N tend to a probability measure having the form in (1) with N replaced by and β = β(x) for some β(x) > 0. For a more physical notion of LTE, one sometimes considers only -chains for 1 N . A. Single-site and single-gap marginals. We consider marginal distributions of the NESS at single sites and single gaps at [x N ], where x ∈ (0, 1) is fixed and N is varied. To show that the system does not tend to LTE (by any definition), it suffices to show that these marginals are not Gibbsian. This is because if marginals on -chains tended to µˆ x,,N , projecting onto the single site or gap at [x N ] one would again obtain a distribution of the form in (1). Figure 2 shows two examples of single-site marginal densities at x = 0.25 and 0.6 for a chain of length N = 1600. The plots are log-linear, so that exponential functions

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Fig. 2. Site energy distributions for the exponential baths model in log-linear scales. The parameters are TL = 1, T R = 10, N = 1600. Empirical data are plotted in open squares; the solid curve represents the mixture of left and right bath distributions that best fits the data (in the total variation norm)

would appear as straight lines. Without a doubt both of the distributions shown are very far from exponential. These distributions are compared to mixtures of the bath distributions, meaning distributions of the form a L(ξ ) + (1 − a)R(ξ ), a = a(x) ∈ (0, 1). In Fig. 2, open squares represent empirical data (from simulations), while the solid curves are given by the formulas for mixtures with a chosen in each of the two cases to fit the data best. Given the numerical cutoffs, etc., we think the fits between empirical data and the mixtures curves are excellent. Plots (not shown) at other locations evenly spaced along the chain show the same phenomenon, with the “knee” moving steadily to the left as x increases. Likewise, single-gap marginals are found to be mixtures: At gaps adjacent to [x N ] 2 for fixed x, instead of Z β−1 e−βv for some β = β(x), we find numerically the marginals to be of the form a·

e−β L v Z βL

2

+ (1 − a) ·

e−β R v Z βR

2

for some a = a(x), in clear violation of the prescription of Gibbs for Hamiltonian systems in equilibrium. We explain how we arrived at the idea of “mixtures”: First, the energies in a chain should reflect those injected, and if there is a nontrivial discrepancy between β L and β R , it is difficult to imagine having an abundance of the energies in the “middle” to constitute all the exponential distributions along the chain. We are also influenced by the following stochastic model, which can be thought of as a “zeroth-order” approximation to our Hamiltonian systems.

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The random swaps model. This is a stochastic model defined by N random variables ξ1 , . . . , ξ N , to be thought of as energies, located at sites 1, . . . , N . As usual there are two baths, situated at sites 0 and N +1. At bonds between sites, i and i +1 for i ∈ {0, 1, . . . , N } are exponential clocks which ring independently at rate 1. When a clock goes off, energies between the 2 sites are swapped. That is to say, for i = 0, N , when the clock between sites i and i + 1 rings, the values of ξi and ξi+1 are interchanged. Swapping energy with the left bath means that ξ1 is replaced by an energy drawn randomly from the left bath distribution L(ξ ), and similarly for ξ N and the right bath, which has distribution R(ξ ). The energies emitted by each bath are assumed to be i.i.d. and independent of emissions by the other bath. A model along similar lines was studied in [10]. Proposition 6.1. In the random swaps model, for each i ∈ {1, . . . , N }, the marginal distribution ρi of the unique NESS is given by

i N +1−i L+ R ρi = N +1 N +1 for arbitrary bath distributions L and R. Sketch of proof. First, we distinguish only between whether an energy is “from the left” or “from the right”. In this reduced system, it is easy to see that there is a unique steady state, the single-site marginals of which are, by definition, mixtures of these two kinds of energies. Since the trajectory of each energy, from the time it enters the system to when it leaves, is that of a simple unbiased random walk, the weights in the formula for ρi follow from basic recursive relations. Finally, each of the energies in this reduced system can be assigned independently any one of the allowed values, so that the marginals are really mixtures of L and R.

Since Proposition 6.1 holds for arbitrary bath distributions L and R, it is natural to ask if a similar result holds for the Hamiltonian chain with non-exponential baths. We investigated this question and found the answer to be negative. For example, when L and R are uniform distributions, marginal distributions are far from mixtures of uniform distributions, see Fig. 3. When the densities of L and R have shapes closer to exponential distributions, single-site marginals are closer to mixtures, but noticeable differences were seen in each of the half-dozen or so cases tested. The numerical results above raise the following questions: (A) In the case of exponential baths, are single-site marginals as N → ∞ genuinely mixtures of the two bath distributions, or are they simply close to mixtures? (B) If the mixtures result here is exact, are exponential baths the only distributions that have this property, and if so, what are the underlying reasons? Independent of the answer to (A), we believe we have shown very definitively that local marginal distributions do not have the form in (1). Hence the concept of LTE does not apply to this class of Hamiltonian chains. B. Vanishing of spatial correlations. We investigate next if, as N → ∞, µˆ x,,N → µρ, where ρ is the mixture found in Paragraph A. Since spatial correlations are expected to be largest between adjacent sites and gaps, we verify only that (i) marginal distributions on two adjacent sites are product measures; (ii) marginal distributions on two adjacent gaps are product measures, and the directions of travel are independent;

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for the exponential-baths model. The 4 curves in each panel show P ξ[N x]+1 ∈ I | ξ[N x] ∈ [2k, 2(k + 1)] for k = 0, 1, 2, · · · , 14 and I = [0, 2], [2, 4], [4, 6], and [6, 8] (top to bottom). In both panels, x = 0.6. Bath parameters are TL = 1, T R = 10

(iii) marginal distributions on a√site and its adjacent gap are product measures, with gap density = const·ρ(η)/ η, where site density = ρ. A sample of simulation results in support of (i) is shown in Fig. 4. In the horizontal axis are values of ξi for i = [0.6N ], and plotted are conditional distributions of ξi+1 given the various values of ξi . Here we group the values of ξi+1 into intervals (0, 2), (2, 4), (4, 6), and (6, 8), the 4 graphs representing the conditional probabilities of ξi+1 being in each one of these intervals given the value of ξi in the horizontal axis. The two plots, for N = 100 and 1600 respectively, show weak dependence of ξi and ξi+1 in the first and close to zero dependence for the longer chain.

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Table 1. Joint distributions of gap energies conditioned on directions of travel. Here, N = 1600 and i = [0.6 N ]. The numbers are raw probabilities of the specified events

A small sample of data is shown in Table 1 to illustrate the combinations of joint densities that need to be checked to verify (ii). Here we focus on two adjacent gaps at x = 0.6, and distinguish only between “high” and “low” energies, referring to energies η > 3 as “high” and η < 3 as “low”. The fractions of time in all configurations of high-low and (σi , σi+1 ), referring to the directions of travel in these gaps, are tabulated. Finally, we have computed energy distributions for adjacent sites and gaps at various sites along the chain. The data confirm property (iii) above. Remark. The “local product structure” finding above may be valid beyond the case of exponential baths. Our simulations show that it holds both for the uniform distribution models in Fig. 3 and for the “two-energy models” discussed in the next section. Since the latter are rather “extreme” from many points of view, it gives reason to conjecture that this phenomenon – which is very natural – may hold widely. 7. Two-Energy Models The bath distributions in this section are sharply peaked Gaussians truncated at 0. In the simulations shown, these Gaussians have means 1 and 5, and their standard deviations are chosen to be 0.2, so that the probability of being within ±0.5 of the means is roughly 99%. Even though these systems are small perturbations of integrable models, we know from Theorem 1 that all of their invariant measures are ergodic. 7.1. Macroscopic energy profiles and local distributions. Figure 5(a) shows site-energy profiles f N for a range of N . Unlike the exponential case, these profiles vary substantially with N : For N = 10 (not shown), the profile is essentially flat; as N increases, it acquires a gradient and appears to be stabilizing, but even at N = 3200, the profile is still moving a little. From these profiles, one suspects – correctly – that for (very) small N , most energies move monotonically along the chain, entering from one end and exiting at the other. As N increases, some of the energies turn around, some doing so a number of times before exiting, creating a gradient in the profile. Figure 5(b) shows profiles for gap energies with specified directions, showing that for the chain with N = 200, f N+ and f N− are substantially different, with more energy

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Fig. 5. Mean energy profiles for two-energy model. Panel (a) shows mean site energies; the curves are N =40, 200, and 3200, in order of increasing values at x = 1. Panel (b) shows mean gap energies f N− , f N , and f N+ (top to bottom) for N = 200. The rules used to determine when a steady state is reached are as Fig. 1. See text for the bath distributions

1s moving to the right and 5s moving to the left. This is to be contrasted with Fig. 1(b), which shows that for exponential baths, this left–right discrepancy has by and large vanished by N = 200. Indeed in the present model, f N+ and f N− remain quite far apart in some of the gaps even at N = 3200, leaving open the question whether or not the distributions conditioned on σi = + and − will eventually equalize. A follow-up investigation is discussed in Section 7.2. With regard to local distributions, computations on site-to-site, gap-to-gap, and siteto-gap correlations similar to those in Section 6 were carried out. Independence was found to be achieved already at relatively small N . Figure 6 captures the two main points: The two curves representing P(ξ[0.6N ]+1 = 1|ξ[0.6N ] = k), k = 1 and 5, are virtually on top of each other for N ≥ 400, illustrating the rapid vanishing of correlations between adjacent sites, while both curves continue to have a slightly negative slope at N = 3200, illustrating the slow convergence of mean site energies. 7.2. Ballistic transport vs. diffusion: a phenomenological explanation. The purpose of this subsection is to examine more closely the way in which energy is transported from one end of the chain to the other under near-integrable dynamics. Since the vast majority of the energies in this system are very close to 1 or 5, let us assume there are only two kinds of energies in the system. For definiteness, we adopt the view in Section 1.3, and focus on the movement of 1s on the right side of the chain. We will demonstrate that modulo certain time changes, the statistics generated by the movements of these energies resemble those of a particle undergoing a 1-D diffusion with a spatially-varying diffusion coefficient. We begin by introducing an especially simple model which will be used for comparison purposes: Model A. Consider a Markov chain with state space {0, 1, · · · , N + 1} defined as follows: Starting from 1, one performs a simple, unbiased random walk until either 0 or N + 1 is reached, then returns to 1 to start over again. For 1 ≤ i ≤ N − 1, let n (i) and rn (i) denote the number of left and right crossings respectively between sites i and i + 1

Prob( ξ[Nx]+1 | ξ[Nx] )

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N Fig. 6. Nearest-neighbor conditional probabilities in the two-energy model. This plot shows P(ξ[N x]+1 < 3 | ξ[N x] < 3) (open squares) and P(ξ[N x]+1 < 3 | ξ[N x] > 3) (solid discs) as functions of N . The location is x = 0.6; N ranges from 100 to 3200

in the first n steps, and let E N ,i = limn→∞ E N [n (i)]/E N [rn (i)]. Here are some easy facts: (i) For all N and i, 0 < E N ,i < 1. (ii) For each N , E N ,i decreases as i increases. (iii) For each x ∈ (0, 1), E N ,[x N ] increases to 1 as N → ∞. These facts follow from the following observations: Fix i and N , and focus only on left and right crossings between sites i and i + 1. Then a left crossing is necessarily followed by a right crossing, while a right crossing can be followed by either. The probability that a right crossing is followed by another right crossing is equal to the probability of starting from site i + 1 and reaching site N + 1 before site i in the random walk. The numbers E N ,i can be explicitly estimated if one so desires. To bring this simple model closer to our Hamiltonian chains, we first identify some relevant features of the chains. The following idea is used in the rigorous part of this paper (Lemmas 4.2 and 4.4): Consider a scenario in which an energy 1 is in the midst of many energy 5s, i.e. in some segment of the chain, ξ j = η j = 5 for all j except for a ˆ where η ˆ = 1. Then the energy 1 will move monotonically in some direction single j, j until the pattern is disrupted by the approach of another (oncoming) energy 1. After such a “collision” a variety of things can happen; the resulting motion of the energies depends on the details of the interaction. Since the energy profiles have nonzero gradient (see Fig. 5), 1s are more sparse on the right side of the chain. Thus the closer to the right end, the larger the “mean free distance” between “collisions” of energy 1s. With regard to the ratio of left to right crossings (and not the actual number of crossings per unit time), the observations above suggest the following modification of Model A: Model B. Let λ : (0, 1) → (0, ∞) be a function that monotonically increases from 1 to ∞ on some subinterval (x∗ , 1) ⊂ (0, 1). We consider a process similar to that in Model A but with transition rules modified as follows: when at site i = [x N ], go X sites to the

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Fig. 7. Statistics on lengths of runs. (a) Mean lengths of runs in different intervals [a, b]. Note that runs that reach the boundary of [a, b] are artificially cut short by our numerical procedure (see “numerical details” below); we do not consider averages based on too many (> 10%) such runs meaningful and have omitted those averages. (*) marks averages for which 5 − 10% of the runs were cut short. (b) Log-linear plot of the histogram of run-lengths in the interval [0.6, 0.8], for N = 3200. (c) The ratio P(ηi ≈ 1, σ = −)/P(ηi ≈ 1, σ = +) as a function of x = i/N . The curves are, from bottom to top, N =100, 400, 1600, 3200. Numerical details for (a) and (b): For a given interval [a, b], we begin with a suitably prepared system and wait until an energy ≈ 1 hits the site at x = (a + b)/2. The energy is tracked until it hits either x = a or x = b; the lengths of runs executed by this energy are recorded. This process is repeated (after a delay to ensure that the energies are not too correlated). Each data point in Table (a) based on 104 – 106 runs collected this way

left/right with probability 21 / 21 , where X is an exponential random variable with mean λ(x), and return as before to 1 to start over once 0 or N + 1 is reached. First we confirm numerically that in terms of the statistics generated, Model B gives a good approximation of movements of energy 1s on the right side of the real chain. (Obviously, we do not claim that the two models are equivalent.) We study the lengths of runs of randomly picked 1s in specified segments of the chain, a run being defined to be a consecutive sequence of moves in the same direction. Some mean length-of-runs are tabulated in Fig. 7(a). The numbers in each row increase as one moves to the right, a trend consistent with the environment becoming increasingly dominated by energy 5s. As N increases, these means stabilize, as one would expect them to when the local marginals tend to a distribution of the form µρ, . Histograms of lengths of runs within specified intervals are plotted and found to have roughly exponential distributions for large N ; one such plot is shown in Fig. 7(b). Returning to Models A and B, observe that they are qualitatively similar if we imagine that the sites in Model B are “packed closer and closer” as x → 1. More precisely, the ratio of left to right crossings at site [x N ] in Model B should resemble that at [y N ] in Model A for some y > x, with (1 − y)/(1 − x) → 0 as x → 1. Properties (i-iii) in

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Model A therefore pass to Model B. Assuming they pass from there to our Hamiltonian chain, one would conclude that for our chain: – left-right traffic will equalize everywhere as N → ∞; – this equalization occurs more slowly than at corresponding sites in Model A, the discrepancy increasing as x → 1. Simulations to compute the ratios of left/right crossings were performed. The results, shown in Fig. 7(c), are very much consistent with the predictions above: for each N , the x → Prob(σ[N x] = −)/Prob(σ[N x] = +) curves are decreasing, and at each x ∈ (0, 1), this ratio increases with N and appears to head toward 1. Remarks on connection with theory. We have used the fact, proven rigorously in Section 4.3, that in regions of the chain occupied by significantly more energies of one kind than the other, the ones with low density will, on average, have relatively long runs. The same is true, in fact, for the energies with high density. Our rigorous work leaves untreated the situation where the two energies occur in roughly comparable proportions. For these regions, we have seen via simulations that the mean lengths of runs tend still to be greater than 2, the value for simple unbiased random walks. For example, when TL = 1 and TR = 5, the region with the most even mix of 1s and 5s is [0.2, 0.4], and there the mean run-length for 1s is > 5 (see Fig. 7(a)). Thus the phenomena discussed in this subsection appear to be valid for both energies on a good part of the chain.

Summary and Conclusion We considered a Hamiltonian chain with many conserved quantities and studied its nonequilibrium steady states when the two ends of the chain are put in contact with unequal heat baths. Our main findings are: Rigorous results. First, under mild restrictions on the bath distributions L(ξ ) and R(ξ ), we proved ergodicity of the invariant measure (assuming existence). Second, we identified a class of equilibrium measures {µρ,N }, where µρ,N is the unique invariant probability on the N -chain with L = R = ρ; the measures µρ,N are product measures. Numerical results. Simulations were performed for chains with two kinds of bath distributions: exponential distributions and sharply peaked Gaussians, the latter giving rise to what is called the “two-energy model”. 1. We demonstrated numerically that (a) NESS exist for finite N and mean energy profiles converge as N → ∞; (b) as N → ∞ local marginals at x ∈ (0, 1) tend to measures of the form µρ, for some distribution ρ = ρ(x). 2. For exponential bath distributions, the limits of local marginals are definitively not Gibbs measures, i.e., this chain violates the concept of LTE. The marginals appear to be weighted averages of Gibbs measures. 3. For the 2-energy model, the paths traced out by energies resemble the sample paths of a random walk with a bias in favor of continuing in the same direction, with this bias increasing as x → 0 or 1. We conclude that the resulting transport behavior is more normal, i.e., more diffusive, than one might have expected given the integrable character of the dynamics.

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References 1. Bernardin, C., Olla, S.: Fourier’s law for a microscopic model of heat conduction. J. Stat. Phys. 118, 271–289 (2005) 2. Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory for stationary non-equilibrium states. J. Statist. Phys. 107, 635–675 (2002) 3. Bonetto, F., Lebowitz, J., Rey-Bellet, L.: Fourier law: a challenge to theorists. In: Mathematical Physics 2000, edited by Fokas, A., Grigoryan, A., Kibble, T., Zegarlinski, B., London: Imp. Coll. Press, 2000 4. Bricmont, J., Kupiainen, A.: Fourier’s law from closure equations. Phys. Rev. Lett. 98, 214301 (2007) 5. Bricmont, J., Kupiainen, A.: Towards a derivation of Fourier’s law for coupled anharmonic oscillators. Commun. Math. Phys. 274, 555–626 (2007) 6. Collet, P., Eckmann, J.-P.: A model of heat conduction. Preprint, http://arxiv.org/abs/0804.3025v1[mathph], 2008 7. Collet, P., Eckmann, J.-P., Mejía-Monasterio, C.: Superdiffusive heat transport in a class of deterministic one-dimensional many-particle Lorentz gases. Preprint, http://arxiv.org/abs/0810.4461v1[cond-mat. stat.-mech], 2008 8. Derrida, B.: Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. J. Stat. Mech. (2007) P07023, doi:10.1088/1742-5468/2007/07/p02023, July 2007 9. Derrida, B., Lebowitz, J.L., Speer, E.R.: Large deviation of the density profile in the steady state of the open symmetric simple exclusion process. J. Stat. Phys. 107, 599–634 (2002) 10. Dhar, A., Dhar, D.: Absence of local thermal equilibrium in two models of heat conduction. Phys. Rev. Lett. 82, 480–483 (1999) 11. de Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. Amsterdam: North-Holland, 1962 12. Eckmann, J.-P., Hairer, M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212, 105–164 (2000) 13. Eckmann, J.-P., Jacquet, P.: Controllability for chains of dynamical scatterers. Nonlinearity 20, 1601– 1617 (2007) 14. Eckmann, J.-P., Mejía-Monasterio, C., Zabey, E.: Memory effects in nonequilibrium transport for deterministic Hamiltonian systems. to appear in J. Stat. Phys., 2006 15. Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Commun. Math. Phys. 201, 657–697 (1999) 16. Eckmann, J.-P., Young, L.-S.: Nonequilibrium energy profiles for a class of 1-D models. Commun. Math. Phys. 262, 237–267 (2006) 17. Gaspard, P., Gilbert, T.: Heat conduction and Fourier’s law in a class of many particle dispersing billiards. New J. Phys. 10, 103004 (2008) 18. Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Berlin: Springer-Verlag, 1999 19. Kipnis, C., Marchioro, C., Presutti, E.: Heat flow in an exactly solvable model. J. Stat. Phys. 27, 65–74 (1982) 20. Larralde, H., Leyvraz, F., Mejía-Monasterio, C.: Transport properties of a modified Lorentz gas. J. Stat. Phys. 113, 197–231 (2003) 21. Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377(1), 1–80 (2003) 22. Li, B., Casati, G., Wang, J., Prosen, T.: Fourier Law in the alternate-mass hard-core potential chain. Phys. Rev. Lett. 92, 254301 (2004) 23. Lin, K.K., Young, L.-S.: Correlations in nonequilibrium steady states of random-halves models. J. Stat. Phys. 128, 607–639 (2007) 24. Olla, S., Varadhan, S.R.S., Yau, H.T.: Hydrodynamical limit for a Hamiltonian system with weak noise. Commun. Math. Phys. 155, 523–560 (1993) 25. Rateitschak, K., Klages, R., Nicolis, G.: Thermostating by deterministic scattering: the periodic Lorentz gas. J. Stat. Phys. 99, 1339–1364 (2000) 26. Ravishankar, K., Young, L.-S.: Local thermodynamic equilibrium for some stochastic models of Hamiltonian origin. J Stat. Phys. 128, 641–665 (2007) 27. Rey-Bellet, L., Thomas, L.E.: Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Commun. Math. Phys. 225, 305–329 (2002) 28. Rieder, Z., Lebowitz, J.L., Lieb, E.: Properties of a harmonic crystal in a stationary nonequilibrium state. J. Math. Phys. 8, 1073 (1967) 29. Spohn, H.: Long range correlations for stochastic lattice gases in a nonequilibrium steady state. J. Phys. A 16, 4275–4291 (1983) Communicated by A. Kupiainen

Commun. Math. Phys. 294, 229–249 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0869-2

Communications in

Mathematical Physics

Ground State Energy of Large Atoms in a Self-Generated Magnetic Field László Erd˝os1, , Jan Philip Solovej2, 1 Institute of Mathematics, University of Munich, Theresienstr. 39,

D-80333 Munich, Germany. E-mail: [email protected]

2 Department of Mathematics, University of Copenhagen, Universitetsparken 5,

DK-2100 Copenhagen, Denmark. E-mail: [email protected] Received: 11 March 2009 / Accepted: 16 April 2009 Published online: 3 July 2009 – © Springer-Verlag 2009

Abstract: We consider a large atom with nuclear charge Z described by non-relativistic quantum mechanics with classical or quantized electromagnetic field. We prove that the absolute ground state energy, allowing for minimizing over all possible self-generated electromagnetic fields, is given by the non-magnetic Thomas-Fermi theory to leading order in the simultaneous Z → ∞, α → 0 limit if Z α 2 ≤ κ for some universal κ, where α is the fine structure constant. 1. Introduction The ground state energy of non-relativistic atoms and molecules with large nuclear charge Z can be described by Thomas-Fermi theory to leading order in the Z → ∞ limit [L,LS]. Magnetic fields in this context were taken into account only as an external field, either a homogeneous one [LSY1,LSY2] or an inhomogeneous one [ES] but subject to certain regularity conditions. Self-generated magnetic fields, obtained from Maxwell’s equation are not known to satisfy these conditions. In this paper we extend the validity of Thomas-Fermi theory by allowing a self-generated magnetic field that interacts with the electrons. This means we look for the absolute ground state of the system, after minimizing for both the electron wave function and for the magnetic field and we show that the additional magnetic field does not change the leading order Thomas-Fermi energy. Apart from finite energy, no other assumption is assumed on the magnetic field. The nonrelativistic model of an atom in three spatial dimensions with nuclear charge Z ≥ 1 and with N electrons in a classical magnetic field is given by the Hamiltonian N 1 1 Z cl H N ,Z (A) = T j (A) − + + B2 (1.1) |x j | |xi − x j | 8π α 2 R3 j=1

i< j

Partially supported by SFB-TR12 of the German Science Foundation. Work partially supported by the Danish Natural Science Research Council and by a Mercator Guest Professorship from the German Science Foundation.

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acting on the space of antisymmetric functions 1N H with a single particle Hilbert space H = L 2 (R3 ) ⊗ C2 . The coordinates of the N electrons are denoted by x = (x1 , x2 , . . . , x N ). The vector potential A : R3 → R3 generates the magnetic field B = ∇ × A and it can be chosen divergence-free, ∇ · A = 0. The last term in (1.1) is the energy of the magnetic field. The kinetic energy of an electron is given by the Pauli operator T (A) = [σ · (p + A)]2 = (p + A)2 + σ · B,

p = −i∇x .

Here σ is the vector of Pauli matrices. We use the convention that for any one-body operator T , the subscript in T j indicates that the operator acts on the j th variable, i.e. T j (A) = [σ j · (−i∇x j + A(x j ))]2 . The term −Z |x j |−1 describes the attraction of the j th electron to the nucleus located at the origin and the term |xi − x j |−1 is the electrostatic repulsion between the i th and j th electron. Our units are 2 (2me2 )−1 for the length, 2me4 −2 for the energy and 2mec−1 for the magnetic vector potential, where m is the electron mass, e is the electron charge and is the Planck constant. In these units, the only physical parameter that appears in (1.1) is the dimensionless fine structure constant α = e2 (c)−1 . We will assume that Z α 2 ≤ κ with some sufficiently small universal constant κ ≤ 1 and we will investigate the simultaneous limit Z → ∞, α → 0. Note that the field energy is added to the total energy of the system and by the condition ∇ · A = 0 we have B2 = |∇ ⊗ A|2 , (1.2) R3

R3

where ∇ ⊗ A denotes the 3 × 3 matrix of all derivatives ∂i A j and |∇ ⊗ A|2 = 3 2 i, j=1 |∂i A j | . We will always assume that the vector potential belongs to the space of divergence free H 1 -vector fields A := {A ∈ H 1 (R3 , R3 ), ∇ · A = 0}. In the analogous nonrelativistic model of quantum electrodynamics, the electromagnetic vector potential is quantized. In the Coulomb gauge it is given by A (x) = A(x) = A− (x) + A− (x)∗ with A− (x) =

α 1/2 2π

R3

g(k) aλ (k)eλ (k)eik·x dk. √ |k| λ=±

Here g(k) is a cutoff function, satisfying |g(k)| ≤ 1 and supp g ⊂ {k ∈ R3 : |k| ≤ } with a constant < ∞ (ultraviolet cutoff). The field operators A(x) depend on the cutoff function g(k) whose precise form is unimportant; the only relevant parameter is . For each k, the two polarization vectors e− (k), e+ (k) ∈ R3 are chosen such that together with the direction of propagation k/|k| they are orthonormal. The operators aλ (k), aλ (k)∗ are annihilation and creation operators acting on the bosonic Fock space F over L 2 (R3 ) and satisfying the canonical commutation relations [aλ (k), aλ (k )] = [aλ (k)∗ , aλ (k )∗ ] = 0, [aλ (k), aλ (k )∗ ] = δλλ δ(k − k ).

Energy of Atoms in a Self-Generated Field

The field energy is given by Hf = α The total Hamiltonian is qed H N ,Z

=

N j=1

−1

231

R3

|k|

aλ (k)∗ aλ (k)dk.

λ=±

1 Z T j (A ) − + + Hf , |x j | |xi − x j |

(1.3)

i< j

and it acts on ( 1N H) ⊗ F. The stability of atoms in a classical magnetic field [F,FLL,LL,LLS] implies that the operator (1.1) is bounded from below uniformly in A, if Z α 2 is small enough. It is known [LY,ES2] that stability fails if Z α 2 is too large. The analogous stability result for quantized field [BFG] states that (1.3) is bounded from below if Z α 2 is small. In particular, we can for each fixed A define the operators in (1.1) and (1.3) as the Friedrichs extensions of these operators defined on smooth functions with compact support. The ground state energy of the operator with a classical field is given by N

cl cl ∞ 3 2 E N ,Z (A) = inf , H N ,Z (A) : ∈ C0 (R ) ⊗ C , = 1 , 1

and after minimizing in A we set cl E cl N ,Z = inf E N ,Z (A). A∈A

We note that it is sufficient to minimize over all A ∈ A0 , where A0 = Hc1 (R3 , R3 ) denotes the space of compactly supported H 1 vector fields. It is easy to see that the Euler-Lagrange equations for the above minimizations in and A correspond to the stationary version of the coupled Maxwell-Pauli system, i.e., the eigenvalue problem 2 H Ncl,Z (A) = E cl N ,Z together with the Maxwell equation ∇ × B = 4π α J , where J is the current of the wave function . It is for this reason that it is natural to refer to B as a self-generated magnetic field in this context. In the case of the quantized field, we define N

qed qed ∞ 3 2 C0 (R ) ⊗ C ⊗ F, = 1 . E N ,Z = inf , H N ,Z : ∈ 1 qed

The stability results of [F,FLL,LL,LLS,BFG] imply that E cl N ,Z > −∞ and E N ,Z > −∞ 2 if Z α is small enough. Finally, we define the ground state energy with no magnetic field as cl E nf N ,Z := E N ,Z (A = 0).

In all three cases we define E #Z := inf E #N ,Z , N ∈N

# ∈ {cl, qed, nf},

for the absolute (grand canonical) ground state energy. The main result of this paper states that the magnetic field does not change the leading term of the absolute ground state energy of a large atom in the Z → ∞ limit. In particular, Thomas-Fermi theory is correct to leading order even with including self-generated magnetic field.

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Theorem 1.1. There exists a positive constant κ such that if Z α 2 ≤ κ, then 7

1

cl nf 3 − 63 . E nf Z ≥ EZ ≥ EZ − C Z

(1.4)

For the quantized case, if we additionally assume 1

7

1

≤ κ 4 Z 12 −γ α − 4 with some 0 ≤ γ ≤

1 63 ,

(1.5)

then qed

2 E nf Z + C Z α ≥ E Z

7

3 −γ . ≥ E nf Z − CZ

(1.6)

We note that Z α2 Z 7/3 if κ −1/4 Z 11/12 in the Z → ∞ limit. Remark 1. The leading term asymptotics of the non-magnetic problem is given by the 7 2 3 Thomas-Fermi theory and E nf Z = −cTF Z + O(Z ) as Z → ∞, where cTF = 3.678 · 2

(3π 2 ) 3 is the Thomas-Fermi constant. The leading order asymptotics was established in [LS] (see also [L]). The correction to order Z 2 is known as the Scott correction and was established in [H,SW1] and for molecules in [IS] (see also [SW2,SW3,SS]). The next term in the expansion of order Z 5/3 was rigorously established for atoms in [FS]. Remark 2. The exponents in the error terms are far from being optimal. They can be improved by strengthening our general semiclassical result Lemma 1.3 for special Coulomb like potentials using multi-scale analysis. Remark 3. For simplicity, we state and prove our results for atoms, but the same proofs work for molecules as well; if the number of nuclei K is fixed, each has a charge Z , and assume that the nuclei centers {R1 , . . . , R K } are at least at distance Z −1/3 away, i.e. |Ri − R j | ≥ cZ −1/3 , i = j. Remark 4. Theorem 1.1 holds for the magnetic Schrödinger operator as well, i.e. if we replace the Pauli operator T (A) = [σ · (p + A)]2 by T (A) = (p + A)2 everywhere. The proof is a trivial modification of the Pauli case. The argument is in fact even easier; instead of the magnetic Lieb-Thirring inequality for the Pauli operator one uses the usual Lieb-Thirring inequality that holds for the magnetic Schrödinger operator uniformly in the magnetic field. We leave the details to the reader. Note that although the condition Z α 2 ≤ κ is not needed in order to ensure stability in the Schrödinger case, we still need it in the statement in Theorem 1.1. In the Schrödinger case this condition is not optimal. The upper bound in (1.4) is trivial by using a non-magnetic trial state. The upper bound in (1.6) is obtained by a trial state that is the tensor product of a non-magnetic electronic trial function with the vacuum | of F. The field energy H f and all terms that are linear in A give zero expectation value in the vacuum. The only effect of the quantized field is in the nonlinear term A2 . A simple calculation shows that |A2 | ≤ Cα2 . The main task is to prove the lower bounds. Using the results from [BFG], the result for the quantized field (1.6) will directly follow from an analogous result for a slightly modified Hamiltonian with a classical field. Let N 1 1 Z + + Ti (A)− |∇ ⊗ A|2 H N ,Z (A) = H N ,Z ,α (A) = |xi | |xi − x j | 8π α 2 |x|≤3r i=1

i< j

(1.7)

Energy of Atoms in a Self-Generated Field

233

with some r = D Z −1/3 with D ≥ 1.

(1.8)

Note that instead of the local field energy, the total local H 1 -norm of A is added in (1.7). By (1.2), we have H Ncl,Z (A) ≥ H N ,Z (A)

(1.9)

for any A ∈ A. We define the ground state energy of the modified Hamiltonian (1.7), N

∞ 3 2 E N ,Z (A) := inf , H N ,Z (A) : ∈ C0 (R ) ⊗ C , = 1 , 1

and set E N ,Z := inf E N ,Z (A), A∈A

E Z := inf E N ,Z , N

where the infimum for A ∈ A can again be restricted to compactly supported vector potentials A ∈ A0 . For the modified classical Hamiltonian we have the following theorem: Theorem 1.2. Let Z α 2 ≤ κ and assume that r = D Z −1/3 with 1 ≤ D ≤ Z 1/63 . Then nf 7/3 −1 E nf D . Z ≥ EZ ≥ EZ − C Z

(1.10)

Taking into account (1.9), Theorem 1.2 immediately implies the lower bound in (1.4). The proof of the lower bound in (1.6) follows from Theorem 1.2 adapting an argument in [BFG] that we will review in Sect. 6 for completeness. One of the key ingredients of the proof of Theorem 1.2 is the following semiclassical statement that is of interest in itself. The first version is formulated under general conditions but without an effective error term. In our proof we actually use the second version that has a quantitative error term. Theorem 1.3. Let Th (A) = [σ · (hp + A)]2 or Th (A) = (hp + A)2 , h ≤ 1, and V ≥ 0. 1) If V ∈ L 5/2 (R3 ) ∩ L 4 (R3 ), then

Tr [Th (A) − V ]−+ h −2 B2 ≥ Tr −h 2 − V +o(h −3 ) as h → 0. (1.11) R3

−

2) Assume that V ∞ ≤ K with some 1 ≤ K ≤ Ch −2 and consider the operators with Dirichlet boundary condition on ⊂ R3 . Let B R√denote the ball of radius R about the origin and let √h := + B√h denote the h-neighborhood of the set . We set |√h | for the Lebesgue measure of √h . Then B2 Tr [(Th (A) − V ) ]− + h −2 3 R

≥ Tr (−h 2 − V ) −

1/2

1/2 −3 5/2 √ 3/2 3/2 1 + hK . (1.12) −Ch K | h | h K

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L. Erd˝os, J. P. Solovej

Remark. Despite the electrons being confined to , their motion generates a magnetic field in the whole R3 , so the magnetic field energy in (1.12) is given by integration over R3 . We use the convention that letters C, c denote positive universal constants whose values may change from line to line. 2. Reduction to the Main Lemmas Proof of Theorem 1.2. We focus on the lower bound, the upper bound is trivial. We start with two localizations, one on scale r ≥ Z −1/3 and the other one on scale d ≤ Z −1/3 . The first one is designed to address the difficulty that the H 1 -norm of A is available only locally around the nucleus. This step would not be needed for the direct proof of (1.4). The second localization removes the “Coulomb tooth”, i.e. the Coulomb singularity near the nucleus. In this section we reduce the proof of the lower bound in Theorem 1.2 to two lemmas. Lemma 2.1 will show that the Coulomb tooth is indeed negligible. Lemma 2.2 shows that the magnetic field cannot substantially lower the energy for the problem without the Coulomb tooth. In the proof of Lemma 2.2 we will use Theorem 1.3. Recall that B R denotes the ball of radius R about the origin. We construct a pair of smooth cutoff functions satisfying the following conditions: θ02 + θ12 ≡ 1, supp θ1 ⊂ B2d , θ1 ≡ 1 on Bd , |∇θ0 |, |∇θ1 | ≤ Cd −1 . We will choose d = δ Z −1/3

(2.13)

with some δ ≤ 1, in particular d ≤ r . We split the Hamiltonian as H N ,Z (A) = H N0 ,Z (A) + H N1 ,Z (A) with H N0 ,Z (A)

N

Z 2 2 θ0 T (A) − − |∇θ0 | + |∇θ1 | θ0 = |x| i i=1 1 1 + + |∇ ⊗ A|2 , |xi − x j | 16π α 2 B3r

(2.14)

i< j

H N1 ,Z (A) =

N

Z θ1 T (A) − − |∇θ0 |2 + |∇θ1 |2 θ1 |x| i i=1 1 + |∇ ⊗ A|2 , 16π α 2 B3r

where we used the IMS localization formula that is valid for the Pauli operator as well as for the Schrödinger operator. In Sect. 3 we deal with H N1 ,Z , to prove that it is negligible:

Energy of Atoms in a Self-Generated Field

235

Lemma 2.1. There is a positive universal constant κ such that for any Z , α with Z α 2 ≤ κ we have inf inf H N1 ,Z (A) ≥ −C Z 7/3 δ 1/2 − Z 2/3 δ −2 N A∈A0

if C Z −2/3 ≤ δ ≤ D with a sufficiently large constant C. Starting Sect. 4 we will treat H N0 ,Z (A) and we prove the following: Lemma 2.2. There is a positive universal constant κ such that for any Z , α with Z α 2 ≤ κ we have

inf inf H N0 ,Z (A) ≥ −cTF Z 7/3 − C Z 7/3 Z −1/30 + D −1 (2.15) N

A

with a sufficiently large constant C if Z −1/6 ≤ δ ≤ 1 and D ≤ Z 1/24 δ 13/16 . The main ingredient in the proof is Theorem 1.3 that will be proven in Sect. 5. The proof of the lower bound in Theorem 1.2 then follows from Lemmas 2.1 and 2.2 after choosing δ = Z −2/63 . 3. Estimating the Coulomb Tooth Proof of the Lemma 2.1. Let χ 0 be a smooth cutoff function supported on B3r such that |∇ χ0 | ≤ Cr −1 and χ 0 ≡ 1 on B2r . Let A := |B3r |−1 B3r A. We define A0 := (A − A ) χ0 ,

B0 := ∇ × A0 ,

(3.16)

0 ∇ ⊗ A + (A − A ) ⊗ ∇ χ0 . Clearly then ∇ ⊗ A0 = χ 2 2 2 2 −2 B0 ≤ |∇ ⊗ A0 | ≤ 2 χ 0 |∇ ⊗ A| + Cr (A − A )2 R3 R3 R3 B3r ≤ C1 |∇ ⊗ A|2 (3.17) B3r

for some universal constant C1 , where in the last step we used the Poincaré inequality. Let ϕ be a real phase such that ∇ϕ = A . Since χ 0 ≡ 1 on the support of θ1 by D ≥ δ, we have θ1 T (A)θ1 = θ1 e−iϕ T (A − A )eiϕ θ1 = θ1 e−iϕ T (A0 )eiϕ θ1 . After these localizations, we have H N1 ,Z (A) ≥

N

θ1 e−iϕ (T (A0 ) − W (x)) eiϕ θ1 + j

j=1

with

W (x) =

Z + Cd −2 1(|x| ≤ 2d). |x|

1 2C1 α 2

B20

(3.18)

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L. Erd˝os, J. P. Solovej

Now we use the “running energy scale” argument in [LLS]. ∞ N

θ1 e−iϕ [T (A0 ) − W ] eiϕ θ1 ≥ − N−e (T (A0 ) − W )de j=1

j

µ

≥−

0 µ

≥− 0

N−e (T (A0 ) − W )de −

0

∞

µ ∞

N0

N−e (T (A0 ) − W )de −

µ

µ

N0

e

T (A0 ) − W + e de

e2 e T (A0 ) − W + µ µ

de,

(3.19)

where N−e (A) denotes the number of eigenvalues of a self-adjoint operator A below −e. In the first term we use the bound T (A0 ) ≥ (p + A0 )2 − |B0 | and the CLR bound: µ µ 3/2 N−e (T (A0 ) − W )de ≤ C de (W + |B0 | − e)+ R3 0 0 µ 3/2 ≤C de (W − e/2)+ 3 0 µ R 3/2 +C de (|B0 | − e2 /2µ)+ 3 R 0 5/2 ≤C W + Cµ1/2 B20 R3 R3 5/2 1/2 −2 1/2 = C Z d + Cd + Cµ B20 . (3.20) R3

In the second term of (3.19) we use 1 e 2eZ 2 (p + A0 ) − 1(|x| ≤ 2d) T (A0 ) − W ≥ µ 2 µ|x| 1 Ce + (p + A0 )2 − |B0 | − 1(|x| ≤ 2d), 2 µd 2 and that (p + A0 )2 − i.e. T (A0 ) −

2eZ 4eZ 2eZ 2 , 1(|x| ≤ 2d) ≥ (p + A0 )2 − ≥− µ|x| µ|x| µ

(3.21)

2 1 e eZ Ce W ≥ (p + A0 )2 − 2 − |B0 | − 1(|x| ≤ 2d). µ 2 µ µd 2

We choose µ = 4Z 2 , then using Ce/µd 2 ≤ e2 /4µ for µ ≤ e (i.e. C ≤ (δ Z 2/3 )2 ), we get ∞ ∞ 1 e2 e e2 2 de ≤ (p + A0 ) − |B0 | + de N0 T (A0 ) − W + N0 µ µ 2 4µ µ µ µ 3/2 ≤C de (|B0 | − e2 /4µ)+ 3 0 R 1/2 ≤ Cµ B20 . (3.22) R3

Energy of Atoms in a Self-Generated Field

237

Note that if Z α 2 ≤ κ with some sufficiently small universal constant κ, then the magnetic energy terms in (3.20) and (3.22) can be controlled by the corresponding term in (3.18). Combining the estimates (3.18), (3.19), (3.20) and (3.22) we obtain H N1 ,Z (A) ≥ −C Z 5/2 d 1/2 − Cd −2 and Lemma 2.1 follows.

(3.23)

4. Removing the Magnetic Field Proof of Lemma 2.2. We start with two preparations. In Sect. 4.1 we give an upper bound for the number of electrons N in the truncated model described by H N0 ,Z (A). In Sect. 4.2 we then reduce the problem to a one-body semiclassical statement on boxes. The semiclassical problem will be investigated in Sect. 5 and this will complete the proof of Lemma 2.2. 4.1. Upper bound on the number of electrons N . Let N

E 0N ,Z (A) := inf , H N0 ,Z (A) : ∈ C0∞ (R3 ) ⊗ C2 , = 1 1

be the ground state energy of the truncated Hamiltonian H N0 ,Z (A) defined in (2.14). The following lemma shows that we can assume N ≤ C Z when taking the infimum over N in (2.15). The proof is a slight modification of the proof of the Ruskai-Sigal theorem as presented in [CFKS]. We note that the original proof was given for the non-magnetic case and it can be trivially extended to the Schrödinger operator with a magnetic field but not to the Pauli operator. This is because a key element of the proof, the standard lower bound on the hydrogen atom, − − Z /|x| ≥ −Z 2 /4, is valid if − = p2 replaced by (p + A)2 but there is no lower bound for the ground state energy of the hydrogen atom with the Pauli kinetic energy that is independent of the magnetic field. However, for the truncated Coulomb potential the trivial lower bound can be used. Lemma 4.1. There exist universal constants c and C such that for any fixed A ∈ A0 and Z we have E 0N ,Z (A) = E 0N −1,Z (A) whenever N ≥ C Z and Z −1/6 ≤ δ ≤ c. In particular inf inf E 0N ,Z (A) = inf N A∈A0

inf E 0N ,Z (A)

N ≤C Z A∈A0

(4.24)

if Z −1/6 ≤ δ ≤ c. Proof. We mostly follow the proof of Theorem 3.15 in [CFKS] and we will indicate only the necessary changes. For any x = (x1 , x2 , . . . x N ) ∈ R3N we define x∞ (x) := max{|xi |, : i = 1, 2, . . . , N }, A0 := {x : |x j | < ∀ j = 1, 2, . . . N }, Ai := x : |xi | ≥ (1 − ζ )x∞ (x), x∞ (x) > 2

238

L. Erd˝os, J. P. Solovej

for some fixed positive and ζ < 1/2 to be chosen later. According to Lemma 3.16 in [CFKS], there is partition of unity {Ji }i=0,1,...N , with i Ji2 ≡ 1, supp Ji ⊂ Ai such that the gradient estimates L(x) =

N

|∇ Ji (x)|2 ≤

C N 1/2 2

|∇ Ji (x)|2 ≤

C N 1/2 x∞ (x)

i=0

L(x) =

N i=0

if x ∈ A0 , if x ∈ A j , j ≥ 1

hold with a suitable universal constant C. Moreover, J0 is symmetric in all variables, while Ji , i ≥ 1, is symmetric in all variables except xi . We subtract the local field energy that is an irrelevant constant, i.e. define 1 0 H N := H N ,Z (A) − |∇ ⊗ A|2 16π α 2 B3r and E N = inf Spec H N . We will show that E N = E N −1 for N ≥ C Z . By removing one electron to infinity, clearly E N ≤ E N −1 ≤ 0; here we used the fact that A is compactly supported. By the IMS localization H N = J0 (H N − L)J0 +

N

Ji (H N − L)Ji .

(4.25)

i=1

In the first term we use that on the support of θ0 we have −Z |x|−1 ≥ −Z d −1 . Hence N (N − 1) C N 1/2 −1 −2 J0 . (4.26) − J0 (H N − L)J0 ≥ J0 −C Z N d − C N d + 4 2 Choosing = 8d we see that J0 (H N − L)J0 ≥ 0 if N ≥ C Z with a constant C if δ ≥ C Z −2/3 . To estimate the terms Ji (H N − L)Ji for i = 0, we define H N(i)−1

N

Z 1 2 2 θ0 T (A) − − |∇θ0 | + |∇θ1 | . θ0 + := |x| |x − xj| k j j=1 k< j j =i

k, j =i

On the support of Ji we have |xi | ≥ /4 = 2d, so ∇θ0 and ∇θ1 vanish. Then we can estimate N −1 C N 1/2 Z (i) + − Ji Ji (H N − L)Ji ≥ Ji H N −1 − |xi | 2x∞ (x) x∞ (x) C N 1/2 Z 1/3 1 N −1 (1 − ζ ) − Z − Ji ≥ Ji E N −1 + |xi | 2 δ ≥ Ji E N −1 Ji (4.27) if N ≥ C Z and N is large. Thus we conclude from (4.25), (4.26) and (4.27) that E N ≥ E N −1 if N ≥ C Z .

Energy of Atoms in a Self-Generated Field

239

4.2. Reduction to a one-body problem. We start by presenting an abstract lemma whose proof is given in Appendix A. Lemma 4.2. Let h be a one-particle operator on H = L 2 (R3 ) and let W be a twoparticle operator defined on H ∧ H. We assume that the domains of h and W include the C0∞ functions. Let θ ∈ C ∞ (R3 ) with compact support := supp θ . Then ⎧ ⎡ ⎫ ⎤ N N ⎨ ⎬ inf , ⎣ θi hi θi + θi θ j Wi j θ j θi ⎦ : ∈ C0∞ (R N ), = 1 ⎩ ⎭ 1 i=1 1≤i< j≤N ⎧ ⎛ ⎫ ⎞ n n ⎨ ⎬ ≥ inf inf , ⎝ hi + Wi j ⎠ : ∈ C0∞ (), = 1 , ⎩ ⎭ n≤N i=1

1

1≤i< j≤n

(4.28) where hi denotes the operator h acting on the component of the tensor product, and similar convention is used for the two-particle operators. The same result holds with obvious changes if H = L 2 (R3 ) ⊗ C2 . i th

To continue the proof of Lemma 2.2, we first localize H N0 ,Z (A) onto a ball Br of radius r = D Z −1/3 (see (1.8)) and we also localize the magnetic field as in Sect. 3. We introduce smooth cutoff functions χ0 and χ1 with χ02 + χ12 ≡ 1, supp χ0 ⊂ B2r , χ0 ≡ 1 on Br , |∇χ0 |, |∇χ1 | ≤ Cr −1 . We get H N0 ,Z (A)

≥

N

θ0 χ0 e

i=1

1 + 2 Cα

−iϕ0

1 Z Ti (A0 ) − eiϕ0 χ0 θ0 + |xi | |xi − x j | i< j

R3

B20 − C N d −2 − C N Zr −1

(4.29)

using that the new localization error |∇χ1 |2 + |∇χ0 |2 ≤ Cr −2 ≤ Cd −2 and that −Z /|x| ≥ −Zr −1 on the support of χ1 . We also used (3.17). Let Ad,r = {x : d ≤ |x| ≤ r } ⊂ R3 . Using (4.24), the positivity of the Coulomb repulsion |xi − x j |−1 > 0 and Lemma 4.2 with θ := θ0 χ0 we obtain ⎧ ⎪ ⎨

⎡

inf inf H N0 ,Z (A) ≥ inf inf inf , ⎣ N A∈A0 N ≤C Z A0 ∈A0 ⎪ ⎩

+

⎫ ⎪ ⎬

⎤ 1 Z ⎦ T (A0 ) − + |x| i |x − x j | i=1 i< j i Ad,r

N

1 B2 − C Z 5/3 δ −2 − C Z 7/3 D −1 , Cα 2 R3 0 ⎪ ⎭

(4.30)

where the infimum is over all antisymmetric wave functions ∈ 1N C0∞ (Ad,r ) ⊗ C2 with 2 = 1. The notation [H ] Q indicates the N -particle operator H with Dirichlet boundary condition on the domain Q N ⊂ R3N . We define f (x)g(y) 1 dxdy. D( f, g) := 2 R3 ×R3 |x − y|

240

L. Erd˝os, J. P. Solovej

Lemma 4.3. There is a universal constant C0 > 0 such that for any ∈ 1N C0∞ (R3 )⊗ C2 with 2 = 1, for any nonnegative function : R3 → R with D(, ) < ∞, for any A ∈ A0 , and for any ε > 0 we have

⎡

⎤

1 ⎦ + C0 Ti (A) + B2 , ⎣ε 3 |xi − x j | R i=1 i< j N

∗ |xi |−1 − Cε−1 N . ≥ −D(, ) + , N

(4.31)

i=1

Proof. By the Lieb-Oxford inequality [LO] and by the positivity of the quadratic form D(·, ·), 1 4/3 ≥ D( , ) − C , 3 |xi − x j | R i< j N 4/3 ≥ −D(, ) + , ( ∗ |xi |−1 ) − C ,

(4.32)

R3

i=1

where (x) is the one-particle density of . The error term is controlled by the following kinetic energy inequality for the Pauli operator

% ,

N

&

T (A)i ≥ c

i=1

5/3

R3

4/3

min{ , γ } − γ

B2

(4.33)

R3

with some positive universal constant c and for any γ > 0. For the proof of (4.33) use the magnetic Lieb-Thirring inequality

%

& N , [T (A) − U ]i ≥ −C i=1

U R3

5/2

− Cγ

−3

U −γ 4

R3

R3

B2 .

With the choice U = β min{ , γ } we can ensure that 21 U ≥ CU 5/2 +Cγ −3 U 4 if β is sufficiently small (independent of γ ) and this proves (4.33). Thus 4/3 5/3 4/3 ≤ γ −1 min{ , γ } + γ R3 R3 R3 % N & −1 ≤ (cγ ) T (A)i + c−1 B2 + γ N (4.34) , 2/3

1/3

i=1

R3

so choosing γ = Cε−1 with a sufficiently large constant C, we obtain (4.31).

Energy of Atoms in a Self-Generated Field

241

Using Lemma 4.3 we can continue the estimate (4.30) (writing A instead of A0 in the infimum) as inf inf H N0 ,Z (A) N

A

⎧ % & N ⎨ [T (A) + W ]i ≥ (1 − ε) inf inf inf , N ≤C Z A∈A0 ⎩ i=1

Ad,r

1 + Cα 2

B2 R3

−D(, ) − Cε−1 Z − C Z 5/3 δ −2 − C Z 7/3 D −1 with W (x) :=

⎫ ⎬ ⎭

(4.35)

Z 1 − + ∗ |x|−1 1−ε |x|

and assuming that α ≤ α0 with some small universal α0 . We now perform a rescaling: x = Z −1/3 X , p = Z 1/3 P and A(X ) = Z −2/3 A(Z −1/3 X ),

B(X ) = ∇ × A(X ) = Z −1 B(Z −1/3 X ).

A) := [(hP + A) · σ ]2 , we obtain that the kinetic energy Introducing h = Z −1/3 and Th ( changes as A) · σ ]2 = Z 4/3 Th ( A) [(p + A) · σ ]2 = Z 4/3 [(Z −1/3 P + and the field energy changes as 2 B (x)dx = Z R3

R3

B2 (X )dX.

The new potential energy is (X ) = Z −4/3 W (Z −1/3 X ) = W

1 1 − + ∗ |X |−1 , 1−ε |X |

where (X ) = Z −2 (Z −1/3 X ) and D( , ) = Z −7/3 D(, ). After rescaling, we get from (4.35), inf inf H N0 ,Z (A) N A∈A0

⎧ % & N ⎨ 4/3 ]i ≥ (1 − ε)Z inf inf inf , [Th ( A) + W N ≤C Z A∈A0 ⎩ i=1

+ Aδ,D

−Z 7/3 D( , ) − Cε−1 Z − C Z 5/3 δ −2 − C Z 7/3 D −1 ,

h −2 C Z α2

B2

R3

⎫ ⎬ ⎭

(4.36)

where Aδ,D = {X : δ ≤ |X | ≤ D} and inf denotes the infimum over all normalized antisymmetric functions. Using (1.12) from Theorem 1.3 and the fact that Z α 2 ≤ κ, we get

) Aδ,D − C Z 13/6 D 3 δ −13/4 inf inf H N0 ,Z (A) ≥ (1 − ε)Z 4/3 Tr (−h 2 + W N

A

−

−Z 7/3 D( , ) − Cε−1 Z − C Z 5/3 δ −2 − C Z 7/3 D −1 , (4.37)

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assuming δ ≥ Z −2/9 . By a standard semiclassical result for Coulomb-like potentials (see e.g. the result in Sect. V.2 of [L]):

2 2 −3 −5/2 − Ch −3+1/10 , ≥ Tr −h + W ≥ −Csc h Tr (−h + W ) Aδ,D W −

−

R3

(4.38) 1 where Csc = 2/(15π 2 ) is the Weyl constant in semiclassics. The 10 exponent in the error term is far from being optimal; the methods developed to prove the Scott correction can yield an exponent up to one (see Remark 1 after Theorem 1.1). Taking the optimal to be the Thomas-Fermi density for Z = 1 = TF (see, e.g. Sect. II of [L]) and defining the Thomas-Fermi constant as

5/2 1 −1 , − + TF ∗ |X | |X | R3 −

cTF := D(TF , TF ) + C SC we get ' inf inf N

A

H N0 ,Z (A)

≥ (1 − ε)

−3/2

Z

7/3

−D( , ) − Csc

5/2 ( 1 −1 − + ∗ |X | |X | R3 −

−C Z 7/3−1/30 − C Z 13/6 D 3 δ −13/4 − Cε−1 Z − C Z 5/3 δ −2 − C Z 7/3 D −1 ≥ −(1 − ε)−3/2 cTF Z 7/3 − C Z 7/3−1/30 − Cε−1 Z − C Z 5/3 δ −2 − C Z 7/3 D −1

(4.39) ≥ −cTF Z 7/3 − C Z 7/3 Z −1/30 + D −1 , where we optimized for ε and we used that D ≤ Z 1/24 δ 13/16 and Z −1/6 ≤ δ ≤ 1. This completes the proof of Lemma 2.2. 5. Semiclassics: Proof of Theorem 1.3 We present the Schrödinger and Pauli cases in parallel. We prove the statement with Dirichlet boundary conditions (1.12) in detail and in Sect. 5.4 we comment on the necessary changes for the proof of the (1.11). The potential V is defined only on , but we extend it to be zero on R3 \; we continue to denote by V its extension. 5.1. Localization onto boxes. We choose a length L with h ≤ L ≤ 13 h 1/2 . Let L = + B L be the L-neighborhood of . Let Q k = {y ∈ R3 : y − k∞ < L/2} with k ∈ (LZ)3 ∩ L denote a non-overlapping covering of with boxes of size L. In this sectionthe index k will always run over the set (LZ)3 ∩ L . Let ξk be a partition of unity, k ξk2 ≡ 1, subordinated to the collection of boxes Q k , such that supp ξk ⊂ (2Q)k ,

|∇ξk | ≤ C L −1 ,

where (2Q)k denotes the cube of side-length 2L with center k. Let ξk be a cutoff funck := (3Q)k and tion such that ξk ≡ 1 on (2Q)k (i.e. on the support of ξk ), supp ξk ⊂ Q |∇ ξk | ≤ C L −1 .

Energy of Atoms in a Self-Generated Field

k |−1 Let A k = | Q Bk := ∇ × Ak , then

k Q

243

A. Similarly to (3.16), we define Ak := (A − A k ) ξk and

R3

B2k ≤ C

k Q

|∇ ⊗ A|2

(5.40)

as in (3.17). From the IMS localization with ψk satisfying h∇ψk = Ak we have ∗ −2 2 −2 Tr [[Th (A) − V ] ]− + h B = inf Tr (γ [Th (A) − V ]) + h |∇ ⊗ A|2 R3

γ

≥ inf γ

R3

∗

Ek (γ )

k∈(LZ)3 ∩

(5.41)

L

with

Ek (γ ) := Tr γ ξk e−iψk [Th (A − A k ) − V ]eiψk ξk − γ |h∇ξk |2 + c0 h −2

k Q

|∇ ⊗ A|2

with some universal constant c0 . Here inf ∗γ denotes infimum over all density matrices 0 ≤ γ ≤ 1 that are supported on , i.e. they are operators on L 2 () ⊗ C2 . We also used R3 B2 = R3 |∇ ⊗ A|2 and we reallocated the second integral. We introduce the notation −2 Fk := c0 h |∇ ⊗ A|2 . k Q

5.2. A priori bound on the local field energy. In case of the Pauli operator, for each fixed k we apply the magnetic Lieb-Thirring inequality [LLS] together with (5.40) and box Q with the bound V ∞ ≤ K to obtain that for any density matrix γ ,

Ek (γ ) ≥ Tr [Th (Ak ) − V − Ch 2 L −2 ] Qk + Fk − ≥ −Ch −3 [V + Ch 2 L −2 ]5/2 k Q

−C

k Q

[V + Ch 2 L −2 ]4

1/4

h −2

3/4

k Q

B2k

+ Fk

c 0 −2 h ≥ −C h −3 K 5/2 L 3 + h 2 L −2 + K 4 L 3 + h 8 L −5 − |∇ ⊗ A|2 + Fk 2 k Q 1 ≥ −Ch −3 K 5/2 L 3 + Fk , (5.42) 2 using h ≤ L and 1 ≤ K ≤ Ch −2 . In the Schrödinger case we use the usual LiebThirring inequality [LT] that holds with a magnetic field as well. The estimate (5.42) is then valid even without the third term in the second line. Let S ⊂ (LZ)3 ∩ L denote the set of those k indices such that Fk ≤ Ch −3 K 5/2 L 3

(5.43)

holds with some large constant C. In particular Ek (γ ) ≥ 0, for all k ∈ S and for any γ .

(5.44)

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5.3. Improved bound. We use the Schwarz inequality in the form Th (A − A k ) ≥ −(1 − εk )h 2 − Cεk−1 (A − A k )2 , with some 0 < εk < 13 . We have for any γ supported on that

Ek (γ ) ≥ Tr 1 ξk [−(1 − 2εk )h 2 − V − Ch 2 L −2 ]ξk 1 −

−1 2 2 +Tr 1 Qk [−εk h − Cεk (A − A k ) ]1 Qk + Fk .

(5.45)

We will show at the end of the section that

Tr 1 ξk [−(1 − 2εk )h 2 − V − Ch 2 L −2 ]ξk 1 −

2 −3 5/2 k |. ≥ Tr 1 ξk (−h − V )ξk 1 − Ch K (εk + h 2 L −2 )| Q

(5.46)

−

−

Using (5.44) and (5.46), ∗ ∗ inf Ek (γ ) ≥ inf Ek (γ ) γ

γ

k

≥

k

≥

k∈S

k

Tr 1 ξk [−h 2 − V ]ξk 1 + Dk −

k∈S

inf Tr ξk γk ξk 1 [−h 2 − V ]1 + Dk γk

≥ Tr (−h 2 − V ) + Dk −

k∈S

(5.47)

k∈S

with

k |(εk + h 2 L −2 ) + Fk . Dk := Tr [−εk h 2 − Cεk−1 (A − A k )2 ] Qk − Ch −3 K 5/2 | Q −

(5.48) In the last step in (5.47) we used that for any collection of density matrices γk , the density matrix k 1 ξk γk ξk 1 is admissible in the variational principle

Tr (−h 2 − V ) = inf Tr γ −h 2 − V : 0 ≤ γ ≤ 1, supp γ ⊂ . −

(5.49) We estimate Dk for k ∈ S as follows: −4 −3 k |(εk + h 2 L −2 ) + Fk Dk ≥ −Cεk h (A − A k )5 − Ch −3 K 5/2 | Q k Q

≥ Fk − Cεk−4 h 2 L 1/2 Fk

5/2

k |(εk + h 2 L −2 ). − Ch −3 K 5/2 | Q

(5.50)

In the first step we used the Lieb-Thirring inequality, in the second step the Hölder and Sobolev inequalities in the form 5/2 5 1/2 2 (A − A k ) ≤ C L |∇ ⊗ A| . k Q

k Q

Energy of Atoms in a Self-Generated Field

245

We choose εk = h L −1/2 K −1/2 Fk , 1/2

and using the a priori bound (5.43), we see that εk ≤ Ch −1/2 L K 3/4 . Thus, assuming L ≤ ch 1/2 K −3/4

(5.51)

with a sufficiently small constant c, we get εk ≤ 1/3. With this choice of εk , and recalling k | = 9|Q k | = 9L 3 , we have |Q Dk ≥ Fk − Ch −2 L 5/2 K 2 Fk − Ch −3 K 5/2 L 3 h 2 L −2

≥ −Ch −3 L 3 K 5/2 h −1 L 2 K 3/2 + h 2 L −2 . 1/2

(5.52)

If we choose L = h 3/4 K −3/8 , then

1/2 Dk ≥ −Ch −3 L 3 K 5/2 h K 3/2 . This choice is allowed by (5.51) if K ≤ ch −2/3 . If ch −2/3 ≤ K ≤ h −2 , then we choose L = ch 1/2 K −3/4 and we get from (5.52), Dk ≥ −Ch −3 L 3 K 5/2 (1 + h K 3/2 ). Combining these two inequalities, we get that

1/2

1/2 Dk ≥ −Ch −3 L 3 K 5/2 h K 3/2 1 + h K 3/2

(5.53)

always holds. Summing up (5.53) for all k and using that L 3 ≤ C|3L | ≤ C|√h | k∈(LZ)3 ∩ L

(recall that √h is a and (5.53), ∗

inf γ

√

k

h-neighborhood of and 3L ≤ h 1/2 ), we obtain from (5.47)

Ek (γ ) ≥ Tr (−h 2 − V )

−

1/2

1/2 1 + h K 3/2 , −Ch −3 K 5/2 |√h | h K 3/2

(5.54)

and this proves (1.12). Finally, we prove (5.46). Let γ be a trial density matrix for the left hand side of (5.46). We can assume that

0 ≥ Tr γ 1 ξk [−(1 − 2εk )h 2 − V − Ch 2 L −2 ]ξk 1 .

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L. Erd˝os, J. P. Solovej

Then

1 1 0 ≥ Tr γ 1 ξk [− h 2 + K ]ξk 1 + Tr γ 1 ξk [− h 2 −V −Ch 2 L −2 − K ]ξk 1 6 6 1 2 ≥ Tr γ 1 ξk [− h + K ]ξk 1 − Ch −3 [V + K + Ch 2 L −2 ]5/2 , (5.55) 6 k Q

where we used the Lieb-Thirring inequality. Thus, using |V | ≤ K , h ≤ L and K ≥ 1, we have 1 2 k |. Tr γ 1 ξk [− h + K ]ξk 1 ≤ Ch −3 K 5/2 | Q 6 Therefore

Tr γ 1 ξk [−(1 − 2εk )h 2 − V − Ch 2 L −2 ]ξk 1

k |. ≥ Tr γ 1 ξk (−h 2 − V )ξk 1 − Ch −3 K 5/2 (εk + h 2 L −2 )| Q

Now (5.46) follows by variational principle.

(5.56)

5.4. Reduction of (1.11) to (1.12). We approximate V ∈ L 5/2 ∩ L 4 by a bounded , V ∞ ≤ K , that is supported on a ball B R/2 and V ≤ V . By choosing K potential V and R sufficiently large, we can make V − V 5/2 +V − V 4 arbitrarily small. We choose a cutoff function χ R that is supported on B R , χ R ≡ 1 on B R/2 and |∇χ R | ≤ C R −1 and R2 ≡ 1. let χ R satisfy χ R2 + χ Borrowing a small part of the kinetic energy, by IMS localization we have ]χ R Th (A) − V ≥ (1 − ε)χ R [Th (A) − V ) − |∇χ R |2 − |∇ +εTh (A) − (V − (1 − ε)V χ R |2 .

(5.57)

Using the magnetic Lieb-Thirring inequality [LLS] to estimate the second term, we get ) B R ]− Tr [Th (A) − V ]− ≥ (1 − ε)Tr [(Th (A) − V −3/2 −3 5/2 −Cε h |U | − C

1 |U | − h −2 3 2 R

R3

4

B2 (5.58) R3

with ) + |∇χ R |2 + |∇ U := (V − (1 − ε)V χ R |2 . For 2the first term in (5.58) we use (1.12) (and that it holds even with a 1/2 in front of B ) and the fact that

) B R Tr (−h 2 − V ≥ Tr −h 2 − V −

−

≤ V . The second and the third terms in (5.58) can be made arbiby monotonicity, V trarily small compared with h −3 for any fixed ε if R and K are sufficiently large and h is small. Finally, choosing ε sufficiently small, we proved (1.11).

Energy of Atoms in a Self-Generated Field

247

6. Proof of the Quantized Field Case For the proof of the lower bound in (1.6), we follow the argument of [BFG] to reduce the problem to the classical bound (1.10). We set |g(k)|2 |k| aλ (k)∗ aλ (k)dk Hg = α −1 R3

λ=±

to be the cutoff field energy, then H f ≥ Hg and only the modes appearing in Hg interact with the electron. By Lemma 3 of [BFG], for any real function f ∈ L 1 (R3 ) ∩ L ∞ (R3 ) we have 1 f (x)|∇ ⊗ A(x)|2 dx ≤ α 2 f ∞ Hg + Cα4 f 1 . 8π R3 Applying it with f being the characteristic function of the ball B3r with r = D Z −1/3 (the radius of the ball is here chosen differently from [BFG]) and using Z α 2 ≤ κ we get 2 Zα Z H f ≥ Hg ≥ Hg ≥ |∇ ⊗ A(x)|2 dx − Cκ −1 α4 D 3 . κ 8π κ B3r Setting α = (κ/Z )1/2 , i.e. Z α 2 = κ, we have for κ sufficiently small, −1 4 3 E N ,Z ≥ E N ,Z , α − Cκ α D , qed

where E N ,Z , α is the ground state energy of the Hamiltonian (1.7) with fine structure constant α . Applying (1.10) to this Hamiltonian, we get qed

EZ

7

−1 3 ≥ E nf − Cκ −1 α4 D 3 Z − CZ D

1

whenever 1 ≤ D ≤ Z 63 . Writing D = Z γ and applying the upper bound (1.5) on , we obtain the lower bound in (1.6). A. Proof of Lemma 4.2 Let the function χ (x) ∈ C ∞ (R3 ) be defined such that θ 2 (x)+χ 2 (x) ≡ 1. For any subset α ⊂ {1, 2, . . . , N } we denote by xα the collection of variables {xi : i ∈ α} and define ) ) α = α (xα ) := θ (xi ), α = α (xα ) := χ (xi ). i∈α

i∈α

We set the notation α c = {1, 2, . . . , N } \ α for the complement of the set α and set n := {1, 2, . . . , n}. Let |α| denote the cardinality of the set α. For an arbitrary function ∈ 1N C0∞ (R3 ), = 1, and for 0 ≤ n ≤ N we define * + , n := n Tr n c n c | |n c n , where Tr n c denotes taking the partial trace for the xn+1 , xn+2 , . . . , x N variables. Define .N the fermionic Fock space as F = n=0 Hn with Hn := n H and we define a density matrix N N n |α| on F. := = n α⊂{1,2,...,N }

n=0

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We first prove that ≤ I on F. It is sufficient to show that ≤ I on the n-particle sectors for each n. Let n ≤ N , choose ∈ Hn , and compute , |α| = dxα dxα (xα ) n (xα , xα )(xα ) α⊂{1,2,...N } |α|=n

=

α⊂{1,2,...N } |α|=n

dxα dxα dyα c (xα )α (xα )α c (yα c ) (xα , yα c )

α

× (xα , yα c )α c (yα c )α (xα )(xα ) ≤ dxα dxα dyα c 2α (xα )2α c (yα c )| (xα , yα c )|2 |(xα )|2 α

=

22

dx| (x)|2

α

= 22 ,

2α (xα )2α c (xα c )

/N

(A.59)

using Schwarz inequality and that 1 ≡ j=1 [θ 2 (x j )+χ 2 (x j )] = α 2α (xα )2α c (xα c ). Second, for a fixed n ≤ N , we compute % & % & N N n n 0 N n Tr F Tr F hi = hi n n=0 i=1 n=0 i=1 = dxα dxα c (xα , xα c )α (xα )α c (xα c )hi (α (xα )α c (xα c ) (xα , xα c )) α

=

N

i∈α

i=1 α : i∈α

=

N

dx (xα , xα c )2α\{i} (xα\{i} )2α c (xα c )θ (xi )hi (θ (xi ) (xα , xα c ))

, θi hi θi ,

(A.60)

i=1

where the trace on the left/hand side is computed on F. In the last step we used that for 2 2 , where the summation any fixed i, we have 1 ≡ j=i [θ 2 (x j ) + χ 2 (x j )] = 1 α 1 α 1 αc c is over all 1 α ⊂ {1, 2, . . . , N }\{i} and 1 α = {1, 2, . . . , N }\{i}\α. A similar calculation for the two-body potential shows that ⎡ ⎤ N 0 2 3 Tr F ⎣ , θi θ j Wi j θ j θi . Wi j ⎦ = n=0 1≤i< j≤n

1≤i< j≤N

Thus, by the variational principle, ⎛ N , ⎝ θi hi θi + i=1

⎞

n=0

i=1

θi θ j Wi j θ j θi ⎠

1≤i< j≤N

⎛ N n 0 ⎝ ≥ inf Tr F ⎣ hi +

⎡

⎞⎤ Wi j ⎠⎦.

1≤i< j≤n

Since is a density matrix supported on , we obtain (4.28).

Energy of Atoms in a Self-Generated Field

249

References [BFG] [CFKS] [ES]

[ES2] [F] [FLL] [FS] [H] [IS] [L] [LL] [LLS] [LO] [LS] [LSY1] [LSY2] [LT]

[LY] [SW1] [SW2] [SW3] [SS]

Bugliaro, L., Fröhlich, J., Graf, G.M.: Stability of quantum electrodynamics with nonrelativistic matter. Phys. Rev. Lett. 77, 3494–3497 (1996) Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Berlin-Heidelberg-New York: Springer-Verlag, 1987 Erd˝os, L., Solovej, J.P.: Semiclassical eigenvalue estimates for the pauli operator with strong non-homogeneous magnetic fields. II. leading order asymptotic estimates. Commun. Math. Phys. 188, 599–656 (1997) Erd˝os, L., Solovej, J.P.: The kernel of dirac operators on S3 and R3 . Rev. Math. Phys. 13, 1247– 1280 (2001) Fefferman, C.: Stability of coulomb systems in a magnetic field. Proc. Nat. Acad. Sci. USA 92, 5006–5007 (1995) Fröhlich, J., Lieb, E.H., Loss, M.: Stability of coulomb systems with magnetic fields. I. the oneelectron atom. Commun. Math. Phys. 104, 251–270 (1986) Fefferman, C., Seco, L.A.: On the energy of a large atom. Bull. AMS 23(2), 525–530 (1990) Hughes, W.: An atomic energy bound that gives scott’s correction. Adv. Math. 79, 213–270 (1990) Ivrii, V.I., Sigal, I.M.: Asymptotics of the ground state energies of large coulomb systems. Ann. of Math. (2) 138, 243–335 (1993) Lieb, E.H.: Thomas-fermi and related theories of atoms and molecules. Rev. Mod. Phys. 65(4), 603–641 (1981) Lieb, E.H., Loss, M.: Stability of coulomb systems with magnetic fields. II. the many-electron atom and the one-electron molecule. Commun. Math. Phys. 104, 271–282 (1986) Lieb, E.H., Loss, M., Solovej, J.P.: Stability of matter in magnetic fields. Phys. Rev. Lett. 75, 985–989 (1995) Lieb, E.H., Oxford, S.: Improved lower bound on the indirect coulomb energy. Int. J. Quant. Chem. 19, 427–439 (1981) Lieb, E.H., Simon, B.: The thomas-fermi theory of atoms. molecules and solids. Adv. Math. 23, 22–116 (1977) Lieb, E.H., Solovej, J.P., Yngvason, J.: Asymptotics of heavy atoms in high magnetic fields: I. lowest landau band region. Commun. Pure Appl. Math. 47, 513–591 (1994) Lieb, E.H., Solovej, J.P., Yngvason, J.: Asymptotics of heavy atoms in high magnetic fields: II. semiclassical regions. Commun. Math. Phys. 161, 77–124 (1994) Lieb, E.H., Thirring, W.: A bound on the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann. E. H. Lieb, B. Simon, A. Wightman, eds., Princeton, NJ: Princeton University Press, 1976, pp. 269–303 Loss, M., Yau, H.T.: Stabilty of coulomb systems with magnetic fields. III. zero energy bound states of the pauli operator. Commun. Math. Phys. 104, 283–290 (1986) Siedentop, H., Weikard, R.: On the leading energy correction for the statistical model of an atom: interacting case. Commun. Math. Phys. 112, 471–490 (1987) Siedentop, H., Weikard, R.: On the leading correction of the thomas-fermi model: lower bound. Invent. Math. 97, 159–193 (1990) Siedentop, H., Weikard, R.: A new phase space localization technique with application to the sum of negative eigenvalues of schrödinger operators. Ann. Sci. École Norm. Sup. (4) 24(2), 215–225 (1991) Solovej, J.P., Spitzer, W.L.: A new coherent states approach to semiclassics which gives scott’s correction. Commun. Math. Phys. 241, 383–420 (2003)

Communicated by I. M. Sigal

Commun. Math. Phys. 294, 251–272 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0903-4

Communications in

Mathematical Physics

The Hermitian Laplace Operator on Nearly Kähler Manifolds Andrei Moroianu1 , Uwe Semmelmann2 1 CMLS, École Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau, France.

E-mail: [email protected]

2 Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany.

E-mail: [email protected] Received: 23 March 2009 / Accepted: 11 May 2009 Published online: 9 August 2009 – © Springer-Verlag 2009

Abstract: The moduli space N K of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1, 1) forms (cf. Moroianu et al. in Pacific J Math 235:57–72, 2008). Using the Hermitian Laplace operator and some representation theory, we compute the space N K on all 6-dimensional homogeneous nearly Kähler manifolds. It turns out that the nearly Kähler structure is rigid except for the flag manifold F(1, 2) = SU3 /T 2 , which carries an 8-dimensional moduli space of infinitesimal nearly Kähler deformations, modeled on the Lie algebra su3 of the isometry group. 1. Introduction Nearly Kähler manifolds were introduced in the 70’s by A. Gray [8] in the context of weak holonomy. More recently, 6-dimensional nearly Kähler manifolds turned out to be related to a multitude of topics among which we mention: Spin manifolds with Killing spinors (Grunewald), SU3 -structures, geometries with torsion (Cleyton, Swann), stable forms (Hitchin), or super-symmetric models in theoretical physics (Friedrich, Ivanov). Up to now, the only sources of compact examples are the naturally reductive 3-symmetric spaces, classified by Gray and Wolf [13], and the twistor spaces over positive quaternion-Kähler manifolds, equipped with the non-integrable almost complex structure. Based on previous work by R. Cleyton and A. Swann [6], P.-A. Nagy has shown in 2002 that every simply connected nearly Kähler manifold is a Riemannian product of factors which are either of one of these two types, or 6-dimensional [12]. Moreover, J.-B. Butruille has shown [5] that every homogeneous 6-dimensional nearly Kähler manifold is a 3-symmetric space G/K , more precisely isometric with S 6 = G 2 /SU3 , S 3 × S 3 = SU2 × SU2 × SU2 /SU2 , CP3 = SO5 /U2 × S 1 or F(1, 2) = SU3 /T 2 , all endowed with the metric defined by the Killing form of G. This work was supported by the French-German cooperation project Procope no. 17825PG.

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A method of finding new examples is to take some homogeneous nearly Kähler manifold and try to deform its structure. In [10] we have studied the deformation problem for 6-dimensional nearly Kähler manifolds (M 6 , g) and proved that if M is compact, and has normalized scalar curvature scalg = 30, then the space N K of infinitesimal deformations of the nearly Kähler structure is isomorphic to the eigenspace for the eigenvalue 12 of the restriction of the Laplace operator g to the space of co-closed (1,1) primitive (1, 1)-forms 0 M. It is thus natural to investigate the Laplace operator on the known 3-symmetric examples (besides the sphere S 6 , whose space of nearly Kähler structures is well-understood, and isomorphic to SO7 /G 2 ∼ = RP7 , see [7] or [5, Prop. 7.2]). Recall that the spectrum of the Laplace operator on symmetric spaces can be computed in terms of Casimir eigenvalues using the Peter-Weyl formalism. It turns out that a similar method can be applied ¯ (called the Hermitian in order to compute the spectrum of a modified Laplace operator Laplace operator) on 3-symmetric spaces. This operator is SU3 -equivariant and coincides with the usual Laplace operator on co-closed primitive (1, 1)-forms. The space of infinitesimal nearly Kähler deformations is thus identified with the space of co-closed ¯ = 12α}. Our main result is that the forms in 0(1,1) (12) := {α ∈ C ∞ (0(1,1) M) | α nearly Kähler structure is rigid on S 3 × S 3 and CP3 , and that the space of infinitesimal nearly Kähler deformations of the flag manifold F(1, 2) is eight-dimensional. The paper is organized as follows. After some preliminaries on nearly Kähler man(1,1) ifolds, we give two general procedures for constructing elements in 0 (12) out of Killing vector fields or eigenfunctions of the Laplace operator for the eigenvalue 12 (Corollary 4.5 and Proposition 4.11). We show that these elements can not be co-closed, thus obtaining an upper bound for the dimension of the space of infinitesimal nearly Kähler deformations (Proposition 4.12). We then compute this upper bound explicitly on the 3-symmetric examples and find that it vanishes for S 3 × S 3 and CP3 , which therefore have no infinitesimal nearly Kähler deformation. This upper bound is equal to 8 on the flag manifold F(1, 2) = SU3 /T 2 and in the last section we construct an explicit isomorphism between the Lie algebra of the isometry group su3 and the space of infinitesimal nearly Kähler deformations on F(1, 2). In addition, our explicit computations (in Sect. 5) of the spectrum of the Hermitian Laplace operator on the 3-symmetric spaces, together with the results in [11] show that every infinitesimal Einstein deformation on a 3-symmetric space is automatically an infinitesimal nearly Kähler deformation. 2. Preliminaries on Nearly Kähler Manifolds An almost Hermitian manifold (M 2m , g, J ) is called nearly Kähler if (∇ X J )(X ) = 0,

∀ X ∈ T M,

(1)

where ∇ denotes the Levi-Civita connection of g. The canonical Hermitian connection ¯ defined by ∇, ∇¯ X Y := ∇ X Y − 21 J (∇ X J )Y,

∀ X ∈ T M, ∀ Y ∈ C ∞ (M)

(2)

¯ = 0 and ∇¯ J = 0) with torsion T¯X Y = −J (∇ X J )Y . is a Um connection on M (i.e. ∇g A fundamental observation, which goes back to Gray, is the fact that ∇¯ T¯ = 0 on every nearly Kähler manifold (see [2]).

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We denote the Kähler form of M by ω := g(J., .). The tensor + := ∇ω is totally skew-symmetric and of type (3, 0) + (0, 3) by (1). From now on we assume that the dimension of M is 2m = 6 and that the nearly Kähler structure is strict, i.e. (M, g, J ) is not Kähler. It is well-known that M is Einstein in this case. We will always normalize the scalar curvature of M to scal = 30, in which case we also have | + |2 = 4 point¯ wise. The form + can be seen as the real part of a ∇-parallel complex volume form + + i − on M, where − = ∗ + is the Hodge dual of + . Thus M carries a SU3 ¯ Notice that Hitchin has shown structure whose minimal connection (cf. [6]) is exactly ∇. + − that a SU3 structure (ω, , ) is nearly Kähler if and only if the following exterior system holds: dω = 3 + (3) d − = −2ω ∧ ω. Let A ∈ 1 M ⊗ EndM denote the tensor A X := J (∇ X J ) = − +J X , where Y+ denotes the endomorphism associated to Y + via the metric. Since for every unit vector X , A X defines a complex structure on the 4-dimensional space X ⊥ ∩ (J X )⊥ , we easily get in a local orthonormal basis {ei } the formulas |A X |2 = 2|X |2 , Aei Aei (X ) = −4X,

∀X ∈ T M, ∀X ∈ T M,

(4) (5)

where here and henceforth, we use Einstein’s summation convention on repeating subscripts. The following algebraic relations are satisfied for every SU3 structure (ω, + ) on T M (notice that we identify vectors and 1-forms via the metric): A X ei ∧ ei + X − + (X ) ∧ + (J X + ) ∧ ω

= −2X ∧ ω, ∀X ∈ T M, + = −J X , ∀X ∈ T M, = X ∧ ω2 , ∀X ∈ T M, = X ∧ +, ∀X ∈ T M.

(6) (7) (8) (9)

The Hodge operator satisfies ∗2 = (−1) p on p M and moreover ∗ (X ∧ + ) = J X + , ∗(φ ∧ ω) = −φ,

∀φ ∈

∗(J X ∧ ω ) = −2X, 2

∀X ∈ T M,

(10)

(1,1) 0 M,

(11)

∀X ∈ T M.

(12)

From now on we assume that (M, g) is compact 6-dimensional not isometric to the round sphere (S 6 , can). It is well-known that every Killing vector field ξ on M is an automorphism of the whole nearly Kähler structure (see [10]). In particular, L ξ ω = 0,

L ξ + = 0,

L ξ − = 0.

(13)

¯ Then the formula (cf. [1]) Let now R and R¯ denote the curvature tensors of ∇ and ∇. RW X Y Z = R¯ W X Y Z − 41 g(Y, W )g(X, Z ) + 41 g(X, Y )g(Z , W ) + 43 g(Y, J W )g(J X, Z ) − 43 g(Y, J X )g(J W, Z ) − 21 g(X, J W )g(J Y, Z ) may be rewritten as R X Y = − X ∧ Y + R CY XY

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and R¯ X Y = − 43 (X ∧ Y + J X ∧ J Y − 23 ω(X, Y )J ) + R CY XY , where R CY X Y is a curvature tensor of Calabi-Yau type. We will recall the definition of the curvature endomorphism q(R) (cf. [10]). Let E M be the vector bundle associated to the bundle of orthonormal frames via a representation π : SO(n) → Aut(E). The Levi-Civita connection of M induces a connection on E M, whose curvature satisfies R XE YM = π∗ (R X Y ) = π∗ (R(X ∧ Y )), where we denote with π∗ the differential of π and identify the Lie algebra of S O(n), i.e. the skew-symmetric endomorphisms, with 2 . In order to keep notations as simple as possible, we introduce the notation π∗ (A) = A∗ . The curvature endomorphism q(R) ∈ End(E M) is defined as q(R) = 21 (ei ∧ e j )∗ R(ei ∧ e j )∗

(14)

for any local orthonormal frame {ei }. In particular, q(R) = Ric on T M. By the same formula we may define for any curvature tensor S, or more generally any endomorphism S of 2 T M, a bundle morphism q(S). In any point q : R → q(R) defines an equivariant map from the space of algebraic curvature tensors to the space of endomorphisms of E. Since a Calabi-Yau algebraic curvature tensor has vanishing Ricci curvature, q(R CY ) = 0 holds on T M. Let R 0X Y be defined by R 0X Y = X ∧ Y + J X ∧ J Y − 23 ω(X, Y )J . Then a direct calculation gives (ei ∧ e j )∗ (ei ∧ e j )∗ + 21 (ei ∧ e j )∗ (J ei ∧ J e j )∗ − 23 ω∗ ω∗ . q(R 0 ) = 21 We apply this formula on T M. The first summand is exactly the SO(n)-Casimir, which acts as −5id. The third summand is easily seen to be 23 id, whereas the second summand acts as −id (cf. [11]). Altogether we obtain q(R 0 ) = − 16 3 id, which gives the following ¯ acting on T M: expression for q( R) ¯ T M = 4 id T M . q( R)|

(15)

3. The Hermitian Laplace Operator In the next two sections (M 6 , g, J ) will be a compact nearly Kähler manifold with scalar curvature normalized to scalg = 30. We denote as usual by the Laplace operator = d ∗ d + dd ∗ = ∇ ∗ ∇ + q(R) on differential forms. We introduce the Hermitian Laplace operator ¯ ¯ = ∇¯ ∗ ∇¯ + q( R),

(16)

which can be defined on any associated bundle E M. In [11] we have computed the ¯ on a primitive (1, 1)-form φ: difference of the operators and ¯ ( − )φ = (J d ∗ φ) + .

(17)

¯ coincide on co-closed primitive (1, 1)-forms. We now compute In particular, and ¯ on 1-forms. Using the calculation in [11] (or directly from (15)) we the difference −

The Hermitian Laplace Operator on Nearly Kähler Manifolds

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¯ = id on T M. It remains to compute the operator P = ∇ ∗ ∇ − ∇¯ ∗ ∇¯ have q(R) − q( R) on T M. A direct calculation using (5) gives for every 1-form θ , P(θ ) = − 41 Aei Aei θ − Aei ∇¯ ei θ = θ − Aei ∇¯ ei θ = θ + 21 Aei Aei θ − Aei ∇ei θ = −θ − Aei ∇ei θ. In order to compute the last term, we introduce the metric adjoint α : 2 M → T M of the bundle homomorphism X ∈ T M → X + ∈ 2 M. It is easy to check that α(X + ) = 2X (cf. [10]). Keeping in mind that A is totally skew-symmetric, we compute for an arbitrary vector X ∈ T M,

Aei (∇ei θ ), X = A X ei , ∇ei θ = A X , ei ∧ ∇ei θ = A X , dθ = − +J X , dθ = − J X, α(dθ ) = J α(dθ ), X , whence Aei (∇ei θ ) = J α(dθ ). Summarizing our calculations we have proved the following Proposition 3.1. Let (M 6 , g, J ) be a nearly Kähler manifold with scalar curvature normalized to scalg = 30. Then for any 1-form θ it holds that ¯ = −J α(dθ ). ( − )θ The next result is a formula for the commutator of J and α ◦ d on 1-forms. Lemma 3.2. For all 1-forms θ , the following formula holds: α(dθ ) = 4J θ + J α(d J θ ). Proof. Differentiating the identity θ ∧ + = J θ ∧ − gives dθ ∧ + = d J θ ∧ − +2J θ ∧ ω2 . With respect to the SU3 -invariant decomposition 2 M = (1,1) M ⊕ (2,0)+(0,2) M, we can write dθ = (dθ )(1,1) + 21 α(dθ ) + and d J θ = (d J θ )(1,1) + 21 α(d J θ ) + . Since the wedge product of forms of type (1, 1) and (3, 0) vanishes we derive the equation + + 1 2 (α(dθ ) ) ∧

= 21 (α(d J θ ) + ) ∧ − + 2J θ ∧ ω2 .

Using (8) and (9) we obtain 2 1 2 α(dθ ) ∧ ω

= 21 J α(d J θ ) ∧ ω2 + 2J θ ∧ ω2 .

Taking the Hodge dual of this equation and using (12) gives J α(dθ ) = −α(d J θ ) − 4θ, which proves the lemma. Finally we note two interesting consequences of Proposition 3.1 and Lemma 3.2. Corollary 3.3. For any closed 1-form θ it holds that ¯ = 0, ( − )θ

¯ θ = 4J θ. ( − )J

¯ coincide on θ . Proof. For a closed 1-form θ Lemma 3.1 directly implies that and For the second equation we use Proposition 3.1 together with Lemma 3.2 to conclude ¯ θ = −J α(d J θ ) = 4J θ − α(dθ ) = 4J θ, ( − )J since θ is closed. This completes the proof of the corollary.

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¯ 4. Special -Eigenforms on Nearly Kähler Manifolds In this section we assume moreover that (M, g) is not isometric to the standard sphere ¯ (S 6 , can). In the first part of this section we will show how to construct -eigenforms on M starting from Killing vector fields. Let ξ be a non-trivial Killing vector field on (M, g), which in particular implies d ∗ ξ = 0 and ξ = 2Ric(ξ ) = 10ξ . As an immediate consequence of the Cartan formula and (13) we obtain d J ξ = L ξ ω − ξ dω = −3ξ ψ +

(18)

so by (4), the square norm of d J ξ (as a 2-form) is |d J ξ |2 = 18|ξ |2 .

(19)

In [9] we showed already that the vector field J ξ is co-closed if ξ is a Killing vector field and has unit length. However it turns out that this also holds more generally. Proposition 4.1. Let ξ be a Killing vector field on M. Then d ∗ J ξ = 0. Proof. Let dv denote the volume form of (M, g). We start with computing the L 2 -norm of d ∗ J ξ . ∗ 2 ∗ ∗ d J ξ L 2 =

d J ξ, d J ξ dv = [ J ξ, J ξ − d ∗ d J ξ, J ξ ]dv

M

[ ∇ ∗ ∇ J ξ, J ξ + 5|J ξ |2 − |d J ξ |2 ]dv

=

M

M

=

[|∇ J ξ |2 + 5|ξ |2 − |d J ξ |2 ]dv =

M

[|∇ J ξ |2 − 13|ξ |2 ]dv. M

Here we used the well-known Bochner formula for 1-forms, i.e. θ = ∇ ∗ ∇θ + Ric(θ ), with Ric(θ ) = 5θ in our case. Next we consider the decomposition of ∇ J ξ into its symmetric and skew-symmetric parts 2∇ J ξ = d J ξ + L J ξ g, which together with (19) leads to |∇ J ξ |2 = 41 (|d J ξ |2 + |L J ξ g|2 ) = 9|ξ |2 + 14 |L J ξ g|2 .

(20)

(Recall that the endomorphism square norm of a 2-form is twice its square norm as a form.) In order to compute the last norm, we express L J ξ g as follows: L J ξ g(X, Y ) = g(∇ X J ξ, Y ) + g(X, ∇Y J ξ ) = g(J ∇ X ξ, Y ) + g(X, J ∇Y ξ ) + + (X, ξ, Y ) + + (Y, ξ, X ) = −g(∇ X ξ, J Y ) − g(J X, ∇Y ξ ) = −dξ (1,1) (X, J Y ), whence L J ξ g2L 2 = 2dξ (1,1) 2L 2 .

(21)

On the other hand, as an application of Lemma 3.2 together with Eq. (18) we get α(dξ ) = 4J ξ + J α(d J ξ ) = −2J ξ , so dξ (2,0) = −J ξ + .

(22)

The Hermitian Laplace Operator on Nearly Kähler Manifolds

257

Moreover, ξ = 10ξ since ξ is a Killing vector field, which yields dξ (1,1) 2L 2 = dξ 2L 2 − dξ (2,0) 2L 2 = 10ξ 2L 2 − 2ξ 2L 2 = 8ξ 2L 2 . This last equation, together with (20) and (21) gives ∇ J ξ 2L 2 = 13ξ 2L 2 . Substituting this into the first equation proves that d ∗ J ξ has vanishing L 2 -norm and thus that J ξ is co-closed. Proposition 4.2. Let ξ be a Killing vector field on M. Then ξ = 10ξ,

and

J ξ = 18J ξ.

In particular, J ξ can never be a Killing vector field. Proof. The first equation holds for every Killing vector field on an Einstein manifold with Ric = 5id. From (18) we know d J ξ = −3ξ + . Hence the second assertion follows from: (10) (12) d ∗ d J ξ = − ∗ d ∗ d J ξ = −3 ∗ d(J ξ ∧ + ) = 9 ∗ (ξ ∧ ω2 ) = 18J ξ. Since the differential d commutes with the Laplace operator , every Killing vector field ξ defines two -eigenforms of degree 2: d J ξ = 18d J ξ

and

dξ = 10dξ.

As a direct consequence of Proposition 4.2, together with formulas (18), (22), and Proposition 3.1 we get: Corollary 4.3. Every Killing vector field on M satisfies ¯ = 12ξ, ξ

¯ ξ = 12J ξ. J

¯ -eigenform. By (22) we have Our next goal is to show that the (1, 1)-part of dξ is a dξ = φ − J ξ + ,

(23)

for some (1, 1)-form φ. Using Proposition 4.1, we can write in a local orthonormal basis {ei }:

dξ, ω = 21 dξ, ei ∧ J ei = ∇ei ξ, J ei = d ∗ J ξ = 0, thus showing that φ is primitive. The differential of φ can be computed from the Cartan formula: (23) (7) dφ = d(J ξ + + dξ ) = −d(ξ − ) (13) = −L ξ − + ξ d − = −2ξ ω2 = −4J ξ ∧ ω. From here we obtain ∗dφ = −4 ∗ (J ξ ∧ ω) = 4ξ ∧ ω,

(24)

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whence (23) d ∗ dφ = 4dξ ∧ ω − 12ξ ∧ + = 4φ ∧ ω − 4(J ξ + ) ∧ ω − 12ξ ∧ + (9) = 4φ ∧ ω − 16ξ ∧ + . Using (10) and (11), we thus get d ∗ dφ = − ∗ d ∗ dφ = 4φ + 16J ξ + . On the other hand, (11) (24) (12) d ∗ φ = − ∗ d ∗ φ = ∗d(φ ∧ ω) = X (−4J ξ ∧ ω2 + 3φ ∧ + ) = 8ξ and finally dd ∗ φ = 8dξ = 8φ − 8J ξ + . The calculations above thus prove the following proposition Proposition 4.4. Let (M 6 , g, J ) be a compact nearly Kähler manifold with scalar curvature scalg = 30, not isometric to the standard sphere. Let ξ be a Killing vector field on (1,1) M and let φ be the (1, 1)-part of dξ . Then φ is primitive, i.e. φ = (dξ )0 . Moreover ∗ + d φ = 8ξ and φ = 12φ + 8J ξ . Corollary 4.5. The primitive (1, 1)-form ϕ satisfies ¯ = 12φ. φ Proof. From (17) and the proposition above we get ¯ = φ − ( − )φ ¯ φ = 12φ + 8J ξ + − (J d ∗ φ) + = 12φ. In the second part of this section we will present another way of obtaining primi¯ tive -eigenforms of type (1, 1), starting from eigenfunctions of the Laplace operator. Let f be such an eigenfunction, i.e. f = λ f . We consider the primitive (1, 1)-form (1,1) η := (d J d f )0 . Lemma 4.6. The form η is explicitly given by η = d J d f + 2d f + +

λ 3

f ω.

Proof. According to the decomposition of 2 M into irreducible SU3 -summands, we can write d J d f = η + γ + + hω for some vector field γ and function h. From Lemma 3.2 we get 2γ = α(d J d f ) = −4d f . In order to compute h, we write (12) 6h dv = hω ∧ ω2 = d J d f ∧ ω2 = d(J d f ∧ ω2 ) = 2d ∗ d f = 2λ f dv.

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We will now compute the Laplacian of the three summands of η separately. First, we ¯ f = λd f . Since ¯ commutes with J , we have d f = λd f and Corollary 3.3 yields d ¯ also have J d f = λJ d f and from the second equation in Corollary 3.3 we obtain ¯ d f + ( − )J ¯ d f = (λ + 4)J d f. J d f = J Hence, d J d f is a -eigenform for the eigenvalue λ + 4. Lemma 4.7. The co-differential of the (1, 1)-form η is given by d ∗ η = 2λ 3 − 4 J d f. Proof. Notice that d ∗ ( f ω) = −d f ω and that d ∗ J d f = − ∗ d ∗ J d f = − 21 ∗ d(d f ∧ ω2 ) = 0, since dω2 = 0. Using this we obtain d ∗ η = J d f + 2d ∗ (d f + ) − λ3 d f ω = (λ + 4)J d f − 2 ∗ d(d f ∧ − ) − λ3 J d f (12) = (λ + 4 − λ3 )J d f − 4 ∗ (d f ∧ ω2 ) = ( 2λ 3 − 4)J d f. In order to compute of the second summand of η we need three additional formulas Lemma 4.8. ¯ ) + . ¯ + ) = (X (X ¯ Since + is ∇-parallel ¯ ¯ = ∇¯ ∗ ∇¯ + q( R). Proof. Recall that we immediately obtain ¯ + ) = −∇¯ ei ∇¯ ei (X + ) = −(∇¯ ei ∇¯ ei X ) + . ∇¯ ∗ ∇(X The map A → A∗ + is a SU3 -equivariant map from 2 to 3 . But since 3 does not (1,1) contain the representation 0 as an irreducible summand, it follows that A∗ + = 0 for any skew-symmetric endomorphism A corresponding to some primitive (1, 1)-form. Hence we conclude ¯ ¯ ) + , ¯ i )∗ (X + ) = (ωi∗ R(ω ¯ i )∗ X ) + = (q( R)X q( R)(X + ) = ωi∗ R(ω where, since the holonomy of ∇¯ is included in SU3 , the sum goes over some ortho(1,1) ¯ + ) = normal basis {ωi } of 0 M. Combining these two formulas we obtain (X ¯ ) + . (X Lemma 4.9. ¯ ( − )(d f + ) = 6(d f + ) −

4λ 3

f ω − 2η.

Proof. From Proposition 3.4 in [11] we have ¯ ¯ ¯ ( − )(d f + ) = (∇ ∗ ∇ − ∇¯ ∗ ∇)(d f + ) f + ) + (q(R) − q( R))(d ¯ = (∇ ∗ ∇ − ∇¯ ∗ ∇)(d f + ) + 4d f + . The first part of the right hand side reads ¯ (∇ ∗ ∇ − ∇¯ ∗ ∇)(d f + ) = − 41 Aei ∗ Aei ∗ d f + − Aei ∗ ∇¯ ei (d f + ).

(25)

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From (5) we get Aei ∗ Aei ∗ d f + = Aei ∗ (Aei ek ∧ + (d f, ek , ·)) = Aei Aei ek ∧ + (d f, ek , ·) + Aei ek ∧ Aei + (d f, ek , ·) = −4ek ∧ ek d+f + Aei ek ∧ Aei e j + (d f, ek , e j ) = −8d+f , where we used the vanishing of the expression E = Aei ek ∧ Aei e j + (d f, ek , e j ): E = A J ei ek ∧ A J ei e j + (d f, ek , e j ) = Aei J ek ∧ Aei J e j + (d f, ek , e j ) = Aei ek ∧ Aei e j + (d f, J ek , J e j ) = −E. It remains to compute the second term in (25). We notice that by Schur’s Lemma, every SU3 -equivariant map from the space of symmetric tensors Sym2 M to T M vanishes, so in particular (since ∇d f is symmetric), one has Aei ∇ei d f = 0. We then compute Aei ∗ ∇¯ ei d+f = Aei ∗ ((∇¯ ei d f ) + ) = (Aei ∇¯ ei d f ) + + (∇¯ ei d f )Aei ∗ + (6) = (Aei ∇ei d f ) + − 21 (Aei Aei d f ) + − 2(∇¯ ei d f )(ei ∧ ω) = 2d+f + 2d ∗ d f ω + Aei d f, ei ω + 2ei ∧ J ∇¯ ei d f = 2d+f +2λ f ω + 2ei ∧ ∇¯ ei J d f = 2d+f +2λ f ω + 2d J d f − ei ∧ Aei J d f = 2d+f + 2λ f ω + 2d J d f + 2 A J d f = 4d+f + 2λ f ω + 2d J d f. Plugging back what we obtained into (25) yields ¯ f + ) = −(2d+f + 2λ f ω + 2d J d f ), (∇ ∗ ∇ − ∇¯ ∗ ∇)(d which together with Lemma 4.6 and the first equation prove the desired formula.

Lemma 4.10. f ω = (λ + 12) f ω − 2(d f + ). Proof. Since d ∗ ( f ω) = −d f ω = −J d f we have dd ∗ ( f ω) = −d J d f . For the second summand of ( f ω) we first compute d( f ω) = d f ∧ ω + 3 f + . Since d ∗ + = 1 ∗ ∗ + + ∗ + + 3 d dω = 4ω, we get d f = −d f + f d = −d f + 4 f ω. Moreover d ∗ (d f ∧ ω) = − ∗ d(J d f ∧ ω) = − ∗ (d J d f ∧ ω − 3J d f ∧ + ) = − ∗ ([η − 2d f + −

λ 3

f ω] ∧ ω) + 3 ∗ (J d f ∧ + )

= η + 2 ∗ ((d f + ) ∧ ω) + = η + 2d f + + Recalling that η = d J d f + 2d f + +

2λ 3

λ 3

2λ 3

f ω − 3d f +

f ω − 3d f + .

f ω, we obtain

f ω = −d J d f − 3d f + + 12 f ω + η − d f + +

2λ 3

f ω = (λ + 12) f ω − 2d f + .

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261

Applying these three lemmas we conclude ¯ f + ) + ( − )(d ¯ f + ) = (λ + 6)(d f + ) − (d f + ) = (d

4λ 3

f ω − 2η,

and thus η = (λ + 4)d J d f + (2λ + 12)(d f + ) − + = λη + 4 − 2λ 3 (d f ).

8λ 3

f ω − 4η + λ3 (λ + 12) f ω −

2λ + 3 (d f )

¯ on Finally we have once again to apply the formula for the difference of and primitive (1, 1)-forms. We obtain ¯ = η − J d ∗ η + = η + 2λ − 4 (d f + ) = λη. η 3 Summarizing our calculations we obtain the following result. Proposition 4.11. Let f be an -eigenfunction with f = λ f Then the primitive (1, 1)form η := (d J d f )0(1,1) satisfies ¯ = λη and d ∗ η = 2λ − 4 J d f. η 3 ¯ ¯ = Let 0 (12) ⊂ C ∞ (M) be the -eigenspace for the eigenvalue 12 (notice that (1,1) ¯ on functions) and let 0 (12) denote the space of primitive (1, 1)-eigenforms of corresponding to the eigenvalue 12. Summarizing Corollary 4.5 and Proposition 4.11, we have constructed a linear mapping (1,1)

: i(M) → 0

(1,1)

(12),

(ξ ) := dξ0 (1,1)

from the space of Killing vector fields into 0 : 0 (12) → 0(1,1) (12),

(12) and a linear mapping

( f ) := (d J d f )0(1,1) .

Let moreover N K ⊂ 0(1,1) (12) denote the space of nearly Kähler deformations, which (1,1) by [10] is just the space of co-closed forms in 0 (12). Proposition 4.12. The linear mappings and defined above are injective and the sum Im() + Im() + N K ⊂ 0(1,1) (12) is a direct sum. In particular, dim(N K) ≤ dim(0(1,1) (12)) − dim(i(M)) − dim(0 (12)).

(26)

Proof. It is enough to show that if ξ ∈ i(M), f ∈ 0 (12) and α ∈ N K satisfy (1,1)

dξ0

(1,1)

+ (d J d f )0

+ α = 0,

(27)

then ξ = 0 and f = 0. We apply d ∗ to (27). Using Propositions 4.4 and 4.11 to express the co-differentials of the first two terms we get 8ξ + 8J d f = 0.

(28)

Since J ξ is co-closed (Proposition 4.1), formula (28) implies 0 = d ∗ J ξ = d ∗ d f = 12 f , i.e. f = 0. Plugging back into (28) yields ξ = 0 too.

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5. The Homogeneous Laplace Operator on Reductive Homogeneous Spaces 5.1. The Peter-Weyl formalism. Let M = G/K be a homogeneous space with compact Lie groups K ⊂ G and let π : K → Aut(E) be a representation of K . We denote by E M := G ×π E the associated vector bundle over M. The Peter-Weyl theorem and the Frobenius reciprocity yield the following isomorphism of G-representations: Vγ ⊗ Hom K (Vγ , E), (29) L 2 (E M) ∼ = γ ∈Gˆ

where Gˆ is the set of (non-isomorphic) irreducible G-representations. If not otherwise stated we will consider only complex representations. Recall that the space of smooth sections C ∞ (E M) can be identified with the space C ∞ (G; E) K of K -invariant E-valued functions, i.e. functions f : G → E with f (gk) = π(k)−1 f (g). This space is made into a G-representation by the left-regular representation , defined by ((g) f )(a) = f (g −1 a). Let v ∈ Vγ and A ∈ Hom K (Vγ , E), then the invariant E-valued function corresponding to v ⊗ A is defined by g → A(g −1 v). In particular, each summand in the Hilbert space direct sum (29) is a subset of C ∞ (E M) ⊂ L 2 (E M). Let g be the Lie algebra of G. We denote by B the Killing form of g, B(X, Y ) := tr(ad X ◦ adY ). The Killing form is non-degenerate and negative definite if G is compact and semi-simple, which will be the case in all examples below. If π : G → Aut(E) is a G-representation, the Casimir operator of (G, π ) acts on E by the formula CasπG = (π∗ X i )2 , (30) where {X i } is a (−B)-orthonormal basis of g and π∗ : g → End(E) denotes the differential of the representation π . Remark 5.1. Notice that the Casimir operator is divided by k if one uses the scalar product −k B instead of −B. If G is simple, the adjoint representation ad on the complexification gC is irreducible, so, by Schur’s Lemma, its Casimir operator acts as a scalar. Taking the trace in (30) for G = −1. π = ad yields the useful formula Casad Let Vγ be an irreducible G-representation of highest weight γ . By Freudenthal’s formula the Casimir operator acts on Vγ by scalar multiplication with ρ2 − ρ + γ 2 , where ρ denotes the half-sum of the positive roots and · is the norm induced by −B on the dual of the Lie algebra of the maximal torus of G. Notice that these scalars are always non-positive. Indeed ρ2 − ρ + γ 2 = − γ , γ + 2ρ B and γ , ρ ≥ 0, since γ is a dominant weight, i.e. it is in the the closure of the fixed Weyl chamber, whereas ρ is the half-sum of positive weights and thus by definition has a non-negative scalar product with γ . 5.2. The homogeneous Laplace operator. We denote by ∇¯ the canonical homogeneous connection on M = G/K . It coincides with the Levi-Civita connection only in the case that G/K is a symmetric space. A crucial observation is that the canonical homogeneous connection coincides with the canonical Hermitian connection on naturally reduc¯ ∈ tive 3-symmetric spaces (see below). We define the curvature endomorphism q( R) ¯ ¯ ¯ π = ∇¯ ∗ ∇+q( End(E M) as in (14) and introduce as in (16) the second order operator R) acting on sections of the associated bundle E M := G ×π E.

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Lemma 5.2. Let G be a compact semi-simple Lie group, K ⊂ G a compact subgroup, and let M = G/K be the naturally reductive homogeneous space equipped with the Riemannian metric induced by −B. For every K -representation π on E, let E M := ¯ acts G ×π E be the associated vector bundle over M. Then the endomorphism q( R) ¯ = −CasπK . Moreover the differential operator ¯ acts on the fibre-wise on E M as q( R) space of sections of E M, considered as G-representation via the left-regular represen¯ = −CasG . tation, as Proof. Consider the Ad(K )-invariant decomposition g = k ⊕ p. For any vector X ∈ g we write X = X k + X p, with X k ∈ k and X p ∈ p. The canonical homogeneous connection is the left-invariant connection in the principal K -fibre bundle G → G/K corresponding to the projection X → X k. It follows that one can do for the canonical homogeneous connection on G/K the same identifications as for the Levi Civita connection on Riemannian symmetric spaces. In particular, the covariant derivative of a section φ ∈ (E M) with respect to ˆ of the corresponding function φˆ ∈ some X ∈ p translates into the derivative X (φ) ∞ K ˆ = C (G; E) , which is minus the differential of the left-regular representation X (φ) ∗ ¯ ¯ ˆ −∗ (X )φ. Hence, if {eµ } is an orthonormal basis in p, the rough Laplacian ∇ ∇ trans¯ it remains ¯ = ∇¯ ∗ ∇¯ +q( R) lates into the sum −∗ (eµ )∗ (eµ ) = (−CasG +CasK ). Since K K ¯ = −Cas = −Casπ in order to complete the proof of the lemma. to show that q( R) We claim that the differential i ∗ : k → so(p) ∼ = 2 p of the isotropy representation i : K → SO(p) is given by i ∗ (A) = − 21 eµ ∧ [A, eµ ] for any A ∈ k. Indeed ( 21 eµ ∧ [A, eµ ])∗ X = − 21 B(eµ , X )[A, eµ ] + 21 B([A, eµ ], X )eµ = −[A, X ]. Next we recall that for X, Y ∈ p the curvature R¯ X,Y of the canonical connection acts by −π∗ ([X, Y ]k) on every associated vector bundle E M, defined by the representation π . Hence the curvature operator R¯ can be written for any X, Y ∈ p as ¯ ∧ Y ) = 1 eµ ∧ R¯ X,Y eµ = − 1 eµ ∧ [[X, Y ]k, eµ ] = i ∗ ([X, Y ]k). R(X 2 2 Let PSO(p) = G ×i SO(p) be the bundle of orthonormal frames of M = G/K . Then any SO(p)-representation π˜ defines a K -representation by π = π˜ ◦ i. Moreover any vector bundle E M associated to PSO(p) via π˜ can be written as a vector bundle associated via π to the K -principle bundle G → G/K , i.e. E M = PSO(p) ×π˜ E = G ×π E. ¯ we have Let { f α } be an orthonormal basis of k. Then by the definition of q( R) ¯ µ ∧ eν )) = 1 π˜ ∗ (eµ ∧ eν ) π∗ ([eµ , eν ]k) ¯ = 1 π˜ ∗ (eµ ∧ eν ) π˜ ∗ ( R(e q( R) 2 2 = −21 B([eµ , eν ], f α )π˜ ∗ (eµ ∧eν ) π∗ ( f α ) = − 21 B(eν , [ f α , eµ ])π˜ ∗ (eµ ∧ eν ) π∗ ( f α ) = 21 π˜ ∗ (eµ ∧ [ f α , eµ ]) π∗ ( f α ) = −π∗ ( f α ) π∗ ( f α ) = −CasπK . ¯ ∈ End(E M) acts fibre-wise as −CasπK . Let Z ∈ k and We have shown that q( R) f ∈ C ∞ (G; E) K , then the K -invariance of f implies π∗ (Z ) f = −Z ( f ) = ∗ (Z ) f and also CasπK = CasK , which concludes the proof of the lemma.

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¯ on sections of E M is the set It follows from this lemma that the spectrum of of numbers λγ = ρ + γ 2 − ρ2 , where γ is the highest weight of an irreducible G-representation Vγ such that Hom K (Vγ , E) = 0, i.e. such that the decomposition of Vγ , considered as a K -representation, contains components of the K -representation E. 5.3. Nearly Kähler deformations and Laplace eigenvalues. Let (M, g, J ) be a compact simply connected 6-dimensional nearly Kähler manifold not isometric to the round sphere, with scalar curvature normalized to scalg = 30. Recall the following result from [10]: Theorem 5.3. The Laplace operator coincides with the Hermitian Laplace operator ¯ on co-closed primitive (1, 1)-forms. The space N K of infinitesimal deformations of the nearly Kähler structure of M is isomorphic to the eigenspace for the eigenvalue 12 ¯ to the space of co-closed primitive (1, 1)-forms on M. of the restriction of (or ) Assume from now on that M is a 6-dimensional naturally reductive 3-symmetric space G/K in the list of Gray and Wolf, i.e. SU2 × SU2 × SU2 /SU2 , SO5 /U2 or SU3 /T 2 . As was noticed before, the canonical homogeneous and the canonical Hermitian connection coincide, since for the later it can be shown that its torsion and its curvature are parallel, a property, which by the Ambrose-Singer-Theorem characterizes the canonical homogeneous connection (cf. [5]). In order to determine the space N K on M we thus ¯ need to apply the previous calculations to compute the -eigenspace for the eigenvalue 12 on primitive (1, 1)-forms and decide which of these eigenforms are co-closed. According to Lemma 5.2 and the decomposition (29) we have to carry out three 1,1 steps: first to determine the K -representation 1,1 0 p defining the bundle 0 T M, then to compute the Casimir eigenvalues with the Freudenthal formula, which gives all pos¯ sible -eigenvalues and finally to check whether the G-representation Vγ realizing the eigenvalue 12 satisfies Hom K (Vγ , 1,1 0 p) = {0} and thus really appears as eigenspace. Before going on, we make the following useful observation Lemma 5.4. Let (G/K , g) be a 6-dimensional homogeneous strict nearly Kähler manifold of scalar curvature scalg = 30. Then the homogeneous metric g is induced from 1 B, where B is the Killing form of G. − 12 Proof. Let G/K be a 6-dimensional homogeneous strict nearly Kähler manifold. Then the metric is induced from a multiple of the Killing form, i.e. G/K is a normal homogeneous space with Ad(K )-invariant decomposition g = k ⊕ p. The scalar curvature of the metric h induced by −B may be computed as (cf. [3]) scalh =

3 2

− 3CasλK ,

where λ : K → so(p) is the isotropy representation. From Lemma 5.2 we know that ¯ which on the tangent bundle was computed in Lemma 15 as q( R) ¯ = CasλK = −q( R), 2scalh 3 2 5 15 id. Hence we obtain the equation scalh = 2 + 5 scalh and it follows scalh = 2 , i.e. 1 the metric g corresponding to − 12 B has scalar curvature scalg = 30. ¯ 5.4. The -spectrum on S 3 × S 3 . Let K = SU2 with Lie algebra k = su2 and G = K × K × K with Lie algebra g = k ⊕ k ⊕ k. We consider the 6-dimensional manifold

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M = G/K , where K is diagonally embedded. The tangent space at o = eK can be identified with p = {(X, Y, Z ) ∈ k ⊕ k ⊕ k | X + Y + Z = 0}. 1 Let B be the Killing form of k and define B0 = − 12 B. Then it follows from Lemma 5.4 that the invariant scalar product

B0 ((X, Y, Z ), (X, Y, Z )) = B0 (X, X ) + B0 (Y, Y ) + B0 (Z , Z ) defines a normal metric, which is the homogeneous nearly Kähler metric g of scalar curvature scalg = 30. The canonical almost complex structure on the 3-symmetric space M, corresponding to the 3rd order G-automorphism σ , with σ (k1 , k2 , k3 ) = (k2 , k3 , k1 ), is defined as J (X, Y, Z ) =

√2 (Z , 3

X, Y ) +

√1 (X, Y, 3

Z ).

The (1, 0)-subspace p1,0 of pC defined by J is isomorphic to the complexified adjoint 2 representation of SU2 on suC 2 . Let E = C denote the standard representation of SU2 ∼ ∼ ¯ (notice that E = E because every SU2 = Sp1 representation is quaternionic). (1,1)

Lemma 5.5. The SU2 -representation defining the bundle 0 ducible summands Sym4 E and Sym2 E.

T M splits into the irre-

Proof. The defining SU2 -representation of (1,1) T M is p1,0 ⊗ p0,1 ∼ = Sym2 E ⊗ 4 E ⊕ Sym 2 E ⊕ Sym 0 E from the Clebsch-Gordan formula. Since we Sym2 E ∼ Sym = are interested in primitive (1, 1)-forms, we still have to delete the trivial summand Sym0 E ∼ = C. Since G = SU2 × SU2 × SU2 , every irreducible G-representation is isomorphic to one of the representations Va,b,c = Syma E ⊗Symb E ⊗Symc E. The Casimir operator of the SU2 -representation Symk E (with respect to B) is − 18 k(k + 2) and the Casimir operator ¯ of G is the sum of the three SU2 -Casimir operators. Hence all possible -eigenvalues with respect to the metric B0 are of the form 3 2 (a(a

+ 2) + b(b + 2) + c(c + 2))

(31)

for non-negative integers a, b, c. It is easy to check that the eigenvalue 12 is obtained only for (a, b, c) equal to (2, 0, 0), (0, 2, 0) or (0, 0, 2). The restrictions to SU2 (diagonally embedded in G) of the three corresponding G-representations are all equal to the SU2 -representation Sym2 E, thus dim HomSU2 (V2,0,0 , 0(1,1) p) = 1, and similarly ¯ on primitive (1, 1)-forms for for the two other summands. Hence the eigenspace of the eigenvalue 12 is isomorphic to V2,0,0 ⊕ V0,2,0 ⊕ V0,0,2 and its dimension, i.e. the multiplicity of the eigenvalue 12, is equal to 9. Since the isometry group of the nearly Kähler manifold M = SU2 ×SU2 ×SU2 /SU2 has dimension 9, the inequality (26) yields (1,1)

dim(N K) ≤ dim(0

(12)) − dim(i(M)) − dim(0 (12)) = − dim(0 (12)) ≤ 0.

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We thus have obtained the following Theorem 5.6. The homogeneous nearly Kähler structure on S 3 × S 3 does not admit any infinitesimal nearly Kähler deformations. Finally we remark that there are also no infinitesimal Einstein deformations either. In [11] we showed that the space of infinitesimal Einstein deformations of a nearly Kähler metric g, with normalized scalar curvature scalg = 30, is isomorphic to the direct sum ¯ of -eigenspaces of primitive co-closed (1, 1)-forms for the eigenvalues 2, 6 and 12. It ¯ is clear from (31) that neither 2 nor 6 can be realized as -eigenvalues. Corollary 5.7. The homogeneous nearly Kähler metric on S 3 × S 3 does not admit any infinitesimal Einstein deformations. ¯ 5.5. The -spectrum on CP 3 . In this section we consider the complex projective space CP 3 = SO5 /U2 , where U2 is embedded by U2 ⊂ SO4 ⊂ SO5 . Let G = SO5 with Lie algebra g and K = U2 with Lie algebra k. We denote the Killing form of G with B. Then we have the B-orthogonal decomposition g = k ⊕ p, where p can be identified with the tangent space in o = eK . The space p splits as p = m ⊕ n, where m resp. n can be identified with the horizontal resp. vertical tangent space at o of the twistor space 1 fibration SO5 /U2 → SO5 /SO4 = S 4 . We know from Lemma 5.4 that B0 = − 12 B defines the homogeneous nearly Kähler metric g of scalar curvature scalg = 30. Let {ε1 , ε2 } denote the canonical basis of R2 . Then the positive roots of SO5 are α1 = ε1 , α2 = ε2 , α3 = ε1 + ε2 , α4 = ε1 − ε2 , with ρ = 23 ε1 + 21 ε2 . Let gα ⊂ gC be the root space corresponding to the root α. Then mC = gα1 ⊕ g−α1 ⊕ gα2 ⊕ g−α2 ,

nC = gα3 ⊕ g−α3 .

The invariant almost complex structure J may be defined by specifying the (1, 0)subspace p1,0 of pC : p1,0 = {X − i J X | X ∈ p} = gα1 ⊕ gα2 ⊕ g−α3 . It follows that J is not integrable, since the restricted root system {α1 , α2 , −α3 } is not closed under addition (cf. [4]). We note that replacing −α3 by α3 yields an integrable almost complex structure. This corresponds to the well-known fact that on the twistor space the non integrable almost complex structure J is transformed into the integrable one by replacing J with −J on the vertical tangent space. Let Ck denote the U1 -representation on C defined by (z, v) → z k v, for v ∈ C and z ∈ U1 ∼ = C∗ . Then, since U2 = (SU2 × U1 )/Z2 , any irreducible U2 -representation is of the form E a,b = Syma E ⊗ Cb , with a ∈ N, b ∈ Z and a ≡ b mod 2. As usual let E = C2 denote the standard representation of SU2 . With this notation we obtain the following decomposition of p1,0 considered as a U2 -representation: p1,0 ∼ = E 0,−2 ⊕ E 1,1

E 0,−2 ∼ = g−α3 and E 1,1 ∼ = gα1 ⊕ gα2 . (32) Since p0,1 is obtained from p1,0 by conjugation we have p0,1 ∼ = E 0,2 ⊕ E 1,−1 . The definwith

ing U2 -representation of (1,1) T M is p1,0 ⊗ p0,1 , which obviously decomposes into 5 irreducible summands, among which, two are isomorphic to the trivial representation E 0,0 . Considering only primitive (1, 1)-forms we still have to delete one of the trivial summands and obtain

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Lemma 5.8. The U2 -representation defining the bundle 0 decomposition into irreducible summands: (1,1)

0

T M has the following

p = E 0,0 ⊕ E 1,3 ⊕ E 1,−3 ⊕ E 2,0 .

Let Va,b be an irreducible SO5 -representation of highest weight γ = (a, b) with a, b ∈ N and a ≥ b ≥ 0, e.g. V1,0 = 1 and V1,1 = 2 . The scalar product induced by the Killing form B on the dual t∗ ∼ = R2 of the maximal torus of SO5 is − 16 times the Euclidean scalar product. By the Freudenthal formula we thus get CasVa,b = γ , γ + 2ρ B = − 16 (a(a + 3) + b(b + 1)).

(33)

G Notice that we have V1,1 = soC 5 and Cas V1,1 = −1, which is consistent with Casad = −1. ¯ It follows (cf. Remark 5.1) that all possible -eigenvalues with respect to the metric induced by B0 are of the form 2(a(a + 3) + b(b + 1)). The eigenvalue 12 is realized if and only if (a, b) = (1, 1). We still have to decide whether the SO5 -representation V1,1 actually appears in the decomposition (29) of L 2 (1,1 0 T M). However this follows from

Lemma 5.9. The SO5 -representation V1,1 restricted to U2 ⊂ SO5 has the following decomposition as U2 -representation: V1,1 ∼ = (E 0,0 ⊕ E 2,0 ) ⊕ (E 0,−2 ⊕ E 1,1 ⊕ E 0,2 ⊕ E 1,−1 ) and in particular C dim HomU2 (V1,1 , 1,1 0 p )=2

and

dim HomU2 (V1,1 , C) = 1.

Proof. We know already that V1,1 = soC 5 is the complexified adjoint representation and C ⊕(p1,0 ⊕p0,1 ). The decomposition of the last two summands is contained = u that soC 2 5 in (32). Hence it remains to make explicit the adjoint representation of U2 on uC 2 . It is clear that its restriction to U1 acts trivially, whereas its restriction to SU2 decomposes C ∼ into C ⊕ suC 2 , i.e. u2 = E 0,0 ⊕ E 2,0 . ¯ on primitive (1, 1)-forms for the eigenvalue 12 is thus isomorThe eigenspace of phic to the sum of two copies of V1,1 , i.e. the eigenvalue 12 has multiplicity 2 · 10 = 20. It is now easy to calculate the smallest eigenvalue and the corresponding eigenspace ¯ which coincides of the Laplace operator on non-constant functions. We do this for , (1,1) with on functions. Then we have to replace 0 p in the calculations above with the trivial representation C and to look for SO5 -representations Va,b containing the zero weight. It follows from Lemma 5.9 and (33) that the -eigenspace on functions 0 (12) is isomorphic to V1,1 and is thus 10-dimensional. Since the dimension of the isometry group of the nearly Kähler manifold SO(5)/U2 is 10, the inequality (26) shows that (1,1)

dim(N K) ≤ dim(0

(12)) − dim(i(M)) − dim(0 (12)) = 20 − 10 − 10 = 0,

so there are no infinitesimal nearly Kähler deformations in this case either. Finally, we remark like before that there are also no other infinitesimal Einstein defor¯ on mations, since by (33), the eigenvalues 2 and 6 do not occur in the spectrum of (1,1) 0 M. Summarizing, we have obtained the following: Theorem 5.10. The homogeneous nearly Kähler structure on CP3 = SO5 /U2 does not admit any infinitesimal nearly Kähler or Einstein deformations.

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¯ 5.6. The -spectrum on the flag manifold F(1, 2). In this section we consider the flag manifold M = SU3 /T 2 , where T 2 ⊂ SU3 is the maximal torus. Let g = su3 and let k = t, the Lie algebra of T 2 . We have the decomposition g=k⊕p

and

p = m ⊕ n.

Denoting by E i j , Si j the “real and imaginary” part of the projection of the vector X i j ∈ gl3 (equal to 1 on i th row and j th column and 0 elsewhere) onto su3 : E i j = X i j − X ji

Si j = i(X i j + X ji ),

the subspaces m and n are explicitly given by m = span{E 12 , S12 , E 13 , S13 } = span{e1 , e2 , e3 , e4 }, n = span{E 23 , S23 } = span{e5 , e6 }. The dual of the Lie algebra t of the maximal torus T 2 can be identified with t∗ ∼ = {(λ1 , λ2 , λ3 ) ∈ R3 | λ1 + λ2 + λ3 = 0}. If {εi } denotes the canonical basis in R3 then the set of positive roots is given as φ + = {αi j = εi − ε j | 1 ≤ i < j ≤ 3} and the half-sum of the positive roots is ρ = ε1 − ε3 . 1 B defines the Let B denote the Killing form of SU3 . By Lemma 5.4, B0 = − 12 homogeneous nearly Kähler metric g of scalar curvature scalg = 30. The almost complex structure J is explicitly defined on p by J (e1 ) = e2 ,

J (e3 ) = −e4 ,

J (e5 ) = e6 .

Alternatively we may define the (1, 0)-subspace of pC : p1,0 = gα12 ⊕ gα31 ⊕ gα23 = span{X 12 , X 31 , X 23 }, where gα is the root space for α. It follows that J is not integrable, since the restricted root system {α12 , α31 , α23 } is not closed under addition (cf. [4]). ¯ Let E = C3 be the standard representation of SU3 with conjugate representation E. Any irreducible representations of SU3 is isomorphic to one of the representations ¯ 0, Vk,l := (Symk E ⊗ Syml E) where the right-hand side denotes the kernel of the contraction map ¯ Symk E ⊗ Syml E¯ → Symk−1 E ⊗ Syml−1 E, ¯ The weights of Symk E are i.e. Vk,l is the Cartan summand in Symk E ⊗ Syml E. aε1 + bε2 + cε3 ,

with a, b, c ≥ 0, a + b + c = k.

If v1 , v2 , v3 are the weight vectors of E, then these weights correspond to the weight vectors v1a · v2b · v3c in Symk E. Since the weights of Syml E¯ are just minus the weights of Syml E, we see that the weights of Vk,l are (a − a )ε1 + (b − b )ε2 + (c − c )ε3 , a, b, c, a , b , c ≥ 0, a + b + c = k, a + b + c = l.

(34)

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From the given definition of the almost complex structure J it is clear that the T 2 -representation on p1,0 splits in three one-dimensional T 2 -representations with the weights α12 , α31 , α23 . Since the weights of a tensor product representation are the sums of weights of each factor and since ε1 + ε2 + ε3 = 0 on the Lie algebra of the maximal torus of SU3 , we immediately obtain Corollary 5.11. The weights of the T 2 -representation on 1,1 p ∼ = p1,0 ⊗ p0,1 are ±3ε1 , ±3ε2 , ±3ε3 , and 0. It remains to compute the Casimir operator of the irreducible SU3 -representations Vk,l . The highest weight of Vk,l is γ = kε1 − lε3 and ρ = ε1 − ε3 , thus CasVk,l = γ , γ + 2ρ B = − 16 (k(k + 2) + l(l + 2)).

(35)

Here we use again the Freudenthal formula and the fact that the Killing form B induces − 16 times the Euclidean scalar product on t∗ ⊂ R3 (easy calculation). Notice that we G have V1,1 = suC 3 and CasV1,1 = −1, which is consistent with Casad = −1 as in the previous cases. ¯ It follows that all possible -eigenvalues (with respect to the metric B0 ) are of the form 2(k(k + 2) + l(l + 2)). Obviously the eigenvalue 12 can only be obtained for k = l = 1. Moreover, the restriction of the SU3 -representation V1,1 contains the zero weight space. In fact, from (34), the zero weight appears in Vk,l if and only if there exist a, b, c, a , b , c ≥ 0, a + b + c = k, a + b + c = l such that (a − a )ε1 + (b − b )ε2 + (1,1) (c − c )ε3 = 0, which is equivalent to k = l. We see that dim Hom T 2 (V1,1 , 0 p) = 2 · 2 = 4. ¯ on primitive (1, 1)-forms for the eigenvalue 12 is isomorHence the eigenspace of phic to the sum of four copies of V1,1 , i.e. the eigenvalue 12 has multiplicity 4 · 8 = 32. Computing the smallest eigenvalue and the corresponding eigenspace of the Laplace operator on non-constant functions we find V0,0 for the eigenvalue 0 and V1,1 for the eigenvalue 12. All other possible representations give a larger eigenvalue. Hence, the -eigenspace on functions 0 (12) is isomorphic to two copies of V1,1 , i.e. the eigenvalue 12 has multiplicity 8 · 2 = 16. Since the dimension of the isometry group of the nearly Kähler manifold SU3 /T 2 is 8, we obtain from (26) (1,1)

dim(N K) ≤ dim(0

(12)) − dim(i(M)) − dim(0 (12)) = 8.

(36)

In the next section we will show by an explicit construction that actually the equality holds, so the flag manifold has an 8-dimensional space of infinitesimal nearly Kähler deformations. Before describing this construction we note that there are no infinitesimal Einstein deformations other than the nearly Kähler deformations. It follows from (35) that the ¯ on (1,1) M. The eigenvalue 6 could eigenvalue 2 does not occur in the spectrum of 0 be realized on the SU3 -representations V = V1,0 or V = V0,1 . However it is easy to (1,1) check that Hom T 2 (V, 0 p) = {0}. Corollary 5.12. Every infinitesimal Einstein deformation of the homogeneous nearly Kähler metric on F(1, 2) = SU3 /T 2 is an infinitesimal nearly Kähler deformation.

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6. The Infinitesimal Nearly Kähler Deformations on SU3 / T 2 In this section we describe by explicit computation the space of infinitesimal nearly Kähler deformations of the flag manifold F(1, 2) = SU3 /T 2 . The Lie algebra u3 is spanned by {h 1 , h 2 , h 3 , e1 , . . . , e6 }, where h 1 = i E 11 , h 2 = i E 22 , h 3 = i E 33 , e1 = E 12 − E 21 , e3 = E 13 − E 31 , e5 = E 23 − E 32 , e2 = i(E 12 + E 21 ), e4 = i(E 13 + E 31 ), e6 = i(E 23 + E 32 ). We consider the bi-invariant metric g on SU3 induced by −B/12, where B denotes the Killing form of su3 . It is easy to check that |ei |2 =1 and |h i − h j |2 = 1 with respect √ to g. We extend this metric to U3 in the obvious way which makes the frame {ei , 2h j } orthonormal. This defines a metric, also denoted by g, on the manifold M = F(1, 2). From now on we identify vectors and 1-forms using this metric and use the notation ei j = ei ∧ e j , etc. An easy explicit commutator calculation yields the exterior derivative of the leftinvariant 1-forms ei on U3 : de1 de2 de3 de4 de5 de6

= = = = = =

−2e2 ∧ (h 1 − h 2 ) + e35 + e46 , 2e1 ∧ (h 1 − h 2 ) + e45 − e36 , 2e4 ∧ (h 3 − h 1 ) − e15 + e26 , −2e3 ∧ (h 3 − h 1 ) − e25 − e16 , −2e6 ∧ (h 2 − h 3 ) + e13 + e24 , 2e5 ∧ (h 2 − h 3 ) + e14 − e23 .

(37)

Let J denote the almost complex structure on M = F(1, 2) whose Kähler form is ω = e12 − e34 + e56 . (It is easy to check that ω, which a priori is a left-invariant 2-form on U3 , projects to M because L h i ω = 0.) J induces an orientation on M with volume form −e123456 . Let + + i − denote the associated complex volume form on M defined by the adT 3 -invariant form (e2 + i J e2 ) ∧ (e4 + i J e4 ) ∧ (e6 + i J e6 ). Explicitly, + = e136 + e246 + e235 − e145 ,

− = e236 − e146 − e135 − e245 .

Using (37) we readily obtain d(e12 ) = −d(e34 ) = d(e56 ) = + ,

(38)

so dω = 3 + ,

and

d − = −2ω2 .

The pair (g, J ) thus defines a nearly Kähler structure on M (a fact which we already knew). We fix now an element ξ ∈ su3 ⊂ u3 , and denote by X the right-invariant vector field on U3 defined by ξ . Consider the functions xi = g(X, ei ),

vi = g(X, h i ).

(39)

The functions vi are projectable to M and clearly v1 + v2 + v3 = 0. Let us introduce the vector fields on U3 , a1 = x6 e5 − x5 e6 ,

a2 = x3 e4 − x4 e3 ,

a3 = x2 e1 − x1 e2 .

The Hermitian Laplace Operator on Nearly Kähler Manifolds

271

One can check that they project to M. Of course, one has J a1 = x5 e5 + x6 e6 ,

J a2 = x3 e3 + x4 e4 ,

J a3 = x1 e1 + x2 e2 .

The commutator relations in SU3 yield dv1 = a2 − a3 ,

dv2 = a3 − a1 ,

dv3 = a1 − a2 .

(40)

Using (37) and some straightforward computations we obtain d(J a1 ) = (−a1 + a2 + a3 ) + + 4(v2 − v3 )e56 , d(J a2 ) = (a1 − a2 + a3 ) + + 4(v1 − v3 )e34 , d(J a3 ) = (a1 + a2 − a3 ) + + 4(v1 − v2 )e12 .

(41)

We claim that the 2-form ϕ = v1 e56 − v2 e34 + v3 e12

(42)

on M is of type (1,1), primitive, co-closed, and satisfies ϕ = 12ϕ. The first two assertions are obvious (recall that v1 + v2 + v3 = 0). In order to prove that ϕ is co-closed, it is enough to prove that dϕ ∧ ω = 0. Using (38) and (40) we compute: dϕ ∧ ω = [(a2 − a3 ) ∧ e56 − (a3 − a1 ) ∧ e34 + (a1 − a2 ) ∧ e12 ] ∧ (e12 − e34 + e56 ) = (a1 − a2 ) ∧ e1256 − (a3 − a2 ) ∧ e1234 + (a1 − a2 ) ∧ e3456 = 0. Finally, using (41), we get ϕ = d ∗ dϕ = − ∗ d ∗ [(a2 − a3 ) ∧ e56 − (a3 − a1 ) ∧ e34 + (a1 − a2 ) ∧ e12 ] = −∗ d[J a2 ∧ e12 + J a3 ∧ e34 + J a3 ∧ e56 − J a1 ∧ e12 − J a1 ∧ e34 − J a2 ∧ e56 ] = − ∗ [d(J a2 ) ∧ (e12 − e56 ) + d(J a3 ) ∧ (e34 + e56 ) − d(J a1 ) ∧ (e12 + e34 )] = − ∗ [(a1 + a2 + a3 ) + ∧ (e12 − e56 + e34 + e56 − e12 − e34 ) −2 (a2 + ) ∧ (e12 − e56 )−2(a3 + ) ∧ (e34 +e56 )+2(a1 + ) ∧ (e12 + e34 ) + 4(v1 − v3 )e34 ∧ (e12 − e56 ) + 4(v1 − v2 )e12 ∧ (e34 + e56 ) − 4(v2 − v3 )e56 ∧ (e12 + e34 )] = − ∗ [4(2v1 − v2 − v3 )e1234 + 4(v1 + v3 − 2v2 )e1256 + 4(2v3 − v1 − v2 )e3456 ] = − ∗ [12v1 e1234 − 12v2 e1256 + 12v3 e3456 ] = 12ϕ. Taking into account the inequality (36), we deduce at once the following Corollary 6.1. The space of infinitesimal nearly Kähler deformations of the nearly Kähler structure on F(1, 2) is isomorphic to the Lie algebra of SU3 . More precisely, every right-invariant vector field X on SU3 defines an element ϕ ∈ N K via the formulas (39) and (42). Acknowledgements. We are grateful to Gregor Weingart for helpful discussions and in particular for suggesting the statement of Lemma 5.4.

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A. Moroianu, U. Semmelmann

References 1. Baum, H., Friedrich, Th., Grunewald, R., Kath, I.: Twistor and Killing Spinors on Riemannian Manifolds. Stuttgart–Leipzig: Teubner–Verlag, 1991 2. Belgun, F., Moroianu, A.: Nearly Kähler 6-manifolds with reduced holonomy. Ann. Global Anal. Geom. 19, 307–319 (2001) 3. Besse, A.: Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10. Berlin: SpringerVerlag, 1987 4. Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces I. 80, 458–538 (1958) 5. Butruille, J.-B.: Classification des variétés approximativement kähleriennes homogènes. Ann. Global Anal. Geom. 27, 201–225 (2005) 6. Cleyton, R., Swann, A.: Einstein metrics via intrinsic or parallel torsion. Math. Z. 247, 513–528 (2004) 7. Friedrich, Th.: Nearly Kähler and nearly parallel G 2 -structures on spheres. Arch. Math. (Brno) 42, 241– 243 (2006) 8. Gray, A.: The structure of nearly Kähler manifolds. Math. Ann. 223, 233–248 (1976) 9. Moroianu, A., Nagy, P.-A., Semmelmann, U.: Unit Killing Vector Fields on Nearly Kähler Manifolds. Internat. J. Math. 16, 281–301 (2005) 10. Moroianu, A., Nagy, P.-A., Semmelmann, U.: Deformations of Nearly Kähler Structures. Pacific J. Math. 235, 57–72 (2008) 11. Moroianu, A., Semmelmann, U.: Infinitesimal Einstein Deformations of Nearly Kähler Metrics. to appear in Trans. Amer. Math. Soc., 2009 12. Nagy, P.-A.: Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 3, 481–504 (2002) 13. Wolf, J., Gray, A.: Homogeneous spaces defined by Lie group automorphisms I, II. J. Differ. Geom. 2, 77–114, 115–159 (1968) Communicated by A. Connes

Commun. Math. Phys. 294, 273–301 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0933-y

Communications in

Mathematical Physics

Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I. Manoussos G. Grillakis 1 , Matei Machedon 1 , Dionisios Margetis 1,2,3 1 Department of Mathematics, University of Maryland, College Park,

MD 20742, USA. E-mail: [email protected]

2 Institute for Physical Science and Technology, University of Maryland,

College Park, MD 20742, USA

3 Center for Scientific Computation and Mathematical Modeling, University of Maryland,

College Park, MD 20742, USA Received: 31 March 2009 / Accepted: 30 July 2009 Published online: 2 October 2009 – © Springer-Verlag 2009

Abstract: Inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schrödinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential v(x) = χ (x)|x|−1 , where is sufficiently small and χ ∈ C0∞ even, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (Part II) of this paper. 1. Introduction An advance in physics in 1995 was the first experimental observation of atoms with integer spin (Bosons) occupying a macroscopic quantum state (condensate) in a dilute gas at very low temperatures [1,4]. This phenomenon of Bose-Einstein condensation has been observed in many similar experiments since. These observations have rekindled interest in the quantum theory of large Boson systems. For recent reviews, see e.g. [23,29]. A system of N interacting Bosons at zero temperature is described by a symmetric wave function satisfying the N -body Schrödinger equation. For large N , this description is impractical. It is thus desirable to replace the many-body evolution by effective (in an appropriate sense) partial differential equations for wave functions in much lower space dimensions. This approach has led to “mean-field” approximations in which the single particle wave function for the condensate satisfies nonlinear Schrödinger equations (in 3 + 1 dimensions). Under this approximation, the N -body wave function is viewed simply as a tensor product of one-particle states. For early related works, see the papers by Gross [15,16], Pitaevskii [28] and Wu [34,35]. In particular, Wu [34,35] introduced a second-order approximation for the Boson many-body wave function in terms of the pair-excitation function, a suitable kernel that describes the scattering of atom pairs

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M. G. Grillakis, M. Machedon, D. Margetis

from the condensate to other states. Wu’s formulation forms a nontrivial extension of works by Lee, Huang and Yang [21] for the periodic Boson system. Approximations carried out for pair excitations [21,34,35] make use of quantized fields in the Fock space. (The Fock space formalism and Wu’s formulation are reviewed in Sects. 1.1 and 1.3, respectively.) Connecting mean-field approaches to the actual many-particle Hamiltonian evolution raises fundamental questions. One question is the rigorous derivation and interpretation of the mean field limit. Elgart, Erd˝os, Schlein and Yau [6–11] showed rigorously how mean-field limits for Bosons can be extracted in the limit N → ∞ by using Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchies for reduced density matrices. Another issue concerns the convergence of the microscopic evolution towards the mean field dynamics. Recently, Rodnianski and Schlein [31] provided estimates for the rate of convergence in the case with Hartree dynamics by invoking the formalism of Fock space. In this paper, inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schrödinger equation describing an improved approximation for the evolution of the Boson system. This approximation offers a second-order correction to the usual tensor product (mean field limit) for the many-body wave function. Our equation yields a corresponding new estimate in Fock space, which complements nicely the previous estimate [31]. The static version of the many-body problem is not studied here. The energy spectrum was addressed by Dyson [5] and by Lee, Huang and Yang [21]. A mathematical proof of the Bose-Einstein condensation for the time-independent case was provided recently by Lieb, Seiringer, Solovej and Yngvanson [22–25]. 1.1. Fock space formalism. Next, we review the Fock space F over L 2 (R3 ), following Rodnianski and Schlein [31]. The elements of F are vectors of the form ψ = (ψ0 , ψ1 (x1 ), ψ2 (x1 , x2 ), . . .), where ψ0 ∈ C and ψn ∈ L 2s (R3n ) are symmetric in x1 , . . . , xn . The Hilbert space structure of F is given by (φ, ψ) = n φn ψn d x. For f ∈ L 2 (R3 ) the (unbounded, closed, densely defined) creation operator a ∗ ( f ) : F → F and annihilation operator a( f¯) : F → F are defined by n ∗ 1 a ( f )ψn−1 (x1 , x2 , . . . , xn ) = √ f (x j )ψn−1 (x1 , . . . , x j−1 , x j+1 , . . . xn ), n j=1 √ a( f )ψn+1 (x1 , x2 , . . . , xn ) = n + 1 ψ(n+1) (x, x1 , . . . , xn ) f (x) d x.

The operator valued distributions ax∗ and ax defined by ∗ f (x)ax∗ d x, a (f) = a( f ) = f (x) ax d x. These distributions satisfy the canonical commutation relations [ax , a ∗y ] = δ(x − y), [ax , a y ] = [ax∗ , a ∗y ] = 0.

(1)

Second-Order Corrections for Weakly Interacting Bosons I

275

Let N be a fixed integer (the total number of particles), and v(x) be an even potential. Consider the Fock space Hamiltonian H N : F → F defined by 1 HN = ax∗ ∆ax d x + v(x − y)ax∗ a ∗y ax a y d x d y 2N 1 (2) =: H0 + V. N This H N is a diagonal operator which acts on each ψn in correspondence to the Hamiltonian H N ,n =

n

∆x j +

j=1

n 1 v(xi − x j ). 2N i, j=1

In the particular case n = N , this is the mean field Hamiltonian. Except for the Introduction, this paper deals only with the Fock space Hamiltonian. The reader is alerted that “PDE” Hamiltonians such as H N ,n will always have two subscripts. The sign of v will not play a role in our analysis. However, the reader is alerted that due to our sign convention, v ≤ 0 is the “good” sign. The time evolution in the coordinate space for Bose-Einstein condensation deals with the function eitHn,n ψ0

(3)

for tensor product initial data, i.e., if ψ0 (x1 , x2 , . . . , xn ) = φ0 (x1 )φ0 (x2 ) . . . φ0 (xn ), where φ0 L 2 (R3 ) = 1. This approach has been highly successful, even for very singular potentials, in the work of Elgart, Erd˝os, Schlein and Yau [6–11]. In this context, the convergence of evolution to the appropriate mean field limit (tensor product) as N → ∞ is established at the level of marginal density matrices γi(N ) in the trace norm topology. The density matrices are defined as (N ) γi (t, x1 , . . . , xi ; x1 , . . . xi ) = ψ(t, x1 , . . . , x N )ψ(t, x1 , . . . , x N ) d xi+1 · · · d x N .

1.2. Coherent states. There are alternative approaches, due to Hepp [17], Ginibre and Velo [13], and, most recently, Rodnianski and Schlein [31] which can treat Coulomb potentials v. These approaches rely on studying the Fock space evolution eit HN ψ 0 , where the initial data ψ 0 is a coherent state, ψ 0 = (c0 , c1 φ0 (x1 ), c2 φ0 (x1 )φ0 (x2 ), · · · ); see (4) below. The evolution (3) can then be extracted as a “Fourier coefficient” from the Fock space evolution, see [31]. Under the assumption that v is a Coulomb potential, this approach leads to strong L 2 -convergence, still at the level of the density matrices (N ) γi , as we will briefly explain below. To clarify the issues involved, let us consider the one-particle wave function φ(t, x) (to be determined later as the solution of a Hartree equation), satisfying the initial condition φ(0, x) = φ0 (x). Define the skew-Hermitian unbounded operator A(φ) = a(φ) − a ∗ (φ)

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and the vacuum state Ω = (1, 0, 0, . . .) ∈ F. Accordingly, consider the operator √

W (φ) = e−

N A(φ)

,

which is the Weyl operator used by Rodnianski and Schlein [31]. The coherent state for the initial data φ0 is √

ψ 0 = W (φ0 )Ω = e− =e

−N φ2 /2

N A(φ0 )

1, . . . ,

Ω n 1/2

N n!

φ0 (x1 ) . . . φ0 (xn ), · · ·

.

(4)

Hence, the top candidate approximation for eitHN ψ 0 reads √

ψ tensor (t) = e−

N A(φ(t,·))

Ω.

(5)

Rodnianski and Schlein [31] showed that this approximation works (under suitable assumptions on v), in the sense that √ √ 1 itHN e ψ 0 , a ∗y ax eit HN ψ 0 − e− N A(φ(t,·)) Ω, a ∗y ax e− N A(φ(t,·)) Ω Tr N eCt ) N → ∞; = O( N the symbol Tr here stands for the trace norm in x ∈ R3 and y ∈ R3 . The first term in the last relation, including N1 , is essentially the density matrix γ1(N ) (t, x, y). For the precise statement of the problem and details of the proof, see Theorem 3.1 of Rodnianski and Schlein [31]. Our goal here is to find an explicit approximation for the evolution in the Fock space. For this purpose, we adopt an idea germane to Wu’s second-order approximation for the N -body wave function in Fock space [34,35].

1.3. Wu’s approach. We first comment on the case with periodic boundary conditions, when the condensate is the zero-momentum state. For this setting, Lee, Huang and Yang [21] studied systematically the scattering of atoms from the condensate to states of opposite momenta. By diagonalizing an approximation for the Hamiltonian in Fock space, these authors derived a formula for the N -particle wave function that deviates from the usual tensor product, as it expresses excitation of particles from zero monentum to pairs of opposite momenta. For non-periodic settings, Wu [34,35] invokes the splitting ax = a0 (t)φ(t, x) + ax,1 (t), where a0 corresponds to the condensate, [a0 , a0∗ ] = 1, and ax,1 corresponds to ∗ ]. Wu applies the following states orthogonal to the condensate, [a0 , ax,1 ] = 0 = [a0 , ax,1 ansatz for the N -body wave function in Fock space: N (t) eP [K 0 ] ψ N0 (t),

(6)

where ψ N0 (t) describes the tensor product, N (t) is a normalization factor, and P[K 0 ] is an operator that averages out in space the excitation of particles from the condensate φ to

Second-Order Corrections for Weakly Interacting Bosons I

277

other states with the effective kernel (pair excitation function) K 0 . An explicit formula for P[K 0 ] is ∗ P[K 0 ] = [2N0 (t)]−1 ax,1 a ∗y,1 K 0 (t, x, y) a0 (t)2 , (7) where N0 is the expectation value of particle number at the condensate. This K 0 is not a-priori known (in contrast to the case of the classical Boltzmann gas) but is determined by means consistent with the many-body dynamics. In the periodic case, (6) reduces to the many-body wave function of Lee, Huang and Yang [21]. Wu derives a coupled system of dispersive hyperbolic partial differential equations for (φ, K 0 ) via an approximation for the N -body Hamiltonian that is consistent with ansatz (6). A feature of this system is the spatially nonlocal couplings induced by K 0 . Observable quantities such that the depletion of the condensate can be computed directly from solutions of this PDE system. This system has been solved only in a limited number of cases [26,27,35].

1.4. Scope and outline. Our objective in this work is to find an explicit approximation for the evolution eitHN ψ 0 in the Fock space norm, where ψ 0 is the coherent state (4). This would imply an approximation for the evolution eitHN ,N ψ0 in L 2 (R3N ) as N → ∞. To the best of our knowledge, no such approximation is available in the mathematics or physics literature. In particular, the tensor product type approximation (5) for φ satisfying a Hartree equation, as in [31], is not known to be such a Fock space approximation (nor do we expect it to be). To accomplish our goal, we propose to modify (5) in two ways. One minor correction is the multiplication by an oscillatory term. A second correction is a composition with a second-order “Weyl operator”. Both corrections are inspired by the work of Wu [34,35]; see also [26,27]. However, our set-up and derived equation are essentially different from these works. We proceed to describe the second order correction. Let k(t, x, y) = k(t, y, x) be a function (or kernel) to be determined later, with k(0, x, y) = 0. The minimum regularity expected of k is k ∈ L 2 (d x d y) for a.e. t. We define the operator 1 (8) B= k(t, x, y)ax a y − k(t, x, y)ax∗ a ∗y d x d y. 2 Notice that B is skew-Hermitian, i.e., iB is self-adjoint. The operator e B could be defined by the spectral theorem; see [30]. However, we prefer the more direct approach of defining it first on the dense subset of vectors with finitely many non-zero components, where it can be defined by a convergent Taylor series if k L 2 (d xd y) is sufficiently small. Indeed, B restricted to the subspace of vectors with all entries past the first N identically zero has norm ≤ C N k L 2 . Then e B is extended to F as a unitary operator.

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Now we have described all ingredients needed to state our results and derivations. The remainder of the paper is organized as follows. In Section 2 we state our main result and outline its proof. In Sect. 3 we study implications of the Hartree equation satisfied by the one-particle wave function φ(t, x). In Sect. 4 we develop bookkeeping tools of Lie algebra for computing requisite operators containing B. In Sect. 5 we study the evolution equation for a matrix K that involves the kernel k. In Sect. 6 we develop an argument for the existence of solution to the equation for the kernel k. In Sect. 7 we find conditions under which terms involved in the error term e B V e−B are bounded. In Sect. 8 we study similarly the error term e B [A, V ]e−B . In Sect. 9 we show that we can control traces needed in derivations. 2. Statement of Main Result and Outline of Proof In this section we state our strategy for general potentials satisfying certain properties. Later in the paper we show that all assumptions of the related theorem are satisfied , : sufficiently small, and χ ∈ C0∞ : even. locally in time for v(x) = χ (x) |x| Theorem 1. Suppose that v is an even potential. Let φ be a smooth solution of the Hartree equation i

∂φ + ∆φ + (v ∗ |φ|2 )φ = 0 ∂t

(9)

with initial conditions φ0 , and assume the three conditions listed below: 1. Assume that we have k(t, x, y) ∈ L 2 (d xd y) for a.e. t, where k is symmetric, and solves (iut + ug T + gu − (1 + p)m) = (i pt + [g, p] + um)(1 + p)−1 u,

(10)

where all products in (10) are interpreted as spatial compositions of kernels, “1” is the identity operator, and u(t, x, y) := sh(k) := k +

1 kkk + · · · , 3!

1 kk + · · · , (11) 2! g(t, x, y) := −∆x δ(x − y) − v(x − y)φ(t, x)φ(t, y) − (v ∗ |φ|2 )(t, x)δ(x − y),

δ(x − y) + p(t, x, y) := ch(k) := δ(x − y) + m(t, x, y) := v(x − y)φ(t, x)φ(t, y). 2. Also, assume that the functions

f (t) := e B [A, V ]e−B ΩF and g(t) := e B V e−B ΩF are locally integrable (V is defined in (2)).

Second-Order Corrections for Weakly Interacting Bosons I

3. Finally, assume that

279

d(t, x, x) d x is locally integrable in time, where d(t, x, y) = ish(k)t + sh(k)g T + gsh(k) sh(k) − (ich(k)t + [g, ch(k)]) ch(k) −sh(k)mch(k) − ch(k)msh(k).

Then, there exist real functions χ0 , χ1 such that √

e−

N A(t) −B(t) −i

t

≤

0

e

e t

f (s)ds + √ N

t

0

0 (N χ0 (s)+χ1 (s))ds

Ω − eitHN ψ 0 F

g(s)ds . N

(12)

Recall that we defined (see Sect. 1) √

ψ 0 = e−

N A(0)

Ω an arbitrary coherent state (initial data),

A(t) = a(φ(t, ·)) − a ∗ (φ(t, ·)), 1 B(t) = k(t, x, y)ax a y − k(t, x, y)ax∗ a ∗y d x d y. 2 A few remarks on Theorem 1 are in order. Remark 1. Written explicitly, the left-hand side of (10) equals ∂ iut + ug T + gu − (1 + p)m = i − ∆x − ∆ y u(t, x, y) ∂t −φ(t, x) v(x − z)φ(t, z)u(t, z, y) dz − φ(t, y) u(t, x, z)v(z − y)φ(t, z) dz −(v ∗ |φ|2 )(t, x)u(t, x, y) − (v ∗ |φ|2 )(t, y)u(t, x, y) −v(x − y)φ(t, x)φ(t, y) −φ(t, y) (1 + p)(t, x, z)v(z − y)φ(t, z) dz. The main term in the right-hand side equals ∂ i pt + [g, p] + um = i p(t, x, y) + −∆x + ∆ y p(t, x, y) ∂t −φ(t, x) v(x − z)φ(t, z) p(t, z, y) dz +φ(t, y) p(t, x, z)v(z − y)φ(t, z) dz −(v ∗ |φ|2 )(t, x) p(t, x, y) + (v ∗ |φ|2 )(t, y) p(t, x, y) + u(t, x, z)v(z − y)φ(t, z)φ(t, x) dz. Remark 2. The algebra, as well as the local analysis presented in this paper do not depend on the sign of v. However, the global in time analysis of our equations would require v to be non-positive.

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Remark 3. Our √techniques would allow us to consider more general initial data of the form ψ 0 = e− N A(0) e−B(0) Ω. For convenience, we only consider the case of tensor products (B(0) = 0) in this paper. √

Proof. Since e ei

NA

and e B are unitary, the left-hand side of (12) equals

t

0 (N χ0 (s)+χ1 (s))ds

e B(t) e

√

√ N A(t) itH N − N A(0)

e

e

Ω − ΩF .

Define Ψ (t) = e B(t) e

√

√ N A(t) it H − N A(0)

e

e

Ω.

In Corollary 1 of Sect. 5 we show that our equations for φ, k insure that 1 ∂ Ψ = LΨ, i ∂t where L = L − N χ0 − χ1 for some L: Hermitian, i.e. L = L ∗ , where L commutes with functions of time, χ0 , χ1 are real functions of time, and, most importantly (see Corollary 1 of Sect. 5 and the remark following it), LΩF ≤ N −1/2 e B [A, V ]e−B ΩF + N −1 e B V e−B ΩF .

(13)

We apply energy estimates to

t 1 ∂

− L (ei 0 (N χ0 (s)+χ1 (s))ds Ψ − Ω) = LΩ. i ∂t

Explicitly, ∂ i t (N χ0 (s)+χ1 )ds (e 0 Ψ − Ω)2F ∂t t ∂ i t (N χ0 (s)+χ1 )ds i 0 (N χ0 (s)+χ1 )ds 0 = 2 (e Ψ − Ω), e Ψ −Ω ∂t t t ∂ i 0 (N χ0 (s)+χ1 )ds i 0 (N χ0 (s)+χ1 )ds

− i L (e Ψ − Ω), e Ψ −Ω = 2 ∂t t = 2 i LΩ, ei 0 (N χ0 (s)+χ1 )ds Ψ − Ω t ≤ 2 N−1/2 e B [A, V ]e−B ΩF + N−1 e B V e−B ΩF (ei 0 (N χ0 (s)+χ1 )ds Ψ −Ω)F . Thus t ∂ (ei 0 (N χ0 (s)+χ1 )ds Ψ − Ω) ≤ N −1/2 e B [A, V ]e−B ΩF + N −1 e B V e−B ΩF ∂t

and (12) holds. This concludes the proof.

Second-Order Corrections for Weakly Interacting Bosons I

281

3. The Hartree Equation In this section we see how far we can go by using only the Hartree equation for the one-particle wave function φ. Lemma 1. The following commutation relations hold (where the t dependence is suppressed, A denotes A(φ) and V is defined by formula (2)): [A, V ] = v(x − y) φ(y)ax∗ ax a y + φ(y)ax∗ a ∗y ax d x d y, [A, [A, V ]] = v(x − y) φ(y)φ(x)ax a y + φ(y)φ(x)ax∗ a ∗y + 2φ(y)φ(x)ax∗ a y d x d y, +2 v ∗ |φ 2 | (x)ax∗ ax d x, A, A, [A, V ] = 6 v ∗ |φ 2 | (x) φ(x)ax∗ + φ(x)ax d x,

A, A, [ A, [A, V ]]

= 12 v ∗ |φ 2 | (x)|φ(x)|2 d x.

(14)

Proof. This is an elementary calculation and is left to the interested reader. √

√

Now, we consider Ψ1 (t) = e N A(t) eit H e− N A(0) Ω for which we have the basic calculation in the spirit of Hepp [17], Ginibre-Velo [13], and Rodnianski-Schlein [31]; see Eq. (3.7) in [31]. Proposition 1. If φ satisfies the Hartree equation i

∂φ + ∆φ + (v ∗ |φ|2 )φ = 0 ∂t

while Ψ1 (t) = e

√

√ N A(t) it H − N A(0)

e

e

Ω,

then Ψ1 (t) satisfies 1 1 ∂ Ψ1 (t) = H0 + [A, [A, V ]] i ∂t 2 N +N −1/2 [A, V ] + N −1 V − v(x − y)|φ(t, x)|2 |φ(t, y)|2 d x d y Ψ1 (t). 2 Proof. Recall the formulas ∂ C(t) −C(t) 1 ˙ + ··· ˙ + 1 C, [C, C] e = C˙ + [C, C] e ∂t 2! 3! and eC H e−C = H + [C, H ] +

1 [C, [C, H ]] + · · · 2!

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Applying these relations to C =

√

N A we get

1 ∂ ψ1 (t) = L 1 ψ1 , i ∂t

(15)

where

√ √ 1 ∂ √ N A(t) −√ N A(t) e e + e N A(t) H e− N A(t) i ∂t 1 N ˙ + H + N 1/2 [A, H0 ] N 1/2 A˙ + [A, A] = i 2 N +N −1/2 [A, V ] + [A, [A, H0 ]] 2 N 1 N 1/2 A, A, [A, V ] + A, A, [ A, [A, V ]] . × [A, [A, V ]] + 2 3! 4! √ Eliminating the terms with a weight of N , or setting 1 1 ˙ A, A, [A, V ] = 0, (16) A + [A, H0 ] + i 3! L1 =

is exactly equivalent to the Hartree equation (9). By taking an additional bracket with A in (16), we have 1 1 ˙ [A, A] + [A, [A, H0 ]] + A, A, [ A, [A, V ]] = 0, i 3! and thus simplify (15) to 1 ∂ 1 ψ1 (t) = H0 + [A, [A, V ]] i ∂t 2 +N This concludes the proof.

−1/2

[A, V ] + N

−1

1 A, A, [A, [A, V ]] ψ1 . V−N 4!

on the right-hand side are the main ones. The next two terms are two terms The first 1 1 and O N . The last term equals O √ N

−

N 2

v(x − y)|φ(t, x)|2 |φ(t, y)|2 d x d y := −N χ0 .

Notice that L 1 (Ω) is not small because of the presence of ax∗ a ∗y in [A, [A, V ]]. In order to eliminate these terms, we introduce B (see (8)) and take ψ = e B ψ1 . Accordingly, we compute 1 ∂ ψ = Lψ, i ∂t

Second-Order Corrections for Weakly Interacting Bosons I

where 1 L= i

283

∂ B −B e e + e B L 1 e−B ∂t

= L Q + N −1/2 e B [A, V ]e−B + N −1 e B V e−B − N χ0 , and LQ

1 = i

∂ B −B 1 B H0 + [A, [A, V ]] e−B e e +e ∂t 2

(17)

contains all quadratics in the operators a, a ∗ . Equation (10) for k turns out to be equivalent to the requirement that L has no terms of the form a ∗ a ∗ . Terms of the form aa ∗ will occur, and will be converted to a ∗ a at the expense of χ1 . In other words, we require that L Q have no terms of the form a ∗ a ∗ . For a similar argument (but for a different set-up), see Wu [35]. 4. The Lie Algebra of “Symplectic Matrices” In this section we describe the bookkeeping tools needed to compute L Q of (17) in closed form. The results of this section are essentially standard, but they are included here for the sake of completeness. We start with the remark that [a( f 1 ) + a ∗ (g1 ), a( f 2 ) + a ∗ (g2 )] = f 1 g2 − f 2 g1 f = − f 1 g1 J 2 , (18) g2 where

0 −δ(x − y) J= . δ(x − y) 0

This observation explains why we have to invoke symplectic linear algebra. We thus consider the infinite-dimensional Lie algebra sp of “matrices” of the form d k S(d, k, l) = l −d T for symmetric kernels k = k(t, x, y) and l = l(t, x, y), and arbitrary kernel d(t, x, y). (The dependence on t will be suppressed when not needed.) This situation is analogous to the Lie algebra of the finite-dimensional complex symplectic group, with x, y playing the role of i and j. We also consider the Lie algebra Quad of quadratics of the form ∗ d k 1 −a y ∗ ax ax Q(d, k, l) := l −d T ay 2 ∗ ∗ ax a y + a y ax 1 = − d(x, y) dx dy + k(x, y)ax a y d x d y 2 2 1 − (19) l(x, y)ax∗ a ∗y d x d y 2

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(k, l and d as before). Furthermore, we agree to identify operators which differ (formally) by a scalar operator. Thus, d(x, y)ax a ∗y is considered equivalent to d(x, y)a ∗y ax . We recall the following result related to the metaplectic representation (see, e.g. [12]). Theorem 2. Let S = S(d, k, l), Q = Q(d, k, l) related as above. Let f , g be functions (or distributions). Denote f := f (x)ax + g(x)ax∗ d x. (ax , ax∗ ) g We have the following commutation relation: f f ∗ ∗ = (ax , ax )S , Q, (ax , ax ) g g where products are interpreted as compositions. We also have f f Q ∗ −Q ∗ S , e e (ax , ax ) = (ax , ax )e g g

(20)

(21)

provided that e Q makes sense as a unitary operator (Q: skew-Hermitian). Proof. The commutation relation (20) can be easily checked directly, but we point out that it follows from (18). In fact, using (18), for any rank one quadratic we have a( f 1 ) + a ∗ (g1 ) a( f 2 ) + a ∗ (g2 ) , a( f ) + a ∗ (g) f1 f2 f ∗ f 1 g1 + f 2 g2 J = − ax ax . g g2 g1 Thus, for any R we have a f ax ax∗ R ∗y , a( f ) + a ∗ (g) = − ax ax∗ R + R T J . ay g Now specialize to R = 21 S J , S ∈ sp, and use S T = J S J to complete the proof. The second part, Eq. (21), follows from the identity e Q Ce−Q = C + [Q, C] +

1 [Q, [Q, C]] + · · · , 2!

or, in the language of adjoint representations, Ad(e Q )(C) = ead(Q) (C), which is applied to C = a( f ) + a ∗ (g).

A closely related result is provided by the following theorem. Theorem 3. 1. The linear map I : sp → Quad defined by S(d, k, l) → Q(d, k, l) is a Lie algebra isomorphism. 2. Moreover, if S = S(t), Q = Q(t) and I(S(t)) = Q(t) is skew-Hermitian, so that e Q is well defined, we have ∂ Q −Q ∂ S −S = e . e e e (22) I ∂t ∂t

Second-Order Corrections for Weakly Interacting Bosons I

285

3. Also, if R ∈ sp, we have I e S Re−S = e Q I(R)e−Q .

(23)

Remark 4. In the finite-dimensional case, this is (closely related to) the “infinitesimal metaplectic representation”; see p. 186 in [12] . In the infinite dimensional case, we must be careful, as some of our operators are not of trace class. For instance, ax ax∗ does not make sense. Proof. First, we point out that (21) implies (23), at least in the case where R is the “rank one” matrix f hi . R= g Notice that (21) can also be written as ax −Q S T ax f g eQ f g e . = e ax∗ ax∗ In conclusion, we find ∗ −a y −Q ∗ e ax ax R e ay f a Q ∗ h i a a J a ∗y e−Q =e x x g y f a Q −Q Q ∗ a h i a e e J a ∗y e−Q =e x x g y f J S J ay h i Je = ax ax∗ e S a ∗y g ∗ −a y , = ax ax∗ e S Re−S ay Q

since S T = J S J if S ∈ sp, and J e J S J = e−S J . We now give a direct proof that (19) preserves Lie brackets. Denote the quadratic 1 ∗ ∗ ∗ ∗ ∗ building blocks by Q x y = ax a y , Q x y = ax a y , N x y = 2 ax a y + a y ax . One can verify the following commutation relations, which will be also needed below: Q x y , Q ∗zw = δ(x − z)N yw + δ(x − w)N yz + δ(y − z)N xw + δ(y − w)N x z , (24) (25) Q x y , Nzw = δ(x − w)Q yz + δ(y − w)Q x z , ∗ ∗ (26) N x y , Q zw = δ(x − z)Q yw + δ(x − w)Q yz , N x y , Nzw = δ(x − w)Nzy − δ(y − z)N xw . (27) Using (24) we compute 1 1 k(x, y)ax a y d xd y, − l(x, y)ax∗ a ∗y d xd y = − (kl)(x, y)N x y d x d y, 2 2

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which corresponds to the relation 0 0 k , l 0 0

0 0

=

kl 0 . 0 −lk

The other three cases are similar. To prove (22), expand both the left-hand side and the right-hand side as ∂ S −S I e e ∂t 1 ˙ ˙ = I S + [S, S] + · · · 2 1 ˙ + ··· = Q˙ + [Q, Q] 2 ∂ Q −Q e . = e ∂t The proof of (23) is along the same lines.

Remark 5. Note on rigor: All the Lie algebra results that we have used are standard in the finite-dimensional case. In our applications, S will be K , where K is a matrix of the form (29), see below, and Q will be B = I(K ). The unbounded operator B is skew-Hermitian and e B ψ is defined by a convergent Taylor series if ψ ∈ F has only finitely many non-zero components, provided k(t, ·, ·) L 2 (d x d y) is small. We then extend e B to all F as a unitary operator. The norm k(t, ·, ·) L 2 (d x d y) iterates under compositions; thus, the kernel e K is well defined by its convergent Taylor expansion. In the expression e B Pe−B = P + [B, P] + · · ·

(28)

for P, a first- or second-order polynomial in a, a ∗ , we point out that the right-hand side stays a polynomial of the same degree, and converges when applied to a Fock space vector with finitely many non-zero components. For our application, we need to know if (28) is true when applied to Ω. The same comment applies to the series ∂ B −B 1 ˙ + ··· . e e = B˙ + [B, B] ∂t 2 5. Equation for Kernel k Now apply the isomorphism of the previous section to the operator B = I(K ) for

0 k(t, x, y) . k(t, x, y) 0

K =

(29)

This agrees to the letter with the isomorphism (19). The next two isomorphisms, (30) and (31), require special treatment because aa ∗ terms mirroring the a ∗ a terms are missing

Second-Order Corrections for Weakly Interacting Bosons I

287

in (2), (14). However, the discrepancy only happens on the diagonal. Once the relevant terms are commuted with B, they fit the pattern exactly. It isn’t quite true that −(∆δ)(x − y) 0 H0 = I 0 (∆δ)(x − y) −∆ 0 =I (30) 0 ∆ since, strictly speaking, ∗ ax ∆ax + ax ∆ax∗ −(∆δ)(x − y) 0 = I dx 0 (∆δ)(x − y) 2 is undefined. However, one can compute directly that [∆x ax , a ∗y ] = (∆δ)(x − y). Using that, we compute 1 [B, H0 ] = (∆x + ∆ y )k(x, y)ax a y + (∆x + ∆ y )k(x, y)ax∗ a ∗y d x d y. 2 This commutator is in agreement with (29), (30), and the result can be represented in accordance with (19), namely 0k −(∆δ)(x − y) 0 [B, H0 ] = I , . 0 (∆δ)(x − y) k0 We also have e B H0 e−B − H0 0 −(∆δ)(x − y) 0 K −(∆δ)(x − y) −K e =I e − , 0 (∆δ)(x − y) 0 (∆δ)(x − y) since e B H0 e−B − H0 = [B, H0 ] + 21 [B, [B, H0 ]] + · · ·. The same comment applies to the diagonal part of 1 v12φ 1 φ 2 −v12 φ 1 φ2 − v ∗ |φ|2 δ12 , (31) [A, [A, V ]] = I −v12 φ1 φ2 v12 φ1 φ 2 + v ∗ |φ|2 δ12 2 where v12 φ1 φ2 is an abbreviation for the product v(x − y)φ(x)φ(y), etc. Formula (31) isn’t quite true either, but becomes true after commuting with B. To apply our isomorphism, we quarantine the “bad” terms in (30) and the diagonal part of (31). Define g 0 0 m , G= and M= −m 0 0 −g T where g = −∆δ12 − v12 φ 1 φ2 − (v ∗ |φ|2 )δ12 , m = v12 φ 1 φ 2 , and split H0 +

1 [A, [A, V ]] = HG + I(M), 2

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where

HG = H0 + v(x − y)φ(y)φ(x)ax∗ a y d x d y + v ∗ |φ 2 | (x)ax∗ ax d x.

(32)

By the above discussion we have [B, HG ] = I([K , G]) and B −B K [e , HG ]e = I([e , G]e−K ). Write

1 ∂ B −B 1 e e + e B H0 + [A, [A, V ]] e−B i ∂t 2 1 ∂ B −B e e + HG + [e B , HG ]e−B + e B I(M)e−B = i ∂t 1 ∂ K −K e e = HG + I + [e K , G]e−K + e K Me−K i ∂t = HG + I(M1 + M2 + M3 ).

LQ =

(33)

Notice that if K is given by (29), then ch(k) sh(k) K e = , sh(k) ch(k) where 1 1 ch(k) = I + kk + kkkk + · · · , (34) 2 4! and similarly for sh(k). Products are interpreted, of course, as compositions of operators. We compute 1 ch(k)t sh(k)t ch(k) −sh(k) M1 = −sh(k) ch(k) i sh(k)t ch(k)t 1 ch(k)t ch(k) − sh(k)t sh(k) −ch(k)t sh(k) + sh(k)t ch(k) = ∗ ∗ i T [ch(k), g] −sh(k)g − gsh(k) [e K , G] = ∗ ∗ and M2 = [e K , G]e−K =

[ch, g] ch + (shg T + gsh)sh −[ch, g]sh − (shg T + gsh)ch , ∗ ∗

where sh is an abbreviation for sh(k), etc, and −shm ch − chmsh shmsh + chmch . M3 = e K Me−K = ∗ ∗ Now define M = M1 + M2 + M3 . We have proved the following theorem.

Second-Order Corrections for Weakly Interacting Bosons I

289

Theorem 4. Recall the isomorphism (19) of Theorem 3. 1. If L Q is given by (17), then

L Q = H0 + v(x − y)φ(y)φ(x)ax∗ a y d x d y + v ∗ |φ 2 | (x)ax∗ ax d x + I (M) .

(35)

2. The coefficient of ax a y in I (M) is −M12 or (ish(k)t + sh(k)g T + gsh(k))ch(k) − (ich(k)t − [ch(k), g])sh(k) −sh(k)msh(k) − ch(k)mch(k). 3. The coefficient of ax∗ a ∗y equals minus the complex conjugate of the coefficient of ax a y . 4. The coefficient of −

ax a ∗y + a ∗y ax 2

is M11 , or d(t, x, y) = ish(k)t + sh(k)g T + gsh(k) sh(k) − (ich(k)t + [g, ch(k)]) ch(k) −sh(k)mch(k) − ch(k)msh(k).

(36)

Corollary 1. If φ and k satisfy (9) and (10) of Theorem 1, then the coefficients of ax a y and ax∗ a ∗y drop out and L Q becomes ∗ v ∗ |φ 2 | (x)ax∗ ax d x L Q = H0 + v(x − y)φ(t, y)φ(t, x)ax a y d x d y + ax a ∗y + a ∗y ax − d(t, x, y) d x d y, 2 where d is given by (36) and the full operator reads ∗ v ∗ |φ 2 | (x)ax∗ ax d x L = H0 + v(x − y)φ(y)φ(t, x)ax a y d xd y + − d(t, x, y)a ∗y ax d x + N −1/2 e B [A, V ]e−B + N −1 e B V e−B − N χ0 − χ1 := L − N χ 0 − χ1 , and 1 χ0 = 2

v(x − y)|φ(t, x)|2 |φ(t, y)|2 d x d y, 1 χ1 (t) = − d(t, x, x)d x. 2 Remark 6. Notice that

LΩ = N −1/2 e B [A, V ]e−B + N −1 e B V e−B Ω,

and therefore we can derive the bound LΩ ≤ N −1/2 e B [A, V ]e−B Ω + N −1 e B V e−B Ω.

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M. G. Grillakis, M. Machedon, D. Margetis

Also, L is (formally) self-adjoint by construction. The kernel d(t, x, y), being the sum of the (1,1) entry of the self-adjoint matrices 1i ∂t∂ e K e−K , [e K , G]e−K = e K Ge−K − G and the visibly self-adjoint term −sh(k)mch(k) − ch(k)msh(k), is self-adjoint; thus, it has a real trace. Hence, L is also self-adjoint. In the remainder of this paper, we check that the hypotheses of our main theorem are satisfied, locally in time, for the potential v(x) = χ (x) |x| . 6. Solutions to Equation 10 0 0 Theorem 5. Let 0 be sufficiently small and assume that v(x) = |x| , or v(x) = χ (x) |x| for χ ∈ C0∞ (R3 ). Assume that φ is a smooth solution to the Hartree equation (16), φ L 2 (d x) = 1. Then there exists k ∈ L ∞ ([0, 1])L 2 (d xd y) solving (10) with initial conditions k(0, x, y) = 0 for 0 ≤ t ≤ 1. The solution k satisfies the following additional properties:

1.

2.

3.

∂ i − ∆x − ∆ y k L ∞ [0,1]L 2 (d xd y) ≤ C, ∂t ∂ i − ∆x − ∆ y sh(k) L ∞ [0,1]L 2 (d xd y) ≤ C, ∂t ∂ i − ∆x + ∆ y p L ∞ [0,1]L 2 (d xd y) ≤ C. ∂t

4. The kernel k agrees on [0, 1] with a kernel k for which k

1 1

X 2,2+

≤ C;

see (38) for the definition of the space X s,δ and, of course, 21 + denotes a fixed number slightly bigger than 21 . Proof. We first establish some notation. Let S denote the Schrödinger operator S=i

∂ − ∆x − ∆ y ∂t

and let T be the transport operator T =i

∂ − ∆x + ∆ y . ∂t

Let : L 2 (d xd y) → L 2 (d xd y) denote schematically any linear operator of operator norm ≤ C0 , where C is a “universal constant”. In practice, will be (composition with)

Second-Order Corrections for Weakly Interacting Bosons I

291

a kernel of the type φ(t, x)φ(t, y)v(x − y), or multiplication by v ∗ |φ|2 . Also, recall the inhomogeneous term m(t, x, y) = v(x − y)φ(t, x)φ(t, y). Then, Eq. (10), written explicitly, becomes Sk = m + S(k − u) + (u) + ( p) + (T p + ( p) + (u))(1 + p)−1 u.

(37)

Note that ch(k)2 − sh(k)sh(k) = 1; thus, 1 + p = ch(k) ≥ 1 as an operator and (1 + p)−1 is bounded from L 2 to L 2 . We plan to iterate in the norm N (k) = k L ∞ [0,1]L 2 (d xd y) + Sk L ∞ [0,1]L 2 (d xd y) . Notice that m L 2 (d xd y) ≤ C0 . Now solve Sk0 = m with initial conditions k0 (0, ·, ·) = 0, where N (k0 ) ≤ C0 . Define u 0 , p0 corresponding to k0 . For the next iterate, solve Sk1 = m + S(k0 − u 0 ) + (u 0 ) + ( p0 ) + (T p0 + ( p0 ) + (u 0 ))(1 + p0 )−1 u 0 ; the non-linear terms satisfy S(u 0 − k0 ) L ∞ [0,1]L 2 (d xd y) 1 (Sk0 )k 0 k0 − k0 (Sk0 )k0 + k0 k 0 Sk0 + · · · L ∞ [0,1]L 2 (d xd y) = 3! = O(N (k0 )3 ). Also, recalling that p0 = ch(k0 ) − 1, we have 1 T ( p0 ) L ∞ [0,1]L 2 (d xd y) = (Sk0 )k 0 − k0 (Sk0 ) + · · · L ∞ [0,1]L 2 (d xd y) 2 = O N (k0 )2 . Thus, N (k1 ) ≤ C0 + C02 . Continuing this way, we obtain a fixed point solution in this space which satisfies the first three requirements of Theorem 5. N a N a In fact, we can apply the same argument to ∂t∂ D k, since ∂t∂ D m ∈ L ∞ [0, 1] 1 2 L (d x d y) for 0 ≤ a < 2 . However, we cannot repeat the argument for D 1/2 k. We would like to have S D 1/2 k L ∞ [0,1]L 2 (d x d y) finite. Unfortunately, this misses “logarithmically” because of the singularity of v. Fortunately, we can use the well-known X s,δ spaces (see [2,18,20]) to show that |S|s D 1/2 u L 2 (dt)L 2 (d x d y) is finite locally in time for (all) 1 > s > 21 . This assertion will be sufficient for our purposes. Recall the definition of X s,δ : δ u L 2 (dτ dξ ) := u X s,δ . (38) |ξ |s |τ − |ξ |2 | + 1 Going back to (37), we write S(k) = m + F,

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where we define the expression F(k) := S(k − u) − (u) + pm + (T ( p) + ( p) + um) (1 + p)−1 u. The idea is to localize in time on the right-hand side: S( k) = χ (t) (m + F), k = k on [0, 1]. where χ ∈ C0∞ (R), χ = 1 on [0, 1]. Then, N a D k L 2 [0,1]L 2 (d x d y) ≤ C for As we already pointed out, we can estimate S ∂t∂ 1

0 ≤ a < 2 . We can further localize k in time to insure that these relations hold globally in time. By using the triangle inequality |τ −|ξ |2 |+|τ | ≥ |ξ |2 , we immediately conclude that 1+ 3 2 |ξ | 2 − |τ − |ξ |2 | + 1 kχ L 2 (dτ dξ ) ≤ C.

7. Error Term e B V e−B The goal of this section is to list explicitly all terms in e B V e−B and to find conditions under which these terms are bounded. Recall that V is defined by V = v(x0 − y0 )Q ∗x0 y0 Q x0 y0 d x0 dy0 . For simplicity, shb(k) denotes either sh(k) or sh(k), and chb(k) denotes either ch(k) or ch(k). Let x0 = y0 ; we obtain e B Q ∗x0 y0 Q x0 y0 e−B = e B Q ∗x0 y0 e−B e B Q x0 y0 e−B . According to the isomorphism (19), we have Q ∗x0 y0 = I

0 −2δ(x − x0 )δ(y − y0 )

0 , 0

where the operator e

B

Q ∗x0 y0 e−B

ch(k) −sh(k) ch(k) sh(k) 0 0 =I −2δ(x − x0 )δ(y − y0 ) 0 −sh(k) ch(k) sh(k) ch(k)

is a linear combination of the terms chb(k)(x, x0 )chb(k)(y0 , y)Q ∗x y d x d y, shb(k)(x, x0 )chb(k)(y0 , y)N x y d x d y, shb(k)(x, x0 )shb(k)(y0 , y)Q x y d x d y.

(39)

Second-Order Corrections for Weakly Interacting Bosons I

A similar calculation shows that e B Q x0 y0 e−B is a linear combination of chb(k)(x, x0 )chb(k)(y0 , y)Q x y d x d y, shb(k)(x, x0 )chb(k)(y0 , y)N x y d x d y, shb(k)(x, x0 )shb(k)(y0 , y)Q ∗x y d x d y.

293

(40)

Thus, e B Q ∗x0 y0 Q x0 y0 e−B is a linear combination of the nine possible terms obtained by combining the above. Now we list all terms in e B V e−B Ω. Terms in e B V e−B ending in Q x y are automatically discarded because they contribute nothing when applied to Ω. The remaining six terms are listed below. chb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 )chb(k)(y0 , y2 )

v(x0 − y0 )Q ∗x1 y1 N x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 ,

(41)

chb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 )shb(k)(y0 , y2 )

v(x0 − y0 )Q ∗x1 y1 Q ∗x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 ,

(42)

shb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 )chb(k)(y0 , y2 )

v(x0 − y0 )N x1 y1 N x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 ,

(43)

shb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 )shb(k)(y0 , y2 )

v(x0 − y0 )N x1 y1 Q ∗x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 ,

(44)

shb(k)(x1 , x0 )shb(k)(y0 , y1 )shb(k)(x2 , x0 )chb(k)(y0 , y2 )

v(x0 − y0 )Q x1 y1 N x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 ,

(45)

shb(k)(x1 , x0 )shb(k)(y0 , y1 )shb(k)(x2 , x0 )shb(k)(y0 , y2 ) v(x0 − y0 )Q x1 y1 Q ∗x2 y2 Ωd x1 dy1 d x2 dy2 d x0 dy0 .

(46)

To compute the above six terms,recall (24) through (27) as well as (1). In general, N x y Ω = 1/2δ(x − y)Ω, while f (x, y)Q ∗x y d xd yΩ = (0, 0, f (x, y), 0, . . .) up to symmetrization and normalization. The resulting contributions (neglecting symmetrization and normalization) follow. From (41): ψ(x1 , y1 ) = chb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 ) ×chb(k)(y0 , x2 )v(x0 − y0 )d x2 d x0 dy0 . From (42): ψ(x1 , y1 , x2 , y2 ) =

(47)

chb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x2 , x0 ) ×shb(k)(y0 , y2 )v(x0 − y0 )d x0 dy0 .

(48)

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From (43): ψ=

shb(k)(x1 , x0 )chb(k)(y0 , x1 )shb(k)(x2 , x0 ) ×chb(k)(y0 , x2 )v(x0 − y0 )d x1 d x2 d x0 dy0 .

(49)

From (44), with the N and Q ∗ reversed, we get ψ(x2 , y2 ) =

shb(k)(x1 , x0 )chb(k)(y0 , x1 )shb(k)(x2 , x0 ) ×shb(k)(y0 , y2 )v(x0 − y0 )d x1 d x0 dy0 ,

(50)

as well as the contribution from [N , Q ∗ ], i.e. ψ(y1 , y2 ) =

shb(k)(x1 , x0 )chb(k)(y0 , y1 )shb(k)(x1 , x0 ) ×shb(k)(y0 , y2 )v(x0 − y0 )d x1 d x0 dy0 .

(51)

The contribution of (45) is zero, and, finally, the contribution of (46), using (24), consists of four numbers, which can be represented by the two formulas ψ=

shb(k)(x1 , x0 )shb(k)(y0 , x1 )shb(k)(x2 , x0 )

(52)

×shb(k)(y0 , x2 )v(x0 − y0 )d x1 d x2 d x0 dy0 and |shb(k)|2 (x1 , x0 )|shb(k)|2 (y0 , y1 )v(x0 − y0 )d x1 dy1 d x0 dy0 .

ψ=

(53)

We can now state the following proposition. Proposition 2. The state e B V e−B Ω has entries on the zeroth, second and fourth slot of a Fock space vector of the form given above. In addition, if ∂ i − ∆x − ∆ y sh(k) L 1 [0,T ]L 2 (d xd y) ≤ C1 , ∂t ∂ i − ∆x + ∆ y p L 1 [0,T ]L 2 (d xd y) ≤ C2 , ∂t and v(x) =

1 |x| ,

1 or v(x) = χ (x) |x| , then

T 0

e B V e−B Ω2F dt ≤ C,

where C only depends on C1 and C2 .

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Proof. This follows by writing ch(k) = δ(x − y)+ p and applying Cauchy-Schwartz and local smoothing estimates as in the work of Sjölin [32], Vega [33]; see also Constantin and Saut [3]. In fact, we need the following slight generalization (see Lemma 2 below): If ∂ i − ∆x1 − ∆x2 ± ∆x3 · · · ± ∆xn f (t, x1 , . . . xn ) L 1 [0,T ]L 2 (dtd x) ≤ C, ∂t with initial conditions 0, then

f (t, x1 , x2 , . . .) L 2 [0,T ]L 2 (d xd y) ≤ C. |x1 − x2 |

(54)

We will check a typical term, (48). This amounts to proving the following three terms are in L 2 . 1. ψ pp (t, x1 , y1 , x2 , y2 ) = p(t, x1 , x0 ) p(t, y0 , y1 )shb(k)(t, x2 , x0 )shb(k)(t, y0 , y2 )v(x0 − y0 ) d x0 dy0 . We use Cauchy-Schwartz in x0 , y0 to get

T

|ψ pp |2 dt d x1 d x2 dy1 dy2 0 ≤ sup | p(t, x1 , x0 ) p(t, y0 , y1 )|2 d x1 d x0 dy1 dy0 t

×

T

|shb(k)(t, x2 , x0 )shb(k)(t, y0 , y2 )v(x0 − y0 )|2 dt d x2 d x0 dy2 dy0 ≤ C.

0

The first term is estimated by energy, and the second one is an application of (54) with f = shb(k)shb(k). Notice that, because of the absolute value, we can choose either sh(k) or sh(k) to insure that the Laplacians in x0 , y0 have the same signs. 2. ψ pδ (t, x1 , y1 , x2 , y2 ) = p(t, x1 , x0 )shb(k)(t, x2 , x0 )shb(k)(t, y1 , y2 )v(x0 − y1 ) d x0 . Here, we use Cauchy-Schwartz in x0 to estimate, in a similar fashion,

T

|ψ pδ |2 dt d x1 d x2 dy1 dy2 ≤ sup | p(t, x1 , x0 )|2 d x1 d x0

0

t

×

T 0

|shb(k)(t, x2 , x0 )shb(k)(t, y1 , y2 )v(x0 − y1 )|2 dt d x2 d x0 dy2 dy0 ≤ C.

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3. ψδδ (x1 , y1 , x2 , y2 ) = shb(k)(t, x2 , x1 )shb(k)(t, y1 , y2 )v(x1 − y1 ), which is just a direct application of (54). All other terms are similar.

We have to sketch the proof of the local smoothing estimate that we used above. Lemma 2. If f : R3n+1 → C satisfies ∂ i − ∆x1 − ∆x2 ± ∆x3 · · · ± ∆xn f (t, x1 , . . . xn ) L 1 [0,T ]L 2 (d xd y) ≤ C ∂t with initial conditions f (0, · · · ) = 0, then

f (t, x1 , x2 , . . .) L 2 [0,T ]L 2 (d x) ≤ C. |x1 − x2 |

Proof. We follow the general outline of Sjölin, [32]. Using Duhamel’s principle, it suffices to assume that ∂ i − ∆x1 − ∆x2 ± ∆x3 · · · ± ∆xn f (t, x1 , · · · xn ) = 0 (55) ∂t with initial conditions f (0, · · · ) = f 0 ∈ L 2 . Furthermore, after the change of variables x1 → x1√+ x2 , x2 → x2√−x1 , it suffices to prove that 2

2

f (t, x1 , x2 , . . .) L 2 [0,T ]L 2 (d x) ≤ C, |x1 |

where f satisfies the same equation (55). Changing notation, denote x = (x2 , x3 , . . .) and let < ξ >2 be the relevant expression ±|ξ2 |2 ± |ξ3 |2 . . .. Write 2 2 f (t, x1 , x) = eit (|ξ1 | + ) ei x1 ·ξ1 +i x·ξ f 0 (ξ1 , ξ ) dξ1 dξ. Thus, we obtain | f (t, x1 , x)|2 dtd x1 d x |x1 |2 i x ·(ξ −η )+i x·(ξ −η) 2 2 2 2 e 1 1 1 = eit (|ξ1 | −|η1 | + − ) |x1 |2 × f (ξ , ξ ) f 0 (η1 , η)dξ1 dξ dη1 dηdt d x d x1 0 1 1 = c δ(|ξ1 |2 − |η1 |2 ) f 0 (η1 , ξ )dξ1 dη1 dξ f 0 (ξ1 , ξ ) |ξ1 − η1 | ≤ | f 0 (ξ1 , ξ )|2 d x1 dξ, because one can easily check that sup δ(|ξ1 |2 − |η1 |2 ) ξ1

1 dη1 ≤ C. |ξ1 − η1 |

1 is bounded from L 2 (dη1 ) to L 2 (dξ1 ). Thus, the kernel δ(|ξ1 |2 − |η1 |2 ) |ξ1 −η 1|

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8. Error Terms e B [ A, V ]e−B We proceed to check the operator e B [A, V ]e−B . The calculations of this section are similar to those of the preceding section with the notable exception of (60)–(63). Recall the calculations of Lemma 1 and write B −B e [A, V ]e = v(x − y) φ(y)e B ax∗ e−B e B ax a y e−B (56) +φ(y)e B ax∗ a ∗y e−B e B ax e−B d x d y. Now fix x0 . We start with the term (56). According to Theorem 2, we have B ∗ −B e ax0 e = sh(k)(x, x0 )ax + ch(k)(x, x0 )ax∗ d x, while e B ax0 a y0 e−B has been computed in (40). The relevant terms are and shb(k)(x, x0 )chb(k)(y0 , y)N x y d x d y shb(k)(x, x0 )shb(k)(y0 , y)Q ∗x y d x d y. Combining these two terms, there are three non-zero terms (which will act on Ω): 1.

v(x0 − y0 )φ(y0 )shb(k)(x1 , x0 )shb(k)(x2 , x0 ) ×shb(k)(y0 , y2 )ax1 Q ∗x2 y2 Ωd x1 d x2 dy2 d x0 dy0 .

(57)

This term contributes terms of the form ψ(t, y2 ) = v(x0 − y0 )φ(t, y0 )(shb(k)(t, x1 , x0 ))2 shb(k)(t, y0 , y2 )d x1 d x0 dy0 (58) as well as the term ψ(t, x2 ) =

v(x0 − y0 )φ(t, y0 )shb(k)(t, x1 , x0 )shb(k)(t, x2 , x0 ) ×shb(k)(t, y0 , x1 )d x1 d x0 dy0 ,

(59)

which we know how to estimate. The second contribution is: 2.

v(x0 − y0 )φ(y0 )chb(k)(x1 , x0 )shb(k)(x2 , x0 )Ω ×chb(k)(y0 , y2 )ax∗1 N x2 y2 d x1 d x2 dy2 d x0 dy0 .

(60)

ax∗1

with ax2 , we find that (60) contributes ψ(t, y2 ) = v(x0 − y0 )φ(t, y0 )chb(k)(t, x1 , x0 )shb(k)(t, x1 , x0 )

Commuting

×chb(k)(t, y0 , y2 )d x1 d x0 dy0 .

(61)

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We expand chb(k)(t, x1 , x0 ) = δ(x1 − x0 ) + p(k)(t, x1 − x0 ). The contributions of p are similar to previous terms, but δ(x1 − x0 ) presents a new type of term, which will be addressed in Lemma 3. These contributions are ψδp (t, y2 ) = v(x1 − y0 )φ(t, y0 )shb(k)(t, x1 , x1 )p(k)(t, y0 , y2 ) d x1 dy0 (62) and

ψδδ (t, y2 ) = φ(t, y2 )

v(x1 − y2 )shb(k)(t, x1 , x1 )d x1 .

(63)

The last contribution of (56) is: 3.

v(x0 − y0 )φ(y0 )chb(k)(x1 , x0 )shb(k)(x2 , x0 )shb(k)(y0 , y2 )ax∗1 Q ∗x2 y2 Ω ×d x1 d x2 dy2 d x0 dy0 ∼ ψ(x1 , x2 , y2 ),

where

ψ(t, x1 , x2 , y2 ) = v(x0 − y0 )φ(t, y0 ) ×chb(k)(t, x1 , x0 )shb(k)(t, x2 , x0 )shb(k)(t, y0 , y2 )d x0 dy0 ,

modulo normalization and symmetrization. This term, as well as all the terms in (57), are similar to previous ones and are omitted. We can now state the following proposition: Proposition 3. The state e B [A, V ]e−B Ω has entries in the first and third slot of a Fock space vector of the form given above. In addition, if ∂ i − ∆x − ∆ y sh(k) L 1 [0,T ]L 2 (d xd y) ≤ C1 , ∂t ∂ i − ∆x + ∆ y p L 1 [0,T ]L 2 (d xd y) ≤ C1 ∂t and shb(k)(t, x, x) L 2 ([0,T ]L 2 (d x)) ≤ C3 , and v(x) =

χ (x) |x|

(64)

for χ a C0∞ cut-off function, then

T 0

e B [A, V ]e−B Ω2F ≤ C,

where C only depends on C1 , C2 , C3 . Proof. The proof is similar to that of Proposition 2, the only exception being the terms (62), (63). It is only for the purpose of handling these terms that the Coulomb potential has to be truncated, since the convolution of the Coulomb potential with the L 2 function shb(k)(x, x) does not make sense. If v is truncated to be in L 1 (d x), then we estimate the convolution in L 2 (d x), and take φ ∈ L ∞ (dydt).

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To apply this proposition, we need the following lemma. 1 1

Lemma 3. Let u ∈ X 2 , 2 + . Then, u(t, x, x) L 2 (dt d x) ≤ Cu

1 1

X 2,2+

.

Proof. As it is well known, it suffices to prove the result for u satisfying ∂ i − ∆x − ∆ y u(t, x, y) = 0 ∂t 1

with initial conditions u(0, x, y) = u 0 (x, y) ∈ H 2 . This can be proved as a “Morawetz estimate”, see [14], or as a space-time estimate as in [19]. Following the second approach, the space-time Fourier transform of u (evaluated at 2ξ rather than ξ for neatness) is u 0 (ξ − η, ξ + η)dη

u (τ, 2ξ ) = c δ(τ − |ξ − η|2 − |ξ + η|2 ) δ(τ − |ξ − η|2 − |ξ + η|2 ) F(ξ − η, ξ + η)dη, =c (|ξ − η| + |ξ + η|)1/2 where F(ξ − η, ξ + η) = (|ξ − η| + |ξ + η|)1/2 u 0 (ξ − η, ξ + η). By Plancherel’s theorem, it suffices to show that u L 2 (dτ dξ ) ≤ CF L 2 (dξ dη) . This, in turn, follows from the pointwise estimate (Cauchy-Schwartz with measures) δ(τ − |ξ − η|2 − |ξ + η|2 ) dη | u (τ, 2ξ )|2 ≤ c |ξ − η| + |ξ + η| × δ(τ − |ξ − η|2 − |ξ + η|2 )|F(ξ − η, ξ + η)|2 dη, and the remark that

δ(τ − |ξ − η|2 − |ξ + η|2 ) dη ≤ C. |ξ − η| + |ξ + η|

9. The Trace

d(t, x, x)d x

This section addresses the control of traces involved in derivations. Recall that d(t, x, y) = ish(k)t + sh(k)g T + gsh(k) sh(k) − (ich(k)t + [g, ch(k)]) ch(k) −sh(k)mch(k) − ch(k)msh(k). Notice that if k1 (x, y) ∈ L 2 (d x d y) and k2 (x, y) ∈ L 2 (d x d y) then |k1 k2 |(x, x)d x ≤ |k1 (x, y)||k2 (y, x)|d y d x ≤ k1 L 2 k2 L 2 .

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Recall from Theorem 5 that if v(x) = |x| or v(x) = χ (x) |x| then ish(k)t + sh(k)g T + gsh(k), ich(k)t + [g, ch(k)] and sh(k) are in L ∞ ([0, 1])L 2 (d xd y). This allows us to control all traces except the contribution of δ(x − y) to the second term. But, in fact, we have ich(k)t + [g, ch(k)] = ikt − ∆x k − ∆ y k k − k ikt − ∆x k − ∆ y k + · · · ,

which has bounded trace, uniformly in [0, 1]. Acknowledgements. The first two authors thank William Goldman and John Millson for discussions related to the Lie algebra of the symplectic group, and Sergiu Klainerman for the interest shown for this work. The third author is grateful to Tai Tsun Wu for useful discussions on the physics of the Boson system. The third author’s research was partially supported by the NSF-MRSEC grant DMR-0520471 at the University of Maryland, and by the Maryland NanoCenter.

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23. Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvanson, J.: The Mathematics of the Bose Gas and its Condensation. Birkhaüser Verlag, Basel, 2005 24. Lieb, E.H., Seiringer, R., Yngvanson, J.: Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev. A 61, 043602 (2006) 25. Lieb, E.H., Seiringer, R., Yngvason, J.: A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Commun. Math. Phys. 224, 17–31 (2001) 26. Margetis, D.: Studies in Classical Electromagnetic Radiation and Bose-Einstein Condensation. Ph.D. thesis, Harvard University, 1999 27. Margetis, D.: Solvable model for pair excitation in trapped Boson gas at zero temperature. J. Phys. A: Math. Theor. 41, 235004 (2008); Corrigendum. J. Phys. A: Math. Theor. 41, 459801 (2008) 28. Pitaevskii, L.P.: Vortex lines in an imperfect Bose gas. Soviet Phys. JETP 13, 451–454 (1961) 29. Pitaevskii, L.P., Stringari, S.: Bose-Einstein Condensation. Oxford: Oxford University Press, 2003 30. Riesz, F., Nagy, B.: Functional analysis. New York: Frederick Ungar Publishing, 1955 31. Rodnianski, I., Schlein, B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291(2), 31–61 (2009) 32. Sjölin, P.: Regularity of solutions to the Schrödinger equation. Duke Math. J. 55, 699–715 (1987) 33. Vega, L.: Schrödinger equations: Pointwise convergence to the initial data. Proc. AMS 102, 874–878 (1988) 34. Wu, T.T.: Some nonequilibrium properties of a Bose system of hard spheres at extremely low temperatures. J. Math. Phys. 2, 105–123 (1961) 35. Wu, T.T.: Bose-Einstein condensation in an external potential at zero temperature: General theory. Phys. Rev. A 58, 1465–1474 (1998) Communicated by H.-T. Yau

Commun. Math. Phys. 294, 303–342 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0952-8

Communications in

Mathematical Physics

Entropic Bounds on Semiclassical Measures for Quantized One-Dimensional Maps Boris Gutkin Fachbereich Physik, Universität Duisburg-Essen, Lotharstrasse 1, 47048 Duisburg, Germany. E-mail: [email protected] Received: 15 April 2008 / Accepted: 16 September 2009 Published online: 24 November 2009 – © Springer-Verlag 2009

Abstract: Quantum ergodicity asserts that almost all infinite sequences of eigenstates of quantized ergodic Hamiltonian systems are equidistributed in phase space. This, however, does not prohibit existence of exceptional sequences which might converge to different (non-Liouville) classical invariant measures. It has been recently shown by N. Anantharaman and S. Nonnenmacher in [20,21] (with H. Koch) that for Anosov geodesic flows the metric entropy of any semiclassical measure µ must satisfy a certain bound. This remarkable result seems to be optimal for manifolds of constant negative curvature, but not in the general case, where it might become even trivial if the (negative) curvature of the Riemannian manifold varies a lot. It has been conjectured by the same authors, that in fact, a stronger bound (valid in the general case) should hold. In the present work we consider such entropic bounds using the model of quantized piecewise linear one-dimensional maps. For a certain class of maps with non-uniform expansion rates we prove the Anantharaman-Nonnenmacher conjecture. Furthermore, for these maps we are able to construct some explicit sequences of eigenstates which saturate the bound. This demonstrates that the conjectured bound is actually optimal in that case. 1. Introduction The theory of quantum chaos deals with quantum systems whose classical limit is chaotic. It is assumed in general that chaotic dynamics induce certain characteristic patterns. For instance, the Random Matrix conjecture predicts that statistical distribution of high-lying eigenvalues in a chaotic system is the same as in certain ensembles of random matrices and depends only on symmetries of the system [1]. In the same spirit, it is believed that eigenstates of chaotic systems are delocalized over the entire available part of the phase space [2,3] which is totally different from the case of quasi-integrable systems, where eigenstates are known to concentrate near KAM tori [4]. The rigorous implementation of that idea is known as Quantum Ergodicity Theorem. It was first proven by

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B. Gutkin

A. I. Schnirelman for Laplacians on surfaces of negative curvature [5] and later generalized [6,7] and extended to other systems e.g., ergodic billiards [8,9], quantized maps [10] and general Hamiltonians [11]. Very generally, the Quantum Ergodicity Theorem states that for a classically ergodic system “almost all” eigenstates ψk become uniformly distributed over the phase space in the semiclassical limit k → ∞. To give a more precise meaning of this statement it is convenient to use the notion of measure. Given some Hamiltonian system, let ψk , k = 1, 2, . . . be a sequence of normalized eigenstates of the corresponding quantum Hamiltonian operator. With any ψk one can associate the distribution µk , such that µk ( f ) = ψk Opk ( f ), ψk ,

f ∈ Cc∞ (X ),

where f (q, p) is a classical observable on the phase space X of the system and Opk ( f ) is the corresponding quantum observable. Here Opk ( · ) is a quantization procedure set at the scale k which in turn is fixed by the state ψk under consideration. As k → ∞, k → 0 and we are looking for possible semiclassical limits of the measure µk . Although the exact form of µk depends on the quantization procedure Opk ( · ) (e.g., Weyl, AntiWick quantization, etc.), a weak limit of µk as k → ∞, in the distributional topology does not depend on the choice of the quantization. Any such limit µ (called semiclassical measure) is a probability measure which is, in addition, invariant under the classical flow. The Quantum Ergodicity theorem asserts that for “almost all” sequences of eigenstates the corresponding semiclassical measure is actually the Liouville measure. Since the Quantum Ergodicity theorem does not exclude the possibility that exceptional sequences of eigenstates produce non-Liouville classically invariant measures, it makes sense to ask whether such measures might actually appear. In the context of Anosov geodesic flows on surfaces of negative curvature it was conjectured [12] that a typical system possesses Quantum Unique Ergodicity property, meaning that all sequences of eigenstates converge to the Liouville measure. However, so far there have been only a limited number of rigorous results supporting this conjecture. One of the most significant results in that direction was obtained by E. Lindenstrauss in [13], where he proved that all Hecke eigenstates of the Laplacian on some compact arithmetic surfaces are equidistributed. If (as widely believed) all the Laplacian eigenstates are non-degenerate, this result would amount to the proof of Quantum Unique Ergodicity for the arithmetic case. On the other hand, it is known that exceptional sequences actually do appear in some quantum systems. For instance, as has been recently shown by A. Hassell [14], Quantum Unique Ergodicity fails for “almost all” stadia billiards. Another example is provided by quantized hyperbolic symplectomorphisms of two-dimensional torus (known as “cat maps” in physics literature) [15,16]. Here the semiclassical measures induced by exceptional sequences of eigenstates can be, for instance, composed of two ergodic components: µ = aµL + (1 − a)µD ,

1 ≥ a ≥ 1/2,

(1)

where the first part µL is the Liouville measure equidistributed over the entire phase space and the second part µD is the Dirac peak concentrated on a single unstable periodic orbit. Similar sequences of eigenstates have been also constructed for the “Walsh quantization” of the baker’s map [17]. For quantized hyperbolic symplectomorphisms of higher-dimensional tori there exists a different type of semiclassical measures which are Lebesgue measures on some invariant co-isotropic subspaces of the torus [18]. As we know that non-Liouville semiclassical measures do appear (at least) in some systems, it would be of great interest to understand which kind might exist in the general

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case. Quite recently, it has been proven by N. Anantharaman and S. Nonnenmacher [19– 21] (with H. Koch) that for the Laplacian on a compact Riemannian manifold X with Anosov geodesic flow on the unit cotangent bundle S ∗ X the metric (Kolmogorov-Sinai) entropy HKS (µ) of any semiclassical measure µ on S ∗ X must satisfy a certain bound. Particularly, in the two-dimensional case the following result holds [21]: 1 HKS (µ) ≥ | log J u (x)|dµ − λmax , (2) 2 S∗ X where J u (x) is the unstable Jacobian of the flow at the point x ∈ S ∗ X and λmax is the maximum expansion rate of the flow. If the maximum expansion rate is close to its average value, this remarkable bound gives valuable information on µ itself. In particular, for surfaces with a constant negative curvature it implies that the fraction of any semiclassical measure concentrated on periodic orbits cannot exceed “half” of the total measure of the phase space. On the other hand, if the expansion rate varies a lot, the above bound might provide little information, as the right hand side of (2) can even become negative. Thus, it is natural to expect that (2) is not an optimal result, and a stronger bound might exist in a general case. Such a bound has been conjectured in [17,19]. The conjecture states that for an Anosov canonical map (resp. Hamiltonian flow) on a compact symplectic manifold (resp. on a compact energy shell) any semiclassical measure must satisfy: 1 HKS (µ) ≥ (3) | log J u (x)|dµ. 2 Assuming that the bound is true, it provides certain restriction on the class of possible semiclassical measures in the general case. Suppose, for instance, that a semiclassical measure takes the form (1). (Note that for Anosov geodesic flows we don’t know whether such semiclassical measures actually exist.) Then the bound (3) implies that the LiouD ville part should be always present and its proportion satisfy a ≥ λavλ+λ , where λav is D the average Lyapunov exponent (with respect to the Liouville measure) and λD is the Lyapunov exponent for the periodic orbit where µD is localized. 2. Model and Statement of the Main Results The central purpose of this paper is to provide support for the conjectured bound (3) using the model of quantized one-dimensional piecewise linear maps. A procedure for quantization of one-dimensional linear maps was originally introduced in [22] in order to generate families of quantum graphs with some special properties. Being much simpler on the technical level, these models still exhibit characteristic properties of typical quantum chaotic Hamiltonian systems. Most importantly, it turns out that the quantum evolution here follows the classical evolution till the (Ehrenfest) time which grows logarithmically with the dimension N of the “quantum” Hilbert space.1 Note that in such a model N is always finite and N −1 plays the role of the Planck’s constant. As will be shown in the body of the paper, the construction of quantized one-dimensional maps is also closely related to the Walsh quantized baker’s maps in [17]. We consider a class of piecewise linear maps T : [0, 1) → [0, 1) =: I which preserve the Lebesgue measure on I . More specifically, let {I j , j = 1, . . . l} be a partition of 1 As we deal in the present paper with a discrete time evolution, the term “time” stands here and after for the number of iterations of either classical or quantum maps.

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the unit interval I = ∪lj=1 I j into l subintervals I j = β− (I j ), β+ (I j ) , j = 1, . . . l, where β− (I j ) and β+ (I j ) denote the left and right endpoints of I j respectively. At each subinterval I j , T is then defined as a simple linear map T : I j → I : T (x) = j (x − β− (I j )),

j = 1/|I j |,

for x ∈ I j , j = 1, . . . l,

(4)

with |I j | standing for the length of I j . Furthermore, we will assume that the slopes j are positive integers satisfying l

−1 j = 1,

j > 1, j = 1, . . . l.

j=1

These conditions guarantee that the map is both Lebesgue-measure preserving and chaotic. Note that each such map T is uniquely determined by the ordered set of its slopes = {1 , . . . l }, so the notation T will be sometimes used to define the corresponding map. We will now briefly describe the procedure introduced by P. Pako´nski et al [22] for quantization of such maps. Let M = {E i , i = 1, . . . N } be a partition of I¯ = [0, 1] into N intervals E i = [(i − 1)/N , i/N ], i = 1, . . . N of equal lengths. For each interval E i we will denote by β+ (E i ) (β− (E i )) the right (resp. left) endpoint of E i and by N β (E ) the set of all endpoints of the partition M. Obviously both M β(M) = ∪i=1 ± i and β(M) are uniquely determined by the size N of the partition. In what follows we will consider an increasingly refined sequence of the above partitions Mk whose sizes Nk , k = 1, . . . ∞ grow exponentially. Conditions 1. Given a map T of the form (4) we impose the following conditions on the sequence of Mk : • Each partition Mk is a refinement of the previous one. That means for each k ≥ 1, Nk+1 /Nk is an integer number greater than one. • The set of the endpoints of the initial partition M1 must include all singular points of T , i.e., β(M1 ) ⊇ β(Ii ) for all i = 1, . . . l. For a sequence of partitions Mk , k = 1, . . . ∞ satisfying Conditions 1 consider the corresponding sequence of the transfer (Frobenius-Perron) operators given by Nk × Nk doubly stochastic matrices Bk , whose elements read as: −1 |E i ∩ T −1 E j | Ei if E i ∩ T −1 E j = ∅ Bk (i, j) = = (5) 0 otherwise, |E i | where Ei is the slope of T at the partition element E i . We will call a piecewise linear map T quantizable if there exists a sequence of partitions Mk , k = 1, . . . ∞ such that for each matrix Bk one can find a unitary matrix Uk of the same dimension satisfying Bk ( j, i) = |Uk (i, j)|2

(6)

for each matrix element ( j, i); j, i ∈ {1, . . . Nk }.2 For quantizable maps the matrices Uk are regarded as “quantizations” of Bk and play the role of quantum evolution operators acting on the Nk -dimensional Hilbert space Hk C Nk equipped with the Nk ¯ ψ(i)ϕ(i) for vectors ψ = (ψ(1), . . . ψ(Nk )), standard scalar product: ψ, ϕ = i=1 ϕ = (ϕ(1), . . . ϕ(Nk )). As an example, consider the following linear map (see Fig. 1): 2 Note that our definition for U matrix corresponds to the adjoint of the corresponding quantum evolution in [22,23].

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1

307

1

1

1/2

0

1

0

1/2

1

0

1/2

1

0

1

Fig. 1. Linear maps with uniform (left) and non-uniform slopes (middle) which allow “tensorial” quantization. On the right is shown the asymmetric Baker map corresponding to the linear map in the middle

T (x) = 2x

mod 1,

x ∈ I.

(7)

Here for the sequence of partitions Mk of the unit interval into Nk = 2k equal pieces, the matrix elements Bk (i, j) of the classical transfer operators take the values 1/2 if j = 2i, j = 2i − 1, j + Nk = 2i, j + Nk = 2i − 1 and 0 otherwise: ⎞ ⎛ 1 1 0 0 1 1 1 1 ⎜0 0 1 1⎟ ··· . , B4 = ⎝ B2 = ⎠, 2 1 1 2 1 1 0 0 0 0 1 1 It is worth mentioning that the structure of Bk , actually, resembles the structure of the map T (rotated clockwise by π/2). It is also easy to see that the map (7) is quantizable. By a permutation of rows Bk can be brought into the block diagonal form such that every block is 2×2 matrix B2 whose elements are all 1/2. Thus the question of the quantization of T reduces to finding a unitary 2 × 2 matrix U satisfying |U(l, m)|2 = 1/2 for all its elements. An appropriate choice for U is provided, for instance, by the two-dimensional matrix F2 , where F p is the p-dimensional discrete Fourier transform: 2πi 1 (l − 1)(m − 1) , l, m = 1, . . . p. (8) F p (l, m) = √ exp − p p This construction can be straightforwardly generalized to all other maps with uniform slopes. For a general piecewise linear map it becomes, however, a non-trivial problem to determine whether the corresponding doubly stochastic matrices Bk allow the representation (6) in terms of unitary matrices Uk (see [22,24] and references there). Nevertheless, we show in the Appendix to the paper that the class of quantizable piecewise linear maps is quite wide and contains many maps with non-uniform slopes. Notice that the above quantization of one-dimensional piecewise linear maps is just a formal procedure for generation of unitary matrices Uk . To turn it to a “meaningful” quantization one needs, in addition, to make a connection between classical observables on the unit interval and the corresponding quantum observables on the Hilbert space Hk . Such a quantization procedure has been introduced in [23]. With a classical observable f ∈ L 2 [0, 1] one associates the sequence of the quantum observables Opk ( f ), defined by the diagonal matrices of the dimension Nk whose components Opk ( f ) j, j equal the average value of f at j’s element of the partition Mk . The key observation making the above quantization interesting is the existence of the semiclassical correspondence (Egorov property) between evolutions of classical and quantum observables. Precisely, for a Lipschitz continuous observable f (x) one has [23]:

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Uk∗ Opk ( f )Uk − Opk ( f ◦ T ) = O

1 Nk

.

(9)

Recall that the size Nk of the partition plays here the role of the inverse Planck constant −1 k and the semiclassical limit corresponds to k → ∞. Equipped with the above quantization procedure we can define now the sequence of the semiclassical measures associated with the eigenstates of Uk . For any normalized eigenstate ψk = (ψ(1), . . . ψ(Nk )) ∈ Hk , Uk ψk = eiθk ψk , let µk be the associated probability measure, whose density is given by the piecewise constant function ρk (x) = Nk |ψ(i)|2 , for x ∈ E i , i = 1, . . . Nk . By this definition, the quantum average of any observable f ∈ L 2 [0, 1] can be written as: ψk Opk ( f )ψk = f (x) ρk (x)d x ≡ f (x) dµk . (10) I

I

We will be concerned with the possible weak limits of µk as k → ∞ and call any such limiting measure µ a semiclassical measure. Speaking informally µ characterizes the possible sets of the localization on I for the eigenstates of quantized maps. (Alternatively (see [23]), one can think that µ determines the limiting eigenstates distribution on the sequence of quantum graphs provided by Uk ’s.) An immediate consequence of the Egorov property is that any semiclassical measure µ must be invariant under the map T . Indeed, since ψk is an eigenstate of Uk : 1 , (11) f (x) dµk (x) = ψk Uk∗ Opk ( f )Uk ψk = f (T (x)) dµk (x) + O N k I I and the invariance of µ follows immediately after taking the limit k → ∞. As there exist many classical measures preserved by T , the invariance alone does not determine all possible outcomes for the semiclassical measures. Similarly to the case of Hamiltonian systems, using the Egorov property one can show by standard methods (see e.g., [26]) that almost any sequence of the eigenstates gives rise to the Lebesgue measure in the semiclassical limit (this was proved in [23] by a somewhat different method). Theorem 1 (Quantum Ergodicity [23, Thm. 2]). Let T be a quantizable piecewise linear map of the form (4) and let Uk , k = 1, . . . ∞ be a sequence of its quantizations (i) with eigenstates ψk , i = 1, . . . Nk . Then for each k there exists a subsequence of Nk (i N )

eigenstates: k := {ψk(i1 ) , . . . ψk k } such that limk→∞ Nk /Nk and for any sequence of eigenstates ψk j ∈ k j , j = 1, . . . ∞ and any Lipschitz continuous function f one has: lim ψk j Opk j ( f )ψk j = f (x) d x. (12) j→∞

I

In the present paper we go beyond the Quantum Ergodicity and ask about the possible exceptional semiclassical measures. In what follows we will restrict our treatment to a special subclass of the piecewise linear maps (4): Definition 1. Let I = ∪li=1 I j be a partition of I into a number of subintervals. We define T p as a piecewise linear map (4) whose precise form at each subinterval is given by: T p (x) = j x

mod 1,

for x ∈ I j , j = 1, 2 . . . l.

(13)

Furthermore, the slopes j = p n j , j = 1, 2 . . . l are given by positive powers n j ∈ N of an integer p > 1, satisfying condition: lj=1 p −n j = 1.

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Using the method developed in [17,20,21] we can easily prove for these maps an analog of the bound (2): Theorem 2 Let Uk , k = 1, . . . ∞ be a sequence of quantizations of T p and let µ be a semiclassical measure induced by a subsequence ψk , k = 1, . . . ∞ of the eigenstates of Uk ’s. Then, the following bound holds for the metric entropy of µ:

1 1 log (x) dµ(x) − log max = µ(I j ) log j − log max , 2 2 l

HKS (T p , µ) ≥ I

j=1

(14) where max := max1≤ j≤l j and µ(I j ) are the measures of the intervals I j . As the right hand side of (14) can in principle be negative, one might suspect that this bound is not optimal for the maps T p with non-uniform slopes. The main result of the present paper is a proof of a stronger bound on the metric entropy of semiclassical measures. Namely, in the body of the paper we show that the maps T p allow a special type of “tensorial” quantizations. For the maps T p quantized in that way we prove the precise analog of the Anantharaman-Nonnenmacher conjecture (3). Theorem 3 Let T p be a map as in Definition 1 and let Uk , k = 1, . . . ∞ be a sequence of “tensorial” quantization of T p . Then for any sequence of eigenstates ψk of Uk , k = 1, . . . ∞ the corresponding semiclassical measure µ satisfies: 1 µ(I j ) log j . 2 l

HKS (T p , µ) ≥

(15)

j=1

Furthermore, for certain “tensorial” quantizations of T p it becomes possible to construct explicit subsequences of eigenstates of Uk . Using these eigenstates we obtain a set of semiclassical measures which can be subsequently analyzed to test (15). It turns out that some of these semiclassical measures, in fact, saturate the bound implying that the result is sharp. Remark 1. In what follows we will distinguish the class of “tensorial” quantizations (which will be introduced in the next section) from general quantizations of one dimensional maps which only need to satisfy the condition (6). Some of the results in the paper are specific for “tensorial” quantizations and it will be always explicitly mentioned that we deal with such class of quantization whenever it is relevant. Let us also notice that the results of the present paper can in fact be proven for a more general class of one dimensional linear maps. For instance, the bound (14) can be proven for all quantizable one dimensional maps, while the main result (15) can be straightforwardly extended to all maps (with general quantizations) whose slopes are given by integer powers of an integer number [33]. However, the restriction of the discussion to the class of maps T p provided by Definition 1 and “tensorial” quantizations, allows to represent the main ideas and results in a particularly transparent way, reducing the technical details to the minimum.

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The paper is organized as follows. In Sect. 3 we introduce a special class of “tensorial” quantizations for the maps T p and demonstrate their connection with the Walsh quantized Baker maps. In Sect. 4 we review the construction in [23] for quantization of observables and prove the Egorov property up to the Ehrenfest time. In Sect. 5 we connect metric entropy of semiclassical measures with a certain type of quantum entropy functions. Based on the method of [20] we then prove Theorem 2 in Sect. 6 using the Entropic Uncertainty Principle. In the next two sections we deal with “tensorial” quantizations of T p . Section 7 is devoted to the proof of Theorem 3. In Sect. 8 we explicitly construct a certain class of semiclassical measures for the maps T p and test the bound (15). The concluding remarks are presented in Sect. 9. 3. “Tensorial” Quantizations of One-Dimensional Piecewise Linear Maps We will consider now in detail the quantization procedure for the piecewise linear maps T p given by Definition 1. As we show below, these maps allow a special type of “tensorial” quantizations mimicking the action of the corresponding classical shift map. This quantization procedure is basically restricted to the class of maps T p . For the sake of completeness, we also consider in Appendix A a different approach suitable for quantization of a more general class of maps (4). Maps with uniform slopes. We will first consider the piecewise linear maps T¯ p := T{ p,... p} with the uniform slope j ≡ p ∈ N i.e, the maps: T¯ p (x) = px mod 1,

x ∈ I.

(16)

(Here and after we will use the bar symbol to distinguish the above uniform maps from non-uniform ones.) For any point x ∈ I it will be convenient to use a p-base numeral system: x = 0.x1 x2 x3 . . ., xi ∈ {0, . . . , p − 1} to represent x. Obviously, each point is then encoded by an infinite sequence (not necessarily unique) of symbols x1 , x2 , x3 . . .. With such representation for the points of I the action of T¯ p becomes equivalent to the shift map: T¯ p : x1 x2 x3 x4 · · · → x2 x3 x4 x5 . . . .

(17)

In the following we will use the symbol x = x1 x2 x3 . . . xm for both finite and infinite sequences with the notation |x| := m reserved for the length of the sequence. So for x with |x| = ∞ the symbol x will stand for the corresponding point x = 0.x in the interval I . For a sequence x, with finite |x| = m we will use notation x to denote the corresponding cylinder set. A point x belongs to x if the first m digits of x after the point coincide with x1 , x2 , . . . xm . For any map T¯ p , there exists a sequence of natural Markov partitions Mk into Nk = p k cylinder sets of the length k: {E x = x, |x| = k}. The corresponding transfer operator is then given by the matrix Bk , whose elements: −1 , i = 1, . . . k − 1 if xi = xi+1 p (18) Bk (x, x ) = 0 otherwise, give the transition probabilities for reaching E x , x = x1 x2 . . . xk starting from E x , x = x1 x2 . . . xk after one step of the classical evolution. These matrices can be now

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“quantized” as follows. Let H C p , be a vector space of the dimension p with a scalar product ·, · and an orthonormal basis {| j, j ∈ {0 . . . p − 1}}. Take U to be a unitary transformation on H such that in the basis above: |Ui, j |2 = 1/ p,

Ui, j := i|U| j.

(19)

(One possible choice for the matrix Ui, j is provided by the p-dimensional discrete Fourier transform.) With each partition Mk we now associate Nk -dimensional Hilbert space: Hk = H ⊗ H ⊗ · · · ⊗ H . k

Using the orthonormal basis in Hk given by the vectors: |x := |x1 ⊗ |x2 ⊗ · · · ⊗ |xk ,

x = x1 . . . xk , xi ∈ {0 . . . p − 1},

one defines the unitary transformation U¯ k as: U¯ k |x = |x2 ⊗ |x3 ⊗ · · · ⊗ |xk ⊗ U|x1 ,

(20)

and the corresponding adjoint: U¯ k∗ |x = U∗ |xk ⊗ |x1 ⊗ |x2 ⊗ · · · ⊗ |xk−1 .

(21)

The action of U¯ k basically mimics the action of the shift map. From this and property (19) of the U matrix it follows immediately that U¯ k satisfies (6) and therefore, indeed, a quantization of Bk . Note that if U∗ is given by the discrete Fourier transform F p , the matrix U¯ k coincides with the evolution operator of the Walsh-quantized Baker map in [17]. In that case U¯ k4 = 1 and the spectrum of U¯ k is highly degenerate. Note also that the U matrix in the definition (20) of U¯ k should not necessarily be constant. More general construction is obtained if one takes U in the form U(x) = exp(iφ(x))U (x2 , x3 . . . xk ), where φ(x) is a real function of x and U (x2 , x3 . . . xk ) is a unitary matrix depending on x2 , x3 . . . xk and satisfying (19). Maps with non-uniform slopes. Let us consider now maps T p given by Definition 1 such that not all n j are equal. For a given p we will use exactly the same representation x = x1 x2 x3 x4 . . ., xi ∈ {0, 1 . . . p − 1} for the point x = 0.x, and the same set of the partitions Mk , k ≥ n max := max j∈{1,...l} n j as for the map T¯ p with the uniform expansion rate. The action of T p in that representation is given by the shift map, where the size of the shift depends on the point itself: T p : x1 x2 x3 x4 · · · → xn j +1 xn j +2 xn j +3 . . . ,

if 0.x ∈ I j , j = 1, 2 . . . l.

(22)

The corresponding classical evolution matrix for the partition Mk is then given by −n p i if x ⊆ Ii and xj = xn i + j , j = 1, . . . k − n i (23) Bk (x, x ) = 0 otherwise.

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Remark 2. It should be emphasized that for the uniform map T¯ p the above p-base encoding (for the points on the interval I ) provides a standard symbolic representation of the dynamical system defined with respect to the initial Markov partition lj=1 I j . That means the sequence x = x1 x2 x3 . . .; xi ∈ {0, . . . p − 1} describes the future history of the point 0.x regarding that partition under the action of T¯ p . On the other hand, for maps T p with non-uniform slopes the above representation possesses no such a dynamical significance. In particular, T p does not act as a simple shift map on the sequences x. Later we will also use the dynamical symbolic representation for the points on I which is defined with respect to the initial Markov partition lj=1 I j . In that case the points on I are represented by sequences of symbols ε = ε0 ε1 ε2 . . .; εi ∈ {1, . . . l} which encode their future histories under the action of T p . In that representation T p acts as the standard shift map on ε. To distinguish between two different representation systems we will always use x and ε letters to denote the corresponding symbols. Note that with this notation ε0 ε1 . . . εn and x1 x2 . . . xn refer, in general, to different types of sets. While the first set is a cylinder defined with respect to the action of the map T p , the second set is a cylinder for the corresponding uniform map T¯ p (and its own Markov partition). Now it is not difficult to “quantize” the matrices (23) using exactly the same Hilbert space as in the uniform case. For each state |x, x = x1 . . . xk such that x ⊆ I j , define the action of Uk on |x by: Uk |x = |xn j +1 ⊗ · · · ⊗ |xk ⊗ Un j |xn j ⊗ Un j −1 |xn j −1 ⊗ · · · ⊗ U1 |x1 , (24) where all the matrices Ui , i = 1, . . . n j satisfy (19). Note that the unitarity of Uk requires the flip in the order of the shifted p-bits on the right hand side of (24). It follows straightforwardly from the definition that Uk is indeed unitary and fulfills (6), thereby it provides a “quantization” of Bk . As for the maps with uniform slopes, the matrices Ui , i = 1, . . . n max should not necessarily be constant and can, for instance, depend on xn max +1 , xn max +2 . . . xk , as well. Example. As an example of the above quantization construction consider the map T2 = T{2,4,4} (see Fig. 1) which will be a principle model for us in what follows. Explicitly, for x = x1 x2 x3 . . ., xi ∈ {0, 1} the action of T2 on x = 0.x is given by 2x mod 1 if 0 ≤ x < 1/2 T2 (x) = (25) 4x mod 1 if 1/2 ≤ x < 1. For the vector space Hk = H ⊗ · · · ⊗ H (k times), H C2 the corresponding quantum evolution acts on |x ∈ Hk as: |x2 ⊗ |x3 ⊗ · · · ⊗ |xk ⊗ U1 |x1 if x1 = 0 (26) Uk |x = |x3 ⊗ · · · ⊗ U2 |x2 ⊗ U1 |x1 if x1 = 1, where U1,2 are constant matrices whose elements satisfy: |U1,2 (i, j)| =

√

1/2.

Connection with Walsh quantized Baker maps. Although one-dimensional maps are not Hamiltonian systems, there exists a close connection between their quantizations and Walsh quantized Baker maps. This is clear for the maps with uniform expansion rates, since in that case Uk defined by Eq. (20) with U∗ = F p , gives precisely the evolution operator of the Walsh quantized standard Baker map. Let us show now that in a similar way some natural quantizations of non-uniformly expanding maps provide

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the time evolution operator for à la Walsh quantized asymmetric Baker maps. For the sake of concreteness consider the map (25). The corresponding asymmetric Baker map T2 : (q, p) → (q, p), q, p ∈ [0, 1) (see Fig. 1) is then defined by: if q ∈ [0, 1/2) (T2 (q), p + 21 ) (27) T2 (q, p) = (T2 (q), p + 41 4q) if q ∈ [1/2, 1). Using the binary representation q = 0.x1 x2 . . ., p = 0.x−1 x−2 . . ., xi ∈ {0, 1} for the space and momentum coordinates of the phase space one can assign to the point x = (q, p) the bi-sequence x = . . . x−2 x−1 · x1 x2 . . .. In this representation the action of the map T2 on the point x takes a simple form . . . x−2 x−1 x1 · x2 x3 . . . if x1 = 0 T2 (. . . x−2 x−1 · x1 x2 . . . ) = (28) . . . x−1 x2 x1 · x3 x4 . . . if x1 = 1. To quantize such a map one can apply the same procedure as in [17]. With each cylinder (rectangle) set x− · x+ m , x− = x−(k−m) . . . x−2 x−1 , x+ = x1 x2 . . . xm , m ∈ {0, . . . k} in the phase space one associates the following coherent state in Hk : |x− · x+ m := |x1 ⊗ |x2 ⊗ · · · ⊗ |xm ⊗ F2∗ |x−(k−m) ⊗ · · · ⊗ F2∗ |x−2 ⊗ F2∗ |x−1 . Note that each coherent state |x− · x+ m is strictly localized in the rectangle x− · x+ m . One then uses these states in order to quantize observables by means of the Anti-Wick quantization procedure (see [17] for details). It is a simple observation that the operator Uk defined by Eq. (26) with U1 = U2 = F2∗ acts on the coherent states in accordance with the action of the classical map (28): Uk |x− · x+ m = |T2 (x− · x+ )m− ,

Uk∗ |x− · x+ m = |T2−1 (x− · x+ )m+ , m ∈ {1, . . . k − 2},

where equals either 1 or 2. Thereby, the operator Uk plays the role of the quantum evolution operator for à la Walsh quantized map T2 . In an analogous way the quantization (24) of the map T p with Ui = F p∗ , i = 1, . . . n max supplies the quantum evolution operator for à la Walsh quantized asymmetric Baker map T p whose expansion rates are given by the powers of p. As both quantizations provide the same set of eigenfunctions, one can utilize the results of the present paper in order to deduce the bound (3) for semiclassical measures of à la Walsh quantized T p . 4. Quantization of Observables We recall now the procedure for the quantization of observables introduced in [23]. Let Mk be the partition of the unit interval I¯ into Nk intervals {E i = [(i − 1)Nk−1 , i Nk−1 ], i = 1, . . . Nk } and let Hk C Nk denote the corresponding Hilbert space. For each function f ∈ L 2 ( I¯) the corresponding quantum observable Op( f ) is given by the matrix, whose elements are Op( f )i, j := δi, j Nk f (x) d x, i, j = 1, . . . Nk . (29) Ei

Let Ic be the circle corresponding to I = [0, 1), where the endpoints 0 and 1 are identified. It will be assumed that Ic is equipped with the standard Euclidean metric

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coming from R. In particular, the distance d(x, y) between two points x, y ∈ Ic is defined by d(x, y) := min{|x − y|, |x − y − 1|}. Below we will deal with a class of observables f ∈ Lip(Ic ) which are Lipschitz continuous on Ic . Recall that the space Lip(Ic ) is equipped with the Lipschitz norm: f Lip = sup | f (x)| + sup x∈I

x= y∈I

| f (x) − f (y)| d(x, y)

(30)

and f ∈ Lip(Ic ) iff f Lip is finite. The definition (29) is strongly motivated by the existence of the correspondence between classical and quantum evolutions of observables (Egorov property). In the context of quantized one-dimensional maps the Egorov property was proved in [23, Thm. 3] for Lipschitz continuous observables undergoing one step evolution. The following theorem is a straightforward extension of that result to larger times. Theorem 4 Let U = Uk be a quantum evolution operator (satisfying (6)) for a quantizable one-dimensional map T of the form (4) and let f be a Lipschitz continuous function on Ic , then U −n Op( f )U n − Op( f ◦ T n ) ≤ D(T ) f Lip

nmax , Nk

(31)

where D(T ) is a constant independent of n and Nk . Proof. For n = 1 the following bound was proved in [23]: U −1 Op( f )U − Op( f ◦ T ) ≤ f Lip

D(T ) . Nk

(32)

From this one immediately gets for n iterations: U −n Op( f )U n − Op( f ◦ T n ) n U −i Op( f ◦ T n−i )U i − U 1−i Op( f ◦ T n−i+1 )U i−1 ≤ i=1 n D(T ) n ≤ f ◦ T i−1 Lip ≤ D(T ) f Lip max , Nk Nk

(33)

i=1

where we used the fact that f ◦ T i ∈ Lip(Ic ) and f ◦ T i Lip ≤ imax f Lip . The inequality (31) implies that the quantum evolution follows the classical one up to the time: n E := log Nk / log max which plays the role of the Ehrenfest time for the model. (Here and after y denotes the largest integer smaller than y.) It is worth noticing that for a certain class of observables the Egorov property turns out to be exact. Let x1 , x2 be two points on the lattice β(Mk ), then with an interval X = [x1 , x2 ] ⊂ I we can associate the projection operator PX := Op(χ X ), where χ X is the characteristic function on the set X . For such operators one has the following result.

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

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Proposition 1. Let X ⊂ I be an interval (or union of intervals) such that all the endpoints β(X ) and β(T −1 X ) belong to β(Mk ), then U −1 PX U = PT −1 X . Proof. Written in matrix form the left side of (34) is given by (U j,l )∗ U j,m , (U ∗ PX U )l,m =

(34)

(35)

{ j|E j ⊆X }

where E j denotes j’s element of the partition Mk . Observe that when E j ⊆ X , the elements (U j,l )∗ = 0, (U j,m = 0) only if T (El ) ⊆ X (resp. T (E m ) ⊆ X ). On the other hand, if the last condition holds, one can extend the summation in (35) to all values of j. By the unitarity of U it gives the right side of (34). For the class of maps T p the proposition above implies the exact correspondence between classical and quantum evolutions of some projection operators up to the times of order nE . Corollary 1. Let T p , be a map of the form (13). Denote U a general quantization of T p (satisfying (6)) acting on the vector space Hk of the dimension Nk = p k . For a cylinder x of the length |x| ≤ k the evolution of the corresponding projection operator Px is given by U −n Px U n = PT p−n x

for all n + |x|/n max ≤ n E ,

(36)

where max = p n max is the maximum slope of T p . 5. Metric Entropy of Semiclassical Measures Let Uk : Hk → Hk , k = 1, · · · ∞ be a sequence of unitary quantizations of a quantizable map T of the form (4). For a given sequence of eigenstates: ψk ∈ Hk , Uk ψk = eiθk ψk , the corresponding measures µk , k = 1, · · · ∞ are defined by Eq. (10) through the Riesz representation theorem. We will be concerned with the possible outcome for semiclassical T -invariant measures µ = limk→∞ µk . Following the approach of [17,19,20] we will consider the metric entropy HKS (T, µ) of µ. Below we recall some basic properties of classical entropies and connect them to a certain type of quantum entropies. s Let π = i=1 Ii be a certain partition of I into s intervals. Given a measure µ on I the entropy function of µ with respect to the partition π is defined by h π (µ) := −

s

µ(Ii ) log(µ(Ii )).

i=1

More generally, one can consider the pressure function: pπ,v (µ) := −

s

µ(Ii ) log(vi2 µ(Ii )),

i=1

where the weights v = {vi |i = 1, . . . s} are given by a set of real numbers fixed for a given partition. Obviously, if all vi equal one, then pπ,v is just the entropy defined

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s above. An important feature of h π (µ) is its subadditivity property. If π = i=1 Ii and s τ = i=1 Ji are two partitions, then for the partition π ∨ τ consisting of the elements Ii ∩ J j and a measure µ one has: h π ∨τ (µ) ≤ h π (µ) + h τ (µ).

(37)

Now consider dynamically generated refinements of π . Define ε = ε0 ε1 . . . εn−1 , and let it be a sequence of the elements εi ∈ {1, . . . s} of the length |ε| = n. For any n ≥ 1 set partition: π (n) = |ε|=n ε of I is the collection of the sets: ε := T −(n−1) Iεn−1 ∩ T −(n−2) Iεn−2 ∩ . . . Iε0 . Each cylinder ε has a simple meaning as the set of the points with the same “ε-future” up to n iteration. One is interested in the entropies of T -invariant measures µ with respect to the partitions π (n) : h n (µ) := h π (n) (µ) = − µ(ε) log(µ(ε)). |ε|=n

If µ is T -invariant, it follows (see e.g., [25]) by the subadditivity (37) that: h n+m (µ) ≤ h n (µ) + h m (µ).

(38)

For the entropy function this implies the existence of the limit: 1 h n (µ). n→∞ n

Hπ (T, µ) = lim

(39)

The metric (Kolmogorov-Sinai) entropy is then defined as the supremum over all finite measurable initial partitions π : HKS (T, µ) = sup Hπ (T, µ). π

In the quantum mechanical framework one needs to define a quantum entropy (preslimit. Note that a measure of sure) reproducing h n (µ) (resp. pn,v (µ)) in the semiclassical each set Ii can be written as the average µ(Ii ) = χIi (x) dµ over the classical observable χIi (x) which is the characteristic function of the set Ii . The quantum observable corresponding to χIi is then simply the projection operatorPi := PIi = Op(χIi ) on the set Ii . Now we need to “quantize” the refined partitions |ε|=n ε. The most straightforward approach would be considering quantization of the observables χε . A different scheme was suggested in [20]. Instead of taking classically refined observables χε and then quantizing them, one considers a natural quantum dynamical refinement of the initial quantum partition. We will say that a sequence of operators πˆ = {πˆ i , i = 1 . . . s} defines a quantum partition of Hk if they resolve the unity operator: 1Hk =

s

πˆ i∗ πˆ i .

i=1

For a quantum partition πˆ the entropy (resp. pressure) of a state ψ ∈ Hk is given by hˆ πˆ (ψ) := −

s i=1

πˆ i ψ2 log(πˆ i ψ2 ),

pˆ π,v ˆ (ψ) := −

s i=1

πˆ i ψ2 log(πˆ i ψ2 vi2 ).

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

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Now with each set ε of π (n) one associates the operator defined by: Pεi ( p) = U − p Pεi U p .

Pε := Pεn−1 (n − 1) . . . Pε1 (1)Pε0 (0),

(40)

As follows immediately from the definition of Pε , the sets of the operators πˆ (n) = {Pε , |ε| = n}, πˆ ∗(n) = {Pε∗ , |ε| = n} define quantum partitions of 1Hk . Note that Pε∗ and Pε differ only by the order of the components Pεi (i) and both πˆ (n) , πˆ ∗(n) correspond to the same classical partition π (n) . For an eigenfunction ψk ∈ Hk of the operator Uk let hˆ πˆ (n) (ψk ), hˆ πˆ ∗(n) (ψk ) be the corresponding entropies. After introducing the weight functions: µˆ ∗k (ε) := Pε∗ ψk 2

µˆ k (ε) := Pε ψk 2 ,

(41)

for the elements ε of the corresponding classical partition π (n) , the quantum entropies of ψk can be written with a slight abuse of notation (in principle, µˆ k , µˆ ∗k are not measures but merely positive weight functions defined only on the elements of the partitions) as the classical entropy function of µˆ k , µˆ ∗k : hˆ πˆ (n) (ψk ) = h n (µˆ k ),

h n (µˆ k ) = −

hˆ πˆ ∗(n) (ψk ) = h n (µˆ ∗k ),

h n (µˆ ∗k ) = −

µˆ k (ε) log µˆ k (ε),

|ε|=n

|ε|=n

µˆ ∗k (ε) log µˆ ∗k (ε).

(42)

Note that the weight function µˆ k satisfies an important compatibility condition: µˆ k (ε0 . . . εn−1 ) =

µˆ k (ε0 . . . εn ).

εn ∈{1...s}

As Pεi (i), i ≤ n = |ε| approximately commute with each other for a finite n by virtue of the Egorov property, the same property also holds (up to semiclassically small errors) for µˆ ∗k . Furthermore, for a finite n the Egorov property guarantees that both µˆ k (ε) and µˆ ∗k (ε) equal (up to semiclassically small errors) the measure µk (ε) induced by the eigenstate ψk . Hence in the semiclassical limit: lim h n (µˆ k ) = lim h n (µˆ ∗k ) = h n (µ),

k→∞

k→∞

(43)

where µ = limk→∞ µk is the corresponding semiclassical measure. To extract from h n (µ) the metric entropy HKS (T, µ) of the measure µ it is necessary to apply the classical limit (39). In complete analogy, the quantum pressures of ψk : pˆ πˆ (n) ,v (ψk ) = pn,v (µˆ k ),

pn,v (µˆ k ) = −

pˆ πˆ ∗(n) ,v (ψk ) = pn,v (µˆ ∗k ),

pn,v (µˆ ∗k ) = −

|ε|=n

|ε|=n

µˆ k (ε) log µˆ k (ε)vε2 , µˆ ∗k (ε) log µˆ ∗k (ε)vε2 (44)

converge in the limit k → ∞ to the classical pressure pn,v (µ) of µ.

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6. Bound on Metric Entropy The main purpose of this section is to prove the bound (14) on the possible values of HKS (T p , µ). In what follows we will closely follow the approach developed in [20,21] for Anosov geodesic flows. The main technical tool is a variant of the entropic uncertainty relation first proposed in [27,28] and later generalized and proved in [29]. Here we will make use of a particular case of the statement appearing in [20,21]. s , τˆ = Theorem 5 (Entropic Uncertainty Principle [20, Thm. 6.5]). Let πˆ = {πˆ i }i=1 s , be two partitions of the unity operator 1 on a complex Hilbert space (H, ., .) {τˆi }i=1 H s , w = {w }s be the families of the associated weights. For any and let v = {vi }i=1 i i=1 normalized ψ ∈ H and any isometry U on H the corresponding pressures satisfy:

pˆ π,v ˆ ∗j ). ˆ (ψ) + pˆ τˆ ,w (Uψ) ≥ −2 log(sup v j wk τˆk U π

(45)

j,k

Let T p be a map satisfying Definition 1 and let Uk be its quantization satisfying (6) which acts on the Hilbert space Hk . In what follows we will use Theorem 5 for the Hilbert space Hk , quantum partitions πˆ = {Pε∗ , |ε| = n}, τˆ = {Pε , |ε| = n}, defined by (40) as “quantizations” of n-times refinement π (n) of the classical partition π = li=1 Ii . For each ε-element of the partitions τˆ , πˆ the corresponding weights are then defined by n−1 1/2 vε = wε = i=0 εi , ε = ε0 . . . εn−1 , where εi is the expansion rate of the map T p at the interval Iεi , εi = 1, . . . l. Finally, the isometry U will be the unitary transformation (Uk )n and the normalized state ψ will be an eigenstate ψk of Uk . With such a choice the left side of (45) reads: µˆ k (ε) log(µˆ k (ε)vε2 ) + µˆ ∗k (ε) log(µˆ ∗k (ε)vε2 ). pn,v (µˆ k ) + pn,v (µˆ ∗k ) = − |ε|=n

Thus, in order to bound pn,v (µk ) from below we need an estimation on the right hand side of (45). This amounts to the control over the elements: Pε U n Pε = U Pεn−1 U Pεn−2 . . . U Pε0 U Pεn−1 . . . U Pε0 ,

where U = Uk . The following proposition gives the required estimation. Proposition 2. Let π = li=1 Ii be the classical partition of I and let πˆ = {Pi := Op(χ Ii ), i = 1 . . . l} be the corresponding quantum partition. Then for any sequence ε = ε0 . . . εn−1 , n > 0, the product Pε = U Pεn−1 U Pεn−2 . . . U Pε0 , satisfies the bound: 1/2

Pε ≤ Nk

n−1

−1/2 . εi

(46)

i=0

Proof. It is easy to understand the source of (46). Since the structure of U basically mimics the structure of the corresponding classical map, each time when U Pε j acts on a −1/2

vector from the Hilbert space, it “decreases” its components by the factor ε j . More precisely, for any v ∈ Hk , the absolute value of j’s component of the vector v = U Pεi v satisfies the bound |v j | ≤ (−1/2 ) εi

max

m=1,...Nk

|vm |.

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

319

Applying this inequality n times one gets for the components of the vector v (n) = Pε v: n−1 −1/2 (n) |v j | ≤ εi max |vm |, j = 1, . . . Nk . m=1,...Nk

i=0

From this the desired estimation follows immediately, since |v j | ≤ v for all j and 1/2 v (n) ≤ Nk max |v (n) j |. The entropic uncertainty principle together with Proposition 2 give now the bound on the pressure of ψk : 1/2 , (47) pn,v (µˆ k ) + pn,v (µˆ ∗k ) ≥ −2 log Nk which can be also written as h n (µˆ k ) + h n (µˆ ∗k ) ≥

1/2 . µˆ k (ε) + µˆ ∗k (ε) log vε2 − 2 log Nk

(48)

|ε|=n

n 1/2 Note that such a bound becomes nontrivial only for times n when vε = i=1 εi is 1/2 comparable with Nk . In other words, n should be of the same order as the Ehrenfest time n E . For shorter times (48) would only imply that h n (µˆ k ) + h n (µˆ ∗k ) > C0 , where C0 < 0 (which is completely redundant as h n is a positive function). It is now tempting to use the inequality (48) for n = n E to get a bound on the metric entropy. Recall, however, that in such a case the relevant partition used to define h n E is of the quantum size Nk−1 . On the other hand, the correct order of the semiclassical and classical limits in the definition of HKS (T, µ) requires a bound on the entropy function for partitions of a finite (classical) size, independent of k. Thus in order to extract useful information from (47,48) it is necessary to connect the pressure pn E ,v (µˆ k ) for the quantum time n E with the pressure pn,v (µˆ k ) for an arbitrary classical time n (independent of k). To this end it has been suggested in [17] to make use of the subadditivity of the metric entropy. More specifically, for a classical invariant measure µ the subadditivity of the entropy function implies: pn+m,v (µ) ≤ pn,v (µ) + pm,v (µ), ⇒ pm,v (µ) ≤ qpn,v (µ) + pr,v (µ), m = qn + r.

(49)

The subadditivity property (49) cannot be utilized straightforwardly, as the weights µˆ ∗k , µˆ k are not T -invariant, in general. However, as follows from the following lemma, by virtue of the Egorov property the measures µk (ε) of cylinders ε turn out to be invariant for sufficiently short times. Lemma 1. Let ε be a cylinder of the size |ε| = m ≤ n E , and let µk (ε) be its measure as defined by Eq. (10). Then µk (ε) = µˆ ∗k (ε) = µˆ k (ε), where µˆ ∗k (ε), µˆ k (ε) as in Eq. (41). Furthermore, for any integer 0 ≤ n such that n + m ≤ n E the measures remain invariant under the action of T p−n : µk (ε) = µk (T p−n ε).

(50)

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B. Gutkin

Proof. Note that any set ε of the size |ε| = m ≤ n E can be written in the x-representation (see Remark 2) as x1 x2 . . . xm , where m ≤ k. The proof of the lemma then straightforwardly follows from Corollary 1 and the fact that all Px commute with each other for |x| ≤ k. From this lemma the desired connection between the pressures for partitions of classical and quantum sizes immediately follows. Proposition 3. Let Uk , k = 1, . . . ∞ be a sequence of unitary quantizations of a map T p (satisfying Definition 1), and let {ψk } be a sequence of their eigenstates with the corresponding measures µk , as defined by eq. (10). Set the pressure for each µk to be pn,v (µk ) := − µk (ε) log µk (ε)vε2 . |ε|=n

Then for n E = qn + r , q, n, r ∈ N, 0 ≤ r < n we have: pn E ,v (µk ) ≤ qpn,v (µk ) + pr,v (µk ).

(51)

Proof. Straightforwardly follows from the subadditivity of h n and (50). Equipped with the above proposition we can prove now the bound (14) on the metric entropy for maps T p . Theorem 6 Let Uk , k = 1, . . . ∞ be a sequence of unitary quantizations of a map T p , and let {ψk } be a sequence of their eigenstates. Assuming that the corresponding limiting invariant measure µ = limk→∞ µk exists, µ must satisfy the following bound: 1 HKS (T p , µ) ≥ µ(I j ) log j − log max . (52) 2 j

Proof. From the bound (48) and Proposition 3 it follows that the pressure for the partition of an arbitrary fixed size 0 < n < n E satisfies the inequality: pn,v (µk ) 1 pr,v (µk ) r pn,v (µk ) ≥ − log max − − . n 2 nE n nE

(53)

Because r , pr,v are bounded for a fixed n, the last two terms in the right hand side of (53) vanish when k → ∞ and one gets: pn,v (µ) 1 ≥ − log max . n 2 To complete the proof it remains to notice that n µ(ε) log εi , pn,v (µ) = h n (µ) − |ε|=n

and

µ(ε) log

|ε|=n

since µ is an invariant measure.

n i=1

i=1

εi

(54)

=

j

µ(I j ) log j ,

(55)

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

321

7. Proof of the Anantharaman-Nonnenmacher Conjecture for T p Maps As we have shown in the previous section, the method of N. Anantharaman and S. Nonnenmacher can be employed for the proof of the bound (14). However, exactly as for Anosov geodesics flows, such an approach does not allow to prove a stronger result (15). Very roughly, the reason for this can be explained in the following way. For a generic map the entropy function h n (µk ) is a “non-homogeneous” quantity which contains contributions from the cylinders ε with different “expansion rates” ε . The domain of validity for subadditivity of the entropy function is determined by an entry (cylinder) with the largest expansion rate and thus, restricted to the times n ≤ n E . On the other max , hand, the bound (47) becomes informative for times n ≥ n, ¯ where n¯ = n E log 2 log l log = i=1 µ(Ii ) log i . When the expansion rate is highly non-uniform one is unable to match long “quantum” times n > n¯ with short “classical” times n < n E , see Fig. 2. This results in the bound (14) which is, in general, weaker than (15) for maps with non-uniform slopes. Below we formulate a certain modification of the original strategy to overcome the problem.

7.1. General idea. Speaking informally, the basic idea here is to “homogenize” the original system, making it uniformly expanding first and only then apply the method used in the previous section. More specifically, we consider the class of maps T = T p , given by Definition 1. In what follows we adopt the tower construction widely used in the theory of dynamical systems (see e.g., [30]). As we show in the next subsection, T can be regarded as the first return map for a certain uniformly expanding dynamical system. Namely, the action of T on I turns out to be equivalent to the action of the : so-called tower map T I → I on a subset (“zero level”) of the tower phase space I. By a standard construction for first return maps, any invariant measure µ for T induces . The corresponding metric entropies HKS (T , a measure µ on I invariant under T µ), HKS (T, µ) are then related to each other by Abramov’s formula and the entropic bound (15) turns out to be equivalent to: 1 , µ) ≥ log p. HKS (T 2

(56)

Thus, in order to prove the conjecture of S. Nonnenmacher and N. Anantharaman for maps T p one needs to show (56) for the measure µ. It turns out that a pure classical construction above can be “lifted” to the quantum level. Recall that µ is a semiclassical measure generated by eigenstates of a sequence {Uk } of unitary quantizations of T . The crucial observation is that µ is actually a semik } of quantizations of T . In Subsect. 7.3 we show classical measure for a sequence {U that for each sequence {ψk } of the eigenstates of {Uk } generating in the semiclassical k } generating the limit the measure µ there exists a sequence { k } of eigenstates of {U measure µ. This is schematically depicted by the following diagram: µψk k→∞

µ = µ ◦ T −1

Quantum

+3 µ k k→∞

Classical3+ −1 µ= µ◦T

(57)

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B. Gutkin

n

classical

quantum

nE classical

I1

n

I0 [0]

quantum

[1]

nE Fig. 2. On the left down (up) the case when (14) provides non-trivial (resp. trivial) bound on the metric entropy HKS (T, µ) is shown. On the right is depicted the tower for the map T{2,4,4}

is a map with a uniform expansion rate one can apply the method used in Since T the previous section in order to prove (56). From this the metric bound (15) follows immediately. As we would like to keep the exposition and notation below as simple as possible, we will first consider in detail the map T2 = T{2,4,4} defined in (25). The results can then be straightforwardly extended to all other maps T p . 7.2. Classical towers. In what follows we construct the tower dynamical system corresponding to the map T := T{2,4,4} as defined by Eq. (25). To this end let us double the original phase space and consider the set I := I × {0, 1}. We will refer to the sets I0 := I × {0} = {(x, 0), x ∈ I }, I1 := I × {1} = {(x, 1), x ∈ I } as zero and first levels of the tower I = I0 ∪ I1 respectively. Let D0 := [0, 1/2), D1 := [1/2, 1). The tower : map T I → I is then defined by: ¯ (x, η) = (T (x), 0) if η = 0, x ∈ D0 or η = 1 and any x, (58) T (T¯ (x), 1) if η = 0, x ∈ D1 where T¯ := T{2,2} is the uniformly expanding map corresponding to T . Consider now I on the set I on the first return map T I0 . It is straightforward to see that the action of T 0 0 I0 ∼ = I coincides with the action of T on I . In other words, T can be regarded as the first return map for the lowest level of the tower (see Fig. 2). I ) one can construct (using Given an invariant measure µ for T (equivalently for T 0 a standard procedure, see e.g., [25,31]) the probability measure µ which is invariant . Precisely, for a set A ⊆ I one defines the measures of the sets under the tower map T (A × {0}), (A × {1}) by µ(A × {0}) = −1 µ(A),

µ(A × {1}) = −1 µ(T¯ −1 A ∩ [1/2, 1]),

with the normalization constant = 1+µ([1/2, 1]). If A = x is a cylinder set (defined with respect to the action of T¯ ) this can be rewritten as: µ(x × {0}) = −1 µ(x),

µ(x × {1}) = −1 µ(1x).

(59)

it makes sense to consider the corresponding metric entropy Since µ is invariant under T , , HKS (T µ). An important observation is that HKS (T µ) is related to HKS (T, µ). As T is −1 the first return map for I0 , and µ( I0 ) = , by Abramov’s formula (see e.g., [25]): , µ). HKS (T, µ) = HKS (T

(60)

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

323

Having an invariant measure µ on I it is possible in turn to construct a measure µ¯ on I which is invariant under the homogeneous map T¯ . Let π I : I → I be a natural projection on the tower: π I (x, η) = x, for all x ∈ I , η ∈ {0, 1}. As = T¯ ◦ π I , πI ◦ T it follows immediately that the measure µ¯ := µ ◦ π ∗I

(61)

is invariant under T¯ . Furthermore, the metric entropy of µ¯ turns out to be equal to the metric entropy of µ: , HKS (T¯ , µ) ¯ = HKS (T µ). (62) This equality can be deduced, from a version of the Abramov-Rokhlin relative entropy formula in [32]. For the sake of completeness we give a simple proof of (62) in Appendix B. The above construction allows a straightforward extension to the case of an arbitrary map T p satisfying Definition 1. The tower phase space here is defined as lt = n max (n max = max j=1,...l {n j }) copies of I : I = I × {0, 1, . . . , lt − 1} ∼ =

l t −1

Ij,

(63)

j=0

p : I → I where the set I j = I × { j} stands for j’s level of the tower. The tower map T is then defined with the help of the uniformly expanding map T¯ p given by Eq. (16). For each level η ∈ {0, 1, . . . , lt − 1} define the set: Ij. Dη := { j|n j =η+1}

p is given by: Then the action of the map T ¯ if x ∈ T¯ η Dη p (x, η) = (T p (x), 0) T (T¯ p (x), η + 1) if x ∈ T¯ η Dη .

(64)

p a point x ∈ Dη × {0} climbs η Such a definition implies that under the action of T steps upstairs in the tower phase space, then it “jumps” downstairs to zero level and the process is repeated. It is now straightforward to see that the map T p coincides with the first return map p for zero level of T I0 of the tower. As a result, starting from an invariant measure µ p . For a set µ for the tower map T for T p one can easily construct the invariant measure A × {η} ⊆ I , with A ⊆ I and level η ∈ {0, . . . lt − 1} the corresponding measure is given by µ(A × {η}) = −1

{k|n k ≥η}

µ(T¯ p−η (A) ∩ Ik ),

=

l

n j µ(I j ),

(65)

j=1

where is the average return time to zero level of the tower. Precisely as for the map T{2,4,4} , one can also construct the measure µ¯ invariant under the action of T¯ p . The corresponding metric entropies are then related by: p , HKS (T p , µ) = HKS (T¯ p , µ) ¯ = HKS (T µ). (66)

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7.3. Quantum towers. We are going now to consider the quantum analog of the above tower construction. Construction. Let U = Uk be a tensorial quantization of the map T = T{2,4,4} , acting on the Hilbert space Hk of the dimension 2k = dim(Hk ). We will assume that U is of the form (26). In that case U allows an obvious decomposition: U = U¯ P0 + V¯ U¯ P1 ,

V¯ = σ U¯ .

(67)

Here U¯ : Hk → Hk , U¯ : Hk → Hk stand for tensorial quantization (20) of the uniformly expanding map T¯ = T{2,2} with U given by U1 and U2 respectively, and σ stands for the unitary transformation reversing the order of the last two symbols in |x1 ⊗ · · · ⊗ |xk−1 ⊗ |xk ∈ Hk : σ |x1 ⊗ · · · ⊗ |xk−1 ⊗ |xk = |x1 ⊗ · · · ⊗ |xk ⊗ |xk−1 . In addition to P0 , P1 it will be also convenient to use the projection operator: P1 = U¯ P1 U¯ ∗ .

(68)

Explicitly its action on the basis states of Hk is given by: P1 |x1 ⊗ · · · ⊗ |xk−1 ⊗ |xk = |x1 ⊗ · · · ⊗ |xk−1 ⊗ U1 P1 U1∗ |xk , where P1 |i = δi,1 |i, i = 0, 1. It worth to notice that P1 commutes with V¯ : V¯ P1 = P1 V¯ .

(69)

=H 0 ⊕ H 1 , dim(H) = 2k + 2k−1 with We define now the “tower” Hilbert space H 0 := Hk , and H 1 := U¯ P1 Hk ≡ P Hk , H 1

(70)

0 , H 1 with the corresponding to zero and the first levels of the tower. As we identify H 0 (resp. φ ∈ H 1 ) can also be regarded space (resp. the subspace of) Hk , any vector φ ∈ H as a vector from Hk . By this identification, operations defined on Hk can be “lifted” to 0 , H 1 , as well. In particular, we can define the scalar product on H the Hilbert spaces H using the scalar product on Hk . Namely for = (φ0 , φ1 ) ∈ H, = (φ0 , φ1 ) ∈ H, 0 and φ1 , φ ∈ H 1 : with φ0 , φ0 ∈ H 1 ( , ) = φ0 , φ0 + φ1 , φ1 . can be easily constructed from an orthonormal basis in Hk . An orthonormal basis in H A convenient choice is provided by the vectors: E(x,0) := (|x1 ⊗ · · · ⊗ |xk−1 ⊗ |xk , 0), x = x1 . . . xk−1 xk , E(x,1) := (0, |x1 ⊗ · · · ⊗ |xk−1 ⊗ |1 ),

x = x1 . . . xk−2 xk−1 ,

(71)

where |0 := U1 |0, |1 := U1 |1 and xi , i = 1, . . . k (resp. i = 1, . . . k − 1) run over all possible sequences of {0, 1}.

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

325

In what follows we will consider a one-parameter family of tower evolution operators →H defined in the following way. For any = (φ0 , φ1 ) ∈ H, with φ0 ∈ H 0 θ : H U and φ1 ∈ H1 : θ := (V¯ P φ1 + U¯ P0 φ0 , eiθ U¯ P1 φ0 ). U 1

(72)

→H is given by ∗ : H Correspondingly, the adjoint operation U θ θ∗ = (e−iθ P1 U¯ ∗ φ1 + P0 U¯ ∗ φ0 , P V¯ ∗ φ0 ). U 1

(73)

and U ∗ ∈ H θ is a unitary θ , U Main properties. It is straightforward to see that U θ operation on H: ∗ be as above, then θ , U Proposition 4. Let U θ θ U θ∗ U θ∗ = U θ = 1. U ∗ U Proof. In the block matrix representation the product U θ θ takes the form P0 + P1 P0 U¯ ∗ V¯ P1 P0 U¯ ∗ e−iθ P1 U¯ ∗ U¯ P0 V¯ P1 = . P1 V¯ ∗ 0 P V¯ ∗ U¯ P0 P eiθ U¯ P1 0 1

1

(74) Using Eqs. (68, 69) it is now straightforward to see that the off-diagonal terms in the right side of (74) vanish. θ . Specifically the short Below we demonstrate that the Egorov property holds for U time evolution of projection operators is prescribed by the classical evolution of the corresponding tower map. Proposition 5. Let x ⊆ I be a cylinder of the length m = |x| < k − 1, then: θ = (P0 P ¯ −1 ⊕ P P ¯ −1 ), θ∗ (Px ⊕ 0) U U T x 1 T x ∗ Uθ (0 ⊕ P1 Px ) Uθ = (P1 PT¯ −1 x ⊕ 0). Proof. In the matrix representation the left side of (75) reads: ¯ P0 U¯ ∗ e−iθ P1 U¯ ∗ U P0 V¯ P1 Px 0 . P1 V¯ ∗ 0 0 0 eiθ U¯ P1 0

(75) (76)

(77)

By Eqs. (68, 69) the off diagonal terms of the above product are zeros, and the diagonal part: P0 U¯ ∗ Px U¯ P0 0 0 P1 V¯ ∗ Px V¯ P1 coincides with the right side of Eq. (75), as follows from Corollary 1 and the equality: σ Px σ = Px . Equation (76) is then proved analogously.

326

B. Gutkin

Let X be a subset of I and let PX be the corresponding projection operator. The projection operator corresponding to the subset X × {0} ∪ X × {1} of the tower, is then defined as: X = PX ⊕ P PX . P 1 The following corollary follows immediately from Proposition 5. Corollary 2. For a given cylinder x ⊂ I of the length |x| = m and an integer n such ¯ −n be the projection operator corresponding to the subset that n + m < k − 1, let P T x −n −n ¯ ¯ T x × {0} ∪ T x × {1} of the tower. Then: ¯ −n U θ∗ P U T x θ = PT¯ −n−1 x .

(78)

Eigenfunctions and semiclassical measures. Given an eigenfunction ψ of the original θ = eiθ evolution operator U , U ψ = eiθ ψ we can construct the eigenfunction , U of the tower evolution operator. Precisely, −1 = ψ, U¯ P1 ψ ψ 2 ,

ψ = 1 + ψ, P1 ψ,

(79)

θ . Indeed, this is so, since by (68), is the normalized eigenstate of the operator U 1 1 θ = (U¯ P0 + U¯ 1 P U¯ P1 )ψ, eiθ U¯ P1 ψ − 2 = U ψ, eiθ U¯ P1 ψ − 2 , U ψ ψ 1 and ψ is the eigenstate of U . It is also easy to verify that ( , )) = 1. For any sequence of quantizations {Uk } of T and their eigenstates {ψk }, let us conθk } of quantizations of the tower map T determined sider the corresponding sequence {U by Eq. (72). (Note that these quantizations depend on the eigenvalues eiθk of ψk ’s.) θk } can be constructed applying Eq. (79). As a result, Then the eigenstates { k } of {U a sequence of semiclassical measures µk on I induces the sequence of semiclassical measures µk on I . For a cylinder x ⊂ I , |x| ≤ k the measures µk of the tower sets x × {0}, x × {1} are defined as: µk (x × {0}) = ( k , (Px ⊕ 0) k) ,

µk (x × {1}) = ( k , (0 ⊕ P1 Px ) k) .

By Eq. (79) these measures are related to the measure µk of the set x: µk (x × {0}) = k−1 µk (x),

µk (x × {1}) = k−1 µk (1x),

(80)

where we set k = ψk . Note that after taking the limit k → ∞ in (80) one obtains Eqs. (59), where µ = limk→∞ µk is precisely the measure of the classical tower obtained from the semiclassical measure µ = limk→∞ µk by the procedure from the previous section. Also, defining the measure µ¯ k on I by x k) = −1 µk (x) + µk (1x) , µ¯ k (x) := ( k , P (81) k one reveals in the semiclassical limit the measure µ¯ = limk→∞ µ¯ k related to µ by Eq. (61).

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

327

Extension to other maps T p . The above construction of quantum towers can be straightforwardly extended to other maps T p . Let U be a quantization of T p acting on the Hilbert space Hk , as defined by Eq. (24). Note that U can be cast in the form: U=

l t −1

U¯ (0) = 1, U¯ (η) := U¯ η U¯ η−1 . . . U¯ 1 for η > 0,

ση U¯ (η+1) PDη ,

η=0

where U¯ i , i = 1, . . . lt are quantizations of the uniformly expanding map T¯ p , σ j is the operator which reverse the order of the last j elements in the products |x1 ⊗· · ·⊗|xk−1 , lt is the height of the corresponding classical tower and PDη is the projection operator corresponding to the set Dη as defined in Sect. 7.2. Mimicking the construction of the classical towers one defines then the tower Hilbert space as the direct sum: = H

lt −1

η , H

η=0

η ∼ H = Pη + P¯η Hk ,

with the projection operators Pη , P¯η defined by Pη := U¯ (η) PDη (U¯ (η) )∗ ,

P¯η := U¯ (η) P¯Dη (U¯ (η) )∗ ,

P¯Dη :=

PD j .

j>η

θ whose action on the states = φ0 , φ1 , . . . Now take the tower evolution operator U is given by: φlt −1 ∈ H ⎛

l t −1

θ = ⎝ U

η=0

⎞ ση U¯ η+1 Pη φη , eiθ U¯ 1 P¯0 φ0 , eiθ U¯ 2 P¯1 φ1 , . . . , eiθ U¯ lt P¯lt −1 φlt −1 ⎠ . (82)

θ follow from It can be easily checked that the unitarity and the Egorov property for U the same properties for the operator U . Finally, if U ψ = eiθ ψ then = ψ, U¯ (1) P¯D0 ψ, U¯ (2) P¯D1 ψ, . . . , U¯ (lt −1) P¯Dlt −2 ψ / ψ

(83)

θ with the eigenvalue eiθ and the normalization constant given by is the eigenstate of U ψ = 1 +

l t −2 j=0

ψ, P¯D j ψ =

l

ψ, PI j ψn j .

j=1

This shows that any semiclassical measure µ = limk→∞ µk on I induced by a sequence of eigenstates {ψk } of Uk , generates through Eq. (83) the semiclassical measures: µ= µk on I and µ¯ = limk→∞ µk ◦ π ∗I on I which are invariant under the action limk→∞ p and T¯ p , respectively. Note also that the corresponding metric entropies are related of T to each other by Eq. (66).

328

B. Gutkin

7.4. Proof of Theorem 3. Let us now prove the bound (15) for the map T{2,4,4} . Theorem 7 Let {Uk }∞ k=1 be a sequence of tensorial quantizations of T = T{2,4,4} . For ∞ a sequence {ψk }k=1 of eigenstates Uk ψk = eiθk ψk let µ = limk→∞ µk be the corresponding semiclassical measure, then HKS (T, µ) ≥

µ(0) + 2µ(1) log 2. 2

(84)

Proof. To prove (84) it is possible, in principle, to follow precisely the scheme described , µ) for the corresponding in the beginning of the section i.e., to prove the bound on HKS (T semiclassical measure µ on the tower and then deduce the bound (84) using Abramov’s formula. From the technical point of view, however, it turns out to be easier to prove an equivalent bound for the metric entropy HKS (T¯ , µ) ¯ of the measure µ. ¯ Let { k }∞ be the sequence of the tower eigenstates corresponding to the sequence k=1 ˆ of ψk ’s, and let h n ( k ) ≡ h n (µ¯ k ) be the entropy function for the corresponding measures µ¯ k : x k 2 log( P x k 2 ). (85) h n (µ¯ k ) = − µ¯ k (x) log µ¯ k (x) = − P |x|=n

|x|=n

Then the metric entropy HKS (T¯ , µ) ¯ is obtained after first applying the semiclassical limit: h n (µ) ¯ = lim h n (µ¯ k ),

(86)

1 HKS (T¯ , µ) ¯ = lim h n (µ). ¯ n→∞ n

(87)

k→∞

and then the classical limit:

¯ corresponds to the “probing” of towers with parAs can be seen from Eq. (85), HKS (T¯ , µ) x ). This explains titions made of “vertical rectangles” (represented by the projections P , µ). ¯ and HKS (T ˜ To prove the source of the equality between metric entropies HKS (T¯ , µ) the bound on HKS (T¯ , µ) ¯ we will make use of the same scheme as in [17]. The first step is to get the bound on the entropy function, when n is of the same order as k. This is provided by the following proposition. Proposition 6. Let h n (µ¯ k ) be as in (85) and set n = k − 1, then k−1 − 1 log 2. h k−1 (µ¯ k ) ≥ 2

(88)

Proof. We will use the Uncertainty Entropic principle (Theorem 5) for the partitions: y , |y| = k − 1}, weights: vy = wy ≡ 1 and the isometry operation π = τ = {P k −1 θk ) . Since k is an eigenstate of U θk it follows immediately from (45): U = (U h k−1 ( k ) ≥ − log(

sup

|y|=|y |=k−1

y (U θk )k−1 P y ). P

(89)

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

329

y (U θk )k−1 P y . To Thus one needs to estimate the norm of the operator C(y , y) = P this end let us calculate the matrix elements of C(y , y): ( E(x ,i ) , C(y , y)E(x,i)) , in the basis of orthogonal states (71) with the parameters: i, i ∈ {0, 1}, |x| = k, (|x | = k) if i = 0 (resp. i = 0) and |x| = k − 1, (|x | = k − 1) if i = 1 (resp. i = 1). The action of the projection operator on the basis states is given by ⎛ y E(x,i) = E(x,i) ⎝ P

k −1

⎞ δxm ,ym ⎠ .

(90)

m=1

Hence for each pair of y, y there exist at most two values of x and two values of x such that the matrix elements ( E(x ,i ) , C(y , y)E(x,i)) are not zeros. From that follows: C(y , y) ≤ 2

max

(x,i),(x ,i )

=2

max

|((E(x ,i ) , C(y , y)E(x,i)) |

(x,i),(x ,i )

θk )k−1 E(x,i)) |. |((E(x ,i ) , (U

(91)

θk )k−1 in the basis of Therefore, it remains to estimate the elements of the operator (U θk on {E(x,i) } up to times k closely {E(x,i) }. To this end, let us notice that the action of U on the sets x × {i} of connected to the action of the corresponding tower map T I. k −1 Specifically, let E = (Uθk ) E(x,i) . Then, as follows from Eq. (72), depending on x, i the state E takes the values (e, 0) or (0, e), where e is of the form: e=

ei Qθk |xk ⊗ Ui1 |xi1 ⊗ Ui2 |xi2 ⊗ · · · ⊗ Uik−1 |xik−1 if i = 0, ei Qθk Ui1 |xi1 ⊗ |1 ⊗ Ui2 |xi2 ⊗ · · · ⊗ Uik−1 |xik−1 if i = 1.

(92)

Here Uim is either U1 or U2 , xi1 , xi2 . . . xik−1 is some permutation of the original sequence √ x1 , x2 . . . xk−1 and Q is an integer number. Since |x j , U1,2 xi | = 1/ 2 for any pair xi , x j ∈ {0, 1}, θk ) |((E(x ,i ) , (U

k−1

E(x,i)) | = |((E(x ,i ) , E))| ≤ 2

− k−1 2

.

Together with (89) and (91) this gives the proof of the proposition. The second necessary step is to connect values h k−1 (µ¯ k ) of the entropy at quantum times of order k to its values h n (µ¯ k ) at short fixed classical times n. Proposition 7. Let h n (µ¯ k ) be as in (85), and let k − 1 = qn + r , r < n, where n < k − 1 is a fixed (classical) time and q, r are integers, then 1 n log 2 1 h n (µ¯ k ) ≥ h k−1 (µ¯ k ) − . n k−1 k−1

(93)

330

B. Gutkin

Proof. The prove of (93) is analogous to the proof of Proposition 3. One makes use of the fact that the measure µ¯ k is invariant under the transformation T¯ j for “short” times. From the definition of µ¯ k and Eq. (78) it follows that for any cylinder x of length |x| = m: (94) µ¯ k (x) = µ¯ k (T¯ −n x), for m + n ≤ k − 1. Let n, q, r be as in the conditions of the proposition. Then the subadditivity property of the entropy function implies: h k−1 (µ¯ k ) ≤ qh n (µ¯ k ) + h r (µ¯ k ).

(95)

Since |h r (µ¯ k )| is bounded from above by n log 2, one immediately obtains the inequality (93). End of the proof of Theorem 7: The final step is to combine Propositions 6 and 7: 1 log 2 (n + 1) log 2 h n (µ¯ k ) ≥ − , for all n < k. n 2 k−1 Taking in (96) first limit k → ∞ and then n → ∞ gives: log 2 HKS (T¯ , µ) , ¯ ≥ 2 which by (62, 60) implies the bound: log 2 HKS (T, µ) ≥ . 2 Since = µ(0) + 2µ(1), this gives the bound (84).

(96)

(97)

Sketch of proof of Theorem 3. All the ingredients of the above construction can be straightforwardly extended from the map T{2,4,4} to a general map T p satisfying Definition 1. Specificaly, given a sequence of eigenstates Uk ψk = eiθk ψk , k = 1, . . . ∞, one k }∞ of eigenstates for quantizations {U θk }∞ of k ∈ H first constructs the sequence { k=1 k=1 the tower map T p . These sets of eigenstates induce then two sequences of related measures (defined as in Eq. (81)): {µk }∞ ¯ k }∞ k=1 , {µ k=1 on I . Assuming that in the semiclassical limit µk ’s converge to an invariant measure µ of T p , the sequence of the measures µ¯ k must converge to the measure µ¯ which remains invariant under the action of the corresponding uniformly expanding map T¯ p . Repeating the same steps as in the proof of Proposition 6 it is strightforward to get the metric bound on the measures µ¯ k : k + 1 − lt h k+1−lt (µ¯ k ) ≥ (98) + 1 − lt log p. 2 Furtheremore, the subadditivity property of h n together with Eq. (98) then imply that the limiting measure µ¯ = limk→∞ µ¯ k must satisfy the bound: log p . HKS (T¯ p , µ) ¯ ≥ 2 Since the metric entropies HKS (T¯ p , µ), ¯ HKS (T p , µ) are connected to each other by Eq. (66) one immediately gets log p HKS (T p , µ) ≥ , (99) 2 where is as in Eq. (65). Finally, it remains to check that the right side of (99) coincides with the right side of (15).

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

331

8. Explicit Sequences of “Non-ergodic” Eigenstates Below we construct some explicit sequences of eigenstates for maps T¯ p , T p quantized as in Sect. 3. Having such sequences we can calculate the induced semiclassical measures and test the bound (15) for the corresponding metric entropies. 8.1. Maps with uniform slopes. Let us first consider the map T¯ p whose quantization is given by Eq. (20). Note that if U∗ is given by the discrete Fourier transform F p , the evolution operator U¯ k and the corresponding eigenstates are precisely the same as for the Walsh-quantized baker’s map treated in [17]. As we show below, the same construction can be carried out for a general U. Similarly to the case of the Walsh-quantized baker’s map, the resulting semiclassical measures on I turn out to be Bernoulli measures. Let w ∈ H be an eigenstate of U; then it is easy to see that (w)

ψk

(w)

= w ⊗ w ⊗ · · · ⊗ w,

ψk

k

∈ Hk = H ⊗ H ⊗ · · · ⊗ H

(100)

k

is the eigenstate of U¯ k . It is now straightforward to compute the semiclassical measure p−1 (w) µw corresponding to the sequence ψk , k = 1, . . . ∞. Assuming that w = i=0 wi |i, where {|i, i = 0, . . . p−1} is an orthonormal basis in H, the µw -measure of the cylinder set x, x = x1 . . . xm , xi ∈ {0, . . . p − 1} is given by: (w)

(w)

µw (x) = lim ψk Px ψk = k→∞

m

|wxi |2 .

(101)

i=1

As this is a simple product measure (i.e., a measure µ which factorizes: µ(x) = m i=1 µ(xi ) with respect to the corresponding Markov partition), one gets for the metric entropy: HKS (T¯ p , µw ) = −

p−1

|wi |2 log(|wi |2 ).

(102)

i=0

Now let us show that for some quantizations of T¯ p there also exist semiclassical measures which are linear combinations of Bernoulli measures constructed in the previous example. Assume that U is such that there exists an eigenstate w(0) ∈ H of Ud : Ud w (0) = eiθ w (0) for some integer d > 0, while w (0) is not an eigenstate of Ui for any integer i < d. (This, for instance, is possible if Ud = 1 for some d > 1, as in the case: U = F p∗ ). Taking then such normalized state w (0) one can form the corresponding sequence w( j) := e−iθ j/d U j w (0) , j = 0, . . . d − 1 of cyclically related states: e−iθ/d Uw ( j) = w ( j+1 mod d) , with |w ( j) |w (i) | < 1 if i = j. Define w0 := w (0) ⊗ w (1) · · · ⊗ w (d−1) and let w := {w0 , w1 , . . . wd−1 } be the sequence of the states obtained from w0 by the cyclic permutation of its components: w j := w ( j

mod d)

⊗ w (1+ j

mod d)

⊗ · · · ⊗ w (d−1+ j

mod d)

,

j = 0, . . . d − 1.

For each k satisfying: k mod d = 1 one looks for normalized eigenstates of U¯ k in the form (w)

ψk

=

d−1 j=0

(k)

Cj

w j ⊗ w j ⊗ · · · ⊗ w j ⊗w j , (k−1)/d

(w)

ψk

∈ Hk .

(103)

332

B. Gutkin (k)

Equation (103) then defines an eigenstate of U¯ k if all C j are equal to some constant C (k) (w)

fixed by the normalisation condition ψk = 1. (Note that the eigenstates provided by Eq. (100) could be seen as a particular case of (103) when d = 1.) The corresponding semiclassical measure is then given by (w)

(w)

µw (x) = lim ψk Px ψk k→∞ = lim

k→∞

d−1

|C (k) |2 w j ⊗ · · · ⊗ w j Px wi ⊗ · · · ⊗ wi .

i, j=0

Since |w ( j) |w (i) | < 1 if i = j, only the diagonal terms survive the limit k → ∞ and one has: d−1 1 ( j) µw (x), µw (x) = d

( j) µw (x)

j=0

=

m

|wx( kj+k−1

| ,

mod d) 2

(104)

k=1

( j)

where wi is i’s component of the vector w( j) in the basis {|i, i = 0, . . . p − 1}. Although µw is not a simple product measure, it is still possible to calculate the metric entropy by noticing that the product measures entering in (104) are cyclically related to ( j+1 mod d) ( j) each other: µw = T¯ p∗ ◦ µw and remain invariant under the action of (T¯ p )d = T¯ pd . Since the metric entropy is an affine function of the measure, one has: 1 HKS (T¯ p , µw ) = HKS (T¯ pd , µw ) d d−1 d−1 p−1 1 1 ( j) 2 ( j) ( j) = 2 HKS (T¯ pd , µw ) = − |wi | log(|wi |2 ), d d j=0

j=0 i=0

(105) ( j)

where we have used the fact that each µw is a simple product measure and all ( j) HKS (T¯ pd , µw ) are equal to each other. Furthermore, by a simple application of the Uncertainty Entropic Principle one has for all j = 0, . . . d − 1, ( j) ˆ ( j) ) + h(Uw ˆ h(w ) ≥ −2 log(max |Uk,m |) = log p, k,m

ˆ ( j) ) = h(w

p−1

( j)

( j)

|wi |2 log(|wi |2 ).

i=0

As e−iθ j/d Uw ( j) = w ( j+1 mod d) , it follows immediately by (105) that HKS (T¯ p , µw ) ≥ 1 2 log p which is precisely the bound (14) (equivalent to (15) in that case). It is worth mentioning that for U = F p∗ and d = 1 there exist vectors w such that the measures µw saturate this bound [17]. ( j)

Remark 3. Note that if all wi = 0 the measures above are supported on the whole I . As has been shown in [17] in the case when U = F p∗ , it is also possible to construct an entirely different class of exceptional sequences of eigenstates, where parts of the

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

333

corresponding semiclassical measures are localized on some periodic trajectories. This construction uses the fact that U¯ kn = 1 for a “short” time n, meaning the spectrum of U¯ k becomes highly degenerate. Since no such degeneracies are expected for quantized maps with non-uniform slopes, it seems that this type of semiclassical measures can be constructed only for the maps T¯ p . We refer the reader to [15, 17] for the details of the construction.

8.2. Maps with non-uniform slopes. For the maps T p we will look for sequences of eigenstates exactly in the same form (103), as in the uniform case. As we will show, for certain quantizations of T p one can, indeed, construct such sequences by choosing the (k) constants Ci in an appropriate way. Rather than consider a general case, we will provide below several concrete examples of such a construction for the map T2 (see Eq. (25)) whose quantization is given by (26) with some choice for U1 , U2 . In order to calculate the metric entropy of a semiclassical measure µ we need, in general, a knowledge of the measures µ(ε0 . . . εn−1 ), where ε0 . . . εn−1 , εi ∈ {1, 2, 3} are cylinder sets defined with respect to the Markov partition j=1,2,3 I j . On the other hand, because of the eigenstates, structure, it turns out to be easier to calculate the measures µ(x1 . . . xn ) of the “binary” cylinders, where xi ∈ {0, 1} (as defined in Sect. 3). The transition between two representations is straightforward. Given some cylinder ε0 . . . εn−1 in ε-representation its binary representation can be obtained by switching every element εi , i = 0, . . . n − 1 to 0 if εi = 1, to 10 if εi = 2 and to 11 if εi = 3 respectively. (For instance the cylinder 123 in the binary representation becomes 01011.) Thus, in order to obtain the measures of the sets ε0 . . . εn−1 , it is convenient first to transform them into the binary form using the above procedure and then find their measures as in the previous subsection. Example 1. Below we construct a semiclassical measure which is totally concentrated on a part of I , where T2 has a uniform slope. Let U1 = U2 = U be a two-by-two √ matrix satisfying U2 = 1, |U(i, j)| = 1/ 2, e.g., the discrete Fourier transform. Let U|1 =: |e+ . Since U|e+ = |1 it can be easily seen that for even k, (1)

ψk = |1 ⊗ |e+ ⊗ · · · ⊗ |1 ⊗ |e+

(106)

k

(1)

is an eigenstate of Uk . For the sequence of states ψk the induced semiclassical measure µ(1) has entire support on the Cantor set: {x = 0.x1 x2 x3 · · · ∈ I, x2i+1 = 1}. Correspondingly, µ(1) (ε0 . . . εn−1 ) = 1/2n if εi ∈ {2, 3} for all i, and µ(1) (ε0 . . . εn−1 ) = 0 otherwise. The metric entropy of µ(1) can be easily calculated and it is given by: HKS (T2 , µ(1) ) = log 2. This saturates the bound (15) coinciding in that case with (14). Let us mention that since µ(1) is supported on the interval [1/2, 1] (where the slope of T2 is 4), it is also an eigenmeasure of T¯4 map, with the same metric entropy as for T2 . Example 2. Let us show now that for some quantisations of T2 one can construct sequences of eigenstates precisely in the form (100). As a result, the corresponding semiclassical measure µ(2) is given by Eq. (101). Although it remains invariant under the action of both T2 and T¯2 , the metric entropies of µ(2) with respect to T2 and T¯2 turn out to be different.

334

B. Gutkin

Consider the following quantization of T2 . Let U be an arbitrary unitary matrix whose √ elements have modules 1/ 2 and let w be one of its eigenvectors with the eigenvalue eiγ . We now fix U1 , U2 by the conditions U2 = e−iγ U, U1 = U. The state (2)

ψk = w ⊗ w ⊗ · · · ⊗ w, k

is then an eigenstate of Uk . (Note that this example allows a straightforward general(2) ization to all maps T p .) Denote µw the corresponding semiclassical measure. Unlike (2) the previous example, in general, µw is supported over all I . For a given state w = w0 |0 + w1 |1 the measures of the sets ε0 , ε0 = {1, 2, 3} are: ⎧ ⎨ p for ε0 = 1 (2) (2) pq for ε0 = 2 µ(2) w (ε0 ) = lim ψk Pε0 ψk = ⎩ q 2 for ε = 3, k→∞ 0 (2)

where we introduced notation |w0 |2 = p, |w1 |2 = q. Since µw is a simple product measure the corresponding metric entropy is given by: (2) µ(2) HKS (T2 , µ(2) w )=− w (ε0 ) log µw (ε0 ) ε0 ={1,2,3}

= −( p log p + pq log( pq) + q 2 log q 2 ). This should be compared to the bound given by the right side of (15). For the measure (2) µw this bound is equal to 21 (1 + q) log 2. Recall that w is an eigenvector of a unitary matrix whose entries have the same This restricts the possible values of q, √ modules. √ p = 1 − q to the interval [(2 − 2)/4, (2 + 2)/4]. As can be easily checked, for all values of q, p in this interval the strict inequality (15) holds. It is worth to mention (2) that µw is, in fact, an eigenmeasure of the map T¯2 , as well. The corresponding metric entropy HKS (T¯2 , µ(2) w ), however, is given by Eq. (102) and it is obviously different from (2) the expression for HKS (T2 , µw ). Example 3. Unlike two previous examples, here we construct semiclassical measures which somewhat differ from the semiclassical measures obtained for the maps with uniform slopes. Most importantly, we demonstrate that for a certain choice of parameters the bound (15) is actually saturated. To construct such semiclassical measures we are looking for normalized eigenstates of Uk in the form of two state products: (3)

(k)

(1) (2) (1) (2) ⊗ w (2) ⊗ w (1) ⊗ w ψk = C1 w ⊗ · · · ⊗ w ⊗ w

k

(2) + C2(k) w

⊗w

(1)

⊗w

(2)

(1) (2) (1) ⊗w ⊗ · · · ⊗ w ⊗ w .

k

Take U2 = U1 = U,

1 U= √ 2

1 eiα , e−iα −1

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

335

HKS 0.7 0.6 0.5 0.4

0

0.5

1

1.5

Re z 2

Fig. 3. Metric entropy (green, solid line) and the corresponding bound (blue, dashed line) (15) for the semiclassical measure from Example 3 as functions of Re(z) when Im(z) = 0 (k)

and for a given z ∈ C set C1

√ (k) = zC2 , c = 1 + |z 2 − 1|2 ,

√ w (1) = c−1/2 |0 + e−iα (z 2 − 1)|1 ,

√ w (2) = c−1/2 z|0 + e−iα ( 2 − z)|1 .

With such a choice (U)2 = 1 and Uw (1) = w (2) , Uw (2) = w (1) . It is easy to check that ψk(3) is an eigenstate of Uk for any z ∈ C. The resulting semiclassical measure (k)

(3)

µz = (C1 )2 µ1z + (C2 )2 µ2z , Ci = limk→∞ |Ci |, i = 1, 2 is a linear combination of two simple product measures µ1z , µ2z = µ1z −1 defined in Eq. (104). Note that this measure has the same structure as in the case of maps with uniform slopes. However, here the con(2) (2) (1) stants C1 , C2 are not equal, in general. Denote p1 = |w0 |2 , p2 = |w1 |2 , q1 = |w0 |2 , (1) 2 (i) (i) q2 = |w1 | , where w0 , w1 are the first and second component of the vectors w(i) , i = 1, 2. As will be shown below, the metric entropy of µ(3) z can be explicitly calculated and it is given by HKS (T2 , µ(3) z )=−

pk log pk + qk log qk , 2 k=1,2

where = 2(µ([10]) + µ([11])) + µ([0]) = 1 + (C1 )2 p1 + (C2 )2 p2 , |z| . C1 = |z|C2 = # |z|2 + 1 The plot in Fig. 3 shows the metric entropy versus the bound of (15) as functions of the real part of z when Im(z) = 0.

2

log 2 given by right side

Some special cases: 1) |z| = 1. In this case p1 = p2 , q1 = q2 and the resulting measures are of a simple product type. Furthermore, both w(1) and w (2) are the eigenvectors of the same unitary matrix whose elements have equal modulus. Thus one actually, (2) gets the measures of the same type as for one vector product states ψk in Example 2. 2) z = 0, z = ∞. In that case either C2 or C1 vanishes and we get the states considered

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√ √ in Example 1. 3) z = 2 (or z −1 = 2). In such a case p1 = q1 = 1/2, p2 = 0, q2 = 1 √ ) = 2 log 2 saturates the bound (15). and the metric entropy HKS (T2 , µ(3) 3 2 Calculation of metric entropy. Using the same approach as in the last example, one can, in principle, construct d-state product eigenstates of the type (103) for other maps T p . For such a construction it is necessary to define the quantum evolution operator Uk by Eq. (24) with Ui = U, i = 1, . . . n max , where Ud = 1 and the vectors w ( j) , j = 0 . . . d − 1 are cyclically related to each other by U (precisely as in Sect. 8.1). (k) Assume that by an appropriate choice of constants Ci one can obtain a sequence of (w) eigenstates ψk of Uk ’s in the form (103). The corresponding semiclassical measures ( j)

µw then have the form of a linear combination of the components µw from Eq. (104). Let us show now how the metric entropy of µw can be computed. First, note that ψk(w) being an eigenstate of Uk , is in addition, an eigenstate for the operator (U¯ k )d , where U¯ k is the quantization (20) of the map T¯ p with the uniform slope p. Since (U¯ k )d is also a quantization of the map T¯ pd , the semiclassical measure µw and all its components ( j) µw turn out to be invariant under T¯ pd . Furthermore, as has been shown in Sect. 7.2, using measure µw one can construct the corresponding measure µ¯ w invariant under the action of T¯ p . An important observation is that this measure has the same structure as µw . Namely, it is a linear combination of the simple product measures given by Eq. (104): µ¯ w =

d−1 j=0

( j)

α j µw ,

d−1

α j = 1, α j ∈ R.

j=0

It is now easy to calculate HKS (T¯ p , µ¯ w ) using the affineness of the metric entropy and ( j) the fact that HKS (T¯ pd , µw ) is equal to HKS (T¯ pd , µw ) for each j: d−1 1 1 1 ( j) HKS (T¯ p , µ¯ w ) = HKS (T¯ pd , µ¯ w ) = α j HKS (T¯ pd , µw ) = HKS (T¯ pd , µw ). d d d j=0

(107) Using now connection (66) between the metric entropies of µw , µ¯ w and expression (105) for HKS (T¯ pd , µw ) one obtains: d−1 p−1 ( j) 2 ( j) ¯ HKS (T p , µw ) = HKS (T pd , µw ) = − |wi | log(|wi |2 ). d d

(108)

j=0 i=0

Note that as the right side of (15) amounts to log2 p the proof of the AnantharamanNonnenmacher conjecture for the measure µw amounts to the proof of the bound log p d , HKS (T¯ pd , µw ) ≥ 2

(109)

for the uniformly expanding map. This bound has been shown already in Sect. 8.1. Note also that the bound (15) saturates if (109) saturates for the corresponding uniformly expanding map T¯ pd .

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Finally, it is instructive to see how the formula (108) can be understood in an intuitive way. Recall that each point ζ ∈ I can be encoded by sequences ε(ζ ) = ε0 ε1 . . ., εi ∈ {1, . . . l} in accordance with the dynamical (forward) “history” of ζ regarding the action of T p . Furthermore, this sequence generates the set of cylinders: G n (ζ ) := {ε0 . . . εn−1 , n = 1, 2, . . . } corresponding to the “partial histories” of the point evolution. Assuming that µw is ergodic, the Shannon-McMillan-Breiman theorem asserts that for almost every (with respect to µw ) point ζ ∈ I the metric entropy of T p is given by: 1 log µw (G n (ζ )). (110) n Speaking informally, this means that on average the measures of cylinders G n = ε0 . . . εn−1 decay exponentially as functions of n: µw (G n ) ∼ exp(−n HKS (T p , µw )) with the exponent given by HKS (T p , µw ). Analogously, one can argue that HKS (T¯ pd , µw ) determines the measures of G n = x1 . . . xm n in the x-representation: µw (G n ) ∼ exp(− mdn HKS (T¯ pd , µw )). On the other hand, by Birkhoff’s ergodic theorem the lengths of G n in both representations can be easily connected. Namely, for almost every ζ ∈ I (with respect to µw ), HKS (T p , µw ) = − lim

n→∞

lim m n /n =

n→∞

l

n i µw (Ii ) = ,

i=1

where m n , n is the length of G n (ζ ) in the x and ε-representation, respectively. Comparing the asymptotics of µw (G n ) in both representations gives Eq. (108). 9. Conclusions and Outlook In the current paper we have proved the Anantharaman-Nonnenmacher conjecture for the “tensorial” quantizations of the one-dimensional piecewise linear maps T p . It should be emphasized that we deal here with the “tensorial” quantizations mostly for the sake of convenience, as these quantizations allow very explicit treatment. Actually the current method with minimal adjustments can be used to prove the result for general quantizations of the maps T p . On the other hand, the present approach is clearly restricted to the class of maps whose slopes are all powers of an integer p > 1, since only such maps can be represented as first return maps for towers with uniform expansion rates. To prove the conjecture for a more general class of maps (or other chaotic systems) a more flexible tower construction is needed. We believe that such a modified tower construction is in fact possible and can be utilized to prove the result for all quantizable piecewise linear (chaotic) maps T , [33]. Note also that in the recent preprint [34] of G. Riviere a related approach has been used to prove the Anantharaman-Nonnenmacher conjecture for geodesics flows on two-dimensional Riemannian manifolds of non-positive curvature. The present application demonstrates that quantized one-dimensional maps (or similarly à la Walsh quantized asymmetric Baker maps) can be useful as toy models for understanding general features of quantum chaotic systems. On the technical level these models are much simpler than generic chaotic Hamiltonian systems, but still exhibit their most important features. A quite rare opportunity (for chaotic systems) to construct explicit sequences of eigenstates make them potentially useful as test systems. Another possibility is to use one dimensional maps as models for scattering systems. By opening a “gap” in the unit interval one can produce quantized one-dimensional maps with

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an “absorption” (in complete analogy with the open Walsh-Baker maps introduced in [35]). Finally, since we know already that various exceptional semiclassical measures appear for the “tensorial” quantizations of the maps T p it would be of interest to identify an opposite class of quantizations for which quantum unique ergodicity holds i.e, no exceptional sequences of eigenstates are present. Acknowledgements. I would like to thank Andreas Knauf and Christoph Schumacher for numerous fruitful discussions. I am particularly grateful to Stephane Nonnenmacher for reading the preliminary version of the manuscript and very helpful comments. Most of the present work was accomplished during my pleasant stay in Erlangen-Nuremberg University. I am grateful to all my colleagues at the Mathematical Department for the hospitality extended to me. The financial support of the Minerva Foundation and SFB/TR12 of the Deutsche Forschungsgemainschaft is acknowledged.

Appendix A: Quantization of General Maps We have shown in Sect. 3 that the maps T p can be quantized by means of the “tensorial” quantization procedure. Here we discuss how a more general class of maps T , = {1 , . . . l } can be quantized. Let i1 , . . . i , > 1 be the maximal set of different slopes in , i.e., in = im for n = m. Assuming that each slope ik has a multiplicity m k ≥ 1, the Lebesgue measure preservation condition mk = 1, ik

(111)

k=1

imposes certain restrictions on the values of ik , m k . In particular, it is clear that the set ik , k = 1, . . . must have a greatest common divisor p larger than one. This means ¯ k, ik = p

¯ k ∈ N for k ∈ {1, . . . }.

¯ i ’s are relatively prime. Then it follows immediately from (111) Assume now that all ¯ k , m¯ k ∈ N, k ∈ {1, . . . l}, where lk=1 m¯ k = p. that m k ’s are of the form m k = m¯ k We are going now to show that the maps T whose slopes satisfy the above conditions are quantizable. ¯ i , p ∈ N, ¯ i+1 ≥ ¯i Proposition 8. Let T be a map (4) with the slopes i = p ¯ i ’s are relatively prime integers, then T is having multiplicities m i , i ∈ {1, . . . l}. If “quantizable”. Proof. As the first step notice that T can be represented as a composition of the uniformly expanding map T¯ p and the “block diagonal” map TBD , whose slopes are uniform at each block. Lemma 2. Let T be a map as defined above, then T = T¯ p ◦ TBD , where T¯ p (x) = px mod 1 and mj for x ∈ [bi , bi+1 ], bi = . TBD (x) = (i x mod 1) / p + bi , j j

Proof. Straightforward calculation.

Entropic Bounds on Semiclassical Measures for Quantized 1-D Maps

1

1

339

1

= 0

1/2

1

0

1/2

1

0

1/2

1

Fig. 4. A “generic” map (112) and its decomposition into the uniformly expanding and the “block diagonal” parts

The parameters entering into the definition of TBD have the following simple meaning. m The points bi , bi+1 mark the position of i’s block which is the square of the size jj . Inside of each such block the map TBD acts as a piecewise linear map with the uniform ¯ i. expansion rate Example. To illustrate the above lemma consider as an example the map with the slopes 6 and 4: 6x mod 1 if x ∈ [0, 1/2) T (x) = (112) 4x mod 1 if x ∈ [1/2, 1). As shown in Fig. 4, it can be decomposed into the uniformly expanding map T¯2 = 2x mod 1 and the “block diagonal” map: TBD (x) =

(6x mod 1)/2 if x ∈ [0, 1/2) (4x mod 1)/2 + 1/2 if x ∈ [1/2, 1).

Let us now define a set of partitions Mk of I by setting their sizes Nk . Take N0 = ¯ i , then Nk = (N0 )k for k = 1, . . . ∞. It is clear that these partitions satisfy p li=1 Conditions 1. For each partition Mk denote by B¯ k , BkBD the corresponding evolution operators for the map T¯ p and TBD respectively. Note that both B¯ k and BkBD are quantizable i.e., one can find unitary matrices U¯ k , UkBD satisfying (6). Indeed, this is completely obvious for B¯ k as T¯ p has the uniform slope. Since BkBD is of the block diagonal form, the corresponding quantum evolution UkBD can be defined as the block diagonal matrix of the same structure where each block is quantized with the help of the discrete Fourier transform. Given matrices B¯ k , BkBD , and the quantizations U¯ k , UkBD one can easily construct the transfer operator for the composition map T = T¯ p ◦ TBD and the corresponding quantization. Lemma 3. Let T , Mk be the map and partition as above and let Bk be the corresponding evolution operator, then Bk = BkBD B¯ k and the matrix Uk = U¯ k UkBD satisfies (6). Proof. Straightforward check. From this the proof of the theorem follows immediately.

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It worth mentioning that using the above quantization procedure it is also possible to obtain the tensorial quantization of the maps T p . For instance, the map T{2,4,4} can be decomposed into the uniformly expanding T¯2 and the map TBD consisting of two “blocks” with the slope two and one, respectively. Correspondingly, the tensorial quantization (67) ¯ BD , where U¯ , U BD = P0 + U¯ ∗ U¯ 1 U¯ P1 of T{2,4,4} can be cast in the form: U = UU give the “uniform” and the “block-diagonal” components.

Appendix B: Proof of Eq. (62) Let T , T¯ be as in Sect. 7.2 and T˜ be the corresponding tower map given by (58). From the Markov partition of I : {ε0 , ε0 ∈ {1, 2}} one can easily construct the Markov partition of I˜: {ε0 × {η}, ε0 ∈ {1, 2} and η ∈ {0, 1}}. The corresponding n-times refined (with respect to T˜ ) partition is given then by the set of cylinders: {˜ε, ε˜ = ε˜ 0 . . . ε˜ n−1 }, where ε˜ i = (εi , ηi ), εi ∈ {1, 2} and ηi ∈ {0, 1}}. The metric entropy HKS (T˜ , µ) ˜ is determined by the corresponding limit of the entropy function: ˜ =− h n (µ)

µ(˜ ˜ ε) log µ(˜ ˜ ε).

(113)

|˜ε |=n

For a cylinder ˜ε let ε = π I ˜ε be the corresponding cylinder in I containing exactly the same sequence of ε as in ε˜ . Note that the time evolution of any point ζ˜ ∈ I˜ is completely determined by the sequence ε and the initial level η0 . Therefore, for a given ε there are precisely two non-empty cylinders ˜ε, ˜ε such that π I ˜ε = π I ˜ε = ε. Furthermore, µ(˜ ˜ ε) = −1 µ(ε), µ(˜ ˜ ε ) = −1 µ(1ε) and h n (µ) ˜ can be rewritten as: h n (µ) ˜ = − −1

µ(ε) log µ(ε) −1 + µ(1ε) log µ(1ε) −1 .

|ε|=n

On the other hand, the entropy of the measure µ¯ is given by h n (µ) ¯ = −

−1

µ(ε) ¯ + µ(1ε) ¯ . µ(ε) ¯ + µ(1ε) ¯ log

|ε|=n

It remains to see that two limits limn→∞ h n (µ)/n, ˜ limn→∞ h n (µ)/n ¯ coincide. By the convexity of the entropy function ¯ ≥ h n (µ) ˜ + log 2. h n (µ)

(114)

Since, log(x + y) ≥ log x one also has: ¯ ≤ h n (µ). ˜ h n (µ) From (114, 115) the claim immediately follows.

(115)

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References 1. Bohigas, O.: Random matrix theory and chaotic dynamics. In: Giannoni, M.J., Voros, A., Zinn-Justin, J., eds., Chaos et physique quantique, (École d’été des Houches, Session LII, 1989), Amsterdam: North Holland, 1991 2. Berry, M.V.: Regular and irregular semiclassical wave functions. J. Phys. A 10, 2083–2091 (1977) 3. Voros, A.: Semiclassical ergodicity of quantum eigenstates in the Wigner representation. In: Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Casati, G., Ford, J., eds., Proceedings of the Volta Memorial Conference, Como, Italy, 1977, Lecture Notes in Phys. 93, Berlin: Springer, 1979, pp. 326–333 4. Lazutkin, V.F.: Semiclassical asymptotics of eigenfunctions. In: Partial Differential Equations V, Berlin: Springer, 1999 5. Schnirelman, A.I.: Ergodic properties of eigenfunctions. Usp. Mat. Nauk 29, no. 6 (180), 181–182 (1974) 6. Zelditch, S.: Uniform distribution of the eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987) 7. Colinde Verdière, Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102, 497–502 (1985) 8. Gérard, P., Leichtnam, É.: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71(2), 559–607 (1993) 9. Zworski, M., Zelditch, S.: Ergodicity of eigenfunctions for ergodic billiards. Commun. Math. Phys. 175, 673–682 (1996) 10. Bouzouina, A., De Bièvre, S.: Equipartition of the eigenfunctions of quantized ergodic maps on the torus. Commun. Math. Phys. 178, 83–105 (1996) 11. Helffer, B., Martinez, A., Robert, D.: Ergodicité et limite semi-classique. Commun. Math. Phys. 109, 313–326 (1987) 12. Rudnick, Z., Sarnak, P.: The behavior of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161, 195–213 (1994) 13. Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. Math. 163, 165–219 (2006) 14. Hassell, A.: Ergodic billiards that are not quantum unique ergodic, with an appendix by A. Hassell, L. Hillairet. Preprint (2008) http://arxiv.org/abs/0807.0666v3[math,AP], 2008, to appear in Ann. of Math 15. Faure, F., Nonnenmacher, S., De Bièvre, S.: Scarred eigenstates for quantum cat maps of minimal periods. Commun. Math. Phys. 239, 449–492 (2003) 16. Faure, F., Nonnenmacher, S.: On the maximal scarring for quantum cat map eigenstates. Commun. Math. Phys. 245, 201–214 (2004) 17. Anantharaman, N., Nonnenmacher, S.: Entropy of semiclassical measures of the Walsh-quantized baker’s map. Ann. H. Poincaré 8, 37–74 (2007) 18. Kelmer, D.: Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus. Preprint (2005), http://arxiv.org/abs/math-ph/0510079v5, 2007, to appear in Ann. of Math. 19. Anantharaman, N.: Entropy and the localization of eigenfunctions. Ann. of Math. 168(2), 435–475 (2008) 20. Anantharaman, N., Nonnenmacher, S.: Half–delocalization of eigenfunctions of the Laplacian on an Anosov manifold. Ann. de l’Inst. Fourier 57(7), 2465–2523 (2007) 21. Anantharaman, N., Nonnenmacher, S., Koch, H.: Entropy of eigenfunctions. http://arXiv.org/abs/0704. 1564v1[math-ph], 2007 ˙ 22. Pako´nski, P., Zyczkowski, K., Ku´s, M.: Classical 1D maps, quantum graphs and ensembles of unitary matrices. J. Phys. A 34(43), 9303–9317 (2001) 23. Berkolaiko, G., Keating, J.K., Smilansky, U.: Quantum Ergodicity for Graphs Related to Interval Maps. Commun. Math. Phys. 273, 137–159 (2007) 24. Zyczkowski, K., Ku´s, M., Słomczy´nski, W., Sommers, H.-J.: Random unistochastic matrices. J. Phys. A 36(12), 3425–3450 (2003) 25. Keller, G.: Equilibrium States in Ergodic Theory. London Mathematical Society Student Texts 42 Cambridge: Cambridge University Press, 1998 26. De Bièvre, S.: Quantum chaos: a brief first visit. In: Second Summer School in Analysis and Mathematical Physics (Cuernavaca, 2000), Vol. 289 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2001, pp. 161–218 27. Deutsch, D.: Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631–633 (1983) 28. Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987) 29. Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103–1106 (1988) 30. Baladi, V.: Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics, Vol 16, Singapore: World Scientific, 2000

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Commun. Math. Phys. 294, 343–352 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0967-1

Communications in

Mathematical Physics

The Vanishing of Two-Point Functions for Three-Loop Superstring Scattering Amplitudes Samuel Grushevsky1 , Riccardo Salvati Manni2 1 Mathematics Department, Princeton University, Fine Hall, Washington Road,

Princeton, NJ 08544, USA. E-mail: [email protected]

2 Dipartimento di Matematica, Università “La Sapienza”,

Piazzale A. Moro 2, Roma, I 00185, Italy. E-mail: [email protected] Received: 16 June 2008 / Accepted: 18 December 2008 Published online: 12 December 2009 – © Springer-Verlag 2009

Abstract: In this paper we show that the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen [2] vanishes. Our proof uses the reformulation of the ansatz given in [8], theta functions, and specifically the theory of the 00 linear system, introduced by van Geemen and van der Geer [6], on Jacobians. At the two-loop level, where the amplitudes were computed by D’Hoker and Phong [11–14,17,18], we give a new proof of the vanishing of the two-point function (which was proven by them). We also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera [3,8,25]. 1. Introduction An investigation of the problem of computing the superstring measure explicitly for arbitrary genus of the worldsheet was begun by the work of Green and Schwarz [7], who gave an explicit formula in genus 1 using operator methods. D’Hoker and Phong in a series of papers [11–14] introduced a gauge-fixing procedure and computed from first principles the genus 2 superstring measure, verifying that it satisfied the physical constraints, e.g. the vanishing of the 1,2,3-point functions. They also proposed in [15,16] to search for an ansatz for the superstring measure in arbitrary genus as the product of the bosonic measure and a modular form. The ansatz for three-loop measure in this form was then proposed by Cacciatori, Dalla Piazza, and van Geemen in [2]. The genus g ≤ 3 ansatze were reformulated in terms of syzygetic subspaces by the first author in [8], where an ansatz for general genus was proposed, under the assumption on holomorphicity of certain 2r -roots. Cacciatori, Dalla Piazza, and van Geemen in [3] give the genus 4 ansatz in terms of quadrics in the theta constants. The second author in [25] showed that the proposed ansatz is holomorphic in genus 5. Dalla Piazza and van Geemen in [4] proved the uniqueness of the Research is supported in part by National Science Foundation under the grant DMS-05-55867.

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modular form in genus 3 satisfying the factorization constraints. Morozov in [23] surveyed this work and gave an alternative proof that factorization constraints are satisfied for the ansatz; in [24] he has also investigated the 1,2,3-point functions of the proposed ansatz, proving under certain non-trivial mathematical assumption that they vanish on the hyperelliptic locus. In this paper we use the techniques of theta functions, and especially the 00 sublinear system of the linear system |2| introduced by van Geemen and van der Geer [6] to prove the vanishing of the 2-point function in genus 3. We also obtain a new proof of the vanishing of the 2-point function in genus 2. 2. Notations and Definitions We denote by Ag the moduli space of complex principally polarized abelian varieties (ppav for short) of dimension g, and by Hg the Siegel upper half-space of symmetric complex matrices with positive-definite imaginary part, called period matrices. The space Hg is the universal cover of Ag , with the deck group Sp(2g, Z), so that we have Ag = Hg / Sp(2g, Z) for a certain action of the symplectic group. A function f : Hg → C is called a (scalar) modular form of weight k with respect to a subgroup ⊂ Sp(2g, Z) if f (γ ◦ τ ) = det(Cτ + D)k f (τ )

∀γ ∈ , ∀τ ∈ Hg ,

where C and D are the lower blocks if we write γ as four g × g blocks. For a period matrix τ ∈ Hg the principal polarization τ on the abelian variety Aτ := Cg /(Zg + τ Zg ) is the divisor of the theta function exp(πi(n t τ n + 2n t z)). θ (τ, z) := n∈Zg

Notice that for fixed τ theta is a function of z ∈ Cg , and its automorphy properties under the lattice Zg + τ Zg define the bundle τ . Given a point of order two on Aτ , which can be uniquely represented as τ ε+δ 2 for g ε, δ ∈ Z2 (where Z2 = {0, 1} is the additive group), the associated theta function with characteristic is ε θ (τ, z) := exp(πi((n + ε)t τ (n + ε) + 2(n + ε)t (z + δ)). δ n∈Zg

ε is odd or even depending on whether the scalar product δ ε · δ ∈ Z2 is equal to 1 or 0, respectively. The theta function with characteristic is the generator of the space of sections of the bundle τ + τ ε+δ 2 (where we have implicitly identified the principally polarized abelian variety with its dual, and think of points as bundles of degree 0). Thus the square of any theta function with characteristic is a section of 2τ , and the basis for the space of sections of this bundle is given by theta functions of the second order ε (2τ, 2z) [ε](τ, z) := θ 0 As a function of z, θ

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g

for all ε ∈ Z2 . Riemann’s addition formula is an explicit expression of the squares of theta functions with characteristics in this basis: ε θ (τ, z)2 = (−1)δ·σ [σ ](τ, 0)[σ + ε](τ, z). (1) δ g σ ∈Z2

Theta constants are restrictions of theta functions to z = 0; thus all theta constants with odd characteristics vanish identically in τ , while theta constants with even characteristics and all theta constants of the second order do not vanish identically. All theta constants with characteristics are modular forms of weight one half with respect to a certain normal subgroup of finite index (4, 8) ⊂ Sp(2g, Z), while all theta constants of the second order are modular forms of weight one half with respect to a bigger normal subgroup (2, 4) ⊃ (4, 8). Theta constants with characteristics are not algebraically independent, and satisfy a host of algebraic identities, some of which follow from Riemann’s addition formula. However, the theta constants of the second order are algebraically independent for g = 1, 2, and the only relation among them in genus 3 is of degree 16, and has been known classically. It is discussed in detail in [6] — here we give the explicit formula for easy reference. Indeed, a special case of Riemann’s quartic addition theorem in genus 3 is the following identity for theta constants (where we suppress the argument τ ): θ

000 000 000 000 θ θ θ 110 010 100 000 001 001 001 000 000 000 000 001 θ θ . =θ θ θ θ +θ θ 100 000 111 011 101 001 110 010

If we denote the three terms in this relation by ri , so that the relation is r1 = r2 + r3 , then multiplying the 4 “conjugate” relations r1 = ±r2 ± r3 yields the identity F := r14 + r24 + r34 − 2r12 r22 − 2r22 r32 − 2r32 r12 = 0.

(2)

Notice that F is a polynomial of degree 8 in the squares of theta constants with characteristics, and thus by applying Riemann’s addition formula (1) F can be rewritten as a polynomial of degree 16 in theta constants of the second order. We refer to [1,19] for details on theta functions and modular forms, and the current knowledge about the ideal of relations among theta constants of the second order for g > 3 (which is not known completely even for g = 4). 3. The Linear System 00 In this section we review the definition and some facts about the linear system 00 ⊂ |2| introduced and studied in [6]. We refer to that paper for details, as well as to [5,9,20] for a review and results on the importance of the linear system 00 for the Schottky problem of characterizing Jacobians. The linear system 00 ⊂ |2| is defined to consist of all sections vanishing to order at least four at the origin. Since all sections of 2 are even, this is equivalent to the value and all the second derivatives ∂zi ∂z j vanishing at zero. These conditions turn out be

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independent, when (Aτ , ) is an indecomposable ppav (i.e. not isomorphic to a product of lower-dimensional ppavs). In this case the matrix ⎛ ⎞ ](τ,0) ](τ,0) [ε1 ](τ, 0) ∂[ε∂τ111 . . . ∂[ε∂τ1gg ⎜ ⎟ .. .. .. .. ⎜ ⎟ . . . . ⎝ ⎠ ∂[ε2g ](τ,0) ∂[ε2g ](τ,0) g [ε2 ](τ, 0) ... ∂τ11 ∂τgg has rank

g(g+1) 2

+ 1, cf. [26] and thus ([6], Prop. 1.1)

dim 00 = dim |2| − 1 −

1 = 2g − 1 −

1≤i≤ j≤g

g(g + 1) . 2

(3)

Thus the linear system 00 is zero for g ≤ 2, has dimension 1 for g = 3, and higher dimension for all other genera. The above description leads to a simple construction of a basis for the space 00 . Proposition 1. Let τ0 be an irreducible point of Hg (i.e. corresponding to indecomposg able ppav). Denote N := 1 + g(g+1) 2 , and choose ε1 , . . . , ε N ∈ Z2 such that the modular form ⎞ ⎛ ](τ,0) ](τ,0) . . . ∂[ε∂τ1gg [ε1 ](τ, 0) ∂[ε∂τ111 ⎟ ⎜ .. .. .. ⎟ gε1 ,...,ε N (τ ) := det ⎜ . . . ⎠ ⎝ ∂[ε N ](τ,0) ∂[ε N ](τ,0) ... [ε N ](τ, 0) ∂τ11 ∂τgg does not vanish at τ0 . Then the sections ⎛ [ε1 ](τ0 , z) [ε1 ](τ0 , 0) ⎜ .. .. ⎜ ⎜ . . f ε (τ0 , z) := det ⎜ ⎜[ε N ](τ0 , z) [ε N ](τ0 , 0) ⎝ [ε](τ0 , z) [ε](τ0 , 0)

∂[ε1 ](τ0 ,0) ∂τ11

.. .

∂[ε N ](τ0 ,0) ∂τ0 τ11 ∂[ε](τ0 ,0) ∂τ11

... .. . ... ...

∂[ε1 ](τ0 ,0) ∂τgg

⎞

⎟ ⎟ ⎟ , ∂[ε N ](τ0 ,0) ⎟ ⎟ ∂τgg ⎠ ∂[ε](τ0 ,0) ∂τgg

g

for ε ∈ Z2 \ {ε1 , . . . , ε N } form a basis of 00 ⊂ |2τ0 |. Proof. The proof is a simple linear algebra argument that we recall for completeness. First note that f ε (τ0 , z) belongs to 00 , as the determinant and all the second z-derivatives (equal to the first τ -derivatives by the heat equation) vanish for z = 0, as two of the columns of the matrix become identical. It thus remains to show that the functions f ε for various ε are linearly independent. Indeed, recall that theta functions of the second order form a basis of sections of 2, and now note that the basis element [ε](τ0 , z) enters only the expression of f ε (τ0 , z), and that with non-zero coefficient gε1 ,...,ε N (τ0 ). Remark 2. It can be shown that on the open sets {gε1 ,...,ε N (τ ) = 0} the coefficients of the basis vectors are in fact modular forms of weight g + 1 + N /2, see [10]. In particular when g = 3 we have a global expression of the unique section f (τ, z) of the space 00 . Remark 3. Observe that if the period matrix τ is decomposable, then the dimension of 00 increases; however, a basis can still be constructed by using the same method.

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There exists another method for constructing elements of 00 — it is described in [6], and is as follows. Suppose I is an algebraic relation among theta constants of the second order (in genus g). This is to say, suppose I ∈ C[x0...0 , . . . , x1...1 ] is a polynomial in 2g variables such that for any τ ∈ Hg we have I ([ε](τ )) = 0. Then the function f I (z) :=

∂I ([0 . . . 0](τ, 0), . . . , [1 . . . 1](τ, 0)) [ε](τ, z) g ∂xε

ε∈Z2

lies in 00 ⊂ |2τ |. Indeed, since I vanishes identically on Hg , by Euler’s formula we have f I (0) = 0. Moreover, by the heat equation 2πi(1 + δ j,k )

∂ I ∂[ε](τ ) ∂2 fI ∂ I ([0 . . . 0], . . . [1 . . . 1]) |z=0 = = , ∂z j ∂z k ∂x ∂τ ∂τ jk ε jk g ε∈Z2

which is zero since I vanishes identically on Hg , and thus its derivative in any direction is also zero. In [6], Prop. 1.2 it is shown that as I ranges over the ideal of relations among theta constants, the functions f I generate the linear system 00 . Since for g ≥ 4 the ideal of algebraic relations among theta constants of the second order is not completely known, for g ≥ 4 this method does not yield a complete description of the basis of 00 . However, the geometry of these relations is intriguing, and this method produces elements of 00 with coefficients algebraic in theta constants, rather than involving their derivatives as well. 4. The Proposed Ansatz for the Superstring Measure An ansatz for the 3-loop superstring measure was proposed in [2]. The reformulation of this ansatz in terms of products of theta constants with characteristics in a syzygetic subspace given in [8] is as follows. For any i = 0 . . . g define (g) Gi

ε (τ ) := δ

2g V ⊂Z2 ; dim V =i

α β

4−i ε+α θ (τ )2 . δ+β

(4)

∈V

Noticethat since any i-dimensional linear subspace contains zero, all products will conε . Since all odd theta constants vanish identically, it is enough to sum over the tain θ δ even cosets of syzygetic i-dimensional subspaces containing [ε, δ], see [8,25]. 2g To simplify notations, we write m := [ε, δ] ∈ Z2 for characteristics and similarly ε . Then the proposed ansatz for the superstring measure is the product write θm := θ δ of the bosonic measure (which is a form on Mg ) and, for any even characteristic m, the expression (g) m :=

g i(i−1) (g) (−1)i 2 2 G i [m] i=0

(5)

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which is a modular form of weight 8 with respect to a subgroup of Sp(2g, Z) conjugate to (1, 2). In particular for genus 3 we have (3)

(3)

(3)

(3)

(3) m := G 0 [m] − G 1 [m] + 2G 2 [m] − 8G 3 [m].

(3) In [25] it is shown that the sum m m is a non-zero multiple of the modular form F given by (2), and thus vanishes identically on H3 . (g) (g) From definition (4) of the summands G i [m] of the measure m it follows that (g) G i [m] is a polynomial in the squares of theta constants with characteristics for i ≤ 3, divisible by θm2 (τ ). Since this is the only kind of summands appearing in the definition (g) of m for g ≤ 3, by applying Riemann’s addition formula (1) we get (g)

(g)

Proposition 4. For g ≤ 3 the modular form m defined by (5) and the ratio m /θm2 (τ, 0) are both polynomials in theta constants of the second order, of degrees 16 and 14, respectively. 5. The Vanishing of the 2-Point Function We recall (see [12] for explicit formulas) that the vanishing of the cosmological constant

(g) reduces to the identity m m = 0 (proven for the proposed ansatz for g ≤ 4 in [25]), and this also implies the vanishing of the 1-point function, while as shown in [18] the vanishing of the two-point function is equivalent to the vanishing of 2 (g) m Sm (a, b) m

for any points a, b on the Riemann surface (thought of as embedded into its Jacobian), where Sm is the Szeg˝o kernel Sm (a, b) :=

θm (a − b) , θm (0)E(a, b)

with E being the prime form on the Riemann surface. Since the prime form does not depend on m, it is a common factor in all summands above, and thus does not matter for the vanishing of the 2-point function, so the vanishing of the 2-point function is equivalent to the vanishing of X 2 (a, b) :=

(g) m (τ ) θ (τ, a − b)2 , 2 (τ, 0) m θ m m

where τ is the period matrix of the Jacobian J ac(C) of a Riemann surface C, and a, b ∈ C ⊂ J ac(C) are arbitrary. We will now relate the vanishing of the 2-point function and the 00 linear system. Set m (τ ) θ (τ, z)2 (6) X 2 (τ, z) := 2 (τ, 0) m θ m m and note that for τ fixed this function is a section of |2τ |. The vanishing of the 2-point function is then equivalent to X 2 (z) vanishing along the surface C − C ⊂ J ac(C). By Proposition 2.1 in [6] and the subsequent remark, a section of |2| vanishes along the surface C − C if and only if it lies in 00 . We thus get

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Theorem 5. The 2-point function for the proposed superstring measure ansatz vanishes (for genus g) if and only if for the period matrix τ of any Jacobian of a Riemann surface of genus g the section X 2 (τ, z) of 2τ defined above lies in 00 . Since for g = 1, 2 the linear system 00 is zero, the vanishing of the two-point function is equivalent to X 2 (τ, z) vanishing identically in z and τ for g ≤ 2. If we write out X 2 (τ, z) as a linear combination of the basis for sections of 2τ given by theta functions of the second order X 2 (τ, z) = cε (τ )[ε](τ, z), (7) then X 2 vanishes identically if and only if each cε (τ ) vanishes identically. This allows us to recover the result of D’Hoker and Phong in genus 2. Proposition 6. The 2-point function for the proposed superstring ansatz vanishes identically for g ≤ 2. Proof. Let us apply Riemann’s addition formula (1) to the definition (6) of the two-point function to rewrite it in terms of theta functions of the second order (notice that the only term depending on z is θm (τ, z)2 , and we apply the addition formula to it as well). Notice that by Proposition 4 the coefficients cε in (7) obtained in this way are explicit polynomials of degree 15 in theta constants of the second order, and since there are no algebraic relations among theta constants of the second order for g ≤ 2, one needs to verify that all polynomials cε are zero. This can be done on a computer (we used Maple). Note that the computation can be made easier by noting that since X 2 (τ, z) has a transformation formula with respect to the entire symplectic group, (i.e. it is a Jacobi form) and the coefficients cε (τ ) are permuted under the action of a suitable subgroup Sp(2g, Z) that acts monomially on the theta constants of the second order, it is enough to check that just one of cε is the zero polynomial. In the case of g = 3, recall from (3) that the space 00 is one-dimensional, and by the results of [6] we know that it is generated by F2 :=

∂F ([000](τ, 0), . . . , [111](τ, 0)) [ε](τ, z), ∂xε 3

ε∈Z2

where we recall that F, given by (2), is the only polynomial relation of degree 16 among the 8 theta constants of the second order for g = 3. Proposition 7. For any τ ∈ H3 the sections F2 and X 2 of 2τ are proportional; more precisely F2 = − 14 5 X 2. Proof. We have explicit expressions for F2 and X 2 as linear combinations of the basis of the sections of 2 given by the second order theta functions. Thus what we need to verify is that the coefficient in 5F2 + 14X 2 of any [ε](z) is equal to zero. This coefficient is a polynomial of degree 15 in theta constants of the second order and can be verified to be zero using Maple (since the only relation among theta constants of the second order is of degree 16, a polynomial in theta constants of the second order of degree 15 vanishes identically only if it is zero). Notice that by modularity it is again enough to verify that the coefficient of [000](z) in 5F2 + 14X 2 is equal to zero.

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Theorem 8. The 2-point function for the proposed ansatz for the 3-loop superstring measure vanishes identically. Proof. By the above proposition we see that for any τ ∈ H3 the function X 2 , being a constant multiple of F2 , lies in the linear system 00 ⊂ |2τ |. By Theorem 5 this is equivalent to the identical vanishing of the two-point function. Remark 9. We note that the global section F2 is proportional also to the global section f (τ, z). Really we have that F2 (τ, z) = c(τ ) f (τ, z) for any irreducible τ ∈ Hg . Moreover c(τ ) results to be a modular function with respect to Sp(6, Z) that is regular on the set of irreducible point, so it is regular everywhere (modular form) and hence it is a non-zero constant. This identity produces eight nontrivial identities expressing each jacobian determinant gε1 ,...,ε7 (τ ) as a polynomial of degree 15 in the theta constants [σ ](τ, 0). 6. Conclusion There are two generalizations that it is natural to try to prove. First, one could ask whether the vanishing of the 2-point function can be obtained for the proposed in [8] ansatz in higher genera. For genus 4 the ansatz is also given in [3] and is manifestly holomorphic in either formulation. The holomorphicity of the ansatz in genus 5 was proven in [25], and thus it is natural to ask whether the 2-point function vanishes for g ≤ 5. By Theorem 5 we know that this is equivalent to X 2 lying in the linear system 00 . However, already for genus 4 the geometry of the situation is much more complicated: instead of just one relation F in genus 3 the ideal of relations among theta constants of the second order in genus 4 is unknown, and an explicit basis for 00 is unknown for g = 4. Moreover, it could be that here the fact that we are working on the moduli space of curves M4 rather than A4 plays a role — the geometry of 00 depends on this, see [20]. Second, one could try to prove the vanishing of the 3-point function. As shown in [18] for genus 2, this is equivalent to proving that the sum (g) m Sm (a, b)Sm (b, c)Sm (c, a) m

vanishes. Using the explicit formula for the Szeg˝o kernel and canceling the m-independent factor, this is equivalent to the function X 3 (a, b, c) :=

m (τ ) θm (a − b)θm (b − c)θm (c − a) θm3 (τ, 0) m

vanishing identically for a, b, c ∈ C. However, in this case we do not know a natural function on J ac(C)×n of which X 3 is a restriction, and there is no analog of the theory of the 00 for more points. It seems that the identity among the third order theta functions obtained by Krichever in his proof of the trisecant conjecture ([21], formula (1.18)) may potentially be useful in relating the 3-point and 2-point functions, but so far we have not been able to find an explicit way to do this.

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Added in proof. The recent paper of Matone and Volpato [22], written after the current manuscript was submitted for publication, uses other identities relating theta functions and abelian differentials to show that the expression X 3 above in fact does not vanish identically in genus 3. Note, however, that as shown by D’Hoker and Phong in their series of papers, the full expression for N -point function involves many more terms resulting from gauge-fixing, and thus the result of [22] suggests that these terms have a non-zero contribution starting from the 3-point function in genus 3. Acknowledgements. We are grateful to Eric D’Hoker and Duong Phong for introducing us to questions about the superstring scattering amplitudes and explanations regarding the conjectured properties of N -point functions. The computations for this paper were done using Maplesoft’s Maple© software. We would like to thank the referee.

References 1. Birkenhake, Ch., Lange, H.: Complex abelian varieties: second, augmented edition. Grundlehren der mathematischen Wissenschaften 302, Berlin: Springer-Verlag, 2004 2. Cacciatori, S.L., Dalla Piazza, F., van Geemen, B.: Modular forms and three loop superstring amplitudes. Nucl. Phys. B. 800, 565–590 (2008) 3. Cacciatori, S.L., Dalla Piazza, F., van Geemen, B.: Genus four superstring measures. Lett. Math. Phys. 85, 185–193 (2008) 4. Dalla Piazza, F., van Geemen, B.: Siegel modular forms and finite symplectic groups. http://arXiv.org/ abs/0804.3769v2[math.AG], 2008 5. van Geemen, B.: The Schottky problem and second order theta functions. In: Taller de variedades abelianas y funciones theta, Sociedad Matemática Mexicana, Aportaciones Matemáticas, Investigación 13, 41–84 (1998) 6. van Geemen, B., van der Geer, G.: Kummer varieties and the moduli spaces of abelian varieties. Amer. J. of Math. 108, 615–642 (1986) 7. Green, M.B., Schwarz, J.H.: Supersymmetrical string theories. Phys. Lett. B 109, 444–448 (1982) 8. Grushevsky, S.: Superstring scattering amplitudes in higher genus. Commun. Math. Phys. 287, 749–767 (2009) 9. Grushevsky, S.: A special case of the 00 conjecture. http://arXiv.org/abs/0804.0525v2[math.AG], 2009 10. Grushevsky, S., Salvati Manni, R.: Two generalizations of Jacobi’s derivative formula. Math. Res. Lett. 12(5-6), 921–932 (2005) 11. D’Hoker, E., Phong, D.H.: Two-loop superstrings I, main formulas. Phys. Lett. B 529, 241–255 (2002) 12. D’Hoker, E., Phong, D.H.: Two-loop superstrings II, the chiral measure on moduli space. Nucl. Phys. B 636, 3–60 (2002) 13. D’Hoker, E., Phong, D.H.: Two-loop superstrings III, slice independence and absence of ambiguities. Nucl. Phys. B 636, 61–79 (2002) 14. D’Hoker, E., Phong, D.H.: Two-loop superstrings IV, The cosmological constant and modular forms. Nucl. Phys. B 639, 129–181 (2002) 15. D’Hoker, E., Phong, D.H.: Asyzygies, modular forms, and the superstring measure I. Nucl. Phys. B 710, 58–82 (2005) 16. D’Hoker, E., Phong, D.H.: Asyzygies, modular forms, and the superstring measure. II. Nucl. Phys. B 710, 83–116 (2005) 17. D’Hoker, E., Phong, D.H.: Two-loop superstrings V, Gauge slice independence of the N-point function. Nucl. Phys. B 715, 91–119 (2005) 18. D’Hoker, E., Phong, D.H.: Two-loop superstrings VI, non-renormalization theorems and the 4-point function. Nucl. Phys. B 715, 3–90 (2005) 19. Igusa, J.-I.: Theta functions. Die Grundlehren der mathematischen Wissenschaften, Band 194. New YorkHeidelberg: Springer-Verlag, 1972 20. Izadi, E.: The geometric structure of A4 , the structure of the Prym map, double solids and 00 -divisors. J. Reine Angew. Math. 462, 93–158 (1995) 21. Krichever, I.: Characterizing Jacobians via trisecants of the Kummer Variety. Ann. of Math., to appear, http://arXiv.org/abs/math/0605625v4[math.AG], 2008 22. Matone, M., Volpato, R.: Superstring measure and non-renormalization of the three-point amplitude. Nucl. Phys. B 806, 735–747 (2009)

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23. Morozov, A.: NSR superstring measures revisited. JHEP 0805, 086 (2008) 24. Morozov, A.: NSR measures on hyperelliptic locus and non-renormalization of 1,2,3-point functions. Phys. Lett. B 664, 116–122 (2008) 25. Salvati Manni, R.: Remarks on Superstring amplitudes in higher genus. Nucl. Phys. B 801, 163–173 (2008) 26. Sasaki, R.: Modular forms vanishing at the reducible points of the Siegel upper-half space. J. Reine Angew. Math. 345, 111–121 (1983) Communicated by N.A. Nekrasov

Commun. Math. Phys. 294, 353–388 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0941-y

Communications in

Mathematical Physics

Escape Rates and Physically Relevant Measures for Billiards with Small Holes Mark Demers1, , Paul Wright2, , Lai-Sang Young3, 1 Department of Mathematics and Computer Science, Fairfield University,

Fairfield, USA. E-mail: [email protected]

2 Department of Mathematics, University of Maryland,

College Park, USA. E-mail: [email protected]

3 Courant Institute of Mathematical Sciences, New York University,

New York, USA. E-mail: [email protected] Received: 4 November 2008 / Accepted: 3 April 2009 Published online: 25 November 2009 – © Springer-Verlag 2009

Abstract: We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map. This paper is about leaky dynamical systems, or dynamical systems with holes. Consider a dynamical system defined by a map or a flow on a phase space M, and let H ⊂ M be a hole through which orbits escape, that is to say, once an orbit enters H , we stop considering it from that point on. Starting from an initial probability distribution µ0 on M, mass will leak out of the system as it evolves. Let µn denote the distribution remaining at time n. The most basic question one can ask about a leaky system is its rate of escape, i.e. whether µn (M) ∼ ϑ n for some ϑ. Another important question concerns the nature of the remaining distribution. One way to formulate that is to normalize µn , and to inquire about properties of µn /µn (M) as n tends to infinity. Such limiting distributions, when they exist, are not invariant; they are conditionally invariant, meaning they are invariant up to a normalization. Comparisons of systems with small holes with the corresponding closed systems, i.e. systems for which the holes have been plugged, are also natural. These are some of the questions we will address in this paper. We do not consider these questions in the abstract, however; for a review paper in this direction, see [DY]. Our context here is that of billiard systems with small holes.

This research is partially supported by NSF grant DMS-0801139. This research is partially supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. This research is partially supported by a grant from the NSF.

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Specifically, we carry out our analysis for the collision map of a 2-dimensional periodic Lorentz gas, and expect our results to be extendable to other dispersing billiards. Our holes are “physical” holes, in the sense that they are derived from holes in the physical domain of the system, i.e., the billiard table: we consider both convex holes away from the scatterers and holes that live on the boundaries of the scatterers. The holes considered in this paper are very small, but their placements are immaterial. For these leaky systems, we prove that there is a common rate of escape and a common limiting distribution for a large class of natural initial distributions including those with densities with respect to Liouville measure. These conditionally invariant measures, therefore, can be viewed as characteristic of the leaky systems in question, in a way that is analogous to physical measures or SRB measures for closed systems. We show, in fact, that as hole size tends to zero, these measures tend to the natural invariant measure of the corresponding closed billiard system. Our proof involves constructing a Markov tower extension with a special property over the billiard map, the new requirement being that it respects the hole. Let us backtrack a little for readers not already familiar with these ideas: In much the same way that Markov partitions have proved to be very useful in the study of Anosov and Axiom A diffeomorphisms, it was shown, beginning with [Y] and continued in a number of other papers, that many systems with sufficiently strong hyperbolic properties (but which are not necessarily uniformly hyperbolic) admit countable Markov extensions. Roughly speaking, these extensions behave like countable state Markov chains “with nonlinearity”; they have considerably simpler structures than the original dynamical system. The idea behind this work is that escape dynamics are much simpler in a Markov setting when the hole corresponds to a collection of “states”; this is what we mean by the Markov extension “respecting the hole.” All this is not for free, however. We pay a price with a somewhat elaborate construction of the tower, and again when we pass the information back to the billiard system, in exchange for having a Markov structure to work with in the treatment of the hole. There are advantages to this route of proof: First, once a Markov extension is constructed for a system, it can be used many times over for entirely different purposes. For the billiard maps studied here, these extensions were constructed in [Y]; our main task is to adapt them to holes. Second, once results on escape dynamics are established on towers, they apply to all Markov extensions. Here, the desired results are already known in a special case, namely expanding towers [BDM]; we need to extend them to the general, hyperbolic setting. What we propose here is a unified, generic approach for dealing with holes in dynamical systems, one that can, in principle, be carried out for all systems that admit Markov towers. Such systems include logistic maps, rank one attractors including the Hénon family, piecewise hyperbolic maps and other dispersing billiards in 2 or more dimensions. Conditionally invariant measures were first introduced in probabilistic settings, namely countable state Markov chains and topological Markov chains, beginning with [V] and more recently in [FKMP and CMS3]. In this setting, such measures are called quasi-stationary distributions and the existence of a Yaglom limit corresponds to the limit µn /µn (M), which we use here to identify a physical conditionally invariant measure for the leaky system. The first works to study deterministic systems with holes took advantage of finite Markov partitions. These include: Expanding maps on Rn with holes which are elements of a finite Markov partition [PY,CMS1,CMS2]; Smale horseshoes [C1,C2]; Anosov diffeomorphisms [CM1,CM2,CMT1,CMT2]; billiards with convex scatterers satisfying a

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non-eclipsing condition [LM,R] and large parameter logistic maps whose critical point maps out of the interval [HY]. In the latter two, the holes are chosen in such a way that the surviving dynamics are uniformly expanding or hyperbolic with Markov partitions. First results which drop Markov requirements on the map include piecewise expanding maps of the interval [BaK,CV,LiM,D1,BDM]; Misiurewicz [D2] and Collet-Eckmann [BDM] maps with generic holes; and piecewise uniformly hyperbolic maps [DL]. The tower construction is used in the one-dimensional studies [D1,D2,BDM]. Typically a restriction on the size of the hole is introduced in order to control the dynamics when a finite Markov partition is absent. General conditions ensuring the existence of conditionally invariant measures are first given in [CMM]. The physical relevance of such measures, however, is unclear without further qualifications. As noted in [DY], under very weak assumptions on the dynamical system, many such measures exist: for any prescribed rate of escape, one can construct infinitely many conditionally invariant densities. This is the reason for the emphasis placed in this paper on the limit µn /µn (M), which identifies a unique, physically relevant conditionally invariant measure. This paper is organized as follows: Our results are formulated in Sect. 1. In Sects. 2 and 3, the geometry of billiard maps and holes are looked at carefully as we modify previous constructions to give a generalized horseshoe that respects the hole. Out of this horseshoe, a Markov tower extension is constructed and results on escape dynamics on it proved; this is carried out in Sects. 4 and 5. These results are passed back to the billiard system in Sect. 6, where the remaining theorems are also proved. 1. Formulation of Results 1.1. Basic definitions. We consider a closed dynamical system defined by a self-map f of a manifold M, and let H ⊂ M be a hole through which orbits escape, i.e., we stop considering an orbit once it enters H . In this paper we are primarily concerned with holes that are open subsets of the phase space; they are not too large and generally not f -invariant. We will refer to the triplet ( f, M, H ) as a leaky system. First we introduce some notation. Let M˚ = M\H . At least to begin with, let us make ˚ : M˚ ∩ f −1 M˚ → M, ˚ and a formal distinction between f and f˚ = f |( M˚ ∩ f −1 M) n ˚ Let η be a probability measure on M. ˚ We define f˚∗ η to write f˚n = f n |( i=0 f −i M). ˚ If η be the measure on M˚ defined by ( f˚∗ η)(A) = η( f˚−1 A) for each Borel set A ⊂ M. (n) n n ˚ ˚ ˚ is an initial distribution on M, then η := f ∗ η/| f ∗ η| is the normalized distribution of points remaining in M˚ after n units of time. Given an initial distribution η, the most basic question is the rate at which mass is leaked out of the system. We define the escape rate starting from η to be − log ϑ(η), where n 1 −i ˚ log ϑ(η) = lim log η assuming such a limit exists. f M n→∞ n i=0

Another basic object is the limiting distribution η(∞) defined to be η(∞) = limn→∞ η(n) if this weak limit exists. Of particular interest is when there is a number ϑ∗ and a probability measure µ∗ with the property that for all η in a large class of natural initial distributions (such as those having densities with respect to Lebesgue measure), we have ϑ(η) = ϑ∗ and η(∞) = µ∗ . In such a situation, µ∗ can be thought of as a physical measure for the leaky system ( f, M, H ), in analogy with the idea of physical measures for closed systems.

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A Borel probability measure η on M is said to be conditionally invariant if it satisfies f˚∗ η = ϑη for some ϑ ∈ (0, 1]. Clearly, the escape rate of a conditionally invariant measure η is well defined and is equal to − log ϑ. Most leaky dynamical systems admit many conditionally invariant measures; see [DY]. In particular, limiting distributions, when they exist, are often conditionally invariant; they are among the more important conditionally invariant measures from an observational point of view. Finally, when a physical measure η for a leaky system ( f, M, H ) has absolutely continuous conditional measures on the unstable manifolds of the underlying closed system ( f, M), we will call it an SRB measure for the leaky system, in analogy with the idea of SRB measures for closed systems. 1.2. Setting of present work. The underlying closed dynamical system here is the billiard map associated with a 2-dimensional periodic Lorentz gas. Let {i : i = 1, · · · , d} be pairwise disjoint C 3 simply-connected curves on T2 with strictly positive curvature, and consider the billiard flow on the “table” X = T2 \ i {interiori }. We assume the “finite horizon” condition, which imposes an upper bound on the number of consecutive tangential collisions with ∪i . The phase space of the unit-speed billiard flow is M = (X × S1 )/ ∼ with suitable identifications at the boundary. Let M = ∪i i × [− π2 , π2 ] ⊂ M be the cross-section to the billiard flow corresponding to collision with the scatterers, and let f : M → M be the Poincaré map. The coordinates on M are denoted by (r, ϕ), where r ∈ ∪i is parametrized by arc length and ϕ is the angle a unit tangent vector at r makes with the normal pointing into the domain X . We denote by ν the invariant probability measure induced on M by Liouville measure on M, i.e., dν = c cos ϕdr dϕ, where c is the normalizing constant. We consider the following two types of holes: Holes of Type I. In the table X , a hole σ of this type is an open interval in the boundary of a scatterer. When q0 ∈ ∪i , we refer to {q0 } as an infinitesimal hole, and let h (q0 ) denote the collection of all open intervals σ ⊂ ∪i in the h-neighborhood of q0 . A hole σ in X of this type corresponds to a set Hσ ⊂ M of the form (a, b) × [− π2 , π2 ]. Holes of Type II. A hole σ of this type is an open convex subset of X away from ∪i i and bounded by a C 3 simple closed curve with strictly positive curvature. As above, we regard {q0 } ⊂ X \ ∪i as an infinitesimal hole, and use h (q0 ) to denote the set of all σ in the h-neighborhood of q0 . In this case, σ ⊂ X does not correspond directly to a set in M. Rather, σ corresponds directly to a set in M, the phase space for the billiard flow, and we must make a choice as to which set in the cross section M will represent the hole for the billiard map. There is a well defined set Bσ ⊂ M consisting of all (r, ϕ) whose trajectories under the billiard flow on M will enter σ × S1 before reaching M again. Thus Hσ = f (Bσ ) is a natural candidate for the hole in M representing σ , and will be taken as such in this work. However, it would also have been possible to take Bσ as the representative set. The geometry of Bσ and Hσ in phase space will be discussed in detail in Sect. 3.1. Also, we note that the requirement that ∂σ be a C 3 simple closed curve with strictly positive curvature can be considerably relaxed. It is even possible to allow some holes σ that are not convex. See the remark at the end of Sect. 3.1. 1.3. Statement of results. Let G = G(Hσ ) denote the set of finite Borel measures η on M that are absolutely continuous with respect to ν with dη/dν being (i) Lipschitz on ∞ f −i M. ˚ Notice that each connected component of M and (ii) strictly positive on ∩i=0 measures on M with Lipschitz dη/dν correspond to measures on M having a Lipschitz density with respect to Liouville measure.

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Standing hypotheses for Theorems 1–3. We assume (1) f : M → M is the billiard map defined in Sect. 1.2, (2) {q0 } is an infinitesimal hole of either Type I or Type II, and (3) σ ∈ h (q0 ), where h > 0 is assumed to be sufficiently small. Theorem 1 (Common escape rate). All initial distributions η ∈ G have a common escape rate − log ϑ∗ for some ϑ∗ < 1; more precisely, for all η ∈ G, ϑ(η) is well defined and is equal to ϑ∗ . Theorem 2 (Common limiting distribution). (a) For all η ∈ G, the normalized surviving distributions f˚∗n η/| f˚∗n η| converge weakly to a common conditionally invariant distribution µ∗ with ϑ(µ∗ ) = ϑ∗ . (b) In fact, for all η ∈ G, there is a constant c(η) > 0 s.t. ϑ∗−n f˚∗n η converges weakly to c(η)µ∗ . Thus from an observational point of view, − log ϑ∗ is the escape rate and µ∗ the physical measure for the leaky system ( f, M, Hσ ). Theorem 3 (Geometry of limiting distribution). (a) µ∗ is singular with respect to ν; (b) µ∗ has strictly positive conditional densities on local unstable manifolds. The precise meaning of the statement in part (b) of Theorem 3 is that there are countably many “patches” (Vi , µi ), i = 1, 2, . . ., where for each i, (i) Vi ⊂ M is the union of a continuous family of unstable curves {γ u }; (ii) µi is a measure on Vi whose conditional measures on {γ u } have strictly positive densities with respect to the Riemannian measures on γ u ; (iii) µi ≤ µ∗ for each i, and i µi ≥ µ∗ . This justifies viewing µ∗ as the SRB measure for the leaky system ( f, M, Hσ ). Our final result can be interpreted as a kind of stability for the natural invariant measure ν of the billiard map without holes. Theorem 4 (Small-hole limit). We assume (1) and (2) in the Standing Hypotheses above. Let σh ∈ h (q0 ), h > 0, be an arbitrary family of holes, and let − log ϑ∗ (σh ) and µ∗ (σh ) be the escape rate and physical measure for the leaky system ( f, M, Hσh ). Then ϑ∗ (σh ) → 1 and µ∗ (σh ) → ν as h → 0. Some straightforward generalizations: Our proofs continue to hold under the more general conditions below, but we have elected not to discuss them (or to include them formally in the statement of our theorems) because keeping track of an increased number of objects will necessitate more cumbersome notation. 1. Holes. Our results apply to more general classes of holes than those described above. For example, we could fix a finite number of infinitesimal holes {q0 }, . . . , {qk } and consider σ = ∪i σi with σi ∈ h (qi ). In fact, we may take more than one σi in each h (qi ) for as long as the total number of holes is uniformly bounded. See Sect. 3.4 for further generalizations on the types of holes allowed. 2. Initial distributions. Theorems 1 and 2 (and consequently Theorems 3 and 4) remain true with G replaced by a broader class of measures. For example, we use only the Lipschitz property of dη/dν along unstable leaves, and it is sufficient for dη/dν to be strictly positive on large enough open sets (see Remark 6.3). Moreover, dη/dν need not be bounded provided it blows up sufficiently slowly near the singularity set for f .

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Finally, we remark that Theorem 2(b) continues to hold without requiring that dη/dν be strictly positive anywhere, except that now c(η) might be 0. 2. Relevant Dynamical Structures Our plan is to show that the billiard maps described in Sect. 1.2 admit certain structures called “generalized horseshoes” which can be arranged to “respect the holes.” The main results are summarized in Proposition 2.2 in Sect. 2.2 and proved in Sect. 3. 2.1. Generalized horseshoes. We begin by recalling the idea of a horseshoe with infinitely many branches and variable return times introduced in [Y] for general dynamical systems without holes. These objects will be referred to in this paper as “generalized horseshoes”. Following the notation in Sect. 1.1 of [Y], we consider a smooth or piecewise smooth invertible map f : M → M, and let µ and µγ denote respectively the Riemannian measure on M and on γ where γ ⊂ M is a submanifold. We say the pair ( , R) defines a generalized horseshoe if (P1)–(P5) below hold (see [Y] for precise formulation): (P1) is a compact subset of M with a hyperbolic product structure, i.e., = (∪ u ) ∩ (∪ s ), where s and u are continuous families of local stable and unstable manifolds, and µγ {γ ∩ } > 0 for every γ ∈ u . (P2) R : → Z+ is a return time function to . Modulo a set of µ-measure zero, is the disjoint union of s-subsets j , j = 1, 2, . . . , with the property that for each j, R| j = R j ∈ Z+ and f R j ( j ) is a u-subset of . There is a notion of separation time s0 (·, ·), depending only on the unstable coordinate, defined for pairs of points in , and there are numbers C > 0 and α < 1 such that the following hold for all x, y ∈ : (P3) For y ∈ γ s (x), d( f n x, f n y) ≤ Cα n for all n ≥ 0. (P4) For y ∈ γ u (x) and 0 ≤ k ≤ n < s0 (x, y), (a) d( f n x, f n y) ≤ Cα s0 (x,y)−n ; n det D f u ( f i x) ≤ Cα s0 (x,y)−n . (b) log i=k det D f u ( f i y)

∞ det D f ( f x) ≤ Cα n for all n ≥ 0. (P5) (a) For y ∈ γ s (x), log i=n det D f u ( f i y) u (b) For γ , γ ∈ , if : γ ∩ → γ ∩ is defined by (x) = γ s (x) ∩ γ , then u

is absolutely continuous and

i

d(−1 ∗ µγ ) (x) dµγ

∞ det D f ( f x) . = i=0 det D f u ( f i x) u

i

The meanings of the last three conditions are as follows: Orbits that have not “separated” are related by local hyperbolic estimates; they also have comparable derivatives. Specifically, (P3) and (P4)(a) are (nonuniform) hyperbolic conditions on orbits starting from . (P4)(b) and (P5) treat more refined properties such as distortion and absolute continuity of s , conditions that are known to hold for C 1+ε hyperbolic systems. We say the generalized horseshoe ( , R) has exponential return times if there exist C0 > 0 and θ0 > 0 such that for all γ ∈ u , µγ {R > n} ≤ C0 θ0n for all n ≥ 0. The setting described above is that of [Y]; it does not involve holes. In this setting, we now identify a set H ⊂ M (to be regarded later as the hole) and introduce a few relevant terminologies. Let ( , R) be a generalized horseshoe for f with ⊂ (M \ H ). We say ( , R) respects H if for every i and every with 0 ≤ ≤ Ri , f ( i ) either does not intersect H or is completely contained in H .

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The following definitions of “mixing” are motivated by Markov-chain considerations: Let s ⊂ be an s-subset. We say s makes a full return to at time n if there are numR +···+Ri j ( s ) ⊂ i j+1 bers i 0 , i 1 , . . . , i k with n = Ri0 + · · · + Rik such that s ⊂ i0 , f i0 n s for j < k, and f ( ) is a u-subset of . (i) We say the horseshoe ( , R) is mixing if there exists N such that for every n ≥ N , some s-subset s (n) makes a full return at time n. (ii) If ( , R) respects H , then when we treat H as a hole, we say the surviving dynamics are mixing if in addition to the condition in (i), we require that f s (n) ∩ H = ∅ for all with 0 ≤ ≤ n. This is equivalent to requiring that s (n) makes a full return to at time n under the dynamics of f˚, where f˚ is the map defined in Sect. 1.1. We note that the mixing of f in the usual sense of ergodic theory does not imply that any generalized horseshoe constructed is necessarily mixing in the sense of the last paragraph, nor does mixing of the horseshoe imply that of its surviving dynamics. 2.2. Main Proposition for billiards with holes. With these general ideas out of the way, we now return to the setting of the present paper. From here on, f : M → M is the billiard map of the 2-D Lorentz gas as in Sect. 1.2. The following result lies at the heart of the approach taken in this paper: Proposition 2.1 (Theorem 6(a) of [Y]). The map f admits a generalized horseshoe with exponential return times. A few more definitions are needed before we are equipped to state our main proposition: We call Q ⊂ M a rectangular region if ∂ Q = ∂ u Q ∪ ∂ s Q, where ∂ u Q consists of two unstable curves and ∂ s Q two stable curves. We let Q( ) denote the smallest rectangular region containing , and define µu ( ) := inf γ ∈ u µγ ( ∩ γ ). Finally, for a generalized horseshoe ( , R) respecting a hole H , we define n( , R; H ) = sup{n ∈ Z+ : no point in falls into H in the first n iterates}. In the rest of this paper, C and α will be the constants in (P3)–(P5) for the closed system f . All notation is as in Sect. 1.2. Proposition 2.2. Given an infinitesimal hole {q0 } of Type I or II, there exist C0 , κ > 0, θ0 ∈ (0, 1), and a rectangular region Q such that for all small enough h we have the following: (a) For each σ ∈ h (q0 ), (i) f admits a generalized horseshoe ( (σ ) , R (σ ) ) respecting Hσ ; (ii) both ( (σ ) , R (σ ) ) and the corresponding surviving dynamics are mixing. (b) All σ ∈ h (q0 ) have the following uniform properties: (i) Q( (σ ) ) ≈ Q 1 , and µu ( (σ ) ) ≥ κ; (ii) µγ {R (σ ) > n} < C0 θ0n for all n ≥ 0; (iii) (P3)–(P5) hold with the constants C and α. ¯ → ∞ as h → 0. Moreover, if n(h) ¯ = inf σ ∈ h (q0 ) n( , R; Hσ ), then n(h) Clarification: 1. Here and in Sect. 3, there is a set, namely Hσ , that is identified to be “the hole,” and a horseshoe is constructed to respect it. Notice that the construction is continued after a set enters Hσ . For reasons to become clear in Sect. 6, we cannot simply disregard those parts of the phase space that lie in the forward images of Hσ . 1 By Q( (σ ) ) ≈ Q, we only wish to convey that both rectangular regions are located in roughly the same region of the phase space, M, and not anything technical in the sense of convergence.

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2. Proposition 2.2 treats only small h, i.e. small holes. The smallness of the holes and the uniformness of the estimates in part (b) are needed for the spectral arguments in Sect. 4 to apply. Without any restriction on h, all the conclusions of Proposition 2.2 remain true except for the following: (a)(ii), where for large holes the surviving dynamics need not be mixing, (b)(i), and (b)(ii), where C0 and θ0 may be σ -dependent. The assertions for large h will be evident from our proofs; no separate arguments will be provided. A proof of Proposition 2.2 will require that we repeat the construction in the proof of Proposition 2.1 – and along the way, to carry out a treatment of holes and related issues. We believe it is more illuminating conceptually (and more efficient in terms of journal pages) to focus on what is new rather than to provide a proof written from scratch. We will, therefore, proceed as follows: The rest of this section contains a review of all the arguments used in the proof of Proposition 2.1, with technical estimates omitted and specific references given in their place. A proof of Proposition 2.2 is given in Sect. 3. There we go through the same arguments point by point, explain where modifications are needed and treat new issues that arise. For readers willing to skip more technical aspects of the analysis not related to holes, we expect that they will get a clear idea of the proof from this paper alone. For readers who wish to see all detail, we ask that they read this proof alongside the papers referenced. 2.3. Outline of construction in [Y]. In this subsection, the setting and notation are both identical to that in Sect. 8 of [Y]. Referring the reader to [Y] for detail, we identify below 7 main ideas that form the crux of the proof of Proposition 2.1. We will point out the use of billiard properties and other geometric facts that may potentially be impacted by the presence of holes. Holes are not discussed explicitly, however, until Sect. 3. Notation and conventions. In [Y], S0 and ∂ M were used interchangeably. Here we use exclusively ∂ M. Clearly, f −1 ∂ M is the discontinuity set of f . (i) u- and s-curves. Invariant cones C u and C s are fixed at each point, and curves all of whose tangent vectors are in C u (resp. C s ) are called u-curves (resp. s-curves). (ii) The p-metric. Euclidean distance on M is denoted by d(·, ·). Unless declared otherwise, distances and derivatives along u- and s-curves are measured with respect to a semi-metric called the p-metric defined by cos ϕdr . These two metrics are 1 related by cp(x, y) ≤ d(x, y) ≤ p(x, y) 2 . By Wδu (x), we refer to the piece of local unstable curve of p-length 2δ centered at x. (P3)–(P5) in Sect. 2.1 hold with respect to the p-metric. See Sect. 8.3 in [Y] for details. (iii) Derivative bounds. With respect to the p-metric, there is a number λ > 1 so that all vectors in C u are expanded by ≥ λ and all vectors in C s contracted by ≤ λ−1 . Furthermore, derivatives at x along u-curves are ∼ d(x, ∂ M)−1 . For purposes of distortion control, homogeneity strips of the form 1 1 π π , k ≥ k0 , Ik = (r, ϕ) : − 2 < ϕ < − 2 k 2 (k + 1)2 are used, with {I−k } defined similarly in a neighborhood of ϕ = − π2 . For convenience, we will refer to M \ (∪|k|≥k0 Ik ) as one of the “Ik ”. Important Geometric Facts (†). The following facts are used many times in the proof: (a) the discontinuity set f −1 ∂ M is the union of a finite number of compact piecewise smooth decreasing curves, each of which stretches from {ϕ = π/2} to {ϕ = −π/2};

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(b) u-curves are uniformly transversal (with angles bounded away from zero) to ∂ M and to f −1 ∂ M. 1. Local stable and unstable manifolds. Only homogeneous local stable and unstable curves are considered. Homogeneity for Wδu , for example, means that for all n ≥ 0, f −n Wδu lies in no more than 3 contiguous Ik . Let δ1 > 0 be a small number to be 1

chosen. We let λ1 = λ 4 , δ = δ14 , and define

for all n ≥ 0}, Bλ+1 ,δ1 = {x ∈ M : d( f n x, ∂ M ∪ f −1 (∂ M)) ≥ δ1 λ−n 1 Bλ−1 ,δ1 = {x ∈ M : d( f −n x, ∂ M ∪ f (∂ M)) ≥ δ1 λ−n for all n ≥ 0}. 1

We require d( f n x, f −1 (∂ M)) ≥ δ1 λ−n 1 to ensure the existence of a local unstable curve through x, while the requirement on d( f n x, ∂ M) is to ensure its homogeneity.2 Similar s (x) is well defined reasons apply to stable curves. Observe that (i) for all x ∈ Bλ+1 ,δ1 , W10δ and homogeneous (this is straightforward since δ δ homogeneous Wloc on both sides of the curves in s . The set , which is defined to be (∪ u ) ∩ (∪ s ), clearly has a hyperbolic product structure. (P5)(b) is standard. This together with the choice of x1 guarantees µγ {γ ∩ } > 0 for γ ∈ u , completing the proof of (P1). A natural definition of separation time for x, y ∈ γ u is as follows: Let [x, y] be the subsegment of γ u connecting x and y. Then f n x and f n y are “not yet separated,” i.e. s0 (x, y) ≥ n, if for all i ≤ n, f i [x, y] is connected and is contained in at most 3 contiguous Ik . With this definition of s0 (·, ·), (P3)–(P5)(a) are checked using previously known billiard estimates. 3. The return map f R : → . We point out that there is some flexibility in choosing the return map f R : Certain conditions have to be met when a return takes place, but when these conditions are met, we are not obligated to call it a return; in particular, R is not necessarily the first time an s-subrectangle of Q u-crosses Q, where Q = Q( ). ˜ n = n \ {R ≤ n}. On ˜ n is a partition P˜ n whose We first define f R on ∞ . Let elements are segments representing distinct trajectories. The rules are different before and after a certain time R1 , a lower bound for which is determined by λ1 , δ1 and the derivative of f .4 2 In fact, provided δ is chosen sufficiently small, one can verify that d( f n x, f −1 (∂ M)) ≥ δ λ−n implies 1 1 1 −(n+1) for all n ≥ 0. This fact, which was not used in [Y], will be used in item 2 that d( f n+1 x, ∂ M) ≥ δ1 λ1

below to simplify our presentation. 3 Later we will impose one further technical condition on the choice of x . See the very end of Sect. 2.4. 1 4 In [Y], properties of R are used in 4 places: (I)(i) in Sect. 3.2, Sublemma 3 in Sect. 7.3, the paragraph 1 following (**) in Sect. 8.4, and a requirement in Sect. 8.3 that stable manifolds pushed forward more than R1 times are sufficiently contracted.

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(a) For n < R1 , P˜ n is constructed from the results of the previous step5 as follows: Let ω ∈ P˜ n−1 , and let ω be a component of ω ∩ n . Inserting cut-points only where necessary, we divide ω into subsegments ωi with the property that f n (ωi ) is homogeneous. These are the elements of P˜ n . No point returns before time R1 . (b) For n ≥ R1 , we proceed as in (a) to obtain ωi . If f n (ωi ) u-crosses the middle of Q with ≥ 1.5δ sticking out on each side, then we declare that R = n on ωi ∩ f −n , and the elements of P˜ n |ωi ∩˜ n are the connected components of ωi \ f −n . Otherwise put ωi ∈ P˜ n as before. This defines R on a subset of ∞ (which we do not know yet has full measure); the defis -curves. nition is extended to the associated s-subset of by making R constant on Wloc −n The s-subsets associated with ωi ∩ f in (b) above are the j in (P2). It remains to check that f R ( j ) is in fact a u-subset of . This is called the “matching of Cantor sets” in [Y] and is a consequence of the fact that ∞ is dynamically defined and that R1 is chosen sufficiently large. It remains to prove that p{R ≥ n} decays exponentially with n. Paragraphs 4, 5 and 6 contain the 3 main ingredients of the proof, with the final count given in 7. 4. Growth of u-curves to “long” segments. This is probably the single most important point, so we include a few more details. We first give the main idea before adapting it to the form it is used. Let ε0 > 0 be a number the significance of which we will explain later. Here we think of a u-curve whose p-length exceeds ε0 > 0 as “long”. Consider a u-curve ω. We introduce a stopping time T on ω as follows. For n = 1, 2, . . ., we divide f n ω into homogeneous segments representing distinguishable trajectories. For x ∈ ω, let T (x) = inf{n > 0 : the segment of f n ω containing f n x has p−length > ε0 }. Lemma 2.3. There exist D1 > 0 and θ1 < 1 such that for any u-curve ω, p(ω \ {T ≤ n}) < D1 θ1n

for all n ≥ 1.

This lemma relies on the following important geometric property of the class of billiards in question. This choice of ε0 > 0 is closely connected to this property: n f −i (∂ M) passing through or (*) ([BSC1], Lemma 8.4) The number of curves in ∪i=1 ending in any one point in M is ≤ K 0 n, where K 0 is a constant depending only on the “table” X . 1 Let α0 := 2 ∞ k=k0 k 2 , where {Ik , |k| ≥ k0 } are the homogeneity strips, and assume 1

that λ−1 + α0 < 1. Choose m large enough that θ1 := (K 0 m + 1) m (λ−1 + α0 ) < 1. u -curve of p-length ≤ ε We may then fix ε0 < δ to be small enough that every Wloc 0 −i has the property that it intersects ≤ K 0 m smooth segments of ∪m 1 f (∂ M), so that the u -curve has ≤ (K m + 1) connected components. f m -image of such a Wloc 0 The proof of Lemma 2.3, which follows [BSC2], goes as follows: Consider a large n, which we may assume is a multiple of m. (Once Lemma 2.3 is proved for multiples of m, the estimate can be extended to intermediate values by enlarging the constant D1 .) 5 In [Y], it was sufficient to allow returns to at times that were multiples of a large fixed integer m. Not only is this not necessary (see Paragraph 4), here it is essential that we avoid such periodic behavior to ensure mixing. Thus we take m = 1 when choosing return times in Paragraph 3. This is the only substantial departure we make from the construction in [Y].

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We label distinguishable trajectories by their Ik -itineraries. Notice that because f i ω is the union of a number of (disconnected) u-curves, it is possible for many distinguishable trajectories to have the same Ik -itinerary. Specifically, by (*), each trajectory of length jm, j ∈ Z+ , gives birth to at most (K 0 m + 1) trajectories of length ( j + 1)m with the same Ik -itinerary. To estimate p(ω \ {T ≤ n}), we assume the worst case scenario, in which the f n -images of subsegments of ω corresponding to all distinguishable trajectories have length ≤ ε0 . We then sum over all possible itineraries using bounds on D f along u-curves in Ik . We now adapt Lemma 2.3 to the form in which it will be used. Let ω = f k ω for some ω ∈ P˜ k in the construction in Paragraph 3. As we continue to evolve ω, f n ω is not just chopped up by the discontinuity set, bits of it that go near f −1 (∂ M) will be lost by intersecting with f k+n k+n , and we need to estimate p(ωn \ {T ≤ n}), where ωn := ω ∩ f k (k+n ) takes into consideration these intersections and T is redefined accordingly. A priori this may require a larger bound than that given in Lemma 2.3: it is conceivable that there are segments that will grow to length ε0 without losing these “bits” but which do not now reach this reference length. We claim that all such segments have been counted, because (i) the deletion procedure does not create new connected components; it merely trims the ends of segments adjacent to cut-points; and (ii) the combinatorics in Lemma 2.1 count all possible itineraries (and not just those that lead to “short” segments). This yields the desired estimate on p(ωn \ {T ≤ n}), which is Sublemma 2 in Sect. 8.4 of [Y]. 5. Growth of “gaps” of . Let ω be the subsegment of some γ u ∈ u connecting the two s-boundaries of Q. We think of this as a return in the construction outlined in Paragraph 3, with the connected components ω of ωc = ω \ being f k -images of elements of P˜ k . We define a stopping time T on ωc by considering one ω at a time and defining on it the stopping time in Paragraph 4. Lemma 2.4. There exist D2 > 0 and θ2 < 1 independent of ω such that p(ωnc \ {T ≤ n}) < D2 θ2n

for all n ≥ 1.

The idea of the proof is as follows. We may identify ω with (see Paragraph 2), so that the collection of ω is precisely the collection of gaps in . We say ω is of generation q if this is the first time a part of ω is removed in the construction of ∞ . There are two separate estimates: (I ) := p(ω ); (I I ) := p(ωn \{T ≤ n}). q>εn gen(ω )=q

q≤εn gen(ω )=q

(I) has exponentially small p-measure: this follows from a comparison of the growth rate of D f along u-curves versus the rate at which these curves get cut (see Paragraph 4). (II) is bounded above by

q≤εn gen(ω )=q

C p(ω ) n−q−1 · D1 θ1 . p( f q−1 ω )

This is obtained by applying the modified version of Lemma 2.3 to f q−1 ω . A lower bound on p( f q−1 ω ) can be estimated as these curves have not been cut by f −1 (∂ M) (though they may have been shortened to maintain homogeneity), reducing the estimate to q gen(ω )=q p(ω ), which is ≤ p(ω).

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6. Return of “long” segments. This concerns the evolution of unstable curves after they have grown “long”, where “long” has the same meaning as in Paragraph 4. The following geometric fact from [BSC2] is used: u -curve ω with p(ω) > ε and (**) Given ε0 > 0, ∃n 0 s.t. for every homogeneous Wloc 0 q every q ≥ n 0 , f ω contains a homogeneous segment which u-crosses the middle half of Q with > 2δ sticking out from each side.

We choose ε0 > 0 as explained in Paragraph 4 above, and apply (**) with q = n 0 to the segments that arise in Paragraphs 4 and 5 when the stopping time T is reached. For example, ω here may be equal to f n ω , where ω is a subsegment of the ω in the last paragraph of Paragraph 4 with T |ω = n. We claim that a fixed fraction of such a segment will make a return within n 0 iterates. To guarantee that, two other facts need to be established: (i) The small bits deleted by intersecting with f n+k n+k before the return still leave a segment which u-crosses the middle half of Q with > 1.5δ sticking out from each side; this is easily checked. (ii) For q ≤ n 0 , ( f q ) is uniformly bounded on f −q -images of homogeneous segments that u-cross Q. This is true because a segment contained in Ik for too large a k cannot grow to length δ in n 0 iterates. 7. Tail estimate of return time. We now prove p{R ≥ n} ≤ C0 θ0n for some θ0 < 1. On , introduce a sequence of stopping times T1 < T2 < · · · as follows: A stopping time T of the type in Paragraph 4 or 5 is initiated on a segment as soon as Tk is reached, and Tk+1 is set equal to Tk + T . In this process, we stop considering points that are lost to deletions or have returned to . The desired bound follows immediately from the following two estimates: (i) There exists ε > 0, D3 ≥ 1, and θ3 < 1 such that p(T[ε n] > n) < D3 θ3n . (ii) There exists ε1 > 0 such that if Tk |ω = n, then p(ω∩{R > n +n 0 }) ≤ (1−ε1 ) p(ω), where n 0 is as in (**) in Paragraph 6. (ii) is explained in Paragraph 6. To prove (i), we let p = [ε n], decompose into sets of the form A(k1 , . . . , k p ) = {x ∈ : T1 (x), . . . , T p (x) are defined with Ti = ki }, apply Lemmas 2.1 and 2.2 to each set and recombine the results. The argument here is combinatorial, and does not use further geometric information about the system. 2.4. Sketch of proof of (**) following [BSC2]. Property (**) is a weaker version of Theorem 3.13 in [BSC2]. We refer the reader to [BSC2] for detail, but include an outline of its proof because a modified version of the argument will be needed in the proof of Proposition 2.2. We omit the proof of the following elementary fact, which relies on the geometry of the discontinuity set including Property (*): Sublemma A. Given any u-curve γ , through µγ -a.e. x ∈ γ passes a homogeneous s (x) for some δ(x) > 0. The analogous statement holds for s-curves. Wδ(x) u -curve as required in (**), the problem is reduced Instead of considering every Wloc to a finite number of “mixing boxes” U1 , U2 , . . . , Uk with the following properties:

(i) U j is a hyperbolic product set defined by (homogeneous) families u (U j ) and s (U j ); located in the middle third of U j is an s-subset U˜ j with ν(U˜ j ) > 0; (ii) ∪ u (U j ) fills up nearly 100% of the measure of Q(U j ); and u -curve ω with p(ω) > ε passes through the middle third of one of the (iii) every Wloc 0 Q(U j ) in the manner shown in Fig. 1 (left).

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Fig. 1. Left: A mixing box U j . Right: The target box U0

That (i) and (ii) can be arranged follows from Sublemma A. That a finite number of U j suffices for (iii) follows from a compactness argument. Next we choose a suitable subset U˜ 0 ⊂ to be used in the mixing. To do that, first pick a hyperbolic product set U0 related to Q( ) as shown in Fig. 1 (right). We require that it meet Q( ) in a set of positive measure, that it sticks out of Q( ) in the u-direction by more than 2δ, and that the curves in u (U0 ) fill up nearly 100% of Q(U0 ). Let 0 > 0 be a small number, and let U˜ 0 ⊂ U0 consist of those density points of U0 ∩ Q( ) with the additional property that if a homogeneous stable curve γ s with p(γ s ) < 0 meets such a point, then p(γ s ∩ U0 )/ p(γ s ) ≈ 1. For 0 small enough, ν(U˜ 0 ) > 0 because the u -curves is absolutely continuous. foliation into Wloc By the mixing property of ( f, ν), there exists n 0 such that for all q ≥ n 0 , ν( f q (U˜ j ) ∩ U˜ 0 ) > 0 for every U˜ j . We may assume also that n 0 is so large that for q ≥ n 0 , if x ∈ U˜ j is such that f q x ∈ U˜ 0 , then p( f q (γ s (x))) < 0 , where γ s (x) is the stable curve in s (U˜ j ) passing through x. Let q ≥ n 0 and j be fixed, and let x ∈ U˜ j be as above. From the high density of unstable curves in both U j and U0 , we are guaranteed that there are two elements γ1u , γ2u ∈ u (U j ) sandwiching the middle third of Q(U j ) such that for each i, a subsegment of γiu containing γ s (x) ∩ γiu is mapped under f q onto some γˆiu ∈ u (U0 ). Let Q ∗ = Q ∗ (q, j) be the u-subrectangle of Q(U0 ) with ∂ u Q ∗ = γˆ1u ∪ γˆ2u . Sublemma B. f −q | Q ∗ is continuous, equivalently, Q ∗ ∩ (∪0 f i (∂ M)) = ∅. q

Sublemma B is an immediate consequence of the geometry of the discontinuity set: q By the choice of x1 in item 2 of Sect. 2.3, Q ∗ ∩∂ M = ∅. Suppose Q ∗ ∩(∪1 f i (∂ M)) = ∅. q i Since ∪1 f (∂ M) is the union of finitely many piecewise smooth (increasing) u-curves each connected component of which stretches from {ϕ = −π/2} to {ϕ = π/2}, and q these curves cannot touch ∂ u Q ∗ , a piecewise smooth segment from ∪1 f i (∂ M) that ∗ s ∗ enters Q through one component of ∂ Q must exit through the other. In particular, it must cross f q γ s (x), which is a contradiction. u -curve with p(ω) > ε . We pick U so that ω passes To prove (**), let ω be a Wloc 0 j through the middle third of U j as in (iii) above. Sublemma B then guarantees that f q (ω∩ f −q Q ∗ ) connects the two components of ∂ s Q ∗ . This completes the proof of (**), except that we have not yet verified that f q (ω ∩ f −q Q ∗ ) is homogeneous. To finish this last point, we modify the above argument as follows: First, we define a u curve γ to be strictly homogeneous if for all n ≥ 0, f −n γ is contained inside one Wloc s curves is defined analogously. The homogeneity strip Ik (n). Strict homogeneity for Wloc conclusions of Sublemma A remain valid if, in its statement, the word “homogeneous” is replaced by “strictly homogeneous.” Thus the mixing boxes U1 , . . . , Uk can be chosen so that their defining families are comprised entirely of strictly homogeneous local manifolds. Furthermore, if x1 is also chosen as a density point of points with sufficiently long strictly homogeneous unstable curves, u (U0 ) can be chosen to be comprised entirely u -curves. Having done this, an argument very similar to of strictly homogeneous Wloc

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the proof of Sublemma B shows that f −i Q ∗ ∩ (∪k ∂ Ik ) = ∅ for 0 ≤ i ≤ q, and this completes the proof of (**). 3. Horseshoes Respecting Holes for Billiard Maps 3.1. Geometry of holes in phase space. We summarize here some relevant geometric properties and explain how we plan to incorporate holes into our horseshoe construction. Holes of Type I. Recall from Sect. 1.2 that for q0 ∈ ∪i and σ ∈ h (q0 ), Hσ ⊂ M is a rectangle of the form (a, b) × [− π2 , π2 ]. We define ∂ Hσ := {a, b} × [− π2 , π2 ], i.e. ∂ Hσ is the boundary of Hσ viewed as a subset of M. It will also be convenient to let H0 ⊂ M denote the vertical line {q0 } × [− π2 , π2 ]. To construct a horseshoe respecting Hσ , it is necessary to view two nearby points as having separated when they lie on opposite sides of ∂ Hσ or on opposite sides of Hσ in M \ Hσ . Thus it is convenient to view f −1 (∂ Hσ ) as part of the discontinuity set of f . For simplicity, consider first the case where q0 does not lie on a line in the table X tangent to more than one scatterer. Then f −1 (∂ Hσ ) is a finite union of pairs of roughly parallel, smooth s-curves. (Recall that s-curves are negatively sloped, with slopes uniformly bounded away from 0 and −∞.) Each of the curves comprising f −1 (∂ Hσ ) begins and ends in ∂ M ∪ f −1 (∂ M), that is to say, the geometric properties of f −1 (∂ Hσ ) ∪ f −1 (∂ M) are similar to those of f −1 (∂ M). Likewise, f (∂ Hσ ) is a finite union of pairs of (increasing) u-curves that begin and end in ∂ M ∪ f (∂ M), and it will be convenient to regard that as part of the discontinuity set of f −1 . Let Nε (·) denote the ε-neighborhood of a set. We will need the following lemma. Lemma 3.1. For each ε > 0 there is an h > 0 such that for each σ ∈ h , Hσ ⊂ Nε (H0 ), f Hσ ⊂ Nε ( f H0 ), and f −1 Hσ ⊂ Nε ( f −1 H0 ). As f is discontinuous, Lemma 3.1 is not immediate. However, it can be easily verified, and we leave the proof to the reader. Points q0 that lie on lines in X with multiple tangencies to scatterers lead to slightly more complicated geometries, and special care is needed when defining what is meant by f H0 and f −1 H0 . For example, consider the case where q0 ∈ 3 lies on a line that

Fig. 2. An infinitesmal hole aligned with multiple tangencies. Left: q0 lies on a line segment in the billiard table X that is tangent to two scatterers. Right: Induced singularity curves in the subset 1 × [−π/2, π/2] of the phase space M

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is tangent to 1 and 2 , but which is not tangent to any other scatterer including 3 . Suppose further that r1 ∈ 1 , r2 ∈ 2 are the points of tangency, that r2 is closer to q0 than r1 is, that no other scatterer touches the line segment [q0 , r1 ], and that 1 and 2 both lie on the same side of [q0 , r1 ]; see Fig. 2 (left). Let σ be a small hole of Type I with q0 ∈ σ . Then in 2 × [−π/2, π/2], f −1 (∂ Hσ ) appears as described above. However, 2 “obstructs” the view of σ from 1 , and so in 1 × [−π/2, π/2], f −1 (Hσ ) is a small triangular region whose three sides are composed of a segment from 1 × {π/2}, a segment from f −1 (2 × {π/2}), and a single segment from f −1 (∂ Hσ ). See Fig. 2 (right). As a consequence, when we write f −1 H0 , we include in this set not just (r2 , π/2), but also f −1 (r2 , π/2) = (r1 , π/2). This is necessary in order for Lemma 3.1 to continue to hold. Aside from such minor modifications, the case of multiple tangencies is no different than when they are not present, and we leave further details to the reader. Holes of Type II. For simplicity, consider first the case where q0 does not lie on a line in the “table” X tangent to more than one scatterer. Recall from Sect. 1.2 that “the hole” Hσ here is taken to be f (Bσ ), where Bσ consists of points in M which enter σ ×S1 under the billiard flow before returning to the section M. As with holes of Type I, we define ∂ Hσ to be the boundary of Hσ viewed as a subset of M. The set Bσ as a subset of M has similar geometric properties as f −1 Hσ for Type I holes, i.e., f −1 (∂ Hσ )\(∂ M ∪ f −1 (∂ M)) consists of pairs of negatively sloped curves ending in ∂ M ∪ f −1 (∂ M). The slopes of these curves are uniformly bounded (independent of σ ) away from −∞ and 0. For the reasons discussed, it will be convenient to view this set as part of the discontinuity set of f . The infinitesimal hole H0 ⊂ M is defined in the natural way, and the analog of Lemma 3.1 can be verified. We will say more about the geometry of Hσ in Sect. 3.3. Points q0 that lie on multiple tangencies lead to slightly more complicated geometries, and special care is needed when defining what is meant by the sets f −1 H0 , H0 , and f H0 as in the case of Type I holes. Further generalizations on holes of Type II: In addition to the generalizations discussed in Sect. 1.3, sufficient conditions on the holes allowed in h for Prop. 2.2 to remain true are the following, as can be seen from our proofs: (1) There exist N and L for which the following hold for all sufficiently small h: (a) f −1 (∂ Hσ ), ∂ Hσ , and f (∂ Hσ ) each consist of no more than N smooth curves, all of which have length no greater than L. (b) For each σ ∈ h , f −1 (∂ Hσ )\(∂ M ∪ f −1 (∂ M)) consists of piecewise smooth, negatively sloped curves (with slopes uniformly bounded away from −∞ and 0), and the end points of these curves must lie on ∂ M ∪ f −1 (∂ M). (2) The analog of Lemma 3.1 holds. Thus it would be permissible to allow a convex hole σ to be in h that did not have a C 3 simple closed curve with strictly positive curvature as its boundary. For example, conditions (a) and (b) above hold if ∂σ is a piecewise C 3 simple closed curve which consists of finitely many smooth segments that are either strictly positively curved or flat. As another generalization, consider the case when any line segment in the table X with its endpoints on two scatterers that passes through the convex hull of σ also intersects σ . Then it is no loss of generality to replace σ by its convex hull. Using this, one can often verify that the set Hσ that arises satisfies properties (a) and (b) above, even if σ is not itself convex. See Fig. 3. In Sect. 3.2, the discussion is for holes of Type I with a single interval deleted. The proof follows mutatis mutandis for holes of Type II, with the necessary minor modifications discussed in Sect. 3.3.

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Fig. 3. Examples of Type II holes that are permissible

3.2. Proof of Proposition 2.2 (for holes of Type I). The idea of the proof is as follows. First we construct a horseshoe ( (0) , R (0) ) with the desired properties for the infinitesimal hole {q0 }. Then we construct ( (σ ) , R (σ ) ) for all σ ∈ h (q0 ), and show that with (σ ) sufficiently close to (0) in a sense to be made precise, ( (σ ) , R (σ ) ) will inherit the desired properties with essentially the same bounds. To ensure that (σ ) can be taken “close enough” to (0) , we decrease the size of the hole, i.e., we let h → 0. Now the constructions of ( (0) , R (0) ) and ( (σ ) , R (σ ) ) are essentially identical. To avoid repeating ourselves more than needed, we will carry out the two constructions simultaneously. It is useful to keep in mind, however, that logically, the case of the infinitesimal hole is treated first, and some of the information so obtained is used to guide the arguments for positive-size holes. As explained in Sect. 3.1, to ensure that the horseshoe respects the hole, it is convenient to include f −1 (∂ Hσ ) as part of the discontinuity set for f . Since Hσ will be viewed as a perturbation of H0 , we include f −1 (H0 ) in this set as well. The following convention will be adopted when we consider a system with hole Hσ : (a) Suppose for definiteness q0 ∈ 1 . The new phase space Mσ is obtained from M by cutting 1 × [− π2 , π2 ] along the lines comprising H0 ∪ ∂ Hσ , splitting it into three connected components. (b) As a consequence, the new discontinuity set of f is f −1 (∂ Mσ ), and the new discontinuity set of f −1 is f (∂ Mσ ). We use the notation “σ = 0” for the infinitesimal hole, so that M0 is obtained from M by cutting along H0 . Notice immediately that this changes the definitions of stable and unstable curves, in the sense that if γ was a stable curve for the system without holes, then γ continues to be a stable curve if and only if (i) γ ∩ ∂ Mσ = ∅, and (ii) f n (γ ) ∩ f −1 ∂ Mσ = ∅ for all n ≥ 0; a similar characterization holds for unstable curves. All objects constructed below will be σ -dependent, but we will suppress mention of σ except where it is necessary. Observe also that the Important Geometric Facts (†) in Sect. 2.3 with f −1 ∂ Mσ instead of f −1 ∂ M as the new discontinuity set remains valid. We now follow sequentially the 7 points outlined in Sect. 2.3 and discuss the modifications needed. These modifications, along with two additional points (8 and 9) form a complete proof of Proposition 2.2. We believe we have prepared ourselves adequately in Sects. 2.3 and 2.4 so that the discussion to follow can be understood on its own, but encourage readers who wish to see proofs complete with all technical detail to read the rest of this section alongside the relevant parts of [Y and BSC2]. The notation within each item below is as in Sect. 2.3.

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369 (σ )±

1. The relationships λ = λ41 and δ = δ14 are as before, and the sets Bλ1 ,δ1 are defined in a manner similar to that in Sect. 2.3. For example, Bλ(σ1 ,δ)+1 = {x ∈ Mσ : d(x, ∂ Mσ ) ≥ δ1 and d( f n x, f −1 ∂ Mσ ) ≥ δ1 λ−n 1 ∀n ≥ 0}. As in Sect. 2.3, the condition on d( f n x, f −1 ∂ Mσ ) is to ensure the existence of stable curves, and the necessity for x to be away from ∂ Mσ is obvious (cf. footnote in item 1 of Sect. 2.3). Properties (i) and (ii) continue to hold for each σ given the geometry of the new discontinuity set. With regard to the choice of δ1 , we let δ1 be as in [Y], (0)+ (0)− and shrink it if necessary to ensure that Bλ1 ,2δ1 ∩ Bλ1 ,2δ1 has positive ν-measure away from f −1 (∂ M0 ) ∪ ∂ M0 ∪ f (∂ M0 ). This is where the sets (σ ) will be located (see Paragraph 2). (σ )± (0)± The following lemma relates Bλ1 ,δ1 and Bλ1 ,δ1 : (σ )±

(0)±

Lemma 3.2. (i) For all σ ∈ h , we have Bλ1 ,δ1 ⊂ Bλ1 ,δ1 . (ii) As h → 0, sup ν(Bλ(0)+ \ Bλ(σ1 ,δ)+1 ) → 0, 1 ,δ1

sup ν(Bλ(0)− \ Bλ(σ1 ,δ)−1 ) → 0. 1 ,δ1

σ ∈ h

σ ∈ h

Proof. (i) follows immediately from ∂ Mσ ⊃ ∂ M0 . As for (ii), let ε > 0 be given. Recall that Nα (·) denotes the α-neighborhood of a set. By Lemma 3.1 we may choose h small enough that for all σ ∈ h , ∂ Hσ ∈ Nε (H0 ) and f −1 (∂ Hσ ) ∈ Nε ( f −1 H0 ). Then if (0)+ (σ )+ x ∈ Bλ1 ,δ1 \ Bλ1 ,δ1 , either x ∈ Nδ1 (∂ Hσ ) \ Nδ1 (H0 ), or x ∈ ∪n≥0 f −n (Nδ1 λ−n ( f −1 ∂ Hσ ) \ Nδ1 λ−n ( f −1 H0 )). 1

1

We estimate the ν-measure of the right side separately for ∪n≥n ε and ∪n 0 is determined by properties of ( (0) , R (0) ) (requirements will appear below, and

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in items 6 and 9). Suffice it to say here that however small δ2 may be, Lemma 3.2 guarantees that this can be done by shrinking h. Once x1(σ ) is chosen, we set = Wδu (x1(σ ) ) and n = {y ∈ : d( f i y, f −1 (∂ Mσ )) ≥ δ1 λ−i 1 for 0 ≤ i ≤ n}. Then the sets ∞ , s , u and (σ ) are constructed as before. (σ ) (0) That Q( (σ ) ) ≈ Q( (0) ) follows immediately from the proximity of x1 to x1 . u (σ ) u (0) Since δ is fixed, µ ( ) ≈ µ ( ) > 0 can be arranged by taking δ2 sufficiently small and using Lemma 3.2 with h sufficiently small. This proves Proposition 2.2(b)(i). With the separation time happening sooner due to the enlarged discontinuity set, (P3)– (P5) remain true with the same C and α for the closed system; in other words, Proposition 2.2(b)(iii) requires no further work. 3. To arrange for mixing properties (not done in [Y]), we will need to delay the return times to by forbidding returns before time R2 for some R2 ≥ R1 determined by ( (0) , R (0) ); see Lemma 3.4. This aside, the construction of f R is as before. The matching of Cantor sets argument should be looked at again since the Cantor sets are different, but the proof goes through as before because the sets are dynamically defined. Notice that for ω ∈ P˜ n , f i ω is either entirely in the hole or outside of the hole, as is i f ( s ), where s is the s-subset of associated with ω, for 0 ≤ i ≤ n; this is a direct consequence of our taking the boundary of the hole into consideration in our definition of the discontinuity set. Together with the fact that is away from ∂ Mσ , it ensures that the generalized horseshoe we are constructing respects the hole. 4. This is where one of the more substantial modifications occurs: Lemma 2.3, which is based largely on the competition between expansion along u-curves and the rate at which they are cut, is clearly affected by the additional cutting due to our enlarged discontinuity set. The condition (*) in Sect. 2.3 must now be replaced by Lemma 3.3. There exists K 1 such that for any m ∈ Z+ , there exists ε0 > 0 with the property that for any u-curve with p(ω) < ε0 , f m (ω) has ≤ (K 1 m 2 + 4) connected components with respect to the enlarged discontinuity set. Proof. Let m ∈ Z+ be given. As in Sect. 2.3, choose ε0 > 0 small enough such that if ω is a u-curve with p(ω) < ε0 , f m (ω) has ≤ (K 0 m + 1) connected components with respect to the original discontinuity set f −1 S0 . Let ω j be the f −m -image of one of these connected components. This means that for 0 ≤ k ≤ m, f k (ω j ) is, in reality, a connected u-curve even though it may not be connected with respect to our enlarged discontinuity set. Since f k ω j is an (increasing) u-curve, it can meet the three vertical lines making up (∂ Hσ ) ∪ H0 in no more than three points. (As the slopes dϕ/dr of u-curves are never less than the curvature of i at r , connected u-curves cannot wrap around the cylinder i × [−π/2, π/2] and meet (∂ Hσ ) ∪ H0 more than once.) Hence the cardinality −k ((∂ H ) ∪ H )} is ≤ 3(m + 1), and as (∂ H ) ∪ H is the additional of {ω j ∩ m σ 0 σ 0 k=0 f set added to ∂ M to create ∂ Mσ , it follows that f m ω has ≤ (K 0 m + 1) · (3(m + 1) + 1) connected components with respect to the enlarged discontinuity set. Using Lemma 3.3, one adapts easily the proof of Lemma 2.3 to the present setup with 1 θ1 = (K 1 m 2 + 4) m (λ−1 + α0 ), where m is chosen large enough so that this number is < 1. The constant D1 depends only on the properties of D f and is unchanged. Hence Lemma 2.3 is valid with D1 and θ1 modified but independent of σ . As in Sect. 2.3, these estimates can then be adapted to estimate p(ωn \{T ≤ n}).

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5. Lemma 2.4 remains valid with modified constants which are independent of σ . Returning to the sketch of the proof provided in Sect. 2.3, we see that both sets of estimates boil down to the geometry of the new discontinuity set and the rates of growth versus cutting, which has been taken care of for the enlarged discontinuity set in Paragraph 4 above. 6. We need to show that there exist n 1 and ε1 > 0 independent of σ such that for every homogeneous u-curve with p-length > ε0 , a fraction ≥ ε1 of ω returns within the next n 1 steps. Before we enlarged the discontinuity set, this property followed from property (**) in Sect. 2.3. We replace (**) here with the following: Lemma 3.4. Given ε0 > 0, provided h and δ2 are sufficiently small, there exists n 1 u -curve ω such that the following holds for each σ ∈ h : for every homogeneous Wloc with p(ω) > ε0 and each q ∈ {n 1 , n 1 + 1}, f q ω contains a homogeneous segment that u-crosses the middle half of Q( (σ ) ) with greater than 2δ sticking out from each side. Once Lemma 3.4 is proved, the fact that a fraction ε1 (independent of σ ) has the desired properties follows from derivative estimates as in Sect. 2.3 and our uniform lower bound on µu ( (σ ) ). The reason we want q to take two consecutive values in the statement of Lemma 3.4 has to do with the mixing property in item 9 below. Proof. Fix ε0 > 0. We first prove the following for the case σ = 0: u -curve ω with p(ω) > (**)’ For σ = 0, there exists n 1 such that any homogeneous Wloc q ε0 and every q ≥ n 1 , f ω contains a homogeneous segment that u-crosses the middle fourth of Q( (0) ) with greater than 4δ sticking out from each side.

The proof of (**)’ is completely analogous to the proof of (**) outlined in Sect. 2.4. Sublemmas A and B continue to hold due to the similar geometry of the discontinuity set. Notice that unlike (**)’, the assertion in Lemma 3.4 is only for q = n 1 and n 1 + 1, so that its proof involves only a finite number of mixing boxes U j and a finite number of iterates. This will be important in the perturbative argument to follow. Consider now σ = 0, and consider a homogeneous unstable curve ω with p(ω) > ε0 . First, ω continues to be an unstable curve with respect to the discontinuity set f −1 ∂ M0 , so by the proof of (**)’, for q ∈ {n 1 , n 1 + 1} and every j, there is a rectangular region Q ∗ = Q ∗ (q, j) such that (i) Q ∗ u-crosses the middle fourth of Q( (0) ) with > 4δ sticking out, (ii) f −q Q ∗ is an s-subrectangle in the middle third of Q(U j ), and (iii) for i = 0, 1, · · · , q, f −i Q ∗ stays clear of f −1 ∂ M0 by some amount. Lemma 3.1 ensures that for h small enough, (iii) continues to hold with f −1 ∂ M0 replaced by f −1 ∂ Mσ . Finally, provided δ2 is small enough, (i) holds for Q( (σ ) ) with > 2δ sticking out on each side. 7. Once steps 4, 5 and 6 have been completed, the argument here is unchanged (as it is largely combinatorial), guaranteeing constants C0 and θ0 independent of σ with p{R ≥ n} ≤ C0 θ0n . This completes the proof of Proposition 2.2(a)(i) and (b)(ii). We have reached the end of the 7 steps outlined in Sect. 2.3. Two items remain: 8. That n(h) ¯ → ∞ as h → 0 is easy: Orbits from (σ ) start away from H0 and cannot −1 approach f H0 faster than a fixed rate. Thus using Lemma 3.1, we can arrange for orbits starting from (σ ) to stay out of Hσ for as long as we wish by taking h small. 9. The mixing of ( (σ ) , R (σ ) ) follows from Lemma 3.5. There exists R2 ≥ R1 (independent of σ ) such that for small enough h, the construction in Step 3 can be modified to give the following:

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(i) no returns are allowed before time R2 , and (ii) at both times R2 and R2 + 1, there are s-subsets of (σ ) making full returns. Proof. Again we first consider the case σ = 0. Here R2 is chosen as follows: Without allowing any returns, let R1 be the smallest time greater than or equal to R1 such that there exists ω ∈ P˜ R1 with p( f R1 ω) > ε0 > 0. With ε0 chosen as before, we take n 1 from Lemma 3.4 and set R2 = R1 + n 1 . Using Lemma 3.4, we find two subsegments ω and ω ⊂ ω such that f R2 ω and f R2 +1 ω are both homogeneous segments that u-cross the middle half of Q( (0) ) with greater than 2δ sticking out from each side. We may suppose that ω and ω are disjoint since f has no fixed points. They give rise to two s-subsets of (0) with the properties in (ii). From time R2 on, returns to (0) are allowed as before. When σ = 0, we follow the same procedure as above to ensure the mixing of ( (σ ) , R (σ ) ). The only concern is that R1 = R1 (σ ) (and hence also R2 = R1 + n 1 ) might not be independent of σ . This is not a problem as the construction above involves only a finite number of steps: With h and δ2 sufficiently small, the elements of P˜ n(σ ) can (0) be defined in such a way that they are in a one-to-one correspondence with those of P˜ n for n ≤ R1 (0). Finally, mixing of the surviving dynamics is ensured by choosing h small enough that n(h) ¯ > R2 + 1. This ensures that the s-subsets s that make full returns at times R2 and R2 + 1 cannot fall into the hole prior to returning. The proof of Proposition 2.2 for holes of Type I is now complete. 3.3. Modifications needed for holes of Type II. The proof for Type II holes is very similar to that for Type I holes. There are, however, some differences due to the more complicated geometry of ∂ Hσ . In the discussion below, we assume the infinitesimal hole {q0 } does not lie on any segment in the table tangent to more than one scatterer. The general situation is left to the reader. From the discussion of the geometry of Type II holes in Sect. 3.1, we see that the Important Geometric Facts (†) in Sect. 2.3 continue to hold with Mσ in the place of M, except that u-curves need not be transversal to the ∂ Hσ ∪ H0 part of ∂ Mσ . Potential problems that may arise are discussed below. The discontinuity set of f , i.e. f −1 ∂ Mσ , has the same geometric properties as before. We now go through the 9 points in Sect. 3.2. No modifications are needed in items 1–3. As expected, item 4 is where the most substantial modifications occur: Modifications in Item 4. Lemma 3.3 is still true as stated, but the geometry is different. In the discussion below related to this lemma, the discontinuity set refers to f −1 ∂ M, not the enlarged discontinuity set f −1 ∂ Mσ , and unstable curves are defined accordingly. For Type I holes, the proof relies on the fact that any (increasing) connected u-curve ω meets ∂ Hσ ∪ H0 , which is the union of three vertical lines, in at most three points. Lemma 3.6. Any unstable curve ω meets ∂ Hσ ∪ H0 in at most three points. Even though Lemma 3.3 is stated for u-curves, we need it only for unstable curves (and the argument here is slightly simpler for unstable curves). Proof. Let us distinguish between two different types of curves that comprise ∂ Hσ : Primary segments, which are the forward images of curves in ∂ Bσ \ f −1 (∂ M),

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Fig. 4. Representative examples of the geometry of Type II holes. Top set: On the left is a configuration on the billiard table X , while on the right is the resulting configuration in the subset 1 × [− π2 , π2 ] of phase space. In this subset, H0 consists of a single primary segment whose endpoints lie on f (2 × { π2 }) and 1 × { π2 }. For σ = 0, ∂ Hσ contains two primary segments and a single secondary segment that lies on f (2 × { π2 }). (Recall that by convention ∂ Hσ does not include subsegments of ∂ M.) Bottom set: The analogous situation when the view of 1 from q0 is obstructed by two scatterers, instead of just one. Observe that now ∂ Hσ contains two secondary segments in 1 × [− π2 , π2 ]. The situation when the view of 1 from q0 is unobstructed by other scatterers is simple and is left to the reader

and secondary segments, which are subsegments of f (∂ M). For examples, see Fig. 4. In general, when q0 does not lie on a line segment with multiple tangencies to the scatterers, secondary segments are absent in H0 , while each component of H0 gives rise to two primary segments in ∂ Hσ for σ = 0. To prove the lemma, observe first that H0 can have no more than one component in any connected component of M \ f (∂ M). Second, ω must also be entirely contained inside one connected component of M \ f (∂ M). This is because unstable curves for f cannot cross the discontinuity set of f −1 . As a consequence, ω also cannot cross any secondary segment as secondary segments of ∂ Hσ are contained in f (∂ M). It remains to show that ω can meet each primary segment in at most one point. Although primary segments are increasing, their tangent vectors lie outside of unstable cones (except at ∂ M where the unstable cone is degenerate). This is because the curves in ∂ Bσ \ f −1 ∂ M are decreasing, while the unstable cones are defined to be the forward images of {0 ≤ dϕ dr ≤ ∞} under D f . Hence primary segments have greater slopes than ω.

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As pointed out in Sect. 2.3, item 4, Lemma 3.3 must be modified to account for the deletions that arise from intersections with forward images of n , and one might be concerned about the absence of uniform estimates on transversality in (†) between ∂ Hσ and unstable curves. This, in fact, is not a problem, because such deletions occur only in neighborhoods of f −1 ∂ Mσ , which are decreasing curves and hence uniformly transversal to u-curves. This completes the modifications associated with item 4. No modifications are required for items 5, 7, 8 and 9. Modifications in Item 6. In the proof of (**)’, the argument needs to be modified, again q due to the difference in geometry: In order to prove that Q ∗ ∩ (∪0 f i (∂ M0 )) = ∅ (Subq i lemma B), in the case of Type I holes we use that ∪1 f (∂ M0 ) is the union of finitely many piecewise smooth increasing curves that stretch from {ϕ = − π2 } to {ϕ = π2 }. For Type II holes, this is not true. However, it can be arranged that Sublemma B will continue to hold as we now explain: First,

q q q−1 ∪1 f i (∂ M0 ) ⊂ (∪1 f i (∂ M)) ∪ (∪0 f i (H0 )) ∪ f q (H0 ). If we write the right side as A ∪ f q (H0 ), then A has the desired geometry, i.e. it is the union of finitely many piecewise smooth increasing curves that stretch from {ϕ = − π2 } to {ϕ = π2 }. Thus the same argument as before shows that this set is disjoint from Q ∗ . One way to ensure that Q ∗ ∩ f q (H0 ) = ∅ is to choose the mixing boxes U j disjoint from H0 , which can easily be arranged given the geometry of primary segments discussed above. This completes the proof of Proposition 2.2 for Type II holes. 4. Escape Dynamics on Markov Towers In this section and the next, we lift the problems from the billiard systems in question to their Markov tower extensions, and solve the problems there. In Sect. 4, we review relevant works and formulate results on towers. Proofs are given in Sect. 5. 4.1. From generalized horseshoes to Markov towers (review). It is shown in [Y] that given a map f : M → M with a generalized horseshoe ( , R) as defined in Sect. 2.1, one can associate a Markov extension F : → which focuses on the return dynamics to (and suppresses details between returns). We first recall some facts about this very general construction, taking the opportunity to introduce some notation. Let = {(x, n) ∈ × N : n < R(x)}, and define F : → as follows: For < R(x) − 1, we let F(x, ) = (x, + 1), and define F(x, R(x) − 1) = ( f R(x) (x), 0). Equivalently, one can view as the disjoint union ∪≥0 , where , the th level of the tower, is a copy of {x ∈ : R(x) > }. This is the representation we will use. There is a natural projection π : → M such that π ◦ F = f ◦ π . In general, π is not one-to-one, but for each ≥ 0, it maps bijectively onto f ( ∩ {R ≥ }). In the construction of ( , R), one usually introduces an increasing sequence of partitions of into s-subsets representing distinguishable itineraries in the first n steps. ˜ .) These partitions (In Sects. 2.3 and 3.2, these partitions were given by P˜ of

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375

induce a partition {, j } of which is finite on each level and and is a (countable) Markov partition for F. We define a separation time s(x, y) ≤ s0 (x, y) by inf{n > 0 : F n x, F n y lie in different , j }. We borrow the following language from ( , R) for use on : For each , j, recall that s (π(, j )) and u (π(, j )) are the stable and unstable families defining the hyperbolic product set π(, j ). We will say γ˜ ⊂ , j is an unstable leaf of , j if π(γ˜ ) = γ ∩ π(, j ) for some γ ∈ u (π(, j )), and use u (, j ) to denote the set of all such γ˜ . Let u () = ∪, j u (, j ) be the set of all unstable leaves of . Stable leaves of , j and the families s (, j ) and s () are defined similarly. Associated with F : → , which we may think of as a “hyperbolic tower”, is its quotient “expanding tower” obtained by collapsing stable leaves to points. Topologically, = /∼, where for x, y ∈ , x ∼ y if and only if y ∈ γ (x) for some γ ∈ s (). Let π : → be the projection defined by ∼, and let F : → be the induced map on satisfying F ◦ π = π ◦ F. We will use the notation = π ( ), , j = π(, j ), and so on. It is shown in [Y] that there is a well defined differential structure on preserved by F. Recall that µγ is the Riemannian measure on γ , and for γ , γ ∈ u ( ), γ ,γ : γ ∩ → γ ∩ is the holonomy map obtained by sliding along stable curves, i.e. γ ,γ (x) = γ s (x) ∩ γ . We introduce the following notation: For x ∈ i ∩ γ , let γ be such that f Ri (γ ) ⊂ γ . Then J u ( f R )(x) = Jm γ ,m γ ( f Ri |(γ ∩ i ))(x) is the Jacobian of f R with respect to the measures m γ and m γ . Lemma 1 of [Y], which we recall below, is key to the differential structure on . Lemma 4.1. There is a function u : → R such that for each γ ∈ u ( ), if m γ is the measure whose density with respect to µγ is eu Iγ ∩ , then we have the following: (1) For all γ , γ ∈ u ( ), (γ ,γ )∗ m γ = m γ . (2) J u ( f R )(x) = J u ( f R )(y) for all y ∈ γ s (x). (3) ∃C1 > 0 (depending on C and α) such that for each i and all x, y ∈ i ∩ γ , u R J ( f )(x) s( f R x, f R y)/2 . J u ( f R )(y) − 1 ≤ C1 α

(1)

1

The properties of u include |u| ≤ C and |u(x) − u(y)| ≤ 4Cα 2 s(x,y) on each γ . (1) and (2) together imply that there is a natural measure m on with respect to which the Jacobian of F, J F, is well defined: First, identify 0 with γ ∩ for any γ ∈ u ( ), and let m|0 be the measure that corresponds to m γ . (1) says that m so R

defined is independent of γ , and (2) says that with respect to m, J F (x) = J u ( f R )(y) for any y ∈ γ s (x). We then extend m to ∪>0 in such a way that J F ≡ 1 on all of −1 \ F ( 0 ). In the rest of Sect. 4.1 we will assume m{R > n} < C0 θ0n for some C0 ≥ 1 and θ0 < 1.6 One of the reasons for passing from the hyperbolic tower to the expanding tower is that the spectral properties of the transfer√operator associated with the latter can be leveraged. We fix β with 1 > β > max{θ0 , α}, and define a symbolic metric on by √ dβ (x, y) = β s(x,y) . Since β > α, Lemma 4.1(3) implies that J F is log-Lipshitz with 6 Our default rule is to use the same symbol for corresponding objects for f, F and F when no ambiguity can arise given context. Thus R is the name of the return time function on , 0 and 0 .

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respect to this metric. A natural function space on is B = {ρ ∈ L 1 ( , m) : ρ < ∞}, where ρ = ρ∞ + ρLip and ρ∞ = sup sup |ρ(x)|β ,

ρLip = sup Lip(ρ| , j )β .

, j x∈ , j

, j

Lip(·) above is with respect to the symbolic metric dβ . The weights β provide the needed contraction from one level to the next, and β > θ0 is needed to maintain exponential tail estimates. 4.2. Towers with Markov holes. Now consider a leaky system ( f, M, H ) as defined in Sect. 2.1, and suppose ( , R) is a generalized horseshoe respecting the hole H . Let F : → be the associated tower map with π : → M, and let H˜ = π −1 (H ). Then (F, , H˜ ) is a leaky system in itself. With the horseshoe respecting H , we have that H˜ is the union of a collection of , j , usually an infinite number of them; we refer to holes of this type as “Markov holes”. The notation H := H˜ ∩ will be used. Projecting and letting H = π ( H˜ ), we obtain the quotient leaky system (F, , H ). Let us say (F, , H˜ ) and (F, , H ) are mixing if the surviving dynamics of the horseshoe that gives rise to these towers are mixing; see Sect. 2.1. ˚ = \ H˜ , we introduce the notation Letting n ˚ = {x ∈ : F i x ∈ n = ∩i=0 F −1 / H˜ for 0 ≤ i ≤ n},

˚ = 0 . Corresponding objects for (F, , H ) are denoted by n . so that in particular 4.2.1. What is known: Spectral properties of expanding towers. Expanding towers (that are not necessarily quotients of hyperbolic towers) with Markov holes were studied in [D1 and BDM]. The following theorem summarizes several results proved in [BDM, Proposition 2.4, Corollary 2.5], under some conditions on the tower that are easily satisfied here. We refer the reader to [BDM] for detail, and state their results in our context of (F, , H ). Let B˚ = {ρ ∈ L 1 ( 0 , m) : ρ < ∞}, where ρ is as above, and let L denote the ˚ i.e., for ρ ∈ B˚ and x ∈ 0 , transfer operator associated with F| 1 defined on B, Lρ(x) = ρ(y)(J F(y))−1 . y∈ 0 ∩F

−1

x

Theorem 4.2 [BDM]. Let (F, , H ) be such that (i) (F, ) has exponential return times and (ii) (F, , H ) is mixing. Assume the following condition on hole size: ≥1

β −(−1) m(H )

β, and it has a unique eigenfunction h ∗ ∈ B˚ with h ∗ dm = 1. In addition, ˚ there exist constants D > 0 and τ < 1 such that for all ρ ∈ B,

Escape Rates and Physically Relevant Measures for Billiards with Small Holes n

ϑ∗−n L ρ − d(ρ)h ∗ ≤ Dρτ n ,

where d(ρ) = lim λ−n n→∞

(2) The eigenvalue ϑ∗ satisfies ϑ∗ > 1 −

1+C1 m( 0 )

≥1 β

377

n

ρ dm < ∞.

−(−1) m(H ).

The spectral property of L as described in Theorem 4.2(1) implies that all ρ except for those in a codimension 1 subspace have d(ρ) = 0. Given the pivotal role played by the base 0 of the tower , one would guess that for a density ρ, if ρ > 0 on 0 , then d(ρ) = 0. A slightly more general condition is given in Corollary 4.3 below. We call , j a surviving element of the tower if some part of , j returns to 0 before entering H . Corollary 4.3 [BDM]. Let ρ ∈ B˚ be a nonnegative function that is > 0 on a surviving , j . Then d(ρ) > 0. 4.2.2. What is desired: Results for hyperbolic towers. Here we formulate a set of results for the hyperbolic tower that connect the results in Sect. 4.2.1 to the stated theorems for billiards. Let G˜ be the class of measures η on with the following properties: (i) η has absolutely continuous conditional measures on unstable leaves; and (ii) π ∗ η = ρdm for some ρ ∈ B˚ with d(ρ) > 0. Let ( (σ ) , R (σ ) ) be a generalized horseshoe with the properties in Proposition 2.2, and let (F, ) be its associated tower. Let n(, H˜ ) := sup{ : H = ∅}, i.e., n(, H˜ ) = n( (σ ) , R (σ ) ; Hσ ) as defined in Sect. 2.2. Theorem 4.4. Assume that n(, H˜ ) is large enough that

β −(−1) m( )

0 such that

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(i) the conditional densities ργ of µ˜ ∗ | with respect to µγ on unstable leaves satisfy C2−1 ϑ∗− ≤ ργ ≤ C2 ϑ∗− ; (ii) µ˜ ∗ (∪>L ) ≤ Kβ −L θ0L ; and (iii) ϑ∗ → 1 as n(, H˜ ) → ∞. 5. Proofs of Theorems on the Tower The following notational abbreviations are used only in this section: – We will sometimes drop the ˜ used to distinguish between objects on M and corresponding objects on ; there can be no ambiguity as long as we restrict ourselves to the towers. ˚ Specifically, F∗n η is to be interpreted as F˚∗n η, and – We will at times drop the ˚ in F. n F ∗ η is to be interpreted the same way. We focus on the stable direction, since that is what lies between Theorem 4.2 and The˚ that are Lipschitz in the stable orem 4.4. The following is a class of test functions on s s s direction. For γ ∈ () and x, y ∈ γ , we denote by d s (x, y) the distance between π(x) and π(y) according to the p-metric, so that d s (F n x, F n y) ≤ λ−n d s (x, y) for some λ > 1 (see Sect. 2.3). Let Fb be the set of bounded, measurable functions on ˚ For ϕ ∈ Fb , we define |ϕ|sLip to be the Lipshitz constant of ϕ restricted to stable . leaves, i.e. |ϕ|sLip =

sup

sup

˚ x,y∈γ s γ s ∈ s ()

ϕ(x) − ϕ(y) , ds (x, y)

˚ = {ϕ ∈ Fb : |ϕ|sLip < ∞}. and let Lips () 5.1. Proof of Theorem 4.4. A. Escape rates. Theorem 4.4(a) follows easily from Theorem 4.2 as (F, , H ) and (F, , H ) have the same escape rate. In more detail, let η ∈ G˜ and notice that since H is a union of , j , we have, for each n, η(n ) = η( n ), ˜ dη = ρ ∈ B˚ with d(ρ) > 0. Theorem 4.2(1) then where η = π ∗ η. By definition of G, dm n implies that ϑ∗−n L ρ converges to d(ρ)h ∗ . Since the convergence is in the · -norm, n we may integrate with respect to m. Noting that L ρ dm = n ρ dm = η( n ), we have lim ϑ −n η(n ) n→∞ ∗

= lim ϑ∗−n η( n ) = d(ρ). n→∞

(4)

Thus − log ϑ∗ , where ϑ∗ is the eigenvalue in Theorem 4.2, is the common escape rate ˜ of (F, , H ) for initial distributions in G. B. Uniqueness of limiting distributions. We first prove uniqueness postponing the proof of existence of limiting distributions. ˜ we define a measure ηs on u (), i.e. a measure transverse to unstaGiven η ∈ G, ble leaves, as follows: Set ηs ( u (, j )) = 0 if η(, j ) = 0. If η(, j ) = 0, then ηs | u (, j ) is the factor measure of η|, j normalized, and {ρdm γ , γ ∈ u (, j )} is the disintegration of η into measures on unstable leaves. We will use the convention that ηs (, j ) = 1, and ρ|γ is the density with respect to m γ , so that ρ|γ dηs (γ ) = ρ, where dπ ∗ η = ρdm.

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˜ Suppose for i = 1, 2, there exists µi∗ such that Lemma 5.1. Let η1 and η2 ∈ G. lim ϑ −n F∗n ηi n→∞ ∗

= d(ρ i )µi∗ ,

where ρ i is the density of π ∗ ηi Then µ1∗ = µ2∗ . The crux of the argument for Lemma 5.1 is contained in ˚ Lemma 5.2. Let η1 and η2 be as above, and assume ρ 1 = ρ 2 . Then for all ϕ ∈ Lips (), −n n n ϑ∗ |F∗ η1 (ϕ) − F∗ η2 (ϕ)| → 0 exponentially fast as n → ∞. Proof. For i = 1, 2, let ηis and ρi be the (normalized) factor measure and (unnormalized) densities on γ ∈ u () of ηi as described above. We consider functions which are constant along stable leaves to be defined on both ˚ and 0 and do not distinguish between the two versions of such functions. For each , j , let γˆ ∈ u (, j ) be a representative leaf. Then |F∗n η1 (ϕ) − F∗n η2 (ϕ)| ≤

, j

γˆ ∩n, j

dm γˆ

γs

ρ1 ϕ ◦ F n dη1s −

γs

ρ2 ϕ ◦ F n dη2s . (5)

Next fix x ∈ γˆ ∩ n and estimate the integrals on γ s (x). Define ϕ n = Then, n s n s s ρ1 ϕ ◦ F dη1 − s ρ2 ϕ ◦ F dη2 γ γ n s n s ≤ ρ1 (ϕ ◦ F − ϕ n )dη1 + ρ2 (ϕ ◦ F − ϕ n )dη2 γs γs + ϕ n ρ1 dη1s − ϕ n ρ2 dη2s . γs

γs

ϕ ◦ F n dη1s .

γs

Since ϕ n is constant on γ s and ρ 1 = ρ 2 , the third term above is 0. For the first two terms, we note that for each y ∈ γ s (x), |ϕ n (y) − ϕ ◦ F n (y)| ≤ |ϕ|sLip λ−n . Thus ϑ∗−n |F∗n µ1 (ϕ) − F∗n µ2 (ϕ)| ≤ ϑ∗−n = n

n, j

2ρ 1 dm |ϕ|sLip λ−n

, j −n n 2ϑ∗ |L ρ 1 |1 |ϕ|sLip λ−n ,

which proves the lemma since ϑ∗−n |L ρ 1 | → d(ρ 1 ) by Theorem 4.2.

(6)

Remark 5.3. We have used in the proof above a property of the billiard maps, namely d s (F n x, F n y) ≤ λ−n d s (x, y). For general towers, one has only the contraction guaranteed by (P3) which is nonuniform. It is not hard to see that the lemma holds in the n more general case with the exponential rate given by max{α 2 , β −n θ0n } in the place of λ−n ; we leave the proof to the interested reader.

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Proof of Lemma 5.1. Let µ∗ = h ∗ m be the conditionally invariant measure given by Theorem 4.2. For i = 1, 2, we have, on the one hand, n lim ϑ −n F ∗ ηi n→∞ ∗

= d(ρ i )µ∗ ,

which follows from Theorem 4.2, and on the other, lim ϑ −n π ∗ F∗n ηi n→∞ ∗

= d(ρ i )π ∗ µi∗ , n

which follows from the hypothesis of the lemma. Since π ∗ F∗n ηi = F ∗ π ∗ ηi for each n ≥ 0, we have π ∗ µ1∗ = µ∗ = π ∗ µ2∗ . Thus ϑ∗−n |F∗n µ1∗ − F∗n µ2∗ | → 0 as n → ∞ by Lemma 5.2. But ϑ∗−n F∗n µi∗ = µi∗ since µi∗ is conditionally invariant. Hence µ1∗ = µ2∗ . ˚ C. Convergence to conditionally invariant measure. For a probability measure η on , n n n −n n ˜ |F∗ η| = η( ) = π ∗ η( ). So for η ∈ G, (4) implies limn→∞ ϑ∗ |F∗ η| = d(ρ) > 0, where ρ is the density of η = π ∗ η. More than that is true: Lemma 5.4. ϑ∗−n F∗n η/d(ρ) converges weakly to a conditionally invariant probability measure µ∗ as n → ∞. This is half of Theorem 4.4(b). Once we have this, it will follow immediately that F∗n η ϑ∗−n F∗n η = lim = µ∗ , n→∞ |F∗n η| n→∞ ϑ∗−n |F n η| ∗ lim

(7)

which is the other half. We will use the following algorithm to “lift” measures from to : Fix a measure µs on u () with µs ( u (, j )) = 1. Given η on with density ρ, we define π −1 ∗ η to η decomposes be the measure on with the property that restricted to each , j , π −1 ∗ into the factor measure µs and leaf measures {ρdm γ }, where ρ|π −1 (x) ≡ ρ(x). Notice that π ∗ π −1 ∗ η = η. ˚ and show that ϑ∗−n F∗n η(ϕ) Proof of Lemma 5.4. Our first step is to fix ϕ ∈ Lips () s is a Cauchy sequence. For a fixed µ as above, let ϕ n (x) = γ s (x) ϕ ◦ F˚ n dµs . Define ˜ η has density ρ ∈ B˚ with d(ρ) > 0. Then by definition of π −1 η = π ∗ η. Since η ∈ G, ∗ , ˚n (π −1 dµs (γ ) ϕ ◦ F˚ n ρ dm γ ∗ π ∗ η)(ϕ ◦ F ) = =

, j

u (, j )

, j

, j

γu

ρ ϕ n dm = π ∗ η(ϕ n ).

(8)

For n, k1 , k2 ≥ 0, write |ϑ∗−n−k1 F∗n+k1 η(ϕ) − ϑ∗−n−k2 F∗n+k2 η(ϕ)

k1 ≤ ϑ∗−n−k1 |F∗n+k1 η(ϕ) − F∗n π −1 ∗ π ∗ F∗ η(ϕ)|

k1 −n−k2 n −1 F∗ π ∗ π ∗ F∗k2 η(ϕ)| +|ϑ∗−n−k1 F∗n π −1 ∗ π ∗ F∗ η(ϕ) − ϑ∗ k2 n+k2 +ϑ∗−n−k2 |F∗n π −1 η(ϕ)|. ∗ π ∗ F∗ η(ϕ)η(ϕ) − F∗

(9)

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The first and third terms of (9) are estimated using Lemma 5.2 since π ∗ (ϑ∗−ki F∗ki η) = ki π ∗ (ϑ∗−ki π −1 ∗ π ∗ F∗ η) for i = 1, 2. Thus by Lemma 5.2, ki s n ϑ∗−n−ki |F∗n+ki η(ϕ) − F∗n π −1 ∗ π ∗ F∗ η(ϕ)| ≤ C d(ρ)(|ϕ|Lip + |ϕ|∞ )ζ

for some C > 0 and ζ < 1. We now fix n and estimate the second term of (9). Due to (8), for any k ≥ 0 we have k −n−k −1 ϑ∗−n−k F∗n π −1 π ∗ π ∗ F∗k η(ϕ ◦ F n · 1˚ n ) = ϑ∗−n−k π ∗ F∗k η(ϕ n · 1 n ) ∗ π ∗ F∗ η(ϕ) = ϑ∗ k k ϕ n · L ρ dm. = ϑ∗−n−k F ∗ η(ϕ n · 1 n ) = ϑ∗−n−k n

˜ we estimate Recalling that ρ ∈ B˚ and d(ρ) > 0 since η ∈ G, k1 −n−k2 n −1 |ϑ∗−n−k1 F∗n π −1 F∗ π ∗ π ∗ F∗k2 η(ϕ)| ∗ π ∗ F∗ η(ϕ) − ϑ∗ −k1 k1 ≤ |ϕ|∞ ϑ∗−n ϑ∗ L ρ − d(ρ)h ∗ dm n −k2 k2 −n +|ϕ|∞ ϑ∗ ϑ∗ L ρ − d(ρ)h ∗ dm.

(10)

n

Both terms of ϑ∗−n

n

(10) are small: By Theorem 4.2(1), −k k −n −k k dm L ρ − d(ρ)h ≤ ϑ L ρ − d(ρ)h ϑ∗ ∗ ∗ ∗ ϑ∗ n

≤ ϑ∗−n |L 1β |1 Dρτ k

n

1β dm

˚ ϑ −n |Ln 1β |1 converges ˚ . Since 1β ∈ B, for k = ki , where 1β (x) = β − for x ∈ ∗ to d(1β ) as n → ∞. Thus (10) can be made arbitrarily small by choosing k1 and k2 sufficiently large. We have shown that ϑ∗−n F∗n η(ϕ)/d(ρ) is a Cauchy sequence and therefore converges to a number Q(ϕ). The functional Q(ϕ) := limn→∞ ϑ∗−n F∗n η(ϕ)/d(ρ) is clearly linear in ϕ, positive and satisfies Q(1) = 1. Also |Q(ϕ)| ≤ |ϕ|∞ Q(1) so that Q extends to a ˚ the set of bounded functions which are continuous bounded linear functional on Cb0 (), ˚ on each , j . By the Riesz representation theorem, there exists a unique Borel probability measure ˚ [H, Sect. 56]. Also, µ∗ satisfying µ∗ (ϕ) = Q(ϕ) for each ϕ ∈ Cb0 () ˚ = lim ϑ∗−n F∗n η(ϕ ◦ F) ˚ = ϑ∗ lim ϑ∗−n−1 F∗n+1 η(ϕ) = ϑ∗ d(ρ)µ∗ (ϕ), d(ρ)µ∗ (ϕ ◦ F) n→∞

n→∞

so that µ∗ is a conditionally invariant measure for F˚ with escape rate − log ϑ∗ .

This completes the proof of parts (a) and (b) of Theorem 4.4. To prove part (c), we must show that µ∗ has absolutely continuous conditional measures on unstable leaves. Proof of a stronger version of this fact is contained in the proof of Proposition 4.6(i) below. Notice that from the proof of Lemma 5.1, we have π ∗ µ∗ = µ∗ so that the density of ˜ π ∗ µ∗ is precisely h ∗ . Since h ∗ ∈ B˚ and d(h ∗ ) = 1 > 0, we conclude µ∗ ∈ G.

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5.2. Proof of Proposition 4.6. Consider the set of holes h (q0 ) for fixed q0 where h is small enough as required in Proposition 2.2. For σ ∈ h (q0 ), let (σ ) be the tower with ) holes induced by the generalized horseshoe, and let µ(σ ∗ be the conditionally invariant measure given by Theorem 4.4. Proof of (i). We drop the superscript (σ ) in what follows and point out that the constants we use are uniform for all σ ∈ h (q0 ) and h sufficiently small. Choose γ0 ∈ u (0 ) and let η0 be the measure supported on γ0 with uniform den˜ It is immediate that π ∗ η0 has density sity with respect to µγ . We claim that η0 ∈ G. −u ˚ ρ = e |γ0 with respect to m, which is in B by Lemma 4.1. To see that d(ρ) > 0, notice that the mixing assumption on (F, ) implies that 0 is necessarily a surviving partition element. Since ρ > 0 on 0 , Corollary 4.3 implies that d(ρ) > 0. (n) By Theorem 4.4(b), η(n) := F∗n η0 /|F∗n η0 | converges to µ∗ . Let ργ denote the density of η(n) with respect to µγ on γ ∈ u (). Notice that inverse branches of F˚ n on γ are well defined. For any x1 , x2 ∈ γ , treating one branch at a time and summing over all branches, we obtain that −1 (n) ˚n Jµγ F˚ n (y2 ) ργ (x1 ) y1 ∈ F˚ −n x1 (Jµγ F (y1 )) = ≤ sup ≤ eC (n) n (y ))−1 n (y ) ˚ ˚ F F (J J −n ργ (x2 ) ˚ −n µ 2 µ 1 ˚ y ∈F x γ γ y2 ∈ F

x2

1

1

by Property (P4)(b), where Jµγ F˚ n is the Jacobian of F˚ n with respect to µγ . Since by Proposition 2.2 the constant C is independent of σ , x and n, we have e−C ≤

(n)

supx∈γ ργ (x) inf x∈γ ργ(n) (x)

≤ eC .

(11)

This estimate plus the minimum length κ of µu ( ) given by Proposition 2.2 yields the desired uniform upper and lower bounds on the conditional densities of η(n) with respect to µγ (and hence to m γ ) on 0 . The uniformity of these bounds in n implies that they pass to the conditional densities of µ∗ in the limit as n → ∞. Since µ∗ is conditionally invariant, µ∗ |˚ = ϑ∗−1 µ∗ | F˚ −1 ˚ . The required bounds on the densities extend easily to ˚ for > 0. Proof of (ii). We decompose µ∗ into a normalized factor measure µs∗ on u ( ) and densities ργ with respect to m γ on γ ∈ u ( ). Then (σ ) µ∗ (∪≥L ) = dµs∗ ργ dm γ ≤ C2 ϑ∗− m( ) ≤ C2 C0 θ0 β − . (σ )

u ≥L ( )

γ

≥L

≥L

Here we have used Proposition 4.6(i) to estimate ργ , Proposition 2.2 and Lemma 4.1 for the uniformity of C0 and θ0 , and the fact that ϑ∗ > β. The sum can be made arbitrarily small since β > θ0 . Proof of (iii). Notice that n(, H˜ ) ≥ n(h) ¯ by definition of n(h) ¯ in Sect. 2.2. From Theorem 4.2(2), we know that the escape rate − log ϑ∗ satisfies 1 + C1 −1 1 + C1 −1 ϑ∗ > 1 − β m(H ∩ ) > 1 − β C0 θ0 . κ κ ≥1

By Proposition 2.2, n(h) ¯ → ∞ as h → 0, so that ϑ∗ → 1.

≥n(h) ¯

Escape Rates and Physically Relevant Measures for Billiards with Small Holes

383

6. Proofs of Theorems for Billiards In the Proofs of Theorems 1–3, we fix a hole σ that is acceptable with respect to Proposition 2.2 and for which n( (σ ) , R (σ ) , Hσ ) is large enough to meet the condition in Theorem 4.4. We suppress mention of σ , and let (F, , H˜ ) be the tower constructed n ˚ M ∞ = ∩n≥0 M n . from ( , R, H ). Define M n = ∩i=0 f −n M, 6.1. Proof of Theorems 1 and 2. The first order of business is to show that each η ∈ G ˚ with can be lifted to a measure η˜ ∈ G˜ in such a way that the escape dynamics on initial distribution η˜ reflect those on M˚ with initial distribution η. Recall that the natural invariant probability measure for the closed billiard system f : M → M is denoted by ν. In [Y, Sect. 2], it is shown that there is a unique invariant probability measure ν˜ for the tower map F : → with absolutely continuous conditional measures on unstable leaves, and this measure has the property π∗ ν˜ = ν. Given η ∈ G, we define η˜ on as follows: By definition, every η ∈ G is absolutely continuous with respect to ν. Let ˜ ˜ where ψ˜ = ψ ◦ π . This ψ = dη dν . We take η˜ to be the measure given by d η˜ = ψd ν, implies in particular that π∗ η˜ = η. ˜ Lemma 6.1. If η ∈ G, then η˜ ∈ G. As before, let Fb denote the set of bounded functions on . For ϕ ∈ Fb and γ ∈ u (), we let Lipu (ϕ|γ ) be the Lipschitz constant of ϕ|γ with respect to the dβ -metric (notice that dβ , the symbolic metric defined on , can be thought of as a metric on unstable leaves). Let |ϕ|uLip =

sup

γ ∈ u ()

Lipu (ϕ|γ ),

and Lipu () = {ϕ ∈ Fb : |ϕ|uLip < ∞}. The first step toward proving Lemma 6.1 is ˜ uLip ≤ Lemma 6.2. Let ϕ : M → R be Lipschitz. Then ϕ˜ := ϕ ◦ π ∈ Lipu () with |ϕ| CLip(ϕ). Proof. Recall that for x, y ∈ M lying in a piece of local unstable manifold, we have d(x, y) ≤ p(x, y)1/2 , where p(·, ·) is the p-metric (see Sect. 2.3). Now for γ ∈ u () and x, y ∈ γ , we have 1

|ϕ(x) ˜ − ϕ(y)| ˜ = |ϕ(π x) − ϕ(π y)| ≤ Lip(ϕ)d(π x, π y) ≤ Lip(ϕ) p(π x, π y) 2 . 1

By (P4)(a), p(π x, π y) 2 ≤ Cα s(π x,π y)/2 , which is ≤ Cdβ (x, y) since s ≤ s0 and √ β ≥ α. ˚ Let ψ = Proof of Lemma 6.1. (i) First we show π ∗ η˜ = ρ m with ρ ∈ B. s u disintegrating η˜ into η˜ and {ργ dm γ , γ ∈ ()}, we obtain ργ := ψ˜ ·

Then

d ν˜ dµγ · . dµγ dm γ

Now ψ˜ is bounded by assumption and is ∈ Lipu () by Lemma 6.2, Lipu ()

dη dν .

dµγ dm γ

d ν˜ dµγ

is bounded

([Y], Sect. 2), as is (Lemma 4.1). Thus we conclude that ργ ∈ and is ∈ u Lip () and is bounded. Recall that ρ(x) = γ s (x) ργ d η˜ s . It follows immediately that |ρ|∞ ≤ supγ |ργ |∞ and Lip(ρ) ≤ supγ Lipu (ργ ).

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(ii) It remains to show d(ρ) > 0. By definition of G, ψ > 0 on M ∞ , the set of points −u are ˚ so ψ˜ > 0 on ∞ . The fact that d ν/dµ which never escape from M, ˜ γ and e ∞ strictly positive implies that ρ > 0 on ; hence it is > 0 on a surviving cylinder set, k i.e. a set E k such that F maps E k onto a surviving , j before any part of it enters the k n hole. By Corollary 4.3, d(L ρ) > 0. Since n g dm = L g dm for each n ≥ 0 and k

g ∈ L 1 (m), we have d(L ρ) = ϑ∗k d(ρ) so that d(ρ) > 0 as well.

Proof of Theorems 1 and 2. Given η ∈ G, let η˜ be as defined earlier. Then η˜ ∈ G˜ by Lemma 6.1. For ϕ ∈ C 0 (M), let ϕ˜ = ϕ ◦ π . Then ϕ˜ ∈ Cb0 () and for n ≥ 0 we have, f˚∗n η(ϕ) = η(ϕ ◦ f n · 1 M n ) = η( ˜ ϕ˜ ◦ F n · 1n ) = F˚∗n η( ˜ ϕ). ˜

(12)

˚ = F˚∗n η( ˚ = η( Setting ϕ ≡ 1 in (12), we have η(M n ) = f˚∗n η( M) ˜ ) ˜ n ) for n > 0, so lim

n→∞

1 1 log η(M n ) = lim log η( ˜ n ) = log ϑ∗ n→∞ n n

by Theorem 4.4(a). This proves Theorem 1. Let µ∗ = π∗ µ˜ ∗ , where µ˜ ∗ is given by Theorem 4.4. Then µ∗ (ϕ) = µ˜ ∗ (ϕ), ˜ and f˚∗ µ∗ (ϕ) = F˚∗ µ˜ ∗ (ϕ) ˜ = ϑ∗ µ˜ ∗ (ϕ) ˜ = ϑ∗ µ∗ (ϕ), proving µ∗ is conditionally invariant. Using (12) again, the fact that the normalizations are equal, and Theorem 4.4(b), we obtain lim

n→∞

f˚∗n η(ϕ) = lim n→∞ ˚ f˚∗n η( M)

F˚∗n η( ˜ ϕ) ˜ ˜ = µ∗ (ϕ). = µ˜ ∗ (ϕ) n ˚ ˚ F∗ η()

Thus f˚∗n η/η(M n ) → µ∗ weakly. Finally, lim ϑ −n f˚∗n η(ϕ) n→∞ ∗

= lim ϑ∗−n F˚∗n η( ˜ ϕ) ˜ = d(ρ) · µ˜ ∗ (ϕ) ˜ = d(ρ) · µ∗ (ϕ), n→∞

˜ This completes the proof of Theorem 2. where d(ρ) > 0 since η˜ ∈ G.

Remark 6.3. In the proof of Lemma 6.1, step (i) holds for any η that has Lipschitz densities on unstable leaves. Thus for this class of measures, Theorem 2(b) holds (with c(η) possibly equal to zero). It is also clear from step (ii) that to show d(ρ) > 0, it suffices to assume ψ > 0 on M ∞ ∩ , or on M ∞ ∩π(, j ), where π (, j ) is any surviving element. 6.2. Proof of Theorem 3. Let µ∗ = π∗ µ˜ ∗ be as above. (a) Since f˚∗ µ∗ = ϑ∗ µ∗ , it follows that µ∗ is supported on M \ ∪n≥0 f n (H ), where H = Hσ . This set has Lebesgue measure zero since by the ergodicity of f , ∪n≥0 f n (H ) has full Lebesgue measure. Thus µ∗ is singular with respect to Lebesgue measure. (b) First, we argue that µ∗ has absolutely continuous conditional measures on unstable leaves (without claiming that the densities are strictly positive). This is true because for each , j, µ˜ ∗ |, j has absolutely continuous conditional measures on γ ∈ u (, j ), and π |, j , which is one-to-one, identifies each γ with a positive Lebesgue measure subset of a local unstable manifold of f .

Escape Rates and Physically Relevant Measures for Billiards with Small Holes

385

The rest of the proof is concerned with showing that the conditional densities of µ∗ are strictly positive. To do that, it is not productive to view µ∗ as π∗ µ˜ ∗ . Instead, we will view µ∗ as the weak limit of ν (n) := f˚∗n ν/| f˚∗n ν| as n → ∞, where ν is the natural invariant measure for f . This convergence of ν (n) is guaranteed by Theorem 2. We will prove that µ∗ has the properties immediately following the statement of Theorem 3 in Sect. 1.3. Step 1: Our first patch is built on V = ∪{γ u : γ u ∈ u ( )}, where u = u ( ) is the defining family of unstable curves for . To understand the geometric properties of ν (n) |V , observe that in backward time, each γ u ∈ u either falls into the hole completely or stays out completely. This is because f (∂ H ) is regarded as part of the discontinuity set for f −1 when we constructed the horseshoe (see Sect. 3.2). Thus there is a decreasing sequence of sets Un = ∪{γ u ∈ u : f −i γ u ∩ H = ∅ for all 0 ≤ i ≤ n} ⊂ V consisting of whole γ u -curves. Assuming ν(Un ) > 0 for now, we have ν (n) |V = cn ν|Un for some constant cn > 0 as ν is f -invariant. Let ζ be a limit point of ν (n) |V , i.e., ζ = limn k ν (n k ) |V . Assuming ζ (V ) > 0, lower bounds for conditional probability densities of ν (n k ) |V , equivalently those of ν|Un , are passed to ζ , and these bounds are strictly positive. To see that ζ (V ) > 0, recall that ν = π∗ ν˜ for some ν˜ on the tower , so that ν (n) = π∗ ν˜ (n) , where ν˜ (n) = F˚∗n ν˜ /| F˚∗n ν˜ |. Since π(0 ) ⊂ V , we have ζ = lim ν (n k ) |V ≥ lim π∗ (˜ν (n k ) |0 ) = π∗ (µ˜ ∗ |0 ). nk

nk

We have written an inequality (as opposed to equality) above because parts of for ≥ 1 may get mapped into V as well. Clearly, µ˜ ∗ (0 ) > 0, thereby ensuring ζ (V ) > 0, hence ν(Un ) > 0 and the strictly positive conditional densities property above. This together with ζ ≤ µ∗ |V (equality is not claimed because it is possible for part of ν (n k ) from outside of V to leak into V in the limit) proves that (V, ζ ) is an acceptable patch. Step 2: Next we use (V, ζ ) to build patches (V, j , ζ, j ) corresponding to partition elements , j of the tower with > 0 and µ˜ ∗ (, j ) > 0. From Sects. 3 and 4, we know that π(, j ) is a hyperbolic product set, and π(, j ) = f ( s ) for some s-subset s ⊂ . Moreover, f i ( s ) ∩ H = ∅ for all 0 < i ≤ . Thus we may assume V, j = ∪{γ u : γ u ∈ u (π(, j ))} ⊂ f (V ). Let ζ, j = ϑ∗− ( f ∗ ζ )|V, j . Then ζ, j has strictly positive conditional densities on unstable curves because ζ does, and ζ, j ≤ µ∗ |V, j as µ∗ satisfies f˚∗ µ∗ = ϑ∗ µ∗ . Finally, since ζ, j ≥ π∗ (µ˜ ∗ |, j ) for each , j, it follows that , j ζ, j ≥ µ∗ , completing the proof of Theorem 3. 6.3. Proof of Theorem 4. Suppose h n is a sequence of numbers tending to 0, σh n ∈ h n (q0 ) is a sequence of holes in the billiard table, and Hn = Hσh n the corresponding holes in M. For each n, let ϑn be the escape rate and µn the physical measure for the leaky system ( f, M, Hn ) given by Theorem 2. By Proposition 4.6(iii), we have ϑn → 1 as n → ∞. To prove µn → ν, we will assume, having passed to a subsequence, that µn converges weakly to some µ∞ , and show that (i) µ∞ is f -invariant, and (ii) it has absolutely continuous conditional measures on unstable leaves. These two properties together uniquely characterize ν. The following notation will be used: (n) is the generalized horseshoe respecting the hole Hn , (n) = ∪ (n) is the corresponding tower, Fn : (n) → n is the tower

386

M. Demers, P. Wright, L.-S. Young

map, πn : (n) → M is the projection, and µ˜ n is the conditionally invariant measure on (n) that projects to µn . (i) Proof of f -invariance: Let S = ∂ M ∪ f −1 ∂ M. Lemma 6.4. µ∞ (S) = 0 Proof. Let δ1 and λ1 be as in Sect. 2.3, and let Nε (S) denote the ε-neighborhood of S. We claim that there exist constants C3 , ς > 0 such that for ε < δ1 and for all n, µn (Nε (S)) ≤ C3 ες . By the construction of = (n), any n, d( f ( ), S) ≥ δ1 λ− 1 . Thus f ( ) ∩ Nε (S) = ∅ for all ≤ − log(ε/δ1 )/ log λ1 . Hence µn (Nε (S)) ≤ µ˜ n ( (n)), >− log(ε/δ1 )/ log λ1

which by Proposition 4.6(ii) is ≤ K (β −1 θ0 )− log(ε/δ1 )/ log λ1 , proving the claim above with C3 = K /δ1 and ς = log(βθ0−1 )/ log λ1 . Since C3 and ς are independent of n, these bounds pass to µ∞ , implying µ∞ (S) = 0. Having established that f is well defined µ∞ -a.e., we now verify that µ∞ is f -invariant: Let ϕ : M → R be a continuous function. Then (ϕ ◦ f )dµ∞ = lim (ϕ ◦ f )dµn = lim ϕ d( f ∗ µn ), n→∞

and

n→∞

ϕ d( f ∗ µn ) =

M\Hn

ϕ d( f ∗ µn ) +

Hn

ϕ d( f ∗ µn ).

(13)

Since( f ∗ µn )| M\Hn = f˚∗ µn = ϑn µn , the first integral on the right side of (13) is equal to ϑn ϕ dµn , while the absolute value of the second is bounded by (1 − ϑn )|ϕ|∞ . Since ϑn → 1 as n → ∞, the right side of (13) tends to ϕ dµ∞ . (ii) Absolutely continuous conditional measures on unstable leaves: Since the measures µ˜ n do not live on the same space for different n, a first task here is to find common domains in M on which (πn )∗ µ˜ n can be compared. In the constructions to follow, the discontinuity set refers to the real discontinuity set of f , not the ones that include boundaries of holes (as was done in Sect. 3). We choose a rectangular region Qˇ slightly larger than Q in Proposition 2.2, large enough that Qˇ ⊃ (n) for all n, and let ˇ u denote the set of all homogeneous unstable ˇ Let Vˇ = ∪{γ u ∈ ˇ u }. Then (n) ⊂ Vˇ curves connecting the two components of ∂ s Q. u u ˇ ˇ for all n, for γ ∩ Q ∈ for every γ ∈ ( (n)) (defined using the enlarged discontinuity set). Now for all n, (πn )∗ (µ˜ n |0 (n) ) is a sequence of measures on Vˇ with absolutely continuous conditional measures on the elements of ˇ u . Moreover, the conditional densities are uniformly bounded from above with a bound independent of n (Proposition 4.6(i)). Let µ∞,0 be a limit point of (πn )∗ (µ˜ n |0 (n) ). Assuming µ∞,0 (Vˇ ) > 0, these density bounds are inherited by µ∞,0 . To show µ∞,0 (Vˇ ) > 0, we will argue there exists b > 0 such that µ˜ n (0 (n)) > b for all n, and that is true because the µ˜ n are probability measures, there is a uniform lower bound on µ˜ n (∪ 0, we define Qˇ to be the finite union of s-subrectangles of Qˇ retained in steps in the construction of when f has no holes, i.e. roughly speaking, Qˇ consists of points that stay away from S = ∂ M ∪ f −1 ∂ M by a distance ≥ δ1 λ−i 1 at step i. Let − ˚ ˇ ˇ ˇ ˇ V = V ∩ Q . Then πn ( Fn (n)) ⊂ V for all n. In fact, for each j, πn ( F˚n− , j (n)) is contained in a connected component of Qˇ . The argument for µ∞,0 can now be repeated to conclude the existence of a limit point of (πn ◦ F˚n− )∗ (µ˜ n | (n) ) with absolutely continuous conditional measures on unstable leaves. Pushing all measures forward by f ∗ , this gives a limit point µ∞, of (πn )∗ (µ˜ n | (n) ) as n → ∞ with the same property. To proceed systematically, we perform a Cantor diagonal argument, choosing a single subsequence n k with the property that for each ≥ 0, (πn k )∗ (µ˜ n k | (n k ) ) converges to a measure µ∞, on f Vˇ . Finally, to conclude µ∞ = µ∞, , we need a tightness condition as the towers are noncompact. This is given by Proposition 4.6(ii). The proof of Theorem 4 is now complete. Acknowledgements. The second author (P.W.) would like to thank The Courant Institute of Mathematical Sciences, New York University, where he was affiliated when this project began. The authors thank MSRI, Berkeley, and ESI, Vienna, where part of this work was carried out.

References [BaK] [BDM] [BSC1] [BSC2] [B] [C1] [C2] [CM1] [CM2] [CM3] [CMT1] [CMT2] [CMM] [CMS1] [CMS2] [CMS3] [CV]

Baladi, V., Keller, G.: Zeta functions and transfer operators for piecewise monotonic transformations. Commun. Math. Phys. 127, 459–477 (1990) Bruin, H., Demers, M., Melbourne, I.: Convergence properties and an equilibrium principle for certain dynamical systems with holes. To appear in Ergod. Th. and Dynam. Sys. Bunimovich, L.A., Sina˘ı, Ya.G., Chernov, N.I.: Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv. 45(3), 105–152 (1990) Bunimovich, L.A., Sina˘ı, Ya.G., Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46(4), 47–106 (1991) Buzzi, J.: Markov extensions for multidimensional dynamical systems. Israel J. of Math. 112, 357–380 (1999) Cenvoca, N.N.: A natural invariant measure on smale’s horseshoe. Soviet Math. Dokl. 23, 87–91 (1981) Cenvoca, N.N.: Statistical properties of smooth Smale horseshoes. In: Mathematical Problems of Statistical Mechanics and Dynamics, R.L. Dobrushin, ed. Dordrecht: Reidel, 1986, pp. 199–256 Chernov, N., Markarian, R.: Ergodic properties of anosov maps with rectangular holes. Bol. Soc. Bras. Mat. 28, 271–314 (1997) Chernov, N., Markarian, R.: Anosov maps with rectangular holes. Nonergodic Cases. Bol. Soc. Bras. Mat. 28, 315–342 (1997) Chernov, N., Markarian, R.: Chaotic Billiards. Number 127 in Mathematical Surveys and Monographs, Providence, RI: Amer. Math. Soc., 2006 Chernov, N., Markarian, R., Troubetzkoy, S.: Conditionally invariant measures for anosov maps with small holes. Ergod. Th. and Dynam. Sys. 18, 1049–1073 (1998) Chernov, N., Markarian, R., Troubetzkoy, S.: Invariant measures for anosov maps with small holes. Ergod. Th. and Dynam. Sys. 20, 1007–1044 (2000) Collet, P., Martínez, S., Maume-Deschamps, V.: On the existence of conditionally invariant probability measures in dynamical systems. Nonlinearity 13, 1263–1274 (2000) Collet, P., Martínez, S., Schmitt, B.: The Yorke-Pianigiani measure and the asymptotic law on the limit cantor set of expanding systems. Nonlinearity 7, 1437–1443 (1994) Collet, P., Martínez, S., Schmitt, B.: Quasi-stationary distribution and Gibbs measure of expanding systems. In: Instabilities and Nonequilibrium Structures. V. E. Tirapegui, W. Zeller, eds. Dordrecht: Kluwer, 1996, pp. 205–219 Collet, P., Martínez, S., Schmitt, B.: The Pianigiani-Yorke measure for topological Markov chains. Israel J. Math. 97, 61–70 (1997) Chernov, N., van dem Bedem, H.: Expanding maps of an interval with holes. Ergod. Th. and Dynam. Sys. 22, 637–654 (2002)

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M. Demers, P. Wright, L.-S. Young

Demers, M.: Markov extensions for dynamical systems with holes: an application to expanding maps of the interval. Israel J. of Math. 146, 189–221 (2005) Demers, M.: Markov extensions and conditionally invariant measures for certain logistic maps with small holes. Ergod. Th. and Dynam. Sys. 25(4), 1139–1171 (2005) Demers, M., Liverani, C.: Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360(9), 4777–4814 (2008) Demers, M., Young, L.-S.: Escape rates and conditionally invariant measures. Nonlinearity 19, 377–397 (2006) Ferrari, P.A., Kesten, H., Martínez, S., Picco, P.: Existence of quasi-stationary distributions. A Renewal Dynamical Approach. Annals of Prob. 23(2), 501–521 (1995) Halmos, P.R.: Measure Theory. University Series in Higher Mathematics, Princeton, NJ: D. Van Nostrand Co., Inc., 1950, 304 p. Homburg, A., Young, T.: Intermittency in families of unimodal maps. Ergod. Th. and Dynam. Sys. 22(1), 203–225 (2002) Katok, A., Strelcyn, J.M.: Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. Volume 1222, Springer Lecture Notes in Math., Berlin-Heidelberg-NewYork: Springer, 1986 Liverani, C., Maume-Deschamps, V.: Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set. Annales de l’Institut Henri Poincaré Probability and Statistics 39, 385–412 (2003) Lopes, A., Markarian, R.: Open billiards: cantor sets, invariant and conditionally invariant probabilities. SIAM J. Appl. Math. 56, 651–680 (1996) Pianigiani, G., Yorke, J.: Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252, 351–366 (1979) Richardson, P.A., Jr.: Natural Measures on the Unstable and Invariant Manifolds of Open Billiard Dynamical Systems. Doctoral Dissertation, Department of Mathematics, University of North Texas, 1999 Sina˘ı, Ya.G.: Dynamical systems with elastic collisions. Ergodic Properties of Dispersing Billiards. Usp. Mat. Nauk 25(2), 141–192 (1970) Vere-Jones, D.: Geometric ergodicity in denumerable Markov chains. Quart. J. Math. 13, 7–28 (1962) Wojtkowski, M.: Invariant families of cones and Lyapunov exponents. Ergod. Th. Dynam. Sys. 5(1), 145–161 (1985) Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Annals of Math. 147(3), 585–650 (1998)

Communicated by G. Gallavotti

Commun. Math. Phys. 294, 389–410 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0938-6

Communications in

Mathematical Physics

Argyres-Seiberg Duality and the Higgs Branch Davide Gaiotto1 , Andrew Neitzke2 , Yuji Tachikawa1 1 School of Natural Sciences, Institute for Advanced Study, Princeton,

New Jersey 08540, USA. E-mail: [email protected]

2 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

Received: 10 November 2008 / Accepted: 20 August 2009 Published online: 1 December 2009 – © Springer-Verlag 2009

Abstract: We demonstrate the agreement between the Higgs branches of two N = 2 theories proposed by Argyres and Seiberg to be S-dual, namely the SU(3) gauge theory with six quarks, and the SU(2) gauge theory with one pair of quarks coupled to the superconformal theory with E 6 flavor symmetry. In mathematical terms, we demonstrate the equivalence between a hyperkähler quotient of a linear space and another hyperkähler quotient involving the minimal nilpotent orbit of E 6 , modulo the identification of the twistor lines. Contents 1. 2. 3. 4.

5.

6. 7.

Introduction . . . . . . . . . . . . . . . 1.1 Argyres-Seiberg duality . . . . . . . 1.2 Higgs branch . . . . . . . . . . . . Rudiments of Hyperkähler Cones . . . . Geometry of the Minimal Nilpotent Orbit SU(3) Side . . . . . . . . . . . . . . . . 4.1 Poisson brackets . . . . . . . . . . . 4.2 Conjugation . . . . . . . . . . . . . 4.3 Constraints . . . . . . . . . . . . . Exceptional Side . . . . . . . . . . . . . 5.1 Poisson brackets . . . . . . . . . . . 5.2 Conjugation . . . . . . . . . . . . . 5.3 Constraints . . . . . . . . . . . . . 5.4 Gauge invariant operators . . . . . . Comparison . . . . . . . . . . . . . . . 6.1 Identification of operators . . . . . . 6.2 Constraints . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . .

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A. Conventions . . . . . . . . . . . . . B. Twistor Spaces of Hyperkähler Cones C. Comparison of the Kähler Potential . C.1 Exceptional side . . . . . . . . . C.2 SU(3) side . . . . . . . . . . . . C.3 Comparison . . . . . . . . . . . D. Mathematical Summary . . . . . . .

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404 405 406 406 407 407 408

1. Introduction 1.1. Argyres-Seiberg duality. In a remarkable paper [1], a new type of strong-weak duality of four-dimensional N = 2 theories was introduced. Consider an N = 2 supersymmetric SU(3) gauge theory with six quarks in the fundamental representation. This theory has vanishing one-loop beta function, and the gauge coupling constant τ=

θ 8πi + 2 π g

(1.1)

is exactly marginal. Argyres and Seiberg carried out a detailed study of the behavior of the Seiberg-Witten curve close to the point τ → 1, where the theory is infinitely strongly-coupled, and were led to conjecture a dual description involving an SU(2) group with gauge coupling τ =

1 . 1−τ

(1.2)

To understand the matter content of the dual theory, one first needs to recall the interacting superconformal field theory (SCFT) with flavor symmetry E 6 first described by [2]. This theory has one-dimensional Coulomb branch parametrized by u whose scaling dimension is 3, and is realized as the low-energy limit of the worldvolume theory on a D3-brane probing the transverse geometry of an F-theory 7-brane with E 6 gauge group. The gauge group on the 7-brane then manifests as a flavor symmetry from the point of view of the D3-brane. We denote this theory by SCFT[E 6 ] following [1]. Now, the theory Argyres and Seiberg proposed as the dual of the SU(3) gauge theory with six quarks consists of the SU(2) gauge bosons, coupled to one hypermultiplet in the doublet representation, and also to a subgroup SU(2) ⊂ E 6 of SCFT[E 6 ]. The SU(2) subgroup is chosen so that the raising operator of SU(2) maps to the raising operator for the highest root of E 6 . In the following, we refer to two sides of the duality as the SU(3) side and the exceptional side, respectively. Argyres and Seiberg provided a few compelling pieces of evidence for this duality. First, the flavor symmetry agrees. On the SU(3) side, there is a U(6) = U(1) × SU(6) symmetry which rotates the six quarks. On the exceptional side, there is an SO(2) symmetry which rotates a pair of quarks in the doublet representation, which can be identified with the U(1) part of U(6). Then, the flavor symmetry of the SCFT with E 6 is broken down to the maximal subgroup commuting with SU(2) ⊂ E 6 , which is SU(6). Second, the scaling dimensions of Coulomb-branch operators agree. Indeed, on the SU(3) side one has tr φ 2 and tr φ 3 , where φ is the adjoint chiral multiplet of SU(3). The dimensions are thus 2 and 3. On the exceptional side, one has tr ϕ 2 (where ϕ is the adjoint chiral multiplet of SU(2)), which has dimension 2, and the Coulomb-branch operator u of SCFT[E 6 ], which has dimension 3.

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391

Third, Argyres and Seiberg studied in detail the deformation of the Seiberg-Witten curve under the SU(6) mass deformation, and found remarkable agreement. Fourth, they computed the current algebra central charge of the SU(6) flavor symmetry on the SU(3) side, which agreed with the central charge of the E 6 symmetry on the exceptional side, inferred from the fact that the beta function of the SU(2) gauge group coupling is zero. This is as it should be, because SU(6) arises as a subgroup of E 6 on the exceptional side. This provided a prediction of the current central charge of SCFT[E 6 ] for the first time, which was later reproduced holographically by [3]. There are generalizations to similar duality pairs involving SCFTs with flavor symmetries other than E 6 [1,4]. Our aim in this note is to present further convincing evidence for this duality, by showing that the Higgs branches of the two sides of the duality are equivalent as hyperkähler cones. Mathematically speaking, we will show the agreement of their twistor spaces as complex varieties with real structure, but we have not been able to prove that they share the same family of twistor lines. Instead we give numerical evidence that their Kähler potentials agree in Appendix C. 1.2. Higgs branch. On the SU(3) side, let us denote the squark fields by Q ia ,

Q˜ ia ,

(1.3)

where i = 1, . . . , 6 are the flavor indices and a = 1, 2, 3 the color indices. The Higgs branch is the locus where the F-term and the D-term both vanish, divided by the action of the gauge group SU(3). As is well known, this space can also be obtained by setting F = 0 without setting D = 0, and dividing by the complexified gauge group SL(3, C). Thus the Higgs branch is parametrized by gauge invariant composite operators M i j = Q ia Q˜ aj ,

j

B i jk = abc Q ia Q b Q kc ,

B˜ i jk = abc Q˜ ia Q˜ bj Q˜ ck

(1.4)

which satisfy various constraints, e.g. B [i jk M l] m = 0

(1.5)

to which we will come back later. The fields Q ia , Q˜ ia have 36 complex components, while the F-term condition imposes 8 complex constraints. The quotient by SL(3, C) reduces the complex dimension further by 8, so the Higgs branch has complex dimension 2 × 3 × 6 − 8 − 8 = 20.

(1.6)

Our problem is to understand how this structure of the Higgs branch is realized on the exceptional side. Firstly, we have one hypermultiplet in the doublet representation, which we denote as vα , v˜α in N = 1 superfield notation. Here α = 1, 2 is the doublet index. We also have the Higgs branch of SCFT[E 6 ], the structure of which is known through the F-theoretic construction of the SCFT. Recall that this theory is the worldvolume theory on one D3-brane probing a F-theory 7-brane of type E 6 . Say the D3-brane extends along the directions 0123, and the 7-brane along the directions 01234567. The onedimensional Coulomb branch of this theory is identified with the transverse directions 89 to the 7-brane. The theory becomes superconformal when the D3-brane hits the 7-brane, at which point the Higgs branch emanates. This is identified as the process where a D3-brane is absorbed into the worldvolume of the 7-brane as an E 6 instanton

392

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along the directions 4567. The real dimension of the N -instanton moduli space of E 6 is 4h E 6 N with the dual Coxeter number h E 6 = 12. The center-of-mass motion of the instanton corresponds to a decoupled free hypermultiplet, and thus the genuine moduli space is the so-called ‘centered’ one-instanton moduli space without the center-of-mass motion, which has complex dimension 22. The SU(2) gauge group couples to the quark fields vα , v˜α , and this instanton moduli space. Imposing the F-term condition and dividing by the complexified gauge group, we find the complexified dimension of the Higgs branch as 2 × 2 + 22 − 3 − 3 = 20,

(1.7)

which correctly reproduces the dimension of the Higgs branch on the SU(3) side. We would like to perform more detailed checks, and for that purpose one needs to have a concrete description of the instanton moduli. It is well known that the ADHM description is available for classical gauge groups, but how can we proceed for exceptional groups? Luckily, there is another description of the 1-instanton moduli spaces, applicable to any group G, which identifies the centered 1-instanton moduli space with the minimal nilpotent orbit of G [5]. Let us now define the minimal nilpotent orbit. G C acts on the complexified Lie algebra gC , which has the Cartan generators H i and the raising/lowering operators E ±ρ for roots ρ. G C also acts on the dual vector space g∗C of gC via the coadjoint action,1 and the minimal nilpotent orbit Omin (G) of G is the orbit of (E θ )∗ , where θ denotes the highest root: Omin (G) = G C · (E θ )∗ .

(1.8)

The minimal nilpotent orbit is known to have polynomial defining equations. Moreover, they can be chosen to be quadratic, transforming covariantly under G C . These relations are known under the name of the Joseph ideal [6]. The simplest example is the case G = SU(2). In this case gC is three-dimensional; denote its three coordinates by a, b and c, which transform as a triplet of SU(2). The minimal nilpotent orbit is then given by a 2 + b2 + c2 = 0

(1.9)

which describes the space C2 /Z2 , and as is well-known, the centered one-instanton moduli space of SU(2) is exactly this orbifold. Let us come back to the case of E 6 . We fix an SU(2) subalgebra generated by E ±θ . The maximal commuting subalgebra is then SU(6). The E 6 algebra can be decomposed under the subgroup SU(2) × SU(6) into Xi j,

Yα[i jk] ,

Z αβ ,

(1.10)

where i, j, k = 1, . . . , 6 are the SU(6) indices, α, β = 1, 2 those for SU(2). Here X ij i jk

and Z αβ are adjoints of SU(6) and SU(2) respectively, and Yα transforms as the threeindex anti-symmetric tensor of SU(6) times the doublet of SU(2). The minimal nilpotent orbit is then given by the simultaneous zero locus of quadratic equations in X , Y and Z which we describe in detail later. 1 One can of course identify g∗ and g using the Killing form, but it is more mathematically natural to C C use the coadjoint representation here.

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393

For now let us see what are the gauge-invariant coordinates of the Higgs branch of the exceptional side. The SU(2) gauge group is identified to the SU(2) ⊂ E 6 just chosen above, i.e. the SU(2) gauge bosons couple to the current of this SU(2) subgroup of the E 6 symmetry. We also have the quarks vα and v˜α in addition to the fields X , Y and Z , and we need to make SU(2)-invariant combinations of them. Moreover, we need to impose the F-term equation, which is Z αβ + v(α v˜β) = 0

(1.11)

as we argue later. Thus, any appearance of Z inside a composite operator can be eliminated in favor of v and v. ˜ Therefore we have the following natural gauge-invariant composites, from which all gauge-invariant operators can be generated as will be shown in Sec. 5.4: (v v), ˜

Xi j,

(Y i jk v),

(Yi jk v). ˜

(1.12)

Here we defined (uw) ≡ u α wβ αβ

(1.13) i jk

for two doublets u α and wα , and Yi jk,α is defined by lowering the indices of Yα by the epsilon tensor, see Appendix A. This suggests the following identifications between the operators on the two sides of the duality: tr M ↔ (v v), ˜ B

i jk

↔ (Y

i jk

v),

Mˆ i j ↔ X i j , B˜ i jk ↔ (Yi jk v), ˜

(1.14) (1.15)

where Mˆ ij is the traceless part of M ij . The identifications preserve the dimensions of the operators if we assign dimensions 2 to the fields X , Y and Z . The SU(6) transformation nicely agrees. The U(1) part of the flavor symmetry can be matched if one assigns charge ˜ and charge ±3 to v, v. ±1 to Q, Q, ˜ This factor of 3 was predicted in the original paper [1] from a totally different point of view, by demanding that the two-point function of two U(1) currents should agree under the duality. Let us quickly recall how it was derived there. The form of the twopoint function of the U(1) current jµ is strongly constrained by the conservation and the conformal symmetry, and we have jµ (x) jν (0) ∝ k

x 2 gµν − 2xµ xν + ··· . x8

(1.16)

k is called the central charge, and · · · stands for less singular terms. Let us normalize k such that one hypermultiplet of charge q contributes q 2 to k. Assign Q, Q˜ the charge ±1, and let the charge of v, v˜ be ±q. Then k calculated from the SU(3) side is 6 × 3 = 18, while k determined from the exceptional side is 2q 2 . Equating these, Argyres and Seiberg concluded that the charge of v, v˜ should be q = ±3. The agreement is already impressive at this stage, but we would like to see how the constraints are mapped. We would also like to study how the hyperkähler structures agree, because so far we considered the Higgs branch only as a complex manifold. For that purpose we need to recall more about the hyperkähler cone.

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The structure of the rest of the paper is as follows: We discuss in Sec. 2 what data are mathematically necessary to show the equivalence of the Higgs branches. Section 3 is devoted to the description of the minimal nilpotent orbit, i.e. the 1-instanton moduli space, as a hyperkähler space. Sections 4 and 5 will be spent in calculating the necessary data on the SU(3) side and the exceptional side, respectively. Then they are compared in Sec. 6 which shows remarkable agreement. We conclude in Sec. 7. We have four Appendices: Appendix A collects our conventions, Appendix B gathers the machinery of twistor spaces required to show the equivalence of hyperkähler cones, and Appendix C compares the Kähler potentials of the duality pair. Appendix D is a summary for mathematicians. 2. Rudiments of Hyperkähler Cones Here we collect the basics of the hyperkähler cones in a physics language. Mathematically precise formulation can be found in [7,8]. The Higgs branch M of an N = 2 gauge theory is a hyperkähler manifold, i.e. one has three complex structures J 1,2,3 satisfying J 1 J 2 = J 3 , compatible with the metric g, and the associated two-forms ω1,2,3 are all closed. We choose a particular N = 1 supersymmetry subgroup of the N = 2 supersymmetry group, which distinguishes one of the complex structures, say J ≡ J 3 . M is then thought of as a Kähler manifold with the Kähler form ω = ω3 . = ω1 + iω2 is a closed (2, 0)-form on M which then defines a holomorphic symplectic structure on M. Physically this means that the N = 1 chiral ring, i.e. the ring of holomorphic functions on M, has a natural holomorphic Poisson bracket [ f 1 , f 2 ] = ( −1 )i j ∂i f 1 ∂ j f 2

(2.1)

for two holomorphic functions f 1,2 on M. Second, we are dealing with the Higgs branch of an N = 2 superconformal theory, which has the dilation and the SU(2) R symmetry built in the symmetry algebra. The dilation makes M into a cone with the metric 2 2 dsM = dr 2 + r 2 dsbase ,

(2.2)

and SU(2) R symmetry acts on the base of the cone as an isometry, rotating the three complex structures as a triplet. These two conditions make M into a hyperkähler cone. K = r 2 is a Kähler potential with respect to any of the complex structures J 1,2,3 , and is called the hyperkähler potential in the mathematical literature. The dilatation assigns the scaling dimensions, or equivalently the weights, to the chiral operators on M. Let us consider an element of SU(2) R which acts on the three complex structures as (J 1 , J 2 , J 3 ) → (J 1 , −J 2 , −J 3 ). This element defines an anti-holomorphic involution σ : M → M because it reverses J ≡ J 3 . This induces an operation σ ∗ on holomorphic functions on M via (σ ∗ ( f ))(x) ≡ f (σ (x)). σ ∗ maps holomorphic functions to anti-holomorphic functions, but is a linear operation, not a conjugate-linear operation. We call this operation the conjugation. As will be detailed in Appendix B, the space M as a complex manifold, with the Poisson brackets, the scaling weights and the conjugation, almost suffices to reconstruct the hyperkähler metric on M. Therefore, our main task in checking the agreement of the Higgs branches of the duality pair is to identify them as complex manifolds, and to show that the extra data defined on them also coincide. In order to complete the proof we need to show that the families of the twistor lines coincide, which we have not been

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able to do. Instead we will give numerical support by calculating the Kähler potential directly on both sides in Appendix C. The Higgs branches M that we treat here are gauge theory moduli spaces. They can be described by the hyperkähler quotient construction [9], which we now review. Let us start with an N = 2 gauge theory with the gauge group G, whose hypermultiplets take value in the hyperkähler manifold X . The action of G on X preserves three Kähler structures, and thus there are three moment maps µas (s = 1, 2, 3; a = 1, . . . , dim G) which satisfy dµas = ιξ a ωs ,

(2.3)

where ξ a is the Killing vector associated to the a th generator of G. The Higgs branch of the gauge theory, in the absence of any non-zero Fayet-Iliopoulos parameter, is then given by M ≡ X ///G ≡ {x ∈ X µas (x) = 0}/G. (2.4) With one complex structure J = J 3 chosen, it is convenient to call D a = µa3 ,

F a = µa1 + iµa2 .

(2.5)

M = {x ∈ X F a = 0}/G C .

(2.6)

Then, as a complex manifold,

It is instructive to note that F a is exactly the Hamiltonian which generates the G action on the chiral ring of X , under the Poisson bracket associated to = ω1 + iω2 . The conjugation σ ∗ and the Poisson bracket [·, ·] on the quotient M are given by the restriction of the corresponding operations on X . It is instructive to see why the Poisson bracket of the quotient is well-defined: two G-invariant holomorphic functions f 1,2 on X lead to the same function on M if and only if f 1 = f 2 + u a F a with holomorphic functions u a . Then we have, for a G-invariant holomorphic function h, [ f 1 , h] − [ f 2 , h] = [u a F a , h] = [u a , h]F a + u a [F a , h]

(2.7)

on X . The first term in the right hand side is zero on M because we set F a = 0, while the second term is zero because h is G-invariant. Therefore [ f 1 , h] and [ f 2 , h] determine the same holomorphic function on M. The Kähler potential of M is similarly the restriction of that of X to the zero locus of the moment maps in our situation, as discussed in Sec. 2B of [9]. To illustrate the procedure, let us consider an N = 1 supersymmetric U(1) gauge theory coupled to chiral fields i of charge qi whose Lagrangian is 4 ∗ 2qi V L= d θ i e i + ξ V , (2.8) i

where ξ is the Fayet-Iliopoulos parameter. The moduli space can be determined by taking the gauge coupling to be formally infinite, i.e. treating the linear superfield V as an auxiliary field. Then V is determined via its equation of motion qi i∗ e2qi V i + ξ = 0, (2.9) i

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i.e. i = eqi V i solve the usual D-term equation. The Kähler potential of the moduli space is then given by plugging the solution to (2.9) into (2.8). It can be generalized to any gauge group, and the result agrees with the mathematical formula given in Sec. 3.1 of [10]. This shows that the Kähler potential is given just by the restriction of the original one if ξ = 0. This analysis does not incorporate quantum corrections, but it is wellknown that for N = 2 theories the quantum effect does not modify the hyperkähler structure, see Sec. 3 of [11]. 3. Geometry of the Minimal Nilpotent Orbit Here we gather the relevant information on the hyperkähler geometry of the minimal nilpotent orbit of any simple group G, which coincides with the centered moduli space of single instantons with gauge group G [5,8]. We hope this section might be useful for anyone who wants to deal with the one-instanton moduli space. In the following G stands for a compact simple Lie group, gR its Lie algebra. We let G C and gC be complexifications of G and gR respectively. The existence of a uniform description of the one-instanton moduli space applicable to any G might be understood as follows: we can construct a one-instanton configuration easily by taking a BPST instanton of SU(2) and regard it as an instanton of G via a group embedding SU(2) ⊂ G. It is known that any one-instanton of G arises in this manner [12]. The one-instanton moduli space is then parameterized by the position, the size, and the gauge orientation of the BPST instanton inside G. This description realizes the one-instanton moduli space as a cone over a homogeneous manifold G/H , where H is the maximal subgroup of G which commutes with the SU(2) used in the embedding. It is however not directly suitable for the analysis of its complex structure. For that purpose we use another realization of the one-instanton moduli space as the minimal nilpotent orbit Omin of G [5]. Let us define Omin . First we decompose gC into the Cartan generators H i and the raising/lowering operators E ±ρ for roots ρ. The minimal nilpotent orbit Omin (G) is then the orbit of (E θ )∗ in g∗C , where θ denotes the highest root: Omin (G) = G C · (E θ )∗ ⊂ g∗C .

(3.1)

We will write Omin without explicitly writing G for the sake of simplicity when there is no confusion. We think of elements of gC as holomorphic functions on Omin , i.e. we have holomorphic functions2 Xa (a = 1, . . . , dim G ) on Omin . The defining equations of Omin are a set of quadratic equations which we call the Joseph relations [6].3 These relations can be studied using a theorem of Kostant [13]: Let V (α) denote the representation space of a semisimple group G with the highest weight α, and let v ∈ V (α)∗ be a vector in the highest weight space. The orbit G C · v is then an affine algebraic variety whose defining ideal I is generated by its degree-two part I2 . Furthermore, I2 is given by the relation Sym2 V (α) = V (2α) ⊕ I2 ,

(3.2)

∗ 2 More mathematically, one has a natural holomorphic g∗ -valued function X : O min → gC given by C the embedding. Then every element t ∈ gC gives a holomorphic function (X, t) on Omin via x ∈ Omin → (X(x), t). Our Xa is (X, T a ) for a generator T a of gC . We take a real basis of gC , so in fact T a ∈ gR ⊂ gC . 3 Strictly speaking, the Joseph ideal is a two-sided ideal in the universal enveloping algebra of g , and C what we use below is its associated ideal in the polynomial algebra.

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where we identify Sym2 V (α) as the space of degree-two polynomials on V (α)∗ . The minimal nilpotent orbit is exactly of this form where V (α) is the adjoint representation, i.e. Omin = {X ∈ g∗C (X ⊗ X)|I2 = 0}. (3.3) For practice, let us apply this to the case G = SU(2). There, V (α) is the triplet representation, so by (3.2) I2 is the singlet representation. Therefore, if we parameterize su(2) by (a, b, c), the minimal nilpotent orbit is given by the equation a 2 + b2 + c2 = 0,

(3.4)

which is C2 /Z2 as it should be. Now that we have given Omin as a complex manifold, let us describe its hyperkähler structure. The main fact we use is that G acts isometrically on Omin , preserving the hyperkähler structure. There is a triplet of moment maps µas for this action where a = 1, . . . , dim G and s = 1, 2, 3. The functions Xa are the holomorphic moment maps of the G action, i.e. Xa = µa1 + iµa2 . It follows that their Poisson bracket is [Xa , Xb ] = f ab c Xc ,

(3.5)

where f ab c are the structure constants of G. Phrased differently, the holomorphic symplectic structure underlying the hyperkähler structure of the nilpotent orbit is the standard Kirilov–Kostant–Souriau symplectic form on the coadjoint orbit [5,8]. The conjugation is given by the SU(2) R action, which sends (µ1 , µ2 , µ3 ) to (µ1 , −µ2 , −µ3 ). Therefore σ ∗ (Xa ) = (Xa )∗ .

(3.6)

The scaling dimension of X is fixed to be two, as it should be for the F-term in an N = 2 supersymmetric theory. Let us next describe a Kähler potential for Omin , which was determined in [14]. The derivation boils down to the following: G acts on Omin with cohomogeneity one, and by averaging over this action we can consider K to be G-invariant; so K is a √ function of tr XX∗ . K should be of scaling dimension two, so that K is proportional to tr XX∗ up to a constant. The constant factor can be fixed by considering a particular element on Omin . For this purpose we again turn to the minimal nilpotent orbit of SU(2), which is C2 /Z2 . The normalization of the Kähler potential of the minimal nilpotent orbit of a general group can then be determined because it contains the minimal nilpotent orbit of SU(2) as a subspace. We parameterize C2 by (u, u) ˜ and divide by the multiplication by −1. We define our conventions for the holomorphic Poisson bracket and the Kähler potential of a flat H as follows: K = |u|2 + |u| ˜ 2,

[u, u] ˜ = 1.

(3.7)

Now, C2 /Z2 is parametrized by Z 11 = u 2 /2,

Z 12 = Z 21 = u u/2, ˜

Z 22 = u˜ 2 /2

(3.8)

which satisfy Z 11 Z 22 = Z 12 2 .

(3.9)

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The Kähler potential is now K = 2 |Z 11 |2 + |Z 22 |2 + 2|Z 12 |2 = 2 Z αβ Z¯ αβ .

(3.10)

Then, the moment map associated to the generator J3 of non-R SU(2) acting on C2 /Z2 can be explicitly calculated, with the result F = Z 12 ,

D=

2 (|Z 11 |2 − |Z 22 |2 ). K

(3.11)

Now that the preparation is done, we move on to the calculation of the Higgs branch on both sides of the duality. 4. SU(3) Side The theory has six quarks in the fundamental representation, Q ia ,

Q˜ ia ,

(4.1)

where a = 1, . . . , 6 and i = 1, 2, 3. As is well known, any SU(3)-invariant polynomial constructed out of these fields is a polynomial in the operators [11]: M i j = Q ia Q˜ aj ,

j

B i jk = abc Q ia Q b Q kc ,

B˜ i jk = abc Q˜ ia Q˜ bj Q˜ ck .

(4.2)

In the following we study the Poisson brackets, the action of the conjugation, and the constraints in turn. 4.1. Poisson brackets. The Poisson bracket of the basic fields is given by [Q ia , Q˜ bj ] = δ i j δ b a .

(4.3)

[M ij , Q ak ] = −δ kj Q ia ,

(4.4)

Then we have, for example,

i.e. M ij is the generator of U(6). We define tr M to be the trace of M i j , and 1 Mˆ ij = M ij − δ ij tr M 6

(4.5)

is its traceless part. Mˆ i j is the SU(6) generator and tr M the U(1) generator. We define the U(1) charge q of an operator O to be given by [tr M, O] = −qO.

(4.6)

The most complicated bracket is [B i jk , B˜ lmn ] = 18M [i [l M j m δ k] n] 1 = 18 Mˆ [i [l Mˆ j m δ k] n] + 6(tr M) Mˆ [i [l δ j m δ k] n] + (tr M)2 δ [i [l δ j m δ k] n] . 2 (4.7)

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4.2. Conjugation. We choose the involution on the elementary fields to be σ ∗ (Q ia ) = ( Q˜ ia )∗ ,

σ ∗ ( Q˜ ia ) = −(Q ia )∗ .

(4.8)

Then the transformation of the composites are σ ∗ (M ij ) = −(M j i )∗ ,

σ ∗ (tr M) = −(tr M)∗ ,

(4.9)

σ ∗ (B i jk ) = ( B˜ i jk )∗ ,

σ ∗ ( B˜ i jk ) = −(B i jk )∗ .

(4.10)

4.3. Constraints. The constraints were studied in [11]. Those which come before imposing the F-term constraint are B i jk B˜ lmn = 6M [i l M j m M k] n , B i j[k B lmn] = 0, B˜ i j[k B˜ lmn] = 0, M [i j B klm] = 0, M i [ j B˜ klm] = 0.

(4.11) (4.12) (4.13)

The F-term constraint 1 ˜ =0 Q ia Q˜ ib − δab (Q Q) 3

(4.14)

Mˆ ij B jkl = 16 (tr M)B ikl ,

(4.15)

Mˆ ij B˜ ikl = 16 (tr M) B˜ jkl ,

(4.16)

Mˆ ij M j k

(4.17)

further imposes

=

1 6 (tr

M)Mki .

We will find it convenient later to have constraints in terms of irreducible representations (irreps) of SU(6). We use the Dynkin labels to distinguish the irreps in the following. The M B = 0 relations (4.13), (4.15), (4.16) give Mˆ {i l B [ jk]}l = 0, Mˆ l {i B[ jk]}l = 0,

Mˆ {i l B˜ [ jk]}l = 0, Mˆ l {i B˜ [ jk]}l = 0.

(4.18) (4.19)

Here we defined the projector from a tensor with the structure Ai[ jk] to the irrep (1, 1, 0, 0, 0) by A{i[ jk]} ≡ Ai[ jk] − A[i[ jk]] . We also have 1 Mˆ [i l B jk]l = (tr M)B i jk , 6

1 Mˆ l [i B˜ jk]l = (tr M) B˜ i jk . 6

(4.20)

The M M = 0 relation (4.17) gives 1 j Mˆ ij Mˆ k = δki Mˆ nm Mˆ mn , 6 1 j i Mˆ j Mˆ i = (tr M)2 . 6

(4.21) (4.22)

The B B = 0 relation (4.12) gives B ikl B jkl = 0,

B˜ ikl B˜ jkl = 0.

(4.23)

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Finally, the decomposition of the B B˜ = M M M relation gives, using (4.21) and (4.22) repeatedly, 2 (tr M)3 , 9 2 = (tr M)2 Mˆ ij , 9 2 = (tr M) Mˆ [i [k Mˆ j] l] 0,1,0,1,0 , 3 j = 6 Mˆ [i l Mˆ m Mˆ k] n 0,0,2,0,0 .

B i jk B˜ i jk = B ikl B˜ jkl adj B i jm B˜ klm 0,1,0,1,0 B i jk B˜ lmn

0,0,2,0,0

(4.24) (4.25) (4.26) (4.27)

5. Exceptional Side 5.1. Poisson brackets. We have chiral fields Xa which transform in the adjoint of E 6 , and satisfy the quadratic Joseph identities. We decompose Xa under the subgroup SU(2) × SU(6) ⊂ E 6 . It gives X ij , Yα[i jk] ,

Z αβ ,

(5.1) i jk

where X ij and Z αβ are the adjoints of SU(6) and SU(2) respectively, and Yα is in the doublet of SU(2) and in the representation (0, 0, 1, 0, 0), i.e. the three-index antisymmetric tensor, of SU(6). The Poisson brackets of the fields X , Y and Z are exactly the Lie brackets as explained above, which we take to be

[Z αβ , Z γ δ ] =

[X ij , X lk ] = δli X kj − δ kj X li ,

(5.2)

1 (αγ Z βδ + βγ Z αδ + αδ Z βγ + βδ Z αγ ) 2

(5.3)

and 1 [X ij , Yαklm ] = −3δ [k j Yαlm]i + δ ij Yαklm , 2 i jk [Z αβ , Yγi jk ] = Y(α β)γ ,

(5.4) (5.5)

and finally [Yαi jk , Yβlmn ] = i jklmn Z αβ −

3 αβ (X [i p jk]lmnp + X [l p mn]i jkp ). 2

(5.6)

The final commutation relation can also be written as [Yαi jk , Ylmn β ] = −18X [i [l δ j m δ k] n] − 6Z αβ δ [i [l δ j m δ k] n] .

(5.7)

As we explained above, X , Y and Z are the holomorphic moment maps of the E 6 action. Therefore the contribution from Omin to the F-term constraint for the SU(2) gauge group is given just by Z αβ . We take the bracket of v and v˜ to be [vα , v˜β ] = αβ .

(5.8)

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401

Then we have [v(α v˜β) , vγ ] = v(α β)γ ,

(5.9)

and [(v v), ˜ vα ] = vα ,

[(v v), ˜ v˜α ] = −v˜α .

(5.10)

Recall that we define (uw) ≡ u α wβ αβ for two doublets u α and wα . It is straightforward to check that v(α v˜β) is the moment map of the SU(2) action on v and v. ˜ Thus the F-term condition is v(α v˜β) + Z αβ = 0.

(5.11)

5.2. Conjugation. We take the conjugation on the variables v, v˜ to be σ ∗ (vα ) = (v˜β )∗ αβ ,

σ ∗ (v˜α ) = (vβ )∗ αβ .

(5.12)

i jk

In terms of our variables (X ij , Yα , Z αβ ), the conjugation acts as follows: σ ∗ (X ij ) = −(X j i )∗ , σ ∗ (Yαi jk ) = (Yi jk β )∗ αβ , ∗

(5.13)

σ ∗ (Yi jk α ) = −(Yβ )∗ αβ , i jk

∗

σ (Z αβ ) = (Z γ δ ) αγ βδ .

(5.14) (5.15)

5.3. Constraints. As explained in Sec. 3, the Joseph relations are given by (X ⊗ X)|I2 = 0,

(5.16)

Sym2 V (adj) = V (2adj) ⊕ I2 .

(5.17)

where I2 is given by the relation

Here, V (adj) is the adjoint representation of E 6 whose Dynkin label is adj = 0 0 01 0 0 . We then have (5.18) I2 = V 1 0 00 0 1 ⊕ V 0 0 00 0 0 . The representations which appear in I2 , decomposed under SU(2) × SU(6), are summarized in Table 1. The table reads as follows: e.g. for relation 4, the fourth column tells us there is one Joseph identity transforming as a doublet in SU(2) and as (0, 0, 1, 0, 0) under SU(6), but the fifth column says one can construct two objects in i jk this representation from bilinears in X ij , Yα and Z αβ . This means the identity has the form 0 = Yαi jk Z βγ αβ + c4 X [i l Yγjk]l ,

(5.19)

where c4 needs to be fixed, which can be done e.g. by explicitly evaluating the right hand side on a few elements on the nilpotent orbit. Elements on the nilpotent orbit can be readily generated, because one knows that the point X ij = 0,

Yαi jk = 0,

Z 11 = 1,

Z 12 = Z 22 = 0

(5.20)

402

D. Gaiotto, A. Neitzke, Y. Tachikawa Table 1. Decomposition of I2 in terms of SU(2) × SU(6) ⊂ E 6

1. 2. 3. 4. 5. 6. 7.

SU(2)

SU(6)

in I2

in Sym2 V (adj)

3 2 2 2 1 1 1

(1,0,0,0,1) (1,1,0,0,0) (0,0,0,1,1) (0,0,1,0,0) (0,1,0,1,0) (1,0,0,0,1) (0,0,0,0,0)

1 1 1 1 1 1 2

2 1 1 2 2 1 3

is on the nilpotent orbit by definition. Then the rest of the points can be generated by the coadjoint action of E 6 , which can be obtained by exponentiating the structure constants. Carrying out this program, we obtain the following full set of Joseph identities: 1. 2. 3.

1 ikl 0 = X ij Z αβ + Y(α Y jklβ) , 4 0 = X l {i Y[ jk]}lα , 0=

4.

0=

5.

0=

6. 7. 7’.

X l Yα[ jk]}l , Yαi jk Z βγ αβ

(5.21) (5.22)

{i

(5.23) [i

+ X l Yγjk]l , (Yαi jm Yklmβ αβ − 4X [i [k X j] l] ) 0,1,0,1,0 ,

1 0 = X ki X k j − δ ij X k l X l k , 6 0 = Yαi jk Yi jkβ αβ + 24Z αβ Z γ δ αγ βδ , 0=

X ij X j i

+ 3Z αβ Z γ δ

αγ βδ

.

(5.24) (5.25) (5.26) (5.27) (5.28)

5.4. Gauge invariant operators. Let us enumerate the generators of the SU(2)-invariant i jk operators constructed out of vα , v˜α , and X ij , Yα , Z αβ , using the F-term equation (5.11) and the Joseph identities (5.21) ∼ (5.28). Suppose we have a monomial constructed from those fields. We first replace every appearance of Z αβ by −v(α v˜β) . All the SU(2) indices are contracted by epsilon tensors of SU(2). Therefore the monomial is a product of X ij , (v v), ˜ (Y i jk v), (Y i jk v) ˜ and (Y i jk Y lmn ). The last of these can be eliminated using the Joseph identities. Indeed, the combination of the relations (5.25), (5.27) and (5.28) gives a Joseph identity of the form Yαi jk Ylmn β αβ = 18X [i [l X j m δ k] n] − 3Z αβ Z γ δ αγ βδ δ [i [l δ j m δ k] n] .

(5.29)

We conclude that any SU(2)-invariant polynomial is a polynomial in ˜ (Y i jk v), and (Y i jk v). ˜ X ij , (v v),

(5.30)

6. Comparison 6.1. Identification of operators. Let us now proceed to the comparison of the structures we studied in Sec. 4 and in Sec. 5. We first make the following identification: Mˆ ij = X ij ,

tr M = −3(v v). ˜

(6.1)

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403

These are the moment maps of the flavor symmetries SU(6) and U(1), so the identification is fixed including the coefficients, and then the Poisson brackets involving either Mˆ ˜ also agrees with that or tr M automatically agree. The conjugation acting on X ij , (v v) i ˆ on M j and tr M. We then set B i jk = c(Y i jk v),

B˜ i jk = c(Y ˜ i jk v). ˜

(6.2)

One has σ ((Y i jk v)) = (Yi jk v) ˜ ∗.

(6.3)

To be consistent with (4.10), we need to have c˜ = c∗ .

(6.4)

Let us then calculate the Poisson bracket of (Y i jk v) and (Ylmn v) ˜ using (5.29). We have 9 ˜ 2 . (6.5) ˜ = −18X [i [l X j m δ k] n] + 18(v v)X ˜ [i [l X j m δ k] n] − (v v) [(Y i jk v), (Ylmn v)] 2 Comparing with the bracket [B i jk , B˜ lmn ] calculated in (4.7), we find they indeed agree if cc˜ = −1. Thus we conclude c = c˜ = i, i.e. B i jk = i(Y i jk v),

B˜ i jk = i(Yi jk v). ˜

(6.6)

6.2. Constraints. Now, let us check using the Joseph relations that the constraints on the SU(3) side, listed in Eqs. (4.18) ∼ (4.27), can be correctly reproduced on the exceptional side. • • • • • •

(4.18): Contract v or v˜ to the relation 2, (5.22). (4.19): Contract v or v˜ to the relation 3, (5.23). (4.20): Contract v or v˜ to the relation 4, (5.24). (4.21): This is exactly the relation 6, (5.26). (4.22): This is exactly the relation 7’, (5.28). (4.23): Contract vα vβ or v˜α v˜β to the relation 1 (5.21).

As for the relation of the type B B˜ = M M M, • • • •

(4.24): The singlet part. Contract vα v˜β to the relation 7 (5.27). (4.25): The adjoint part. Contract vα v˜β to the relation 1 (5.21). (4.26): The (0, 1, 0, 1, 0) part. Contract vα v˜β to the relation 5 (5.25). (4.27):This is the (0, 0, 2, 0, 0) part and is slightly trickier, but it follows from a cubic Joseph identity 0 = αγ βδ Z αβ Yγi jk Ylmn,δ 0,0,2,0,0 −6X [i l X j m X k] n 0,0,2,0,0 (6.7) upon replacing Z αβ with v(α v˜β) . This cubic Joseph identity itself can be derived from the quadratic Joseph identities, as it should be. First, we use the relation 4 (5.24) to show αγ βδ Z αβ Yγi jk Ylmn δ 0,0,2,0,0 ∝ X [i p Yαjk] p Ylmn β αβ . (6.8)

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Now, the antisymmetric product of two Y ’s contain both the singlet and the (0, 1, 0, 1, 0) part. One sees the singlet drops out inside the projector to the (0, 0, 2, 0, 0) part, so we have ∝ X [i p (Y jk] p Ylmn ) 0,1,0,1,0 0,0,2,0,0 . (6.9) Then we use the relation 5 (5.25) to transform this to ∝ X li X j m X k n 0,0,2,0,0 .

(6.10)

The proportionality constant can be fixed, e.g. by evaluating on a few points on the orbit. This concludes the comparison of the constraints. 7. Conclusions In the previous three sections, we determined the Higgs branches both on the SU(3) side and on the exceptional side. We demonstrated that their defining equations agree, and furthermore exhibited that the Poisson bracket and the conjugation are the same on both sides. As was stated in Sec. 2 and will be detailed in Appendix B, these are (almost) sufficient to conclude that they are the same as hyperkähler manifolds. To remove any remaining doubts, we compare the Kähler potentials of the two sides in Appendix C. Again, they show remarkable agreement with one another. Thus we definitely showed the agreement of the Higgs branches of the new S-duality pair proposed by Argyres and Seiberg in [1], which provides a convincing check of their conjecture. In this paper we only dealt with the example involving E 6 , but there are more examples of similar dualities in [1 and 4]. It would be interesting to carry out the same analysis of the Higgs branches to those examples. A pressing issue is to understand the Argyres-Seiberg duality more fully. For example, it would be nicer to have an embedding of this duality in string/M-theory. We hope to revisit these problems in the future. Acknowledgements. The authors thank Alfred D. Shapere for collaboration at an early stage of the project. They would also like to thank S. Cherkis, C. R. LeBrun, H. Nakajima for helpful discussions. They also relied heavily on the softwares LiE4 and Mathematica. DG is supported in part by the DOE grant DE-FG0290ER40542 and in part by the Roger Dashen membership in the Institute for Advanced Study. AN was supported in part by the Martin A. and Helen Chooljian Membership at the Institute for Advanced Study, and in part by the NSF under grant numbers PHY-0503584 and PHY-0804450. YT is supported in part by the NSF grant PHY-0503584, and in part by the Marvin L. Goldberger membership in the Institute for Advanced Study.

A. Conventions Greek indices α, β are for the doublets of SU(2), a, b, c, . . . for the triplets of SU(3) and i, j, k, . . . for the sextets of SU(6). We define (uw) ≡ u α wβ αβ for two doublets u α and wα , 4 It can be downloaded from http://www-math.univ-poitiers.fr/~maavl/LiE/.

(A.1)

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405

We use the following sign conventions for the epsilon tensors of SU(2), SU(3) and SU(6): αβ = −αβ , abc = abc , i jklmn = i jklmn .

(A.2)

We normalize the antisymmetrizer [abc...] and the symmetrizer (abc...) so that they are projectors, i.e. T i jk = T [i jk]

(A.3)

for the antisymmetric tensors T i jk , etc. We raise and lower three antisymmetrized indices of SU(6) via the following rule: T i jk =

1 i jklmn Tlmn , 6

Tlmn =

1 i jk T i jklmn . 6

(A.4)

Our convention for the placement of the indices of the complex conjugate is e.g. Z¯ αβ ≡ (Z αβ )∗ ,

(A.5)

i.e. the complex conjugation is always accompanied by the exchange of subscripts and superscripts, as is suitable for the action of SU groups. We take the Kähler potential of a flat C parameterized by z with the standard metric to be K = |z|2 .

(A.6)

B. Twistor Spaces of Hyperkähler Cones Recall that a hyperkähler manifold M admits a continuous family of complex structures Jζ , parameterized by ζ ∈ CP1 . The full information in the hyperkähler metric is captured by this family of complex structures and their Poisson brackets. It can be encoded into purely holomorphic data on a complex manifold Z, the twistor space of M, as we now review. Topologically Z = M × CP1 . Its complex structure can be specified by specifying which functions on Z are holomorphic: they are f (x, ζ ) which are holomorphic in ζ for fixed x ∈ M, and also holomorphic in x with respect to complex structure Jζ for fixed ζ . Hence we may view Z as a holomorphic fiber bundle over CP1 , where the fiber over ζ is just a copy of M, equipped with complex structure Jζ . The Poisson brackets on the holomorphic functions in each fiber glue together globally to give a bracket operation on Z. This bracket operation is globally twisted by the line bundle O(−2): i.e. given local holomorphic functions f 1 , f 2 we get a local section { f 1 , f 2 } of O(−2), and more generally if f 1 , f 2 are sections of O(d1 ), O(d2 ) then { f 1 , f 2 } is a section of O(d1 + d2 − 2). Finally there is an involution σ on Z, simply defined by (x, ζ ) → (x, −1/ζ¯ ). This is an antiholomorphic involution, since the complex structure Jζ is opposite to J−1/ζ¯ . As a complex manifold Z is a fibration over CP1 , and (x, ζ ) with x fixed gives a holomorphic section of this fibration, which is invariant under σ . The normal bundle to this section is isomorphic to the line bundle O(1)⊕n , where n is the complex dimension of M. Conversely, a holomorphic section of Z which is invariant under σ and whose normal bundle is isomorphic to O(1)⊕n is called a twistor line. Therefore, the points on M give rise to a n-dimensional family of twistor lines on Z.

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It was shown in [9] that given Z, together with its Poisson brackets and antiholomorphic involution, one can canonically reconstruct a hyperkähler metric on the space of twistor lines. Therefore, to check that our two hyperkähler cones are the same is essentially to check that their twistor spaces Z are the same. Now, the twistor space of a hyperkähler cone can be constructed from the data we described in Sec. 2, i.e. the Poisson bracket, the dilatation and the conjugation on M. We pick one complex structure induced from the hyperkähler structure, and regard M as a complex manifold. We then form Z as a complex manifold as Z = ((C2 \ (0, 0)) × M)/C× ,

(B.1)

where C× acts on the first factor by multiplication, and on the second factor as the natural

complexification of the action of the dilatation. Then the Poisson bracket on M naturally induces one on Z. We define σ on Z to send (z, w, x) ∈ C2 ×M to (−w, ¯ z¯ , σ (x)). Then it is straightforward to check that this Z is the twistor space of M, using the SU(2) R action on M rotating three complex structures. There is a subtle problem remaining, however. Namely, the theorem in [9] asserts that there is a component of the space of the twistor lines of Z which agrees metrically with the original hyperkähler manifold M, but does not exclude the possibility that the space of twistor lines has many components, each of which is a hyperkähler manifold with the same complex structure but with a different metric. Mathematicians the authors consulted know no concrete example where this latter possibility is realized, so the authors think it quite unlikely that our two hyperkähler manifolds are the same as holomorphic symplectic manifolds but not as hyperkähler manifolds. To dispel this last possibility, in the next appendix we directly compare the Kähler potential of our two hyperkähler manifolds. C. Comparison of the Kähler Potential In this Appendix, we describe the method to calculate and compare the Kähler potential of the Higgs branches on the two sides of the duality. C.1. Exceptional side. The invariant norm of E 6 in our notation is 1 Z αβ Z¯ αβ + Yαi jk Y¯iαjk + X ij X¯ j i . 6 Therefore the correctly normalized Kähler potential is

1 i jk K E 6 = 2 Z αβ Z¯ αβ + Yα Y¯iαjk + X ij X¯ j i , 6

(C.1)

(C.2)

and the D-term for the SU(2) ⊂ E 6 is 2 1 i jk ¯ γ i jk ¯ γ (E 6 ) γδ γδ ¯ ¯ Z αγ Z δβ + Z βγ Z δα + (Yα Yi jk γβ + Yβ Yi jk γ α ) . Dαβ = K E6 12 (C.3) We also have quarks vα , v˜α which have (|vα |2 + |v˜α |2 ) K v,v˜ = α

(C.4)

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407

and (v,v) ˜

Dαβ

=

1 (vα v¯ γ γβ + vβ v¯ γ γ α + v˜α v¯˜ γ γβ + v˜β v¯˜ γ γ α ). 2

(C.5)

The Kähler potential of the exceptional side is thus given by K v,v˜ + K E 6

(C.6)

restricted to the locus v(α v˜β) + Z αβ = 0,

(v,v) ˜

Dαβ

(E )

+ Dαβ 6 = 0

(C.7)

expressed as a function of M ij , B i jk , B˜ i jk and their complex conjugates. C.2. SU(3) side. We start from the Kähler potential |Q ia |2 + | Q˜ a |2 . K =

(C.8)

i

i,a

Using the analysis in [11], the Kähler potential on the quotient was determined in [15] as

ν2 (C.9) K =2 m i2 + , 4 i=1,2,3

where (m 21 , m 22 , m 23 , 0, 0, 0) are the eigenvalues of M ij M¯ j k , and ν is defined by 3ν =

|Q ia |2 − | Q˜ ia |2 ,

(C.10)

i,a

i.e. 1/3 of the U(1) D-term. In terms of gauge invariants we have ⎛ ⎞ 2 ν ν ⎝ m2 + + ⎠ = 16 B i jk B¯ i jk , i 4 2 i=1,2,3 ⎛ ⎞ 2 ν ν ⎝ m2 + − ⎠ = 16 B˜ i jk B¯˜ i jk . i 4 2

(C.11)

(C.12)

i=1,2,3

C.3. Comparison. Now, the Kähler potentials of the two sides, (C.6) and (C.9) should ˜ but we have not been able to check that analytically. agree as functions of M, B and B, Instead, one can check it numerically on as many points on the quotient as computer time allows. The algorithm is as follows: i jk

1. Generate a point X = (X ij , Yα , Z αβ ) on the nilpotent orbit of E 6 , by applying an i jk

element of the group E 6 to the point (Z 11 , Z 12 , Z 22 ) = (1, 0, 0), X ij = Yα

= 0.

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2. Find vα , v˜α which satisfy v(α v˜β) + Z αβ = 0.

(C.13)

˜ This is more or less unique up to C× action on v, v. 3. Apply SL(2, C) action to (v, v, ˜ X) to find the solution of the D-term equation, (v,v) ˜

Dαβ

(E )

+ Dαβ 6 = 0.

(C.14)

This is equivalent to the minimization of ˜ + K E6 (g(X)), K v,v˜ (g(v), g(v))

(C.15)

where g is an SL(2, C) action. 4. Form M, B, B˜ from v, v˜ and X thus obtained, and calculate ν and m i . At this point, two checks of the sanity of the calculation are possible. One is to see that three eigenvalues of M M¯ are zero. Another is to see that ν determined from (C.11), (C.12) is equal to ν= |vα |2 − |v˜α |2 . (C.16) α

The latter fact follows from the identification of ν as 1/3 of the U(1) moment map on the quotient. 5. Evaluate the Kähler potential of the SU(3) side using (C.9) and compare it to that of the exceptional side (C.6). We implemented the algorithm above in Mathematica, and found that the value of the Kähler potential at any points agrees on both sides of the duality to arbitrary accuracy.5 An analytic proof of the agreement of the Kähler potential will be welcomed. D. Mathematical Summary Let us summarize briefly in the language of mathematics what was done in this paper. Let M(m, n) be M(m, n) = Hom(V, W ) ⊕ Hom(W, V )

where V = Cm , W = Cn

(D.1)

which is a flat hyperkähler space of quaternionic dimension mn. It has a natural triholomorphic action of U(m) × U(n) induced from its action on V and W . Let N (m, n) be the flat hyperkähler space N (m, n) = Rm ⊗R Hn

(D.2)

of quaternionic dimension mn, which has a natural triholomorphic action of SO(m) × Sp(n). One then defines a hyperkähler quotient A by A1 = M(6, 3)///SU(3). 5 We thank H. Elvang for improvement of the accuracy in the calculation.

(D.3)

Argyres-Seiberg Duality and the Higgs Branch

409

We consider another hyperkähler quotient A2 = (N (2, 1) × Omin (E 6 ))///Sp(1)

(D.4)

where Omin (G) is the minimal nilpotent orbit of the group G, and the Sp(1) action on Omin (E 6 ) is given by considering the maximally compact subgroup Sp(1)×SU(6) ⊂ E 6 . One sees easily that A1,2 are both of quaternionic dimension 10, both carry a natural triholomorphic action of SU(6) × U(1). Our claim is that A1 = A2 as hyperkähler cones. We demonstrated that A1 and A2 match as holomorphic symplectic varieties by explicitly showing that their defining equations and the holomorphic symplectic forms are the same. We also found that the twistor spaces of A1 and A2 are the same as complex manifolds with antiholomorphic involution, but could not show that A1 and A2 correspond to the same family of twistor lines. Instead we directly compared the Kähler potentials of A1 and A2 . Again we could not rigorously prove the equivalence, but we performed numerical calculations of the Kähler potential which convinced us that they agree. The equivalence of A1,2 was suggested by the analysis of a new type of S-duality in four-dimensional N = 2 supersymmetric gauge theories in [1]. In [1,4], more examples of the same type of duality were described, of which we record two more here. Now consider B1 = N (12, 2)///Sp(2)

(D.5)

B2 = Omin (E 7 )///Sp(1).

(D.6)

and

Here Sp(1) acts on Omin (E 7 ) through the maximal subgroup Sp(1)×SO(12) ⊂ E 7 . The quaternionic dimension of B1,2 is 14, and both have triholomorphic actions of SO(12). We believe B1 = B2 as hyperkähler cones. For an example which involves Omin (E 8 ), consider C1 = (Z ⊕ N (11, 3))///Sp(3).

(D.7)

Here Z is a pseudoreal irreducible representation of Sp(3) of quaternionic dimension 7, which arises as ∧3C X = Z ⊕ X,

(D.8)

where X = C6 is the defining representation of Sp(3). Let us take another hyperkähler quotient C2 = Omin (E 8 )///SO(5),

(D.9)

where SO(5) acts via embedding SO(5) × SO(11) ⊂ SO(16) ⊂ E 8 .

(D.10)

It is easy to check that C1,2 are both of quaternionic dimension 19, and SO(11) acts triholomorphically on both C1 and C2 . We predict that C1 = C2 as hyperkähler cones.

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References 1. Argyres, P.C., Seiberg, N.: S-duality in N = 2 supersymmetric gauge theories. JHEP 0712, 088 (2007) 2. Minahan, J.A., Nemeschansky, D.: An N = 2 superconformal fixed point with E 6 global symmetry. Nucl. Phys. B 482, 142 (1996) 3. Aharony, O., Tachikawa, Y.: A holographic computation of the central charges of d = 4, N = 2 SCFTs. JHEP 0801, 037 (2008) 4. Argyres, P.C., Wittig, J.R.: Infinite coupling duals of N = 2 gauge theories and new rank 1 superconformal field theories. JHEP 0801, 074 (2008) 5. Kronheimer, P.B.: Instantons and the geometry of the nilpotent variety. J. Diff. Geom. 32, 473 (1990) 6. Joseph, A.: The minimal orbit in a simple Lie algebra and its associated maximal ideal. Ann. Sci. École Norm. Sup. Ser. 4 9, 1 (1976) 7. Swann, A.: Hyperkähler and quaternionic Käher geometry. Math. Ann. 289, 421 (1991) 8. Brylinski, R.: Instantons and Kähler geometry of nilpotent orbits. In: Representation Theories and Algebraic Geometry. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Dordrecht: Kluwer, 1998, pp. 85–125 9. Hitchin, N.J., Karlhede, A., Lindström, U., Roˇcek, M.: Hyperkähler metrics and supersymmetry. Commun. Math. Phys. 108, 535 (1987) 10. Biquard, O., Gauduchon, P.: Hyper-Kähler metrics on cotangent bundles of Hermitian symmetric spaces. In: Lecture Notes in Pure and Appl. Math. 184, Newyork: Dekker, 1997, pp. 287–298 11. Argyres, P.C., Plesser, M.R., Seiberg, N.: The Moduli Space of N = 2 SUSY QCD and Duality in N = 1 SUSY QCD. Nucl. Phys. B 471, 159 (1996) 12. Vainshtein, A.I., Zakharov, V.I., Novikov, V.A., Shifman, M.A.: ABC of instantons. Sov. Phys. Usp. 25, 195 (1982) [Usp. Fiz. Nauk 136, 553 (1982)] 13. Garfinkle, D.: A new construction of the Joseph ideal. MIT thesis, 1982. Available on-line at the service ‘MIT Theses in DSpace.’ http://dspace.mit.edu/handle/1721.1/15620, 1982 (see chap. III) 14. Kobak, P., Swann, A.: The hyperkähler geometry associated to Wolf spaces. Boll. Unione Mat. Ital. Serie 8, Sez. B Artic. Ric. Mat. 4, 587 (2001) 15. Antoniadis, I., Pioline, B.: Higgs branch, hyperkähler quotient and duality in SUSY N = 2 Yang-Mills theories. Int. J. Mod. Phys. A 12, 4907 (1997) Communicated by A. Kapustin

Commun. Math. Phys. 294, 411–437 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0955-5

Communications in

Mathematical Physics

Isometric Immersions and Compensated Compactness Gui-Qiang Chen1 , Marshall Slemrod2 , Dehua Wang3 1 Department of Mathematics, Northwestern University, Evanston, IL 60208, USA.

E-mail: [email protected]

2 Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA.

E-mail: [email protected]

3 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA.

E-mail: [email protected] Received: 27 December 2008 / Accepted: 10 September 2009 Published online: 27 November 2009 – © Springer-Verlag 2009

Abstract: A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold M2 which can be realized as isometric immersions into R3 . This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in R3 . The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in R3 . As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a C 1,1 isometric immersion of the two-dimensional manifold in R3 satisfying our prescribed initial conditions. To achieve this, we introduce a vanishing viscosity method depending on the features of initial value problems for isometric immersions and present a technique to make the a priori estimates including the L ∞ control and H −1 –compactness for the viscous approximate solutions. This yields the weak convergence of the vanishing viscosity approximate solutions and the weak continuity of the Gauss-Codazzi system for the approximate solutions, hence the existence of an isometric immersion of the manifold into R3 satisfying our initial conditions. The theory is applied to a specific example of the metric associated with the catenoid. 1. Introduction A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold M2 which can be realized as isometric immersions into R3 (cf. Yau [40]; also see [21,34,36]). Important results have been achieved

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for the embedding of surfaces with positive Gauss curvature which can be formulated as an elliptic boundary value problem (cf. [21]). For the case of surfaces of negative Gauss curvature where the underlying partial differential equations are hyperbolic, the complimentary problem would be an initial or initial-boundary value problem. Hong in [23] first proved that complete negatively curved surfaces can be isometrically immersed in R3 if the Gauss curvature decays at a certain rate in the time-like direction. In fact, a crucial lemma in Hong [23] (also see Lemma 10.2.9 in [21]) shows that, for such a decay rate of the negative Gauss curvature, there exists a unique global smooth, small solution forward in time for prescribed smooth, small initial data. Our main theorem, Theorem 5.1(i), indicates that in fact we can solve the corresponding problem for a class of large non-smooth initial data. Possible implication of our approach may be in existence theorems for equilibrium configurations of a catenoidal shell as detailed in Vaziri-Mahedevan [39]. When the Gauss curvature changes sign, the immersion problem then becomes an initial-boundary value problem of mixed elliptic-hyperbolic type, which is still under investigation. The purpose of this paper is to introduce a general approach, which combines a fluid dynamic formulation of balance laws with a compensated compactness framework, to deal with the isometric immersion problem in R3 (even when the Gauss curvature changes sign). In Sect. 2, we formulate the isometric immersion problem for two-dimensional Riemannian manifolds in R3 via solvability of the Gauss-Codazzi system. In Sect. 3, we introduce a fluid dynamic formulation of balance laws for the Gauss-Codazzi system for isometric immersions. Then, in Sect. 4, we provide a compensated compactness framework and present one of our main observations that this framework is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in R3 . A generalization of this approach to higher dimensional immersions has been given in Chen-Slemrod-Wang [8]. In Sect. 5, as a first application of this approach, we focus on the isometric immersion problem of two-dimensional Riemannian manifolds with strictly negative Gauss curvature. Since the local existence of smooth solutions follows from the standard hyperbolic theory, we are concerned here with the global existence of solutions of the initial value problem with large initial data. The metrics gi j we study have special structures and forms usually associated with the catenoid of revolution when g11 = g22 = cosh(x) and g12 = 0. For these cases, while Hong’s theorem [23] applies to obtain the existence of a solution for small smooth initial data, our result yields a large-data existence theorem for a C 1,1 isometric immersion. To achieve this, we introduce a vanishing viscosity method depending on the features of the initial value problem for isometric immersions and present a technique to make the a priori estimates including the L ∞ control and H −1 –compactness for the viscous approximate solutions. This yields the weak convergence of the vanishing viscosity approximate solutions and the weak continuity of the Gauss-Codazzi system for the approximate solutions, hence the existence of a C 1,1 –isometric immersion of the manifold into R3 with prescribed initial conditions. We remark in passing that, for the fundamental ideas and early applications of compensated compactness, see the classical papers by Tartar [38] and Murat [32]. For applications to the theory of hyperbolic conservation laws, see for example [4,10,13,18,37]. In particular, the compensated compactness approach has been applied in [3,6,11,12, 25,26] to the one-dimensional Euler equations for unsteady isentropic flow, allowing for cavitation, in Morawetz [29,30] and Chen-Slemrod-Wang [7] for two-dimensional

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413

steady transonic flow away from stagnation points, and in Chen-Dafermos-SlemrodWang [5] for subsonic-sonic flows. 2. The Isometric Immersion Problem for Two-Dimensional Riemannian Manifolds in R3 In this section, we formulate the isometric immersion problem for two-dimensional Riemannian manifolds in R3 via solvability of the Gauss-Codazzi system. Let ⊂ R2 be an open set. Consider a map r : → R3 so that, for (x, y) ∈ , the two vectors {∂x r, ∂y r} in R3 span the tangent plane at r(x, y) of the surface r() ⊂ R3 . Then n=

∂x r × ∂y r |∂x r × ∂y r|

is the unit normal of the surface r() ⊂ R3 . The metric on the surface in R3 is ds 2 = dr · dr

(2.1)

ds 2 = (∂x r · ∂x r) (dx)2 + 2(∂x r · ∂y r) dxdy + (∂y r · ∂y r) (dy)2 .

(2.2)

or, in local (x, y)–coordinates,

Let gi j , i, j = 1, 2, be the given metric of a two-dimensional Riemannian manifold M parameterized on . The first fundamental form I for M on is I := g11 (dx)2 + 2g12 dxdy + g22 (dy)2 .

(2.3)

Then the isometric immersion problem is to seek a map r : → R3 such that dr · dr = I, that is, ∂x r · ∂x r = g11 , ∂x r · ∂y r = g12 , ∂y r · ∂y r = g22 ,

(2.4)

so that {∂x r, ∂y r} in R3 are linearly independent. The equations in (2.4) are three nonlinear partial differential equations for the three components of r(x, y). The corresponding second fundamental form is II := −dn · dr = h 11 (dx)2 + 2h 12 dxdy + h 22 (dy)2 ,

(2.5)

and (h i j )1≤i, j≤2 is the orthogonality of n to the tangent plane. Since n · dr = 0, then d(n · dr) = 0 implies −II + n · d 2 r = 0, i.e.,

2 II = (n · ∂x2 r) (dx)2 + 2(n · ∂xy r) dxdy + (n · ∂y2 r) (dy)2 .

The fundamental theorem of surface theory (cf. [14,21]) indicates that there exists a surface in R3 whose first and second fundamental forms are I and II if the smooth coefficients (gi j ) and (h i j ) of the two given quadratic forms I and II with (gi j ) > 0 satisfy the Gauss-Codazzi system. It is indicated in Mardare [28] (Theorem 9; also see [27]) that this theorem holds even when (h i j ) is only in L ∞ for given (gi j ) in C 1,1 , for

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which the immersion surface is C 1,1 . This shows that, for the realization of a two-dimensional Riemannian manifold in R3 with given metric (gi j ) > 0, it suffices to solve for (h i j ) ∈ L ∞ determined by the Gauss-Codazzi system to recover r a posteriori. The simplest way to write the Gauss-Codazzi system (cf. [14,21]) is as (2)

(2)

(2)

∂x M − ∂y L = 22 L − 212 M + 11 N , (1)

(1)

(1)

(2.6)

∂x N − ∂y M = −22 L + 212 M − 11 N , with L N − M 2 = κ.

(2.7)

Here h 11 L=√ , |g|

h 12 M=√ , |g|

h 22 N=√ , |g|

2 , κ(x, y) is the Gauss curvature that is determined by the |g| = det (gi j ) = g11 g22 − g12 relation: R1212 (m) (m) (n) (m) (n) (m) κ(x, y) = , Ri jkl = glm ∂k i j − ∂ j ik + i j nk − ik n j , |g|

Ri jkl is the curvature tensor and depends on (gi j ) and its first and second derivatives, and (k)

i j =

1 kl g ∂ j gil + ∂i g jl − ∂l gi j 2

is the Christoffel symbol and depends on the first derivatives of (gi j ), where the summation convention is used, (g kl ) denotes the inverse of (gi j ), and (∂1 , ∂2 ) = (∂x , ∂y ). Therefore, given a positive definite metric (gi j ) ∈ C 1,1 , the Gauss-Codazzi system gives us three equations for the three unknowns (L , M, N ) determining the second fundamental form II . Note that, although (gi j ) is positive definite, R1212 may change sign and so does the Gauss curvature κ. Thus, as we will discuss in Sect. 3, the Gauss-Codazzi system (2.6)–(2.7) generically is of mixed hyperbolic-elliptic type, as in transonic flow (cf. [2,7,9,31]). In §3–4, we introduce a general approach to deal with the isometric immersion problem involving nonlinear partial differential equations of mixed hyperbolic-elliptic type by combining a fluid dynamic formulation of balance laws in §3 with a compensated compactness framework in §4. As an example of direct applications of this approach, in §5 we show how this approach can be applied to establish an isometric immersion of a two-dimensional Riemannian manifold with negative Gauss curvature in R3 . 3. Fluid Dynamic Formulation for the Gauss-Codazzi System From the viewpoint of geometry, the constraint condition (2.7) is a Monge-Ampère equation and the equations in (2.6) are integrability relations. However, our goal here is to put the problem into a fluid dynamic formulation so that the isometric immersion problem may be solved via the approaches that have shown to be useful in fluid dynamics

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for solving nonlinear systems of balance laws. To achieve this, we formulate the isometric immersion problem via solvability of the Gauss-Codazzi system (2.6)–(2.7), that is, solving first for h i j , i, j = 1, 2, via (2.6) with constraint (2.7) and then recovering r a posteriori. To do this, we set L = ρv 2 + p,

M = −ρuv,

N = ρu 2 + p,

and set q 2 = u 2 + v 2 as usual. Then the Codazzi equations in (2.6) become the familiar balance laws of momentum: (2)

(2)

(2)

(1)

(1)

(1)

∂x (ρuv) + ∂y (ρv 2 + p) = −(ρv 2 + p)22 − 2ρuv12 − (ρu 2 + p)11 ,

(3.1)

∂x (ρu 2 + p) + ∂y (ρuv) = −(ρv 2 + p)22 − 2ρuv12 − (ρu 2 + p)11 , and the Gauss equation (2.7) becomes ρpq 2 + p 2 = κ.

(3.2)

From this, we can see that, if the Gauss curvature κ is allowed to be both positive and negative, the “pressure” p cannot be restricted to be positive. Our simple choice for p is the Chaplygin-type gas: 1 p=− . ρ Then, from (3.2), we find −q 2 +

1 = κ, ρ2

and hence we have the “Bernoulli” relation: 1 ρ= . 2 q +κ

(3.3)

This yields p = − q 2 + κ,

(3.4)

and the formulas for u 2 and v 2 : u 2 = p( p − M),

v 2 = p( p − L),

M 2 = (N − p)(L − p).

The last relation for M 2 gives the relation for p in terms of (L , M, N ), and then the first two give the relations for (u, v) in terms of (L , M, N ). We rewrite (3.1) as ∂x (ρuv) + ∂y (ρv 2 + p) = R1 , ∂x (ρu 2 + p) + ∂y (ρuv) = R2 , where R1 and R2 denote the right-hand sides of (3.1).

(3.5)

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We now find the corresponding “geometric rotationality–continuity equations”. Multiplying the first equation of (3.5) by v and the second by u, and setting ∂x v − ∂y u = −σ, we see v R1 1 div(ρu, ρv) − ∂y κ = + σ u, ρ 2 ρ R2 1 u div(ρu, ρv) − ∂x κ = − σ v, ρ 2 ρ and hence 1ρ ∂y κ + 2v 1ρ div(ρu, ρv) = ∂x κ + 2u

div(ρu, ρv) =

R1 ρuσ + , v v R2 ρvσ − . u u

Thus, the right hand sides of (3.6) are equal, which gives a formula for σ : 1 1 1 σ = v − u . ρ∂ ρ∂ κ + R κ + R x 2 y 1 ρq 2 2 2

(3.6)

(3.7)

If we substitute this formula for σ into (3.6), we can write down our “rotationality-continuity equations” as 1 1 1 ∂x v − ∂y u = u − v , (3.8) κ + R κ + R ρ∂ ρ∂ y 1 x 2 ρq 2 2 2 1 ρu 1 ρv v u ∂x (ρu) + ∂y (ρv) = ∂x κ + ∂y κ + 2 R 1 + 2 R 2 . (3.9) 2 2 2q 2q q q In summary, the Gauss-Codazzi system (2.6)–(2.7), the momentum equations (3.1)– (3.4), and the rotationality-continuity equations (3.3) and (3.8)–(3.9) are all formally equivalent. However, for weak solutions, we know from our experience with gas dynamics that this equivalence breaks down. In Chen-Dafermos-Slemrod-Wang [5], the decision was made (as is standard in gas dynamics) to solve the rotationality-continuity equations and view the momentum equations as “entropy” equalities which may become inequalities for weak solutions. In geometry, this situation is just the reverse. It is the Gauss-Codazzi system that must be solved exactly and hence the rotationality-continuity equations will become “entropy” inequalities for weak solutions. The above issue becomes apparent when we set up “viscous” regularization that preserves the “divergence” form of the equations, which will be introduced in §5.3. This is crucial since we need to solve (3.8)–(3.9) exactly, as we have noted. To continue further our analogy, let us define the “sound” speed: c2 = p (ρ),

(3.10)

which in our case gives c2 =

1 . ρ2

(3.11)

Isometric Immersions and Compensated Compactness

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Since our “Bernoulli” relation is (3.3), we see c2 = q 2 + κ.

(3.12)

Hence, under this formulation, (i) when κ > 0, the “flow” is subsonic, i.e., q < c, and system (3.1)–(3.2) is elliptic; (ii)