Annales Henri Poincaré - Volume 3

Ann. Henri Poincar´e 3 (2002) 1 – 17 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/010001-17 $ 1.50+0.20/0

Annales Henri Poincar´ e

On the Gap in the Spectrum of the Translations H. J. Borchers Abstract. The spectrum of the translations in local quantum field theory will be analyzed in order to show that in a positive energy representation without vacuum vector and with lowest mass m1 there is no gap in the spectrum which is larger than 2m1 . In particular in a zero mass representation there is no hole at all. These results are obtained with methods of analytic functions of several complex variables.

1 Introduction Let π be a representation of quantum field theory of local observables in the sense of Araki, Haag and Kastler [Ha92], which fulfils the spectrum condition. We assume that the representation of the translation is the minimal one, which implies that its spectrum is located on a Lorentz invariant set of the closed forward light cone V + (see [BB85] and [Bch85]). If the representation is generated from a vacuum state then Wightman’s result [Wi64] implies that the spectrum is an additive set, i.e. it fulfils S + S ⊂ S, (1.1) where S denotes the support of the spectrum. If the representation space does not contain a vacuum vector then I conjectured in [Bch96] that the spectrum fulfils the relation S + S + S ⊂ S. (1.2) In particle physics factor representations are also called super selection sectors because a superposition of vectors in two different sectors does not lead to observable consequences. Therefore, the charges, describing the different sectors, must commute with every observable. Since the time development belongs to the weak closure of the observable algebra it acts separately in every sector. In such a situation one can characterize the factor representations of the observable algebra by charge quantum numbers. Dealing with vectors carrying a charge and adding to this a particle with a non-zero charge, then the total charge changes. This implies that we obtain another sector. Therefore, one stays in the same sector only if we add particles carrying the total charge zero. Let us associate to every sector a charge quantum number Q, then the structure of the spectrum of the translation is known, provided the representation in a charged sector is generated from the vacuum sector by a charged field, in the sense of Doplicher, Haag and Roberts [DHR69a,b]. In this situation one has in the case of Abelian charges the relation SQ1 + SQ2 = SQ1 +Q2 ,

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which has been shown in [Bch65]. SQ denotes the spectrum of the sector characterized by Q. We are interested in a statement concerning only one sector. If the above picture with the charged fields is correct, then the adjoint of a charged field carries the opposite charge. Therefore, one must find to every sector a sector with opposite charge. Since the spectrum does not change by this charge conjugation one has: S−Q = SQ . More generally this relation holds for particles and anti-particles. Since a particle together with its anti-particle form a neutral pair we remain in the same sector if we add to a vector in this sector such a pair. If one takes the particle from the given sector, one obtains Eq. (1.2). In general theories one cannot expect that the representations obtained from a vacuum sector are all representations. In the presence of zero-mass particles the investigations of Doplicher and Spera [DS82], Borek [Bk82] in the non-interacting, and of Buchholz and Doplicher [BD84] in the interacting situation show, that most positive energy representations are not connected with vacuum representations. A similar situation might hold also in a gauge field theory. Therefore, Eq. (1.2) needs a separate investigation. One special result is known for arbitrary sectors [Bch86], which indicates that the conjecture is probably correct. Theorem. Let π be a covariant factor representation of a theory of local observables fulfilling the spectrum condition, and let T (a) be the unique minimal representation of the translations. Assume the spectrum of T (a) consists of two parts: (a) Isolated hyperboloids with the masses m1 < m2 · · · mi < · · · . (b) The rest starting at mc > mi where mc denotes the beginning of the continuous spectrum or the first accumulation point of the mi  s. Then 3m1 ≥ mc . One would like to show that above 3m1 there is no gap in the spectrum at all. Unfortunately techniques with which one can derive such result are missing. Since we know that there can be a gap between m1 and 3m1 , such method must distinguish between the regions below 3m1 and above 3m1 . Our procedure does not do it, therefore, we only obtain that there is no gap larger than 2m1 . The technique we are using is that of analytic functions of several complex variables, in particular the edge of the wedge theorem (Brehmermann, Oehme and Taylor [BOT56]). In this case one has two analytic functions holomorphic in the forwardand backward tube respectively. These functions coincide on some real coincidence domain and the edge of the wedge theorem implies that both functions are different branches of one analytic function. The problem consists in calculating the envelope of holomorphy or of parts of it.

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In the next section the details of the problem and the technique to be used will be presented. In our situation the coincidence domain consists of three parts each of which can be handled with help of the Jost-Lehmann-Dyson representation [Dy58]. We will compute the holomorphy domains of the three parts in section 3. In the last section the influence of the different parts on each other will be discussed with help of the double cone theorem. In favorable situations there is a simpler proof of the main result. It is using the ”reentrant nose theorem”, which is a consequence of the Jost-Lehmann-Dyson representation. The method will be described at the end of section 2.

2 Representation of the problem Quantum field theory of local observables in the sense of Araki, Haag and Kastler [Ha92] is concerned with C ∗ -algebras A(O), associated with bounded open regions O ⊂ Rd , which have a common unit. These algebras shall fulfil isotony, i.e. O1 ⊂ O2 implies A(O1 ) ⊂ A(O2 ), and locality, i.e. if O1 and O2 are space-like separated, then the algebras A(O1 ) and A(O2 ) commute element-wise. In order that this statement makes sense the space Rd must be furnished with the Minkowski metric. The global algebra A is defined as the C ∗ -inductive limit of {A(O)}. We also assume that the net {A(O)} is covariant under a representation of the translation group of Rd by automorphisms, i.e. to a ∈ Rd exists an automorphism αa with αa A(O) = A(O + a). A covariant representation π of A in the representation space H is such that there exists a continuous unitary representation T (a) of the translation group of Rd implementing αa T (a)π(A)T (−a) = π(αa (A)),

A ∈ A.

In addition T (a) shall fulfil the spectrum condition, i.e. spectrum T (Rd ) is contained in the closed forward light-cone. As a consequence of the spectrum condition one can choose the representation T (a) of the translation as elements of the von Neumann algebra generated by π(A) [Bch96]. Since T (a) leaves the center of π(A)” point-wise invariant [Ar64] it is no restriction assuming that π is a factor representation. The spectrum of the translations shall have the following structure: It starts at m1 > 0 and in addition there shall be a gap in the spectrum between m2 and m3 , m2 < m3 . We want to show that these assumptions lead to a contradiction if m3 − m2 > 2m1 holds. Since the statement we are looking for is scale invariant we may assume m1 = 1. For the investigation one looks at matrix elements of the form F + (a) = (ψ, BT (a)Aψ)

(2.1)

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where ψ belongs to the representation space Hπ and A, B are operators belonging to π(A(D)). D denotes a double cone with center at the origin. One also looks at the expression (2.2) F − (a) = (ψ, T (a)AT (−a)BT (a)ψ). Locality implies that F + (a) and F − (a) coincide for a ∈ (2D) , where the prime denotes the space-like complement. We are interested in the Fourier transform of (2.1) and (2.2). The support of the Fourier transform of (2.1) is contained in the support of the spectrum S described above. In order that the Fourier transform of (2.2) has good support property we must choose ψ in a suitable manner. The best way is to choose the spectral support of ψ close to the bottom of the spectrum. Let t = (1 + , 0) then we choose the support of ψ in S ∩ (t + V − ). Because of the curvature of the hyperboloid the support of the vector is contained in the double 1 , 0). This choice allows to describe the support of the cone Dt
,t with t = ( 1+ − Fourier transform of F (a). If we denote this support by S  then we get S  = 2t + (V1− \ {crescent moon}), − − crescent moon = set below 2t + Hm and above 2t + Hm . 2 3

(2.3)

+ In this formula Vm+ characterizes the set {p ∈ V + ; p2 ≥ m2 }, Vm− = −Vm+ . Hm 2 2 − denotes the upper sheet of the hyperboloid p = m and Hm its lower sheet. For the future investigations we choose  so small that the crescent moon domain is sufficiently big. The details will be discussed later. About the spectrum of the translations we know definitely only the values mi , i = 1, 2, 3. Whether or not there exists other gaps in the spectrum than that between m2 and m3 is not determined by the formulation of the result Thm. 4.1. Therefore, one knows only vectors with energy-momentum support in the neighbourhood of the masses mi . This motivates the choice of the vector ψ. If m1 = 0, then one has to choose for the support of ψ a small double cone such that its closure is in the interior of the forward light cone. Such vector exists because the theorem in the introduction implies that m1 = 0 is not an isolated hyperboloid. Let F (a) = F + (a)− F − (a) be the commutator function. It vanishes in (2D) . Therefore, we can write F (a) = G+ (a) − G− (a) with

supp G+ (a) ⊂ −b + V + ,

supp G− (a) ⊂ b + V − ,

(2.4)

where b is the upper tip of the double cone 2D. Looking at the Fourier transform we obtain F˜ (p) = F˜ + (p) − F˜ − (p) and

supp F˜ (p) ⊂ S ∪ S  .

(2.5)

On the other hand one can write ˜ + (p) − G ˜ − (p), F˜ (p) = G

(2.6)

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˜ + (p) is the boundary value of a function holomorphic in the forward where G + tube T = {z = x + iy ∈ Cd ; x ∈ Rd , y ∈ V + }. Here V + denotes the interior ˜ − (p) is the boundary value of an of the forward light cone. Correspondingly G − + analytic function holomorphic in T = −T . Since the two functions coincide on Rd \ {S ∪ S  } we have to deal with an edge of the wedge problem. If one has an edge of the wedge problem where the coincidence domain is bounded by two space-like hyper-surfaces then the domain of holomorphy can be computed with help of the Jost-Lehmann-Dyson formula [Dy58] (see also Bros, Messiah and Stora [BMS61] [Bch96]), i.e. the complement of the holomorphy domain is the union of hyperboloids not entering the coincidence domain. If we assume that the same method would also work in our situation then the coincidence domain could be enlarged provided m3 > 3 holds. But one does not know whether or not also for time-like disconnected coincidence domains a formula like the Jost-Lehmann-Dyson representation holds. Therefore, we have to use other methods of enlarging the domain of holomorphy. If the domain of holomorphy can be enlarged into S \ {2t − V1 } then one can conclude that the spectrum is smaller than assumed. The reason is the following: Let ∆ be the set Dt
,t and E(∆) the corresponding spectral projection of the translation, then we can choose an arbitrary vector in E(∆)H. Moreover, the matrix elements are formed with operators in π(A(D)). Since the sets D are arbitrary, except for the requirement that their center is the origin, any operator in ∪A(D) can be chosen. These are norm-dense in A. Therefore, the conclusion for the spectrum holds for any vector in the closure of π(A)E(∆)H. The projection F on this space belongs to π(A) . Let Z be the central carrier of F then F π(A) and Zπ(A) are isomorphic, and hence we obtain a change of ZS. In the coming investigations we are dealing with factor representations. So one has Z = 1l and the analyticity methods show directly the change of the spectrum.

2.a A simple plausibility argument If one disregards the problem of the support of the vector ψ then there is a direct argument showing that there is no gap of width larger than 2m1 . This method uses the reentrant nose theorem, which follows from the Jost-Lehmann-Dyson representation. Since in this subsection we do not present a correct proof we look at the enlargement of the domain of holomorphy, which cuts into the support of −S convoluted with the double of the support of ψ. For a correct proof one would have to interchange the gap with the crescent moon domain. At the end of this subsection an example will be given showing that this method does not work in all cases, and that one has to use the more complicated methods described in section 3 and 4. Let τ be the vector (τ, 0) and Dτ − ,τ be the double cone {τ −  + V + } ∩ {τ + V − } and assume that for τ > m1 the spectral projection E(Dτ − ,τ ) of the translations does not vanish. Choosing τ then the support of F − (p) is (2τ +

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Vm−1 ) \ crescent moon. (The crescent moon part is unimportant for the coming considerations.) Therefore, for m2 < 2τ − m1 < m3

(2.7)

The support of F − (p) cuts into the gap between m2 and m3 . The gap itself is a Jost-Lehmann-Dyson domain. Therefore, one can ask for conditions such that the part of (2τ + Vm−1 ) which cuts into the gap is a reentrant nose such that part of it can be erased by holomorphic completion. This means (gap) \ (2τ + Vm−1 ) is not the complement of hyperboloids not entering this domain. If this is the case then holomorphic completion shows that (2τ + Vm−1 ) is not a stable set, contradicting the assumptions that m1 is the lower boundary of the spectrum. The critical hyperboloid is Hm1 centered at 2τ . Its upper sheet shall not cut the hyperboloid Hm3 . The upper sheets of the two hyperboloids hit each-other if m3 − 2τ > m1

or

m3 − m1 > 2τ

holds. Together with Eq. (2.7) this implies m3 − m1 > m2 + m1 ,

(2.8)

which shows that there is no gap with width larger than 2m1 . The difficulty of this approach is the existence of suitable vectors ψτ which we have to assume in the above approach. That this assumption is not always satisfied shows the following Example: Look at the situation: (i) There is a gap between m1 and 3m1 . (ii) There is a gap between 3m1 and 6m1 . Trying to apply the above method we find that ψτ must be located at the + . Therefore, 2τ is a little above 6m1 and the reentrant nose hyperboloid H3m 1 appears only if m3 > 7m1 . Therefore, the existence of the second gap can not be excluded with this method.

3 The holomorphy domain for subsets of the coincidence region The coincidence domain of our problem consists of three parts. The envelope of holomorphy of each part can be computed separately with help of the JostLehmann-Dyson representation. The union of the three domains will be the starting point for the investigation of the whole problem. The three domains in question are: 1. The complement of {V1+ ∪ (2t + V1− )}.

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2. The gap, i.e. the domain between the two upper sheets of the hyperboloids + + Hm and Hm . 2 3 3. The crescent moon shaped domain, i.e. the domain between the hyperboloids − − − 2t + Hm and 2t + Hm . This means the points below 2t + Hm and above 2 3 2 − 2t + Hm3 . We start with the first coincidence domain and notice that it is invariant under rotations. The complement of the domain of holomorphy is given by the union of d ; (p − u)2 = µ2 } the real or complex points of the hyperboloids H(µ, √ u) = {p ∈ C √ which have their center u at the surface u0 = t2 and u = eλ t2 , where e is a real unit vector in the plane p0 = 0 and 0 ≤ λ ≤ 1. The possible values of µ are determined by the condition that the real part of H(µ, u) is contained in {V1+ ∪ (2t + V1− )}. Fixing u then the possible values of µ are larger or equal than a minimal value µ0 (u). Since the zero-component of u is fixed the function µ0 (u) depends only on the norm of the space-like components of u. √ √ First let us compute the function µ0 (u). To this end set τ = t2 , τ  = t2 , which implies the representation u = τ (1, λe). The hyperboloids are given by the formula
(p0 − τ )2 − (p − λτ e)2 = µ2 or p0 = τ ± µ2 + (p − λτ e)2 . In order that the upper sheet of this hyperboloid belongs to V1+ there must hold

τ + µ2 + (p − λτ e)2 − 1 + p2 ≥ 0. (3.1) Since we want to look at the minimum we can choose p and e parallel and set p = πe. To obtain an equation for minimum one has to differentiate (3.1) with respect to π. π π − λτ −√ . 0=
2 2 1 + π2 µ + (π − λτ ) This leads to a second degree equation for π (π − λτ )2 (1 + π 2 ) (π − λτ )2 (π −

2 λτ 1−µ2 )

with the solution π=

= π 2 (µ2 + (π − λτ )2 ), = π 2 µ2 , =

λ2 τ 2 µ2 (1−µ2 )2 ,

λτ λτ , (1 ± µ) = 2 1−µ 1∓µ

which leads to the expressions
1 + π2
µ2 + (π − λτ )2

= =

1
(1 ∓ µ)2 + λ2 τ 2 , 1∓µ µ
(1 ∓ µ)2 + λ2 τ 2 . 1∓µ

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The minimal value of (3.1) shall be zero, which implies τ+

µ − 1
(1 ∓ µ)2 + λ2 τ 2 = 0. 1∓µ

Hence we have to take the − sign. This implies for the minimal µ0 (λ)
µ0 (λ) = max{1 − τ 1 − λ2 , 0}.

(3.2)

Since µ has to be non-negative one must choose µ = 0 if the first expression becomes negative. This first expression is the equation of a semi-circle with radius τ. For the holomorphy domain one must find the points p + iq such that the equation (p − u + iq)2 = µ2 ≥ µ20 (3.3) cannot be fulfilled. Eq. (3.3) reads (p − u)2 − q 2 + 2i(q, p − u) = µ2 . Therefore, points of analyticity are: Either one has (q, p − u) = 0 ∀ u, or if (q, p − u) = 0 for one u then (p − u)2 < µ20 (u) + q 2 must hold.

(3.4)

Next we look at the gap as coincidence domain. This set is invariant under Lorentz transformations. Therefore, the surface for the center of the hyperboloids is also invariant, and the same holds for the minimal µ. The centers of 2 2 ) and the minimal µ is the hyperboloids are located on the surface u2 = ( m3 +m 2 m3 −m2 µ0 = . Let (p, q) be given, then exists always a u with (q, p − u) = 0. In 2 order that p + iq belongs to the domain of holomorphy (q, p − u) = 0 and (p − u)2 < µ20 + q 2

∀ u fulfilling the first eqation

(3.5)

must hold. Since (p − u) is time-like this equation can only be fulfilled if −q 2 < 2 2 ) holds. ( m3 −m 2 Finally we look at the crescent moon problem. The interesting part of the surface for the centers of the hyperboloids must be inside this coincidence domain. The maximal value of v := u, which we denote by vm is obtained if (u0 , u) belongs to both hyperboloids, i.e. if   2 = 2τ − 2 m23 + vm u0 = 2τ  − m22 + vm holds. With this value for u0 we obtain for vm a second order equation
2 2 2 )(m2 + v 2 ), 4(τ − τ  )2 = (m23 + vm ) + (m22 + vm ) − 2 (m23 + vm m 2 2 2 2 (4(τ − τ  )2 − 2vm − (m22 + m23 ))2 = 4(m23 + vm )(m22 + vm ),

2 vm 16(τ − τ  )2 − 16(τ − τ  )4 + 8(τ − τ  )2 (m22 + m23 ) − (m23 − m22 )2 = 0,

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which implies 2 vm =

16(τ − τ  )4 + (m23 − m22 )2 − 8(τ − τ  )2 (m22 + m23 ) . 16(τ − τ  )2

(3.6)

2

2) This formula is valid for (τ − τ  )2 < (m3 −m as should be. One concludes this 4 2 because vm has to be positive. The corresponding value of u0 is restricted to    2 2 (3.7) 2τ − m3 + v < u0 < 2τ − m22 + v 2 ,

where we have set v = u. The exact value of u0 will be determined together with the minimal value µ0 of µ by the requirement that the hyperboloid H(µ0 , u) has to touch both H − (m2 , 2t ) and H − (m3 , 2t). With π = p one obtains two equations: 
(3.8a) u0 + µ2 + (π − v)2 − 2τ  + m22 + π 2 > 0, 
u0 − µ2 + (π − v)2 − 2τ + m23 + π 2 < 0. (3.8b) One finds the minimal value of (3.8a) by differentiating this equation with respect to π. π−v
2 µ + (π − v)2

+

(π − v)2 (m22 + π 2 ) =

π
= 0, 2 m2 + π 2

π 2 (µ2 + (π − v)2 ).

2 This implies π = mvm . The sign of the square root has to be chosen in such a 2 ±µ manner that π ≤ v holds. The minimal value of (3.8a) is   2 )2 . (3.9a) u0 − 2τ  + µ2 + v 2 ( m2µ+µ )2 + m22 + v 2 ( mm 2 +µ

The maximal value of (3.8b) is obtained by π = u0 − 2τ +

vm3 m3 −µ

which leads to

  3 µ2 + v 2 ( m3µ−µ )2 + m23 + v 2 ( mm )2 . 3 −µ

(3.9b)

The minimal value of µ is obtained by setting (3.9a) resp. (3.9b) equal to zero. This implies (u0 − 2τ  )2

=

= =

2 (µ2 + v 2 ( m2µ+µ )2 ) + (m22 + v 2 ( mm )2 ) + 2 +µ  2 2 2 (µ2 + v 2 ( mµ2 +µ )2 )(m22 + v 2 ( mm )2 ), 2 +µ  2 2 (m22 + µ2 )(1 + (m2v+µ)2 ) + 2 µ2 m22 (1 + (m2v+µ)2 )2 ,

(m2 + µ)2 (1 +

v2 (m2 +µ)2 )

= (m2 + µ)2 + v 2 ,

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and hence µa =

Ann. Henri Poincar´e


(u0 − 2τ  )2 − v 2 − m2 .

(3.10a)


(u0 − 2τ )2 − v 2 .

(3.10b)

In the same manner one finds µb = m3 −

Now we can determine u0 from the equation µa = µb . This leads to (m2 + m3 )2 = ((u0 − 2τ )2 − v 2 ) + ((u0 − 2τ  )2 − v 2 )+
2 ((u0 − 2τ )2 − v 2 )((u0 − 2τ  )2 − v 2 ), {(m2 + m3 )2 − ((u0 − 2τ )2 − v 2 ) − ((u0 − 2τ  )2 − v 2 )}2 = 4((u0 − 2τ )2 − v 2 )((u0 − 2τ  )2 − v 2 ), u20 [−4(m2 + m3 )2 + 16(τ − τ  )2 ] + u0 [8(m2 + m3 )2 (τ + τ  ) − 32(τ + τ  )(τ − τ  )2 ] + (m2 + m3 )4 + 4(m2 + m3 )2 v 2 − 8(m2 + m3 )2 (τ 2 + τ 2 ) + 16(τ 2 − τ 2 )2 = 0, 0 = [u0 − (τ + τ  )]2 − (τ + τ  )2 − {(m2 + m3 )4 + 4(m2 + m3 )2 v 2 − 8(m2 +

1 4[(m2 +m3 )2 −4(τ −τ
)2 ] m3 )2 (τ 2 + τ 2 ) + 16(τ 2

− τ 2 )2 },

and hence [u0 − (τ + τ  )]2 =

(m2 +m3 )2 4[(m2 +m3 )2 −4(τ −τ
)2 ] [(m2

For v = 0 we have to find u0 = (τ + τ  ) − take the negative square root, i.e. u0 = (τ + τ  ) −

(m2 +m3 ) 2

+ m3 )2 − 4(τ − τ  )2 + 4v 2 ].

(m2 +m3 ) . 2

 1+

(3.11)

This implies that one has to

4v 2 [(m2 +m3 )2 −4(τ −τ
)2 ] .

(3.12)

Inserting this expression into (3.10a) we obtain µ0

= =

  (τ  − τ ) − 

(m2 +m3 ) 2

 1+

2 4v 2 [(m2 +m3 )2 −4(τ −τ
)2 ]

 3)  )) 1 + { (m2 +m − (τ − τ 2

− v 2 − m2 ,

4v 2 2 [(m2 +m3 )2 −4(τ −τ
)2 ] }

− m2 ,

which implies µ0 =

m3 −m2 2

 − (τ − τ  ) 1 +

4v 2 [(m2 +m3 )2 −4(τ −τ
)2 ] .

(3.13)

For v larger than vm we take the surface u0 (v) = u0 (vm ), v > vm and µ0 (v) = 0.

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4 The case of a big gap In this section we want to treat the gap problem in a situation where techniques of enlarging the domain of holomorphy at real points apply. The method which shall be used is the double cone theorem discovered by Vladimirov [Vl60] and Borchers [Bch61]. We need the local version of this theorem explained in [Bch96]. Let the domain D ⊂ Cd be invariant under the map z → z ∗ . For x ∈ Rd we define the local cone Cx as follows: Cx = {y ∈ Rd ; ∃ ρ > 0

with x + iρy ∈ D}.

By the symmetry the set Cx is either Rd or it contains an even number of components. We are interested in the situation of two components. In the case where x belongs to D ∩ Rd one has Cx = Rd . Since the domain of holomorphy is open, the cone Cx is open too. If D contains real points, then they are called the coincidence ˜ = H(D)∩Rd . Asdomain K. Let H(D) be the envelope of holomorphy of D and K ˜ has a tangent hyper-plane at a point x ∈ ∂ K, ˜ sume that the coincidence domain K then the double cone theorem requires that this hyper-plane is not allowed to enter the local cone Cx . In other terms if for the original coincidence domain K at a point x ∈ ∂K the tangent hyper-plane enters the local cone Cx , then D can be enlarged through this point. In our situation the local cones Cx contain at least the open light-cones V + ∪ V − . With help of this double cone theorem we want to show the following result:

4.1

Theorem

Let π be a translational covariant factor representation of the theory of local observables fulfilling the spectrum condition for the translations. Assume that the representation of T (a) is the unique minimal one. Let the spectrum of the translations start at m1 , then there is no gap with the width larger than 2m1 . Before proving the theorem we draw the following conclusion:

4.2

Corollary

Use the assumptions of the last theorem and assume the spectrum starts at m1 = 0, then there is no gap in the spectrum. This is a simple consequence of the theorem because one has in this case 2m1 = 0. Proof of the Theorem. If m2 coincides with m1 , then the result follows from the theorem mentioned in the introduction. Therefore, it is no restriction if m2 > m1 is assumed in the proof. The result will be shown by contradiction, i.e. the assumption that there is a gap of width larger than 2m1 is not in accordance with the value of the lowest mass m1 . Since the support of the spectrum is Lorentz invariant

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[BB85],[Bch85], it is sufficient to show that one can enlarge the coincidence domain at some parts of the hyperboloid H + (m1 , 0). This will be done by computing the local cones at some points of H + (m1 , 0). Since the complement of Vm+1 ∪ (2t + Vm−1 ) and also the gap between H + (m2 , 0) and H + (m3 , 0) does not lead to an instability of H + (m1 , 0) the interesting contribution for the local cone must come from the crescent moon domain. Our domain is invariant under rotations, therefore, we can restrict our attention to the (0, 1)-plane. Since the domain of analyticity we are starting with is open it is sufficient to compute Cp ∩ R2 , where the plane in question is the (0, 1)-plane, because Cp is open in Rd . For the calculations we use again m1 = 1. We start with the local cone obtained from the complement of {V1+ ∪ (2t + − V1 )}, and afterwards we look at the enlargement of this cone influenced by the crescent moon domain. The local cone contains always the interior of the light cone, therefore, one is interested in light-like and space-like directions. In order to find them we recall first that the hyperboloids H(m, u), filling the complement of the holomorphy domain, are real. Therefore, one must have (p − u, q) = 0 if (p − u + iq) belongs to such hyperboloid. In addition one must have (p − u)2 − q 2 = m2 . So we have to look at points (p − u + iq) where one of the two conditions is violated for every of the hyperboloid filling the complement of the domain of holomorphy. The hyperboloids of interest are those which are centered in the hyper-surface described in the last section. Let p be the point of interest then there are two possibilities of finding directions of the local cone Cp . 1. The set of directions of p − u form a cone C. Then (p − u + iq) belongs to the domain of holomorphy if q is not perpendicular to any of these directions, ˆ i.e. if q or −q belongs to the interior of the dual cone C. 2. If (p − u) belongs to C and q ⊥ (p − u) then q belongs to the local cone Cp if the equations

(p − u)2 − q 2 < µ0 (u)2 ,

(p − u, q) = 0,

(4.1)

can be fulfilled. Altogether the local cone at p will consist of three parts. Two of them are consequences of the coincidence domain {complement of Vm+1 ∪ (2t + Vm−1 )} and the third part is the contribution of the crescent moon domain. (See Fig.1) We choose a point p ∈ H + (1, 0) with p1 > 0 and p21 > 1 − τ12 . This condition implies that the set p − V + does not meet the intersection of V1+ with 2t − V1+ . With this choice p − V + meets the surface for the centers of the hyperboloids in the interval [λ0 τ, τ ] where λ0 is obtained from the relation  ( 1 + p21 , p1 ) − α(1, 1) = (τ, λ0 τ ),

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On the Gap in the Spectrum of the Translations

13

(2) (1)

tangent

(3)

Figure 1: Structure of the local cone (1) y ⊥ (p − u) ∀ u (2) y ⊥ (p − u) and (p − u)2 − y 2 < µ20 (u) (3) Contribution of the crescent moon domain. which implies  1 + p21 − α = τ, (1 − λ0 )τ

=

p1 − α = λ0 τ.  1 + p21 − p1 ,

and consequently

 λ0 τ = τ + p1 − 1 + p21 . (4.2)
If we take the vector u = √ τ 2 ( 1 + p21 , p1 ) in the u-hyper-surface, then we 1+p1

obtain (p − u)2 = (1 − √ τ 2 ). In this situation one has λ1 = √ p1 2 and √ 1 2 = 1+p1 1+p1 1+p1
2 1 − λ1 . This implies (p − u(λ1 ))2 = µ0 (λ1 ). (4.3) If λ0 ≤ λ < λ1 , then the second case can be fulfilled because one finds with p1 = √ λ1 2 the relation 1−λ1

(p − u(λ)) = eτ (λ1 − λ) + (p − u(λ1 )),

e2 = −1.

This implies with Eq. (3.2), (4.3) and the above value of p1 (p − u(λ))2

= (p − u(λ1 ))2 − τ 2 (λ1 − λ)2 − 2τ (λ1 − λ)(p1 − τ λ1 )
= µ0 (λ)2 − (1 + τ 2 (1 − λ2 ) − 2τ 1 − λ2 ))(1 + τ 2 (1 − λ21 )  − 2τ 1 − λ21 ) − τ 2 (λ21 + λ2 − 2λλ1 ) − 2τ (λ1 − λ)( √ λ1 2 − τ λ1 ) 1−λ1

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1 (λ1 −λ) = µ0 (λ)2 − 2τ ( 1 − λ21 − 1 − λ2 + λ√ ) 1−λ21
= µ0 (λ)2 − √ 2τ 2 (1 − λ1 λ − (1 − λ1 λ)2 − (λ1 − λ)2 ) 1−λ1  λ1 −λ 2 ) ) = µ0 (λ)2 − √ 2τ 2 (1 − λ1 λ − (1 − λ1 λ) 1 − ( 1−λ 1λ 1−λ1

2

< µ0 (λ) −

2 1 (λ1 −λ) √ 2τ 2 2 1−λ λ 1−λ 1

< µ0 (λ)2 .

Next we turn to the crescent moon domain. Since  can be chosen arbitrarily small we take first  = 0 in order to see what can be expected. In this situation the u-hyper-plane is met by every time-like direction starting from p. Looking for conditions implying that the local cone Cp contains the tangent plane at p one takes p − u0 = (1 + α)p. Since p2 = 1 and q 2 = −β 2 equation (4.1) reads (1 + α)2 + β 2 < µ20 . The quantity α is a function of p1 . In order that the tangent direction belongs to the local cone one must have µ0 > 1 + min{α(p1 )}. Later we 2 will show that α → 0 for p1 → ∞. Since µ0 = m3 −m , one obtains an enlargement 2 of the local cone for µ0 > 1. With µ0 − 1 = δ and the choice α(p1 ) < 2δ one can  vary β in the interval (0, δ + 34 δ 2 ). Since (p − u) depends continuously on u one can still vary u in a small neighbourhood around u0 without violating Eq. (4.1), i.e. the local cone at p contains the tangent direction at p as an interior direction. It remains to show α(p1 ) → 0 for p1 → ∞ and to look at the variation for small 3 . With m2 +m = M , the point u0 must lie on the hyperboloid H − (M, 2t) which 2 leads to the equations
−αp = 2t − ( M 2 + r2 , r), 
−αp1 = −r, −α 1 + p21 = 2 − M 2 + r2 , 

and hence 2=

 M 2 + α2 p21 − α 1 + p21 .

This implies 4 = M 2 + α2 (1 + 2p21 ) − 4(M 2 + α2 p21 )α2 (1 + p21 ) = −α4 + α2 {2M 2 + α2 = (M 2 + 4(1 + 2p21 )) ±

 2 (M 2 + α2 p21 )α2 (1 + p21 ), {(M 2 − 4) + α2 (1 + 2p21 )}2 , 8(1 + 2p21 )} = (M 2 − 4)2 ,  (M 2 + 4(1 + 2p21 ))2 − (M 2 − 4)2 .

Since α is bounded by 1 for all p1 we must choose the minus sign and get   1 (M 2 −4)2  2 −4)2 ≈ 2 M 2 +4(1+2p2 ) . α2 = (M 2 + 4(1 + 2p21 )) 1 − 1 − (M 2(M +4(1+2p2 ))2 1

This tends to zero for p1 → ∞ as claimed before.

1

(4.4)

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On the Gap in the Spectrum of the Translations

15

Finally we must look at the approximation in . For the future calculations 1 1 one sets σ = +( 1+ −1), δ = −( 1+ −1) which implies τ +τ  = 2+δ, τ −τ  = σ. The crescent moon domain becomes smaller for δ = 0, which simplifies the coming calculations. With these notations we obtain with (3.12) and (3.13) the following equations for α and µ0 (v).  −α 1 + p21

= −

 2 2 − M 2 + v 2 MM −αp1 = −v, 2 −σ 2 ,   2 α 1 + p21 = 2 − M 2 + α2 p21 MM 2 −σ 2 ,

and µ0 =

m3 −m2 2

 −σ

1+

α2 p21 M 2 −σ2 .

(4.5)

(4.6)

In order that the tangent at p belongs to the local cone one has to choose  small enough such that  m3 − m2 α2 p2 − 1 > σ 1 + M 2 −σ1 2 2 holds. To show that this is possible we have to calculate p1 as a function of α. 2 With Eq. (4.5) and with Γ = MM 2 −σ 2 = 1 − γ one obtains as equation for α  4 = M 2 + α2 (1 + p21 (1 + Γ)) − 2 (M 2 + α2 p21 Γ)α2 (1 + p21 ), 4(M 2 + α2 p21 Γ)α2 (1 + p21 ) = {(M 2 − 4) + α2 (1 + p21 (1 + Γ))}2 , 4{M 2 α2 (1 + p21 ) + α4 (p21 Γ(1 + p21 ))} = (M 2 − 4)2 + 2(M 2 − 4)α2 (1 + p21 (1 + Γ))+ α4 (1 + 2p21 (1 + Γ) + p41 (1 + Γ)2 ), p41 {−α4 γ 2 } + p21 {2α4 γ + 8α2 (1 + Γ) − 2M 2 α2 γ} + {−α4 + α2 (2M 2 + 8) − (M 2 − 4)2 } = 0,   16α2 +γ(2α4 −α2 (2M 2 −8)) (+α4 −α2 (2M 2 +8)+(M 2 −4)2 )α4 γ 2 2 1 − . (4.7) 1 − p1 = α4 γ 2 {16α2 +γ(2α4 −α2 (2M 2 −8))}2 One must choose p1 at least as big as stated in the formula. This is not too different from the value of the case  = 0. With (p − u) = (1 + α)p one chooses α such that the inequality (4.1) is fulfilled. Then one can find an appropriate value for p1 using Eq. (4.7) and finally a finite value for  with help of (4.6). Hence the local cone Cp contains the tangent at p, which leads to the desired contradiction.
Acknowledgment. I like to thank Jacques Bros for discussions.

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References [Ar64]

H. Araki, On the algebra of all local observables, Progr. Theor. Phys. 32, 844–854 (1964).

[Bch61]

¨ H. J. Borchers, Uber die Vollst¨andigkeit lorentzinvarianter Felder in einer zeitartigen R¨ohre, Nuovo Cimento 19, 787–796 (1961).

[Bch65]

H. J. Borchers, Local Rings and the Connection of Spin with Statistics, Commun. Math. Phys. 1, 281–307 (1965).

[Bch85]

H. J. Borchers, Locality and covariance of the spectrum, Fizika 17, 289– 304 (1985).

[Bch86]

H. J. Borchers, A Remark on Antiparticles, in Lecture Notes in Physics Vol 257, 268–280 (1986).

[Bch96]

H. J. Borchers, Translation Group and Particle Representations in Quantum Field Theory, Lecture Notes in Physics m40 Springer, Heidelberg (1996).

[BB]

H. J. Borchers and D. Buchholz, The Energy-Momentum Spectrum in Local Field Theories with Broken Lorentz-Symmetry, Commun. Math. Phys. 97, 169–185 (1985).

[Bk82]

R. Borek, Kovariante Darstellungen Masseloser Fermi Felder, Dissertation, G¨ ottingen (1982).

[BOT]

H. J. Brehmermann, R. Oehme and J. G. Taylor, Proof of dispersion relation in quantized field theories, Phys. Rev. 109, 2178–2190 (1958).

[BMS]

J. Bros, A. Messiah and R. Stora, A problem of analytic completion related to the Jost-Lehmann-Dyson formula, J. Math. Phys. 2, 639–651 (1961).

[BD84]

D. Buchholz and S. Doplicher, Exotic Infrared Representations of interacting Systems, Ann. Inst. Henri Poincar´e 40, 157–184 (1994).

[DHR69a] S. Doplicher, R. Haag and J. E. Roberts, Fields, observables and gauge transformations I, Commun. Math. Phys. 13, 1 (1969). [DHR69b] S. Doplicher, R. Haag and J. E. Roberts, Fields, observables and gauge transformations II, Commun. Math. Phys. 15, 173 (1969). [DS82]

S. Doplicher and M. Spera, Representations obeying the spectrum condition, Commun. Math. Phys. 84, 505–513 (1982).

[Dy58]

F. J. Dyson, Integral representation of causal commutators, Phys. Rev. 110, 1460–1464 (1958).

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On the Gap in the Spectrum of the Translations

17

[Ha92]

R. Haag, Local Quantum Physics, Springer Verlag, Berlin-HeidelbergNew York (1992).

[Vl]

V. S. Vladimirov, The construction of envelopes of holomorphy for domains of special type, Doklady Akad. Nauk SSSR 134, 251 (1960).

[Wi64]

A. S. Wightman, La th´eorie quantique locale et la th´eorie quantique des champs, Ann. Inst. Henri Poincar´e A, I, 403–420 (1964).

Hans J¨ urgen Borchers Institut f¨ ur Theoretische Physik Universit¨ at G¨ ottingen Bunsenstrasse 9 D-37073 G¨ ottingen Germany Communicated by Klaus Fredenhagen submitted 19/04/01, accepted 02/10/01

Ann. Henri Poincar´e 3 (2002) 19 – 27 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/010019-9 $ 1.50+0.20/0

Annales Henri Poincar´ e

Singular Vectors of the N = 1 Superconformal Algebra K. Iohara and Y. Koga Abstract. We give two singular vector formulae of the N = 1 superconformal algebra.

1 Introduction Singular vectors of Verma modules over the Virasoro algebra play important roles both in mathematics and physics. In fact, many studies on their explicit forms have been done e.g., [Ke] etc. In particular, two formulae were obtained by B. L. Feigin and D. B. Fuchs [FeFu1], and these formulae played an essential role when they determined the structure of the fusion rings of the minimal models [FeFu2]. In this paper, we will derive similar formulae for the N = 1 superconformal algebra, i.e., the Neveu-Schwarz algebra. In our case, one technical difficulty arises, viz. factorizations in a non-commutative ring are required (see Remark 3.2 and Theorem 3.5). We note that one of our formulae, i.e. Theorem, 3.1 has been partially obtained in [BS] in a special case. Namely, they obtained their formula explicitly in the case when one of a and b is one and provided an algorithm to compute the general case via fusion procedure. Our method is based on the embedding diagram given in Proposition 2.3 and the result is given for the general cases. This paper is organized as follows: In §2, we will recall the definition of the N = 1 superconformal algebra and its Verma modules. Further, we will collect some properties of them. In §3, we will derive two singular vector formulae.

2 N = 1 Superconformal Algebra Here we recall the definition of the N = 1 superconformal algebra. Further we introduce Verma modules and state their properties.

2.1

Definitions

Let V ir 12 be the N = 1 superconformal algebra, the Neveu-Schwarz algebra, i.e., V ir 12 is the Lie superalgebra V ir 12 =


i∈Z

CLi ⊕


j∈Z+ 12

CGj ⊕ CC,

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and these generators satisfy the following commutation relations: deg Li = deg C = 0, deg Gi = 1, 1 [Li , Lj ] = (i − j)Li+j + (i3 − i)δi+j,0 C, 12 1 [Gi , Lj ] = (i − j)Gi+j , 2 1 1 [Gi , Gj ] = 2Li+j + (i2 − )δi+j,0 C, 3 4 [C, V ir ] = {0}. Note that V ir 12 admits the following triangular decomposition: − 0 V ir 12 = V ir+ 1 ⊕ V ir 1 ⊕ V ir 1 , 2

2

where

V ir± 1 := 2

For later use, we set




CLi ⊕

±i>0




V ir01 := CL0 ⊕ CC. 2

±i>0

+ 0 V ir≥ 1 := V ir 1 ⊕ V ir 1 . 2

2.2

CGi ,

2

2

2

Verma Modules

Next, we introduce Verma modules over V ir 12 and collect some basic properties of them. For (z, h) ∈ C2 , let Cz,h = C1z,h be the 1-dimensional representation of V ir≥ 1 defined by 2

x.1z,h = 0

if x ∈ V ir+ 1 , 2

C.1z,h = z1z,h , L0 .1z,h = h1z,h . The Verma module M (z, h) over V ir 12 with highest weight (z, h) is defined as follows: M (z, h) := U (V ir 12 ) ⊗U(V ir≥ ) Cz,h . 1 2

The values z and h are called central charge and conformal weight, respectively. For simplicity, we set |z, h := 1 ⊗ 1z,h . Since the choice of the parity of a highest weight vector is not essential in the sequel, we do not specify it. Note that M (z, h) has the following weight space decomposition:
M (z, h) = M (z, h)n , M (z, h)n := {v ∈ M (z, h)|L0 .v = (h + n)v}. n∈ 12 Z≥0

A non-zero vector in {M (z, h)n }

Vir+ 1 2

is called a singular vector of level n.

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Here we recall two basic facts on singular vectors. The first fact is the existence of singular vectors. As a consequence of the determinant formula for M (z, h) [KW], we see the following: If there exist t ∈ C \ {0} and a, b ∈ Z>0 satisfying a ≡ b (mod 2) and 15 − 3(t + t−1 ), 2 1 1 1 h = ha,b (t) := (a2 − 1)t − (ab − 1) + (b2 − 1)t−1 , 8 4 8 z = z(t) :=

then M (z, h) has a singular vector of level 12 ab. The second fact is the uniqueness of singular vectors. Indeed, we have Proposition 2.1 ([IK1]) For any highest weight (z, h) ∈ C2 , the following hold: dim{M (z, h)n}

2.3

Vir + 1 2

≤1

(∀n ∈

1 Z≥0 ). 2

Embedding Diagrams of Verma Modules

To prove singular vector formulae, we will use embedding diagrams of Verma modules [IK1]: Here we recall them. It should be noted that non-trivial homomorphisms between Verma modules over a Lie superalgebras are not injective in general. But for the N = 1 superconformal algebra, the injectivity holds. Proposition 2.2 ([IK1]) Suppose that M (z, h) has a singular vector of level n. Then the following map is an embedding: M (z, h + n) −→ M (z, h). When z = z(t) for t ∈ Q \ {0}, we have the following embedding diagrams of Verma modules. Let p and q be integers such that z = z(± pq ), p, q ≥ 2, p − q ∈ 2Z and ( p−q 2 , q) = 1. For the integers p and q, we set ± ± ± ± ± := Kp,q (1) ∪ Kp,q (2) ∪ Kp,q (3) ∪ Kp,q (4), Kp,q ± (i) are give by where Kp,q ± (1) := {(r, s) ∈ Z2 |0 < r < q, 0 < ±s < p, rp + sq ≤ pq}, Kp,q ± (2) := {(q, s)|0 < ±s < p}, Kp,q ± (3) := {(r, −p)|0 < r < q}, Kp,q ± Kp,q (4) := {(0, ±p), (q, ±p)}.

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± For (r, s) ∈ Kp,q , we also set    h(i−1)q+r,−s ± p i ≡ 1 (mod 2)   q . hi := p hiq+r,s ± i ≡ 0 (mod 2) q

Then we have the following proposition: Proposition 2.3 Let us take integers p, q, r and s as above. Then there exist the following embedding diagrams of Verma modules: + For (r, s) ∈ Kp,q (1),

/ M (z, h3 ) / M (z, h2 ) / M (z, h1 ) · · · M (z, h4 ) UUU* 9 KKK KKK MMM q8 s9 q s ss K K q MM s s q K K s s M (z, h0 ). qqMMMM ssKKKK ssKKKK qqq sss sss & % % iii4 · · · M (z, h−4 ) / M (z, h−3 ) / M (z, h−2 ) / M (z, h−1 ) + (2), For (r, s) ∈ Kp,q

· · · M (z, h4 )

/ M (z, h3 )

/ M (z, h2 )

/ M (z, h3 )

/ M (z, h−2 )

/ M (z, h1 )

/ M (z, h0 ).

+ (3), For (r, s) ∈ Kp,q

· · · M (z, h−4 )

/ M (z, h1 )

/ M (z, h0 ).

+ For (r, s) = (p, q) ∈ Kp,q (4)

· · · M (z, h8 )

/ M (z, h6 )

/ M (z, h4 )

/ M (z, h2 )

/ M (z, h4 )

/ M (z, h−2 )

/ M (z, h0 ).

+ (4) For (r, s) = (0, q) ∈ Kp,q

· · · M (z, h8 )

/ M (z, h−6 )

/ M (z, h0 ).

− (1), For (r, s) ∈ Kp,q

M (z, h3 ) o · · · M (z, h4 ) o M (z, h2 ) o M (z, h1 ) jU fMMM eKKK eKKK UU q s q s s MM KKs KKsKs s q s q K M q s s M (z, h0 ). K K M q s s KK KK MM tiii xqqq ysss ysss · · · M (z, h−4 ) o M (z, h−3 ) o M (z, h−2 ) o M (z, h−1 ) − For (r, s) ∈ Kp,q (2),

· · · M (z, h4 ) o

M (z, h3 ) o

M (z, h2 ) o

M (z, h1 ) o

M (z, h0 ).

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− For (r, s) ∈ Kp,q (3),

· · · M (z, h−4 ) o

M (z, h3 ) o

M (z, h−2 ) o

M (z, h1 ) o

M (z, h0 ).

− For (r, s) = (p, −q) ∈ Kp,q (4),

· · · M (z, h8 ) o

M (z, h6 ) o

M (z, h4 ) o

M (z, h2 ) o

M (z, h4 ) o

M (z, h−2 ) o

M (z, h0 ).

− For (r, s) = (0, −q) ∈ Kp,q (4),

· · · M (z, h8 ) o

M (z, h−6 ) o

M (z, h0 ).

3 Singular Vector Formulae In this section, we will present our main results, singular vector formulae.

3.1

Notation

Suppose that z = z(t) and h = ha,b (t) for some a, b ∈ Z>0 such that a ≡ b (mod 2). As stated before, there exists a unique element (up to a scalar multiple) −1 ] Sa,b (t) ∈ U (V ir− 1 ) ⊗C C[t, t 2

such that Sa,b (t)|z, h is a non-zero singular vector of level 12 ab. From now on, we will give two formulae on Sa,b (t).

3.2

Loop Modules

The first formula is concerned with the representations over V ir 12 called loop modules. Although, this formula has already been considered in [IK2], we have stated there without proof. Here, we will show a way to prove this formula. Let θ be the Grassmann variable i.e. θ2 = 0. For λ, µ ∈ C and σ ∈ {0, 1}, we set

σ Fλ,µ = CFi ⊕ CFi θ, i∈Z+σ 12

i∈Z+(1−σ) 12

and regard it as a V ir -module via C.Fi θγ = 0, 1 Ln .Fi θγ = {−i + µ + (n − 1)λ − nγ}Fi+n θγ , 2 Gm .Fi θγ = δγ,1 Fi+m + δγ,0 {−i + µ + (2m − 1)λ}Fi+m θ, where γ = 0, 1, n ∈ Z and m ∈ Z + 12 . σ Here we describe the action of Sa,b (t) on Fλ,µ explicitly. To do it, we set  1 if i ≡ j (mod 2), (2) δi,j = 0 if i ≡ j (mod 2).

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σ Then Sa,b (t) acts on F0 θγ ∈ Fλ,µ as follows:

Theorem 3.1 ([IK2]) Suppose that a, b ∈ Z>0 such that a ≡ b (mod 2). Then we have (2) γ Sa,b (t).F0 θγ = Pa,b (λ, µ, t)F− 12 ab θδab,1−γ , γ where Pa,b (λ, µ, t)2 is, up to a scalar depending on the normalization, given by



Qk,l a,b (λ, µ, t)

0≤k≤a−1 0≤l≤b−1 k−l≡γ (mod 2)

and Qk,l a,b (λ, µ, t)

1 1 1 1 1 = (µ − 2λ) − (kt 2 − lt− 2 ){(a − k)t 2 − (b − l)t− 2 } 2

1 1 1 1 1 × (µ − 2λ) − {(k + 1)t 2 − (l + 1)t− 2 }{(a − k − 1)t 2 − (b − l − 1)t− 2 } 2 2 1 1 1 + (a − 1 − 2k)t 2 − (b − 1 − 2l)t− 2 λ. 2 (1)

σ Proof. Since every weight space of the loop module Fλ,µ is of dimension 1, there γ −1 exists Φa,b (λ, µ, t) ∈ C[λ, µ, t, t ] which satisfies (2)

Sa,b (t).F0 θγ = Φγa,b (λ, µ, t)F− 12 ab θδab,1−γ . γ From now on, we verify Φγa,b (λ, µ, t) = Pa,b (λ, µ, t). To do it, we first compute the γ degree of Φa,b (t) as a polynomial in t. Following [AsFu], one obtains

degΦγa,b (t) ≤ 

ab + 1 , 2

where x denotes the maximal integer that does not exceed x. Let us prove γ Φγa,b (λ, µ, t) = Pa,b (λ, µ, t) by induction on the level of singular vectors. It follows from Proposition 2.3 that there exists t0 ∈ Q \ {0} such that the singular vector of level 12 ab factorize as Sa1 ,b1 (t0 )Sa2 ,b2 (t0 ) · · · Sak ,bk (t0 )|z(t0 ), ha,b (t0 ) ,

(2)

k where k > 1 and ai , bi ∈ Z>0 satisfy ai ≡ bi (mod 2) and 12 ab = i=1 21 ai bi . By the uniqueness of singular vectors and induction hypothesis, we have Φγa,b (λ, µ, t0 ) ∝

k  i=1

Paγi ,bi (λ, µ, t0 ).

(3)

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25

γ Here, one can show that the right hand side of (3) coincides with Pa,b (λ, µ, t) by direct computation. On the other hand, by Proposition 2.3, we can check that the number ({t0 } such that the singular vector of level 12 ab factorize as (2) is greater than the γ γ degree of Pa,b (λ, µ, t). This implies that Φγa,b (λ, µ, t) = Pa,b (λ, µ, t) holds for any t ∈ C \ {0}. We have completed the proof.


3.3

Projections of Singular Vectors

Next we describe an explicit form of the image of Sa,b (t) under the projection − − − − − π : U (Vir− 1 ) −→ U (Vir 1 )/U (Vir 1 )[Vir 1 , [Vir 1 , Vir 1 ]]. 2

2

2

2

2

2

(4)

Remark 3.2 In [FeFu1], a similar projection was computed for the Virasoro algebra. One of the differences between their case and our case is whether the quotient space is a commutative ring or not. Indeed, the quotient space in (4) is noncommutative, and, to derive the formula, we have to consider factorizations in this non-commutative space. We introduce Xp,q , Yr ∈ U (Vir− 1 ) for p, q, r ∈ C as follows: 2

Xp,q :=

L2−1

− pG− 12 G− 32 − qG− 32 G− 12 ,

Yr := L−1 G− 12 − rG− 32 . Then we have the following lemma: Lemma 3.3

1. π(Xp,r )π(Xq,s ) = π(Xp,s )π(Xq,r ) = π(Xq,r )π(Xp,s ).

2. π(Xp,q )π(Yr ) = π(Xp,r )π(Yq ) = π(Yr )π(Xp,q ). To describe the image of π(Sa,b (t)), we introduce some notation. Let us first introduce an index set Pσa,b for σ ∈ {0, 1} and a, b ∈ Z>0 such that a − b ≡ 0 (mod 2). Set     (i) (r, s) ∈ (Z>0 × Z) ∪ ({0} × Z≥σ ),         (ii) 0 ≤ r ≤ a − 1 ∧ −(b − 1) ≤ s ≤ b − 1, σ  , Pa,b := (r, s)      (iii) a − 1 − r ≡ 0 (mod 2) ∧ b − 1 − s ≡ 0 (mod 2),     (iv) a + b − (r + s) ≡ 2(1 − σ) (mod 4) and nσa,b := (Pσa,b . We remark that  n0a,b − n1a,b =

0 1

if a ≡ b ≡ 0 mod 2 . if a ≡ b ≡ 1 mod 2

Moreover, let us take a sequence {(r1σ , sσ1 ), (r2σ , sσ2 ), · · · , (rnσ , sσn )}

(n := nσa,b )

of elements Pσa,b , which satisfies the following conditions:

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1. (riσ , sσi ) = (rjσ , sσj ) for any i, j (1 ≤ i < j ≤ n), 2. if (0, 0) ∈ Pσa,b , i.e., σ = 0, a ≡ b ≡ 1 (mod 2) and a + b ≡ 2 (mod 4), then (rnσ , sσn ) = (0, 0). Next, we define a subgroup Ga,b of the symmetric group Sn , where n := nσa,b , by  Ga,b :=

{τ ∈ Sn |τ (n0a,b ) = n0a,b } Sn

if (0, 0) ∈ Pσa,b . otherwise

For i ∈ Z such that 1 ≤ i ≤ n0a,b and τ ∈ Ga,b , we define an element mτi ∈ P0a,b × (P1a,b ∪ {vac}) as follows:  mτi := Further, we set

[(rτ0 −1 (i) , s0τ −1 (i) ), (ri1 , s1i )]

if 1 ≤ i ≤ n1a,b

[(rτ0 −1 (n0 ) , s0τ −1 (n0 ) ), vac] a,b a,b

if n0a,b = n1a,b ∧ i = n1a,b

.

Ωτa,b := {mτi |1 ≤ i ≤ n0a,b }.

Finally, for m ∈ Ωτa,b , we set

Sm a,b

and

 1 1 1 1 X    14 (r t 2 +s t− 2 )2 , 14 (r t 2 +s t− 2 )2 := Y 1 12 − 12 2 (rt +st )    4 1 G− 2 mτ

if m = [(r, s), (r , s )] if m = [(r, s), vac] for (r, s) = (0, 0) , if m = [(0, 0), vac] mτ

mτ

Sa,b;τ := Sa,b1 Sa,b2 · · · Sa,bn , where n := n0a,b . By Lemma 3.3, we have Lemma 3.4 Sa,b;τ does not depend on the choice of τ ∈ Ga,b . Consequently, the image of Sa,b (t) under the projection π in (4) is described as follows: Theorem 3.5 Suppose that a, b ∈ Z>0 such that a ≡ b (mod 2). Then we have π(Sa,b (t)) ∝ Sa,b;τ up to scalar multiplication depending on the normalization. Proof. The proof of this formula is essentially the same as the proof of Theorem 3.1. Hence we omit it.
Acknowledgment. We would like to thank the referee for pointing out the reference [BS], where a similar result was obtained.

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27

References [AsFu]

A. Astashkevich and D. B. Fuchs, Asymptotics for singular vectors in Verma modules over the Virasoro algebra, Pacific J. Math. 177, no. 2, 201–209 (1997).

[BS]

L. Benoit and Y. Saint-Aubin, Fusion and the Neveu-Schwarz Singular Vectors, Int. Jour. Mod. Phys. A 9, 547–566 (1994).

[FeFu1]

B. L. Feigin and D. B. Fuchs, Verma modules over the Virasoro algebra, Lecture Notes in Math., 1060, 230–245 (1982).

[FeFu2]

B. L. Feigin and D. B. Fuchs, Cohomology of some nilpotent subalgebras of the Virasoro and Kac-Moody Lie algebras, J. Geom. Phys. 5, 209–235 (1988).

[IK1]

K. Iohara and Y. Koga, Representation Theory of Neveu-Schwarz and Ramond Algebras I: Verma Modules, preprint.

[IK2]

K. Iohara and Y. Koga, Fusion algebras for N=1 superconformal field theories through coinvariants II: N = 1 super Virasoro symmetry, J. Lie Theory 11, 305–337 (2001).

[Ke]

A. Kent, Singular vectors of the Virasoro algebra, Phys. Lett. B 273, 56–62 (1991).

[KW]

V. G. Kac and M. Wakimoto, Unitarizable highest weight representation of the Virasoro, Neveu-Schwarz and Ramond algebras, Lect. Notes in Phys., 261, 345–372, (1986).

Kenji Iohara Department of Mathematics Faculty of Science Kobe University Kobe 657-8501 Japan email: [email protected] Communicated by Tetsuji Miwa submitted 26/01/01, accepted 08/09/01

Yoshiyuki Koga Department of Mathematics Faculty of Science and Technology Sophia University Tokyo 102-8554 Japan email: [email protected]

Ann. Henri Poincar´e 3 (2002) 29 – 86 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/010029-58 $ 1.50+0.20/0

Annales Henri Poincar´ e

Front Fluctuations in One Dimensional Stochastic Phase Field Equations L. Bertini∗, S. Brassesco†, P. Butt` a‡ and E. Presutti‡

Abstract. We consider a conservative system of stochastic PDE’s, namely a weakly coupled, one dimensional phase field model with additive noise. We study the fluctuations of the front proving that, in a suitable scaling limit, the front evolves according to a non–Markov process, solution of a linear stochastic equation with long memory drift.

Part I. Introduction 1 General setting, model, and results The term “sharp interface limit” denotes a scaling procedure aimed at the derivation of interfaces as geometric objects, e.g. surfaces of codimension one with bounded variation, that is, enough regular for the area measure to be well defined. Of course this makes only sense in the context of systems which undergo phase transitions and of states where different phases coexist. In the limit the other degrees of freedom are lost and we are left with the interface alone. Rigorous proofs are hard, yet a great variety of models has been successfully worked out. The mathematics involved is correspondingly rich, e.g. the theory of Γ–convergence (to study the sharp interface limit of Ginzburg–Landau like free energy functionals in relation with the equilibrium shape of the interface, as in the Wulff problem) and correspondingly the theory of Gibbsian large deviations (to study the same problems at the more microscopic level of statistical mechanics); singular limit in PDE’s, like in the Allen–Cahn, Cahn–Hilliard and phase field equations, and correspondingly, at the microscopic level, hydrodynamic limits of spin or particle systems. This paper deals with fluctuations. Here again the questions are, first, whether in a sharp interface limit the system is described by a [fluctuating] interface with closed equations of motion and, secondly, the nature of such equations. The problem greatly simplifies in one space dimension where the limit interface is represented by a point which separates the two phases (one to its left and the other one ∗ Partially

supported by Cofinanziamento MURST 1999. supported by agreement CONICIT–CNR, and Proyecto F–97 000004 (CONICIT). ‡ Partially supported by Cofinanziamento MURST 1999 and by NATO Grant PST. CLG. 976552. † Partially

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to its right). Indeed, until now, most fluctuation results are restricted to d = 1. Such a state is marginally stable (w.r.t. displacements of the interfaces, see Section 2 for a precise statement) while it becomes stable when there is a conservation law (which imposes that the mass of each phase is preserved). By the conservation law, each displacement of the interface must then necessarily come with a deformation of the profile, hence the intuition that other degrees of freedom may then enter into play. This is confirmed, but in a sense also infirmed, by our results: indeed we will see that extra degrees of freedom become relevant, but their effect can be represented in the limit by terms which depends on the previous history of the interface evolution. The system we consider is a phase field equation with additive stochastic noise, see (1.1)–(1.2) below. Without noise the interface is given by a special instantonic profile connecting the two phases; in the presence of noise, after suitable rescalings, the limit state is still represented by the same instantonic profile which is however randomly displaced. The displacement obeys an ordinary stochastic equation driven by a white noise forcing term and with a long memory drift, whose effect is to force the interface back toward its initial position, thus restoring the equilibrium of the two coexisting phases. The evolution is defined by the following stochastic equations 
√ 1
∆m(t) − V (m(t)) + λh(t) dt + γ dW (a) (t) dm(t) = 2 1 d [h(t) + m(t)] = ∆h(t)dt 2

(1.1) (1.2)

where the unknowns, m(t) = m(t, x), h(t) = h(t, x), (t, x) ∈ R+ × R, are two scalar random fields. In (1.1), λ and γ are positive parameters; ∆ is the Laplacian on R and W (a) (t) is a white noise with a cutoff in the spatial covariance. This means W (a) (t) is the canonical process in the filtered probability space (Ω, F , Ft , P) where Ω := C(R+ ; S
(R)) (S
(R) the space of tempered distributions), F its Borel σ– algebra, Ft the canonical filtration, and P the Gaussian measure with mean zero and covariance     aγ (x) := a γ β/2 x (1.3) E W (a) (t), ϕW (a) (t
), ϕ
 = t ∧ t
ϕ, a2γ ϕ
, where t ∧ t
:= min{t, t
}, ·, · denotes the inner product in L2 (R, dx) as well as the canonical pairing between S and S
. Above β > 0 and a ∈ C02 (R), i.e. a twice differentiable function with compact support. We assume the normalization a(0) = 1. Finally V
(m) is the derivative of a symmetric, smooth double well potential V (m); for simplicity we assume m2 m4 − . (1.4) 4 2 Note we are omitting to write explicitly the dependence on the randomness ω ∈ Ω. This will be done throughout the whole paper without further mention. V (m) =

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In a companion paper, [2], we prove global existence and uniqueness in a space of H¨older continuous functions for the system  t  t √
m(t) = pt m(0) + ds pt−s [−V (m(s)) + λh(s)] + γ pt−s dW (a) (s) (1.5) 0 0  1 t ds (∆pt−s )m(s) (1.6) h(t) + m(t) = pt [h(0) + m(0)] − 2 0 where pt = et∆/2 is the heat semigroup, namely the integral operator with kernel 2

pt (x, y) =

e−(x−y) /2t √ . 2πt

(1.7)

Observe that the integral on the r.h.s. of (1.6) is well defined because m(s) is H¨ older continuous. The system (1.1)–(1.2) is defined in terms of the integral equations (1.5)–(1.6) and called “stochastic phase field equations”. General background and physical interpretation are discussed in the sequel, here we proceed by stating our main result, presented in the next theorem. We consider the initial condition m(0) = m, ¯

h(0) = 0

(1.8)

where m ¯ ξ (x) := tanh(x − ξ),

m ¯ := m ¯0

(1.9)

is a standing wave (that we call “instanton”) with “center” ξ ∈ R. We suppose that √ (1.10) λ= γ   (λ) (λ) and denote by m (t), h (t) the solution of (1.1)–(1.2) with (1.10) and initial condition (1.8). Our main result is

Theorem 1.1 There exists a process x(λ) (t) in C(R+ ), adapted to Ft , such that for each τ, ε > 0 

(λ) sup m (t) − m ¯ x(λ) (t) > ε = 0 . (1.11) lim P λ↓0

t≤λ−2 τ

∞

Furthermore, denoting weak   convergence in C(R+ ) by =⇒ and defining, after scaling, xλ (τ ) := x(λ) λ−2 τ , we have that xλ =⇒ x as λ ↓ 0, where x(τ ) is the unique solution of  τ x(s) x(τ ) = b(τ ) − 3 (1.12) ds  2π(τ − s) 0 in which b(τ ) is a one dimensional Brownian motion with diffusion coefficient D = 3/4.

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The coefficient 3 in (1.12) and the value D = 3/4 above are related to the specific choice of the potential V . Existence and properties of the process solution of (1.12) are discussed in [3]. For the physical interpretation, we start from the equation without noise and with λ = 0, namely (1.1) with λ = γ = 0. This is the well known Allen–Cahn (AC) equation with double well potential V (m), which arises as the gradient flow associated to the Ginzburg–Landau free energy functional
    1 dx (1.13) F (m) = |∇m(x)|2 + V m(x) 4 R (F decreases along the solutions of the AC equation). The minimizers of F (m) are the two constant functions m+ (x) = 1 and m− (x) = −1, therefore the values of the order parameter m = ±1 correspond to pure phases and the interface for (1.13) is (up to translations) the profile m(x) which minimizes F (m) under the condition that asymptotically as x → ±∞ it converges to ±1. The associated Euler–Lagrange equation is the stationary AC equation 1 ∆m − V
(m) = 0 2

(1.14)

which, imposing the above conditions at ±∞, has the instanton m ¯ of (1.9) as its unique solution (modulo translations). Therefore m ¯ ξ is the equilibrium state which has the two phases coexisting to the right and to the left of ξ, it represents the “mesoscopic interface” with ξ its “mesoscopic location” (mesoscopic instead of macroscopic because the interface is “diffuse” and the transition from one phase to the other, even though exponentially fast, is not sharp; mesoscopic instead of microscopic because the AC equation and the Ginzburg–Landau functional can be derived by a scaling procedure from particle or spin systems, i.e. from an underlying more microscopic structure). The next step is with λ > 0, but still γ = 0. Then (1.1) is coupled to (1.2) and the two together give an example of phase field equations (PFE). Here h is a thermodynamic potential conjugated to the order parameter m: if m is a magnetization density, then h is a magnetic field, our notation is inspired by such an interpretation. More commonly however, m is a relative concentration of one species in a binary alloy and h is a relative temperature. The effective potential will then depend on the relative temperature h, our choice is simply to replace V (m) by V (m) − hm. At the critical temperature, which corresponds to h = 0, the alloy can exist in two different concentrations m = ±1, but, as the temperature h changes, the equilibrium concentrations vary, one becomes stable and the other one metastable. In the presence of a given temperature profile h, the AC rate of change of the concentration density at x at time t, i.e. dm(t, x)/dt, is given by (1.1) (with γ = 0). Due to latent heat, there is a corresponding change of temperature given (in the proper units) by dh(t, x)/dt = −dm(t, x)/dt. Simultaneously, by the

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Fourier law, the temperature diffuses according to the r.h.s. of (1.2). This has a feedback in (1.1) so that (1.1) and (1.2) are coupled. In conclusion the PFE describe a change of phase including latent heat effects and, because of that, unlike for the AC equation, there is a conservation law: the integral of m + h is in fact invariant under PFE. The noise term in (1.1) takes into account some external fluctuating force and the resulting equation is known in the physical literature as model C of Hohenberg and Halperin, [14]. Since m ¯ solves (1.14), the state m = m, ¯ h = 0 is a stationary solution of PFE, which is therefore interpreted, like in AC, as the mesoscopic interface. We are then studying in Theorem 1.1 what happens to the interface when there are small (because γ → 0 in Theorem 1.1), external perturbations which produce a √ random change γdW (a) (t) of magnetization (or, in the other interpretation, of concentration), the analysis including latent heat effects. The small parameter γ in the noise term has the meaning of a ratio between mesoscopic and microscopic space units, the former referring to (1.1), the latter to some microscopic model, as for instance the Glauber + Kawasaki process introduced in [12]. A formal comparison with the microscopic model in [12] would lead to a more complex structure for the additive noise; however we stick to (1.1) (which catches the correct behavior of critical fluctuations, see [5]) to make our analysis simpler. In conclusion the scaling γ → 0 has a natural justification in terms of the microscopic origin of the noise term, the scaling of time in Theorem 1.1 is on the other hand justified a posteriori: it is the correct scaling for observing finite displacements of the interface. On the contrary, the equality (1.10) has no clear physical interpretation; it is true that a scaling with λ → 0 is widely used in the PFE literature to stress “kinetic undercooling effects”, see [11], [17], but relating λ to the noise as in (1.10) is just a matter of technical convenience. We will come back to this in the next section in the paragraph “the role of the assumptions”.

2 Heuristic analysis and outline of proofs By Theorem 1.1, with probability going to 1, the process m(λ) (t), t ≤ λ−2 τ , is always close to the manifold of instantons M = {m ¯ ξ , ξ ∈ R}, see (1.11); the theorem then identifies the motion along M, (1.12). It is then natural to describe m(λ) (t) in terms of coordinates along and transversal to M, these are the Fermi coordinates that we are going to define. Stability of instantons, Fermi coordinates. Closeness to M is a consequence of the stability of M under the AC evolution. Under AC, in fact, M attracts exponentially fast all data which are in a small neighborhood, in sup norm,  · ∞ , of M: namely there are δ, c and a all positive so that if for some ξ, m − m ¯ ξ ∞ < δ, then there is a ξ
for which, for all t, m(t) − m ¯ ξ ∞ ≤ ce−at , m(t) being the solution of the AC equation starting from m. This obviously fails if we add noise (thus considering (1.1) with λ = 0 and γ > 0), but the noise, in a polynomial scale, cannot drive too far away from M because of the exponential attraction of

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the deterministic part of the equation: in [10] it is shown that for any τ > 0 and ζ > 0, 

lim P

γ→0

sup dist(m(t), M) ≥ γ 1/2−ζ

=0

(2.1)

t≤γ −1 τ

where “dist” denotes distance in sup norm. The analysis extends to our case with √ λ = γ and (1.1)–(1.2) coupled, as stated in (1.11) and proved in Section 8. Going back to AC without noise, observe that stability of M does not mean stability of the single instanton: let m be a small deviation from m ¯ ξ , then from what we said above it will relax under AC to some m ¯ ξ , with ξ
close but not necessarily equal to ξ. In the space of all profiles m, m ¯ ξ is marginally stable along the direction M while all the other directions are stable. It is then natural to associate to each m (as above) the value ξ
of the center of the limit instanton. In practice, however, it is better to work with a more “geometrical” definition. Following [10, 9], we define a center ξ of m as a real number which minimizes the ¯ x (as a function of x). Then ξ is such that L2 –norm of m − m ¯
ξ  = 0. m − m ¯ ξ, m

(2.2)

The center ξ has also a dynamical interpretation. Let Lξ be the operator 1 ∆v + (1 − 3m ¯ 2ξ )v (2.3) 2 obtained by linearizing the AC equation around m ¯ ξ . It is readily seen that Lξ is ¯
ξ while the rest of self-adjoint in L2 (R, dx), it has eigenvalue 0 with eigenvector m the spectrum is on the negative axis strictly away from 0. Then, if m has center ξ, the deviation v = m − m ¯ ξ by (2.2) has no component along m ¯
ξ , hence the linearized evolution starting from v decays exponentially fast and correspondingly m converges to m ¯ ξ , so that the center of m is also the center of the instanton to which the linearized AC evolution converges. The pair {ξ, m − m ¯ ξ } is known as the Fermi coordinates of m. This notion of a center of a function plays an important role also in our proofs, so we will spend a few more words, recalling Proposition 3.2 in [9], which gives a sufficient condition for m = m(x) to have a center. Lξ v =

Proposition 2.1 There is a constant δ > 0 such that if there exists x0 ∈ R so that m− m ¯ x0 ∞ ≤ δ then the following holds for some constant C = C(δ) independent of x0 and m. (i) The function m has a unique center x and ¯ x0  ∞ |x − x0 | ≤ Cm − m

(2.4)

(ii) the center x has the expansion 3
9 ¯ x0 , m − m ¯
,m− m x = x0 − m ¯ x0  − m ¯ x0 m ¯

x0 , m − m ¯ x0  + R(m − m ¯ x0 ), 4 16 x0 ¯ x0 3∞ . (2.5) |R(m − m ¯ x0 )| ≤ Cm − m

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Thus, by (2.1) and (1.11) and with the help of Proposition 2.1, we can talk unambiguously, with probability going to 1, of the center ξ(t) of m(t) both for the stochastic AC equation and PFE. This gives us a precise definition of the location of the interface even without sharp interface limits; we will prove convergence to (1.12) by studying the asymptotic law of ξ(t) as γ → 0. Heuristics of the AC equation and PFE with noise. We start from the simpler (and instructive) case of the AC equation with noise, i.e. (1.1) with λ = 0 but with γ > 0. This is well studied in the literature, [9, 10, 13], even though in slightly different contexts. The scaling procedure is the same as in Theorem 1.1 and it leads to the same limit law but without drift, i.e. a Brownian motion. To √ explain heuristically the result, let us regard the forcing term γdW (a) (t) as a “source of small kicks” which we decompose in a component along M and another one orthogonal to M. The latter fights against the AC drift which pushes back √ toward M, and because of the small factor γ, to a first order, we forget about orthogonal components. On the contrary the kicks along M are not contrasted and they sum up: thus we are approximating dm(t) ≈

3√
γm ¯ ξ(t) m ¯
ξ(t) , dW (a) (t), 4

m ¯
, m ¯
 =

4 3

(2.6)

where ξ(t) is the center of m(t). Also m(t) ≈ m ¯ ξ(t) , hence dm(t) ≈ m ¯
ξ(t) dξ(t)

(2.7)

and, in conclusion,

3√ γm ¯
ξ(t) , dW (a) (t) (2.8) 4 namely ξ(γ −1 t) is a Brownian motion with diffusion 3/4, which is what is proved in [10]. The argument for the system (1.1)–(1.2) is similar, the only difference from the stochastic AC equation in (1.1) lies in the simple, innocent looking term λh, which is however the source of all problems. The same heuristics leading to (2.6) √ applies to (1.1) by simply adding λh(t) to the noise; writing γ = λ according to (1.10), we then get

3 ¯
ξ(t) m dm(λ) (t) ≈ λm ¯
ξ(t) , dW (a) (t) (2.9) ¯
ξ(t) , h(t)dt + m 4 dξ(t) ≈

and, using (2.7), dξ(t) ≈

3
λ m ¯ ξ(t) , h(t)dt + m ¯
ξ(t) , dW (a) (t) . 4

(2.10)

On the other hand, writing (1.2) in integral form and recalling that h(0) = 0, we have  t pt−s dm(λ) (s) (2.11) h(t) = − 0

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Using again (2.7) and taking the scalar product with m ¯
ξ(t) we get from (2.11) m ¯
ξ(t) , h(t)

 ≈−

0

t

m ¯
ξ(t) , pt−s m ¯
ξ(s) dξ(s) .

(2.12)

The approximate system (2.10)–(2.12) (with ≈ replaced by equality) is not as easy to study as the one which approximates the stochastic AC equation, but it can be seen to give the correct result (1.12) for the limit motion of the center, [3]. To see the relation with (1.12) we make another approximation whose validity will be justified in the course of the proof of Theorem 1.1. The approximation consists in ¯
(i.e. ξ(t) → 0) in the scalar products in (2.10) and (2.12) replacing m ¯
ξ(t) by m (the reason for its validity is that the displacements of the center are finite while the field h becomes flat because the diffusion in (1.2) acts for long times). The new system is then (forgetting about the noise cutoff) 3 λm ¯
, h(t) + λdb(t) 4  t m ¯
, pt−s m ¯
dξ(s) m ¯
, h(t) = −

(2.13)

dξ(t) =

(2.14)

0

where b(t) is the Brownian motion with diffusion 3/4 of Theorem 1.1. Using (2.14) to rewrite the first term on the r.h.s. of (2.13) we get  t  s 3 ξ(t) = λb(t) − λ ds m ¯
, ps−s m ¯
dξ(s
) . 4 0 0 Integrating by parts, after some simple algebra,  t 3 ξ(t) = λb(t) − λ ds
ξ(s
)m ¯
, pt−s m ¯
 . 4 0

(2.15)

Approximating 







¯
(y) m ¯
(x)m dy  2π(t − s
) (2.16) 
and recalling that m ¯ = 2, (2.15) becomes (1.12), in the above approximation, which can be made rigorous in the limit λ → 0, having set t = λ−2 τ . ¯
 = m ¯
, pt−s m

dx

dy m ¯
(x)pt−s (x, y)m ¯
(y) ≈

dx

Main difficulties and outline of proof. The heuristic arguments outlined above are essentially based on a linear approximation, their validity therefore rests on a rigorous proof that the non linear effects are negligible. Since the strength of √ the noise is γ, we cannot hope to improve the a priori bounds beyond m(t) − √ m ¯ ξ(t) ∞ ≈ γ. Then the non linear terms which are, to lowest order, quadratic have order γ; since they act for a time λ−2 τ (that is γ −1 τ ), a naive estimate gives a non–negligible contribution. The fact that they are indeed negligible must then

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come from cancellations, and the true problem is to find them and take them into proper account. This is more clearly seen in the stochastic AC equation. Following [10], we split the time axis into intervals of length T , taking T long, yet very small as compared to the macroscopic time γ −1 τ , for instance T = γ −1/10 . The crucial point is an iterative procedure for which the problem reduces to the analysis of only one of these intervals, a clear advantage, because in such a “short time” the non linear effects are under control. Couplings are used for this crucial reduction. We compare in fact, in the generic interval [nT, (n + 1)T ], the true process m(t) which starts from m(nT ) and the new process m(t) ˆ which starts from m ¯ ξ(nT ) , ξ(nT ) the center of m(nT ), the coupling is simply to use for both processes the √ ¯ ξ(nT ) ∞ ≤ γ 1/2−ζ , same noise γdW (a) (t). Under the assumption that m(nT )−m see (2.1), it can be seen that, with probability going to 1 as γ → 0, m((n + 1)T ) − m((n ˆ + 1)T )∞ ≤ Cγ 1−2ζ , (C a constant). By Proposition 2.1, the displacements of the center in the interval [nT, (n + 1)T ], as computed with the two evolutions, differ proportionally to γ 1−2ζ : since the number of intervals grows proportionally to γ −1 τ /T , the sum of all these differences goes like γ 1−2ζ γ −1 τ /T , which vanishes, after choosing 2ζ < 1/10. We can then study in each interval the process which starts from an instanton. Neglecting for simplicity the cutoff on the noise (with the cutoff some extra computations are needed), then the displacement of the center in a time interval T does not depend on the initial center and it is therefore independent of the past. The displacements of the centers (each time restarting from an instanton) are thus independent variables with mean 0 (by the symmetry between right and left) and, using classical arguments on convergence to Brownian motion, in the end, we need to sum their variances; since we are already with squares, it turns out that the linear approximation is sufficiently accurate and this explains the validity of the linear approximation in the previous heuristic analysis. While the above approach works well in the stochastic AC equation, Theorem 1.1 tell us that it fails, as there are long memory effects in the limit law. More bad news: the last term in (1.12), responsible for these effects, according to the heuristic analysis of the previous paragraph, comes from the term λh(t) and since it produces a finite drift in a time λ−2 , the order of magnitude of h(t) must be ≈ λ. Therefore we need an accuracy of order λ, which is comparable with the √ deviations of m(λ) from M (recall λ = γ) that has been neglected so far. Our approach to the problem, since when we began the present work, has been “to trust” [10] and to consider the non linear terms that are left as being negligible when we linearize (1.1) around m ¯ ξ(nT ) , ξ(nT ) the center of m(λ) (nT ). It is evidently not possible to use the coupling argument of [10], yet in some maybe more complicated way, in the end we “must” see that they are not relevant. The extra term λh(t) is new w.r.t. [10] and has to be dealt anew. According to the final result and for what said before, we expect h(t) ≈ λ, but let us even suppose, pessimistically, that h(t) is of the order of unity. Its effect for a time T will then be of order λT , hence still infinitesimal. Moreover, by (2.11), h(t) can be large only if dm(λ) (s) is large, but dm(λ) (s) is under control except for the term λh(s). Due

38

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Ann. Henri Poincar´e

to the presence of the small factor λ this gives a virtuous feedback which allows to control the magnitude of h(t). This is done in Sections 3 and 4. We first write down an integral equation, ¯ ξ(nT ) , the superscript n recalls that we are con(3.7), for v (n) (t) = m(λ) (t) − m sidering t ∈ [nT, (n + 1)T ]. v (n) (t) is written as the sum of 8 terms, two of them, (n) (n) called Γ2 (t) and Λ3 (t) are the important ones, which give contributions to the limit, respectively the Brownian motion and the drift. All the others vanish, but at this stage this is not yet established as they depend on the unknown m(λ) (t). In Section 4 we study by iterations the integral equation (3.7) to prove bounds on v (n) (t) and on the displacements of the center. We derive in this way (1.11) and establish even sharper bounds that will be used in the successive proofs. The proof of the convergence to (1.12) is reported in the remaining sections. (n) We begin Section 5 by splitting the term Λ3 (t) into the sum of 3 other terms, see (5.1)–(5.5), which foresees the approximation done in (2.16). We then use (2.5) to deduce an equation for the displacements of the center, which are called ψn , see (5.7). The equation is (5.9), which is a sort of linear integral equation in the ψn with kernel An,k , k < n, and known data ηn : “sort of” because the ηn still depend on the unknowns v (n) (·). The elements An,k decay as (n − k)−1/2 . It is then convenient to reduce to the matrix A2 , so that we iterate once (5.9) obtaining (5.17), where the kernel is now A2n,k and the “known terms” are ηn and (Aη)n . In Section 6 we study these “known terms” which are splitted into four groups. The first one consists of truly known terms which survive in the limit (n) (they come from Γ2 (t)). The terms in the second group, which instead may de(n) pend on v (t), are all directly proved to be negligible using the a priori bounds of Section 4. For those in the third group we cannot proceed in this way, but we need to use the integral equation for v (n) (t) and only after sufficiently many iterations, we can show that they are negligible. Finally, the last group collects terms which become negligible because of stochastic cancellations. The latter are studied in Section 7, the others in Section 6. We draw the conclusions of our analysis in Section 8 where we complete the proof of Theorem 1.1. Role of assumptions. We start from the assumption (1.10) which is conceptually the most important one, the others are of a more technical nature. As already remarked, there are several studies of sharp interface limits on PFE where λ is scaled to 0. This describes an intermediate regime (called kinetic undercooling) where thermodynamical equilibrium is not fully reached. Thus our model should be regarded as kinetic undercooling in the presence of stochastic perturbations. As said, the relation between λ and γ stated in (1.10) does not have a straight physical interpretation, it is just the right way to scale (1.1)–(1.2) and have a nice limit law. One may however wonder what would happen if we took a different relation than (1.10). We have not worked out the details, but we can at least present some educated guess. If we multiply h(t) in (1.1) by a constant θ, we would then derive a limit law with such a factor multiplying the last term in (1.12). Let us then

Vol. 3, 2002

Front Fluctuations in One Dimensional Stochastic Phase Field Equations

39

consider the scaling behavior of  x(t) = b(t) − 3θ

t

0

x(s) ds  2π(t − s)

(2.17)

both when θ vanishes and when it diverges. Set y(t) = θa x(θc t), then a c/2

y(t) = θ θ

b(t) − 3θθ

c/2

 0

t

y(s) . ds  2π(t − s)

(2.18)

Imposing a + c/2 = 0,

1 + c/2 = 0;

a = 1,

c = −2

(2.19)

we have (2.18) equal to (1.12). Thus if we take (1.1) with λ → λ1+α and call θ = λα , we believe that our analysis extends, at least for small |α|, and that the center ξ(t) of the solution of the corresponding equation is such that λα ξ(λ−2−2α t)

(2.20)

converges in law to (1.12). Concluding remarks and perspectives. A forthcoming paper, [3], is devoted to the analysis of the limit process (1.12). This can be characterized in terms of a Brownian motion with absorption at the origin, which in turn is reduced to the well studied one dimensional Schr¨ odinger equation with Dirac’s delta potentials. A quite explicit expression for the solution of (1.12), is then available, in √ particular it shows that the displacements ξ(τ ) of the front have typical size log τ for τ large; “aging phenomena” are also present. The “cluster fluctuations” have instead the usual Brownian structure. Consider an initial state with h0 = 0 and m0 (x) the symmetric function which coincides with m ¯ −ξ (x), ξ > 0, for x ≤ 0. We interpret it as a “plus cluster” in the region (−ξ, ξ) with the minus phase outside. To make it sharp we set ξ = λ−1 -, - > 0, and √ consider the process (1.1)–(1.2) (always assuming λ = γ). Preliminary results (see also [3]) indicate that, in proper units, the coordinates ξ1 (τ ) and ξ2 (τ ) of the two centers, evolve in the limit according to a system of two stochastic equations. The system can be diagonalized into the variables ξG (τ ) := [ξ1 (τ ) + ξ2 (τ )]/2 and ξ(τ ) := ξ1 (τ ) − ξ2 (τ ). The variable ξG (τ ) has the law of a Brownian motion, ξ(τ ) is independent of ξG (τ ) and obeys an equation like (1.12). We hope to be able to accomplish the above program including the analysis of the case with several clusters (and suitable scalings), the ultimate goal being the study of interface fluctuations in many dimensions.

40

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Ann. Henri Poincar´e

Part II. Stability of fronts 3 The iterative scheme Before starting with the proofs we introduce some notation that will be used throughout the paper: Notation. C will denote a generic constant whose numerical value may change from line to line (from the statements it will appear clear on which parameters it depends). For p ∈ [1, ∞] we denote by  · p the norm in Lp (R, dx). We will study the problem (1.1)–(1.10) by an iterative procedure. To this end, we divide the microscopic time–line R+ into intervals [Tn , Tn+1 ], Tn = nT , where n ∈ N and T = λ−(1∧β)/20 (β as in (1.3)). We then associate to any macroscopic time interval [0, τ ], τ > 0, the set of microscopic time intervals [Tn , Tn+1 ], with n ≤ nλ (τ ), where nλ (τ ) is the integer part of (λ2 T )−1 τ , namely nλ (τ ) = [(λ2 T )−1 τ ]. To simplify notation, we write m(t) = {m(t, x), x ∈ R} and h(t) = {h(t, x), x ∈ R} for the solution of (1.1)–(1.10) (omitting the dependence on λ). We next define, by induction on n ≥ 0, the numbers xn and the functions v (n) (t) = {v (n) (t, x), x ∈ R}, t ∈ [Tn , Tn+1 ]. They will have the property that for each t ∈ [Tk , Tk+1 ] ¯ xk v (k) (t) = m(t) − m

if v (h) (Th+1 )∞ ≤ δ for all 0 ≤ h ≤ k − 1

(3.1)

with δ as in Proposition 2.1. For n = 0 we set x0 = 0 and v (0) (t) = m(t) − m ¯ x0 , 0 ≤ t ≤ T , so that (3.1) holds. Suppose that, by the induction hypothesis, we have already defined for all k ≤ n − 1 both xk and v (k) (t) and that (3.1) holds for such k’s. If there is k ≤ n − 1 such that v (k) (Tk+1 )∞ > δ, we set xn = 0 and v (n) (t) = 0, ¯ xn−1 + v (n−1) (Tn ), which t ∈ [Tn , Tn+1 ]. Otherwise we define xn as the center of m (n−1) by Proposition 2.1 is well defined, as v (Tn )∞ ≤ δ. We then set, according to ¯ xn , t ∈ [Tn , Tn+1 ], and have (always under the assumption (3.1), v (n) (t) = m(t)− m that v (k) (Tk+1 )∞ ≤ δ for all k ≤ n − 1) ¯ xn−1 + v (n−1) (Tn ) = m ¯ xn + v (n) (Tn ), m(Tn ) = m Let

h(n) (t) = h(t),

m ¯
xn , v (n) (Tn ) = 0 . (3.2)

t ∈ [Tn , Tn+1 ],

introduce the stopping time w.r.t. the discrete filtration FTn+1

Nδ := inf k : v (k) (Tk+1 )∞ > δ (n)

(δ as in Proposition 2.1) and define gt (0) (2.3). We abbreviate gt = gt .

(3.3)

(3.4)

:= exp{tLxn }, t ∈ R+ , with Lxn as in

Vol. 3, 2002

Front Fluctuations in One Dimensional Stochastic Phase Field Equations

41

After expanding V
(m(t)) in (1.8) and (1.10) about m ¯ xn , for t ∈ [Tn , Tn+1 ] as long as n ≤ Nδ , v (n) (t) and h(n) (t) are given as the solution of   dv (n) (t) = Lxn v (n) (t) + λh(n) (t) − 3m ¯ xn v (n) (t)2 − v (n) (t)3 dt + λdW (a) (t)

h(n) (t) = pt−Tn h(n−1) (Tn ) − v (n) (t) + pt−Tn v (n) (Tn ) +



(3.5) t

ds Tn

∂pt−s (n) v (s) ∂s (3.6)

with initial condition v (n) (Tn ) for the first equation. Recall that v (n) (Tn ) is determined from v (n−1) (Tn ) via (3.2), and that from (3.3) we have h(n−1) (Tn ) = h(n) (Tn ). We will write in the next section v (n) (t) as a sum of 10 terms, v (n) (t) = 10 (n) i=1 Γi (t), but for the moment it is more convenient to keep some of the terms together. We are going to prove that for n < Nδ , (n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

v (n) (t) = Γ1 (t) + Γ2 (t) + Λ3 (t) + Γ4 (t) + Γ5 (t) + Γ6 (t) + Γ9 (t) + Γ10 (t) (3.7) where (n)

Γ1 (t) (n)

Γ2 (t) (n)

Λ3 (t) (n) Γ4 (t) (n)

Γ5 (t) (n) Γ6 (t) (n)

Γ9 (t) (n)

Γ10 (t)

(n)

:= gt−Tn v (n) (Tn )  (n) := λz (t) := λ n  

:= −λ

k=1 t

 := λ

Tn

:= −λ

k=1  t

 := −

t

Tn

Tn t

Tn

gt−s dW (a) (s)

(3.9)

(n)

ds gt−s ps−Tk [m ¯ xk − m ¯ xk−1 ] 

s

ds


Tn

∂ps−s (n)
v (s ) ∂s


(n)

Tn

(n) ds gt−s



Tk

(3.12) ds


Tk−1

∂ps−s (k−1)
v (s ) ∂s 

 (n) ¯ xn v (n) (s)2 ds gt−s m

(3.15)

−1

to (3.5) we get  (n) (n) (n) (n) v (n) (t) = Γ1 (t) + Γ2 (t) + Γ9 (t) + Γ10 (t) + λ

(3.13) (3.14)

  (n) ds gt−s v (n) (s)3 .

Proof of (3.7). By applying (∂t − Lxn )

(3.10) (3.11)

ds gt−s v (n) (s)

Tn n  t 

:= −λ := −3

t

(n)

Tn

(n) ds gt−s



(3.8) t

t Tn

(n)

ds gt−s h(n) (s) .

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L. Bertini, S. Brassesco, P. Butt` a and E. Presutti

Ann. Henri Poincar´e

We use for h(n) (s) the expression given by (3.6), and get v (n) (t) =

(n)

(n)

(n)

(n)

(n)

(n)

Γ1 (t) + Γ2 (t) + Γ4 (t) + Γ5 (t) + Γ9 (t) + Γ10 (t)  t   (n) ds gt−s ps−Tn h(n−1) (Tn ) + v (n) (Tn ) . +λ Tn

To complete the proof of (3.7) we thus need to show 

t

λ Tn

  (n) (n) (n) ds gt−s ps−Tn h(n−1) (Tn ) + v (n) (Tn ) = Λ3 (t) + Γ6 (t) .

(3.16)

From (3.2), we have ¯ xn − m ¯ xn−1 ) h(n−1) (Tn ) + v (n) (Tn ) = h(n−1) (Tn ) + v (n−1) (Tn ) − (m

(3.17)

¯ xn−1 ) reproduces We use this identity in the l.h.s. of (3.16). The difference (m ¯ xn − m (n) the last term of the sum in (3.10) (i.e. in the definition of Λ3 (t)). For the term with h(n−1) (Tn ) + v (n−1) (Tn ), we use (3.6) to write h(n−1) (Tn ) + v (n−1) (Tn )

 (n−1) (n−1) (Tn−1 ) + v (Tn−1 ) + = pTn −Tn−1 h

Tn

Tn−1

ds


∂pTn −s (n−1)
v (s ). ∂s


The contribution of the last integral to the l.h.s. of (3.16) gives the last term of (n) the sum in (3.13), i.e. the definition of Γ6 (t). By iteration we then get (3.16) and (3.7) is therefore proved.


4 A priori bounds We will use the representation (3.7) to prove in Proposition 4.1 below some a priori bounds on v (n) (t) and other quantities. We need first some more notation; recalling Proposition 2.1, we set ξn := −

n−1 3
m ¯ xk , v (k) (Tk+1 ) 4

(4.1)

k=0

We should think of ξn as a linear approximation to xn , since the increment ξk −ξk−1 is, according to (2.5), the linear approximation to the displacement xk − xk−1 of the center in the time interval [Tk−1 , Tk ]. We set (n) sup Vn,∗ := sup Vk , V∗ (τ ) := Vnλ (τ ),∗ (4.2) Vn := v (t) , t∈[Tn ,Tn+1 ]

∞

k≤n

Vol. 3, 2002

Front Fluctuations in One Dimensional Stochastic Phase Field Equations

and

δ ∗ (τ ) :=

sup |xk+1 − xk |

43

(4.3)

k≤nλ (τ )

Let also Zn :=

sup s∈[Tn ,Tn+1 ]

(n) z (s)

∞

,

Zn,∗ := sup Zk k≤n

(4.4)

(1)

and Bλ,τ ⊂ Ω be the event (recall the definition of (Ω, F , Ft , P) given in Section 1) √ (1) (4.5) Bλ,τ := ω ∈ Ω : Znλ (τ ),∗ ≤ λ−ζ T . We will prove in Appendix B the following Gaussian estimate: for each τ, ζ, q > 0 there is a constant C = C(τ, ζ, q) such that for any λ > 0   (1) (4.6) P Bλ,τ ≥ 1 − Cλq . The next proposition is the key ingredient for the bound (1.11), see the beginning of §8 for the conclusion of the proof. Proposition 4.1 For each τ, ζ > 0 there is a constant C = C(τ, ζ) such that, for any λ > 0, √ √ (1) V∗ (τ ) ≤ Cλ1−ζ T , δ ∗ (τ ) ≤ Cλ1−ζ T on the set Bλ,τ . (4.7) In particular, for λ > 0 small enough, Nδ > nλ (τ )

(1)

on the set Bλ,τ .

(4.8)

Proof. Let (mλ , tλ ), tλ ∈ [Tmλ , Tmλ +1 ], mλ ≤ nλ (τ ), be the stopping times (mλ w.r.t. FTn+1 , tλ w.r.t. Ft ) so that (mλ , tλ ) = (nλ (τ ), Tnλ (τ )+1 ) if V∗ (τ ) < λT , otherwise (mλ , tλ ) = (n, t), where n is the first index such that Vn ≥ λT and t the first time in [Tn , Tn+1 ] for which v (n) (t)∞ ≥ λT . In the following we may assume λ is small enough, otherwise (4.7) holds trivially. We claim there is a constant C so that v (mλ ) (tλ )∞ ≤ CλT .

(4.9)

In fact, if tλ ∈ (Tmλ , Tmλ +1 ], (4.9) follows from the continuity of v (mλ ) (t)∞ ; otherwise, i.e. if tλ = Tmλ , v (mλ ) (Tmλ )∞ ≤ v (mλ −1) (Tmλ )∞ + m ¯ mλ − m ¯ mλ −1  ≤ λT + |xmλ − xmλ −1 | ≤ CλT from (3.2) and (2.4), since m ¯
∞ = 1 and v (mλ −1) (Tmλ )∞ ≤ λT .

44

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Ann. Henri Poincar´e

We will study the evolution till time tλ , so that our next considerations are tacitly restricted to n ≤ mλ and t ≤ tλ . By (4.9), Nδ > mλ (for λ small enough) so that we can use (3.7), regarding it as an integral equation in v (n) (t). We are going to bound one after the other all eight terms on the r.h.s. of (3.7). We will use in the sequel the following bounds (see e.g. [9] and references therein). There is C < ∞ so that, for any measurable function ϕ, gt ϕ∞ ≤ Cϕ∞ .

(4.10)

Moreover, by the Perron–Frobenius Theorem, there are α > 0 and C < ∞ so that for any ϕ orthogonal to m ¯
, m ¯
, ϕ = 0, gt ϕ∞ ≤ Ce−αt ϕ∞ .

(4.11)

Then, by the last identity in (3.2), for t ∈ [Tn , Tn+1 ] (n) ≤ Ce−α(t−Tn ) v (n) (Tn )∞ Γ1 (t) ∞   ≤ Ce−α(t−Tn ) v (n−1) (Tn )∞ + m ¯ xn − m ¯ xn−1 ∞ ≤

Ce−α(t−Tn ) v (n−1) (Tn )∞ .

(4.12)

The second inequality follows from (3.2) and the last one follows from (2.4), proceeding as in the proof of (4.9). By definition (4.4) we get (n) sup (4.13) Γ2 (t) ≤ CλZn . ∞

t∈[Tn ,Tn+1 ]

(n)

We next bound Λ3 . We have 2 1 pt m ¯
z ∞ ≤ √ ≤ √ t 2πt √ ¯
z 1 = 2 for any z ∈ R. Then because pt (x, y) ≤ 1/ 2πt and m    |x − x  xk   k k−1 |
pt (m √ ¯ xk − m ¯ xk−1 ) ∞ ≤  dz pt m ¯ z ∞  ≤   xk−1 t so that, by (4.10), we get (n) Λ3 (t)

∞

≤ Cλ

n   k=1

t

|xk − xk−1 | ds √ . s − Tk Tn

(4.14)

(4.15)

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Front Fluctuations in One Dimensional Stochastic Phase Field Equations

45

By (2.4), we then get (n) Λ3 (t)

n √  Vk−1 √ ≤ Cλ T . n −k+1 k=1

∞

(4.16)

(n)

We will next bound Γ4 (t). We write 

(n)

gt−s = pt−s −

t

ds



s

 d  (n) gt−s ps −s = pt−s +

ds



t

s

(n)

ds

gt−s [1 − 3m ¯ 2xn ]ps −s

having used (2.3) in the second equality. Then (n)

Γ4 (t)





t

= λ

s

ds


ds Tn



Tn



t

+λ

∂pt−s (n)
v (s ) ∂s


t

ds

ds

Tn





s

(n) gt−s [1

−

3m ¯ 2xn ]



s

ds


Tn

∂ps −s (n)
v (s ) . ∂s


By using (4.10) and (see Appendix A) ∂ pt ϕ ≤ 1 ϕ∞ ∂t t ∞ since v (n) (s
)∞ ≤ Vn , we get (n) Γ4 (t)

∞

≤ CλT 2 Vn .

This is not optimal but good enough for our purposes. Analogously  t (n) ds v (n) (s)∞ ≤ CλT Vn Γ5 (t) ≤ Cλ ∞

(n) Γ6 (t)

∞

≤ Cλ

n   k=1

Tn

(4.18)

(4.19)

Tn



t

(4.17)

Tk

ds

ds


Tk−1

Finally, using again (4.10), we have (n) Γ9 (t) ∞ (n) Γ10 (t) ∞

n  v (k−1) (s
)∞ Vk−1 . ≤ CλT s − s
n−k+1 k=1 (4.20)

≤

CT Vn2

(4.21)

≤

CT Vn3

(4.22)

hence, recalling the definition of (mλ , tλ ) and (4.9), (n) (n) Γ9 (t) + Γ10 (t) ≤ CλT 2 Vn ∞

∞

(4.23)

46

L. Bertini, S. Brassesco, P. Butt` a and E. Presutti

Ann. Henri Poincar´e

After collecting all these bounds, we have, for t ∈ [Tn , Tn+1 ], n ≤ mλ , t ≤ tλ , (n) ≤ C e−α(t−Tn ) v (n−1) (Tn ) + λZn + λT 2 Vn v (t) ∞

∞

n n  √  V Vk−1 √ k−1 . + λT +λ T n−k+1 n−k+1 k=1 k=1

(4.24)

By iterating once the above inequality we get
Vn ≤ C e−αT Vn−2 + λ(Zn + Zn−1 ) + λT 2 (Vn + Vn−1 )  n n  √  Vk−1 Vk−1 √ +λ T + λT . n−k+1 n−k+1 k=1 k=1 Recalling (4.4), we have Vn

≤ C e−αT Vn,∗ + λZn,∗ + λT 2 Vn,∗ n n  √  Vk−1,∗ Vk−1,∗ √ +λ T + λT n−k+1 n−k+1 k=1 k=1

and since the r.h.s. is an increasing function of n, Vn,∗ ≤ C e−αT Vn,∗ + λZn,∗ + λT 2 Vn,∗ n n  √  V Vk−1,∗ √ k−1,∗ + λT . +λ T n−k+1 n−k+1 k=1 k=1

(4.25)

Inequality (4.25) yields, for n ≤ mλ ≤ nλ (τ ), √ n−1  Cλ T   2  −αT √ 1 − C λT + e + λT log nλ (τ ) Vn,∗ ≤ CλZn,∗ + Vk,∗ . n−k k=0 For λ small enough the square bracket term is larger than 1/2, so that (provided we double the value of the constant C) Vn,∗ ≤ CλZn,∗ +

n−1  k=0

√ Cλ T √ Vk,∗ . n−k

(4.26)

By iteration of (4.26) we get Vn,∗ ≤ CλZn,∗ +

n−1  k=0

√ n−2  C 2 λ2 T 2 2 √ α(n, k)Vk,∗ Zk,∗ + C λ T n−k k=0

(4.27)

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where α(n, k) =

n−1 

1 1 √ √ n−h h−k h=k+1

47

(4.28)

is a bounded function of n and k. We bound the second term on the r.h.s. of (4.27) by Zn,∗

n−1  k=0

√ √ C 2 λ2 T √ ≤ CZn,∗ λ2 T n ≤ CλZn,∗ n−k

since n ≤ mλ ≤ (λ2 T )−1 τ . We then get from (4.27) Vn,∗ ≤ CλZn,∗ + Cλ2 T

n−2 

Vk,∗

(4.29)

k=0

from which, by Gronwall Lemma, there is C = C(τ ) such that Vn,∗ ≤ CλZn,∗ ,

for all n ≤ mλ .

(4.30)

By (4.5), choosing λ small enough, we have, for n ≤ mλ , √ (1) on the set Bλ,τ Vn,∗ ≤ Cλ1−ζ T (1)

which implies mλ = nλ (τ ) on the set Bλ,τ . Thus the first bound in (4.7) follows from (4.5) and (4.30). Since Nδ > mλ , (4.8) also follows. Since m ¯
∞ = 1, the second estimate in (4.7) follows directly from the first one and Proposition 2.1.
¯
xn , is We are going to prove that the component of v (n) (Tn+1 ) orthogonal to m 1−ζ (n) bounded by Cλ , thus considerably improving the bound on the full v (Tn+1 ). Let   3
(n,⊥) (n)
¯ m gt := gt ¯ xn | 1 − |m 4 xn the operator whose kernel is (n,⊥)

gt

3
(n) ¯ (x)m (x, y) = gt (x, y) − m ¯
xn (y) . 4 xn

(4.31) (n)

The superscript ⊥ recalls L2 –orthogonality w.r.t. the eigenvector m ¯
xn of gt , i.e. (n)
¯ xn = m ¯
xn . It follows from (4.11) that there are constants α > 0 and C < ∞ gt m so that, for any ϕ, (n,⊥) ϕ ≤ Ce−αt ϕ∞ . (4.32) gt ∞

Let also

3
¯ , z (n) (t)m z (n,⊥) (t) := z (n) (t) − m ¯
xn 4 xn

(4.33)

48

L. Bertini, S. Brassesco, P. Butt` a and E. Presutti

Ann. Henri Poincar´e

be the component of z (n) (t) orthogonal to m ¯
xn and introduce the event   (2)

Bλ,τ :=

ω∈Ω :

sup

sup

n≤nλ (τ ) t∈[Tn ,Tn+1 ]

z (n,⊥)(t)∞ ≤ λ−ζ

.

(4.34)

In Appendix B we will prove that for each τ, ζ, q > 0 there exists a constant C = C(τ, ζ, q) such that for any λ > 0   (2) P Bλ,τ ≥ 1 − Cλq . (4.35) Define now

V∗⊥ (τ ) :=

sup v (n,⊥) (Tn+1 )

n≤nλ (τ )

∞

,

3
¯ , v (n) (Tn+1 )m v (n,⊥) (Tn+1 ) := v (n) (Tn+1 ) − m ¯
xn . 4 xn

(4.36)

Proposition 4.2 Recalling (4.5) and (4.34), set (1,2)

(1)

(2)

Bλ,τ := Bλ,τ ∩ Bλ,τ .

(4.37)

Then, for each τ, ζ > 0 there is a constant C = C(τ, ζ) such that, for any λ > 0, V∗⊥ (τ ) ≤ Cλ1−2ζ

(1,2)

on the set Bλ,τ

(4.38)

and, recalling (4.1) for the definition of ξn , sup n≤nλ (τ )+1

|ξn − xn | ≤ CT −1/2+ζ

(1,2)

on the set Bλ,τ

.

(4.39)

Proof. Let (n,⊥)

Λ3

(Tn+1 ) := =

3
(n) (n) ¯ , Λ (Tn+1 )m Λ3 (Tn+1 ) − m ¯
xn 4 xn 3 n  Tn+1  (n,⊥) −λ dt gTn+1 −t pt−Tk [m ¯ xk − m ¯ xk−1 ] k=1

(4.40)

Tn

where we used (4.31). By using (4.14), (4.32), m ¯
∞ ≤ 1, and recalling (4.3) we have n  Tn+1  1 (n,⊥) ∗ (Tn+1 ) ≤ Cδ (τ )λ dt e−α(Tn+1 −t) √ Λ3 ∞ t − Tk k=1 Tn  λ ≤ Cδ ∗ (τ ) √ nλ (τ ) ≤ Cλ1−ζ T −1/2 (4.41) T (1)

the last inequality being true, by (4.7), on the set Bλ,τ .

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49

Shorthanding by R(n) the sum of all the terms on the r.h.s. of (3.7), except (n) and Λ3 , and calling R(n,⊥) its orthogonal projection,

(n) Γ2

(n,⊥)

v (n,⊥) (Tn+1 ) = λz (n,⊥) (Tn+1 ) + Λ3

(Tn+1 ) + R(n,⊥) .

(4.42)

The last term is bounded by   |R(n,⊥) | ≤ C e−αT + λT 2 + λT log nλ (τ ) V∗ (τ )

(4.43)

as it follows from (4.12) and (4.18)–(4.20), and (4.23). The bound (4.38) now follows from the definition (4.34), and equations (4.41)–(4.43). It remains to prove (4.39). Recalling that x0 = 0, from (2.5) and (4.1), xn =

n−1 

(xk+1 − xk ) = ξn −

k=0

n−1 9 
m ¯ xk , v (k) (Tk+1 )m ¯

xk , v (k) (Tk+1 ) 16 k=0

+

n−1 

R(v (k) (Tk+1 )) (4.44)

k=0

then, for n ≤ nλ (τ ), since m ¯
1 = 2, 

  

 ¯ xn , v (n) (Tn+1 ) + |ξn − xn | ≤ nλ (τ ) 2V∗ (τ ) sup m n≤nλ (τ )

sup R(v (n) (Tn+1 ))

∞

n≤nλ (τ )

By (2.5)

sup R(v (n) (Tn+1 ))

∞

n≤nλ (τ )

≤ CV∗ (τ )3 .

.

(4.45)

(4.46)

¯
 = 0, |m ¯

xn , v (n) (Tn+1 )| ≤ CV∗⊥ (τ ), (4.39) follows from (4.7), Since m ¯

, m (4.38), (4.45), and (4.46).
¯
xn , v (n) (Tn ) = 0, The bound in (4.38) holds also for v (n) (Tn ). Indeed, since m we have the following proposition. (1,2)

Proposition 4.3 Let Bλ,τ be as in (4.37). Then, for each τ, ζ > 0 there is a constant C = C(τ, ζ) such that, for any λ > 0 (1,2) on the set Bλ,τ . (4.47) sup v (n) (Tn ) ≤ Cλ1−2ζ n≤nλ (τ )

Proof. By (3.2)

∞

v (n) (Tn ) = v (n−1) (Tn ) + m ¯ xn−1 − m ¯ xn .

(4.48)

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L. Bertini, S. Brassesco, P. Butt` a and E. Presutti

Ann. Henri Poincar´e

By Taylor expansion up to fourth order, we have ¯ xn−1 m ¯ xn − m

1

¯ −m ¯
xn−1 (xn − xn−1 ) + m (xn − xn−1 )2 2 xn−1 1


− m (xn − xn−1 )3 + an−1 (xn − xn−1 )4 ¯ 6 xn−1

=

(4.49)

where an−1 is bounded. Thus, by Proposition 2.1, we have m ¯ xn − m ¯ xn−1 =

3
m ¯ , v (n−1) (Tn )m ¯
xn−1 + αn 4 xn−1

(4.50)

where 1

1


m ¯ ¯ (xn − xn−1 )2 − m (xn − xn−1 )3 + an−1 (xn − xn−1 )4 2
xn−1 6 xn−1  9


(n−1)
(n−1) (n−1) m ¯ +m ¯ xn−1 ,v (Tn )m ¯ xn−1 , v (Tn ) − R(v (Tn )) . (4.51) 16 xn−1 αn =

From (4.48) and (4.50) we get v (n) (Tn ) = v (n−1,⊥) (Tn ) − αn .

(4.52)

√ (1,2) Note that, from (4.7), on the set Bλ,τ we have supn≤nλ (τ ) |xn −xn−1 | ≤ Cλ1−ζ T ; by (4.38) and (4.46) we have |αn | ≤ Cλ2−2ζ T (a better bound is proved in Section 6). The bound (4.47) follows.


Part III. Limit motion 5

A new integral equation (1,2)

Unless otherwise stated, we will work in the set Bλ,τ which appears in Propositions 4.2 and 4.3. In particular we can use the integral representation (3.7) for all n ≤ nλ (τ ) (if λ is small enough) and the bounds of Section 4. It is now convenient (n) to decompose the term Λ3 (t) into the sum of three new terms; we thus use (4.50) to write (n) (n) (n) (5.1) Λ3 (t) = Γ3+8 (t) + Γ7 (t) where (n)

Γ3+8 (t) := (n)

Γ7 (t) :=

n

3λ 
m ¯ xk−1 , v (k−1) (Tk ) 4 k=1 n  t  (n) −λ ds gt−s ps−Tk αk −

k=1

Tn



t

Tn

(n)

ds gt−s ps−Tk m ¯
xk−1

(5.2) (5.3)

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(n)

51

(n)

and next we decompose Γ3+8 (t) = Γ3 (t) + Γ8 (t) where (n)

Γ3 (t) := −

 t n−1 3λ 
2 (n) m ¯ xk , v (k) (Tk+1 ) ds gt−s  4 2π(s − Tk+1 ) Tn

(5.4)

k=0

   t n−1 3λ 
2 (n) (k)
:= − m ¯ xk , v (Tk+1 ) ds gt−s ps−Tk+1 m ¯ xk −  . 4 2π(s − Tk+1 ) Tn k=0 (5.5) (n) (n) Note that Γ3 (t) is obtained from Γ3+8 (t) replacing pt (x, y) by (2πt)−1/2 (recall that dx m ¯
(x) = 2). In conclusion we have (n) Γ8 (t)

v (n) (t) =

10  i=1

Let us define

(n)

Γi (t) .

3
ψn := − m ¯ , v (n) (Tn+1 ) 4 xn

(5.6)

(5.7)

so that, recalling (4.1), ξn =

n−1 

ψk .

(5.8)

k=0

We then set t = Tn+1 in (5.6) and project it on m ¯
xn , getting ψn = ηn −

n−1 

An,k ψk

(5.9)

3
(n) ¯ , Γ (Tn+1 ), i = 1, . . . , 10 ηn (i) = − m 4 xn i

(5.10)

k=0

where ηn :=

10  i=1 i=3

and

ηn (i),

√ 2 T 3 √ 1k 0 there is a constant C = C(τ ) such that for any λ > 0 Cj . sup Aj n ≤ j! n≤nλ (τ )

(5.15)

Hence, from (5.13), ψ=

∞ 

(−A)j η = (1 + A)−1 η,

sup (1 + A)−1 n ≤ C .

(5.16)

n≤nλ (τ )

j=0

It will be convenient to consider also one iteration of (5.13), i.e. ψ = η − Aη + A2 ψ .

(5.17)

Explicitly we have (A2 )n,k =

n−1 

An,j Aj,k 10≤k 0 there is a constant C = C(τ ) such that     2  sup  A n,k  ≤ Cλ2 T .

(5.19)

0≤k 0, lim

sup

λ↓0 0≤τ2 δ

  2 −1  2  (λ T ) A n (τ1 ),n (τ2 ) −  λ λ

 9  =0. 2

(5.20)

Vol. 3, 2002

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Proof. By (5.11)

53

n−1     1   2 √ √  A n,k  ≤ Cλ2 T n − j j−k j=k+1

which proves (5.19). By (5.11) we have   9 9 (λ2 T )−1 A2 n (τ ),n (τ ) − = λ 1 λ 2 2 2π  1 )−1  nλ (τ  2   ×  nλ (τ1 ) − j + nλ (τ1 ) − j − 1 j=n (τ )+1 λ

2

2  −π ×  j − nλ (τ2 ) + j − nλ (τ2 ) − 1

 .

(5.21)

1 Since π = 0 dx [x(1 − x)]−1/2 , by letting σ = τ1 − τ2 and changing the index of summation the r.h.s. of (5.21) equals    1 nλ (σ)−2  2 2 1 9   √   √ − dx  2π  j=0 nλ (σ) − j + nλ (σ) − j − 1 j + j + 1 x(1 − x)  0 (5.22) which converges, as λ ↓ 0, to 0 uniformly for σ ∈ [δ, τ ], as can be easily checked. Proposition 5.2 is proved.


6 Bounds on η Recalling (5.8)–(5.10), a term η(i) (as in the previous section, we are here using vectorial notation) contributing to ψ whose seminorm is |η(i)|n = o(λ2 T ) (i.e. such that (λ2 T )−1 |η(i)|n vanishes as λ ↓ 0) does not contribute to the limiting equation for ξn since n ≤ (λ2 T )−1 τ . This is the case for some of the η(i)’s, i.e. η(1), η(7), and η(10), as we shall see. √ Clearly ηn (2) is not negligible because its typical magnitude is λ T . It will be examined in the next section, where we shall see that , summed over n it gives a finite contribution because of cancellations related to its martingale nature.The other terms, i.e. ηn (i), with i from 4 to√9, live on an intermediate stage: they are smaller than the a priori bound λ1−ζ T , yet not small enough to be directly negligible. We shall rewrite the factors v (n) (t) via (5.6), with the idea that if we get two η(i), i = 4, . . . , 10 then the corresponding terms become negligible. We will use the following notation. Definition 6.1 Let ηn (i1 , . . . , ik ), k > 1, ij ∈ {3, 4, 5, 6, 8} when j < k, ik ∈ {1, . . . , 10}, be the term which is obtained from ηn (i1 ) by replacing the v () (·)–  () function in its expression by Γi2 (·); then, the new v ( ) (·) function which appears

54

L. Bertini, S. Brassesco, P. Butt` a and E. Presutti

Ann. Henri Poincar´e

( )

is replaced by Γi3 (·) and so forth till the last one, which is not changed. Instead of an index ij may appear the symbol 3 + 8 or 4 + 5, it means that, at that stage () () of the iteration, we replace the v () (·)–function by Γ3 (·) + Γ8 (·), or respectively () () by Γ4 (·) + Γ5 (·). We will also consider i1 = 9, in which case there is a product of two v– functions. Then, ηn (9|i2 , . . . , ik ; i
2 , . . . , i
h ) is defined by doing the branching i2 , . . . , ik (with the previous rules) for the first v function and the branching i
2 , . . . , i
h for the second one (the second branching may be absent). We define σn := ηn (2) + ηn (4 + 5, 2) + ηn (6, 2) + ηn (9|2; 2)

(6.1)

which will be studied in the next section where we will bound its sum over n using probabilistic arguments, σn is in fact a truly stochastic term. The difference between ηn and σn is negligible, this being the main result in this section. Let         (3) ω ∈ Ω : sup  An,k ηk (2) ≤ λ2−ζ T (6.2) Bλ,τ :=  n≤nλ (τ ) k 0, there is a constant C = C(τ ) so that for any λ > 0 and ζ small enough |ηn − σn | ≤ Cλ2 T 3/4 (1 + x2n+1,∗ )

on the set Bλ,τ .

(6.7)

Proof. We will call negligible a term which is bounded by the r.h.s. of (6.7). We will next examine one by one all the terms which contribute to ηn − σn and show that they are all negligible, thus proving Proposition 6.2.

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55

Step 1. The terms ηn (1) and ηn (10). We have that ηn (1) = 0; indeed by (3.8), (n) (n)
since gt is self–adjoint and gt m ¯ xn = m ¯
xn , 3
3
(n) ¯ , g v (n) (Tn ) = − m ¯ , v (n) (Tn ) = 0 ηn (1) = − m 4 xn T 4 xn

(6.8)

by (3.2). To show ηn (10) is negligible we use the bound (4.22); since we are in the set Bλ,τ , we get, from (4.7) (n)

sup t∈[Tn ,Tn+1 ]

Γ10 (t)∞ ≤ CT 5/2 λ3−3ζ

hence

|ηn (10)| ≤ CT 5/2 λ3−3ζ . (6.9)

Step 2. The term ηn (7). By (5.3) and (4.51), (n)

Γ7 (t) = I1 + I2 + I3 + I4

(6.10)

where  t n λ (n) 2 I1 := − (xk − xk−1 ) ds gt−s ps−Tk m ¯

xk−1 2 T n k=1  n "  (xk − xk−1 )3 t (n) ds gt−s ps−Tk m ¯


I2 := −λ xk−1 6 T n k=1 #  t (n) + (xk − xk−1 )4 ds gt−s ps−Tk ak−1 n

I3 := −

Tn

9λ 

m ¯ xk−1 , v (k−1) (Tk )m ¯
xk−1 , v (k−1) (Tk ) 16 k=1

I4 := λ

n  t  k=1

Tn

(6.11)

(6.12) 

t Tn

(n)

ds gt−s ps−Tk m ¯
xk−1 (6.13)

  (n) ¯
xk−1 R(v (k−1) (Tk )) . ds gt−s ps−Tk m

(6.14)

To bound I1 we first consider the term with k = n. We write, for t > Tn + 1,   t  t  Tn +1 (n) (n)

ds gt−s ps−Tn m ¯ xn−1 = + ¯

xn−1 . ds gt−s ps−Tn m Tn

Tn

Tn +1

The first integral is bounded by a constant. In the second integral, as well as in the integrals in I1 when k < n, we write ps−Tk m ¯

xk−1 = 2

∂ps−Tk m ¯ xk−1 ∂s

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L. Bertini, S. Brassesco, P. Butt` a and E. Presutti

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and then use (4.17). We thus get recalling the definition (4.3) of δ ∗ (τ ), and using (4.10),    ∗ 2 n  δ (τ ) 1 ∗ 2 √ ≤C I1 ∞ ≤ Cλδ (τ ) log T + λ3 T | log λ| . n−k+1 λ T k=1

−1/2 Since m ¯


¯


and as ak−1 is bounded, using xk−1 ∈ L1 (R, dx), pt m xk−1 ∞ ≤ Ct again (4.10) n  t n   ∗ 3 −1/2 ∗ 4 ds (s − Tk ) + Cλδ (τ ) (t − Tn ) I2 ∞ ≤ Cλδ (τ )

 ≤ C

k=1

∗

δ (τ ) √ λ T

3

Tn

λ3 T 3/2 + C



δ ∗ (τ ) √ λ T

4

k=1

λ3 T 2 .

Recalling the definition (4.2) of V∗ (τ ) and the definition (4.36) of V∗⊥ (τ ), we get    √ V∗ (τ )V∗⊥ (τ ) √ I3 ∞ ≤ CλV∗ (τ )V∗⊥ (τ ) T nλ (τ ) ≤ C λ2 T . 2 λ T Finally, recalling (2.5),  I4 ∞ ≤ Cλ T nλ (τ )V∗ (τ )3 ≤ C



V∗ (τ ) √ λ T

3

λ3 T 3/2 .

√ Since we are √restricting our considerations to the set where δ ∗ (τ ) ≤ λ1−ζ T , V∗ (τ ) ≤ λ1−ζ T and V∗⊥ (τ ) ≤ λ1−ζ , we get √ √ (n) Γ7 (t)∞ ≤ Cλ2−2ζ T hence |ηn (7)| ≤ Cλ2−2ζ T . (6.15) sup t∈[Tn ,Tn+1 ]

Step 3. The term ηn (8). In order to show ηn (8) is negligible (as specified at the beginning of the present proof) we cannot use directly the a priori bounds, but we use equation (5.6) to get ηn (8) =

10 

ηn (8, i) + ηn (8, 3 + 8)

(6.16)

i=1 i=3,8

and show each of the terms on the r.h.s. above are negligible. We have n−1  (1) ηn (8) = An,k ψk k=0

where, recalling 2 = 1, m ¯
,  Tn+1xk 3λ (1) An,k := − ds 4 Tn

(6.17)

Vol. 3, 2002

Front Fluctuations in One Dimensional Stochastic Phase Field Equations

 m ¯
xn , ps−Tk+1 m ¯
xk 

−

 1
1, m ¯ xk  1k 0)

 

57

(6.18)

  
:= λ ds dx dy − y) −  ¯ xk (y) . m  2π(s − Tk+1 )  Tn (6.19) We are going to prove that for any τ > 0 there is a constant C = C(τ ) such that for any 0 ≤ k < n ≤ nλ (τ ) 
    1  (1)  (1) 2 √ + 1k=n−1 . (6.20) An,k  ≤ CBn,k ≤ Cλ 1 + xn+1,∗ T (n − k)3/2 (1) Bn,k



Tn+1





 m ¯
xn (x) ps−Tk+1 (x

1

Proof of (6.20). Let xn,k := xn − xk and define, for any θ ∈ R and t > 0,      e−(y−x−θ)2 /2t − 1    √ f (t, θ) = dx dy m ¯
(x)m ¯
(y)   .   2πt Then, after a change of variables in (6.19), its r.h.s. becomes equal to λ f (s − Tk+1 , xn,k ).

 Tn+1 Tn

ds

The case k < n − 1. After changing variables in the time integral, we get, for k < n − 1,  1   (1) (6.21) Bn,k = λT dt f T (t + n − 1 − k), xn,k . 0

Since |e

−|ξ|




− 1| ≤ |ξ| and m ¯ (x) decays exponentially to 0 as |x| → ∞, we get f (t, θ) ≤ C

1 + θ2 t3/2

(6.22)

and, from (6.21), (1) Bn,k

≤ CλT

−1/2

[1 +

x2n,k ]



1

dt 0

1 (t + n − 1 − k)3/2

which proves (6.20) when k < n − 1. The case k = n − 1. We have, after a change of variables in the time integral,  T (1) Bn,n−1 = λ dt f (t, xn,n−1 ) . 0

Using the inequality f (t, θ) ≤ 4(2πt)−1/2 when t ≤ 1 and (6.22) when t > 1, we get  T  1 1 + x2n,n−1 4 (1) √ + Cλ dt dt ≤ Cλ(1 + x2n,∗ ) . Bn,n−1 ≤ λ t3/2 2πt 0 1

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The proof of (6.20) is concluded. Recall now that ηn (8, i) = −

n−1 3  (1)
An,k m ¯ xk , Γki (Tk+1 ), 4

i = 1, 2, 3 + 8, 4, 5, 6, 7, 9, 10.

k=0

The term ηn (8, 1) in (6.16) vanishes by the same argument as in (6.8). Since (1) we are working in the set Bλ,τ , see (4.37), we have    √  n−2  1 2 √ + 1 λ1−ζ T |ηn (8, 2)| ≤ Cλ 1 + xn+1,∗ T (n − k)3/2 k=0   ≤ Cλ2−ζ 1 + x2n+1,∗

(6.23)

which is negligible for ζ small. To bound ηn (8, 3 + 8) use (6.17), (5.2), and (5.7) to write ηn (8, 3 + 8) = λ

n−1  k=0

(1) An,k

k−1 

 Bk,h ψh ,

Bk,h :=

h=0

Tk+1

Tk

ds m ¯
xk , ps−Th+1 m ¯
xh  . √

(6.24)

We note that, by (5.7), |ψ|n ≤ CVn,∗ ; it is also easy to verify Bk,h ≤ C T (k − h)−1/2 . Plugging this bounds, together with (6.20), into (6.24), we get  √   n−1 |ηn (8, 3 + 8)| ≤ Cλ2 T Vn,∗ 1 + x2n+1,∗ k=0

×




n−1 


k−1 

  1 √ ≤ CλVn,∗ 1 + x2n+1,∗ k−h h=0 k=0

1 √ + 1k=n−1 T (n − k)3/2



 1 √ + 1k=n−1 T (n − k)3/2   ≤ CλVn,∗ 1 + x2n+1,∗ (6.25)

which shows ηn (8, 3 + 8) is negligible. It remains to bound ηn (8, i), i = 4, 5, 6, 7, 9, 10. The negligibility of those terms follows directly from (6.17), (6.20), and the bounds (4.18)–(4.22), (6.15). We thus conclude √   |ηn (8)| ≤ Cλ2−ζ T 1 + x2n+1,∗ . (6.26) Step 4. The term ηn (4 + 5). We claim ηn (4 + 5) =

3λ 4



Tn+1

Tn

ds m ¯
xn , pTn+1 −s v (n) (s)

(6.27)

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Indeed   s 3λ Tn+1 ∂ps−s (n)
ηn (4) = − ds ds
m ¯
xn , v (s ) 4 Tn ∂s
Tn   3λ Tn+1
Tn+1 ∂ps−s (n)
= v (s ) ds ds m ¯
xn , 4 Tn ∂s  s  3λ Tn+1

= ds m ¯ xn , pTn+1 −s v (n) (s
) − ηn (5) 4 Tn which proves (6.27). In order to bound ηn (4 + 5), we use equation (5.6) to get ηn (4 + 5) =

ηn (4 + 5, 2) + ηn (4 + 5, 3, 2) + ηn (4 + 5, 3, 3) 10  + [ηn (4 + 5, i) + ηn (4 + 5, 3, i)] .

(6.28)

i=1 i=2,3

The term ηn (4 + 5, 2) is in σn . Postponing the analysis of ηn (4 + 5, 3, 2) let us first show that ηn (4 + 5, i), i = 2, 3 are negligible. The terms ηn (4 + 5, i), i = 2, 3. We have ηn (4 + 5, 1) =

3λ 4



Tn+1

Tn

(n)

ds m ¯
xn , pTn+1 −s gs−Tn v (n) (Tn )

¯
xn , see (3.2), we can apply (4.11) to deduce that since v (n) (Tn ) is orthogonal to m  |ηn (4 + 5, 1)| ≤ Cλ

Tn+1

ds e−α(s−Tn ) λ1−2ζ

Tn

having used that v (n) (Tn )∞ ≤ Cλ1−2ζ , see (4.47). Thus |ηn (4 + 5, 1)| ≤ Cλ2−2ζ which for ζ small is negligible. We next show ηn (4 + 5, 8) is negligible. We have n−1

ηn (4 + 5, 8) =

3 2 λ ψk 4 k=0

$ ×

(n) m ¯
xn , pTn+1 −t gt−s





Tn+1

t

dt Tn

 ¯
xk ps−Tk+1 m

ds Tn

2 − 2π(s − Tk+1 ) (n)

% .

(6.29)

We now use the decomposition (4.31) for gt−s above. Recalling (6.19), the (n) term obtained by replacing gt−s (x, y) by (3/4)m ¯
xn (x)m ¯
xn (y) in (6.29) is bounded

60

L. Bertini, S. Brassesco, P. Butt` a and E. Presutti

Ann. Henri Poincar´e

by Cλ

n−1  k=0

(1) Bn,k |ψk |

×



Tn+1

Tn

n−1 
k=0

√   dt m ¯
xn , pTn+1 −t m ¯
xn  ≤ C T λ2 Vn,∗ 1 + x2n+1,∗

1 √ + 1k=n−1 T (n − k)3/2



√   ≤ C T λ2 Vn,∗ 1 + x2n+1,∗ (n)

where we used (6.20). We next consider the case when we replace gt−s in (6.29) (n,⊥) by gt−s . Since, by (4.14) and (4.32), (n,⊥) ¯
xk ≤ Ce−α(t−s) ps−tk+1 m ¯
xk ∞ pTn+1 −t gt−s ps−Tk+1 m ∞

1 ≤ Ce−α(t−s)  s − Tk+1 (n)

(n,⊥)

we can bound (6.29) (with gt−s replaced by gt−s ) as n−1 





Tn+1

t

1 ds e−α(t−s)  s − Tk+1 Tn Tn k=0  n−1  
√  √ 1 ≤ C T λ2 Vn,∗ ≤ C λ3−ζ T + λ2−ζ T 1k=n−1 + √ n−k k=0 Cλ2

|ψk |

dt

which is negligible for ζ small enough. It remains to show ηn (4 + 5, i), i = 4, 5, 6, 7, 9, 10 is negligible. This follows from the bounds (4.18)–(4.22), (6.15), and (6.27). The terms ηn (4 + 5, 3, i). We write ηn (4 + 5, 3) =

n−1  k=0

(2)

An,k ψk

(6.30)

where (2) An,k

3λ2 = 4



Tn+1

ds Tn







$

Tn+1

ds s




(n) m ¯
xn , pTn+1 −s gs −s

2

 2π(s

− Tk+1 )

% 1k 1. We now write the ρ term in (4.7) as Vρ(G(E0 , ω, λ) + 1 − t) = ρ (G(E0 , ω, λ) + 1 − t)VG(E0 , ω, λ) = ρ (G(E0 , ω, λ) + 1 − t)(G(E0 , ω, λ) + O(λ)) We note that ρ ≤ 0, and that on suppρ , one has G(E0 , ω, λ) ≤ (−1 + 2κ). So, for λ sufficiently small, we obtain −ρ (G(E0 , ω, λ) + 1 − t) ≤ −

1 2(1 − 2κ)


ω∈Zd 2n+1

ω ˜γ

∂ρ (G(E0 , ω, λ) + 1 − t). (4.8) ∂ωγ

d ρ(x + 1 − t) = −ρ (x + 1 − t), the right With this estimate, and the fact that dt hand side of (4.7) is bounded from above by

−

1 2(1 − 2κ)


γ∈Zd 2n+1



3κ/2

−3κ/2

E{˜ ωγ

∂ tr[ρ(G(E0 , ω, λ) + 1 − t)]}dt. ∂ωγ

(4.9)

In order to evaluate the expectation, we select one random variable, say ωγ (γ ∈ Zd2n+1 ), and first integrate with respect to this variable using hypothesis (H3). Let h0 be the common density of the random variables (ωγ )γ∈Zd . By assumption (H3), −1 there is a decomposition supp(ω0 ) = ∪N l=0 (Ml , Ml+1 ) so that h0 is absolutely ˜ 0 be the function h ˜ 0 (x) := xh0 (x). As h ˜ 0 is continuous on each subinterval. Let h locally absolutely continuous, we can integrate by parts and obtain $ $ $ $ ˜ 0 (ωγ ) ∂ tr{ρ(G(E0 , ω, λ) + 1 − t) − ρ(G(E0 , ω, λ)M0 ,γ + 1 − t)}$ $ dωγ h $ $ ∂ωγ R $ −1  Ml+1 $N
$ ˜ 0 (ωγ ) ∂ tr{ρ(G(E0 , ω, λ) + 1 − t) =$ dωγ h $ ∂ωγ l=0 Ml $ −ρ(G(E0 , ω, λ)M0 ,γ + 1 − t)}$ ˜ 0 ∞ |tr{ρ(G(E0 , ω, λ)MN ,γ + 1 − t) − ρ(G(E0 , ω, λ)M0 ,γ + 1 − t)}| ≤ h ˜ ∞ + h 0

sup |tr{ρ(G(E0 , ω, λ)x,γ + 1 − t) − ρ(G(E0 , ω, λ)M0 ,γ + 1 − t)}|,

x∈supp˜ ω

where G(E0 , ω, λ)x,γ is the operator G(E0 , ω, λ) with the coupling constant ωγ at the γ th -site fixed at the value ωγ = x. As G(E0 , ω, λ) is of rank at most Cnd , we

732

F. Klopp

Ann. Henri Poincar´e

get that, for some C > 0, one has $ $ $ $ ˜ 0 (ωγ ) ∂ tr{ρ(G(E0 , ω, λ) + 1 − t) − ρ(G(E0 , ω, λ)M0 ,γ + 1 − t)}$≤ Cnd . $ dωγ h $ $ ∂ωγ R

Pluging this in turn into (4.9), (4.8), (4.7), and finally (4.6), for λ sufficiently small, we obtain n ˜ 2d ελ−1 . E(#{eigenvalues of Hω,λ in [E0 − ε, E0 + ε]}) ≤ Cn2d κ = Cn

This completes the proof of Proposition 3.1.




5 Appendix 5.1

The dependence of Σλ on λ

In this section, we study the dependence in λ of Σλ (i.e. of E− (λ)) in a neighborhood of a gap of Σ0 . In the case of a nonnegative single site potential V , such a study was done in [17]. As E(0) is simple, there exists a unique Floquet eigenvalue of H, say En (·), such that, for some θ ∈ T∗ , E− (0) = En (θ). Let Z ⊂ T∗ be the set of points θ such that En (θ) = E− (0). This set is discrete by assumption (H1), thus, finite, as T∗ is compact. To fix ideas, set Z = {θj ; 1 ≤ j ≤ nz }. As E− (0) is simple, for 1 ≤ j ≤ nz , the Floquet eigenspace associated to E− (0) and θj is one-dimensional. Hence, by standard analytic perturbation theory (cf [14, 31]), for θ sufficiently close to θj , the Floquet eigenvalue En (θ) is analytic in θ, and one can find a Floquet eigenvector associated to En (θ), say ϕn (x, θ), that is normalized and analytic in θ. Applying now analytic perturbation theory to H λ (θ) (i.e. H λ with θ-periodic boundary conditions) for small λ, we obtain that, there exists δ > 0 and λ0 > 0 such that, for |λ| < λ0 and 1 ≤ j ≤ nz , • in the interval ]E− (0) − δ, E− (0) + δ[, H λ (θ) has a unique Floquet eigenvalue for the Floquet parameter |θ − θj | < δ; let En (θ, λ) be this eigenvalue; • En (θ, λ) is real analytic in (θ, λ) ∈ {|θ − θj | < δ}×]E− (0) − δ, E− (0) + δ[, and, there exists a Floquet eigenvector, say ϕn (x, θ, λ), that is normalized and analytic in (θ, λ); • the rest of the spectrum of H λ (θ) lies outside of ]E0 , E− (0) + δ[. This proves that E(λ) is equal to one of the numbers (En (θ˜j (λ), λ))1≤j≤nz where θ˜j (λ) is unique minimum of En (θ, λ) in {|θ−θj | < δ}. The points θ˜j (λ) are analytic in λ; they satisfy the equation ∇θ En (θ˜j (λ), λ) = 0 and the Hessian matrix of θ → En (θ, λ) is non degenerate in a neighborhood of θj for λ small. This is a

Vol. 3, 2002

Weak Disorder Localization and Lifshitz Tails : Continuous Hamiltonians

733

consequence of perturbation theory as it holds for λ = 0 by assumption. One computes ∂λ En (θ˜j , λ)
λ=0 = * ) ˜ λ)∂λ θ˜j (λ) V ϕn (x, θ˜j (λ), λ), ϕn (x, θ˜j (λ), λ) + ∇θ En (θ(λ),
λ=0  = V (x)|ϕn (x, θj )|2 dx. Rd

(5.1) Hence, assumption (H2) is satisfied if and only if  ∀1 ≤ j ≤ nz , V (x)|ϕn (x, θj )|2 dx = 0 Rd   or ∃1 ≤ j < j ≤ nz , V (x)|ϕn (x, θj )|2 dx · Rd

Rd

V (x)|ϕn (x, θj )|2 dx < 0. (5.2)

For a given operator H, this condition is satisfied for a generic V . The computation (5.1) and the analyticity of the (En (θ˜j (λ), λ))1≤j≤nz implies that, for some 1 ≤ j ≤ nz ,  E(λ) = E− (0) + λ V (x)|ϕn (x, θj )|2 dx + O(λ2 ) (5.3) Rd

On the other hand, using the characterization of Σλ , the almost sure spectrum of Hω,λ in terms of admissible periodic spectra (see [15, 30]), we know that, for λ sufficiently small and t ∈ess-supp(ω0 ), we have 1 (E− (0) + E+ (0)) ≤ E− (λ) ≤ E(λt). 2 Let ω+ and ω− respectively be the essential supremum and infimum of the random variables (ωγ )γ∈Zd . As the random variables are not trivial, using (5.2) and the computation (5.1), one sees that, for some C > 0 and for λ sufficiently small |E(λω) − E(λω+ )| ≥ |λ|/C and |E(λω) − E(λω− )| ≥ |λ|/C. This implies that (0.2) holds for some C > 0 and λ sufficiently small.

5.2

A useful lemma

We prove Lemma 5.1 Pick ε > 0 and u ∈ L2 (Rd ) such that supp(u) ⊂ {|x| ≤ ε}. Then, for any η ∈ (0, 1), there exists u ˜ ∈ L2 (Rd ) such that

734

F. Klopp

Ann. Henri Poincar´e

• ˜ uL2 = uL2 , • u − u ˜L2 ≤ CηuL2 , η , γ ∈ Zd where • u ˆη is constant on each cube Cγ, 2ε

Cγ,r = {x = (x1 , . . . , xd ); −πr ≤ xi − 2πrγi < πr}. Lemma 5.1 is a very simple quantitative version of the Uncertainty Principle. It is the analogue of Lemma 6.2 in [21] developed for the discrete setting. Proof. Pick ε, η and u as in Lemma 5.1. Consider u as a (2ε/η)Zd -periodic function (continue it by 0 on the fundamental cell of (2ε/η)Zd and periodically to the rest of Rd ). Expand it in a Fourier series to get ) η *d 
πη πη u(x) = u ˆγ ei ε γx where uˆγ = u(x)e−i ε γx dx. 2ε Rd d γ∈Z

Parseval’s identity then reads
 2ε d |ˆ uγ |2 . η d

u2L2 =

(5.4)

γ∈Z

Define the function v : Rd → Rd by  v(ξ) =

2ε η

d

d η , γ ∈ Z . u ˆγ for ξ ∈ Cγ, 2ε

˜ be the inverse Fourier transform of v. Check that it By (5.4), v ∈ L2 (Rd ). Let u satisfies the properties stated in Lemma 5.1. First, (5.4) yields ˜ u2L2

1 = v2L2 = (2π)d



2ε2 πη 2

d
 γ∈Zd

C0,

|ˆ uγ |2 dξ = u2L2 . η 2ε

Let u ˆ be the Fourier transform of u; we compute  Rd

2

|ˆ u(ξ) − v(ξ)| dξ =


 γ∈Zd

C0,

η 2ε

$  d $$2 $ 2ε πη $ $ ˆ(ξ + γ ) − uˆγ $ dξ. $u $ $ ε η

On the other hand, we note u ˆ(ξ + γ

πη )− ε



2ε η

d

 u ˆγ =

Rd

u(x)(eixξ − 1)e−i

πη ε γx

dx.

(5.5)

Vol. 3, 2002

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As we did for u, we consider x → u(x)(eixξ − 1) as a (2ε/η)Zd -periodic function; then, Parseval’s identity yields $  d $$2  d 
$$ 2ε 2ε πη $ uˆγ $ = |u(x)(eixξ − 1)|2 dx $uˆ(ξ + γ ) − $ $ ε η η |x|≤ε d

γ∈Z

Substituting in (5.5), we obtain  Rd

that is

2

|ˆ u(ξ) − v(ξ)| dξ =  Rd

 |x|≤ε

2



|u(x)|

|ˆ u(ξ) − v(ξ)|2 dξ ≤ Cη 2



This completes the proof of Lemma 5.1.

|x|≤ε

2ε η



d  |e C0,

ixξ

2

− 1| dξ

dx

η 2ε

|u(x)|2 dx = Cη 2 u2L2 .


References [1] J. M. Barbaroux, J. M. Combes and P. D. Hislop, Localization near band edges for random Schr¨ odinger operators. Helv. Phys. Acta, 70(1-2), 16–43 (1997). Papers honouring the 60th birthday of Klaus Hepp and of Walter Hunziker, Part II (Z¨ urich, 1995). [2] M. Sh. Birman, Perturbations of periodic Schr¨ odinger operators. Lectures given at the Mittag-Leffler Insitute during the programm “Spectral Problems in Mathematical Physics”, 1992. [3] J. M. Combes, P. D. Hislop and E. Mourre, Spectral averaging, perturbation of singular spectra and localization, Transactions of the American Mathematical Society, 348, 4883–4895 (1996). [4] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon. Schr¨ odinger Operators. Springer Verlag, Berlin, 1987. [5] Y Colin de Verdi`ere, Sur les singularit´es de Van Hove g´en´eriques, in Analyse globale et physique math´ematique (Lyon, 1989), volume 46 of M´emoire de la Soci´et´e Math´ematique de France, pages 99–110, 1991. Colloquium en l’honneur d’E. Combet. [6] A. Dembo and O. Zeitouni, Large deviation techniques and applications. Jones and Bartlett Publi-shers, Boston, 1992. [7] J.-M. Deuschel and D. Stroock, Large deviations, volume 137 of Pure and applied Mathematics, Academic Press, 1989. [8] M. Eastham, The spectral theory of periodic differential operators. Scottish Academic Press, Edinburgh, 1973.

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[9] F. Germinet and A. Klein, Bootstrap multiscale analysis and Hilbert-Schmidt dynamical localization, Technical report, UCI, 2001. to appear in Comm. Math. Phys. [10] F. Ghribi. Asymptotique de Lifshitz pour des op´erateurs de Schr¨ odinger a ` champ magn´etique al´eatoire. PhD thesis, Universit´e Paris 13, Villetaneuse. en pr´eparation. [11] P. Hislop, Exponential decay of two-body eigenfunctions: A review. Available on mp-arc, 2001. [12] P. Hislop and F. Klopp. The integrated density of states for some random operators with nonsign definite potentials. To appear in Jour. Func. Anal. [13] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schr¨ odinger operators. Annals of Mathematics 121, 463–494 (1984). [14] T. Kato, Perturbation Theory for Linear Operators. Springer Verlag, Berlin, 1980. [15] W. Kirsch and F. Martinelli, On the spectrum of Schr¨ odinger operators with a random potential, Communications in Mathematical Physics 85, 329–350 (1982). [16] W. Kirsch and B. Simon, Comparison theorems for the gap of Schr¨ odinger operators, J. Funct. Anal. 75(2), 396–410 (1987). [17] W. Kirsch, P. Stollmann and G. Stolz, Localization for random perturbations of periodic Schr¨ odinger operators, Random Oper. Stochastic Equations 6(3), 241–268 1998. [18] M. Klaus, Some applications of the Birman-Schwinger principle, Helv. Phys. Acta 55(1), 49–68 (1982/83). [19] F. Klopp, Localization for some continuous random Schr¨ odinger operators, Communications in Mathematical Physics 167, 553–570 (1995). [20] F. Klopp, Lifshitz tails for random perturbations of periodic schr¨ odinger operators, To appear in the proceedings of the conference “Schr¨ odinger operators”, Goa, Dec. 2000.. [21] F. Klopp, Weak disorder localization and Lifshitz tails, Technical report, Universit´e Paris-Nord, 2001. [22] F. Klopp and J. Ralston, Endpoints of the spectrum of periodic operators are generically simple, Methods and Applications of Analysis 7(3), 459–464 (2000). [23] F. Klopp, Internal Lifshits tails for random perturbations of periodic Schr¨ odinger operators, Duke Math. J. 98(2), 335–396 (1999). [24] F. Klopp, Internal Lifshitz tails for Schr¨ odinger operators with random potentials, To appear in Jour. Math. Phys.

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Weak Disorder Localization and Lifshitz Tails : Continuous Hamiltonians

737

[25] F. Klopp and L. Pastur, Lifshitz tails for random Schr¨ odinger operators with negative singular Poisson potential, Comm. Math. Phys. 206(1), 57–103 (1999). [26] P. Kuchment, Floquet theory for partial differential equations, volume 60 of Operator Theory: Advances and Applications, Birkh¨auser, Basel, 1993. [27] I. M. Lifshitz, Structure of the energy spectrum of impurity bands in disordered solid solutions, Soviet Physics JETP 17, 1159–1170 (1963). [28] I.M. Lifshitz, S.A. Gredeskul and L.A. Pastur, Introduction to the theory of disordered systems, Wiley, New-York, 1988. [29] A. Outassourt, Comportement semi-classique pour l’op´erateur de Schr¨ odinger a potentiel p´eriodique, Journal of Functional Analysis 72, 65–93 (1987). ` [30] L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer Verlag, Berlin, 1992. [31] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol IV: Analysis of Operators, Academic Press, New-York, 1978. [32] B. Simon, Semi-classical analysis of low lying eigenvalues III. width of the ground state band in strongly coupled solids, Annals of Physics 158, 415–420 (1984). [33] J. Sj¨ ostrand, Microlocal analysis for periodic magnetic Schr¨ odinger equation and related questions, In Microlocal analysis and applications, volume 1495 of Lecture Notes in Mathematics, Berlin, 1991. Springer Verlag. [34] P. Stollman, Caught by disorder. Birkh¨ auser, 2001. [35] E.C. Titschmarch, Eigenfunction expansions associated with second-order differential equations. Part II, Clarendon Press, Oxford, 1958. [36] I. Veselic, Localization for random perturbations of periodic Schr¨ odinger operators with regular Floquet eigenvalues, Technical report, Universt¨ at Bochum, 1998. Fr´ed´eric Klopp Universit´e de Paris-Nord D´epartement de Math´ematique Institut Galil´ee U.M.R. 7539 C.N.R.S 99 Avenue J.-B. Cl´ement F-93430 Villetaneuse France email: [email protected] Communicated by Bernard Helffer submitted 12/10/01, accepted 05/02/02

Ann. Henri Poincar´e 3 (2002) 757 – 772 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/040757-16

Annales Henri Poincar´ e

Remarks on Nonlinear Schr¨ odinger Equations with Harmonic Potential R. Carles Abstract. Bose-Einstein condensation is usually modeled by nonlinear Schr¨ odinger equations with harmonic potential. We study the Cauchy problem for these equations. We show that the local problem can be treated as in the case with no potential. For the global problem, we establish an evolution law, which is the analogue of the pseudo-conformal conservation law for the nonlinear Schr¨ odinger equation. With this evolution law, we give wave collapse criteria, as well as an upper bound for the blow up time. Taking the physical scales into account, we finally give a lower bound for the breaking time. This study relies on two explicit operators, suited to nonlinear Schr¨ odinger equations with harmonic potential, already known in the linear setting.

1 Introduction This paper is devoted to existence and blow up results for the nonlinear Schr¨ odinger equation with isotropic harmonic potential,  2 2  i
∂ u
+
∆u
= ω x2 u
+ λ|u
|2σ u
, t 2 2  u
|t=0 = u
0 ,

(t, x) ∈ R+ × Rn ,

(1.1)

where
> 0, λ ∈ R, and ω, σ > 0. The notation x2 stands for |x|2 . Similar equations are considered for Bose-Einstein condensation (see for instance [8], [15], [16]), with σ = 1; the real λ may be positive or negative, according to the considered chemical element, and is proportional to
2 . With the operators used in [3] and [4] (see Eq. (1.3)), we prove existence results which are analogous to the wellknown results for the nonlinear Schr¨odinger equation with no potential (see for instance [7]). These operators simplify the proof of some results of [13], [15] and [16], and provide more general results (in particular, for the case of Bose-Einstein condensation in space dimension three). In addition, we state two evolution laws (Lemma 3.1), which can be considered as the analogue of the pseudo-conformal evolution law of the free nonlinear Schr¨ odinger field, and allow us to prove blow up results. Precisely, if we assume that λ is negative (attractive nonlinearity) and σ ≥ 2/n, then under the condition λ 1 
∇u
0 2L2 + u
2σ+2 2σ+2 ≤ 0, 2 σ+1 0 L

758

R. Carles

Ann. Henri Poincar´e

π the wave collapses at time t
∗ ≤ 2ω (Prop. 3.2). In particular, blow up occurs for focusing cubic nonlinearities (λ < 0 and σ = 1) in space dimensions two and three, but not in space dimension one (for other reasons; see Sect. 2). Sect. 4 is devoted to estimates from below of the time t
∗ , under the  breaking 
2 n assumption that the initial data u0 is bounded in u ∈ L (R ); xu,∇u ∈ L2 (Rn ) , uniformly with respect to
∈]0, 1] (in particular, this means that u
0 is not
oscillatory). We prove that if λ is negative and proportional to
2 , σ = 1 (the physical case), and n = 2 or 3, then the wave collapse time can be bounded from π − Λ
α , for some constant Λ and positive number α (Prop. 4.1). When below by 2ω n = 1, we consider the case of a quintic nonlinearity (σ = 2), which should be the right model for Bose-Einstein Condensation in low dimension (see [12]), and we π − Λ
, for some constant Λ. Notice that all these results are prove that t
∗ ≥ 2ω proved for fixed
, with constants independent of
∈]0, 1]. The following quantities are formally independent of time,

N
=u
(t)2L2 , 1 ω2 λ xu
(t)2L2 + u
(t)2σ+2 E
= 
∇x u
(t)2L2 + L2σ+2 . 2 2 σ+1

(1.2)

If N
and E
are defined at time t = 0, we prove that the solution u
is defined locally in time, with the conservation of N
and E
, provided that σ < 2/(n − 2) when n ≥ 3. If λ ≥ 0, then the solution u
is defined globally in time. If λ < 0, several cases occur. • If σ < 2/n, then the solution is defined globally in time. • If σ ≥ 2/n, then the solution is defined globally in time if u
0 is sufficiently small. • If σ ≥ 2/n and E
≤

ω2
2 2 xu0 L2 ,

then the solution collapses at time t
∗ ≤

π 2ω .

The operators on which our analysis relies are Jj
(t) =

ω xj sin(ωt) − i cos(ωt)∂j ;


Hj
(t) = ωxj cos(ωt) + i
sin(ωt)∂j . (1.3)

We denote J
(t) (resp. H
(t)) the operator-valued vector with components Jj
(t) (resp. Hj
(t)). Lemma 1.1 J
and H
satisfy the following properties. • The commutation relation,     ω2 2 ω2 2
2
2 x = H
(t), i
∂t + ∆ − x = 0. J
(t), i
∂t + ∆ − 2 2 2 2

(1.4)

Vol. 3, 2002

Remarks on Nonlinear Schr¨ odinger Equations with Harmonic Potential x2

759

x2

• Denote M
(t) = e−iω 2
tan(ωt) , and Q
(t) = eiω 2
cot(ωt) , then J
(t) = −i cos(ωt)M
(t)∇x M
(−t),

(1.5)

H
(t) = i
sin(ωt)Q
(t)∇x Q
(−t). • The modified Sobolev inequalities. For n ≥ 2, and 2 ≤ r < by  1 1 − . δ(r) ≡ n 2 r

2n n−2 ,

define δ(r) (1.6)

2n Let 2 ≤ r < n−2 (2 ≤ r ≤ ∞ if n = 1); there exists Cr independent of
such that, for any ϕ ∈ Σ, 1−δ(r)

ϕLr ≤ Cr
−δ(r) ϕL2


J
(t)ϕ)L2 + H
(t)ϕL2

δ(r)

.

(1.7)

• For any function F ∈ C 1 (C, C) of the form F (z) = zG(|z|2 ), we have, π Z, ω π π + Z. J
(t)F (v) = ∂z F (v)J
(t)v − ∂z¯F (v)J
(t)v, ∀t ∈ 2ω ω

H
(t)F (v) = ∂z F (v)H
(t)v − ∂z¯F (v)H
(t)v, ∀t ∈

(1.8)

Remark. Property (1.8) is a direct consequence of (1.5). Property (1.7) is a consequence of the usual Sobolev inequalities and (1.5). These operators are well-known in the linear theory (see e.g. [14] p. 108, [3]), they are the quantization of momentum and position, hence (1.4). Their action in the nonlinear setting, as stated in the above lemma, proves to be very efficient to analyze (1.1). Notations. We work with initial data which belong to the space   Σ := u ∈ L2 (Rn ) ; xu, ∇u ∈ L2 (Rn ) . Notice that Σ = D( −∆ + |x|2 ): we work in the same space as in [13]. The notation r stands for the H¨older conjugate exponent of r. The paper is organized as follows. In Sect. 2, we study the local Cauchy problem for (1.1), and we give sufficient conditions for the solution of (1.1) to be defined globally in time. In Sect. 3, we give a sufficient condition under which the solution blows up in finite time, and provide an upper bound for the breaking time. In Sect. 4, we give a lower bound for the breaking time, that shows that the upper bound underscored in Sect. 3 is the physical breaking time in the semi-classical limit, provided that no rapid oscillation is present in the initial data. The results of Sections 2 and 3 were announced in [6].

760

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Ann. Henri Poincar´e

2 Existence results The solution of (1.1) with λ = 0 is given by Mehler’s formula (see e.g. [9]),  x2 +y2  n/2 

iω ω cos(ωt)−x.y
2
sin(ωt) e u
0 (y)dy =: U
(t)u
0 (x). u (t, x) = 2iπ
sin ωt Rn This formula defines a group U
(t), unitary on L2 , for which Strichartz estimates are available, that is, mixed time-space estimates, which are exactly the same as t
for U0
(t) = ei 2 ∆ . Recall the main properties from which such estimates stem (see [7], or [11] for a more general argument). • The group U
(t) is unitary on L2 , U
(t)L2 →L2 = 1. • For 0 < t ≤

π 2ω ,

the group is dispersive, with U
(t)L1 →L∞ ≤ C|
t|−n/2 .

We postpone the precise statement of Strichartz estimates to Sect. 4. Duhamel’s formula associated to (1.1) reads u
(t, x) = U
(t)u
0 (x) − iλ
−1

 0

t



U
(t − s) |u
|2σ u
(s, x)ds.

Replacing U
(t) by U0
(t) yields Duhamel’s formula associated to  2  i
∂ u +
∆u = λ|u
|2σ u
, t 2  u
|t=0 = u
0 .

(2.1)

The local Cauchy problem for this equation is now well-known in many cases (see for instance [7] for a review). In particular, the local well-posedness in Σ is established thanks to the operators
∇x and x/
+ it∇x (Galilean operator). This result is proved thanks to Strichartz inequalities, and to the following properties. • The above two operators commute with i
∂t +


2 2 ∆.

• They act on the nonlinearity |u
|2σ u
like derivatives. • Gagliardo-Nirenberg inequalities. From Lemma 1.1, the operators H
and J
meet all these requirements. Mimicking the classical proofs for (2.1) easily yields, Proposition 2.1 Let u
0 ∈ Σ. If n ≥ 3, assume moreover σ < 2/(n − 2). Then there exists T
> 0 such that (1.1) has a unique maximal solution u
∈ C([0, T
[, Σ). u
is maximal in the sense that if T
is finite, then u
(t)Σ → ∞ as t ↑ T
. Moreover N
and E
defined by (1.2) are constant for t ∈ [0, T
[.

Vol. 3, 2002

Remarks on Nonlinear Schr¨ odinger Equations with Harmonic Potential

761

Remark. This result was proved in [13], for more general potentials. We want to underscore the fact that in the case of the harmonic potential, there is essentially nothing to prove, when using J
and H
. If λ > 0, the conservations of mass and energy provide a priori estimates on the Σ-norm of u
(t), and prove global existence in Σ. If λ < 0 and σ < 2/n, then the energy E controls the Σ-norm of u
(t). Indeed, from Gagliardo-Nirenberg inequalities (1.7), 1−δ(2σ+2)

u
(t)L2σ+2 ≤ Cu
(t)L2

δ(2σ+2) 
J
(t)u
L2 + H
(t)u
L2 .

Notice that the following identity holds point-wise, |ωxu
(t, x)|2 + |
∇x u
(t, x)|2 = |
J
(t)u
(t, x)|2 + |H
(t)u
(t, x)|2 , and one can rewrite the energy as E
=

1 1 λ 
J
(t)u
2L2 + H
(t)u
2L2 + u
(t)2σ+2 L2σ+2 . 2 2 σ+1

(2.2)

Therefore, using the conservation of mass N
yields 
J
(t)u
2L2 + H
(t)u
2L2 ≤ 2E
+ C(
J
(t)u
L2 + H
(t)u
L2 )nσ , and if σ < 2/n, then the quantity 
J
(t)u
2L2 + H
(t)u
2L2 remains bounded for all times (for any fixed
). Similarly, global existence can be proved for small data. Proposition 2.2 Let u
0 ∈ Σ, and if n ≥ 3, assume σ < 2/(n − 2). Then u
is defined globally in time and belongs to C([0, +∞[, Σ) in the following cases. • λ ≥ 0 (repulsive nonlinearity). • λ < 0 (attractive nonlinearity) and σ < 2/n. • λ < 0, σ ≥ 2/n and u
0 Σ sufficiently small. Remark. In particular, in space dimension one, the solution u
is always globally defined for cubic nonlinearities (σ = 1).

3 Wave collapse Split the energy E
into E1
+ E2
, with 1 λ 
J
(t)u
2L2 + cos2 (ωt)u
(t)2σ+2 L2σ+2 , 2 σ+1 1 λ sin2 (ωt)u
(t)2σ+2 E2
(t) = H
(t)u
2L2 + L2σ+2 . 2 σ+1

E1
(t) =

762

R. Carles

Ann. Henri Poincar´e

Lemma 3.1 The quantities E1
and E2
satisfy the following evolution laws, ωλ dE1
= (nσ − 2) sin(2ωt)u
(t)2σ+2 L2σ+2 , dt 2σ + 2 ωλ dE2
= (2 − nσ) sin(2ωt)u
(t)2σ+2 L2σ+2 . dt 2σ + 2 Remark. This lemma can be regarded as the analogue of the pseudo-conformal conservation law, discovered by Ginibre and Velo ([10]) for the case with no potential (ω = 0). Sketch of the proof. Expanding |
Jj
(t)u
(t, x)|2 yields, |
Jj
(t)u
(t, x)|2 =ω 2 x2j sin2 (ωt)|u
(t, x)|2 +
2 cos2 (ωt)|∂j u
(t, x)|2 +
ωxj sin(2ωt) Im(u∂j u). When differentiating the above relation with respect to time and integrating with respect to the space variable, one is led to computing the following quantities,   ∂t |xj u
(t, x)|2 dx =2
Im xj u
∂j u
,    ω2 λ Im xj u
∂j u
− 2 Im ∂j2 u
|u
|2σ u
, ∂t |∂j u
(t, x)|2 dx = − 2

   2 
ω λ (3.1) ∂t Im (xj u
∂j u
) = |∇x u
|2 + x2 |u
|2 + |u
|2σ+2 2 2

  ω2 Re xj ∂j u
x2 u
−
Re xj ∂j u
∆u
+
 λ
2σ + 2 Re xj ∂j u
|u | u
.
It follows, d dt



|
J
(t)u
(t, x)|2 dx =

ωσλ sin(2ωt) σ+1



|u|2σ+2  2 − 2λ
cos (ωt) Im ∂j2 u|u|2σ u.

Notice that it is sensible that the right hand side is zero when λ = 0; from the commutation relation (1.4), the L2 -norm of J
(t)u
is conserved when λ = 0, since odinger equation. J
(t)u
then solves a linear Schr¨ Finally, the first part of Lemma 3.1 follows from the identity,  d
u (t)2σ+2 = −
(σ + 1) Im |u|2σ u∆u. L2σ+2 dt The second part of Lemma 3.1 follows from the relation E1
+ E2
= E
= cst. The justification of these formal computations relies on a regularizing technique, which can be found for instance in [7], Lemma 6.4.3.


Vol. 3, 2002

Remarks on Nonlinear Schr¨ odinger Equations with Harmonic Potential

763

As an application of this lemma, we can prove wave collapse when E1
(0) ≤ 0. Proposition 3.2 Let u
0 ∈ Σ be nonzero, and if n ≥ 3, assume σ < 2/(n − 2). Assume that the nonlinearity is attractive (λ < 0) and σ ≥ 2/n. Then under the condition 1 λ 
∇u
0 2L2 + u
2σ+2 2σ+2 ≤ 0, 2 σ+1 0 L u
collapses at time t
∗ ≤ π/2ω, π , lim ∇x u
(t)L2 = ∞, ∃t
∗ ≤ and lim u
(t)L∞ = ∞.
2ω t→t
t→t ∗ ∗ Proof. From our assumptions, if u
∈ C([0, T ]; Σ) with T ≤ π/2ω, 1 dE1
E1
(0) = E
− ωxu
0 2L2 ≤ 0, and ≤ 0, ∀t ∈ [0, T ]. 2 dt

(3.2)

So long as ∇x u
remains bounded in L2 , so does xu
. This follows from the conservations of mass and energy, along with Gagliardo-Nirenberg inequality. Assume u
∈ C([0, π/2ω]; Σ). Then letting t go to π/2ω yields

π  1

π 2   ≥ ωxu
,x  , E1 2ω 2 2ω L2 which is impossible from (3.2) and the conservation of the L2 -norm of u
. Thus, there exists t
∗ ≤ π/2ω such that lim ∇x u
(t)L2 = ∞.

t→t
∗

From the conservation of energy, lim u
(t)2σ+2 L2σ+2 = ∞,

t→t
∗

and the last part of the proposition stems from the conservation of mass.




Remark. Notice that the blow up condition also reads ω2 xu
0 2L2 . 2 In term of energy, this means that the blow up occurs for higher values of the Hamiltonian than in the case with no potential, where the similar condition reads E
< 0. This condition was found independently by Zhang [16], in the particular case σ = 2/n. In particular, our approach can treat the case of Bose-Einstein condensation in space dimension three, where the cubic nonlinearity is supercritical (σ = 1 > 2/n = 2/3). E
≤

Corollary 3.3 Assume σ ≥ 2/n, λ < 0. Let v0
∈ Σ. For k ∈ R, define u
0 = kv0
. Then for |k| sufficiently large, u
(t, x) collapses at time t
∗ ≤ π/2ω, as in Prop. 3.2. Proof. For |k| large, E1
(0) becomes negative, and one can use the results of Prop. 3.2.


764

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Ann. Henri Poincar´e

4 Lower bound for the breaking time In this section, we specify the dependence of the coupling constant λ upon physical constants, and assume λ = a
2 . We first assume that the nonlinearity is cubic, σ = 1. Physically, a is the s-wave scattering length. It is negative in the case of Bose-Einstein condensation for 7 Li system ([1], [2]). We prove that if the space dimension n is two or three, then the nonlinear term a
2 |u
|2 u
in (1.1) is negligible in the semi-classical limit
→ 0, up to some time depending on
. This will give us a lower bound for the breaking time t
∗ when
→ 0, and prove that under the assumptions of Prop. 3.2, π . t
∗ −→
→0 2ω As previously noticed, no blow up occurs for σ = 1 and n = 1, that is why we restrict our attention to n = 2 or 3. In the one-dimensional case, it has been proved in [12] that the right model for Bose-Einstein consists in replacing the cubic nonlinearity |u
|2 u
by the quintic nonlinearity |u
|4 u
. This case is critical for global existence issues (see Prop. 2.2, Prop. 3.2), and is treated at the end of this section. Define the function v
as the solution of the linear Cauchy problem,  2 2  i
∂ v
+
∆v
= ω x2 v
, t 2 2 
= u
0 . v|t=0

4.1

(4.1)

The case n = 2 or 3

When n = 2 or 3, recall that we consider now the initial value problem for u
,  2 2  i
∂ u
+
∆u
= ω x2 u
+ a
2 |u
|2 u
, t 2 2 (4.2)  u
|t=0 = u
0 , where a is fixed. Our first result is independent of the sign of a. Proposition 4.1 Assume n = 2 or 3. Let u
0 ∈ Σ be such that u
0 L2 , ∇x u
0 L2 and xu
0 L2 are bounded, uniformly with
∈]0, 1]. Then there exist C, Λ, α > 0 and a finite real q such that the following holds. Let
0 > 0 be such that π/2ω −
α Λ
α 0 > 0. Then for any
∈]0,
0 ], u is defined in Σ at least up to time π/2ω−Λ
, and satisfies  
A (t)(u
− v
)(t) 2 ≤ C
1/q , sup L 0≤t≤π/2ω−Λ
α

where A
(t) can be either of the operators Id, J
(t) or H
(t). In particular, if a < 0 and u
collapses at time t
∗ , then π − Λ
α , ∀
∈]0,
0 ]. t
∗ ≥ 2ω

Vol. 3, 2002

Remarks on Nonlinear Schr¨ odinger Equations with Harmonic Potential

765

Remark. Notice that the assumption ∇x u
0 L2 be bounded uniformly with
means that u
0 has no
-dependent oscillation. This is crucial, for quadratic oscilπ lations could lead for instance to t
∗ = 4ω , see [5]. To prove Prop. 4.1, we first state precisely the Strichartz estimates we will use. Recall the classical definition (see e.g. [7]), 2n (resp. 2 ≤ r ≤ ∞ if n = 1, Definition 1 A pair (q, r) is admissible if 2 ≤ r < n−2 2 ≤ r < ∞ if n = 2) and  1 1 2 = δ(r) ≡ n − . q 2 r

Strichartz estimates provide mixed type estimates (that is, in spaces of the form Lqt (Lrx ), with (q, r) admissible) of quantities involving the unitary group t

U0 (t) = ei 2 ∆ . t


A simple scaling argument yields similar estimates when U0 is replaced with ei 2 ∆ , with precise dependence upon the parameter
. As noticed in Sect. 2, the same t
Strichartz estimates hold when ei 2 ∆ is replaced by U
(t) (provided that only finite time intervals are involved). Proposition 4.2 Let I be a interval contained in [0, π/2ω]. For any admissible pair (q, r), there exists Cr such that for any f ∈ L2 , 
 −1/q U (t)f  q f L2 . r ≤ Cr
L (I;L )

For any admissible pairs (q1 , r1 ) and (q2 , r2 ), there exists Cr1 ,r2 such that for F = F (t, x),      
U (t − s)F (s)ds ≤ Cr1 ,r2
−1/q1 −1/q2 F Lq2 (I;Lr2 ) . (4.3)   q  I∩{s≤t} r L

1 (I;L 1 )

The above constants are independent of I ⊂ [0, π/2ω] and
∈]0, 1]. We now state two technical lemmas on which the proof of Prop. 4.1 relies. The first one is easy, and we leave out the proof. Lemma 4.3 If n = 2 or 3, there exists q, r, s and k satisfying  1 1 2    r = r + s , 1 1 2     = + , q q k and the additional conditions:

(4.4)

766

R. Carles

Ann. Henri Poincar´e

• The pair (q, r) is admissible, • 0
0 such that for any T < π/2ω, and any 0 ≤ t ≤ T ,  

F (a )(t) r ≤

L

π 2ω


 C0 2δ(s) a (t)Lr . −t

Then there exist C, Λ > 0 independent of
∈ [0, 1[ such that the following holds. Let
0 > 0 be such that π/2ω − Λ
α 0 > 0. Then for any
∈]0,
0 ],   a
(t)L2 ≤ C
1−1/q S
Lq (0,π/2ω−Λ
α ;Lr ) , sup π 0≤t≤ 2ω −Λ
α

where α =

1 kδ(s)−1 .

Proof of Lemma 4.4. From (4.3) with q1 = q2 = q, for any t < π/2ω, a
Lq (0,t;Lr ) ≤ C
1−2/q S
Lq (0,t;Lr ) + C
1−2/q F
(a
)Lq (0,t;Lr ) .

(4.5)

From our assumptions,





F (a )Lq (0,t;Lr )

   ≤ 



π 2ω

  C0 
r a (s) . 2δ(s) Lx  q −s L (0,t)

Apply H¨ older’s inequality in time with (4.4), F
(a
)Lq (0,t;Lr ) ≤ C



≤ C

0

π 2ω

t



ds π 2ω

−t

−s

2/k kδ(s)

a
Lq (0,t;Lr )

1
2δ(s)−2/k a Lq (0,t;Lr ) .

Plugging this estimate into (4.5) yields, for t ≤ π/2ω − Λ
α , a
Lq (0,t;Lr ) ≤ C
1−2/q S
Lq (0,t;Lr ) + C
1−2/q (Λ
α )2/k−2δ(s) a
Lq (0,t;Lr ) .

Vol. 3, 2002

Remarks on Nonlinear Schr¨ odinger Equations with Harmonic Potential

767

1 From (4.4), the power of
in the last term is canceled for α = kδ(s)−1 . If in addition Λ is sufficiently large, the last term of the above estimate can be absorbed by the left hand side (up to doubling the constant C for instance),

a
Lq (0,t;Lr ) ≤ C
1−2/q S
Lq (0,t;Lr ) . The last three estimates also imply, F
(a
)Lq (0,t;Lr ) ≤ CS
Lq (0,t;Lr ) .

(4.6)

The lemma then follows from Prop. 4.2, (4.3), with this time q1 = ∞ and q2 = q, along with (4.6).
Proof of Proposition 4.1. Denote w
= u
− v
the remainder we want to assess. It solves the initial value problem,  2 2  i
∂ w
+
∆w
= ω x2 w
+ a
2 |u
|2 u
, t 2 2 (4.7) 
= 0. w|t=0 We first want to apply Lemma 4.4 with a
= w
. Since u
= v
+ w
, we can take F
(w
) = a|u
|2 w
,

S
= a|u
|2 v
.

The point is now to control the Ls -norm of u
. Notice that we can easily control the Ls -norm of v
. Indeed, as we already emphasized, for any time t, v
(t)L2 = u
0 L2 ,

J
(t)v
L2 = ∇u
0 L2 .

From Lemma 1.1, (1.5), and Gagliardo-Nirenberg inequality, we also have, C 1−δ(s) δ(s) v
(t)L2 J
(t)v
L2 δ(s) | cos(ωt)| C 1−δ(s) δ(s)
≤

J
(t)v
L2 . δ(s) v (t)L2 π 2ω − t

v
(t)Ls ≤

Therefore, the assumptions of Prop. 4.1 imply that there exists C0 > 0 independent of
such that for any t < π/2ω, v
(t)Ls ≤

C0 π 2ω

−t

δ(s) .


= 0 and we know from Prop. 2.1 that there exists T
such that the Now w|t=0 Σ-norm of w
is continuous on [0, T
]. In particular, there exists t
> 0 such that the following inequality,

w
(t)Ls ≤

C0 π 2ω

−t

δ(s) ,

(4.8)

768

R. Carles

Ann. Henri Poincar´e

holds for t ∈ [0, t
]. So long as (4.8) holds, we have obviously u
(t)Ls ≤

2C0 δ(s) . −t

π 2ω

This estimate allows us to apply Lemma 4.4, which yields, along with (4.4), and provided that t ≤ π/2ω − Λ
α , w
L∞ (0,t;L2 ) ≤ C
1−1/q |u
|2 v
Lq (0,t;Lr ) ≤ C
1−1/q u
2Lk (0,t;Ls ) v
Lq (0,t;Lr )

(4.9)

≤ CΛ−2/k
1/q . Now apply the operator J
to (4.7). From Lemma 1.1, J
w
solves the same equation as w
, with |u
|2 u
replaced by J
(|u
|2 u
). From (1.8), |J
(t)(|u
|2 u
)(t, x)| ≤ 4|u
(t, x)|2 |J
(t)u
(t, x)|. Writing J
u
= J
v
+ J
w
and proceeding as above yields, so long as (4.8) holds, (4.10) J
w
L∞ (0,t;L2 ) ≤ CΛ−2/k
1/q . Combining (4.9) and (4.10), along with Gagliardo-Nirenberg inequality, yields, so long as (4.8) holds, w
(t)Ls ≤ C

1 π 2ω

−t

δ(s) Λ

−2/k 1/q




.

(4.11)

Possibly enlarging the value of Λ, (4.11) shows that (4.8) remains valid up to time π/2ω − Λ
α . This proves Prop. 4.1 when A
(t) = Id or J
(t), from (4.9) and
(4.10). The case A
(t) = H
(t) is then an easy by-product.

4.2

The case n = 1

We finally prove the analogue of the above results in space dimension one. When n = 1, one can do without Strichartz estimates, and simply use the Sobolev embedding H 1 ⊂ L∞ , 1/2 1/2 f L∞ ≤ Cf L2 ∂x f L2 . The wave u
now solves  2 2  i
∂ u
+
∂ 2 u
= ω x2 u
+ a
2 |u
|4 u
, t 2 x 2  u
|t=0 = u
0 . We start with the analogue of Lemma 4.4.

(4.12)

Vol. 3, 2002

Remarks on Nonlinear Schr¨ odinger Equations with Harmonic Potential

769

Lemma 4.5 Assume n = 1, and let a
∈ C(0, T ; Σ) defined for some positive T , solution of  2 2  i
∂ a
+
∂ 2 a
= ω x2 a
+
2 F
(a
) +
2 S
, t x 2 2 (4.13)  a
|t=0 = 0. Assume that there exists C0 > 0 such that for any T < π/2ω, and any 0 ≤ t ≤ T , 

 F (a )(t) 2 ≤

L

C0 π 2ω

−t


 2 a (t)L2 .

Then there exists C > 0 independent of
∈ [0, 1[ such that for any Λ ≥ 1, the following holds. Let
0 > 0 be such that π/2ω − Λ
0 > 0. Then for any
∈]0,
0 ],  π/2ω−Λ

 S (t) 2 dt. sup a
(t)L2 ≤ C
L π 0≤t≤ 2ω −Λ


0

Proof. Multiply (4.13) by a
, integrate with respect to x, and take the imaginary part of the result. This yields, from Cauchy-Schwarz inequality, d
a (t)L2 ≤ 2
F
(a
)(t)L2 + 2
S
(t)L2 dt  2C0
 a
(t) 2 + 2
S
(t)L2 . ≤

 2 L π 2ω − t The lemma then follows from the Gronwall lemma.




We can now prove the analogue of Prop. 4.1. Proposition 4.6 Assume n = 1. Let u
0 ∈ Σ be such that u
0 L2 , ∂x u
0 L2 and xu
0 L2 are bounded, uniformly with
∈]0, 1]. Then there exist C, Λ > 0 such that the following holds. Let
0 > 0 be such that π/2ω − Λ
0 > 0. Then for any
∈]0,
0 ], u
is defined in Σ at least up to time π/2ω − Λ
, and satisfies 
 A (t)(u
− v
)(t) 2 ≤ C, sup L 0≤t≤π/2ω−Λ


where A
(t) can be either of the operators Id, J
(t) or H
(t).In particular, if a < 0 and u
collapses at time t
∗ , then π − Λ
, ∀
∈]0,
0 ]. t
∗ ≥ 2ω Proof. The proof follows the proof of Prop. 4.1 very closely, if we take q = ∞, (s, k) = (∞, 4). Denote w
= u
− v
the remainder we want to assess. It solves the initial value problem,  2 2  i
∂ w
+
∂ 2 w
= ω x2 w
+ a
2 |u
|4 u
, t x 2 2 
w|t=0 = 0.

770

R. Carles

Ann. Henri Poincar´e

We first want to apply the above lemma with a
= w
. Since u
= v
+ w
, we can take F
(w
) = a|u
|4 w
, S
= a|u
|4 v
. The point is now to control the L∞ -norm of u
. Notice that we can easily control the L∞ -norm of v
. Indeed, as we already emphasized, for any time t, v
(t)L2 = u
0 L2 ,

J
(t)v
L2 = ∂x u
0 L2 .

From Lemma 1.1, (1.5), and Gagliardo-Nirenberg inequality, we also have, C 1/2 1/2 v
(t)L2 J
(t)v
L2 | cos(ωt)|1/2 C 1/2


1/2 ≤

1/2 v (t)L2 J (t)v L2 . π 2ω − t

v
(t)L∞ ≤

Therefore, the assumptions of Prop. 4.6 imply that there exists C0 > 0 independent of
such that for any t < π/2ω, v
(t)L∞ ≤

C0 π 2ω

−t

1/2 .

So long as w
(t)L∞ ≤

C0 π 2ω

−t

1/2 ,

(4.14)

holds, we have obviously u
(t)L∞ ≤

π 2ω

2C0 1/2 . −t

This estimate allows us to apply the above lemma, which yields, provided that t ≤ π/2ω − Λ
, w
L∞ (0,t;L2 ) ≤ C
|u
|4 v
L∞ (0,t;L2 ) ≤ C
u
2L4 (0,t;L∞ ) v
L∞ (0,t;L2 ) ≤ CΛ

−1

(4.15)

.

Similarly, applying the operator J
to (4.7) yields, so long as (4.8) holds, J
w
L∞ (0,t;L2 ) ≤ CΛ−1 .

(4.16)

Combining (4.15) and (4.16), along with Gagliardo-Nirenberg inequality, yields, so long as (4.14) holds, w
(t)L∞ ≤ C

1 π 2ω

−t

1/2 Λ

−1

.

(4.17)

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Remarks on Nonlinear Schr¨ odinger Equations with Harmonic Potential

771

Taking Λ large enough, (4.17) shows that (4.14) remains valid up to time π/2ω−Λ
. This proves Prop. 4.6 when A
(t) = Id or J
(t), from (4.15) and (4.16). The case A
(t) = H
(t) is then an easy by-product.
Acknowledgement. The results in this paper were improved thanks to remarks made by T. Colin.

References [1] C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet, Evidence of Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions, Phys. Rev. Lett. 75, 1687–1690 (1995). [2] C. C. Bradley, C. A. Sackett and R. G. Hulet, Bose-Einstein Condensation of Lithium: Observation of Limited Condensate Number, Phys. Rev. Lett. 78, 985–989 (1997). ´ [3] R. Carles, Equation de Schr¨ odinger semi-classique avec potentiel harmonique et perturbation non-lin´eaire, S´eminaire X-EDP, 2001–2002, Exp. No. III, ´ 12p., Ecole Polytechnique, Palaiseau, (2001). [4] R. Carles, Semi-classical Schr¨odinger equations with harmonic potential and nonlinear perturbation, preprint, to appear in Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 2001. [5] R. Carles, Critical nonlinear Schr¨ odinger equations with and without harmonic potential, to appear in Math. Mod. Meth. Appl. Sci., 2002. [6] R. Carles, Remarques sur l’´equation de Schr¨odinger non lin´eaire avec potentiel harmonique, Comptes Rendus de l’Acad´emie des Sciences. S´erie I. Math´ematique 334, 763–766 (2002). [7] T. Cazenave, An introduction to nonlinear Schr¨ odinger equations, Text. Met. Mat. 26, Univ. Fed. Rio de Jan., (1993). [8] C. Cohen-Tannoudji, Cours du Coll`ege de France, 1998–99, available at www.lkb.ens.fr/˜laloe/PHYS/cours/college-de-france/ . [9] R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (International Series in Pure and Applied Physics), Maidenhead, Berksh.: McGraw-Hill Publishing Company, Ltd., 365 p., 1965. [10] J. Ginibre and G. Velo, On a class of nonlinear Schr¨ odinger equations. II Scattering theory, general case, J. Funct. Anal. 32, 33–71 (1979). [11] M. Keel and T. Tao, Endpoint Strichartz Estimates, Amer. J. Math. 120, 5, 955–980 (1998).

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[12] E. B. Kolomeisky, T. J. Newman, J. P. Straley and X. Qi, Low-Dimensional Bose Liquids: Beyond the Gross-Pitaevskii Approximation, Phys. Rev. Lett. 85, 6, 1146–1149 (2000). [13] Yong-Geun Oh, Cauchy problem and Ehrenfest’s law of nonlinear Schr¨ odinger equations with potentials, J. Diff. Eq. 81, 2, 255–274 (1989). [14] Walter Thirring, A course in mathematical physics. Vol. 3, Springer-Verlag New York, 1981, Quantum mechanics of atoms and molecules, Translated from the German by Evans M. Harrell, Lecture Notes in Physics 141, MR 84m:81006. [15] Takeya Tsurumi and Miki Wadati, Stability of the D-dimensional nonlinear Schr¨ odinger equation under confined potential, J. Phys. Soc. Japan 68, 5, 1531–1536 (1999). [16] Jian Zhang, Stability of attractive Bose-Einstein condensates, J. Statist. Phys. 101, n 3-4, 731–746 (2000). R´emi Carles Math´ematiques Appliqu´ees de Bordeaux UMR 5466 CNRS 351 cours de la Lib´eration F-33405 Talence cedex France email: [email protected] Communicated by Rafael D. Benguria submitted 04/12/01, accepted 21/05/02

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 3 (2002) 773 – 792 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/040773-20

Annales Henri Poincar´ e

The Wave Function of the Lyman–Alpha Photon Part I. The Wave Function in the Angular and Linear Momentum Representations H. E. Moses Abstract. To the best of the writer’s knowledge no one has given the wave function of a photon emitted in an atomic, molecular, or nuclear transition. In the present paper we derive the wave function in the angular momentum and linear momentum representations for the photon emitted by a non-relativistic hydrogen atom, when the electron of the atom falls from the first excited state to the ground state. This is the simplest transition which produces a photon. A two level model for the atom is used, in which the lower level (the ground state energy) is associated with a nondegenerate wave function and the upper level (the energy of the first excited state) is associated with wave functions corresponding to the four–fold degeneracy of the that state). We use a generalization of Dirac’s method for finding the eigenfunctions in resonance scattering. We find the exact solution of the two-level problem using the exact matrix elements of the interaction. The calculations are finite without renormalization. In the next paper we shall introduce the
x–representation and thereby obtain the “position”, “shape”, and “trajectory” of the photon.

1 Introduction As far as the author is aware, no one has ever given the wave function of a photon emitted in an atomic transition or other process which releases a photon. Possibly part of the problem is the fact that the description of the photon, though known from Wigner’s monumental work [1], seems not to be used generally. We have introduced the wave function of the photon in an angular momentum basis [2, 3] to find the exact electromagnetic matrix elements for the hydrogen atom and to obtain from first principles the resonance scattering cross section of Lyman–α radiation from a hydrogen atom in the ground state. Since angular momentum is conserved in atomic transitions, this basis for the photon wave functions is a natural one for calculations. We use a generalization of the Dirac [4] procedure1 for finding the exact eigenvectors of the truncated Hamiltonian, which Dirac uses to describe resonance scattering. Friedrichs [5] has studied a simple model of the Dirac procedure in a mathematically rigorous fashion. He has obtained the Wigner–Weisskopf exponential decay as a limit of vanishing interaction. Closely related problems have been treated in [6]. The author is grateful to the referee for pointing out these papers. The approach in these papers is closer to 1 Dirac

uses a somewhat different language than ours to describe his procedure.

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the traditional one than ours. We believe our results are more explicit and avoid some hazards.

2 The Hamiltonian of the Hydrogen Atom Interacting with the Photon Field 2.1

The Full Hamiltonian

The full Hamiltonian of the non–relativistic atom interacting with the photon field  x), which we take to be in the Coulomb gauge and whose vector potential is A( hence such that  · A(  x) = 0 , ∇ (1) is 1 H= 2m




2 ¯ h e e2 ∇ − A(x) − + Hp , i c r

(2)

   x) 2 and where Hp is the Hamiltonian of the free photon field. On ignoring A( using Eq. (1) H

i¯h  ¯ 2 2 e2 h  ∇ − + Hp + A( x) · ∇ 2m r m = H0 + HI ,

= −

(3)

where H0 is the unperturbed Hamiltonian operating in the direct product space of the hydrogen Hamiltonian HH = −

¯2 2 h e2 ∇ − 2m r

(4)

and the free photon field whose Hamiltonian is Hp . Thus H 0 = HH + H p .

(5)

 x) · ∇  is the interaction between the hydrogen atom The operator HI = (i¯ h/m)A( and the photon field. We shall simplify the interaction HI by restricting the number of matrix elements of HI between eigenstates of H0 . Unlike most workers in the field, we shall use the exact matrix elements of [2, 3] instead of the approximate elements of the long wave–length limit. The use of the exact matrix elements leads to finite results instead of the infinite results of the approximate elements. 2.1.1 The Free Photon Field. Questions of Gauge The free photon field2 , based on Wigner’s definition of a photon as a relativistic massless particle of spin 1, is treated in great detail [2, 3]. We shall briefly review 2 In our treatment, as in the treatments of atomic transitions by most other workers, we ignore the contribution to the photon field due to the electron as a source. This contribution leads to a self–energy problem whose contribution should be small.

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The Wave Function of the Lyman–Alpha Photon. Part I

775

some properties of the quantized field in the angular momentum–energy basis.  x; t) We shall work for the while in the Heisenberg picture where the operators E(  x; t) are time–dependent Hermitian operators. We introduce the operator and H( G(E, j, m, λ) and its Hermitian adjoint G∗ (E, j, m, λ) as annihilation and creation operators which satisfy the usual commutation rules   G(E, j, m, λ), G(E  , j  , m , λ ) = 0  ∗  G (E, j, m, λ), G∗ (E  , j  , m , λ ) = 0   (6) G(E, j, m, λ), G∗ (E  , j  , m , λ ) = E δ(E − E  ) δj,j
δm,m
δλ,λ
. The eigenvalue λ = ±1 describes the circular polarization of the photon: λ = −1 refers to a photon whose circular polarization is in the direction of propagation, while λ = 1 describes a photon whose circular polarization is opposite to the direction of propagation. The quantity |λ| = 1 is the spin of the photon. The eigenvalue E is in the continuous spectrum 0 < E < ∞ and gives the energy of the photon. The discrete variable j describes the total angular momentum of the photon, while m is the z– component of the angular momentum or “magnetic” quantum number of the photon. The ranges of these eigenvalues are: j = 1, 2, · · ·, and m = −j, −j + 1, −j + 2, · · · j − 1, j.  x; t) and H(  x; t) are required to transform in the usual The quantized fields E( relativistic fashion and satisfy Maxwell’s equations  × H(  x; t) = ∇  × E(  x; t) = ∇  x; t) = 0 ∇ · H(

,

 x; t) 1 ∂ E( c ∂t  x; t) 1 ∂ H( − c ∂t  x; t) = 0 . ∇ · E(

(7)

Then the only Hermitian solutions of Maxwell’s equations, which satisfy the usual commutation rules and the relativistic transformations rules, are obtained as follows. Define the operator Akmλ,n (r; t) by  ∞ 
E 1 r e−i(Et/¯h) . Akmλ,n (r; t) = dE G(E, k, m, λ)jn (8) hc ¯ ¯hc 0

In Eq. (8) jn (r) is the usual spherical Bessel function. Let us now define the  1 (x; t) as follows: operator A  1 (x; t) A

∞ k

√ kkm (θ, φ) Akmλ,k (r; t) = − 2 ik Y



λ=±1 k=1 m=−k

k Yk,k+1,m (θ, φ) Akmλ,k+1 (r; t) − iλ 2k + 1   k+1  Yk,k−1,m (θ, φ)Akmλ,k−1 (r; t) . + iλ 2k + 1

(9)

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j,k,m (θ, φ) are the usual vector spherical harmonics in the In Eq. (9) the vectors Y notation of [7], for example. The angles θ and φ are the spherical polar angles of x as usually defined. x = r(sin θ cos φ, sin θ sin φ, cos θ) .  0 (x; t) is defined by The Hermitian operator A  0 (x; t) = A  1 (x; t) + A  ∗1 (x; t). A

(10)

 ·A  0 (x; t) = 0. ∇

(11)

It is seen that Then the the only Hermitian operator solution of Maxwell’s equations which transform relativistically is (within unitary transformations)  x; t) = E(  x; t) = H(

 0 (x; t) 1 ∂A c ∂t  ×A  0 (x; t) . ∇ −

(12)

 0 (x; t) is a vector potential in the Coulomb From Eq. (11) and (12) it is clear that A gauge. In this gauge the scalar potential is zero. It behooves us now to find the  x; t) general vector and scalar (operator) potentials. The general vector potential A( and scalar potential V (x; t) are required to be Hermitian solutions of the equations  x; t) = E(  x; t) = H(

 x; t) 1 ∂ A(  (x; t) − ∇V c ∂t  × A(  x; t) , ∇ −

(13)

 x; t) and H(  x; t) satisfy Maxwell’s equations. The general vector and where E( scalar potentials are  x; t) A( V (x; t)

 0 (x; t) + ∇F  (x; t) = A 1 ∂F (x; t) = − , c ∂t

(14)

where F (x; t) is any real function or Hermitian operator. It sets the gauge. The vector potential A0 is the minimal vector potential needed to obtain a solution of Maxwell’s equations. For the general Coulomb gauge, F (x; t) is independent of t and satisfies Laplace’s equation ∇2 F (x) = 0. If the vector and scalar potentials are to satisfy the Lorentz condition, then F (x; t) satisfies the wave equation ∇2 F (x; t) −

1 ∂ 2 F (x; t) = 0. c2 ∂t2

The vector and scalar potentials in any gauge can be represented as in Eq. (14).  0 (x; t). We shall show later that we need deal only with the vector potential A

Vol. 3, 2002

The Wave Function of the Lyman–Alpha Photon. Part I

777

We shall now give the field Hamiltonian Hp . The classical Hamiltonian of the electromagnetic field is  

1  x; t)2 .  x; t)2 + H( (15) dx E( Hp = 8π As most other investigators do, we take this definition also for the quantized fields.  x; t) and H(  x; t), the field Hamiltonian Despite the appearance of the time t in E( operator Hp is independent of time. From Eq. (12)–(14), using normal re-ordering of annihilation and creation operators, we obtain the field Hamiltonian as Hp =

∞ j ∞

dE G∗ (E, j, m, λ)G(E, j, m, λ) .

(16)

λ=±1 j=1 m=−j 0

It will be convenient to use the symbol s for the set {E, j, m, λ}: s → {E, j, m, λ} , and



(17)

ds to denote the sums and integrals 

∞ j ∞ dE ··· . ds · · · = E j=1 m=−j λ=±1

(18)

0

Later we shall use the symbol Ep for E when only one photon is present to distinguish the energy of the photon from the energy of the system. It is also useful to have a notation when there are sets of variables {Ei , ji , mi , λi } labeled by an index i. We use the symbol si to indicate the set: 

si dsi · · ·

→ {Ei , ji , mi , λi } =

∞ ji ∞ dEi ··· . Ei m j

λi =±1

i=1

i=−ji

(19)

0

Moreover, if we are dealing with a function or operator f (Ei , ji , mi , λi ), we shall write 

f (si ) ≡ f (Ei , ji , mi , λi ) dsi f (si ) ≡

∞

ji

∞

λi =±1 ji=1 mi =−ji 0



Thus Hp =

dEi f (Ei,ji ,mi ,λi ) , Ei

ds E G∗ (s)G(s) .

(20)

(21)

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Ann. Henri Poincar´e

We also introduce the (total) number operator  N= ds G∗ (s)G(s) .

(22)

From the commutation rules Eq. (6), the expansion Eq. (9), Eq. (10), and Eq. (12), we can obtain the usual commutation rules for the components of the electric and magnetic fields and potentials [8]. 2.1.2 The Free Photon Field. The Hilbert Space We now specify the Hilbert space Hp upon which the operators G(s), Hp , N act. We shall use the Fock realization3 . One assumes the existence of a normalizable vacuum state |V > for which G(s)|V > = 0 , < V |V > = 1

(23)

N |V > = 0 , Hp |V > = 0 .

(24)

It follows that Thus |V > is an eigenstate of both N and Hp with the point eigenvalue 0. In addition there are non-normalizable n–particle states4 which with |V > span the space Hp . Such states are denoted by |E1 , j1 , m1 , λ1 ; E2 , j2 , m2 , λ2 ; · · · ; En , jn , mn , λn >≡ |s1 , s2 , · · · , sn > . They are defined as |s1 , s2 , · · · , sn >=

n

G∗ (si )|V > ,

n = 1, 2, · · · .

(25)

i=1

The asterisk now means Hermitian adjoint instead of complex conjugate, as before. The kets are symmetric in the arguments si . To show how the operators G(s) and G∗ (s) act in this basis, we shall define the symmetrization operator Sy acting on a function or operator f (s1 , s2 , · · · , sn ) by 1 Sy f (s1 , s1 , · · · , sn ) = P f (s1 , s2 , · · · , sn ) , (26) s 1 , s2 · · · , sn n! P

where the right– hand side of Eq. (26) means the sum over all permutations of the arguments s1 , s2 , · · · , sn . Then G∗ (s)|s1 , s2 , · · · , sn > G(s)|s1 , s2 , · · · , sn > 3 In

= |s1 , s2 , · · · , sn , s > Sy = E δ(s, sn )|s1 , s2 , · · · , sn−1 > , (27) s 1 , s2 , · · · , sn

K. O. Friedrichs, Mathematical Aspects of the Quantum Theory of Fields, Interscience, New York (1953), it is shown that the Fock realization is equivalent to the usual occupation number representation and oscillator representation even in a more general context than “quantization of fields in a box.” 4 These states are not in H . Suitable superpositions, however, are. p

Vol. 3, 2002

The Wave Function of the Lyman–Alpha Photon. Part I

In Eq. (27)

779

δ(s, s ) = δ(E − E  )δjj
δmm
δλλ
.

(28)

Moreover the kets satisfy the completeness and orthonormality conditions |V >< V | +

∞ 



 ds1

ds2 · · ·

dsn |s1 , s2 , · · · , sn >< s1 , s2 , · · · , sn | = Ip

n=1

and < V |s1 , s2 , · · · , sn >


= 0 for any n = δn,m E1 E2 · · · En × δ(s1 , s1 )δ(s2 .s2 ) · · · δ(sn , sn ) .

(28a)

In Eq.(28a) Ei is the energy variable in si and Ip is the identity operator in photon space. One sees that N |s1 , s2 , · · · , sn > =

n|s1 , s2 , · · · , sn > n ( Ei )|s1 , s2 , · · · , sn > .

Hp |s1 , s2 , · · · , sn > =

(29)

i=1

Thus the kets |s1 , s2 , · · · , sn > are eigenkets of N and Hp . 2.1.3 Eigenstates of the Unperturbed Hydrogen Hamiltonian The unperturbed Hamiltonian HH for the hydrogen atom is given by Eq. (4). We shall denote the eigenstates of HH corresponding to point eigenvalues by |nH , jH , mH >, where nH , (nH = 1, 2 · · ·) is the principal quantum number, jH , (0 ≤ jH < nH ) is the quantum number describing the total angular momentum, and mH is the quantum number that describes the z–component of the angular momentum, (−jH ≤ mH ≤ jH ). These eigenfunctions are nomalizable and are usually normalized to unity.








< nH , jH , mH |nH , jH , mH >= δnH ,n
δjH ,j
δmH ,m
. H

H

H

(30)

The energy associated with the principal quantum number nH will be denoted by E(nH ). Thus HH |nH , jH , mH >= E(nH )|nH , jH , mH > . (31) The continuous spectrum lies above the discrete spectrum. Its eigenstates are not normalizable. They are labeled by the energy EH , (EI < EH < ∞) where EI is the ionization energy, jH , (0 ≥ jH < ∞), mH , (−jH ≤ mH ≤ jH ) The (improper) eigenstates are denoted by |EH , jH , mH > they are orthogonal to the eigenstates corresponding to discrete values of the energy.

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2.1.4 Elimination of the Gauge Terms We now go back to the complete Hamiltonian for the hydrogen atom Eq. (2) and write the Schr¨ odinger equation for the combined hydrogen–photon system. The Schr¨ odinger equation is:   2
h ¯ e ¯h ∂ 1 ∇ − A(x; t) + eV ( ψ(x; t) , (32) q x; t) ψ(x; t) = − 2m i c i ∂t  0 and the gauge where the vector and scalar potentials are given in terms of A function F (x; t) by Eq. (14). Let us introduce the wave function Ψ(x; t) through ψ(x; t) = ei(e/¯hc)F (x;t) Ψ(x; t) .

(33)

 Then Ψ(x; t) satisfies the Schr¨odinger equation Eq. (32) with V (x; t) ≡ 0 and A  replaced by A0 . The gauge is eliminated by a unitary transformation. Thus we can work with the wave function Ψ(x; t) and, having solved for it, can find the solution in any other gauge5 . Henceforth, we shall assume that we are working in a gauge–free representation for which the Hamiltonian is given by Eq.(3)-(5)  0 (x))2 –term6 , the Hamiltonian H is given by which, after ignoring the (A H = H0 + H I .

(34)

The perturbation HI is given by HI =

i¯ h   . A0 (x) · ∇ mc

(35)

3 The Eigenfunctions of the Hamiltonian H 3.1

The Eigenfunctions of H0

The eigenfunctions of the unperturbed Hamiltonian H0 are the direct product of the eigenfunctions of the hydrogen Hamiltonian HH and the unperturbed photon Hamiltonian Hp . Using Dirac’s notation for such products, we denote the eigenfunctions of H0 for which the hydrogen atom is in a bound state by |nH , jH , mH , s1 , s2 , · · · , sn >= |nH , jH , mH > |s1 , s2 , · · · , sn > ,

(36)

when there are n photons and |nH , jH , mH , V >= |nH , jH , mH > |V > ,

(37)

5 Possibly one can find the significance of the gauge by performing interference experiments when the gauge is not zero. 6 We are now working in the Schr¨ odinger picture so that operators are not time–dependent.
0. A
0 (

x; 0) We therefore drop the time t in the argument of A x)Schr¨ odinger = A0 (
Heisenberg .

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The Wave Function of the Lyman–Alpha Photon. Part I

781

when there are no photons. When the energy of the atom is in the continuum, the analogous eigenfunctions of H0 are |EH , jH , mH , s1 , s2 , · · · , sn >= |EH , jH , mH > |s1 , s2 , · · · , sn > ,

(38)

|EH , jH , mH , V >= |EH , jH , mH > |V > .

(39)

It follows that: n   Ei |nH , jH , mH , s1 , s2 , · · · , sn > , H0 |nH , jH , mH , s1 , s2 , · · · , sn > = E(nH ) +

 H0 |EH , jH , mH , s1 , s2 , · · · , sn > = EH +

n

i=1

 Ei |EH , jH , mH , s1 , s2 , · · · , sn > ,

i=1

H0 |nH , jH , mH , V > = E(nH )|nH , jH , mH , V > , H0 |EH , jH , mH , V > = EH |EH , jH , mH , V > .

3.2

(40)

The Splitting of the Hilbert Space

The Hermitian operators H0 and H are defined in the same Hilbert space H. The two–level assumption is that the perturbation HI acting on any eigenstate7 of H0 is zero except for the following eigenstates: |1, 0, 0, s >, |2, 0, 0, s >, |2, 1, mH , V >, |2, 1, mH , s >. (For a complete two–level system we should also include the eigenstates |1, 0, 0, V > and |2, 0, 0, V >. It can be shown, however, that these eigenstates do not contribute to the photon wave function [3] because of angular momentum conservation). The eigenstates which we use span a subspace of H which we shall call HD 8 . All the eigenstates of H0 except the four listed above are also eigenstates of H = H0 + HI . The interaction HI now denotes the truncated interaction. For simplicity and in order to be able to use the notation of [3] we rename the eigenstates of H0 which span HD in the following way: |1, s > |2, s > |2, M > |2, M, s >

≡ |1, 0, 0, Ep , j, m, λ > ≡ |2, 0, 0, Ep , j, m, λ > , ≡ |2, 1, M, V > ≡ |2, 1, M, Ep , j, m, λ >,

(41)

for M = 0, ±1 and s = {Ep , j, m, λ}. We denote by E1 and E2 the energies of the ground state and the first excited state, respectively9 . Then H0 |i, s > = 7 We

(Ei + Ep )|i, s >

use the terms “eigenstates”, “eigenkets”, and “eigenvectors” interchangeably. subscript D on HD refers to Dirac who uses this subspace in his resonance scattering theory. It is also the subspace used by Wigner and Weisskopf. 9 These definitions of E (i = 1, 2) represent a slight change of notation. Moreover, the helicity i variable β of [3] is called λ in the present paper. 8 The

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H0 |2, M > = H0 |2, M, s > =

Ann. Henri Poincar´e

E2 |2, M > , (E2 + Ep )|2, M, s > .

(42)

The eigenstates of H in HD have been calculated exactly in [3] using the exact matrix elements, not the long wave–length limit customarily used in such calculations. There are only eigenstates which correspond to eigenvalues E in the continuous spectrum of H0 + HI . There are no discrete eigenvalues [3, 5]. The point eigenvalue E1 of H0 is “swallowed” by the continuous spectrum, to use the language of [5]. We denote these eigenstates by |1, s)10 . We do not use perturbation theory to find it, as in the Wigner–Weisskopf theory. Instead we use Dirac’s method of solution which is an exact method for the truncated perturbation. The eigenket|1, s) is an eigenket of H with the eigenvalue E1 + Ep . H|1, s) = (E1 + Ep )|1, s) . Among unitarily equivalent sets, they have been specified by requiring11   H0  H  t exp − i t |1, s) = lim exp i h ¯ ¯h

t→−∞

|1, s > .

(43)

It is shown in [3] that the eigenstates |1, s) span the space HD . Hence, the eigenstates of H0 which do not span HD together with the eigenstates |1, s) span the entire Hilbert space H. They are also eigenkets of the truncated Hamiltonian H. Hence if | > is any one of the eigenkets of H0 which does not span HD   H0  H  t exp − i t | >= | > . lim exp i h ¯ ¯h From scattering theory it is shown that t→±∞

  H0  H  lim exp i t exp − i t |1, s) = |1, s > −2π i δ(E − H0 )HI |1, s) . h ¯ h ¯

t→+∞

(44)

(45)

In the above equation E = E1 + Ep .

3.3

The Initial Value Problem and the Final Value Problem

Our objective is to start at time t = 0 in the eigenstate |2, M > of the Hamiltonian H0 (the first excited p–state of hydrogen with no photon present) and find the final state as t → ∞. We expect the final state to correspond to the direct product of the ground state of the hydrogen atom and the photon wave function for Lyman–α radiation12 . 10 We use the round bracket { ) } to distinguish it from the eigenstate of H , which uses the 0 angular bracket { > }. The notation used here differs from the notations of [3] but we do not think this will cause difficulty in referring to [3] for some of the results needed. 11 We are motivated by traditional quantum scattering theory, e.g. [9]. 12 If the initial state of hydrogen were the first excited S–state, one can show that in this approximation there would be no transition to the ground state. This result is, of course, also in the Wigner–Weisskopf treatment of transitions and is a consequence of the conservation of angular momentum.

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3.3.1 The Orthogonality and Completeness Relations. Matrix Elements of HI The kets of the H0 -basis in HD , namely |1, s >, |2, s >, and |2, M, s > satisfy the orthogonality relations < 1, s|1, s > < 2, s|2, s >

= =

Ep δ(s, s ) Ep δ(s, s )

< 2, M, s|2, M , s > < 1, s|2, s >

= =

Ep δ(s, s )δM,M
< 2, s|1, s >= 0

< 1, s|2, M, s > < 2, s|2, M, s >

= =

< 2, M, s|1, s >= 0 < 2, M, s|2, s >= 0 .

(46)

In Eq. (45) Ep ∈ s. The identity operator ID in HD is    ID = |1, s > ds < 1, s|+ |2, s > ds < 2, s|+ |2, M, s > ds < 2, M, s|. M=0,±1

(47) The kets |1, s) which are the complete set of eigenkets of H in HD satisfy the orthogonality relations (48) (1, s|1, s ) = Ep δ(s, s ) . This equation is proved from scattering theory (see [9]). The identity operator ID of Eq. (46) is also given by  (49) ID = |1, s)ds(1, s| . The eigenkets |1, s) of H can be expanded in terms of the eigenkets |1, s >, |2, s >, |2, M, s >. |1, s) = = +

ID |1, s)      |1, s > ds < 1, s |1, s) + |2, s > ds < 2, s |1, s)  |2, M, s > ds < 2, M, s |1, s) .

(50)

M=0,±1

We shall now give the components of |1, s) from [3]. They will be expressed in terms of a set of functions which come from the matrix elements of HI . A(x) G1 (x)

x2 √ 2(x2 + 1)3  x 2 = − 2 3 [x + ( 32 )2 ]2 =

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G2 (x) I1 (x)

Ann. Henri Poincar´e

x3 1 = −√ 2 12 (x + 1)3 ∞ dξ = [G1 (ξ)]2 γ− (x − ξ) ξ 0

I2 (x)

∞ dξ = [G2 (ξ)]2 γ− (x − ξ) ξ 0

I3 (x)

∞ dξ = [A(ξ)]2 γ− (x − ξ) . ξ

(51)

0

In the last three of Eq. (50) the function γ− (x) is the distribution defined by γ− (x) = lim

→0+

P 1 = −iπδ(x) + . x + i0 x

The real parts of the functions Ii (x) are13

5 5 1 5 4 2 6 π(− + x2 + x + x ) Re I1 (x) = 9 32 24 162 179 (x2 + 4 )4  8 64 7 2 11 4 2 + ( x + x3 + x5 + x ) + x log( |x|) 18 9 81 6561 3 3

1 25 2 1 25 4 25 6 π(− Re I2 (x) = − x − x + x (x2 + 1)6 2048 6144 1024 1024 1 25 8 1 10 1 1 x + x ) + (− x − x3 + x5 + 6144 2048 480 48 72  1 7 1 9 1 11 1 5 x + x + x ) + x log |x| + 24 96 720 12

105 2 105 4 1 35 6 7 − x + x + x Re I3 (x) = π(− 2 6 (x + 1) 1024 24 512 512 13 21 8 3 10 1 1 x + x ) + (− x + x3 + x5 + 1024 1024 20 48 2  1 7 1 1 1 x + x9 + x11 ) + x3 log |x| . + 4 12 80 2 The imaginary parts of Ii are Im I1 (x)

∞ dξ [G1 (x)]2 = −π [G1 (ξ)]2 δ(x − ξ) = −π H(x) ξ x 0

Im I2 (x)

∞ [G2 (x)]2 dξ = −π H(x) = −π [G2 (ξ)]2 δ(x − ξ) ξ x 0

13 These

results correct some misprints in [3].

(52)

(53)

Vol. 3, 2002

The Wave Function of the Lyman–Alpha Photon. Part I

Im I3 (x)

= −π

785

∞ [A(x)]2 dξ = −π H(x) . [A(ξ)]2 δ(x − ξ) ξ x

(54)

0

In Eq. (52) δ(x) is the usual Dirac δ–function and the symbol P means principal part when used in an integral. Also H(x) is the Heaviside function, defined by H(x)

= 1 if x ≥ 0 = 0 if x < 0 .

Furthermore we use α as the fine structure constant, a as the Bohr radius, and κ as the wave number of the Lyman–α radiation given by the Bohr formula: κ=

E2 − E1 3 α = . hc ¯ 8 a

Finally, we define functions γ1 (k), γ2 (k), γ3 (k), δ1 (k), δ2 (k), δ3 (k), δ(k) = γ(k) =

3  i=1

(55) 3  i=1

δi (k),

γi (k), which give the line shape and energy level shift, by14

δ1 (k)

=

δ2 (k)

=

δ3 (k)

=

γ1 (k)

=

γ2 (k)

=

γ3 (k)

=

γ4 (k)

=

2α3 Re I1 (ka) πa   2α3 Re I2 (k − κ)a πa   2α3 Re I3 (k − κ)a 3πa 2α3 − 2 |G1 (ka)|2 H(k) ka   2α3 − |G2 (k − κ)a |2 H(k − κ) (k − κ)a2   2α3 − |A (k − κ)a |2 H(k − κ) 3(k − κ)a2 γ1 (k) − γ2 (k) − γ3 (k) .

(56)

The matrix elements of the interaction HI which we shall need are < n, s|HI |2, M >

< n|HI |2, M, s >

= < 2, M |HI |n, s >∗  e2 α δM,m δj,1 Gn (ka) = −iλ a π = < 2, M, s|HI |n >∗

14 δ(k), as defined here, is not to be confused with the Dirac–δ. Context will make clear which δ is being used.

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 e2 α δM,−m δj,1 Gn (ka) a π = < 2, M, s|HI ||2, M  >∗ 
 2 α 1 1 1 M
e = −(−1) δj,1 A(ka) . (57) a π m − M M = −(−1)m iλ

< 2, M  |HI |2, M, s >

In Eq. (56) the quantity k in given in terms of the energy of the photon Ep contained in the set s by Ep . (58) k= hc ¯ Thus k is the wave number associated with the photon energy Ep . Then h ¯ k = Ep /c is the absolute value of the momentum p of the photon. The symbol 
j1 j2 j3 m1 m2 m3 is the usual symmetric Wigner form of Clebsch-Gordan coefficients used in discussions of the reduction of the direct product of irreducible representations of the rotation group [8]. The components of |1, s) in terms of the H0 representation (see Eq. (49)) are < 1, s |1, s) = kδ(k − k  )δj,j
δm,m
δλ,λ
+ × < 2, M |1, s) = < 2, s |1, s) = < 2, M  , s |1, s) = ×

3.4

1 δj,1 δj,j
δm,m
γ− (k − k  ) 2π

 λλ kk  γ1 (k)γ1 (k  ) k − κ − δ(k) − iγ(k)  λ −(k/2π)γ1 (k) −iδj,1 δM,m k − κ − δ(k) − iγ(k)  kk  γ1 (k)γ2 (k  + κ) 1   δj,1 δj,j
δm,m
λλ γ− (k − k − κ) 2π k − κ − δ(k) − iγ(k)
 1 1 1 i m  (−1) δj,1 δm,(m
+M
) λ γ− (k − k − κ) m − m M 2π  3kk  γ1 (k)γ3 (k  + κ) . (59) k − κ − δ(k) − iγ(k)

Statement of the Problem and Its Solution

We are now in a position to state the problem in mathematical terms. As mentioned earlier, the system consisting of the hydrogen and photon field will initially be in an eigenstate of H0 , namely |2, M > which describes the atom being in the 2 − p state with no photons present. This state is in the Hilbert spaces H and HD . If there were no interaction HI , the system would remain in this eigenstate. The time variation of this eigenstate would be   E2  H0  t |2, M >= exp − i t |2, M > . exp − i h ¯ ¯h

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787

The state remains the same with only a phase change dependent on time t as given above15 . However, with the interaction HI present, the state various in a more complicated way with time. Denoting the time–dependent state by |2, M ; t >, one has  H  (60) |2, M ; t >= exp − i t |2, M > . ¯h The time–dependent state is a time–varying superposition of eigenstates of H0 . Our objective is to find the eigenstate of H0 which |2, M ; t > approaches as t → ∞. Toward this end we use the tools of scattering theory. We evaluate   H0  H  t exp − i t |2, M >= |Φ > , lim exp i t→+∞ h ¯ ¯h

(61)

where the state |Φ > depends on the eigenvalue M . Eq. (61) means   H0  H  t |Φ > . exp − i t |2, M >≈ exp − i h ¯ ¯h That is, the solution of the perturbed time–dependent Schr¨ odinger equation approaches a solution of the unperturbed time–dependent Schr¨ odinger equation for large times. The quantity    HH  H0  Ep  < 1, 0, 0, Ep, j, m, λ| exp i t exp − i t |Φ >= exp − i t ψ(Ep , j, m, λ) h ¯ ¯h ¯h (62) is the time-dependent wave function of the emitted photon in the energy–angular momentum representation16. We recall that < 1, s| ≡< 1, 0, 0, Ep , j, m, λ|. We shall now evaluate the state |Φ >. Using the resolution of the identity ID given by Eq. (49)   H0  H  t exp − i t |2, M > lim exp i t→+∞ h ¯ h ¯   H0  H  = lim exp i t exp − i t ID |2, M > t→+∞ h ¯ ¯h
∞  +j ∞   H0  dEp H  t exp − i t = lim exp i  Ep t→+∞ ¯h ¯h



λ =±1 j =1 m =−j 0 |1, 0, 0, Ep , j  , m , λ )

(1, 0, 0, Ep , j  , m , λ |2, M > .

In the above equation and later we shall show explicitly the variables which up to now have been designated collectively by the set s. We use Eq. (45) to evaluate   H0  H  t exp − i t |1, 0, 0, Ep , j  , m , λ ). lim exp i t→+∞ h ¯ h ¯ 15 In

older quantum texts such a state would be described as a “standing wave.” are abridging a somewhat long argument.

16 We

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H. E. Moses

Ann. Henri Poincar´e

Thus the ket |Φ > of Eq. (61) is |Φ > = × ×

∞




j

∞

dEp

|1, 0, 0Ep , j  , m , λ > Ep

λ
=±1 j
=1 m
=−j
0 (1, 0, 0, Ep , j  , m , λ |2, M

> −2πiδ(E1 + Ep − H0 )HI  |1, 0, 0, Ep , j  , m , λ )(1, 0, 0, Ep , j  , m , λ |2, M > .

(63)

The wave function of the photon is given by Eq. (62). We therefore multiply both sides of Eq. (64) by  Ep  t < 1, 0, 0, Ep , j, m, λ| . exp − i h ¯ We note < 1, 0, 0, Ep, j, m, λ|1, 0, 0, Ep , j  , m , λ >= Ep δ(Ep − Ep )δj,j
δm,m
δλ,λ
, < 1, 0, 0, Ep , j, m, λ|δ(E1 + Ep − H0 ) = δ(Ep − Ep ) < 1, 0, 0, Ep , j, m, λ| . Then the wave function of the Lyman–α photon is

(64)

17

  Ep  Ep 

t ψ(Ep , j, m, λ) = exp − i t (1, 0, 0, Ep , j, m, λ|2, M > exp − i h ¯ ¯h j
∞ 1 −2πi < 1, 0, 0, Ep , j, m, λ|HI |1, 0, 0, Ep , j  , m , λ ) Ep


j =1 m =−j λ=±1

× (1, 0, 0, Ep , j  , m , λ |2, M > .

(65)

The quantity −

2πi < 1, 0, 0, Ep , j, m, λ|HI |1, 0, 0, Ep , j  , m , λ ) Ep

is evaluated in Eq. (50) of [3]. Moreover, (1, 0, 0, Ep , j, m, λ|2, M > is just the complex conjugate of < 2, M |1, s) of Eq. (59). Thus we can evaluate the wave function explicitly in the energy–angular momentum representation.

4 The Wave Function of the Lyman–α Photon in the Energy–Angular Momentum Representation We are now able to give the wave function for the Lyman–α photon.    Ep  Ep  t ψ(Ep , j, m, λ) = exp − i t λ δj,1 δm,M −(k/2π) γ1 (k) exp − i h ¯ ¯h 17 We

are departing from the bra and ket notation in describing the wave function of the photon.

Vol. 3, 2002

The Wave Function of the Lyman–Alpha Photon. Part I

×



789

 γ1 (k) i −2  2  2 k − κ − δ(k) + iγ(k) k − κ − δ(k) + γ(k)

  Ep  k − κ − δ(k) + iγ4 (k) t iλ δj,1 δm,M (k/2π) (−γ1 (k))  = exp − i 2  2 . h ¯ k − κ − δ(k) + γ(k) (66) In the above equation k = Ep /¯ hc, as before. Furthermore, since γ1 (k) < 0, the argument of the square root is positive as is the square root itself. As is usual in quantum mechanics, probability densities are constructed from wave functions. From the theory of scattering it can be shown that ∞ 

|ψ(Ep , 1, M, λ)|2

λ=±1 0

dEp = 1. Ep

(67)

We have verified Eq. (68) by numerical integration. The probability that the total angular momentum of the photon is given by j = 1 is 0. Likewise the probability that magnetic quantum number m does not equal M , the magnetic quantum number of the atom initially, is likewise 0. The probability that the photon has circular polarization λ = 1 is 12 . The probability that the photon has circular polarization λ = −1 is also 12 . The probability that the energy of the photon is in the interval E0 − ∆ < Ep < E0 + ∆ is

E 0 +∆

|ψ(Ep , 1, M, λ)|2

λ=±1 E −∆ 0

dEp . Ep

(68)

If we consider the “aperture” ∆ to be very small, the expression (69) becomes

|ψ(E0 , 1, M, λ)|2

λ=±1

∆ . E0

(69)

5 Examination of the Resonance We see from Eq. (67) that the probability reaches a maximum when k − κ − δ(k) is a minimum. Most workers using the resonance formula replace k in δi (k) and γi (k) by κ, reasoning that functions δi (k) and γi (k) are slowly varying functions of their arguments. We can plot both   γ1 (κ)γ4 (κ) (70) v(k) =  2  2 (k − κ − δ(κ) + γ(κ)   γ1 (k)γ4 (k) (71) w(k) =  2  2 (k − κ − δ(k) + γ(k)

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H. E. Moses

Ann. Henri Poincar´e

against k in the vicinity of the approximate resonance ≈ κ + δ(κ). We use a

=

5.29167 × 10−9 cm.

α =

7.29720 × 10−3

c

=

κ

=

2.997925 × 1010 cm./sec. 3 α = 5.17124 × 105 1/cm. 8 a

(72)

The curve for v(k) is very close to that for w(k) near the resonance18 . Thus the assumption is correct in the vicinity of the resonance and we shall replace δ(k) and γi (k) by δ = δ(κ), γi = γi (κ) there. For numerical values we have δ γ γ2

= −0.98498 1/cm. = γ1 = −0.0164536 1/cm = γ3 = 0.

(73)

Thus the wave function Eq. (67) simplifies considerably. We are now in a position to find the average energy of the photon Ep . We shall not use the approximation which leads to Eq. (74).

Ep

=

=

∞ j ∞

Ep |ψ(Ep , j, m, λ)|2

λ=±1 j=1 m=−j 0  ∞  k − γ1 (k) k hc ¯

π

0

dEp Ep

2  2  − κ − δ(k) + γ4 (k) dk .  2  2 2 k − κ − δ(k) + γ(k)

(74)

This integral diverges logarithmically for large wave number k if k is replaced by κ in γ1 (k), γ(k), γ4 (k) and δ(k)19 . On the other hand, if this approximation is not used, γ1 (k) provides a strong cut–off for large k and the integral converges strongly. We have evaluated the integral numerically. Our result is Ep = 1.63479 × 10−11 ergs.

(75)

This is also just the value of the energy associated with the resonance peak of the wave function. Hence Ep = h ¯ c[κ + δ(κ)] . This relation was proved numerically. However, it may be possible to prove it analytically. The sharpness of the resonance causes |ψ|2 to behave like a δ–function. 18 The

two curves agree to less than on part in 10−8 in the region. it possible that this approximation is responsible for some of the divergences found in quantum electrodynamics? 19 Is

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The Wave Function of the Lyman–Alpha Photon. Part I

791

6 Linear Momentum Representation of the Wave Function of the Lyman–α Photon Let us define f ( p, M, λ) by f ( p, M, λ) =

c1/2 M,λ Y (θ, φ)ψ(cp, 1, M, λ) , p 1

(76)

where p = Ep /c = h ¯ k and Y1M,λ (θ, φ) is a generalized surface harmonic [10, 2]. The absolute value of the vector p is p and the direction of p is given by the usual spherical polar angles θ, φ. Explicitly the surface harmonics Y1M,λ are given by [11].   1 3 1 3 2iφ 1,1 1,−1 (1 + cos θ), Y1 e (1 − cos θ) (θ, φ) = Y1 (θ, φ) = 4 π 4 π   3 −iφ 3 iφ Y10,1 (θ, φ) = − e e sin θ sin θ, Y10,−1 (θ, φ) = − 8π 8π   1 3 −2iφ 1 3 e (1 + cos θ) . (77) (1 − cos θ), Y1−1,−1 (θ, φ) = Y1−1,1 (θ, φ) = 4 π 4 π The function f ( p, M, λ) is the wave function in the linear momentum basis. It can be shown that  d p = 1, (78) |f ( p, M, λ)|2 cp λ=±1

where the integration is taken over the entire 3-dimensional p–space. The quantity  d p | f ( p, M, λ)|2 cp ∆V

gives the probability that the momentum vector p lies in the volume ∆V of p–space when the circular polarization is given by λ and the higher state of the atom is in the angular momentum state j = 1 with the magnetic quantum number given by M initially. For example, when M = 0  3 f ( p, 0, 1) = −i g(p/¯h) e−iφ sin θ . (79) 8π The direction–independent function g(k) is    k − κ − δ(k) + iγ4 (k) g(k) = (k/2π) − γ1 (k)  2 . 2  (k − κ − δ(k) + γ(k)

(80)

This is the resonance factor first encountered in Eq. (67). The probability density for the state Eq. (77) when M = 0 is |f ( p, 0, λ)|2 =

3 |g(p/¯h)|2 sin2 θ . 8π

(81)

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H. E. Moses

Ann. Henri Poincar´e

The probability densities for M = ±1 are calculated similarly. Thus |f ( p, 1, 1)|2

=

|f ( p, −1, 1)|2

=

θ 3 |g(p/¯ h)|2 cos4 = |f ( p, −1, −1)|2 4π 2 θ 3 |g(p/¯ h)|2 sin4 = |f ( p, 1, −1)|2 . 4π 2

(82)

For a given value of p = | p| = Ep /c, i.e. the absolute value of the momentum, the probability that the photon will pass through the solid angle sin θ dθ dφ depends only on the angle θ. The “radiation pattern” for the various cases are readily understood and we shall not belabor the subject.

References [1] E. P. Wigner, Ann. Math., 40, 149 (1939). [2] H. E. Moses, Phys. Rev. A, 8, 1710 (1973). [3] H. E. Moses, Phys. Rev. A, 22, 2069 (1980). [4] P. A. M. Dirac, Principles of Quantum Mechanics, Oxford (1949), p. 199. [5] K. O. Friedrichs, Comm. Pure and App.Math., 1, 361 (1948). [6] G. Compagno, R. Passante and F. Persico, Jour. Mod Optics 37 1377 (1990) and L. Maiani and M. Testa ,Physics Letters B 356, 319 (1995). [7] A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press (1960). [8] H.E. Moses, Nuovo Cimento, Serie X, 42, 110 (1966). [9] H.E. Moses, Nuovo Cimento, 1, 103 (1955). [10] H.E. Moses, Ann. Phys., 41, 166 (1967). [11] H.E. Moses and A.F. Quesada.Archive Rat. Mech. and Anal., 50, 194 (1971).

Harry E. Moses Applimath Company 150 Tappan Street Brookline, MA 02445 USA email: [email protected] Communicated by Vincent Rivasseau submitted 28/07/01, accepted 26/11/01

Ann. Henri Poincar´e 3 (2002) 793 – 813 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/040793-21

Annales Henri Poincar´ e

The Wave Function of the Lyman–Alpha Photon Part II. The Shape, Position, and Trajectory of the Photon H. E. Moses

Abstract. In Part I of the present paper we derived the wave function of the Lyman– α photon in both the linear and angular momentum bases using relativistic concepts  for the photon wave function. In the present paper, Part II, we derive two X– representations. In the first we assume one–particle theory for the photon wave function and the usual commutation rules for the position operators Xi and linear momentum operators Pi . The second representation employs the quantized photon  representation. The stress, energy tensor density is used to field to derive an X– provide a probability density in  x–space which is relativistic. The two methods of defining  x–space are compared. It is found in the present case that, despite the use of particle operators, the photon resembles a field far more than than it does a particle.

1 The Use of Commutation Rules to Define an
x–Representation 1.1

The Coordinate Operators in a Linear Momentum Basis

We shall assume that Part I of the present paper is available to the reader. Equations in Part I which are referred to in Part II will have a prime (’) attached to the equation number. Likewise footnotes or references of Part I which are referred to will also have a prime attached to the footnote or reference number. The coordinate operators Xi are defined by the usual commutation relations with the linear momentum operators Pi , where the momentum operators operate on the photon wave functions f ( p, M, λ) of Eq. (77’) as follows: p, M, λ) = pi f ( p, M, λ) , Pi f (

(1)

where −∞ < pi < ∞ is the i
th–component of p. The commutation rules are the same as for non-relativistic particle theory. They were first proposed for the Dirac electron [1]. The commutation rules are h ¯ [Xj , Pk ] = − δj,k , [Xj , Xk ] = 0 . i

(2)

794

H. E. Moses

Ann. Henri Poincar´e

We require the operators Pi and Xi to be Hermitian operators with respect p, M, λ) and f ( p, M, λ) 1 . to the inner product of two photon wave functions f (1) ( (f (1) , f ) =




p, M, λ)f ( p, M, λ) f (1)∗ (

λ=±1

d p where p = | p| . cp

(3)

This prescription for obtaining coordinate operators is not really relativistic. For a relativistic treatment would require the existence of a time operator T which satisfies a commutation equation similar to Eq. (2) where Pi is replaced by the Hamiltonian operator H. However, von Neumann’s theorem says that if two operators Xi and Pi satisfy the commutation rules Eq. (2), each has a continuous spectrum ranging from −∞ to ∞. This condition on the spectra is acceptable and even expected for momentum and coordinate operators. If a time operator and the Hamiltonian operator satisfied a commutation relation such as Eq. (2), then each of them too would have a continuous spectrum ranging over the entire real axis. But we know that photons have a minimum energy of 0. Hence, a time operator, which is analogous to a coordinate operator, and the Hamiltonian cannot satisfy  set of the commutation relation and the relativistic symmetry is broken2 . The X operators must be defined anew in each frame of reference. We have the following theorem: The coordinate operators Xi are given in the linear momentum representation by (p)

p, M, λ) = − Xj f (

¯ ∂ h 1 pj  f ( p, M, λ) . − i ∂pj 2 p2

(4)

(p)

The superscript p on Xi means that the operator is expressed in the p− representation. This realization is unique (within unitary transformations) and the operators are Hermitian.

1.2

The Trajectory of the Photon

We are now able to give the trajectory of the photon. That is, we can calculate the mean value of the coordinate operators Xi as a function of time and quantum 1 We are assuming that the set of operators {X , P } are an irreducible set in the Hilbert space i i of photon wave functions. Roughly speaking, this means that any other operators in the space are functions of Xi , Pi . 2 Wigner in his treatment of relativistic particles Ref. (3’) avoids the use of the coordinate representation and coordinate operators and instead uses the 4–momentum and the 4 ×4 relativistic tensor which is a generalization of the angular momentum vector.

Vol. 3, 2002

The Wave Function of the Lyman–Alpha Photon. Part II

795

number M of the 2–p state. We shall denote the mean values by X i (t, M ).
  Ep  ∗  Ep  d p X j (M ; t) = p, M, λ) Xj e−i h¯ t f ( p, M, λ) e−i h¯ t f ( cp λ=±1  ¯h  ∂
 p 1 pj  = p, M, λ)ei h¯ ct − − f ∗ ( i ∂pj 2 p2 λ=±1 p

× e−i h¯ ct f ( p, M, λ)

d p . cp

(5)

Integrating over the angular variables we find X j (M ; t) = 0

(6)

for all M and t. The interpretation of this result is that for any given direction, indicated by a unit vector drawn from the origin, there are as many photons going the direction of the vector as those going in the opposite direction. This result is unsurprising.

1.3

The x–Representation E

The set of functions exp[−i h¯p t]f ( p, M, λ) ≡ f ( p, M, λ; t) can be considered as a p representation, that is, a representation in which the operators Pi are represented by multiplication by pi . We now look for a set of functions χ(x, M, λ; t) which are in a one–to–one correspondence with the functions f ( p, M, λ; t) such that (x)

Xj χ(x, M, λ; t) = xj χ(x, M, λ; t)

(7)

instead of Eq. (4) . The superscript x on the operator Xj indicates that the x– representation is being used. The consequences below result:  p · x)] −i E t exp[ h¯i ( 1 e h¯ f ( χ(x, M, λ; t) = p, M, λ) . (8) d p √ 3 cp (2π¯ h) 2  √ cp i E t i h ¯ f ( p, M, λ) = p · x)]χ(x, M, λ; t) . (9) dx exp[− ( 3 e ¯h (2π¯ h) 2

 d p |f ( p, M, λ)|2 = 1 . (10) dx |χ(x, M, λ; t)|2 = cp λ=±1

λ=±1

 about x at time The probability that the photon is in the volume element dx t is given by P r(x; M ; t) dx where P r(x, M ; t) is the probability density given by
|χ(x, M, λ; t)|2 . P r(x, M ; t) = λ=±1

(11)

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Thus P r(x, M ; t) gives the shape of the photon. The time dependence gives its trajectory. We shall now calculate χ(x, M, λ; t) from Eq. (8) and Eq. (77’). Actually, we shall consider the asymptotic wave function χasy (x, M, λ; t). This wave function is defined by lim

r→∞,r−ct=const.

r χ(x, M, λ; t) = g(r − ct, u, M, λ) .

(12)

In the above equation u = x/r and is the unit vector in the direction of x. We shall use methods that we have previously used in discussing wave propagation [2]. Indeed, the wave-function χ is a solution of the three–dimensional wave equation. The condition Eq. (12) states that χ(x, M, λ; t) ≈ χasy (x, M, λ; t) ≡

g(r − ct, u, M, λ) r

(13)

for large values of r and t. It is shown in [2] that the function g(w, u, M, λ) always exists. It is also shown that the exact function χ(x, M, λ; t) can be reconstructed from the asymptotic value. The space–time region in which Eq. (13) holds is called the wave zone. Eq. (12) and (13) state that, except for a factor 1/r, the wave function in x–space is a wave moving along each ray in x–space. The function g(w, u, M, λ) is found using the methods of [2]. g(w, u, M, λ)

λ M,λ ˆ ˆ (θ, φ) Y 2π 1 ∞ −γ1 (k) (k − κ − δ(k) + iγ4 (k)) ikw e dk . ×

2

2 k − κ − δ(k) + γ(k) =

(14)

0

In Eq. (14) θˆ and φˆ are the polar angles which fix the direction of u = x/r. The probability that a photon is lies between r and dr and in the solid angle r2 sin θˆ dθˆ dφˆ at time t is |g(r − ct, u, M, λ)|2 dr sin θˆ dθˆ dφˆ . The shape of the photon is given by |g(w, u, M, λ)|2 sin θˆ3 . It is seen from Eq. (13) and (14) that the probability density or shape function P r(r, u, M ; t) can also be written P r(r, u, M ; t) =


|Y M,λ (θ, ˆ φ)| ˆ 2 P (r − ct) 1 where r2

λ=±1

P (w)

=

∞ 2 −γ1 (k) (k − κ − δ(k) + iγ4 (k)) ikw 1 e dk . (15)

2

2 2 4π k − κ − δ(k) + γ(k) 0

3 The

factor sin θˆ which appears above is contained in the volume element ˆ d x ≡ r 2 sin θˆ dr dθˆ dφ.

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Thus the shape function is a wave which moves outward along all rays u with the velocity c. The amplitude of the wave varies along each ray as determined

of light ˆ φ)| ˆ 2 . The quantum number M and the angles θ and φ appear by |Y1M,λ (θ, λ=±1

in this factor and determine this factor as a function of θˆ 4 . Thus to finish the problem of finding the shape function of the photon requires us merely to evaluate the one–dimensional Fourier transform in Eq. (8) and (9). We shall first study the radiation pattern. Let us define the radiation pattern ˆ by function S(M ; θ)
ˆ φ)| ˆ 2. ˆ = |Y1M,λ (θ, (16) S(M ; θ) λ=±1

Let us first consider the case where the atom is initially in the M = 0 state. Then from Eq(78’) ˆ = S(0; θ)

3 sin2 θˆ . 4π

(17)

When the magnetic quantum number M is 1 or -1 the radiation pattern functions are ˆ = 3 (1 + cos2 θ) ˆ . S(±1; θ) (18) 8π Thus the probability of finding the photon in the coordinate element dr dθˆ dφˆ is Q(r, u, M ; t)dr dθˆ dφˆ where Q(r, u, 0; t) = Q(r, u, ±1; t) =

3 ˆ (r − ct) sin3 θP 4π 3 ˆ + cos2 θ)P ˆ (r − ct) sin θ(1 8π

We have evaluated P numerically. The results are given graphically below. We have scaled the variable w = r − ct using the wave number κ = 2π/λα , whereλα is the wavelength of the Lyman-α radiation. Then the probability density ˆ The graph is depends upon r for a given time t and on the cone defined by θ. remarkable in that approximate causality is preserved in the quantum pictures of light. One might think that in Figure 1 the wriggles corresponding to wave propagation have been suppressed by the coarseness of the graphics. We shall show later that this suppression is not present. 4 The

ˆ use of the absolute value eliminates the φ–dependence.

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P 0.012 0.010 0.008 0.006 0.004 0.002 −1 × 107

−5 × 106

5 × 106

1 × 107

(r − ct) × κ

Figure 1. The Radial Shape of the Photon (Unscaled.) The probability has its peak for r slightly less than ct. This peak travels with the velocity of light outward along each radius. From exact causality one expects the probability to be 0 when r is greater than ct. Instead, the is a very small “precursor.” This precursor may be due to the approximations used in the analysis and that the starting points of r and t must be reset in the far field. Another possibility is that small cumulative errors in the numerical integrations are making themselves felt. Still another possibility is that Figure 1 is essentially correct and reflects the uncertainty principle in a particle picture of light. In any case the calculations must be almost correct to have causality so nearly satisfied. It will be noticed that we have not considered the regime for which (r−t)×κ < −107 . The reason for this is that our computer required an inordinate length of time to complete the calculations when longer intervals in this variable were considered5 .

1.4

The Electromagnetic Field Associated with the Lyman–α Photon

We are working with a single–photon theory. Equivalently, we are working with a first–quantized electromagnetic field (Ref 7’). In first quantized theory the operators G(E, j, m, λ) of Eq. (8’)6 are replaced by the wave function of the photon which in our case will be given by Eq. (67’), Then the electric vector  x; t) = − 1 ∂ A  0 (x; t) E( c ∂t 5 Our

comments on the precursor and the range of variables also apply to later calculations. are replacing the variable k of Eq. (8’) by j so that we can use k as a wave number. Moreover, we are showing the velocity of light c explicitly. 6 We

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becomes7   x; t) E(

= −2

∞
 ¯c h  iλ Y1,1,M (θ, φ) Re dk k −k γ1 (k) π λ=±1

0

k − κ − δ(k) + iγ4 (k) −ikct ×

2

2 j1 (kr) e k − κ − δ(k) + γ(k)  ∞ 1 Y1,2,M (θ, φ) + dk k −k γ1 (k) 3 0

k − κ − δ(k) + iγ4 (k) −ikct ×

2

2 j2 (kr) e k − κ − δ(k) + γ(k)  ∞ 2 Y1,0,M (θ, φ) − dk k −k γ1 (k) 3 0

×

 k − κ − δ(k) + iγ4 (k) −ikct .

2

2 j0 (kr) e k − κ − δ(k) + γ(k)

(19)

In Eq. (19) Re means the real part . The summation over λ gives zero for the first term of the expression in curly  x; t) becomes brackets. The remaining two terms are doubled. The expression for E(   x; t) = − E(

 16¯ hc 1,2,M (θ, φ) Re Y 3π

∞ dk k

−k γ1 (k)

0

k − κ − δ(k) + iγ4 (k) −ikct ×

2

2 j2 (kr) e k − κ − δ(k) + γ(k) ∞ √  − 2 Y1,0,M (θ, φ) dk k −k γ1 (k) 0

×

 k − κ − δ(k) + iγ4 (k) −ikct . j (kr) e 2 0

2

k − κ − δ(k) + γ(k)

(20)

Eq. (20) can be used to obtain the electric field of the photon for all values of x  x; t). and time t. A similar expression can be used to find the magnetic field H( We shall find the fields for large values of the radius r and t. By “large values of the radius” we mean that r should be several times the wavelength of the Lyman– α radiation λα = 1.215 × 10−5 cm. Similarly, the length ct should also be several times the length λα . 7 We

ˆ now use θ and φ as the polar angles which give the direction of  x, instead of θˆ and φ.

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 x; t) In [2] it is shown that at such distances and time the electric field E(  x; t) of any solution of Maxwell’s equations without sources and magnetic field H( must have the asymptotic forms  − ct, u) G(r r  − ct, u) G(r  x; t) ≈ u × H( r x . u = r  x; t) ≈ E(

(21)

 In Eq. (21) G(w, u) is a real vector function of its arguments. which is orthogonal to u:  u · G(w, u) = 0 . (22) We have called the space–time domain in which Eq. (2) holds “the wave zone” and  the vector G(w, u) “the wave zone vector”. For a fixed direction u the asymptotic electromagnetic field satisfies the one–dimensional Maxwell’s equations if we ignore the factor r in the denominator. The wave zone vector for any solution of Maxwell’s equations without currents or sources is obtained from  − ct, u) = G(r

lim

r→∞,r−ct=const.

 x; t) . r E(

(23)

 The vector G(w, u) always exists if the energy of the field is finite.8 We propose to take the solution of the electric field of the photon given by Eq. (20) and obtain the wave zone vector from Eq. (23). As a preliminary calculation we shall calculate limr→∞ r e−ikr jn (kr) for n = 0, 2. First we consider n = 2:  3 1  1 − 2ikr r e−ikr j2 (kr) = − k 3 r2 k 2i 3(1 + e−2ikr ) . (24) − k2 r As r → ∞ all the terms on the right hand side of Eq. (24) vanish except −(1 − e−2ikr )/2ik. However, as a distribution lim e−2ikr = 0 ,

r→∞

by the Riemann–Lebesgue theorem which says, roughly speaking, that if f (k) is a test function which is sufficiently smooth and integrable in the interval 0 < k < ∞, then ∞ dk f (k) e−2ikr = 0 . lim r→∞

8 In

0

the Gaussian cgs system it is measured in abvolts.

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(One could be more careful in taking the limits but we do not wish to belabor the point.) Thus 1 . (25) lim re−ikr j2 (kr) = − r→∞ 2ik Similarly, 1 . (26) lim re−ikr j0 (kr) = r→∞ 2ik From Eq. (20)  lim

r→∞,r−ct=const.

 x; t) = r E(

−

  ∞ 16¯h  Re Y1,2,M (θ, φ) dk k 3π 0

×

k − κ − δ(k) + iγ4 (k) −k γ1 (k)

2

2 k − κ − δ(k) + γ(k)

×

 eik(r−ct) lim r e−ikr j2 (kr)

−

 ∞ √  2 Y1,0,M (θ, φ) dk k

r→∞

0

× ×

k − κ − δ(k) + iγ4 (k) −k γ1 (k)

2

2 k − κ − δ(k) + γ(k)

 eik(r−ct) lim r e−ikr j0 (kr) . r→∞

(27)

On using Eq. (25) and (26) in Eq. (27) we obtain a relatively simple result for  G(w, u).   G(w, u) =

−2 ∞

×

dk 0

   √ ¯c h 1,0,M (θ, φ) Re i Y1,2,M (θ, φ) + 2 Y 3π  k − κ − δ(k) + iγ4 (k) ikw . −kγ1 (k)

2

2 e k − κ − δ(k) + γ(k)

(28)

The dependence on u is given by the factor which involves the angles and by a second factor which is a Fourier transform of the resonance term. The factor which gives the dependence on w requires numerical integration. However, the factor which depends on u can expressed as simple functions of the angles θ and φ. Let u, as before, be the unit vector in the direction of x and let aθ and aφ be the unit θ and φ, respectively. The components  vectors in the√direction of increasing    of Y1,2,M (θ, φ) + 2 Y1,0,M (θ, φ) can be obtained from Eq. (109)–(111) of [3] which give the components of the vector spherical harmonics in terms of spherical

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Ann. Henri Poincar´e

coordinates. One finds that   √ 1,0,M (θ, φ) = 0 1,2,M (θ, φ) + 2 Y u · Y

(29)

as required. Also for M = 0   √ 1,2,0 (θ, φ) + 2 Y 1,0,0 (θ, φ) aθ · Y   √ 1,2,0 (θ, φ) + 2 Y 1,0,0 (θ, φ) aφ · Y

 = −

9 sin θ 8π

= 0.

(30)

For M = ±1,   √ 1,2,M (θ, φ) + 2 Y 1,0,M (θ, φ) aθ · Y   √ 1,2,M (θ, φ) + 2 Y 1,0,M (θ, φ) aφ · Y

3 = −M √ eiMφ cos θ 4 π 3 = i M √ eiMφ . 4 π

(31)

Thus for M = 0 there is only a θ–component: 

 G(w, u) = ×

 ∞ 3¯ hc k − κ − δ(k) + iγ4 (k) sin θ aθ Re i dk −kγ1 (k)

2 2

2 2π k − κ − δ(k) + γ(k) 0  ikw . (32) e

 On the other hand, for M = ±1, G(w, u) has both θ and φ components.Then     3¯ hc  G(w, u) = − M Re i eiMφ iaθ cos θ − aφ 2 4π ∞  k − κ − δ(k) + iγ4 (k) ikw × . (33) dk −kγ1 (k)

2

2 e k − κ − δ(k) + γ(k) 0

 Or on using Gθ (w, u) and Gφ (w, u) as the θ and φ components of G(w, u) we have for M = 0  Gθ (w, u)

= −

 ∞ 3¯ hc k − κ − δ(k) sin θ dk −kγ1 (k)

2

2 2 2π k − κ − δ(k) + γ(k) 0

∞ ×

sin kw +

dk 0

Gφ

≡ 0.

 γ4 (k) −kγ1 (k)

2

2 cos kw k − κ − δ(k) + γ(k) (34)

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For M = ±1  Gθ (w, u) =

  ∞ 3¯ hc M cos θ cos φ dk −kγ1 (k) 2π 2 0

×

k − κ − δ(k)  2  2 cos kw k − κ − δ(k) + γ(k) ∞ dk

+



−kγ1 (k) 

0

−

 ∞ dk

M sin φ 0

∞ +

dk



0

 Gφ (w, u) = ×

−

k − κ − δ(k) −kγ1 (k)  2  2 sin kw k − κ − δ(k) + γ(k)

 γ4 (k) −kγ1 (k)  2  2 cos kw , k − κ − δ(k) + γ(k)

  3¯ hc M cos φ 2π 2

∞ dk

(35)

−kγ1 (k)

0

k − κ − δ(k) 2 sin kw  2  k − κ − δ(k) + γ(k) ∞

+

dk

−kγ1 (k) 

0

 ∞ +

 γ4 (k) 2 sin kw 2  k − κ − δ(k) + γ(k)

M sin φ

dk 0

∞ +

dk

k − κ − δ(k) −kγ1 (k)  2  2 cos kw k − κ − δ(k) + γ(k)

−kγ1 (k) 

0

 γ(k) 2  2 cos kw k − κ − δ(k) + γ(k)

 γ(k) 2 sin kw . 2  k − κ − δ(k) + γ(k)

(36)

We have computed the θ–component9 of the electric field for the case that M = 0 and θ = π/2. The electric field has a wave–group–like structure with wavelength roughly equal to that of Lyman-α radiation. The shape of the envelope is similar to that of the probability density P . The radiation appears as a long pulse moving outward 9 It

is the only component.

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107 × Gθ

−1 × 107

−5 × 106

5 × 106

1 × 107

(r − ct) × κ

Figure 2. Gθ Component of Wave Zone Vector for M = 0 and θ = π/2. −107 < (x − ct) × κ < 107

107 × Gθ 4 2 −10

−5

5

10

(r − ct) × κ

2 4

Figure 3. Within the Wave Zone Vector −10 < (x − ct) × κ < 10. along the radius with the velocity of light. For a given time t the envelope has its maximum near r = ct. The computations are clearly trying to indicate a wave–like structure of the pulse, unlike the situation for Figure 1 which is free of the highfrequency component. To see more clearly the wave – Figure 3 shows the form of the wave zone vector within the pulse of Figure 2 we have expanded the horizontal axis by a factor of 106 . For a general value of θ the wave zone vector components of Figures 2 and 3 are multiplied by sin θ. For θ = π/2, i.e. on the “equator” of the sphere surrounding

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the atom, where the field is a maximum, the electric field is polarized parallel to the z–axis. On the poles of the sphere there is no field. The electric fields for M = ±1 can be evaluated similarly.

2 Energy Density Definition of Probability in
x–Space. Shape of the Photon 2.1

The Reason for Second Quantization

Since the introduction of the coordinate operators is fraught with contradictions with respect to the requirements of relativity, we are obliged to compare the consequences of alternative definitions of probability in x–space with the results just obtained. As explained in Part I, a candidate for an alternative definition of probability is the normalized energy density P r1 (x; t) =

 x; t)]2 + [H(  x; t)]2 1 [E( . 8π Ep

(37)

In Eq. (38) E p is the expectation value of the energy of the photon Eq. (75’) and (76’)10 . A reason to define this as a probability density in x–space is that if the energy density had a sharp peak at x0 , it would be reasonable to say that the photon is at x0 . The four–vector {x, ct} is now a label instead of a set of spectral variables. Thus we have no need to define Xj operators as we did earlier. Moreover, the energy density is part of the stress–energy tensor and the total energy is the time–component of the energy-momentum four-vector, each of whose transformation properties are relativistic11 .  x; t) and H(  x; t) We shall use second quantization of the fields. In this case E( are Hermitian operators. We shall first derive some results using first quantization. We consider Eq. (9’) –(12’) in first quantized theory where G(E, j, m, λ) is a general photon wave function in first quantization. We rewrite Eq. (9’) as12  ∞ j ∞ Et 2

j
 A1 (x; t) = − i dE G(E, j, m, λ) exp[−i ] hc ¯ ¯h m=−j −∞ λ=±1 j=1  

E  j j,j+1,m (θ, φ) jjm (θ, φ)jj r − iλ Y × Y hc ¯ 2j + 1 

E 

E  j+1  Yj,j−1,m (θ, φ) jj−1 × jj+1 r + iλ r . (38) hc ¯ 2j + 1 ¯hc R 1  x; t)]2 + expectation value is equal to the total energy of the field 8π { d x [E( 2  H( x; t)] }.(See [2]) . 11 The transformation properties of the photon wave function are given in [4]. 12 We hope that no one will confuse the spherical Bessel function j with the angular momentum n quantum number also designated by j. 10 This

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 1 (x; t) which we shall denote by A  (w) (x; t). We want the wave zone form of A 1 By definition (w)

 1 (x; t) = r A  rA 1

lim

r→∞,r−ct=const.

(x; t) ≡ G1 (r − ct, u)

(39)

 0 (x; t), E(  x; t), From G1 (r − ct, u) we shall be able to get the wave zone forms of A  x; t). and H( lim

 0 (x; t) = r A  (w) (x; t) ≡ G0 (r − ct, u) , rA 0

(40)

 x; t) = r E  (w) (x; t) ≡ G(r  − ct, u) , r E(

(41)

 x; t) = r H  (w) (x; t) ≡ u × G(r  − ct, u) . r H(

(42)

r→∞,r−ct=const.

lim

r→∞,r−ct=const.

lim

r→∞,r−ct=const.

As before, u = xr . From G1 (w, u) one can obtain G0 (w, u): G0 (w, u) = G1 (w, u) + G1∗ (w, u)

(43)

∂   G0 (w, u) . G(w, u) = ∂w

(44)

 and G(w, u):

We shall first derive an expression for the wave zone vector G1 , since the other  are derived from it. The generalization of Eq. (24) wave zone vectors G0 and G and (26) is (−i)(n+1) (45) lim r e−ikr jn (kr) = r→∞ 2k as a distribution. On substituting Eq. (45) into Eq. (38) we obtain G1 (w, u)) as  G1 (w, u) =

−

 j ∞ ¯c


h dE i G(E, j, m, λ) ei kw 2 E j=1 m=−j ∞

λ=±1

×  j,m,λ (u) = O +

0

 j,m,λ (u) where k = E , O  ¯hc  j  j,j,m (θ, φ) + λ Yj,j+1,m (θ, φ) −Y 2j + 1   j+1  λ Yj,j−1,m (θ, φ) . 2j + 1

(46)

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Thus  G0 (w, u)

= −i

 j ∞ ¯c 


h dE G(E, j, m, λ) ei kw 2 E j=1 m=−j ∞

λ=±1

0

∞ j ∞


dE ∗  × Oj,m,λ (u) − G (E, j, m, λ) e−i kw E λ=±1 j=1 m=−j 0  ∗  × O u) . j,m,λ (

(47)

Moreover,   G(w, u) =

 j ∞ 1 


∞ dE G(E, j, m, λ) ei kw 2¯ hc j=1 m=−j λ=±1

×

 j,m,λ (u) + O

×

∗ O u) j,m,λ (

0

∞ j ∞




dE G∗ (E, j, m, λ) e−i kw

λ=±1 j=1 m=−j 0

 .

(48)

 u · G(w, u) = 0 .

(49)

It should be noted that This result is a general form for the wave zone vector in terms of photon wave functions in an energy–angular momentum representation. The total energy of the photon field is Ep =

 j ∞


λ=±1 j=1 m=−j

∞

dE G∗ (E, j, m, λ)G(E, j, m, λ) .

0

The energy density in the wave zone is 2 (w)   1   (w)  (x; t) 2 E (x; t) + H 8π 2 1  = G(r − ct, u) 2 4πr ∞ j j
∞
∞




1 = dE 8π¯ hc r2


j=1 m=−j

D(x; t) =

λ=±1

∞ 0

0

λ =±1 j =1 m =−j


 dE
G(E, j, m, λ)G(E
, j
, m
, λ
)

(50)

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H. E. Moses

Ann. Henri Poincar´e


 j
,m
,λ
(u)  j,m,λ (u)O × ei(r−ct)(k+k ) O


+ G∗ (E, j, m, λ)G∗ (E
, j
, m
, λ
)e−i(r−ct)(k+k ) ∗  ∗


(u) + 2G∗ (E, j, m, λ)G(E
, j
, m
, λ
)e−i(r−ct)(k−k
) ×O u)O j,m,λ ( j ,m ,λ  ∗  j
,m
,λ
(u) . ×O u)O (51) j,m,λ ( It can be shown that13

 dx D(x; t) = E p

2.2

(52)

The Effect of Second Quantization

To second-quantize the theory we replace the wave functions G(E, j, m, λ) by annihilation operators and its complex conjugate G∗ (E, j, m, λ) by creation operators whose actions are described in Part I of the present paper. We use the same symbol for the operators as the wave functions which they replace. Now E p and D(x; t) are operators in the Heisenberg picture. The wave functions in the second quantized theory will be independent of time in this picture. Let < E p > , < D(x; t) > be the expectation values of these operators between the wave function of the photon which is emitted from the atom. If in Dirac bra and ket notation |χ(M ) > is that state, then

Of course,

< Ep >

=

< χ(M )|E p |χ(M ) > ,

< D(x; t) >

=

< χ(M )|D(x; t)|χ(M ) > .

(53)

 dx < D(x; t) >=< E p > .

(54)

We propose to define a relative probability density for the position of the photon as14 Prel (x; t) =< D(x; t) > . (55) As in the first quantized theory, the quantity Prel is a truly relativistic quantity. It transforms as a component of the expectation relativistic stress– energy density. Now, the wave function of the photon |χ(M ) > in the second quantized 13 It is perhaps a surprising fact that the total energy of the exact fields equals the integrated energy density constructed from the asymptotic fields in the wave zone. 14 The probability density will be a function of M the magnetic quantum number of the initial p-state of the hydrogen atom.

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formalism is ∞ |χ(M ) > = 0

×

j ∞ dE


|E, j, m, λ > E j=1 m=−j λ= pm1

ψ(E, j, m, λ) .

(56)

In Eq. (56) k = E/¯ hc and ψ(E, j, m, λ) is the wave function of the photon as given by Eq. (67’). Moreover, the ket |E, j, m, λ > is the one–particle ket |s > of Eq. (25’) with s written in expanded form s → {E, j, m, λ}. The wave function of the photon represented as a ket has neither a vacuum component |V > nor components |s1 , s2 · · · , sn > for n > 1. From Eq. (67’) ∞ |χ(M ) >

=i 0

× ×

dE
λ|E, 1, M, λ > E λ=±1





 k/2π − γ1 (k) k − κ − δ(k) + iγ4 (k)

2

2 . k − κ − δ(k) + γ(k)

(57)

The expectation value < D(x; t) > contains (see Eq. 51) terms of the form < χ(M )|G(E, j, m, λ)G(E
, j
, m
, λ
)|χ(M ) > . But these terms are zero because G(E
, j
, m
λ
)|χ(M ) > is proportional to the vacuum state of the photon and G(E, j, m, λ) acting on the vacuum state gives zero. Similarly, < χ(M )|G∗ (E, j, m, λ)G∗ (E
, j
, m
, λ
)|χ(M ) >= 0. Thus < D(x; t) >

=

×



1 2 4π¯ hcr





λ=±1 λ =±1 λ

=±1 λ


=±1 ∞ ∞ j
j ∞
∞




dE

j=1 j
=1 m=−j m
=−j
0

× ×




∗ O u) j,m,λ (

dE

0




∞ 0

dE

E



∞ 0

dE


E




−i(r−ct)(k−k
)

 j
,m
,λ
(u)e ·O     λ

λ


k

/2π k


/2π γ1 (k

)γ1 (k


)

×

k

− κ − δ(k

) − iγ4 (k

) k


− κ − δ(k


) + iγ4 (k


)

2

2

2

2 k

− κ − δ(k

) + γ(k

) k


− κ − δ(k


) + γ(k


)

×

< E

, 1, M, λ

|G∗ (E, j, m, λ)G(E
, j
, m
, λ
)|E


, 1, M, λ


> . (58)

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H. E. Moses

Ann. Henri Poincar´e

In Eq. (64), k
= E
/¯ hc, k

= E

/¯ hc, k


= E


/¯hc. We first evaluate < E

, 1, M, λ

|G∗ (E, j, m, λ)G(E
, j
, m
, λ
)|E


, 1, M, λ


> using Eq. (27’), (28’ ) and (28a’). We find that < E

, 1, M, λ

|G∗ (E, j, m, λ)G(E
, j
, m
, λ
)|E


, 1, M, λ


> =

EE
δ(E
− E


)δ(E − E

)δj
,1 δj,1 δm
,M δm,M δλ
,λ


δλ,λ

.

(59)

We substitute into Eq. (58) and sum and integrate over the δ–functions. Thus15 < D(x; t) > = ×


2  ∞ 1  1,M,λ λ O dE −kγ1 (k) 2 2 4π ¯ hcr 0 λ=±1 2 k − κ − δ(k) − iγ4 (k) ik(r−ct) e .

2

2 k − κ − δ(k) + γ(k)

(60)

But dE = h ¯ c dk and


2   2 √ 1,2,M (θ, φ) + 2 Y1,0,M (θ, φ) .  1,M,λ (u) = 2 Y λO 3

(61)

λ=±1

The quantity in square brackets is given by Eq. (30)-(32) in terms of components. For M = 0
2  1,0,λ (u) = 3 sin2 θ , λO (62) 4π λ=±1

while for M = ±1
2  1,M,λ (u) = 3 (1 + cos2 θ) . λO 8π

(63)

λ=±1

The “radiation pattern” is identical to that when the coordinate operators are used to define “position” (Eq. (17) and (18)). For M = 0 3¯ hc sin2 θ P0 (r − ct) (64) < D(x; t) >= 16π 3 r2 and for M = ±1 < D(x; t) >= 15 If

3¯ hc (1 + cos2 θ) P0 (r − ct) . 32π 3 r2

a is a vector with complex components, we define |a|2 to be a∗ · a.

(65)

Vol. 3, 2002

The Wave Function of the Lyman–Alpha Photon. Part II

811

In Eq. (64) and (65) the function P0 (w) is given by P0 (w)

=

∞ dk −kγ1 (k) 0

×

2 k − κ − δ(k) + iγ4 (k) ikw .

2

2 e k − κ − δ(k) + γ(k)

(66)

The function < D(x; t) > gives the relative probability density of finding a photon at x, t. The function P0 (w) gives the radial distribution.

Energy Density

200000 150000 100000 50000

−1 × 107

−5 × 106

5 × 106

1 × 107

(r − ct) × κ

Figure 4. The Radial Probability Density Function P0 (Unscaled) The radial probability density function P0 is very similar to the radial probability density P of Eq. (15) obtained by assigning coordinate operators to the one particle photon wave function. From Eq.(64) it is seen that the density and hence probability of finding a photon is greatest at θ = π/2. or at the equator of the sphere whose north pole is the angle θ = 0 and where the probability for finding a photon is 0. Thus for the transition from the atomic state for which M = 0 the maximum probability for finding a photon is at the equator of an expanding sphere, just inside the radius r = ct. There are fainter rings of probability for smaller values of r, which correspond to local maxima in Figure 4. These remarks also hold for the coordinate definition of probability. When the transition from the excited state for which M = ±1 causes a photon to be emitted, one has the same expanding sphere, which has a non– zero probability over the entire surface. The probability is greatest at the north

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H. E. Moses

Ann. Henri Poincar´e

and south poles of the sphere, θ = 0 , π. The probability is a minimum at the equator, θ = 0. The graph Figure 4 shows some wriggles. One must ask the question whether the graph hides fine wriggles as in Figure 3. In Figure 5 we expand the horizontal axis.

Energy Density

52815.5 52815 52814.5 52814 52813.5 −10

−5

5

10

(r − ct) × κ

52812.5 Figure 5. The Radial Probability Density Function P0 on an Expanded Horizontal Axis (Unscaled) The probability density P0 shows none of the structure (wave forms) shown by the wave zone vector in Figure 3. The probability density P obtained by using  do not show this structure either. Our notion of the the coordinate operators X photon shape as being a particle which carries a wave form which reproduces the wave form of the radiated field is not valid. The photon is a rather smooth particle which expands like a non-spherical but highly symmetric balloon enclosed by a “soft” layer (the precursor) on the a hard layer near r = ct. On the other hand, the field does have the wave-like structure after the hard layer of the balloon has passed beyond the region of observation.

3 Conclusion The notion of photon as a particle of light having momentum and energy, and, presumably, having position and shape, was introduced by Einstein almost a hundred years ago. Our computation using quantum mechanical ideas and relativistic electromagnetic fields, shows that the field associated with a photon emitted by an atom leads to results much more in consonance with field notions, despite the

Vol. 3, 2002

The Wave Function of the Lyman–Alpha Photon. Part II

813

introduction of position variables and other dynamical variables associated with particles. This result appears to be true for any photon emitted by an atom, molecule, or excited nucleus16 . Perhaps this result explains a paradox that when one uses classical tools, such as gratings and prisms, and classical wave theory to measure radiations due to quantum transitions, one gets very good results, despite the inconsistency of the procedure. However, the photon shape and trajectory are quite different than those which arise from our notion of what a particle should be.

References [1] H.L. Pryce, Proc. Roy. Soc. Lond., A 150, 165 (1935). [2] H.E. Moses and R.T. Prosser, Proc. Roy. Soc. Lond. A, 422, 351 (1989). [3] H. E. Moses, SIAM Jour. App. Math., 21, 114 (1971). [4] H.E. Moses, Annals of Physics, 41, 158 (1967).

Harry E. Moses Applimath Company 150 Tappan Street Brookline, MA 02445 USA email: [email protected] Communicated by Vincent Rivasseau submitted 28/07/01, accepted 26/11/01

To access this journal online: http://www.birkhauser.ch

16 The

sharpness of the resonance is responsible.

Ann. Henri Poincar´e 3 (2002) c Birkh¨
auser Verlag, Basel, 2002

Annales Henri Poincar´ e

Editorial Note

In view of some recent public exchanges, the Editors of Annales Henri Poincar´e wish to reaffirm the journal’s steadfast commitment to the inclusive exchange of scientific ideas, filtered by merit, and to the spirit of nurturing science and its institutions and culture everywhere in the world.

The Editorial Board and the Editor in Chief

Ann. Henri Poincar´e 3 (2002) 817 – 845 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/050817-29

Annales Henri Poincar´ e

Hierarchy of (2 + 1)-Dimensional Nonlinear Schr¨ odinger Equation, Self-Dual Yang-Mills Equation, and Toroidal Lie Algebras S. Kakei, T. Ikeda and K. Takasaki Abstract. The hierarchy structure associated with a (2 + 1)-dimensional Nonlinear Schr¨ odinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using the language of free fermions. A relation to the self-dual Yang-Mills equation is also discussed.

1 Introduction There have been many studies on multi-dimensional integrable evolution equations. An example of such equations was given by Calogero [C], which is a (2 + 1)dimensional extension of the Korteweg-de Vries equation,
x 1 1 uy dx. (1.1) ut = uxxy + uuy + ux 4 2 Bogoyavlensky [Bo1] showed that there is a hierarchy of higher-order integrable equations associated with (1.1). In the previous paper [IT1], two of the present authors generalized the Bogoyavlensky’s hierarchy based on the Sato theory of the Kadomtsev-Petviashvili (KP) hierarchy [Sa, SS, DJKM, JM, JMD, UT], and discussed the relationship to toroidal Lie algebras. We note that the relation between integrable hierarchies and toroidal algebras has been discussed also by Billig [Bi], Iohara, Saito and Wakimoto [ISW1, ISW2] by using vertex operator representations. In this paper, we shall consider a (2 + 1)-dimensional extension of the nonlinear Schr¨ odinger (NLS) equation [Bo2, Sc, St1, St2],
X (|u|2 )Y dX = 0, (1.2) iuT + uXY + 2u and the hierarchy associated with this equation. In the case X = Y , this equation is reduced to iuT + uXX + 2|u|2 u = 0, (1.3) which is the celebrated Nonlinear Schr¨ odinger (NLS) equation. Equation (1.2) is related to the self-dual Yang-Mills (SDYM) equation, and has been studied

818

S. Kakei, T. Ikeda and K. Takasaki

Ann. Henri Poincar´e

by several researchers from various viewpoints: Lax pairs [Bo2, Sc, St1], Hirota bilinear method [SOM, St2], twistor approach [St2], Painlev´e analysis [JB], and so on. Strachan [St2] pointed out that (1.2) is transformed to Hirota-type equations, (iDT + DX DY )G · F = 0,

2 ¯ DX F · F = 2GG,

(1.4)

with the transformation u = G/F . Here we have used the Hirota’s D-operators, Dxm · · · Dyn f · g = (∂x − ∂x
)m · · · (∂y − ∂y
)n f (x, . . . , y)g(x
, . . . , y
)|x
=x,y
=y , (1.5) and the bar ¯· denotes complex conjugation. Based on the bilinear equations (1.4), Sasa, Ohta and Matsukidaira [SOM] constructed determinant-type solutions. Their work strongly suggests that equation (1.2) may be related to the KP hierarchy. The main purpose of the present paper is to generalize the results of the previous work [IT1] so that we can treat equation (1.2) and the SDYM equation. We shall use the language of formal pseudo-differential operators (PsDO for short) that have matrix coefficients. In other words, we will generalize the theory of the multi-component KP hierarchy [Di, Sa, UT] to the (2 + 1)-dimensional NLS hierarchy. We will also use the free fermion operators [DJKM, JM, JMD] to clarify the relation to the toroidal Lie algebras. This paper is organized as follows: In Section 2, we introduce SDYM-type time evolutions to the 2-component KP hierarchy and show that the resulting hierarchy contains the (2 + 1)-dimensional NLS equation (1.2). We also discuss bilinear identity for the τ -functions, and relation to the SDYM equation. In Section 3, we present two ways to construct special solutions. Relation to toroidal Lie algebras is explained in Section 4. Based on the Fock space representation, we derive the bilinear identities from the representation-theoretical viewpoint. Section 5 is devoted to the concluding remarks.

2 Formulation of the (2 + 1)-dimensional NLS hierarchy 2.1

2-component KP hierarchy

We first review the theory of the multi-component KP hierarchy [Di, Sa, UT] in the language of formal pseudo-differential operators with (N × N )-matrix coefficients. Let ∂x denote the derivation ∂/∂x. A formal PsDO is a formal linear combiˆ =  an ∂ n , of integer powers of ∂x with matrix coefficients an = an (x) nation, A x n that depend on x. The index n ranges over all integers with an upper bound. The least upper bound is called the order of this PsDO. The first non-vanishing coefficient aN is called the leading coefficient. If the leading coefficient is equal to I, the unit matrix, the PsDO is said to be monic. It is convenient to use the following notation:   ˆ 0). (4.24) There exists a unique linear map (the vacuum expectation value),

Ψ(α) s

(s < 0), (s = 0), (s > 0),

def

(2)∗

s1 , s2 | = vac|Ψ(1)∗ s1 Ψs2 ,

Ψ(α)∗ s

F ∗ ⊗A F −→ C

(4.25)

such that vac| ⊗ |vac → 1. For a ∈ A we denote by vac|a|vac the vacuum expectation value of the vector vac|a ⊗ |vac (= vac| ⊗ a|vac ) in F ∗ ⊗A F . Using the expectation value, we prepare another important notion of the normal ordering: (α) (β)∗ def (α) (β)∗ (α) (β)∗ : ψi ψj : = ψi ψj − vac|ψi ψj |vac .

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Lemma 1 ([DJKM, JM, JMD]) The operators   E(z) = ψ (1) (z)ψ (2)∗ (z),   F (z) = ψ (2) (z)ψ (1)∗ (z),    H(z) = : ψ (1) (z)ψ (1)∗ (z) : − : ψ (2) (z)ψ (2)∗ (z) : ,

(4.26)

 2 on the fermionic satisfy the OPE (4.5) with c = 1, i.e., give a representation of sl Fock space F . ¿From Lemma 1 and Proposition 4, we have a representation of gtor on F ⊗ Fy . We will use this representation in what follows to derive bilinear identities. Note that the operators E(z), F (z) and H(z) are invariant under the following automorphism of fermions: def Fytor =

(a)

(a)

(a)∗

ι(ψj ) = ψj+1 ,

4.2

ι(ψj

(a)∗

) = ψj+1

(j ∈ Z, a = 1, 2).

(4.27)

Derivation of the bilinear identity from representation theory

We first introduce the following operator acting on Fytor ⊗ Fytor
: Ω

tor def

=



 

m∈ZM α=1,2

dλ (α) ψ (λ)Vm (y; λ) ⊗ ψ (α)∗ (λ)V−m (y
; λ). 2πiλ

(4.28)

Lemma 2 The operator Ωtor enjoys the following properties: (i) (ii)

tor [Ωtor , sltor 2 ⊗ 1 + 1 ⊗ sl2 ] = 0,

Ω

tor

⊗2

(|s2 , s1 ⊗ 1)

= 0.

(4.29) (4.30)

Proof. Since the representation of sltor under consideration is constructed from 2 Lemma 1, it is enough to show   Ωtor , ψ (α) (p)ψ (β)∗ (p)Vn (y; p) ⊗ 1 + 1 ⊗ ψ (α) (p)ψ (β)∗ (p)Vn (y
; p) = 0, (4.31) for α, β = 1, 2 and n ∈ ZM . From (4.20), we have   ψ (α) (p)ψ (β)∗ (q), ψ (γ) (λ) = δβγ δ(q/λ)ψ (α) (p),   ψ (α) (p)ψ (β)∗ (q), ψ (γ)∗ (λ) = −δαγ δ(p/λ)ψ (β) (q).

(4.32)

These equations and the relation Vm (y; λ)Vn (y; λ) = Vm+n (y; λ) give the commutativity above.

Vol. 3, 2002

Hierarchy of (2 + 1)-Dimensional Nonlinear Schr¨ odinger Equation

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If we translate Lemma 2 into bosonic language, then it comes out a hierarchy of Hirota bilinear equations. To do this, we present a summary of the boson(α) fermion correspondence in the 2-component case. Define the operators Hn as  def (α) (α) (α)∗ Hn = j∈Z ψj ψj+n for n = 1, 2, . . . , α = 1, 2, which obey the canonical (α)

(β)

(α)

commutation relation [Hm , Hn ] = mδm+n,0 δαβ · 1. The operators Hn generate the Heisenberg subalgebra (free bosons) of A, which is isomorphic to the algebra (α) (α) with the basis {nxn , ∂/∂xn (α = 1, 2, n = 1, 2, . . . )}. Lemma 3 ([DJKM, JM, JMD]) For any |ν ∈ F and s1 , s2 ∈ Z, we have the following formulas, ( ' (1) (2) s1 , s2 |eH(x ,x ) ψ (1) (λ)|ν (1)

=

(−)s2 λs1 −1 eξ(x

=

(−)s2 λ−s1 e−ξ(x

,λ)

' ( (1) (2) s1 , s2 |eH(x ,x ) ψ (1)∗ (λ)|ν (1)

' ( (1) −1 (2) s1 − 1, s2 |eH(x −[λ ],x ) |ν ,

,λ)

'

s1 + 1, s2 |eH(x

(1)

+[λ−1 ],x(2) )

( ' (1) (2) s1 , s2 |eH(x ,x ) ψ (2) (λ)|ν ' ( (2) (1) (2) −1 = λs2 −1 eξ(x ,λ) s1 , s2 − 1|eH(x ,x −[λ ]) |ν , ( ' (1) (2) s1 , s2 |eH(x ,x ) ψ (2)∗ (λ)|ν ' ( (2) (1) (2) −1 = λ−s2 e−ξ(x ,λ) s1 , s2 + 1|eH(x ,x +[λ ]) |ν ,

( |ν ,

(4.33)

(4.34)

(4.35)

(4.36)

where the “Hamiltonian” H(x(1) , x(2) ) is defined as H(x

(1)

,x

(2)

def

) =

∞  

(α) x(α) n Hn .

(4.37)

α=1,2 n=1

We prepare one more lemma due to Billig [Bi].  nj Pj , where Pj Lemma 4 ([Bi], Proposition 3. See also [ISW2]) Let P (n) = j≥0 n are differential operators that may not depend on z. If n∈Z z P (n)f (z) = 0 for some function f (z), then P ( − z∂z )f (z)|z=1 = 0 as a polynomial in . Now we are in position to state the bilinear identity for the (2+1)-dimensional denote a group of invertible linear transformations on NLS hierarchy. Let SLtor 2 Fytor generated by the exponential action of the elements in sl2 ⊗ R acting locally nilpotently. Define the τ -function associated with g ∈ SLtor 2 as s
,s


def tor

τs21,s12 (x(1) , x(2) , y) = def

s
1 , s
2 |eH(x

(1)

def

,x(2) )

g(y)|s2 , s1 tor ,

(4.38)

where |s2 , s1 tor = |s2 , s1 ⊗ 1 and tor s
1 , s
2 | = s
1 , s
2 | ⊗ 1. Hereafter we shall omit the superscripts “tor” if it does not cause confusion. Since g ∈ SLtor 2 , the

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S. Kakei, T. Ikeda and K. Takasaki

Ann. Henri Poincar´e

τ -function (4.38) have the following properties [JM]: ++1,s2 −+1 +,s2 − τss21+1,s = (−1) τss21,s , 1 1 +1

∂ ∂ s
,s
+ (2) τs21,s12 = 0, (1) ∂xj ∂xj def

(1)

(4.39) (4.40)

(2)

i.e., the τ -function depends only on {xj = xj − xj } and {yj }. Proposition 5 For non-negative integers k, l1 and l2 , the τ -functions satisfy 


dλ s
1 −s

1 +k−2 ξ((x−x
)/2,λ) λ (−1)s2 +s2 e 2πi s
−1,s


s

+1,s

2

(x
+ [λ−1 ], y + bλ )

s
,s
−1

s

,s

+1

(x
− [λ−1 ], y + bλ ) = 0.

× τs 1+l ,s2 +l (x − [λ−1 ], y − bλ )τs21,s1  2 2 1 1 dλ s
2 −s

2 +k−2 ξ((x
−x)/2,λ) λ e + 2πi

× τs21+l22 ,s1 +l1 (x + [λ−1 ], y − bλ )τs21,s12

(4.41)

Proof. This is the direct consequence of Lemmas 2, 3, 4. Setting 0,0 F = τ0,0 ,

1,−1 G = τ0,0 ,

˜ = −τ −1,1 , G 0,0

(4.42)

one can show that (4.41) contains the bilinear equations (2.44)–(2.46) with the condition ˇb + cˇ = 0. We now turn to the 2-dimensional derivative NLS (DNLS) equation [St1],


X 2 iuT + uXY + 2i u (|u| )Y dX = 0. (4.43) X

This equation can also be treated in terms of the bilinear formulation [SOM]. Following Sasa et al., we set u=

fg , f˜2

u=−

f˜g˜ , f2

(4.44)

where we have assumed f = f˜,

g = −˜ g.

(4.45)

The validity of this assumption will be discussed in the next section. These u and u solve (4.43) if the variables f and g obey the Hirota equations, (iDX DY − DT )f · g = 0, (iDX DY + DT )˜ g · f˜ = 0,

(4.46)

(iDX DY + 2DT )f · f˜ = DY g˜ · g, iDX f · f˜ = g˜ g.

(4.48)

(4.47) (4.49)

Vol. 3, 2002

Hierarchy of (2 + 1)-Dimensional Nonlinear Schr¨ odinger Equation

837

We note that our bilinearization is slightly different from that of Sasa et al. The first two equations can be obtained from (4.41) by making the change of the variables, X = ix1 , f=

1,0 τ0,1 ,

g=

Y = y0 , T = y1 , f˜ = τ 0,0 , g˜ = τ 1,−1 .

0,1 τ0,1 ,

0,0

0,0

(4.50)

The bilinear identity including the rest two can be obtained in the same way as Proposition 5: Proposition 6 For non-negative integers k, the τ -functions satisfy  dλ s
1 −s

1 +k−2 ξ((x−x
)/2,λ) s
2 +s

2 (−1) λ e 2πi s
−1,s


s

+1,s



s
,s
−1

(x + [λ−1 ], y − bλ )τs21,s12+1 (x
− [λ−1 ], y + bλ )

× τs21,s1 2 (x − [λ−1 ], y − bλ )τs21,s1 +12 (x
+ [λ−1 ], y + bλ )  dλ s
2 −s

2 +k−2 ξ((x
−x)/2,λ) + λ e 2πi × τs21,s12


s

,s

+1












s ,s s ,s = τs21,s12+1 (x, y0 , yˇ − ˇb)τs21,s12 (x
, y0 , yˇ + ˇb).

(4.51)

Proof. Using    
Ωtor |s2 , s1 tor ⊗ |s2 , s1 + 1 tor = (|s2 , s1 + 1 ⊗ emy0 ) ⊗ |s2 , s1 ⊗ e−my0 (4.52) instead of (4.30), we can derive the desirous result. Expanding (4.51), we can obtain Hirota-type differential equations including the following ones: 1,0 0,0 1,−1 0,1 (Dx1 Dy0 − 2Dy1 )τ0,1 · τ0,0 = Dy0 τ0,0 · τ0,1 , 1,0 Dx1 τ0,1

·

0,0 τ0,0

+

0,1 1,−1 τ0,1 τ0,0

= 0.

(4.53) (4.54)

These equations agree with (4.48) and (4.49).

4.3

Reality conditions and soliton-type solutions

In this section, we consider an algebraic meaning of the reality condition (3.21). To this aim, we introduce an automorphism ρ of the fermion algebra as   (α)∗ ρ ψn(α) = ψ−n−1 ,

  (α) ρ ψn(α)∗ = ψ−n−1

which have the following properties:

(n ∈ Z, α = 1, 2),

(4.55)

838

S. Kakei, T. Ikeda and K. Takasaki

• ρ2 = id,  (α)  (α) • ρ Hn = −Hn

Ann. Henri Poincar´e

(α = 1, 2), ∀

• vac|ρ(g)|vac = vac|g|vac ,

g ∈ SLtor 2 .

We note that the similar automorphism have been discussed by Jaulent, Manna and Martinez-Alonso [JMM]. Assuming the conditions x(i) n ∈ iR (n ∈ N, i = 1, 2),

ρ(g) = g,

(4.56)

we find that ' ( (1) (2)

0, 0|eH(x(1) ,x(2) ) g|0, 0 = 0, 0|eH(x ,x ) g|0, 0 , ' ( (1) (2)

1, −1|eH(x(1) ,x(2) ) g|0, 0 = − 1, 1|eH(x ,x ) g|0, 0 .

(4.57) (4.58)

˜ of (4.42) satisfy the reality conUnder these conditions, the τ -functions F , G, G dition, ˜ F = F, G = −G. (4.59) Next we introduce another automorphism σ to treat the (2 + 1)-dimensional DNLS equation (4.43): σ(ψn(1) ) = ψn(2) , σ(ψn(1)∗ ) = ψn(2)∗ , (1)

(1)∗

σ(ψn(2) ) = ψn+1 , σ(ψn(2)∗ ) = ψn+1 ,

(4.60)

which have the following properties, • If ι(g) = g, then σ 2 (g) = g,  (1)  (2) • σ Hn = H n ,

 (2)  (1) σ Hn = H n ,

• 1, 0|σ(g)|0, 1 = vac|g|vac ,

∀

g ∈ SLtor 2 .

Imposing the conditions (2)

x(1) n = xn

(n ∈ N),

σ(g) = g,

(4.61)

we find that ' ( (1) (2)

0, 0|eH(x(1) ,x(2) ) g|0, 0 = 1, 0|eH(x ,x ) g|0, 1 , ' ( (1) (2)

1, −1|eH(x(1) ,x(2) ) g|0, 0 = − 0, 1|eH(x ,x ) g|0, 1 .

(4.62) (4.63)

In this case, the τ -functions f , g, f˜, g˜ of (4.50) satisfy the reality condition (4.45).

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Hierarchy of (2 + 1)-Dimensional Nonlinear Schr¨ odinger Equation

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As an example of special solutions, we consider soliton-type solutions given by  ' ( (1) (2) s
,s
 (4.64) τs21,s12 x(1) , x(2) = s
1 , s
2 |eH(x ,x ) g N (y)|s2 , s1 , def

N

g N (y) =

 exp aj ψ (1) (pj )ψ (2)∗ (pj )Vmj (y; pj )

j=1

+bj ψ

(2)

(qj )ψ

(1)∗

 (qj )Vnj (y; qj ) .

(4.65)

In the NLS case ρ(g N ) = g N , the parameters should obey the conditions, qj = pj ,

bj = −aj

(j = 1, . . . , N ).

(4.66)

(j = 1, . . . , N ).

(4.67)

In the DNLS case σ(g N ) = g N , we have qj = pj ,

b j = aj p j

We conclude that the NLS case and the DNLS case have different complex structure that correspond to different real forms of the toroidal Lie algebra sltor 2 .

4.4

Bilinear identity for the SDYM hierarchy

The SDYM equation can also be treated also by the Hirota’s bilinear method [SOM]. Toward this aim, we shall take so-called “Yang’s R-gauge” defined as fol¯ such that lows: Due to (2.25), there exist matrix-valued functions G and G ) ) ¯ = GA ¯ y¯, ∂y¯G ∂y G = GAy , (4.68) ¯ ¯ z¯. ∂z G = GAz , ∂z¯G = GA def ¯ −1 , the self-duality equation (2.25) takes If we define the matrix J as J = GG the form     (4.69) ∂y¯ J −1 ∂y J + ∂z¯ J −1 ∂z J = 0.

We then consider the gauge field J of the form,   1 1 τ2 τ1 −g J= , e= , f = , 2 − eg e f f τ5 τ5

g=

τ3 . τ5

(4.70)

The gauge field J of (4.70) solves (4.69) if the τ -functions satisfy the following seven Hirota-type equations [SOM], τ52 + τ2 τ8 − τ4 τ6 = 0, Dy τ1 · τ5 = Dz¯τ4 · τ2 ,

(4.71) (4.72)

Dy τ2 · τ6 = Dz¯τ5 · τ3 ,

(4.73)

Dy τ4 · τ8 = Dz¯τ5 · τ7 , Dz τ1 · τ5 = Dy¯τ2 · τ4 ,

(4.74) (4.75)

Dz τ2 · τ6 = Dy¯τ3 · τ5 , Dz τ4 · τ8 = Dy¯τ7 · τ5 ,

(4.76) (4.77)

where we have introduced auxiliary dependent variables τ4 , τ6 , τ7 , τ8 .

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The bilinear identity associated with these equations is given as follows: Proposition 7 For non-negative integers k, the τ -functions satisfy 


dλ s
1 −s

1 +k−2 ξ((x−x
)/2,λ) (−1)s2 +s2 λ e 2πi s
−1,s


s

+1,s



× τs21,s1 2 (x − [λ−1 ], y − bλ )τs21+1,s12+1 (x
+ [λ−1 ], y + bλ )  dλ s
2 −s

2 +k−2 ξ((x
−x)/2,λ) + λ e 2πi

  s

,s

2 +1 
 s ,s −1 x + [λ−1 ], y − bλ τs21+1,s x − [λ−1 ], y + bλ × τs21,s12 1 +1 s
,s


s

,s



2 = τs21+1,s (x, y0 , yˇ − ˇb)τs21,s12+1 (x
, y0 , yˇ + ˇb) 1













s ,s s ,s2 − τs21,s12+1 (x, y0 , yˇ − ˇb)τs21+1,s (x
, y0 , yˇ + ˇb). 1

(4.78)

Proof. This can be proved in the same fashion as Proposition 5; Use   Ωtor |s1 , s2 tor ⊗ |s1 + 1, s2 + 1 tor  
= (|s1 + 1, s2 ⊗ emy0 ) ⊗ |s1 , s2 + 1 ⊗ e−my0  
− (|s1 , s2 + 1 ⊗ emy0 ) ⊗ |s1 + 1, s2 ⊗ e−my0

(4.79)

instead of (4.30). Expanding (4.78) and applying (4.39), we can obtain the following Hirotatype equations, +1,s2 s1 ,s2 +1 +1,s2 s1 ,s2 +1 ,s2 2 ) + τss21+1,s τ − τss21,s τ = 0, (τss21,s 1 1 s2 ,s1 +1 1 +1 s2 +1,s1 +1,s2 −1 Dy0 τss21,s 1

·

,s2 τss21,s 1

=

−1,s2 +1 ,s2 · τss21,s = Dy0 τss21,s 1 1

+1,s2 Dy1 τss21,s 1 +1 s1 ,s2 +1 Dy1 τs2 ,s1 +1

· ·

+1,s2 τss21+1,s , 1 ,s2 +1 τss21+1,s , 1

(4.80) (4.81) (4.82)

which agree with (4.71)–(4.74) if we set y¯ = y0 , 0,0 τ1 = τ1,−1 , 0,0 τ5 = τ0,0 ,

0,1 τ2 = iτ1,0 , 0,1 τ6 = iτ0,1 ,

z = y1 , −1,1 τ3 = τ0,0 ,

τ7 =

1,−1 τ0,0 ,

1,0 τ4 = iτ1,0 ,

τ8 =

(4.83)

1,0 iτ0,1 .

If we introduce another set of variables {zj (j = 0, 1, . . .)} that play the same role as {yj } and set z¯ = z0 , y = −z1 , the corresponding τ -functions solve (4.71)–(4.77) simultaneously. We remark that the introduction of the variables {zj } corresponds to the symmetry of the 3-toroidal Lie algebra as mentioned in Section 4.1. To consider the reality condition for the SU (2)-gauge fields, we introduce an anti-automorphism κ as     (4.84) κ ψn(α) = ψn(α)∗ , κ ψn(α)∗ = ψn(α) (n ∈ Z, α = 1, 2), which have the following properties:

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Hierarchy of (2 + 1)-Dimensional Nonlinear Schr¨ odinger Equation

841

• κ2 = id, • vac|κ(g)|vac = vac|g|vac ,

∀

g ∈ SLtor 2 .

Using κ, we impose the following condition on g = g(y, z): κ(g(y, z)) = g(y, z).

(4.85)

Then we find that the τ -function (4.38) with x(1) = x(2) = 0 obeys

s
1 , s
2 |g(y, z)|s2 , s1 = s1 , s2 |g(y, z)|s
2 , s
1 ,

(4.86)

and that e, f and g of (4.70) satisfies f = −f, ˜ as If we define J ˜= J

 ω 0

  0 ω J ω −1 0

e = g.  0 , ω −1

(4.87)

1+i ω= √ , 2

(4.88)

˜ = tJ ˜ (See, for example, [Pr]). ˜ satisfies (4.69) and the reality condition J then J

5 Concluding remarks We have described the hierarchy structure associated with the (2 + 1)-dimensional NLS equation (1.2) based on the theory of the KP hierarchy, and discussed several methods to construct special solutions. Using the language of the free fermions, we have obtained the bilinear identities from the representation of the toroidal Lie algebras. The solutions constructed explicitly in this paper are limited in the class of soliton-type. In case of the hierarchy of the (2 + 1)-dimensional KdV equation (1.1), an algebro-geometric construction of the Baker-Akhiezer function is indeed possible [IT2]. It may be also possible to discuss algebro-geometric (“finite-band”) solutions for the (2 + 1)-dimensional NLS hierarchy by extending our construction of the soliton-type solutions. Furthermore, by extending our theory, it may be possible to consider (2 + 1)dimensional generalizations of other soliton equations, such as the sine-Gordon equation, the Toda lattice, and so on. We will discuss the subjects elsewhere.

Acknowledgments The authors would like to thank Dr. Yasuhiro Ohta, Dr. Yoshihisa Saito, Dr. Narimasa Sasa for their interests and discussions. The first author is partially supported by Waseda University Grant for Special Research Project 2000A-155, and the Grant-in-Aid for Scientific Research (No. 12740115) from the Ministry of Education, Culture, Sports, Science and Technology.

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K. Iohara, Y. Saito and M. Wakimoto, “Hirota bilinear forms with 2-toroidal symmetry”, Phys. Lett. A254, 37–46 (1999); Erratum, ibid. A268, 448–449 (2000).

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M. Jimbo, T. Miwa and E. Date, Solitons, Cambridge Tracts in Mathematics 135, Cambridge University Press (2000).

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V.G. Kac, Infinite dimensional Lie algebras, 3rd ed., Cambridge University Press (1990).

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V.G. Kac, Vertex algebras for beginners, 2nd ed., University lecture series 10, AMS (1998).

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C. Kassel, K¨ahler differentials and coverings of complex simple Lie algebras extended over commutative algebras, J. Pure Appl. Algebra 34, 265–275 (1985).

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I.A.B. Strachan, Some integrable hierarchies in (2 + 1) dimensions and their twistor description, J. Math. Phys. 34, 243–259 (1993).

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N. Sasa, Y. Ohta and J. Matsukidaira, Bilinear form approach to the self-dual Yang-Mills equations and integrable systems in (2 + 1)dimension, J. Phys. Soc. Jpn. 67, 83–86 (1998).

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Vol. 3, 2002

Hierarchy of (2 + 1)-Dimensional Nonlinear Schr¨ odinger Equation

Saburo Kakei Department of Mathematical Sciences School of Science and Engineering Waseda University Ohkubo 3-8-1, Shinjyuku-ku Tokyo 169-8555 Japan (Present address: Department of Mathematics Rikkyo University Nishi-ikebukuro 3-34-1, Toshima-ku Tokyo 171-8501 Japan email: [email protected]) Takeshi Ikeda Department of Applied Mathematics Okayama University of Science Ridaicho 1-1 Okayama 700-0005 Japan email: [email protected] Kanehisa Takasaki Department of Fundamental Sciences Faculty of Integrated Human Studies Kyoto University Yoshida, Sakyo-ku Kyoto 606-8501 Japan email: [email protected] Communicated by Tetsuji Miwa submitted 30/07/01, accepted 29/04/02

To access this journal online: http://www.birkhauser.ch

845

Ann. Henri Poincar´e 3 (2002) 847 – 881 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/050847-35

Annales Henri Poincar´ e

Boundary WZW, G/H, G/G and CS theories K. Gaw¸edzki

Abstract. We extend the analysis [19] of the canonical structure of the Wess-ZuminoWitten theory to the bulk and boundary coset G/H models. The phase spaces of the coset theories in the closed and in the open geometry appear to coincide with those of a double Chern-Simons theory on two different 3-manifolds. In particular, we obtain an explicit description of the canonical structure of the boundary G/G coset theory. The latter may be easily quantized leading to an example of a twodimensional topological boundary field theory.

1 Introduction Bidimensional boundary conformal field theory is a subject under intense study in view of its applications to boundary phenomena in 1+1- or two-dimensional critical systems and to the brane physics in string theory. Although much progress has been achieved in understanding boundary CFT’s since the seminal paper of Cardy [6], much more remains to be done. The structure involved in the boundary theories is richer than in the bulk ones and their classification program involves new notions and an interphase with sophisticated mathematics [28][30]. One approach that offered a conceptual insight into the properties of correlation functions of boundary conformal models consisted of relating them to boundary states in threedimensional topological field theories [12][13]. In the simplest case of the boundary Wess-Zumino-Witten (WZW) models (conformal sigma models with a group G as a target [34]), the topological three-dimensional model appears to be the group G Chern-Simons (CS) gauge theory [32][35]. In [19] it has been shown how the relation between the boundary WZW model and the CS theory arises in the canonical approach. The purpose of the present paper is to extend the analysis of [19] to the case of the coset G/H models of conformal field theory obtained by gauging in the group G WZW model the adjoint action of a subgroup H ⊂ G. In the WZW model the simplest class of boundary conditions is obtained by restricting the boundary values of the classical G-valued field g to fixed conjugacy classes in the group labeled by weights of the Lie algebra g of G. Such boundary conditions reduce to the Dirichlet conditions for toroidal targets. It was shown in [19] that the phase space of the WZW model on a strip with such boundary conditions is isomorphic to the phase space of the CS theory on the time-line R times a disc D with two time-like Wilson lines. In the coset models we shall use more general boundary conditions requiring that field g belongs on the boundary components to

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pointwise product of group G and subgroup H conjugacy classes. The phase space of the coset theory on a strip with such boundary conditions becomes isomorphic to the phase space of the double CS theory on R × D with group G and group H gauge fields, both coupled to two time-like Wilson lines. In particular, the phase space of the boundary G/G coset model1 becomes isomorphic to the moduli space of flat connections on the 2-sphere with four punctures. The latter case lends itself easily to quantization giving rise to an example of the two-dimensional boundary topological field theory, a structure that promises to play the role of the K-theory of loop spaces [28]. Much of the motivation for the present work stemmed from interaction with Volker Schomerus who generously shared his insights with the author. The discussions with Laurent Freidel are also greatfully acknowledged. Special thanks are due to the Erwin Schr¨ odinger Institute in Vienna were this work was started. After the present work was finished, we have received the paper [9] which discussed the same boundary conditions in the coset theories that the ones proposed here.

2 Action functionals of the WZW and coset theories The Wess-Zumino-Witten model is a specific two-dimensional sigma model with a group manifold G as the target. For simplicity, we shall assume that G is compact connected and simply connected. We shall denote by g the Lie algebra of G. The G-valued fields of the WZW model are defined on two-dimensional surfaces Σ (the “worldsheets”) that we shall take oriented and equipped with a conformal or pseudo-conformal structure. The action of the model in the Euclidean signature is the sum of two terms:
k ¯ + S W Z (g) . (2.1) S(g) = 4πi tr (g −1 ∂g)(g −1 ∂g) Σ

Above, tr stands for the Killing form normalized so that the long coroots have length squared equal to 2. The second (Wess-Zumino) term in the action is related to the canonical closed 3-form χ(g) = 13 tr (g −1 dg)3 on G. Informally, it may be written as
k WZ (g) = 4πi g ∗ ω (2.2) S Σ

where ω is a 2-form on G such that dω = χ. This definition is, however, problematic since there is no global ω with the last property. If Σ has no boundary then the 1 As discussed in detail in [11], there are other ways to impose boundary conditions in the G/G theory relating it to the boundary topological Poisson sigma models of [7]

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Boundary WZW, G/H, G/G and CS Theories

problem may be solved by setting [34] S W Z (g) =

k 4πi




849

g∗ χ ,

(2.3)

B

where B is a 3-manifold such that ∂B = Σ and g extends g to B. It is well known that this determines S W Z (g) modulo 2πik so that the amplitudes exp[−S W Z (g)] are well defined if k is integer. By the Stokes Theorem, the definition (2.3) reduces to the naive expression (2.2) whenever g maps into the domain of a local form ω. The variation δS W Z (g) involves only the 3-form χ so that the classical equations are determined unambiguously. For surfaces with boundary, one should impose proper boundary conditions on fields g. Let ∂Σ = Sn1 where Sn1 are disjoint circles always considered with the orientation inherited from Σ. Let, for µ in the Cartan subalgebra of g, Cµ denote the conjugacy class {γ e2πiµ γ −1 | γ ∈ G}. We shall require that g(Sn1 ) ⊂ Cµn .

(2.4)

These are the so called fully symmetric conformal boundary conditions. When restricted to a conjugacy class Cµ , the 3-form χ becomes exact. In particular, the 2-form ωµ (g) = tr (γ −1 dγ) e2πiµ (γ −1 dγ) e−2πiµ

(2.5)

on Cµ satisfies dωµ = χ|Cµ . Let Σ = Σ#(Dn ) be the surface without boundary obtained from Σ by gluing discs Dn to the boundary components Sn1 of Σ, see Fig. 1.

Σ

Σ’ Fig. 1

Each field g satisfying the boundary conditions (2.4) may be extended to g  : Σ → G in such a way that g  (Dn ) ⊂ Cµn (the conjugacy classes are simply connected). Following [3][16], we shall define the WZ-action of the field g satisfying the boundary conditions (2.4) by setting 
∗ k WZ WZ  (g) = S (g ) − 4πi g  ωµn . (2.6) S n D n

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This again reduces to the naive definition whenever g maps into the domain of a form ω such that dω = χ, provided that the restrictions of ω to Cµn coincide with ωµn . A different choice of the restrictions would change the boundary contributions to the classical equations. As explained in [16], (for k = 0) the right hand side of (2.6) is well defined modulo 2πi iff k is an integer and Cµn = Cλn /k for integrable weights2 λn . The boundary conditions are thus labeled by the same set as the bulk primary fields of the current algebra also corresponding to integrable weights. This is an illustration of the Cardy’s theory of boundary conditions [6]. The G/H coset theories [21] may be realized as the versions of the group G WZW theory where the adjoint action of the subgroup H ⊂ G has been gauged [5][17][23][18]. Let A denote a 1-form with values in ih, where h stands for the Lie algebra of H. For ∂Σ = ∅, the action of the theory coupled to the gauge field A = A10 + A01 is S(g, A) = S(g) −

k 2πi




 ¯ tr (g∂g −1 )A01 + A10 (g −1 ∂g)

Σ

 +gA10 g −1A01 − A10 A01 .

(2.7)

In fact, getting rid of the so called “fixed point problem” [31][14] (that obstructs factorization properties of the theory) requires considering the WZW theory coupled to gauge fields in non-trivial H/Z-bundles, where Z is the intersection of the center of G with H [29][22]. For simplicity, we shall not do it here. For the surfaces with boundary ∂Σ = Sn1 , we shall use the same formula (2.7) to include the coupling to the gauge field, but we shall admit more general boundary conditions for the field g than the ones considered before. Namely, we shall assume that g|Sn1 = gn h−1 n

with gn : Sn1 → CµGn ,

hn : Sn1 → CνHn .

(2.8)

In other words, we shall admit fields g that, on each boundary component, are a pointwise product of loops in conjugacy classes of, respectively, group G and group H, keeping also track of the decomposition factors3 . We shall label such conditions by pairs (µn , νn ) ≡ Mn . For νn = 0, they reduce to the conditions considered in the previous section. We still need to generalize the definition of the Wess-Zumino term of the action to fields g satisfying (2.8). Such fields may be extended to maps g  : Σ → G in such a way that g  |Dn = gn hn−1 and 2 The

3 The

with

gn (Dn ) ⊂ CµGn ,

g  |Sn1 = gn ,

hn |Sn1 = hn

hn (Dn ) ⊂ CνHn .

weights integrable at level k are the ones lying in the positive Weyl alcove inflated by k decomposition of elements of the pointwise product CµGn (CνHn )−1 might not be unique.

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We shall define then S W Z (g) = S W Z (g  ) −

k 4πi


 n

∗

∗

gn ωµGn − hn ωνHn

 −1 −1 + tr (gn dgn )(hn dhn ) . Dn

(2.9)

−1

The form in the brackets has (gn hn )∗ χ as the exterior derivative which assures invariance of the right hand side under continuous deformations of gn and hn inside discs Dn . If k = 0 and H is simply connected then a slight extension of the argument in [16] shows that S W Z (g) given by (2.9) is well defined modulo 2πi iff k is integer and CµGn = CλGn /k , CνHn = CηHn /k , where λn and ηn are integrable weights of g and h, respectively. We shall use (2.1), (2.9) and (2.7) to define the complete gauged action S(g, A). With the above choices of the boundary labels, the gauge invariance exp[−S(hgh−1 , hAh−1 + hdh−1 )] = exp[−S(g, A)]

(2.10)

holds for h : Σ → H. If H has an abelian factor, then the same selection of boundary conditions is imposed if we add to the demand that exp[−S W Z (g)] be well defined the requirement of the gauge-invariance (2.10). For example, for the parafermionic SU (2)/U (1) coset theory, this restricts the boundary labels to pairs (λn , ηn ) = (jn σ3 , mn σ3 ) with jn = 0, 12 , . . . , k2 and mn = 0, 12 , . . . , k − 12 . The labels of the parafermionic primary states (j, m) have additional selection rule j = m mod 1 and the identification (j, m) ∼ ( k2 − j, m + k2 mod k). For the boundary labels, the first may be imposed by requiring the gauge invariance with respect to h : Σ → U (1)/Z2 and the second by identifying the decompositions −1 gn h−1 . Similarly, in the general case we may impose the local n and (−gn )(−hn ) H/Z gauge invariance and identify the decompositions differing by an element in Z [20]. Such restrictions lead to the same labeling of the boundary conditions and of the primary fields, but is not obligatory if we ignore the fixed point problem. Since the gauge field A enters quadratically into the action (2.7), it may be eliminated classically (and also quantum mechanically) from the equations of motion. What results is a sigma model with the space G/Ad(H) of the orbits of the adjoint action of H on G as the target. The target space G/Ad(H) (that may be singular) comes equipped with a specific metric, a non-meric volume form (“dilaton field”) and a 2-form. Let [g] denote the projection of g to G/Ad(H). The boundary conditions (2.8) restrict the boundary values of [g] to the projection to G/Ad(H) of the rotated G-group conjugacy class CµGn e−2πiνn (but contain more data if the decomposition gn h−1 n is not unique). For example, for G = SU (2) with elements ( z¯z


−z
z¯

), where |z|2 + |z  |2 = 1, and for H = U (1), the coset space

G/Ad(H) may be identified with the unit disc D = {z | |z| ≤ 1}. The boundary conditions (2.8) with (λn , νn ) corresponding to (jn , mn ) restrict the boundary

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values of [g] to the intervals [e 2πi(j−m)/k , e−2πi(j+m)/k ] ⊂ D with 2k end-points on the disc boundary. Since the conjugacy classes of U (1) are composed of single points, the decomposition gn h−1 n is unique in this case, given the conjugacy class labels. Imposing the restriction j = m mod 1 eliminates half of the interval endpoints [26].

3 Canonical structure of the WZW and coset theories The classical field theory studies the solutions of the variational problem δS = 0 determined by the action functional S. The space of solutions on a worldsheet with the product structure R × N and Minkowski signature admits a canonical closed 2-form Ω, see e.g. [15] or [19]. If the latter is degenerate (a situation in gauge theories, where the degenerate directions correspond to local gauge transformations), one passes to the space of leaves of the degeneration distribution. By definition, the resulting space is the phase space of the theory and it carries the canonical symplectic structure4 .

3.1

Bulk WZW model

Let us start with the well known case of the WZW model on the cylinder Σ = R × S 1 = {(t, x mod 2π)}. The variational equation δS = 0 becomes a non-linear version of the wave equation ∂+ (g −1 ∂− g) = 0 ,

(3.1)

where ∂± = ∂x± with x± = x±t. The solutions may be labeled by the Cauchy data g(t, · ) and (g −1 ∂t g)(t, · ). The space of solutions forms the phase space P W ZW of the bulk WZW model. Its canonical symplectic form is given by the expression [15] Ω

WZ

=

k 4π


2π  tr − δ(g −1 ∂t g) g −1 δg 0

 + 2 (g −1 ∂+ g) (g −1 δg)2 (t, x) dx

(3.2)

which is t-independent5 . Similarly as for the wave equation, the general solution of (3.1) may be decomposed as g(t, x) = g (x+ ) gr (x− )−1 .

(3.3)

The left-right movers g,r : R → G are not necessarily periodic but satisfy g,r (y + 2π) = g,r (y) γ for the same γ ∈ G. They are determined uniquely up 4 We ignore here the eventual problems with the infinite-dimensional character of the spaces and singularities that may be usually dealt with in concrete situations 5 We use the symbol δ for the exterior derivative on the space of classical solutions

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to the simultaneous right multiplication by an element of G. The expression of the symplectic form in terms of the left-right movers is described in Appendix A. The currents J = ik g ∂+ g −1 = ik g ∂+ g−1 ,

Jr = ik g −1 ∂− g = ik gr ∂+ gr−1

(3.4)

generate the current algebra symmetries of the theory. The conformal symmetries are generated by the components T =

1 2k

tr J2 ,

Tr =

1 2k

tr Jr2

(3.5)

of the energy momentum tensor.

3.2

Bulk G/H model

In the same cylindrical worldsheet geometry Σ = R × S 1 , the classical equations for the coset G/H model take the form: D+ (g −1 D− g) = 0 , E g −1 D− g = 0 = E g D+ g −1 , F (A) = 0 ,

(3.6)

where D± = ∂± + [A± , · ] are the light-cone covariant derivatives, E is the orthogonal projection of g onto h and F (A) = dA + A2 is the curvature of A. The equations are preserved by the H-valued gauge transformations of the fields. The gauge transformations provide for the degeneration of the canonical closed 2-form on the space of solutions so that the phase space P G/H of the bulk coset theory is composed of the gauge-orbits of the solutions of the classical equations (3.6). The gauge-orbits of solutions may be parametrized in a more effective way. The flat gauge field A may be expressed as h−1 dh for h : R2 → H such that h(t, x + 2π) = ρ−1 h(t, x) for some ρ ∈ H. The map h is determined uniquely up to the left multiplication by an element of H. Let us set g = hgh−1 . Note that g : R2 → G with g(t, y + 2π) = ρ−1 g(t, y) ρ. In terms of field  g, the classical equations reduce to ∂+ ( g −1 ∂−  g) = 0 ,

E g−1 ∂− g = 0 = E g ∂+ g−1 .

(3.7)

The gauge-orbits of the classical solutions of (3.6) are in ono-to-one correspondence with the orbits of pairs ( g , ρ) under the simultaneous conjugation by elements of H. In terms of these data, the canonical symplectic form on P G/H , obtained following the general prescriptions of [15], is given by Ω

G/H

=

k 4π


2π   tr − δ( g −1 ∂t  g) g−1 δ g + 2 ( g −1 ∂+ g) ( g −1 δ g )2 (t, x) dx 0

  k g(t, 0)−1 (δρ)ρ−1  + 4π tr (δρ)ρ−1  g(t, 0)

 g)(t, 0) − ((δ g ) g −1 )(t, 0) − ( g −1 δ

(3.8)

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for any fixed t. The solutions of the classical equations (3.7) may be expressed again by the left-right movers:  g(t, x) = g (x+ )gr (x− )−1 , where g,r : R → G are such that −1 = 0 E g,r ∂y g,r

and

g,r (y + 2π) = ρ−1 g,r (y) γ .

(3.9)

Given  g , the one-dimensional fields g,r are determined up to the simultaneous right multiplication by an element of G. The expression for the symplectic form ΩG/H in terms of the left-right movers is given in Appendix A. The left-right components of the energy-momentum tensor T = − 2 tr (g D+ g −1 )2 = − 2 tr (g ∂+ g−1 )2 , k

k

(3.10) Tr =

k −2

tr (g

−1

2

D− g)

=

k −2

tr (gr ∂+ gr−1 )2

generate the conformal symmetries of the bulk coset model.

3.3

Bulk G/G model

For the topological coset G/G theory, the classical equations (3.7) reduce to g−1 d g = 0, i.e.  g is constant and it commutes with the monodromy ρ. The phase space P G/G may be identified with the space of commuting pairs ( g , ρ) in G modulo simultaneous conjugations. It is finite-dimensional, in agreement with the topological character of the theory. It comes equipped with the symplectic form    k (3.11) g − g −1 δ g − (δ g ) g −1 . ΩG/G = 4π tr (δρ)ρ−1 g−1 (δρ)ρ−1  Up to a simultaneous conjugation, g = e 2πi µ and ρ = e 2πi ν for µ and ν in the Cartan algebra and the symplectic form becomes a a constant form on the product of two copies of the Cartan algebra. ΩG/G = 2π k tr [dν dµ] .

(3.12)

In particular, conjugation-invariant functions of g Poisson-commute and so do those of ρ.

3.4

Boundary WZW model

The canonical treatment of the boundary theories is quite analogous to that of the bulk ones, except for the necessity to treat the boundary contributions. We consider the strip geometry Σ = R × [0, π] with Minkowski signature and impose on the field g : Σ → G of the WZW model the boundary conditions discussed in Sect. 2: g(t, 0) ∈ Cµ0 ,

g(t, π) ∈ Cµπ .

(3.13)

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For variations δg tangent to the space of fields respecting conditions (3.13), the classical equations δS(g) = 0 reduce to the bulk equation (3.1) supplemented with the boundary equations g −1 ∂− g + g ∂+ g −1 = 0

for x = 0, π .

(3.14)

The classical solutions obeying (3.13) form the phase space PµW0 µZπ of the boundary WZW model. Its symplectic form is given by [19] Z ΩW µ0 µπ =


π

k 4π

  tr − δ(g −1 ∂t g) g −1 δg + 2(g −1 ∂+ g) (g −1 δg)2 (t, x) dx

0

+

k 4π



ωµ0 (g(t, 0)) − ωµπ (g(t, π))

 (3.15)

for any fixed t. As in the bulk, the classical equations may be solved explicitly, as was described in [19]. We have6 g(t, x) = g (x+ ) m0 g (−x− )−1 = g (x+ ) mπ g (2π − x− )−1 ,

(3.16)

where m0 ∈ Cµ0 , mπ ∈ Cµπ and g : R → G satisfy g (y + 2π) = g (y) γ

for

γ = m−1 0 mπ .

(3.17)

Note that the boundary conditions (3.13) are fulfilled. The orbits of the triples (g , m0 , mπ ) under the right multiplication of g by elements of G accompanied by the inverse adjoint action on m0 and mπ are in one-to-one correspondence with the classical solutions. The expression of the symplectic form in terms of these data is given in Appendix A. The boundary WZW theory has a single current 1 tr J 2 . J = ik g ∂+ g−1 with the corresponding energy-momentum tensor T = 2k

3.5

Boundary G/H model

For the boundary coset G/H model with the G-valued field g and ih-valued gaugefield A defined on the strip R × [0, π], we shall impose the boundary conditions g(t, 0) = g0 (t) h0 (t)−1 ,

g(t, π) = gπ (t) hπ (t)−1

(3.18)

with g0 , h0 , gπ and hπ mapping the boundary lines into the conjugacy classes CµG0 , CνH0 , CµGπ and CνHπ , respectively, see (2.8). The gauge fields A will not be restricted. We shall label such boundary conditions by the pairs (M0 , Mπ ), where M0 = (µ0 , ν0 ) and Mπ = (µπ , νπ ). The variational equations δS(g, A) = 0 reduce now to the bulk equations (3.6) supplemented with the boundary equations (g

−1

−1 h−1 0 D t h0 = 0 = hπ D t hπ , −1 D− g)( · , 0) + h0 (g D+ g )( · , 0) h−1 = 0, 0

(g −1 D− g)( · , π) + hπ (g D+ g −1 )( · , π) h−1 = 0, π where Dt = D+ − D− is the covariant derivative along the boundary. 6 We

use here a slightly different parametrization of the solutions than in [19].

(3.19) (3.20) (3.21)

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The flat gauge field A may be gauged away by representing it as h−1 dh for h mapping the strip to G. Setting as in the bulk geometry g = hgh−1 and, on the boundary components,  g0 = hg0 h−1 ,  h0 = hh0 h−1 , and similarly for gπ amd  hπ , we reduce the bulk equations to (3.7) and the boundary equations to g)( · , 0) + n0 ( g ∂+ g−1 )( · , 0) n−1 = 0, ( g −1 ∂−  0

(3.22)

g ∂+ g−1 )( · , π) n−1 = 0, ( g −1 ∂− g)( · , π) + nπ ( π

(3.23)

with  h0 and  hπ equal, respectively, to constant elements n0 ∈ CνH0 and nπ ∈ CνHπ . The boundary conditions (3.18) become: g(t, 0) = g0 (t) n−1 0

g(t, π) =  gπ (t) n−1 π

(3.24)

for g0 mapping the line into CµG0 and gπ into CµGπ . As in the bulk case, the G/H phase space PM of the G/H coset model with the boundary conditions (3.18) is 0 Mπ composed of the gauge-orbits of the classical solutions. The latter are in one-to-one correspondence with the orbits of the triples ( g, n0 , nπ ) under the simultaneous conjugation by elements of H. In this parametrization, the symplectic form of the boundary theory is given by G/H

ΩM0 Mπ =

k 4πi

+

k 4πi

−

k 4πi


π

  tr − δ( g −1 ∂t  g) g−1 δ g + 2( g −1 ∂+ g) ( g −1 δ g )2 (t, x) dx

0  ωµG0 ( g0 (t)) − ωνH0 (n0 ) + tr ( g0−1 δ g0 )(t) n−1 0 δn0   ωµGπ ( gπ (t)) − ωνHπ (nπ ) + tr ( gπ−1 δ gπ )(t) n−1 π δnπ

(3.25)

for any fixed t. Similarly as in the case of the WZW model, see (3.16) and (3.17), the twodimensional fields  g satisfying the classical equations (3.22), (3.23) and the boundary conditions (3.24) may be rewritten in terms of a one-dimensional field g as + − −1 −1 nπ g(t, x) = g (x+ ) m0 g (−x− )−1 n−1 0 = g (x ) mπ g (2π − x )

(3.26)

with m0 ∈ CµG0 , mπ ∈ CµGπ and g : R → G satisfying E g ∂y g−1 = 0

and

g (y + 2π) = ρ−1 g (y) γ

−1 for ρ = n−1 0 n π , γ = m0 mπ .

(3.27)

Given ( g , n0 , nπ ), the triple (g , m0 , mπ ) is determined up to the right multiplication of g by an element of G accompanied by the adjoint action of its inverse may be rewritten in terms of the on m0 and mπ . The symplectic form ΩG/H M0 Mπ data (g , m0 , n0 , mπ , m0 , nπ ). The result is given by formula (A.2) in Apendix A. The single energy-momentum component of the boundary G/H model is T = − k2 tr (g ∂+ g−1 )2 .

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Boundary G/G model

In the boundary topological coset G/G theory, the classical equations imply that h−1 h−1 g0 ,  h0 ,  gπ ,  hπ and  g = g0 gπ  are all constant so that the phase space π 0 =  G/G PM0 Mπ of the boundary G/G theory is composed of the orbits under simultaneous conjugations of the quadruples h0 , gπ ,  hπ ) ∈ Cµ0 × Cν0 × Cµπ × Cνπ ⊂ G4 ( g0 ,  with

h−1 h−1 g0  = gπ  π . 0

(3.28)

The symplectic form is given by   k  = 4πi ωµ0 ( g0 ) − ων0 ( h0 ) + tr ( g0−1 δ g0 )  h−1 ΩG/G 0 δ h0 M0 Mπ   k  − 4πi ωµπ ( gπ ) − ωνπ ( hπ ) + tr ( gπ−1 δ gπ )  h−1 π δ hπ .

(3.29)

Using this expression, one may check that conjugation invariant functions of g Poisson commute. Below, we shall find a more transparent interpretation for the phase spaces of the two-dimensional theories described above, including the last example.

4 Canonical structure of the CS theory The classical Chern-Simons theory [32, 35] is determined by the action functional of ig-valued 1-forms A (gauge fields, connections) on an oriented 3-manifold M
  k 2 CS S (A) = − 4π tr A dA + 3 A3 (4.1) M

that does not require a metric on M for its definition.

4.1

Case without boundary

If M has no boundary then, under the G-valued gauge transformations g : M→ G
k g∗χ . (4.2) S CS (gAg −1 + g dg −1 ) = S CS (A) − 4π M

In particular, the action is invariant under gauge transformations homotopic to CS 1 and, for integer k, e−S (A) is invariant under all gauge transformations. The classical equations δS CS = 0 are well known to require that F (A) = 0 with the solutions corresponding to flat connections. In the cylindrical geometry M = R × Σ with ∂Σ = ∅, the canonical closed 2-form on the space of solutions is

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degenerate along the gauge directions. Writing A = A + A0 dt where A is tangent to Σ and t is the coordinate of R, we may use the gauge freedom to impose the condition A0 = 0. In this gauge, the classical equations reduce to ∂t A = 0 ,

F (A) = 0

(4.3)

with the solutions given by static flat connections on the surface Σ. The canonical closed 2-form on the space of solutions becomes
k tr (δA)2 . (4.4) ΩCS = 4π Σ

Its degeneration is given by the static gauge transformations A → gAg −1+ g dg −1 . The phase space P CS of the CS theory is then composed of the gauge-orbits of flat gauge fields A. In other words, P CS coincides with the moduli space of flat connections on Σ. Formula (4.4) defines the canonical symplectic structure on P CS . Below, we shall need several refinements of the above well known scheme.

4.2

Wilson lines

First of all, the CS theory may be coupled to a Wilson line C ⊂ M marked with a label µ belonging to the Cartan subalgebra of g. Let γ be a G-valued map defined on the line C. In the presence of these data, the action functional is modified to
CS CS S (A, γ) = S (A) + ik tr µ γ −1 (d + A)γ C

= S CS (gAg −1 + g dg −1 , gγ) for g homotopic to 1. The corresponding classical equations read   F (A) = 2πi γµγ −1 C , d(γµγ −1 ) + [A, γµγ −1 ] C = 0 ,

(4.5)

(4.6)

where C is viewed as a singular current. They imply that A is a flat connection with a singularity on C. In the cylindrical geometry M = R × Σ with the Wilson line R × {ξ} we may still go to the A0 = 0 gauge in which the classical equations reduce to ∂t A = 0 ,

∂t (γµγ −1 ) = 0 ,

F (A) = 2πi γµγ −1 δξ .

(4.7)

The canonical 2-form ΩCS on the space of solutions has now the form µ = ΩCS − ik tr µ(γ −1 dγ)2 ΩCS µ

(4.8)

where the last term is the Kirillov-Kostant 2-form on the (co)adjoint orbit Oµ in g passing through µ. The orbits of pairs (A, γµγ −1 ) solving (4.7) under the time-

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independent gauge transformations7 form the phase space PµCS of the theory. ΩCS µ defines on PµCS the canonical symplectic structure. Of course, one may consider the CS theory with several Wilson lines.

4.3

Boundaries

If the 3-manifold M has a boundary then one needs to impose boundary conditions on the gauge fields A. In the cylindrical geometry M = R × Σ where ∂Σ = ∅, we may require that A0 = 0

on

R × ∂Σ .

(4.9)

The classical equations are still given by F (A) = 0 and the closed 2-form by (4.4). The only modification is that the degeneration of the latter is given now by the gauge transformations equal to 1 on ∂Σ. The same remarks pertain to the case with time-like Wilson lines.

4.4

Double CS theory

The last modification of the CS theory on a 3-manifold M = R × Σ that we shall need is the double theory [29] with a pair (A, B) of the, respectively, group G and group H ⊂ G gauge fields. The action functional of the double theory is the difference of the CS actions for group G and H : S 2CS (A, B) = S S (A) − S CS (B) .

(4.10)

On the boundary R × ∂Σ we shall impose the boundary conditions (1 − E) A0 = 0 ,

E A0 = B0 ,

E Aτ = Bτ ,

(4.11)

where Aτ denotes the component of A tangent to ∂Σ. The phase space of the double theory P 2CS is composed of the pairs (A, B) of flat connections on Σ satisfying the last condition of (4.11), modulo G-valued gauge transformations of A and H-valued ones of B that coincide on the boundary of Σ. The symplectic form
  k 2CS (4.12) = 4π tr (δA)2 − (δB)2 . Ω Σ

Clearly, both gauge fields may be coupled to time-like Wilson lines with labels in the Cartan subalgebras of g and h, respectively. In the particular case when  with H = G, the double CS theory reduces to the single one on the space R × Σ  = Σ#(−Σ) obtained by gluing Σ along the boundary to its the double surface Σ copy with reversed orientation. The phase spaces reduce accordingly. 7 In fact, the singular terms in (4.7) require some care. A possible way is to consider only solutions of (4.7) that around ξ are of the form A = i γµγ −1 dϕ , where ϕ denotes the argument of a local complex parameter and to admit the gauge transformations that are constant around ξ. Different choices of local parameters lead then to canonically isomorphic phases spaces.

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5 Symplectic relations between the WZW, coset and CS theories The symplectic structure of the phase spaces of the WZW and coset theories is given by the complicated expressions, see (3.2), (3.8), (3.11), (3.15), (3.25), (3.29), (A.1) and (A.2). Although obtained by applying the general procedures of [15][19], these expressions are far from being transparent. On the other hand, the interpretation of the symplectic structure of the phase spaces of the CS theory determined by the standard constant symplectic form on the space of two-dimensional gauge fields and by the Kirillov-Kostant form on the coadjoint orbits, see (4.4), (4.8) or (4.12), has a clear interpretation. In the present section, we shall describe symplectic isomorphisms between the phase spaces of the WZW and coset theories and those of the CS theory, elucidating this way the canonical structure of the first ones. The existence of such isomorphisms for the bulk WZW and coset theories has been known for long time, see [35][8][29]. We only give their slightly more explicit realization. The isomorphism of the boundary WZW theory phase space with a moduli space of flat connections on a twice punctured disc has been first described in [19]. It represents another aspect of the relations between the boundary conformal theories and the topological three-dimensional theories developed in [12][13].

5.1

Bulk WZW model

The bulk WZW model on R × S 1 corresponds to the CS theory on R × Z, where Z = {z | 12 ≤ |z| ≤ 1}, with the boundary condition A0 = 0. The isomorphism I between the phase spaces P CS and P W Z of the two theories is defined by the formula giving the classical solution of the WZW model on R × S 1 in terms of a flat connection on Z :  A

g(t, x) = P e x,t 

,

(5.1)

···

where P e  stands for the path-ordered (from left to right) exponential and x,t is an appropriate contour, see Fig. 2. In particular, x,0 is a radial segment from eix to 12 eix and x,t is obtained from x,0 by rotating continuously the beginning of the segment by angle t and its end by angle −t. It is not difficult to see that I is a symplectic isomorphism, see Appendix B. In terms of the CS gauge field A, + − the currents (3.4) become J (x+ ) = ik Aϕ (eix ) and Jr (x− ) = −ik Aϕ ( 12 eix ), where Aϕ denotes the angular component of A.

5.2

Bulk G/H model

For the bulk G/H coset model, the corresponding CS theory is the double one on R × Z. Recall that the phase space of the latter is formed by the gauge-orbits of pairs (A, B) of, respectively, group G and group H flat connections on Z whose components tangent to the boundary are related by EAϕ = Bϕ . Choose a base

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ei(x+t)

1 2

1

1ei(x−t) 2

l x,t

Fig. 2 point 1 ∈ Z. We may consider w =

1 i

ln z =

z 1

dz iz

as the coordinate on the

covering space Z of Z. Let us inroduce two maps gA and hB from Z to G and H , respectively, 1 gA (w) = P e

z

−1

1

A

,

hB (w) = P e

z

B

.

(5.2)

−1

Clearly A = gA dgA and B = hB dhB . We shall set g(t, x) = hB (x + t)−1 gA (x + t) gA (x − t + w0 )−1 hB (x − t + w0 ) for w0 = i ln 2. Note that g(t, x + 2π) = ρ ρ = P e

−1



(5.3)

g (t, x) ρ where 

B

(5.4)

C

with C the clock-wise contour around the unit circle from 1 to 1. It is straightforward to check that  g satisfies the classical equations (3.7) of the bulk G/H coset model. Under the G-valued gauge transformations of A and H-valued ones of B that agree on the boundary of Z, the pair ( g , ρ) undergoes a simultaneous conjugation by a fixed element of H. We infer that (5.3) defines an injective map I  from the phase space P 2CS of the double CS theory on R × Z to the phase space P G/H of the bulk coset G/H model. Using the parametrization of the solutions g of the coset model by the left-right movers g (y) = h−1 (y)gA (y) , B

gr (y) = hB (y + w0 )−1 gA (y + w0 ) ,

satisfying (3.9) for ρ given by (5.4) and γ = P e

 C

(5.5)

A

,

(5.6)

it is easy to see that the map I  is also onto. The main result is that it defines a symplectic isomorphism, see Appendix B.

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Bulk G/G model

In the special case of H = G where the phase space P 2CS reduces to that of the gauge-orbits of flat connections on the torus represented as the double surface Z#(−Z), the field g of (5.3) is (t, x)-independent and it describes the parallel transport around the a-cycle, see Fig. 3.

1

a b

Fig. 3 Similarly, the monodromy ρ, commuting with g, describes the parallel transport around the b-cycle. Equations (3.11) and (3.12) express then the symplectic form (4.4) in terms of the holonomy of the gauge field and is a special case of the result of [2]. Note that conjugation-invariant functions of the holonomy around a fixed cycle on the torus Z#(−Z) Poisson-commute.

5.4

Boundary WZW model

The case of the boundary WZW model with the boundary conditions (3.13) has been analyzed in [19]. The corresponding CS theory is the one on the solid cylinder R × D where D is the unit disc in the complex plane, with two time-like Wilson lines, say R × { 21 } with label µ0 and R × {− 21 } with label −µπ , see Fig. 4. i(x+t) e

l x,t

1 2

lπ

l0

1 2

i(t−x) e

Fig. 4

Vol. 3, 2002

Boundary WZW, G/H, G/G and CS Theories

863

Let A be a flat connection on D with F (A) = 2πi γ0 µ0 γ0−1 δ 12 − 2πi γπ µπ γπ−1 δ− 12 .

(5.7)

Its holonomy around the contour 0 of Fig. 4 lies then in the conjugacy class Cµ0 and around π in Cµπ . To each such connection, we may associate a classical solution of the WZW theory on a strip R × [0, π] by setting  A

g(t, x) = P e x,t

,

(5.8)

with the contour x,t as in Fig. 4. In particular, for t = 0, x,0 goes from eix to e−ix crossing once the interval (− 12 , 12 ). For other times, x,t is obtained by rotating both ends of x,0 by the angle t. Note how the boundary conditions (3.13) are assured. The bulk equations (3.1) and the boundary ones (3.14) are also satisfied. The right hand side of (5.8) is clearly invariant under the gauge transformations of A equal to 1 on the boundary of the disc. The decomposition (3.16) of the solution in terms of the one-dimensional field g is obtained by setting    A

g (y) = P e y

A

,

m0 = P e  0

A

mπ = P e  π

,

.

(5.9)

Here for y ∈ [0, π], the contour y coincides with the interval [eiy , 0] and it is deformed continuously for other values of y. Contours 0 and π are as in Fig. 4. We obtain this way an isomorphism Iµ0 µπ from the phase space PµCS(−µπ ) of the CS 0 of the theory on R×D with two time-like Wilson lines onto the phase space PµW0 µZW π boundary WZW theory. As was explained in [19], Iµ0 µπ preserves the symplectic structure. We sketch in Appendix B the idea of the proof. In [19], this result was used to quantize the boundary WZW theory.

5.5

Boundary G/H model

Finally, let us consider the coset G/H theory on the strip R × [0, π] with the (M0 , M1 ) boundary conditions (3.18). It corresponds to the double CS theory on R×D coupled to Wilson lines. The group G gauge field is coupled to lines R×{ 21 } and R × {− 21 } with labels µ0 and −µπ and the group H gauge field to the same lines with labels ν0 and −νπ , respectively. Let us define   A

gA (y) = P e y

B

,

hB (y) = P e y

with the contour y as in (5.9), and   A

m0 = P e

0



A

, mπ = P e

π

, n0 = P e

0

,

(5.10) 

B

, nπ = P e

π

B

.

(5.11)

864

K. Gaw¸edzki

Ann. Henri Poincar´e

The monodromy of gA and hB is given by: gA (y + 2π) = gA (y) γ

for

γ = m−1 0 mπ ,

hB (y + 2π) = hB (y) ρ

for

ρ = n−1 0 nπ .

(5.12)

Setting g (y) = hB (y)−1 gA (y) we obtain a one-dimensional field satisfying (3.27) and describing via (3.26) a classical solution  g(t, x) of the boundary G/H coset theory with the (M0 , Mπ ) boundary conditions. Clearly, g is invariant under the gauge transformations of A and B equal on the boundary of the disc. We obtain this  2CS between the phase space PM of the double CS way an isomorphism IM (−Mπ ) 0 Mπ 0

G/H of the theory on R × D with two pairs of Wilson lines and the phase space PM 0 Mπ  boundary coset model. The proof that IM0 Mπ preserves the symplectic structure is similar to the one in the case of the boundary WZW model, see Appendix B.

5.6

Boundary G/G model

In the special case H = G, the phase space of the double CS theory reduces to that of the single theory on the 2-sphere S 2 = D#(−D) with four Wilson lines: R × { 21 } and R × {− 21 } in R × D with labels µ0 and −µπ and their images in h0 ∈ Cν0 ,  hπ ∈ Cνπ R × (−D) with labels −ν0 and νπ . The group elements and  −1 −1   and g =  g0 h0 = gπ hπ , see (3.28), are given by the contour integrals    A


 h0 = P e 0

A

,


 hπ = P e π

A

,

g = P e



,

(5.13)

where 0 and π are the copies in −D of 0 and π , see Fig. 4, and is the closed contour as in Fig. 5 starting and ending at the center of −D, with the broken

l 1 2

1 2

Fig. 5 pieces contained in −D and the solid ones in D. The equality of the symplectic form on the moduli space of flat connections A on S 2 with four punctures to the form of (3.29) is essentially again a special case of the result of [2].

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Boundary WZW, G/H, G/G and CS Theories

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6 Quantization of the boundary G/G coset model The description of the canonical structure of the two-dimensional WZW and coset theories in terms of the moduli spaces of flat connections on surfaces with boundary may be further reduced by the “topological fusion” to the case of a disc with a single insertion and of a closed surface with multiple insertions. This provides a good starting point for the quantization of the theory. Indeed, quantization of the moduli spaces on a disc with a single puncture is an example of the orbit method in the representation theory [25]. It gives rise to the highest weight representations of the current algebra [1][8]. On the other hand, quantization of the moduli spaces of flat connections on closed surfaces leads to the finite-dimensional spaces of conformal blocks of the WZW theories [35] that may be also viewed as spaces of invariant tensors of the iπ quantum group Uq (g) for q = e k+g∨ (g ∨ stands for the dual Coxeter number of G). This was described in detail for the boundary WZW theory with G = SU (2) in [19]. Here we shall carry out the quantization program for the boundary coset theory G/G. Recall that the canonical structure of the G/G model has been described directly in terms of the moduli spaces of flat connections on closed surfaces so that no topological fusion will be needed. As a result, we shall obtain an example of a two-dimensional boundary topological field theory, a structure that has recently attracted some attention [24][27][28]. Let us start from the well known case of the bulk G/G coset model. As explained above, the phase space of the theory coincides with the moduli space of flat connections on the torus Z#(−Z). The quantization of this moduli space gives rise to the space of the conformal blocks H of the the WZW theory on the torus8 . The space is spanned by the affine characters χkλ of level k of the associated to Lie algebra g or by the characters χλ of irreducible current algebra g representations of group G of integrable highest weights λ restricted to the points9 ζ+ρ ζ ≡ 2π k+gW∨ for ζ running through the integrable weights and ρW standing for the Weyl vector. As is well known, the restricted characters induce under pointwise multiplication a commutative ring Rk , the fusion ring of the WZW theory. This way, the space H of the conformal blocks on the torus becomes a commutative algebra with unity H ∼ = Rk ⊗ C. The unit element is given by the character χ0 ≡ 1 of the trivial representation. H may be viewed as the algebra of functions on the discrete set {ζ | ζ integrable} = Spec(H). The operator of multiplication g ) on the by the restricted character χλ is the quantizations of the function χλ ( phase space P G/G . We may equip H with a non-degenerate symmetric bilinear 8 One way to proceed is to use the natural K¨ ahler structure of the moduli space given by its identification with the moduli space of holomorphic vector bundles, see e.g. [10] 9 We view the characters as functions on the Cartan algebra identified with functions on the group via the exponential map

866

K. Gaw¸edzki

Ann. Henri Poincar´e

a1 a2 , a3 = a1 , a2 a3

(6.1)

form such that10 χλ , χλ
= δλ¯ λ
satisfying 1, 1 = 1 ,

for ai ∈ H. These are the data of a two-dimensional bulk topological field theory [4]. Such a theory assigns to each compact, not-necessarily connected, oriented surface Σ with the boundary ∂Σ = Sn1 a linear functional IΣ : ⊗ H → C, the n

amplitude of Σ. This assignment is supposed to have three properties. First, I is supposed to be multiplicative under disjoint products: IΣ1 Σ2 = IΣ1 ⊗ IΣ2 .

(6.2)

Second, it should be be covariant with respect to surface homeomorphisms preserving orientation. This means that IΣ = IΣ π for any permutation π of factors in ⊗ H within the connected components of Σ so that IΣ depends only on the coln

lection of the numbers of boundary circles and handles in each component. Third, I is required to be consistent with the gluing of boundary components: IΣnm = IΣ Pnm

(6.3)

if Σnm is obtained from Σ by gluing together the n-th and the m-th boundary components of opposite orientation. Pnm = id ⊗ . . . ⊗ pγ ⊗ . . . ⊗ pγ . . . ⊗ id γ n m where  pγ ⊗ pγ ∈ H ⊗ H (6.4) P = γ

is the dual of the bilinear form · , · (we may take pγ = χλ , pγ = χλ¯ ). It is easy to see that if Σ is connected and has g handles then

 (6.5) an , P g . IΣ ⊗ an = n

n

In particular, the surfaces of Fig. 6 (with the orientation inherited from the plane) correspond to the amplitudes a → a, 1 ,

a1 ⊗ a2 → a1 , a2 ,

a1 ⊗ a2 ⊗ a3 → a1 a2 , a3 .

(6.6)

They permit to reconstruct the amplitudes for all surfaces. For the bulk G/G theory and an = χλn , expression (6.5) is equal to an integer N(λn ) (g), the Verlinde dimension of the space of conformal blocks on Σ with insertions of the primary fields with labels λn . Explicitly [33],  ζ 2−2g ζ ζ N(λn ) (g) = (S0 ) (Sλn /S0 ) , (6.7) ζ

n

10 λ ¯ denotes the highest weight of the representation of G complex conjugate to the one with the highest weight λ.

Vol. 3, 2002

Boundary WZW, G/H, G/G and CS Theories

867

Fig. 6 ζ

λ

where S λ = S ζ = S ζλ¯ are the elements of the matrix giving the modular transformation of the affine characters χkλ . For closed surfaces, IΣ is an integer N (g). It is equal to 1 for the sphere and to dim(H), i.e. the number of integrable weights, for the torus. In short, the bulk topological G/G coset theory is the theory of the Verlinde dimensions. They may all be obtained from the fusion coefficients

ζ ζ Nλη = Nληζ¯ (0) which define the product in H: χλ χη = Nλη χζ . We would like to extend this structure to the case of the boundary G/G coset theory with the (M0 , Mπ ) boundary conditions where M0 = (µ0 , ν0 ) and G/G Mπ = (µπ , νπ ). Recall that we have identified the phase space PM of this 0 M1 2CS 2 of flat connections on S = D#(−D) theory with the moduli space PM 0 (−Mπ ) with four punctures labeled by µ0 , −ν0 , −µπ and νπ , all in the Cartan algebra of G. For k a positive integer and µ0 = λ0 /k, ν0 = η0 /k, µπ = λπ /k, νπ = ηπ /k, 2CS where λ0 , η0 , λπ , ηπ are integrable weights, the phase space PM gives upon 0 (−Mπ ) quantization the space of conformal blocks of the WZW theory on D#(−D) with ¯ π in D and by η¯0 and ηπ in insertions of the primary fields labeled by λ0 and λ −D. We shall denote this space by HL0 Lπ with L0 = (λ0 , η0 ) and Lπ = (λπ , ηπ ). By the factorization properties of the spaces of conformal blocks, HL0 Lπ ∼ = ⊕ Hom(Hλπ η¯π ζ , Hλ0 η¯0 ζ )

(6.8)

ζ

where Hληζ denotes the space of conformal blocks on S 2 with insertions of three primary fields labeled by the integrable weights λ, η and ζ. In particular, HLL is an associative (in general, non-commutative) algebra with unity, a direct sum of matrix algebras. More generally, there is a natural bilinear product HL1 L2 × HL2 L3 → HL1 L3 defined by composition of homomorphisms in each ζ-component. It gives HL1 L2 the structure of a left HL1 L1-module and of a right HL2 L2-module. It is useful to consider the direct sum of the boundary spaces Hb = ⊕ HL1 L2 . L1 ,L2

The product in Hb defined by ab = =

 L

 aL1L bLL2

(6.9)

868

K. Gaw¸edzki

Ann. Henri Poincar´e

for a = (aL1 L2 ) and b = (bL1 L2 ) makes Hb an associative algebra with unity 1 = (δL1 L2 ). Each space HL1 L2 is, additionally, a module of the commutative algebra H with the character χλ ∈ Rk acting diagonally in the decomposition (6.8) as the This action quantizes the classical observables χ ( multiplication by χλ (ζ). g ), λ where, in the CS description, g is is the holonomy around the countour on Fig. 5, see (5.13). The induced structure of the H-module on Hb satisfies a (b c) = (a b) c = b (a c)

(6.10)

for a ∈ H and b, c ∈ Hb which is equivalent to the statement that a b = (a 1) b and that elements a1 are in the center of Hb . We shall equip Hb with a non-degenerate symmetric bilinear form · , · b with the only non-vanishing matrix elements between subspaces with permuted boundary labels, i.e. such that  a, b b = aL1 L2 , bL2 L1 b . (6.11) L1 ,L2

Explicitly, we shall set: aL1 L2 , bL2 L1 b =



ζ

(±S 0 ) tr [ aL1 L2 (ζ) aL2 L1 (ζ) ] ,

(6.12)

ζ

where the sign is fixed once for all. It is easy to see that the bilinear form · , · b satisfies a b, c b = a, b c b .

(6.13)

The last relation, together with the symmetry of the form implies the cyclic symmetry a b, c b = b c, a b = c a, b b . Let PL1 L2 =



A

(6.14)

A

pL1 L2 ⊗ pL2 L1 ∈ HL1 L2 ⊗ HL2 L1

(6.15)

A

be the dual of the bilinear form (6.12) on HL1 L2×HL2 L1 . We may take A = (ζ, i, j) and A

ζ

−1 2

pL1 L2 = (±S 0 )

eiL

1ζ

A

e∗j , L ζ

ζ

−1 2

pL2 L1 = (±S 0 )

2

ejL

2ζ

e∗i , L ζ 1

(6.16)

where, for L = (λ, η), (eiLζ ) is a basis of Hληζ and (e∗i ) is the dual basis. The ¯ Lζ bilinear forms on Hb and on H are tied together by the relation   A A aL1 L1 pL1 L2 bL2 L2 , pL2 L1 b = aL1 L1 , pγ 1 b pγ 1 , bL2 L2 b . (6.17) A

γ

Vol. 3, 2002

Boundary WZW, G/H, G/G and CS Theories

Indeed, with the use of (6.16), the left hand side may be rewritten as  tr [ aL1 L1 (ζ)] tr [ bL2 L2 (ζ)]

869

(6.18)

ζ

and the right hand side is  ζ
S ζ χ ¯ (ζ  ) tr [ a S 0 χλ (ζ) (ζ)] tr [ bL2 L2 (ζ  )] . L1 L1 λ 0

(6.19)

λ,ζ,ζ


Note that the sign ambiguity in the definition (6.12) of the bilinear form on Hb disappears from both expressions. The equality of the two sides is inferred by using
ζ = S ζ , S ζ χ ¯ (ζ) = S ζ
and the unitarity of the modular the relations S 0 χλ (ζ) λ 0 λ λ ζ matrix (Sλ ). We may abstract from the above construction an algebraic structure   H, · , · , Hb , · , · b (6.20) such that 1. H is a finite-dimensional associative commutative algebra with unity equipped with the non-degenerate symmetric bilinear form · , · , 2. Hb = ⊕HL1 L2 is a finite-dimensional associative algebra with unity (in general non-commutative) equipped with the non-degenerate symmetric bilinear form · , · b , 3. Hb is an H-module with each HL1 L2 being a submodule, 4. relations (6.1), (6.9), (6.11), (6.13), (6.10) and (6.17) hold. Such a structure defines a boundary two-dimensional topological field theory [24] [27] [28]. The amplitudes of such a theory correspond11 to compact oriented surfaces Σ with boundary where the boundary components Sn1 may contain distinguished closed disjoint subintervals (possibly the whole component) marked with labels L of the boundary conditions, see Fig. 7. Let, for each Sn1 with labeled subintervals, (Ins ) be the collection of the remaining subintervals of Sn1 . The amplitude assigned to such a labeled surface is a linear functional     (6.21) IΣ : ⊗ H ⊗ ⊗ Hb → C , n

n,s

where the first tensor product is over the boundary components without labeled subintervals. IΣ is required to vanish on the all the components HL1 L2 of Hb except those with (L1 , L2 ) given by the labels of the intervals adjacent to Ins . For example, the amplitude of the labeled surface of Fig. 8 is a linear functional on 11 There are minor differences between our formulation and that of the above references, mostly a matter of convenience. In particular, we consider only boundary orientations induced from the bulk.

870

K. Gaw¸edzki

Ann. Henri Poincar´e

L3 L1

Σ

L2

Fig. 7 L4

..

.

L3

LS

L2 L1

Fig. 8 H ⊗ HL1 L2⊗ HL2 L3⊗ . . . ⊗ HL L1 . The amplitude assignment I is still required S to obey (6.2) and to be covariant under orientation and label preserving homeomorphisms. The latter means that IΣ = IΣ π for cyclic permutations of boundary intervals and their labels within boundary circles. The amplitudes depend this way on the collections of boundary labels with the cyclic order within each boundary circle (including the empty collection). The consistency with gluing (6.3) is now generalized to include the gluing along two unlabeled intervals of opposite orientation and permuted labels of the adjacent intervals as in Fig. 9. In the latter case the dual bilinear form P ∈ H ⊗ H should be replaced in (6.3) by the dual form PL1 L2 ∈ HL1 L2 ⊗ HL2 L1 inserted in the appropriate factors of the tensor product ⊗ Hb . n,s

It is not difficult to construct the amplitudes IΣ from the data (6.20). For completeness, we shall describe the argument. First, besides the bulk amplitudes already discussed, it is enough to know only the amplitudes corresponding to labeled surfaces of Fig. 10 aLL → aLL , 1 b ,

aL1 L2 ⊗ bL2 L1 → aL1 L2 , bL2 L1 b ,

a ⊗ aL1 L2 ⊗ bL2 L3 ⊗ cL3 L1 → aL1 L2 bL2 L3 , a cL3 L1 b .

Vol. 3, 2002

Boundary WZW, G/H, G/G and CS Theories

... L 2

...

L1

L2 . . .

...

...

L2

L1

...

L1 . . .

871

...

Fig. 9

L

L3

1

L

L2

1

L

L2 Fig. 10

Gluing the unlabeled disc to the inner boundary of the annulus gives the disc with three labeled boundary intervals and the amplitude aL1 L2 ⊗ bL2 L3 ⊗ cL3 L1 → aL1 L3 bL3 L2 , cL2 L1 b .

(6.22)

We may subsequently glue such a disc to the annulus as in Fig. 11 to obtain the amplitude a ⊗ aL1 L2 ⊗ bL2 L3 ⊗ cL3 L4 ⊗ dL4 L1  A A

→ aL1 L2 bL2 L3 , a pL3 L1 b pL1 L3 cL3 L4 , dL4 L1 b A

= aL1 L2 bL2 L3 cL3 L4 , a dL4 L1 b ,

(6.23)

where the last equality follows from the trivial identity  A

A

A

aL1 L2 pL2 L1 b pL1 L2 bL2 L1 b = aL1 L2 , bL2 L1 b .

(6.24)

872

K. Gaw¸edzki

Ann. Henri Poincar´e

L4

L1

L3

L2 Fig. 11 One obtains similarly the amplitudes of the general annuli of Fig. 8. They are given by the linear functionals 1

S

a ⊗ aL 1 L 2 ⊗ · · · ⊗ aL

S

L1

→

1

S−1

aL 1 L 2 · · · aL 1

S−1

S

L

S

, a aL

S

L1

b

(6.25)

S

invariant under the cyclic permutations of aL1 L2 · · · aL L1 due to (6.10) and (6.14). S The formula extends to the cases with S = 1 and S = 0 corresponding to the surfaces depicted in Fig. 12 if we interpret it as giving the linear maps a ⊗ aLL →

L

L

Fig. 12 aLL , a 1LL and a → 1LL , a 1LL b , respectively, where 1LL stands for the unity of HLL . For a general surface, we may obtain its amplitude by first cutting off the labeled boundary circles around nearby unlabeled ones as in Fig. 13, and then composing the amplitude from that of the annuli of Fig. 8 and of the ones for the surface with unlabeled boundary. It is easy to show that the resulting amplitudes are consistent with the gluing of surfaces. For unlabeled surfaces, this is a well known fact. For surfaces glued along two unlabeled boundary intervals in boundary components with labels, there are two different cases.

Vol. 3, 2002

Boundary WZW, G/H, G/G and CS Theories

873

Σ L2

L3

L5

L4

L1 Fig. 13 If the glued intervals are in two different boundary components, then the consistency boils down to the case of Fig. 14 and it follows with the use of (6.24).

L4

L3

L’1

=

L’3

L1

L’1

... ...

L1

L’2

L2

...

...

L2

L3

...

L4

L’2

L’4

L’3

L’4

Fig. 14 The case when one glues two intervals in the same boundary components may be similarly reduced to the check that the gluings of Fig. 15 give the same result. The first one leads to the amplitude 1

S

S


1

→ a ⊗ aL 1 L 2 ⊗ · · · ⊗ aL L 1 ⊗ b L
L
⊗ · · · ⊗ b L
L
S 1 2 S
1  1 S A 1 S
aL 1 L 2 · · · aL L p L L
b L
L
· · · b L
L
, a p A

b . L
L S

A

1

1

1

1

S


2

1

1

1

(6.26)

The second one results in 1

S

S


1

a ⊗ aL 1 L 2 ⊗ · · · ⊗ aL L 1 ⊗ b L
L
⊗ · · · ⊗ b L
L


→ S 1 2 S
1  1 S−1 S 1 S
−1 S
aL1 L2 · · · aL L , pγ aL L b · bL
L
· · · , bL , pγ a bL L γ

S−1

S

S

1

1

2

S
−1

S


S


L
1

b . (6.27)

874

K. Gaw¸edzki

L3’

L3

L’S’ L’1

L6

=

L2

LS

L4

L3’

...

...

L’2

L’4

...

L6

L5

L4

L’2

L’4 L’1

LS

L1

L3

’ LS’

...

L5

Ann. Henri Poincar´e

L2 L1

Fig. 15

=

Fig. 16

The equality of both expressions follows from (6.17). Conversely, the amplitudes (6.21) of a two-dimensional boundary topological field theory define the data (6.20). First, the amplitudes of Fig. 6 determine the unity, the bilinear form and the product in H. The commutativity of the latter follows from the homeomorhism covariance of the amplitudes that allows to permute the two inner discs of the third surface of Fig. 6. The associativity of the product in H results from the equality of the two ways to glue the amplitudes for the sphere without four discs presented in Fig. 16. The first of the relations (6.1) is equivalent to the normalization of the amplitude of the sphere S 2 to 1 and the second follows again from the homeomorphismcovariance of the amplitudes. Similarly, the amplitudes of the first two surfaces of Fig. 10 give the unit elements 1LL ∈ HLL and the bilinear form pairing HL1 L2 and HL2 L1 . The amplitude of the third surface applied to 1 ∈ H, together with the bilinear form · , · b , determine the product HL1 L2 × HL2 L3 → HL1 L3 in such a way that the cyclic invariance (6.14) holds. The associativity is proved similarly as before by equating two ways of gluing a disc with four labeled boundary intervals from pairs of discs with three labeled boundary intervals, see Fig. 17. The action of the elements of the bulk space H on the boundary space Hb is obtained from the amplitude of the annulus of Fig. 18 with the use of the bilinear

Vol. 3, 2002

Boundary WZW, G/H, G/G and CS Theories

L

875

L

4

4

L

L

1

3

=

L

L3

1

L2

L2 Fig. 17

L1

L2

Fig. 18 form · , · b . By definition, this action preserves the subspaces HL1 L2 ⊂ Hb . The proof that it defines a representation of the commutative algebra H in Hb follows from Fig. 19. Similarly, relations (6.10) follow from Fig. 20 and (6.17) from Fig. 15 with S = S  = 1. We obtain this way the algebraic structure (6.20) possessing all the four properties listed. We shall call a two-dimensional topological field theory unitary if there exist anti-linear involutions C : H → H and Cb : Hb → Hb with Cb (HL1 L2 ) = HL2 L1 such that the sesqui-linear forms C · , · and Cb · , · b define scalar products on H and Hb and that 5. C(ab) = (Ca)(Cb) , Cb (a b) = (Cb b)(Cb a) , Cb (a b) = (Ca)(Cb b) for a, b ∈ H and a, b ∈ Hb . The last three properties guarantee that I−Σ = IΣ



   ⊗ C ⊗ ⊗ Cb , n

n,s

(6.28)

where −Σ denotes the surface with the reversed orientation and, conversely, they follow from (6.28). For the G/G theory, one may take for C the complex conjugation of functions of integrable weights and for Cb the hermitian conjugation of

876

K. Gaw¸edzki

L

L2

1

Ann. Henri Poincar´e

L1

L2

=

Fig. 19

L3

L2

L1

L3

L2

L1

L2

L3

L1

Fig. 20 linear transformations in HL0 Lπ , see (6.8), relative to some scalar product in the spaces Hληζ of three-point conformal blocks. One obtains then a unitary topologζ ical field theory provided the sign in (6.12) is chosen so that ±S 0 > 0. Recall that, due to (6.10), the elements a1LL for a ∈ H are in the center of HLL . Following [24], we shall call the boundary condition L irreducible if all the elements of the center of HLL are of this form. This is the case in the G/G theory. To each boundary condition L one may associate a state aL ∈ H using the amplitude of the second surface of Fig. 12 and the bilinear form on H. Explicitly, aL is defined by demanding that 1LL , a 1LL b = aL , a

(6.29)

for all a ∈ H. We shall call the family of boundary conditions (L) complete if the states (aL ) span H. In the G/G theory, for L = (λ, η), = N (S ) aL (ζ) 0 λη ¯ ζ

ζ

−1

(6.30)

and the completeness is easy to see by taking, for example, the conditions with L = (λ, 0). On the other hand, the diagonal subfamily of boundary conditions corresponding to L = (λ, λ) is, in general, not complete since not all integrable weights appear in the fusion of pairs of complex conjugate weights (e.g. for G = SU (2), a( j,j) (j  ) vanishes for half-integer spins j  ).

Vol. 3, 2002

Boundary WZW, G/H, G/G and CS Theories

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The bulk topological theories may be perturbed by “massive” topological perturbations. For example, in the SU (2)/SU (2) model such perturbations permit to establish a relation with twisted minimal N = 2 topological theories. One of the interesting open problems for future research is how to extend such relations to the case of the boundary G/G theory.

A

Appendix

When expressed in terms of the left and right movers, the symplectic form of the bulk G/H coset theory becomes:

Ω

G/H

k 4π

=


2π   tr (g−1 δg ) ∂y (g−1 δg ) − (gr−1 δgr ) ∂y (gr−1 δgr ) dy    tr (δρ)ρ−1 ((δg )g−1 )(0) − ((δgr )gr−1 )(0)

0

−

k 4π

 + g (0)−1 (δρ)ρ−1 g (0) − gr (0)−1 (δρ)ρ−1 gr (0)   − (g−1 δg )(0) + (gr−1 δgr )(0) (δγ)γ −1 .

(A.1)

The expression for the bulk WZW model symplectic form ΩW ZW may be obtained from the latter by setting ρ identically to 1. Similarly, the expression in terms of the left-mover g for the boundary G/H model symplectic form becomes: G/H

ΩM0 Mπ =

k 4π


2π   tr (g−1 δg ) ∂y (g−1 δg ) dy +

k 4π

 −1 tr (δn0 )n−1 0 (δnπ )nπ

0

−1 −1 − (δm0 )m−1 ((δg ) g−1 )(0) 0 (δmπ )mπ − (δρ)ρ

+ (g−1 δg )(0) (δγ)γ −1 − (δρ)ρ−1 g (0) (δγ)γ −1 g (0)−1  + ωµG0 (m0 ) − ωνH0 (n0 ) − ωµGπ (mπ ) + ωνHπ (nπ )

(A.2)

ZW is and the expression for the boundary WZW model symplectic form ΩW µ0 µπ obtained by setting ρ, n0 and nπ identically to 1.

B Appendix  Our proof of the fact that the isomorphisms I , I  , Iµ0 µπ and IM between the 0 Mπ WZW and G/H phase spaces and the CS ones preserve the symplectic structure is  based on a direct calculation of the form tr (δA)2 and of its counterpart for the Σ

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gauge field B, very much in the spirit of a similar calculation [2] for closed surfaces. Consider first the bulk case that is somewhat simpler. It is enough to examine the case of the G/H coset theory which for H = {1} reduces to the WZW model. With gA given by (5.2), we have   (B.1) tr (δA)2 = d tr (gA−1 δgA ) d(gA−1 δgA ) . Integrating the last expression over the annulus Z cut along the interval [ 12 , 1],

z

z

1 2

1

1 2

0

1 2

1

Fig. 21 see Fig. 21, and using the Stokes theorem, we infer that k 4π




2

tr (δA) Z

=

k 4π


2π  tr (gA−1 δgA )(y) ∂y (gA−1 δgA )(y)

 − (gA δgA )(y + w0 ) ∂y (gA−1 δgA )(y + w0 ) dy 0

−1

   k − 4π tr (δγ)γ −1 (gA−1 δgA )(0) − (gA−1 δgA )(w0 ) ,

(B.2)

where the second line is the contribution from the integrals along the cut. Similar expression holds for B, hB and ρ replacing A, gA and γ, respectively. Subtracting both formulae, we obtain an expression for the symplectic form of the double CS theory on Z which may be shown to coincide with the right hand side of (A.1) by using (5.5) and the second equality of (3.9). The case of the boundary G/H coset model may be treated similarly. We define for z in the unit disc D cut along the sub-interval [− 21 , 1] of the real axis 



+i0

gA (z) = P e

z

+i0

A

,

 hB (z) = P e

z

B

.

(B.3)

hB (eiy ) = Note that for y ∈ (0, 2π) we have the equalities  gA (eiy ) = gA (y) and  hB (y) for gA and hB given by (5.10).

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Similarly as before,

  −1 tr (δA)2 = lim tr ( gA δ gA ) d( gA−1 δ gA , D

→0 ∂D

879

(B.4)

where D is the cut unit disc without -discs around ± 12 , see Fig. 21. A tedious but straightforward calculation results in the formula for the symplectic structure of the double CS phase space that coincides with equation (A.2). We leave the details to the reader just stressing that a more direct and conceptual proof of equality between the canonical structures of two-dimensional CFT’s and three-dimensional topological field theories would be welcome.

References [1]

A. Alekseev and S. Shatashvili, Quantum groups and WZNW models, Commun. Math. Phys. 133, 353–368 (1990).

[2]

A. Yu. Alekseev and A. Z. Malkin, Symplectic structure of the moduli space of flat connections on a Riemann surface, Commun. Math. Phys. 169, 99–120 (1995).

[3]

A. Yu. Alekseev and V. Schomerus, D-branes in the WZW model, Phys. Rev. D 60, R061901-R061902 (1999).

[4]

M. F. Atiyah, Topological quantum field theory, Publ. IHES 68, 175–186 (1989).

[5]

K. Bardak¸ci, E. Rabinovici and B. S¨ aring, String models with c < 1 components, Nucl. Phys. B 299, 151–182 (1988).

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J. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324, 581–596 (1989).

[7]

A. S. Cattaneo and G. Felder, A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys. 212, 591–611 (2000).

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S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nucl. Phys. B 326, 104–134 (1989).

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S. Elitzur and G. Sarkissian, D-branes on a gauged WZW model, Nucl. Phys. B 625, 166–178 (2002).

[10] F. Falceto and K. Gaw¸edzki, Chern-Simons States at Genus One, Commun. Math. Phys. 159, 549–579 (1994). [11] F. Falceto and K. Gaw¸edzki, Boundary G/G theory and topological PoissonLie model, Lett. Math. Phys. 59, 61–79 (2002).

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[12] G. Felder, J. Fr¨ ohlich, J. Fuchs and C. Schweigert, Conformal boundary conditions and three-dimensional topological field theory, Phys. Rev. Lett. 84, 1659–1662 (2000). [13] G. Felder, J. Fr¨ ohlich, J. Fuchs and C. Schweigert, Correlation functions and boundary conditions in RCFT and three-dimensional topology, Compos. Math. 131, 189–237 (2002). [14] J. Fuchs, B. Schellekens and C. Schweigert, The resolution of field identification fixed points in diagonal coset theories, Nucl.Phys. B 461, 371–406 (1996). [15] K. Gaw¸edzki, Classical origin of quantum group symmetries in Wess-ZuminoWitten conformal field theory, Commun. Math. Phys. 139, 201–213 (1991). [16] K. Gaw¸edzki, Conformal field theory: a case study. In: Conformal Field Theory, Frontiers in Physics 102, eds. Nutku, Y., Sa¸clioglu, C., Turgut, T., Perseus Publishing, Cambridge Ma. 2000, pp. 1–55. [17] K. Gaw¸edzki and A. Kupiainen, G/H conformal field theory from gauged WZW model, Phys. Lett. B 215, 119–123 (1988). [18] K. Gaw¸edzki and A. Kupiainen, Coset construction from functional integrals, Nucl. Phys. B 320, 625–668 (1989). [19] K. Gaw¸edzki, I. Todorov and P. Tran-Ngoc-Bich, Canonical quantization of the boundary Wess-Zumino-Witten model, arXiv:hep-th/0101170. [20] K. Gaw¸edzki and N. Reis, in preparation. [21] P. Goddard, A. Kent and D. Olive, Unitary representations of the Virasoro and super-Virasoro algebras, Commun. Math. Phys. 103, 105–119 (1986). [22] K. Hori, Global aspects of gauged Wess-Zumino-Witten models, Commun. Math. Phys. 182, 1–32 (1996). [23] D. Karabali, Q. Park, H. J. Schnitzer and Z. Yang, A GKO construction based on a path integral formulation of gauged Wess-Zumino-Witten actions, Phys. Lett. B 216, 307–312 (1989). [24] C. I. Lazaroiu, On the structure of open-closed topological field theory in two-dimensions, Nucl. Phys. B 603, 497–530 (2001). [25] A. Kirillov, Elements of the Theory of Representations, Berlin, Heidelberg, New York, Springer 1975. [26] J. Maldacena, G. Moore and N. Seiberg, Geometrical interpretation of Dbranes in gauged WZW models, arXiv:hep-th/0105038.

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[27] G. Moore, Some Comments on Branes, G-flux, and K-theory, Int. J. Mod. Phys. A 16, 936–944 (2001). [28] G. Moore, Santa Barbara lectures, http://online.itp.ucsb.edu/online /mp01. [29] G. Moore and N. Seiberg, Taming the conformal Zoo, Phys. Lett. B 220, 422–430 (1989). [30] V. B. Petkova and J.-B. Zuber, Conformal boundary conditions and what they teach us, arXiv:hep-th/0103007. [31] A. N. Schellekens and S. Yankielowicz, Field identification fixed points in the coset construction, Nucl. Phys. B 334, 67–102 (1990). [32] A. Schwarz, The partition function of degenerate quadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2, 247–252 (1978). [33] E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 [FS 22], 360–376 (1988). [34] E. Witten, Non-abelian bosonization in two dimensions, Commun. Math. Phys. 92, 455–472 (1984). [35] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121, 351–399 (1989). Krzysztof Gaw¸edzki C.N.R.S. Laboratoire de Physique ENS-Lyon 46, All´ee d’Italie F-69364 Lyon France email: [email protected] Communicated by Jean-Bernard Zuber submitted 03/12/01, accepted 08/04/02

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 3 (2002) 883 – 894 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/050883-12

Annales Henri Poincar´ e

Sign of the Monodromy for Liouville Integrable Systems R. Cushman and V˜ u Ngoc San Abstract. In this note we show that the monodromy of a two degree of freedom integrable Hamiltonian system has a universal sign in the case of a focus-focus singularity. We also show how to extend the monodromy index to several focusfocus fibers when the integrable system has an S 1 symmetry.

1 Introduction The Hamiltonian monodromy of integrable systems has a surprisingly recent history dating back to Duistermaat’s 1980 article [8]. Its application to quantum spectra was suggested in 1988 [5]. But it was not before 1998 – with the rigorous quantum formulation [17] and several examples [3], [7], [14], [10] (and others) – that it became a common tool for the analysis of spectra of many mathematically and physically relevant models (eg. [19]). (Quantum) Hamiltonian monodromy is usually used to demonstrate the nonexistence of global action variables (or good quantum numbers). This can be detected by a sort of “point defect” in the lattice of joint eigenvalues. The goal of our note is to sharpen this analysis by showing that this point defect can be attributed a sign, and in the generic case this sign is always positive (theorem 1). Moreover, as a first step in the study of systems with several isolated singularities, in theorem 3 we show how to compute the global monodromy in case of an S 1 symmetry (ie. one global action). A consequence of this sign for general systems without S 1 symmetry is that the global monodromy can cancel only for systems with complicated topology (proposition 5). We apply our results to a simple example with two points of monodromy: the quadratic spherical pendulum, for which we have also numerically computed the joint spectrum.

2 General Setup Let M be a 4-dimensional connected symplectic manifold with symplectic form ω, let B be a 2-dimensional manifold, and let F : M → B be a smooth proper surjective Lagrangian fibration with singularities which has connected fibers. We

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assume that the set of critical values ci of F is discrete and that each critical point of F in F −1 (ci ) is a focus-focus singularity. Recall that a point m ∈ M with dF (m) = 0 is called a focus-focus singularity if there exist local canonical coordinates (x, y, ξ, η) ∈ (T ∗ R2 , ω = dξ ∧ dx + dη ∧ dy) near m and a local chart of B at F (m) such that the vector space spanned by the Hessians D2 F1 (m) and D2 F2 (m) (where (F1 , F2 ) are the components of F ) is generated by the standard focus-focus quadratic forms (q1 , q2 ): q1 = xξ + yη

q2 = xη − yξ.

Recall that any critical point of F of Morse-Bott type (=“non-degenerate” in the sense of [9]) whose critical value is isolated in B is of focus-focus type. We are mainly interested in the case where F comes from a Liouville integrable system. Here B is a connected subset of R2 and F = (H1 , H2 ), where Hi are Poisson commuting Hamiltonians. Typically, M is a connected open subset of
where F may have non focus-focus critical points, see a symplectic manifold M [9].

3 Monodromy Let Br = B \ {ci } be the set of regular values of F and denote by Fr the restriction of F to Mr = F −1 (Br ). Then Fr is a regular Lagrangian fibration over Br with compact connected fibers. In a local chart of Br the fibration Fr = (H1 , H2 ) is a Liouville integrable system. By the Arnold-Liouville theorem, the fibers of F are affine 2-torii on which the flows of the Hamiltonian vector fields XH1 and XH2 define a linear action of T2 . The 2-torus bundle Fr : Mr → Br obtained this way is locally trivial. In fact it is locally a principal 2-torus bundle. The obstruction for it to be globally a principal bundle is the monodromy µ. More precisely, monodromy is the holonomy of a Z2 -bundle over Br whose fiber is the lattice of 2π-periodic vector fields, which in a local chart on Br about c are given by linear combinations of XH1 and XH2 whose flow on F −1 (c) is 2π-periodic. For more details, see [8], [4, Appendix D]. Let P → Br be this bundle of period lattices. Then the monodromy µ ∈ Hom(π1 (Br ), Aut(P)). Given a point c ∈ Br , a period lattice Pc with basis {X1 , X2 } and a loop γ in Br passing through c, the monodromy µc (γ) is a matrix in Gl(2, Z), whose conjugacy class in Gl(2, Z) is invariant under a change of basis. If γ encircles a single critical value  c of Fr , then there is a basis B such that the monodromy is the unipotent matrix   1 0 , (1) k 1 see [20], [6]. Here k is a nonzero integer called the monodromy index of γ relative to the basis B. The absolute value |k| is invariant under conjugation by elements

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885

of Gl(2, Z) and hence is independent of the choice of basis B. We call |k| the absolute monodromy index. In [2], [12] and [20] it was shown that this latter index is precisely the number of focus-focus critical points in F −1 ( c). Moreover, F −1 ( c) is homeomorphic to a |k|-pinched 2-torus.

4 Oriented monodromy Suppose now that Br is oriented, which is indeed the case when Br is an open subset of R2 . Then there is a induced orientation on the Liouville torii and hence on the bundle of period lattices P. This induced orientation is determined as follows. Let {α1 , α2 } be a positively oriented ordered basis of Tc∗ Br , which is dual to a positively oriented basis of Tc Br . Then the ordered basis of tangent vectors to F −1 (c) given by the set of vector fields {ω
(F ∗ (α1 ), ω
(F ∗ (α2 )} is said to be positively oriented. In the case of our two degree of freedom Liouville integrable system, if we use the standard orientation on R2 , then {XH1
F −1 (c) , XH2
F −1 (c) } gives the induced positive orientation for the 2-torus F −1 (c). We define the oriented monodromy index of the oriented loop γ in Br around the focus-focus critical value  c to be the integer k in (1) when the basis chosen to compute it is positively oriented. The number k is invariant under conjugation by orientation preserving automorphisms. When referring to the oriented monodromy index of a focus-focus critical value  c we assume that γ is positively oriented. Remark In this article we use the convention of (1) to write the monodromy matrix as a lower triangular matrix (instead of an upper triangular one), which amounts to a sign convention for k. Theorem 1 The oriented monodromy index of a focus-focus critical value is positive and hence is equal to the number of focus-focus critical points in the critical fiber. Proof. Using Eliasson’s theorem [9] one can find a chart near a focus-focus critical point (which corresponds to the critical value 0) so that F = g(q1 , q2 ), where g is a local diffeomorphism of R2 , and q1 = xξ + yη, q2 = xη − yξ, where (x, ξ, y, η) are coordinates for R4 with symplectic form dx ∧ dξ + dy ∧ dη. Using the symplectomorphism (x, ξ, y, η) → (−x, −ξ, y, η) one may change the sign of q2 , if necessary, to ensure that the ordered basis {Xq1 , Xq2 } is positively oriented. In other words, we can ensure that the local diffeomorphism g is orientation preserving, that is, det Dg(0) > 0. Following [18] we can choose a point c near the critical value 0 and an ordered basis B of the form {α Xq
1 + β Xq2 , Xq2 }, where α, β > 0, for which the monodromy matrix is

1 1

0 1

.

Since B has the same orientation as the ordered basis {Xq1 , Xq2 }

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and hence as the ordered basis {XH1 , XH2 }, we see that the monodromy index is positive.  Note that theorem 1 is purely local, since a small enough neighborhood of a focus-focus critical value is always orientable. Making no orientability assumptions on Br , theorem 1 can be phrased as follows. Theorem 1 (bis) The monodromy index k of a loop in Br around a single focusfocus critical value is positive if and only if the loop and the basis chosen to compute k have the same orientation.

5 Parallel transport The fibration Fr : Mr → Br endows Br with an integral affine structure, whose charts are the action coordinates. This affine structure induces a parallel transport on T Br , whose holonomy is the contragredient of the holonomy of the 2-torus bundle P → Br , that is, the monodromy. For more details see [1]. Suppose that  c1 and  c2 are two critical values of F that can be joined by a path Γ : [0, 1] → B such that Γ : (0, 1) → Br . Assume that a neighborhood of Γ ci in the positive sense. in B is orientable and fix a small loop γi which encircles  We obtain Corollary 2 The monodromy index of γ1 with respect to some basis B has the same sign as the monodromy index of γ2 computed with respect to a basis obtained by parallel transport of B. Proof. The holonomy of the affine manifold B being dual to the monodromy, has determinant 1. Hence parallel transport is orientation preserving. 

6 Case of S 1 symmetry Locally, a focus-focus singularity always admits an S 1 symmetry. However this symmetry does not in general extend globally, in particular when several critical fibers are present. This issue will be discussed in section 7. We show in this section how to extend the oriented monodromy index to several focus-focus points when the fibration F has a global S 1 symmetry. Here B is oriented an connected. Let G be the monodromy group of the regular fibration (= the image under µ of the fundamental group π1 (Br )). For any c ∈ Br , G acts on the lattice H1 (F −1 (c), Z)  Z2 .

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Theorem 3 Suppose that B is oriented, connected and simply connected. Then the following properties are equivalent 1. each element of G has a non-trivial fixed point in H1 (F −1 (c), Z); 2. there is a non-trivial X ∈ H1 (F −1 (c), Z) that is fixed by G; 3. G is Abelian; 4. there is a symplectic S 1 action on (M, ω) that preserves the fibration F ; 5. there is a Hamiltonian S 1 action on (M, ω) that preserves the fibration F ; 6. there is a unique group homomorphism µ ¯ : π1 (Br ) → Z such that for any γ ∈ π1 (Br ) the monodromy µ(γ)
with respect to a positively oriented basis is 1 k

conjugate in Sl(2, Z) to

0 1

with k = µ ¯(γ).

Proof. First note that properties 1 and 2 are of course independent of the choice of the base point c. We choose an oriented basis of H1 (F −1 (c), Z), which allows 2 us to identify G with a subgroup of
Sl(2, Z) acting on Z . In this proof we shall denote by Mk the matrix

1 k

0 1

.

The first three assertions are simple properties of Sl(2, Z). Proof of 1=⇒2. Let g0 be a non-trivial element of G, and g be any element of G. Since g0 , g and g0 g have all 1 in their spectrum, they all have trace equal to 2. Because we can find an integral eigenvector
of g0 , there is an integral basis of Z2 in which g0 = Mk (k = 0) and g = ac db . But Tr(g0 g) = a + kb + d = 2 + kb

which implies b = 0. Then g must have the form Mc . In other words the second element of our basis is necessarily a common eigenvector for all g ∈ G. Proof of 2=⇒3. Complete X into an integral basis of Z2 . Then all g ∈ G have the form Mk(g) in this basis. Hence they commute, by virtue of the formula Mk Mk
= Mk+k
.

(2)

Proof of 3=⇒1. The fundamental group π1 (Br ) is generated by the set γ1 , . . . , γn , where γi is a small loop around a single focus-focus critical value. Since Br is connected these loops can be deformed in Br to pass through the point c. Hence the corresponding monodromy transformations µi = µ(γi ) generate G. Since they are all trigonalizable (they are conjugate to Mk for some k) and G is Abelian, they are simultaneously trigonalizable. Now the product law (2) implies property 1. Proof of 2=⇒4. Recall that H1 (F −1 (c), Z) is isomorphic to the period lattice Pc : in a local chart of Br where F = (H1 , H2 ), the periodic vector fields on the torus F −1 (c) of the form xXH1 + yXH2 for constant x and y are determined uniquely by the homology class of any of their orbits.

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Thus we identify X with its representant in Pc . By parallel transport it locally extends to a flat local section of P, that is, a 2π-periodic vector field X on F −1 (U ), where U is a small neighborhood of c ∈ Br . The 1-form iX ω is invariant under the joint flow of F , and hence is of the form F ∗ β, for a 1-form β on U . By Liouville-Arnold theorem, dβ = 0, hence X is symplectic. Since by hypothesis the action of the monodromy group G on X is trivial, X can be extended to a global section of the bundle of period lattices P over Br . The 2π-flow of this vector field defines a symplectic S 1 action on F −1 (Br ) preserving the fibration. At the focus-focus singularity m, the period lattice no longer exists. However near m there is a unique 2π-periodic Hamiltonian vector field (with prescribed orientation) that is tangent to the Lagrangian foliation (see for instance [17]). Hence the above S 1 action extends uniquely to a global S 1 action on M preserving the fibration F . Note that this shows that the 1-form iX ω is the pull-back by F of a global closed 1-form β on B. Proof of 4=⇒5. Let Φ be the symplectic S 1 action and let X be the infinitesimal generator of Φ. Since Φ is symplectic, X is locally Hamiltonian. Since F is preserved by Φ, X is locally constant on the leaves in any action-angle coordinates. Hence X is actually a section of P above Br . Hence we are in the situation of the proof above, and there is a closed 1-form β on B such that iX ω = F ∗ β. Since H 1 (B) = 0, β is exact, namely β = dL. Hence X = XF ∗ L is a Hamiltonian vector field on (M, ω). Thus the S 1 action Φ is Hamiltonian on (M, ω) with momentum map L ◦ F . Proof of 5=⇒6. As in the proof above, we let L be a smooth function on B such that X = XL◦F , where X is the generator of the S 1 action. Since L is a global action, the 1-form dL is invariant under parallel transport on T ∗ Br defined by the integral affine structure on Br . Thus X is fixed by the monodromy group G: hence the hypotheses of assertion 2 are satisfied. Recall the choice of generators γi in the proof of 3=⇒1. Then X can be completed to an integral basis of Pc in which for all i, µ(γi ) = Mki for some ki ∈ Z. We define µ ¯ to be the homomorphism that ¯ realizes assigns to a loop γ = γi1 · · · γip the integer k = ki1 + · · · + kip . Note that µ an isomorphism between G and d Z where d is the gcd of (k1 , . . . , kn ). Proof of 6=⇒1. Obvious, since any matrix of the form Mk has a fixed point.  Corollary 4 Suppose that there is a global Hamiltonian S 1 action on (M, ω) preserving F . Then the monodromy index along an embedded, positively oriented loop γ in Br increases with the number of focus-focus critical values inside γ. In particular it can never cancel out. Proof. Each each focus-focus critical value adds a positive integer to the global monodromy index. 

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7 Vanishing of the monodromy Some integrable systems do not have an S 1 action. For instance if B is a sphere this would contradict corollary 4, since a loop around all focus-focus critical values would be contractible. However, even without an S 1 action, it is not easy to have the monodromy cancel along an embedded loop, as shown in the following proposition. Proposition 5 Assume that B is oriented, connected and simply connected. Let γ be an embedded loop in Br such that the monodromy along γ is trivial. Let n be the number of focus-focus critical values inside γ, and suppose that they are all simple: their index is 1. Then n is a multiple of 12. Proof. This is a consequence of the following lemma. See also Moishezon [13, p.179].  Lemma 6 Suppose that there are matrices A1 , A2 , . . . , An in Sl(2, Z) such that n 

where F =

1 1

0 1

(Ai F A−1 i ) = id,

(3)

i=1




. Then n is a multiple of 12.

Proof. (In order to stick to the usual conventions for the modular group, we shall use T = tF instead of F . The result follows by transposing (3).) It is well known (see [15]) that the modular group G = Sl(2, Z)/{±I} admits the following presentation G = S, T ; S 2 = (ST )3 = I, where S =

0 1

−1 0




. From this it easily follows that Sl(2, Z) admits the following

presentation Sl(2, Z) = S, T ;

S 4 = I, S 2 = (ST )3 .

Therefore the abelianization K of Sl(2, Z) is the group K = S, T ;

S 4 = I, S 2 = (ST )3 , ST = T S,

which yields K = S, T ; T 12 = I, S = T −3 . Hence K  Z/12Z and T is a generator of K. The image of the formula (3) in K gives T n = I, which implies that n is a multiple of 12.  As pointed to us by V. Matveev and O. Khomenko [11], from the data in the hypothesis of lemma 6, one can construct an integrable system with 12k focusfocus fibers and whose local monodromy around each critical value ci is equal in

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some fixed basis to Ai F A−1 i . Hence the monodromy around all critical values is the identity. This is done by pasting together a chain of fibrations with one focus-focus fiber, where the gluing maps between two tori are given by the Ai ’s. We therefore obtain a singular torus fibration over an open disc in R2 that cannot admit any S 1 symmetry, due to corollary 4. To have an example of a sequence of matrices in Sl(2, Z) satisfying the hy−1 potheses of
lemma 6, take A2j = I and A2j+1 = S. Then the product T ST S = 0 −1

1 1

is of order 6 in Sl(2, Z).

If one constructs a singular Lagrangian fibration over the open disc in R2 using these gluing matrices Aj , with 12 focus-focus critical values, we see that we can obtain as monodromy matrices of oriented loops the following ones: T (loop around one critical value), T −1 (because of (3)), S −1 (which is obtained by looping around the first three critical values, since T ST S −1T = S −1 ), and finally S (again because of (3)). Therefore by arbitrarily composing the corresponding loops together, we obtain any matrix of Sl(2, Z). When B is a Riemann surface, one can show further that the number of focus-focus points (if they are all simple) is equal to 12k, where k is the Euler characteristic of B. See [16] for more details. For example in [21] Tien Zung constructs an integrable system on a K3 surface which yields a singular Lagrangian fibration over S 2 with 24 simple focus-focus points.

8 Example with S1 symmetry Consider the quadratic spherical pendulum. This is a Hamiltonian system on T S 2 ⊆ T R3 (with coordinates (x, ξ)) defined by x, x = 1 and x, ξ = 0, where  ,  is the usual Euclidean inner product. The symplectic form on T S 2 is 3 the restriction of i=1 dxi ∧ dξi to T S 2 . The Hamiltonian is H(x, ξ) =

1 2

ξ, ξ + V (x3 ),

where V (x3 ) = 2(x3 − α)2 with α ∈ (0, 1). H is invariant under the lift of rotation around the x3 axis to T S 2 . Hence H Poisson commutes with the angular momentum K(x, ξ) = ξ × x, e3 . Thus the quadratic spherical pendulum is Liouville integrable with energy momentum mapping F : T S 2 → R2 : (x, ξ) → (H(x, ξ), K(x, ξ)), that is, F = (H, K). The set of critical values of F (see figure 1) is composed of two points A = (2(1 − α)2 , 0) and B = (2(1 + α)2 , 0) and a smooth parabola-like

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Sign of the Monodromy for Liouville Integrable Systems

curve parametrized by  h = 2z −1 (α − z)(1 + zα − 2z 2 )  k = ±2(1 − z 2 ) α/z − 1

891

for z ∈ (0, α].

K z→0 2

1

z=

1 4 0

A 1

B 2

H

3

4

–1

–2

z→0

Figure 1: critical values of the momentum map F . Here α = 1/4. It is straightforward to check that each point on the above curve corresponds to a relative equilibrium of the quadratic spherical pendulum, whose image under the tangent bundle projection is a horizontal circle on S 2 with x3 = ±z. The isolated points are unstable equilibria namely, the poles of S 2 , which are of focusfocus type. Since the fibers F −1 (A) and F −1 (B) contain each a single critical point, both A and B have oriented monodromy index 1. Hence the global index around both points is 2.

9 Semiclassical quantization The constancy of the sign of the monodromy is easily seen on a semiclassical joint spectrum. The latter has a local lattice structure admitting a discrete parallel transport, which is an asymptotic version of the integral affine structure on Br . For more details see [17]. This shows Theorem 7 Let a positively oriented basis B of the quantum lattice around a focusfocus point evolve in the positive sense. Then we obtain a final
basis by applying 1 0 to B a 2 × 2 matrix which is conjugate in Sl(2, Z) to k 1 with k ≥ 0.

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We illustrate theorem 7 with the quantum quadratic spherical pendulum. Let ˆ and K ˆ be the self-adjoint operators acting on L2 (S 2 ) defined as follows: H ˆ H ˆ K

= =


2 2 2 ∆S
∂ − i ∂θ ,

+ V (x3 )

where ∆S 2 is the Laplace-Beltrami operator on S 2 (with positive eigenvalues), ˆ et K ˆ V = 2(x3 − α)2 and θ is the polar angle around the vertical axis (Ox3 ). H ˆ K] ˆ = 0 and hence define a quantum are
-differential operators that commute: [H, integrable system. Their classical limit is given by the principal symbols H and K in C ∞ (T ∗ S 2 ), which are of course the Hamiltonians of section 8. K

2

1

A 0

1

B 2

3

4

H

−1

−2

Figure 2: Joint spectrum for the quadratic spherical pendulum. The quantum monodromy is represented by the deformation of a small cell of the asymptotic lattice. ˆ and K ˆ for α = 1/4 and
= 0.1. For Figure 2 shows the joint spectrum of H such “large” values of
the easiest way to compute the spectrum globally is to ˆ in the basis of standard spherical harmonics express the matrix associated to H ˆ (they are also eigenfunctions of K). The action of the potential V is obtained from the recurrence relation of the Legendre polynomials. This matrix can be cut to a finite size without any important loss in the accuracy of the computation, due to the fact that the modes we are looking at are microlocalized in a region of bounded energy H ≤ Hmax , which is compact. We have used this method of calculation to produce figure 2.

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The best way to have precise results near critical values for small
would be to use the singular Bohr-Sommerfeld rules of [18].

Acknowledgments The authors wish to thank Prof. B. Zhilinskii and Dr. D. Sadovskii of the Universit´e du Littoral in Dunkerque for telling us their monodromy conjectures which resulted in corollary 4 and assertion 6 in theorem 3. We also wish to thank Drs. Nguyˆen Tiˆen Zung and V.S. Matveev of the Universit´e de Montpellier and the Universit¨ at Freiburg, respectively, for their valuable remarks and criticisms of an early draft of this note. Finally, we acknowledge interesting remarks by the referee. The authors were (partially) supported by European Commission funding for the Research Training Network “Mechanics and Symmetry in Europe” (MASIE), Contract No. HPRN-CT-2000-00113.

References [1] L. Bates, Monodromy in the Champagne bottle, Z. Angew. Math. Phys. 42, 837–847 (1991). [2] A.V. Bolsinov and A.T. Fomenko, Application of classification theory for integrable hamiltonian systems to geodesic flows on 2-sphere and 2-torus and to the description of the topological structure of momentum mappings near singular points, J. Math. Sci. (Dynamical systems, 1) 78, 542–555 (1996). [3] M.S. Child, Quantum states in a Champagne bottle, J. Phys. A. 31, 657–670 (1998). [4] R. Cushman and L. Bates, Global aspects of classical integrable systems, Birkh¨auser, 1997. [5] R. Cushman and J.J. Duistermaat, The quantum spherical pendulum, Bull. Amer. Math. Soc. (N.S.) 19, 475–479 (1988). [6]

, Non-hamiltonian monodromy, J. Differential Equations 172, 42–58 (2001).

[7] R. Cushman and D.A. Sadovski´ı, Monodromy in the hydrogen atom in crossed fields, Phys. D 142, no. 1–2, 166–196 (2000). [8] J.J. Duistermaat, On global action-angle variables, Comm. Pure Appl. Math. 33, 687–706 (1980). [9] L.H. Eliasson, Hamiltonian systems with Poisson commuting integrals, Ph.D. thesis, University of Stockholm, 1984.

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Ann. Henri Poincar´e

[10] L. Grondin, D. Sadovskii and B. Zhilinski´ı, Monodromy in systems with coupled angular momenta and rearrangement of bands in quantum spectra, Phys. Rev. A (3) 65, no. 012105, 1–15 (2001). [11] V. Matveev, Private communication, email 5 November 2001. [12]

, Integrable hamiltonian systems with two degrees of freedom. topological structure of saturated neighborhoods of saddle-saddle and focus points, Mat. Sb. 187, 29–58 (1996).

[13] B. Moishezon, Complex surfaces and connected sums of complex projective planes, Lecture Notes in Mathematics, no. 603, Springer-Verlag, 1977. [14] D.A. Sadovski´ı and B.I. Zhilinski´ı, Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A 256, no. 4, 235–244 (1999). [15] J.-P. Serre, Cours d’arithm´etique, Presses Universitaires de France, 1970. [16] M. Symington, in preparation. [17] S. V˜ u Ngo.c, Quantum monodromy in integrable systems, Commun. Math. Phys. 203, no. 2, 465–479 (1999). [18]

, Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type, Comm. Pure Appl. Math. 53, no. 2, 143–217 (2000).

[19] H. Waalkens and H.R. Dullin, Quantum monodromy in prolate ellipsoidal billiards, Ann. Physics 295, 81–112 (2002). [20] Nguyˆen Tiˆen Zung, A note on focus-focus singularities, Diff. Geom. Appl. 7, no. 2, 123–130 (1997). [21]

, Symplectic topology of integrable hamiltonian systems, II: Topological classification, Preprint Univ. Montpellier, math.DG/0010181, 2000.

Richard Cushman Mathematics Institute University of Utrecht 3508TA Utrecht The Netherlands email: [email protected] Communicated by Eduard Zehnder submitted 27/12/01, accepted 24/05/02

V˜ u Ngoc San Institut Fourier Universit´e Joseph Fourier, BP 74 38402-Saint Martin d’H`eres Cedex France email: [email protected]

Ann. Henri Poincar´e 3 (2002) 895 – 920 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/050895-26

Annales Henri Poincar´ e

Some Upper Bounds on the Number of Resonances for Manifolds with Infinite Cylindrical Ends T. Christiansen Abstract. We prove some sharp upper bounds on the number of resonances associated with the Laplacian, or Laplacian plus potential, on a manifold with infinite cylindrical ends.

The purpose of this note is to bound the number of resonances, poles of the meromorphic continuation of the resolvent, associated to a manifold with infinite cylindrical ends. These manifolds have an infinity which is in some sense onedimensional, even though the manifold is n-dimensional. The bounds we obtain on the resonances reflect this dichotomy- we obtain one bound like that for onedimensional scattering, and other bounds of the type expected for n-dimensional manifolds. As part of our study of resonances, we relate poles of the resolvent to L2 eigenvalues and to poles of appropriately defined “scattering matrices.”

1 Introduction A smooth Riemannian manifold X is said to be a manifold with cylindrical ends if it can be decomposed as X = Xcomp ∪ X∞ , where Xcomp is a compact manifold with boundary Y , X∞  [a, ∞)t × Y , (Y, g) is a compact Riemannian manifold (possibly disconnected), and the metric on X∞ is (dt)2 + g. We may also allow X itself to be a manifold with boundary, and then the ends take the form [a, ∞)t × Y , with Y a smooth, compact manifold with boundary. In this case, we require that the boundary of Xcomp be compact and smooth except for a finite number of corners corresponding to a × ∂Y . An example of such a manifold is a waveguide, a domain with smooth boundary in the plane, with one or more infinite straight ends. If we allow X to have a boundary, we will consider the Laplacian with Dirichlet or Neumann boundary conditions. Let ∆Y be the Laplacian on (Y, g). Let {σj2 }, σ12 ≤ σ22 ≤ σ32 ≤ · · · be the set of all eigenvalues of ∆Y , repeated according to their multiplicity, and let ν12 < ν22 < ν32 < · · · be the distinct eigenvalues of ∆Y . Then the resolvent of the Laplacian ∆ on X, or of ∆ + V , for V ∈ L∞ comp (X) real-valued, has a meromorphic continuation ˆ to the Riemann surface Z on which (z − νj2 )1/2 is a single-valued function for all j ([11, 16]). Thus, the resonances, poles of the meromorphic continuation of the ˆ The complicated nature of this Riemann resolvent, are associated to points in Z.

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surface makes more difficult the question of bounding the number of resonances, and makes necessary the restrictions we place on {νj2 }. Part of our study of resonances is to understand the relationship between poles of the resolvent and their multiplicities and poles of a “scattering matrix.” For a manifold with cylindrical ends, there are several reasonable objects to call the scattering matrix. One is an infinite dimensional matrix. In Section 3 we define this matrix and associated finite-dimensional matrices which contain the information about poles which we desire. We show that if z0 ∈ Zˆ is not a ramification point of Zˆ and is a pole of the resolvent, then there is an associated L2 eigenfunction with an appropriate expansion at infinity or there is a pole of a “scattering matrix” at z0 . We make this precise and address the issue of multiplicities in Theorem 3.1 and Proposition 3.3. In [2] such a relationship is noted, though to our knowledge no proof appears in the literature, so we include it here for completeness. We will often require that there exists an α > 0 such that 2 2 − νm−1 ≥ ανm νm

(H1)

for all sufficiently large m. Examples of cross-sectional manifolds that satisfy such requirements are spheres, an interval with Dirichlet or Neumann boundary conditions, and projective space. We can easily construct a manifold that satisfies (H1) with m ends by taking Y to be the disjoint union of m copies of the same manifold from the previous list. Other situations are possible as well, of course. If n > 2, then assumption (H1) is not a generic condition. It would be interesting to know if results analogous to our Theorems 1.1 and 1.2 hold in greater generality. Let P be the operator ∆, the Laplacian, or ∆ + V , for real-valued V ∈ 2 −1 is bounded on L2 (X) L∞ comp (X). Then, for z ∈ C \ [ν1 , ∞), R(z) = (P − z) except, perhaps, for a finite number of z. It has a meromorphic continuation to the Riemann surface Zˆ described earlier. We bound the number of resonances in ˆ In doing so, we take the view that resonances near the physical certain regions of Z. sheet, the sheet of Zˆ on which the resolvent is bounded, are more interesting, as they have greater physical relevance. Let rj (z) = (z − νj2 )1/2 . Assuming the hypothesis (H1), we simplify the study of the resonances somewhat and are then able to better bound them. Theorem 1.1 Assume X satisfies the hypothesis (H1) and let β < 1. Then, in the √ connected components of {z ∈ Zˆ : |rm (z)| < β ανm } that meet the physical sheet, n−1 there are at most Oβ (m ) resonances. We remark that we count all poles with their multiplicities and that the poles of the resolvent include eigenvalues. In [5, 18] there is an example of a family of manifolds that has lim inf λ→∞ N (λ)λ−n > 0, where N (λ) is the number of eigenvalues of the Laplacian with norm less than λ2 . Since the cross-sectional manifolds in the example can be taken to be n − 1-dimensional unit spheres, this shows that the order appearing in this theorem is optimal. This type of bound is indicative of the n-dimensional nature of the manifold. In a simpler case, we can say a bit more.

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Theorem 1.2 Let X = R × Y and suppose X satisfies (H1), and let ρ > 0 be fixed. Consider the operator ∆ + V , for real-valued V ∈ L∞ comp (X). Then, on the ˆ connected components of {z ∈ Z : |rm (z)| < ρ} that meet the physical sheet, the number of poles is bounded by C(1 + mn−2 ). In case V = V (t), t ∈ R, this theorem is easy to prove, and this example shows that the bound is of optimal order. Our proofs of these two theorems involve an adaptation of techniques developed by Melrose ([15]), Zworski ([24, 25]), and Vodev ([21]) to bound the number of poles. (See [12] and [22] for further references and results.) It involves constructing an approximation to the resolvent to find a holomorphic function whose zeros include the poles of the resolvent. Then we bound the function and apply Jensen’s theorem. This requires some knowledge of the function at a “base point.” For us, that will mean a lower bound. Since we will be changing the base point, we need some kind of uniform lower bound and that is different from these other applications of this technique. In order to do this, we will construct approximations of the resolvent especially well-suited to the regions where we work. The following theorem does not require the hypothesis (H1), and its proof uses a different technique. ˆ and let {zk } be the resonances of P on this sheet. Theorem 1.3 Fix a sheet of Z, Then
| Im r1 (zk )| < ∞. |r1 (zk )|2 This theorem is an analogue of what one finds in one-dimensional scattering theory (see [9, 23]), where the natural variable to consider is λ = z 1/2 . The problem of obtaining upper bounds on the resonance-counting function has been widely studied for Euclidean (e.g. [9, 15, 21, 23, 25]) and hyperbolic scattering (e.g. [12, 13, 17, 19]). For a survey and further references, see [22] or [26]. In this paper we use results of [16], which studied the Laplacian on compact manifolds with boundary and exact b-metrics. Under a change of variable, a special case of such manifolds is the class of manifolds considered here (see also [11]). The papers [5, 18] independently obtained that the number of eigenvalues less than λ2 of the Laplacian grows at most like cλn . The existence of eigenvalues or complex resonances has been studied in, for example, [1, 2, 3, 5, 6, 7, 18] and references. In finishing the paper, the author received a copy of [8]. There Edward obtains a result similar to our Theorem 1.1 for the Laplacian on waveguides, that is, domains in the plane which outside of a compact set coincide with (−∞, ∞) × π, and thus fall in the category of manifolds which we consider. The waveguides 2 are very regularly distributed for either the satisfy hypothesis (H1) and their νm Dirichlet or Neumann Laplacian. Use the metric induced on Zˆ by the pull-back of 2 ˆ Then summing over the νm the metric on C to define distance on Z. ≤ r we can

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obtain from our Theorem 1.1 an upper bound #{zj : zj is a resonance of ∆, dist(zj , physical sheet) < c1

 |π(zj )|, |π(zj )| < r} = O(r)

for some c1 > 0, where π : Zˆ → Z is projection. This may be compared to Theorem 2 of [8], where for c1 = 1/2, Edward obtains a bound O(r3+ ), any > 0. Remark. We note that the “black-box” formalism of [20] can be adapted to this situation. Thus we could replace P by P˜ , a more general self-adjoint, compactly supported perturbation of the Laplacian than the ones considered here. If P˜ satisfies the assumptions of [20], properly interpreted for this setting, and is bounded below, then Theorem 1.3 will hold for P˜ . Let P˜ # = P˜|{t 0 for all j and all z and on which R(z) is bounded. Other sheets will be identified, when necessary, by indicating for which values of j Im rj (z) < 0. Each sheet can be identified with C \ [ν12 , ∞). With this convention, there are points in Zˆ which belong to no sheet but which belong to the boundary of the closure of two sheets, and the ramification points, which correspond to {νj2 } and belong to the closure of four sheets (except for ramification points corresponding to ν12 ). We note that sheets that meet the physical sheet are characterized by the existence of a J ∈ N such that Im rj (z) < 0 for all z on that sheet if and only if j ≤ J. Let {φj } be an orthonormal set of eigenfunctions of ∆Y associated with {σj2 }. On an end, we use the coordinates (t, y), with t ∈ (a, ∞) and y ∈ Y . Let r˜l (z) = rj (z) if σl2 = νj2 . We define an operator on [0, ∞)t × Yy . Using the same notation for an operator and its Schwartz kernel, let Rel (z) =



i (ei|t−t |˜rl (z) − ei|t+t |˜rl (z) )φl (y)φl (y  ). 2˜ rl (z)

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Then, for z ∈ C \ [σ12 , ∞), RY (z) =




Rel (z)

899

(1)

l

is the resolvent of the Dirichlet Laplacian on [0, ∞) × Y . As an operator from 2 ([0, ∞) × Y ) it has a holomorphic continuation to the L2comp([0, ∞) × Y ) to Hloc ˆ Riemann surface Z and we will see that it is this that determines the surface to which the resolvent of P has a meromorphic continuation. In general, we shall use z to stand for a point in Zˆ and π(z) to represent its projection to C. For w ∈ Rm , w = (1 + |w|2 )1/2 .

3 The Scattering Matrix and Poles of the Resolvent We emphasize that the results of this section, about the correspondence between the poles of the resolvent and the poles of the scattering matrix, do not require hypothesis (H1). There are several reasonable definitions of the scattering matrix in this setting. We recall some results of [4, 16]. Fix a coordinate t on the ends, so that t = 0 lies on the ends. Then, for all but a finite number of z in the physical space, there are functions Φj (z, p) with (P − π(z))Φj (z, p) = 0

(2)

and, on the ends, Φj (z, t, y) = e−i˜rj (z)t φj (y) +




Smj (z)ei˜rm (z)t φm (y).

(3)

The Φj have a meromorphic continuation to all of Zˆ and thus, so do the Smj . The Smj (z) depend on the choice of the coordinate t in a fairly straightforward way (see [5]). This dependence is not important here as it does not change the location of the scattering poles, so we ignore it but we do consider the coordinate t to be fixed throughout, and chosen so that {p ∈ X : t = 0} ⊂ X∞ . There are several reasonable choices of objects to call the scattering matrix. One possibility is the infinite matrix of the Sij (z), as in [16]. Another, which is well-defined for z on the boundary of the physical sheet, is a normalized, finitedimensional matrix of the Sij (z), where the dimension changes as z crosses a νl2 . This is used in [4] and has the advantage of being unitary (though we note that the variable used in [4] is λ = z 1/2 ). Here, however, this is unnecessarily complicated as it requires the introduction of (z − νi2 )1/4 . We shall work with finite-dimensional matrices, of the form (Sij (z))i,j∈E˜ for some set E˜ ⊂ N, where E˜ is chosen, depending on z, to be most helpful for our purposes.

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Definitions and Some Properties

In order to describe the poles and their multiplicities of the resolvent and matrices of the Sij (z), we introduce some notation. Definition 3.1 We define the multiplicity of a pole of the resolvent R(z) at z0 to be mz0 (R) = dim Image Ξz0 (R), where Ξz0 (R) is the singular part of R at the point z0 . ˆ then It follows as in [13, Lemma 2.4] that, if z0 is not a ramification point of Z, mz0 (R) is also the rank of the residue of R at z0 . In order to define the multiplicity of the pole of a matrix, we shall use the following lemma. Though it may be well known, we include it and a proof for the convenience of the reader. Lemma 3.1 Suppose A(z) is a d × d-dimensional meromorphic matrix, invertible for some value of z. Then, near z0 , it can be put into the form   p p


A(z) = E(z)  (z − z0 )−kj Pj + (z − z0 )lj Pj + P0  F (z) j=1

j=p+1

where E(z), F (z), and their inverses are holomorphic near z0 , and Pi Pj = δij Pi , tr P0 = d − p , tr Pi = 1, i = 1, . . . , p . The kj and lj are, up to rearrangement, uniquely determined. Proof. We outline a proof. Without loss of generality we may assume that z0 = 0. First we show the existence of such a decomposition. Choose k such that z k A(z) = B(z) is holomorphic at 0. Now the proof of the existence of such a decomposition follows much like a proof from [10, Section VI.2.4]. Let B(z) = (bij (z)). Choose an element bij (z) that vanishes to the lowest order at 0, and by permuting rows and columns make this element b11 (z). By subtracting from the kth row the first row multiplied by bk1 (z)/b11 (z) (which is holomorphic near z = 0), we can make all the entries in the first column, other than the first one, zero. Similar column operations reduce B(z) to the form     

b11 (z) 0 ··· 0 c22 (z) · · · .. .. . . ··· 0 cd2 (z) · · ·

0 c2d (z) .. . cdd (z)

     .

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Repeating the procedure on the (d − 1) × (d − 1) dimensional matrix of the cij , we obtain a matrix of the form   b11 (z) 0 0 ··· 0   0 c22 (z) 0 ··· 0    0 0 α33 (z) · · · α3d (z)      .. .. .. ..   . . . ... . 0

0

αd3 (z)

···

αdd (z)

.

Repeating this procedure a finite number of times, we obtain a matrix with only the diagonal entries, hii (z), nonzero. Since hii (z) = z mi gi (z), with gi (0) = 0 and gi (z) holomorphic near 0, by multiplying by a diagonal matrix with nonzero entries 1/gi (z) we obtain a diagonal matrix with diagonal entries of the form z mi . Finally, multiplying the whole thing by z −k I we obtain a matrix equivalent to A(z) near z = 0. We note that the construction ensures that E and F are holomorphic near 0. Moreover, since each of E and F is the product of elementary matrices and one diagonal matrix (holomorphic near 0, with nonzero determinant), E and F are both invertible near 0. To see the uniqueness of the ki , li , we prove it for B(z) = z k A(z), where B is holomorphic at 0, and use the straightforward relationship between A and B. Suppose there are two such decompositions: B(z) = E1 (z)D1 (z)F1 (z) and B(z) = E2 (z)D2 (z)F2 (z), where Ei (z) and Fi (z) are as in the statement of the lemma and the Di are diagonal matrices with nonzero entries dmm,i = z lm,i . By row and column operations we can make 0 ≤ l1,i ≤ l2,i ≤ · · · ≤ ld,i . We have E(z)D1 (z) = D2 (z)F (z)

(4)

for new matrices E(z), F (z), holomorphic and invertible near 0. Thus it is easy to see that rank(D1 (0)) = rank(D2 (0)) ≡ r0 . Using (4) and the fact that lj,i = 0 if and only if j ≤ r0 , we obtain that, if E(z) = (eij (z)), F (z) = (fij (z)), eij (0) = 0 = fji (0) if j ≤ r0 and i > r0 .

(5) (j)

We finish the proof of the uniqueness by induction. Let rj,i = rank(D i (0)), and notice it suffices to prove that rj,1 = rj,2 for all j, and that we have j rj,i = d. Suppose we have shown that rq,1 = rq,2 ≡ rq for q ≤ N , and that, if RN = r0 + r1 + · · · + rN , eij (0) = 0 = fji (0) for j ≤ RN , i > RN .

(6)

If v = (v1 , v2 , . . . , vd )t , let Πs v = (0, . . . , 0, vs+1 , . . . , vd )t for s ∈ N, s ≤ d. Then rank(ΠRN E(0)ΠRN ) = d − RN = rank(ΠRN F (0)ΠRN ),

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using (6) and the fact that E(0) and F (0) are invertible. Since, for j ≤ N , (j) (j) D1 (0)ΠRN = 0 and ΠRN D2 (0) = 0, we get 

dN +1 (N +1) (Π E(z)D (z)Π ) E(0)D (0)Π = rank Π RN 1 RN |z=0 RN RN 1 dz N +1



dN +1 (N +1) = rank D1 (0) = rank (ΠRN D2 (z)F (z)ΠRN )|z=0 dz N +1

(N +1) (0) = rank D2

rank

and thus r1,N +1 = r2,N +1 ≡ rN +1 . If RN + rN +1 < d, from (4) we obtain eij (0) = 0 = fji (0) for j ≤ RN + rN +1 , i > RN + rN +1 . The induction need only continue until RN +1 = d, finishing the proof.




Definition 3.2 Let A(z) be a meromorphic matrix, invertible for some values of z, and let k1 , k2 , . . . , kp , lp+1 , lp+2 , . . . , lp
be as in Lemma 3.1. Set µmz0 (A) =

p


kj ,

j=1

the “maximal multiplicity” of the pole of A at z0 . Set µdz0 (A) =

p
j=1




kj −

p


lj ,

j=p+1

the “determinantal multiplicity” of the pole of A at z0 . We note that µdz0 (A) = min{j ∈ Z : (z − z0 )j det A(z) is regular at z0 }. Each of these measures of multiplicity will be useful. Suppose E ⊂ N is a finite subset. Let E˜ = {l ∈ N : σl2 = νj2 for some j ∈ E}. Define the matrix SE (z) = (Sij (z))i,j∈E˜. ˆ let For z ∈ Z,

Ez = {j ∈ N : Im rj (z) ≤ 0}

and Jz = {j ∈ N : j ≤ max Ez }.

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If E ⊂ N is a finite set, define wE : Zˆ → Zˆ as follows. To z we may associate the set of square roots {rj (z)}. Then wE (z) may be determined by saying it is the element of Zˆ associated to the set {rj (wE (z))}, with  −rj (z), if j ∈ E rj (wE (z)) = if j ∈ E. rj (z), ˆ where For J ∈ N, define a similar operator wJ : Zˆ → Z,  −rj (z), if j ≤ J rj (wJ (z)) = rj (z), if j > J. This operator appears in the relations (14) and (15). The following Proposition generalizes similar results of [16] and [4]. ˜ Proposition 3.1 Let E ⊂ N be a finite set. For j ∈ E,
Φj (wE (z)) = Skj (wE (z))Φk (z) k∈E˜

and

(SE (z))−1 = SE (wE (z)).

˜ consider Proof. For j ∈ E, Ψj (z) = Φj (wE (z)) −




Skj (wE (z))Φk (z).

k∈E˜

Then (P − π(z))Ψj (z) = 0 and, on the ends, Ψj has an expansion Ψj (z, t, y) = ei˜rj (z)t φj (y) −

+









Skj (wE (z))Slk (z)ei˜rl (z)t φl (y)

k,l∈E˜

Slj (wE (z)) −

l∈E˜




 Skj (wE (z))Slk (z) ei˜rl (z)t φl (y).

(7)

k∈E˜

Since, for z on the physical sheet, Im r˜l (z) > 0 for all l, Ψj (z) ∈ L2 (X) there and thus Ψj (z) ≡ 0 for all z in the physical space. By analytic continuation, Ψj (z) ≡ 0 ˆ This proves the first part of the Proposition. It also shows that, for for all z ∈ Z. ˜ j, l ∈ E,
Skj (wE (z))Slk (z) = δjl ; k∈E˜

that is, (SE (z))−1 = SE (wE (z)).




904

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Ann. Henri Poincar´e

Relation between Poles of the Resolvent and Poles of SJ z0 (z)

First, we recall some results of [16, Sections 6.7, 6.8] on the nature of poles of the resolvent on the boundary of the physical sheet. Proposition 3.2 Suppose z lies on the boundary of the physical sheet. Then, if z ˆ the multiplicity of z as a pole of the resolvent is not a ramification point of Z, is equal to the dimension of the L2 null space of P − π(z). If z is a ramification point, then the resolvent has a double pole at z, with coefficient the projection onto the L2 null space of P − π(z), if there is any. The residue has rank equal to the dimension of {f : (P − π(z))f = 0, f ∈ L2 (X),

∂ f ∈ L2 (X)}. ∂t

Since Φj (z, p) = χ(t)e−i˜rj (z)t φj − (P − z)−1 (P − π(z))χ(t)e−i˜rj (z)t φj ,

(8)

where χ(t) ∈ C ∞ (R) is supported in t > a and is one in a neighborhood of infinity, it is clear by their definition that the Sij (z) cannot have a pole unless R(z) has a pole. More can be said, and we begin our study of the relationship between the poles of the resolvent and poles of the scattering matrix with the following ˆ and z0 is not a ramification point. Then Theorem 3.1 Suppose z0 ∈ Z, mz0 (R) = µmz0 (SJz0 ) + dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼




cj ei˜rj (z0 )t φj }

j∈J˜z0

where the expansion must be valid on any end. The notation f∼




cj ei˜rj (z0 )t φj

j∈E˜

means that f|t>0 (t, y) =




cj ei˜rj (z0 )t φj (y).

j∈E˜

In proving this theorem, we will use some techniques from [13, Section 2]. We will call
{f : (P − π(z0 ))f = 0, f ∼ cj ei˜rj (z0 )t φj }

(9)

j∈J˜z0

the set of eigenfunctions of type I. This depends on z0 of course, and we will note the dependence in cases of possible confusion. We first show that

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ˆ then Lemma 3.2 If z0 is not a ramification point of Z, µmz0 (SJz0 ) ≤ mz0 (R) − dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼




cj ei˜rj (z0 )t φj }.

j∈J˜z0

Proof. Just as in Lemma 2.4 of [13], whose notation we use, one can prove that if ˆ then near z0 R(z) has a pole at z0 , for z0 not a ramification point of Z, R(z) =

p
Ak (z0 ) + H(z0 , z) (z − z0 )k

(10)

k=1

where H(z0 , z) is holomorphic near z0 , q


Ak (z0 ) =

alm k (z0 )ϕl ⊗ ϕm

l,m=1



and (ϕl ⊗ ϕm )f (p) = ϕl (p)

p
∈X

ϕm (p )f (p )dvX .

As in [13], we have (P − π(z0 ))Ak (z0 ) = Ak+1 (z0 ) = Ak (z0 )(P − π(z0 )). If ak (z0 ) = (alm k (z0 ))1≤l,m≤q , then a1 (z0 ) is symmetric with rank q, d(z0 ) = a1 (z0 )−1 a2 (z0 ) is nilpotent, and ak (z0 ) = a1 (z0 )dk−1 (z0 ), k > 1. Moreover, ϕl has an expansion on the ends of the form

ϕl (t, y)|t>a = cljm tm ei˜rj (z0 )t φj (y) (11) j

m≤p

([16]) and the ϕl are linearly independent. Let B(z) be the matrix B(z) = (blm (z))l,m≤q with blm (z) =

p
alm k (z0 ) . (z − z0 )k

k=1

Then B(z) can be written, as in Proposition 2.11 of [13], as  
p
B(z) = E # (z)  (z − z0 )−kj Pj + P0  F # (z)

(12)

j=1

where E # (z), F # (z) and their inverses are holomorphic near z0 , Pi Pj = δij Pi , tr Pi = 1, i = 0, tr P0 = q − p , and k1 + k2 + · · · + kp
= q.

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Ann. Henri Poincar´e

First assume that there are no type I eigenfunctions of P with eigenvalue π(z0 ). We recall that we can construct the generalized eigenfunctions Φj , j ∈ J˜z0 , as indicated in (8). We then obtain the entries of SJz0 (z) by first restricting Φj to t = 0 and then extracting the coefficients of φk . This, taken with the representation (12), shows that the singular part of SJz0 (z) near z0 will be given by  
p
E  (z)  (z − z0 )−kj Pj + P0  F  (z) j=1

where E  (z), F  (z) are holomorphic matrices near z0 which may not be invertible, or even square, and this proves the lemma in this special case. To handle the case where π(z0 ) is an eigenvalue of P with
ne (z0 ) = dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼ cj ei˜rj (z0 )t φj } > 0, (13) j∈J˜z0

ˆ R(z)ψ we need to be a bit more careful. We first remark that for a general z ∈ Z, is well-defined if ψ ∈ e−t max(0,− Im rj (z)) L2 (X). This can be seen from the construction of the analytic continuation of the resolvent in [16]. Now suppose ψ is an L2 eigenfunction of P . Then it is exponentially decreasing, and, for J ∈ N, if z and wJ (z) both lie on the boundary of the physical space, (14) (R(z) − R(wJ (z))) ψ = 0. By analytic continuation, if z, wJ (z) are such that ψ ∈ e−t max(0,− Im rj (z),− Im rj (wJ (z))) L2 (X), then (14) holds. By repeatedly applying (14) and using our knowledge of the structure of the resolvent on the closure of the physical space, we obtain that if
ψ ∈ {f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼ cj ei˜rj (z0 )t φj }, j∈J˜z0

then there is a neighborhood of z0 so that in this neighborhood R(z) =

1 ψ ⊗ ψ + B1 (z, z0 , ψ) (π(z0 ) − π(z))ψ2

with B1 (z, z0 , ψ)ψ = 0. That is, each L2 eigenfunction with an expansion at infinity of this type contributes to the singularities of R(z) in the expected way. Using the fact that if j ∈ Jz0  ψ(P − π(z0 ))χ(t)e−i˜rj (z0 )t φj (y) = 0, X

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we see that the singularities of R(z) at z0 corresponding to L2 eigenfunctions of type I do not contribute to the singularities of SJz0 (z) at z0 . After making this observation, the proof for the case ne (z0 ) > 0 follows just as in the case
ne (z0 ) = 0. The proof of Theorem 3.1 will be completed by ˆ then Lemma 3.3 If z0 is not a ramification point of Z, µmz0 (SJz0 ) + dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼




cj ei˜rj (z0 )t φj }

j∈J˜z0

≥ mz0 (R). Proof. We use, for J ∈ N, R(z) − R(wJ (z)) =

i
1 Φj (z) ⊗ Φj (wJ (z)) . 2 2 2 r˜j (z)

(15)

σj ≤νJ

Equation (15) holds, by Stone’s formula, [16, Section 6.8], and [4, Section 2.2], for 2 z on the boundary of the physical space, with νJ2 < π(z) < νJ+1 , and then holds on the rest of Zˆ by analytic continuation. Using Proposition 3.1, we may write (15) as R(z) − R(wJ (z)) =

i

1 . Smj (z)Φm (wJ (z)) ⊗ Φj (wJ (z)) 2 2 2 2 2 r˜j (z)

(16)

σj ≤νJ σm ≤νJ

Recalling equations (10) and (11), we see that if ψ is in the image of the singular part of R(z) at z0 , then ψ is a linear combination of eigenfunctions of type I (see (9)) and {f : (P − π(z0 ))k f = 0 for some k ∈ N; f ∼



j

bjm tm ei˜rj (z0 )t φj (y),

m≤k

bjm = 0 for some j ∈ J˜z0 , some m}. (17) We call functions of the form (17) type II. Now take J = max Ez0 . If g is in the image of the singular parts of R(z) at both z0 and at wJ (z0 ), then it must be of type I. It is the appearance of the type II functions which we must understand. Suppose g is in the image of the residue of R(z) at z0 and is of type II. Then, since it is not in the image of the residue of R(z) at wJ (z0 ), it must be in the image of the residue of

1 (18) Smj (z)Φm (wJ (z)) ⊗ Φj (wJ (z)) r ˜ j (z) 2 2 2 2 σj ≤νJ σm ≤νJ

908

T. Christiansen

Ann. Henri Poincar´e

at z = z0 . First assume that Φj (wJ (z)), j ∈ J˜z0 has no poles at z0 . Recalling that we may write SJz0 (z) near z0 as in Lemma 3.1, we can write (18) as p


(z − z0 )−kj Ψ1j (z) ⊗ Ψ2j (z) + H(z, z0 )

(19)

j=1

where k1 + · · · + kp = µmz0 (SJz0 (z)), and Ψ1j (z), Ψ2j (z), and H(z, z0 ) are holomorphic near z0 . Then it is easy to see that the rank of the image of the residue of (19) cannot exceed µmz0 (SJz0 (z)), and the total rank of the residue of R(z) at z0 cannot exceed
µmz0 (SJz0 ) + dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼ cj ei˜rj (z0 )t φj }. j∈J˜z0

To finish the proof, we need only understand what happens if Φj (wJ (z)) has a pole at z0 for some j ∈ J˜z0 . However, a pole of Φj (wJ (z)) cannot contribute to the singularity of R(z) at z0 because the expansion on the ends of the singular part is of the wrong form. Therefore, the proof of this case follows much as the proof of the previous one.
The following proposition will also be useful. Proposition 3.3 Suppose z0 ∈ Zˆ is such that π(z0 ) is not in the spectrum of P . Then mz0 (R) = µdz0 (SEz0 ). Proof. We sketch the proof of this proposition, as it is very similar to the proof of Theorem 3.1. Just as in Lemma 3.2, one can show that µmz0 (SEz0 ) ≤ mz0 (R). Here, of course,
dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼ cj ei˜rj (z0 )t φj } = 0, j∈J˜z0

by our assumption on π(z0 ). Moreover, SEz0 (z) has no zeros at z0 as a zero would imply the existence of an L2 eigenfunction, and thus µmz0 (SEz0 ) = µdz0 (SEz0 ). Finishing the proof thus requires showing that mz0 (R) ≤ µmz0 (SEz0 ). This can be done as in Lemma 3.3, first noting that, if ψl ≡ 0 is in the image of the singular part of R at z0 , then on the ends

ψl (t, y) = bljm tm ei˜rj (z0 )t φj (y) (20) j

m≤mj

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where bljm = 0 for some j ∈ Ez0 and some m. In fact, because π(z0 ) is not an eigenvalue of P , the linear independence of ψ1 , ψ2 , . . . , ψq in the image of the singular part of R(z) at z0 is equivalent to the linear independence of

bljm tm ei˜rj (z0 )t φj (y), j∈E˜z0 m≤mj

l = 1, . . . , q. Using this and the fact that Ξz0 (R) is symmetric, where Ξz0 is the singular part at z0 , we obtain that dim Image Ξz0 (R)



≤ dim Image Ξz0 


j,m∈E˜z0

 1  . Smj (z)Φm (wJ (z)) ⊗ Φj (wJ (z)) r˜j (z)

Then, if Φj (wJ (z)) is regular at z0 for all j ∈ J˜z0 , it is easy to see that mz0 (R) ≤ µmz0 (SEz0 ). Again, if Φj (wJ (z)) has a pole at z0 , it does not contribute to the singularities of
R(z) at z0 but corresponds instead to a singularity of R(z) at wJ (z0 ).

4 Proof of Theorem 4.1 In this section we bound the number of poles of the resolvent in neighborhoods of the ramification points on the boundary of the physical sheet, with the size of the neighborhoods increasing. We will use the fact that if χ = 1 for t < max(a, 0), then dim Image Ξz0 (R) = dim Image Ξz0 (Rχ), where Ξz0 (T ) is the singular part of T at z0 . We recall Theorem 1.1 Theorem. Assume X satisfies the hypothesis (H1) and let β < 1. Then, in the √ connected components of {z ∈ Zˆ : |rm (z)| < β ανm } that meet the physical sheet, there are at most Oβ (mn−1 ) resonances. A corollary to this is Theorem 4.1 Assume X satisfies the hypothesis (H1) and let β < 1. Then, on the closure of the sheet with Im rj (z) < 0 if and only if j ≤ m, there are at most √ Oβ (mn−1 ) poles of the resolvent of P with |rm (z)| < β ανm . We shall use the Fredholm determinant method used in, for example, [12, # 15, 21, 24, 25]. We will find a trace class operator Km (z) so that, in the desired # (z), and thus region, the poles of the resolvent are contained in the zeros of I + Km # # in the zeros of det(I + Km (z)). In addition, Km (z0 ) = 0, where z0 ∈ Zˆ is a “base point” which depends on m. To do this, we first construct an approximation of the

910

T. Christiansen

Ann. Henri Poincar´e

resolvent adapted to this problem and valid in this region, obtaining a compact operator Km (z) so that the poles of the resolvent in this region are contained in the zeros of I + Km (z). Further manipulations simplify I + Km (z). For m > 1, there are two ramification points on the boundary of the physical 2 , and thus, for large m, two connected comsheet of Zˆ which correspond to νm √ ˆ ponents of {z ∈ Z : |rm (z)| < β ανm } which meet the physical sheet. We shall work near the ramification point which is reached by taking a limit as Im π(z) ↓ 0, 2 , with z in the physical sheet. We will designate this ramification Re π(z) → νm 2 2 point by (νm )+ . A similar analysis works near the other point, (νm )− , obtained 2 by taking a limit for z in the physical sheet, Im π(z) ↑ 0, Re π(z) → νm . We assume m > 1. For c > max(a, 0), let Xc = Xcomp ∪ (X∞ ∩ [a, c] × Y ). For ζ ∈ C, let Rc (ζ) = (P|Xc − ζ)−1 be the resolvent of P on Xc with Dirichlet boundary conditions. (If X has a boundary and we are considering Neumann boundary conditions on X, we use Neumann boundary conditions on Xc here.) Recall that RY (z) = (Dt2 + ∆Y − z)−1 is defined by (1). For i = 1, 2, 3, choose χi ∈ Cc∞ (X) so that χi χi+1 = χi , i = 1, 2, χ1 ≡ 1 on Xmax(a,0) , ∇χi only depends on t, and |∇χi | < γ, |∆χi | < γ. It suffices to take −1 γ ≤ α(1 − β)2 (1000(1 + α)) . Choose c0 so that the support of χ3 is properly contained in Xc0 . 2 + 14 α(1 − β)2 νm i and let Choose z0 in the physical plane so that π(z0 ) = νm Em (z) = χ3 Rc0 (π(z))Πm χ2 +χ3 Rc0 (π(z0 ))(1−Πm )χ2 +(1−χ1 )RY (z)(1−χ2 ). 2 Here Πm projects off of the eigenfunctions of P|Xc0 with eigenvalues in (νm − 2 5ανm , νm + 5ανm ). This of course depends on c0 , but we omit this dependence in our notation. Then ˜ m (z), (21) (P − π(z))Em (z) = I + K

where ˜ m (z) = [P, χ3 ] (Rc0 (π(z))Πm + Rc0 (π(z0 ))(1 − Πm )) χ2 K − χ3 (π(z) − π(z0 ))Rc0 (π(z0 ))(1 − Πm )χ2 − [P, χ1 ]RY (z)(1 − χ2 ), ˜ m (z) is meromorphic on Z, ˆ with and, if the domain is restricted to L2comp (X), K the poles corresponding to poles of Rc0 (π(z))Πm . Let χ ∈ Cc∞ (X) be 1 on the support of χ3 . Then ˜ m (z)χ) = χ(I + Km (z)) (P − π(z))Em (z)χ = χ(I + K and Km (z) is compact.

(22)

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Our choice of z0 and γ guarantee that I + Km (z0 ) is invertible, with norm bounded by 2. Then, by analytic Fredholm theory, I + Km (z) is invertible at all ˆ The points where it is not invertible correspond to but a discrete set of points of Z. zeros of I + Km (z), and the poles of the cut-off resolvent (P − z)−1 χ are included in the union of the zeros of I + Km (z) and the poles of Em (z). However, we will restrict our attention to a region that does not include any poles of Em (z). Using the fact that I + Km (z0 ) is invertible, with norm bounded by 2, we obtain that in the region in question, the poles of the resolvent are contained in the zeros of (23) I + (I + Km (z0 ))−1 (Km (z) − Km (z0 )). √ Now we restrict our attention to a region where |rm (z) − rm (z0 )| < ρ ανm , where ρ = (β 2 /4 + 3/4)1/2 2 and z lies on one of the four sheets that meet the ramification point (νm )+ , with m large. In this region, we have

(I + Km (z0 ))−1 [P, χ3 ] (Rc0 (π(z)) − Rc0 (π(z0 )) Πm χ2  ≤

1 , 3

and ((I + Km (z0 ))−1 [P, χ1 ]




(Rej (z) − Rej (z0 )) (χ − χ2 ) ≤

2 σj2 >νm+1

1 . 3

(24)

For (24), we are using the fact that [P, χ1 ] depends only on t. Therefore, in this region the poles of the resolvent are contained in the zeros of # (z), I + Km

where

(25)

# Km (z) = Lm (z) (K1m (z) + K2m (z))

with K1m (z) = −χ3 (π(z) − π(z0 ))Rc0 (π(z0 ))(1 − Πm )χ2 ,
K2m (z) = −[P, χ1 ] (Rej (z) − Rej (z0 )) (χ − χ2 ),

(26) (27)

2 σj2 ≤νm+1

and the norm of Lm (z) is bounded, independent of m and z, as long as we stay in the region described. # (z) is trace class. We consider the function Now Km # (z)), h(z) = det(I + Km

and note that h(z0 ) = 1 and that h is holomorphic on Zˆ when |rm (z) − rm (z0 )| < √ 2 ανm . We will apply Jensen’s theorem to h to obtain an upper bound on the number of zeros of the resolvent in this region, and to do this we need the following lemma.

912

T. Christiansen

Ann. Henri Poincar´e

√ Lemma 4.1 If |rm (z)− rm (z0 )| ≤ ρ ανm and z lies in the connected component of n−1 √ 2 {z  ∈ Zˆ : |rm (z  )−rm (z0 )| ≤ ρ ανm } containing (νm )+ , then |h(z)| ≤ CeCm . Proof. We remark that because of Weyl’s law and the hypothesis (H1), for large m, νm is bounded above and below by constant multiples of m. We use the property that | det(I + A + B)| ≤ det(I + |A|)2 det(I + |B|)2

(28)

(e.g. [12, Lemma 6.1]). Since (I − Πm )χ2 has rank bounded by Cmn−1 and √ Rc0 (π(z0 )) ≤ νCm  , we obtain, in the region with |rj (z) − rj (z0 )| ≤ ρ ανm , and thus |π(z) − π(z0 )| ≤ C νm , | det(1 + |Lm (z)K1m (z)|)| ≤ CeCm

n−1

.

Consider next Lm (z)K2m (z) = Lm (z)[P, χ1 ]




(Rej (z) − Rej (z0 )) (χ − χ2 )

2 σj2 ≤νm+1

= Lm (z)[P, χ1 ]

m+1






i ei|t−t |rj (z) − ei|t+t |rj (z) − ei|t−t |rj (z0 ) + ei|t+t |rj (z0 )

j=1

2rj (z)




×

φl (y)φl (y  )(χ − χ2 ).

σl2 =νj2

Since [P, χ1 ] and χ − χ2 have disjoint supports,





[P, χ1 ] ei|t−t |rj (z) − ei|t+t |rj (z) − ei|t−t |rj (z0 ) + ei|t+t |rj (z0 ) φl (y)φl (y  )(χ − χ2 ) is a rank four operator with norm bounded by CeC max(0,− Im rj (z)) . We need, therefore, to bound m


2 | Im rj (z)|#{σl2 : σl2 = νj2 } + #{σl2 : σl2 = νm+1 }.

(29)

1

We have

#{σl2 : σl2 = νj2 } ≤ Cj n−2 .

2 − νj2 |)−1/2 , if On the region of interest, | Im rj (z)| can be bounded by 2ανm (|νm √ j < m, and ανm , if j = m. Therefore, m


2 | Im rj (z)|#{σl2 : σl2 = νj2 } + #{σl2 : σl2 = νm+1 }

1

=

m−2
1

| Im rj (z)|#{σl2 : σl2 = νj2 } + O(mn−1 ). (30)

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913

To bound (29), then, it suffices to bound  νm−1  2 −1/2 n−2 2ανm νm − l2 l dl 0

 n−1 = 2ανm

νm−1 /νm

0

n−1 (1 − s2 )−1/2 sn−2 ds ≤ αCνm . (31)


Proof of Theorem 1.1. The poles of the resolvent in the region in question are 2 )+ , we use rm (z) as contained in the zeros of h(z). Working near the point (νm a coordinate, which we may do as long as we keep away from other ramification points or regions where rm (z) fails to be one-to-one. We apply Jensen’s Theorem to √ h, using a circle centered at z0 and having radius ρ ανm . Then the theorem follows √ because the disk |rm (z)| ≤ β ανm is properly contained in the disk |rm (z) − √ rm (z0 )| ≤ ρ ανm , with the ratio of the distance between the two boundaries and the radius of the larger disk bounded from below, independent of m.


5 Proof of Theorem 1.2 We recall the statement of Theorem 1.2: Theorem. Let X = R × Y and suppose X satisfies (H1), and let ρ > 0 be fixed. Consider the operator ∆ + V , for real-valued V ∈ L∞ comp (X). Then, on the conˆ nected components of {z ∈ Z : |rm (z)| < ρ} that meet the physical sheet, the number of poles is bounded by C(1 + mn−2 ). In this section we continue to use the {σj2 } to denote the eigenvalues of ∆Y , repeated according to their multiplicity, though our manifold X actually has two ends isomorphic to (1, ∞) × Y . We use the other notation introduced earlier as well. We prove this theorem by the Fredholm determinant method as in the previous theorem. We assume that ρ > 1. Let R0 (z) = (∆ − z)−1 ; its Schwartz kernel is given by R0 (z) =

∞
l=1


i ei|t−t |˜rl (z) φl (y)φl (y  ), 2˜ rl (z)

(32)

and let RV (z) = (∆ + V − z)−1 . We have (∆ + V − π(z))R0 (z) = I + V R0 (z).

(33)

Let χ(t) ∈ Cc∞ (X) be one on the support of V , with |χ| ≤ 1. Then, if z0 ∈ Zˆ is not a ramification point and RV (z)χ has a pole at z0 , then I + V R0 (z0 )χ has nontrivial null space. For r > 0, m ∈ N, set Bm,r to be the connected components of {z ∈ Zˆ : |rm (z)| < r} that meet the physical sheet.

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1/2

Let ρ˜ = max(ρ, 2V ∞ ), and restrict z to Bm,4ρ˜. Take m sufficiently large 2 that the only ramification points in Bm,4ρ˜ correspond to νm . Then I + V R0 (z)χ = (I + K1 (z))(I + (I + K1 (z))−1 K2 (z)) where K1 (z) has Schwartz kernel


i ei|t−t |rj (z) K1 (z) = V (t, y) φl (y)φl (y  )χ(t ) 2rj (z) 2 2 j=m

σl =νj

and K2 (z) K2 (z) = V (t, y)

i




2rm (z)

ei|t−t |rm (z)




φl (y)φl (y  )χ(t ).

2 σl2 =νm

By choosing m sufficiently large, so that

 A1/2 16ρ˜2 1 , exp A √ < ανm 2V  |ανm − 16ρ˜2 |1/2 where A = maxt,t
∈supp χ |t − t |, we have (I + K1 (z))−1  ≤ 2 on Bm,4ρ˜. The poles of RV (z) in Bm,4ρ˜, other than the ramification point, correspond to values of z for which I + (I + K1 (z))−1 K2 (z) has non-trivial null space. We remark that 2 2 R0 (z) has a pole of rank MY (νm ) at the ramification point, where MY (νm ) is the 2 multiplicity of νm as an eigenvalue of ∆Y , and this can contribute a pole of up to the same multiplicity to RV (z), even if I + (I + K1 (z))−1 K2 (z) is invertible there. Let K3 (z) = (I + K1 (z))−1 K2 (z). The poles of RV (z) in Bm,4ρ˜ are contained in the zeros of I + K3 (z) in the same region, except, possibly, at the ramification points, as discussed above. Now we 2 )+ as in the proof of the previous theorem, as a similar analysis shall work near (νm will work for the other connected component of Bm,4ρ˜. Choose z0 in the physical 2 space with π(z0 ) = νm + 4iV ∞ . Then K2 (z0 ) ≤ 1/4, and  −1     I + (I + K1 (z0 ))−1 K2 (z0 )  = (I + K3 (z0 ))−1  ≤ 2. Let

h(z) =

rm (z) rm (z0 )

2 2MY (νm )

  det I + (I + K3 (z0 ))−1 (K3 (z) − K3 (z0 )) .

Then h(z0 ) = 1, and, except, possibly, for some at the ramification point, the poles 2 of RV (z) in the connected component of Bm,4ρ˜ that includes (νm )+ , are contained 2 n−2 in the zeros of h(z) in the same region. This misses at most 2MY (νm ) = O(νm ) poles of RV at the ramification point. An application of Jensen’s theorem on a circle centered at z0 and with |rm (z) − rm (z0 )| ≤ 3ρ˜ will then finish the proof, after we have proved the following lemma.

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2 Lemma 5.1 On the connected component of Bm,4ρ˜ that contains (νm )+ ,

|h(z)| ≤ exp(C(1 + |m|n−2 )). Proof. Let Rems (z) be the operator with Schwartz kernel i




2rm (z)

χ(t)(ei|t−t |rm (z) − 1)

and let Rem0 (z0 ) be the operator with Schwartz kernel
i χ(t)ei|t−t |rm (z0 ) . 2rm (z0 )

Let A1 (z) = (I + K1 (z))−1 V




i 2rm (z)

φl ⊗ φl χ.

2 σl2 =νm

Then   K3 (z) − K3 (z0 ) = (I + K1 (z))−1 V Rems (z) − (I + K1 (z0 ))−1 V Rem0 (z0 )
φl ⊗ φl χ + A1 . × 2 σl2 =νm

Now we shall use (28) and | det(I + |BT |)| ≤ det(I + B|T |) (e.g. [12, Lemma 6.1]). Then | det(I + (I + K3 (z0 ))−1 (K3 (z) − K3 (z0 )))|




≤ det(I + (I + K3 (z0 ))−1 (I + K1 (z))−1 V |Rems (z)

φl ⊗ φl χ|)4

2 σl2 =νm

× det(I + (I + K3 (z0 ))−1 (I + K1 (z0 ))−1 V |Rem0 (z0 )




φl ⊗ φl χ|)4

2 σl2 =νm

× det(I + |(I + K3 (z0 ))−1 A1 |)2 . On Bm,4ρ˜, (I + K3 (z0 ))−1 (I + K1 (z))−1 V  ≤ C. For a compact operator A, let µ1 (A) ≥ µ2 (A) ≥ µ3 (A) ≥ · · · be the characteristic values of A; that is, the eigenvalues of |A∗ A|1/2 . Then, if A is trace class,  det(I + |A|) = (I + µj (A)). For p = 0, 1, 2, . . . , 2 )+j (Rems (z) µpMY (νm


2 σl2 =νm

2 φl ⊗ φl χ) = µ ˜p (rm (z)), j = 1, . . . , MY (νm ).

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Here we use the fact that Rems (z) depends on z only through rm (z). The µ ˜p (w) are independent of m. Therefore,


| det(I + (I + K3 (z0 ))−1 (I + K1 (z))−1 V |Rems (z)

φl ⊗ φl χ|)|

2 σl2 =νm 2

≤ C MY (νm ) ≤ C m

n−2

.

The same argument gives | det(I + (I + K3 (z0 ))−1 (I + K1 (z0 ))−1 V |Rem0 (z0 )




φl ⊗ φl χ|)| ≤ C m

n−2

.

2 σl2 =νm

A similar argument bounds 2

rm (z)2MY (νm ) det(I + |(I + K3 (z0 ))−1 A1 |)2 . 2 The rank of (I + K3 (z0 ))−1 A1 is MY (νm ), and (I + K3 (z0 ))−1 A1  is bounded for 4ρ˜ ≥ |rm (z)| > c > 0. Since rm (z)(I + K3 (z0 ))−1 A1 is bounded on Bm,4ρ˜, 2 M (νm )

rm Y

2

det(I + |(I + K3 (z0 ))−1 A1 |) ≤ C MY (νm ) ≤ C m

n−2




on Bm,4ρ˜. A consequence of Theorem 1.2 is

Corollary 5.1 Let X = R × Y and suppose X satisfies (H1). Let V ∈ L∞ comp (X) be real-valued, and let N (λ) = #{λ2j ≤ λ2 : λ2j is an eigenvalue of ∆ + V }. Then N (λ) = O(λn−1 ). Proof. Suppose τ ∈ R+ . Then, using (32) and (33), we see that if 1 V (length supp V + 1) ≤ minj |rj (τ )| 2 then τ cannot be an eigenvalue of ∆ + V . Here length supp V =

max

a,b∈supp V

|a − b|.

This means that any eigenvalue must lie within a fixed distance of some νj2 . Theorem 1.2 provides a bound on the number of eigenvalues within such a ball; summing over the νj2 we obtain the corollary.


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6 Proof of Theorem 1.3 We recall Theorem 1.3: ˆ and let {zk } be the resonances of P on this sheet. Then Theorem. Fix a sheet of Z,
| Im(r1 (zk ))| |r1 (zk )|2

k

< ∞.

In the proof of this theorem we shall use Proposition 3.3 and Carleman’s Theorem, which we recall (e.g. [14, Section V.i]). Theorem (Carleman). If F (ζ) is a holomorphic function in the region Im ζ ≥ 0, F (0) = 1, and if ak = rk eiθk (k = 1, 2, . . . ) are its zeros in this region, then


1 rk 1 ( − 2 ) sin θk = rk R πR

rk ≤R

+

1 2π

 

π

0

0

R

ln |F (Reiθ )| sin θdθ

1 1 − 2 x2 R

 ln |F (x)F (−x)|dx +

1 Im F  (0). 2

We note that Carleman’s Theorem also holds for a function F (ζ) which is holomorphic in Im ζ > 0 and continuous in Im ζ ≥ 0. In order to see this, apply Carleman’s Theorem to F (ζ) = F (ζ + i )/F (i ), > 0. Then, since both sides of the equation are continuous in for small ≥ 0, the theorem holds in this case as well. Proof of Theorem 1.3. ˆ and let Fix a sheet of Z, E = {j ∈ N : Im rj (z) < 0 on this sheet}. By Proposition 3.3, the poles of the resolvent on this sheet (but not on its boundary) correspond, with multiplicity, to the poles of det SE (z) on this sheet. We have, by Proposition 3.1, SE−1 (z) = SE (wE (z)), and, if z lies on the sheet with Im rj (z) < 0 if and only if j ∈ E, then wE (z) lies on the physical sheet. Therefore, we reduce the problem to a question about the zeros of det SE (z) for z on the physical sheet. It is helpful to identify the physical sheet with the upper half plane using the variable ζ = r1 (z). Let Ψij (ζ) = Sij (z(ζ)) and Ψ(ζ) = SE (z(ζ)). Using the fact that SE (z) is meromorphic on Zˆ we can extend Ψ to the closed upper half plane by continuity, except, perhaps, for a finite number of points corresponding

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to poles of SE (z). We shall also call these points poles of Ψ(ζ). The matrix Ψ(ζ) has at most a finite number of poles in the closed upper half plane. To prove the theorem, we shall use Carleman’s theorem applied to a multiple of det Ψ(ζ), chosen so that the product is holomorphic in the upper half plane and continuous on its closure. In order to do this, we need bounds on det Ψ. Let J = max{j ∈ E}. If ζ ∈ R, ζ 2 + ν12 > νJ2 , then |Ψij | ≤ C ([4]). To bound |Ψij (ζ)| away from the real axis, we need to bound |Sij (z)|, where Sij is determined by the expansion on the ends of Φj , see (2) and (3). We recall that Φj (z(ζ)) = χ(t)e−i˜rj (z(ζ))t φj − (P − z(ζ))−1 (P − ζ 2 − ν12 )χ(t)e−i˜rj (z(ζ))t φj (34) where χ(t) ∈ C ∞ (R) is supported in t > a and is one in a neighborhood of infinity. ˜ We note that for j ∈ E, (P − ζ 2 − ν12 )χ(t)e−i˜rj (z(ζ))t φj  ≤ C ζ 2 + ν12 − σj2 1/2 eC| Im r˜j (z(ζ))| ≤ C ζ 2 + ν12 − σj2 1/2 eCIm ζ

 ei˜rl (z(ζ))t φl

and Since

|t>a

 = (2| Im r˜l (z(ζ))|)−1/2 e−a Im r˜l (z(ζ)) .

(35) (36)

(P − ζ 2 − ν12 )−1  ≤ (dist(ζ 2 + ν12 , σ(P )))−1 ,

we have, using (3), (34), (35) and (36), |Ψlj (ζ)| ≤

C ζ 2 + ν12 − σl2 1/2 CIm ζ e dist(ζ 2 + ν12 , σ(P ))

when |ζ| is large. This proves | det Ψ(ζ)| ≤ CeCIm ζ if Im ζ > > 0 and |ζ| large. To obtain a bound in the closure of the upper half-plane, apply the Phragmen-Lindel¨ of theorem to h(ζ) =

k0  1

ζ − ζj det Ψ(ζ) ζ − ζj + M i

where ζ1 , ζ2 , . . . , ζk0 are the poles of det Ψ(ζ) in the closed upper half plane and M is chosen sufficiently large that M > Im kj , j = 1,. . . ,k0 . Then h(ζ) is holomorphic in the upper half plane, continuous in the closed upper half plane, and |h(ζ)| ≤ CeCIm ζ in the closed upper half plane. An application of Carleman’s Theorem to h(ζ) finishes the proof.
Acknowledgments. It is a pleasure to thank Maciej Zworski for suggesting the problem of counting resonances in this setting. I am grateful to him and to Dan Edidin for helpful discussions. Thanks to Julian Edward and the referee for helpful comments.

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References [1] A. Aslanyan and E.B. Davies, Separation of variables in perturbed cylinders, preprint. [2] A. Aslanyan, L. Parnovski and D. Vassiliev, Complex resonances in acoustic waveguides, Quart. J. Mech. Appl. Math. 53, no. 3, 429–447 (2000). [3] W. Bulla, F. Gesztesy, W. Renger and B. Simon, Weakly coupled bound states in quantum waveguides, Proc. Amer. Math. Soc. 125, no. 5, 1487–1495 (1997). [4] T. Christiansen, Scattering theory for manifolds with asymptotically cylindrical ends, J. Funct. Anal. 131, 2, 499–530 (1995). [5] T. Christiansen and M. Zworski, Spectral asymptotics for manifolds with cylindrical ends, Ann. Inst. Fourier 45, 1, 251–263 (1995). [6] E.B. Davies and L. Parnovski, Trapped modes in acoustic waveguides, Quart. J. Mech. Appl. Math. 51, no. 3, 477–492 (1998). [7] P. Duclos, P. Exner and B. Meller, Exponential bounds on curvature-induced resonances in a two-dimensional Dirichlet tube, Helv. Phys. Acta 71, no. 2, 133–162 (1998). [8] J. Edward, On the resonances of the Laplacian on waveguides. To appear, Journ. Math. Anal. and Appl. [9] R. Froese, Asymptotic distribution of resonances in one dimension, J. Differential Equations 137, no. 2, 251–272 (1997). [10] F.R. Gantmacher, The Theory of Matrices, Volume I. Chelsea Publishing Company, New York, 1959. [11] L. Guillop´e, Th´eorie spectrale de quelques vari´et´es `a bouts, Ann. Scient. Ec. Norm. Sup. 22, 4, 137–160 (1989). [12] L. Guillop´e and M. Zworski, Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Functional Analysis 129, No. 2, 364–389 (1995). [13] L. Guillop´e and M. Zworski, Scattering asymptotics for Riemann surfaces, Annals of Mathematics 145, 597-660 (1997). [14] B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I. 1964 viii+493 pp. [15] R.B. Melrose, ‘Polynomial bounds on the distribution of poles in scattering ´ by an obstacle’, Journ´ees “Equations aux D´eriv´ees partielles” Saint-Jean-deMonts, 1984.

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[16] R.B. Melrose, The Atiyah-Patodi-Singer Index Theorem, A.K. Peters, Wellesley, MA 1993. [17] W. M¨ uller, Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math. 109, 265–305 (1992). [18] L. Parnovski, Spectral asymptotics of the Laplace operator on manifolds with cylindrical ends, Int. J. Math. 6, 911-920 (1995). [19] A. Selberg, G¨ ottingen lectures in: Collected Works, Vol. I, 626–674, SpringerVerlag, Berlin, 1989. [20] J. Sj¨ ostrand and M. Zworski, Complex scaling and the distribution of scattering poles, J. of the AMS, 4, No. 4, 729–769 (1991). [21] G. Vodev, Sharp bounds on the number of scattering poles for perturbations of the Laplacian, Comm. Math. Phys. 146, 205–216 (1992). [22] G. Vodev, Resonances in the Euclidean scattering, Cubo Matem´ atica Educacional 3 no. 1, 317–360 (2001). [23] M. Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73 (2), 277–296 (1987). [24] M. Zworski, Sharp polynomial bounds on the number of scattering poles of radial potentials, J. Funct. Anal. 82, 370–403 (1989). [25] M. Zworski, Sharp polynomial bounds on the number of scattering poles, Duke Math. Jour. 59 (2), 311–323 (1989). [26] M. Zworski, Counting scattering poles, Spectral and scattering theory (Sanda, 1992), 301–331, Lecture Notes in Pure and Appl. Math. 161, Dekker, New York, (1994). T. Christiansen1 Department of Mathematics University of Missouri Columbia, Missouri 65211 USA email: [email protected] Communicated by Bernard Helffer submitted 21/12/02, accepted 18/05/02

1 Partially

supported by the NSF grant DMS 0088922.

Ann. Henri Poincar´e 3 (2002) 921 – 938 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/050921-18

Annales Henri Poincar´ e

Analyticity of the Gibbs State for a Quantum Anharmonic Crystal: No Order Parameter R. A. Minlos, E. A. Pechersky and V. A. Zagrebnov Abstract. We prove that for a ν-dimensional quantum crystal model of interacting anharmonic oscillators of mass m there exists m0 such that in the light-mass domain 0 < m < m0 the corresponding Gibbs state is analytic with respect to external field (conjugate to site displacements) for all temperatures T ≥ 0, i.e. including the ground state. This means that for the model with harmonic interaction and a symmetric double-well one-site potential, the light-mass quantum fluctuations suppress the symmetry breaking structural phase transition known in this model for ν ≥ 3 and m > M0 ≥ m0 , where M0 is large enough.

1 Introduction In the paper [1] the existence of Gibbs states for a ν-dimensional quantum anharmonic crystal model was proved for all temperatures T : 0 ≤ T < ∞, and for small masses m < m0 of one-dimensional oscillators. This result has been recently extended to the multi-dimensional oscillators in [2]. The proof in [1] is based on a reduction of this quantum model to a classic one for ensemble of continuous trajectories (paths, or ”spins”) as a configuration space with help of the FeynmanKac formula. Using this reduction the cluster expansion method was developed in [1] for the random field of paths to construct the corresponding infinite-volume quantum Gibbs state. The standard question of Statistical Mechanics is whether this state is unique? In this particular case it splits into two questions : the first concerns the uniqueness of the classical counterpart (field of paths), whereas the second question concerns the uniqueness of the quantum state. It is known that in our model the later is not unique at low temperatures as soon as ν ≥ 3, and the oscillator mass m is sufficiently large : m > M0 (”classical” limit), see [3] - [6]. For a symmetric double-well anharmonic oscillators this nonuniqueness is due to a symmetry breaking structural phase transition with a nonzero displacement order parameter at low temperatures T < Tc (m). On the other hand, one can prove that there exists m0 > 0 such that for m < m0 (”quantum” limit) this phase transition is suppressed for all temperatures T ≥ 0 by microscopic quantum fluctuations (tunneling effect) [7]. In the conjecture concerning our model it is crucial that on the phase diagram (m, T ) the line of the critical temperatures Tc (m) decreases for decreasing m and becomes Tc (m0 ) = 0 for some non-zero threshold mass m0 > 0. This conjecture about the critical line Tc (m), and the

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Ann. Henri Poincar´e

phenomenon of suppression of the phase transition by quantum fluctuations can be checked explicitly for the spherical approximation of this quantum anharmonic crystal model [8], [9]. There it was shown that (m = m0 > 0, T = Tc (m0 ) = 0) is indeed the critical point of a quantum phase transition from non-zero to zero displacement order parameter when decreasing mass crosses m0 > 0 along the line T = 0. Similar phenomenon is known in the quantum X − Y (or the Ising) model in a transverse magnetic field, see e.g.[10]. Moreover, it is also known that with decreasing mass m the macroscopic quantum fluctuations change their properties. In the ”quantum” limit they all become normal, see [11]-[13]. Whereas in the phase transition regime (i.e. on the critical line Tc (m) ≥ 0) they could be abnormal or even supernormal (squeezed), which is known for the moment only for the corresponding spherical approximation, see [8],[9]. It is noteworthy that the uniqueness problem of the classical counterpart of our model can be posed in the spirit of the DLR-theory [14],[15]-[18], since definition of the classical ensemble of paths introduces specifications, which are a set of consistent measures [1]. Hence, uniqueness for the classical counterpart has a clear sense and means the existence only one Gibbs random field of paths, see [15]-[18], where it is proved in the light-mass limit for temperatures strictly separated from zero. By virtue of noncompactness of the path (”spin”) variables, one has to take a precaution to consider only tempered Gibbs measures [19]. We have to mention that in fact, the paper [1] proves the uniqueness of the Gibbs state of the ν-dimensional quantum crystal model for all temperatures including zero 0 ≤ T < ∞, in the following restricted sense : Let be boundary condition fixed by the ”freezing” of particle positions on the surface of the crystal. This corresponds to the straight-line configurations of quantum paths on the whole space-time boundary. If these positions belong to a fixed bounded interval, then the corresponding infinite random Gibbs field of paths is unique, i.e. independent of these boundary conditions, in the class of tempered Gibbs measures in the light-mass domain, i.e., as soon as m < m0 . The same is true if we fix boundary conditions by the ”freezing” of the surface particle quantum paths in such a way that they would have some uniformly bounded amplitudes. This manner to fix boundary conditions (i.e. quantum Gibbs measure) seems not to be acceptable from the physical point of view, since we manipulate with the particles as with classical objects. More acceptable is to fix the Gibbs state by a (conjugate to displacements) external field. We shall follow this strategy below . As far as concerns the uniqueness problem of quantum states, at the present time there is no conventional setting of the question that would have a clear physical sense, except the KMS uniqueness [20]. In the present paper we prove (Theorem 1 ) analytic properties of the model with respect to the external field in the light-mass domain m < m0 for any T ≥ 0. This means in particular that for symmetric double-well anharmonic potential we can rule out (Theorem 2 ) symmetry breaking phase transitions in ”direction” of the external field for all temperatures including T = 0, cf. [7]. It is relevant to em-

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phasize that similar to [15]-[18], our method does not solve the uniqueness problem in the DLR-sense in the light-mass domain at T = 0, since it does not prove the uniqueness of the Gibbs field of paths for this temperature, if the boundary trajectories do not have limited amplitudes. The main obstacle to making this conclusion is the noncompactness of the path (”spin”) variables. However, since our Theorem 2 shows that in the light-mass domain m < m0 there is no order parameter corresponding to the symmetry breaking structural phase transition that exists for m > M0 , one could anticipate in this domain the uniqueness of the quantum Gibbs state for all temperatures including T = 0. For the proof in the case of the compact ”spins” see [21]. Notice that this makes a striking difference between the quantum model and its classical analog corresponding formally to m → ∞ (or the Planck constant
→ 0): in the quantum case the suppression of any ordering is expected even at zero temperature, due to the tunneling microscopic quantum fluctuations for sufficiently light masses m < m0 .

2 The model and main results We study a quantum model of interacting one-dimensional anharmonic oscillators on the lattice Zν , ν ≥ 1. For any finite Λ ⊂ Zν the Hamiltonian of the model is the operator: HΛh = −


2
∂ 2 + 2m ∂qx2 x∈Λ




(qx − qy )2 +




V (qx ) − h

x∈Λ

x,y∈Λ,x−y=1




qx

(1)

x∈Λ

which acts in the Hilbert space HΛ = L2 (RΛ , dΛ q) of functions defined on the set of oscillator configurations QΛ = {qx ∈ R1 : x ∈ Λ} ∈ RΛ . In (1)  > 0 is a parameter of the nearest-neighbor harmonic interaction, and V is a one-site potential , which is a real polynomial function of the form V (q) = a0 q 2s + a1 q 2s−1 + · · · + a2s−2 q 2 ,

(2)

where s > 1 and a0 > 0. This means that below we consider empty boundary conditions. It is semibounded from below and For real h the operator HΛh isself-adjoint.  h −βHΛ . it generates the Gibbs semigroup e β≥0

Remark 1 It is worth to stress that below we consider the integer s > 1, that leads s−1 to a small scaled interaction   = m s+1  in the light-mass limit. This is a key to the cluster expansion constructed in [1]. On the other hand the case s ≤ 1 leads in the light-mass limit to harmonic-term domination, see discussion in [7]. We extracted the last (linear) term in (1) from V in order to find an explicit expression of cluster coefficients as functions of the external field h conjugated to

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Ann. Henri Poincar´e

displacements {qx }x∈Λ . Let B(HΛ ) be algebra of bounded operators on HΛ . The finite-volume Gibbs state on B(HΛ ) is the continuous linear functional AhΛ,β =

h 1 Tr(e−βHΛ A), A ∈ B(HΛ ), h ZΛ,β

(3) h

where 0 < β < ∞ is the inverse temperature : β −1 = kB T . Since e−βHΛ is a traceh h class (nuclear) operator for β > 0, the partition function ZΛ,β = Tre−βHΛ exists. Therefore, the functional (3) is well-defined on B(HΛ ) for 0 < β < ∞. Moreover, there exists the extension of ·hΛ,β to the case β = ∞. Namely, h h , ψΛ,0 ), AhΛ,∞ = lim AhΛ,β = (AψΛ,0

(4)

β→∞

h is the ground-state eigenvector corresponding to the minimal eigenvalue where ψΛ,0 h

of HΛh . Notice that for any bounded Λ ⊂ Zν the Gibbs semigroup {e−βHΛ }β≥0 is h defined for any complex h ∈ C , and e−βHΛ ∈ T r − class(HΛ ) of operators on HΛ h for β > 0. Moreover, ZΛ,β is a holomorphic function of h ∈ C , see e.g. [22] and h=0 [23]. Since ZΛ,β > 0 , there exists a real h0 (m, Λ) > 0 such that the states (3) and (4) are defined on B(HΛ ) for complex h : |h| < h0 (m, Λ). In fact, we shall show that h0 does not depend on Λ in the light-mass domain m < m0 . ν Let Λ1 ⊂ Λ2 ⊂ · · · and ∪∞ n=1 Λn = Z , and let A∞ be the closure of the 0 inductive limit A∞ of increasing the subalgebras B(HΛ1 ) ⊂ B(HΛ2 ) ⊂ · · · , where the embedding B(HΛn ) ⊂ B(HΛn+1 ) is defined by the representation HΛn+1 = HΛn ⊗ HΛn+1 \Λn . If for any A ∈ A0∞ there exists the limit Ahβ = lim AhΛn ,β , 0 ≤ β −1 < ∞, n→∞

(5)

then ·hβ is called a (limiting) Gibbs state. Since the state (5) is norm-continuous, it can be extended to a state on A∞ . Recall that 1 ln ZΛn (β, h) f (β, h) = lim − (6) n→∞ β|Λn | is the thermodynamic limit of the free-energy density. The following theorem is the first main result of the present paper : Theorem 1 There exists a mass m0 > 0 such that for all 0 < m < m0 : (a) There is h0 (m) = inf Λ h0 (m, Λ) > 0 such that the limits (5) and (6) exist in the domain {0 ≤ β −1 < ∞} × {h ∈ C : |h| < h0 (m)}.

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(b) The limit f (β, h) is an analytic function in the circle {h ∈ C : |h| < h0 (m)} for any β −1 ≥ 0. (c) For any β −1 ≥ 0 and A ∈ A0∞ the function Ahβ is analytic in the circle {h ∈ C : |h| < h0 (m)}. Remark 2 It is an open question, whether h0 (m) does depend on m in the light mass domain 0 < m < m0 . Remark 3 Again it is unknown, whether (c) is true for the state ·hβ . The positive     answer would follow from the uniform estimate: Ahβ  ≤ CA, for A ∈ A0∞ and {0 ≤ β −1 < ∞} × {h ∈ C : |h| < h0 (m)}. In our estimates below (see (29)) the coefficient C = CA is A-dependent.

3 Proofs The Hamiltonian (1) can be represented as HΛh = −

1
∂2 + 2m ∂qx2 x∈Λ




(qx − qy )2 +




Vh (qx )

x∈Λ

x,y∈Λ,x−y=1

where Vh (q) := V (q) − hq, and we put for simplicity
= 1. Therefore, the Hamiltonian HΛh satisfies all conditions required in [1] for real h. Notice that in [1] the proof of existence of the state is based on cluster expansions. However, to study the analyticity of the state (5), or the limiting free-energy (6) for complex h, we need another form of the cluster expansions, which explicitly takes into account the cluster coefficient dependence on h. To this end we first modify the technique h of [1] by giving a new form of the cluster expansion of the partition function ZΛ,β for non-zero temperatures β < ∞ as well as for the case β = ∞, which requires some special attention. With the new cluster expansion in hands one can apply the construction of [1] to obtain the free-energy density and the limiting Gibbs states, when Λ ↑ Zν . Therefore, below we sketch a particular generalization of [1] to Vh stressing in the proofs only new aspects related to h−analyticity, and omitting details which the reader can find in [1].

3.1

Dilatation and reduction to classical ensemble of trajectories

Let U : L2 (RΛ , dΛ q)→L2 (RΛ , dΛ q) be the unitary mapping  |Λ|/(4s+4) 1 f ({qx }x∈Λ ) (U f )({ qx }x∈Λ ) = m generated by the dilatation:  qx =

1 m

1/(2s+2) qx .

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R. A. Minlos, E. A. Pechersky and V. A. Zagrebnov

Ann. Henri Poincar´e

Then taking into account the transformations 2s−1

 β = m s+1 β,

h = m 2s+2  h

s

we obtain for the state (3) the following representation: 1

h

·Λ,β =

Zh Λ,β

h

Tr{e−βHΛ (·)}.

(7)

Here  Λh = − 1 H 2 where


∂2 +  ∂ qx2

x∈Λ




( qx − qy )2 +




V ( qx ) −  h

x∈Λ

x,y∈Λ,x−y=1

1 2 V ( q ) = a0 q2s + a1 m 2s+2 q2s−1 + a2 m 2s+1 q2s−2 + · · · ,

 =m

s−1 s+1

.




qx ,

(8)

x∈Λ

(9) (10)

Remark 4 Since below we consider exclusively the operators and the states (7) after dilatation U , the hat ”  ” will be systematically omitted. Let the self-adjoint one-dimensional Schr¨ odinger operator with the polynomial potential (2) 1 ∂2 + V (q) h=− 2 ∂q 2 be defined in L2 (R1 , dq). Let ψ0 be its ground state, i.e. the unique positive normalized eigenvector corresponding to the minimal eigenvalue E0 . Then using the kernel of the Gibbs semigroup generated by h: Gt (q1 , q2 ) = e−th (q1 , q2 ), t ≥ 0, one can define a stationary diffusion process ξ with the invariant measure dν(q) = ψ02 (q)dq and with the transition probability density (with respect to dν) equal to : ρt (q1 /q2 ) =

Gt (q1 , q2 ) . ψ0 (q1 )ψ0 (q2 )e−E0 t

It is known that (with the probability 1) the sample paths of this process are continuous. Notice that for s > 1 (2) the transition probability density ρt (q1 /q2 ) → 1, as t → ∞, uniformly and at exponential rate, see [1, Corollary 5.1] or [2, Lemma 3.1] for a simpler proof. Let µβ be a restriction of the measure of the process ξ on the set C[−β, β] (see [1] for details). Let Cper [−β, β] = {ω(·) ∈ C[−β, β] : ω(−β) = ω(β)}, and let µper be the conditional measure of µβ conditioned by ω(−β) = ω(β). Notice β per that µβ can be interpreted as a measure corresponding to the stationary (with

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Analyticity of the Gibbs State for a Quantum Anharmonic Crystal

927

respect to rotations) diffusive process ξβ on the circle Oβ of the length 2β. By dνβ we denote a related to the process ξβ invariant measure, which has density ρ2β (q/q) with respect to the measure dν. The process ξβ is Markovian: if one fixes it in two points ξt1 = x1 and ξt2 = x2 on the circle, 0 < t1 − t2 < 2β, then its values on two complementary arcs are independent. ν Zν Let Mβ = µZβ and Mβper = (µper be product-measures on the spaces β ) ν ν (C[−β, β])Z and (Cper [−β, β])Z correspondingly. Then we can represent the partition function of the model with the Hamiltonian (8) using the Feynman–Kac formula: h,per h = aΛ ZΛ,β , (11) ZΛ,β where h,per := Z Λ,β (Cper [−β,β])Λ

(12) e−




x,y∈Λ,x−y=1

β

−β

(ωx (t)−ωy (t))2 dt+h




x∈Λ

β

−β

ωx (t)dt

Mβper (dωΛ ),

Here ωΛ = {ωx ∈ Cper [−β, β], x ∈ Λ} and aΛ = (Tre−βh )|Λ| . Notice that by virtue of our definition of the process ξ the potential V does not appear in the formal Hamiltonian corresponding to ensemble of trajectories ωΛ in (12). Since we are going to consider below the low temperature regime, it is convenient to use another representation of (12). Following [1] we make a partition of the interval [−β, β] into subintervals having the length a > 0 such that β = aN with an integer N . As in [1] we put a = a() to be a function of the parameter  involved in the interaction term of (8), (12). Similar to [1] we shall choose the value of a() in dependence on properties of the stationary process ξ , see Remark 6 below. Notice that for any trajectory ωx = {ωx (t), −β ≤ t ≤ β} ∈ Cper [−β, β], x ∈ Zν , one can define restrictions : ω(x,k) := {ω(x,k) (t) = ωx (t + ka), 0 ≤ t ≤ a}

(13)

on the intervals [ka, (k + 1)a], k ∈ {−N, −N + 1, . . . , N − 1} := [−N, N − 1] of the partition of the interval [−β, β] . For those k  s and any x ∈ Zν the restrictions verify the continuity conditions : q(x,k+1) := ω(x,k+1) (0) = ω(x,k) (a).

(14)

In the case of periodic trajectories we consider the residue group z2N , i.e. k ∈ z2N . Then we have q(x,N ) = q(x,−N ) . (15) Therefore, in fact the configurations {ωx ∈ Cper [−β, β], x ∈ Zν } coincides with a collection of ”elementary” trajectories defined by the restrictions (13) satisfying conditions (14) and (15). We denote this collection by ωβ := {ω(x,k) , x ∈ Zν , k ∈ z2N } ∈ Ωβ ,

(16)

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Ann. Henri Poincar´e

where Ωβ is the set of those trajectories, and the conditions (14) and (15) are assumed. Let Λ ⊆ Zν and U = Λ × z2N . Then we define on the configurations of ”elementary” trajectories (16) the interaction Hamiltonian by a
HU,h (ωβ ) :=  (ω(x,k) (t) − ω(y,k) (t))2 dt (17) 0

(x,k),(y,k)∈U x−y=1
a

−h

ω(x,k) (t)dt.

(x,k)∈U

0

Since the product measure Mβper is also defined on the space Ωβ of the ”elementary”

,h,per trajectories (16), the corresponding to HΛ,h (ωβ ) perturbed Gibbs measure Mβ,Λ is defined by the Radon-Nikodim derivative: ,h,per dMβ,Λ

dMβper

,h

e−HU (ωβ ) (ωβ ) = . Zh,per

(18)

Λ,β

Let Q = {q(x,k) ∈ R1 : (x, k) ∈ Zν × z2N }. Then by ·Q we denote the conditional expectation with respect to Mβper given the configuration Q. Then for the partition function (12) we get the following representation : h,per = Z Λ,β



= RU



a

2 exp − (ω(x,k) (t) − ω(y,k) (t)) dt ×



0

(x,k),(y,k)∈U, x−y=1

 (x,k)∈U

exp h

(x,k),(x,k+1)∈U

×

ω(x,k) (t)dt

0





a

Q

ρa (q(x,k+1) /q(x,k) )



ν(dq(x,k) ). (19)

(x,k)∈U

Notice that ·Q depends only on QU = {q(x,k) ∈ R1 : (x, k) ∈ U = Λ × z2N }. As above ρa (q  /q) denotes the transition probability density for the time-interval a with respect to the invariant measure dν(q) = ψ02 (q)dq of the stationary diffusion process ξβ on the circle Oβ . The similar formulae hold for the case of the measure Mβ , i.e. nonperiodic trajectories, when U = Λ × [−N, N − 1] and the condition (15) is not required.

3.2

Cluster expansion

Notice that in [1] we first obtained cluster expansion for the partition function for the nonperiodic measure Mβ . Then we extended it to the case of the periodic

Vol. 3, 2002

Analyticity of the Gibbs State for a Quantum Anharmonic Crystal

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measure Mβper via some routine transformation, see [1, Section 5.3] for details. Below we proceed in the similar way. Having the cluster expansion for the partition function one can obtain then the corresponding expansions for observables using a standard method of [19], see also [1]. h for the Therefore, our next step is to obtain the cluster expansion for ZΛ,β nonperiodic measure Mβ . To this end we start (cf.(19)) with partition function ZUh (Q) conditioned by the configuration Q:

a

 h ZU (Q) := exp − (ω(x,k) (t) − ω(y,k) (t))2 dt × (x,k),(y,k)∈U,x−y=1

0

exp h

 (x,k)∈U



a

ω(x,k) (t)dt

0

. (20) Q

In fact this expression is conditioned only by the subconfiguration QU = {q(x,k) ∈ R1 : (x, k) ∈ U = Λ × [−N, N − 1]}. By virtue of the Markovian property of the process ξ one gets from (20) that

a

  h ZU (Q) = exp − (ω(x,k) (t) − ω(y,k) (t))2 dt × −N ≤k 0 and h0 > 0 such that for || < 0 , |h| < h0 , and Γ = (γ1 , . . . , γs , τ1 , . . . , τl ) one has the estimate: |KΓ (h)| < C0

 γ∈Γ

p(γ) r(γ)

λ()

h

 τ ∈Γ

λ()

q(τ )

.

(28)

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Ann. Henri Poincar´e

Here λ() is such that lim→0 λ() = 0. Similarly one has :    h K (AL ) < AL  |KΓ (h)|, ξ

(29)

Γ∈ξ

where AL  = supR|L| {|AL ({qx }x∈L )|}. The both KΓ (h) and Kξh (AL ) are analytic functions of h in the circle {h ∈ C : |h| < h0 }. Remark 5 The coefficients DB (η) are such that if (28) holds, then 
|DB (η)| |KΓ (h)| < C |B| , η: ∂η∩B =∅

(30)

Γ∈η

where B ⊂ Zν+1 and C > 0 (see [19]). Proof of Proposition 1. To get the estimates (28) and (29) we follow the method of [1]. Our main tool will be Lemma 5.2 [1]. For reader’s convenience we formulate it below : Lemma 1 Let {(Ex , µx )}x∈T be a family of probability spaces indexed by a finite space T. LetS = {S} be a family of subsets of T. For any S ∈ S let functions fS : ET = x∈T Ex → R be such that fS depends only on variables of Ex for x ∈ S. At last let nS be such that for every x ∈ T
1 ≤1 nS

S: x∈S

Then

    n1    S      nS fS dµx  ≤ |fS | dµx   ET  ET S

x∈T

S

x∈T

 γ (Q), see (23). To this end we use With help of Lemma 1 we first estimate K Lemma 1 by taking all objects as follows: T is the set of time-edges included in γ with taking in account their possible multiplicity. S = {
:
= {1 (
), 2 (
)} ∈ γ} ∪ { :  ∈ γ}. The probability 1 2 spaces are E = {ω ∈ C[0, a] : q = q(x,k) , q = q(x,k+1) } with a measure µ which is the conditional measure on E generated by µβ under boundary 1 2 conditions q = q(x,k) , q = q(x,k+1) . We put a

f
(ω1 , ω2 ) = exp − (ω1 (
) (t) − ω2 (
) (t))2 dt − 1, (31) 0

for
∈ S , and

f (ω ) = exp h 0

a

ω (t)dt − 1,

(32)

Vol. 3, 2002

Analyticity of the Gibbs State for a Quantum Anharmonic Crystal

933

for  ∈ S . Let integers n
and n be attributed correspondingly to all plaquettes and time-edges in such a way that 2ν 1 + ≤ 1. n
n

(33)

Then Lemma 1 implies for (23) that       F
(Q
) F (Q ), Kγ (Q) ≤
∈γ

(34)

∈γ

where

 n
 (n
)−1 a

  exp − F
(Q
) = (ω1 (
) (t) − ω2 (
) (t))2 dt − 1 ,  Q

0

(35) and

 n  (n )−1

  exp h ω (t)dt − 1 F (Q ) = .  

(36)

Q

Here Q
and Q are values conditioned by the configuration Q at the end-points of
and  correspondingly. Now, to estimate KΓ (h) (24) we again apply Lemma 1. We put T = ∂Γ, and   S= [{
:
∈ γ} ∪ { :  ∈ γ}]  { :  ∈ τ }. γ∈Γ

τ ∈Γ

The sign  means that the edge  will be counted twice, if it belongs to the both sets on the left- and right-hand sides of . The probability spaces are E(x,k) = R with the measures µ(x,k) = ν. We have three kinds of functions with supports on the end-points of plaquettes and edges : F
(Q
), F (Q) for  ∈ γ, and 2 1 /q ) − 1|, for  ∈ τ . Then we choose the integer n
, n satisG (Q) = |ρa (q fying 4ν 4 + ≤ 1, n
n that implies (33). Then applying Lemma 1 we obtain the bound   (n
)−1      F
(Q
)n
|KΓ (h)| ≤ ν(dq(x,k) )   γ∈Γ
∈γ (x,k)∈(
)  (n )−1    1 2 × )ν(dq ) F (Q )n ν(dq  ∈γ

×

   τ ∈Γ ∈τ

n

G (Q )

(n )−1 1 2 ν(dq )ν(dq )

.

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R. A. Minlos, E. A. Pechersky and V. A. Zagrebnov

Ann. Henri Poincar´e

Remark 6 Let a = a() := −A ln , where A > 0. Then following the same line reasoning as in [1, Section 5.2] the integrals   F
(Q
)n


(n
)−1



ν(dq(x,k) )

 and

(n )−1 1 2 Gn ν(dq )ν(dq )

(x,k)∈(
)

can be estimated by λ() = R2/3 for n
= 8ν and n = 8. Therefore, it rests only to estimate the integral: 1 2 F (Q )n ν(dq )ν(dq ), which gives dependence on the external field h. Our estimate below is based on Lemma 5.3 of [1]. Since distribution density of the random variable q 1 = ω (t = 0) with respect to ν is unit, this lemma claims that there exist b > 0 such that ρa (q 1 , q 2 ) = ρa (q 2 /q 1 ) > b for any q 1 , q 2 , and a ≥ a0 (). Here ρa (q 1 , q 2 ) is the density of the stationary distribution of the pair of random variables (ω (0), ω (a)) with the respect of the measure ν × ν. Then 1 2 )ν(dq )= F (Q )n ν(dq n  a

 1 2   )ν(dq ) ν(dq 2 1  exp h ≤ ω (t)dt − 1 ρ (q /q )  a     2 1 ρa (q /q ) 1 ,q 2 ) 0 R2 (q  n a

  1   µβ (dω ) =: J, exp h ω (t)dt − 1 ∆   b 0 where µβ is the measure corresponding to the process ξ. The representation  a    1 a

a      exp h   = |h| ω (t)dt − 1 ω (t)dt ehs 0 ω (t)dt ds       0

0

0

gives the bound n

J ≤ |h|

1 b

   

a 0

n  a  ω (t)dt en |h| 0 |ω (t)|dt

(37)

µβ

Using results of [1, Section 5.2] we conclude that the expectation with respect to µβ in the right-hand side of (37) is bounded by some C > 0 as soon as |h| < h0 = h0 () and a < a(). Therefore we obtain (28). The estimate of coefficients Kξh (AL ) goes through verbatim of the above arguments, that gives (29). Since the same line of

Vol. 3, 2002

Analyticity of the Gibbs State for a Quantum Anharmonic Crystal

935

reasoning as above yields similar estimates for derivatives ∂h KΓ (h) and ∂h Kξh (AL ) in the circle {h ∈ C : |h| < h0 ()}, the functions KΓ (h) and Kξh (AL ) are analytic in this circle. This finishes the proof of Proposition 1.
Proof of Theorem 1. Now the statements (a),(b) and (c) of Theorem 1 follow from the Proposition 1, representations (25),(26), and relation (10) between ( ) and m.


3.3

Order parameter

We recall that the order parameter for the model (1) is defined by : σ(β) := lim lim  h→0 n→∞

1
qx hΛn ,β . |Λn |

(38)

x∈Λn

If the one-site potential V (q) in (1) is double-well and symmetric, then for ν ≥ 3 (see [3]-[6]) there is M0 such that for m > M0 there exists a critical temperature Tc (m) > 0. At Tc (m) one has a symmetry breaking phase transition with a nonzero displacement order parameter at low temperatures: σ± (β > βc (m)) := lim lim  h→±0 n→∞

1
qx hΛn ,β = ±σ+ (β > βc (m)), |Λn |

(39)

x∈Λn

and one gets zero order parameter for T ≥ Tc (m) : σ± (β ≤ βc (m)) = 0.

(40)

Now we can proof our second main result : Theorem 2 Let V (q) be a double-well symmetric one-site potential with s > 1. Then there exists a mass m0 > 0 such that for any 0 < m ≤ m0 and for all temperatures, including β = ∞, the order parameter is trivial for empty boundary conditions: (41) σ± (β) = 0. Proof. This statement is in fact a corollary of some general results of [19] and assertion (b) of our Theorem 1. Indeed, one has that: σΛn (β, h) :=

1
qx hΛn ,β = ∂h fΛn (β, h), |Λn |

(42)

x∈Λn

where fΛn (β, h) := −

1 ln ZΛn (β, h) β|Λn |

(43)

is the finite-volume free-energy density, cf (6). By virtue of cluster representation of partition function (25) one gets that ZΛn (β, h) = 0, the function fΛn (β, h) is

936

R. A. Minlos, E. A. Pechersky and V. A. Zagrebnov

Ann. Henri Poincar´e

analytic in h in some (independent of the volume) circle in the complex plane. Moreover, the limit (6) exists and it is analytic in this circle , see [19, Ch.III,§4]. Then by Theorem 1(b) the limit σ(β, h) := limn→∞ σΛn (β, h) is an analytic function in the circle {h ∈ C : |h| < h0 (m)}. Since by symmetry of the one-site potential we have σΛn (β, h = 0) = 0, one gets limh→±0 σ(β, h) = 0 for any temperature as soon as 0 < m < m0 .


Acknowledgments This paper was originated during E.A.P.’s stay at CPT-CNRS-Luminy and V.A.Z.’s stay at IITP-Moscow funded by the French-Russian Project 7787. The manuscript reached its final form during R.A.M.’s visit to CPT-CNRS-Luminy funded by the same Project and grants of RFFI (Grant 99-01-00284) and CRDF (Grant RM1-2085). E.A.P. also would like to acknowledge the latter Grant as well as RFFI (Grant 99-01-00003) and A.N.Lyapunov Center at MSU (Project 98-02). We thank the referee for numerous useful remarks.

References [1] R.A. Minlos, A. Verbeure and V.A. Zagrebnov, A Quantum Crystal Model in the Light-Mass Limit: Gibbs States, Rev. Math. Phys. 12, 981–1032 (2000). [2] W.G. Faris and R.A. Minlos, A Quantum Crystal with Multidimensional Anharmonic Oscillators, J. Stat. Phys. 94, 365–387 (1999). [3] W. Dressler, L. Landau and J.F. Perez, Estimates of Critical Lenght and Critical Temperatures for Classical and Quantum Lattice Systems, J. Stat. Phys. 20, 123–162 (1979). [4] L.A. Pastur and V.A. Khoruzhenko, Phase Transistion in Quantum Models of Rotators and Ferroelectrics, Theor. and Math. Phys. 73, 111–124 (1987). [5] Yu. G. Kondratiev, Phase Transitions in Quantum Models of Ferroelectrics, in Stochastic Processes, Physics and Geometry pp.465–475, World Scientific, Singapore, 1994. [6] S. Albeverio, A. Yu. Kondratiev and A.L. Rebenko, Peierls Argument and Long-Range Order Behavior of Quantum Lattice Systems with Unbounded Spins, J. Stat. Phys. 92, 1137–1152 (1998). [7] A. Verbeure and V.A. Zagrebnov, No-go Theorem for Quantum Structural Phase Transistions, J. Phys. A 28, 5415–5421 (1995). [8] A. Verbeure and V.A. Zagrebnov, Phase Transistions and Algebra of Fluctuation Operators in an Exactly Soluble Model of a Quantum Anharmonic Crystal, J. Stat. Phys. 69, 329–359 (1992).

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Analyticity of the Gibbs State for a Quantum Anharmonic Crystal

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[9] A. Verbeure and V.A. Zagrebnov, Dynamics of Quantum Fluctuations Operators in an Anharmonic Crystal Model, J. Stat. Phys. 79, 377–393 (1995). [10] P. Pfeuty and R.J. Elliott, The Ising Model with a Transverse Field - II Ground State Properties, J. Phys. C 4, 2370–2385 (1971). [11] S. Albeverio, Yu. G. Kondratiev,and Yu. Kozitsky, Suppression of Critical Fluctuations by Strong Quantum Effects in Quantum Lattice Systems, Commun. Math. Phys. 194, 493–512 (1998). [12] Yu. Kozitsky, Quantum Effects in a Lattice Model of Anharmonic Vector Oscillators, Lett. Math. Phys. 51, 71–81 (2000). [13] Yu. Kozitsky, Scalar Domination and Normal Fluctuations in N -vector Quantum Anharmonic Crystal, Lett. Math. Phys. 53, 289–303 (2000). [14] R.L. Dobrushin, Prescribing a system of random variables by conditional distributions, Teor. Veroatn. Primen. 15, 458–486 (1970) (transl. from the Russian). [15] S. Albeverio, Yu. G. Kondratiev, M. R¨ ockner and T.V. Tsikalenko, Uniqueness of Gibbs States for Quantum Lattice Systems, Prob. Theory Relat. Fields 108, 193–218 (1997). [16] S. Albeverio, Yu. G. Kondratiev, M. R¨ ockner and T.V. Tsikalenko, Dobrushin’s Uniqueness for Quantum Lattice Systems with Nonlocal Interaction, Commun. Math. Phys. 189, 621–630 (1997). [17] S. Albeverio, Yu. Kondratiev, Yu. Kozitsky and M. R¨ ockner, Uniqueness for Gibbs Measures of Quantum Lattices in Small Mass Regime, Ann. Inst. H. Poincar´e: Probab.Statist. 137, 43–69 (2001). [18] S. Albeverio, Yu. G. Kondratiev, R.A. Minlos and A.L. Rebenko, Small-Mass Behavior of Quantum Gibbs States for Lattice Models with Unbounded Spins, J. Stat. Phys. 92, 1153–1172 (1998). [19] V.A. Malyshev and R.A. Minlos, Gibbs Random Fields, Cluster Expansions, (Kluwer Acad.Publ., Dordrecht, 1991). [20] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics,Vol.2, Second edition (Springer, Berlin, 1996). [21] S. Albeverio, Yu. G. Kondratiev, R.A.Minlos and G.V.Shchepan’uk, Uniqueness Problem for Quantum Lattice Systems Models with Compact Spins, Lett. Math. Phys. 52, 185–195 (2000). [22] H.D. Maison, Analyticity of the Partition Function for Finite Quantum Systems, Commun. Math. Phys. 22, 166–172 (1971).

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[23] V.A. Zagrebnov, Perturbations of Gibbs Semigroups, Commun. Math. Phys. 120, 653–664 (1989). Robert A. Minlos and Eugene A. Pechersky Institute for Information Transmissions Problems Bolshoj Karetny per.19, GSP-4 Moscow 101447 Russia Valentin A. Zagrebnov Universit´e Aix-Marseille II Centre de Physique Th´eorique CNRS-Luminy-Case 907 F-13288 Marseille Cedex 09 France email: [email protected] Communicated by Joel Feldman submitted 30/10/01, accepted 24/05/02

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 3 (2002) 939 – 965 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/050939-27

Annales Henri Poincar´ e

Effective N -Body Dynamics for the Massless Nelson Model and Adiabatic Decoupling without Spectral Gap S. Teufel Abstract. The Schr¨ odinger equation for N particles interacting through effective pair potentials is derived from the massless Nelson model with ultraviolet cutoffs. We consider a scaling limit where the particles are slow and heavy, but, in contrast to earlier work [7], no “weak coupling” is assumed. To this end we prove a spaceadiabatic theorem without gap condition which gives, in particular, control on the rate of convergence in the adiabatic limit.

1 Introduction The physical picture underlying nonrelativistic quantum electrodynamics is that of charged particles which interact through the exchange of photons and dissipate energy through emission of photons. In situations where the velocities of the particles are small compared to the propagation speed of the photons the interaction is given through effective, instantaneous pair potentials. If, in addition, also accelerations are small, then dissipation through radiation can be neglected in good approximation. Instead of full nonrelativistic QED we consider the massless Nelson model. This model describes N spinless particles coupled to a scalar Bose field of zero mass. The content of this work is a mathematical derivation of the time-dependent Schr¨ odinger equation for N particles with Coulombic pair potentials from the massless Nelson model with ultraviolet cutoffs. The key mechanism in our derivation is adiabatic decoupling without a spectral gap. Before we turn to a more careful discussion of the type of scaling we shall consider, notice that the coupling of N noninteracting particles to the radiation field has three effects. • The effective mass, or more precisely, the effective dispersion relation of the particles is modified. The term “effective” refers to the reaction of the particles to weak external forces. The physical picture is that each particle now carries a cloud of photons with it, which makes it heavier. • The particles feel an interaction mediated through the field. If the propagation speed of the particles is small compared to the one of the photons, then retardation effects should be negligible and the interaction between the particles can be described in good approximation by instantaneous pair potentials.

940

S. Teufel

Ann. Henri Poincar´e

• Energy is dissipated through photons moving freely to infinity. The motion of the particles is, in general, no longer of Hamiltonian type. The rate of energy emitted as photons is proportional to the acceleration of a particle squared. The scaling to be studied is most conveniently explained on the classical level. The classical equations of motion for N particles with positions qj , masses mj and rigid “charge” distributions ρj coupled to the scalar field φ(x, t) with propagation speed c are 1 ¨ φ(x, t) c2

=

mj q¨j (t)

=

∆x φ(x, t) −  −

R3

N


ρj (x − qj (t))

(1)

j=1

dx (∇x φ)(x, t) ρj (x − qj (t)) ,

1≤j≤N.

(2)

One can think of ρj (x) = ej ϕ(x) as a smeared out point charge ej with a form factor ϕ ≥ 0 satisfying R3 dx ϕ(x) = 1. Taking the limit c → ∞ in (1) yields the Poisson equation for the field and thus, after elimination of the field, (2) describes N particles interacting through smeared Coulomb potentials. Mass renormalization for the particles in not visible at leading order. Instead of taking c → ∞ one can as well explore for which scaling of the particle properties one obtains analogous effective equations. Since retardation effects should be negligible, the initial velocities of the particles are now assumed to be O(ε) compared to the fixed propagation speed c = 1 of the field, ε  1. In order to see motion of the particles over finite distances, we have to follow this dynamics at least over times of order O(ε−1 ). To make sure that the velocities are still of order O(ε) after times of order O(ε−1 ), the accelerations must be at most of order O(ε2 ). The last constraint also guarantees that the energy dissipated over times of order O(ε−1 ) is at most of order O(ε3 ). The natural procedure would now be to consider such initial data, for which the velocities stay of order O(ε) over sufficiently long times. The problem simplifies if we assume, as we shall do in this work, that the mass of a particle is of order O(ε−2 ). As a consequence accelerations are and stay of order O(ε2 ) uniformly for all initial conditions. In this scaling limit mass renormalization is not visible at leading order. Indeed, if we substitute t
= εt and m
j = ε2 mj in (1) and (2), we find that the limit ε → 0 is equivalent to the limit c → ∞. After quantization, however, the two limiting procedures are no longer equivalent. The limit c → ∞ for the Nelson model was analyzed by Davies [7] and later also by Hiroshima [10], who removed the ultraviolet cutoff. A comparison of their results with ours can be found at the end of this introduction. We will adopt the point of view that it is more natural to explore the regime of particle properties which gives rise to effective equations than to take the limit c → ∞. The deeper reason for our choice is that the more natural procedure of restricting to appropriate initial conditions gives rise to a similar mathematical struc-

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ture. If the bare mass of the particles of order O(1), then the proper scaling which yields effective equations with renormalized masses was introduced and analyzed for the classical Abraham model by Kunze and Spohn, see [12, 21] and references therein. Denoting again the ratio of the velocities of the particles and the field as ε, they consider charges initially separated by distances of order O(ε−2 ) in units of their diameter. Hence the forces are O(ε4 ) initially. For times up to order O(ε−3 ) and for appropriate initial conditions – excluding head on collisions – the separation of the particles remains of order O(ε−2 ) and thus the velocities remain of order O(ε). In particular, the rescaled macroscopic position q
(t
) = ε−2 q(t
/ε3 ) satisfies (d/dt
)2 q
(t
) = O(ε4 ), which matches the order of the forces. As a consequence one obtains a sensible limiting dynamics for the macroscopic variables. One would expect that the same scaling limit applied to the quantum mechanical model yields in a similar fashion effective dynamics with renormalized dispersion. However, inserting this scaling into the massless Nelson model, one faces mathematical problems beyond those in the simpler m = O(ε−2 ) scaling. Without going into details we remark that the main problem is that for massless bosons the Hamiltonian at fixed total momentum does not have a ground state in Fock space, cf. [8, 6]. (As a consequence it is not even clear how to translate the result in [23] for a single quantum particle coupled to a massive quantized scalar field and subject to weak external forces to the massless case.) Nevertheless, the simpler scaling with m = O(ε−2 ) provides at least a first step in the right direction, since the mechanism of adiabatic decoupling without gap will certainly play a crucial role also in a more refined analysis. In the remainder of the introduction we briefly present the massless Nelson model, explain our main result and compare it to Davies’ “weak coupling limit” [7]. Up to a modified dispersion for the particles, the following model is obtained through canonical quantization of the classical system (1) and (2). The state space for N spinless particles is L2 (R3N ) and as Hamiltonian we take Hp =

N 
−c2max ∆xj + c4max m2 ,

(3)

j=1

where cmax is the maximally attainable speed of the particles and m their mass,
= 1. As explained before, we consider the scaling limit ε  1 with

cmax = O(ε)

and m = O(ε−2 ) .

(4)

It might seem somewhat artificial to have a relativistic dispersion relation for the particles which does not contain the speed of light, but some other maximal speed cmax . This is done only for the sake  of simple presentation. We could as well 1 consider the quadratic dispersion Hp = − N j=1 2m ∆xj for the particles. However, there would be no maximal speed and we would be forced to either introduce a cutoff for large momenta or to change the topology in (17). While both strategies

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are technically straightforward by using exactly the same methods as in [22] in the context of Born-Oppenheimer approximation, they would obscure the simple structure of our result. We insert the scaling (4) into (3) and change units such that the particle Hamiltonian is now given through Hpε =

N 
−ε2 ∆xj + 1 .

(5)

j=1

The particles are coupled to a scalar field whose state is an element of the bosonic Fock space over L2 (R3 ) given as m 2 3 F = ⊕∞ m=0 ⊗(s) L (R ) ,

(6)

0 2 3 where ⊗m (s) is the m-times symmetric tensor product and ⊗(s) L (R ) := C. The Hamiltonian for the free bosonic field is

Hf = dΓ(|k|) ,

(7)

where k is the boson momentum. In our units the propagation speed of the bosons is equal to one. The reader who is not familiar with the notation is asked to consult the beginning of Section 3, where the model is introduced in full detail. In the standard Nelson model the coupling between the j th particle and the field is given through  HI,j =

R3

dy φ(y) ρj (y − xj ) ,

(8)

where φ is the field operator in position representation and xj the position of the j th particle. The charge density ρj ∈ L1 (R3 )∩L2 (R3 ) of the j th particle is assumed to be spherically symmetric and its Fourier transform is denoted by ρˆj . For the moment we also assume an infrared condition, namely that N
ρˆj (k) ∈ L2 (R3 ) . 3/2 |k| j=1

(9)

Condition (9) constrains the total charge of the system but not that of an individual particle to zero. The state of the combined particles + field system is an element of H = L2 (R3N ) ⊗ F and its time evolution is generated by the Hamiltonian H ε = Hpε ⊗ 1 + 1 ⊗ dΓ(|k|) +

N
j=1

HI,j .

(10)

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σ( H (x) ) 0

x

e0 E (x) 0

Figure 1: The spectrum of H0 (x) for N = 2. The thick line indicates the eigenvalue E0 (x) sitting at the bottom of continuous spectrum. Note that H contains no terms which directly couple different particles. All interactions between the particles must be mediated through the boson field. Our goal is the construction of approximate solutions of the time dependent Schr¨ odinger equation iε

d Ψ(t) = H ε Ψ(t) , dt

Ψ(0) = Ψ0 ∈ H

(11)

from solutions of an effective Schr¨ odinger equation iε

d ε ψ(t) = Heff ψ(t) , dt

ψ(0) = ψ0 ∈ L2 (R3N )

(12)

for the particles only. Notice the factor ε in front of the time derivative in (11) and (12), which means that we switched to a time scale of order ε−1 in microscopic units. As explained before, this is necessary in order to see nontrivial dynamics of the particles, since their speed is O(ε). We remark that the scaling (4) coincides with the one in time-dependent Born-Oppenheimer approximation, where m = O(ε−2 ) is the mass of the nuclei and where, at fixed kinetic energy, the velocities of the nuclei are also of order O(ε). The Hamiltonian (10) has the same structure as the molecular Hamiltonian and the role of the electrons in the Born-Oppenheimer approximation is now played by the bosons. The key observation for the following is that the interaction Hamiltonian depends only on the configuration x of the particles and that the operator H0 (x) = dΓ(|k|) +

N
j=1

HI,j (x) ,

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which acts on F for fixed x ∈ R3N , has a unique ground state Ω(x) with ground state energy j−1 N

E0 (x) = Vij (xi − xj ) + e0 , (13) j=2 i=1



where Vij (z) = − and

dv dw R3 ×R3

N

e0 = −

1
2 j=1

ρi (v − z)ρj (w) 4π|v − w|

(14)

ρj (v)ρj (w) . 4π|v − w|

(15)

 dv dw R3 ×R3

Vij (z) is the electrostatic interaction energy of the charge distributions ρi and ρj at distance z, however, with the “wrong” sign. It is a peculiarity of the electromagnetic field that the interaction between charges with equal sign is repulsive. e0 is the sum of all self energies. The remainder of the spectrum is purely absolutely continuous and the ground state eigenvalue is not isolated, cf. Figure 1. Let P∗ (x) = |Ω(x) Ω(x)|, then the states in  Ran P∗ =

⊕ R3N

dx ψ(x)Ω(x) : ψ ∈ L (R 2

3N

 ) ⊂H

(16)

correspond to wave packets without free bosons. These states are sections of ground states, not sections of vacua, and as such contain what is called “virtual” bosons in physics. States in RanP∗ could be called dressed many-particle states, since the particles carry a cloud of virtual bosons. If the particles are moving at small speeds and if the accelerations are also small, one expects that no free bosons are created, i.e. that RanP∗ is approximately invariant under the dynamics generated by H ε . Moreover the wave function ψ(x) of the particles should approximately be governed by the effective Schr¨ odinger equation (12) with ε Heff =

j−1 N  N


−ε2 ∆xj + 1 + Vij (xi − xj ) . j=1

j=2 i=1

⊕ dx ψ0 (x)Ω(x) ∈ RanP∗ we Our main result, Theorem 7, states that for Ψ0 = have  ⊕  

ε  −iH ε t/ε  Ψ0 − dx e−iHeff t/ε ψ0 (x) e−ie0 t/ε Ω(x) (17) e 3N R  = O(ε ln(1/ε)) (1 + |t|) Ψ0  . Notice that in the approximate solution of the full Schr¨ odinger equation the state of the field is, up to a fast oscillating global phase e−ie0 t/ε , adiabatically following

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the motion of the particles. In particular, there are no bosons traveling back and forth between the particles and the phrase that the particles “interact through the exchange of bosons”, which comes from perturbation theory, should not be taken literally in the present setting. Let us stress the physical relevance of the allowed initial conditions in our result. Heuristically one would expect that any initial state for the Nelson model radiates its free bosons to infinity and thus effectively becomes a state in RanP∗ . To make even precise this statement is not completely trivial and it is a hard problem in scattering theory to prove it. In case of the Nelson model for a single particle and with an infrared cutoff in the interaction, asymptotic completeness for Compton scattering was proven only recently in [9]. In particular, for a single particle the heuristic expectation formulated above holds true and every initial state, after a sufficiently long time, follows the effective dynamics. As mentioned before, there is a strong similarity to the time-dependent BornOppenheimer approximation, where one obtains an effective Schr¨ odinger equation for the nuclei in a molecule with an effective potential generated by the electrons [22]. In both cases the physical mechanism which leads to the approximate invariance of the subspace RanP∗ is adiabatic decoupling. I.e. the separation of time scales for the motion of the different parts of the system lets the fast degrees of freedom, in our case the bosons, instantaneously adjust to the motion of the slow degrees of freedom, the particles. However, for massless bosons – in contrast to the Born-Oppenheimer approximation – there is no spectral gap which pointwise separates the energy band E0 = {(x, E0 (x)) : x ∈ R3N } from the remainder of the spectrum of H0 (x), but E0 lies at the bottom of continuous spectrum. Hence we need a space-adiabatic theorem, cf. [22, 23, 17], without gap condition. The prefix space in space-adiabatic is used to distinguish this type of result from the standard adiabatic theorem of quantum mechanics, to which we refer as the time-adiabatic theorem. In the timeadiabatic theorem one considers Hamiltonians with an explicit time-dependence on a slow time scale, while in the space-adiabatic setting the separation of time scales has a dynamical origin. Only recently time-adiabatic theorems without a gap condition were established in [2, 5, 24]. In Section 2 a general space-adiabatic theorem without gap condition is formulated and proved. The proof is based on ideas developed in [24] and our approach gives, in particular, good control on the rate of convergence in the adiabatic limit. As an application of the result from Section 2 we consider in Section 3 the scaling limit ε → 0 of the massless Nelson model as described above. We emphasize at this  point that, in view of the missing gap condition, the rate of convergence O(ε ln(1/ε)) in (17) is surprisingly fast, since it is almost as good as in the case with a gap. Moreover, if all particles have individually total charge equal to zero, then the rate is exactly O(ε) as in the case with a gap. Hence, the logarithmic correction must be attributed to the Coulombic long range character of the interparticle interaction. In the situation with gap it is known [15, 17] that the wave function stays in a subspace RanP∗ε which is ε-close to the band subspace RanP∗

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up to an error of order O(ε∞ ). However, in the situation without gap, we expect that a piece of order εα , α < ∞, of the wave function is “leaking out” of RanP∗ . Physical considerations suggest that α = 3/2 for the present problem, see Remark 9. As a consequence, the ε2 corrections to the effective Hamiltonian are still dominating dissipation and can be formally derived from the results in [17]. The effective Hamiltonian (55) then contains a renormalized mass term and the momentum dependent Darwin interaction. In Section 3 we also consider an infrared-renormalized model suggested in [1] and [14], which allows us to do without the global infrared condition (9). The results are exactly the same as in the standard Nelson model. We remark that in [3] the time-adiabatic theorem without gap condition was applied to the Dicke model with a constant magnetic field whose direction changes slowly in time. The Dicke model is a simplified version of the spin-boson model, i.e. a two-level system coupled to the quantized massless scalar field. In the Dicke model one drops the anti-resonant terms in the interaction and can, as a consequence, explicitly calculate the ground state as a function of time. This is very similar to our setting, where we will obtain a rather explicit expression for the ground state of H0 (x) asa function of x ∈ R3N . In [3] the order of the error in the adiabatic limit is O(ε ln(1/ε)), exactly as in (17). As to be explained at the end of Section 2.1, in both cases the specific form of the error can be traced back to the spectral density of the massless scalar field at zero energy. Finally let us compare our results to those obtained by Davies [7], who considers the limit c → ∞ for the Hamiltonian √ H c = Hp ⊗ 1 + 1 ⊗ dΓ(c|k|) + cHI . (18) Notice that H c is obtained through canonical quantization of (1) and (2) if one does not set c = 1 as we did before. Davies proves that for all t ∈ R c

s − lim e−iH t (ψ ⊗ Ω) = (e−i(Heff +e0 )t ψ) ⊗ Ω , c→∞

where Ω = {1, 0, 0, . . .} denotes the Fock vacuum and Hp := Hpε=1 and Heff := ε=1 . This shows that although the limit c → ∞ is equivalent to our scaling on Heff the classical level, the results for the quantum model differ qualitatively. While we obtain effective dynamics for states which contain a nonzero number of bosons independent of ε, cf. (17), the c → ∞ limit yields effective dynamics for states which contain no bosons at all. Furthermore, the limit ε → 0 is a singular limit as no limiting dynamics for ε = 0 exists.

2 A space-adiabatic theorem without gap condition Generalizing from the time-adiabatic theorem of quantum mechanics [11], we consider perturbations of self-adjoint operators H0 , which are fibered over the base space Rn , where, for better readability, we use M := Rn to denote this base space.

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Let Hf be a separable Hilbert space and let dx denote Lebesgue measure. Recall that H0 acting on H = L2 (M, dx) ⊗ Hf = L2 (M, dx; Hf ) is called fibered, cf. [20], if there is a measurable map M  x → H0 (x) with values in the self-adjoint operators on Hf such that  H0 =

⊕

M

dx H0 (x) .

There seems to be no standard name for the set

Σ = (x, s) ∈ M × R, s ∈ σ(H0 (x)) and we propose to call it the fibered spectrum of H0 . Let σ∗ ⊂ Σ be such that x → P∗ (x) is measurable, where P∗ (x) denotes ⊕ the spectral projection of H0 (x) associated with σ∗ (x). Then P∗ = M dx P∗ (x) is an orthogonal projection which commutes with H0 , but which is in general not a spectral projection of H0 . We consider perturbations of H0 , which mix the fibers, in a sense, slowly. As a prototype consider for a sufficiently regular real valued function h on “momentum space” Rn the self-adjoint operator hε = h(−iε∇x ) on L2 (M ). Here ε > 0 is the adiabatic parameter and [hε ⊗ 1, A] = O(ε) for any operator A which is fibered over M . Let H ε = H0 + h ε ⊗ 1 , then the invariant subspaces for H0 constructed above are still “approximately” ε invariant for H ε with ε small, since [H ε , P∗ ] = O(ε) and thus [e−iH s , P∗ ] = ε O(ε|s|). But the relevant time scale for the dynamics generated by h is t/ε with ε t = O(1). Thus the unitary group of interest is e−iH t/ε . However, according to −iH ε t/ε , P∗ ] = O(|t|) and the subspaces RanP∗ seem to be the naive argument, [e not even approximately invariant as ε → 0. It is well known [22, 23] that the failure of the naive argument can be cured if σ∗ is separated by a gap from the remainder of the fibered spectrum Σ. Then ε [e−iH t/ε , P∗ ] = O(ε) (1+|t|), a result that was baptized space-adiabatic theorem in [22]. The object of this section is to establish an analogous result without assuming a gap condition. We remark that the general setup for space-adiabatic theory are Hamiltonians which are “fibered” over phase space, in the sense that they can be written as quantizations of operator valued symbols [17].

2.1

Assumptions and results

Let H0 (x), x ∈ M , be a family of self-adjoint operators on some common dense domain D ⊂ Hf , Hf a separable Hilbert space. Let  · H0 (x) denote the graph norm of H0 (x) on D, i.e., for ψ ∈ D, ψH0 (x) = H0 (x)ψ + ψ. We assume

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that all the H0 (x)-norms are equivalent in the sense that there is an x0 ∈ M and constants C1 , C2 < ∞ such that C1 ψH0 (x0 ) ≤ ψH0 (x) ≤ C2 ψH0 (x0 ) . Then  H0 =

⊕

M

dx H0 (x)

with domain D(H0 ) = L2 (M ) ⊗ D is self-adjoint, where here and in the following D resp. D(H0 ) are understood to be equipped with the  · H0 (x0 ) resp.  · H0 norm. For k ∈ N0 and E some Banach space let   k n k n α n Cb (R , E) = f ∈ C (R , E) : sup ∂x f (x)E < ∞ ∀ α ∈ N with |α| ≤ k . x∈Rn

L(H1 , H2 ) denotes the space of bounded linear operators from H1 to H2 and Lsa (H) denotes the set of bounded self-adjoint operators on H. Let | · | be the Euclidean norm on Rn and denote the Hessian of a function A on Rn by ∇(2) A(x). For the resolvent we write Rλ (A) = (A − λ)−1 . Let m ≥ 2. m Assumption Hm 0 . Let H0 (·) ∈ Cb (M, L(D, Hf )) and for all x ∈ M let P∗ (x) be an orthogonal projection such that H0 (x) P∗ (x) = E(x) P∗ (x) with P∗ (·) ∈ Cbm+1 (M, L(Hf )) and E(·) ∈ Cbm (M, R). In addition one of the following assertions holds:

(i) For 1 ≤ j ≤ n lim ess sup  δ RE(x)−iδ (H0 (x)) (∂xj P∗ )(x)P∗ (x)L(Hf ) = 0 .

δ→0

x∈M

(19)

(ii) There is a constant δ0 > 0 and a function η : [0, δ0 ] → [0, δ0 ] with η(δ) ≥ δ and a constant C < ∞ such that for δ ∈ (0, δ0 ] and 1 ≤ j ≤ n ess sup  RE(x)−iδ (H0 (x)) (∂xj P∗ )(x)P∗ (x)L(Hf ) ≤ C δ −1 η(δ) . x∈M

(iii) In addition to (20) for 1 ≤ k, j ≤ n also 

   ess sup ∂xk RE(x)−iδ (H0 (x))(∂xj P∗ )(x)P∗ (x)  x∈M

L(Hf )

(20)

≤ C δ −1 η(δ) (21)

holds. A few remarks concerning Assumption Hm 0 are in order: • It is not assumed that P∗ (x) is the spectral projection of H0 (x) corresponding to the eigenvalue E(x). However, (19) holds pointwise in x whenever P∗ (x) is the spectral projection and has finite rank, cf. Proposition 2.

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• Inequality (20) is always satisfied with η(δ) = 1. For Assumption Hm 0 (ii) and (iii) to have nontrivial consequences on the rate of convergence in the adiabatic theorem, η(δ) must satisfy limδ→0 η(δ) = 0. These assumptions might look rather artificial at first sight, but turn out to be very natural in the proof and also in our application. For the simpler time-adiabatic setting, which gives rise to similar conditions, we refer the reader to [24]. • The regularity of P∗ (x) has to be assumed, since it does not follow from the regularity of H0 (x) without the gap condition, even if P∗ (x) is spectral. The regularity of E(x) follows from the one of H0 (x) and P∗ (x) whenever P∗ (x) has finite rank, as can be seen by writing E(x) = tr(H0 (x)P∗ (x))/trP∗ (x). ⊕ The “band subspace” RanP∗ defined through P∗ = M dx P∗ (x) is invariant under the dynamics generated by H0 , since [H0 , P∗ ] = 0 holds by construction. We will consider perturbations hε of H0 satisfying Assumption hm . For ε ∈ (0, 1] let hε be a self-adjoint operator with domain D(h) ⊂ H independent of ε such that H0 + hε is essentially self-adjoint on D(h) ∩ D(H0 ). There exists an operator (Dh)ε ∈ Lsa (H)⊕n with sup  | (Dh)ε | L(H) < ∞

ε∈(0,1]

satisfying: (i) There is a constant C < ∞ such that for each A ∈ Cbm (M, L(Hf )) m


[hε , A] + i ε ∇x A · (Dh)ε L(H) ≤ C

j=2

εj

sup x∈M, |α|=j

∂xα A(x)L(Hf ) .

(ii) There is a constant C < ∞ such that  | [(Dh)ε , H0 ] | L(D(H0 ),H) +  | [(Dh)ε , hε ] | L(H) ≤ ε C . By assumption, H ε = H0 + hε is essentially self-adjoint on D(h) ∩ D(H0 ) and we use its closure, again denoted by H ε , to define for t ∈ R U ε (t) = e−iH

ε

t/ε

.

Since, according to Assumption hm (i), [H ε , P∗ ] = [hε , P∗ ] = O(ε), the naive argument gives [U ε (t), P∗ ] = |t|O(1). Indeed, our aim is to cure the failure of the naive argument and to show that RanP∗ is invariant for U ε (t) in the limit ε → 0. To this end we will compare U ε (t) with the unitary group generated by ε = H0 + P∗ hε P∗ + P∗⊥ hε P∗⊥ . Hdiag

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ε Also Hdiag is self-adjoint on D(H ε ) since P∗ (·) ∈ Cbm (M, L(Hf )) and thus H ε − ε ⊥ ε Hdiag = P∗ [h , P∗ ]P∗ − P∗ [hε , P∗ ]P∗⊥ is bounded according to hm (i). Again we abbreviate for t ∈ R ε ε (t) = e−iHdiag t/ε , Udiag

and we have by construction that  ε (t)] = 0 , P∗ , Udiag i.e. RanP∗ and RanP∗⊥ are invariant subspaces for the dynamics generated by ε Hdiag . m Theorem 1. Assume Hm for some m ≥ 2. Let ε ∈ (0, δ0 ], then 0 and h

• Hm 0 (i) implies that for t ∈ R     ε lim U ε (t) − Udiag (t) ε→0

L(H)

= 0,

(22)

• Hm 0 (ii) implies that for some constant C < ∞ and all t ∈ R   1   ε ε (t) ≤ C η(ε 2 ) (1 + |t|) , U (t) − Udiag

(23)

• Hm 0 (iii) implies that for some constant C < ∞ and all t ∈ R     ε ε (t) ≤ C η(ε) (1 + |t|) . U (t) − Udiag

(24)

L(H)

L(H)

Note that in Theorem 1 the whole spectrum of possible rates of convergence between o(1) and O(ε) as in the case with gap is covered. The estimates for the massless Nelson model as an application of Theorem 1 will show that, in principle, all rates can occur. The following proposition shows that, assuming the first part of Hm 0 but (i) always holds pointwise in x if neither (i), (ii) or (iii), then Assumption Hm 0 P∗ (x) is the spectral projection and has finite rank. The proof is analogous to the one of Lemma 4 in [2]. Proposition 2. Assume H10 without (i), (ii) or (iii). If P∗ (x) is the spectral projection of H0 (x) corresponding to the eigenvalue E(x) and has finite rank, then lim  δ RE(x)−iδ (H0 (x)) (∇x P∗ )(x)P∗ (x)L(Hf ) = 0 .

δ→0

(25)

Proof. Since P∗ (x) has finite rank, the uniform statement (25) follows if we can show that limδ→0  δ RE(x)−iδ (H0 (x)) ψ = 0 for all ψ ∈ Ran(∇x P∗ )(x)P∗ (x). We have  δ2 lim  i δRE(x)−iδ (H0 (x)) ψ2Hf = lim µψ (dλ) = µψ (E(x)) , δ→0 δ→0 R (λ − E(x))2 + δ 2 (26)

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where µψ denotes the spectral measure of H0 (x) for ψ. Since P∗ (x) is the spectral projection on {E(x)} and since, according to (29), Ran(∇x P∗ )(x)P∗ (x) ⊂ RanP∗⊥ (x) we have µψ (E(x)) = 0 and thus (25). It is clear from (26) that additional information on the regularity of the spectral measure µψ provides some control on the rate of convergence in (26). E.g., if µψ (dλ) = ρψ (λ)dλ with ρψ ∈ L∞ (R, dλ), then   δ2 δ2 µψ (dλ) ≤ ρ  dλ = O(δ) ψ ∞ (λ − E(x))2 + δ 2 (λ − E(x))2 + δ 2 R R and  hence (20) would hold pointwise in x with η(δ) = δ 1/2 . In a sense, the rate O(ε ln(1/ε)) for the massless Nelson model (17) is a consequence of the relevant spectral measure having a density ρ(λ) ∼ λ − E(x) for 0 < λ − E(x)  1. This explains why in [3] for the Dicke model in R3 the same error estimates are obtained: the same spectral density enters the proof. Finally we emphasize that (25) for all x ∈ M does not imply Hm 0 (i), even in the case of compact M . This is because for pointwise convergence to imply uniform convergence one would need uniform equicontinuity of a sequence of functions. However, in the time-adiabatic setting it is indeed sufficient to have (25) for almost all x ∈ I, where I ⊂ R is the relevant time interval, see [24].

2.2

Proof of Theorem 1

We start with the standard argument and find that on D(H ε )  t  d  ε ε ε (t) = − U ε (t) ds (s) U ε (t) − Udiag U (−s) Udiag ds 0  t  ε  i ε ε = − U (t) Udiag (s) , ds U ε (−s) H ε − Hdiag ε 0 where

  ε H ε − Hdiag = P∗⊥ hε P∗ + P∗ hε P∗⊥ = P∗⊥ hε , P∗ P∗ + adj. .

(27)

In (27) and in the following “± adj.” means that the adjoint operator of the first term in a sum is added resp. subtracted. Inserting hm (i) into (27) and the result back into (27) one obtains   ε ε U (t) − Udiag (t)L(H) = (28)    

0

t

    ε ds U ε (−s) P∗⊥ (∇x P∗ ) P∗ · (Dh)ε P∗ + adj. Udiag (s) 

L(H)

+ O(ε)|t| .

In (28) we also used that (∇x P∗ )(x) = P∗⊥ (x)(∇x P∗ )(x)P∗ (x) + adj. ,

(29)

which follows from (∇x P∗ )(x) = (∇x P∗2 )(x) = (∇x P∗ )(x)P∗ (x) + P∗ (x)(∇x P∗ )(x).

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The nontrivial part in adiabatic theorems is to show that also the remaining term on the right hand side of (28) vanishes as ε → 0. Assuming a gap condition, the basic idea is to express the integrand, which is O(1), as the time-derivative of a function that is O(ε) plus a remainder that is O(ε) and integrate, cf. [23, 22]. The key ingredient in this case would be the operator F (x) = RE(x) (H0 (x)) (∇x P∗ )(x) P∗ (x) ,

(30)

which is, according to (29), well defined and bounded if the eigenvalue E(x) is separated from the rest of the spectrum of H0 (x) by a gapand ifP∗ (x) is spectral. The definition (30) is made to give [H0 , F ] = P∗⊥ ∇x P∗ P∗ . However, in absence of a gap (30) is not well defined as an operator on Hf and, following [24], we shift the resolvent into the complex plane and define Fδ (x) = RE(x)−iδ (H0 (x)) P∗⊥ (x) (∇x P∗ )(x) P∗ (x) . One now obtains     H0 (x) , Fδ (x) = P∗⊥ (x) ∇x P∗ (x) P∗ (x) + Yδ (x)

(31)

with Yδ (x) = − i δRE(x)−iδ (H0 (x)) (∇x P∗ )(x) P∗ (x) .

(32)

Hm 0

(i), (ii) and (iii) each imply that limδ→0 Yδ L(H) = 0. To see Assumptions this recall that for A(·) ∈ L∞ (M, L(Hf )) one has AL(H) = ess sup A(x)L(Hf ) . x∈Rn

Note that for better readability we omit the Euclidean norm | . . . | in the notation and understand that A always includes also the Euclidean norm if A is an operator with n components. Thus with (32) we can make the remainder in (31) arbitrarily small by choosing δ small enough. However, for the time being we let δ > 0 but carefully keep track of the dependence of all errors on δ. By assumption H0 (·) ∈ Cbm (M, L(D, Hf )) and P∗ (·) ∈ Cbm+1 (M, L(Hf )), which implies Fδ (·) ∈ Cbm (M, L(Hf )⊕n ) and hence, according to hm (i),  ε   h , Fδ 

L(H)

≤ C

m
j=1

εj sup  ∂xα Fδ L(H) =: f1 (ε, δ) .

(33)

|α|=j

Combining (31) and (33) we obtain  ε    H , Fδ = P∗⊥ ∇x P∗ P∗ + O(Yδ , f1 (ε, δ)) ,

(34)

where in (34) and in the following O(a, b, c, . . .) stands for a sum of operators whose norm in L(H) is bounded by a constant times a + b + c + . . .. Defining Bδ = Fδ · (Dh)ε P∗ − adj. ,

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one finds with hm (ii) and f2 (δ) = Fδ L(H,D(H0 )) that  ε      H , Bδ = H ε , Fδ · (Dh)ε P∗ + Fδ · H ε , (Dh)ε P∗   (35) + Fδ · (Dh)ε H ε , P∗ + adj.   ⊥ ε = P∗ ∇x P∗ P∗ · (Dh) + adj. + O(ε, Yδ , f1 (ε, δ), εf2 (δ)) . Now the integrand in (28) can be written as the time-derivative of Aδ (s) = − i ε U ε (−s) Bδ U ε (s) , plus a remainder: d Aδ (s) = ds =

U ε (−s) [H ε , Bδ ] U ε (s)     U ε (−s) P∗⊥ ∇x P∗ P∗ · (Dh)ε P∗ + adj. U ε (s) + O(ε, Yδ , f1 (ε, δ), εf2 (δ)) .

(36)

Inserting (36) into (28) enables us to do integration by parts,   ε U (t) − U ε (t) ≤ diag L(H)  t      d ε ε  ≤  A ds (s) U (−s) U (s) δ diag   ds 0 L(H)

+ |t| O ε, Yδ , f1 (ε, δ), εf2 (δ) ≤ Aδ (t)L(H) + Aδ (0)L(H)  t     d ε ε  U + ds A (s) (−s) U (s) δ diag   ds 0 L(H)

+ |t| O ε, Yδ , f1 (ε, δ), εf2 (δ)

≤ C ε (2 + |t|) Fδ L(H) + |t| O ε, Yδ , f1 (ε, δ), εf2 (δ) .

(37)

For the last inequality in (37) we used that Aδ (t)L(H) ≤ C εFδ L(H) uniformly for t ∈ R and that   ε i d ε ε ε U (t0 , s) Udiag (s, t0 ) = − U ε (t0 , s) H ε (s) − Hdiag (s) Udiag (s, t0 ) ds ε is bounded uniformly, according to (27) and hm (i). Writing out the various terms in (37) explicitly, we conclude that there is a constant C < ∞ such that   ε U (t) − U ε (t) ≤ C ε F  + C |t| ε + Yδ L(H) δ L(H) diag L(H) +ε Fδ L(H,D(H0 )) + ε Fδ L(H) +

m
j=1

εj sup ∂xα Fδ L(H) . |α|=j

(38)

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Hence we are left to establish bounds on Fδ , on its derivatives and on Yδ in terms of δ, which is the content of the following Lemma. m ⊕n ) and there is a constant Lemma 3. Assume Hm 0 , then Fδ (·) ∈ Cb (M, L(Hf ) C < ∞ such that for δ ∈ (0, δ0 ]

Fδ L(H,D(H0 )) ≤

C η(δ) , δ

C η(δ) δ j+1

sup  ∂xα Fδ L(H) ≤

|α|=j

for 1 ≤ j ≤ m .

(39) (40)

m In case Hm 0 (i) holds, we have (39) and (40) with η(δ) = 1. Furthermore, if H0 m (i) holds, then limδ→0 Yδ L(H) = 0 and if H0 (ii) or (iii) holds, then Yδ L(H) ≤ C η(δ). If Hm 0 (iii) holds, then (40) can be improved to

sup  ∂xα Fδ L(H) ≤

|α|=j

C η(δ) δj

for 1 ≤ j ≤ m .

(41)

Before we turn to the proof of Lemma 3 we finish the proof of Theorem 1. Assuming Hm 0 (i), (22) follows by inserting the bounds from Lemma 3 into (38) and choosing δ = δ(ε) such that limε→0 δ(ε) = 0 and limε→0 ε/δ(ε)2 = 0. If Hm 0 (ii) holds, then the bounds (39) and (40) inserted into (38) yield m  ε  η(δ)
j η(δ)

η(δ) ε U (t) − Udiag + C ε + η(δ) + ε + (t)L(H) ≤ C ε ε j+1 |t| . (42) δ δ δ j=1 1

In (42) the optimal choice is δ(ε) = ε 2 , which gives (23). Finally, the bounds (39) and (41) inserted into (38) yield m  ε  η(δ)
j η(δ)

η(δ) U (t) − U ε (t) |t| , + C + ε + η(δ) + ε ≤ C ε ε diag L(H) δ δ δj j=1

where the optimal choice δ(ε) = ε gives (24). Proof of Lemma 3. We abbreviate RE(x)−iδ (H0 (x)) as R(δ, x) in this proof and note that R(δ, ·) ∈ Cbm (M, L(Hf )) and thus Fδ (·) ∈ Cbm (M, L(Hf )⊕n ) follow from H0 (·) ∈ Cbm (M, L(D, Hf )) together with P∗ (·) ∈ Cbm+1 (M, L(Hf )). We start with the case Hm 0 (ii), where (i) is included by making the obvious changes for η(δ) = 1. Assumption Hm 0 (ii) immediately yields Fδ L(H) ≤

C η(δ) δ

(43)

and the bound on Yδ . (39) follows from H0 (x)R(δ, x) = 1 + (E(x) − iδ)R(δ, x) and (43) together with the assumption that E(x) is uniformly bounded.

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For (40) we start by observing that ∇x Fδ (x)





= R(δ, x) ∇x (∇x P∗ )(x)P∗ (x) + ∇x R(δ, x) (∇x P∗ )(x)P∗ (x) (44)

= R(δ, x)∇x (∇x P∗ )(x)P∗ (x) − R(δ, x)(∇x H0 (x) − ∇x E(x))Fδ (x).

Using (43) and the fact that |∇x E(x)| is uniformly bounded by assumption, we infer from (44) that  |∇x Fδ | L(H) = sup  |∇x Fδ (x)| L(Hf ) ≤ C(δ −1 + δ −2 η(δ)) . x∈M

Hence (40) follows for j = 1, since δ ≤ η(δ) by assumption. By differentiating (44) again, we find, using a reduced notation with obvious meaning, that

∇(2) Fδ = −2Rδ (∇H0 − ∇E) ∇Fδ + Rδ ∇(2) (∇P∗ )P∗ − Rδ (∇(2) H0 − ∇(2) E) Fδ . (45) Hence ∇(2) Fδ L(H) ≤ C(δ −3 η(δ) + δ −1 + δ −2 η(δ)) which proves (40) for j = 2. By repeated differentiation one finds inductively (40) for j ≤ m. To show (41) assuming Hm 0 (iii), note that (41) holds by assumption for j = 1 and inserted into (45) it gives ∇(2) Fδ L(H) ≤ C(δ −2 η(δ) + δ −1 + δ −2 η(δ)). Analogously the estimates for all larger j ≤ m are improved by a factor of δ.

3 Effective dynamics for the Nelson model As explained in the introduction, we consider N spinless particles coupled to a scalar, massless, Bose field with an ultraviolet regularization in the interaction. This class of models is nowadays called Nelson’s model [1, 4, 13, 14] after E. Nelson [16], who studied the ultraviolet problem. We briefly complete the introduction of the model and collect some basic, well known facts. A point in the configuration space R3N of the particles is denoted by x = (x1 , . . . , xN ) and the Hamiltonian Hpε for the particles is defined in (5). Hpε is self-adjoint on the domain H 1 (R3N ), the first Sobolev space. 2 3 The Hilbert space √ for the scalar field is the bosonic Fock space over L (R ) defined in (6). On D( N ), N the number operator, the annihilation operator a(f ) acts for f ∈ L2 (R3 ) as  √ (m) (a(f )ψ) (k1 , . . . , km ) = m + 1 dk f¯(k) ψ (m+1) (k, k1 , . . . , km ) , R3

√  (m) 2 only if ∞  < ∞. where ψ = (ψ (0) , ψ (1) , ψ (2) , . . .) ∈ D( N ) if and m=0 mψ √ ∗ The adjoint a (f ), which is also defined on D( N ), is the creation operator and

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for f, g ∈ L2 (R3 ) the operators a(f ) and a∗ (g) obey the canonical commutation relations (CCRs)  dk f¯(k) g(k) =: f, g , [a(f ), a(g)] = [a∗ (f ), a∗ (g)] = 0 . [a(f ), a∗ (g)] = R3

(46)  It is common to write a(f ) = R3 dk f¯(k) a(k). The Hamiltonian of the field as defined in (7) can formally be written as  dk |k| a∗ (k) a(k) . Hf = R3

More on the m-particle sector the action of Hf is (Hf ψ)(m) (k1 , . . . , km ) = m explicitly, (m) (k1 , . . . , km ), and Hf is self-adjoint on its maximal domain. For f ∈ j=1 |kj |ψ L2 (R3 ) the Segal field operator

1 Φ(f ) = √ a(f ) + a∗ (f ) 2 √ is essentially self-adjoint on D( N ). The field operator φ as used in (8) is related to Φ through φ(f ) = Φ(f / |k|). For the following it turns out to be more convenient to write the interaction Hamiltonian in terms of Φ, where

HI = Φ |k| v(x, k) acts on the Hilbert space H = L2 (R3N ) ⊗ F of the full system. We will consider two different choices for v(x, k) in more detail. For the standard Nelson model (SN), as discussed in the introduction, one has vSN (x, k) =

N
j=1

eik·xj

ρˆj (k) . |k|3/2

(47)

For the infrared-renormalized models (IR), as considered by Arai [1] and, more generally, by L¨ orinczi, Minlos and Spohn [14], one has vIR (x, k) =

N
 ik·xj  ρˆj (k) e −1 . |k|3/2 j=1

(48)

In both cases, the charge distribution ρj ∈ L1 (R3 ) of the j th particle is assumed to be real-valued and spherically symmetric. As to be discussed below, cf. Remarks 4 and 6, we have to assume the infrared condition (9) for the (SN) model, but not for the (IR) model. The infrared condition implies, in particular, that the total charge of the system of N particles must be zero. The full Hamiltonian is given as the sum H ε = Hpε ⊗ 1 + 1 ⊗ Hf + HI + VIR ⊗ 1

(49)

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and is essentially self-adjoint on D(Hp ⊗1)∩D(1⊗Hf ) if supx  |k|s v(x, k) L2 (R3 ) < ∞ for s ∈ { 21 , 1}. Only in the (IR) model a potential VIR is added, which acts as multiplication with the bounded, real-valued function VIR (x) =

N 
j,i=1

ρi (k)∗ −ik·xj 1 ρˆj (k)ˆ dk e + |k|2 2 R3

2  N    j=1 ρˆj (k)  dk   .   |k| R3



(50)

Remark 4. If the charge distributions satisfy the infrared condition (9), then the (SN) Hamiltonian and the (IR) Hamiltonian are related by the unitary transformation    N
ρˆj∗ (k)  , UG = exp −iΦ i 3/2 |k| j=1 cf. [1], which is related to the Gross transformation [16] for x = 0. If the infrared condition is not satisfied, the (SN) model and the (IR) model carry two inequivalent representations of the CCRs for the field operators. Physically speaking, the transformation UG removes the mean field that the N charges would generate, if all of them would be moved to the origin. The vacuum in the (IR) representation corresponds to this removed mean field in the original representation, a fact which has to be taken care of in the interaction: for each particle the interaction term is now evaluated relative to the interaction at x = 0, cf. (48), which makes also necessary the counter terms VIR . If the total charge of the system is different from zero, then the mean field is long range and, as a consequence, the corresponding transformation is no longer unitarily implementable. Indeed, it was shown that the (SN) Hamiltonian with confining potential does not have a ground state, cf. [13], while the (IR) Hamiltonian with the same confining potential does have a ground state, cf. [1]. ♦ In order to apply Theorem 1 we observe that HI (x) acts for fixed x ∈ R3N (∼ = M ) on F (∼ = Hf ) and with H0 (x) = Hf + HI (x) + VIR (x) we have ε

H =

Hpε

 ⊗1+



⊕ M

dx H0 (x)

∼ = h ε + H0 .

The following proposition collects some results about H0 (x) and its ground state. Its proof is postponed to after the presentation of the the main theorem. Proposition 5. Assume that v(x, ·) ∈ L2 (R3 ) for all x ∈ R3N and that for some n≥1 (i) | · |∂xα v(x, ·) ∈ L2 (R3 ) for all x ∈ R3N and 0 ≤ |α| ≤ n,  (ii) supx∈R3N  | · | ∂xα v(x, ·) L2 (R3 ) < ∞ for 0 ≤ |α| ≤ n, (iii) supx∈R3N  ∂xα v(x, ·) L2 (R3 ) < ∞ for 1 ≤ |α| ≤ n.

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Let Im v(x, ·), ∇x v(x, ·)L2 (R3 ) = 0 for all x ∈ R3N and VIR (·) ∈ Cb3 (R3N ). Then 1. H0 (x) is self adjoint on D = D(Hf ) for all x ∈ R3N and H0 (·) ∈ Cbn (R3N , L(D, F )), where D is equipped with the graph-norm of Hf . 2. H0 (x) has a unique ground state Ω(x) for all x ∈ R3N and, in particular, H0 (x)Ω(x) = E(x)Ω(x) for  1 E(x) = − dk |k| |v(x, k)|2 + VIR (x). (51) 2 R3 Furthermore Ω(·) ∈ Cbn (R3N , F ). It is straightforward to check that Im v(x, ·), ∇x v(x, ·)L2 (R3 ) = 0 for vSN defined in (47) and vIR defined in (48). For the (SN) model as well as for the (IR) model (51) is easily evaluated and one finds E(x) = E0 (x) as given in (13). Remark 6. For the (SN) model the assumptions made on v(x, k) in Proposition 5 are satisfied, if ρˆj (k) decays sufficiently fast for large |k| and each j = 1, . . . , N , or, equivalently, if ρj (x) is sufficiently smooth. This is an ultraviolet condition individually for each particle. But v(x, ·) ∈ L2 (R3 ) follows from v(0, ·) ∈ L2 (R3 ), which is exactly the global infrared condition (9). While the necessity for an ultraviolet regularization remains in the (IR)  − 32 ik·xj model, the infrared condition is replaced − 1) ∈ L2 (R3 ),  by j |k| ρj (k)(e which can be satisfied without having j ρj (0) = 0. Thus the (IR) model allows us to consider particles with total charge different from zero. ♦ Let P∗ (x) = |Ω(x) Ω(x)|, then P∗ (·) ∈ Cbn (R3N , L(F )) and RanP∗ is a candidate for an adiabatically decoupled subspace. Indeed, we will show that hε = ε m ε Hp ⊗ 1 satisfies Assumption h with (Dh)xj = −iε∇xj / −ε2 ∆xj + 1 and that  H0 and P∗ satisfy Assumption Hm 0 (iii) with η(δ) = δ ln(1/δ) if particles with charges different from zero are present and η(δ) = δ if all particles have total charge zero. Hence we can apply Theorem 1 to conclude that for some constant C 0 sufficiently small  ε   ∗ ε  H P∗ L(H) ≤ ε2 C . diag − U Heff U Proof. In order to apply h4 (i) to U(x), we have to extend U(x) : RanP∗ (x) → C   = defined in (53) to a map U(·) ∈ Cb4 (R3N , L(F )) first. To this end let U(x) ∗ |Ω0  Ω(x)| and note that U U = P∗ . With this definition one finds ε Hdiag P∗

=

H0 P∗ + P∗ hε P∗ = E P∗ + P∗ U∗ hε U P∗ + P∗ [ hε , U∗ ] U P∗

=

ε U ∗ Heff U P∗ + P∗ [ hε , U∗ ] U P∗ ,

and we are left to show that P∗ [ hε , U∗ ] U P∗  = O(ε2 ). Using h4 (i) with A = U∗ we find that P∗ [ hε , U∗ ] U P∗ = −iε P∗ (∇x U∗ ) · (Dh)ε U P∗ + O(ε2 ) .

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However, according to (60) P∗ (x)(∇x U∗ )(x) = |Ω(x) Ω(x), ∇x Ω(x) Ω0 | = 0 , and thus the desired result follows. Proof of Lemma 10. Heuristically h4 (i) and (ii) hold, because they are just special cases of the expansion of a commutator of pseudodifferential operators. However, since hε is unbounded and A is only 4-times differentiable, we need to check the estimates “by hand”. For notational simplicity we restrict ourselves to the case N = 1, from which the general case follows immediately. Let g(p) = 1/ p2 + 1, g ε = g(−iε∇x ) ⊗ 1 and A ∈ Cb4 (R3 , Hf ), then | · |s g ∈ L1 (R3 ) for s ∈ {0, 4} and thus for ψ ∈ S

ε





g Aψ (x) = dy  g(y) A(x − εy) ψ(x − εy)    1  2 (2) ds y, ∇ A(x − sεy) y ψ(x − εy) = dy g(y) A(x) − εy · ∇A(x) + ε 0



= Ag ε ψ (x) − i ε ∇A · ∇g ε ψ (x)  1  ds y, ∇(2) A(x − sεy) yψ(x − εy) . (65) + dy  g(y) ε2 0

From (65) one concludes after a lengthy but straightforward computation involving several integrations by parts that ε2 ∆x [g ε , A] = −i ε∇A · (∇g)ε (ε2 ∆x ) + R with R  ≤ C

4
j=2

εj

sup x∈R3N , |α|=j

∂xα A(x)L(Hf ) .

Hence we find [hε , A] = [(1 − ε2 ∆x )g ε , A] = (1 − ε2 ∆x )[g ε , A]− [ε2 ∆x , A]g ε = −i ε∇A·(Dh)ε + R
with R
 ≤ C


4
j=2

εj

sup x∈R3N , |α|=j

∂xα A(x)L(Hf ) .

This proves h4 (i). By the same type of arguments one shows also h4 (ii).

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Acknowledgments I am grateful to Herbert Spohn for suggesting the massless Nelson model as an application for a space-adiabatic theorem without gap, as well as for numerous valuable discussions, remarks and hints concerning the literature. Parts of this work developed during a stay of the author at the Universit´e de Lille and I thank Stephan De Bi`evre and Laurent Bruneau for hospitality and for a helpful introduction to Reference [1]. For critical remarks which lead to an improved presentation I thank Detlef D¨ urr and the referee.

References [1] A. Arai, Ground state of the massless Nelson model without infrared cutoff in a non-Fock representation, Rev. Math. Phys. 13, 1075–1094 (2001). [2] J. E. Avron and A. Elgart, Adiabatic theorem without a gap condition, Commun. Math. Phys. 203, 445–463 (1999). [3] J. E. Avron and A. Elgart, Adiabatic theorem without a gap condition: Twolevel system coupled to quantized radiation field, Phys. Rev. A 58, 4300 (1998). [4] V. Betz, Gibbs measures relative to Brownian motion and Nelson’s model, Dissertation, TU M¨ unchen (2002). [5] F. Bornemann, Homogenization in time of singularly perturbed mechanical systems, Lecture Notes in Mathematics 1687, Springer, Heidelberg, 1998. [6] T. Chen, Operator-theoretic infrared renormalization and construction of dressed 1-particle states in non-relativistic QED, Dissertation, ETH Z¨ urich No. 14203 (2001). [7] E. B. Davies, Particle-boson interactions and the weak coupling limit, J. Math. Phys. 20, 345–351 (1979). [8] J. Fr¨ ohlich, On the infrared problem in a model of scalar electrons and massless scalar bosons, Ann. Inst. Henri Poincar´e 19, 1–103 (1973). [9] J. Fr¨ ohlich, M. Griesemer and B. Schlein, Asymptotic completeness for Compton scattering, preprint, mp arc 01-420 (2001). [10] F. Hiroshima, Weak coupling limit with a removal of an ultraviolet cutoff for a Hamiltonian of particles interacting with a massive scalar field, Inf. Dim. Anal., Quant. Prob. and Related Topics 1, 407–423 (1998). [11] T. Kato, On the adiabatic theorem of quantum mechanics, Phys. Soc. Jap. 5, 435–439 (1958).

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[12] M. Kunze and H. Spohn, Slow motion of charges interacting through the Maxwell field, Commun. Math. Phys. 203, 1–19 (2000). [13] J. L¨ orinczi, R. A. Minlos and H. Spohn, The infrared behavior in Nelson’s model of a quantum particle coupled to a massless scalar field, Ann. Henri Poincar´e 3, 1–28 (2002). ˙ Minlos and H. Spohn, Infrared regular representation of [14] J. L¨orinczi, R.A. the three dimensional massless Nelson model, to appear in Lett. Math. Phys. (2001). [15] A. Martinez and V. Sordoni, On the time-dependent Born-Oppenheimer approximation with smooth potential, Comptes Rendus Acad. Sci Paris 334, 185–188 (2002). [16] E. Nelson, Interaction of nonrelativistic particles with a quantized scalar field, Jour. Math. Phys. 5, 1190–1197 (1964). [17] G. Panati, H. Spohn and S. Teufel, Space-adiabatic perturbation theory, ePrint ArXive math-ph/0201055 (2002). [18] G. Panati, H. Spohn and S. Teufel, Space-adiabatic perturbation theory in quantum dynamics, Phys. Rev. Lett. 88, 250405 (2002). [19] M. Reed and B. Simon, Methods of modern mathematical physics II, Academic Press (1975). [20] M. Reed and B. Simon, Methods of modern mathematical physics IV, Academic Press (1978). [21] H. Spohn, Dynamics of charged particles and their radiation field, in preparation. [22] H. Spohn and S. Teufel, Adiabatic decoupling and time-dependent BornOppenheimer theory, Commun. Math. Phys. 224, 113–132 (2001). [23] S. Teufel and H. Spohn, Semiclassical motion of dressed electrons, Rev. Math. Phys. 4, 1–28 (2002). [24] S. Teufel, A note on the adiabatic theorem without gap condition, Lett. Math. Phys. 58, 261–266 (2001). Stephan Teufel Zentrum Mathematik Technische Universit¨ at M¨ unchen 80290 M¨ unchen Germany email: [email protected] Communicated by Gian Michele Graf submitted 25/03/02, accepted 17/06/02

Ann. Henri Poincar´e 3 (2002) 967 – 981 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/050967-15

Annales Henri Poincar´ e

Curvature-Induced Bound States for a δ Interaction Supported by a Curve in R3 P. Exner and S. Kondej Abstract. We study the Laplacian in L2 (Ê3 ) perturbed on an infinite curve Γ by a δ interaction defined through boundary conditions which relate the corresponding generalized boundary values. We show that if Γ is smooth and not a straight line but it is asymptotically straight in a suitable sense, and if the interaction does not vary along the curve, the perturbed operator has at least one isolated eigenvalue below the threshold of the essential spectrum.

1 Introduction Relations between the geometry and spectral properties are one of the vintage topics of mathematical physics. In the last decade they attracted attention also in the context of quantum mechanics. A prominent example is the curvature-induced ˇ GJ, DE, RB]. This effect appears to be binding in infinite tube like regions [ES, a robust one: it has been demonstrated recently that bends can produce localized states not only if the transverse confinement is hard, i.e. realized by a Dirichlet condition, but also when it is weaker corresponding to a potential well or a δ interaction [EI]. The result is appealing, not only because it concerns an interesting mathematical problem, but also in view of applications in mesoscopic physics where such operators are used as a natural model for semiconductor “quantum wires”. Since in the latter electrons are trapped due to interfaces between two different materials representing finite potential jumps, by tunneling effect they can be found outside the wire, albeit not too far because the exterior is (for the energies in question) the classically forbidden region. The main result of the paper [EI] concerns nontriviality of the discrete spectrum for a class of operators in L2 (R2 ) which can be formally written as −∆ − αδ(x − Γ) with α > 0, where Γ is a curve which is not a straight line but it is asymptotically straight in a suitable sense. A question naturally arises whether a similar result is valid for a curve in R3 . Such an extension is not trivial, because ˇ representing in a the argument in [EI] relies on the resolvent formula of [BEKS] sense a generalization of the Birman-Schwinger theory. The said formula is valid for singular perturbations of the Laplacian which can be treated by means of a quadratic-form sum, i.e. as long as the codimension of the manifold supporting the perturbation is one. Thus if we want to address the stated question, we are forced to look for

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other tools. One possibility is to employ the resolvent formula for a curve in R3 derived in [Ku]. However, since it uses rather strong regularity hypotheses about the curve we take another route and begin instead with an abstract formula for strongly singular perturbations due to A. Posilicano [Po1]. When it is specified to our particular case, it contains again an embedding operator into a space of functions supported on the curve Γ, however, this time it is not the “naive” L2 but rather a suitable element from the scale of Sobolev spaces. Of course, one can regard it as a generalization of Krein’s formula; recall that such a way of expressing the resolvent can be used not only to describe δ interaction perturbations but also more general dynamics supported by zero measure sets [Ka, KK, Ko]. Another aspect of the absence of a description in terms of the quadratic-form sum concerns the very definition of the operator we want to study. We have to employ boundary conditions which relate the corresponding generalized boundary values in the normal plane to the curve modeled after the usual two-dimensional δ interaction [AGHH], which requires us to impose stronger regularity conditions on Γ. Furthermore, a modification of the Birman-Schwinger technique used in [EI] demands stronger restrictions on the regularity of the curve. On the other hand, apart of these technical hypotheses our main result – stated in Theorem 5.6 below – is analogous to that of [EI], namely that for any curve which is asymptotically straight but not a straight line the corresponding operator has at least one isolated eigenvalue. This conclusion is by no means obvious having in mind how different are the point interactions in one and two dimensions.

2 The resolvent formula As a preliminary let us show how self-adjoint extensions of symmetric operators can be characterized in terms of a Krein-type formula derived in [Po1]; we refer to this paper for the proof and a more detailed discussion. With a later purpose on mind we do not strive for generality and restrict ourselves to the case of the Hilbert space H := L2 (R3 ) ≡ L2 and the Laplace operator, −∆ : D(∆) → L2 , which is well known to be self-adjoint on the domain D(∆) which coincides with the usual Sobolev space H 2 (R3 ) ≡ H 2 . For any z belonging to the resolvent set (−∆) = C \ [0, ∞) we define the resolvent as the bounded operator Rz := (−∆ − z)−1 : L2 → H 2 . Consider a bounded operator τ : H2 → X into a complex Banach space X and its adjoint in the dual space X
. Recall that for a closed linear operator A : X → Y the adjoint is defined by (A∗ l)(x) = l(Ax) for all x ∈ D(A) and l ∈ D(A∗ ) ⊆ Y
. Then we can introduce the operators Rτz = τ Rz : L2 → X ,

˘ τz = (Rτz¯ )∗ : X
→ L2 , R

which are obviously bounded too. Let Z be an open subset of (−∆) symmetric w.r.t. the real axis, i.e. such that z ∈ Z implies z¯ ∈ Z. Suppose that for any

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z ∈ Z there exists a closed operator Qz : D ⊆ X
→ X satisfying the following conditions, ˘ τz , Qz − Qw = (z −w)Rτw R z¯ l1 (Q l2 ) = l2 (Qz l1 ) . ∀l1 , l2 ∈ D ,

(2.1) (2.2)

It will be used to construct a family of self-adjoint operators which coincide with −∆ when restricted to ker τ . They can be parametrized by symmetric operators Θ : D(Θ) ⊆ X
→ X . To this end, we define QzΘ = Θ + Qz : D(Θ) ∩ D ⊆ X
→ X , ¯ −1 ) exist and are bounded } . ZΘ := { z ∈ ρ(−∆) : (QzΘ )−1 , (QzΘ With this notation we can state the result we want to borrow from [Po1]. Theorem 2.1 Assume that the conditions

and

ZΘ = ∅

(2.3)

Ran τ ∗ ∩ L2 = {0}

(2.4)

are satisfied. Then the bounded operator z ˘ z (Qz )−1 Rz , Rτ,Θ := Rz − R τ Θ τ

z ∈ ZΘ ,

is the resolvent of the self-adjoint operator −∆τ,Θ defined by ˘ z (Qz )−1 τ fz , fz ∈ D(∆) } , D(∆τ,Θ ) = { f ∈ L2 : f = fz − R τ Θ (−∆τ,Θ − z)f := (−∆ − z)fz , which coincides with −∆ on the ker τ . z by The above formula allows us to study the singularities of the resolvent Rτ,Θ z −1 means of those of (QΘ ) in full analogy with the usual Birman-Schwinger method. Indeed, using the argument of [Po2] one derives easily the equivalence

z ∈ σdisc (AΘ ) ⇔ 0 ∈ σdisc (QzΘ ) .

3 Singular perturbation on a curve in R3 Henceforth, we will be interested in a specific class of perturbations of the Laplacian on H = L2 (R3 ). The free resolvent Rz = (−∆ − z)−1 : L2 (R3 ) → H 2 (R3 ) ,

z ∈ (−∆) ,

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is an integral operator with the kernel √

ei z|x−y| . G (x−y) = 4π |x−y| z

Let Γ ⊂ R3 be a curve defined as a graph of a continuous function which is assumed to be piecewise C 1 . Recall that Γ admits a natural parametrization by the arc length which is unique up to a choice if the reference point; we denote the parameter as s and use the symbol γ(s) : R → R3 for the corresponding function. Then we have (3.1) |γ(s)−γ(s
)| ≤ |s−s
| . To specify further the family of curves which we will consider, we introduce for any ω ˜ ∈ (0, 1) and ε˜ > 0 the set
s Sω˜ ,˜ε := (s, s
) : ω ˜<
ξ(˜ ω )˜ ε, s  and |s−s
| < ε˜ if |s+s
| < ξ(˜ ω )˜ ε , where ξ(˜ ω ) :=

1+˜ ω 1−˜ ω

. We adopt the following assumptions:

(a1) there exists a c ∈ (0, 1) such that |γ(s)−γ(s
)| ≥ c |s−s
|, (a2) there are ω ∈ (0, 1), µ ≥ 0 and positive ε, d such that the inequality 1−

|γ(s)−γ(s
)| |s−s
| ≤ d |s−s
| (|s−s
| + 1)(1 + (s2 +s
2 )µ )1/2

holds for all (s, s
) ∈ Sω,ε . The first condition means, in particular, that Γ has no cusps and self-intersections. The second assumption is basically a requirement of asymptotic straightness (see Remark 5.7), but in contrast to [EI] it restricts also the behaviour of |γ(s)−γ(s
)| at small distances; it is straightforward to check that the bound cannot be satisfied unless Γ is C 1 -smooth. To make use of Theorem 2.1 we take X = L2 (R) and denote the corresponding scalar product by (·, ·)l (see also Remark 3.1 below). The operator τ : H 2 (R3 ) → L2 (R) which we will employ in our construction is a trace map defined in the following way: τ φ(s) := φ(γ(s)) ; it is a standard matter to check that the definition makes sense and the operator τ is bounded [BN]. The adjoint operator τ ∗ : L2 (R) → H −2 (R3 ) is determined by the relation τ ∗ h, ω = (h, τ ω)l ,

h ∈ L2 (R) ,

ω ∈ H −2 (R3 ) ,

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where ·, · stands for the duality between H −2 (R3 ) and H 2 (R3 ), in other words, we can write τ ∗ h = hδΓ , where δΓ is the Dirac measure supported by Γ. Since δΓ ∈ / L2 (R3 ) we get Ran τ ∗ ∩ L2 (R3 ) = {0} , so condition (2.4) is satisfied. Remark 3.1 Notice that the map τ as introduced above is not surjective. Indeed, since γ(s) is a Lipschitz function we have Ran τ = H 1 (R) – cf. [BN]. However, we lose nothing by keeping X = L2 (R) in the further discussion. The problem at hand is to define an operator Qz : D ⊆ L2 (R) → L2 (R) satisfying the conditions (2.1) and (2.2). To this end some preliminaries are needed. Since our considerations concern spectral properties at the negative halfline, it suffices for further discussion to restrict ourselves to z = −κ2 with κ > 0. In such a case it is convenient to modify slightly the used notation by introducing 2

Qκ := Q−κ ,

2

Rκτ := Rτ−κ ,

˘ −κ2 . ˘ κτ := R R τ

and similarly e−κ|s−s | , 4π |s−s
|


Gκ (s−s
) :=

e−κ|γ(s)−γ(s )| . 4π |γ(s)−γ(s
)|


Gκ (γ(s)−γ(s
)) =

The difference of these two kernels, Bκ (s, s
) = Gκ (γ(s)−γ(s
)) − Gκ (s−s
) , defines the integral operator Bκ : D(Bκ ) → L2 (R) with the domain D(Bκ ) = {f ∈ L2 (R) : Bκ f ∈ L2 (R)}. A key observation is that this operator has a definite sign: −κξ in view of (3.1) and of the fact that the function ξ → e ξ decreases monotonically for κ, ξ positive, we have (3.2) Bκ (s, s
) ≥ 0 . The operator Bκ is related obviously with the deviation of Γ from a straight line; below we shall demonstrate that properties for a curve satisfying the assumptions (a1) and (a2) with any µ ≥ 0 is bounded (see Remark 5.4). Next we need to show how the free resolvent kernel behaves when one of the three dimensions is integrated out. By a direct computation one can show that for all κ, κ
> 0 and f1 , f2 ∈ L2 (R) the following relation, 
f1 (s)f2 (s
) [Gκ (s−s
) − Gκ (s−s
)] ds ds
R2  = f1 (s)f2 (s
) [Tˇκ (s−s
) − Tˇκ
(s−s
)] ds ds
, R2

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is valid, where Tˇκ (s−s
) := −

1 (2π)2



1/2 ip(s−s
)  ln p2 +κ2 e dp .

This result means, in particular, that   κ


f1 (s)f2 (s ) G (s−s ) ds ds − R2

(3.3)

R

R2

f1 (s)f2 (s
) Tˇκ (s−s
) ds ds


(3.4)

is κ-independent. Let Tκ : D(Tκ) → L2 (R) be the integral operator with the domain D(Tκ ) = {f ∈ L2 (R) : R Tˇκ (s − s
)f (s
) ds
∈ L2 (R)} and the kernel 1 Tκ (s−s
) := Tˇκ (s−s
) + 2π (ln 2 + ψ(1)) where −ψ(1) ≈ 0.577 is Euler’s number. Then Tk is self-adjoint and we can define the operator Qκ f = (Tκ +Bκ )f : D ≡ D(Tκ ) → L2 (R) , which is also self-adjoint and has the needed properties: 2

Lemma 3.2 The operators Q−κ ≡ Qκ satisfy the conditions (2.1), (2.2). Proof. Let f1 , f2 ∈ D, then a direct computation yields
˘ κτ f2 )l (κ2 − κ
2 )(f1 , Rκτ R 
= f1 (s)f2 (s
) [Gκ (γ(s)−γ(s
)) − Gκ (γ(s)−γ(s
))] ds ds
.

R2

On the other hand, by definition of Qκ and the κ-independence of the expression
(3.4) we find that (f1 , (Qκ − Qκ )f2 )l is also given by the right-hand side of the last formula, which proves (2.1). Since Qκ is self-adjoint, the condition (2.2) is satisfied too.
The operator Θ : L2 (R) → L2 (R) appearing in Theorem 2.1 will be identified here with the multiplication by a real number, Θf = −αf with α ∈ R and the sign convention made with a later purpose on mind. Then the operator QκΘ = Θ + Qκ : D → L2 (R) is self-adjoint for any κ > 0. For simplicity we identify in the following the symbols of the operators τ, Θ with γ, α, respectively. In this notation Theorem 2.1 says the following: if κ ∈ Zα , i.e. if the operator (Qκα )−1 = (Qκ−α)−1 : L2 (R) → L2 (R) exists and is bounded, then ˘ κγ (Qκ −α)−1 Rκγ Rκγ,α = Rκ − R

(3.5)

is the resolvent of a self-adjoint operator which we denote as −∆γ,α . In Section 5 below we will show that the real part of Zα is non-empty being equal to the interval

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(sκ , ∞) with the exception of a discrete set, thus verifying a posteriori that the assumption (2.3) is satisfied. Furthermore, the operator −∆γ,α coincides with −∆ on ker τ = {g ∈ H 2 (R3 ) : g(x) = 0, x ∈ Γ} and ˘ κγ (Qκ −α)−1 τ fk , fκ ∈ D(∆)} , D(−∆γ,α ) = {f ∈ L2 : f = fκ − R (−∆γ,α + κ2 )f = (−∆ + κ2 )fκ .

4 The interaction in terms of boundary conditions To proceed further we have to impose slightly stronger regularity requirement on the curve Γ . Specifically, we assume that it is given by a function γ(s) : R → R3 which is C 1 everywhere and piecewise C 2 , and satisfies the condition (a1). Then we can introduce, apart of a discrete set, the Frenet’s frame for Γ, i.e. the triple (t(s), b(s), n(s)) of the tangent, binormal and normal vectors, which are by assumption piecewise continuous functions of s. Given ξ, η ∈ R we denote r = (ξ 2 +η 2 )1/2 and define the set the “shifted” curve ξη Γr ≡ Γξη r := { γr (s) ≡ γr (s) := γ(s) + ξb(s) + ηn(s) } .

It follows from the smoothness of γ in combination with (a1) that there exists an r0 > 0 such that Γr ∩ Γ = ∅ holds for each r < r0 . 2 (R3 \ Γ) is continuous on R3 \ Γ its restriction to Since any function f ∈ Hloc Γr , r < r0 is well defined; we denote it as f Γr (s). In fact, we can regard f Γr (s) as a distribution from D
(R) with the parameter r. We shall say that a function 2 f ∈ Hloc (R3 \ Γ) ∩ L2 (R3 ) belongs to Υ if the following limits Ξ(f )(s) := − lim

r→0

1 f  (s) , ln r Γr

  Ω(f )(s) := lim f Γr (s) + Ξ(f )(s) ln r , r→0

exist a.e. in R, are independent of the direction 1r (ξ, η), and define functions from L2 (R). The limits here are understood in the sense of the D
(R) topology. With these prerequisites we are able now to characterize the operator −∆γ,α discussed above in terms of (generalized) boundary conditions, postponing the proof to the appendix. Theorem 4.1 With the assumption stated above we have D(−∆γ,α ) = Υα := { g ∈ Υ : 2παΞ(g)(s) = Ω(g)(s) } , −∆γ,α f = −∆f

for

x ∈ R3 \ Γ .

(4.1)

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5 Curvature-induced bound states Let us first find the spectrum of −∆γ0 ,α where γ0 is a linear function describing a straight line. Since Bκ = 0 holds in this case we have Qκ = Tκ . Then the resolvent formula (3.5) yields σ(−∆γ0 ,α ) = { −κ2 : α ∈ σ(Tκ ) = σac (Tκ ) } . Using the momentum representation of Tκ we immediately get σac (Tκ ) = (−∞, sκ ] , where sκ :=

1 2π (ψ(1)

− ln(κ/2)). Hence the spectrum of −∆γ0 ,α is given by σ(−∆γ0 ,α ) = σac (−∆γ0 ,α ) = [ζ0 , ∞) ,

where ζ0 = −4e2(−2πα+ψ(1)) as we expect with the spectrum of a two-dimensional δ interaction [AGHH] and the natural separation of variables in mind. To find the spectrum of −∆γ,α for a non-straight curve we treat the respective operator Qκ as a perturbation of the one corresponding to a straight line. First we have to localize the essential spectrum. Following step by step the argument given in the proof of Proposition 5.1 of Ref. [EI] we get Lemma 5.1 Let Γ be a curve given by a function γ(s) satisfying (a1) and (a2) with µ > 1/2. Then σess (−∆γ,α ) = [ζ0 , ∞). Next we observe that a nontrivial bending pushes the upper bound of the spectrum of Qκ up. Lemma 5.2 If Γ is not a straight line we have sup σ(Qκ ) > sκ .

(5.1)

Proof. Let φ be a non-negative function from C0∞ (R) such that φ(0) = 0. Given λ > 0 we set φλ (s) := λ1/2 φ(λs). To show (5.1) it suffices to check the following inequality (Qκ φλ , φλ )l − sκ (φλ , φλ )l > 0 , which is easily seen to be equivalent to 1 − 2π







λu ln 1+ κ R

2 1/2 

ˆ 2

φ(u) du + λ

R2

Bκ (s, s
)φ(λs)φ(λs
) ds ds
> 0 ,

(5.2) ˆ where φ stands for the Fourier transform of φ. The first term in the last expression can expanded as

2 

1 λ

ˆ 2 − u2 φ(u)

du + O(λ4 ) . 4π κ R

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Since Γ is not straight by assumption the inequality (3.2) is sharp in an open subset of R2 , so there is D > 0 such that λ R2 Bκ (s, s
)φ(λs)φ(λs
) ds ds
≥ Dλ as λ → 0. Consequently, for all sufficiently small λ the inequality (5.2) is satisfied.
On the other hand, the part of the spectrum in (sκ , ∞) added in this way is at most discrete provided the curve has the asymptotic straightness properties expressed by the assumption (a2) with µ large enough. Lemma 5.3 If µ > 1/2 then Bk are Hilbert-Schmidt operators. Moreover, norms Bκ HS are uniformly bounded with respect κ ≥ κ0 = |ζ0 |1/2 . Proof. Denote ρ ≡ ρ(s, s
) := |γ(s)−γ(s
)| and σ ≡ σ(s, s
) := |s−s
|. In this notation the assumptions (a1), (a2) can be written as (a1) there is a c ∈ (0, 1) such that ρ(s, s
) ≥ cσ(s, s
), (a2) there are ω ∈ (0, 1), µ ≥ 0 and ε, d > 0 s.t. for all (s, s
) ∈ Sω,ε we have 1−

dσ(s, s
) ρ(s, s
) ≤ .

σ(s, s ) (σ(s, s )+1)(1 + (s2 +s
2 )µ )1/2

Next we notice that the perturbation kernel is monotonous with respect to the spectral parameter, Bκ (s, s
) ≤ Bκ
(s, s
)

for κ
< κ ,

thus to prove lemma it suffices to show that Bκ0 is a Hilbert-Schmidt operator. −κ0 υ Since the function υ → e υ is strictly decreasing and convex in (0, ∞), we have the following estimate,  −κ0 σc 
e−κ0 σ e e−κ0 ρ − ≤− (σ − ρ) , 0≤ ρ σ σc where σc := cσ and c is the constant appearing in (a1). Thus we get 0≤

e−κ0 σ σ−ρ e−κ0 ρ − ≤ (κ0 σc + 1) 2 e−κ0 σc , ρ σ σc

and moreover, the assumption (a1) gives the bound σ−ρ ≤ 1 − c. σ In view of (a2), there exists a positive c˜ such that σ(s, s
) ≥ c˜ .

(5.3)

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holds for any (s, s
) ∈ R2 \ Sω,ε . Combining the last three inequalities we have in R2 \ Sω,ε the estimate   e−κ0 σ 1 e−κ0 ρ − (5.4) ≤ M1 e−κ0 σc 4π ρ σ with M1 := (4π)−1 (1−c) c−2(κ0 c+ c˜−1 ). On the other hand using (5.3) and (a2) we get   1 e−κ0 ρ e−κ0 σ 1 − (5.5) ≤ M2 e−κ0 σc 2 4π ρ σ (1 + (s +s
2 )µ )1/2 for (s, s
) ∈ Sω,ε , where M2 := (4π)−1 dc−2 max{1, κ0 c}. Putting now the estimates (5.4), (5.5) together we find  Bk (s, s
)2 dsds
R2

 ≤ M12 ≤

R2 \Sω,ε

1 2 1+ω M 2 1 1−ω



e−2κ0 c|s−s | ds ds
+ M22


 0

∞

e−κ0 cu u du + M22



Sω,ε

Sω,ε


e−2κ0 c|s−s | ds ds
1 + (s2 +s
2 )µ

e−2κ0 c|s−s | ds ds
< ∞ , 1 + (s2 +s
2 )µ


which proves the result because the last integral converges for µ > 1/2.




Remark 5.4 As we have said, the assumption (a2) includes a decay of the quantity characterizing the non-straightness at large distances within Sω,ε as well as a restriction for s close to s
. The latter (which is independent of µ) ensures the boundedness of Bκ uniformly w.r.t. κ. As in the proof of the above lemma the uniformity is easy; it suffices to check that Bκ0 is bounded. To this end we employ the Schur-Holmgren bound: we have Bκ0 l ≤ Bκ0 SH , where the right-hand side of the last inequality is for integral operators with symmetric positive kernels defined as  Bκ0 SH = sup Bκ0 (s, s
) ds
. s∈R

R

Let us use the notation from the previous proof. If σ ≤ ε, then by assumption (a2) there exists for any µ ≥ 0 a C1 > 0 such that Bκ0 (s, s
) ≤ C1 . On the other hand, if σ > ε then by (5.3) we can find C2 > 0 such that Bκ0 (s, s
) ≤ C2 e−κ0 σc . Combining the above two inequalities we get the following estimate,

   s+ε  ∞
e−κ0 cε Bκ0 (s, s
)ds
≤ C1 ds
+ 2C2 e−κ0 c|s−s | ds
= 2 C1 ε + C2 , κ0 c s−ε s+ε R which shows that Bκ0 SH is finite.

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Lemma 5.5 Let Γ be defined as before. Then the function κ → Qκ is continuous in the norm operator in (κ0 , ∞), and moreover, lim sup σ(Qκ ) = −∞ .

κ→∞

(5.6)

Proof. First we observe that the function κ → Tκ is continuous in the norm operator. Indeed, for any f ∈ D we have 2 

1 p 2 + κ2 = |fˆ(p)|2 dp ln 2 4(2π)3 R p + κ
2 1  κ 2 ln
f 2l → 0 ≤ 4(2π)3 κ

(Tκ −Tκ
)f l

(5.7)

as κ
→ κ. On the other hand, in analogy with [EI] we can estimate |(Bκ −Bκ
)(s, s
)| ≤ 2(Bκ (s, s
)2 +Bκ
(s, s
)2 ) ≤ 4Bκ˜ (s, s
)2 , 2

where κ ˜ := min{κ, κ
} arriving therefore at lim Bκ −Bκ
HS → 0 ;

κ
→κ

(5.8)

from (5.7) and (5.8) we get the norm-operator continuity. Let further f ∈ D. The limiting relation (5.6) follows directly from the bound     2  1 κ 2 1/2 (Q f, f )l = − ln p +κ + ln 2 + ψ(1) |fˆ(p)|2 dp (2π)3/2 R 1 κ 2 2 +(Bκ f, f )l ≤ (− ln + ψ(1)) f l + S f l , 2 (2π)3/2 where S := supκ≥κ0 Bk l < ∞.




Now we are in position to state and prove our main result. Theorem 5.6 Let Γ be a curve determined by a function γ : R → R3 which is C 1 and piecewise C 2 , and satisfies the conditions (a1), (a2) with µ > 1/2. Then the operator −∆γ,α has at least one isolated eigenvalue in (−∞, ζ0 ). Proof. By Lemma 5.2 we have sup σ(Qκ ) > sκ , while by Lemma 5.3 this operator has only isolated eigenvalues of a finite multiplicity in (sκ , ∞). Let λ(κ) be such an eigenvalue of Qκ . Using then Lemma 5.5 we conclude that the function λ(·) is 1/2 such continuous and λ(κ) → −∞ as κ → ∞. Consequently, there is κ ˜ > |ζ0 | that λ(˜ κ) = α. From the resolvent formula (3.5) we then infer that −˜ κ2 ∈ (−∞, ζ0 ) is an eigenvalue of −∆γ,α .


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Remarks 5.7 (a) It is clear that the claim holds without the C 2 assumption, however, the latter is needed if we want to interpret the δ interaction on the curve in the spirit of Theorem 4.1. Furthermore, we see that any deviation from a straight Γ pushes the spectrum threshold below the value ζ0 but without the assumption (a2) we cannot be sure about the nature of this added part of the spectrum. (b) One may ask what the requirement of asymptotic straightness expressed by (a2) means. Suppose that γ is C 2 smooth. Then the curvature of Γ is every1/2  3 2 where defined and can expressed as k(s) = , where ki (s) := i=1 ki (s)

εijk γj
(s)γk

(s) with the summation convention for the indices of the Levi-Civita tensor. It allows us estimate the distance between γ(s) and γ(s
) in the following way,  1

 s1 2 1/2   3  s πν
|γ(s)−γ(s )| = cos ki (s2 ) ds2 + ds1 i=1 s
2 s
ν=0 2

 s1 3  s 1 ≥ ki (s2 ) ds2 1− ds1 , i=1 s
2 s


where we assume without loss of generality that s > s
. Suppose that there are positive β, ci such that |ki (s)| ≤ ci |s|−β . Then |k(s)| ≤ 3c |s|−β , where c = maxi {ci } and one can estimate 2  s  s1 |γ(s)−γ(s
)| 1 1− ≤ k(s ) ds ds1 2 2 |s−s
| 2 |s−s
| s
s
 s 2 1 1 3c2 c2 |s
−s| c2 2
≤ |s −s | ds ≤ ≤ . 1 1 2 |s−s
| |s
|2β s
2 |s
|2β 2 |s
|2β−2 Thus the conclusion is the same as in the two-dimensional case discussed in [EI]: the assumption (a2) with µ > 1/2 is satisfied if β > 5/4.

Appendix: proof of Theorem 4.1 First we check the inclusion D(−∆γ,α ) ⊆ Υα . Suppose that f ∈ D(−∆γ,α ), i.e. that there is fκ ∈ D(∆) such that ˘ κγ (Qκ −α)−1 τ fκ . f = fκ − R

(A.1)

Denote h := (Qκ −α)−1 τ fκ ∈ L2 (R), so f = fκ − Rκ τ ∗ h. Since fκ ∈ H 2 (R3 ) and 2 τ ∗ h ∈ H −2 (R3 ) is a measure supported by Γ we can conclude that f ∈ Hloc (R3 \Γ) – see [RS]. Using properties of the Macdonald function K0 (ς) and the following relation  2
2 1/2
1 e−κ(r +(s−s ) ) 1 = K0 ((p21 + κ2 )1/2 r) eip1 (s−s ) dp1 4π (r2 +(s−s
)2 )1/2 (2π)2 R

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Curvature-Induced Bound States

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we can check that    −κ(r2 +(s−s
)2 )1/2 1 1 e lim ln rh(s) = Tκ h(s). h(s
)ds
+ r→0 4π R (r2 +(s−s
)2 )1/2 2π Now it is easy to demonstrate that the function ˘ κγ h)(x) = (R

1 4π

 R

e−κ|x−γ(s)| h(s) ds |x−γ(s)|

satisfies the limiting relation   ˘ κ h) Γr (s) + 1 ln rh(s) = Tκ h(s) + Bκ h(s) lim (R r→0 2π

(A.2)

with respect to families of “shifted” curves described in Sec. 4. The above limits are understood in distributional sense. It follows from (A.1) and (A.2) that Ξ(f )(s) = −

1 h(s). 2π

(A.3)

On the other hand, since fκ ∈ D(∆) = H 2 (R3 ) the same relations (A.1) and (A.2) yield Ω(f )(s) = (τ fκ )(s) − (Qκ h)(s) = −αh(s).

(A.4)

Combining (A.3) and (A.4) we obtain that f ∈ Υ and 2παΞ(f )(s) = Ω(f )(s). Conversely, one can show by analogous considerations that any function from Υα ˘ κγ (Qκ − α)−1 τ fκ with fκ ∈ D(∆), so can be represented in the form f = fκ − R D(−∆γ,α ) = Υα . Moreover, since (−∆γ,α + κ2 )f = (−∆ + κ2 )fκ and τ ∗ h ∈ H −2 (R3 ) is a measure supported by Γ we infer that −∆γ,α f (x) = −∆f (x),

x ∈ R3 \Γ .

This completes the proof.

Acknowledgments The work was supported by GAAS under the contract #1048101. The authors are obliged to A. Posilicano for making his results available to them prior to publication and to the referee for useful comments.

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References [AGHH] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer, Heidelberg 1988. [BN]

O.V. Besov, V.P. Il’in and S.M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems, Winston & Sons, Washington 1978–9.

ˇ J.F. Brasche, P. Exner, Yu.A. Kuperin and P. Seba, ˇ [BEKS] Schr¨ odinger operators with singular interactions, J. Math. Anal. Appl. 184 (1994), 112–139. [DE]

P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), 73– 102.

[EI]

P. Exner and T. Ichinose, Geometrically induced spectrum in curved leaky wires, J. Phys. A34 (2001), 1439–1450.

ˇ [ES]

ˇ P. Exner and P. Seba, Bound states in curved quantum waveguides, J.Math. Phys. 30 (1989), 2574–2580.

[GJ]

J. Goldstone and R.L. Jaffe, Bound states in twisting tubes, Phys. Rev. B45 (1992), 14100–14107.

[Ka]

W. Karwowski, Hamiltonians with additional kinetic energy terms on hypersurfaces, in Applications of Self-Adjoint Extensions in Quantum Physics, Springer, LNP 324, Berlin 1989; pp. 203–217.

[KK]

W. Karwowski and V. Koshmanenko, Schr¨odinger operator perturbed by dynamics of lower dimension, in Differential Equations and Mathematical Physics, American Math. Society, Providence, R.I., 2000; pp. 249–257.

[Ko]

S. Kondej, Perturbation of the dynamics by objects supported by small sets, PhD Thesis, Wroclaw 2001.

[Ku]

Y.V. Kurylev, Boundary condition a curve for a three-dimensional Laplace operator, J. Sov. Math. 22 (1983), 1072–1082.

[Po1]

A. Posilicano, A Krein-like formula for singular perturbations of selfadjoint operators and applications, J. Funct. Anal. 183 (2001), 109–147.

[Po2]

A. Posilicano, Self-adjoint extensions by additive perturbations, submitted for publication

[RS]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, II. Fourier Analisis, Self-adjointness, Academic Press, New York 1978.

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[RB]

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W. Renger and W. Bulla, Existence of bound states in quantum waveguides under weak conditions, Lett. Math. Phys. 35 (1995), 1–12.

P. Exner and S. Kondej Department of Theoretical Physics Nuclear Physics Institute Academy of Sciences ˇ z near Prague 25068 Reˇ Czech Republic email: [email protected], [email protected] Communicated by Gian Michele Graf submitted 19/03/02, accepted 17/05/02

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 3 (2002) 983 – 1002 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/050983-20

Annales Henri Poincar´ e

Relations between the Hepp-Lieb and the Alli-Sewell Laser Models F. Bagarello Abstract. In this paper we show that the dissipative version of the laser model proposed by Alli and Sewell can be obtained by considering the stochastic limit of the (open system) hamiltonian introduced by Hepp and Lieb in their seminal work. We also prove that the Dicke-Haken-Lax hamiltonian produces, after the stochastic limit is considered, the generator of a semigroup with equations of motion very similar to those of Alli-Sewell, and coinciding with these under suitable conditions.

I Introduction In two recent papers, [1, 2], a dissipative laser model has been introduced and analyzed in some details. In particular in [1] (AS in the following) the rigorous definition of the unbounded generator of the model, which consists of a sum of a free radiation and a free matter generator plus a matter-radiation term, is given and the existence of the thermodynamical limit of the dynamics of some macroscopic observables is deduced. Moreover, the analysis of this dynamics shows that two phase transitions occur in the model, depending on the value of a certain pumping strenght. In [2] the analysis has been continued paying particular attention to the existence of the dynamics of the microscopic observables, which are only the ones of the matter since, in the thermodynamical limit, we proved that the field of the radiation becames classical. Also, the existence of a transient has been proved and an entropy principle has been deduced. On the other hand, in a series of papers [3, 4] culminating with the fundamental work by Hepp and Lieb [5] (HL in the following) many conservative models of matter interacting with radiation were proposed. In particular, in [5] the authors have introduced a model of an open system of matter and of a single mode of radiation interacting among them and with their (bosonic) reservoirs, but, to simplify the treatment, they have considered a simplified version in which the matter bosonic reservoir is replaced by a fermionic one. In this way they avoid dealing with unbounded operators. This is what they call the Dicke-Haken-Lax model (DHL model in the following). In [1, 2] the relation between the AS model and a many mode version of the HL model is claimed: of course, since no reservoir appear in the semigroup formulation as given by [1], this claim is reasonable but it is not clear the explicit way in which HL should be related to AS. In this paper we will prove that the relation between the two models is provided by (a slightly modified version of)

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the stochastic limit (SL), [6] and reference therein. In particular, if we start with the physical AS system (radiation and matter) and we introduce in a natural way two reservoirs (one is not enough!) for the matter and another reservoir for the radiation, then the SL of the hamiltonian for the new system constructed in this way returns back the original AS generator, under very reasonable hypotheses. Moreover, the model which we have constructed ad hoc to get this generator surprisingly coincides with the HL laser model, [5]. This is the content of Section 3, which follows a section where we introduce all the models we will deal with, to keep the paper self-contained. In Section 4 we will consider the SL of the fermionic version of the HL model, known as the DHL model, [5, 7]. We will find that, even if the form of the generator apparently differs from the one by AS, under certain conditions on the coefficients which define the model, the equations of motion for the observables of the matterradiation system coincide with the ones given in AS. In Section 5 we give our conclusions while the Appendix is devoted to summarize few results on SL which are used everywhere in this paper. Before concluding this section we wish to remark that we call the procedure proposed here stochastic limit even if a minor difference exists between the original approach, [6], and the one we will use here, namely the appearance of different powers of the over-all coupling constant λ which appear in our hamiltonian operators. The final remark concerns our notation which we try to keep as simple as possible by neglecting the symbol of tensor product (almost) everywhere in the paper.

II The Physicals Models In this section we will discuss the main characteristics of the three physical models which will be considered in this paper. In particular, we will only give the definition of the hamiltonians for the HL and the DHL models and the expression of the generator for the AS model, without even mentioning mathematical details like, for instance, those related to the domain problem intrinsic with all these models due to the presence of bosonic operators. We refer to the original papers for these and further details which are not relevant in this work. We begin with the AS model. This model is a dissipative quantum system, Σ(N ) , consisting of a chain of 2N + 1 identical two-level atoms interacting with an n−mode radiation field, n fixed and finite. We build the model from its constituent parts starting with the single atom. This is assumed to be a two-state atom or spin, Σat . Its algebra of observables, Aat , is that of the two-by-two matrices, spanned by the Pauli matrices (σx , σy , σz ) and the identity, I. They satisfy the relations σx2 = σy2 = σz2 = I; σx σy = iσz , etc.

(2.1)

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We define the spin raising and lowering operators σ± =

1 (σx ±iσy ). 2

(2.2)

We assume that the atom is coupled to a pump and a sink, and that its dynamics is given by a one-parameter semigroup {Tat (t)|t∈R+ } of completely positive, identity preserving contractions of Aat , whose generator, Lat , is of the following form. Lat σ± = −(γ1 ∓i)σ± ; Lat σz = −γ2 (σz − ηI),

(2.3)

where (> 0) is the energy difference between the ground and excited states of the atom, and the γ’s and η are constants whose values are determined by the atomic coupling to the energy source and sink, and are subjected to the restrictions that 0 < γ2 ≤2γ1 ; −1≤η≤1.

(2.4)

The matter consists of 2N + 1 non-interacting copies of Σat , located at the sites r = −N, . ., N of the one-dimensional lattice Z. Thus, at each site, r, there is a copy, Σr , of Σat , whose algebra of observables, Ar , and dynamical semigroup, Tr , are isomorphic with Aat and Tat , respectively. We denote by σr,u the copy of σu at r, for u = x, y, z, ±. We define the algebra of observables, A(N ) , and the dynamical semigroup, N (N ) of the matter to be ⊗N is the r=−N Ar and ⊗r=−N Tr , respectively. Thus, A algebra of linear transformations of C4N +2 . We identify elements Ar of Ar with those of A(N ) given by their tensor products with the identity operators attached to the remaining sites. Under this identification, the commutant, A
r , of Ar is the tensor product ⊗s=r As . The same identification will be implicitly assumed for the other models. (N ) Tmat ,

(N )

(N )

It follows from these specifications that the generator, Lmat , of Tmat is given by the formula
(N ) Lmat = Ll , (2.5) l∈IN

where IN = {−N, . . . , −1, 0, 1, . . . , N }. Here Lr σr,± = −(γ1 ∓i)σr,± ; Lr σr,z = −γ2 (σr,z − ηI); and Lr (Ar A
r ) = (Lr Ar )A
r ∀Ar ∈Ar , A
r ∈A
r

(2.6)

We assume, furthermore, that the radiation field consists of n(< ∞) modes, represented by creation and destruction operators {a
l , al |l = 0, . ., n − 1} in a Fock-Hilbert space Hrad as defined by the standard specifications that (a) these operators satisfy the CCR, [al , a
m ] = δlm I; [al , am ] = 0,

(2.7)

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and (b) Hrad contains a (vacuum) vector Φ, that is annihilated by each of the a’s and is cyclic w.r.t. the algebra of polynomials in the a
’s. The formal generator of the semigroup Trad of the radiation is
n−1   iωl [a
l al , .] + 2κl a
l (.)al − κl {a
l al , .} , (2.8) Lrad = l=0

where {., .} denotes anticommutator, and the frequencies, ωl , and the damping constants, κl , are positive. We refer to [1] for a rigorous definition of Lrad . The composite (finite) system is simply the coupled system, Σ(N ) , comprising the matter and the radiation. We assume that its algebra of observables, B (N ) , is the tensor product A(N ) ⊗R, where R is the
−algebra of polynomials in the a’s, a
’s and the Weyl operators. Thus, B (N ) , like R, is an algebra of both bounded and unbounded operators in the Hilbert space H(N ) := C4N +2 ⊗Hrad . We shall identify elements A, R, of A(N ) , R, with A⊗Irad and Imat ⊗R, respectively. We assume that the matter-radiation coupling is dipolar and is given by the interaction Hamiltonian
(N ) ) Hint = (σr,+ φ(N + h.c.), (2.9) r r∈IN

where we have introduced the so-called radiation field, φ(N ) , whose value at the site r is
n−1 ) = −i(2N + 1)−1/2 λl al exp(2πilr/n). (2.10) φ(N r l=0

Here the λ’s are real-valued, N −independent coupling constants. Among the other results contained in [1], one of the most relevant is that the map

(N )

(N )

L(N ) = Lmat + Lrad + i[Hint , .] is really the generator of a N -depending semigroup, T (N ) , regardless of the un(N ) bounded nature of both Lrad and Hint . This is the starting point for a successive analysis, see [1, 2]. Now we introduce the HL model, changing a little bit the notations with respect to the original paper, [5], and introducing n modes for the radiation instead of the only one considered by HL. The HL hamiltonian for the 2N + 1 atoms and for the n modes of the radiation can be written as follows: H = H (S) + H (R) ,

(2.11)

where ”S” refers to the system (radiation+matter) and ”R” to the reservoir. The hamiltonian of the system is H (S) = ωR

n−1
j=0

a†j aj + µ


l∈IN

σl,z

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n−1

α +√ (σl,+ aj e2πijl/n + σl,− a†j e−2πijl/n ) + 2N + 1 j=0 l∈I N

β +√ 2N + 1

n−1





(σl,+ a†j e−2πijl/n + σl,− aj e2πijl/n ),

(2.12)

j=0 l∈IN

which differs from the one in [5] for the phases introduced in the last two terms, phases which are related to the presence of many modes in this hamiltonian with respect to the original one. Notice that the presence of β means that we are not restricting our model to the rotating wave approximation, (RWA). The hamiltonian for the reservoir contains two main contributions, one related to the two reservoirs of the matter and one to the reservoir of the radiation. We have:
(A) H (R) = H (P ) + Hl , (2.13) l∈IN

where H

(P )

=

n−1



† √ n−1 (rj (g j )aj + rj (gj )a†j ), dk ωr,j (k)rj (k) rj (k) + α †

j=0

(2.14)

j=0

and (A)

Hl

=

2 
s=1

dk ωms (k)m†s,l (k)ms,l (k) +

√ α(m†1,l (h1 )σl,− + h.c.)

√ + α(m†2,l (h2 )σl,+ + h.c.)

(2.15)

Few comments are necessary in order to clarify the formulas above.  1) first of all we are using the notation: rj (gj ) = dk rj (k)gj (k) and rj† (g j ) =  dk rj† (k)g j (k). Here dk is a shortcut notation for dk 3 . 2) the functions gj and h1,2 are introduced by HL to regularize the bosonic fields rj (k) and m(1,2),l (k). 3) we notice that in this model two indipendent reservoirs, m1,l (k) and m2,l (k), are introduced for (each atom of) the matter, while only one, rj (k), is used for (each mode of) the radiation. This result will be recovered also in our approach. 4) it should be pointed out that the hamiltonian above is really only one of the possible extentions of the HL original one to the n-modes situation, and, in fact, is quite a reasonable extension. In particular we are introducing different dispersion laws and different regularizing functions hj for each mode of the radiation, while we use the same ω and the same h for the atoms localized in different lattice sites. This ”non-symmetrical” choice is motivated by the

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AS model itself, where we see easily that the free evolution of the matter observables does not dipend on the lattice site, while in the form of the generator (2.8) a difference is introduced within the different modes of the radiation. The reason for that is, of course, the mean field approximation which is being used to deal with the model. √ 5) the coupling constant α is written explicitly for later convenience. The role of each term of the hamiltonian above is evident. Instead of using the above expression for H, where we explicitly consider the effect of the system and the effect of the reservoirs, we divide H as a free and an interaction part, in the following way: √ (2.16) H = H0 + αHI , where H0 = ω R

n−1





a†j aj + µ

j=0

σl,z +

2 



dkωms (k)m†s,l (k)ms,l (k)

l∈IN s=1

l∈IN

+

n−1


dkωr,j (k)rj (k)† rj (k)

(2.17)

j=0

and HI =

n−1


(rj† (g j )aj +rj (gj )a†j )

j=0

+




[(m†1,l (h1 )σl,− + h.c.) + (m†2,l (h2 )σl,+ + h.c.)] +

l∈IN

√ n−1

α (σl,+ aj e2πijl/n + σl,− a†j e−2πijl/n ) + +√ 2N + 1 j=0 l∈I N

β + α(2N + 1)

n−1





(σl,+ a†j e−2πijl/n + σl,− aj e2πijl/n ). (2.18)

j=0 l∈IN

The only non trivial commutation relations, which are different from the ones already given in (2.1,2.7), are: [rj (k), rl (k
)† ] = δj,l δ(k − k
),

[ms,l (k), m†s
,l
(k
)] = δs,s
δl,l
δ(k − k
)

(2.19)

We end this section by introducing the DHL model. The main difference, which is introduced to avoid dealing with unbounded operators, consists in the use of a fermionic reservoir for the matter, and for this reason the Pauli matrices of both AS and HL are replaced by fermionic operators as described in details, for instance, in [7].

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The idea for introducing these operators is quite simple: since we are considering only two-levels atoms (this is the reason why Pauli matrices appear!) a possible description of one such atom could consist in using two pairs of independent fermi operators, (b− , b†− ) which annihilates and creates one electron in the lowest energy level Ψ− , with energy E− , and (b+ , b†+ ) which annihilates and creates one electron in the upper energy level Ψ+ , with energy E+ . If we restrict the Hilbert space of the single atom to the states in which exactly one electron is present, in the lower or in the upper level, it is clear that b†+ b− behaves like σ+ , that is when it acts on a vector with one electron in the lowest state (b†− Ψ0 , Ψ0 being the state with no electrons), it returns a state with an electron in the upper level (b†+ Ψ0 ), and so on. Moreover b†+ b+ − b†− b− has eigenvectors b†± Ψ0 with eigenvalues ±1, so that it can be identified with σz . Going back to the finite system we put σ+,l = b†+,l b−,l ,

σ−,l = b†−,l b+,l ,

σz,l = b†+,l b+,l − b†−,l b−,l .

(2.20)

where l ∈ IN . The only non trivial anti-commutation relations for operators localized at the same lattice site are: {b±,l , b†±,l } = 1.

(2.21)

Moreover, see [7], two such operators commute if they are localized at different lattice site. For instance we have [b±,l , b†±,s ] = 0 if l = s. Since the number of the atomic operators is now doubled with respect to the HL model, it is not surprising that also the number of the matter reservoir operators is doubled as well: from 2 × (2N + 1) we get 4 × (2N + 1) operators, each one coupled with a b±,l 1 operator. On the other hand, the part of the radiation is not modified passing from the HL to the DHL model. Let us write the hamiltonian for√ the open system in the form which is more convenient for us and using λ instead of α. We have H = H0 + λHI ,

(2.22)

where H0 = ω R

n−1


a†j aj + µ

j=0

+






l∈IN

(b†+,l b+,l − b†−,l b−,l ) +

n−1


dk ωr,j (k)rj (k)† rj (k) +

j=0

† † (k)Bs,l (k) + Cs,l (k)Cs,l (k)) dk (k)(Bs,l

(2.23)

l∈IN s=± 1 We use here x
to indicate one of the two possibilities: x or x† , x being a generic operator of the physical system.

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and HI =

n−1



(rj† (g j )a + rj (gj )a†j ) + λ

j=0

+





l∈IN

(N ) † b+,l b−,l

(φl

+ h.c.) +

† † [b†s,l (Bs,l (gBs ) + Cs,l (gCs )) + (Bs,l (gBs ) + Cs,l (gCs ))bs,l ]. (2.24)

l∈IN s=±

Here gB± and gC± are real function. The commutation rules for the radiation operators (system and reservoir) coincide with the ones of the HL model. For what concerns the matter operators (system and reservoirs) the first remark is that any two operators localized at different lattice sites commutes, as well as any operator of the radiation with any observable of the matter. As for operators localized at the same lattice site, the only non trivial anticommutators are † † (k
)} = {C±,l (k), C±,l (k
)} = δ(k − k
), {B±,l (k), B±,l

(2.25)

together with the (2.21), while all the others are zero. Finally, to clarify the different roles between the B and the C fields it is enough to consider their action on the ground state of the reservoir ϕ0 : † rj (k)ϕ0 = B±,l (k)ϕ0 = C±,l (k)ϕ0 = 0.

(2.26)

These equations, together with what has been discussed, for instance, in [7], show that B is responsible for the dissipation, while C is the pump. Again in reference [7] it is discussed which kind of approximations, other than using a fermionic reservoir, are introduced to move from ”real life” to the DHL model. Here we mention only a few: the atom is considered as a two level system; only n modes of radiation are considered (n=1 in the original model, [5]); the electromagnetic interaction is written in the dipolar approximation and within the RWA; the model is mean field, etc. It is worth remarking that since all the contributions in H0 above are quadratic in the various creation and annihilation operators, they all commute among them. This fact will be used in the computation of the SL of this model.

III Alli-Sewell versus Hepp-Lieb We begin this section with a pedagogical note on the single-mode single-atom version of the AS model. This will be useful in order to show that two reservoirs must be introduced to deal properly with the matter. After that we will consider the full AS model and we will show that the hamiltonian which produces the AS generator after considering its SL is nothing but the HL hamiltonian in the RWA. We will conclude this section proving that adding the counter-rotating term (the one proportional to β in (2.12)) does not affect this result, since its contribution disappear rigorously after the SL.

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The starting point is given by the set of equations (2.3)-(2.10) restricted to n = 1 and N = 0, which means only one mode of radiation and a single atom. (N ) With this choice the phases in φl disappear so that the interaction hamiltonian (2.9) reduces to Hint = i(σ− a† − h.c.), (3.1) and the total generator is L = Lmat + Lrad + i[Hint , .]. Let us suppose that the atom is coupled not only to the radiation by means of Hint , but also to a bosonic background m(k) with the easiest possible dipolar interaction: (3.2) HMm = σ+ m(h) + h.c. Of course this background must have a free dynamics and the natural choice is  (3.3) H0,m = dk ωm (k)m† (k)m(k). For what concerns the radiation background the situation is completely analogous:  HR,r = ar† (g) + h.c. (3.4) H0,r = dk ωr (k)r† (k)r(k), are respectively the free hamiltonian and the radiation-reservoir interaction. We take the complete hamiltonian as simply the sum of all these contributions, with the coupling constant λ introduced as below:  † H = H0 + λHI = {µσz + ωR a a + dk ωm (k)m† (k)m(k)  + dk ωr (k)r† (k)r(k)} +λ{(ar† (g) + h.c.) + (σ+ m(h) + h.c.) + λi(σ− a† − h.c.)}.

(3.5)

Taking the SL of this model simply means, first of all, considering the free evolution of the interaction hamiltonian, HI (t) = eiH0 t HI e−iH0 t , see Appendix and reference [6]. It is a simple computation to obtain that, if ωR = 2µ, †

i(ωr −ωR )t

(3.6)

i(2µ−ωm )t

†

) + h.c.) + λi(σ− a − h.c.). (3.7) In this case the SL produces, see Appendix, the following effective time-depending interaction hamiltonian: HI (t) = (ar (ge

(sl)

HI

) + h.c.) + (σ+ m(he

(t) = (arg† (t) + h.c.) + (σ+ mh (t) + h.c.) + i(σ− a† − h.c.),

(3.8)

where the dependence on λ disappears and the operators rg (t), mh (t) and their hermitian conjugates satisfy the following commutation relations for t ≥ t
, (g)

[rg (t), rg† (t
)] = Γ− δ(t − t
),

(h)

[mh (t), m†h (t
)] = Γ− δ(t − t
).

(3.9)

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Here we have defined the following complex quantities: 

(g)

Γ− = (h) Γ−



0

dk|g(k)|2 e−i(ωr (k)−ωR )τ ,

dτ −∞





0

=

dk|h(k)|2 e−i(2µ−ωm (k))τ .

dτ

(3.10)

−∞

We want to stress that the restriction t > t
does not prevent to deduce the commutation rules (3.13) below, which are the main ingredient to compute the SL. However, the extension to t < t
can be easily obtained as discussed in [6]. It is clear now what should be the main analytical requirement for the regularizing functions h and g: they must be such that the integrals above exist finite! In order to obtain the generator of the model we introduce the wave operator Ut (in the interaction representation) which satisfy the following operator differential equation: (sl)

∂t Ut = −iHI

(t)Ut , with

U0 = 1 .

(3.11)

In [6], and reference therein, it is proven that for a large class of quantum mechanical models, the equation above can be obtained as a suitable limit of differential equations for a λ-depending wave operator. Analogously, rg (t) and mh (t) can be considered as the limit (in the sense of the correlators) of the rescaled operators 2 1 −i(ωr −ωR )t/λ2 ) and λ1 m(hei(2µ−ωm )t/λ ). It is not surprising, therefore, that λ r(ge not only the operators but also the vectors of the Hilbert space of the theory are affected by the limiting procedure λ → 0. In particular, the vacuum η0 for the operators rg and mh , mh (t)η0 = rg (t)η0 = 0, does not coincide with the vacuum ϕ0 for m(k) and r(k), r(k)ϕ0 = m(k)ϕ0 = 0, see [6] for more details. Equation (3.11) above can be rewritten in the more convenient form  Ut = 1 − i

0

t

HIsl (t
)Ut
dt
,

(3.12)

which is used, together with the time consecutive principle, see Appendix and [6], and with equation (3.9), to obtain the following useful commutation rules (g)

(h)

[rg (t), Ut ] = −iΓ− aUt ,

[mh (t), Ut ] = −iΓ− σ− Ut .

(3.13)

If we define the flow of a given observabe X of the system as jt (X) = Ut† XUt , the generator is simply obtained by considering the expectation value of ∂t jt (X) (ξ) on a vector state η0 = η0 ⊗ ξ, where ξ is a generic state of the system. Using formulas (3.11,3.13) and their hermitian conjugates, together with the properties of the vacuum η0 , the expression for the generator follows by identifying L in the equation < ∂t jt (X) >η(ξ) =< jt (L(X)) >η(ξ) . 0

0

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The result is L(X) = L1 (X) + L2 (X) + L3 (X), (g)

(g)

L1 (X) = Γ− [a† , X]a − Γ− a† [a, X], (h)

(h)

L2 (X) = Γ− [σ+ , X]σ− − Γ− σ+ [σ− , X], L3 (X) = i2 [σ− a† − σ+ a, X]

(3.14)

It is evident that both L1 and L3 can be rewritten in the same form of the radiation and interaction terms of the AS generator but this is not so, in general, for L2 which has the form of the AS radiation generator only if the pumping parameter η is equal to −1. This is not very satisfactory and, how we will show in the following, is a consequence of having introduced a single reservoir for the atom. We will show that the existence of a second reservoir allows for the removal of the constraint η = −1 we have obtained in the simplified model above. With all of this in mind it is not difficult to produce an hamiltonian which should produce the full AS generator for the physical system with 2N + 1 atoms and n modes of radiation. With respect to the one discussed above, it is enough to double the number of reservoirs for the matter and to sum over l ∈ IN for the matter and over j = 0, 1, . . . , n − 1 for the radiation. The resulting hamiltonian is therefore necessarily very close to the HL one: H = H0 + λHI ,

(3.15)

with H0 = ω R

n−1


a†j aj

j=0

+

+µ

n−1





σl,z +

l∈IN

2 



dk ωms (k)m†s,l (k)ms,l (k)

l∈IN s=1

dk ωr,j (k)rj (k)† rj (k)

(3.16)

j=0

and HI =

n−1


(rj† (g j )aj + rj (gj )a†j ) +

j=0

+(m†2,l (h2 )σl,+

+ h.c.)] + λ







[(m†1,l (h1 )σl,− + h.c.)

l∈IN (N )

(φl

σl,+ + h.c.),

(3.17)

l∈IN

where the radiation field has been introduced in (2.10). It is clear that, but for the RWA which we are assuming here, there are not many other differences between this hamiltonian and the one in (2.11)-(2.15). It is worth mentioning that λ appears both as an overall coupling constant, see (3.15), and as a multiplying factor of

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√ + h.c.) and plays the same role as α in the HL hamiltonian. As for the commutation rules they are quite natural: but for the spin operators, which satisfy their own algebra, all the others operators satisfy the CCR and commute whenever they refer to different subsystems. In particular, for instance, all the m1,l (k) commute with all the m2,l
(k
), for all k, k
and l, l
. The procedure to obtain the generator is the same as before: we first compute HI (t) = eiH0 t HI e−iH0 t , which enters in the differential equation for the wave operator. Taking the limit λ → 0 of the mean value in the vector state defined by (ξ) ϕ0 = ϕ0 ⊗ ξ of the first non trivial approximation of the rescaled version of Ut (sl) we deduce the form of an effective hamiltonian, HI (t), which is simply (N ) σl,+ l∈IN (φl

(sl)

HI

(t) =

n−1





j=0

l∈IN

† (aj rg,j (t) + h.c.) +

+




(σl,+ m1,l (t) + h.c.)

(σl,− m2,l (t) + h.c.) +

l∈IN




(N )

(φl

σl,+ + h.c.).

(3.18)

l∈IN

Again, we are assuming that ωR = 2µ, which is crucial in order not to have a time (sl) dependence in the last term of HI (t) in (3.18). The only non trivial commutation rules for t > t
for the new operators are: (g)

†

[rg,j (t), rg,j
(t )] = Γ−,j δj,j
δ(t − t ), (h )

[m1,l (t), m†1,l
(t
)] = Γ− 1 δl,l
δ(t − t
), [m2,l (t), m†2,l
(t
)]

=

(h ) Γ− 2 δl,l
δ(t

(3.19)

− t
),

where we have defined the following complex quantities: 

(g)

Γ−,j = (h ) Γ− 1 (h )



0

dτ −∞  0

=

Γ− 2 =

 dτ

−∞  0

 dτ

−∞

dk|gj (k)|2 ei(ωr,j (k)−ωR )τ , dk|h1 (k)|2 ei(ωm1 (k)−ωR )τ , dk|h2 (k)|2 ei(ωm2 (k)+ωR )τ .

(3.20)

The above commutators, given for t > t
, are sufficient to compute the commutation relations between the fields of the reservoir and the wave operator  (sl) Ut = 1 − i HI (t
)Ut
dt
, as it is obtained after the SL. We get (g)

[rg,j (t), Ut ] = −iΓ−,j aj Ut , (h )

[m1,l (t), Ut ] = −iΓ− 1 σl,− Ut , (h )

[m2,l (t), Ut ] = −iΓ− 2 σl,+ Ut .

(3.21)

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The expression for the generator can be obtained as for the N = 0, n = 1 model described before, that is computing the mean value < ∂t jt (X) >η(ξ) . Here, as 0

before, jt (X) is the flux of the system observable X, jt (X) = Ut† XUt , and η0 is the vacuum of the operators rg,j (t) and ms,l (t), s = 1, 2. The computation gives the following result, which slightly generalize the one in (3.14): L(X) = L1 (X) + L2 (X) + L3 (X), n−1
(g) † (g) L1 (X) = (Γ−,j [aj , X]aj − Γ−,j a†j [aj , X]), j=0

L2 (X) =




(h )

(h1 )

(h )

(h2 )

(Γ− 1 [σ+,l , X]σ−,l − Γ− σ+,l [σ−,l , X]

l∈IN

+Γ− 2 [σ−,l , X]σ+,l − Γ− σ−,l [σ+,l , X],
(N ) [(φl σ+,l + h.c.), X]. L3 (X) = i

(3.22)

l∈IN

It is not difficult to compare this generator with the one proposed by AS, see formulas ((2.3),(2.10)), and the conclusion is that the two generators are exactly the same provided that the following equalities are satisfied: (g)

(g)

(h )

(h )

(h )

(h )

Γ−,j = ωj , Γ−,j = kj , (Γ− 1 + Γ− 2 ) = γ1 , (Γ− 1 − Γ− 2 ) =  1 1 (h ) (h ) Γ− 1 = γ2 (1 − η), Γ− 2 = γ2 (1 + η). (3.23) 4 4 Here the lhs all contain variables of the hamiltonian model while the rhs are related to the AS generator. Due to the fact that the hamiltonian in (3.15)-(3.17) essentially coincides with the one in (2.11)-(2.15) with β = 0, that is in the RWA, we can conclude that if we start with the HL hamiltonian, choosing the regularizing functions in such a way that the equalities (3.23) are satisfied, the SL produces a generator of the model which is exactly the one proposed in [1, 2], with the only minor constraint γ2 = 2γ1 , which is a direct consequence of (3.23). From a physical point of view the implications of this result are quite interesting: the original model, [5], was not (easily) solvable and for this reason a certain number of approximations were introduced. Among these, the crucial ones are the replacement of the original reservoir with what HL call a singular reservoir which, moreover, is made of fermions. Under these assumptions the model can be discussed in some details, and this was done in [8], where the thermodynamic limit for the intensive and the fluctuation observables was discussed. What we have shown here is that all these approximations can be avoided using another kind of perturbative approach, that is the one provided by the SL. The resulting model is exactly the one proposed and studied in [1, 2]. The role of the singular reservoir, or the need for a fermionic reservoir, is therefore not crucial and can be avoided. However, we will consider the DHL model in the next section in order to complete our analysis.

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We conclude this section with a remark concerning the role of the RWA and its relation with the SL. In particular, this is a very good approximation after the SL is taken. To show why, we first notice that adding a counter-rotating term (extending the one in (2.18)) to the interaction hamiltonian HI in (3.17), considering the same coupling constant for both the rotating and the counterrotating term (β = α), simply means to add to HI in (3.17) a contribution like  (N ) λ l∈IN (φl σl,− + h.c.). While the rotating term, if 2µ = ωR , does not evolve freely, the free time evolution of this other term is not trivial. However, the differences with respect to the previous situation all disappear rigorously after the SL, because these extra contributions to the mean value of the wave operator go to zero with λ, so that at the end the expression for the generator is unchanged. This allows us to conclude that the full HL hamiltonian is equivalent to the AS generator, where the equivalence relation is provided by the SL. We want to end this section with a final remark concerning the different number of phase transitions in the two different situations, 1 for the HL and 2 for the AS model. In view of the above conclusion, we can guess that the SL procedure produces some loss of information and, as a consequence, a difference between the original and the approximated system. This is not so surprising since, thought being a powerful tool, nevertheless the SL is nothing but a perturbative method!

IV The SL of the DHL model In this section we consider the SL of the DHL model as introduced in Section 2. In particular we find the expression of the generator and we show that, under some conditions on the quantities defining the model, the equations of motion do not differ from the ones in AS. The free evolved interaction hamiltonian HI in (2.24) is, HI (t) = eiH0 t HI e−iH0 t =
+λ

n−1


(aj rj† (g j ei(ωr,j −ωR )t ) + h.c.)

j=0 (N ) (φl b†+,l b−,l l∈IN

+C+,l (gC+ e

it(µ− )

)) +

+ h.c.) +




[b†+,l (B+,l (gB+ eit(µ− ) )

l∈IN

† (B+,l (gB+ e−it(µ− ) )

† + C+,l (gC+ e−it(µ− ) ))b+,l

+b†−,l (B−,l (gB− e−it(µ+ ) ) + C−,l (gC− e−it(µ+ ) )) † † +(B−,l (gB− eit(µ+ ) ) + C−,l (gC− eit(µ+ ) ))b−,l ].

(4.1)

Following the usual strategy discussed in the Appendix and in [6], we conclude t that (the rescaled version of) the wave operator Uλ (t) = 1 − iλ 0 HI (t
)Uλ (t
)dt
converges for λ → 0 to another operator, which we still call the wave operator,

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997

satisfying the equation  t (ls) (ls) Ut = 1 − i HI (t
)Ut
dt
, or, equivalently ∂t Ut = −iHI (t)Ut , with U0 = 1. 0

Here

(ls) HI (t) (ls)

HI

(4.2) is an effective time dependent hamiltonian defined as (t)

=

n−1


† (aj rg,j (t) + h.c.) +

j=0

+





l∈IN

(N ) † b+,l b−,l

(φl

+ h.c.)

† † [b†+,l (β+,l (t) + γ+,l (t)) + (β+,l (t) + γ+,l (t))b+,l

l∈IN † † +b†−,l (β−,l (t) + γ−,l (t)) + (β−,l (t) + γ−,l (t))b−,l ].

(4.3)

(ls)

The operators of the reservoir which appear in HI are the stochastic limit of the original (rescaled) time evoluted operators of the reservoir and satisfy (anti)commutation relations which are related to those of the original ones. In particular, after the SL, any two operators of the matter (system and reservoirs) localized at different lattice sites commutes, as well as any operator of the radiation with any observable of the matter. As for operators localized at the same lattice site, the only non trivial anticommutators are (B±)

† (t
)} = δ(t − t
)Γ− {β±,l (t), β±,l

(C±)

† {γ±,l (t), γ±,l (t
)} = δ(t − t
)Γ−

,

,

(4.4)

which should be added to (g)

[rg,j (t), rg,j
(t
)] = δj,j
δ(t − t
)Γ−,j .

(4.5)

In all these formulas the time ordering t > t
has to be understood and the following quantities are defined:  0  (g) dτ dk |gj (k)|2 ei(ωr,j (k)−ωR )τ , Γ−,j = (B±) Γ− (C±) Γ−

−∞  0

=

 dτ



−∞ 0

=

 dτ

−∞

dk (gB± (k))2 ei( (k)∓µ)τ , dk (gC± (k))2 e−i( (k)∓µ)τ .

(4.6)

We call now η0 the vacuum of these limiting operators. We have † rg,j (t)η0 = β±,l (t)η0 = γ±,l (t)η0 = 0.

(4.7)

Paying a little attention to the fact that here commutators and anti-commutators simultaneously appear, we can compute the commutators between the operators

998

F. Bagarello

Ann. Henri Poincar´e

  rg,j (t) , γ±,l (t), β±,l (t) and the wave operator Ut by making use of ((4.4),(4.5)). We give here only those commutation rules which are used in the computation of the generator: (g)

[rg,j (t), Ut ] = −iΓ−,j aj Ut , (B±)

[β±,l (t), Ut ] = −iΓ−

b±,l Ut ,

(C±) † b±,l Ut .

† (t), Ut ] = iΓ− [γ±,l

(4.8)

We are now ready to compute the generator following exactly the same strategy as in the previous section and in Appendix. However, we must remark that, due to the presence of fermionic operators, to make the computation simpler, we will focus on system operators X which are quadratic (or quartic, . . . ) in the matter † b†−,l , operators localized in a given lattice site (e.g. X = b+,l b†−,l , X = b†+,l β+,l γ−,l . . . ) in such a way to ensure the commutativity between X and any of the matter (ls) operators entering in the operator HI (t), (4.3). We get for the full generator the following sum of three different contributions: L(X) = L1 (X) + L2 (X) + L3 (X), L1 (X) =

n−1


(g)

(g)

(Γ−,j [a†j , X]aj − Γ−,j a†j [aj , X]),

j=0

L2 (X) =




(B+)

(Γ−

[b†+,l , X]b+,l

l∈IN (B+) −Γ− b†+,l [b+,l , X] + (B−)

+Γ−

(C+)

Γ−

(C+)

[b+,l , X]b†+,l − Γ−

(B−) † b−,l [b−,l , X]

[b†−,l , X]b−,l − Γ−

b+,l [b†+,l , X]

(C−)

+ Γ−

[b−,l , X]b†−,l

(C−)

b−,l [b†−,l , X]),
(N ) † L3 (X) = i [(φl b+,l b−,l + h.c.), X].

−Γ−

(4.9)

l∈IN

We see that the first and the last terms exactly coincide with the analogous contributions of the AS generator, but for a purely formal difference which is due to the different matter variables which are used in the two models. The second contribution, on the other hand, cannot be easily compared with the free AS matter generator. What is convenient, and sufficient, to get full insight about L2 , is to compute its action on a basis of the local algebra, that is on b†+,l b−,l (≡ σ+,l ) and

on b†+,l b+,l − b†−,l b−,l (≡ σz,l ), all the others being trivial or an easy consequence of these ones. It is not hard to find the result: (B+)

L2 (b†+,l b−,l ) = −b†+,l b−,l ( [Γ− (B+) −i[Γ−

−

(B−) Γ−

−

(C+) Γ−

+

(B−)

+ Γ−

(C−) Γ− ]),

(C+)

+ Γ−

(C−)

+ Γ−

]−

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Relations between the Hepp-Lieb and the Alli-Sewell Laser Models (B+)

L2 (b†+,l b+,l − b†−,l b−,l ) = 2(−b†+,l b+,l ( [Γ− (B−) +b†−,l b−,l ( [Γ−

+

(C−) Γ− )

−

(C+)

+ Γ−

(C−) Γ− ).

(C+)

) + Γ−

999

+ (4.10)

The equation for σ+,l is recovered without any problem, modulo some identification (B+) (B−) (C+) (C−) ( [Γ− + Γ− + Γ− + Γ− ] = γ1 , . . . ), while to recover the equation for σz,l it is necessary to choose properly the regularizing functions which define the different Γ− . In particular we need to have the following equality fulfilled: (B+)

(Γ−

(C+)

+ Γ−

(B−)

) = (Γ−

(C−)

+ Γ−

).

(4.11)

Under this condition we can conclude that the SL of the DHL model produces the same differential equations as the AS generator, as for the HL model. It is also easy to check that, as a consequence of our approach, we must have γ1 = γ2 in the generator we obtain. Of course this result is not surprising since already in the HL paper, [5], the fact that the two models are quite close (under some aspects) was pointed out. Here we have learned also that the SL of both these models, at least under some conditions, give rise to the same dynamical behaviour.

V

Outcome and Future Projects

We have proved that the relation between the HL and the AS model, whose existence is claimed in [1], is provided by the SL. This result is quite interesting since it shows that the approximations introduced by HL in their paper [5], in particular the use of the fermionic reservoir for the matter which produces the DHL model, together with the so called singular reservoir approximation, can be avoided by using the original HL model with no approximation, taking its SL and finally using the results in [1, 2] to analyze, e.g., the thermodynamical limit of the model. It is interesting to remark that while in the AS model two phase transitions occur, in the HL model we only have one. This could be a consequence of the SL procedure, which is nothing but a perturbative approach simplifying the study of the quantum dynamics, so that some of the original features of the model can be lost after the approximation. We want to conclude this paper by remarking that this is not the first time the HL model is associated to a dissipative system, as in the AS formulation. A similar strategy was discussed by Gorini and Kossakowski already in 1976, [9]. It would be interesting to study their generator again in the connection with the SL to see if any relation between their generator and the HL original hamiltonian appears.

Acknowledgments I am indebted with Prof. Lu for a suggestion which is at the basis of this paper. I also would like to aknowledge financial support by the Murst, within the

1000

F. Bagarello

Ann. Henri Poincar´e

project Problemi Matematici Non Lineari di Propagazione e Stabilit` a nei Modelli del Continuo, coordinated by Prof. T. Ruggeri.

A

Appendix: Few results on the stochastic limit

In this Appendix we will briefly summarize some of the basic facts and properties concerning the SL which are used all throughout the paper. We refer to [6] and references therein for more details. Given an open system S+R we write its hamiltonian H as the sum of two contributions, the free part H0 and the interaction λHI . Here λ is a (sort of) coupling constant, H0 contains the free evolution of both the system and the reservoir, while HI contains the interaction between the system and the reservoir and, for composite systems, amoung the different buildind blocks of the whole physical system. Working in the interaction picture, we define HI (t) = eiH0 t HI e−iH0 t and the so called wave operator Uλ (t) which satisfies the following differential equation ∂t Uλ (t) = −iλHI (t)Uλ (t),

(A.1)

with the initial condition Uλ (0) = 1. Using the van-Hove rescaling t → λt2 , see [7, 6] for instance, we can rewrite the same equation in a form which is more convenient for our perturbative approach, that is ∂t Uλ (

t i t t ) = − HI ( 2 )Uλ ( 2 ), λ2 λ λ λ

with the same initial condition as before. Its integral counterpart is  t i t t
t
HI ( 2 )Uλ ( 2 )dt
, Uλ ( 2 ) = 1 − λ λ 0 λ λ

(A.2)

(A.3)

which is the starting point for a perturbative expansion, which works in the following way: let ϕ0 be the ground state of the reservoir and ξ a generic vector of (ξ) the system. Then we put ϕ0 = ϕ0 ⊗ ξ. We want to compute the limit, for λ going to 0, of the first non trivial order of the mean value of the perturbative expansion (ξ) of Uλ (t/λ2 ) above in ϕ0 , that is the limit of  t  t1 i t1 t2 Iλ (t) = (− )2 dt1 dt2 < HI ( 2 )HI ( 2 ) >ϕ(ξ) , (A.4) 0 λ λ λ 0 0 for λ → 0. Under some regularity conditions on the functions which are used to smear out the (typically) bosonic fields of the reservoir, this limit is shown to exist for many relevant physical models, see [6] and [10] for a recent application to many body theory. At this stage all the complex quantities like the various Γ− we have introduced in the main body of this paper appear. We call I(t) the limit limλ→0 Iλ (t). In the same sense of the convergence of the (rescaled) wave operator Uλ ( λt2 ) (the convergence in the sense of correlators), it is possible to check that also

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Relations between the Hepp-Lieb and the Alli-Sewell Laser Models

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the (rescaled) reservoir operators converge and define new operators which do not satisfy canonical commutation relations but a modified version of these. Moreover, these limiting operators depend explicitly on time and they live in a Hilbert space which is different from the original one. In particular, they annihilate a vacuum vector, η0 , which is no longer the original one, ϕ0 . It is not difficult to write down, as we have done several times in this paper, (ls) the form of a time dependent self-adjoint operator HI (t), which depends on the system operators and on the limiting operators of the reservoir, such that the first  t (ls) non trivial order of the mean value of the expansion of Ut = 1 − i 0 HI (t
)Ut
dt
(ξ) on the state η0 = η0 ⊗ ξ coincides with I(t). The operator Ut defined by this integral equation is called again the wave operator. The form of the generator follows now from an operation of normal ordering. ˜ = More in details, we start defining the flux of an observable of the system X ˜ t . Then, using ˜ = Ut† XU X ⊗1 1res , where 1res is the identity of the reservoir, as jt (X) ˜ = iU † [H (ls) (t), X]U ˜ t. the equation of motion for Ut and Ut† , we find that ∂t jt (X) t I (ξ) In order to compute the mean value of this equation on the state η0 , so to get rid of the reservoir operators, it is convenient to compute first the commutation relations between Ut and the limiting operators of the reservoir. At this stage the so called time consecutive principle is used in a very heavy way to simplify the computation. This principle, which has been checked for many classes of physical models, and certainly holds in our case where all the interactions are dipolar, states that, if β(t) is any of these limiting operators of the reservoir, then [β(t), Ut
] = 0, for all t > t
.

(A.5)

Using this principle and recalling that η0 is annihiled by the limiting annihilation operators of the reservoir, it is now a technical exercise to compute < ∂t jt (X) >η(ξ) 0 and, by means of the equation < ∂t jt (X) >η(ξ) =< jt (L(X)) >η(ξ) , to identify the 0 0 form of the generator of the physical system.

References [1] G. Alli and G. L. Sewell, New methods and structures in the theory of the multi-mode Dicke laser model, J. Math. Phys. 36, 5598 (1995). [2] F. Bagarello and G. L. Sewell, New Structures in the Theory of the Laser Model II: Microscopic Dynamics and a Non-Equilibrim Entropy Principle, J. Math. Phys. 39, 2730–2747 (1998). [3] R. H. Dicke, Coherence in Spontaneous Radiation Processes, Phys. Rev. 93, 99–110 (1954.) [4] R. Graham and H. Haken, Laserlight- First example of a second order phase transition far away from thermal equilibrium, Z. Phys. 237, 31 (1970); and

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F. Bagarello

Ann. Henri Poincar´e

H. Haken, Handbuch der Physik, Bd. XXV/2C, Springer, Heidelberg, Berlin, New York, 1970. [5] K. Hepp and E. H. Lieb, Phase transitions in reservoir-driven open systes with applications to lasers and superconductors, Helv. Phys. Acta 46, 573 (1973). [6] L. Accardi, Y. G. Lu and I. Volovich, Quantum Theory and its Stochastic Limit, Springer (2001). [7] P. A. Martin, Mod`eles en M´ecanique Statistique des Processus Irr´eversibles, Lecture Notes in Physics 103, Springer-Verlag, Berlin. [8] K. Hepp and E. H. Lieb, in Constructive Quantum Field Theory, G. Velo and A. S. Wightman Eds., Lect. Notes in Phys. 25, Springer (1973). [9] V. Gorini and A. Kossakowski, N-level system in contact with a singular reservoir, J. Math. Phys., 17, 1298–1305 (1976). [10] L. Accardi and F. Bagarello, The stochastic limit of the Fr¨ ohlich Hamiltonian: relations with the quantum Hall effect, submitted to Int. Jour. Phys., Preprint N. 443 del Centro Vito Volterra. Fabio Bagarello Dipartimento di Matematica ed Applicazioni Fac. Ingegneria, Universit` a di Palermo Viale delle Scienze I-90128 Palermo Italy email: [email protected] Communicated by Gian Michele Graf submitted 5/02/02, accepted 16/04/02

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´e 3 (2002) 1003 – 1018 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/051003-16

Annales Henri Poincar´ e

Pure Point Dynamical and Diffraction Spectra J.-Y. Lee, R. V. Moody∗ and B. Solomyak†

Abstract. We show that for multi-colored Delone point sets with finite local complexity and uniform cluster frequencies the notions of pure point diffraction and pure point dynamical spectrum are equivalent.

1 Introduction The notion of pure pointedness appears in the theory of aperiodic systems in two different forms: pure point dynamical spectrum and pure point diffraction spectrum. The objective of this paper is to show that these two widely used notions are equivalent under a type of statistical condition known as the existence of uniform cluster frequencies. The basic objects of study here are Delone point sets of Rd . The points of these sets are permitted to be multi-colored, the colors coming from a finite set of colors. We also assume that our point sets Λ have the property of finite local complexity (FLC), which informally means that there are only finitely many translational classes of clusters of Λ with any given size. Under these circumstances, the orbit of Λ under translation gives rise, via completion in the standard Radin-Wolff type topology, to a compact space XΛ . With the obvious action of Rd , we obtain a dynamical system (XΛ , Rd ). The dynamical spectrum refers to the spectrum of this dynamical system, that is to say, the spectrum of the unitary operators Ux arising from the translational action on the space of L2 -functions on XΛ . On the other hand, the diffraction spectrum (which is the idealized mathematical interpretation of the diffraction pattern of a physical experiment) is obtained by first assigning weights to the various colors of the multiset and then determining the autocorrelation, if it exists, of this weighted multiset. The Fourier transform of the autocorrelation is the diffraction measure whose pure pointedness is the question. There is a well known argument of S. Dworkin ([4], [6]) that shows how to deduce pure pointedness of the diffractive spectrum from pure pointedness of the dynamical system. Our main result (Theorem 3.2) shows that, under the additional assumption that Λ has uniform cluster frequencies (or equivalently, that ∗ RVM acknowledges on-going support from the Natural Sciences and Engineering Research Council of Canada. † BS acknowledges support from NSF grants DMS 9800786 and DMS 0099814.

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the dynamical system XΛ is uniquely ergodic), the process can be reversed, so in fact the two notions of pure pointedness are equivalent. The present understanding of diffractive point sets is very limited. One of the important consequences of this result is that it allows the introduction of powerful spectral theorems in the study of such sets. Our forthcoming paper [8] on diffractive substitution systems makes extensive use of this connection. In the last section we discuss what can be salvaged when there are no uniform cluster frequencies. Then the equivalence of pure point dynamical and pure point diffraction spectra still holds – perhaps, not for the original Delone set Λ, but for almost every Delone set in XΛ , with respect to an ergodic invariant measure. The prototype of the dynamical system (XΛ , Rd ) is a symbolic dynamical system, that is, the Z-action by shifts on a space of bi-infinite sequences. In the symbolic setting, the equivalence of pure point dynamical and diffraction spectra has been established by Queffelec [11, Prop. IV.21], and our proof is largely a generalization of her argument. When the dynamical spectrum is not pure point, its relation to the diffraction spectrum is not completely understood. It follows from [4] that the latter is essentially a “part” of the former. So, for instance, if the dynamical spectrum is pure singular/absolutely continuous, then the diffraction spectrum is pure singular/absolutely continuous (apart from the trivial constant eigenfunction which corresponds to a delta function at 0). However, the other direction is more delicate: Van Enter and Mi¸ekisz [5] have pointed out that, in the case of mixed spectrum, the non-trivial pure point component may be “lost” when passing from dynamical spectrum to diffraction spectrum. The presentation below contains a number of results that are essentially wellknown, though not always quite in the form needed here. For the convenience of the reader we have attempted to make the paper largely self-contained.

2 Multisets, dynamical systems, and uniform cluster frequencies A multiset or m-multiset in Rd is a subset Λ = Λ1 × · · · × Λm ⊂ Rd × · · · × Rd (m copies) where Λi ⊂ Rd . We also write Λ = (Λ1 , . . . , Λm ) = (Λi )i≤m . We say that Λ = (Λi )i≤m is a Delone multiset in Rd if each Λi is Delone and supp(Λ) :=
m d i=1 Λi ⊂ R is Delone. Although Λ is a product of sets, it is convenient to think of it as a set with types or colors, i being the color of points in Λi . A cluster of Λ is, by definition, a m-multiset P = (Pi )i≤m where Pi ⊂ Λi is finite for all i ≤ m. The cluster P is non-empty if supp(P) is non-empty. Many of the clusters that we consider have the form A ∩ Λ := (A ∩ Λi )i≤m , for a bounded set A ⊂ Rd . There is a natural translation Rd -action on the set of Delone multisets and their clusters in Rd . The translate of a cluster P by x ∈ Rd is x + P = (x + Pi )i≤m . We say that two clusters P and P are translationally equivalent if P = x + P for some x ∈ Rd .

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We write BR (y) for the closed ball of radius R centered at y and use also BR for BR (0). Definition 2.1 The Delone multiset Λ has finite
local complexity (FLC) if for every m R > 0 there exists a finite set Y ⊂ supp(Λ) = i=1 Λi such that ∀x ∈ supp(Λ), ∃ y ∈ Y : BR (x) ∩ Λ = (BR (y) ∩ Λ) + (x − y). In plain language, for each radius R > 0 there are only finitely many translational classes of clusters whose support lies in some ball of radius R. In this paper we will usually assume that our Delone multisets have FLC. Let Λ be a Delone multiset and X be the collection of all Delone multisets each of whose clusters is a translate of a Λ-cluster. We introduce a metric on Delone multisets in a simple variation of the standard way: for Delone multisets Λ1 , Λ2 ∈ X, ˜ 1 , Λ2 ), 2−1/2 } , d(Λ1 , Λ2 ) := min{d(Λ

(2.1)

where ˜ 1 , Λ2 ) = d(Λ

inf{ε > 0 : ∃ x, y ∈ Bε (0), B1/ε (0) ∩ (−x + Λ1 ) = B1/ε (0) ∩ (−y + Λ2 )} .

Let us indicate why this is a metric. Clearly, the only issue is the triangle inequality. Suppose that d(Λ1 , Λ2 ) ≤ ε1 , d(Λ2 , Λ3 ) ≤ ε2 ; we want to show that d(Λ1 , Λ3 ) ≤ ε1 + ε2 . We can assume that ε1 , ε2 < 2−1/2 , otherwise the claim is obvious. Then (−x1 + Λ1 ) ∩ B1/ε1 (0) = (−x2 + Λ2 ) ∩ B1/ε1 (0) for some x1 , x2 ∈ Bε1 (0), (−x2 + Λ2 ) ∩ B1/ε2 (0) = (−x3 + Λ3 ) ∩ B1/ε2 (0) for some x2 , x3 ∈ Bε2 (0). It follows that (−x1 − x2 + Λ1 ) ∩ B1/ε1 (−x2 ) = (−x2 − x2 + Λ2 ) ∩ B1/ε1 (−x2 ). Since B1/ε1 (−x2 ) ⊃ B(1/ε1 )−ε2 (0), this implies (−x1 − x2 + Λ1 ) ∩ B(1/ε1 )−ε2 (0) = (−x2 − x2 + Λ2 ) ∩ B(1/ε1 )−ε2 (0).

(2.2)

Similarly, (−x2 − x2 + Λ2 ) ∩ B(1/ε2 )−ε1 (0) = (−x2 − x3 + Λ3 ) ∩ B(1/ε2 )−ε1 (0). A simple computation shows that ε11 − ε2 ≥ ε1 , ε2 < 2−1/2 , so by (2.2) and (2.3),

1 ε1 +ε2

and

1 ε2

− ε1 ≥

1 ε1 +ε2

(−x1 − x2 + Λ1 ) ∩ B1/(ε1 +ε2 ) (0) = (−x2 − x3 + Λ3 ) ∩ B1/(ε1 +ε2 ) (0), hence d(Λ1 , Λ3 ) ≤ ε1 + ε2 .

(2.3) when

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We define XΛ := {−h + Λ : h ∈ Rd } with the metric d. In spite of the special role played by 0 in the definition of d, any other point of Rd may be used as a reference point, leading to an equivalent metric and more importantly the same topology on XΛ . The following lemma is standard. Lemma 2.2 ([12], [13]) If Λ has FLC, then the metric space XΛ is compact. The group Rd acts on XΛ by translations which are obviously homeomorphisms, and we get a topological dynamical system (XΛ , Rd ). Definition 2.3 Let P be a non-empty cluster of Λ or some translate of Λ, and let V ⊂ Rd be a Borel set. Define the cylinder set XP,V ⊂ XΛ by XP,V := {Λ ∈ XΛ : −g + P ⊂ Λ for some g ∈ V }. Let η(Λ) > 0 be chosen so that every ball of radius η(2Λ) contains at most one point of supp(Λ), and let b(Λ) > 0 be such that every ball of radius b(2Λ) contains at least a point in supp(Λ). These exist by the Delone set property. The following technical result will be quite useful. Lemma 2.4 Let Λ be a Delone multiset with FLC. For any R ≥ b(2Λ) and 0 < δ < η(Λ), there exist Delone multisets Γj ∈ XΛ and Borel sets Vj with diam(Vj ) < δ, Vol(∂Vj ) = 0, 1 ≤ j ≤ N , such that XΛ =

N 

XPj ,Vj

j=1

is a disjoint union, where Pj = BR (0) ∩ Γj . Proof. For any R ≥ b(2Λ) consider the clusters {BR (0) ∩ Γ : Γ ∈ XΛ }. They are non-empty, by the definition of b(Λ). By FLC, there are finitely many such clusters up to translations. This means that there exist Γ1 , . . . , ΓK ∈ XΛ such that for any Γ ∈ XΛ there are unique n = n(Γ) ≤ K and u = u(Γ) ∈ Rd satisfying BR (0) ∩ Γ = −u + (BR (0) ∩ Γn ). For j = 1, . . . , K let Wj = {u(Γ) : Γ ∈ XΛ such that n(Γ) = j}.
By construction, XΛ = K j=1 XPj ,Wj , and this is a disjoint union. Next we show that the sets Wj are sufficiently “nice,” so that they can be obtained from a finite number of closed balls using operations of complementation, intersection, and union.

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Let b = b(Λ) and fix j. Since every ball of radius b/2 contains a point in supp(Λ), we have that Wj ⊂ Bb (0). Indeed, shifting a cluster of points in BR (0) by more than b would move at least one point out of BR (0). Let Pj := BR (0) ∩ Γj . The set Wj consists of vectors u such that −u + Pj is a BR (0)-cluster for some Delone multiset in XΛ . Thus u ∈ Wj if and only if the following two conditions are met. The first condition is that for each x ∈ supp(Pj ), we have −u + x ∈ BR (0). The second condition is that no points of Γj outside of BR (0) move inside after the translation by −u. Since Wj ⊂ Bb (0), only the points in BR+b (0) have a chance of moving into BR (0). Thus we need to consider the BR+b (0) extensions of Pj . By FLC, in the space XΛ there are finitely many BR+b (0)-clusters that extend the cluster Pj . Denote these clusters by Q1 , . . . , QL . Summarizing this discussion we obtain       Wj = (−BR (0) + x) ∩ (−(Rd \ BR (0)) + x) . x∈supp(Pj )

i≤L

x∈supp(Qi )\BR (0)

This implies that Wj is a Borel set, with Vol(∂Wj ) =
0. nj It remains to partition each Wj such that Wj = k=1 Vjk , where diam(Vjk ) ≤ δ, 0 < δ < η(Λ). To this end, consider, for example, a decomposition of the cube [−b, b]d into a disjoint union of (half-open and closed) grid boxes of diameter less than δ < η(Λ). Let Q denote the (finite) collection of all these grid boxes. Then Wj =



(Wj ∩ D) =

D∈Q

nj 

Vjk ,

k=1

where Vjk ’s are disjoint and Vol(∂Vjk ) = 0. Note that the union XPj ,Wj =
nj k=1 XPj ,Vjk is disjoint, from the definition of Wj and diam(Vjk ) < η(Λ) for
all k ≤ nj . So the lemma is proved. For a non-empty cluster P and a bounded set A ⊂ Rd denote LP (A) = {x ∈ Rd : x + P ⊂ A ∩ Λ}, where  means the cardinality. In plain language, LP (A) is the number of translates of P contained in A, which is clearly finite. For a bounded set F ⊂ Rd and r > 0, let F +r := {x ∈ Rd : dist(x, F ) ≤ r}, F −r := {x ∈ F : dist(x, ∂F ) ≥ r} ⊃ F \ (∂F )+r . A van Hove sequence for Rd is a sequence F = {Fn }n≥1 of bounded measurable subsets of Rd satisfying lim Vol((∂Fn )+r )/Vol(Fn ) = 0, for all r > 0.

n→∞

(2.4)

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Definition 2.5 Let {Fn }n≥1 be a van Hove sequence. The Delone multiset Λ has uniform cluster frequencies (UCF) (relative to {Fn }n≥1 ) if for any non-empty cluster P, the limit freq(P, Λ) = lim

n→∞

LP (x + Fn ) ≥0 Vol(Fn )

exists uniformly in x ∈ Rd . Recall that a topological dynamical system is uniquely ergodic if there is a unique invariant probability measure. Theorem 2.6 Let Λ be a Delone multiset with FLC and {Fn }n≥1 be a van Hove sequence. The system (XΛ , Rd ) is uniquely ergodic if and only if for all continuous functions f : XΛ → C (f ∈ C(XΛ )),  1 (In )(Γ, f ) := f (−g + Γ) dg → const, n → ∞, (2.5) Vol(Fn ) Fn uniformly in Γ ∈ XΛ , with the constant depending on f . This is a standard fact (both directions, see e.g. [16, Th. 6.19], [3, (5.15)], or [11, Th. IV.13] for the case of Z-actions); we include a (well-known) elementary proof of the needed direction for the reader’s convenience. Proof of sufficiency in Theorem 2.6. For any invariant measure µ, exchanging the order of integration yields   In (Γ, f ) dµ(Γ) = f dµ, XΛ

XΛ

so by the Dominated Convergence Theorem, the constant in (2.5) is X f dµ. If Λ there is another invariant measure ν, then X f dµ = X f dν for all f ∈ C(XΛ ), Λ Λ hence µ = ν.
Now we prove that FLC and UCF imply unique ergodicity of the system (XΛ , Rd ). This is also a standard fact, see e.g. [11, Cor. IV.14(a)] for the case of Z-actions. Theorem 2.7 Let Λ be a Delone multiset with FLC. Then the dynamical system (XΛ , Rd ) is uniquely ergodic if and only if Λ has UCF. Proof. Let XP,V be a cylinder set with diam(V ) ≤ η(Λ) and f be the characteristic function of XP,V . Then we have by the definition of the cylinder set:  f (−x − h + Λ) dx Jn (h, f ) := Fn

=

Vol{x ∈ Fn : −x − h + Λ ∈ XP,V }

=

Vol{x ∈ h + Fn : −y + P ⊂ −x + Λ for some y ∈ V } 

 ((h + Fn ) ∩ (xν + V )) Vol

=

ν

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where xν are all the vectors such that xν + P ⊂ Λ. It is clear that the distance between any two vectors xν is at least η(Λ), so the sets xν + V are disjoint. Let r = max{|y| : y ∈ V } + max{|x| : x ∈ supp(P)}. Then Vol(V )LP (h + Fn−r ) ≤ Jn (h, f ) ≤ Vol(V )LP (h + Fn+r ).

(2.6)

Note that LP (h + Fn+r ) − LP (h + Fn−r ) ≤ LP (h + ∂Fn +2r ) ≤

Vol(∂Fn +2r ) . Vol(B η(Λ) ) 2

So

lim

n→∞

Jn (h, f ) Vol(V ) · LP (h + Fn ) − Vol(Fn ) Vol(Fn )

= 0 uniformly in h ∈ Rd .

(2.7)

If (XΛ , Rd ) is uniquely ergodic, Jn (h, f  ) exists uniformly in h ∈ Rd n→∞ Vol(Fn ) for continuous functions f  approximating the characteristic function f of the cylinder set. Thus for any cluster P, lim

lim

n→∞

LP (h + Fn ) exists uniformly in h ∈ Rd , Vol(Fn )

i.e. Λ has UCF. On the other hand, we assume that Λ has UCF. By Lemma 2.4, f ∈ C(XΛ ) can be approximated in the supremum norm by linear combinations of characteristic functions of cylinder sets XP,V . Thus, it is enough to check (2.5) for f the characteristic function of XP,V with diam(V ) < η(Λ). We can see in the above (2.7) that (2.5) holds for all −h + Λ uniformly in h ∈ Rd under the assumption that Λ has UCF. Then we can approximate the orbit of Γ ∈ XΛ on Fn by −hn +Λ as closely as we want, since the orbit {−h + Λ : h ∈ Rd } is dense in XΛ by the definition of XΛ . So we compute all those integrals (2.5) of −hn + Λ over Fn and use the fact that independent of hn they are going to a constant. Since each of these is uniformly close to (In )(Γ, f ) in (2.5), we get that (In )(Γ, f ) too goes to a constant. Therefore (XΛ , Rd ) is uniquely ergodic.
Denote by µ the unique invariant probability measure on XΛ . As already mentioned, the constant in (2.5) must be X f dµ. Thus, the proof of unique Λ ergodicity yields the following result. Corollary 2.8 Let Λ be a Delone multiset with FLC and UCF. Then for any Λcluster P and any Borel set V with diam(V ) < η(Λ), we have µ(XP,V ) = Vol(V ) · freq(P, Λ).

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3 Pure-pointedness and Diffraction 3.1

Dynamical spectrum and diffraction spectrum

Suppose that Λ = (Λi )i≤m is a Delone multiset with FLC and UCF. There are two notions of pure pointedness that appear in this context. Although they are defined very differently, they are in fact equivalent. Given a translation-bounded measure ν on Rd , let γ(ν) denote its autocorrelation (assuming it is unique), that is, the vague limit 1 γ(ν) = lim (3.1) (ν|Fn ∗ ν|Fn ) , n→∞ Vol(Fn ) where {Fn }n≥1 is a van Hove sequence 1 . In particular, for the Delone multiset Λ we see that the autocorrelation is unique for any measure of the form   ai δΛi , where δΛi = δx and ai ∈ C . (3.2) ν= i≤m

x∈Λi

Indeed, a simple computation shows γ(ν) =

m  i,j=1

ai aj



freq((y, z), Λ)δy−z .

(3.3)

y∈Λi ,z∈Λj

Here (y, z) stands for a cluster consisting of two points y ∈ Λi , z ∈ Λj . The measure  is γ(ν) is positive definite, so by Bochner’s Theorem the Fourier transform γ(ν) a positive measure on Rd , called the diffraction measure for ν. We say that the  is a pure point or discrete measure ν has pure point diffraction spectrum if γ(ν) 2 measure . On the other hand, we also have the measure-preserving system (XΛ , µ, Rd ) associated with Λ. Consider the associated group of unitary operators {Ux }x∈Rd on L2 (XΛ , µ): Ux f (Λ ) = f (−x + Λ ). Every f ∈ L2 (XΛ , µ) defines a function on Rd by x → (Ux f, f ). This function is positive definite on Rd , so its Fourier transform is a positive measure σf on Rd called the spectral measure corresponding to f . We say that the Delone multiset Λ has pure point dynamical spectrum if σf is pure point for every f ∈ L2 (XΛ , µ). We recall that f ∈ L2 (XΛ , µ) is an eigenfunction for the Rd -action if for some α = (α1 , . . . , αd ) ∈ Rd , Ux f = e2πix·α f,

for all x ∈ Rd ,

where · is the standard inner product on Rd . 1 Recall



that if f is a function in d , then f˜ is defined by f˜(x) = f (−x). If µ is a measure, µ ˜ is defined by µ ˜(f ) = µ(f˜) for all f ∈ C0 ( d ). In particular for ν in (3.2), ν˜ = i≤m ai δ−Λi . 2 We


also say that Λi (resp Λ) has pure point diffraction spectrum if γ(δ Λi ) (resp each


γ(δ Λi ), i = 1, . . . , m) is a pure point measure.

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Theorem 3.1 σf is pure point for every f ∈ L2 (XΛ , µ) if and only if the eigenfunctions for the Rd -action span a dense subspace of L2 (XΛ , µ). This is a straightforward consequence of the Spectral Theorem, see e.g. Theorem 7.27 and §7.6 in [17] for the case d = 1. The Spectral Theorem for unitary representations of arbitrary locally compact Abelian groups, including Rd , is discussed in [10, §6].

3.2

An equivalence theorem for pure pointedness

In this section we prove the following theorem. Theorem 3.2 Suppose that a Delone multiset Λ has FLC and UCF. Then the following are equivalent: (i) Λ has pure point dynamical spectrum;  (ii) The measure ν = i≤m ai δΛi has pure point diffraction spectrum, for any choice of complex numbers (ai )i≤m ; (iii) The measures δΛi have pure point diffraction spectrum, for i ≤ m. The theorem is proved after  a sequence of auxiliary lemmas. Fix complex numbers (ai )i≤m and let ν = i≤m ai δΛi . For Λ = (Λi )i≤m ∈ XΛ let  νΛ
= ai δΛ
i , i≤m

so that ν = νΛ . To relate the autocorrelation of ν to spectral measures we need to do some “smoothing.” Let ω ∈ C0 (Rd ) (that is, ω is continuous and has compact support). Denote ρω,Λ
:= ω ∗ νΛ
and let

fω (Λ ) := ρω,Λ
(0) for Λ ∈ XΛ .

Lemma 3.3 fω ∈ C(XΛ ). Proof. We have fω (Λ ) =

 ω(−x) dνΛ
(x) =

 i≤m



ai

ω(−x).

x∈−supp(ω)∩Λ
i

The continuity of fω follows from the continuity of ω and the definition of topology
on XΛ . Denote by γω,Λ the autocorrelation of ρω,Λ . Since under our assumptions there is a unique autocorrelation measure γ = γ(ν), see (3.1) and (3.2), we have  ) ∗ γ. γω,Λ = (ω ∗ ω

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Ann. Henri Poincar´e

Lemma 3.4 ([4], see also [6]) σfω = γ ω,Λ . Proof. We provide a proof for completeness, following [6]. By definition, fω (−x + Λ) = ρω,Λ (x). Therefore, γω,Λ (x)

=

 1 ρω,Λ (x + y)ρω,Λ (y) dy n→∞ Vol(Fn ) F  n 1 fω (−x − y + Λ)fω (−y + Λ) dy lim n→∞ Vol(Fn ) F n  fω (−x + Λ )fω (Λ ) dµ(Λ )

=

(Ux fω , fω ) ,

= =

lim

XΛ

(3.4)

where {Fn }n≥1 is a van Hove sequence. Here the third equality is the main step; it follows from unique ergodicity and the continuity of fω , see Theorem 2.6. Thus,  γ
ω,Λ = (U(·) fω , fω ) = σfω ,


and the proof is finished.

The introduction of the function fω and the series of equations (3.4) is often called Dworkin’s argument. Fix ε with 0 < ε < b(1Λ) . Consider all the non-empty clusters of diameter ≤ 1/ε in Γ ∈ XΛ . There are finitely many such clusters up to translation, by FLC. Thus, there exists 0 < θ1 = θ1 (ε) < 1 such that if P, P are two such clusters, then ρH (P, P ) ≤ θ1 ⇒ P = −x + P Here

for some x ∈ Rd .

(3.5)

ρH (P, P ) = max{ρH (Pi , Pi ) : i ≤ m},

where ρH (Pi , Pi )

 =

max{dist(x, Pi ), dist(y, Pi ) : x ∈ Pi , y ∈ Pi }, 1, if Pi = ∅ and Pi = ∅ (or vice versa),

if Pi , Pi = ∅;

with P = (Pi )i≤m and P = (Pi )i≤m . Let

θ = θ(ε) := min{ε, θ1 , η(Λ)}

(3.6)

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and fi,ω (Λ ) = (ω ∗ δΛ
i )(0)

for Λ = (Λi )i≤m ∈ XΛ .

Denote by Ei the cluster consisting of a single point of type i at the origin; formally, Ei = (∅, . . . , ∅, {0} , ∅, . . . , ∅).  i

Let χEi ,V be the characteristic function for the cylinder set XEi ,V . Lemma 3.5 Let V ⊂ Rd be a bounded set with diam(V ) < θ, where θ is defined by (3.6), and 0 < ζ < θ/2. Let ω ∈ C0 (Rd ) be such that  x ∈ V −ζ ;  ω(x) = 1, ω(x) = 0, x ∈ Rd \ V ;  0 ≤ ω(x) ≤ 1, x ∈ V \ V −ζ . Then fi,ω − χEi ,V 22 ≤ freq(Ei , Λ) · Vol((∂V )+ζ ). Proof. We have by the definition of Ei and Definition 2.3:  1, if Λi ∩ (−V ) = ∅; χEi ,V (Λ ) = where Λ ∈ XΛ . 0, otherwise, On the other hand, since ω is supported in V and there is at most one point of Λi in V ,   ω(−x), if ∃ x ∈ Λi ∩ (−V ); fi,ω (Λ ) = ω(−x) dδΛ
i (x) = 0, otherwise. It follows that fi,ω (Λ ) − χEi ,V (Λ ) = 0 if

Λi ∩ (−V −ζ ) = ∅.

Thus, fi,ω − χEi ,V

22



|fi,ω (Λ ) − 1|2 dµ(Λ )

≤ XE

i

,V \V −ζ

≤ µ(XEi ,V \V −ζ ) = freq(Ei , Λ) · Vol(V \ V −ζ ) ≤ freq(Ei , Λ) · Vol((∂V )+ζ ), as desired.




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Ann. Henri Poincar´e

Lemma 3.6 Let P = (Pi )i≤m = B1/ε (0) ∩ Γ with Γ ∈ XΛ , and diam(V ) < θ, where θ is defined by (3.6). Then the characteristic function χP,V of XP,V can be expressed as   χP,V = χx+Ei ,V . i≤m x∈Pi

Proof. We just have to prove that  

XP,V =

Xx+Ei ,V .

i≤m x∈Pi

A Delone multiset Γ is in the left-hand side whenever −v + P ⊂ Γ for some v ∈ V . A Delone multiset Γ is in the right-hand side whenever for each i ≤ m and each x ∈ Pi there is a vector v(x) ∈ V such that −v(x) + x ⊂ Γ, where x = (∅, . . . , ∅, {x} , ∅, . . . , ∅) stands for a single element cluster. Thus, “⊂” is trivial.  i

The inclusion “⊃” follows from the fact that diam(V ) < θ, see (3.6) and (3.5).
Denote by Hpp the closed linear span in L2 (XΛ , µ) of the eigenfunctions for the dynamical system (XΛ , µ, Rd ). The following lemma is certainly standard, but since we do not know a ready reference, a short proof is provided. Lemma 3.7 If φ and ψ are both in L∞ (XΛ , µ) ∩ Hpp , then their product φψ is in L∞ (XΛ , µ) ∩ Hpp as well. Proof. Fix arbitrary  > 0. Since φ ∈ Hpp , we can find a finite linear combination  of eigenfunctions φ = ai fi such that  2< φ − φ

 . ψ∞

Since the dynamical system is ergodic, the eigenfunctions have constant modulus, hence φ ∈  L∞ . Thus, we can find another finite linear combination of eigenfunctions ψ = bj fj such that  2< ψ − ψ

 .  φ∞

Then  2 φψ − φψ

 − ψ)  2 + (φ − φ)ψ  ≤ φ(ψ 2    2 ≤ φ∞ ψ − ψ2 + ψ∞ φ − φ ≤ 2.

It remains to note that φψ ∈ Hpp since the product of eigenfunctions for a dynamical system is an eigenfunction. Since  is arbitrarily small, φψ ∈ Hpp , and the lemma is proved.


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Proof of Theorem 3.2. (i) ⇒ (ii) This is essentially proved by Dworkin in [4], see also [6] and [1]. By Lemma 3.4, pure point dynamical spectrum implies that γ
ω,Λ is pure point for any ω ∈ C0 (Rd ). Note that ω |2  γ. γω,Λ = |


(3.7)

Choosing a sequence ωn ∈ C0 (Rd ) converging to the delta measure δ0 in the vague topology, we can conclude that  γ is pure point as well, as desired. (This approximation step requires some care; it is explained in detail in [1].) (ii) ⇒ (iii) obvious. (iii) ⇒ (i) This is relatively new, although it is largely a generalization of Queffelec [11, Prop. IV.21].  We are given that δΛi has pure point diffraction spectrum, that is, γi := γ(δ Λi ) is pure point, for all i ≤ m. In view of (3.7) and Lemma 3.4, we obtain that σfi,ω is pure point for all i ≤ m and all ω ∈ C0 (Rd ). So fi,ω ∈ Hpp for all i ≤ m and all ω ∈ C0 (Rd ). Fix ε > 0 and let V be a bounded set with diam(V ) < θ = θ(ε), where θ is defined by (3.6), and Vol(∂V ) = 0. Find ω ∈ C0 (Rd ) as in Lemma 3.5. Since Vol((∂V )+ζ ) → Vol(∂V ) = 0 in Lemma 3.5, as ζ → 0, we obtain that χEi ,V ∈ Hpp . Therefore, also Ux χEi ,V = χx+Ei ,V ∈ Hpp . Then it follows from Lemma 3.6 and Lemma 3.7 that χP,V ∈ Hpp where P = B1/ε (0) ∩ Γ for any Γ ∈ XΛ , diam(V ) < θ, and Vol(∂V ) = 0. Our goal is to show that Hpp = L2 (XΛ , µ). Since (XΛ , µ) is a regular measure space, C(XΛ ) is dense in L2 (XΛ , µ). Thus, it is enough to show that all continuous functions on XΛ belong to Hpp . Fix f ∈ C(XΛ ). Using the decomposition XΛ =
N j=1 XPj ,Vj from Lemma 2.4 we can approximate f by linear combinations of characteristic functions of cylinder sets XPj ,Vj . So it suffices to show that these characteristic functions are in Hpp , which was proved above. This concludes the proof of Theorem 3.2.


4 Concluding remarks: what if the UCF fails? Here we present a version of the main theorem for Delone multisets which do not necessarily have uniform cluster frequencies. For this we must assume that in addition to the van Hove property (2.4) our averaging sequence {Fn } is a sequence of compact neighbourhoods of 0 satisfying the Tempel’man condition: (i) ∪Fn = Rd (ii) ∃ K ≥ 1 so that Vol(Fn − Fn ) ≤ K · Vol(Fn ) for all n.

(4.1)

Let Λ be a Delone multiset with FLC in Rd . Consider the topological dynamical system (XΛ , Rd ) and an ergodic invariant Borel probability measure µ (such measures always exist). The ergodic measure µ will be fixed throughout the section. Theorem 4.1 Suppose that a Delone multiset Λ has FLC. Then the following are equivalent:

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Ann. Henri Poincar´e

(i) The measure-preserving dynamical system (XΛ , µ, Rd ) has pure point spectrum;  (ii) For µ-a.e. Γ ∈ XΛ , the measure ν = i≤m ai δΓi has pure point diffraction spectrum, for any choice of complex numbers (ai )i≤m ; (iii) For µ-a.e. Γ ∈ XΛ , the measures δΓi have pure point diffraction spectrum, for i ≤ m. In fact, this formulation is closer to the work of Dworkin [4] who did not assume unique ergodicity. The proof is similar to that of Theorem 3.2, except that we have to use the Pointwise Ergodic Theorem instead of the uniform convergence of averages in the uniquely ergodic case (2.5). Theorem 4.2 (Pointwise Ergodic Theorem for Rd -actions (see, e.g. [15], [2]) 3 ) Suppose that a Delone multiset Λ has FLC and {Fn } is a van Hove sequence satisfying (4.1). Then for any f ∈ L1 (XΛ , µ),   1 f (−x + Γ) dx → f (Λ ) dµ(Λ ), as n → ∞, (4.2) Vol(Fn ) Fn for µ-a.e. Γ ∈ XΛ . For a cluster P ⊂ Λ, a bounded set A ⊂ Rd , and a Delone set Γ ∈ XΛ , denote LP (A, Γ) = #{x ∈ Rd : x + P ⊂ A ∩ Γ}. Lemma 4.3 For µ-a.e. Γ ∈ XΛ and for any cluster P ⊂ Λ, freq (P, Γ) := lim

n→∞

LP (Fn , Γ) , Vol(Fn )

(4.3)

exists for µ-a.e. Γ ∈ XΛ . Moreover, if diam(V ) < η(Λ), then the cylinder set XP,V satisfies, for µ-a.e. Γ ∈ XΛ : µ(XP,V ) = Vol(V ) · freq (P, Γ).

(4.4)

Note that we no longer can claim uniformity of the convergence with respect to translation of Γ. Sketch of the proof. Fix a cluster P ⊂ Λ and let XP,V be a cylinder set, with diam(V ) < η(Λ). Applying (4.2) to the characteristic function of XP,V and arguing as in the proof of Theorem 2.7 (with −h + Λ replaced by Γ), we obtain (4.3) and (4.4) for µ-a.e. Γ. Since there are countably many clusters P ⊂ Λ, we can find a set of full µ-measure on which (4.3) and (4.4) hold for all P.
3 For recent developments of this theorem in the direction of general locally compact amenable groups, see [9].

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 For a Delone set Γ = (Γi )i≤m , let ν = m i=1 ai δΓi . Then, for µ-a.e. Γ, the autocorrelation γ(ν) exists as the vague limit of measures Vol1(Fn ) (ν|Fn ∗ ν|Fn ), and γ(ν) =

m  i,j=1

ai aj



freq ((y, z), Γ)δy−z ,

y∈Γi ,z∈Γj

 is a positive measure, called for µ-a.e. Γ, which is the analogue of (3.3). Again, γ(ν) the diffraction measure, giving the meaning to the words “pure point diffraction spectrum” in Theorem 4.1. Sketch of the proof of Theorem 4.1. For µ-a.e. Γ ∈ XΛ , the Pointwise Ergodic Theorem 4.2 holds for all functions f ∈ C(XΛ ) (since the space of continuous functions on XΛ is separable). The νΛ
, ρω,Λ
, and fω are defined the same way as in Section 3. Lemma 3.3 applies to our situation. Next we can show that σfω = γ ω,Γ

(4.5)

for µ-a.e. Γ. This is proved by the same chain of equalities as in (3.4), except that we average over Fn defined in (4.1) and use Theorem 4.2 instead of Theorem 2.7. Lemma 3.5 goes through, after we replace freq(Ei , Λ) by freq (Ei , Γ), for µ-a.e. Γ. There are no changes in Lemmas 3.6 and 3.7, since we did not use UCF or unique ergodicity in them. The proof of Theorem 4.1 now follows the scheme of the proof of Theorem 3.2. We only need to replace Λ by µ-a.e. Γ, for which hold all the “typical” properties discussed above.


References [1] M. Baake and R. V. Moody, Weighted Dirac combs with pure point diffraction, math.MG/0203030, Preprint, 2002. [2] J. Chatard, Sur une g´en´eralisation du th´eor`eme de Birkhoff, C.R. Acad. Sc. Paris, t.275, Serie A, 1135–1138 (1972). [3] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Springer Lecture Notes in Math. 527, Springer, 1976. [4] S. Dworkin, Spectral theory and X-ray diffraction, J. Math. Phys. 34, 2965– 2967 (1993). [5] A. C. D. van Enter and J. Mi¸ekisz, How should one define a (weak) crystal? J. Stat. Phys. 66, 1147–1153 (1992). [6] A. Hof, Diffraction by aperiodic structures, in The Mathematics of LongRange Aperiodic Order, (R. V. Moody, ed.), 239–268, Kluwer, 1997.

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[7] J.-Y. Lee and R. V. Moody, Lattice Substitution Systems and Model Sets. Discrete and Computational Geometry 25, 173–201 (2001). [8] J.-Y. Lee, R. V. Moody and B. Solomyak, Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems, Discrete and Computational Geometry (to appear), 2002. [9] E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math. 146, 259-295 (2001). [10] G. W. Mackey, Unitary Group Representations in Physics, Probability, and Number Theory, Addison-Wesley, 1978,1989. [11] M. Queffelec, Substitution dynamical systems – spectral analysis, Springer Lecture Notes in Math. 1294, Springer, 1987. [12] C. Radin and M. Wolff, Space tilings and local isomorphism, Geometriae Dedicata 42, 355–360 (1992). [13] M. Schlottmann, Generalized model sets and dynamical systems, in: Directions in Mathematical Quasicrystals, eds. M. Baake and R. V. Moody, CRM Monograph series, AMS, Providence RI (2000), 143–159. [14] B. Solomyak, Dynamics of self-similar tilings, Ergodic Th. Dynam. Sys. 17, 695–738 (1997). [15] A. A. Tempel’man, Ergodic theorems for general dynamical systems, Dokl. Akad. Nauk SSSR, vol. 176 (1967), no. 4, 790–793 (English translation: Soviet Math. Dokl., vol. 8 (1967), no. 5, 1213–1216). [16] P. Walters, An introduction to ergodic theory, Springer Graduate Texts in Mathematics, Springer-Verlag, New York, 1982. [17] J. Weidmann, Linear Operators in Hilbert Space, Springer Graduate Texts in Mathematics, Springer-Verlag, New York, 1980. Jeong-Yup Lee and Robert V. Moody Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta T6G 2G1 Canada email: [email protected] email: [email protected] Communicated by Jean Bellissard submitted 03/01/02, accepted 04/04/02

Boris Solomyak Department of Mathematics University of Washington Seattle, WA 98195 USA email: [email protected]

Ann. Henri Poincar´e 3 (2002) 1019 – 1047 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/051019-29

Annales Henri Poincar´ e

Linearly Recurrent Circle Map Subshifts and an Application to Schr¨ odinger Operators B. Adamczewski and D. Damanik∗

Abstract. We discuss circle map sequences and subshifts generated by them. We give a characterization of those sequences among them which are linearly recurrent. As an application we deduce zero-measure spectrum for a class of discrete onedimensional Schr¨ odinger operators with potentials generated by circle maps.

1 Introduction and Results 1.1

Introduction

The concept of linear recurrence or linear repetitivity, LR in short, has been recently discussed and investigated by quite a number of researchers within various frameworks. For example, the articles [15, 17, 19] study the LR property from the point of view of combinatorics on words, whereas [14, 32, 38] discuss its implications within the theory of tilings. In both cases one considers structures (e.g., an infinite word or a tiling of Euclidean space), or families of structures (e.g., a subshift or a family of tilings), and their local patterns (e.g., subwords or patches occurring in the given tiling) which are equivalence classes modulo translations. Fixing such a local pattern, one may look at the set of occurrences of the pattern in the structure and compare the distance between two “consecutive” occurrences with the size of the pattern. If the distance is bounded by a fixed linear function of the size, the structure is said to have the LR property. Although the concepts are the same in spirit, applied to words it is usually referred to as linear recurrence, whereas among tiling theorists this concept is usually called linear repetitivity. Since this article will be concerned with a class of words and subshifts, we will henceforth use the term linear recurrence. The usefulness of the LR property has been independently realized by numerous people who had quite different applications in mind. LR has been shown to have consequences in mathematical disciplines as diverse as combinatorics [15, 19], ergodic theory [14, 32, 34], and spectral theory of Schr¨ odinger operators [35]. ∗ D. D. was supported in part by the National Science Foundation through Grant DMS– 0010101.

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Our present study is motivated by the paper [35]. Consider discrete onedimensional Schr¨ odinger operators (Hψ)(n) = ψ(n + 1) + ψ(n − 1) + V (n)ψ(n)

(1)

in 2 (Z), where the potential V : Z → R is given by V (n) = λχ[0,β) (nα + θ

mod 1).

(2)

Here, λ = 0 is the coupling constant, α ∈ (0, 1) irrational is the rotation number, and β ∈ (0, 1) and θ ∈ [0, 1) are arbitrary numbers. These potentials are called circle map potentials in the mathematical physics community (cf. [23, 24, 25]) and codings of rotations by people working in combinatorics on words or symbolic dynamics. The operator (1) with potential (2) has been studied in many papers; for example, [3, 4, 6, 10, 11, 12, 16, 23, 24, 25, 26, 27, 28, 29, 39, 40]. One expects the following picture to be true (cf. [9]): The operator H has purely singular continuous spectrum which is supported on a Cantor set of Lebesgue measure zero. To establish this, one has to prove the following three properties of H: (i) The spectrum σ(H) of H has Lebesgue measure zero. (ii) The absolutely continuous spectrum σac (H) of H is empty. (iii) The point spectrum σpp (H) of H is empty. Actually, it is easy to see that (i) implies (ii). However, (ii) is known in great generality while (i) is not. Namely, it follows from Kotani [31] and Last and Simon [33] that for all parameter values allowed above (recall λ = 0 and α irrational), (ii) holds. Moreover, (iii) is known in many cases. For example, Delyon and Petritis showed that the point spectrum is empty for every λ and β, almost every α, and almost every θ [16]. Hof et al., on the other hand, prove (iii) for every λ, α, and β, and generic θ (i.e., for a dense Gδ set) [24]. Thus, properties (ii) and (iii) are well understood. This is not the case for property (i). Until very recently, there was only one approach to (i). This approach is based on trace maps and it allowed Bellissard et al. to prove the zero measure property in the case where α = β, that is, in the Sturmian case [3] (see also S¨ ut˝ o [40] for the Fibonacci case). Their results were extended to the quasi-Sturmian case in [13]. (A quasi-Sturmian sequence is essentially a morphic image of a Sturmian sequence.) In the non-(quasi-)Sturmian case, very little is known. The only result, due to H¨ornquist and Johansson [25], concerns a small class which can be shown to be generated by substitutions so that the adaptation [5] of [3] to potentials generated by substitutions applies. Essentially, the absence of a trace map is the reason that no other results are known for the non-Sturmian case. A new approach to zero-measure Cantor spectrum, which is not based on trace maps, was recently developed by Lenz [35]. It is therefore natural, and was in fact suggested in [35], to try to apply this new approach to the potentials in (2). This new approach shows that linear recurrence allows one to deduce (i). Thus, we are led to the following question: For which

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Linearly Recurrent Circle Map Subshifts

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choices of parameter values is V in (2) linearly recurrent? It is the aim of this paper to answer this question. In fact, we shall characterize this set of parameter values. We note that the examples considered by H¨ornquist and Johansson are linearly recurrent so that our result contains theirs. For convenience, we will slightly change the setting from individual sequences to subshifts. However, at the end of Section 5 we shall clearly state for which parameter values we get property (i). The organization of the article is as follows. In the remainder of this section we will recall some key notions and state our main result which provides a characterization of the circle map sequences/subshifts which are linearly recurrent. In Section 2 we will develop the general setup and in particular recall the connection between LR subshifts and primitive S-adic subshifts. The link between circle map sequences and interval exchange transformations, and particularly the results of [1] which will be crucial to our paper, will be explained in Section 3. Section 4 contains the proof of our main result. The application of this theorem to Schr¨ odinger operators is discussed in Section 5. Appendix A explains how to prove a finite index for some circle map sequences which are not LR. Finally, in Appendix B we discuss possible generalizations of the approach presented in this paper. Acknowledgments. We would like to thank Julien Cassaigne for useful discussions and particularly for his contributions to what is presented in Appendix A. Moreover, D. D. would also like to express his gratitude for the hospitality at CPT and IML at CNRS, Luminy where this work was initiated.

1.2

Circle maps

Definition 1 Let (α, β) ∈ (0, 1)2 . The circle map corresponding to the parameters (α, β) is the symbolic sequence U = (un )n≥0 defined over the binary alphabet {0, 1} by:
1 if {nα} ∈ [0, β[, un = 0 else. We will restrict our attention to circle maps where α is irrational and β ∈ Z + αZ. The case α rational is not interesting since the associated circle map is periodic (and hence, in this case, the corresponding Schr¨ odinger operator has purely absolutely continuous spectrum which is supported on a finite union of closed intervals). The case β = α gives a Sturmian sequence and, more generally, the case β ∈ Z + αZ corresponds to quasi-Sturmian sequences and will be not considered in this paper (see [7, 37]). (Zero-measure spectrum for chr¨ odinger operators with quasi-Sturmian potentials was shown in [13]). Definition 2 A circle map is called nondegenerate if its parameters satisfy: • α is irrational, • β ∈ Z + αZ. Such a circle map is called admissible if in addition we have α < β.

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The D-expansion

In a previous paper [1], one of us has investigated the links between nondegenerate circle maps and three-interval exchange transformations. An algorithm was introduced which can be regarded as a speed-up of the Rauzy induction for threeinterval exchange transformations. This algorithm is also a generalization of the classical continued fraction algorithm. Let us introduce a map D : [0, 1[×R∗+ −→ [0, 1[×R∗+ given by

  { x }  y−1 1  ,  1 x 1 x  y−1 − y−1  y−1 − y−1        y x x (x, y) −→ { 1−y }, 1−y −  1−y         (0, 1)

if

y > 1,

if

y < 1,

if

y = 1.

Definition 3 Given an admissible circle map with parameters (α, β), the associated D-expansion is given by the sequence (an , in )n∈N which is defined as follows:

an

=

in

=



 xn yn − 1  1 if yn < 1 0 if yn > 1

where    α−β 1 − 1−β α α   ,   .   (xn , yn ) = Dn (x0 , y0 ) and (x0 , y0 ) =  + 1 α 1 − 1−β +1 α 1 − 1−β α α 

For a circle map corresponding to (α, β) ∈ [0, 1[2 which is nondegenerate and not admissible (i.e., α > β), its D-expansion is given by the D-expansion associated with the admissible circle map corresponding to (1 − α, 1 − β). Conversely, for any sequence (an , in )n∈N with (an )n∈N not ultimately vanishing and (in )n∈N not ultimately constant, and any k ∈ N, there is exactly one nondegenerate pair (α, β) such that  1−β α  = k and the corresponding circle map sequence has D-expansion (an , in )n∈N . We also want to mention the recent paper [22] of Ferenczi, Holton, and Zamboni which introduces a generalized continued fraction algorithm for three-interval exchange transformations which is based on a different induction process.

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Results

Our main result is Theorem 4 which gives a characterization of linearly recurrent nondegenerate circle map subshifts. Theorem 4 A nondegenerate circle map subshift is linearly recurrent if and only if its D-expansion (an , in )n∈N satisfies the following: there exists an integer M such that for every integer n, (i) an ≤ M , (ii) in = in+1 = . . . = in+t ⇒ t ≤ M , (iii) an = an+1 = . . . = an+t = 0 ⇒ t ≤ M . In the following, we will call this condition the (∗)-condition. In particular, the class of LR nondegenerate circle map subshifts contains, but is not equal to, the circle map subshifts corresponding to parameters (α, β), where α and β lie in the same quadratic field. This follows directly from the fact proved in [17] that a primitive substitutive subshift is linearly recurrent and Theorem 5 (Adamczewski [1]) For a subshift associated with a nondegenerate circle map corresponding to parameters (α, β), the following are equivalent: (i) It is primitive substitutive, that is, it can be generated by the morphic image of a fixed point of a primitive substitution. (ii) The associated D-expansion is ultimately periodic. (iii) α and β lie in the same quadratic field. In terms of interval exchange transformations, Theorem 4 is a full geometric generalization of the following theorem. Theorem 6 (Durand [20]) A Sturmian subshift associated with an irrational number α is linearly recurrent if and only if the coefficients of the continued fraction expansion of α are bounded.

2 Definitions and Background 2.1

Symbolic sequences and substitutions

A finite and nonempty set A is called alphabet. The elements of A are called letters. A finite word on A is a finite sequence of letters and an infinite word on A is a sequence of letters indexed by N. The length of a finite word ω, denoted by |ω|, is the number of letters it is built from. The empty word, ε, is the unique

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word of length 0. We denote by A∗ the set of finite words on A and by AN the set of sequences over A. Let U = (uk )k∈N be a symbolic sequence defined over the alphabet A. A factor of U is a finite word of the form ui ui+1 . . . uj , 0 ≤ i ≤ j. If ω is a factor of U and a a letter, then |w|a is the number of occurrences of the letter a in the word ω. We denote by L(U ) the set of all the factors of the sequence U , L(U ) is called the language of U . A sequence in which all the factors have an infinite number of occurrences is called recurrent. When these occurrences have bounded gaps, the sequence is called uniformly recurrent. A sequence U is called K-power free if uK ∈ L(U ) implies u = ε. A sequence U is called power free if there exists an integer K such that U is K-power free. Endowed with concatenation, the set A∗ is a free monoid with unit element ε. A map from A to A∗ \{ε}, called substitution on A, can be extended by concatenation to an endomorphism of the free monoid A∗ and then to a map from AN to itself. Given a substitution σ defined on A, we call the matrix Mσ = (|σ(j)|i )(i,j)∈A2 the incidence matrix associated with σ. The composition of substitutions corresponds to the multiplication of incidence matrices. A substitution is called primitive if there exists a power of its incidence matrix for which all the entries are positive.

2.2

Return words

We present here the main definitions concerning the notion of return words introduced in [18]. Let U be a uniformly recurrent sequence over the alphabet A and let u = u1 u2 . . . un be a nonempty prefix of U . A return word to u of U is a factor u[i,j−1] (= ui ui+1 . . . uj−1 ) of U such that i and j are two consecutive occurrences of u. The sequence U can be written in a unique way as a concatenation of return words to u. Let RU,u be the set of return words to u in U . Then U = ω0 ω1 . . . ωi . . ., where ωi ∈ RU,u . The fact that U is uniformly recurrent implies that RU,u is a finite set. We can therefore consider a bijective map ΛU,u from RU,u to the finite set {1, 2, . . . , Card(RU,u )} = AU,u , where, for definiteness, the return words are ordered according to their first occurrence (i.e., Λ−1 U,u (1) is the first return word ω0 , −1 ΛU,u (2) is the first ωi which is different from ω0 , and so on). The derived sequence of U on u is the sequence with values in the alphabet AU,u given by Du (U ) = ΛU,u (ω0 )ΛU,u (ω1 ) . . . ΛU,u (ωi ) . . . . To such a sequence we can associate a morphism ΘU,u from AU,u to A∗ defined by: ΘU,u (i) = ωi . We obtain ΘU,u (Du (U )) = U . The morphism ΘU,u is called the return morphism to u of U . When AU,u = A, we will call it return substitution to u of U . When it

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does not create confusion, we will suppress the “U ” in the symbols RU,u , ΘU,u , and AU,u . Proposition 7 (Durand [18]) Let u be a nonempty prefix of U . Then the following holds. (i) The set Ru is a code and the map Θu is one to one. (ii) Let v be a nonempty prefix of Du (U ). Then there exists a nonempty prefix w of U such that Dv (Du (U )) = Dw (U ). Moreover, we have Θu ◦ Θv = Θw . A derived sequence of a derived sequence is hence a derived sequence. Definition 8 Let U be a symbolic sequence defined over the alphabet A starting with the symbol 1 ∈ A. We introduce the following notation: D(0) (U ) = U and, for n ∈ N, D(n+1) (U ) = D1 (D(n) (U )); Θ0 is the identity map and, for n ∈ N, Θn+1 = Θn ◦ ΘD(n) (U),1 . Remark 9 According to Proposition 7, we obtain that (D(n) )n∈N is a sequence of derived sequences of U and (Θn )n∈N is a sequence of return morphisms of U .

2.3

LR sequences

Definition 10 Let A be an alphabet, K a positive integer, and U a sequence over A. The sequence U is called K-linearly recurrent (K-LR) if it is uniformly recurrent and for all ω ∈ Ru , we have |ω| ≤ K|u|. A sequence is called linearly recurrent (LR) if it is K-LR for some K. Proposition 11 (DHS [17]) Let U be an aperiodic K-LR sequence over an alphabet A. Then: 1. For every n, each factor of U of length n has at least one occurrence in each factor of U of length (K + 1)n. 2. U is (K + 1)-power free. 3. For every nonempty prefix u of U and for all ω ∈ Ru , we have

2.4

1 K |u|

< |ω|.

Subshifts and LR subshifts

Let A be an alphabet. The topology of AN is given by the product of the discrete topologies on A. We denote by T the standard shift transformation which associates to each symbolic sequence U = (uk )k≥0 the sequence T (U ) = (uk )k≥1 . To a sequence U in AN we associate the dynamical system (O(U ), T ), where O(U ) is the closure of the orbit of U under the shift. This dynamical system is called the subshift associated with U . A dynamical system is minimal if it has no nontrivial invariant closed set. For a subshift associated with a sequence U , minimality of the subshift is equivalent to uniform recurrence of U .

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Definition 12 A subshift is called primitive substitutive if it contains a primitive substitutive sequence (i.e., a sequence which is the morphic image of a fixed point of a primitive substitution). A minimal subshift associated with a sequence U is called linearly recurrent (LR) if and only if U is LR.

2.5

S-adic sequences and S-adic subshifts

Let A be an alphabet, a a letter of A, and S a finite set of substitutions from A to A∗ . We will say that a sequence U ∈ AN is an S-adic sequence (generated by (σn )n∈N ∈ S N and a) if there exists a sequence (σn )n∈N ∈ S N such that U = limn→∞ σ0 σ1 . . . σn (aa . . .). Let U be such a sequence. If there exists an integer s0 such that for all b ∈ A and all c ∈ A, the letter b has an occurrence in σr+1 σr+2 . . . σr+s0 (c), then U is called a primitive S-adic sequence (with constant s0 ). The subshift associated with an S-adic sequence (resp., a primitive S-adic sequence) is called an S-adic subshift (resp., a primitive S-adic subshift). These notions were introduced by S. Ferenczi in [21] and by F. Durand in [19]. It was claimed in [19] that a subshift is LR if and only if it is primitive S-adic. In [20], Durand provides a counterexample and exhibits a primitive S-adic subshift which is not LR. However, LR does imply primitive S-adic and with an additional condition we can obtain a partial converse given in Proposition 14 below. Definition 13 Let A be an alphabet and σ a substitution on A. The substitution σ is called (b, c)-proper if for any letter i in A, σ(i) begins with b and ends with c. An S-adic sequence is called proper if there exist two letters b and c in A such that any substitution in S is a (b, c)-proper substitution. A subshift which contains a proper and primitive S-adic sequence is called a proper primitive S-adic subshift. Proposition 14 (Durand [20]) A subshift (X, T ) is LR if and only if it is a proper primitive S-adic subshift.

2.6

Interval exchange transformations

Interval exchange transformations are classical examples of dynamical systems. Definition 15 Let s ∈ N, s ≥ 2. Let σ be a permutation of the set {1, 2, . . . , s} and let λ = (λ1 , λ2 , . . . , λs ) be a vector in Rs with strictly positive entries. Let I = [0, |λ|[, where |λ| =

s  i=1

    λi and for 1 ≤ i ≤ s, Ii =  λj , λj  . j

j≤i

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The interval exchange transformation associated with (λ, σ) is the map E from I into itself, defined as the piecewise isometry which arises from ordering the intervals Ii with respect to σ. More precisely, if x ∈ Ii ,   E(x) = x + ai , where ai = λσk − λk . k β. The proof is based on a study of an induction process for three-interval exchange transformations close to that of Rauzy. We also obtain an analog to Proposition 17 in the case of nondegenerate circle map subshifts. Figure 3 is the analog of Figure 1 and Figure 4 is the analog of Figure 2. To a nondegenerate circle map we can associate an I.D.O.C. three-interval exchange transformation. The orbit of such an interval exchange transformation under the Rauzy induction does not ultimately remain in one of the primitivity subgraphs G1 , G2 , or G3 represented in Figure 4. σ4

σ1

σ2 312

321 σ3

σ1 231

σ4

Figure 3: The Rauzy induction graph for three-interval exchange transformations. Moreover, an I.D.O.C. three-interval exchange transformation is LR if and only if its orbit under the Rauzy induction can stay in any of the primitivity subgraphs G1 , G2 , and G3 only for a bounded number of consecutive induction steps. This last remark provides a geometric interpretation of the (∗)-condition in Theorem 4 and will be proved in Section 4. A similar study could clearly be carried out in the general case of an I.D.O.C. interval exchange transformation. However, the results quickly become hard to read since the complexity of the equivalent to the (∗)-condition increases rapidly (cf. Appendix B). In this section we have exhibited some similarities between the Sturmian and the circle map cases. On the other hand, some aspects of the two cases do not have mutual counterparts. The strategy used to prove Theorem 6 is the following: • Exhibit a primitive S-adic expression for Sturmian subshifts generated by an irrational α when the coefficients of the continued fraction expansion of α are bounded and use this to establish linear recurrence in this case.

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G1 :

312

321

, G2 :

312

321

231

, G3:

Ann. Henri Poincar´e

321

231

Figure 4: The primitivity subgraphs for three-interval exchange transformations. • Show that otherwise a Sturmian sequence contains arbitrarily high powers. We thus obtain that a Sturmian sequence is LR if and only if it is power free. However, such an equivalence does not hold for circle maps. We can therefore not mimic the strategy used in the Sturmian case. For example, the circle map sequences with D-expansion (an , in )n∈N , where (in )n∈N is the periodic sequence (10)ω (i.e., in = 0 if n is even and in = 1 if n is odd), an = 1 if n is a power of 2 and 0 otherwise, are both non-LR and power free (see Appendix A).

4 Proof of Theorem 4 The proof of Theorem 4 is based on Theorem 18 and Proposition 14 which states that a proper primitive S-adic subshift is LR. Our strategy to prove this theorem is the following: • We exhibit a proper primitive S-adic expression for three-interval exchanges associated with circle maps whose D-expansion satisfies the (∗)-condition (Proposition 20). • We prove the existence of a uniform upper bound of the gaps between successive occurrences of letters in the different derived sequences of an LRsequence (Lemma 24). • Finally, we show that such a uniform bound does not exist for a circle map whose D-expansion does not satisfy the (∗)-condition (Proposition 23). For i ∈ {1, 2, 3, 4}, let Ai denote the incidence matrix of the substitution σi which has been defined in the previous section. For every integer k, we write Fk = (σ1 σ2k σ3 ) and Gk = (σ4 σ1k σ4 ),

(3)

and for the associated incidence matrices, we write Bk = (A1 Ak2 A3 ) and Ck = (A4 Ak1 A4 ).

(4)

Definition 19 Let (C, D) ∈ M3 (R)2 , C = (ci,j ), D = (di,j ). We say that C ≥ D if ci,j ≥ di,j , ∀(i, j) ∈ {1, 2, 3}2. Similarly, we say that C > D holds if ci,j > di,j , ∀(i, j) ∈ {1, 2, 3}2. Proposition 20 A nondegenerate circle map whose D-expansion satisfies the (∗)condition is the image by a morphism of a proper primitive S-adic sequence.

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Lemma 21 If C is a nonnegative matrix in M3 (Z), then for every integer k, the following four inequalities hold: Bk C ≥ C, CBk ≥ C, Ck C ≥ C, and CCk ≥ C. Proof. This follows directly from Bk = I3 + A k with A k ≥ 0 and Ck = I3 + Bk with Bk ≥ 0.
Lemma 22 Let (an , in )n∈N be a D-expansion satisfying the (∗)-condition with an ij 1−ij (k+1)M0 integer M0 and let S = , k ∈ N . Then S is a finite set of j=kM0 Faj ◦ Gaj substitutions and each of its element is (1, 3)-proper. Proof. The set S is clearly finite because the sequence (an )n∈N is bounded by M0 . In view of (3), we obtain for every integer k 1 2 3

Fk  → 13 − −→ 2k+1 3 −→ 2k 3

1 2 3

Gk −→ 12k −→ 12k+1 −→ 13

Let k be an integer and i ∈ {1, 2, 3}. Then Fk (i) ends with 3 and Fk (1) begins with 1. Moreover Gk (i) begins with 1 and Gk (1) ends with 3. It follows thus that each composition of substitutions of types Fk and Gk in which the two types both appear is (1, 3)-proper. Part (ii) of the (∗)-condition allows us to conclude.
Proof of Proposition 20. Let us consider a circle map U whose D-expansion satisfies the (∗)-condition with some integer M0 . Theorem 18 provides us with an S-adic expression for this circle map. Our goal is now to prove that we can extract a proper primitive S-adic expression for U from this representation. We can suppose that U is admissible in order to simplify the notation. We have  n    ij  1−ij  a a j j σ1 ◦ σ2 ◦ σ3 ◦ σ4 ◦ σ1 ◦ σ4 (1) . U = lim Φ 1−β   

n→∞

α

j=0

Let 

 n        ij 1−ij a a V = lim  σ1 ◦ σ2 j ◦ σ3 ◦ σ4 ◦ σ1 j ◦ σ4 (1) . n→∞

(5)

j=0

Thus, U = Φ 1−β  (V ) α

(6)

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B. Adamczewski and D. Damanik



and

V = lim  n→∞



We have then

V = lim  n→∞

Let S=

n 

n 

j=0

 

j (1). Faijj ◦ Ga1−i j



 j  (1). Faijj ◦ Ga1−i j

(7)

j=kM0

 (k+1)M  0 



(k+1)M0

k=0

Ann. Henri Poincar´e

j Faijj ◦ Ga1−i , k∈N j

j=kM0

  

.

Then Lemma 22 implies that (7) gives us a proper S-adic representation of V . We have now to prove that this representation is primitive or more precisely that there exists an integer s0 such that for every integer r and all b ∈ {1, 2, 3} and c ∈ {1, 2, 3}, the letter b has an occurrence in    k+s 0 (r+1)M  0 j    (c). Faijj ◦ Ga1−i j r=k

j=rM0

Or similarly, we have to show that the corresponding product of matrices    k+s 0 (r+1)M  0   Baij ◦ Ca1−ij  j

r=k

j

j=rM0

is positive, where the matrices Bl and Cl are defined in (4). Let us consider the matrix   (r+1)M0  j Mr =  Baijj ◦ Ca1−i . j j=rM0

By the fact that the D-expansion associated with U satisfies the (∗)-condition with the integer M0 , we get ∃j1 ∈ {1, 2, . . . , l} such that ij1 = 0, ∃j2 ∈ {1, 2, . . . , l} such that ij2 = 1, ∃j3 ∈ {1, 2, . . . , l} such that aj3 ≥ 1. The previous remark and Lemma 21 show that at least one of the following inequalities holds: Mr ≥ B 0 C1 , Mr ≥ B 1 C0 , Mr ≥ C1 B 0 , Mr ≥ C0 B 1 .

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Now we just have to remark that each element of {B0 C1 , B1 C0 , C1 B0 , C0 B1 }2 is positive. Therefore, we obtain primitive S-adicity of our representation with constant s0 = 2. We therefore obtain that U is the image under the morphism Φ 1−β  of α the proper primitive S-adic sequence V , concluding the proof.
Proposition 23 A nondegenerate circle map subshift whose D-expansion does not satisfy the (∗)-condition is not linearly recurrent. Since we will work with the derived sequences of a given circle map sequence in our proof of Proposition 23, we start off by discussing LR properties of derived sequences of an LR sequence. Lemma 24 Let U be a K-linearly recurrent sequence defined over an alphabet A and let ω be a nonempty prefix of U . Then every word of length at least K 2 (K + 1) in Dω (U ) contains all the elements of Aω . Proof. Let ω be a factor of U and i ∈ Aω = {1, 2, . . . , d}. Then there exists a unique word ωi such that Θω (i) = ωi . By definition we have ∀j ∈ Aω , |ωj | ≤ K|ω|. This inequality implies that ωi appears in each word of length at least (K + 1)(K|ω|), in view of Proposition 11. Moreover, again by Proposition 11, we have ∀j ∈ Aω ,

1 |ω| ≤ |ωj | ≤ K|ω|. K

The set Rω is a code. We thus obtain that the letter i occurs in each word of
length at least K 2 (K + 1) in Dω (U ). Lemma 25 Let U be a K-linearly recurrent sequence. Then, for every integer n, we have ∀i ∈ An , |Θn (i)| ≤ K 2 (K + 1), where the maps Θn are introduced in Definition 8. Proof. Let i be an element of An and Θn (i) = ωi . By definition of the return words and the sequence D(n) , the letter 1 has just one occurrence in ωi and 1 is the first letter of ωi . Then, 1 does not appear in the maximal proper suffix of ωi . Lemma 24 implies that the length of this suffix is at most K 2 (K + 1) − 1.
Lemma 26 Let U be a K-linearly recurrent sequence defined over an alphabet A and let ω be a nonempty prefix of U . Then the sequence Dω (U ) is K 3 -linearly recurrent.

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Proof. This statement and its proof are very similar in spirit to the previous two lemmas. Let x be a factor of Dω (U ). Consider any occurrence of x in Dω (U ) and the length of the corresponding return word to x in Dω (U ) (i.e., the length of the gap between this occurrence of x and the next, plus the length of x). We use again that Rω is a code. Namely, to this occurrence of x in Dω (U ) corresponds a word of length at most K · |ω| · |x| in U whose return words have length at most K 2 · |ω| · |x|. Choose the one that corresponds to this particular occurrence and go back via ΛU,ω to factors of Dω (U ). We conclude that the length of the return word to x in question is bounded by K 3 · |x|. In the previous steps, we have made repeated use of Proposition 11. This shows that Dω (U ) is K 3 -linearly recurrent since x and its occurrence were arbitrary.
Lemma 27 Let r be a positive integer. Then for every (i1 , i2 , . . . , ir ) ∈ {1, 2, 3, 4}r, we have |σi1 ◦ σi2 ◦ · · · ◦ σir (123)| ≥ σi1 ◦ σi2 ◦ · · · ◦ σir−1 (123) + 1. Proof. We just have to remark that for each k ∈ {1, 2, 3, 4}, there exists a letter
b ∈ {1, 2, 3} such that |σk (b)| ≥ 2 and that 1, 2, and 3 occur in σk (123). Lemma 28 Let r be a positive integer and (i1 , i2 , . . . , i3r+1 ) ∈ {1, 2, 3, 4}3r+1. Then there exists at least one letter b ∈ {1, 2, 3} such that σi1 ◦ σi2 ◦ · · · ◦ σi3r+1 (b) > r. Proof. According to Lemma 27, it follows by induction that σi1 ◦ σi2 ◦ · · · ◦ σi3r+1 (123) ≥ 3r + 1. The assertion follows immediately.




Lemma 29 Let n be an integer, (m0 , m1 , . . . , mn ) ∈ Nn , and (l0 , l1 , . . . , ln ) ∈ {0, 1}n. Then, for each b ∈ {1, 2, 3}, we have   n  m (i) j=0 σ1 σ2 j σ3 (b) ≤ 1, 1    n l 1−l (ii) j=0 (σ1 σ3 ) j ◦ (σ4 σ4 ) j (b) ≤ 1, 2   n  mj (iii) j=0 σ4 σ1 σ4 (b) ≤ 1. 3

Here, |w|i denotes the number of occurrences of the symbol i in the word w.  n  m Proof. (i) The incidence matrix associated with the substitution j=0 σ1 σ2 j σ3 n is j=0 Bmj , where the matrices Bmj are defined in (4). For each integer k, the matrix Bk is of the form   1 0 0  × × × , × × ×

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 and so the matrix nj=0 Bmj is of course of the same form. Then, the definition of the incidence matrix allows us to conclude.  n  l 1−l (ii) The incidence matrix associated with j=0 (σ1 σ3 ) j ◦ (σ4 σ4 ) j is equal to n  lj 1−lj  . The matrices B0 and C0 are of the form j=0 B0 C0 

×  0 × and so the matrix

n

j=0

 × × 1 0 , × ×

  l 1−l B0j C0 j is of the same form.

 n  m (iii) The incidence matrix associated with the substitution j=0 σ4 σ1 j σ4 is n equal to j=0 Cmj , where the matrices Cmj are defined in (4). For each integer k, the matrix Ck is of the form   × × ×  × × × , 0 0 1 and so the matrix

n

j=0

Cmj is of the same form, concluding the proof.




Proof of Proposition 23. Let U be a circle map whose D-expansion (an , in )n∈N does not satisfy the (∗)-condition. Let V be as in (5) so that we have (6). Let us assume for the moment that 1 − β > α so that V is the derived sequence corresponding to the prefix 1 of U . We will comment later on the case 1 − β < α. Now assume there exists an integer K such that U is K-LR. We consider four cases. (i) Let us suppose that the sequence (an )n∈N is unbounded. Then a direct consequence of the fact that σ2an (3) = 2an 3, σ1an (1) = 13an , and that powers propagate by substitution is that U cannot be (K +1)-power free. Proposition 11 thus yields a contradiction. (ii) Let us suppose that the sequence (in )n∈N contains arbitrarily long blocks of 1’s. In particular, there exists an integer n0 such that in0 = in0 +1 = . . . = in0 +12K 2 (K+1) = 1.

(8)

We recall that there exists an increasing sequence of integers (kN )N ∈N such that kN   ij  1−ij  a a σ1 σ2 j σ3 ◦ σ4 σ1 j σ4 , ΘN = j=kN −1 +1

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where ΘN is introduced in Definition 8. This follows from Remark 16 and the fact that, as was already observed in [1], certain steps of our induction process correspond to induction on the first interval of three-interval exchange transformations associated with U . According to Lemmas 25 and 28, the fact that U is K-LR implies that for each integer N , kN +1 − kN < 3K 2 (K + 1) + 1.

(9)

Now, let us consider two particular elements of the sequence (kN )N ∈N : ! kN1 = min kN , n0 ≤ kN ≤ n0 + 12K 2(K + 1) and

! kN2 = max kN , n0 ≤ kN ≤ n0 + 12K 2 (K + 1) .

By the inequality (9), we obtain that kN1 and kN2 are well-defined and kN2 − kN1 ≥ 6K 2 (K + 1) + 1.

(10)

Let us introduce the substitution Θ = ΘN1 +1 ΘN1 +2 . . . ΘN2 . Then, kN2

Θ=

 ij  1−ij  a a σ1 σ2 j σ3 ◦ σ4 σ1 j σ4 .



j=kN1 +1

More precisely, using condition (8), we have kN2

Θ=



  a σ1 σ2 j σ3 .

(11)

j=kN1 +1

Proposition 7 implies that Θ is a return substitution for U since it is a composition of return substitutions. Thus there exists a nonempty prefix ω of U such that Θ = ΘU,ω . According to the inequality (10) and Lemma 28, we obtain that there exists a letter b in the alphabet {1, 2, 3} such that |Θ(b)| ≥

kN2 − kN1 > 2K 2 (K + 1), 3

and it follows from the equality (11) and Lemma 29 that |Θ(b)|1 ≤ 1. But Θ(b) is necessarily a factor of Dω (U ). Hence there exists a factor of Θ(b) of length greater or equal than K 2 (K + 1) in which the letter 1 does not occur. We obtain finally that there exists a factor of Dω (U ) of length greater than or equal to K 2 (K + 1) in which the letter 1 does not occur. This last remark is in contradiction with the K-LR property of U in view of Lemma 24.

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(iii) Let us suppose that the sequence (in )n∈N contains arbitrarily long blocks of 0’s. Then, we just have to mimic the above arguments in order to find a return substitution Θ for U and a letter b in {1, 2, 3} such that |Θ (b)|3 ≤ 1 and |Θ (b)| > 2K 2 (K + 1). We thus obtain a nonempty prefix ω of U such that Dω
(U ) contains a factor of length greater than or equal to K 2 (K + 1) in which the letter 3 does not occur. (iv) Let us suppose that the sequence (an )n∈N contains arbitrarily long blocks of 0. Then, analogous reasoning gives a return substitution Θ for U and a letter b in {1, 2, 3} such that: |Θ (b)|2 ≤ 1 and |Θ (b)| > 2K 2 (K + 1). We find a nonempty prefix ω of U such that Dω

(U ) contains a factor of length greater than or equal to K 2 (K + 1) in which the letter 2 does not occur. Thus we arrive at a contradiction in each case. Recall that we assumed 1−β > α at the beginning of the proof. Let us now discuss the case where 1 − β < α. In this case V in (5) is not the derived sequence corresponding to the prefix 1 of U , that is, D1 (U ) = V . In fact, V takes three values, while 1 has only two return words, 1 and 10. However, for sufficiently large n, it is relatively easy to see that D(n) (U ) is one of the sequences obtained in the induction process of [1] (leading to the representation (6)) and hence there is a morphism Ψ such that V = Ψ(D(n) (U )). If we now again assume that U is LR, then so is D(n) (U ), by Lemma 26, and hence we get that V is LR. Now we can derive a contradiction following the steps given above.
Proof of Theorem 4. In view of Proposition 14, Theorem 4 follows directly from Propositions 20 and 23.


5 Application of Theorem 4 to Schr¨ odinger Operators In this section we apply our main result, Theorem 4, to discrete one-dimensional Schr¨ odinger operators with potentials given by circle maps. As explained in the introduction, this is in part motivated by previous results on their Sturmian counterparts and a recent result of Lenz which relates aspects of their spectral theory to LR properties. A discrete one-dimensional Schr¨ odinger operator acts in the Hilbert space 2 (Z). If φ ∈ 2 (Z), then Hφ is given by (Hφ)(n) = φ(n + 1) + φ(n − 1) + V (n)φ(n), where V : Z → R. The map V is called the potential.

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If A is an alphabet, T : Z → Z is the standard shift, Ω ⊆ AZ is T -invariant (i.e., T Ω = Ω) and closed (discrete topology on A and product topology on AZ ), then Ω is called a (two-sided) subshift. Given such a subshift and a function f : A → R, we define, for ω ∈ Ω, a potential V = Vω by Vω (n) = f (ωn ) and an operator Hω (as above, with this particular potential). It is a standard result that if Ω is minimal, then the spectrum of Hω does not depend on ω, that is, there is a set Σ ⊆ R such that σ(Hω ) = Σ for every ω ∈ Ω (see, e.g., [9]). A special case of a recent result of Lenz is given in the following theorem. Theorem 30 (Lenz [35]) If Ω is a linearly recurrent subshift and Ω and f are such that the resulting potentials Vω are aperiodic, then Σ has Lebesgue measure zero. Note in particular that the result is essentially independent of the function f . Moreover, it suffices that at least one Vω is aperiodic. This implies that all Vω are aperiodic. Our goal is to apply this theorem to circle map subshifts. A circle map generates a two-sided subshift as follows. If u ∈ {0, 1}N is a circle map corresponding to parameters (α, β), the associated subshift is given by Ω = Ωα,β = {ω ∈ {0, 1}Z : every factor of ω is a factor of u}. If we restrict the sequences in Ω to the right half-line, we get exactly the one-sided subshift that was introduced and discussed above. By recurrence, the languages associated with the one-sided and two-sided subshifts are the same. In particular, LR-properties are the same for both subshifts. Combining our Theorem 4 and the theorem of Lenz, we obtain the following result. Theorem 31 Suppose that u is a nondegenerate circle map corresponding to parameters (α, β) whose D-expansion (an , in )n∈N satisfies the (∗)-condition. Consider the associated subshift Ω = Ωα,β and, for a nonconstant function f : {0, 1} → R, the operators (Hω )ω∈Ω . Then we have that for every ω ∈ Ω, the spectrum of Hω has Lebesgue measure zero. It is easy to see that for every θ, the sequence ωn = χ[0,β) (nα + θ mod 1) is an element of Ωα,β . In other words, Theorem 31 says that if α, β are such that their D-expansion (an , in )n∈N satisfies the (∗)-condition, then the potential V in (2) is linearly recurrent for every choice of θ and λ = 0, and in this case, the operator H satisfies property (i) from the introduction.

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Appendix A In this section we give a proof (and a little bit more) of the power freeness of the sequence we consider in the end of the Section 3. This proof was suggested by J. Cassaigne [8]. Let us introduce the following two substitutions, defined over {1, 2, 3}, given by: g = σ1 σ2 σ3 σ4 σ4 f = σ1 σ3 σ4 σ4 1 2 3

−→ −→ −→

13 1323 133

1 2 3

−→ −→ −→

13 13223 1323

where the substitutions σi are defined in Section 3. We denote by F the largest language defined over the alphabet {1, 2, 3} which satisfies the following three conditions: • ∀ω ∈ {1, 2, 3}∗, ω 4 ∈ F ⇒ ω = ε, • ∀ω ∈ {1, 2, 3}∗ and ∀z ∈ {1, 2, 3}, (ωz)3 ω ∈ F, • 11 ∈ F. The language F is naturally obtained as the union of all the languages defined over the alphabet {1, 2, 3} which satisfy these three conditions. Lemma 32 If ω ∈ F, then f (ω) and g(ω) are two elements of F . Proof. Let ω be an element of F . We consider three cases to prove that f (ω) ∈ F. 1. Assume there exists a nonempty word M such that M 4 is a factor of f (ω). Then, M could be decomposed in a unique way in xf (v)y, where (x, v, y) ∈ {ε, 3, 23, 33, 323} × {1, 2, 3}∗ × {ε, 1, 13, 132} and the length of v is maximal with the convention that if v ends with the letter 1, then y = ε. We consider two subcases. (a) Let us suppose that v = ε. Then M = xy and thus M ∈ {3, 33, 3313, 32313} ∪ {31, 313, 3132, 2313, 331, 3231} ∪ {23, 33132, 323, 323132} ∪ {23132} ∪ {231}. But M ∈ {3, 33, 3313, 32313} because 33 is always followed by a 1 in f (ω). If M ∈ {31, 313, 3132, 2313, 331, 3231}, we obtain that there exists a letter z ∈ {1, 2, 3} such that z 3 is a factor of ω. This gives a contradiction because ω ∈ F. The word M cannot belong to the set {23, 33132, 323, 323132} because 23 is always followed by a 1 in f (ω). M cannot belong to {23132} because the letter 2 is always followed by a 3 in f (ω). Finally, M cannot belong to {231} because the letter 1 is never followed by a 2 in f (ω).

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(b) Let us suppose that v = ε. Then M 4 = xf (v)yxf (v)yxf (v)yxf (v)y and necessarily yx = f (z) with z ∈ {ε, 1, 2, 3}. If z = ε, then M 4 = f (v 4 ). The fact that v does not end with a 1 allows us to infer that v 4 is a factor of ω. We obtain a contradiction because ω ∈ F. If z is a letter, then f ((vz)3 v) is a factor of f (ω). The fact that v does not end with a 1 shows that (vz)3 v is a factor of ω. We obtain a contradiction because ω ∈ F. 2. Let us suppose that there exist a word M and a letter z such that (M z)3 M is a factor of f (ω). Then, M can be decomposed in a unique way in xf (v)y, where (x, v, y) ∈ {ε, 3, 23, 33, 323} × {1, 2, 3}∗ × {ε, 1, 13, 132} and the length of v is maximal with the convention that if v ends with the letter 1, then y = ε. We obtain that (M z)3 M = xf (v)yzxf (v)yzxf (v)yzxf (v)y, and necessarily xzy = f (m) with m ∈ {1, 2, 3} and |m| ≤ 2 because |xzy|1 ≤ 2 and the letter 1 has exactly one occurrence in the image of each letter. Again we consider two subcases. (a) Let us suppose that |m| = 2. Then there exist two letters a and b such that yzx = f (ab). But |y|1 ≤ 1 and |x|1 = 0 imply that y = f (a) and z = 1. We get (M z)3 M = xf ((vab)3 va). If a = 1, then (vab)3 va is a factor of f (ω) and we obtain a contradiction because ω ∈ F. If a = 1, xf ((v1b)3 v) is a factor of f (ω). We recall that zx = 1x = f (b). It follows that if b = 2 or b = 3, then x = 323 or x = 33 and thus x is always preceded by the letter 1 in f (ω). This implies that 1xf ((v1b)3 v) is a factor of f (ω). But since 1xf ((v1b)3 v) = f ((bv1)3 bv) and v does not end with the letter 1, it follows that (bv1)3 bv is a factor of ω. This is in contradiction with ω ∈ F. Finally, if b = 1, then f ((v11)3 v) is a factor of f (ω). The fact that v does not end with the letter 1 gives that (v11)3 v is a factor of ω and thus 11 is a factor of ω. We get a contradiction since 11 ∈ F. (b) Let us suppose that |m| = 1, then (M z)3 M = xf ((vm)3 v)y. In particular, f ((vm)3 v) is a factor of f (ω). But since v does not end with the letter 1, (vm)3 v is a factor of ω. We obtain a contradiction because m is a letter and ω ∈ F. 3. Let us suppose that 11 is a factor of f (ω). This yields a contradiction immediately because the letter 1 is always followed by a 3 in f (ω) by definition of f. The proof for g is exactly the same.




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Proposition 33 A circle map whose D-expansion (an , in )n∈N satisfies • (in )n∈N = (10)ω , • in = 0 implies an = 0, and • in = 1 implies an ∈ {0, 1} is power free. Proof. Let U be such a circle map and V be the natural coding of the three-interval exchange transformation associated with U . Theorem 18 says that there exists a sequence of integers (bn )n∈N such that   U = lim Φ 1−β  f b0 g b1 f b2 g b3 . . . f b2n (1) n→∞

and thus

α

V = lim f b0 g b1 f b2 g b3 . . . f b2n (1). n→∞

With the previous notation, 1 ∈ F. Then Lemma 32 implies that f b0 g b1 f b2 g b3 . . . f b2n (1) ∈ F for every integer n. We thus obtain L(V ) ⊂ F. This implies that V is 4-power free. Then, in view of the definition of the morphisms Φk , U is clearly power free if  1−β α  > 0 (i.e., 1 − β > α). In the case where 1 − β < α, we can use an argument similar to the one used in the proof of Proposition 23. It is relatively easy to see that if a sequence is not power free, then all of its derived sequences are not power free, either. We have already noticed at the end of the proof of Proposition 23 that for sufficiently large n, there is a morphism Ψ such that V = Ψ(D(n) (U )). Now, if we assume that U is not power free, then D(n) (U ) is not power free and hence V is not power free because morphisms propagate powers. We therefore obtain a contradiction to the 4-power freeness of V obtained above.
In particular, we obtain the power freeness of the sequences mentioned in Section 3. These sequences are of course not LR in view of Theorem 4 and hence they are both power free and not LR. To the best of our knowledge, these are the first examples of sequences with these two properties. We end this appendix with the following conjecture concerning the power freeness of circle maps. Conjecture. A nondegenerate circle map is power free if and only if its D-expansion (an , in )n∈N satisfies the following: there exists an integer M such that for every integer n, we have • an ≤ M , • in = in+1 = . . . = in+M ⇒ ∃k, n ≤ k ≤ n + M such that ak = 0.

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Appendix B We present here what would be the analog of the geometric considerations of Section 3 in the case of I.D.O.C. four-interval exchange transformations which lie in the Rauzy class of (4321). The notion of Rauzy class for an interval exchange transformation was introduced in [36]. Let us introduce the following substitutions, defined on the alphabet {1, 2, 3, 4}, given by 1 2 3 4

σ1 −→ −→ −→ −→

1 14 2 3

1 2 3 4

σ2 −→ −→ −→ −→

14 2 3 4

1 2 3 4

σ3 −→ −→ −→ −→

1 2 3 34

1 2 3 4

σ4 −→ −→ −→ −→

1 2 34 4

1 2 3 4

σ5 −→ −→ −→ −→

1 2 24 3

1 2 3 4

σ6 −→ −→ −→ −→

1 24 3 4

The Rauzy induction graph for the Rauzy class of (4321) is given in Figure 5. The orbit of an I.D.O.C. four-interval exchange transformation in the Rauzy class of (4321) under the Rauzy induction cannot be ultimately confined to one of its primitivity subgraphs G1 , G2 , G3 or G4 represented in Figures 6, 7, 8, and 9, respectively. Moreover, an I.D.O.C. four-interval exchange in the Rauzy class of (4321) is LR if and only if its orbit under the Rauzy induction can stay in any of the primitivity subgraphs G1 , G2 , G3 , and G4 only for a bounded number of consecutive induction steps. 2431

4132

σ5 σ6

σ2

σ2

σ1 σ3

σ5

σ4

3142

σ4 4213

σ1

4321

σ6

2413

σ1

3241

σ3

σ2

Figure 5: The Rauzy induction graph for the Rauzy class of (4321).

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4132

σ5

σ2

σ6

σ5 3142

4321

σ6

σ4 4213

σ3

Figure 6: The primitivity subgraph G1 for the Rauzy class of (4321).

2431

4132

σ2

σ2

σ1 σ4

σ4 4213

σ1

4321

σ6

2413

σ1

3241

σ3

Figure 7: The primitivity subgraph G2 for the Rauzy class of (4321).

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2431

4132

σ5 σ6

σ2

σ1

σ5 3142

σ1

4321

σ6

σ1

σ4 4213

3241

σ2

Figure 8: The primitivity subgraph G3 for the Rauzy class of (4321).

2431

σ2

σ1 σ3 σ4 σ1

4321

2413

σ1

3241

σ2

Figure 9: The primitivity subgraph G4 for the Rauzy class of (4321).

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References [1] B. Adamczewski, Codages de rotations et ph´enom`enes d’autosimilarit´e, to appear in J. Th´eor. Nombres Bordeaux. [2] P. Arnoux, S. Ferenczi and P. Hubert, Trajectories of rotations, Acta Arith. 87, 209–217 (1999). [3] J. Bellissard, B. Iochum, E. Scoppola and D. Testard, Spectral properties of one-dimensional quasi-crystals, Commun. Math. Phys. 125, 527–543 (1989). [4] J. Bellissard, B. Iochum and D. Testard, Continuity properties of the electronic spectrum of 1D quasicrystals, Commun. Math. Phys. 141, 353–380 (1991). [5] A. Bovier and J.-M. Ghez, Spectral properties of one-dimensional Schr¨ odinger operators with potentials generated by substitutions, Commun. Math. Phys. 158, 45–66 (1993); Erratum: Commun. Math. Phys. 166, 431–432 (1994). [6] M. Casdagli, Symbolic dynamics for the renormalization group of a quasiperiodic Schr¨ odinger equation, Commun. Math. Phys. 107, 295–318 (1986). [7] J. Cassaigne, Sequences with grouped factors, Developments in Language Theory III, Aristotle University of Thessaloniki, 211–222 (1998). [8] J. Cassaigne, private communication [9] D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in Directions in Mathematical Quasicrystals, M. Baake, R. V. Moody, eds., CRM Monograph Series 13, AMS, Providence, RI, 277–305 (2000). [10] D. Damanik, R. Killip and D. Lenz, Uniform spectral properties of onedimensional quasicrystals, III. α-continuity, Commun. Math. Phys. 212, 191– 204 (2000). [11] D. Damanik and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues, Commun. Math. Phys. 207, 687–696 (1999). [12] D. Damanik and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent, Lett. Math. Phys. 50, 245–257 (1999). [13] D. Damanik and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, IV. Quasi-Sturmian potentials, preprint, Caltech and TU Chemnitz (2001). [14] D. Damanik and D. Lenz, Linear repetitivity, I. Uniform subadditive ergodic theorems and applications, Discrete Comput. Geom. 26, 411–428 (2001).

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[15] D. Damanik and D. Zare, Palindrome complexity bounds for primitive substitution sequences, Discrete Math. 222, 259–267 (2000). [16] F. Delyon and D. Petritis, Absence of localization in a class of Schr¨ odinger operators with quasiperiodic potential, Commun. Math. Phys. 103, 441–444 (1986). [17] F. Durand, B. Host and C. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam. Systems 19, 953–993 (1999). [18] F. Durand, A characterization of substitutive sequences using return words, Discrete Math. 179, 89–101 (1998). [19] F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam. Systems 20, 1061–1078 (2000). [20] F. Durand, Corrigendum and appendum to: Linearly recurrent subshifts have a finite number of non-periodic subshift factors, preprint, LAMFA, Facult´e de Math´ematiques et d’Informatique, Universit´e de Picardie Jules Verne (2001). [21] S. Ferenczi, Rank and symbolic complexity, Ergodic Theory Dynam. Systems 16, 663–682 (1996). [22] S. Ferenczi, C. Holton and L. Zamboni, Structure of three interval exchange transformations. I. An arithmetic study, Ann. Inst. Fourier (Grenoble) 51, 861–901 (2001). [23] A. Hof, Some remarks on discrete aperiodic Schr¨ odinger operators, J. Statist. Phys. 72, 1353–1374 (1993). [24] A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schr¨ odinger operators, Commun. Math. Phys. 174, 149–159 (1995). [25] M. H¨ ornquist and M. Johansson, Singular continuous electron spectrum for a class of circle sequences, J. Phys. A 28, 479–495 (1995). [26] B. Iochum, L. Raymond and D. Testard, Resistance of one-dimensional quasicrystals, Physica A 187, 353–368 (1992). [27] B. Iochum and D. Testard, Power law growth for the resistance in the Fibonacci model, J. Stat. Phys. 65, 715–723 (1991). [28] S. Jitomirskaya, Singular spectral properties of a one-dimensional discrete Schr¨ odinger operator with quasiperiodic potential, Adv. Sov. Math. 3, 215– 254 (1991). [29] M. Kaminaga, Absence of point spectrum for a class of discrete Schr¨ odinger operators with quasiperiodic potential, Forum Math. 8, 63–69 (1996).

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[30] M. Keane, Interval exchange transformations, Math. Z. 141, 25–31 (1975). [31] S. Kotani, Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys. 1, 129–133 (1989). [32] J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, to appear in Ergodic Theory Dynam. Systems [33] Y. Last and B. Simon, Eigenfunctions, transfer matrices and absolutely continuous spectrum of one-dimensional Schr¨ odinger operators, Invent. Math. 135, 329–367 (1999). [34] D. Lenz, Uniform ergodic theorems on subshifts over a finite alphabet, Ergodic Theory Dynam. Systems 22, 245–255 (2002). [35] D. Lenz, Singular spectrum of Lebesgue measure zero for quasicrystals, Commun. Math. Phys. 227, 119–130 (2002). ´ [36] G. Rauzy, Echanges d’intervalles et transformations induites, Acta Arith. 34, 315–328 (1979). [37] G. Rote, Sequences with subword complexity 2n, J. Number Theory 46, 196– 213 (1994). [38] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom. 20, 265–279 (1998). [39] A. S¨ ut˝ o, The spectrum of a quasiperiodic Schr¨ odinger operator, Commun. Math. Phys. 111, 409–415 (1987). [40] A. S¨ ut˝ o, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Stat. Phys. 56, 525–531 (1989).

Boris Adamczewski Institut de Mathmatique de Luminy CNRS UPR 9016 13288 Marseille cedex 09 France email: [email protected]

Communicated by Jean Bellissard submitted 31/05/02, accepted 11/07/02

David Damanik Department of Mathematics 253–37 California Institute of Technology Pasadena, CA 91125 USA email: [email protected]

Ann. Henri Poincar´e 3 (2002) 1049 – 1111 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/0601049-63

Annales Henri Poincar´ e

Kasner-Like Behaviour for Subcritical Einstein-Matter Systems T. Damour, M. Henneaux, A.D. Rendall and M. Weaver Abstract. Confirming previous heuristic analyses ` a la Belinskii-Khalatnikov-Lifshitz, it is rigorously proven that certain “subcritical” Einstein-matter systems exhibit a monotone, generalized Kasner behaviour in the vicinity of a spacelike singularity. The D-dimensional coupled Einstein-dilaton-p-form system is subcritical if the dilaton couplings of the p-forms belong to some dimension-dependent open neighborhood of zero [1], while pure gravity is subcritical if D ≥ 11 [13]. Our proof relies, like the recent Theorem [15] dealing with the (always subcritical [14]) Einstein-dilaton system, on the use of Fuchsian techniques, which enable one to construct local, analytic solutions to the full set of equations of motion. The solutions constructed are “general” in the sense that they depend on the maximal expected number of free functions.

1 Introduction 1.1

The problem

In recent papers [1, 2, 3], the dynamics of the coupled Einstein-dilaton-p-form system in D spacetime dimensions, with action (in units where 8πG = 1), (j)

S[gαβ , φ, Aγ1 ···γnj ] = SE [gαβ ] + Sφ [gαβ , φ] + 

k


(j)

Sj [gαβ , φ, Aγ1 ···γnj ] + “more”,

(1.1)

j=1

√ (1.2) R −g dD x,  √ 1 (1.3) Sφ [gαβ , φ] = − ∂µ φ ∂ µ φ −g dD x, 2  √ 1 (j) (j) Fµ1 ···µnj +1 F (j) µ1 ···µnj +1 eλj φ −g dD x, Sj [gαβ , φ, Aγ1 ···γnj ] = − 2(nj + 1)! (1.4) SE [gαβ ] =

1 2

was investigated in the vicinity of a spacelike (“cosmological”) singularity along the lines initiated by Belinskii, Khalatnikov and Lifshitz (BKL) [4]. In (1.1), gαβ is the spacetime metric, φ is a massless scalar field known as the “dilaton”, while the (j) Aγ1 ···γnj are a collection of k exterior form gauge fields (j = 1, . . . , k), with exponential couplings to the dilaton, each coupling being characterized by an individual

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constant λj (“dilaton coupling constant”). The F (j) ’s are the exterior derivatives F (j) = dA(j) , whereas “more” stands for possible coupling terms among the pforms which can be either of the Yang-Mills type (1-forms), Chern-Simons type [5] or Chapline-Manton type [6, 7]. The degrees of the p-forms are restricted to be smaller than or equal to D − 2 since a (D − 1)-form (or D-form) gauge field carries no local degree of freedom. In particular, scalars (nj = 0) are allowed among the A(j) ’s but we then require that the corresponding dilaton coupling λj be non-zero, so that there is only one “dilaton”. Similarly, we require λj = 0 for the (D − 2)forms (if any), since these are “dual”1 to scalars. This restriction to a single dilaton is mostly done for notational convenience: if there were other dilatons among the 0-forms, then, these must be explicitly treated on the same footing as φ and separated off from the p-forms because they play a distinct rˆ ole. In particular, they would appear explicitly in the generalized Kasner conditions given below and in the determination of what we call the subcritical domain. The discussion would proceed otherwise in the same qualitative way. The main motivation for studying actions of the class (1.1) is that these arise as bosonic sectors of supergravity theories related to superstring or M-theory. In fact, in view of various no-go theorems, p-form gauge fields appear to be the only massless, higher spin fields that can be consistently coupled to gravity. Furthermore, there can be only one type of graviton [8]. With this observation in mind, the Action (1.1) is actually quite general. The only restriction concerns the scalar sector: we assume the coupling to the dilaton to be exponential because this corresponds to the tree-level couplings of the dilaton field of string theory. Note, however, that string-loop effects are expected to generate more general couplings exp(λφ) → B(φ) which can exhibit interesting “attractor” behaviours [9]. We also restrict ourselves by not including scalar potentials; see, however, the end of the article for some remarks on the addition of a potential for the dilaton, which can be treated by our methods. Two possible general, “competing” behaviours of the fields in the vicinity of the spacelike singularity have been identified2 : 1. The simplest is the “generalized Kasner behaviour”, in which the spatial scale factors and the field exp(φ) behave at each spatial point in a monotone, power-law fashion in terms of the proper time as one approaches the singularity, while the effect of the p-form fields A(j) ’s on the evolution of gµν and φ can be asymptotically neglected. In that regime the spatial curvature terms can be also neglected with respect to the leading order part of the extrinsic curvature terms. In other words, as emphasized by BKL, time derivatives asymptotically dominate over space derivatives so that one sometimes uses the terminology “velocity-dominated” behaviour [11], instead of “general1 We

recall that the Hodge duality between a (nj + 1)-form and a (D − nj − 1)-form allows
one to replace (locally) a nj -form potential A(j) by a (D − nj − 2)-form potential A(j ) (with
dilaton coupling λj = −λj ). 2 For a recent extension of these ideas to the brane-worlds scenarios, see [10].

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ized Kasner behaviour”. We shall use both terminologies indifferently in this paper, recalling that in the presence of p-forms, which act as potentials for the evolution of the spatial metric and the dilaton (as do the spatial curvature terms), “velocity-dominance” means not only that the spatial curvature terms can be neglected, but also that the p-forms can be neglected in the Einstein-dilaton evolution equations.3 2. The second regime, known as “oscillatory” [4], or “generalized mixmaster” [12] behaviour, is more complicated. It can be described as the succession of an infinite number of increasingly shorter Kasner regimes as one goes to the singularity, one following the other according to a well-defined “collision” law. This asymptotic evolution is presumably strongly chaotic. It is expected that, at each spatial point, the scale factors of a general inhomogeneous solution essentially behave as in certain homogeneous models. For instance, for D = 4 pure gravity this guiding homogeneous model is the Bianchi IX model [4, 12], while for D = 11 supergravity it is its naive one-dimensional reduction involving space-independent metric and three-forms [2]. Whether it is the first or the second behaviour that is relevant depends on: (i) the spacetime dimension D, (ii) the field content (presence or absence of the dilaton, types of p-forms), and (iii) the values of the various dilaton couplings λj . Previous work reached the following conclusions: • The oscillatory behaviour is general for pure gravity in spacetime dimension 4 [4], in fact, in all spacetime dimensions 4 ≤ D ≤ 10, but is replaced by a Kasner-like behaviour in spacetime dimensions D ≥ 11 [13]. (The sense in which we use “general” will be made precise below.) • The Kasner-like behaviour is general for the gravity-dilaton system in all spacetime dimensions D ≥ 3 (see [14, 15] for D = 4). • The oscillatory behaviour is general for gravity coupled to p-forms, in absence of a dilaton or of a dual (D − 2)-form (0 < p < D − 2) [2]. In particular, the bosonic sector of 11-dimensional supergravity is oscillatory [1]. Particular instances of this case have been studied in [16, 17, 18]. • The case of the gravity-dilaton-p-form system is more complicated to discuss because its behaviour depends on a combination of several factors, namely the dimension D, the menu of p-forms, and the numerical values of the dilaton couplings. For a given D and a given menu of p-forms there exists a “subcritical” domain D (an open neighborhood of the origin λj = 0 for all j’s) such that: (i) when the λj belong to D the general behaviour is Kasner3 The

Kasner solution is generalized in two ways: first, the original Kasner exponents include a dilaton exponent (if there is a dilaton), which appears in the Kasner conditions; second, the exponents are not assumed to be constant in space. We shall shorten “exhibits generalized Kasner behaviour” to Kasner-like. We stress that we do not use this term to indicate that the solution becomes asymptotically homogeneous in space.

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like, but (ii) when the λj do not belong to D the behaviour is oscillatory. Note that D is open. Indeed, the behaviour is oscillatory when the λj are on the boundary of D, as happens for instance for the low-energy bosonic sectors of type I or heterotic superstrings [1]. For a single p-form, the subcritical domain D takes the simple form |λj | < λcj , where λcj depends on the formdegree and the spacetime dimension. (λcj can be infinite.) For a collection of p-forms, D is more complicated and not just given by the Cartesian product of the subcritical intervals associated with each individual p-form. The above statements were derived by adopting a line of thought analogous to that followed by BKL. Now, as understood by BKL themselves, these arguments, although quite convincing, are somewhat heuristic. It is true that the original arguments have received since then a considerable amount of both numerical and analytical support [19, 20, 21, 22, 23]. Yet, they still await a complete proof. One notable exception is the four-dimensional gravity-dilaton system, which has been rigorously demonstrated in [15] to be indeed Kasner-like, confirming the original analysis [14]. Using Fuchsian techniques, the authors of [15] have proven the existence of a local (analytic) Kasner-like solution to the Einstein-dilaton equations in four dimensions that contains as many arbitrary, physically relevant functions of space as there are local degrees of freedom, namely 6 (counting q and q˙ independently). To our knowledge, this was the first construction, in a rigorous mathematical sense, of a general singular solution for a coupled Einstein-matter system. Note in this respect several previous works in which formal solutions had been constructed near (Kasner-like) cosmological singularities by explicit perturbative methods, to all orders of perturbation theory [24, 25]. The situation concerning the more complicated (and in some sense more interesting) generalized mixmaster regime is unfortunately – and perhaps not surprisingly – not so well developed. Rigorous results are scarce (note [26]) and even in the case of the spatially homogeneous Bianchi IX model only partial results exist in the literature [27]. The purpose of this paper is to extend the Fuchsian approach of [15] to the more complicated class of models described by the Action (1.1) and to prove that those among the above models that were predicted in [13, 1, 2] to be Kasner-like are indeed so. This provides many new instances where one can rigorously construct a general singular solution for a coupled Einstein-matter (or pure Einstein, in D ≥ 11) system. In fact, our (Fuchsian-system-based) results prove that the formal perturbative solutions that can be explicitly built for these models do converge to exact solutions. This provides a further confirmation of the general validity of the BKL ideas. We shall also explicitly determine the subcritical domain D for a few illustrative models. For all the relevant systems, we construct local (near the singularity) analytic solutions, which are “general” in the sense that they contain the right number of freely adjustable arbitrary functions of space (in particular, these solutions have generically no isometries), and which exhibit the generalized (monotone) Kasner time dependence.

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Strategy and outline of the paper

Our approach is the same as in [15], and results from that work will be used frequently here without restating the arguments. Here is an outline of the key steps. A d + 1 decomposition is used, for d spatial dimensions, d = D − 1. A Gaussian time coordinate, t, is chosen such that the singularity occurs at t = 0. The first step in the argument consists of identifying the leading terms for all the variables. This is accomplished by writing down a set of evolution equations which is obtained by truncating the full evolution equations, and then solving this simpler set of evolution equations. This simpler evolution system is called the Kasnerlike4 evolution system (or, alternatively, the velocity-dominated system). It is a system of ordinary differential equations with respect to time (one at each spatial point) which coincides with the system that arises when investigating metricdilaton solutions that depend only on time. The precise truncation rules are given in Subsection 2.2 below. The second step is to write down constraint equations for the Kasner-like system (called “velocity-dominated” constraints) and to show that these constraints propagate, i.e., that if they are satisfied by a solution to the Kasner-like evolution equations at some time t0 > 0, then they are satisfied for all time t > 0. In the set of Kasner-like solutions, one expects that there is a subset, denoted by V , of solutions which have the property of being asymptotic to solutions of the complete Einstein-dilaton-p-form equations as t → 0, i.e., as one goes to the singularity. This subset is characterized by inequalities on some of the initial data, which, however, are not always consistent. The existence of a non-empty V requires the dilaton couplings to belong to some range, the “subcritical range”. When V is non-empty and open, the solutions in V involve as many arbitrary functions of space as a “general solution” of the full Einstein equations should. On the other hand, if V is empty the construction given in this paper breaks down and the dynamical system is expected to be not Kasner-like but rather oscillatory. To show that indeed, the solutions in V (when it is non-empty) are asymptotic to true solutions, the third step is to identify decaying quantities such that these decaying quantities along with the leading terms mentioned above uniquely determine the variables, and to write down a Fuchsian system for the decaying quantities which is equivalent to the Einstein-matter evolution system. As the use of Fuchsian systems is central to our work let us briefly recall what a Fuchsian system is and how such a system is related to familiar iterative methods. For a more detailed introduction to Fuchsian techniques see [15, 28, 29, 30] and references therein. Note that we shall everywhere restrict ourselves to the analytic case. We expect that our results extend to the C ∞ case, but it is a non-trivial task to prove that they do. 4 Note that we use the terms “Kasner-like solutions” to label both exact solutions of the truncated system and solutions of the full system that are asymptotic to such solutions. Which meaning is relevant should be clear from the context.

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The general form of a Fuchsian system for a vector-valued unknown function u is t ∂t u + A(x) u = f (t, x, u, ux),

(1.5)

where the matrix A(x) is required to satisfy some positivity condition (see below), while the “source term” f on the right-hand side is required to be “regular.” (See [15] for precise criteria allowing one to check when the positivity assumption on A(x) is satisfied and when f is regular.) A key point is that f is required to be bounded by terms of order O(tδ ) (with t → 0, δ > 0) as soon as u and their space derivatives ux are in a bounded set (a simple, concrete example of a source term satisfying this condition is f = tδ1 + tδ2 u + tδ3 ux , with δi ’s larger than δ). A convenient form of positivity condition to be satisfied by the matrix A(x) is that the operator norm of τ A(x) be bounded when 0 < τ < 1 (and when x varies in any open set). Essentially this condition restricts the eigenvalues of the matrix A(x) to have positive real parts. The basic property of Fuchsian systems that we shall use is that there is a unique solution to the Fuchsian equation which vanishes as t tends to zero [28]. One can understand this theorem as a mathematically rigorous version of the recursive method for solving the Equation (1.5). Indeed, when confronted with Equation (1.5), it is natural to construct a solution by an iterative process, starting with the zeroth order approximation u0 = 0 (which is the unique solution of (1.5) with f ≡ 0 that tends to zero as t → 0), and solving (n−1) a sequence of equations of the form t∂t u(n) + A(x)u(n) = f (t, x, u(n−1) , ux ). At each step in this iterative process the source term is a known function which essentially behaves (modulo logarithms) like a sum of powers of t (with spacedependent coefficients). The crucial step in the iteration is then to solve equations of the type t∂t u + A(x)u = C(x)tδ(x) . The positivity condition on A(x) guarantees the absence of homogeneous solutions remaining bounded as t → 0, and ensures the absence of “small denominators” in the (unique bounded) inhomogeneous solution generated by each partial source term: uinhom = (δ + A)−1 Ctδ . (See, e.g., [25] for a concrete iterative construction of a Kasner-like solution and the proof that it extends to all orders.) This link between Fuchsian systems and “good systems” that can be solved to all orders in a formal iteration makes it a priori probable that all cases which the heuristic approach `a la BKL has shown to be asymptotic to a Kasner-like solution (by checking that the leading “post-Kasner” contribution is asymptotically sub-dominant) can be cast in a Fuchsian form. The main technical burden of the present work will indeed be to show in detail how this can be carried out for the evolution systems corresponding to all the sub-critical (i.e., non-oscillatory) Einstein-matter systems. Our Fuchsian formulation proves that (in the analytic case) the formal all-orders iterative solutions for the models we consider do actually converge to the unique, exact solution having a given leading Kasner asymptotic behaviour as t → 0. Finally, the fourth step of our strategy is to prove that the constructed solution does satisfy also all the Einstein and Gauss-like constraints so that it is

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a solution of the full set of Einstein-matter equations. We shall deal successively with the matter (Gauss-like) constraints, and the Einstein constraints. Our paper is organized as follows. In Section 2, we first consider the paradigmatic example of gravity coupled to a massless scalar field and to a Maxwell field in 4 spacetime dimensions. The Action (1.1) reads in this case S[gαβ , φ, Aγ ] =

1 2



√ 1 {R − ∂µ φ ∂ µ φ − Fµν F µν eλφ } −g d4 x. 2

(1.6)

For this simple example, we shall explicitly determine the subcritical domain D, i.e., the critical value λc such that the system is Kasner-like when −λc < λ < λc . Because this case is exemplary of the general situation, while still being technically rather simple to handle, we shall describe in some detail the explicit steps of the Fuchsian approach. In Section 3, vacuum solutions governed by the pure Einstein Action (1.2) with D ≥ 11 are considered. This system was argued in [13] to be Kasner-like and we show here how this rigorously follows from the Fuchsian approach. Note that, contrary to what happens when a dilaton is present, Fuchsian techniques apply here even though not all Kasner exponents can be positive. In Sections 4–8, the results of the previous sections are generalized to the wider class of systems (1.1). First, in Section 4, to solutions of Einstein’s equation with spacetime dimension D ≥ 3 and a matter source consisting of a massless scalar field, governed by the action SE [gαβ ] + Sφ [gαβ , φ]. This is the generalization to any D ≥ 3 of the case D = 4 treated in [15]. In Section 5, we turn to the general situation described by the Action (1.1), without, however, including the additional terms represented there by “more”. We then give some general rules for computing the subcritical domain of the dilaton couplings guaranteeing velocity-dominance (Section 6). The inclusion of interaction terms is considered in the last sections. It is shown that they do not affect the asymptotic analysis. This is done first for the Chern-Simons and Chapline-Manton interactions in Section 7, and next, in Section 8, for the Yang-Mills couplings (for some gauge group G), for which the action reads  √ 1 1 {R − ∂µ φ ∂ µ φ − Fµν · F µν eλφ } −g dD x. (1.7) S[gαβ , φ, Aγ ] = 2 2 Here the dot product, F · F , is a time-independent, Ad-invariant, non-degenerate scalar product on the Lie algebra of G (such a scalar product exists if the algebra is compact, or semi-simple). Contrary to what is done in Sections 2, 5 and 7, we must work now with the vector potential (and not just with the field strength), since it appears explicitly in the coupling terms. In Section 9 we show that self-interactions of a rather general type for the scalar field can be included without changing the asymptotics of the solutions.

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Explicitly, we add a (nonlinear) potential term,  √ SNL [gαβ , φ] = − V (φ) −g dD x,

Ann. Henri Poincar´e

(1.8)

to the Action (1.1), where V (φ) must fulfill some assumptions given in Section 9. V (φ) may, for example, be an exponential function of φ, a constant, or a suitable power of φ. Similar forms for V (φ) were considered with D = 4 in [31]. Finally, in Section 10, we state two theorems that summarize the main results of the paper and give concluding remarks.

1.3

On the generality of our construction

As we shall see the number of arbitrary functions contained in solutions to the velocity-dominated constraint equations is equal to the number of arbitrary functions for solutions to the Einstein-matter constraints. In this function-counting sense, our construction describes what is customarily called a “general” solution of the system. Intuitively speaking, our construction concerns some “open set” of the set of all solutions (indeed, the Kasner-like behaviour of the solution is unchanged under arbitrary, small perturbations of the initial data, because this simply amounts to changing the integration functions). Note that, in the physics literature, such a “general” solution is often referred to as being a “generic” solution. However, in the mathematics literature the word “generic” is restricted to describing either an open dense subset of the set of all solutions, or (when this can be defined) a subset of measure unity of the set of all solutions. In this work we shall stick to the mathematical terminology. We shall have nothing rigorous to say about whether our general solution is also generic. However, we wish to emphasize the following points. First, let us mention that the set V of solutions to the velocity-dominated equations that are asymptotic to solutions of the complete equations is not identical to the set U of all solutions to the velocity-dominated constraint equations. The subset V ⊂ U is defined by imposing some inequalities on the free data. These inequalities do not change the number of free functions. Therefore the solutions in V are still “general”. One can wonder whether there could be a co-existing general behaviour, corresponding to initial data that do not fulfill the inequalities. For instance, could such “bad” initial data lead to a generalized mixmaster regime? This is a difficult question and we shall only summarize here what is the existing evidence. There are heuristic arguments, supported by numerical study, [14, 32, 33, 34] that suggest that if one starts with initial data that do not fulfill the inequalities, one ends up, after a finite transient period (with a finite number of “collisions” with potential walls), with a solution that is asymptotically velocity-dominated, for which the inequalities are fulfilled almost everywhere. In that sense, the inequalities would not represent a real restriction since there is a dynamical mechanism that drives the solution to the regime where they are

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satisfied. For the subcritical values of the dilaton couplings that make the inequalities defining V consistent, there is thus no evidence for an alternative oscillatory regime corresponding to a different (open) region in the space of initial data5 . It has indeed been shown that the inequalities defining V are no restriction in a large spatially homogeneous class [27]. Such rigorous results are, however, lacking in the inhomogeneous case. In fact, an interesting subtlety might take place in the inhomogeneous case. The heuristic arguments and numerical studies of [33, 34] suggest the possibility that the mechanism driving the system to V may be suppressed at exceptional spatial points in general spacetimes, with the result that the asymptotic data at the exceptional spatial points are not consistent with the inequalities we assume and lead to so-called “spikes”. This picture has been given a firm basis in a scalar field model with symmetries [35] but the status of the spikes in a general context remains unclear. Finally, since we only deal with spacelike singularities, the classes of solutions we consider do not contain all solutions governed by the Action (1.1). Other types of singularities (e.g. timelike or null ones) are known to exist. Whether these other types of singularities are general is, however, an open question.

1.4

Billiard picture

At each spatial point, the solution of the coupled Einstein-matter system can be pictured, in the vicinity of a spacelike singularity, as a billiard motion in a region of hyperbolic space [36, 37, 3, 38]. Hyperbolic billiards are chaotic when they have finite volume and non chaotic otherwise. In this latter case, the “billiard ball” generically escapes freely to infinity after a finite number of collisions with the bounding walls. Subcritical Einstein-matter systems define infinite-volume billiards. The velocity-dominated solutions correspond precisely to the last (as t → 0) free motion (after all collisions have taken place), in which the billiard ball moves to infinity in hyperbolic space.

1.5

Conventions

We adopt a “mostly plus” signature (− + + + . . .). The spacetime dimension is D ≡ d + 1. Greek indices range from 0 to d, while Latin indices ∈ {1, . . . , d}. The spatial Ricci tensor is labeled R and the spacetime Ricci tensor is labeled (D) R. Our curvature conventions are such that the Ricci tensor of a sphere is positive definite. Einstein’s equations read Gαβ = Tαβ , where Gαβ = Rαβ − Rgαβ /2 denotes the√Einstein tensor and Tαβ denotes the matter stress-energy tensor, Tαβ = −(2/ −g)δSmatter /δg αβ , and units such that 8πG = 1. The spatial metric compatible covariant derivative is labeled ∇a and the spacetime metric compatible covariant derivative is labeled (D) ∇α . The velocity-dominated metric compatible 5 The oscillatory regime may however be present for peculiar initial data, presumably forming a set of zero measure. For instance, gravity + dilaton is generically Kasner-like, but exhibits an oscillatory behaviour for initial data with φ = 0 (in D < 11 spacetime dimensions).

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covariant derivative is labeled 0 ∇a . According to the context, g denotes the (positive) determinant of gab in d + 1-decomposed expressions, and the (negative) determinant of gµν in spacetime expressions. Whenever tδ or t−δ appears, δ denotes a strictly positive number, arbitrarily small. We use Einstein’s summation convention on repeated tensor indices of different variances. (When the need arises to suspend the summation conventions for some non-tensorial indices, we shall explicitly mention it.) In expressions where there is a sum that the indices do not indicate, all sums in the expression are indicated explicitly by a summation symbol. Indices on the velocity-dominated metric and the velocity-dominated extrinsic curvature are raised and lowered with the velocity-dominated metric.

1.6

d + 1 decomposition

Consider a solution to the Einstein’s equations following from (1.1), consisting of a Lorentz metric and matter fields on a D-dimensional manifold M which is diffeomorphic to (0, T ) × Σ for a d-dimensional manifold, Σ, such that the metric induced on each t = constant hypersurface is Riemannian, for t ∈ (0, T ). Here D is an integer strictly greater than two. Furthermore, consider a d + 1 decomposition of the Einstein tensor, Gαβ , and the stress-energy tensor, Tαβ , with a Gaussian time coordinate, t ∈ (0, T ), and a local frame {ea } on Σ. Note that the frame ea = eia (x)∂i is time-independent. The spacetime metric reads ds2 = −dt2 + gab (t, x)ea eb , where ea = eai (x)dxi (with eai eib = δba ) is the co-frame. Let ρ = T00 , ja = −T0a and Sab = Tab . Define C

=

2G00 − 2T00

=

−k

a

b

k

b

a

(1.9) 2

+ (tr k) + R − 2ρ.

C = 0 is the Hamiltonian constraint. Similarly, Ca = 0 is the momentum constraint, where Ca

= −G0a + T0a = ∇b k

b

a

(1.10)

− ∇a (tr k) − ja .

In Gaussian coordinates, the relation between the metric and the extrinsic curvature is ∂t gab = −2kab . (1.11) The evolution equation for the extrinsic curvature is obtained by setting E a b = 0, with 1 T δab (1.12) E a b = (D) Ra b − T a b + (D − 2) (1.13) ⇒ ∂t k a b = Ra b + (tr k) k a b − M a b . Here M ab = Sab −

1 ((tr S) − ρ)δ a b . D−2

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2 Scalar and Maxwell fields in four dimensions 2.1

Equations of motion

As said above, let us start by considering in detail, as archetypal system, the system defined by the Action (1.6), i.e., the spacetime dimension is D = 4 and the matter fields are a massless scalar field exponentially coupled to a Maxwell field, with the magnitude of the dilaton coupling constant smaller in magnitude than some positive real number determined below, 0 ≤ |λ| < λc . The stress-energy tensor of the matter fields is 1 1 Tµν = (4) ∇µ φ (4) ∇ν φ − gµν (4) ∇α φ (4) ∇α φ + [Fµα Fν α − gµν Fαβ F αβ ]eλφ . 2 4 The matter fields satisfy the following equations. (4) (4)

∇α (4) ∇α φ =

∇µ (F µν eλφ ) = (4)

∇[α Fβγ]

=

λ Fαβ F αβ eλφ , 4 0, 0.

The 3+1 decomposition of the stress-energy tensor is best expressed in terms of the √ electric spatial vector density E a = g F 0a eλφ and the magnetic antisymmetric spatial tensor Fab . 1 1 1 {(∂t φ)2 + g ab ea (φ)eb (φ) + gab E a E b e−λφ + g ab g ch Fac Fbh eλφ }, 2 g 2 1 ja = −∂t φ ea (φ) + √ E b Fab , g 1 1 M a b = g ac eb (φ) ec (φ) − {gbc E a E c − δ a b gch E c E h }e−λφ g 2 1 a ch ij ac hi +{g g Fch Fbi − δ b g g Fci Fhj }eλφ . (2.1) 4 The matter constraint equations are ρ

=

b ea (E a ) + fba Ea h Fc]h e[a (Fbc] ) + f[ab

= =

0 0.

(2.2) (2.3)

c are the (time-independent) structure functions of the frame, [ea , eb ] = Here fab c fab ec . The matter evolution equations are

λ λ gab E a E b e−λφ − g ab g ch Fac Fbh eλφ , 2g 4 √ 1 a ic √ bh i ac ∂t E a = eb ( gg ac g bh Fch eλφ ) + (fib g + fbi g ) gg Fch eλφ , 2 1 c 1 ∂t Fab = −2e[a ( √ gb]c E c e−λφ ) + fab √ gch E h e−λφ . g g

∂t2 φ − (trk)∂t φ = g ab ∇a ∇b φ +

(2.4) (2.5) (2.6)

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Velocity-dominated evolution equations and solution

The Kasner-like, or velocity-dominated, evolution equations are obtained from the full evolution equations by: (i) dropping the spatial derivatives from the rightc -terms count as derivatives hand sides of (1.13), (2.4), (2.5) and (2.6) (note that fab and that we keep the time derivatives of the magnetic field in (2.6) even though Fab = ∂a Ab − ∂b Aa ); and (ii) dropping the p-form terms in both the Einstein and dilaton evolution equations. This is a general rule and yields in this case ∂t 0 gab ∂t 0 k a b

= =

∂t2 0 φ − (tr 0 k) ∂t 0 φ = 0 a

∂t E ∂t 0 Fab

= =

−2 0 kab , (tr 0 k) 0 k a b ,

(2.7) (2.8)

0,

(2.9)

0, 0.

(2.10) (2.11)

(As we shall see below, interaction terms of Yang-Mills or other types – if any – should also be dropped.) It is easy to find the general analytic solution of the evolution system (2.7)– (2.11) since the equations are the same as for “Bianchi type I” homogeneous models (one such set of equations per spatial point). Taking the trace of (2.8) shows that −1/tr 0 k = t+C(x). By a suitable redefinition of the time variable one can set C(x) to zero. Then (2.8) shows that −t 0 k a b ≡ K a b is a constant matrix (which must satisfy trK = K a a = 1, and be such that 0 gac (t0 )K c b is symmetric in a and b), 0 a

k b (t) = −t−1 K a b .

(2.12)

Injecting this information into (2.7) leads to a linear evolution system for 0 gab : t ∂t 0 gab = 2 0 gac K c b , which is solved by exponentiation,   c 2K t 0 0 gab (t) = gac (t0 ) . (2.13) t0 b

The other evolution equations are also easy to solve, 0

φ(t) E (t)

= =

Fab (t)

=

0 a 0

A ln t + B, E

0 a 0

Fab .

(2.14) (2.15) (2.16)

In (2.13) (t/t0 )2K denotes the exponentiation of a matrix. Quantities on the lefthand side of (2.12)–(2.16) may be functions of both time and space, while all the time dependence of the right-hand side is made explicit. For instance, (2.16) is saying that the spacetime dependence of the general magnetic field 0 Fab (t, x) (solution of the velocity-dominated evolution system) is reduced to a simple space dependence 0 Fab (x) (where 0 Fab is an antisymmetric spatial tensor). Let pa denote the eigenvalues of K a b , ordered such that p1 ≤ p2 ≤ p3 . Since trK = 1, we have

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the constraint p1 + p2 + p3 = 1.

(2.17)

In the works of BKL the matrix Solution (2.13) is simplified by using a special frame {ea } with respect to which the matrices 0 gab (t0 ) and K a b are diagonal. However, as emphasized in [15], this choice can not necessarily be made analytically on neighborhoods where the number of distinct eigenvalues of K a b is not constant. To obtain an analytic solution, while still controlling the relation of the solution to the eigenvalues of K a b , a special construction was introduced in [15]. This construction is based on some (possibly small) neighborhood U0 of an arbitrary spatial point x0 ∈ Σ and uses a set of auxiliary exponents qa (x). These auxiliary exponents remain numerically close to the exact “Kasner exponents” pa (x), are analytic and enable one to define an analytic frame (see below). To construct the auxiliary exponents qa (x) one distinguishes three cases: Case I (near isotropic): If all three eigenvalues are equal at x0 , choose a number  > 0 so that for x ∈ U0 , maxa,b |pa (x) − pb (x)| < /2. In this case define qa = 1/3 on U0 , a = 1, 2, 3. Case II (near double eigenvalue): If the number of distinct eigenvalues at x0 is two, choose  > 0 so that for x ∈ U0 , maxa,b |pa − pb | > /2, and |pa
− pb
| < /2 for some pair, a , b , a = b , shrinking U0 if necessary. Denote by p⊥ the distinguished exponent not equal to pa
, pb
. In this case define q⊥ = p⊥ and qa
= qb
= (1 − q⊥ )/2 on U0 . Case III (near diagonalizable): If all eigenvalues are distinct at x0 , choose  > 0 so that for x ∈ U0 , min a,b |pa (x) − pb (x)| > /2, shrinking U0 if necessary. a=b

In this case define qa = pa on U0 . The frame {ea }, called the adapted frame, is required to be such that the related (time-dependent) frame {˜ ea (t) ≡ t−qa ea } is orthonormal with respect to the a velocity-dominated metric at some time t0 > 0, i.e., such that 0 gab (t0 ) = t2q 0 δab . (Here and in the rest of the paper, the Einstein summation convention does not apply to indices on qa and pa . These indices should be ignored when determining sums. Furthermore, quantities with a tilde will refer to the frame {˜ ea (t)}.) In addition, in Case II it is required that e⊥ be an eigenvector of K corresponding to q⊥ and that ea
, eb
span the eigenspace of K corresponding to the eigenvalues pa
, pb
. In case III it is required that the ea be eigenvectors of K corresponding to the eigenvalues qa (≡ pa ). In all cases it is required that {ea } be analytic. The auxiliary exponents, qa , are analytic, satisfy the Kasner rela tion qa = 1, are ordered (q1 ≤ q2 ≤ q3 ), and satisfy q1 ≥ p1 , q3 ≤ p3 and maxa |qa − pa | < /2. If qa = qb , then 0 gab , 0 g ab , 0 g˜ab and 0 g˜ab all vanish, and the same is true with g replaced by k. Equations (2.12)–(2.16), with the form of gab (t0 ) and K a b specialized as given just above, are the general analytic solution to the velocity-dominated evolution equations in the sense that any analytic solution to the velocity-dominated evolution equations takes this form near any x0 ∈ Σ by choice of (global) time coordinate and (local) spatial frame.

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Velocity-dominated constraint equations

When written in terms of the velocity-dominated variables, the velocity-dominated constraints take the same form as the full constraint equations, except the Hamiltonian constraint, which is obtained by dropping spatial gradients and electromagnetic contributions to the energy-density. This is a general rule, valid also for the more general models considered below. Thus, if we define 0 0

ρ

ja

=

1 (∂t 0 φ)2 , 2

1 = −∂t 0 φ ea ( 0 φ) + 

0g

0 b 0

E

Fab ,

we get 0 C = 0 and 0 Ca = 0 for the velocity-dominated constraints corresponding to the Hamiltonian and momentum constraints, with 0 0

C

=

Ca

=

− 0 k a b 0 k b a + (tr 0 k)2 − 2 0 ρ, 0

∇b 0 k b a − ea (tr 0 k) − 0 ja .

(2.18) (2.19)

The velocity-dominated matter constraint equations read b 0 a ea ( 0 E a ) + fba E

=

0,

h 0 e[a ( 0 Fbc] ) + f[ab Fc]h

=

0.

For the Solution (2.12)–(2.14) the velocity-dominated Hamiltonian constraint equation is equivalent to
(2.20) pa 2 + A2 = 1. The conditions (2.17) and (2.20) are the famous Kasner conditions when the dilaton is present. While p1 is necessarily non-positive when A = 0, this is no longer the case when the dilaton is nontrivial (A = 0): all pa ’s can then be positive. This is the major feature associated with the presence of the dilaton, which turns the mixmaster behaviour of (4-dimensional) vacuum gravity into the velocity-dominated behaviour. We shall call (pa , A) the Kasner exponents (because they are the exponents of the proper time in the solution for the scale factors and exp φ) and refer to (2.17) and (2.20) as the Kasner conditions (note that A is often denoted pφ to emphasize its relation to the kinetic energy of φ, and its similarity with the other exponents). A straightforward calculation shows that ∂t 0 C − 2(tr 0 k) 0 C

= 0,

(2.21)

1 = − ea ( 0 C). (2.22) 2 Thus if the velocity-dominated Hamiltonian and momentum constraints are satisfied at some t0 > 0, then they are satisfied for all t > 0. Similarly, since 0 E a and 0 Fab are independent of time, if the matter constraints are satisfied at some time t0 > 0, then they are clearly satisfied for all time. ∂t 0 Ca − (tr 0 k) 0 Ca

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Critical value of dilaton coupling λc

Our ultimate goal is to show that the velocity-dominated solutions asymptotically approach (as t → 0) solutions of the original system of equations. We shall prove that this is the case provided the Kasner exponents pi , A, subject to the Kasner conditions
2

pa 2 − pa = 1 (2.23) pa + A2 = 0, obey additional restrictions. These restrictions are inequalities on the Kasner exponents and read explicitly 2p1 − λA > 0,

p1 > 0,

2p1 + λA > 0.

(2.24)

As explained in [1], and rigorously checked below, these restrictions are necessary and sufficient to ensure that the terms that are dropped when replacing the full Einstein-dilaton-Maxwell equations by the velocity-dominated equations become indeed negligible as t → 0. More precisely, the first condition (respectively the third) among (2.24) guarantees that one can neglect the electric (respectively, magnetic) part of the energy-momentum tensor of the electromagnetic field in the Einstein equations, whereas the condition p1 > 0 is necessary for the spatial curvature terms to be asymptotically negligible. The conditions (2.24) define the set V of velocity-dominated solutions referred to in the introduction. It is clear that if |λ| is small enough – in particular, if λ = 0 – the Inequalities (2.24) can be fulfilled since the Kasner exponents can be all positive when the dilaton is included. But if |λ| is greater that some critical value λc , it is impossible to fulfill simultaneously the Kasner conditions (2.23) and the Inequalities (2.24), because one of the terms ±λA becomes more negative than 2p1 is positive. In that case, the set V is empty and our construction breaks down. For |λ| < λc , however, the set V is non-empty and, in fact, stable under small perturbations of the Kasner exponents since (2.24) defines an open region on the Kasner sphere. We determine in this subsection the critical value λc such that (2.23) and (2.24) are compatible whenever |λ| < λc . To that end, we follow the geometric approach of [3, 39]. In the 4-dimensional space of the Kasner exponents (pa , A), we consider the “wall chamber” W defined to be the conical domain where p1 ≤ p2 ≤ p3 , 2pa − λA ≥ 0, pa ≥ 0, 2pa + λA ≥ 0.

(2.25)

These inequalities are not all independent since the four conditions p1 ≤ p2 ≤ p3 , 2p1 − λA ≥ 0, 2p1 + λA ≥ 0

(2.26)

imply all others. The quadratic Kasner condition (2.23) can be rewritten Gµν pµ pν = 0,

(pµ ) ≡ (pa , A)

where Gµν defines a metric in “Kasner-exponent space”

2

dS 2 = Gµν dpµ dpν = dpa 2 − dpa + (dA)2

(2.27)

(2.28)

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Ann. Henri Poincar´e

The metric (2.28) has Minkowskian signature (−, +, +, +). An example of timelike direction is given by p1 = p2 = p3 , A = 0. Inside or on the light cone, the function  pa does not vanish. The upper light cone (in the space of the Kasner exponents) is conventionally defined by (2.27) and the extra condition pa > 0. It is clear from our discussion that the Kasner conditions (2.23) and the Inequalities (2.24) are compatible if and only if there are light like directions in the interior  of the pa = 1 wall chamber W (by rescaling pµ → αpµ , α > 0, one can always make for any point in the interior of the wall chamber so that this condition does not bring a restriction). The problem amounts accordingly to determining the relative position of the light cone (2.27) and the wall chamber (2.26). This is most easily done by computing the edges of (2.26), i.e., the onedimensional intersections of three faces among the four faces (2.26) of W. There are four of them: (i) p1 = p2 = A = 0, p3 = α; (ii) p1 = A = 0; p2 = p3 = α; (iii) 2p1 = 2p2 = 2p3 = λA = α; and (iv) 2p1 = 2p2 = 2p3 = −λA = α, where in each case, α ≥ 0 is a parameter along the edge (α = 0 being the origin). The vectors eµA (A = 1, 2, 3, 4) along the edges corresponding to α = 1, namely (0, 0, 1, 0), (0, 1, 1, 0), (1/2, 1/2, 1/2, 1/λ) and (1/2, 1/2, 1/2, −1/λ) form a basis in Kasner-exponent space. Any vector v µ can thus be expanded along the eµA , v µ = v A eµA . A point P in Kasner-exponent space is on or inside the wall chamber W if and only if its coordinates pA in this basis fulfill pA ≥ 0 with P inside when pA > 0 for all A s. Thus, if all the edge vectors eµA are timelike or lightlike, the Kasner conditions are incompatible with the Inequalities (2.24) since any linear combination of causal vectors with non-negative coefficients is on or inside the forward light cone (the eµA ’s are future-directed since p1 + p2 + p3 > 0 for all of them). If, however, one (or more) of the edge vectors lies outside the light cone, then, the Kasner conditions and the Inequalities (2.24) are compatible. The nature of some of the edge vectors depends on the value of the dilaton coupling λ: while the first one is always lightlike and the second one always timelike, the squared norm of the last two is −3/2 + 1/λ2 = (2 − 3λ2 )/(2λ2 ). This determines the critical value  2 (2.29) λc = 3 such that the edge vectors are timelike or null (incompatible inequalities) if |λ| ≥ λc , but spacelike (compatible inequalities) if |λ| < λc . Note that the value of λ that arises from dimensionally reducing 5-dimensional vacuum gravity down to 4 √ dimensions is λ = 6 and exceeds the critical value. This “explains” the conclusion reached in [14] that the gravity-dilaton-Maxwell system obtained by Kaluza-Klein reduction of 5-dimensional gravity is oscillatory. We shall assume from now on that |λ| < λc and that the Kasner exponents fulfill the above inequalities. For later use, we choose a number σ > 0 so that, for all x ∈ U0 , σ < 2p1 −λA, σ < 2p1 +λA and σ < p1 /2. Reduce  if necessary so that  < σ/7. If  is reduced, it may be necessary to shrink U0 so that the conditions imposed in Section 2.2 remain satisfied. In Section 2.5 it is assumed that  and U0

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are such that the conditions imposed in Section 2.2 and the conditions imposed in this paragraph are all satisfied.

2.5

Fuchsian system which is equivalent to the Einstein-matter evolution equations

2.5.1 Rewriting of equations Theorem 3 in [15] (Theorem 4.2 in preprint version), on which we rely for our result, states that a Fuchsian equation (i.e., as we mentioned above, an equation of the form (1.5) where A satisfies a positivity condition and f is regular, which includes a boundedness property) has a unique solution u that vanishes as t ↓ 0, and furthermore spatial derivatives of u of any order vanish as t ↓ 0, as shown in [28]. Our goal is to recast the Einstein-matter evolution equations as a Fuchsian equation for the deviations from the velocity-dominated solutions. Thus, we denote the unknown vector u as u = (γ a b , λa bc , κa b , ψ, ωa , χ, ξ a , ϕab )

(2.30)

where the variables γ a b etc. are related to the Einstein-matter variables by gab

=

ec (γ a b ) = kab = φ

=

ea (ψ) t ∂t ψ + β ψ

= =

Ea Fab

= = c

0

gab + 0 gac tα b γ c b , c

t−ζ λa bc , c gac ( 0 k c b + t−1+α b κc b ), 0

φ + tβ ψ,

−ζ

t ωa , χ, 0 a

E + tβ ξ a , 0 Fab + tβ ϕab .

(2.31) (2.32) (2.33) (2.34) (2.35) (2.36) (2.37) (2.38)

In the first of these equations tα b is not the exponentiation of a matrix with c components αc b such as occurs in (2.13). The expression tα b is for each fixed value of c and b the number which is t raised to the power given by the number αc b (defined below). In Equations (2.31) and (2.33) there is no summation on the index b (but there is a summation on c). In (2.38) ϕab is a totally antisymmetric spatial tensor, which contributes three independent components to u. This assumption is consistent with the form of the evolution equation for ϕab , Equation (2.46) below. The exponents appearing in (2.31)–(2.38) are as follows. Define α0 = 4, β = /100 and ζ = /200 (where  is the same (small) quantity which entered the definition of the auxiliary exponents qa in Section 2.2 and which was further restricted at the end of Section 2.4). All of these quantities are independent of t and x. Finally define αa b = 2 max(qb − qa , 0) + α0 = 2qmax{a,b} − 2qa + α0 .

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Ann. Henri Poincar´e

Note that the numbers αa b are all strictly positive. In the second definition of αa b we have used the fact that the qa ’s are ordered. The role of αa b is to shift the spectrum of the Fuchsian-system matrix A, in Equation (1.5), to be positive. It is not clear to what extent the choice of αa b is fixed by the requirement of getting a Fuchsian system. It seems that the (triangle-like) Inequality (42) of [15] (Inequality (5.9) in preprint version) is a key property of these coefficients. We shall further comment below on the specific choice of αa b and its link with the BKL-type approach to the cosmological behaviour near t = 0. When writing the first-order evolution system for u we momentarily abandon the restriction that gab and kab be symmetric, as in [15]. Thus we need to define g ab , and we do so by requiring that gab g bc = δa c . This implies that g ab gbc = δ a c . We lower indices on tensors by contraction with the second index of gab and also raise indices on tensors by contraction with the second index of g ab . This choice is so that raising and then lowering a given index results in the original tensor, and the same for lowering and then raising an index. The position of the indices on quantities appearing in u and other such quantities is fixed. Repeated indices on these quantities imply a summation. On the other hand, as we already mentioned above, one qualifies the summation convention by insisting that indices repeated only because of their occurrence on pa , qa , αa b and other such nontensorial quantities should be ignored when determining sums. Substituting (2.31)–(2.38) in the evolution equations yields equations of motion for u of the form (1.5) t ∂t γ a b + αa b γ a b + 2κa b − 2(t 0 k a c )γ c b + 2γ a c (t 0 k c b ) = a

−2 tα

c a c +α b −α b

γ a c κc b

(2.39)

t ∂t λa bc = tζ ec (t ∂t γ a b ) + ζ tζ ec (γ a b ) t ∂t κ

a

b

a

+ α bκ

a

b

0 a

(2.40)

− (t k b )(trκ) = t (trκ)κ α0

a

b

+t

2−αa b S

( R

a

b

− M b ) (2.41) a

t ∂t ψ + βψ − χ = 0

(2.42)

t ∂t ωa = t {ea (χ) + (ζ − β)ea (ψ)} ζ

t ∂t χ + βχ = t

α0 −β

β

(tr κ)(A + t χ) + t +t2−β {

(2.43) 2−β ab

0

g ∇a ∇b φ + t

2−ζ

∇ ωa a

λ λ gab E a E b e−λφ − g ab g ch Fac Fbh eλφ } 2g 4

(2.44)

√ t ∂t ξ a + βξ a = t1−β {eb ( gg ac g bh Fch eλφ )

t ∂t ϕab + βϕab

1 a ic √ bh i ac +(fib g + fbi g ) gg Fch eλφ } 2 1 c 1 = t1−β {−2e[a ( √ gb]c E c e−λφ ) + fab √ gch E h e−λφ } g g

(2.45) (2.46)

All the quantities entering these equations have been defined, except S Ra b . This is done by taking the Ricci tensor of the symmetric part g(ab) of gab [15]. More

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explicitly, S Ra b = g ac S Rcb , with S

Rab

˜ ahb h = tqa +qb S R (2.47) qa +qb S ˜h S ˜h S ˜i S ˜h S ˜i S ˜h i S ˜ h }, {˜ eh ( Γab ) − e˜a ( Γhb ) + Γab Γhi − Γhb Γai + f˜ah Γ = t ib

and the connection coefficients in the frame {˜ ea },

1 1 S ˜c i i c Γab = S g˜ch e˜a (˜ + f˜ab g(bh) ) + e˜b (˜ g(ha) ) − e˜h (˜ g(ab) ) − g˜(ia) f˜bh − g˜(bi) f˜ah . 2 2 (2.48) Here, S g˜ab is defined as the inverse of g˜(ab) . Once it is shown that the tensor gab in Equation (2.31) is symmetric, then it follows that S Ra b = Ra b and that Equations (2.39)–(2.46) are equivalent to the Einstein-matter equations. 2.5.2 The system (2.39)–(2.46) is Fuchsian A good deal of the work needed to show that Equation (1.5) (as written out in Equations (2.39)–(2.46)) is Fuchsian was done in [15], in the massless scalar field case considered there. The form of the velocity-dominated evolution and the form of the function u are the same in the two cases except for the crucial addition of new source terms and new evolution equations involving the Maxwell field. The presence of the new components does not alter already existing parts of the matrix A, nor already existing terms in f . The difference between A here and A in the massless scalar field case considered in [15] is that here there are additional rows and columns, such that the only non-vanishing new entries are on the diagonal and strictly positive. Therefore the argument in [15] that their A satisfies the appropriate positivity condition implies that our A satisfies the appropriate positivity condition. On the other hand, it is crucial to control in detail the new source terms in f , connected to the Maxwell field, which were absent in [15]. It is for the study of these terms that the results of [1], and in particular the Inequalities (2.24) which were shown there to guarantee that Maxwell source terms become asymptotically subdominant near the singularity, become important. Recall that the crucial criterion for the source f (t, x, u, ux) is that it be O(tδ ), for some strictly positive δ. In regard to this estimate, we use the notation “big O,” “ ” and “small o” as follows. Given two functions F (t, x, u, ux ) and G(t, x, u, ux ) we use the notation F G, to denote that, for every compact set K, there exists a constant C and a number t0 > 0 such that |F (t, x, u, ux )| ≤ C|G(t, x, u, ux )| when (x, u, ux ) ∈ K and 0 < t ≤ t0 (see Definition 1 in [15]). If G is a function only of t (e.g. a power of t), then we replace F G with F = O(G). If f (t, x, u, ux ) = O(tδ ), then by reducing the value of δ (keeping it positive) we have that f (t, x, u, ux ) = o(tδ ) with a “small o” which denotes that f /tδ tends to zero uniformly on compact sets K as t → 0. The new source terms involving the Maxwell field are: the last four terms in M a b (see Equation (2.1)), the last two terms on the right-hand side of Equation (2.44) and the terms of the right-hand sides of Equations (2.45) and (2.46).

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Ann. Henri Poincar´e

The calculation of the estimates starts in the frame, {˜ ea }, defined in Section 2.2. For more details concerning the basic estimates, we refer the reader to [15]. In the frame {˜ ea } the Kasner-like metric is (cf. (2.13))    2(K−Q) c t 0 g˜ab = 0 g˜ac (t0 ) , (2.49) t0 b    −2(K−Q) a t 0 ab 0 cb g˜ = g˜ (t0 ), (2.50) t0 c

where the matrix Q is the diagonal matrix Q b ≡ qa δ a b which commutes with K. With our choice of frame, 0 g˜ab (t0 ) = δab and 0 g˜ab (t0 ) = δ ab . In Lemma 2 in [15] (Lemma 5.1 in preprint version), the form of (2.49) and (2.50) is considered and it is shown that 0 g˜ab = O(t− ) and 0 g˜ab = O(t− ). It is useful to write down expressions for the proposed metric and extrinsic curvature in the frame {˜ ea }. The components in terms of this frame are a

g˜ab ˜ kab

= =

0

g˜ab + 0 g˜ac tα˜ b γ c b , c g˜ac ( 0 k˜c + t−1+α˜ b κc b ). c

b

Here, α ˜ b = α b + qa − qb = |qa − qb | + α0 is symmetric in a, b, α ˜ab = α ˜b a . To get 0 cb an estimate for the inverse metric, we note first that the inverse of gac g is given by g ca0 gcb . Thus it is possible to express the latter quantity algebraically in terms of 0 gab and γ a b . Now define a

a

γ¯a b = −t−α˜ b (δba − g˜ac0 g˜cb ), a

(2.51)

which, from what we just observed, can be expressed algebraically in terms of known quantities and γ a b . Then one has g˜ab =0 g˜ab + tα˜ c γ¯ ac 0 g˜cb . a

(2.52)

As a consequence of an argument given in [15] which uses the (triangle-like) Inequality (42) of that paper ((5.9) in preprint version) and the matrix identity preceding it, this exhibits γ¯a b as a regular function of γ a b . In particular, if it is known that γ a b is o(1) then the same is true of γ¯a b . To better grasp the usefulness of the introduction of the exponents αab and a α ˜ b , and the link of the Fuchsian estimates with the approximate estimates used in the BKL-like works, let us consider more closely the simple case where all the Kasner exponents are distinct (Case III). In this case pa = qa and one can diagonalize the Kasner-metric, so that, in the rescaled frame e˜a , we have simply (for all t ≤ t0 ) 0 g˜ab (t) = δab . In such a case, the BKL-type estimates would be obtained (in the time-dependent rescaled frame e˜a ) by approximating the exact BKL metric by its Kasner limit, i.e., simply g˜ab (t) = δab . By contrast, the estimates a of the Fuchsian analysis are made with the exact metric, g˜ab (t) = δab + tα˜ b γ ab ,

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in which γ ab , being part of u, is considered to be in a compact set and hence is bounded. As the diagonal α ˜aa = α0 > 0, we see that (in the frame e˜a ) the diagonal components of the “Fuchsian” metric asymptote those of the “BKL” metric, and that both are close to one. Concerning the non-diagonal components (in the frame e˜a ) of the “Fuchsian” metric we see that they are constrained, by construction (i.e., by the choice α ˜ ab = |qa − qb | + α0 ), to tend to zero faster than t|qa −qb | . This closeness between the metrics used in the two types of estimates explains the parallelism between the rigorous results derived here and the heuristic estimates used in BKL-type works. If we come back to the general case where the Kasner metric cannot be diagonalized in an analytic fashion, the optimal estimates become worse by a negative power of t (coming from the estimate of the matrix difference 2(K − Q) in Equations (2.49), (2.50) above). The proposed metric in the frame {˜ ea } satisfies then g˜ab t|qa −qb |−

g˜ab t|qa −qb |− .

and

The proposed inverse metric in the adapted frame is g ab = 0 g ab + tα c γ¯ ac 0 g cb . a

The proposed metric in the adapted frame satisfies gab t2qmax{a,b} −

and

g ab t−2qmin{a,b} − .

(2.53)

Estimates of spatial derivatives of the proposed metric are also needed. ec (˜ gab ) t|qa −qb |−δ−

and

ec (˜ g ab ) t|qa −qb |−δ− ,

ec (gab ) t2qmax{a,b} −δ−

and

ec (g ab ) t−2qmin{a,b} −δ− (2.54)

for some strictly positive δ. The determinant of the proposed metric also appears in some of the new source terms. From (2.13), the form of 0 gab (t0 ) and tr K = 1, one gets 0 g = t2 . From (2.49) and 0 g˜ab (t0 ) = δab one gets 0 g˜ = 1. The expression for the determinant is a sum of terms of the form gab gcd gef , such that in each term, each index, 1, 2, 3, occurs exactly twice. From the Kasner relation for the qa ’s and the relation between the two frames, it follows that g = t2 g˜. Considering the form of the various √ √ expressions, one then obtains 1/g = O(t−2 ), g = O(t), 1/ g = O(t−1 ), and  √ 1/ g−1/ 0 g = O(t−1+α0 −3 ) = O(t−1+ ). Spatial derivatives of the determinant also appear in f . Considering the form of g˜ − 0 g˜ and that ea ( 0 g˜) = 0, it follows that ea (˜ g ) = O(tα0 −δ−3 ), and ea (g) = O(t2+α0 −δ−3 ). Finally, ea (g −1/2 ) = −

ea (g) = O(t−1+α0 −δ−3 ). 2g 3/2

(2.55)

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Ann. Henri Poincar´e

Let us now consider the new source terms in f , beginning with the last four a terms of t2−α b M a b . To estimate the contributions of E a and Fab it is sufficient to note from (2.37) and (2.38) that E a = O(1) and Fab = O(1). Then we get, using the definition of αa b and (2.53), t2−α

a

b

1 1 {gbc E a E c − δ a b gch E c E h }e−λφ g 2

t−2qmax{a,b} +2qa +2qmax{b,c} −λA−α0 − + t+2qmax{c,h} −λA−α0 − c

t

c,h

2q1 −λA−α0 −

= O(t

−α0 − +σ

δ

) = O(t ),

a 1 t2−α b {g ac g hi Fch Fbi − δ a b g ch g ij Fci Fhj }eλφ 4
t2−2qmax{a,b} +2qa −2qmin{a,c} −2qmin{h,i} +λA−α0 −2

c,h=c,i=b

+




t2−2qmin{c,h} −2qmin{i,j} +λA−α0 −2

c,h,i=c,j=h

t2q1 +λA−α0 −2 = O(t−α0 −2 +σ ) = O(tδ ) for some strictly positive δ. The crucial inputs in getting these estimates are the Inequalities (2.24). We recall also that the quantity σ (linked to (2.24) being satisfied) was introduced at the end of Subsection 2.4. The estimate of the last two terms on the right-hand side of (2.44) is 1 t2−β gab E a E b e−λφ g

= O(t−β− +σ ) = O(tδ ),

t2−β g ab g ch Fac Fbh eλφ

= O(t−β−2 +σ ) = O(tδ )

The right-hand side of (2.45) is O(tα0 −β−δ−5 +σ ) = O(tδ ). The right-hand side of (2.46) is O(tα0 −β−δ−4 +σ ) = O(tδ ). The other terms which occur in f were estimated in [15], resulting in f = O(tδ ). To show that we indeed have a Fuchsian equation, we need to check not only that f = O(tδ ), but also that ∂u f = O(tδ ) and ∂ux f = O(tδ ), along with other regularity conditions [15, 28]. In [15] it is shown that f is regular with Equation (31) in that paper and the remarks following Equation (31). In our case there is a factor involving the determinant of the metric in various of the terms in f which are not present in the case considered in [15]. The discussion surrounding Equation (31) in [15] applies to our case as well, even for terms in f containing g ±1/2 . The Kasner-like contribution is the leading term, and this function of t and x can be factored out. What is left is of the form w(t, x, u, ux )(1 + h(t, x, u))±1/2 , which is analytic in h at h = 0. The conditions listed following Equation (31) hold. Thus we conclude that (1.5) as written out in (2.39)–(2.46) is a Fuchsian equation.

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2.5.3 Symmetry of metric It remains to show that gab is symmetric, so that Equation (1.5) as written out in (2.39)–(2.46) is equivalent to the Einstein-matter evolution equations. The structure of the argument is the same in any dimension and so it will be written down for general d6 . The number of distinct eigenvalues of K a b is maximal almost everywhere. Thus it is enough to show that g[ab] and k[ab] vanish in the case that the Kasner-like metric is diagonal, since then by analytic continuation they vanish on the entire domain. We therefore consider the case that the Kasner-like metric is diagonal. The redefinitions (2.31), (2.33) from the variables gab , kab to the variables γ a b , a κ b were viewed in the previous subsections as a change between variables with no particular symmetry properties in their indices (18 on each side). One can, however enforce g[ab] = 0 by assuming that γ a b is symmetric and vice versa. Indeed, under a our diagonality assumption for the Kasner-like metric, g˜ab (t) = δab +tα˜ b γ ab where α ˜ ab = |qa − qb | + α0 is symmetric in (a, b). Accordingly, imposing the symmetry γ a b = γ b a algebraically ensures the symmetry of gab . Similarly, one can enforce k[ab] to vanish by imposing consistent constraints on κa b : inserting (2.31) into (2.33) (with the velocity-dominated solution diagonal) and writing out the constraint kab − kba = 0 gives the following condition on κa b κa b − κb a − γ a b pb + γ b a pa + tα(ab)c (γ a c κc b − γ b c κc a ) = 0,

(2.56)

with α(ab)c = 2pmax{a,c} +2pmax{b,c} −2pmax{a,b} −2pc +α0 . These conditions show that there are only six independent components among the κa b , which can be taken to be those with a ≤ b. This is because, the relation (2.56) can be solved uniquely for the components κa b with a > b, given the other ones, at least for t small. That this is true can be seen as follows. Rearrange the Equations (2.56) so that the terms containing κa b with a > b are on the left-hand side and all other terms are on the right-hand side. The result is an inhomogeneous linear system of the form A(t, x)v(t, x) = w(t, x) where A(t, x) and w(t, x) are known quantities and v denotes the components κa b with a > b which we want to determine. Furthermore A(t, x) = I + o(1), where I denotes the identity matrix. It follows that A(t) is invertible for t small, which is what we wanted to show. The solution κa b (a > b) remains moreover bounded when γ b a and κa b are in a compact set. We shall assume from now on that γ a b is symmetric and κa b constrained by (2.56), so that symmetry of the metric is automatic. The redefinitions (2.31), (2.33) from gab , kab to γ a b , κa b can now be viewed as an invertible change of variables, from 12 6 The argument for the symmetry of the metric in [15] is not valid as written since some terms were omitted in the evolution equation for the antisymmetric part of the extrinsic curvature. The correct equation is

∂t (kab − kba ) = (trk)(kab − kba ) − 2(kac k c b − kbc k c a ). In the following a proof of the symmetry of the tensors gab and kab is supplied with the help of a different method.

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independent variables to 12 independent variables. We can also clearly assume λa bc in (2.32) to be symmetric in a, b. With these conventions, there are less components in u than in the previous subsections. The independent components can be taken to be γ a b , κa b and λa bc with a ≤ b, together with the matter variables. An independent system of evolution equations is given by (2.39)–(2.41) with a ≤ b for the gravitational variables, and the same evolution equations as before for the matter variables. These evolution equations are equivalent to all the original evolution equations, since the Equations (2.39)–(2.41) with a > b are then automatically fulfilled, as can be shown using the fact that the Einstein tensor and the stress-energy tensor are symmetric for symmetric metrics. To see this it must be shown that given a symmetric tensor Sab , the vanishing of S a b = g ac Scb for a ≤ b implies that Sab = 0. Consider the linear map which takes a symmetric tensor Sab , raises an index, and keeps the components of the result with a ≤ b. This is a mapping between vector spaces of dimension d(d+ 1)/2 and can be shown to be an isomorphism by elementary linear algebra. This proves the desired result. Now, this reduced evolution system is also Fuchsian. This follows from the same reasoning as above, which still holds because all components of u, including the non-independent ones, can still be assumed to be bounded. Therefore, there is a unique u that goes to zero, which must be equal to the one considered in the previous subsections. The metric considered previously is thus indeed symmetric. 2.5.4 Unique solution on a neighborhood of the singularity Given an analytic solution to the velocity-dominated evolution equations on (0, ∞) × Σ, such that Inequalities (2.24) are satisfied, we now have a solution u to a Fuchsian equation (and a corresponding solution to the Einstein-matter evolution equations) in the intersection of a neighborhood of the singularity with (0, ∞) × U0 where U0 is a neighborhood of an arbitrary point on Σ. These local solutions can be patched together to get a solution to the Einstein-matter evolution equations everywhere in space near the singularity. It may seem like there could be a problem patching together the solutions obtained on distinct neighborhoods with non-empty intersection because the Fuchsian equation is not the same for different allowed choices of  and adapted local frame. The construction is possible because different allowed choices of  and local frame result in a well-defined relationship between the different solutions u which are obtained, such that the corresponding Einstein-matter variables agree on the intersection (up to change of basis). It therefore follows that given an analytic solution to the velocity-dominated evolution equations on (0, ∞) × Σ, such that Inequalities (2.24) are satisfied, our construction uniquely determines a solution to the Einstein-matter evolution equations everywhere in space, near the singularity.

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Einstein-matter constraints

2.6.1 Matter constraints The time derivative of the matter constraint quantities (the left-hand side of Equations (2.2) and (2.3)) vanishes. If the velocity-dominated matter constraints are satisfied, the matter constraint quantities are o(1). A quantity which is constant in time and o(1) must vanish. Therefore the matter constraints are satisfied. 2.6.2 Diagonal Kasner metrics It remains to show that the Hamiltonian and momentum constraints are satisfied, that C and Ca , defined in (1.9) and (1.10), vanish. Since we now have a metric, gµν , it follows that ∇µ Gµν = 0. Since the matter evolution and constraint equations are satisfied, it follows that ∇µ T µν = 0. From the vanishing of the right-hand side of (1.12) and the vanishing of the covariant divergence of both the Einstein tensor and the stress-energy tensor, it follows that ∂t C

=

∂t Ca

=

2(trk)C − 2∇a Ca 1 (trk) Ca − ∇a C. 2

(2.57) (2.58)

Now define C¯ = t2−η1 C and C¯a = t1−η2 Ca , with 0 < η2 < η1 < β. t ∂t C¯ + η1 C¯

=

t ∂t C¯a + η2 C¯a

=

2(1 + t tr k)C¯ − 2t2−η1 +η2 ∇a C¯a 1 (1 + t tr k)C¯a − tη1 −η2 ∇a C¯ 2

(2.59) (2.60)

On the right-hand side of (2.59) and (2.60) C¯ and C¯a are to be considered as ¯ C¯a ). If it is shown that (2.59) and (2.60) is a Fuchsian components of u = (C, system, then there is a unique solution u such that u = o(1). It is clear that u = 0 is a solution to (2.59) and (2.60). If it is shown that C¯ = o(1) and C¯a = o(1), (i.e., that C = o(t−2+η1 ) and Ca = o(t−1+η2 )), then they must be this unique solution. Furthermore, it is sufficient to consider the case that the Kasner-like metric is diagonal, since the number of distinct eigenvalues of K a b is maximal on an open set of Σ. If the constraints vanish on an open set of their domain, then by analytic continuation they vanish everywhere on their domain. Therefore we consider the case that the Kasner-like metric is diagonal and show first that 1 + t tr k ∇a C¯a

= =

O(tδ ) O(t

−2+δ+η1 −η2

(2.61) )

(2.62)

(when C¯a is bounded) so that the system (2.59), (2.60) is Fuchsian (the complete regularity of f (t, x, u, ux) defined by (2.59) and (2.60) can be easily verified); and

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second, that = o(t−2+η1 ) = o(t−1+η2 ).

C Ca

(2.63) (2.64)

Some facts which will be used to show this follow. Consider indices a ∈ {1, 2, 3}. The following inequalities hold for some positive integer n and for real numbers, qa , ordered such that if a < b, then qa ≤ qb . (In later sections we define ordered auxiliary exponents, qa , for a ∈ {1, . . . , d}, for arbitrary fixed d ≥ 2. Then (2.65)–(2.67) hold more generally for indices in {1, . . . , d}.) n


|qai−1 − qai | − qan ≥ 0

(2.65)

|qai−1 − qai | + qan ≥ 2qmax{ak ,aj }

(2.66)

|qai−1 − qai | − qan ≥ −2qmin{ak ,aj }

(2.67)

qa0 +

i=1

qa0 +

n
i=1

−qa0 +

n
i=1

The latter two inequalities hold for any k, j in {0, . . . , n}. In the case that the Kasner-like metric is diagonal, qa = pa . The metric in the frame {˜ ea } is 0 g˜ab = δab , g˜ab g˜ab

= δab + tα˜ b γ a b t|pa −pb | , a = δ ab + tα˜ b γ¯ ab t|pa −pb | . a

The extrinsic curvature satisfies t 0 k a b = −δ a b pb , t ka b t (k˜ab − 0 k˜ab )

a

= −δ a b pb + tα b κa b , = tα˜ b κa b , a

and t tr 0 k = −1, t tr k 2 2 0 2 t {(tr k) − (tr k) }

= −1 + tα0 trκ, = O(tα0 ).

(2.68) (2.69)

The following estimates will also be useful. −k a b k b a + 0 k a b 0 k b a

and

= −2t−2+α0 κa a pa − t−2+α = O(t−2+α0 ),

a

b b +α a

ea (tr k − tr 0 k) = ea (t−1+α0 tr κ) = O(t−1+α0 ).

κa b κb a (2.70)

(2.71)

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The structure functions of the frame {˜ ea } are c f˜ab

=

c tpc −pa −pb fab − ln t ea (pb ) t−pa δ c b + ln t eb (pa ) t−pb δ c a



tpc −pa −pb −δ .

˜ c , the connection coefficients (2.48) in It is convenient to have an estimate of Γ ab the frame {˜ ea }, term by term.
Term A: g˜ch e˜a (˜ gbh ) t|pc −ph |−pa +|pb −ph |−δ , (2.72) h

g˜ e˜b (˜ gha ) ch

Term B:




t|pc −ph |−pb +|ph −pa |−δ ,

(2.73)

t|pc −ph |−ph +|pa −pb |−δ ,

(2.74)

h

gab ) g˜ch e˜h (˜

Term C:


h

h g˜ci g˜ha f˜bi

Term D:




t|pc −pi |+|ph −pa |+ph −pb −pi −δ , (2.75)

h,i=b ci

h g˜bh f˜ai






t|pc −pi |+|pb −ph |+ph −pa −pi −δ , (2.76)

Term E:

g˜

Term F:

c f˜ab tpc −pa −pb −δ .

h,i=a

(2.77)

The difference between the connection coefficients for the metric g˜ab and those for ˜ c . It is useful to have the estimates ˜c − 0Γ ˜c = Γ the Kasner-like metric is ∆Γ ab ab ab a ˜ aac = 1 g˜ab e˜c (˜ Γ gab ) + f˜ac t−pc −δ , 2

and ˜a = ∆Γ ac

1 ab g˜ e˜c (˜ gab ) t−pc +α0 −δ . 2

(2.78)

2.6.3 Momentum and Hamiltonian constraints First, we show (2.61) and (2.62). From equation (2.68), 1 + t tr k = O(tα0 ). Similarly, we can estimate ∇a C¯a , ∇a C¯a

=

˜¯ ˜ aC g˜ab ∇ b

=

˜ cab C¯c t−pc } g˜ab {t−pa ea (C¯b t−pb ) − Γ

The first term is g˜ab t−pa ea (C¯b t−pb ) t|pa −pb |−pa −pb −δ t−2pmin{a,b} −δ .

(2.79)

From (2.72)–(2.76) the second term is ˜ cab C¯c t−pc t−2p3 −δ . g˜ab Γ

(2.80)

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From (2.79) and (2.80), the desired estimate, ∇a C¯a = O(t−2+η1 −η2 ) is obtained. Thus, the system (2.59), (2.60) is Fuchsian. Next we turn to (2.63) and (2.64). A term that appears in the momentum constraint is ∇a k a b . The estimate is needed in the adapted frame, and the covariant derivative is calculated in the frame {˜ ea }. This adds a factor of tpb , ˜ aac k˜c b − Γ ˜ cab k˜ac } tpb ea (k˜ ab ) + Γ ∇a k a b = {˜ Furthermore, the quantity whose estimate will be required is the difference between this term and the corresponding term in the velocity-dominated constraint, ˜ a 0 k˜c + Γ ˜ a (k˜ c − 0 k˜c )}tpb ea (k˜ ab − 0 k˜ab ) + ∆Γ ∇a k a b − 0 ∇a 0 k a b = {˜ ac b ac b b ˜ cab k˜ac tpb + 0 Γ ˜ cab 0 k˜ac tpb −Γ

(2.81) (2.82)

The right-hand side of (2.81) is O(t−1+α0 −δ ). The terms in line (2.82) originating from Term E of the connection coefficients (see (2.76)) are cancelled in h and the symmetry of k˜ai and 0 k˜ai . For the sum, due to the antisymmetry of f˜ai estimating the rest of the terms in line (2.82), it is convenient to rewrite this line as, ˜ cab 0 k˜ac tpb = −∆Γ ˜ cab 0 k˜ac tpb − Γ ˜ cab (k˜ ac − 0 k˜ac ) tpb , ˜ cab k˜ac tpb + 0 Γ −Γ

(2.83)

with ˜ cab ∆Γ

=

b c a 1 {˜ ea (tα˜ c γ b c ) + e˜b (tα˜ a γ c a ) − e˜c (tα˜ b γ a b ) 2
c b h a + tα˜ h γ¯ c h [˜ ea (tα˜ h γ b h ) + e˜b (tα˜ a γ h a ) − e˜h (tα˜ b γ a b )]

h

−




i tα˜ a γ i a f˜bc − i

i

−t

α ˜b i b

i γ i f˜ac −


h




a tα˜ h γ¯ c h f˜bh − c

h

t

α ˜c h c

b γ¯ h f˜ah −







(2.84) (2.85)

i tα˜ h γ¯ c h tα˜ a γ i a f˜bh (2.86) c

i

hi

t

α ˜c h c

i γ¯ h tα˜ i γ b i f˜ah }. b

(2.87)

hi

The terms in line (2.87) need not be considered since they originate from Term E of the connection coefficients and as stated above the contribution from this term ˜ c 0 k˜ac tpb . So considering only lines (2.84)–(2.86), is cancelled by terms in Λ = Γ ab the first term on the right-hand side of (2.83) is ˜ a pa t−1+pb = O(t−1+α0 −δ ) + terms which are cancelled by Λ. ∆Γ ab

(2.88)

Since the terms in the sum come with different weights, pa , (2.78) cannot be used in (2.88). But the estimate is straightforward.  For example, the term in (2.88) originating from the 3rd term in line (2.86) a,h,i t−1+|pa −ph |+|pi −pa |+pi −ph +2α0 −δ = O(t−1+α0 −δ ). Finally consider the rest of the right-hand side of (2.83),  a 1 i i c tα˜ c κa c t−1+pb . − g˜ch {˜ ea (˜ gbh ) + e˜b (˜ gha ) − e˜h (˜ gab ) − g˜ia f˜bh − g˜bi f˜ah } + f˜ab 2 (2.89)

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For all terms except the 5th term in (2.89), the estimate, O(t−1+α0 −δ ) can be obtained from (2.72)–(2.77). The fifth term originates from Term E of the connection coefficients, and was already considered above. Therefore ∇a k a b − 0 ∇a 0 k a b = O(t−1+α0 −δ ).

(2.90)

Next the matter terms are estimated. For the Hamiltonian constraint, an estimate of ρ − 0 ρ is needed. (∂t φ)2 − (∂t 0 φ)2

g ab ea (φ)eb (φ) 1 gab E a E b e−λj φ g g ab g ch Fac Fbh eλj φ Therefore,

= =

{2 ∂t 0 φ + t−1+β (βψ + t∂t ψ)} t−1+β (βψ + t∂t ψ) {2A + tβ (βψ + t∂t ψ)} t−2+β (βψ + t∂t ψ)

=

o(t−2+η1 ), t−2p3 −δ− = o(t−2+η1 ),

=

O(t−2− +σ ) = o(t−2+η1 ),

=

O(t−2−2 +σ ) = o(t−2+η1 ). ρ − 0 ρ = o(t−2+η1 ).

(2.91)

The difference between the matter terms in the momentum constraint and in 0 Ca is −∂t φ ea (φ) + ∂t 0 φ ea ( 0 φ) = −∂t 0 φ ea (tβ ψ) − ∂t (tβ ψ)ea (φ) = O(t−1+β−δ ), 1 1 1 ( √ −  ) 0 E b 0 Fab + √ (E b Fab − 0 E b 0 Fab ) = o(t−1+η2 ). (2.92) 0g g g Estimates related to the determinant which are relevant to (2.92) can be found immediately preceding Equation (2.55). From the estimates just obtained, ja − 0 ja = o(t−1+η2 ).

(2.93)

From R = O(t−2+α0 ) (shown in [15]) and from 0 C = 0, (2.70), (2.69), (2.91) and the relative magnitude of the various exponents, it follows that C = o(t−2+η1 ). From 0 Ca = 0, (2.90), (2.71), (2.93) and the relative magnitude of the various exponents, it follows that Ca = o(t−1+η2 ). Since (2.63)–(2.64) are satisfied, the Hamiltonian and momentum constraints are satisfied.

2.7

Counting the number of arbitrary functions

The number of degrees of freedom of the Einstein-Maxwell-dilaton system in 4 spacetime dimensions is 5 : 2 for the gravitational field, 2 for the electromagnetic field and 1 for the dilaton. Hence, a general solution to the equations of motion should contain 10 freely adjustable, physically relevant, functions of space (each degree of freedom needs two initial data, q and q). ˙ This is exactly the number that appears in the above Kasner-like solutions.

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• The metric carries four, physically relevant, distinct functions of space. This is the standard calculation [4]. • The scalar field carries two functions of space, A and B. • The electromagnetic field carries six functions of space, 0 E a and 0 Fab . These are physically relevant because they are gauge invariant, but they are subject to two constraints, leaving four independent functions. A different way to arrive at the same conclusions is to observe that the respective number of fields, dynamical equations and (first class) constraints are the same for the velocity-dominated system and the full system. Hence, a general solution of the velocity-dominated system (in the sense of function counting) will contain the same number of physically distinct, arbitrary functions as a general solution of the full system. This general argument applies to all systems considered below and hence will not be repeated. In [15] a different way of assessing the generality of the solutions constructed was used. This involved exhibiting a correspondence between solutions of the velocity-dominated constraints and solutions of the full constraints using the conformal method. That method starts with certain free data and shows the existence of a unique solution of the constraints corresponding to each set of free data. It is a standard method for exploring the solution space of the full Einstein constraints [42] and in [15] it was shown how to modify it to apply to the velocity-dominated constraints. While it is likely that the conformal method can be applied in some way to all the matter models considered in this paper, the details will only be worked out in two cases which suffice to illustrate the main aspects of the procedure. These are the Einstein-Maxwell-dilaton system with D = 4 (this section) and the Einstein vacuum equations with arbitrary D ≥ 4 (next section). Even in those cases no attempt will be made to give an exhaustive treatment of all issues arising. It will, however, be shown that the strategies presented for solving the velocity-dominated constraints are successful in some important situations. The procedure presented in the following is slightly different from that used in [15]. Even for the case of the Einstein-scalar field system with D = 4 it gives results which are in principle stronger than those in [15] since they are not confined to solutions which are close to isotropic ones. In the presence of exponential dilaton couplings a change of method seems unavoidable. One part of the conformal method concerns the construction of symmetric second rank tensors which are traceless and have prescribed divergence from the truly free data. In this step there is no difference between the full constraints and the velocity-dominated ones. An account of the methods applied to the full constraints in the case D = 4 can be found in [42]. (These arguments generalize in a straightforward way to other D. It is merely necessary to find the correct conformal rescalings. For D ≥ 4 and vacuum these are written down in the next section.) In view of this we say, with a slight abuse of terminology, that the free data consists of a collection g˜ab , k˜ab , H, φ, φ˜t , E a , Fab where g˜ab is a Riemannian metric, k˜ab is a symmetric tensor with

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vanishing trace and prescribed divergence with respect to g˜ab , H is a non-zero constant, φ and φ˜t are scalar functions and E a and Fab are objects of the same kind as elsewhere in this section. All these objects are defined on a three-dimensional manifold. Next we introduce a positive real-valued function ω which is used to construct solutions of the constraints from the free data. Define gab = ω 4 g˜ab , kab = ω −2 k˜ab + Hgab , φt = ω −6 φ˜t . The objects gab , kab , φ, φt , E a and Fab satisfy the constraints provided the divergence of k˜ab is prescribed as ω 6 ja and ω satisfies a nonlinear equation which in the case of the full Einstein equations is known as the Lichnerowicz equation. In the case of the velocity-dominated constraints it is an algebraic equation. The Lichnerowicz equation is of the form 3
1 3 ai ω αi − H 2 ω 5 = 0 ∆g˜ ω − Rg˜ ω + 8 4 i=1

(2.94)

Here α1 = 1, α2 = −3 and α3 = −7. The functions ai depend on the free data and their exact form is unimportant. All that is of interest are that each ai is non-negative and that at any point of space a1 = 0 iff ∇a φ = 0 , a2 = 0 iff the electromagnetic data vanish and a3 = 0 iff φt and k˜ab vanish. Next consider the velocity-dominated constraints for d = 3. The analogue of the elliptic Equation (2.94) is the algebraic equation 3 bω −7 − H 2 ω 5 = 0 4

(2.95)

Here b is a non-negative function which vanishes at a point of space iff φt and k˜ab vanish. This can be solved trivially for ω > 0 provided b does not vanish at any point since the mean curvature H is non-zero. For each choice of free data satisfying this non-vanishing condition there is a unique solution ω of (2.95). In order to compare the sets of solutions of the full and velocity-dominated constraints in these two cases it remains to investigate the solvability of the elliptic Equation (2.94) for ω. A discussion of this type of problem in any dimension can be found in [43]. We would like to show that for suitable metrics on a compact manifold the equation for ω always has a unique solution, i.e., the situation is exactly as in the case of the velocity-dominated equations. The problem can be simplified by the use of the Yamabe theorem, which says that any metric can be conformally transformed to a metric of constant scalar curvature −1, 0 or 1. In the following only the cases of negative and vanishing scalar curvature of the metric supplied by the Yamabe theorem will be considered. A key role in the existence and uniqueness theorems for Equation (2.94) is played by the positive zeros of 3 3 the algebraic expressions x + 8 i=1 ai xαi − 6H 2 x5 and 8 i=1 ai xαi − 6H 2 x5 . 3 Provided a=1 ai does not vanish anywhere it is possible to show that each of the algebraic expressions has a unique positive zero for each value of the parameters. The significance of the information which has been obtained concerning the zeros of certain algebraic expressions is that it guarantees the existence of a positive solution of the corresponding elliptic equations for any set of free data satisfying

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the inequalities already stated using the method of sub- and supersolutions (cf. [43]). In the case of Equation (2.94) uniqueness also holds. For in that case the equation has a form considered in [44] for which uniqueness is demonstrated in that paper. The advantage of the three-dimensional case is that there the problem reduces to the analysis of the roots of a cubic equation, a relatively simple task compared to the analysis of the zeros of the more complicated algebraic expressions occurring in higher dimensions. For the purpose of analyzing Kasner-like (monotone) singularities it is not enough to know about producing just any solutions of the constraints. What we have shown is that (i) if the Kasner constraints are satisfied at time t0 , then they are propagated at all times by the velocity-dominated evolution equations; and (ii) if the Kasner constraints are satisfied, the exact constraints are also satisfied. It is also necessary to verify, however, that one can satisfy simultaneously the Kasner constraints and the inequalities necessary for applying the Fuchsian arguments, i.e., we must make sure that we can produce a sufficiently large class of solutions which satisfy the inequalities necessary to make them consistent with Kasner behaviour. Because of the indirect nature of the way of solving the momentum constraint (which has not been explained here) it is not easy to control the generalized Kasner exponents of the resulting spacetime. There is however, one practical possibility. Choose a spatially homogeneous solution with Abelian isometry group (for d = 3 this means Bianchi type I) which satisfies the necessary inequalities. Take the free data from that solution and deform it slightly. Then the generalized Kasner exponents of the final solution of the velocity-dominated equations will also only be changed slightly. If the homogeneous solution is defined on the torus T 3 then it is known that any other metric of constant scalar curvature has non-positive scalar curvature. Therefore we are in the case for which existence and uniqueness is discussed above. We could also start with a negatively curved Friedmann model.

3 Vacuum solutions with D ≥ 11 The second class of solutions we consider is governed by the Action (1.2), with D ≥ 11. The d + 1 decomposition is as in Section 1.6, with the matter terms vanishing. The Kasner-like evolution equations are (2.7) and (2.8). The general analytic solution of these equations is given by (2.12) and (2.13). To obtain this form, we again adapt a global time coordinate such that the singularity is at t = 0. We label the eigenvalues of K, p1 , . . . , pd , such that pa ≤ pb if a < b. The d eigenvalues again satisfy i=1 pi = 1, coming from tr K = 1. As in the D = 4 case, in order to preserve analyticity even near the points where some of the eigenvalues coincide, while retaining control of the solution in terms of the eigenvalues, we introduce a special construction involving auxiliary exponents and an adapted frame. In higher dimensions, there are more possibilities to take care of, but the idea is the same as in the D = 4 case. Consider an arbitrary point x0 ∈ Σ. Let

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n be the number of distinct eigenvalues of K at x0 . Let mi be the multiplicity of pAi , i ∈ {1, . . . , n}, with pAi such that pb is strictly less than pAi if b < Ai . Thus pAi , . . . , pAi +mi −1 are equal at x0 . For each integer a ∈ {Ai , . . . , Ai + mi − 1}, define A +m −1 1 i
i qa = pj mi j=Ai

on a neighborhood of x0 , U0 . Note that if mi = 1, then qAi = pAi . Shrinking U0 if necessary, choose  > 0 such that for x ∈ U0 , for a ∈ {Ai , . . . , Ai + mi − 1} and for b ∈ {Aj , . . . , Aj + mj − 1}, if i = j, then |pa − pb | < /2, while if i = j, |pa − pb | > /2. The adapted frame {ea } is again required to be analytic and such that the related frame {˜ ea } is orthonormal with respect to the Kasner-like metric at some time t0 > 0, with e˜a = t−qa ea . In addition, it is required that eAi , . . . , eAi +mi −1 span the eigenspace of K corresponding to the eigenvalues pAi , . . . , pAi +mi −1 . Note qAi . that if mi = 1 then eAi is an eigenvector of K corresponding to the eigenvalue  The auxiliary exponents, qa , are analytic, satisfy the Kasner relation ( qa = 1), are ordered (qa ≤ qb for a < b), and satisfy q1 ≥ p1 , qd ≤ pd and maxa |qa − pa | < /2. If qa = qb , then 0 gab , 0 g ab , 0 g˜ab and 0 g˜ab all vanish, and the same is true with g replaced by k. The velocity-dominated constraints corresponding to the Hamiltonian and momentum constraints are 0 C = 0 and 0 Ca = 0, with 0 C and 0 Ca as in Equations (2.18) and (2.19), with the matter terms vanishing. For the Solution (2.12)– (2.13)  2 the velocity-dominated Hamiltonian constraint equation is equivalent to pa = 1. Equations (2.21)–(2.22) are satisfied, so if the velocity-dominated constraints are satisfied at some t0 , then they are satisfied for all t > 0. For this class of solutions, the Inequality [13], 2p1 + p2 + · · · + pd−2 > 0, or equivalently, (3.1) 1 + p1 − pd − pd−1 > 0, defines the set V which was referred to in the introduction. As shown in [13], this inequality can be realized when the spacetime dimension D is equal to or greater than 11. As in our Maxwell archetypal example above, we expect that this inequality will be crucial to control the effect of the source terms (here linked to the spatial curvature) near the singularity. It is again convenient to introduce a number σ > 0 so that, for all x ∈ U0 , 4σ < 1 + p1 − pd − pd−1 . Reduce  if necessary so that  < σ/(2d + 1). If  is reduced, it may be necessary to shrink U0 so that the conditions imposed above remain satisfied. It is assumed that  and U0 are such that the conditions imposed above and the condition imposed in this paragraph are all satisfied. We again recast the evolution equations in the form (1.5) and show, for D ≥ 11, that (1.5) is Fuchsian and equivalent to the vacuum Einstein equation, with quantities u, A and f as follows. Let u = (γ a b , λc ef , κh i ) be related to the Einstein variables by (2.31)–(2.33). For general d define α0 = (d + 1) and define αa b

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in terms of α0 as in Section 2. Let A and f , be given by Equations (2.39)–(2.41), with M a b = 0. The argument that A in Equation (1.5) satisfies the appropriate positivity condition is analogous to the part of the argument concerning the submatrix of A corresponding to (γ, κ) in [15]. A transformation to a frame in which 0 gab is diagonal induces a similarity transformation of A. The eigenvalues of the submatrix are calculated in this representation in [15], and the generalization of the calculation to integer d ≥ 2 is straightforward. To obtain f = O(tδ ) requires the estimate t2−α b S Ra b = O(tδ ). The strategy used here is different from that used to estimate the curvature in [15]. The general problem is one of organization. There are many terms to be estimated, each of which on its own is not too difficult to handle. The difficulty is to maintain an overview of the different terms. The procedure in [15] made essential use of the fact that the indices only take three distinct values and in the case of higher dimensions, where this simplification is not available, an alternative approach had to be developed. a

First S Ra b is estimated by considering each of the five terms in the Expression (2.47). These five terms are expanded by considering each of the six terms in (2.48) if the indices on S Γcab are distinct, but carrying out the summation before estimating S Γaab . There are thus 55 terms to estimate. While many of these terms are actually identical up to numerical factors, the ease with which each term can be estimated, using the Inequalities (2.65)–(2.67), led to estimation of all 55 terms, i =0 rather than first combining terms. We do however, take into account that fjk if j = k for obtaining the estimates. Once an equation such as (1.5) is shown to be Fuchsian, then it follows that spatial derivatives of u of any order are o(1). At the stage of the argument we are at here, we cannot assume uxx = O(1). This means that t−ζ λa bc must be used for eb (γ a c ) in places where a spatial derivative of eb (γ a c ) occurs. This makes a slight difference, compared to Section 2.6, in what estimate of the terms in the connection coefficients is used for the first and second terms of (2.47) (t−δ is replaced by t−ζ ). There are additional differences from (2.72)–(2.77), because there it is assumed that the Kasner-like metric is diagonal. The estimates 0 g˜ab = O(t− ) and 0 g˜ab = O(t− ), obtained in Lemma 2 of [15], hold in the case we are considering, so that g˜(ab) t|qa −qb |− and (see [15]) S g˜ab t|qa −qb |− . This adds factors of t− to the estimate of terms in the connection coefficients. With these considerations, from (2.48), S ˜a Γac

=

1 S ab a g˜ e˜c (˜ g(ab) ) + f˜ac t−qc −2 −ζ . 2

(3.2)

Here we do not write out the estimates of all 55 terms, but instead give some examples, with a number designating which term of (2.47) is being considered (1–5), and a letter designating which term of (2.48) is being considered (A–F).

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Thus, for example, term 1C is g(cb) )) t−qa +qb g˜ac e˜h ( S g˜hi e˜i (˜
t−qa +qb +|qa −qc |−qh +|qh −qi |−qi +|qc −qb |−δ−3 −ζ c,h,i

t−2qa +2qmax{a,b} −2qd −δ−3 −ζ .

(3.3)

Term 3E is j S ˜h Γhi t−qa +qb g˜ac S g˜ik g˜(bj) f˜ck
t−qa +qb +|qa −qc |+|qi −qk |+|qb −qj |+qj −qc −qk −qi −δ−5 c,i,j,k=c






t−2qmin{a,c} +2qb −2qmin{d,k} −δ−5 .

(3.4)

c,k=c

˜h S Γ ˜ i , the terms resulting from expanding S Γ ˜ h are desIn term 4, t−qa +qb g˜ac S Γ ib ch ib S ˜i ignated by small letters a–f, and those from Γch are designated by capital letters A–F. Term 4dA is g(hk) ) t−qa +qb g˜ac S g˜hl g˜(ji) f˜blj S g˜ik e˜c (˜
−qa +qb +|qa −qc |+|qh −ql |+|qj −qi |+qj −qb −ql +|qi −qk |−qc +|qh −qk |−δ−5 t c,h,i,j,k,l

t−2qa −δ−5 .

(3.5)

Term 4eD is j S il k g˜ g˜(kc) f˜hl t−qa +qb g˜ac S g˜hn g˜(bj) f˜in
c,h,i,j,k,l=h,n=i

t−qa +qb +|qa −qc |+|qh −qn |+|qb −qj |+qj −qi −qn +|qi −ql |+|qk −qc |+qk −qh −ql −δ−5 t2qb −2qd −2qd−1 −δ−5

(3.6)

Term 5D is i S jk h t−qa +qb g˜ac f˜cj g˜ g˜(hi) f˜bk
t−qa +qb +|qa −qc |+qi −qc −qj +|qj −qk |+|qh −qi |+qh −qb −qk −δ−3 c,h,i,j=c,k=b






t−2qmin{a,c} +2q1 −2qmin{j,k} −δ−3 .

(3.7)

c,j=c,k=b

The estimates of the remaining terms are obtained as these. The examples include one of the terms which limits the estimate for each possible choice of indices

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a and b. The result is, S

Ra b t2qb −2qd −2qd−1 −δ−5 +




t−2qmin{a,c} +2q1 −2qmin{j,k} −δ−5 .

(3.8)

c,j=c,k=b

And t2−α

a

b

S

Ra b



{t2qmin{a,b} −2qd −2qd−1
+ t−2qmax{a,b} +2q1 −2qmin{j,k} }t2−α0 −δ−5 c≥a,j=c,k=b



t

2−2qd −2qd−1 +2q1 −(d+7)

t8σ−(d+7) = O(tδ ).

(3.9)

The estimate of the rest of the terms in f is obtained straightforwardly by checking that the exponent of t in each case is strictly positive. The other regularity conditions that f should satisfy are shown to hold by Equation (31) in [15] and the remarks following Equation (31). The symmetry of gab is shown for all d’s in Subsection 2.5.3. That the Hamiltonian and momentum constraints are satisfied is shown by the direct analogue of argument made in Section 2.6 and the estimate R = o(t−2+η1 ) obtained from Equation (3.8). The only change is that Equation (2.80) is replaced by ˜ c C¯c t−pc t−2pd −δ . g˜ab Γ ab

(3.10)

To conclude this section we discuss the solution of the velocity-dominated constraints for the vacuum equations and D ≥ 4. The case D = 3 could be discussed in a similar way but the analogue of the Lichnerowicz equation has a different form and so for brevity that case will be omitted. The discussion proceeds in a way which is parallel to that of the last section. As already indicated there, the essential task is the analysis of the Lichnerowicz equation. In the present case we start with free data g˜ab , k˜ab and H where k˜ab has zero divergence. The actual data are defined by gab = ω 4/(d−2) g˜ab and kab = ω −2 k˜ab + Hgab . The constraints will be satisfied is ω satisfies the following analogue of the Lichnerowicz equation: ∆g˜ ω +

3d−2 d+2 d−2 d(d − 2) 2 d−2 (−Rg˜ ω + k˜ ab k˜ab ω d−2 ) − H ω =0 4(d − 1) 4

(3.11)

The corresponding equation in the velocity-dominated case is 3d−2 d+2 d − 2 ˜ ab ˜ d(d − 2) 2 d−2 H ω k kab ω d−2 − =0 4(d − 1) 4

(3.12)

As in the case of (2.95) it is trivial to solve (3.12) provided k˜ab does not vanish at any point. To determine the solvability of Equation (3.11) it is necessary to study 3d−2 d+2 3d−2 the positive zeros of the algebraic expressions x + bx d−2 − ax d−2 and bx d−2 − d+2 ax d−2 where a > 0 and b > 0. The second expression is very close to what we

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had in the velocity-dominated case and clearly has a unique positive zero for any values of a and b satisfying the inequalities assumed. Looking for positive zeros of the first algebraic expression is equivalent to looking for positive solutions of d+2 x− d−2 + ax−2 − b = 0. Note that the function on the left-hand side of this equation is evidently decreasing for all positive x, tends to infinity as x → 0 and tends to −b as x → ∞. Hence as long as the constant b is non-zero this function has exactly one positive zero, as desired. This is what is needed to obtain an existence theorem. It would be desirable to also obtain a uniqueness theorem for the solution of (3.11). To obtain solutions of the velocity-dominated constraints of the right kind to be consistent with Kasner-like behaviour we can use the same approach as in the last section, starting with Kasner solutions with an appropriate set of Kasner exponents.

4

Massless scalar field, D ≥ 3

Consider Einstein’s equations, D ≥ 3, with a massless scalar field as source, the action given by SE [gαβ ] + Sφ [gαβ , φ], and d + 1 decomposition as in Section 1.6. The stress-energy tensor is Tµν =

(D)

1 ∇µ φ (D) ∇ν φ − gµν (D) ∇α φ (D) ∇α φ. 2

(4.1)

Thus ρ = 12 {(∂t φ)2 + g ab ea (φ)eb (φ)}, ja = −∂t φ ea (φ), and M a b = g ac eb (φ) ec (φ). A crucial step in the generalization to arbitrary D ≥ 3 is that the cancellation of terms involving ∂t φ in the expression for M a b is not particular to D = 4. The scalar field satisfies (D) ∇α (D) ∇α φ = 0, which has d + 1 decomposition ∂t2 φ − (trk)∂t φ = g ab ∇a ∇b φ.

(4.2)

Let the Kasner-like evolution equations be Equations (2.7)–(2.9), with Solutions (2.12)–(2.14) for time coordinate as in Section 3. Given a point x0 ∈ Σ, let the neighborhood U0 , the (local) adapted frame and the constant  be as in Section 3. Define 0 ρ = 12 (∂t 0 φ)2 and 0 ja = −∂t 0 φ ea ( 0 φ). The velocity-dominated constraints corresponding to the Hamiltonian and momentum constraints are 0 C = 0 and 0 Ca = 0, with 0 C and 0 Ca given by Equations (2.18) and (2.19). For the Solution the velocity-dominated Hamiltonian constraint is equivalent  (2.12)–(2.14) to pa 2 + A2 = 1. Equations (2.21) and (2.22) are satisfied so if the velocitydominated constraints are satisfied at some t0 , then they are satisfied for all t > 0. The restriction defining the set V is the Inequality (3.1). (If D < 11, then satis 2 pa + A2 = 1 requires A = 0. Note fying simultaneously (3.1), pa = 1 and that conversely, for D = 3, the restrictions defining V are simply equivalent to A = 0, since (3.1) is in this case a consequence of p1 + p2 = 1 and p21 + p22 < 1). The constant σ > 0 is chosen so that, for all x ∈ U0 , 4σ < 1 + p1 − pd − pd−1 from which it follows that σ < 2 − 2pd . (4.3)

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Now reduce  if necessary so that  < σ/(2d + 1). As before, this may in turn require shrinking U0 . The unknown u = (γ a b , λa bc , κa b , ψ, ωa , χ) is related to the Einstein-matter variables by (2.31)–(2.36). The quantities A and f appearing in Equation (1.5) are given by the evolution Equations (2.39)–(2.43) and t ∂t χ + βχ = tα0 −β (tr κ)(A + tβ χ) + t2−β S g ab S ∇a S ∇b 0 φ + t2−ζ S ∇a ωa . (4.4) The argument that the matrix A satisfies the appropriate positivity condition is analogous to the argument in [15]. Regarding the estimate f = O(tδ ), the estimate a a t2−α b S Ra b = O(tδ ) was obtained in Equation (3.9). The estimate t2−α b M a b = O(tδ ) follows from the Inequality (4.3) and from qd < pd . The only other terms in f whose estimates are not immediate from the estimates made in [15] are the last two terms on the right-hand side of Equation (4.4). The covariant derivative compatible with the symmetrized metric is used in Equation (4.4) for convenience. From the estimate S g˜ab t|qa −qb |− [15], Equations (2.53) and (2.54), S ab

g

t−2qmin{a,b} −

and

ec (g(ab) ) t2qmax{a,b} −δ− .

Therefore, S ab S

g

Γcab



=

S ab S ch



t−2qd −δ−3

g

g

1 a ea (g(bh) ) − eh (g(ab) ) − S g ch fah 2

and t2−β S g ab S ∇a S ∇b 0 φ = t2−ζ S ∇a ωa

=

t2−β S g ab {ea (eb ( 0 φ)) − S Γcab ec ( 0 φ)} t2−2qd −β−δ−3 = O(tδ ), t2−ζ S g ab {ea (wb ) − S Γcab wc } t2−2qd −ζ−δ−3 = O(tδ )

The other regularity conditions that f should satisfy are again shown to hold by Equation (31) in [15] and the remarks following Equation (31). That gab is symmetric (so that Equation (4.4) and Equation (4.2) are equivalent) is shown as in Subsection 2.5.3. That the Hamiltonian and momentum constraints are satisfied is shown by the analogue of the argument made in Section 2.6 and the estimate R = o(t−2+η1 ) obtained from Equation (3.8). Note that the case D = 3 of this result has an interesting connection to the Einstein vacuum equations in D = 4. As it follows from standard Kaluza-Klein lines, the solutions of the latter with polarized U (1) symmetry are equivalent to the Einstein-scalar field system in D = 3 (see e.g. [40] and [41], Section 5). Hence the result of this section implies that we have constructed the most general known class of singular solutions of the Einstein vacuum equations in four spacetime dimensions. These spacetimes have one spacelike Killing vector.

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5 Matter fields derived from n-form potentials 5.1

Equations of motion

We now turn to the general system (1.1), but without the interaction terms “more”. These are considered in Section 7 below. The action is the sum of (1.2), (1.3) and k additional terms, each of the form (1.4). The argument is based on that of Section 4. It is enough here to note the differences. Furthermore, since there is no coupling between additional matter fields, the differences from the argument made in Section 4 can be noted for each additional matter field independently of the others. Therefore consider the jth additional matter field, Fµ0 ···µnj = (nj + 1)∇[µ0 Aµ1 ···µnj ] , with A an nj -form. This matter field contributes the following additional terms to the stress-energy tensor, Equation (4.1),   1 1 Tµν = · · · + Fµα1 ···αnj Fν α1 ···αnj − gµν Fα0 ···αnj F α0 ···αnj eλj φ . nj ! 2(nj + 1)! √ Define E a1 ···anj = g F 0a1 ···anj eλj φ . If nj = 0, E is a spatial scalar density. Throughout this section and the next we use the following conventions. If nj = 0, then Pa1 ···anj is a scalar, ga1 b1 · · · ganj bnj = 1, etc. The d + 1 decomposition of the contribution of this matter field to the stress-energy tensor is ρ

=

ja

=

M ab

=

1 ga b · · · ganj bnj E a1 ···anj E b1 ···bnj e−λj φ 2 g nj ! 1 1 1 g a0 b0 · · · g anj bnj Fa0 ···anj Fb0 ···bnj eλj φ , + (5.1) 2(nj + 1)! 1 E b1 ···bnj Fab1 ···bnj , ···+ √ (5.2) g nj ! 1 nj gbh1 gc2 h2 · · · gcnj hnj E ac2 ···cnj E h1 ···hnj ···− g nj !

nj δ a b gc1 h1 · · · gcnj hnj E c1 ···cnj E h1 ···hnj e−λj φ − (d − 1)nj ! 1 g ac g h1 i1 · · · g hnj inj Fch1 ···hnj Fbi1 ···inj + nj !

nj − δ a b g c0 h0 · · · g cnj hnj Fc0 ···cnj Fh0 ···hnj eλj φ . (d − 1)(nj + 1)! ···+

The jth matter field satisfies (D)

∇µ (F µν1 ···νnj eλj φ ) = (D)

∇[µ Fν0 ···νnj ]

=

0,

(5.3)

0,

(5.4)

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with d + 1 decomposition into constraint equations, ea (E

ab2 ···bnj

)+

c fca E ab2 ···bnj

e[a (Fb0 ···bnj ] ) −

1
bi ab2 ···c···bnj + f E 2 i=2 ac

=

0,

(5.5)

(nj + 1) c f[ab0 F|c|b1 ···bnj ] 2

=

0,

(5.6)

nj

and evolution equations, √ h bc0 a1 c1 g g · · · g anj cnj ∂t E a1 ···anj = −eb ( gg bc0 g a1 c1 · · · g anj cnj Fc0 ···cnj eλj φ ) − {fhb 1
ai bc0 a1 c1 √ fbh g g · · · g hci · · · g anj cnj } gFc0 ···cnj eλj φ , 2 i=1 nj

+

1 ∂t Fa0 ···anj = −(nj + 1)e[a0 ( √ ga1 |b1 | · · · ganj ]bnj E b1 ···bnj e−λj φ ) g (nj + 1)nj c f[a0 a1 g|c||b1 | ga2 |b2 | · · · ganj ]bnj E b1 ···bnj e−λj φ . + √ 2 g

(5.7)

(5.8)

The jth matter field contributes the following terms to the evolution Equation (4.2) for φ. λj ga b · · · ganj bnj E a1 ···anj E b1 ···bnj e−λj φ 2 g nj ! 1 1 λj g a0 b0 · · · g anj bnj Fa0 ···anj Fb0 ···bnj eλj φ − 2(nj + 1)!

∂t2 φ − (trk)∂t φ = · · · +

5.2

(5.9)

Velocity-dominated system

The Kasner-like evolution equations corresponding to this matter field are ∂t 0 E a1 ···anj = 0 and ∂t 0 Fa0 ···anj = 0. The quantities 0 E a1 ···anj and 0 Fa0 ···anj are constant in time with analytic spatial dependence and both are totally antisymmetric. The velocity-dominated matter constraint equations are Equations (5.5) and (5.6) with 0 E and 0 F substituted for E and F . Since all quantities in the velocitydominated matter constraints are independent of time, if the matter constraints are satisfied at some time t0 > 0, then they are satisfied for all t > 0. This matter field does not contribute to 0 ρ. Its contribution to 0 ja is the term shown on the right-hand side of Equation (5.2) with 0 g, 0 E and 0 F substituted for g, E and F . The velocity-dominated constraints corresponding to the Hamiltonian and momentum constraints are 0 C = 0 and 0 Ca = 0, with 0 C and 0 Ca given by Equations (2.18) and (2.19). Equations (2.21) and (2.22) are satisfied, so as before, if the velocity-dominated constraints are satisfied at some t0 > 0, then they are satisfied for all t > 0.

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The presence of the matter field A(j) puts the following restrictions on the set V [1]. 2p1 + · · · + 2pnj − λj A > 0

and

2p1 + · · · + 2pd−nj −1 + λj A > 0. (5.10)

The restrictions generalize the Inequalities (2.24) found for a Maxwell field in 4 dimensions and, like them, guarantee that one can asymptotically neglect the pform A(j) in the Einstein-dilaton dynamical equations. (For nj = 0, the inequality on the left of (5.10) is −λj A > 0 while for nj = 1 it is 2p1 −λj A > 0. For nj = d−1, the inequality on the right is λj A > 0, while for nj = d − 2 it is 2p1 + λj A > 0.) The constant σ is reduced from its value in Section 4, if necessary, so that, for all x ∈ U0 , σ < 2p1 + · · · + 2pnj − λj A and σ < 2p1 + · · · + 2pd−nj −1 + λj A. If σ is reduced, it may be necessary to reduce , and in turn shrink U0 , so that the conditions imposed in Section 4 are still all satisfied.

5.3

Fuchsian property – estimates

The jth matter field contributes the following components to the unknown u in the Fuchsian Equation (1.5). E a1 ···anj Fa0 ···anj

= =

0 a1 ···anj

E + tβ ξ a1 ···anj , 0 Fa0 ···anj + tβ ϕa0 ···anj .

(5.11) (5.12)

a1 ···anj Here, β = /100 as above, is a totally antisymmetric spatial tensor den ξ d sity, so contributes nj independent components to u, and ϕa0 ···anj is a totally   antisymmetric spatial tensor, so contributes njd+1 components to u. This is consistent with the form of the evolution equations. Note that E a1 ···anj = O(1) and Fa0 ···anj = O(1). This matter field contributes additional rows and columns to the matrix A such that the only non-vanishing new entries are on the diagonal and strictly positive. Therefore, the presence of this matter field does not alter that A satisfies the appropriate positivity condition. The terms in the source f which must be estimated on account of the jth matter field are the following. It contributes terms to the components of f corresponding to κ through its contribution to M a b .

1 gbh gc h · · · gcnj hnj E ac2 ···cnj E h1 ···hnj e−λj φ g 1 2 2
−2qmax{a,b} +2qa +2qmax{b,h } +···+2qmax{c ,h } −λj A−α0 −nj nj nj 1 t

t2−α

a

b

t2q1 +···+2qnj −λj A−α0 −nj = O(t−α0 −nj +σ ) = O(tδ ) t

2−αa b ac h1 i1






g g t

···g

hnj inj

Fch1 ···hnj Fbi1 ···inj e

(5.13)

λj φ

2−2qmax{a,b} +2qa −2qmin{a,c} −2qmin{h1 ,i1 } −···−2qmin{hn

j

,in } +λj A−α0 −(nj +1) j

t2q1 +···+2qd−nj −1 +λj A−α0 −(nj +1) = O(t−α0 −(nj +1) +σ ) = O(tδ )

(5.14)

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Here it is used that both E a1 ···anj and Fa0 ···anj are totally antisymmetric, so that the sums indicated by a summation symbol are not over all indices. Note that the Inequalities (5.10) have been crucially used in getting the Estimates (5.13) and (5.14). The desired estimates for the other two terms are obtained similarly. The terms contributed to the component of f corresponding to χ by the jth matter field are obtained by multiplying the right-hand side of Equation (5.9) by t2−β . 1 t2−β ga1 b1 · · · ganj bnj E a1 ···anj E b1 ···bnj e−λj φ = O(t−β−nj +σ ) = O(tδ ) g

(5.15)

t2−β g a0 b0 · · · g anj bnj Fa0 ···anj Fb0 ···bnj eλj φ = O(t−β−(nj +1) +σ ) = O(tδ ).

(5.16)

a1 ···anj

The terms in f corresponding to ξ for the jth matter field are obtained by multiplying the right-hand side of Equation (5.7) by t1−β . These terms are O(t−β−δ−(nj +1) +σ ) = O(tδ ). The terms in f corresponding to ϕa0 ···anj for the jth matter field are obtained by multiplying the right-hand side of Equation (5.8) by t1−β . These terms are O(t−β−δ−nj +σ ) = O(tδ ). Thus the terms which occur in f due to the jth matter field are O(tδ ). The time derivative of the matter constraint quantities for the jth field (the left-hand side of Equations (5.5) and (5.6)) vanishes. If the velocity-dominated matter constraints are satisfied, the matter constraint quantities are o(1). A quantity which is both constant in time and o(1) must vanish. Therefore the matter constraints for the jth field are satisfied. Next the matter terms due to the jth field in the Einstein constraints are estimated, in order to verify that they are consistent with Equations (2.63) and (2.64). The contribution to the Hamiltonian constraint is, from Equation (5.1), 1 ga b · · · ganj bnj E a1 ···anj E b1 ···bnj e−λj φ = O(t−2−nj +σ ) = o(t−2+η1 ), g 11 g a0 b0 · · · g anj bnj Fa0 ···anj Fb0 ···bnj eλj φ = O(t−2−(nj +1) +σ ) = o(t−2+η1 ).

(5.17) (5.18)

The contribution to the momentum constraint is 1 1 (5.19) ja −0 ja = · · · + ( √ −  ) 0 E b1 ···bnj 0 Fab1 ···bnj 0g g 1 + √ (E b1 ···bnj Fab1 ···bnj − 0 E b1 ···bnj 0 Fab1 ···bnj ) = o(t−1+η2 ). g Estimates related to the determinant which are relevant to (5.19) are analogues of the estimates for d = 3 immediately preceding Equation (2.55). The form  of √ these estimates for general d will now be presented. These are 1/ g − 1/ 0 g = O(t−1+α0 −d ), ea (˜ g) = O(tα0 −δ−d ), ea (g) = O(t2+α0 −δ−d ), and ea (g −1/2 ) = −

ea (g) = O(t−1+α0 −δ−d ). 2g 3/2

(5.20)

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6 Determination of subcritical domain The explicit determination of the subcritical range of the dilaton couplings for which the inequalities on the Kasner exponents are consistent so that V exists may be a complicated matter. We consider in this section a few cases and give some general rules. As in Subsection 2.4, we introduce the metric

dpa 2 − ( dpa )2 + (dA)2 (6.1) dS 2 = Gµν dpµ dpν = in the D-dimensional space of the Kasner exponents (pa , A) ≡ (pµ ). This metric has again Minkowskian signature (−, +, +, · · · , +). The forward light cone is defined by
pa > 0. (6.2) Gµν pµ pν = 0, The Kasner conditions met in the previous section are equivalent  to the conditions that the Kasner exponents be on the forward light cone (since pa = 1 can always be achieved by positive rescalings). The wall chamber W is now defined by p1 ≤ p2 ≤ · · · ≤ pd 2p1 + p2 + · · · + pd−2 ≥ 0

(6.3) (6.4)

and, for each p-form, λj A≥0 2 λj p1 + p2 + · · · + pd−nj −1 + A ≥ 0. 2 p1 + p2 + · · · + pnj −

(6.5) (6.6)

These inequalities may not be all independent. The question is to determine the “allowed” values of the dilaton couplings for which the wall chamber contains in its interior future-directed lightlike vectors. It is clear that this set is non-empty since the inequalities can be all fulfilled when the couplings are zero (the pa ’s can be chosen to be positive in the presence of a dilaton).

6.1

Einstein-dilaton-Maxwell system in D dimensions

We consider first the case of a single 1-form in D ≥ 4 dimensions. This case is simple because the Inequalities (6.4) are then consequences of (6.5) and (6.6), which read λ λ p1 + p2 + · · · + pd−2 + A ≥ 0. (6.7) p1 − A ≥ 0, 2 2 Furthermore, the number of faces of the wall chamber (defined by these inequalities and (6.3)) is exactly D and the edge vectors form a basis. Thus, the analysis of Subsection 2.4 can be repeated.

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A basis of edge vectors can be taken to be (0,0,···,0,1,0)  d−k −2 d−k −2 2(d − k − 2)  − ,···,− ,1,···,1,− , k = 1,2,···,d − 2 k+1 k+1 λ(k + 1) 2 (1,1,···,1, ) λ 2(d − 2) ) (1,1,···,1,− λ

(6.8) (6.9) (6.10) (6.11)

In (6.9), the first k components are equal to − d−k−2 k+1 and the next d−k components are equal to 1. The first vector is lightlike. The kth vector in the group (6.9) has squared norm −

(d − 1)[k 2 − k(d − 3) + d] 4(d − k − 2)2 + , (k + 1)2 λ2 (k + 1)2

k = 1, 2, · · · , d − 2

(6.12)

while (6.10) and (6.11) have norm squared equal to −d(d − 1) +

4 λ2

(6.13)

and

4(d − 2)2 , (6.14) λ2 respectively. The subcritical values of λ must (by definition) be such that at least one of the Expressions (6.12), (6.13) or (6.14) is positive. To determine the boundaries ±λc of the subcritical  interval, we first note that (6.13) is positive whenever 2/ d(d − 1). Similarly, (6.14) is positive whenever |λ| < Λ2 |λ| < Λ1 , with Λ1 =  with Λ2 = 2(d − 2)/ d(d − 1). To analyze the sign of (6.12), we must consider two cases, according to whether k 2 − k(d − 3) + d is positive or negative. If d < 9, the factor k 2 − k(d − 3) + d is always positive (for any choice of k, k = 1, 2, · · · , d − 2) and the Expression (6.12) is positive provided |λ| < Πk , with −d(d − 1) +

2(d − k − 2) Πk =  . (d − 1)[k 2 − k(d − 3) + d]

(6.15)

The critical value λc is equal to the largest number among Λ1 , Λ2 and Πk . This largest number is Λ2 for d = 3, 4, 5, 6, Π1 for d = 7 and Π2 for d = 8. We thus have the following list of critical couplings:  2 , d=3 λc = 3 2 λc = √ , d=4 3

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d=5 d=6 d=7 d = 8.

(6.16)

Note that the value of the dilaton coupling that comes from dimensional reduction of vacuum gravity in one dimension higher  d λKK = 2 (6.17) d−1 is always strictly greater than the critical value, except for d = 8, where λKK = λc . (The corresponding values of the Kasner exponents are those of the point on the Kasner sphere exhibited in [13] for D = 10, where all gravitational inequalities are marginally fulfilled.) If d ≥ 9, the factor k 2 − k(d − 3) + d is non-positive for   d − 3 + (d − 9)(d − 1) d − 3 − (d − 9)(d − 1) ≤k≤ (6.18) 2 2 (this always occurs for k = 3). Thus, the Expression (6.12) is positive for such k’s no matter what λ is. This implies that the critical value of λ is infinite, λc = ∞,

d ≥ 9.

(6.19)

The fact that D = 10 appears as a critical dimension for the Einstein-dilatonMaxwell system, above which the system is velocity-dominated no matter what the value of the dilaton coupling is in the line of the findings of [13], since the edges (6.9) differ from those of the pure gravity wall chambers only by an additional component along the spacelike dilaton direction.

6.2

Einstein-dilaton system with one p-form (p = 0, p = D − 2)

The same geometrical procedure for determining the critical values of the dilaton couplings can be followed when there is only one p-form in the system (p = 0, p = D − 2), because in that case the wall chamber has exactly D faces and the edge vectors form a basis. Indeed, the gravitational Inequalities (6.4) are always consequences of the symmetry Inequalities (6.3) and the form Inequalities (6.5) and (6.6) (for nj = 0 and nj = D − 2), 2p1 +p2 +· · ·+pd−2 = (p1 +· · ·+pnj −

λj λj A)+(p1 +pnj +1 +· · ·+pd−2 + A). (6.20) 2 2

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So, if there is only one p-form (with p = 0 and p = D − 2), the D − 2 symmetry Inequalities (6.3) together with the two form Inequalities (6.5) and (6.6) completely define the wall chamber, which has D faces. We shall not provide an explicit example of a calculation of λc for such a system, since it proceeds as for a 1-form. When there is more than one exterior form, one can still drop the gravitational inequalities (if there is at least one p-form with p = 0 and p = D − 2), but the situation is more involved because the inequalities corresponding to different forms are usually independent, so that  the wall chamber has more than D faces (its intersection with the hyperplane pa = 1 is not a simplex). The calculation is then more laborious. The same feature arises for a 0-form, which we now examine.

6.3

0-form in 4 dimensions

We consider the case of a 0-form in 4 spacetime dimensions. As explained above, we impose the condition λ = 0 to the corresponding dilaton coupling7 . Without loss of generality (in view of the φ → −φ symmetry), we can assume λ > 0. The inequalities defining the subcritical domain relevant to the 0-form case can be brought to the form p1 > 0 A>0 λ A>0 2 p2 − p1 > 0

p1 + p2 −

p3 − p2 > 0

(6.21) (6.22) (6.23) (6.24) (6.25)

We denote by α, β, γ, δ and  the corresponding border hyperplanes (i.e., α : p1 = 0, β : A = 0 etc). The Inequalities (6.21)–(6.25) guarantee that all potential walls are negligible asymptotically. They are independent. The five faces α, β, γ, δ and  intersect along the 7 one-dimensional edges generated by the vectors: e1 = (0, 0, 1, 0) ∈ α ∩ β ∩ γ = α ∩ β ∩ δ = α ∩ γ ∩ δ = β ∩ γ ∩ δ (6.26) e2 = (0, 1, 1, 0) ∈ α ∩ β ∩  2 e3 = (0, 1, 1, ) ∈ α ∩ γ ∩  λ e4 = (0, 0, 0, 1) ∈ α ∩ δ ∩  e5 = (−1, 1, 1, 0) ∈ β ∩ γ ∩  e6 = (1, 1, 1, 0) ∈ β ∩ δ ∩  4 e7 = (1, 1, 1, ) ∈ γ ∩ δ ∩  λ

(6.27) (6.28) (6.29) (6.30) (6.31) (6.32)

7 The case λ = 0 is clearly in the subcritical region but must be treated separately because there are then two dilatons. The Kasner conditions read p1 +p2 +· · ·+pd = 1 and p21 +· · · p2d +A21 +A22 = 1, where the scalar fields behave as φ1 ∼ A1 ln t, φ2 ∼ A2 ln t. This allows positive pi ’s, which enables one to drop spatial derivatives as t → 0. The system is velocity-dominated.

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Among these vectors, neither e4 nor e5 bound the subcritical domain since e4 is such that p1 + p2 − (λ/2)A < 0 (changing its sign would make A < 0), while e5 is such that p1 < 0 (changing its sign would make p2 − p1 < 0). The edge-vectors {e1 , e2 , e3 , e6 , e7 } form a complete (but not linearly independent) set. Any vector can be expanded as v = v1 e1 + v2 e2 + v3 e3 + v6 e6 + v7 e7

(6.33)

The coefficients v1 , v2 , v3 , v6 , v7 are not independent but can be changed as v2 → v2 + 2k, v3 → v3 − 2k, v6 → v6 − k, v7 → v7 + k

(6.34)

For (6.33) to be interior to the wall chamber, the coefficients v1 , v2 , v3 , v6 and v7 must fulfill v1 > 0, v2 + v3 > 0, v2 + 2v6 > 0, v3 + 2v7 > 0, v6 + v7 > 0.

(6.35)

Using the above redefinitions, which leave the inequalities invariant, we can make vA ≥ 0, A = 1, 2, 3, 6, 7, with at most two vA ’s equal to zero. Indeed, let s = min(v2 , v3 , 2v6 , 2v7 ). Assume for definiteness that s = v2 (the other cases are treated in exactly the same way). One has then v2 ≤ 2v7 . Take 2k = −s in the redefinitions (6.34). This makes v2 equal to zero and makes v7 equal to v7 −(v2 /2) ≥ 0. Because of (6.35), the new v3 and v6 are strictly positive, as claimed. Thus, one sees that any vector in the wall chamber can be expanded as in (6.33) with nonnegative coefficients. But the vectors e1 , e2 , e3 , e6 and e7 are all future-pointing and timelike or null when λ ≥ 8/3. It follows that for such λ’s, there  is no lightlike direction in the interior of the wall chamber. Conversely, if λ < 8/3, the vector e7 is spacelike and one can find an interior vector αe1 + βe2 + e7 (α, β > 0) that is lightlike. We can thus conclude:  8 for a 0-form in 4 dimensions, (6.36) λc = 3  i.e., the system is velocity-dominated for |λ| < 8/3. The action for the matter fields in the case of a 0-form A coupled to a dilaton φ is  √ 1 (6.37) Sφ [gαβ , φ, A] = − (∂µ φ ∂ µ φ + eλφ ∂µ A ∂ µ A) −g d4 x 2 Note that this is the action for a wave map (also known as a nonlinear σ-model or hyperbolic harmonic map) with values in a two-dimensional Riemannian manifold of constant negative curvature. Its curvature is proportional to λ2 . Thus we obtain an interesting statement on velocity-dominated behaviour for the Einstein equations coupled to certain wave maps. Note for comparison that wave maps in flat space occurring naturally in the context of solutions of the vacuum Einstein equations with symmetry, for instance in Gowdy spacetimes (cf. [34]), are defined by a Lagrangian of the above type (using the flat metric) with λ = 2.

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Collection of 1-forms

We now turn to a system of several 1-forms. It is clear that if these have all the same dilaton coupling, as in the Yang-Mills Action (1.7), then, the critical value of λ is just that computed in (6.16) and (6.19) since each form brings in the same walls. The situation is more complicated if the dilaton couplings are different. One could naively think that the subcritical domain is then just the Cartesian product (j) (j) of the individual subcritical intervals [−λc , λc ], but this is not true because the intersection of the wall chambers associated with each 1-form may have no interior lightlike direction, even if each wall chamber has some. This is best seen on the example of two 1-forms in D spacetime dimensions with opposite dilaton couplings. The relevant inequalities, from which all others follow, are in this case λ λ A > 0, p1 + A > 0 2 2 p1 < p2 < · · · < pd

p1 −

(6.38) (6.39)

and can be easily analyzed because they determine, in this particular instance, a simplex in the hyperplane pa = 1. It follows from (6.38) that p1 > 0. The edgevectors can be taken to be (0, . . . , 0, 1, . . . , 1, 0) (k zeros, d−k ones, k = 1, . . . , d−1) and (1, 1, . . . , 1, ±2/λ). The first d − 1 edge-vectors are timelike or null, while the last two are spacelike provided −d(d − 1)λ2 + 4 > 0. This yields 2 λc =  d(d − 1)

for two 1-forms with opposite dilaton couplings

(6.40)

Accordingly, λc is finite for any spacetime dimension (and in fact, tends to zero as d → ∞), even though λc = ∞ for a single 1-form whenever d > 8.

7 Coupling between the matter fields The actions for the bosonic sectors of the low-energy limits of superstring theories or M-theory contain coupling terms between the p-forms, indicated by “more” in (1.1). These coupling terms are of the Chern-Simons or the Chapline-Manton type. In this section, we show that these terms are consistent with the results obtained in Section 5, in that they are also asymptotically negligible in the dynamical equations of motion when the Kasner exponents are subject to the above Inequalities (6.3)– (6.6). More precisely, the form of the velocity-dominated evolution equations and solutions are in each case exactly as in Section 5. The velocity-dominated matter constraints have additional terms, but as before, the velocity-dominated matter variables (besides the dilaton) are constant in time, so if the constraints are satisfied at some t > 0 they are satisfied for all t > 0. The quantities 0 ρ and 0 ja are defined exactly as in Section 5. Since the velocity-dominated evolution equations

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are also the same, there is nothing additional to check concerning the velocitydominated Hamiltonian and momentum constraints. Turning now to the exact equations, the restrictions defining the set V are unchanged from Section 5. The form of the evolution equation for the dilaton is unchanged. The form of the stress-energy tensor is also unchanged, and so the form of the Einstein evolution equations and the Einstein constraints is unchanged. The additional matter field variables considered in Section 5 are still all O(1), so estimates of terms involving the matter fields do not need to be reconsidered, as long as their form has not changed, for instance, in the argument that the Einstein constraints are satisfied. That the matter constraints are satisfied follows as in the other cases, once it is verified that their time derivative vanishes and that they are o(1). Since so much of the argument is identical to that of Section 5, we only point out the few places where there are differences.

7.1

Chern-Simons terms

First we consider the coupling of i of the additional matter fields via a ChernSimons term in the action. These additional matter fields should be such that i−1+

i


nj = D.

(7.1)

j=1

The Chern-Simons term which is added to the action is  (1) (i) SCS [Aγ1 ···γn1 , · · · , Aγ1 ···γni ] = A(1) ∧ dA(2) ∧ · · · ∧ dA(i) .

(7.2)

The variation of this term with respect to both the metric and the dilaton field, φ, vanishes. The matter Equation (5.4) is unchanged, since it is still the case that F (j) = dA(j) for all j. But Equation (5.3) for each of the i coupled matter fields acquires a non-vanishing right-hand side. (D)

√ ∇µ (F (j)µν1 ···νnj eλj φ ) −g (1)

(j−1)

= Cj ···ν1 ···νnj ··· F··· · · · F···

(j+1)

F···

(i)

· · · F···

(7.3)

Here 0...d = 1 and Cj is a numerical factor. Next, considering the d + 1 decomposition of Equation (7.3), the constraint Equation (5.5), for the jth coupled matter field, acquires the following term on its right-hand side, (1)

(j−1)

−Cj ···0b1 ··· F··· · · · F···

(j+1)

F···

(i)

· · · F···

(7.4)

Here all indices which are not explicit are spatial. So, only magnetic fields appear in (7.4). The following term is added to the right-hand side of the evolution

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Equation (5.7) for the jth coupled matter field. −Cj

j−1


(1)

(nm + 1)···0c1 ···cnm ···a1 ···anj ··· F··· · · ·

m=1

1

(j−1) (j+1) (i) × √ gc1 h1 · · · gcnm hnm E (m)h1 ···hnm e−λm φ · · · F··· F··· · · · F··· g i
(1) (j−1) (j+1) −Cj (nm + 1)···a1 ···anj ···0c1 ···cm ··· F··· · · · F··· F··· ··· (7.5) m=j+1



1 (i) × √ gc1 h1 · · · gcnm hnm E (m)h1 ···hnm e−λm φ · · · F··· . g Again, all indices which are not explicit are spatial. There is in each term only one electric field. The velocity-dominated matter constraint equations for the jth coupled matter field can be obtained from the “full” matter constraint equations for the same field by substituting the velocity-dominated quantities for all variables. The only additional terms occurring in f are due to Equation (7.5). The form of the mth term on the right-hand side of Equation (7.5) is just like the form of the terms on the right-hand side of Equation (5.8) for the mth coupled field. The factors which differ, comparing the mth term of (7.5) to Equation (5.8) for the mth field, are 0(1). Since in both cases a factor of t1−β is added in order to obtain the terms appearing in f , the estimate that the additional terms in f due to the Chern-Simons coupling are O(tδ ) is obtained just as the corresponding previously obtained estimates.

7.2

Chapline-Manton couplings

Next we consider Chapline-Manton couplings. For definiteness, we treat two explicit examples, leaving to the reader the task of checking that the general case works in exactly the same way. The first coupling is between an n-form A and an (n + 1)-form B and is equivalent to making B massive. Let F = dA + B and H = dB. The gauge transformations are B → B + dη, for arbitrary n-form η, and A → A − η + dγ, for arbitrary (n − 1)-form γ. (If n = 0, then dγ is replaced by a constant scalar and we require that the corresponding constant, λA , in the coupling to the dilaton be nonzero.) The form of the action is the same as in Section 5, but since F now depends on B and not just on A, the variation of the action with respect to B acquires an additional term. Also, it is now the case that dF = H. The matter Equation (5.3) is unchanged for F and Equation (5.4) is unchanged for H. Equation (5.3) for H and Equation (5.4) for F are now as follows. (D)

∇µ (H µν0 ···νn eλB φ ) = (D)

∇[µ Fν0 ···νn ]

=

F ν0 ···νn eλA φ , 1 Hµν0 ···νn . (n + 2)

(7.6) (7.7)

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√ √ Define E a1 ···an = g F 0a1 ···an eλA φ and Da0 ···an = g H 0a0 ···an eλB φ . The matter constraint equations which are affected are 1
bi ab1 ···c···bn f D = −E b1 ···bn , 2 i=1 ac n

c Dab1 ···bn + ea (Dab1 ···bn ) + fca

e[a (Fb0 ···bn ] ) −

(n + 1) c 1 f[ab0 F|c|b1 ···bn ] = Hab0 ···bn , 2 n+2

(7.8) (7.9)

The additional term which appears on the right-hand side of Equation (5.7) for Da0 ···an is √ a0 b0 gg · · · g an bn Fb0 ···bn eλA φ . (7.10) The additional term which appears on the right-hand side of Equation (5.8) for Fa0 ···an is −1 (7.11) √ ga0 b0 · · · gan bn Db0 ···bn e−λB φ . g The velocity-dominated matter constraint equations which are affected can be obtained from Equation (7.8) and (7.9) by substituting the corresponding velocitydominated quantities for all variables. The only additional terms occurring in f are due to Equations (7.10) for D and (7.11) for F . The form of the additional terms in these equations is just like the form of the terms which appear in Equations (5.7) for E and in (5.8) for H. Therefore the estimate that the additional terms are O(tδ ) is obtained just as the corresponding previously obtained estimates. The second Chapline-Manton type coupling is between an n-form A and a (2n)-form B. Let F = dA and H = dB + A ∧ F . The gauge transformations are A → A + dγ, for arbitrary (n − 1)-form γ, and B → B + dη − γ ∧ F , for arbitrary (2n − 1)-form η. (If n = 0 the gauge transformations are A → A + C and B → B + D − CA for constant scalars C and D and we require both λA = 0 and also λB = 0.) The form of the action is again the same as in Section 5. √ √ Define E a1 ···an = g F 0a1 ···an eλA φ and Da1 ···a2n = g H 0a1 ···a2n eλB φ . The matter Equations (5.3) for F and (5.4) for H are affected, only if n is odd. The equation for F which is affected (if n is odd) and its d + 1 decomposition are (D)

∇µ (F µν1 ···νn eλA φ ) =

2 H µν1 ···νn σ1 ···σn Fµσ1 ···σn eλB φ , (n + 1)!

(7.12)

1
bi ab2 ···c···bn 2 Dab2 ···bn h1 ···hn Fah1 ···hn , f E = 2 i=2 ac (n + 1)! (7.13) n

c E ab2 ···bn + ea (E ab2 ···bn ) + fca

∂t E a1 ···an

2 ···− √ (7.14) Da1 ···an b1 ···bn gb1 c1 · · · gbn cn E c1 ···cn g n! 2 √ bh0 a1 h1 + gg g · · · g an hn g c1 hn+1 · · · g cn h2n Hh0 ···h2n Fbc1 ···cn eλB φ . (n + 1)! =

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The equation for H which is affected (if n is odd) and its d + 1 decomposition are (D)

∇[µ0 Hµ1 ···µ2n+1 ] =

(2n + 1)! F[µ ···µ Fµ ···µ ], (n + 1)!(n + 1)! 0 n n+1 2n+1

(7.15)

(2n + 1) c (2n + 1)! f[ab0 H|c|b1 ···b2n ] = F[ab0 ···bn−1 Fbn ···b2n ] , 2 (n + 1)!(n + 1)! (7.16) (2n + 2)! b1 ···bn −λA φ g[a |b | · · · gan−1 |bn | E e Fan ···a2n ] . ∂t Ha0 ···a2n = · · · + √ g (n + 1)!(n + 1)! 0 1 (7.17) The velocity-dominated matter constraint equations which are affected can be obtained from Equation (7.13) and (7.16) by substituting the corresponding velocitydominated quantities for all variables. The only additional terms occurring in f are due to Equations (7.14) for E and (7.17) for H. Here again, the estimate that the additional terms in f are O(tδ ), is just as the estimate of terms appearing already in Section 5, either in Equation (5.7) for D or in Equation (5.8) for F . e[a (Hb0 ···b2n ] ) −

8 Yang-Mills We complete our analysis by proving that Yang-Mills couplings also enjoy the property of not modifying the conclusions. The action is (1.7), with a Yang-Mills field as source in addition to the scalar field considered in Section 4 and with |λ| < λc . The argument is again based on that of Sections 2–5 and it is enough here to note differences. The main one is that one must work with the vector potential instead of the fields themselves, because bare A’s appear in the equations. We could, in fact, have developed the entire previous analysis in terms of the vector potentials, thereby reducing the number of matter constraint equations. We followed a manifestly gauge-invariant approach for easing the physical understanding, but this was not mandatory. The stress-energy tensor is 1 1 ∇µ φ (D) ∇ν φ − gµν (D) ∇α φ (D) ∇α φ + [Fµα · Fν α − gµν Fαβ · F αβ ]eλφ . 2 4 (8.1) We work in the temporal gauge, A0 = 0. The matter fields satisfy the following equations. λ (D) ∇α (D) ∇α φ − Fαβ · F αβ eλφ = 0 (8.2) 4 Tµν =

(D)

(D)

∇µ (F µν eλφ ) + [Aµ , F µν ]eλφ = 0,

Fµν = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ].

(8.3) (8.4)

The Lie Bracket has no intrinsic time dependence. The d+1 decomposition of the stress-energy tensor is expressed in terms of the spatial tensor density

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√ 0a λφ g F e and the antisymmetric spatial tensor Fab . 1 1 1 {(∂t φ)2 + g ab ea (φ)eb (φ) + gab E a · E b e−λφ + g ab g ch Fac · Fbh eλφ }, 2 g 2 (8.5) 1 b = −∂t φ ea (φ) + √ E · Fab , (8.6) g 1 1 = g ac eb (φ) ec (φ) − {gbc E a · E c − δ a b gch E c · E h }e−λφ g 2 1 +{g ac g hi Fch · Fbi − δ a b g ch g ij Fci · Fhj }eλφ . (8.7) 4 =

The matter constraint equation is b ea (E a ) + fba E a + [Aa , E a ] = 0.

(8.8)

The matter evolution equations are λ λ gab E a · E b e−λφ − g ab g ch Fac · Fbh eλφ , (8.9) 2g 4 1 √ i ac a ic √ bh = eb ( gg ac g bh Fch eλφ ) + (fib g + fbi g ) gg Fch eλφ (8.10) 2 1 = − √ gab E b e−λφ . (8.11) g

∂t2 φ − (trk)∂t φ = g ab ∇a ∇b φ + ∂t E a ∂t Aa

Note that we use as basic matter variables Aa and E b (the quantity Fab being then defined in terms of Aa as Fab = ∂a Ab − ∂b Aa + [Aa , Ab ]). The Kasner-like evolution equations are Equations (2.7)–(2.10) and ∂t 0 Aa = 0. We consider analytic solutions of the Kasner-like evolution equations of the form (2.12)–(2.15) along with the quantity 0 Aa which is constant in time. Given a point x0 ∈ Σ, we use an adapted spatial frame on a neighborhood of x0 , U0 , as in Section 3. Thus, 0 gab (t0 ) and K a b are specialized as in that section. There is one velocity-dominated matter constraint equation, obtained from Equations elconstraintym) by replacing E a and Aa with 0 E a and 0 Aa . If the velocity-dominated matter constraint is satisfied at some time t0 > 0, then it is satisfied for all t > 0. Define 0 0

ρ

ja

=

1 (∂t 0 φ)2 , 2

(8.12)

1 = −∂t 0 φ ea ( 0 φ) + 

0g

0 b

E · 0 Fab .

(8.13)

The velocity-dominated Einstein constraints are defined as in the other cases. Equations (2.21) and (2.22) are again satisfied, so if the velocity-dominated constraints are satisfied at some t0 , then they are satisfied for all t > 0. The restrictions

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defining the set V are as in Section 5, with nj = 1. The relation of the unknown, u, in Equation (1.5) to the Einstein-matter variables is given by Equations (2.31)– (2.37) and (8.14) Aa = 0 Aa + tβ ϕa . The quantities A and f in Equation (1.5) are given by Equations (2.39)–(2.43) and t ∂t χ + βχ = tα0 −β (tr κ)(A + tβ χ) + t2−β S g ab S ∇a S ∇b 0 φ + t2−ζ S ∇a ωa λ λ +t2−β { gab E a · E b e−λφ − g ab g ch Fac · Fbh eλφ }, (8.15) 2g 4 √ t ∂t ξ a + βξ a = t1−β {eb ( gg ac g bh Fch eλφ ) 1 a ic √ bh i ac +(fib g + fbi g ) gg Fch eλφ }, (8.16) 2 1 t ∂t ϕa + βϕa = −t1−β √ gab E b e−λφ . (8.17) g The estimate that f = O(tδ ) is obtained as before, using E a = O(1) and Fab = O(1). The matter constraint quantity, the left-hand side of Equation (8.8), is o(1) and its time derivative vanishes, so the matter constraint is satisfied. The estimate of the matter terms in the Einstein constraints is obtained as in Section 5 for nj = 1. To conclude: the whole analysis goes through even in the presence of the Yang-Mills coupling terms and the system is asymptotically Kasner-like provided |λ| < λc , where λc is the same as in the abelian case and explicitly given by (6.16) and (6.19).

9 Self-interacting scalar field Consider Einstein’s equations, D ≥ 3, with sources as in Sections 4, 5, 7 or 8, except that the massless scalar field, φ, is replaced by a self-interacting scalar field. That is, the Expression (1.8) is added to the action. Solutions with a monotone singularity can be constructed as in Sections 4–8, with assumptions regarding the function V (φ) which appears in (1.8) given below. There is no change in the velocity-dominated evolution equations and solutions, nor in the velocitydominated constraints. The only change to Equation (1.5) is that two new terms appear in f . There is a new term, t2−α0 δ a b 2 V (φ)/(D − 2), on the right-hand side of the evolution equation for κa b (through M a b ). There is also a new term, −t2−β V  (φ), on the right-hand side of the evolution equation for χ. For Equation (1.5) to be Fuchsian, it must be the case that f = O(tδ ) and, in addition, that f satisfy other regularity conditions [15, 28]. Some examples were considered in [31]. A trivial example is obtained by taking V to be a constant. Then the equation for the scalar field is not changed by the potential while its effect on the Einstein equations is equivalent to the

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addition of a cosmological constant. Thus we see that the analysis of [15] generalizes directly to the case of the Einstein-scalar field system with non-zero cosmological constant. Of course the analogous statement applies to the other dimensions and matter fields considered in previous sections. To get another simple example take V (φ) = λφp for a constant λ and an integer p ≥ 2. Showing that the equation is Fuchsian involves examining the expression V (A ln t + B + tβ ψ) = λ(A ln t + B + tβ ψ)p

(9.1)

and corresponding expressions for the first and second derivatives of V . Of course in this particular case these are given by multiples of smaller powers of t. The aim is to estimate these quantities by suitable powers of t. In this case a Fuchsian system is always obtained. A linear massive scalar field is obtained by choosing p = 2. Another interesting possibility is to choose V (φ) = eλφ for a constant λ, in which case the derivatives of V are also exponentials. Then V (A ln t + B + tβ ψ) = eλB tλA exp(λtβ ψ)

(9.2)

Note that such an exponential potential can be (formally) generated by adding, as matter field, a d-form Aµ1 ···µd with dilaton coupling λd = −λ. Indeed, eliminating the field-strength F = dA (which satisfies eλd φ F = Cη, where C is a constant and η the volume form), leads to a term in the action proportional to e−λd φ C 2 . A Fuchsian system is obtained provided the general “electric” p-form condition (5.10) (with nj = d), 2p1 +· · ·+2pd −λd A > 0 is satisfied, i.e., (after using p1 +· · ·+pd = 1 and λd = −λ) provided λA > −2. This therefore yields a restriction on the data. More generally, it is enough to have a function V on the real line which has an analytic continuation to the whole complex plane and which satisfies estimates of the form ˜ = O(1), ˜ + tβ ψ) t2−c1 V˜ (A˜ ln t + B ˜ = O(1), ˜ + tβ ψ) t2−c2 V˜  (A˜ ln t + B 2−c3 ˜  ˜ β ˜ = O(1), ˜ + t ψ) V (A ln t + B t

(9.3)

˜ are the analytic continuafor some positive numbers c1 , c2 and c3 . Here A˜ and B tions of A(x) and B(x), to some (small, simply connected) complex neighborhood of the range of a coordinate chart. And ψ˜ lies in some region of the complex plane containing the origin. For f to be regular, it must be the case that c1 ≥ α0 and c2 ≥ β, which can be achieved by reducing , if necessary, and also possibly U0 , so that previous assumptions are satisfied. By taking suitable account of the domains of the functions involved it is also possible to obtain an analogue of this result when the functions V and V˜ are only defined on some open subsets of R and C. The only other change to the construction given in Sections 4–8 is that ρ → ρ + V (φ). It is still the case that (D) ∇µ T µν = 0, so Equations (2.59) and (2.60) are satisfied. Equation (2.63) is satisfied due to the assumptions concerning V (φ), so the Einstein constraints are satisfied.

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10 Conclusions Our paper establishes the Kasner-like behaviour for vacuum gravity in spacetime dimensions greater than or equal to 11, as well as the Kasner-like behaviour for the Einstein-dilaton-matter systems with subcritical dilaton couplings. Our results can be summarized as follows Theorem 10.1 Let Σ be a d-dimensional analytic manifold, d ≥ 10 and let ( 0 gab , 0 kab ) be a C ω solution of the Kasner-like vacuum Einstein equations on (0, ∞) × Σ such that t tr 0 k = −1 and such that the ordered eigenvalues of −t 0 kab satisfy 1 + p1 − pd − pd−1 > 0. Then there exists an open neighborhood U of {0} × Σ in [0, ∞) × Σ and a C ω solution (gab , kab ) of the Einstein vacuum field equations on U ∩ ((0, ∞) × Σ) such that for each compact subset K ⊂ Σ there are positive real numbers αab for which the following estimates hold uniformly on K: 1. 0 g ac gcb = δ ab + o(tα b ) a

2. k ab = 0 k ab + o(t−1+α b ) a

Theorem 10.2 Let Σ be a d-dimensional analytic manifold, d ≥ 2 and let (j)

X = ( 0 gab , 0 kab , 0 φ, 0 E (j)a1 ···anj , 0 Fa0 ···anj ), with j taking on values 1 through k for some non-negative integer k (possibly 0, in which case j takes on no values), 0 ≤ nj ≤ d−1. Let λj be constants in the subcritical range. Let X be a C ω solution of the Kasner-like Einstein-matter equations on (0, ∞) × Σ such that t tr 0 k = −1, and such that the ordered eigenvalues of −t 0 kab satisfy 1 + p1 − pd − pd−1 > 0 and, for each j, 2p1 + · · · + 2pnj − λj t ∂t 0 φ > 0 and 2p1 + · · · + 2pd−nj −1 + λj t ∂t 0 φ > 0. Then there exists an open neighborhood U of {0} × Σ in [0, ∞) × Σ and a C ω (j) solution (gab , kab , φ, E (j)a1 ···anj , Fa0 ···anj ) of the Einstein-matter field equations on U ∩ ((0, ∞) × Σ) such that for each compact subset K ⊂ Σ there are positive real numbers β, αab , with β < αab , for which the following estimates hold uniformly on K: 1. 0 g ac gcb = δ ab + o(tα b ) a

2. k ab = 0 k ab + o(t−1+α b ) a

3. φ = 0 φ + o(tβ ) 4. E (j)a1 ···anj = 0 E (j)a1 ···anj + o(tβ ) (j)

(j)

5. Fa0 ···anj = 0 Fa0 ···anj + o(tβ )

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Remarks 1. Corresponding estimates hold for certain first order derivatives of the basic unknowns in Theorems 10.1 and 10.2 (cf. Theorem 2.1 in [15]). These are the derivatives which arise in the definition of new unknowns when second order equations are reduced to first order so as to produce a first order Fuchsian system. 2. Our analysis shows that a solution of the full subcritical Einstein-matter equations satisfying the estimates given in the theorems and the corresponding estimates for first order derivatives just mentioned is uniquely determined by the solution of the velocity-dominated equations (the integration functions are included in the zeroth order, Kasner-like solutions; the deviation from them is uniquely determined). 3. The Einstein-matter field equations may include interaction terms of ChernSimons, Chapline-Manton and Yang-Mills type, and the scalar field may be self-interacting, with assumptions on V (φ) as stated in Section 9. If the (j) (j) (j) jth field is a Yang-Mills field, then Fab is obtained from Aa and 0 Fab is (j) obtained from 0 Aa through Equation (8.4). Note that the condition on tr 0 k which is assumed in both theorems can always be arranged by means of a time translation. 4. The spacetimes of the class whose existence is established by these theorems have the desirable property that it is possible to determine the detailed nature of their singularities by algebraic calculations. This allows them to be checked for consistency with the cosmic censorship hypothesis. What should be done from this point of view is to check that some invariantly defined physical quantity is unbounded as the singularity at t = 0 is approached. This shows that t = 0 is a genuine spacetime singularity beyond which no regular extension of the spacetime is possible. For this purpose it is common to examine curvature invariants but in fact it is just as good if an invariant of the matter fields can be found which is unbounded in the approach to t = 0. This is particularly convenient in the cases where a dilaton is present. Then ∇α φ∇α φ is equal in leading order to the corresponding velocity-dominated quantity and the latter is easily seen to diverge like t−4 for t → 0. The vacuum case is more difficult. It will be shown below that the approximation of the full solution by the velocity-dominated solution is sufficiently good that it is enough to do the calculation for the velocity-dominated metric. This means that it is enough to do the calculation for the Kasner metric in D dimensions. Note that the Kasner metric is invariant under reflection in each of the spatial coordinates. Hence curvature components of the form R0abc vanish, as do components of the form R0a0b with a = b. Hence the Kretschmann scalar Rαβγδ Rαβγδ is a sum of non-negative terms of the form Rabcd Rabcd and (Ra 0a0 )2 . In order to show that the Kretschmann scalar

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is unbounded it is enough to show that one of these terms is unbounded. A simple calculation shows that (Ra 0a0 )2 = p2a (1 − pa )2 t−4 in a Kasner spacetime. Thus the curvature invariant under consideration can only be bounded as t → 0 if all Kasner exponents are zero or one, which does not occur for the solutions we construct. To see that the approximation of the full solution by the velocity-dominated solution is valid for determining the asymptotics of the Kretschmann scalar it is enough to note that all terms appearing in the Kretschmann scalar which were not just considered are o(t−4 ). Only two estimates additional to those already obtained are needed ˜ h = O(t−2+ ) and ∇ ˜ a k˜b = O(t−2+ ) are – for these, the estimates R c abc sufficient. Both of these estimates are straightforward to obtain. The main ˜ c = O(t−1+4σ−2 −δ ) (i.e., the connection coefficients do not need input is Γ ab to be expanded). The expression for the Kretschmann scalar is 4((tr k)k a b − k a c κc b )((tr k)k b a − k b h k h a ) +(k a b k c h − k a h k c b )(k b a k h c − k h a k b c ) +4{(Ra b − M a b )(Rb a − M b a ) + 2(Ra b − M a b )((tr k)k b a − k b h k h a ) ˜ jR ˜ i ˜ a k˜b )(∇ ˜ h k˜c )˜ ˜ a k˜b )(∇ ˜ b k˜a )˜ −2(∇ g ah − 2(∇ g ch } − g˜ab g˜ch R c

b

c

h

aci

bhj

˜ abc h (k˜ ai k˜b h − k˜ah k˜b i )˜ +2R g ci . Apart from the Kasner terms (which can each be written as two factors, with each factor O(t−2 )), the remaining terms can each be written as two factors, with each factor O(t−2 ) and at least one of the two factors o(t−2 ). 5. We have constructed large classes of solutions of the Einstein-matter equations with velocity-dominated singularities for matter models defined by those field theories where the BKL picture predicts that solutions of this kind should exist. No symmetry assumptions were made. When symmetry assumptions are made there are more possibilities of finding specialized classes of spacetimes with velocity-dominated singularities. See for instance [45], where there are results for the Einstein-Maxwell-dilaton and other systems under symmetry assumptions. There are also results for the case where the Einstein equations are coupled to phenomenological matter models such as a perfect fluid and certain symmetry assumptions are made. For one of the most general results of this kind so far see [46]. 6. When solutions are constructed by Fuchsian methods as is done is this paper there is the possibility of algorithmically constructing an expansion of the solution about the singularity to all orders which is convergent when the input data are analytic, as in this paper. (If the input data are only C ∞ the expansion is asymptotic in a rigorous sense when Fuchsian techniques can be applied.) At the same time, there is the possibility of providing a rigorous confirmation of the reliability of existing expansions such as those of [24] and [25]. This is worked out for the case of [24] in [28].

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Acknowledgments We thank Mme Choquet-Bruhat for comments which led to clarifications in the exposition. The work of MH and MW is supported in part by the “Actions de Recherche Concert´ees” of the “Direction de la Recherche Scientifique – Communaut´e Fran¸caise de Belgique”, by a “Pˆ ole d’Attraction Interuniversitaire” (Belgium) and by IISN-Belgium (convention 4.4505.86). The research of MH is also supported by Proyectos FONDECYT 1970151 and 7960001 (Chile) and by the European Commission RTN programme HPRN-CT-00131, in which he is associated to K. U. Leuven. MW would also like to thank the organizers of the Mathematical Cosmology Program at the Erwin Schr¨ odinger Institute, Summer 2001, where a portion of this work was completed.

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[24] B. Grubiˇsi´c and V. Moncrief, Asymptotic behaviour of the T 3 × R Gowdy spacetimes, Phys. Rev. D 47 2371–2382 (1993). [25] A. Buonanno, T. Damour and G. Veneziano, Pre-big bang bubbles from the gravitational instability of generic string vacua, Nucl. Phys. B 543, 275 (1999) [arXiv:hep-th/9806230]. [26] B.K. Berger and V. Moncrief, Exact U(1) symmetric cosmologies with local Mixmaster dynamics, Phys. Rev. D 62 02359 (2000) [arXiv:gr-qc/0001083]. [27] H. Ringstr¨ om, The Bianchi IX attractor, Ann. H. Poincar´e 2, 405–500 (2001) [arXiv:gr-qc/0006035]. [28] S. Kichenassamy, and A.D. Rendall, Analytic description of singularities in Gowdy spacetimes, Class. Quantum Grav. 15 1339–1355 (1998). [29] A.D. Rendall, Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity, Class. Quantum Grav. 17 3305-3316 (2000) [arXiv:gr-qc/0004044]. [30] A.D. Rendall, Applications of the theory of evolution equations to general relativity, arXiv:gr-qc/0109028. [31] A.D. Rendall, Blow-up for solutions of hyperbolic PDE and spacetime singularities, in Proceedings of Journees EDP Atlantique, 2000 [arXiv:grqc/0006060]. [32] J. Demaret, J.L. Hanquin, M. Henneaux, P. Spindel and A. Taormina, The Fate Of The Mixmaster Behavior In Vacuum Inhomogeneous Kaluza-Klein Cosmological Models, Phys. Lett. B 175, 129 (1986). [33] B.K. Berger, Influence of scalar fields on the approach to a cosmological singularity, Phys. Rev. D 61, 023508–1-6 (1999) [arXiv:gr-qc/9907083]. [34] B.K. Berger and D. Garfinkle, Phenomenology of the Gowdy universe on T 3 × R, Phys. Rev. D 57, 4767–4777 (1998) [arXiv:gr-qc/9710102]. [35] A.D. Rendall and M. Weaver, Manufacture of Gowdy spacetimes with spikes, Class. Quantum Grav. 18, 2959–2975 (2001) [arXiv:gr-qc/0103102]. [36] D.M. Chitre, Ph. D. thesis, University of Maryland, 1972. [37] C.W. Misner, in: D. Hobill et al. (Eds), Deterministic chaos in general relativity, Plenum, 1994, pp. 317–328 [gr-qc/9405068]. [38] A.A. Kirillov and V.N. Melnikov, Dynamics of Inhomogeneities of Metric in The Vicinity of a Singularity in Multidimensional Cosmology Phys. Rev. D 52, 723 (1995) [gr-qc/9408004]; V.D. Ivashchuk and V.N. Melnikov, Billiard Representation for Multidimensional Cosmology with Multicomponent Perfect Fluid Near the Singularity, Class. Quantum Grav. 12, 809 (1995).

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[39] T. Damour, M. Henneaux, B. Julia and H. Nicolai, Hyperbolic Kac-Moody algebras and chaos in Kaluza-Klein models, Phys. Lett. B 509, 323 (2001) [arXiv:hepn-th/0103094]. [40] H. Nicolai, Two-dimensional gravities and supergravities as integrable system, DESY-91-038 Lectures presented at 30th Schladming Winter School, Schladming, Austria, Feb 27–Mar 5, 1991. [41] L. Andersson, The global existence problem in general relativity, arXiv:grqc/9911032. [42] Y. Choquet-Bruhat and J.W. York, The Cauchy problem, in A. Held (ed.), General Relativity Plenum, New York, 1980. [43] Y. Choquet-Bruhat, J. Isenberg and J.W. York, Einstein constraints on asymptotically Euclidean manifolds, Phys. Rev. D 61, 084034 (2000). ´ Murchadha and J.W. York, Existence and uniqueness of solutions of the [44] N.O. Hamiltonian constraint of general relativity on compact manifolds, J. Math. Phys. 14, 1551–1557, 1973. [45] M. Narita, T. Torii and K. Maeda, Asymptotic singular behaviour of Gowdy spacetimes in string theory, Class. Quantum Grav 17, 4597–4613 (2000) [arXiv:gr-qc/0003013]. [46] K. Anguige, A class of perfect-fluid cosmologies with polarized Gowdy symmetry and a Kasner-like singularity, arXiv:gr-qc/0005086.

Thibault Damour Institut des Hautes Etudes Scientifiques 35, Route de Chartres F-91440 Bures-sur-Yvette France email: [email protected] Marc Henneaux Physique Th´eorique et Math´ematique Universit´e Libre de Bruxelles C.P. 231 B-1050, Bruxelles Belgium and Centro de Estudios Cient´ıficos Casilla 1469 Valdivia

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Chile email: [email protected] Alan D. Rendall Max-Planck-Institut f¨ ur Gravitationsphysik Albert-Einstein-Institut Am M¨ uhlenberg 1 D-14476 Golm Germany email: [email protected] Marsha Weaver Physique Th´eorique et Math´ematique Universit´e Libre de Bruxelles C.P. 231 B-1050, Bruxelles Belgium email: [email protected] Communicated by Sergiu Klainerman submitted 19/02/02, accepted 15/07/02

To access this journal online: http://www.birkhauser.ch

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Ann. Henri Poincar´e 3 (2002) 1113 – 1181 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/0601113-69

Annales Henri Poincar´ e

Adiabatic Vacuum States on General Spacetime Manifolds: Definition, Construction, and Physical Properties W. Junker and E. Schrohe Abstract. Adiabatic vacuum states are a well-known class of physical states for linear quantum fields on Robertson-Walker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the Klein-Gordon field on globally hyperbolic spacetime manifolds (factoriality, quasiequivalence, local definiteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface.

1 Introduction It has always been one of the main problems of quantum field theory on curved spacetimes to single out a class of physical states among the huge set of positive linear functionals on the algebra of observables. One prominent choice for linear field theories is the class of Hadamard states. It has been much investigated in the past, but only recently gained a deeper understanding due to the work of Radzikowski [41]. He showed that the Hadamard states are characterized by the wavefront set of their two-point functions (see Definition 3.1). This characterization immediately allows for a generalization to interacting fields [8] and puts all the techniques of microlocal analysis at our disposal [27, 28, 29]. They have made possible the construction of the free field theory [31] and the perturbation theory [7] on general spacetime manifolds. On the other hand, there is another well-known class of states for linear field theories on Robertson-Walker spaces, the so-called adiabatic vacuum states. They were introduced by Parker [38] to describe the particle creation by the expansion of cosmological spacetime models. Much work has also been devoted to the investigation of the physical (for a review see [18]) and mathematical [35] properties of these states, but it has never been known how to extend their definition to field theories on general spacetime manifolds. Hollands [24] recently defined these states for Dirac fields on Robertson-Walker spaces and observed that they are in general not of the Hadamard form (correcting an erroneous claim in [31]). It has been the aim of the present work to find a microlocal definition of adiabatic vacuum states which makes sense on arbitrary spacetime manifolds and

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can be extended to interacting fields, in close analogy to the Hadamard states. It turned out that the notion of the Sobolev (or H s -) wavefront set is the appropriate mathematical tool for this purpose. In Appendix B we review this notion and the calculus related to it. After an introduction to the structure of the algebra of observables of the KleinGordon quantum field on a globally hyperbolic spacetime manifold (M, g) in Section 2 we present our definition of adiabatic states of order N (Definition 3.2) in Section 3. It contains the Hadamard states as a special case: They are adiabatic states “of infinite order”. To decide which order of adiabatic vacuum is physically admissible we investigate the algebraic structure of the corresponding GNS-representations. Haag, Narnhofer & Stein [23] suggested as a criterion for physical representations that they should locally generate von Neumann factors that have all the same set of normal states (in other words, the representations are locally primary and quasiequivalent). We show in Section 4.1 (Theorem 4.5 and Theorem 4.7) that this is generally the case if N > 5/2. For the case of pure states on a spacetime with compact Cauchy surface, which often occurs in applications, we improve the admissible order to N > 3/2. In addition, in Section 4.2 we show that adiabatic vacua of order N > 5/2 satisfy the properties of local definiteness (Corollary 4.13) and those of order N > 3/2 Haag duality (Theorem 4.15). These results extend corresponding statements for adiabatic vacuum states on Robertson-Walker spacetimes due to L¨ uders & Roberts [35], and for Hadamard states due to Verch [49]; for their discussion in the framework of algebraic quantum field theory we refer to [21]. In Section 5 we explicitly construct pure adiabatic vacuum states on an arbitrary spacetime manifold with compact Cauchy surface (Theorem 5.10). In Section 6 we show that our adiabatic states are indeed a generalization of the well-known adiabatic vacua on Robertson-Walker spaces: Theorem 6.3 states that the adiabatic vacua of order n (according to the definition of [35]) on a Robertson-Walker spacetime with compact spatial section are adiabatic vacua of order 2n in the sense of our microlocal Definition 3.2. We conclude in Section 7 by summarizing the physical interpretation of our mathematical analysis and calculating the response of an Unruh detector to an adiabatic vacuum state. It allows in principle to physically distinguish adiabatic states of different orders. Appendix A provides a survey of the Sobolev spaces which are used in this paper.

2 The Klein-Gordon field in globally hyperbolic spacetimes We assume that spacetime is modeled by a 4-dimensional paracompact C ∞ -manifold M without boundary endowed with a Lorentzian metric g of signature (+ − −−) such that (M, g) is globally hyperbolic. This means that there is a 3-dimensional smooth spacelike hypersurface Σ (without boundary) which is intersected by each inextendible causal (null or timelike) curve in M exactly once. As a consequence M is time-orientable, and we fix one orientation once and for all defining

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“future” and “past”. Σ is also assumed to be orientable. Our units are chosen such that
= c = G = 1. In this work, we are concerned with the quantum theory of the linear KleinGordon field in globally hyperbolic spacetimes. We first present the properties of the classical scalar field in order to introduce the phase space that underlies the quantization procedure. Then we construct the Weyl algebra and define the set of quasifree states on it. The material in this section is based on the papers [36, 13, 32]. Here, all function spaces are considered to be spaces of real-valued functions. Let us start with the Klein-Gordon equation (2g + m2 )Φ

= =

(g µν ∇µ ∇ν + m2 )Φ √ 1 √ ∂µ (g µν g ∂ν Φ) + m2 Φ = 0 g

(1)

for a scalar field Φ : M → R on a globally hyperbolic spacetime (M, g) where g µν is the inverse matrix of g = (gµν ), g := | det(gµν )|, ∇µ the Levi-Civita connection associated to g and m > 0 the mass of the field. Since (1) is a hyperbolic differential equation, the Cauchy problem on a globally hyperbolic space is well posed. As a consequence (see e.g. [13]), there are two unique continuous linear operators E R,A : D(M) → C ∞ (M) with the properties (2g + m2 )E R,A f = E R,A (2g + m2 )f = f supp (E A f ) ⊂ J − (supp f ) supp (E R f ) ⊂ J + (supp f ) for f ∈ D(M) where J +/− (S) denotes the causal future/past of a set S ⊂ M, i.e., the set of all points x ∈ M that can be reached by future/past-directed causal (i.e., null or timelike) curves emanating from S. They are called the advanced (E A ) and retarded (E R ) fundamental solutions of the Klein-Gordon equation (1). E := E R − E A is called the fundamental solution or classical propagator of (1). It has the properties (2g + m2 )Ef = E(2g + m2 )f = 0 supp (Ef ) ⊂ J + (supp f ) ∪ J − (supp f )

(2)

for f ∈ D(M). E R , E A and E can be continuously extended to the adjoint operators E R , E A , E  : E  (M) → D (M) by E R = E A , E A = E R , E  = −E. Let Σ be a given Cauchy surface of M with future-directed unit normal field nα .

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Then we denote by ρ0 : C ∞ (M) → C ∞ (Σ) u → u|Σ ρ1 : C (M) → C ∞ (Σ) ∞

(3)

→ ∂n u|Σ := (n ∇α u)|Σ α

u

the usual restriction operators, while ρ0 , ρ1 : E  (Σ) → E  (M) denote their adjoints. Dimock [13] proves the following existence and uniqueness result for the Cauchy problem: Proposition 2.1 (a) Eρ0 , Eρ1 restrict to continuous operators from D(Σ) (⊂ E  (Σ)) to E(M) (⊂ D (M)), and the unique solution of the Cauchy problem (1) with initial data u0 , u1 ∈ D(Σ) is given by u = Eρ0 u1 − Eρ1 u0 .

(4)

(b) Furthermore, (4) also holds in the sense of distributions, i.e., given u0 , u1 ∈ D (Σ), there exists a unique distribution u ∈ D (M) which is a (weak) solution of (1) and has initial data u0 = ρ0 u, u1 = ρ1 u (the restrictions in the sense of Proposition B.7). It is given by u(f ) = −u1 (ρ0 Ef ) + u0 (ρ1 Ef ) for f ∈ D(M). (c) If u is a smooth solution of (1) with supp u0,1 contained in a bounded subset O ⊂ Σ then, for any open neighborhood U of O in M, there exists an f ∈ D(U) with u = Ef . Inserting u = Ef into both sides of Equation (4) we get the identity E = Eρ0 ρ1 E − Eρ1 ρ0 E

(5)

on D(M). Proposition 2.1 allows us to describe the phase space of the classical field theory and the local observable algebras of the quantum field theory in two different (but equivalent) ways. One uses test functions in D(M), the other the Cauchy data with compact support on Σ. The relation between them is then established with the help of the fundamental solution E and Proposition 2.1: ˜ σ ˜ := D(M)/ker E, Let (Γ, ˜ ) be the real linear symplectic space defined by Γ ˜ is independent of the choice of representatives f1 , f2 ∈ σ ˜ ([f1 ], [f2 ]) := f1 , Ef2 . σ ˜ For any open D(M) and defines a non-degenerate symplectic bilinear form on Γ. ˜ ˜ ˜ U ⊂ M there is a local symplectic subspace (Γ(U), σ ˜ ) of (Γ, σ ˜ ) defined by Γ(U) := ˜ D(U)/ker E. To a symplectic space (Γ, σ ˜ ) there is associated (uniquely up to ∗˜ σ isomorphism) a Weyl algebra A[Γ, ˜ ], which is a simple abstract C ∗ -algebra gen˜ that satisfy erated by the elements W ([f ]), [f ] ∈ Γ, W ([f ])∗ = W ([f ])−1 = W ([−f ]) (unitarity) i

W ([f1 ])W ([f2 ]) = e− 2 σ˜ ([f1 ],[f2 ]) W ([f1 + f2 ])

(Weyl relations)

(6)

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˜ (see e.g. [3]). The Weyl elements satisfy the “field equation” for all [f ], [f1 ], [f2 ] ∈ Γ 2 W ([(2g + m )f ]) = W (0) = 1. (In a regular representation we can think of the ˆ ˆ ]) is the usual field elements W ([f ]) as the unitary operators eiΦ([f ]) where Φ([f operator smeared with test functions f ∈ D(M) and satisfying the field equation 2 ˆ ]) = Φ([(2 ˆ (2g + m2 )Φ([f g + m )f ]) = 0. (6) then corresponds to the canonical commutation relations.) A local subalgebra A(U) (U an open bounded subset of ˜ M) is then given by A[Γ(U), σ ˜ ]. It is the C ∗ -algebra generated by the elements W ([f ]) with supp f ⊂ U and contains the
quantum observables measurable in the ˜ σ spacetime region U. Then A[Γ, ˜ ] = C ∗ ( U A(U)). Dimock [13] has shown that U → A(U) is a net of local observable algebras in the sense of Haag and Kastler [22], i.e., it satisfies (i) (ii) (iii) (iv) (v)

U1 ⊂ U2 ⇒ A(U 1 ) ⊂ A(U 2 ) (isotony). U1 spacelike separated from U2 ⇒ [A(U 1 ), A(U 2 )] = {0} (locality). There is a faithful irreducible representation of A (primitivity). U1 ⊂ D(U2 ) ⇒ A(U 1 ) ⊂ A(U 2 ). For any isometry κ : (M, g) → (M, g) there is an isomorphism ακ : A → A such that ακ [A(U)] = A(κ(U)) and ακ1 ◦ ακ2 = ακ1 ◦κ2 (covariance).

In (iv), D(U) denotes the domain of dependence of U ⊂ M, i.e., the set of all points x ∈ M such that every inextendible causal curve through x passes through U. Since we are dealing with a linear field equation we can equivalently use the time zero algebras for the description of the quantum √ field theory. To this end we pick a Cauchy surface Σ with volume element d3 σ := h d3 x, where h := det(hij ) and hij is the Riemannian metric induced on Σ by g, and define a classical phase space (Γ, σ) of the Klein-Gordon field by the space Γ := D(Σ)⊕D(Σ) of real-valued initial data with compact support and the real symplectic bilinear form σ :Γ×Γ (F1 , F2 )

→ R → − [q1 p2 − q2 p1 ] d3 σ,

(7)

Σ

Fi := (qi , pi ) ∈ Γ, i = 1, 2. In this case, the local subspaces Γ(O) := D(O) ⊕ D(O) are associated to bounded open subsets O ⊂ Σ. The next proposition establishes the equivalence between the two formulations of the phase space: ˜ Proposition 2.2 The spaces (Γ(O), σ) and (Γ(D(O)), σ˜ ) are symplectically isomorphic. The isomorphism is given by ˜ ρΣ : Γ(D(O))

→ Γ(O)

[f ] → (ρ0 Ef, ρ1 Ef ). The proof of the proposition is a simple application of Proposition 2.1 and Equation (5). It shows in particular that the symplectic form σ, Equation (7), is independent of the choice of Cauchy surface Σ.

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Now, to (Γ, σ) we can associate the Weyl algebra A[Γ, σ] with its local subalgebras A(O) := A[Γ(O), σ]. By uniqueness, A(O) is isomorphic (as a C ∗ -algebra) to A(D(O)) which should justify our misuse of the same letter A. The ∗-isomorphism is explicitly given by α : A(D(O)) → A(O),

αW ([f ]) := W (ρΣ ([f ])).

In the rest of the paper we will only have to deal with the net O → A(O) of local time zero algebras, since they naturally occur when one discusses properties of a linear quantum field theory. Nevertheless, by the above isomorphism, one can translate all properties of this net easily into statements about the net U → A(U) and vice versa. Let us only mention here that locality of the time zero algebras means that [A(O1 ), A(O2 )] = {0} if O1 ∩ O2 = ∅. The states on an observable algebra A are the linear functionals ω : A → C satisfying ω(1) = 1 (normalization) and ω(A∗ A) ≥ 0 ∀A ∈ A (positivity). The set of states on our Weyl algebra A[Γ, σ] is by far too large to be tractable in a concrete way. Therefore, for linear systems, one usually restricts oneself to the quasifree states, all of whose truncated n-point functions vanish for n = 2: Definition 2.3 Let µ : Γ × Γ → R be a real scalar product satisfying 1 |σ(F1 , F2 )|2 ≤ µ(F1 , F1 )µ(F2 , F2 ) 4

(8)

for all F1 , F2 ∈ Γ. Then the quasifree state ωµ associated with µ is given by 1

ωµ (W (F )) = e− 2 µ(F,F ) . If ωµ is pure it is called a Fock state. The connection between this algebraic notion of a quasifree state and the usual notion of “vacuum state” in a Hilbert space is established by the following proposition which we cite from [32]: Proposition 2.4 Let ωµ be a quasifree state on A[Γ, σ]. (a) There exists a one-particle Hilbert space structure, i.e., a Hilbert space H and a real-linear map k : Γ → H such that (i) kΓ + ikΓ is dense in H, (ii) µ(F1 , F2 ) = RekF1 , kF2 H ∀F1 , F2 ∈ Γ, (iii) σ(F1 , F2 ) = 2ImkF1 , kF2 H ∀F1 , F2 ∈ Γ. The pair (k, H) is uniquely determined up to unitary equivalence. Moreover: ωµ is pure ⇔ k(Γ) is dense in H. (b) The GNS-triple (Hωµ , πωµ , Ωωµ ) of the state ωµ can be represented as (F s (H), ρµ , ΩF ), where (i) F s (H) is the symmetric Fock space over the one-particle Hilbert space H,

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(ii) ρµ [W (F )] = exp{−i[a∗ (kF ) + a(kF )]}, where a∗ and a are the standard creation and annihilation operators on F s (H) satisfying [a(u), a∗ (v)] = u, v H and a(u)ΩF = 0 for u, v ∈ H. (The bar over a∗ (kF ) + a(kF ) indicates that we take the closure of this operator initially defined on the space of vectors of finite particle number.) (iii) ΩF := 1 ⊕ 0 ⊕ 0 ⊕ . . . is the (cyclic) Fock vacuum. Moreover: ωµ is pure ⇔ ρµ is irreducible. Thus, ωµ can also be represented as ωµ (W (F )) = exp{− 21 ||kF ||2H } (by (a)) or ˆ ) := a∗ (kF ) + a(kF ) is the usual field ωµ (W (F )) = ΩF , ρµ (F )ΩF (by (b)). Φ(F operator on F s (H) and we can determine the (“symplectically smeared”) two-point function as λ(F1 , F2 ) = = =

ˆ 1 )Φ(F ˆ 2 )ΩF ΩF , Φ(F kF1 , kF2 H i µ(F1 , F2 ) + σ(F1 , F2 ) 2

for F1 , F2 ∈ Γ, resp. the Wightman two-point function Λ as     ρ0 Ef1 ρ0 Ef2 Λ(f1 , f2 ) = λ , ρ1 Ef1 ρ1 Ef2

(9)

(10)

for f1 , f2 ∈ D(M). The fact that the antisymmetric (= imaginary) part of λ is the symplectic form σ implies for Λ:  1 [f1 E  ρ0 ρ1 Ef2 − f1 E  ρ1 ρ0 Ef2 ] d3 σ Im Λ(f1 , f2 ) = − 2 Σ 1 f1 , Ef2 = (11) 2 by Equation (5). All the other n-point functions can also be calculated, one finds that they vanish if n is odd and that the n-point functions for n even are sums of products of two-point functions. Once a (quasifree) state ω on the algebra A has been chosen the GNSrepresentation (Hω , πω , Ωω ) of Proposition 2.4 allows us to represent all the algebras A(O) as concrete algebras πω (A(O)) of bounded operators on Hω . The weak closure of πω (A(O)) in B(Hω ), which, by von Neumann’s double commutant theorem, is equal to πω (A(O)) (the prime denoting the commutant of a subalgebra of B(Hω )), is denoted by Rω (O). It is the net of von Neumann algebras O → Rω (O) which contains all the physical information of the theory and is therefore the main object of study in algebraic quantum field theory (see e.g. [21]). One

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of the most straightforward properties is the so-called additivity. It
states that if an open bounded subset O ⊂ Σ is the union of open subsets O = i Oi then the von Neumann algebra Rω (O) is generated by the subalgebras Rω (Oi ), i.e., Rω (O) =

 

 Rω (Oi )

.

(12)

i

Additivity expresses the fact that the physical information contained in Rω (O) is entirely encoded in the observables that are localized in arbitrarily small subsets of O. The following result is well known: Lemma 2.5 Let ω be a quasifree state of the Weyl algebra, O an open bounded subset of Σ. Then Rω (O) is additive. Proof. Let (k, H) be the one-particle Hilbert space structure of ω (Proposition 2.4). According to results of Araki [1, 34] Equation (12) holds iff kΓ(O) = span kΓ(Oi )

(13)

where the closure is taken w.r.t. the norm in H. With the help of a partition of unity {χi ; supp χi  ⊂ Oi } it is clear that any u = k(F ) ∈ kΓ(O), F ∈ Γ(O), can be written as u = i k(χi F ) ∈ span kΓ(Oi ) (note that the sum is finite since F has compact support in O), and therefore kΓ(O) ⊂ span kΓ(Oi ). The converse inclusion is obvious, and therefore also (13) holds.
(More generally, additivity even holds for arbitrary states since already the Weyl algebra A(O) has an analogous property, cf. [3].) Other, more specific, properties of the net of von Neumann algebras will not hold in such general circumstances, but will depend on a judicious selection of (a class of) physically relevant states ω. For the choice of states we make in Section 3 we will investigate the properties of the local von Neumann algebras Rω (O) in Section 4.

3 Definition of adiabatic states As we have seen in the last section, the algebra of observables can easily be defined on any globally hyperbolic spacetime manifold. This is essentially due to the fact that there is a well-defined global causal structure on such a manifold, which allows to solve the classical Cauchy problem and formulate the canonical commutation relations, Equations (6) and (11). Symmetries of the spacetime do not play any role. This changes when one asks for the physical states of the theory. For quantum field theory on Minkowski space the state space is built on the vacuum state which is defined to be the Poincar´e invariant state of lowest energy. A generic spacetime manifold however neither admits any symmetries nor the notion of energy, and it has always been the main problem of quantum field theory on curved spacetime to find a specification of the physical states of the theory in such a situation.

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Using Hadamard’s elementary solution of the wave equation DeWitt & Brehme [11] wrote down an asymptotic expansion of the singular kernel of a distribution which they called the Feynman propagator of a quantum field on a generic spacetime manifold. Since then quantum states whose two-point functions exhibit these prescribed local short-distance singularities have been called Hadamard states. Much work has been devoted to the investigation of the mathematical and physical properties of these states (for the literature see e.g. [32]), but only Kay & Wald [32] succeeded in giving a rigorous mathematical definition of them. Shortly later, in a seminal paper Radzikowski [41] found a characterization of the Hadamard states in terms of the wavefront set of their two-point functions. This result proved to be fundamental to all ensuing work on quantum field theory in gravitational background fields. Since we do not want to recall the old definition of Hadamard states (it does not play any role in this paper) we reformulate Radzikowski’s main theorem as a definition of Hadamard states: Definition 3.1 A quasifree state ωH on the Weyl algebra A[Γ, σ] of the KleinGordon field on (M, g) is called a Hadamard state if its two-point function is a distribution ΛH ∈ D (M × M) that satisfies the following wavefront set condition W F  (ΛH ) = C + .

(14)

Here, C + is the positive frequency component of the bicharacteristic relation C = ˙ − that is associated to the principal symbol of the Klein-Gordon operator C + ∪C 2g + m2 (for this notion see [16]), more precisely C C

±

:=

{((x1 , ξ1 ; x2 , ξ2 ) ∈ T ∗ (M × M) \ 0; g µν (x1 )ξ1µ ξ1ν = 0,

:=

g µν (x2 )ξ2µ ξ2ν = 0, (x1 , ξ1 ) ∼ (x2 , ξ2 )} {(x1 , ξ1 ; x2 , ξ2 ) ∈ C; ξ10 ≷0, ξ20 ≷0}

(15) (16)

where (x1 , ξ1 ) ∼ (x2 , ξ2 ) means that there is a null geodesic γ : τ → x(τ ) such that x(τ1 ) = x1 , x(τ2 ) = x2 and ξ1ν = x˙ µ (τ1 )gµν (x1 ), ξ2ν = x˙ µ (τ2 )gµν (x2 ), i.e., ξ1 , ξ2 are cotangent to the null geodesic γ at x1 resp. x2 and parallel transports of each other along γ. The fact that only positive frequencies occur in (14) can be viewed as a remnant of the spectrum condition in flat spacetime, therefore (14) (and its generalization to higher n-point functions in [8]) is also called microlocal spectrum condition. However, condition (14) does not fix a unique state, but a class of states that generate locally quasiequivalent GNS-representations [48]. Now to which extent is condition (14) also necessary to characterize locally quasiequivalent states? In [31] one of us gave a construction of Hadamard states by a microlocal separation of positive and negative frequency solutions of the KleinGordon equation. From these solutions we observed that a truncation of the corresponding asymptotic expansions destroys the microlocal spectrum condition (14) but preserves local quasiequivalence, at least if the Sobolev order of the perturbation is sufficiently low (for Dirac fields an analogous observation was made by

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Hollands [24]). In other words, the positive frequency condition in (14) is not necessary to have local quasiequivalence, but can be perturbed by non-positive frequency or even non-local singularities of sufficiently low order. We formalize this observation by defining a new class of states with the help of the Sobolev (or H s -) wavefront set. For a definition and explanation of this notion see Appendix B. Definition 3.2 A quasifree state ωN on the Weyl algebra A[Γ, σ] of the KleinGordon field on (M, g) is called an adiabatic state of order N ∈ R if its twopoint function ΛN is a distribution that satisfies the following H s -wavefront set condition for all s < N + 32 W F s (ΛN ) ⊂ C + .

(17)

Note, that we did not specify W F s for s ≥ N + 32 in the definition. Hence every adiabatic state of order N is also one of order N  ≤ N . In particular, every Hadamard state is also an adiabatic state (of any order). Now the task is to identify those adiabatic states that are physically admissible, i.e., generate the same local quasiequivalence class as the Hadamard states. In [31, Section 3.6] an example of an adiabatic state of order −1 was given that does not satisfy this condition. In Theorem 4.7 we will prove that for N > 5/2 (and in the special case of pure states on a spacetime with compact Cauchy surface already for N > 3/2) the condition is satisfied (and the gap in between will remain unexplored in this paper). For this purpose the following simple lemma will be fundamental: Lemma 3.3 Let ΛH and ΛN be the two-point functions of an arbitrary Hadamard state and an adiabatic state of order N , respectively, of the Klein-Gordon field on (M, g). Then (18) W F s (ΛH − ΛN ) = ∅ ∀s < N + 32 . Proof. From Lemma 5.2 it follows that ∅, s < − 12 s W F (ΛH ) = + C , − 21 ≤ s and therefore W F s (ΛH − ΛN ) ⊂ W F s (ΛH ) ∪ W F s (ΛN ) ⊂ C + , s < N + 32 .

(19)

On the other hand, since ΛH and ΛN have the same antisymmetric part σ ˜ , ΛH − ΛN must be a symmetric distribution, and thus also W F s (ΛH − ΛN ) must be a symmetric subset of T ∗ (M × M), i.e., W F s (ΛH − ΛN ) antisymmetric. However, the only antisymmetric subset of the right-hand side of (19) is the empty set and
hence W F s (ΛH − ΛN ) = ∅ for s < N + 32 . In the next section we will use this lemma to prove the result mentioned above and some other algebraic properties of the Hilbert space representations generated by our new states. In Section 5 we will explicitly construct these states

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and in Section 6 we will show that the old and well-known class of adiabatic vacuum states on Robertson-Walker spacetimes satisfies our Definition 3.2 (the comparison with the order of these states led us to the normalization of s chosen in Definition 3.2). Contrary to an erroneous claim in [31], these states are in general no Hadamard states, but in fact “adiabatic states” in our sense. This justifies our naming of the new class of quantum states on curved spacetimes in Definition 3.2.

4 The algebraic structure of adiabatic vacuum representations 4.1

Primarity and local quasiequivalence of adiabatic and Hadamard states

Let A := A[Γ, σ] be the Weyl algebra associated to our phase space (Γ, σ) introduced in Section 2 and A(O) := A[Γ(O), σ] the subalgebra of observables localized in an open, relatively compact subset O ⊂ Σ. Let ωH denote some Hadamard state on A and ωN an adiabatic vacuum state of order N . It is the main aim of this section to show that ωH and ωN are locally quasiequivalent states for all sufficiently large N , i.e., the GNS-representations πωH and πωN are quasiequivalent when restricted to A(O), or, equivalently, there is an isomorphism τ between the von Neumann algebras πωH (A(O)) and πωN (A(O)) such that τ ◦ πωH = πωN on A(O) (see e.g. [6, Section 2.4]). To prove this statement we will proceed as follows: We first notice that πωH A(O) ˜ is quasiequivalent to πωN A(O) ˜ for is quasiequivalent to πωN A(O) if πωH A(O) ˜ ⊃ O. Since to any open, relatively compact set O we can find an open, some O ˜ containing O and having a smooth boundary we can relatively compact set O assume without loss of generality that O has a smooth boundary. Under this assumption we first show that πωN (A(O)) is a factor (for N > 3/2, Theorem 4.5). Now we note that the GNS-representation (πω˜ , Hω˜ , Ωω˜ ) of the partial state ω ˜ := ωN A(O) is a subrepresentation of (πωN A(O), HωN , ΩωN ). This is easy to see: K := {πωN (A)ΩωN ; A ∈ A(O)} is a closed subspace of HωN which is left invariant by πωN (A(O)). Since for all A ∈ A(O) (Ωω˜ , πω˜ (A)Ωω˜ ) = ω ˜ (A) = ωN (A) = (ΩωN , πωN (A)ΩωN ), the uniqueness of the GNS-representation implies that πω˜ and πωN A(O) coincide on K and (πω˜ , Hω˜ , Ωω˜ ) can be identified with (πωN A(O), K, ΩωN ) (up to unitary equivalence). We recall that a primary representation (which means that the corresponding von Neumann algebra is a factor) is quasiequivalent to all its (non-trivial) subrepresentations (see [14, Prop. 5.3.5]). Therefore, πωN A(O) is quasiequivalent to πω˜ = π(ωN
A(O)) , and analogously πωH A(O) is quasiequivalent to π(ωH
A(O)) . To prove that πωN A(O) and πωH A(O) are quasiequivalent it is therefore sufficient to prove the quasiequivalence of the GNS-representations π(ωN
A(O)) and π(ωH
A(O)) of the partial states. This will be done in Theorem 4.7 for N > 5/2.

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To get started we have to prove in a first step that the real scalar products µN and µH associated to the states ωN and ωH , respectively, induce the same topology on Γ(O) = C0∞ (O) ⊕ C0∞ (O). Let us denote by HµN (O) and HµH (O) the completion of Γ(O) w.r.t. µN and µH , respectively. R. Verch showed the following result [49, Prop. 3.5]: Proposition 4.1 For every open, relatively compact set O ⊂ Σ there exist positive constants C1 , C2 such that    

 q q 2 2 C1 qH 1/2 (O) + pH −1/2 (O) ≤ µH , p p

 ≤ C2 q2H 1/2 (O) + p2H −1/2 (O) for all pq ∈ Γ(O). Theorem 4.2 The topology of HµN (O) coincides with that of HµH (O) whenever ΛN satisfies (17) for N > 3/2. Proof. If (Σ, h) is not a complete Riemannian manifold we can find a function ˜ := f h) is complete [12, f ∈ C ∞ (Σ), f > 0, with f |O = const. such that (Σ, h Ch. XX.18, Problem 6]. Then the Laplace-Beltrami operator ∆h˜ associated with ˜ is essentially selfadjoint on C ∞ (Σ) [9]. The topology on Γ(O) will not be affected h 0 ˜ Without loss of generality we can therefore assume that by switching from h to h. ∆ is selfadjoint. Lemma 3.3 shows that 3 s (M × M) ∀ s < N + . ΛH − ΛN ∈ Hloc 2 In view of the fact that Σ is a hyperplane, Proposition B.7 implies that, for 1 < s < N + 3/2, (ΛH − ΛN )|Σ×Σ

∈

∂n1 (ΛH − ΛN ), ∂n2 (ΛH − ΛN )|Σ×Σ

∈

∂n1 ∂n2 (ΛH − ΛN )|Σ×Σ

∈

s−1 Hloc (Σ × Σ)

(20)

s−2 Hloc (Σ s−3 Hloc (Σ

× Σ)

(21)

× Σ).

(22)

Here, ∂n1 and ∂n2 denote the normal derivatives with respect to the first and second variable, respectively. We denote by λH and λN the scalar products on Γ induced via Equation (10) by ΛH and ΛN , respectively. Since ΛH and ΛN have the same antisymmetric parts we have         q1 q1 q2 q2 (µH − µN ) , = (λH − λN ) , p1 p2 p1 p2     q1 q2 = (23) ,M p1 p2 L2 (Σ)⊕L2 (Σ)

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q1 q2 p1 , p2 ∈ Γ, where M is the integral operator with the kernel function   ∂n1 ∂n2 (ΛH − ΛN )|Σ×Σ −∂n1 (ΛH − ΛN )|Σ×Σ M (x, y) = . (24) −∂n2 (ΛH − ΛN )|Σ×Σ (ΛH − ΛN )|Σ×Σ

˜ of O and a function Note that M (x, y) = M (y, x)∗ . We next fix a neighborhood O ∞ K = K(x, y) ∈ C0 (Σ × Σ) taking values in 2 × 2 real matrices such that K(x, y) = K(y, x)∗ , x, y ∈ Σ, and the entries Kij of K and Mij of M satisfy the relations K11 − M11 L2 (O× ˜ O) ˜ < K21 − M21 H 1/2 (O× ˜ O) ˜
0 is to be specified lateron. By K we denote the integral operator induced by K. We let µN

:=

λN

:=

µH + ·, (K − M)· = µN + ·, K· i µN + σ. 2

By ΛN we denote the associated bilinear form on C0∞ (M) × C0∞ (M)      ρ0 ρ0   ΛN (f, g) := λN Ef, Eg ρ1 ρ1

(26)

(note that, in spite of our notation, ΛN is not the two-point function of a quasifree state in general). Recall from (3) that ρ0 , ρ1 are the usual restriction operators. The definition of ΛN makes sense, since both ρ0 Eg and ρ1 Eg have compact support in Σ so that λN can be applied. In view of the fact that K is an integral operator with a smooth kernel, also λN − λN = µN − µN is given by a smooth kernel. We claim that also ΛN − ΛN is smooth on M × M: In fact,      ρ0 ρ0 (ΛN − ΛN )(f, g) = Ef, K Eg ρ1 ρ1 is given by the Schwartz kernel   ∗   ρ0 ρ0 E K E. ρ1 ρ1 Since E is a Lagrangian distribution of order µ = −3/2 (for more details see Section 5 below), while K is a compactly supported smooth function, the calculus of Fourier integral operators [28, Theorems 25.2.2, 25.2.3] show that the composition is also smooth. It follows from an argument of Verch [48, Prop. 3.8] that there are functions φj , ψj ∈ C0∞ (M), j = 1, 2, . . . , such that ΛN (f, g) − ΛN (f, g) =

∞  j=1

σ(f, φj )σ(g, ψj )

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for all f, g ∈ C0∞ (D(O)), satisfying moreover ∞ 

ΛN (φj , φj )1/2 ΛN (ψj , ψj )1/2 < ∞.

j=1

(An inspection of the proof of [48, Prop. 3.8] shows that it is sufficient for the validity of these statements that ΛN is the two-point function of a quasifree state, ΛN need not be one.) It follows that |ΛN (f, f ) − ΛN (f, f )| ≤



|σ(f, φj )σ(f, ψj )|

j

≤



4ΛN (f, f )1/2 ΛN (φj , φj )1/2 ΛN (f, f )1/2 ΛN (ψj , ψj )1/2

j

= 4ΛN (f, f )



ΛN (φj , φj )1/2 ΛN (ψj , ψj )1/2

j

≤ CΛN (f, f ). Therefore

|ΛN (f, f )| ≤ (1 + C)ΛN (f, f ).

Given q, p ∈ C0∞ (O), we can find f ∈ C0∞ (D(O)) such that q = ρ0 Ef, p = ρ1 Ef (cf. Proposition 2.1). Hence         q q q q   µN , = λN , = ΛN (f, f ) ≤ (1 + C)ΛN (f, f ) p p p p         q q q q = (1 + C)λN , = (1 + C)µN , . (27) p p p p We next claim that for all pq ∈ Γ(O)    

  q  q 2   ≤ C3 q2 1/2 + p , M −1/2 H H (O) (O)  p p  and

   

   q q 2  ≤ C q2 1/2  , (K − M) + p −1/2 H H (O) (O) ,  p p 

(28)

(29)

where C3 and C are positive constants and C can be made arbitrarily small by taking small in (25). Indeed, in order to see this, we may first multiply the kernel functions M and K − M , respectively, by ϕ(x)ϕ(y) where ϕ is a smooth function ˜ of O and ϕ ≡ 1 on O. The above expressions supported in the neighborhood O (28) and (29) will not be affected by this change. We may then localize the kernel functions to R3 × R3 noting that the Sobolev regularity is preserved. Now we can apply Lemma 4.3 and Corollary 4.4 to derive (28) and (29).

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We finally obtain the statement of the theorem from the estimates

C1 qH 1/2 (O) + pH −1/2 (O) ≤ (C1 − C ) qH 1/2 (O) + pH −1/2 (O) 2 if is sufficiently small         q q q q , + , (K − M) ≤ µH p p p p by Prop. 4.1 and (29)     q q = µN , p p     q q ≤ (1 + C)µN , by (27) p p          q q q q , − ,M = (1 + C) µH p p p p

≤ (1 + C)(C2 + C3 ) qH 1/2 (O) + pH −1/2 (O) by Prop. 4.1 and (28).
Lemma 4.3 Let k ∈ H 1/2 (Rn × Rn ). Then the integral operator K with kernel k induces an operator in B(H 1/2 (Rn ), H 1/2 (Rn )) and B(H −1/2 (Rn ), H −1/2 (Rn )). If we even have k ∈ H 1 (Rn × Rn ), then K induces an operator in B(H −1/2 (Rn ), H 1/2 (Rn )). In both cases, the operator norm of K can be estimated by the Sobolev norm of k. Proof. The boundedness of K : H ±1/2 → H ±1/2 is equivalent to the boundedness of L := D ±1/2 KD ∓1/2 on L2 (Rn ). (Here D ±1/2 := (1 − ∆)±1/4 , where ∆ is the Euclidean Laplacian.) This in turn will be true, if its integral kernel l(x, y) := Dx ±1/2 Dy ∓1/2 k(x, y) is in L2 (Rn × Rn ). In this case LB(L2 (Rn )) ≤ lL2(Rn ×Rn ) . We know that Dx ±1/2 Dy ∓1/2 are pseudodifferential operators on Rn × Rn with 1/2 on and Vaillancourt’s Theorem (cf. [33, symbols in S0,0 (R2n × R2n ). By Calder´ Theorem 7.1.6]), they yield bounded maps H 1/2 (Rn × Rn ) → L2 (Rn × Rn ). Hence l ∈ L2 (Rn ×Rn ) and we obtain the first assertion. For the second assertion we check that Dx 1/2 Dy 1/2 k(x, y) ∈ L2 (Rn × Rn ). Since the symbol of Dx 1/2 Dy 1/2 is 1 (R2n × R2n ), this holds whenever k ∈ H 1 .
in S0,0 Corollary 4.4 If

 k=

k11 k21

k12 k22



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with k11 ∈ L2 (Rn × Rn ),

k12 , k21 ∈ H 1/2 (Rn × Rn ),

k22 ∈ H 1 (Rn × Rn ),

then the integral operator K with kernel k induces a bounded map H 1/2 (Rn ) ⊕ H −1/2 (Rn ) → H −1/2 (Rn ) ⊕ H 1/2 (Rn ). Given (q, p) ∈ C0∞ (Rn ) ⊕ C0∞ (Rn ) we can estimate             q   q   q    K q   ,K ≤     p p H −1/2 ⊕H 1/2 p H 1/2 ⊕H −1/2 p L2 ⊕L2   2  q   ≤ K   p  1/2 −1/2 . H ⊕H (Here we used the fact that, for u ∈ H s (Rn ) and v ∈ H −s (Rn ), u, v can be understood as the extension of the L2 bilinear form and |u, v | ≤ uH s vH −s .) We now apply Theorem 4.2 to show that adiabatic vacua (of order N > 3/2) generate primary representations. The proof is a modification of the corresponding argument for Hadamard states due to Verch [48]. Theorem 4.5 Let ωN be an adiabatic vacuum state of order N > 3/2 on the Weyl algebra A[Γ, σ] of the Klein-Gordon field on (M, g) and πωN its GNSrepresentation. Then, for any open, relatively compact subset O ⊂ Σ with smooth boundary, πωN (A(O)) is a factor. In the proof of the theorem we will need the following lemma. Recall that the ˜ introduced in the proof of Theorem 4.2 differs from h only by a conformal metric h factor which is constant on O. ˜ Lemma 4.6 C0∞ (O) + C0∞ (Σ \ O) is dense in C0∞ (Σ) w.r.t. the norm of H 1/2 (Σ, h) −1/2 ˜ (Σ, h)). (and hence also w.r.t. the norm of H Proof. Using a partition of unity we see that the problem is local. We can therefore confine ourselves to a single relatively compact coordinate neighborhood and work on Euclidean space. In view of the fact that ˜h is positive definite, the topology of the Sobolev spaces on Σ locally yields the usual Sobolev topology. The problem therefore reduces to showing that every function in C0∞ (Rn ), n ∈ N, can be approximated by functions in C0∞ (Rn+ ) + C0∞ (Rn− ) in the topology of H 1/2 (Rn ). Following essentially a standard argument [45, 2.9.3] we proceed as follows. We choose a function χ ∈ C ∞ (R) with χ(t) = 1 for |t| ≥ 2 and χ(t) = 0 for |t| ≤ 1, 0 ≤ χ ≤ 1. We define χ : Rn → R by χ (x) := χ(xn / ). Given f ∈ C0∞ (Rn ) we have √ f − χ f L2 (Rn ) ≤ C1 C2 f − χ f H 1 (Rn ) ≤ √ .

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Interpolation shows that {f − χ f }0 0 on O, we have  ˜ 1 , kF ˜ 2 ˜ = σ ˜ (F1 , F2 ) = 2ImkF d3 σh˜ (p1 q2 − q1 p2 ) H O  3/2 = C d3 σh (p1 q2 − q1 p2 ) = C 3/2 σ(F1 , F2 ) O

for all Fi = (qi , pi ) ∈ Γ(O), i = 1, 2.

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∨ ˜ ˜ Imu, v ˜ = 0 ∀v ∈ k(Γ(O))} ˜ Define now k(Γ(O)) := {u ∈ H; and let u ∈ H ∨ ˜ ˜ ˜ k(Γ(O)) ∩ k(Γ(O)) . Then Imu, k(F ) H˜ = 0 for all F ∈ Γ(O) (by the definition of ∨ ∨ ˜ ˜ ) ˜ = 0 for all F ∈ Γ(Σ\ O) (since k(F ˜ ) ∈ k(Γ(O)) ˜ k(Γ(O)) ) and also Imu, k(F H for F ∈ Γ(Σ\O)). This, together with the density statement of Lemma 4.6, implies ˜ ) ˜ = 0 for all F ∈ Γ, and, since k(Γ) ˜ ˜ it follows that that Imu, k(F is dense in H, H u = 0, i.e., (31) is proven for the auxiliary state given by k˜ on A[Γ, σ ˜ ]. Let us now show (31) for an adiabatic vacuum state ωN , N > 3/2, on A[Γ, σ]. Let u ∈ kN (Γ(O)) ∩ kN (Γ(O))∨ , then there is a sequence {Fn , n ∈ N} ⊂ Γ(O) with kN (Fn ) → u in HN . Of course, kN (Fn ) is in particular a Cauchy sequence in HN , i.e., µN (Fn − Fm , Fn − Fm ) = kN (Fn ) − kN (Fm )2HN → 0.

By Theorem 4.2, the norm given by µN , N > 3/2, on Γ(O) is equivalent to the norm given by µ ˜, namely that of H 1/2 (O) ⊕ H −1/2 (O). Therefore we also have ˜ n ) − k(F ˜ m )2˜ = µ k(F ˜(Fn − Fm , Fn − Fm ) → 0 H ˜ n ) → v in H ˜ ˜ for some v ∈ k(Γ(O)). and it follows that also k(F For all G ∈ Γ(O) we have the equalities 0 = = =

Imu, kN (G) HN = lim ImkN (Fn ), kN (G) HN n→∞

1 1 lim σ(Fn , G) = C −3/2 lim σ ˜ (Fn , G) n→∞ n→∞ 2 2 −3/2 ˜ n ), k(G) ˜ ˜ C −3/2 lim Imk(F Imv, k(G) ˜ = C ˜, H H n→∞

∨ ˜ n ) → 0 in ˜ ˜ = {0} and therefore k(F ∩ k(Γ(O)) which imply that v ∈ k(Γ(O)) ˜ ˜ H. Since the norms given by kN and k are equivalent on Γ(O) we also have u = limn→∞ kN (Fn ) = 0 in HN , which proves the theorem.
Our main theorem is the following:

Theorem 4.7 Let ωN be an adiabatic vacuum state of order N and ωH a Hadamard state on the Weyl algebra A[Γ, σ] of the Klein-Gordon field in the globally hyperbolic spacetime (M, g), and let πωN and πωH be their associated GNSrepresentations. (i) If N > 5/2, then πωN A(O) and πωH A(O) are quasiequivalent for every open, relatively compact subset O ⊂ Σ. (ii) If ωN and ωH are pure states on a spacetime with compact Cauchy surface and N > 3/2, then πωN and πωH are unitarily equivalent. As explained at the beginning of this section it is sufficient to prove the quasiequivalence of the GNS-representations of the partial states ωN A(O) and ωH A(O) for part (i) of the theorem, for part (ii) we can take O = Σ. To this end we shall use a result of Araki & Yamagami [2]. To state it we first need some notation.

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Given a bilinear form µ on a real vector space K we shall denote by µC the extension of µ to the complexification K C of K (such that it is antilinear in the first argument): µC (F1 + iF2 , G1 + iG2 ) := µ(F1 , G1 ) + µ(F2 , G2 ) + iµ(F1 , G2 ) − iµ(F2 , G1 ). The theorem of Araki & Yamagami gives necessary and sufficient conditions for the quasiequivalence of two quasifree states ωµ1 and ωµ2 on the Weyl algebra A[K, σ] of a phase space (K, σ) in terms of the complexified data K C , σ C , and C C C µC i , i = 1, 2. Assuming that µ1 and µ2 induce the same topology on K , denote i i C C C C C C C C C ¯ the nletion. Then µ , µ , and λ := µ + σ , λ := µ + σ extend to by K 1 2 1 1 2 2 2 2 ¯ C by continuity (σ C extends due to (8)). We define bounded positive selfadjoint K ¯ C by operators S1 , S2 , and S2 on K λC j (F, G) λC 2 (F, G)

=

2µC j (F, Sj G),

=

 2µC 1 (F, S2 G),

j = 1, 2, ¯ C. F, G ∈ K

(34)

Note that Sj is a projection operator if and only if ωµj is a Fock state. The theorem of Araki & Yamagami [2] then states that the corresponding GNS-representations πω1 and πω2 are quasiequivalent if and only if both of the following two conditions are satisfied: C C (AY1) µC 1 and µ2 induce the same topology on K , 1/2

(AY2) S1

1/2

− S2

¯ C , µC is a Hilbert-Schmidt operator on (K 1 ).

Proof of Theorem 4.7. (i) We choose K = Γ(O) = C0∞ (O) ⊕ C0∞ (O), σ our real symplectic form (7), µH and µN the real scalar products on K defining a Hadamard state and an adiabatic vacuum state of order N > 5/2, respectively, C and check (AY1) and (AY2) for the data K C , σ C , µC H and µN . From Theorem 4.2 we know that the topologies induced by µH and µN on Γ(O) coincide. In view of the fact that µC H (F1 + iF2 , F1 + iF2 ) = µH (F1 , F1 ) + µH (F2 , F2 ),

F1 , F2 ∈ Γ(O),

(and the corresponding relation for µN ), we see that the topologies coincide also on the complexification. Hence (AY1) holds. 1/2 1/2 In order to prove (AY2), we first note that the difference SH − SN for the  operators SH and SN induced by µH and µN via (34) will be a Hilbert-Schmidt  operator provided that SH −SN is of trace class, cf. [40, Lemma 4.1]. By definition,  µC H (F, (SH − SN )G) =

1 C 1 C λH − λC µH − µC N (F, G) = N (F, G). 2 2

(35)

As in (23), (24), our assumption N > 5/2 and Lemma 3.3 imply that there is an integral kernel M = M (x, y) on O × O, given by (24) with entries satisfying (20)–(22), such that

1 C µH − µC N (F, G) = F, MG L2 (O)⊕L2 (O) , 2

F, G ∈ Γ(O),

(36)

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where M is the integral operator with kernel M . We may multiply M by ϕ(x)ϕ(y) ˜ where ϕ is a smooth function supported in a relatively compact neighborhood O of O with ϕ ≡ 1 on O. Equality (36) is not affected by this change. Moreover, as ˜ have smooth we saw in the beginning of this section we may suppose that O and O ˜ boundary. Using a partition of unity it is no loss of generality to assume that O 3 is contained in a single coordinate neighborhood. We then denote by O∗ ⊂ R the C image of O under the coordinate map. We shall use the notation µC H , µN , and M, M also for the push-forwards of these objects. We note that the closure of Γ(O∗ ) with 1/2 −1/2 respect to the topology of H 1/2 (R3 ) ⊕ H −1/2 (R3 ) is H0 (O∗ ) ⊕ H0 (O∗ ) =:  H, cf. Appendix A for the notation. The dual space H w.r.t. the extension of ·, · L2 (O∗ )⊕L2 (O∗ ) , denoted by ·, · , is H −1/2 (O∗ ) ⊕ H 1/2 (O∗ ). The inner product C  ˜ µC H extends to H. By Riesz’ theorem, µH induces an antilinear isometry θ : H → H C ˜ by θF, G = µH (F, G). Defining instead F, θG = µC H (F, G)

(37)

˜  of antilinear functionals we obtain a linear isometry θ from H to the space H ˜, on H. Complex conjugation provides a (real-linear) isometry between H and H  −1/2 1/2 ˜ hence H = H (O∗ ) ⊕ H (O∗ ) as a normed space (and hence as a Hilbert space). We deduce from Lemma 4.8 and Corollary 4.9 below, in connection with the continuity of the extension operator H → H 1/2 (R3 ) ⊕ H −1/2 (R3 ) and the restriction operator H −1/2 (R3 ) ⊕ H 1/2 (R3 ) → H −1/2 (O∗ ) ⊕ H 1/2 (O∗ ), that M induces a mapping M : H → H −1/2 (O∗ ) ⊕ H 1/2 (O∗ ) ˜  by which is trace class. In particular, for G ∈ H, MG defines an element of H F → F, MG . Combining (35)–(37), we see that, for F, G ∈ H,  F, MG = F, θ(SH − SN )G .

Hence  )=M θ(SH − SN

˜  ), in B(H, H

so that  SH − SN = θ−1 M in B(H).

˜  → H is an isometry while M : H → H ˜ As a consequence of the fact that θ−1 : H  is trace class, this implies that SH − SN is trace class. (ii) To prove (ii) we apply the technique of Bogoljubov transformations (we follow  be the [35] and [50, p. 68f.]). Assume that Σ is compact and let SH , SN , and SN operators induced by a pure Hadamard state ωH resp. a pure adiabatic state ωN of order N > 3/2 via (34). As remarked above, SH and SN are projection operators ¯ C , the closure of the complexification of K := Γ(Σ) w.r.t. µC or µC (since Σ on K H N C is compact, µC H and µN are equivalent on all of Γ(Σ), Theorem 4.2). We make a

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¯ C into direct sum decomposition of K + + HN HH K = ⊕ = ⊕ − − HN HH

¯C

(38)

+ − and 0 on HH/N , and the first such that SH/N has the eigenvalue 1 on HH/N C C decomposition is orthogonal w.r.t. µN , the second w.r.t. µH . We also denote the − + ¯ C onto H+ corresponding orthogonal projections of K H/N resp. HH/N by PH/N := − := 1 − SH/N . From Equations (9) and (34) we obtain for j ∈ SH/N resp. PH/N {H, N }

2µC j (F, Sj G) = ⇒ σ C (F, G)

=

i C C λC j (F, G) = µj (F, G) − σ (F, G) 2 C 2µC j (F, i(2Sj − 1)G) = 2µj (F, Jj G)

(39)

¯ C with the properties J 2 = where Jj := i(2Sj − 1) is a bounded operator on K j + ∗ C −1, Jj = −Jj (w.r.t. µj ). It has eigenvalue +i on Hj and −i on Hj− and is called the complex structure associated to µj . Because of (39) both decompositions in (38) are orthogonal w.r.t. σ C . We now define the Bogoljubov transformation 

A B

C D



+ + HN HH : ⊕ → ⊕ − − HN HH

(40)

by the bounded operators + − + − A := PH |H+ , B := PH |H+ , C := PH |H− , D := PH |H− . N

N

N

N

Taking into account Equation (39) and the fact that the decomposition (38) is + orthogonal w.r.t. σ C we obtain for F, G ∈ HN µC N (F, G)

i C = µC N (F, −iJN G) = − σ (F, G) 2 i i + + − − = − σ C (PH F, PH G) − σ C (PH F, PH G) 2 2 i i = − σ C (AF, AG) − σ C (BF, BG) 2 2 C = −iµC H (AF, JH AG) − iµH (BF, JH BG) C = µC H (AF, AG) − µH (BF, BG),

− similarly for F, G ∈ HN C C µC N (F, G) = µH (DF, DG) − µH (CF, CG),

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+ − and for F ∈ HN , G ∈ HN

0 = µC N (F, G)

=

i C σ (F, G) 2 i C + i + − − σ (PH F, PH G) + σ C (PH F, PH G) 2 2 i C i σ (AF, CG) + σ C (BF, DG) 2 2 C iµC H (AF, JH CG) + iµH (BF, JH DG)

=

C −µC H (AF, CG) + µH (BF, DG),

= = =

µC N (F, iJN G) =

hence + + A∗ A − B ∗ B = 1 in B(HN , HN ) − − ∗ ∗ ) D D − C C = 1 in B(HN , HN ∗

∗

B D − A C = 0 in

(41)

− + B(HN , HN ).

In a completely analogous way we can define the inverse Bogoljubov transformation 

by

A˜ ˜ B



C˜ ˜ D

+ + HH HN : ⊕ → ⊕ − − HH HN

(42)

˜ := P − | + , C˜ := P + | − , D ˜ := P − | − . A˜ := PN+ |H+ , B N H N H N H H

H

H

H

+ These operators satisfy relations analogous to (41). Moreover, for F ∈ HN ,G ∈ + HH

˜ µC N (F, AG) = = =

i C i C + ˜ ˜ µC N (F, −iJN AG) = − σ (F, AG) = − σ (F, PN G) 2 2 i i + − σ C (F, G) = − σ C (PH F, G) = −iµC H (AF, JH G) 2 2 µC H (AF, G),

+ + i.e. A˜ = A∗ : HH → HN ˜ = −C ∗ : H+ → H− and similarly B H N C˜ = −B ∗ : H− → H+ H

˜ D

N

− − = D∗ : HH → HN .

(43)

From (41) and (43) one easily finds that AA∗ − CC ∗ = 1 DD∗ − BB ∗ = 1 ∗

∗

AB − CD = 0

+ + in B(HH , HH ) − − in B(HH , HH )

in

− + B(HH , HH )

(44)

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and that (42) is the inverse of (40). Moreover, A is invertible with bounded inverse: + It follows from the first Equations in (41) and (44) that A∗ A ≥ 1 on HN and + ∗ ∗ ∗ AA ≥ 1 on HH , hence A and A are injective. Since {0} = Ker(A ) = Ran(A)⊥ , + . For F = AG ∈ Ran(A) we have A−1 F 2H+ = G2H+ ≤ A has dense range in HH N

N

G, A∗ AG H+ = AG2H+ = F 2H+ , i.e., A−1 is bounded and can be defined on N

H

H

+ . all of HH 1/2 1/2 We are now prepared to show that SH − SN is a Hilbert-Schmidt operator on ¯ C , µC ): We write F ∈ K ¯ C as a column vector w.r.t. the decomposition of K ¯C (K H C w.r.t. µH : +  +   +  HH PH F F F = ∈ ⊕ . =: − F− PH F − HH



Then SH F =

+ PH F

=

1 0

0 0



F+ F−

 .

(45)

 ¯C we get by the basis transformation (42) for all F, G ∈ K For SN

1 C λ (G, F ) = µC N (G, SN F ) 2 N     +  1 0 PN F PN+ G C = µN , 0 0 PN− G PN− F  +     A˜ C˜ PH G 1 0 C = µN , − ˜ D ˜ 0 0 PH G B  +   ∗    ˜∗ A˜ B G 1 0 = µC , H ˜∗ G− 0 0 C˜ ∗ D

 µC H (G, SN F ) =

and hence, utilizing (43),  +     F A −C 1 0 A∗  SN = − F −C ∗ −B D 0 0     F+ AA∗ −AB ∗ . = ∗ ∗ −BA BB F−

A˜ ˜ B

C˜ ˜ D

A˜ ˜ B

C˜ ˜ D

−B ∗ D∗



 + PH F − PH F   +  F , F−



F+ F−



(46)

+ − ⊕ HH From (45) and (46) we have now on HH 1/2 SH

−

1/2 SN

 =  =

1 0 0 0 1 0 0 0

1/2

 − 

 −

AA∗ −BA∗

−AB ∗ BB ∗

AZ −1/2 A∗ −BZ −1/2 A∗

1/2

−AZ −1/2 B ∗ BZ −1/2 B ∗

 ,

(47)

where Z := A∗ A + B ∗ B = 1 + 2B ∗ B is a bounded selfadjoint positive operator + on HN , which has a bounded inverse due to the fact that Z ≥ 1. In Lemma 4.11

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+ − below we will show that (47) is a Hilbert-Schmidt operator on HH ⊕ HH if and only if the operator     AA∗ −AB ∗ 1 0 Y := − 0 0 −BA∗ BB ∗

is Hilbert-Schmidt. From (46) and (34) we see that

1 C λ (G, F ) − λC N (G, F ) 2 H

1 C µ (G, F ) − µC N (G, F ) . 2 H

C  µC H (G, Y F ) = µH (G, (SH − SN )F ) =

=

C Now we argue as in the proof of part (i): µC H − µN is given by an integral operator M with kernel M , where M has the form (24) with entries satisfying (20)–(22). Using a partition of unity we can transfer the problem to Rn with the Sobolev regularity of the entries preserved. For N > 3/2 the conditions in Remark 4.10 are satisfied. Hence M and also Y are Hilbert-Schmidt operators, i.e., ωH and ωN are quasiequivalent on A[Γ, σ], which, in turn, is equivalent to the unitary equivalence of the representations πωH and πωN , if ωH and ωN are pure states.
s Lemma 4.8 Let M ∈ Hcomp (Rn × Rn ), s ≥ 0, and consider the integral operator M with kernel M , defined by  (Mu)(x) = M (x, y)u(y) dy, u ∈ C0∞ (Rn ).

If s >

n−1 n n+1 2 , 2, 2 ,

and

n 2

+ 1, respectively, then M yields trace class operators in

B(H 1/2 (Rn ), H −1/2 (Rn )), B(H −1/2 (Rn )), B(H 1/2 (Rn )), B(H −1/2 (Rn ), H 1/2 (Rn )), respectively. Proof. For the first case, s > n−1 2 , write 



1 1 1 1 x s+ 2 D s+ 2 M M = D −s− 2 x −s− 2 where x s is the operator of multiplication by x s and D s = op(ξ s ) w.r.t. the flat Euclidean metric of Rn . The first factor is known to be a Hilbert-Schmidt operator on H −1/2 (Rn ). Since M has compact support, it is sufficient to check 1 the Hilbert-Schmidt property of D s+ 2 M in B(H 1/2 (Rn ), H −1/2 (Rn )) or, equivs −1/2 2 n on L (R ). This operator, however, has the integral alently, of D MD kernel Dx s Dy −1/2 M (x, y). We may consider Dx s as the pseudodifferential s operator Dx s ⊗ I on Rn × Rn with a symbol in the class S0,0 (R2n × R2n ) and Dy −1/2 as the pseudodifferential operator I ⊗ Dy −1/2 on Rn × Rn with sym-

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0 bol in S0,0 (R2n × R2n ). By Calder´ on and Vaillancourt’s Theorem, Dx s Dy −1/2 s n n 2 n n maps H (R × R ) to L (R × R ), hence D s MD −1/2 is an integral operator with a square integrable kernel, hence Hilbert-Schmidt, and M is the composition of two Hilbert-Schmidt operators, hence trace class. The proofs of the other cases are similar.


Corollary 4.9 It is well known that the operator  M=

M11 M21

M12 M22



H 1/2 (Rn ) H −1/2 (Rn ) ⊕ ⊕ : → H −1/2 (Rn ) H 1/2 (Rn )

is trace class if and only if each of the entries Mij of the matrix is a trace class operator between the respective spaces, cf. e.g., [42, Section 4.1.1.2, Lemma 2]. Denoting by Mij the integral kernel of Mij , M will be trace class if n−1 2 n M12 ∈ H s (Rn × Rn ), s > 2 n+1 s n n M21 ∈ H (R × R ), s > 2 n s n n M22 ∈ H (R × R ), s > + 1. 2 Remark 4.10 In the situation of Corollary 4.9, M will be a Hilbert-Schmidt operator if each of its entries has this property. Using the fact that an integral operator on L2 (Rn ) is Hilbert-Schmidt if its kernel is in L2 (Rn × Rn ), we easily see that it is sufficient for the Hilbert-Schmidt property of M that M11

∈ H s (Rn × Rn ),

M11 M12 , M21 M22

s>

∈ L2 (Rn × Rn ) ∈ H 1/2 (Rn × Rn ) ∈ H 1 (Rn × Rn ).

Lemma 4.11 In the notation of above, the following statements are equivalent: (i) The operator  X :=

1 − AZ −1/2 A∗ BZ −1/2 A∗

AZ −1/2 B ∗ −BZ −1/2 B ∗



+ + HH HH : ⊕ → ⊕ − − HH HH

is Hilbert-Schmidt. (ii) The operator  Y :=

1 − AA∗ BA∗

AB ∗ −BB ∗



+ + HH HH : ⊕ → ⊕ − − HH HH

is Hilbert-Schmidt. − − (iii) The operator BB ∗ : HH → HH is of trace class.

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Proof. Using again the fact that a 2 × 2-matrix of operators is trace class if and only if each of its entries is a trace class operator [42, Sec. 4.1.1.2, Lemma 2] it is sufficient to show the equivalence of the following statements: (i) Each of the entries of the operator 

∗

X X=

1 − A(2Z −1/2 − 1)A∗ B(Z −1/2 − 1)A∗

A(Z −1/2 − 1)B ∗ BB ∗



+ + HH HH : ⊕ → ⊕ − − HH HH

is trace class. (ii) Each of the entries of the operator ∗

Y Y =



1 − A(2 − Z)A∗ B(1 − Z)A∗

A(1 − Z)B ∗ BZB ∗



+ + HH HH : ⊕ → ⊕ − − HH HH

is trace class. − − (iii) BB ∗ : HH → HH is trace class. Remember that a compact operator T : H1 → H2 acting between two (possibly different) Hilbert spaces H1 andH2 is said to be trace class, T ∈ B1 (H1 , H2 ), ∞ if it has finite trace norm T 1 := i=1 si < ∞, where si are the eigenvalues of |T | := (T ∗ T )1/2 on H1 . − − + − Note first, that BB ∗ ∈ B1 (HH , HH ) ⇔ B : HN → HH is Hilbert-Schmidt − + + + ⇔ B ∗ : HH → HN is Hilbert-Schmidt ⇔ B ∗ B ∈ B1 (HN , HN ). + + → HN is a bounded operator with bounded inverse we Since Z := 1 + 2B ∗ B : HN have + + + + + + , HN ) ⇔ ZB ∗ B ∈ B1 (HN , HN ) ⇔ BZB ∗ ∈ B1 (HN , HN ) B ∗ B ∈ B1 (HN

which proves the assertion for the 22-components of X ∗ X and Y ∗ Y . For the 12-components we note that −2B ∗ B = 1 − Z = (Z −1/2 − 1)(Z 1/2 + Z)

(48)

+ where Z 1/2 + Z is a bounded operator on HN with bounded inverse. As shown + + after Equation (44), also A : HN → HH is a bounded operator with bounded inverse, therefore + + + + , HN ) ⇔ A(1 − Z) = −2AB ∗ B ∈ B1 (HN , HH ) B ∗ B ∈ B1 (HN − + ∗ ⇒ A(1 − Z)B ∈ B1 (HH , HH ) − + ⇒ A(Z −1/2 − 1)B ∗ = A(1 − Z)(Z 1/2 + Z)−1 B ∗ ∈ B1 (HH , HH ).

The argument for the 21-component is analogous. As for the 11-component of Y ∗ Y we note, using the invertibility of A, the identity Z = 1 + 2B ∗ B, and (41), that + + + + , HH ) ⇔ A∗ A − A∗ A(2 − Z)A∗ A ∈ B1 (HN , HN ) 1 − A(2 − Z)A∗ ∈ B1 (HH + + + + ⇔ (1 + B ∗ B)B ∗ B(1 + 2B ∗ B) ∈ B1 (HN , HN ) ⇔ B ∗ B ∈ B1 (HN , HN ).

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Similarly, using A∗ A = 1 + B ∗ B = 12 (1 + Z), we rewrite the 11-component of X ∗ X in terms of Z and obtain + + 1 − A(2Z −1/2 − 1)A∗ ∈ B1 (HH , HH ) + + ⇔ A∗ A − A∗ A(2Z −1/2 − 1)A∗ A ∈ B1 (HN , HN ) + + , HN ). ⇔ (1 + Z)(1 − Z −1/2 )(2 − Z 1/2 + Z) ∈ B1 (HN

(49)

Taking into account (48) and the identity (2 − Z 1/2 + Z)(Z 1/2 + Z + 2) = 4 + 3Z + Z 2 , where both Z 1/2 + Z + 2 and 4 + 3Z + Z 2 are bounded operators with bounded inverse, we note that (49) is equivalent to + + , HN ) (1 + Z)(Z − 1)(4 + 3Z + Z 2 ) ∈ B1 (HN

+ + + + ⇔ Z − 1 ∈ B1 (HN , HN ) ⇔ B ∗ B ∈ B1 (HN , HN ).

This finishes the proof.
 Theorem 4.5 and Theorem 4.7 imply that, for N > 5/2, πωN (A(O)) and πωH (A(O)) , and, if Σ is compact, for N > 3/2, πωN (A[Γ, σ]) and πωH (A[Γ, σ]) are isomorphic von Neumann factors. Therefore it follows from the corresponding results for Hadamard representations due to Verch [49, Theorem 3.6] that πωN (A(O)) is isomorphic to the unique hyperfinite type III1 factor if Oc is nonempty, and is a type I∞ factor if Oc = ∅ (i.e., Σ = O is a compact Cauchy surface). Our Theorem 4.7 is the analogue of Theorem 3.3 of L¨ uders & Roberts [35] extended to our definition of adiabatic states on arbitrary curved spacetime manifolds. The loss of order 3/2 + in the compact case and 1/2 + in the non-compact case ( > 0 arbitrary) compared to their result is probably due to the fact that we use the regularity of ΛH − ΛN rather generously in the part of the proof of Theorem 4.2 between Equations (20) and (25).

4.2

Local definiteness and Haag duality

The next property of adiabatic vacua we check is that of local definiteness. It says that any two adiabatic vacua (of order > 5/2) get indistinguishable upon measurements in smaller and smaller spacetime regions. In a first step let us show that in the representation πωN generated by an adiabatic vacuum state ωN (of order N > 3/2) there are no nontrivial observables which are localized at a single point, more precisely: Theorem 4.12 Let x ∈ Σ. Then, for N > 3/2,  πωN (A(O)) = C1, O x

where the intersection is taken over all open bounded subsets O ⊂ Σ.

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Before we prove the theorem let us recall how this, combined with Theorem 4.7, implies the property of local definiteness: Corollary 4.13 Let ωN be an adiabatic vacuum state of order N > 5/2 and ωH a Hadamard state. Let On , n ∈ N0 , be a sequence  of open bounded subsets of Σ shrinking to a point x ∈ Σ, i.e., On+1 ⊂ On and n∈N0 On = {x}. Then (ωN − ωH )|A(On )  → 0

as n → ∞.

Proof. Let (πωN , HωN , ΩωN ) be the GNS-triple generated by ωN , and let RN (On ) := πωN (A(On )) be the corresponding von Neumann algebras associated to the regions On ⊂ Σ. Due to Theorem 4.7 and the remarks at the beginning of Section 4.1 π(ωH
A(O0 )) is quasiequivalent to πωN A(O0 ). This implies [6, Theorem 2.4.21] that ωH A(O0 ) can be represented in HωN as a den ∈ H with ψ 2 = 1 such that sity matrix, i.e., there is a sequence ψ m ω m N m  ωH (A) = m ψm , Aψm for all A ∈ A(O0 ). Let now An ∈ RN (On ) ⊂ RN (O0 ) be a sequence of observables with An  = 1. From Theorem 4.12 it follows that An → c1 in the topology of RN (O0 ) for some c ∈ C. In particular, An → c1 in the weak topology, thus |ΩωN , (An − c1)ΩωN | → 0 as n → ∞, and An → c1 in the σ-weak topology, thus  |ψm , (An − c1)ψm | → 0 as n → ∞. m

From this we can now conclude |(ωN − ωH )(An )|

=

|ΩωN , An ΩωN −



ψm , An ψm |

m

=

|ΩωN , (An − c1)ΩωN −



ψm , (An − c1)ψm |

m

≤

|ΩωN , (An − c1)ΩωN | +



|ψm , (An − c1)ψm |

m

→ 0

as n → ∞,

(50)

i.e., (ωN − ωH )(An ) converges to 0 pointwise for each sequence An . To show the uniform convergence we note that due to A(On+1 ) ⊂ A(On ) rn := sup{|(ωN − ωH )(A)|; A ∈ A(On ), A = 1} is a bounded monotonically decreasing sequence in n ∈ N0 with values in R+ 0. Hence rn → r for some r ∈ R+ . To show that r = 0 let

> 0. For all n ∈ N there 0 0 is an An ∈ A(On ) with An  = 1 such that 0 ≤ rn − |(ωN − ωH )(An )| ≤ .

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Furthermore, due to (50) there is an no ∈ N0 such that for all n ≥ no |(ωN − ωH )(An )| ≤ . From these inequalities we obtain for n ≥ no 0 ≤ r ≤ rn ≤ + |(ωN − ωH )(An )| ≤ 2 and hence r = 0. This proves the assertion. To prove Theorem 4.12 we show an even stronger statement, namely  πωN (A(O)) = C1




(51)

O⊃S

for any smooth 2-dimensional closed submanifold S of Σ. The statement of Theorem 4.12 then follows if we choose x ∈ S. In the proof we will need the following lemma: 

Lemma 4.14

C0∞ (O) = {0},

O⊃S

where the closure is taken w.r.t. the norm of H −1/2 (Σ) (and hence it also holds w.r.t. the norm of H 1/2 (Σ)). Note that we can confine the intersection to all sets O contained in a suitable compact subset of Σ. Hence we can assume that (Σ, h) is a complete Riemannian manifold (otherwise we modify h as in the proof of Theorem 4.2), so that H ±1/2 (Σ) is well defined. Proof of Lemma 4.14. The problem is local, so it suffices to consider the case Σ = Rn , S = Rn−1 ×{0}. Suppose the above intersection contains some f ∈ H −1/2 (Rn ), say f H −1/2 = 1. Fix 0 < < 1/2. Since f H −1/2 = sup{|f (F )|; F ∈ H 1/2 , F H 1/2 = 1} we find some F ∈ H 1/2 (Rn ) such that F H 1/2 = 1 and f (F ) > 1− . According to Lemma 4.6 there exists an F0 ∈ C0∞ (Rn \ (Rn−1 × {0})) such that F −F0 H 1/2 < and therefore f (F0 ) = f (F ) + f (F0 − F ) > 1 − 2 . Clearly there is a δ > 0 such that |xn | > 2δ for each x = (x , xn ) ∈ supp F0 . H −1/2

On the other hand, f ∈ C0∞ (Rn−1 × (−δ, δ)) , hence supp f ⊂ Rn−1 × [−δ, δ] (in order to see this, use the fact that the closure of C0∞ (Rn+ ) in the topology of H s (Rn ) is equal to {u ∈ H s (Rn ); supp u ⊂ Rn+ } for s ∈ R, cf. [45, 2.10.3]). Denoting by χδ a smooth function, equal to 1 on Rn−1 × [−δ, δ] and vanishing outside Rn−1 × (−2δ, 2δ), we have f = χδ f and therefore 1 − 2 < f (F0 ) = (χδ f )(F0 ) = f (χδ F0 ) = f (0) = 0, a contradiction.




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Ann. Henri Poincar´e

Proof of Theorem 4.12. Let (kN , HN ) be the one-particle Hilbert space structure of ωN . According to results of Araki [1, 34] (51) holds iff  kN (Γ(O)) = {0}, (52) O⊃S

where the closure is taken w.r.t. the norm in HN . As in the proof of Theorem 4.5, let us define a one-particle Hilbert space structure ˜ H) ˜ of an auxiliary pure quasifree state on A[Γ, σ] by (k, ˜ k˜ : Γ → L2 (Σ, h) =: H  

 q 1 → √ iD 1/2 q + D −1/2 p ; p 2 as before, we may change h near infinity to obtain completeness. Note that the norm given by   ˜ kF ˜ ˜ = 1 D 1/2 q2 2 + D −1/2 p2 2 , F := (q, p) ∈ Γ, µ ˜(F, F ) := kF, L L H 2 is equivalent to the norm of H 1/2 (Σ) ⊕ H −1/2 (Σ). Let u ∈ kN (Γ(O)) for all O ⊃ S. Thus for every O there is a sequence {FnO , n ∈ N} ⊂ Γ(O) with kN (FnO ) → u in HN . By Theorem 4.2 the norm given by µN , N > 3/2, on Γ(O) is equivalent to the norm given by µ ˜, namely that of H 1/2 (O) ⊕ ˜ O ) → v O in H ˜ ˜ for some v O ∈ k(Γ(O)). H −1/2 (O). Therefore it follows that also k(F n Moreover, v O must be independent of O: To see this, suppose that O1 and O2 are ˜ ⊂ Σ, and let > 0. Then there is an contained in a common open, bounded set O n ∈ N such that ˜ O1 ) ˜ + k(F ˜ O1 ) − k(F ˜ O2 ) ˜ + k(F ˜ O2 ) − v O2  ˜ v O1 − v O2 H˜ ≤ v O1 − k(F n n n n H H H O1 O2 O1 O2 O1 O2 1/2 ˜ ≤ 2 + k(F − F ) ˜ = 2 + µ ˜(F − F , F − F ) n

≤ = ≤ ≤

n

H

n

n

n

n

˜ µN (FnO1 − FnO2 , FnO1 − FnO2 )1/2 2 + C(O) ˜ N (F O1 ) − kN (F O2 )HN 2 + C(O)k n n

˜ kN (FnO1 ) − uHN + u − kN (FnO2 )HN 2 + C(O) ˜ 2 (1 + C(O)),

˜ by v. hence v O1 = v O2 , and we denote this unique element of H ˜ H  ˜ Since v ∈ O⊃S k(Γ(O)) it follows from Lemma 4.14 that v = 0 and therefore O ˜ ˜ k(Fn ) → 0 in H. Since the norms given by kN and k˜ are equivalent on Γ(O) we also have kN (FnO ) → 0 in HN and thus u = limn→∞ kN (FnO ) = 0, which proves the theorem.
In the following theorem we show that the observable algebras RN (O) := πωN (A(O)) generated by adiabatic vacuum states (of order N > 3/2) satisfy a

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certain maximality property, called Haag duality. Due to the locality requirement it is clear that all observables localized in spacelike separated regions of spacetime commute. If O is some open, relatively compact subset of the Cauchy surface Σ with smooth boundary, this means that RN (Oc ) ⊂ RN (O) ,

(53)

where Oc := Σ \ O and  RN (Oc ) := 



 πωN (A(O1 ))

(54)

O1 ⊂O c

is the von Neumann algebra generated by all πωN (A(O1 )) with O1 bounded and O1 ⊂ Oc . One says that Haag duality holds for the net of von Neumann algebras generated by a pure state if (53) is even an equality. For mixed states (i.e., reducible GNS-representations) this can certainly not be true, because in this case, by Schur’s lemma [6, Prop. 2.3.8], there is a set S of non-trivial operators commuting with the representation πωN , i.e., S ⊂ RN (O) ∩ RN (Oc ) .

(55)

If equality held in (53) then the right-hand side of (55) would be equal to RN (O) ∩ RN (O), i.e., to the centre of RN (O), which, however, is trivial due to the local primarity (Theorem 4.5) of the representation πωN , hence S ⊂ C1, a contradiction. Therefore, in the reducible case one has to take the intersection with RN := πωN (A[Γ, σ]) on the right-hand side of (53) to get equality1 (in the irreducible case, again by Schur’s lemma, πωN (A) = C1 ⇒ πωN (A) = B(HωN ), hence the intersection with RN is redundant). Haag duality is an important assumption in the theory of superselection sectors [21] and has therefore been checked in many models of physical interest. For our situation at hand, Haag duality has been shown by L¨ uders & Roberts [35] to hold for the GNS-representations of adiabatic vacua on Robertson-Walker spacetimes and by Verch [47, 49] for those of Hadamard Fock states. He also noticed that it extends to all Fock states that are locally quasiequivalent to Hadamard states, hence, by our Theorem 4.7, to pure adiabatic states of order N > 5/2. Nevertheless, we present an independent proof of Haag duality for adiabatic states that does not rely on quasiequivalence but only on Theorem 4.2 and also holds for mixed states. Theorem 4.15 Let ωN be an adiabatic state of order N > 3/2. Then, for any open, relatively compact subset O ⊂ Σ with smooth boundary, RN (Oc ) = RN (O) ∩ RN , where Oc := Σ \ O and RN (Oc ) is defined by (54). 1 We are grateful to Fernando Lled´ o for pointing out to us this generalization of Haag duality and discussions about this topic.

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Proof. Denoting again by (kN , HN ) the one-particle Hilbert space structure of ωN , it follows from results of Araki [1, 34] that the assertion is equivalent to the statement kN (Γ(Oc )) = kN (Γ(O))∨ ∩ kN (Γ), where the closure has to be taken w.r.t. HN and kN (Γ(O))∨ was defined in Equation (30). Since kN (Γ(Oc )) ⊂ kN (Γ(O))∨ ∩ kN (Γ) (due to the locality of σ, compare (53) above), we only have to show that kN (Γ(Oc )) is dense in kN (Γ(O))∨ ∩ kN (Γ). This in turn is the case iff kN (Γ(O)) + kN (Γ(Oc )) is dense in kN (Γ)

(56)

(for the convenience of the reader, the argument will be given in Lemma 4.16 below). (56) will follow if we show that every element u = kN (F ) ∈ kN (Γ), F = (q, p) ∈ Γ, can be approximated by a sequence in kN (Γ(O)) + kN (Γ(Oc )). To this end we fix a bounded open set O0 ⊂ Σ with smooth boundary such that supp p and supp q ⊂ O0 . According to Lemma 4.6 we find sequences {qn }, {pn } ⊂ C0∞ (O), {qnc }, {pcn} ⊂ C0∞ (Oc ) such that q − (qn + qnc ) p − (pn + pcn )

→ 0 in H 1/2 (O0 ) → 0 in H −1/2 (O0 ).

(57) (58)

Note that it is no restriction to ask that the supports of all functions are contained ˜ H) ˜ the one-particle Hilbert space structure introduced in O0 . Let us denote by (k, in (32) with the real scalar product µ ˜ given by (33). The relations (57) and (58) imply that Γ(O0 )  Fn := (q − (qn + qnc ), p − (pn + pcn )) → 0 with respect to the norm induced by µ ˜. According to Theorem 4.2 it also tends to zero with respect to the norm induced by µN , in other words kN (Fn ) → 0 in HN . This completes the argument.




Lemma 4.16 kN (Γ(O)) + kN (Γ(Oc )) is dense in kN (Γ) ⇔ kN (Γ(Oc )) is dense in kN (Γ(O))∨ ∩ kN (Γ). Proof. ⇒. Let u ∈ kN (Γ(O))∨ ∩ kN (Γ), and choose vn ∈ kN (Γ(O)), wn ∈ kN (Γ(Oc )) such that vn + wn → u in HN . (59) In view of the fact that kN (Γ(Oc )) ⊂ kN (Γ(O))∨ we have kN (Γ(O)) ∩ kN (Γ(Oc )) ⊂ kN (Γ(O)) ∩ kN (Γ(O))∨ = {0}.

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Indeed, the last equality is a consequence of Theorem 4.5, cf. (31). We can therefore define a continuous map π : kN (Γ(O)) ⊕ kN (Γ(Oc )) → HN v ⊕ w → v. Now (59) implies that {vn + wn } is a Cauchy sequence in HN , hence so are {vn } = {π(vn + wn )} and {wn }. Let v0 := lim vn ∈ kN (Γ(O)), w0 := lim wn ∈ kN (Γ(Oc )). By (59), u − w0 = v0 ∈ kN (Γ(O))∨ ∩ kN (Γ(O)) = {0}. Therefore u = w0 ∈ kN (Γ(Oc )). ⇐. Denoting by ⊥ the orthogonal complement in kN (Γ), we clearly have from the ∨ ∩ kN (Γ). Since  kN (Γ(O)) + definition (30) of ∨ that kN (Γ(O))⊥ ⊂ kN (Γ(O))

kN (Γ(O))⊥ = kN (Γ) it follows that kN (Γ(O)) + kN (Γ(O))∨ ∩ kN (Γ) is dense in

kN (Γ). From the assumption that kN (Γ(Oc )) is dense in kN (Γ(O))∨ ∩ kN (Γ) the assertion follows.


5 Construction of adiabatic vacuum states The existence of Hadamard states on arbitrary globally hyperbolic spacetimes has been proven by Fulling, Narcowich & Wald [19] using an elegant deformation argument. Presumably, the existence of adiabatic vacuum states could be shown in a similar way employing the propagation of the Sobolev wavefront set, Proposition B.4. Instead of a mere existence argument, however, we prefer to explicitly construct a large class of adiabatic vacuum states as it is indispensable for the extraction of concrete information in physically relevant situations to have available a detailed construction of the solutions of the theory. In Section 5.1 we first present a parametrization of quasifree states in terms of two operators R and J acting on the L2 -Hilbert space w.r.t. a Cauchy surface Σ, Theorem 5.1. The main technical result is Theorem 5.3 which gives a sufficient condition on R and J such that the associated quasifree states are adiabatic of a certain order. In Section 5.2 we construct operators R and J satisfying the above assumptions, Theorem 5.10.

5.1

Criteria for initial data of adiabatic vacuum states

We recall the following theorem from [31, Theorem 3.11]: Theorem 5.1 Let (M, g) be a globally hyperbolic spacetime with Cauchy surface Σ. Let J, R be operators on L2 (Σ, d3 σ) satisfying the following conditions: (i) (ii) (iii) (iv)

C0∞ (Σ) ⊂ dom(J), J and R map C0∞ (Σ, R) to L2R (Σ, d3 σ), J is selfadjoint and positive with bounded inverse, R is bounded and selfadjoint.

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Then k:Γ

→ H := k(Γ) ⊂ L2 (Σ, d3 σ)

(q, p)

→ (2J)−1/2 [(R − iJ)q − p]

(60)

is the one-particle Hilbert space structure of a pure quasifree state. Note that we can define the inverse square root by  1 ∞ −1/2 λ (λ + 2J)−1 dλ. (2J)−1/2 = π 0

(61)

The integral converges since λ + 2J ≥ λ and hence (λ + 2J)−1 ≤ λ−1 for λ ≥ 0. Therefore (2J)−1/2 is a bounded operator on L2 (Σ, d3 σ). Moreover, (2J)−1/2 maps L2R (Σ, d3 σ) to itself since λ + 2J and therefore (λ + 2J)−1 commutes with complex conjugation (λ ≥ 0). Proof. A short computation shows that for Fj = (qj , pj ) ∈ Γ, j = 1, 2, we have σ(F1 , F2 )

= −q1 , p2 + p1 , q2 = 2Im kF1 , kF2 .

Here, ·, · is the scalar product of L2 (Σ, d3 σ). We then let µ(F1 , F2 ) := Re kF1 , kF2 . We note that |σ(F1 , F2 )|2

≤ =

4|kF1 , kF2 |2 ≤ 4kF1 , kF1 kF2 , kF2 4µ(F1 , F1 )µ(F2 , F2 );

hence k defines the one-particle Hilbert space structure of a quasifree state with real scalar product µ (Definition 2.3) and one-particle Hilbert space H = kΓ + ikΓ (Proposition 2.4). Let us next show that the state is pure, i.e., kΓ is dense in H (see Proposition 2.4). We apply a criterion by Araki & Yamagami [2] and check that the operator S : Γ → L2 (Σ, d3 σ) ⊕ L2 (Σ, d3 σ) defined by kF1 , kF2 = 2µ(F1 , SF2 ) is a projection (cf. Equation (34)). Indeed, this relation implies that   1 −iJ −1 iJ −1 R + 1 S= . iRJ −1 R + iJ −iRJ −1 + 1 2
Therefore S 2 = S, and the proof is complete. The distinguished parametrices of the Klein-Gordon operator In the following we shall use the calculus of Fourier integral operators of Duistermaat & H¨ ormander [16] in order to analyze the wavefront set of certain bilinear forms related to fundamental solutions of the Klein-Gordon operator P = 2g +m2 . We recall from [16, Theorem 6.5.3] that P on a globally hyperbolic spacetime

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(which is known to be pseudo-convex w.r.t. P , see [41]) has 22 = 4 orientations ˙ ν2 of the bicharacteristic relation C, Equation (15). Here, C \ diag (C) = Cν1 ∪C ν is one of the four sets {∅, N+ , N− , N+ ∪ N− } of components of the lightcone 1(2) N := char P , with N± := N ∩ {ξ0 ≷0}. Cν are the subsets of the bicharacteristic +(−) in [16]. Associated to these orientations there relation which are denoted by Cν 1 2 are four pairs Eν , Eν of distinguished parametrices with W F  (Eν1 ) = ∆∗ ∪ Cν1 ,

W F  (Eν2 ) = ∆∗ ∪ Cν2

where ∆∗ is the diagonal in (T ∗ X \ 0) × (T ∗ X \ 0). Moreover, Duistermaat & H¨ormander show that every parametrix E with W F  (E) contained in ∆∗ ∪ Cν1 resp. ∆∗ ∪ Cν2 must be equal to Eν1 resp. Eν2 modulo C ∞ . In addition, Eν1 − Eν2 ∈ I 1/2−2 (M × M, C  ) and Eν1 − Eν2 is noncharacteristic at every point of C  . Here, I µ (X, Λ) denotes the space of Lagrangian distributions of order µ over the manifold X associated to the Lagrangian submanifold Λ ⊂ T ∗ X \ 0, cf. [28, Def. 25.1.1]. We shall need three particular parametrices: For the forward light cone N+ we 1 obtain EN = E R (mod C ∞ ), the retarded Green’s function, for the backward + 1 light cone N− we have the advanced Green’s function EN = E A (mod C ∞ ) while − 1 F EN+ ∪N− is the so-called Feynman parametrix E (mod C ∞ ). We deduce that 1 2 = EN (mod C ∞ ), in particular EN + − E = E R − E A ∈ I −3/2 (M × M, C  ). We next write E = E + + E − with E + := E F − E A , E − := E R − E F . We deduce from [16, Theorem 6.5.7] that E−

1 1 = E R − E F = EN − EN ∈ I −3/2 (M × M, (C − ) ) + + ∪N−

(62)

E+

1 1 = E F − E A = EN − EN ∈ I −3/2 (M × M, (C + ) ) + ∪N− −

(63)

where C + = C ∩ (N+ × N+ ), C − = C ∩ (N− × N− ) as in Equation (16). It follows from [41, Theorem 5.1] that the two-point function ΛH of every Hadamard state coincides with iE + (mod C ∞ ). (We define the physical Feynman propagator by F (x, y) := −iT Φ(x)Φ(y) , i.e., −i × the expectation value of the time ordered product of two field operators. From this choice it follows that iF = ΛH +iE A and hence F = E F (mod C ∞ ) and ΛH = iE + (mod C ∞ ).) Lemma 5.2 For every Hadamard state ΛH we have  ∅, s < − 12 s s + . W F (ΛH ) = W F (E ) = C + , s ≥ − 12

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Proof. The statement for s < −1/2 follows from Equation (63) and Proposition B.10. For s ≥ −1/2 we rely on [16, Section 6]. According to [16, Equation (6.6.1)] 1 1 1 EN + E∅1 = EN + EN mod C ∞ , + ∪N− + −

so that, in the notation of [16, Equation (6.6.3)], 1 E + = EN − E∅1 = SN+ . +

The symbol of SN+ is computed in [16, Theorem 6.6.1]. It is non-zero on the diagonal ∆N in N × N . Moreover, it satisfies a homogeneous first order ODE along the bicharacteristics of P in each pair of variables, so that it is non-zero everywhere on C + . Hence E + is non-characteristic at every point of C + . Now Proposition B.10 gives the assertion.
We fix a normal coordinate t which allows us to identify a neighborhood of Σ in M with (−T, T ) × Σ =: MT . We assume that Rl = {Rl (t); −T < t < T } and Jl = {Jl (t); −T < t < T }, l = 1, 2, are smooth families of properly supported pseudodifferential operators on Σ with local symbols rl = rl (t) ∈ C ∞ ((−T, T ), S 0 (Σ × R3 )) and jl = jl (t) ∈ C ∞ ((−T, T ), S 1(Σ × R3 )). Moreover, let H = {H(t); −T < t < T } be a smooth family of properly supported pseudodifferential operators of order −1 on Σ. We can then also view Rl , Jl , and H as operators on, say, C0∞ ((−T, T ) × Σ). Theorem 5.3 Let Rl , Jl , and H be as above, and let Ql be a properly supported first order pseudodifferential operator on (−T, T ) × Σ such that (N )

Ql (Rl − iJl − ∂t )E − = Sl (N )

E−,

l = 1, 2,

(64)

(N )

with Sl = Sl (t) ∈ C ∞ ((−T, T ), L−N (Σ)) a smooth family of properly supported pseudodifferential operators on Σ of order −N . Moreover, we assume that Ql has a real-valued principal symbol such that char Ql ∩ N− = ∅. Then the distribution DN ∈ D (M × M), defined by DN (f1 , f2 ) = [(R1 − iJ1 )ρ0 − ρ1 ] Ef1 , H [(R2 − iJ2 )ρ0 − ρ1 ] Ef2 satisfies the relation s

W F (DN ) ⊂



∅, s < −1/2 C + , −1/2 ≤ s < N + 3/2.

(65)

Note that DN will in general not be a two-point function unless R1 = R2 and J1 = J2 = H −1 are selfadjoint and J is positive (compare Theorem 5.1).

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Proof. Since ρ0 commutes with Rl , Jl and H we have DN (f1 , f2 ) = ρ0 [R1 − iJ1 − ∂t ] Ef1 , ρ0 H [R2 − iJ2 − ∂t ] Ef2 .

(66)

Denoting by K1 and K2 the distributional kernels of (R1 − iJ1 − ∂t )E and H(R2 − iJ2 − ∂t )E, respectively, we see that DN = (ρ0 K1 )∗ (ρ0 K2 ). We shall apply the calculus of Fourier integral operators in order to analyze the composition (ρ0 K1 )∗ (ρ0 K2 ). The following lemma is similar in spirit to [28, Theorem 25.2.4]. Lemma 5.4 Let X ⊂ Rn1 , Y ⊂ Rn2 be open sets and A ∈ Lk (X) be a properly supported pseudodifferential operator with symbol a(x, ξ). Assume that C is a homogeneous canonical relation from T ∗ Y \ 0 to T ∗ X \ 0 and that a(x, ξ) vanishes on a conic neighborhood of the projection of C in T ∗ X \ 0. If B ∈ I m (X × Y, C  ) then AB ∈ I −∞ (X × Y, C  ). Proof. The problem is microlocal, so we may assume that B has the form  Bu(x) = eiφ(x,y,ξ) b(x, y, ξ)u(y) dy dN ξ, where φ is a non-degenerate phase function on X × Y × (RN \ {0}) and b ∈ S m+(n1 +n2 −2N )/4 (X × Y × RN ) an amplitude. We know that C = Tφ (Cφ ), where Cφ := {(x, y, ξ) ∈ X × Y × (RN \ {0}); dξ φ(x, y, ξ) = 0}, and Tφ is the map Tφ : X × Y × (RN \ {0}) → T ∗ (X × Y ) \ 0 (x, y, ξ)

→ (x, dx φ; y, dy φ).

We recall that ess supp b is the smallest closed conic subset of X × Y × (RN \ {0}) outside of which b is of class S −∞ and that the wavefront set of the kernel of B is contained in the set Tφ (Cφ ∩ ess supp b), cf. [15, Theorem 2.2.2]. Hence we may assume that b vanishes outside a conic neighborhood N of Cφ in X × Y × RN . In fact we can choose this neighborhood so small that a(x, ξ  ) = 0 whenever (x, ξ  ) lies in the projection of Tφ (N ) ⊂ T ∗ X × T ∗ Y onto the first component (we call this projection π1 ). Then  ABu(x) = eiφ(x,y,ξ) c(x, y, ξ)u(y) dy dN ξ

1150

where

W. Junker and E. Schrohe

Ann. Henri Poincar´e

c(x, y, ξ) = e−iφ(x,y,ξ) A(b(·, y, ξ)eiφ(·,y,ξ) ).

According to [44, Ch. VIII, Equation (7.8)], c has the asymptotic expansion  c(x, y, ξ) ∼ Dξα a(x, dx φ(x, y, ξ))Dxβ b(x, y, ξ)ψαβ (x, y, ξ) (67) α≥0 β≤α

where ψαβ is a polynomial in ξ of degree ≤ |α − β|/2. Now from our assumptions on a and b it follows that in Equation (67) b(x, y, ξ) = 0 if a(x, ξ  ) = 0 if

(x, y, ξ) ∈ /N (x, ξ  ) ∈ π1 Tφ (N ) ⇒ a(x, dx φ(x, y, ξ)) = 0 if (x, y, ξ) ∈ N ,

and hence c ∼ 0. This proves that AB ∈ I −∞ (X × Y, C  ). ∞




Lemma 5.5 Let A ∈ C ((−T, T ), L (Σ)) be properly supported and B ∈ I (M × M, (C ± ) ). Then AB ∈ I m+k (MT × M, (C ± ) ). k

m

Proof. Choosing local coordinates and a partition of unity we may assume that M = R4 , Σ = R3 ∼ = R3 × {0} ⊂ R4 and that A is supported in a compact set. We let X = op χ where χ = χ(τ, ξ) ∈ C ∞ (R4 ) vanishes near (τ, ξ) = 0 and is homogeneous of degree 0 for |(τ, ξ)| ≥ 1 with χ(τ, ξ) = 1 for (τ, ξ) in a conic neighborhood of {ξ = 0}, and χ(τ, ξ) = 0 for (τ, ξ) outside a larger conic neighborhood of {ξ = 0}, such that, in particular, χ(τ, ξ) = 0 in a neighborhood of π1 (C ± ) (by π1 we denote the projection onto the first component in T ∗ M × T ∗ M, i.e., π1 (x, ξ; y, η) := (x, ξ)). We have AB = AXB + A(1 − X)B. Denoting by a(t, x, ξ) the local symbol of A, the operator A(1 − X) has the symbol a(t, x, ξ)(1 − χ(τ, ξ)) ∈ S k (R4 × R4 ). (Here we have used the fact that (1−χ(τ, ξ)) is non-zero only in the area where τ can be estimated by ξ .) Hence A(1−X) is a properly supported pseudodifferential operator of order k on MT . We may apply [28, Theorem 25.2.3] with excess equal to zero and obtain that A(1 − X)B ∈ I m+k (MT × M, (C ± ) ). On the other hand, X is a pseudodifferential operator with symbol vanishing in a neighborhood of π1 (C ± ). According to Lemma 5.4, XB ∈ I −∞ (M × M, (C ± ) ). Hence XB is an integral operator with a smooth kernel on M × M, and so is AXB, since A is continuous on C ∞ (M).
Lemma 5.6 (i) (Rl − iJl − ∂t )E + ∈ I −1/2 (MT × M, (C + ) ), l = 1, 2; (ii) H(R2 − iJ2 − ∂t )E + ∈ I −3/2 (MT × M, (C + ) ); (iii) (Rl − iJl − ∂t )E − ∈ I −N −5/2 (MT × M, (C − ) ), l = 1, 2; (iv) H(R2 − iJ2 − ∂t )E − ∈ I −N −7/2 (MT × M, (C − ) ).

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Proof. (i) It follows from (63) and Lemma 5.5 that (Rl − iJl )E + ∈ I −1/2 (MT × M, (C + ) ). Since ∂t is a differential operator, the assumptions of the composition theorem for Fourier integral operators [28, Theorem 25.2.3] are met with excess equal to zero, and we conclude from (63) that also ∂t E + ∈ I −1/2 (MT ×M, (C + ) ). Since, by assumption, H ∈ C ∞ ((−T, T ), L−1 (Σ)) is properly supported we also obtain (ii). (iii) We know from (64) that (N )

Ql (Rl − iJl − ∂t )E − = Sl

E−.

(68)

Applying Lemma 5.5 and (62), the right-hand side is an element of I −3/2−N (MT × (N ) M, (C − ) ) (note that Sl is properly supported). We next observe that the question is local, so that we can focus on a small neighborhood U of a point x0 ∈ M. (1) (2) Here, we write Ql = Ql +Ql as a sum of two pseudodifferential operators, where (1) (2) Ql is elliptic, and the essential support of Ql is contained in the complement of N− . To this end choose a real-valued function χ ∈ C ∞ (T ∗ U ) with the following properties: (α) (β) (γ) (δ)

χ(x, ξ) = 0 for small |ξ|, χ is homogeneous of degree 1 for |ξ| ≥ 1, χ(x, ξ) ≡ 0 on a conic neighborhood of N− , χ(x, ξ) ≡ |ξ| on a neighborhood of char Ql ∩ {|ξ| ≥ 1}.

We denote the local symbol of Ql by ql and let (1)

Ql

:= op (ql (x, ξ) + iχ(x, ξ)),

(2)

Ql

:= op (−iχ(x, ξ)).

By the Lemmata 5.4 and 5.5 we have (2)

Ql (Rl − iJl − ∂t )E − ∈ I −∞ (MT × M, (C − ) ). (1)

Moreover, Ql is elliptic of order 1, since ql is real-valued and χ(x, ξ) = |ξ| on char Ql . We conclude from (68) that (1)

Ql (Rl − iJl − ∂t )E − ∈ I −3/2−N (MT × M, (C − ) ). (1)

Multiplication by a parametrix to Ql

shows that

(Rl − iJl − ∂t )E − ∈ I −5/2−N (MT × M, (C − ) ). (iv) follows from (iii) and Lemma 5.5.
We next analyze the effect of the restriction operator ρ0 . We recall from [15, p. 113] that ρ0 ∈ I 1/4 (Σ × M, R0 ) (69) where R0 := {(xo , ξo ; x, ξ) ∈ (T ∗ Σ × T ∗ M) \ 0; xo = x, ξo = ξ|Txo Σ }.

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Lemma 5.7 ρ0 H(R2 − iJ2 − ∂t )E − ρ0 (R1 − iJ1 − ∂t )E −

∈ ∈

I −N −13/4 (Σ × M, (R0 ◦ C − ) ) I −N −9/4 (Σ × M, (R0 ◦ C − ) )

ρ0 (R1 − iJ1 − ∂t )E + ρ0 H(R2 − iJ2 − ∂t )E +

∈ ∈

I −1/4 (Σ × M, (R0 ◦ C + ) ) I −5/4 (Σ × M, (R0 ◦ C + ) ).

Proof. All these statements follow from (69), Lemma 5.6 and the composition formula for Fourier integral operators [28, Theorem 25.2.3], provided that the compositions R0 ◦ C − and R0 ◦ C + of the canonical relations are clean, proper and connected with excess zero (cf. [27, C.3] and [28, p. 18] for notation). We note that (R0 × C + ) ∩ (T ∗ Σ × diag (T ∗ M) × T ∗ M) = {(xo , ξo ; x, ξ; x, ξ; y, η); x = xo , ξo = ξ|Txo Σ , (x, ξ; y, η) ∈ C + }.

(70)

Given (xo , ξo ) ∈ T ∗ Σ \ 0 there is precisely one (x, ξ) ∈ N+ such that x = xo and ξ|Txo Σ = ξo ; given (x, ξ) ∈ N+ there is a 1-parameter family of (y, η) such that (x, ξ; y, η) ∈ C + . We deduce that codim (R0 × C + ) + codim (T ∗ Σ × diag (T ∗ M) × T ∗ M) = 6 dim M − 1 = codim (R0 × C + ) ∩ (T ∗ Σ × diag (T ∗ M) × T ∗ M); here the codimension is taken in T ∗ Σ × (T ∗M)3 . Hence the excess of the intersection, i.e., the difference of the left- and the right-hand side, is zero. In particular, the intersection is transversal, hence clean. Moreover, the fact that in (70) the (x, ξ) is uniquely determined as soon as (xo , ξo ) and (y, η) are given shows that the associated map (xo , ξo ; x, ξ; x, ξ; y, η) → (xo , ξo ; y, η) is proper (i.e., the pre-image of a compact set is compact). Indeed, the pre-image of a closed and bounded set is trivially closed; it is bounded, because |ξ| ≤ C|ξo | for some constant C. Finally, the pre-image of a single point (xo , ξo ; y, η) is again a single point, in particular a connected set.
An analogous argument applies to R0 ◦ C − . Lemma 5.8 (i) (ρ0 (R1 − iJ1 − ∂t )E − )∗ (ρ0 H(R2 − iJ2 − ∂t )E − ) ∈ I −2N −11/2 (M × M, (C − ) ), (ii) (ρ0 (R1 − iJ1 − ∂t )E + )∗ (ρ0 H(R2 − iJ2 − ∂t )E + ∈ I −3/2 (M × M, (C + ) ). Denoting by D± the relation (R0 ◦ C ∓ )−1 ◦ (R0 ◦ C ± ) we have (iii) (ρ0 (R1 − iJ1 − ∂t )E + )∗ (ρ0 H(R2 − iJ2 − ∂t )E − ) ∈ I −N −7/2 (M × M, (D− ) ), (iv) (ρ0 (R1 − iJ1 − ∂t )E − )∗ (ρ0 H(R2 − iJ2 − ∂t )E + ) ∈ I −N −7/2 (M × M, (D+ ) ).

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Proof. (i) According to [28, Theorem 25.2.2] and Lemma 5.7 (ρ0 (R1 − iJ1 − ∂t )E − )∗ ∈ I −N −9/4 (M × Σ, ((R0 ◦ C − )−1 ) ). We first note that the composition (R0 ◦ C − )−1 ◦ (R0 ◦ C − ) equals C − : In fact, (R0 ◦ C − )−1 is the set of all (y, η; xo , ξo ), where (xo , ξo ) ∈ T ∗ Σ, y is joined to xo by a null geodesic γ, η ∈ N− is cotangent to γ at y and the projection Pγ (η)|Txo Σ of the parallel transport of η along γ coincides with ξo . The codimension of (R0 ◦ C − )−1 × (R0 ◦ C − ) in T ∗ M × T ∗ Σ × T ∗ Σ × T ∗ M therefore equals 4 dim Σ + 2, and we have codim ((R0 ◦ C − )−1 × (R0 ◦ C − )) + codim (T ∗ M × diag (T ∗ Σ) × T ∗ M) = 6 dim Σ + 2 = codim ((R0 ◦ C − )−1 × (R0 ◦ C − )) ∩ (T ∗ M × diag (T ∗ Σ) × T ∗ M). In particular, the intersection of (R0 ◦ C − )−1 × (R0 ◦ C − ) and T ∗ M × diag (T ∗ Σ) × T ∗ M is transversal in T ∗ M × T ∗ Σ × T ∗ Σ × T ∗ M, hence clean with excess 0. Given (y, η; xo , ξo ; xo , ξo ; y˜, η˜) in the intersection, the element (xo , ξo ) is uniquely determined by (y, η) and (˜ y , η˜). The mapping (y, η; xo , ξo ; xo , ξo ; y˜, η˜) → (y, η; y˜, η˜) therefore is proper. The pre-image of each element is a single point, hence a connected set. We can apply the composition theorem [28, Theorem 25.2.3] and obtain the assertion. The proof of (ii), (iii) and (iv) is analogous.
We can now finish the proof of Theorem 5.3. According to (66) and the following remarks we have to find the wavefront set of (ρ0 (R1 − iJ1 − ∂t )(E + + E − ))∗ (ρ0 H(R2 − iJ2 − ∂t )(E + + E − )). By Proposition B.10 we have, for an arbitrary canonical relation Λ, I µ (M × M, Λ) ⊂ H s (M × M) if µ + 12 dim M + s < 0; moreover, the wavefront set of elements of I µ (M × M, Λ) is a subset of Λ. Lemma 5.8 therefore immediately implies (65).


5.2

Construction on a compact Cauchy surface

Following the idea in [31] we shall now show that one can construct adiabatic vacuum states on any globally hyperbolic spacetime M with compact Cauchy surface Σ. In Gaußian normal coordinates w.r.t. Σ the metric reads   1 gµν = −hij (t, x)

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Ann. Henri Poincar´e

and the Klein-Gordon operator reduces to  1 2g + m2 = √ ∂t ( h∂t ·) − ∆Σ + m2 , h where hij is the induced Riemannian metric on Σ, h its determinant and ∆Σ the Laplace-Beltrami operator w.r.t. hij acting on Σ. Following [31, Equation (130)ff.] (N ) (N ) there exist operators P1 , P2 , N = 0, 1, 2, . . . , of the form (N )

P1

(N )

P2

1  = −a(N ) (t, x, Dx ) − √ ∂t h h = a(N ) (t, x, Dx ) − ∂t

with a(N ) = a(N ) (t, x, Dx ) ∈ C ∞ ([−T, T ], L1(Σ)) such that (N )

P1

(N )

◦ P2

− (2g + m2 ) = sN (t, x, Dx )

(71)

with sN ∈ C ∞ ([−T, T ], L−N (Σ)). In fact one gets a(N ) (t, x, Dx ) = −iA1/2 +

N +1 

b(ν) (t, x, Dx );

ν=1

here A is the selfadjoint extension of −∆Σ + m2 on L2 (Σ), so that A1/2 is an elliptic pseudodifferential operator of order 1. The b(ν) are elements of C ∞ ([−T, T ], L1−ν (Σ)) defined recursively so that (71) holds. One then sets similarly as in [31, Equation (134)] j (N ) (t, x, ξ) r

(N )

:=

(t, x, ξ)

:=

J(t)

:=

R(t) :=

−

N +1  1   (ν) b (t, x, ξ) − b(ν) (t, x, −ξ) ∈ S 0 2i ν=1

N +1  1   (ν) b (t, x, ξ) + b(ν) (t, x, −ξ) ∈ S 0 2 ν=1  1  (N ) j (t, x, Dx ) + j (N ) (t, x, Dx )∗ ∈ L1 A1/2 + 2  1  (N ) r (t, x, Dx ) + r(N ) (t, x, Dx )∗ ∈ L0 . 2

(72)

(73)

Lemma 5.9 We can change the operator J defined above by a family of regularizing operators such that the assumptions of Theorem 5.1 are met. Proof. It is easily checked that a pseudodifferential operator on Rn with symbol a(x, ξ) maps C0∞ (Rn , R) to L2R (Rn ) (i.e., commutes with complex conjugation) iff a(x, ξ) = a(x, −ξ). The symbols j (N ) and r(N ) have this property

by construction. The operator family R(t) = 12 r(N ) (t, x, Dx ) + r(N ) (t, x, Dx )∗ ∈ L0 is bounded

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and symmetric, hence selfadjoint. Moreover, it commutes with complex conjugation: If v ∈ L2R (Σ, d3 σ), then R(t)v is real-valued, since for every u ∈ L2R (Σ, d3 σ) 2u, R(t)v = u, (r(N ) + r(N )∗ )v = u, r(N ) v + r(N ) u, v ∈ R. The operator A1/2 maps C0∞ (Σ, R) to L2R (Σ, d3 σ) by (61); it is selfadjoint on D(A1/2 ) = H 1 (Σ). Hence J defines a selfadjoint family of pseudodifferential operators of order  1; it is invariant under complex conjugation. Moreover, its principal symbol is hij ξi ξj > 0. According to [44, Ch. II, Lemma 6.2] there exists a family of regularizing operators J∞ = J∞ (t) such that J + J∞ is strictly positive. Replacing J∞ by 12 (J∞ + CJ∞ C), where C here denotes the operator of complex conjugation, we obtain an operator which is both strictly positive and invariant under complex conjugation. It differs from J by a regularizing family.
Theorem 5.10 For N = 0, 1, 2, . . . we let ΛN (f1 , f2 ) =

 1 [(R − iJ)ρ0 − ρ1 ] Ef1 , J −1 [(R − iJ)ρ0 − ρ1 ] Ef2 2

with J modified as in Lemma 5.9. Then ΛN is the two-point function of a (pure) adiabatic vacuum state of order N . Proof. By Theorem 5.1 and Lemma 5.9, ΛN defines the two-point function of a (pure) quasifree state. We write R(t) = J(t) =

1 1 (N ) r (t, x, Dx ) + r(N ) (t, x, Dx )∗ 2 2   1

1 1/2 (N ) A + j (t, x, Dx ) + A1/2 + j (N ) (t, x, Dx )∗ + j∞ (t, x, Dx ) 2 2

with r(N ) , j (N ) as in (72), (73) and j∞ the regularizing modification of Lemma 5.9. We shall use Theorem 5.3 to analyze the wavefront set of ΛN . We decompose ΛN (f1 , f2 ) =   1  (N ) r (t, x, Dx ) − i(A1/2 + j (N ) (t, x, Dx ) + 2j∞ (t, x, Dx )) ρ0 − ρ1 Ef1 8 

 

r(N ) (t, x, Dx )∗ − i(A1/2 + j (N ) (t, x, Dx )∗ ) ρ0 − ρ1 Ef1 , 

  J(t)−1 r(N ) (t, x, Dx ) − i(A1/2 + j (N ) (t, x, Dx ) + 2j∞ (t, x, Dx )) ρ0 − ρ1 Ef2 

  (74) +J(t)−1 r(N ) (t, x, Dx )∗ − i(A1/2 + j (N ) (t, x, Dx )∗ ) ρ0 − ρ1 Ef2 .

+

Now we let ˜ 1 (t) := Q =

  i A1/2 + i r(N ) (t, x, Dx ) − ij (N ) (t, x, Dx ) + √ ∂t h h    1 (N ) i a(N ) (t, x, Dx ) + √ ∂t h = −iP1 h

(75)

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and 

 ˜ 2 (t) := A1/2 + i r(N ) (t, x, Dx )∗ − ij (N ) (t, x, Dx )∗ + √i ∂t h. Q h

(76)

Equation (71) implies that

 ˜ 1 (t) r(N ) (t, x, Dx ) − i(A1/2 + j (N ) (t, x, Dx )) − 2ij∞ (t, x, Dx ) − ∂t iQ

 (N ) (N ) P2 − 2ij∞ (t, x, Dx ) = P1 = 2g + m2 + s˜N (t, x, Dx )

(77)

where s˜N differs from sN in (71) by an element in C ∞ ([−T, T ], L−∞(Σ)). Next we note that (71) is equivalent to the identity    1  (N ) r − i(A1/2 + j (N ) ) − ∂t −r(N ) + i(A1/2 + j (N ) ) − √ ∂t h h = 2g + m2 + sN which in turn is equivalent to



 −r(N ) + i(A1/2 + j (N ) ) r(N ) − i(A1/2 + j (N ) )



 1 h r(N ) − i(A1/2 + j (N ) ) − √ ∂t h 2 = −∆Σ + m + sN or – taking adjoints and conjugating with the operator C of complex conjugation – 



−C r(N )∗ + i(A1/2 + j (N )∗ ) CC r(N )∗ + i(A1/2 + j (N )∗ ) C



  1 hC r(N )∗ + i(A1/2 + j (N )∗ ) C − √ ∂t h (78) = C(−∆Σ + m2 + s∗N )C = −∆Σ + m2 + Cs∗N C. Here we have used the fact that  



∗ ∗ ∂t h r(N ) − i(A1/2 + j (N ) ) = ∂t h r(N ) − i(A1/2 + j (N ) ) . Using that r(N )∗ , j (N )∗ and A1/2 commute with C, (78) reads 



− r(N )∗ − i(A1/2 + j (N )∗ ) r(N )∗ − i(A1/2 + j (N )∗ )



 1 − √ ∂t h r(N )∗ − i(A1/2 + j (N )∗ ) h 2 ∗ = −∆Σ + m + CsN C.

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Adding the time derivatives, it follows that 

˜ 2 (t) r(N )∗ − i(A1/2 + j (N )∗ ) − ∂t iQ    1  (N )∗ (N )∗ 1/2 (N )∗ = −r + i(A + j ) − √ ∂t h r − i(A1/2 + j (N )∗ ) − ∂t h = 2g + m2 + Cs∗N C. (79) ˜ 1 and Q ˜ 2 , defined by Equations (75) and (76), are not Note that the operators Q yet pseudodifferential operators since their symbols will not decay in the covariable of t, say τ , if we take derivatives w.r.t. the covariables of x, say ξ. To make them pseudodifferential operators we choose a finite number of coordinate neighborhoods {Uj } for Σ, which yields finitely many coordinate neighborhoods for (−T, T ) × Σ. As (t, x) varies over (−T, T ) × Uj , the negative light cone will not intersect a fixed conic neighborhood N of {ξ = 0} in T ∗ ((−T, T ) × Uj ). We choose a realvalued function χ which is smooth on T ∗ ((−T, T ) × Uj ), zero for |(τ, ξ)| ≤ 1/2, homogeneous of degree zero for |(τ, ξ)| ≥ 1 such that χ(t, x, τ, ξ) χ(t, x, τ, ξ)

= 0 = 1

on a conic neighborhood of {ξ = 0} outside N .

(80)

We now let X := op χ. Then ˜1 Q1 := X Q

˜2 and Q2 := X Q

are pseudodifferential operators due to (80). Their principal symbols are ((hij ξi ξj )1/2 − τ )χ(t, x, τ, ξ), so that their characteristic set does not intersect N− . Equations (77), (79) and the fact that (2g + m2 )E − = 0 show that the assumption of Theorem 5.3 is satisfied for each of the four terms arising from the
decomposition of ΛN in (74). This yields the assertion. Lemma 5.8 explicitly shows that the non-Hadamard like singularities of the two-point function ΛN in Theorem 5.10 (i.e., those not contained in the canonical relation C + ) are either pairs of purely negative frequency singularities lying on a common bicharacteristic (C − ) or pairs of mixed positive/negative frequency singularities (D± ) which lie on bicharacteristics that are “reflected” by the Cauchy surface. They may have spacelike separation. For the states constructed in Theorem 5.10 one can explicitly find the Bogoljubov B-operator, which was introduced in the proof of Theorem 4.7(ii), in terms of the operators R and J. Applying the criterion of Lemma 4.11(iii) one can check the unitary equivalence of the GNS-representations generated by these states. A straightforward (but tedious) calculation shows that unitary equivalence already holds for N ≥ 0, thus improving the statement of Theorem 4.7(ii) for these particular examples.

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6 Adiabatic vacua on Robertson-Walker spaces By introducing adiabatic vacua on Robertson-Walker spaces Parker [38] was among the first to construct a quantum field theory in a non-trivial background spacetime. A mathematically precise version of his construction and an analysis of the corresponding Hilbert space representations along the same lines as in our Section 4 were given by L¨ uders & Roberts [35]. Relying on their work we want to show in this section that these adiabatic vacua on Robertson-Walker spaces are indeed adiabatic vacua in the sense of our Definition 3.2. This justifies our naming and gives a mathematically intrinsic meaning to the “order” of an adiabatic vacuum. In [31] one of us had claimed to have shown that all adiabatic vacua on RobertsonWalker spaces are Hadamard states. This turned out to be wrong in general, when the same question was investigated for Dirac fields [24]2 . So the present section also serves to correct this mistake. Our presentation follows [31]. In order to be able to apply our Theorem 5.3 without technical complications we restrict our attention to Robertson-Walker spaces with compact spatial sections. These are the 4-dimensional Lorentz manifolds M = R × Σ where Σ is regarded as being embedded in R4 as Σ = {x ∈ R4 ; (x0 )2 +

3 

(xi )2 = 1} ∼ = S3,

i=1

and M is endowed with the homogeneous and isotropic metric " ! dr2 2 2 2 2 2 2 2 + r (dθ + sin θ dϕ ) ; ds = dt − a(t) 1 − r2

(81)

here ϕ ∈ [0, 2π], θ ∈ [0, π], r ∈ [0, 1) are polar coordinates for the unit ball in R3 , and a is a strictly positive smooth function. In [31] it was shown that an adiabatic vacuum state of order n (as defined in [35]) is a pure quasifree state on the Weyl algebra of the Klein-Gordon field on the spacetime (81) given by a one-particle Hilbert space structure w.r.t. a fixed Cauchy surface Σt = {t = const.} = Σ × {t} (equipped with the induced metric from (81)) kn : Γ → Hn := kn (Γ) ⊂ L2 (Σt ) (q, p) → (2Jn )−1/2 [(Rn − iJn )q − p] of the form (60) of Theorem 5.1, where the operator families Rn (t), Jn (t) acting on L2 (Σ, d3 σ) are defined in the following way:    ˙ (n) (t) Ω 1 a(t) ˙ k + f˜(t, k)φ k (x) (Rn f )(t, x) := − dµ(k) 3 2 a(t) Ω(n) (t) k  (n) (Jn f )(t, x) := dµ(k) Ωk (t)f˜(t, k)φ k (x), (82) 2 We

want to thank S. Hollands for discussions about this point.

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with t in some fixed finite interval I ⊂ R, say. Here, {φ k , k := (k, l, m), k = 0, 1, 2, . . . ; l = 0, 1, . . . , k; m = −l, . . . , l} are the t-independent eigenfunctions of the Laplace-Beltrami operator ∆Σ w.r.t. the Riemannian metric  1  sij = 

1−r 2

r2

 2

2

(83)

r sin θ on the hypersurface Σ: # 2 2 − 3r2 ∂ 1 2 ∂ + ∆(θ, ϕ) φ k = −k(k + 2)φ k , ∆Σ φ k ≡ (1 − r ) 2 + ∂r r ∂r r2 2

2

∂ ∂ 1 ∂ 2 where ∆(θ, ϕ) := ∂θ 2 + cot θ ∂θ + sin2 θ ∂ϕ2 is the Laplace operator on S . They form an orthonormal basis of L2 (Σ, d3 σ) with d3 σ := r2 (1 − r2 )−1/2 dr sin θdθ dϕ. ˜denotes the generalized Fourier transform

˜: L2 (Σ, d3 σ) f

˜ dµ(k)) → L2 (Σ,  ˜  → f (k) := d3 σ φ k (x)f (x),

(84)

Σ

˜ dµ(k)) where Σ ˜ is the space of which is a unitary map from L2 (Σ, d3 σ) to L2 (Σ, $    ∞ values of k = (k, l, m) equipped with the measure dµ(k) := k=0 kl=0 lm=−l [35]. (Note that (84) is defined w.r.t. Σ with the metric sij , Equation (83), and not w.r.t. Σt with the metric a2 (t)sij .) The inverse is given by  f (x) = dµ(k) f˜(k)φ k (x). Using duality and interpolation of the Sobolev spaces one deduces Lemma 6.1



H s (Σ) = {f =

 dµ(k) f˜(k)φ k ;

dµ(k) (1 + k 2 )s |f˜(k)|2 < ∞}.

The Klein-Gordon operator associated to the metric (81) is given by 2g + m2 =

1 ∂2 a˙ ∂ − ∆Σ + m2 . +3 ∂t2 a ∂t a2

(85)

(n)

The functions Ωk (t), n ∈ N0 , in (82) were introduced by L¨ uders & Roberts. (n) Ωk (t) is strictly positive and plays the role of a generalized frequency which is determined by a WKB approximation to Equation (85). It is iteratively defined by the following recursion relations (0)

(Ωk )2 (n+1) 2

(Ωk

)

k(k + 2) + m2 a2  2  2 ˙ (n) ¨ (n) Ω a ˙ a ¨ 3 3 3 1Ω k k + ωk2 − − − . 4 a 2 a 4 Ω(n) 2 Ω(n) k k

:= ωk2 ≡ =

(86)

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In the following we shall study the analytic properties of these functions. We shall (n) see that, using (86), we may express Ωk as a function of t and ωk . As a function of these two variables, it turns out to behave like a classical pseudodifferential symbol. In Lemma 6.2 below we shall derive the corresponding estimates and expansions to consider a ‘covariable’ in R+ ; in later for (t, ωk ) ∈ I × R+ . It is a little unusual  applications, however, we will have ωk = k(k + 2)/a2 + m2 bounded away from zero, so that the behaviour of ωk near zero is irrelevant. (n+1) 2 We first observe that (Ωk ) can be determined by an iteration involving only (n) (Ωk )2 and its time derivatives: Since, for an arbitrary F , we have ∂t (F 2 )/F 2 = 2F˙ /F , we obtain  (n)  2 d  2 (Ωk )2 a ˙ a ¨ 1 3 3 (n+1) 2  dt  + ) = ωk2 − − (Ωk (n) 2 4 a 2 a 16 Ωk  (n)   d 2 1 d  dt (Ωk )  − . (87) (n) 4 dt (Ω )2 k

(n)

An induction argument shows that (Ωk )2 − ωk2 is a rational function in ωk of degree ≤ 0 with coefficients in C ∞ (I). Indeed this is trivially true for n = 0. Suppose it is proven for some fixed n. We write (n)

(Ωk )2 − ωk2 = r(t, ωk ) =

p(t, ωk ) q(t, ωk )

(88)

with polynomials p and q in ωk such that deg (p) ≤ deg (q) and the leading coefficient of q is 1. Then

 (n) 2 d ) (Ω k dt 2ωk ω˙ k + r˙ . (89) = (n) 2 ωk2 + r (Ωk ) In view of the fact that   a˙ m2 ω˙ k = − ωk − a ωk

and r˙ =

q∂t p − p∂t q q∂ωk p − p∂ωk q ω˙ k + , q2 q2

(89) is again rational of degree 0 and the leading coefficient of the polynomial in the denominator again equals 1. The same is true for the time derivatives of (89). The recursion formula (87) then shows the assertion for n + 1.  (n) 2 dl 2 (Ω is a rational function of ωk with Next we observe that also dt ) − ω l k k

coefficients in C ∞ (I) of degree ≤ 0. Moreover, this shows that, for sufficiently (n) large ωk (equivalently for sufficiently large k), (Ωk )2 is a strictly positive function (n) (uniformly in t ∈ I). We may redefine (Ωk )2 for small values of ωk so that it is strictly positive and bounded away from zero on I × R+ . It makes sense to take its (n) square root, and in the following we shall denote this function by Ωk . We note:

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(n)

Lemma 6.2 Ωk , considered as a function of (t, ω) ∈ I × R+ , is an element of 1 Scl (I × R+ ), i.e., (n) ∂tl ∂ωm Ωk (t, ω) = O(ω 1−m ) (90) ∞ (n) (n) (n) and, in addition, Ωk has an asymptotic expansion Ωk ∼ j=0 (Ωk )j into (n)

symbols (Ωk )j ∈ S 1−j which are positively homogeneous for large ω. Its principal symbol is ω. With the same understanding ˙ (n) Ω k (n)

Ωk (n+1) 2

(n)

) − (Ωk )2

(Ωk

0 ∈ Scl (I × R+ ),

(91)

−2n ∈ Scl (I × R+ ).

(92)

Proof. By induction, (90) is immediate from (88) together with the formulae  1 2ω + ∂ω r(t, ω) . ω 2 + r(t, ω) =  2 ω 2 + r(t, ω)

 ˙ (n) /Ω(n) = d (Ω(n) )2 Relation (91) is immediate from (89), noting that 2Ω k k k dt ∂t

 1 ∂t r(t, ω) ω 2 + r(t, ω) =  2 ω 2 + r(t, ω)

and ∂ω

(n)

/(Ωk )2 . In both cases the existence of the asymptotic expansion follows from [44, Ch. II, Theorem 3.2] and the expansion %  j ∞    1/2 r(t, ω) r(t, ω) ω 2 + r(t, ω) = ω 1 + = ω j ω2 ω2 j=0 (n)

(n)

valid for large ω. The principal symbol of Ωk is ω since (Ωk )2 = ω 2 modulo rational functions of degree ≤ 0, as shown above (cf. Equation (88)). In order to show (92) we write, following L¨ uders & Roberts, (n+1) 2

(Ωk

(n)

) = (Ωk )2 (1 + n+1 );

this yields [35, Equation (3.9)]

n+1

=

 1 ω˙ ˙n 1

˙n 1 1 ˙1 + + ··· 2 ω (1 + 1 ) · · · (1 + n ) 4 ω 1 + n 8 1 + 1 1 + n  1 ˙n−1

˙n 5 ˙2n 1 ¨n + + − . 8 1 + n−1 1 + n 16 1 + n 4 1 + n (1)

(0)

(1)

We know already that (Ωk )2 − (Ωk )2 = (Ωk )2 − ω 2 is rational in ω of degree ≤ 0, hence 1 is rational of degree −2. Noting that ˙n = (∂ω n )ω˙ + ∂t n , we deduce from the above recursion that n is rational of degree −2n. This completes the argument.
The operators Rn and Jn , Equation (82), are unitarily equivalent to mul˜ dµ(k)). From the fact that Ω ˙ (n) /Ω(n) is bounded tiplication operators on L2 (Σ, k k

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(n)

and Ωk is strictly positive with principal symbol ωk (Lemma 6.2) we can immediately deduce that the assumptions of Theorem 5.1 are satisfied if we let dom J(t) = H 1 (Σ) for t ∈ I. We are now ready to state the theorem that connects the adiabatic vacua of L¨ uders & Roberts [35] to our more general Definition 3.2: Theorem 6.3 For fixed t let Λn (f, g) := (Rn − iJn − ∂t )Ef, Jn−1 (Rn − iJn − ∂t )Eg L2 (Σt ) be the two-point function of a pure quasifree state of the Klein-Gordon field on the Robertson-Walker spacetime (81) with Rn , Jn given by Equations (82) and (86). Then ∅, s < − 21 s , W F (Λn ) ⊂ C + , − 12 ≤ s < 2n + 32 i.e., Λn describes an adiabatic vacuum state of order 2n in the sense of our Definition 3.2. To prove the theorem we shall need the following observations: Lemma 6.4 Let m ∈ R. Let M be a compact manifold and A : D(M ) → D (M ) a linear operator. Suppose that, for each k ∈ N, we can write A = Pk + Rk

(93)

where Pk is a pseudodifferential operator of order m and Rk is an integral operator with a kernel function in C k (M × M ). Then A is a pseudodifferential operator of order m. Proof. Generalizing a result by R. Beals [4], Coifman & Meyer showed the following: A linear operator T : D(M ) → D (M ) is a pseudodifferential operator of order 0 if and only if T as well as its iterated commutators with smooth vector fields are bounded on L2 (M ) [10, Theorem III.15]. As a corollary, T is a pseudodifferential operator of order m if and only if T and its iterated commutators induce bounded maps L2 (M ) → H −m (M ). Given the iterated commutator of A with, say, l vector fields V1 , . . . , Vl , we write A = Pk + Rk with k ≥ l + |m|. The iterated commutator [V1 , [. . . [Vl , Pk ] . . .]] is a pseudodifferential operator of order m and hence induces a bounded map L2 (M ) → H −m (M ). The analogous commutator with Rk has an integral kernel in C k−l (M × M ). As k − l ≥ |m|, it furnishes even a bounded operator L2 (M ) → H |m| (M ).
µ Lemma 6.5 Let µ ∈ Z and b = b(t, τ ) ∈ Scl (I × R) with principal symbol b−µ (t)τ µ . 1/2

2 Replacing τ by ωk (t) = k(k+2) , b defines a family {B(t); t ∈ I} of a2 (t) + m

operators B(t) : D(Σ) → D (Σ) by

 )(t, k) := b(t, ωk (t))f˜(k). (B(t)f

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We claim that this is a smooth family of pseudodifferential operators of order µ with principal symbol σ (µ) (B(t)) = b−µ (t)(|ξ|Σ /a(t))µ ,

(94)

where the length |ξ|Σ of a covector ξ is taken w.r.t. the (inverse of the) metric (83). Note that for the definition of B(t) we only need to know b(t, τ ) for τ ≥ m. We may therefore also apply this result to the symbols that appear in Lemma 6.2. Proof. The fact that b is a classical symbol allows us to write, for each N , b(t, τ ) =

N 

bj (t)τ −j + b(N ) (t, τ ),

(95)

j=−µ

where bj ∈ C ∞ (I) and |∂tj ∂τl b(N ) (t, τ )| ≤ Cjl (1 + |τ |)−N for all t ∈ I and τ ≥ ,

> 0 fixed. (Note that we will not obtain the estimates for all τ , since we have a fully homogeneous expansion in (95), but as we shall substitute τ by ωk and ωk is bounded away from 0, this will not be important.) Equation (95) induces an analogous decomposition of B: B(t) =

N 

Bj (t) + B (N ) (t),

j=−µ

where Bj (t) is given by −j ˜   (B j (t)f )(t, k) = bj (t)ωk (t) f (k)

and B (N ) (t) by (N ) (t)f )(t,  k) = b(N ) (t, ωk (t))f˜(k). (B 

In view of the fact that ∆Σ φ k = −k(k + 2)φ k , we have

−j/2 . Bj (t) = bj (t) m2 − ∆Σ /a2 (t) According to Seeley [43], Bj is a smooth family of pseudodifferential operators of order −j. Next, we observe that, by (90), ∂tl ωk = O(ωk ) and hence, for each l ∈ N,

 |∂tl b(N ) (t, ωk (t)) | ≤ C(1 + ωk (t))−N ≤ C  (1 + k)−N for all t ∈ I. Lemma 6.1 therefore shows that, for each s ∈ R ∂tl B (N ) (t) : H s (Σ) → H s+N (Σ)

(96)

is bounded, uniformly in t ∈ I. On the other hand, it is well known that a linear operator T which maps H −s−k (Σ) to H s+k (Σ) for some s > 3/2 (dim Σ = 3) has

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an integral kernel of class C k on Σ× Σ. It is given by K(x, y) = T δy , δx . Choosing N > 3 + 2k, the family B (N ) will therefore have integral kernels of class C k . Now we apply Lemma 6.4 to conclude that B is a smooth family of pseudodifferential operators of order µ. Since Bj is of order −j the principal symbol is that of

−µ/2 . This yields (94).
B−µ = b−µ (t) m2 − ∆Σ /a2 (t) Now let us define the family of operators An (t) acting on L2 (Σ, d3 σ) by  (An f )(t, x) := dµ(k) a(n) (t, k)f˜(t, k)φ k (x), with a(n) given by the function a

(n)

˙ (n) 1Ω 3 a˙ (n) k − (t, k) := − iΩk . 2 a 2 Ω(n) k

Moreover let Qn := iX(∂t + An (t)), where X := op χ and χ = χ(t, x, τ, ξ) is as in (80). Lemma 6.6 An ∈ C ∞ (I, L1cl (Σ)) with principal symbol σ (1) (An (t)) = i|ξ|Σ /a(t). Qn is a pseudodifferential operator of order 1 on I × Σ with real-valued principal symbol whose characteristic does not intersect N− . Proof. We apply Lemma 6.5 in connection with Lemma 6.2 to see that An ∈ C ∞ (I, L1cl (Σ)). The operator Qn clearly is an element of L1cl (I × Σ). Outside a small neighborhood of {ξ = 0} its characteristic set is {(t, x, τ, ξ) ∈ T ∗ (I × Σ); −τ + |ξ|Σ /a(t) = 0}. Since N− = {τ = −|ξ|Σ /a(t)}, the intersection is empty.




Proof of Theorem 6.3. In view of Theorem 5.3 we only have to check that Qn (Rn − iJn − ∂t )E − = S (2n) E −

(97)

for a pseudodifferential operator S (2n) of order −2n. A straightforward computation shows that (∂t + An )(Rn − iJn − ∂t ) is the operator defined by   a˙ (n) 2 (n+1) 2 2 2 − ∂t + 3 ∂t + ωk + (Ωk ) − (Ωk ) . a Now ∂t2 + 3 aa˙ ∂t + ωk2 induces 2g + m2 , Equation (85), while, by Lemma 6.2 com(n) (n+1) 2 bined with Lemma 6.5, (Ωk )2 − (Ωk ) induces an element of C ∞ (I, L−2n (Σ)). Composing with the operator X from the left and noting that (2g + m2 )E − = 0, we obtain (97).


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1165

7 Physical interpretation Using the notion of the Sobolev wavefront set (Definition B.1) we have generalized in this paper the previously known positive frequency conditions to define a large new class of quantum states for the Klein-Gordon field on arbitrary globally hyperbolic spacetime manifolds (Definition 3.2). Employing the techniques of pseudodifferential and Fourier integral operators we have explicitly constructed examples of them on spacetimes with a compact Cauchy surface (Theorem 5.10). We call these states adiabatic vacua because on Robertson-Walker spacetimes they include a class of quantum states which is already well known under this name (Theorem 6.3). We order the adiabatic vacua by a real number N which describes the Sobolev order beyond which the positive frequency condition may be perturbed by singularities of a weaker nature. Our examples show that these additional singularities may be of negative frequency or even non-local type (Lemma 5.8). Hadamard states are naturally included in our definition as the adiabatic states of infinite order. To decide which orders of adiabatic vacua are physically admissible we have investigated their corresponding GNS-representations: Adiabatic vacua of order N > 5/2 generate a quasiequivalence class of local factor representations (in other words, a unique local primary folium). For pure states on a spacetime with compact Cauchy surface – a case which often occurs in applications – this holds true already for N > 3/2 (Theorems 4.5 and 4.7). Physically, locally quasiequivalent states can be thought of as having a finite energy density relative to each other. Primarity means that there are no classical observables contained in the local algebras. Hence there are no local superselection rules, i.e., the local states can be coherently superimposed without restriction. For N > 3/2 the local von Neumann algebras generated by these representations contain no observables which are localized at a single point (Theorem 4.12). Together with quasiequivalence this implies that all the states become indistinguishable upon measurements in smaller and smaller spacetime regions (Corollary 4.13). This complies well with the fact that the correlation functions have the same leading short-distance singularities, whence the states should have the same high energy behaviour. Finally, the algebras are maximal in the sense of Haag duality (Theorem 4.15) and additive (Lemma 2.5). For a more thorough discussion of all these properties in the framework of algebraic quantum field theory we refer to [21]. Taken together, all these results suggest that adiabatic vacua of order N > 5/2 are physically meaningful states. Furthermore we expect that the energy momentum tensor of the Klein-Gordon field can be defined in these states by an appropriate regularisation generalizing the corresponding results for Hadamard states [8, 50] and adiabatic vacua on Robertson-Walker spaces [39]. However, all the mentioned physical properties of the GNS-representations are of a rather universal nature and therefore cannot serve to distinguish between different types of states. How can we physically discern an adiabatic state of order

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N from one of order N  , say, or from a Hadamard state? To answer this question we investigate the response of a quantum mechanical model detector (a so-called Unruh detector [5, 46]) to the coupling with the Klein-Gordon field in an N -th order adiabatic vacuum state. So let us assume we are given an adiabatic state ˆ on the spacetime M and its ωN of order N of the Klein-Gordon quantum field Φ associated GNS-triple (HωN , πωN , ΩωN ) as in Proposition 2.4(b). We consider a detector that moves on a wordline γ : R → M, τ → x(τ ), in M and is described as a quantum mechanical system by a Hilbert space HD and a free time evolution w.r.t. proper time τ . It shall be determined by a free Hamiltonian H0 with a discrete energy spectrum E0 < E1 < E2 < · · · , E0 being the groundstate energy of H0 (e.g. a harmonic oscillator). We assume that the detector has negligible ˆ via the interaction Hamiltonian extension and is coupled to the quantum field Φ ˆ ))χ(τ ) HI := λM (τ )Φ(x(τ

(98)

acting on HD ⊗HωN , where λ ∈ R is a small coupling constant, M (τ ) = eiH0 τ M (0) e−iH0 τ the monopole moment operator characterizing the detector and χ ∈ C0∞ (R) a cutoff function that describes the adiabatic switching on and off of the interaction. To calculate transition amplitudes between states of HD ⊗ HωN under the interaction (98) one uses most conveniently the interaction picture, in which the ˆ and the operator M evolve with the free time evolution (but the full coufield Φ pling to the gravitational background) whereas the time evolution of the states is determined by the interaction HI . In this formulation the perturbative S-matrix is given by [5, 7] S

=

=

1+ 1+

 ∞  (−i)j j=1 ∞  j=1

j! (−iλ)j j!

 dτ1 . . . dτj T [HI (τ1 ) . . . HI (τj )] 



dτ1 χ(τ1 ) . . . dτj χ(τj ) T [M (τ1 ) . . . M (τj )]

ˆ ˆ T [Φ(x(τ 1 )) . . . Φ(x(τj ))],

(99)

where T denotes the operation of time ordering. Let us assume that the detector is prepared in its ground state |E0 prior to switching on the interaction, and calculate in first order perturbation theory (j = 1 in (99)) the transition amplitude between the incoming state ψin := |E0 ⊗ ΩωN ∈ HD ⊗ HωN and some outgoing state ψout := |En ⊗ψ, n = 0, where |En ∈ HD is the eigenstate of H0 corresponding to the energy En and ψ some one-particle state in the Fock space HωN (the ˆ scalar products of Φ(x)Ω ωN with other states vanish in a quasifree representation):  ˆ ψout , Sψin = −iλEn |M (0)|E0 dτ χ(τ )ei(En −E0 )τ ψ|Φ(x(τ ))ΩωN . From this we obtain the probability P (En ) that a transition to the state |En occurs in the detector by summing over a complete set of (unobserved) one-particle

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states in HωN : λ2 |En |M (0)|E0 |2

P (En ) =



 dτ1

dτ2 e−i(En −E0 )(τ1 −τ2 )

χ(τ1 )χ(τ2 )ΛN (x(τ1 ), x(τ2 )) λ |En |M (0)|E0 | F (En − E0 ). 2

=

2

Here, |En |M (0)|E0 |2 describes the model dependent sensitivity of the detector, whereas  ∞  ∞ F (E) := dτ1 dτ2 e−iE(τ1 −τ2 ) χ(τ1 )χ(τ2 )ΛN (x(τ1 ), x(τ2 )) −∞

−∞

is the well-known expression for the detector response function depending on the two-point function ΛN of the adiabatic state ωN . Inspection of the formula shows that it is in fact obtained from ΛN ∈ D (M × M) by restricting ΛN to γ × γ ⊂ M × M, multiplying this restricted distribution pointwise by χ ⊗ χ and taking the Fourier transform at (−E, E): ∧

F (E) = 2π ((ΛN |γ×γ ) · (χ ⊗ χ)) (−E, E). It follows from the very definition of ΛN (Definition 3.2) and Proposition B.7 that ΛN |γ×γ is a well-defined distribution on R × R if N > 3/2, since N ∗ (γ) consists only of space-like covectors. It holds W F s (ΛN |γ×γ ) ⊂ ϕ∗ (C + ) for s < N − 3/2, where ϕ∗ is the pullback of the embedding ϕ : γ × γ → M × M. We now observe that (100) {(τ1 , −E; τ2 , E) ∈ R4 ; E ≥ 0} ∩ ϕ∗ (C + ) = ∅ (this observation was already made by Fewster [17] in the investigation of energy mean values of Hadamard states). Hence there is an open cone Γ in R2 \ {0} containing (−E, E), E > 0, such that W F s (ΛN |γ×γ )∩Γ = ∅. By (102) we can write s (R2 ) for s < N −3/2 and W F (u2 )∩Γ = (ΛN |γ×γ )·(χ⊗χ) = u1 +u2 with u1 ∈ Hloc ∅. Since (ΛN |γ×γ ) · (χ ⊗ χ) has compact support we can assume without loss of generality that also u1 and u2 have compact supports. From W F (u2 ) ∩ Γ = ∅ it follows then that u ˆ2 (ξ) = O(ξ −k ) ∀k ∈ N ∀ξ ∈ Γ, s (R2 ) implies that whereas u1 ∈ Hcomp

Dα u1 ∈ L2comp (R2 ) ∧

for |α| ≤ s < N − 3/2, cf. Prop. B.3

⇒ (D u1 ) (ξ) = ξ u ˆ1 (ξ) is bounded α

α

⇒ u ˆ1 (ξ) = O(ξ −|α| ).

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Taken together, we find that ((ΛN |γ×γ ) · (χ ⊗ χ))∧ (ξ) = O(ξ −[N −3/2] ) for ξ ∈ Γ, where [N − 3/2] := max{n ∈ N0 ; n < N − 3/2}. Since (−E, E) ∈ Γ, E > 0, we can now conclude that F (E) = O(E −[N −3/2] ) for an adiabatic vacuum state of order N > 3/2. (Note that this estimate could be improved for the states constructed in Section 5 by taking into account that for them the singularities of lower order are explicitly known, cf. Lemma 5.8, and the sub-leading singularities also satisfy relation (100).) This means that the probability of a detector, moving in an adiabatic vacuum of order N , to get excited to the energy E decreases like E −[N −3/2] for large E, in a Hadamard state it decreases faster than any inverse power of E. We can therefore interpret adiabatic states of lower order as higher excited states of the quantum field. One should however keep in mind that all the states usually considered in elementary particle physics (on a static spacetime, say) are of the Hadamard type: ground states and thermodynamic equilibrium states are Hadamard states [31], particle states satisfy the microlocal spectrum condition (the generalization of the Hadamard condition to higher n-point functions) [8]. We do not know by which physical operation an adiabatic state of finite order could be prepared. Although all results in this paper are concerned with the free Klein-Gordon field, it is clear that our Definition 3.2 is capable of a generalization to higher npoint functions of an interacting quantum field theory in analogy to the microlocal spectrum condition of Brunetti et al. [8]. In order to treat the pointwise product (ΛN )2 we write (ΛN )2 = (ΛH )2 + (ΛN − ΛH )2 + 2(ΛN − ΛH )ΛH , where ΛH is the two-point function of any Hadamard state. It follows from Proposition B.6, Lemma 3.3, and Lemma 5.2 that these pointwise products are well defined if N > −1. It is well known that W F  ((ΛH )2 ) ⊂ C + ⊕ C + , where C + ⊕ C + := {(x1 , ξ1 + η1 ; x2 , ξ2 + η2 ) ∈ T ∗ (M × M) \ 0; (x1 , ξ1 ; x2 , ξ2 ), (x1 , η1 ; x2 , η2 ) ∈ C + } [29, Theorem 8.2.10]. The regularity of the other two terms can be estimated by Theorems 8.3.1 and 10.2.10 in [30] such that finally W F s ((ΛN )2 ) ⊂ C + ⊕ C +

if s < N − 3.

Higher powers of ΛN can be treated similarly. Thus, for sufficiently large adiabatic order N , finite Wick powers (with finitely many derivatives) can be defined. This should be sufficient for the perturbative construction of a quantum field theory with an interaction Lagrangian involving a fixed number of derivatives and powers of the fields. Obviously one has to require more and more regularity of the states if one wants to define higher and higher Wick powers. This complies with a recent result of Hollands & Ruan [25].

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It is also clear that the notion of adiabatic vacua can be extended to other field theory models than merely the scalar field. A first step in this direction has been taken by Hollands [24] for Dirac fields. Finally we want to point out that, although the whole analysis in this paper has been based on a given C ∞ -manifold M with smooth Lorentz metric g, the notion of adiabatic vacua should be particularly relevant for manifolds with C k -metric. Typical examples that occur in general relativity are star models: here the metric outside the star satisfies Einstein’s vacuum field equations and is matched on the boundary C 1 to the metric inside the star where it satisfies Einstein’s equations with an energy momentum tensor of a suitable matter model as a source term. In such a situation Hadamard states cannot even be defined on a part of the spacetime that contains the boundary of the star, whereas adiabatic states up to a certain order should still be meaningful. This remark could e.g. be relevant for the derivation of the Hawking radiation from a realistic stellar collapse to a black hole.

A

Sobolev spaces

H s (Rn ), s ∈ R, is the set of all tempered distributions u on Rn whose Fourier transforms u ˆ are regular distributions satisfying  2 u(ξ)|2 dn ξ < ∞. uH s (Rn ) := ξ 2s |ˆ For a domain U ⊂ Rn we let H s (U) := {rU u; u ∈ H s (Rn )} be the space of all restrictions to U of H s -distributions on Rn , equipped with the quotient topology uH s (U ) := inf{U H s (Rn ) ; U ∈ H s (Rn ), rU U = u}. Moreover, we denote by H0s (U ) the space of all elements in H s (Rn ) whose support is contained in U. If U is bounded with smooth boundary, then it follows from [28, Theorem B.2.1] that C0∞ (U) is dense in H0s (U ) for every s and that H0s (U ) is the dual space of H s (U) with respect to the extension of the sesquilinear form  u ¯v dn x, u ∈ C0∞ (U), v ∈ C ∞ (U). If Σ is a compact manifold without boundary we choose a covering by coordinate neighborhoods with associated coordinate maps, say {Uj , κj }j=1,...,J with a subordinate partition of unity {ϕj }j=1,...,J . Given a distribution u on Σ, we shall say that u ∈ H s (Σ) if, for each j, the push-forward of ϕj u under κj is an element of

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H s (Rn ). It is easy to see that this definition is independent of the choices made for Uj , κj and ϕj . The space H s (Σ) is a Hilbert space with the norm uH s (Σ)

1/2  J  :=  (κj )∗ (ϕj u)2H s (Rn )  . j=1

We denote by ∆ the Laplace-Beltrami operator with respect to an arbitrary metric on Σ. Then we have H 2k (Σ) = {u ∈ L2 (Σ); (1 − ∆)k u ∈ L2 (Σ)} for k = 0, 1, 2, . . .: Clearly, the left-hand side is a subset of the right-hand side. Conversely, we may assume that u has support in a single coordinate neighborhood, so that we can look at the push-forward u∗ under the coordinate map. The fact that both u∗ and ((1 − ∆)k u)∗ belong to L2 (Rn ) implies that u∗ ∈ H 2k (Rn ), hence u ∈ H 2k (Σ). Moreover, this consideration shows that the two topologies are equivalent (and in particular independent of the choice of metric on Σ). We may identify H −s (Σ) with the dual of H s (Σ) with respect to the L2 -inner product in Σ. Now let Σ be a (possibly) non-compact Riemannian manifold which is geodesically complete. The Laplace-Beltrami operator ∆ : C0∞ (Σ) → C0∞ (Σ) is essentially selfadjoint by Chernoff’s theorem [9]. We can therefore define the powers (1−∆)s/2 for all s ∈ R. By H s (Σ) we denote the completion of C0∞ (Σ) with respect to the norm uH s (Σ) := (1 − ∆)s/2 uL2 (Σ) . For s ∈ 2N0 , this shows that H 2k (Σ) is the set of all u ∈ L2 (Σ) for which (1 − ∆)k u ∈ L2 (Σ). We deduce that this definition coincides with the previous one if Σ is compact and s = 2k; using complex interpolation, cf. [44, Ch. I, Theorem 4.2], equality holds for all s ≥ 0. Moreover, we can define a sesquilinear form ., . : H −s (Σ) × H s (Σ) → C by letting

 u, v := (1 − ∆)−s/2 u, (1 − ∆)s/2 v

L2 (Σ)

.

This allows us to identify H −s (Σ) with the dual of H s (Σ), as in the compact case. In particular, the definition of the Sobolev spaces on compact manifolds coincides also for negative s. Now suppose that O is a relatively compact subset of Σ. We let H s (O) := rO H (Σ), the restriction to O of H s -distributions on Σ, endowed with the quotient topology uH s (O) := inf{U H s (Σ) ; U ∈ H s (Σ), rO U = u}. s

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This definition is local: If O is another relatively compact subset with smooth boundary containing O, then we can find a function f ∈ C0∞ (O ) with f ≡ 1 on O. Hence, whenever there exists a U ∈ H s (Σ) with rO U = u, then there is a U1 ∈ H s (Σ) with supp U1 ⊂ O and rO U1 = u, namely U1 = f U . We therefore obtain the same space and the same topology, if we replace the right-hand side by inf{U H s (Σ) ; U ∈ H s (Σ), supp U ⊂ O , rO U = u}. Indeed, both definitions yield the same space, which also is a Banach space with respect to both norms. As the first norm can be estimated by the second, the open mapping theorem shows that both are equivalent. Note that H s (O) is independent of the particular choice of O . On C0∞ (O) the topology of H s (Σ) is independent of the choice of the Riemannian metric; moreover it coincides with that induced from H s (Rn ) via the coordinate maps: This follows from the fact that, for s = 0, 2, 4, . . . , the spaces H s (Σ) are the domains of powers of the Laplacian, together with interpolation and duality. As a consequence, H s (O) does not depend on the choice of the metric, and its topology is that induced by the Euclidean H s -topology. Finally we define the local Sobolev spaces  s  2 n Hloc (Σ) := {u ∈ D (Σ); ξ 2s |κ ∗ (ϕu)(ξ)| d ξ < ∞ for all coordinate maps s (Σ) Hcomp

κ : U → Rn , U ⊂ Σ, and all ϕ ∈ C0∞ (U)} s := {u ∈ Hloc (Σ); supp u compact}.

We have the following inclusions of sets s s s s (O) ⊂ Hcomp (Σ) ⊂ Hloc (Σ) ⊂ H s (O) ⊂ Hloc (O) Hcomp

for any relatively compact subset O of Σ.

B Microlocal analysis with finite Sobolev regularity The C ∞ -wavefront set W F of a distribution u characterizes the directions in Fourier space which cause the appearance of singularities of u. It does however not specify the strength with which the different directions contribute to the singularities. To give a precise quantitative measure of the strength of singular directions of u the notion of the H s -wavefront set W F s was introduced by Duistermaat & H¨ ormander [16]. It is the mathematical tool which we use in the main part of the paper to characterize the adiabatic vacua of a quantum field on a curved spacetime manifold. To make the paper reasonably self-contained we present the definition of W F s and collect some results of the calculus related to it which are otherwise spread over the literature. They are mainly taken from [16, 20, 27, 30, 44]. All other notions from microlocal analysis which we use can also be found there or, in a short synopsis, in [31]. In the following let X denote an open subset of Rn .

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Definition B.1 Let u ∈ D (X), x0 ∈ X, ξ0 ∈ Rn \ {0}, s ∈ R. We say that u is H s (microlocally) in (x0 , ξ0 ) or that (x0 , ξ0 ) is not in the H s -wavefront set of u ((x0 , ξ0 ) ∈ / W F s (u)) if there is a test function ϕ ∈ C0∞ (X) with ϕ(x0 ) = 0 and an open conic neighborhood Γ of ξ0 in Rn \ {0} such that  2 n ξ 2s |ϕu(ξ)| & d ξ < ∞, (101) Γ

2 1/2

where ξ := (1 + |ξ| )

.

Note that, since ϕu ∈ E  (X), there is for all (x, ξ) ∈ X × Rn \ 0 a sufficiently small s ∈ R such that (x, ξ) ∈ / W F s (u). From the definition the following properties of s W F are immediate: (i) W F s (u) is a local property of u, depending only on an infinitesimal neighborhood of a point x0 , in the following sense: If u ∈ D (X), ϕ ∈ C0∞ (X) with ϕ(x0 ) = 0 then (x0 , ξ0 ) ∈ W F s (u) ⇔ (x0 , ξ0 ) ∈ W F s (ϕu) (ii) W F s (u) is a closed cone in X × (Rn \ {0}), i.e., in particular (x, ξ) ∈ W F s (u) ⇒ (x, λξ) ∈ W F s (u)

∀λ > 0.

(iii) s (X) W F s (u) = ∅ ⇔ u ∈ Hloc

(iv) s (X) : (x, ξ) ∈ W F (u − v) (x, ξ) ∈ W F s (u) ⇔ ∀v ∈ Hloc

(102)

(v) W F s1 (u) ⊂ W F s2 (u) ⊂ W F (u) ∀ s1 ≤ s2 (vi) W F s (u1 + u2 ) ⊂ W F s (u1 ) ∪ W F s (u2 ) (vii) W F (u) =



W F s (u)

s∈R

As an example consider the δ-distribution in D (Rn ). One easily calculates from the criterion of the definition ∅, s < −n/2 s W F (δ) = (103) {(0, ξ); ξ ∈ Rn \ {0}}, s ≥ −n/2. The following proposition gives an important characterization of the H s -wavefront m (X × Rn ) is the set in terms of pseudodifferential operators. Remember that Sρ,δ space of symbols of order m and type ρ, δ (m ∈ R, 0 ≤ δ, ρ ≤ 1), and Lm ρ,δ (X) the corresponding space of pseudodifferential operators on X.

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Proposition B.2 Let u ∈ D (X). Then  W F s (u) = char A = A ∈ L01,0 s Au ∈ Hloc (X)



char A,

1173

(104)

A ∈ Ls1,0 Au ∈ L2loc (X)

where the intersection is taken over all properly supported classical pseudodifferential operators A (having principal symbol a(x, ξ)) and char A := a−1 (0) \ 0 is the characteristic set of A. Also the pseudolocal property of pseudodifferential operators can be stated in a refined way taking into account the finite Sobolev regularity: Proposition B.3 If A ∈ Lm ρ,δ (X) is properly supported, with 0 ≤ δ < ρ ≤ 1, and u ∈ D (X), then W F s−m (Au) ⊂ W F s (u) for all s ∈ R, in particular s−m s A : Hloc (X) → Hloc (X).

From Propositions B.2 and B.3 we can draw the following important conclusions: (i) Since the principal symbol of a pseudodifferential operator is an invariant function on the cotangent bundle T ∗ X we see from (104) that W F s (u) is well defined as a subset of T ∗ X \0, i.e., does not depend on a particular choice of coordinates. By a partition of unity one can therefore define W F s (u) for any paracompact smooth manifold M and u ∈ D (M) as a subset of T ∗ M\0 and all results in this appendix remain valid when replacing X by M. s (ii) If Au ∈ Hloc (X) for some properly supported A ∈ Lm 1,0 (X) then

W F s+m (u) ⊂ char A.

(105)

This follows from Proposition B.2 because, choosing some elliptic B ∈ s+m 0 L−m 1,0 (X), we have BA ∈ L1,0 (X) and, by Proposition B.3, BAu ∈ Hloc (X), s+m and therefore, by (104), W F (u) ⊂ char(BA) = char(A). (iii) If A ∈ L−∞ (X), then W F (Au) = ∅ and hence W F s (Au) = ∅ for all s ∈ R. (iv) If A ∈ Lm ρ,δ (X), 0 ≤ δ < ρ ≤ 1, is a properly supported elliptic pseudodifferential operator, u ∈ D (X), then W F s−m (Au) = W F s (u) for all s ∈ R. This is a consequence of the fact that an elliptic pseudodifferential operator has a parametrix, i.e., there is a properly supported Q ∈ L−m ρ,δ (X) with

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QAu = u + Ru and AQu = u + R u for some R, R ∈ L−∞ (X). Therefore, by Proposition B.3, W F s (u) = W F s (QAu) ⊂ W F s−m (Au) ⊂ W F s (u). The behaviour of W F s (u) for hyperbolic operators (like the Klein-Gordon operator, which plays an important role in this work) is determined by the theorem of propagation of singularities due to Duistermaat & H¨ ormander [16, Thes (X) for A ∈ orem 6.1.1’]. It states in particular that, if u satisfies Au ∈ Hloc m L1,0 (X) with real principal symbol a(x, ξ) which is homogeneous of degree m, then W F s+m−1 (u) consists of complete bicharacteristics of A, i.e., complete integral curves in a−1 (0) ⊂ T ∗ X of the Hamiltonian vector field " n !  ∂a(x, ξ) ∂ ∂a(x, ξ) ∂ − Ha (x, ξ) := . ∂xi ∂ξi ∂ξi ∂xi i=1 The precise statement is as follows: Proposition B.4 Let A ∈ Lm 1,0 (X) be a properly supported pseudodifferential operator with real principal symbol a(x, ξ) which is homogeneous of degree m. If u ∈ D (X) and Au = f it follows for any s ∈ R that W F s+m−1 (u) \ W F s (f ) ⊂ a−1 (0) \ 0 and W F s+m−1 (u) \ W F s (f ) is invariant under the Hamiltonian vector field Ha . It is well known that the wavefront set gives sufficient criteria when two distributions can be pointwise multiplied, composed or restricted to submanifolds. We reconsider these operations from the point of view of finite Sobolev regularity and obtain weaker conditions in terms of W F s . We start with the regularity of the tensor product of two distributions: Proposition B.5 Let X ⊂ Rn , Y ⊂ Rm be open sets and u ∈ D (X), v ∈ D (Y ). Then the tensor product w := u ⊗ v ∈ D (X × Y ) satisfies W F r (w) ⊂ W F s (u) × W F (v) ∪ W F (u) × W F t (v) (supp u × {0}) × W F (v) ∪ W F (u) × (supp v × {0}), r = s + t ∪ (supp u × {0}) × W F t (v) ∪ W F s (u) × (supp v × {0}), r = min{s, t, s + t}. The proof of this proposition can be adapted from the proof of Lemma 11.6.3 in [30]. The pointwise product of two distributions u1 , u2 ∈ D (X) – if it exists – is defined by convolution of Fourier transforms as the distribution v ∈ D (X) such that ∀x ∈ X ∃f ∈ D(X) with f = 1 near x such that for all ξ ∈ Rn  1 2 v(ξ) = u2 (ξ − η) dn η f' u1 (η)f' f' (2π)n/2 Rn

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with absolutely convergent integral. It is clear that for the integral to be absolutely convergent it is sufficient that f' u1 (η) and f' u2 (ξ − η) decay sufficiently fast in the opposite directions η resp. −η, i.e., that u1 and u2 are in Sobolev spaces of sufficiently high order at (x, η) resp. (x, −η). The precise condition is the following: Proposition B.6 Let u1 , u2 ∈ D (X). Suppose that ∀(x, ξ) ∈ T ∗ X \ 0 ∃s1 , s2 ∈ R with s1 + s2 ≥ 0 such that (x, ξ) ∈ / W F s1 (u1 ) and (x, −ξ) ∈ / W F s2 (u2 ). Then the pointwise product u1 u2 exists. For a proof see [37]. Next we consider the restriction of distributions to submanifolds. Let Σ be an (n − 1)-dimensional hypersurface of X (i.e., there exists a C ∞ -embedding ϕ : Σ → X) with conormal bundle N ∗ Σ := {(ϕ(y), ξ) ∈ T ∗ X; y ∈ Σ, ϕ∗ (ξ) = 0}. We can define the restriction uΣ ∈ D (Σ) of u ∈ D (X) to Σ – if it exists – as the mapping f →$ u · (f δΣ ), 1 , where f δΣ : C ∞ (X) → C is the distribution given by (f δΣ )(g) := Σ f g, f ∈ D(Σ). If Σ is locally given by x0 = 0 then f δΣ is locally given by f (x)δ(x0 ), where δ(x0 ) is the delta-function in the x0 -variable. By a consideration analogous to (103) we see that ∅, s < −1/2 W F s (f δΣ ) ⊂ . (106) N ∗ Σ, s ≥ −1/2 We obtain Proposition B.7 Let u ∈ D (X) with W F s (u) ∩ N ∗ Σ = ∅ for some s > 1/2. Then the restriction uΣ of u is a well-defined distribution in D (Σ), and W F r−1/2 (uΣ ) ⊂ ϕ∗ W F r (u) := {(y, ϕ∗ (ξ)) ∈ T ∗ Σ; (ϕ(y), ξ) ∈ W F r (u)} for all r > 1/2. Proof. Let s > 1/2 and W F s (u) ∩ N ∗ Σ = ∅. It follows from (106) and Proposition B.6 that the product u·f δΣ is defined. Suppose that (y, η) ∈ W F r−1/2 (uΣ ) for some r−1/2 r > 1/2. By (102) we have (y, η) ∈ W F (uΣ − w) for each w ∈ Hloc (Σ). Since r−1/2 r the restriction operator Hloc (X) → Hloc (Σ) is onto [44, Ch. I, Theorem 3.5], r there exists a v ∈ Hloc (X) for each w such that w = vΣ . Hence we have for every r (X) v ∈ Hloc (y, η) ∈ W F (uΣ − vΣ ) = W F ((u − v)Σ ) ⊂ ϕ∗ W F (u − v) where we have used the standard result on the wavefront set of a restricted distribution [26, Theorem 2.5.11’]. Applying (102) again we obtain the assertion.


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The proposition can easily be generalized to submanifolds of higher codimension by repeated projection. From Proposition B.5 and B.7 one can get an estimate for the H s -wavefront set of the pointwise product in Proposition B.6 when noticing that u1 u2 is the pull-back of u1 ⊗ u2 under the map ϕ : X → X × X, x → (x, x) and that ϕ∗ (ξ1 , ξ2 ) = ξ1 + ξ2 . This estimate, however, is rather poor and we will not present it here, better information on the regularity of products can be gained e.g. from [30, Theorem 8.3.1 and Theorem 10.2.10]. Proposition B.8 Let X ⊂ Rn , Y ⊂ Rm be open sets, u ∈ C0∞ (Y ) and let K ∈ D (X × Y ) be the kernel of the continuous map K : C0∞ (Y ) → D (X). Then we have for all s ∈ R s W F s (Ku) ⊂ W FX (K) := {(x, ξ) ∈ T ∗ X \ 0; (x, ξ; y, 0) ∈ W F s (K) for some y ∈ Y }. Proof. Assume that (x, ξ; y, 0) ∈ / W F s (K) for some (x, ξ) ∈ T ∗ X \ 0, y ∈ Y . s (X × Y ) and (x, ξ; y, 0) ∈ / By (102) we can write K = K1 + K2 with K1 ∈ Hloc W F (K2 ). Since Ku = K1 u + K2 u and W F (K2 u) ⊂ W FX (K2 ) it follows that s (X), because then it (x, ξ) ∈ / W F (K2 u). It remains to be shown that K1 u ∈ Hloc s s s follows from (102) that (x, ξ) ∈ / W F (Ku), i.e., W F (Ku) ⊂ W FX (K). ∞ To this end we localize K1 with test functions ϕ ∈ C0 (X) and ψ ∈ C0∞ (Y ) such $  that ψ = 1 on supp u and estimate for ϕ(K1 u) = ϕ(K1 ψu) = K1 (x, y)u(y) dm y ∈ E  (X), where K1 (x, y) := ϕ(x)K1 (x, y)ψ(y): 

 2   m ˆ  = d ξ (1 + |ξ| )  d η K1 (ξ, −η)ˆ u(η)   ˆ 1 (ξ, −η)|2 ≤ dn ξ (1 + |ξ|2 )s dm η (1 + |η|2 )t |K  u(θ)|2 dm θ (1 + |θ|2 )−t |ˆ   ˆ  (ξ, −η)|2 = C dn ξ dm η (1 + |ξ|2 )s (1 + |η|2 )t |K 1   ˆ 1 (ξ, −η)|2 ≤ C dn ξ dm η (1 + |ξ|2 + |η|2 )s |K

2  d ξ (1 + |ξ| ) |ϕ(K 1 u)(ξ)| n

2 s



n

2 s

s which is finite since K1 ∈ Hcomp (X ×Y ). The last estimate was obtained by putting t := 0 if s ≥ 0, and t := s if s < 0.


In the next proposition we generalize this result to the case where u is a distribution in E  (Y ). Then Ku – if it exists – is defined as the distribution in D (X) such that, for ϕ ∈ C0∞ (X), Ku, ϕ = K(1 ⊗ u), ϕ ⊗ 1 .

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Proposition B.9 Let X ⊂ Rn , Y ⊂ Rm be open sets, K ∈ D (X × Y ) be the kernel of the continuous map K : C0∞ (Y ) → D (X), u ∈ E  (Y ) and denote W FYs (K) := {(y, η) ∈ T ∗ Y \ 0; (x, 0; y, −η) ∈ W F s (K) for some x ∈ X}. If ∀(y, η) ∈ T ∗ Y \ 0 ∃s1 , s2 ∈ R with s1 + s2 ≥ 0 such that (y, η) ∈ / W FYs1 (K) ∩ W F s2 (u), s then Ku exists. If, in addition, W FY (K) = ∅ and K(Hcomp (Y )) ⊂

(107) s−µ Hloc (X),

then

W F s−µ (Ku) ⊂ W F  (K) ◦ W F s (u) ∪ W FX (K), where W F  (K) := {(x, ξ; y, −η) ∈ T ∗ X × T ∗ Y ; (x, ξ; y, η) ∈ W F (K)} is to be regarded as a relation mapping elements of T ∗ Y to elements in T ∗ X. Proof. For the first part of the statement we only have to check that the product K(1 ⊗ u) exists. Indeed, by Proposition B.5 we have W F s2 (1 ⊗ u) ⊂ (X × {0}) × W F s2 (u) and, because of (107), for no point (y, η) ∈ T ∗ Y \ 0 is (x, 0; y, −η) in W F s1 (K) and at the same time (x, 0; y, η) in W F s2 (1 ⊗ u). Therefore, according to Proposition B.6, the pointwise product K(1 ⊗ u) exists. Given an open conic neighborhood Γ of W F s (u) in T ∗ Y , we can write u = u1 + u2 s with u1 ∈ Hloc (Y ) and W F (u2 ) ⊂ Γ. This is immediate from (102) with the help s−µ (Y ), and of a microlocal partition of unity. By assumption we have Ku1 ∈ Hloc hence, by [29, Theorem 8.2.13], W F s−µ (Ku) ⊂ ⊂

W F (Ku2 ) ⊂ W F  (K) ◦ W F (u2 ) ∪ W FX (K) W F  (K) ◦ Γ ∪ W FX (K).

Since Γ was arbitrary, we obtain W F s−µ (Ku) ⊂ W F  (K) ◦ W F s (u) ∪ W FX (K).
The assumptions in the last proposition are tailored for application to the case that K is the kernel of a Fourier integral operator. Indeed, if K ∈ Iρµ (X × Y, C  ), 1/2 < ρ ≤ 1, where C is locally the graph of a canonical transformation from T ∗ Y \ 0 to T ∗ X \ 0, then W F (K) ⊂ C  [26, Theorem 3.2.6] and s−µ s K(Hcomp (Y )) ⊂ Hloc (X) [28, Cor. 25.3.2] and the proposition applies. For pseudodifferential operators we have C = id and hence we get back the result of Proposition B.3. In the next proposition we give information about the smoothness of the kernel K itself: Proposition B.10 Let K ∈ Iρµ (X × Y, Λ), 1/2 < ρ ≤ 1, where Λ is a closed Lagrangian submanifold of T ∗ (X × Y ) \ 0, and K ∈ D (X × Y ) its kernel. Then W F s (K) ⊂ W F (K) ⊂ Λ, more precisely n+m , W F s (K) = ∅ if s < −µ − 4 n+m and λ ∈ Λ is a non-characteristic point of K. λ ∈ W F s (K) if s ≥ −µ − 4

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K ∈ Iρµ (X × Y, Λ) is said to be non-characteristic at a point λ ∈ Λ if the principal symbol has an inverse (as a symbol) in a conic neighborhood of λ. A proof of the proposition can be found in [16, Theorem 5.4.1].

Acknowledgments We want to thank Stefan Hollands, Fernando Lled´ o, J¨ org Seiler and Ingo Witt for helpful discussions. W.J. is grateful to the DFG for financial support, to Bernd Schmidt for moral support, and to Prof. B.-W. Schulze for the hospitable reception in his “Arbeitsgruppe Partielle Differentialgleichungen und komplexe Analysis” at Potsdam University.

References [1] H. Araki, A lattice of von Neumann algebras associated with the quantum theory of a free Bose field, J. Math. Phys. 4, 1343–1362 (1963). [2] H. Araki and S. Yamagami, On quasi-equivalence of quasifree states of the canonical commutation relations, Publ. RIMS, Kyoto Univ. 18, 283–338 (1982). [3] H. Baumg¨ artel and M. Wollenberg, Causal Nets of Operator Algebras. Mathematical Aspects of Algebraic Quantum Field Theory, Akademie Verlag, Berlin, 1992. [4] R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44, 45–57 (1977) and 46, 215 (1979). [5] N.D. Birrell and P.C.W. Davis, Quantum Fields in Curved Space. Cambridge Univ. Press, Cambridge, 1982. [6] O. Bratteli and R.W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1. Springer Verlag, Berlin, 2nd edition, 1987. [7] R. Brunetti and K. Fredenhagen, Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds, Comm. Math. Phys. 208, 623–661 (2000). [8] R. Brunetti, K. Fredenhagen, and M. K¨ ohler, The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes, Comm. Math. Phys. 180, 633–652 (1996). [9] P.R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Funct. Anal. 12, 401–414 (1973). [10] R. Coifman and Y. Meyer, Au del` a des op´erateurs pseudodiff´erentiels. Ast´erisque 57. Soc. Math. de France, Paris, 1978.

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[11] B.S. DeWitt and R.W. Brehme, Radiation damping in a gravitational field, Ann. Phys. (NY) 9, 220–259 (1960). [12] J. Dieudonn´e, Treatise on Analysis, volume IV. Academic Press, New York, 1972. [13] J. Dimock, Algebras of local observables on a manifold, Comm. Math. Phys. 77, 219–228 (1980). [14] J. Dixmier, Les C ∗ -Alg`ebres et Leurs Repr´esentations. Gauthier-Villars, Paris, deuxi`eme ´edition, 1969. [15] J.J. Duistermaat, Fourier Integral Operators. Birkh¨ auser, Boston, 1996. [16] J.J. Duistermaat and L. H¨ ormander, Fourier integral operators II. Acta Math. 128, 183–269 (1972). [17] Ch.J. Fewster, A general worldline quantum inequality, Class. Quant. Grav. 17, 1897–1911 (2000). [18] S.A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time. Cambridge University Press, Cambridge, 1989. [19] S.A. Fulling, F.J. Narcowich, and R.M. Wald, Singularity structure of the two-point function in quantum field theory in curved spacetime, II, Ann. Phys. (NY) 136, 243–272 (1981). ◦

[20] L. Garding, Singularities in Linear Wave Propagation. Lecture Notes in Mathematics 1241. Springer Verlag, Berlin, 1987. [21] R. Haag, Local Quantum Physics: Fields, Particles, Algebras. Springer Verlag, Berlin, 2nd edition, 1996. [22] R. Haag and D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys. 5, 848–861 (1964). [23] R. Haag, H. Narnhofer, and U. Stein, On quantum field theory in gravitational background, Comm. Math. Phys. 94, 219–238 (1984). [24] S. Hollands, The Hadamard condition for Dirac fields and adiabatic states on Robertson-Walker spacetimes, Comm. Math. Phys. 216, 635–661 (2001). [25] S. Hollands and W. Ruan, The state space of perturbative quantum field theory in curved spacetimes. Preprint gr-qc/0108032, 2001. [26] L. H¨ormander, Fourier integral operators I. Acta Math. 127, 79–183 (1971). [27] L. H¨ormander, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Grundlehren der mathematischen Wissenschaften 274. Springer Verlag, Berlin, 1985.

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[28] L. H¨ormander, The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators. Grundlehren der mathematischen Wissenschaften 275. Springer Verlag, Berlin, 1985. [29] L. H¨ormander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Grundlehren der mathematischen Wissenschaften 256. Springer Verlag, Berlin, 2nd edition, 1990. [30] L. H¨ormander, Lectures on Nonlinear Hyperbolic Differential Equations. Math´ematiques & Applications 26. Springer Verlag, Berlin, 1997. [31] W. Junker, Hadamard states, adiabatic vacua and the construction of physical states for scalar quantum fields on curved spacetime. Rev. Math. Phys 8, 1091–1159 (1996) and 14, 511–517 (2002). [32] B.S. Kay and R.M. Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Phys. Rep. 207, 49–136 (1991). [33] H. Kumano-go, Pseudo-Differential Operators, The MIT Press, Cambridge (Mass.), 1981. [34] P. Leyland, J. Roberts, and D. Testard, Duality for quantum free fields, Preprint, Marseille, 1978. [35] C. L¨ uders and J.E. Roberts, Local quasiequivalence and adiabatic vacuum states, Comm. Math. Phys. 134, 29–63 (1990). [36] J. Manuceau and A. Verbeure, Quasi-free states of the C.C.R.-algebra and Bogoliubov transformations, Comm. Math. Phys. 9, 293–302 (1968). [37] M. Oberguggenberger, Products of distributions, J. reine angew. Math. 365, 1–11 (1986). [38] L. Parker, Quantized fields and particle creation in expanding universes. I, Phys. Rev. 183, 1057–1068 (1969). [39] L. Parker and S.A. Fulling, Adiabatic regularization of the energy-momentum tensor of a quantized field in homogeneous spaces, Phys. Rev. D 9, 341–354 (1974). [40] R.T. Powers and E. Størmer, Free states of the canonical anticommutation relations, Comm. Math. Phys. 16, 1–33 (1970). [41] M.J. Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Comm. Math. Phys. 179, 529–553 (1996). [42] S. Rempel and B.-W. Schulze, Index Theory of Elliptic Boundary Problems, Akademie-Verlag, Berlin, 1982.

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[43] R.T. Seeley, Complex powers of an elliptic operator, Amer. Math. Soc. Proc. Symp. Pure Math. 10, 288–307 (1968). [44] M.E. Taylor, Pseudodifferential Operators, Princeton, New Jersey, 1981.

Princeton University Press,

[45] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publ. Company, Amsterdam, 1978. [46] W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D. 14, 870–892 (1976). [47] R. Verch, Nuclearity, split property, and duality for the Klein-Gordon field in curved spacetime, Lett. Math. Phys. 29, 297–310 (1993). [48] R. Verch, Local definiteness, primarity and quasiequivalence of quasifree Hadamard quantum states in curved spacetime, Comm. Math. Phys 160, 507–536 (1994). [49] R. Verch, Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields on curved spacetime, Rev. Math. Phys. 9, 635–674 (1997). [50] R.M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, The University of Chicago Press, Chicago, 1994. [51] K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften 123. Springer Verlag, Berlin, 6th edition, 1980. Wolfgang Junker Max-Planck-Institut f¨ ur Gravitationsphysik Albert-Einstein-Institut Am M¨ uhlenberg 1 D-14476 Golm Germany email: [email protected] Elmar Schrohe Universit¨ at Potsdam Institut f¨ ur Mathematik Am Neuen Palais 10 D-14415 Potsdam Germany email: [email protected] Communicated by Klaus Fredenhagen submitted 25/03/02, accepted 15/04/02

Ann. Henri Poincar´e 3 (2002) 1183 – 1213 c Birkh¨
auser Verlag, Basel, 2002 1424-0637/02/0601183-31

Annales Henri Poincar´ e

Marginal Fermi Liquid Behaviour in the d = 2 Hubbard Model with Cut-Off V. Mastropietro Abstract. We consider the half-filled Hubbard model with a cut-off forbidding momenta close to the angles of the square shaped Fermi surface. By renormalization group methods we find a convergent expansion for the Schwinger function up to exponentially small temperatures. We prove that the system is not a Fermi liquid, but on the contrary it behaves like a Marginal Fermi liquid, a behaviour observed in the normal phase of high Tc superconductors.

1 Main results 1.1

Motivations

The notion of Fermi liquids, introduced by Landau, refers to a wide class of interacting fermionic systems whose thermodynamic properties (like the specific heat or the resistivity) are qualitatively the same of a gas of non-interacting fermions. While there is an enormous number of metals having Fermi liquid behaviour, in recent times new materials has been found whose properties are qualitatively different. In particular the high-temperature superconducting materials (so anisotropic to be considered essentially bi-dimensional) in their normal phase have a non Fermi liquid behaviour, in striking contrast with previously known superconductors, which are Fermi liquids above the critical temperature. While in Fermi liquids the wave function renormalization Z is Z = 1 + O(λ2 ), where λ is the strength of the interaction, in such metals it was found Z
1+O(λ2 log T ) for temperatures T above the critical temperature, see [VLSAR] (see also [VNS] for a review); metals behaving in this way were called Marginal Fermi liquids. Such results stimulated an intense theoretical research. It was found by a perturbative analysis, see for instance [AGD] or [Sh], that in a system of weakly interacting fermions in d = 2 Z is essentially temperature independent, at least for circular or “almost” circular Fermi surfaces. Despite doubts appeared about the reliability of results obtained by perturbative expansions [A], such results were indeed confirmed recently by rigorous renormalization group methods. It was proved in [FMRT] and [DR] that indeed a weakly interacting Fermi system with a circular Fermi surface is a Fermi liquid, up to exponentially small temperatures. Such result was extended in [BGM] to all possible weakly interacting d = 2 fermionic systems with symmetric, smooth and convex Fermi surfaces, up to exponentially small temperatures. These results cannot be obtained by dimensional power counting arguments as such arguments

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give a bound |Z − 1| ≤ Cλ2 | log T | from which one cannot distinguish Fermi or non Fermi liquid behaviour; for obtaining Z = 1 + O(λ2 ) one has instead to use delicate volume improvements in the integrals expressing Z, based on the geometrical constraints to which the momenta close to the Fermi surface (assumed convex, regular and symmetric) are subjected. As Fermi liquid behaviour is found in systems with symmetric, smooth and convex Fermi surfaces, in order to find non Fermi liquid behaviour one has to relax some of such conditions. It was pointed out, see for instance [VR] and [ZYD], that the presence in the Fermi surface of flat regions in opposite sides could produce a non Fermi liquid behaviour; flat regions are indeed present in the Fermi surfaces of high Tc superconductors [S]. The simplest model exhibiting a Fermi surface with flat pieces is the half-filled Hubbard model, describing a system of spinning d = 2 fermions with local interaction and dispersion relation given by ε(kx , ky ) = cos kx + cos ky . The Fermi surface is the set of momenta such that ε(kx , ky ) = 0 and it is a square with corners (±π, 0) and (0, ±π). However this model has the complicating feature of vanishing Fermi velocity at the points (±π, 0) and (0, ±π), i.e., at the corners of the Fermi surface; this originates to the so-called Van Hove singularities in the density of states. In order to investigate the possible non Fermi liquid behaviour of interacting fermions with a Fermi surface with flat pieces, independently from the presence of Van Hove singularities, one can introduce in the half-filled Hubbard model a cut-off forbidding momenta near the corners of the Fermi surface. The half-filled Hubbard model with cut-off (or the essentially equivalent, but slightly simpler, problem of fermions with the linearized dispersion relation ε(kx , ky ) = |kx | + |ky | − π) has been extensively studied in literature, see for instance [M], [L], [ZYD], [VR], [FSW], [DAD], [FSL]. The cut-off is somewhat artificially introduced but the idea is that the model, at least for same values of the parameters, belongs to the same university class of models with “almost” squared and smooth Fermi surface, like the anisotropic Hubbard models [Sh], the Hubbard model with nearest and next to nearest neighbor interaction [Me], or the half-filled Hubbard model close to half-filling. Aim of this paper is to compute in a rigorous way the asymptotic behaviour of the Schwinger functions of the half-filled Hubbard model with cut-off up to exponentially small temperatures. We will show that such a system is indeed a Marginal Fermi liquid, and our result furnishes indeed the first example rigorously established of such behaviour in d = 2. k +k For our convenience, we will consider new variables k+ = x 2 y and k− = kx −ky so that the dispersion relation of the half-filled Hubbard model is given by 2 ε(k+ , k− ) = 2 cos k+ cos k− and the Fermi surface is the set k+ = ± π2 or k− = ± π2 .

(1.1)

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The model

Given a square [0, L]2 ∈ R2 , the inverse temperature β and the (large) integer M , we introduce in Λ = [0, L]2 × [0, β] a lattice ΛM , whose sites are given by the space-time points x = (x0 , x+ , x− ) with (x+ , x− ) ∈ Z2 and x0 = n0 β/M , n0 = 0, 1, . . . , M − 1. We also consider the set D of space-time momenta k = 2 ± (k0 , k+ , k− ) ≡ (k0 , k), with k± , = 2πn L , (n+ , n− ) ∈ Z , [−L/2] ≤ n± ≤ [L − 1/2]; 1 k0 = 2π β (n0 + 2 ), n0 = 0, 1, . . . , M − 1. With each k ∈ D we associate four Grassε , ε, s ∈ {+, −}; s is the spin. The lattice ΛM is introduced manian variables ψˆk,s only for technical reasons so that the number of Grassmann variables is finite, and eventually the (essentially trivial) limit M → ∞ is taken. We introduce also a linear functional P (dψ) on the Grassmanian algebra generated by the variables ε ψˆk,σ , such that
P (dψ)ψˆk−1 ,s1 ψˆk+2 ,s2 = L2 βδs1 ,s2 δk1 ,k2 gˆ(k1 ) , (1.2) where g(k) is defined by gˆ(k) =

χ(k) , −ik0 + 2 cos k+ cos k−

(1.3)

where χ(k) is a cut-off function χ(k) = H(a20 sin2 k+ )C0−1 (k) + H(a20 sin2 k− )C0−1 (k)  C0−1 (k) = H( k02 + 4 cos2 (k+ ) cos2 (k− )) √ and, if γ > 1 and a0 ≥ 2  1 if |t| < γ −1 H(t) = , 0 if |t| > 1

(1.4)

where

(1.5)

(1.6)

The function C0−1 (k) acts as an ultraviolet cut-off forcing the momenta k to be not too far from the Fermi surface, and k0 not too large; the cut-off on k0 is imposed only for technical convenience and it could be easily removed. The functions H(a20 sin2 k± ) forbid momenta near the corners of the Fermi surface, i.e., the points (±π/2, ±π/2). The Grassmanian field ψxε is defined by 1  ˆ± ±ik·x ± ψx,s = 2 . (1.7) ψk,s e L β k∈D

The “Gaussian measure” P (dψ) has a simple representation in terms of the “Lebesgue Grassmanian measure” Dψ =

∗  k∈D,s=±

+ − dψˆk,s dψˆk,s ,

(1.8)

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defined as the linear functional on the Grassmanian algebra, such that, given a − + monomial Q(ψˆ− , ψˆ+ ) in the variables ψˆk,s , ψˆk,s , its value is 0, except in the case  ∗ − ˆ+ − ˆ+ ˆ ˆ Q(ψ , ψ ) = k,s ψk,s ψk,s , up to a permutation of the variables, in which case ∗ its value is 1. Finally k∈D,s=± means a product over the k such that χ(k) > 0. We define P (dψ) = N −1 Dψ · exp[−

1 L2 β

∗  k∈D,σ=±

+ ˆ− χ−1 (k)(−ik0 + 2 cos k+ cos k− )ψˆk,s ψk,s ] ,

(1.9) ∗ with N is a renormalization constant and again k means a sum over k such that χ(k) > 0. The two point Schwinger function is defined by the following Grassman functional integral  − + P (dψ)e−V(ψ) ψx,s ψy,s  S(x − y) = lim lim , (1.10) L→∞ M→∞ P (dψ)e−V(ψ) where, if we use



dx as a shorthand for V(ψ) = λ




β M



x∈ΛM ,

+ − + − ψx,s ψx,−s ψx,−s . dxψx,s

(1.11)

s

ˆ We call S(k) the Fourier transform of S(x − y).

1.3

Main theorem

Our main results are summarized by the following Theorem, which will be proved in the following sections. Theorem. Given a0 large enough, there exist two positive constants ε and c¯ such π ≤ that, for all |λ| ≤ ε and T ≥ exp{−(¯ c|λ|)−1 }, for all k ∈ D such that 2β 2 3π 2 2 2 2 k0 + 4 cos k+ cos k− ≤ 2β and H(a0 sin k− ) = 1 then (k 2 + 4 cos2 k+ cos2 k− )η(k− ) ˆ S(k) = 0 (1 + λ2 AI (k)) , (1.12) −ik0 + 2 cos k+ cos k− 2 π 2 ≤ k02 + 4 cos2 k+ cos2 k− ) ≤ 3π and for k ∈ D such that 2β 2β and H(a0 sin k+ ) = 1 then (k 2 + 4 cos2 k+ cos2 k− )η(k+ ) ˆ S(k) = 0 (1 + λ2 AII (k)) , (1.13) −ik0 + 2 cos k+ cos k− where |Ai (k)| ≤ c, where c > 0 is a constant, and η(k± ) = a(k± )λ2 + O(λ3 ) is a critical index expressed by a convergent series with a(k± ) ≥ 0 a not identically vanishing smooth function.

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Remarks

The above theorem describes the behaviour of the two point Schwinger function up to exponentially small temperatures, i.e., T ≥ exp{−(¯ c|λ|)−1 }; the constant c¯ is essentially given by the second order terms of the perturbative expansion. A straightforward consequence of (1.12), (1.13) is that the wave function renormalization is Z = 1 + O(λ2 log β), which means that the half-filled Hubbard model with cut-off is a marginal Fermi liquid up to exponentially small temperatures. From (1.12), (1.13) we see that the behaviour of the Schwinger function close to the Fermi surface is anomalous and described by critical indices which are functions of the projection of the momentum on the Fermi surface. Critical indices which are momentum dependent were found for the same model also in [FSL] by heuristic bosonization methods. The presence of the critical indices makes the Schwinger function quite similar to the one for d = 1 interacting spinless fermionic systems, characterized by Luttinger liquid behaviour (see for instance [A]). However an important difference is that the critical exponent η in a Luttinger liquid is a number, while here is a function of the momenta. Another crucial difference ˆ is that in a Luttinger liquid S(k)
gˆ(k)|k|η , with η = aλ2 + O(λ3 ) up to T = 0; hence a Luttinger liquid is a Marginal Fermi liquid for high enough temperatures but not all the marginal Fermi liquids are Luttinger liquids. The paper is organized in the following way. In §2 we implement renormalization group ideas by writing the Grassman integration in (1.10) as the product of many integrations at different scales. The integration of a single scale leads to new effective interactions, and the renormalization consists in subtracting from the kernels of the effective interaction (which are not dimensionally irrelevant) their value computed at the Fermi surface. One obtains an expansion for the Schwinger functions as power series of a set of running couplings functions (depending from the momentum on the Fermi surface and the scale). In §3 we prove that this series is convergent if the running coupling functions are small enough; the convergence radius is finite and temperature independent, and this means that the theory is renormalizable. In the proof of convergence one uses the Gram-Hadamard inequality. In §4 we show that the running coupling functions obey to a recursive set of integral equation, called beta function, and we show that the running coupling functions remain small up to exponentially small temperatures T ≥ exp{−(¯ c|λ|)−1 }. Moreover we show that the wave function renormalization has an anomalous flow, with a non-vanishing exponent (contrary to what happens for instance in the case of circular Fermi surfaces), and this essentially concludes the proof of the Theorem. It would be possible to use our beta function to detect (at least numerically) the main instabilities of the system at very low temperatures. At the moment, this kind of numerical analysis was done for this model only in [ZYD] in the parquet approximations, with no control on higher orders which are simply neglected. Finally in §5 we compare the Marginal Fermi liquid behaviour we find in this model with the Luttinger liquid behaviour, and we discuss briefly what happens in the Hubbard model with cut-off close to half-filling.

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It is very likely that the half-filled Hubbard model with cut-off can work as a paradigm for a large class of systems, in which the Fermi surface is flat or almost flat but there are no Van Hove singularities. Marginal Fermi liquid behaviour can be surely found in the Hubbard model with cut-off and close to half-filling, up to temperatures above the inverse of the radius of curvature of the Fermi surface. Another model in which one could possibly find Marginal Fermi liquid behaviour is the anisotropic Hubbard model introduced in [Sh] with dispersion relation cos k1 + t cos k2 , with t = 1 + ε. Such model has a Fermi surface with no van Hove singularities and four “almost” flat and parallel pieces, and one can c|λ|)}]. Another expect Z = 1 + O(λ2 | log(|ε|)| log β) for β ≤ O(min[ε−1 , exp{(¯ interesting question is the possibility of Marginal Fermi liquid behaviour in the Hubbard model close to half-filling (with no cut-off). At half-filling it is believed Z
1 + O(λ2 log2 β), so a different behaviour with respect to Marginal Fermi liquid behaviour. A renormalization group analysis for this problem was begun in [R], and it was proved the convergence of the series not containing subgraphs with −1 two external lines for T ≥ exp{−(c0 |λ|) 2 }.

2 Renormalization group analysis 2.1

The scale decomposition

As the spin index will play no role in the following analysis (on the contrary it is expected to have an important role at lower temperatures) we simply omit it. The cut-off function χ(k) defined in (1.4) has a support in the k space which is given by four disconnected regions, each one containing only one flat side of the Fermi surface. It is natural then to write each Grassman variable as a sum of four independent Grassman variables, with momentum k having value in one of the four disconnected regions; each field will be labeled by a couple of indices, σ = I, II and ω = ±1, so that each field has spatial momenta with values in the region containing (ωpF , 0) if σ = I or (0, ωpF ) if σ = II. We write the Grassman integration as



P (dψ)F (ψ) =





Pσ,ω (dψ)F (

σ=I,II ω=±1





ψσ,ω ) ,

(2.1)

σ=I,II ω=±1

where F is any monomial, ω = ±1 and


− ˆ+ PI,ω (dψ)ψˆI,ω,k
+ω F,I pF,I ψI,ω
,k
+ω
p 1

= δω,ω
δk
1 ,k
H(a20 sin2 k− )

 , k− ) Cω−1 (k0 , k+  cos k −ik0 + 2ω sin k+ −

(2.2)

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− ˆ+ PII,ω (dψ)ψˆII,ω,k
+ω F,II pF,II ψII,ω
,k
+ω
p 1

= δk
1 ,k
δω,ω
H(a20 sin2 k+ )

 , k+ ) Cω−1 (k0 , k− , (2.3)  cos k −ik0 + 2ω sin k− +

    cos k ) , k− ) = θ(ωk+ + pF )H( k02 + 4 sin2 k+ Cω−1 (k0 , k+ −     cos k ) , k+ ) = θ(ωk− + pF )H( k02 + 4 sin2 k− Cω−1 (k0 , k− +

where

(2.4) (2.5)

and pF,σ is defined such that pF,I = ( π2 , 0) and pF,II = (0, π2 ); moreover pF = π2 and k = k  + ω pF,σ ( k  is the momentum measured from the Fermi surface). It is convenient, for clarify reasons, to start by studying the “free energy” of the model, defined as
1 (2.6) − 2 log P (dψ)e−V , L β where, calling with a slight abuse of notation ψˆσ,ω,k
+ωpF,σ ≡ ψˆσ,ω,k
, V is equal to 

λ



ω1 ,dots,ω4 σ1 ,dots,σ4 =I,II




4  εi (ki + ωi pF,σi ))ψˆσ+1 ,ω1 ,k
ψˆσ+2 ,ω2 ,k
ψˆσ−3 ,ω3 ,k
ψˆσ−4 ,ω4 ,k
, (2.7) dk1 . . . dk4 δ( i=1

1

2

3

4

  where dk = L12 β k and δ(k − k ) = L2 βδk,k
. We will evaluate the Grassman integral (2.6) by a multiscale analysis based on (Wilsonian) renormalization group ideas. The starting point is the following decomposition of the cut-off functions (2.4), (2.5)  H( k02 + 4 cos2 kˆσ sin2 k σ )  0 0   = fk (k0 , k σ , kˆσ ) , (2.8) f¯k ( k02 + 4 cos2 kˆσ sin2 k σ ) ≡ k=−∞

k=−∞

with f¯k (t) = H(γ −k t) − H(γ −k+1 t) is a smooth compact support function, with support γ k−1 ≤ |t| ≤ γ k+1 ; moreover: a) kˆσ = k− if σ = I and kˆσ = k+ if σ = II; kˆσ is the projection of k in the direction parallel to the Fermi surface.   b) k σ = k+ if σ = I and k σ = k− if σ = II; k σ + ωpF,σ is the projection of k in the direction normal to the Fermi surface. For each σ, the function fk (k0 , kσ , kˆσ ) has a support in two regions of thickness O(γ k ) around each flat side of the Fermi surface, at a distance O(γ k ) from it. We

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will assume L = ∞ for simplicity and it follows that there is a hβ = O(log β) such that fk = 0 for k < hβ , while fk is not identically vanishing for k ≥ hβ . The integration of (2.6) will be done iteratively integrating out the fields with momenta closer and closer to the Fermi surface. We will prove by induction that  ) and a sequence of effective it is possible to define a sequence of functions Zh (k¯σ,ω (h) potentials V such that


PI (dψ)PII (dψ)e−V




√ ( Zh ψ (≤h) )

, (2.9)

   )ψ ˆ(≤h)
, Zh (k¯  )ψˆ(≤h)
, ( Zh (k¯I,1 I,1,k I,−1,k I,−1   (≤h) (≤h)   Zh (k¯II,1 )ψˆII,1,k
, Zh (k¯II,−1 )ψˆII,−1,k
)

(2.10)

= e−L

2

PZh ,I (dψ (≤h) )PZh ,II (dψ (≤h) )e−V

βEh

where Eh is a constant and

(h)

√ Zh ψˆ(≤h) equal to

and PZh ,σ (dψ (≤h) ) is the fermionic integration with propagator (≤h)  gσ,ω (k )

−1 (k0 , kσ , kˆσ ) H(a20 sin2 kˆσ )Ch,ω 1 = θ(ωk σ + ωpF )  ) Zh (k¯σ,ω −ik0 + 2ω cos kˆσ sin k σ

with −1 Ch,ω (k0 , k σ , kˆσ ) =

h 

fk (k0 , k σ , kˆσ ).

(2.11)

(2.12)

k=−∞

The θ-function in (2.11) can be omitted by the definition of the variables k σ .  We define k¯σ,ω = ( πβ , ωpF , k− ) if σ = I and k¯σ,ω = ( πβ , k+ , ωpF ) if σ = II;   moreover k¯σ,ω = ( πβ , 0, k− ) if σ = I and k¯σ,ω = ( πβ , k+ , 0) if σ = II; moreover we π   ¯ ¯ call kσ,ω = (− β , 0, k− ) if σ = I and kσ,ω = (− πβ , k+ , 0) if σ = II. If ε = ± ∞     V (h) (ψ ≤h ) = n=1 ω1 ,...,ω2n σ1 ,...,σ2n ε1 ,...,εn




dk1



. . . dk2n δ(

i

εi (ki

+ ωi pF,σi ))

2n 

i=1



(≤h)ε ψˆσi ,ωi ,ki
i

ˆ (h) (k , . . . , k W 1 2n−1 ) , (2.13) 2n

where h h ˆ 2n ˆ 2n W (k1 . . . k2n−1 ) = W (k1 + ω1  pF,σ1 . . . k2n−1 + ω2n−1 pF,σ2n−1 ) h ˆ 2n (k1 . . . k2n−1 ) . (2.14) =W

Vol. 3, 2002

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Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off

1191

The renormalization procedure

Let us show that (2.9) is true for h − 1, assuming that it is true for h. We define an L operator acting linearly on the kernels of the effective potential (2.13): ˆ (h) = 0 if n ≥ 2 1) LW 2n 2) If n = 1 1 h ¯ ˆ h (k¯  ) + sin k¯ ∂σ W ˆ h (k¯  ) , ˆ h (k¯  )] + k0 ∂k0 W [W (k ) + W 2 σ,ω 2 σ,ω σ 2 σ,ω 2 2 σ,ω (2.15) where ∂k0 means the discrete derivative and ∂σ = ∂k+ is σ = I and ∂σ = ∂k− ˆ h (k¯  ) + W ˆ h (k¯  )] = 0. is σ = II. We will prove in §4 that [W 2 σ,ω 2 σ,ω 3) If n = 2 ˆ 4h (k1 , k2 , k3 ) = W ˆ 4h (k¯σ ,ω , k¯σ ,ω , k¯σ ,ω ) . LW (2.16) 1 1 2 2 3 3 ˆ h (k ) = LW 2

    ˆ 2h (k¯ω,σ ˆ 2h (k¯ω,σ Calling ∂0 W ) = −iah (k¯ω,σ ), ∂σ W ) = 2ω cos kˆσ zh (k¯ω,σ ) and

ˆ 4h (k¯σ ,ω , k¯σ ,ω , k¯σ ,ω ) , lh (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 ) = W 1 1 2 2 3 3

(2.17)

we can write 
 +(≤h) −(≤h) h   )2ω cos kˆσ sin k σ − ik0 ah (k¯ω,σ )]ψˆk
,σ,ω ψˆk
,σ,ω + dk [zh (k¯ω,σ LV = {ω},{σ}

σ=I,II




+(≤h) +(≤h) −(≤h) −(≤h) dk1 . . . dk4 lh (k¯ω 1 ,σ1 , k¯ω 2 ,σ2 , k¯ω 3 ,σ3 )ψˆk
,σ1 ,ω1 ψˆk
,σ2 ,ω2 ψˆk
,σ3 ,ω3 ψˆk
,σ4 ,ω4 1 2 3 4   δ( εi (ki + pF,σi )) . (2.18) i




We write the right-hand side of (2.9) as
√ √ (h) (≤h) )−RV (h) ( Zh ψ (≤h) ) PI,Zh (dψ (≤h) ) PII,Zh (dψ (≤h) )e−LV ( Zh ψ

(2.19)

with R = 1 − L.

2.3

Remark 1.

The non-trivial action of R on the kernel with n = 2 can be written as   ˆ h (k , k , k ) = [W ˆ h (k , k , k ) − W ˆ h (k¯  RW 4 1 2 3 4 1 2 3 4 σ1 ,ω1 , k2 , k3 )] ˆ h (k¯  ˆ h (k¯  , k , k ) − W , k¯ , k )] + [W 4

σ1 ,ω1

2

3

4

σ1 ,ω1

σ2 ,ω2

3

ˆ 4h (k¯σ ,ω , k¯σ ,ω , k¯σ ,ω )] . (2.20) ˆ 4h (k¯σ ,ω , k¯σ ,ω , k3 ) − W + [W 1 1 2 2 1 1 2 2 3 3

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The first addend can be written as, if σ1 = I (say), in the limit L → ∞ π (k0,1 − ) β




1

ˆ h ( π + t(k0,1 − π ), k  , k−,1 ; k , k ) dt∂k0,1 W 4 +,1 2 3 β β 0
1  ˆ h (0, tk  , k−,1 ; k , k ) .
W dt∂k+,1 +k+,1 4 +,1 2 3

(2.21)

0




 The factors k0,1 − π/β and k+,1 are O(γ h ), for the compact support properties +(≤h)

of the propagator associated to ψI,ω1 ,k
, with h ≤ h, while the derivatives are 1




dimensionally O(γ −h−1 ); hence the effect of R is to produce a factor γ h −h−1 < 1. Similar considerations can be done for the other addenda and for the action of R on the n = 1 terms.

Remark 2. From (2.16) we see that the effect of the L operation is to replace in W2h (k) the momentum k with its projection on the closest flat side of the Fermi surface. Hence the fact that the propagator is singular over an extended region (the Fermi surface) and not simply in a point has the effect that the renormalization point cannot be fixed but it must be left moving on the Fermi surface.

2.4

The anomalous integration

In order to integrate the field ψ (h) we can write



√ (h) √ (≤h) )−RV (h) ( Zh ψ (≤h) ) PI,Zh (dψ (≤h) ) PII,Zh (dψ (≤h) )e−LV ( Zh ψ

√ √ (≤h) ˜h )−RV (h) ( Zh ψ (≤h) ) = PI,Zh−1 (dψ (≤h) ) PII,Zh−1 (dψ (≤h) )e−LV ( Zh ψ (2.22)

where Pσ,Zh−1 (dψ (≤h) ) is the fermionic integration with propagator H(a20 sin2 kˆσ )Ch−1 (k0 , k σ , kˆσ ) Zh−1 (k ) −ik0 + 2ω cos kˆσ sin k  1

(2.23)

σ

and   Zh−1 (k ) = Zh (k¯σ,ω )[1 + H(a20 sin2 kˆσ )Ch−1 (k0 , k σ , kˆσ )ah (k¯σ,ω )] .

(2.24)

Moreover LV˜ h = LV h −


σ=I,II

+(≤h)

−(≤h)

 )[2ω cos kˆσ sin k σ − ik0 ]ψˆk
,σ,ω ψˆk
,σ,ω . (2.25) dk zh (k¯σ,ω

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Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off

1193

We rescale the fields by rewriting the right-hand side of (2.22) as
√ √ (≤h) ˆh (≤h) )−RV (h) ( Zh−1 ψ (≤h) ) ) PII,Zh−1 (dψ (≤h) )e−LV ( Zh−1 ψ , PI,Zh−1 (dψ (2.26)

where ˆh
LV =  dk [δh,ω (k¯σ,ω )2ω cos kˆσ sin kσ )]ψˆk+
,σ,ω ψˆk−
,σ,ω + σ=I,II






dk1 . . . dk4

σ1 ,...,σ4 =I,II

 λh (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 )ψˆk+
,σ1 ,ω1 ψˆk+
,σ2 ,ω2 ψˆk−
,σ3 ,ω3 ψˆk−
,σ4 ,ω4 δ( εi (ki + pF,σi ) 1

2

3

4

i

(2.27) and

 Zh (k¯σ,ω )  )) (2.28) (z (k¯  ) − ah (k¯σ,ω  ) h σ,ω Zh−1 (k¯σ,ω 4  Zh (k¯σ i ,ωi )    λh (k¯σ1 ,ω1 , k¯σ2 ,ω2 , k¯σ3 ,ω3 ) = [ ]lh (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 ) . ¯ Z ( k ) h−1 σi ,ωi i=1  )= δh (k¯ω,σ

We will call δh and λh running coupling functions; the above procedure allow to write a recursive equation for them, see §5. Then we write



(≤h−1) (≤h−1) (h) ) PII,Zh−1 (dψ ) PI,Zh−1 (dψ ) PII,Zh−1 (dψ (h) ) PI,Zh−1 (dψ ˆ (h) (

e−LV

√

Zh−1 ψ (≤h) )−RV (h) (

√

Zh−1 ψ (≤h) )

(2.29)

and the propagator of Pσ,Zh−1 (dψ) is (h) (k ) = H(a20 sin2 kˆσ ) gˆω,σ

1 f˜h (k0 , k σ , kˆσ )  ¯ Zh−1 (kω,σ ) −ik0 + 2ω cos kˆσ sin k σ

(2.30)

and  −1 C −1 (k0 , k σ , kˆσ ) Ch−1 (k0 , k σ , kˆσ )  − f˜h (k0 , kσ , kˆσ ) = Zh−1 (k¯ω,σ )[ h ]  ) Zh−1 (k ) Zh−1 (k¯σ,ω

(2.31)

with H(a20 sin2 kˆσ )f˜h (k0 , k σ , kˆσ ) having the same support that H(a20 sin2 kˆσ ) fh (k0 , k σ , kˆσ ). We integrate then the field ψ (h) and we get e−L

2


βEh−1

PI,Zh−1 (dψ (≤h−1) )




PII,Zh−1 (dψ (≤h−1) )e−V

(h−1)

(

√

Zh−1 ψ (≤h−1) )

(2.32) and the procedure can be iterated.

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We will see in the following section that if the running coupling functions are small  Zk−1 (k¯σ,ω )  )| ≤ 2|λ| sup sup ≤ e2|λ| sup sup |δk (k¯σ,ω  ¯ Z ( k )

¯ ¯ k≥h k k≥h k k σ,ω σ,ω σ,ω sup

¯
k≥h k σ

sup

¯
¯
,k σ2 ,ω2 ,kσ3 ,ω3 1 ,ω1

|λk (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 )| ≤ 2|λ| ,

(2.33)

then the effective potential is given by a convergent series. In §4 we will show that up to exponentially small temperatures this is indeed true.

3 Analyticity of the effective potential 3.1

Coordinate representation

ε It is convenient to perform bounds to introduce the variables ψx,ω,σ . We define ε iεω pF,σ  x ˜ε ψx,ω,σ , or more explicitly the fields ψx,ω,σ = e ε ε ψx,ω,I = eiεωpF x1 ψ˜x,ω,I

ε ε ψx,ω,II = eiεωpF x2 ψ˜x,ω,II

(3.1)

and the propagators of such fields is (h) g˜ω,σ (x − y) =




dk

 ˆ 2ˆ 2 ˜ 1 −ik
(x−y) H(a0 sin kσ )fh (k0 , k σ , kσ ) e  ) Zh−1 (k¯ω,σ −ik0 + ω2 sin k σ cos kˆσ

(3.2)

It is easy to prove, by integration by parts, that for any integer N , for L → ∞ (h)

|∂xn00 ∂xn++ ∂xn−− g˜I,ω (x − y)| ≤

(h)

|∂xn00 ∂xn++ ∂xn−− g˜II,ω (x − y)| ≤ where d(x0 ) =

β π

1+

[γ h |d(x0

Cn0 ,n+ ,n− ,N γ h(1+n0 +n+ ) − y0 )| + γ h |x+ − y+ | + |x− − y− |]N (3.3)

Cn0 ,n+ ,n− ,N γ h(1+n0 +n− ) 1 + [γ h |d(x0 − y0 )| + |x+ − y+ | + γ h |x− − y− |]N (3.4)

sin xβ0 π .

Proof. The above formula can be derived by integration by parts; note that, if for instance σ = I ∂k−

1 1  =  cos k  cos k )2 2ω sin k+ sin k− −ik0 + 2ω sin k+ (−ik0 + 2ω sin k+ − −

(3.5)

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which is O(γ −h ); in the same way the n-th derivative with respect to k− is still n O(γ −h ). On the other hand ∂kn00 ∂k++ is bounded by γ −h−n0 h−n+ h ; finally the integration gives a volume factor γ 2h . We define h W2n (x1 , . . . , x2n ) = 2n   1 −i εr k
r xr ˆ h r=1 W2n (k1 , . . . k2n−1 )δ( e εi (ki + ωi pF,σi )) . (3.6) 2 2n (L β)

i k1 ,...,k2n

Hence (2.13) can be written as V (h) (ψ ≤h ) =

∞ 








dx1 . . . dx2n

n=1 ω1 ,...,ω2n σ1 ,...,σ2n ε1 ,...,ε2n

2n 

 i ψ˜σ(≤h)ε i ,xi ,ωi

(h)

W2n (x1 , . . . , x2n ) .

i=1

(3.7) We now discuss the action of the operator L and R = 1 − L on the effective potential in the x-space representation. Noting that from (3.6), if ε1 = ε2 = −ε3 = −ε4 = + W4h (x1 , x2 , x3 , x4 ) ˜ 4h (x1 − x4 , x2 − x4 , x3 − x4 ) , (3.8) = eix4 (ω1 pF,σ1 +ω2 pF,σ2 −ω3 pF,σ3 −ω4 pF,σ4 ) W we can write the action of R (2.16) as R


 4

dxi

i=1

4 

˜ 4h ψ˜xεii ,σi ,ωi eix4 (ω1 pF,σ1 +ω2 pF,σ2 −ω3 pF,σ3 −ω4 pF,σ4 ) W

i=1

=


 4 i=1

dxi

4 

ψ˜xεii ,σi ,ωi eix4 (ω1 pF,σ1 +ω2 pF,σ2 −ω3 pF,σ3 −ω4 pF,σ4 )

i=1

˜ 4h (x1 −x4 , x2 −x4 , x2 −x4 )−δ(x0,1 −x0,4 )δ(x0,2 −x0,4 )δ(x0,3 −x0,4 )δ(x [W σ1 ,1 −xσ1 ,4 )
˜ (h) (t1 , t2 , t3 )] , δ(xσ2 ,2 −xσ2 ,4 )δ(xσ3 ,3 −xσ3 ,4 ) dt0,1 dt0,2 dt0,3 dt1,σ1 dt2,σ2 dt3,σ3 W 4 (3.9)  β  ˆ where dx = M x∈Λ and ti = (t0,i , tσ,i , tσ,i ) where tI,i = t+,i ; tII,i = t−,i and tˆI,i = t−,i ; tˆII,i = t+,i . On the other hand we can equivalently write the R operation as acting on the fields, and such two representations of the R operation will be used in the

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following. It holds that, by simply integrating the deltas in (3.9) R


 4 1=1

=

 dxi W4h ({x})ψ˜x+1 ,σ1 ,ω1 ψ˜x+2 ,σ2 ,ω2 ψ˜x−3 ,σ3 ,ω3 ψ˜x−4 ,σ4 ,ω4


 4 1=1



dxi W4h ({x}) Dx+1 ,¯x4,σ1 ,σ1 ,ω1 ψ˜x+2 ,σ2 ,ω2 ψ˜x−3 ,σ3 ,ω3 ψ˜x−4 ,σ4 ,ω4 + ˜− ˜− + ψ˜x+ ¯ 4 ,σ1 ,ω1 Dx2 ,¯ x4,σ2 ,σ2 ,ω2 ψx3 ,σ3 ,ω3 ψx4 ,σ4 ,ω4 +

 − ˜+ ˜− ψ˜x+ ¯ 4 ,σ1 ,ω1 ψx ¯ 4 ,σ2 ,ω2 Dx3 ,¯ x4,σ3 ,σ3 ,ω3 ψx4 ,σ4 ,ω4 , (3.10)

¯ 4,σi = (x0,4 , x+,i , x−,4 ) if σi = II; ¯ 4,σi = (x0,4 , x+,4 , x−,i ) if σi = I and x where x moreover Dxε i ,¯x4,σi ,σi ,ωi = ψ˜xε i ,σi ,ωi − ψ˜xε¯ 4,σi ,σi ,ωi . (3.11) This means that the action of the renormalization operator R can be seen as the ε(≤h) replacement of a ψ ε(≤h) field with a Dxi ,¯x4,σi ,σi ,ωi field and some of the other ε(≤h) ψ˜(≤h) fields are “translated” in the localization point. The field Dx ,¯x ,σ ,ω is i

4,σi

i

i

¯ 4 . We can write Dε(≤h) as sum antiperiodic in the time components of xi , and x of two terms (if σi = I for instance): ε(≤h)

Dxi ,¯x4,σi ,σi ,ωi = [ψ˜xε i ,I − ψ˜xε 0,4 ,x+,i ,x−,i ,I ]+[ψ˜xε 0,4 ,x+,i ,x−,i ,I − ψ˜xε 0,4 ,x+,4 ,x−,i ,I ] (3.12) and the second addend can be written as, for L → ∞ ψ˜xε 0,4 ,x+,i ,x−,i ,I − ψ˜xε 0,4 ,x+,4 ,x−,i ,I = (x+,i − x+,4 )




1 0

ε dt∂x+ ψ˜I,ω,x 0,4 ,x+,i −t(x+,i −x+,4 ),x−,i

(3.13)

and x+,i − t(x+,i − x+,4 ) ≡ x+,i,4 (t) is called interpolated point. This means that it is dimensionally equivalent to the product of the “zero” (x+,i − x+,4 ) and the derivative of the field, so that the bound of its contraction
with another field variable on a scale h < h will produce a “gain” γ −(h−h ) , see (≤h)σ (3.3), (3.4), with respect to the contraction of ψ˜x,ω . Similar considerations can be repeated for the first addend of (3.3); some care has to be done as β is finite, and we refer §3.5 of [BM]. If there are two external lines.
R

dx1 dx2 W2h (x1 , x2 )ψ˜x+1 ,σ1 ,ω1 ψ˜x−2 ,σ2 ,ω2
= dx1 dx2 W2h (x1 , x2 )ψ˜x+1 ,σ1 ,ω1 Tx−2 ,¯x1,σ2 ,σ2 ,ω2

(3.14)

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where Txε2 ,¯x1,σ2 ,σ2 ,ω = ψ˜xε 2 ,σ2 ,ω − ψ˜xε¯ 1,σ2 ,ω,σ2 − (x0,2 − x0,1 )∂0 ψ˜xε¯ 1,σ2 ,ω,σ2 − (xσ,2 − x1,σ )∂σ ψ˜x− ¯ 1,σ2 ,ω,σ2

(3.15)

and ∂σ = ∂x+ if σ = I and ∂σ = ∂x− if σ = II. In this case the “gain” produced
by the R operation is γ −2(h−h ) . We can write the local part of the effective potential (2.18) in the following way 
+ xσ,2 [ωδh,ω ((ˆ x1 − x ˆ2 )σ1 )eix2 (ω1 pF,σ1 −ω2 pF,σ2 ) ψ˜ω,σ ∂ ψ˜− x1,σ2 dx1 d˜ LV h = 1 ;x1 σ ω,σ2 ;¯ σ1 =σ2

+




xσ1 ,1 d˜ xσ2 ,2 d˜ xσ3 ,3 λh;ω1 ,...ω4 ((ˆ x1 − x ˆ4 )σ1 ,(ˆ x2 − x ˆ4 )σ1 ,(ˆ x3 − x ˆ4 )σ3 ) dx4 d˜

σ1 ,...,σ4 =I,II

˜+ ˜− ˜− eix4 (ω1 pF,σ1 +ω2 pF,σ2 −ω3 pF,σ3 −ω4 pF,σ4 ) ψ˜x+ ¯ 4,σ1 ,σ1 ,ω1 ψx ¯ 4,σ2 ,σ2 ,ω2 ψx ¯ 4,σ3 ,σ3 ,ω3 ψx4 ,σ4 ,ω4 (3.16) where δh (x) is the Fourier transform of δh (kˆσ )2 cos kˆσ with respect to kˆσ and xi − x ˆj )σ = x+,i − x+,j if σ = II; moreover (ˆ xi − xˆj )σ = x−,i − x−,j if σ = I and (ˆ x ˜σi = x− if σ = I and x ˜σi = x+ if σ = II.

3.2

Tree expansion

By using iteratively the “single scale expansion” we can write the effective potential V (h) (ψ (≤h) ), for h ≤ 0, in terms of a tree expansion. For a tutorial introduction to the tree formalism we will refer to the review [GM].

v r

v0

h

h+1

hv

We need some definitions and notations.

−1 0

+1

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V. Mastropietro

Ann. Henri Poincar´e

1) Let us consider the family of trees which can be constructed by joining a point r, the root, with an ordered set of n ≥ 1 points, the endpoints of the unlabeled tree (see Fig. 1), so that r is not a branching point. n will be called the order of the unlabeled tree and the branching points will be called the non-trivial vertices. The unlabeled trees are partially ordered from the root to the endpoints in the natural way; we shall use the symbol < to denote the partial order. Two unlabeled trees are identified if they can be superposed by a suitable continuous deformation, so that the endpoints with the same index coincide. It is then easy to see that the number of unlabeled trees with n endpoints is bounded by 4n . We shall consider also the labeled trees (to be called simply trees in the following); they are defined by associating some labels with the unlabeled trees, as explained in the following items. 2) We associate a label h ≤ −1 with the root and we denote Th,n the corresponding set of labeled trees with n endpoints. Moreover, we introduce a family of vertical lines, labeled by an integer taking values in [h, 1], and we represent any tree τ ∈ Th,n so that, if v is an endpoint or a non-trivial vertex, it is contained in a vertical line with index hv > h, to be called the scale of v, while the root is on the line with index h. There is the constraint that, if v is an endpoint, hv > h + 1. The tree will intersect in general the vertical lines in set of points different from the root, the endpoints and the non-trivial vertices; these points will be called trivial vertices. The set of the vertices of τ will be the union of the endpoints, the trivial vertices and the non-trivial vertices. Note that, if v1 and v2 are two vertices and v1 < v2 , then hv1 < hv2 . We will call sv the number of subtrees coming out from v. Moreover, there is only one vertex immediately following the root, which will be denoted v0 and cannot be an endpoint; its scale is h + 1. 3) To each endpoint of scale +1 we associate V (1.11). With each endpoint v of scale hv ≤ 0 we associate one of the two terms appearing in (3.16), with coupling λhv −1 or δhv −1 . Moreover, we impose the constraint that, if v is an endpoint and hv ≤ 0, hv = hv
+ 1, if v  is the non-trivial vertex immediately preceding v. 4) We introduce a field label f to distinguish the field variables appearing in the terms associated with the endpoints as in item 3); the set of field labels associated with the endpoint v will be called Iv . Analogously, if v is not an endpoint, we shall call Iv the set of field labels associated with the endpoints following the vertex v; x(f ), ε(f ) and ω(f ) will denote the space-time point, the ε index and the ω index, respectively, of the field variable with label f . If hv ≤ 0, one of the field variables belonging to Iv carries also a derivative ∂σ if the corresponding local term is of type δ, see (3.16). Hence we can associate with each field label f an integer m(f ) ∈ {0, 1}, denoting the order of the derivative.

Vol. 3, 2002

Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off

1199

If h ≤ −1, the effective potential can be written in the following way: V (h) (ψ (≤h) ) + LβEh+1 =

∞  

V (h) (τ, ψ (≤h) ) ,

(3.17)

n=1 τ ∈Th,n

where, if v0 is the first vertex of τ and τ1 , . . . , τs (s = sv0 ) are the subtrees of τ with root v0 , V (h) (τ, ψ (≤h) ) is defined inductively by the relation V (h) (τ, ψ (≤h) ) =

(−1)s+1 T ¯ (h+1) Eh+1 [V (τ1 , ψ (≤h+1) ); . . . ; V¯ (h+1) (τs , ψ (≤h+1) )] , (3.18) s!

and V¯ (h+1) (τi , ψ (≤h+1) ) a) is equal to RV (h+1) (τi , ψ (≤h+1) ) if the subtree τi is not trivial, with R defined as acting on kernels according to (3.9) and its analogous for n = 1; b) if τi is trivial and h < −1, it is equal to LV (h+1) (ψ (≤h+1) ) (3.16) or, if h = −1, to V. T Eh+1 denotes the truncated expectation with respect to the measure Pσ,Zh (dψ (h+1) ), that is

T (X1 ; . . . ; Xp ) ≡ Eh+1

p

∂ log ∂λ1 . . . ∂λp






   Pσ,Zh (dψ (h+1) )eλ1 X1 +···λp Xp   σ=I,II

σ=I,II



. λi =0

(3.19)

We write (3.18) in a more explicit way. If h = −1, the right-hand side of (3.18) can be written in the following way. Given τ ∈ T−1,n , there are n endpoints of scale 1 and only another one vertex, v0 , of scale 0; let us call v1 , . . . , vn the endpoints. We choose, in any set Ivi , a subset Qvi and we define Pv0 = ∪i Qvi . We have  V (−1) (τ, ψ (≤−1) ) = V (−1) (τ, Pv0 ) , (3.20) V (−1) (τ, Pv0 ) = (0)

Kτ,Pv (xv0 ) = 0




Pv0 (0) dxv0 ψ˜≤−1 (Pv0 )Kτ,Pv (xv0 ) , 0

n  1 T ˜(0) E0 [ψ (Pv1 \Qv1 ), . . . , ψ˜(0) (Pvn \Qvn )] Kv(1) (xvi ) , i n! i=1

(3.21) (3.22)

where we use the definitions ψ˜(≤h) (Pv ) =

 f ∈Pv

m(f ) (≤h)ε(f ) ∂ˆσ(f ) ψ˜x(f )

,

h ≤ −1 ,

(3.23)

1200

V. Mastropietro



ψ˜(0) (Pv ) =

f ∈Pv

Kv(1) (xvi ) i

=e

i

 f ∈Iv

i

Ann. Henri Poincar´e

(0)σ(f ) ψ˜x(f ) ,

εf  x(f )ω(f ) pF,σ(f )

λ

(3.24)

xvi = x

(3.25)

It is not hard to see that, by iterating the previous procedure, one gets for V (h) (τ, ψ (≤h) ), for any τ ∈ Th,n , the representation described below. We associate with any vertex v of the tree a subset Pv of Iv , the external fields of v. These subsets must satisfy various constraints. First of all, if v is not an endpoint and v1 , . . . , vsv are the vertices immediately following it, then Pv ⊂ ∪i Pvi ; if v is an endpoint, Pv = Iv . We shall denote Qvi the intersection of Pv and Pvi ; this definition implies that Pv = ∪i Qvi . The subsets Pvi \Qvi , whose union Iv will be made, by definition, of the internal fields of v, have to be non-empty, if sv > 1. Moreover, we associate with any f ∈ Iv a scale label h(f ) = hv . Given τ ∈ Th,n , there are many possible choices of the subsets Pv , v ∈ τ , compatible with all the constraints; we shall denote Pτ the family of all these choices and P the elements of Pτ . Then we can write V (h) (τ, ψ (≤h) ) =



V (h) (τ, P) .

(3.26)

P∈Pτ

V (h) (τ, P) can be represented as V (h) (τ, P) =




(h+1) dxv0 ψ˜(≤h) (Pv0 )Kτ,P (xv0 ) ,

(3.27)

(h+1)

with Kτ,P (xv0 ) defined inductively (recall that hv0 = h + 1) by the equation, valid for any v ∈ τ which is not an endpoint, (h ) Kτ,Pv (xv )

sv 1  = [K (hv +1) (xvi )] EhTv [ψ˜(hv ) (Pv1 \Qv1 ), . . . , ψ˜(hv ) (Pvsv \Qvsv )] , sv ! i=1 vi

(3.28) where ψ˜(hv ) (Pv ) is defined as in (3.23), with (hv ) in place of (≤ hv ), if hv ≤ −1, while, if hv = 0, it is defined as in (3.24). (1)

Moreover, if v is an endpoint and hv = 0, Kv (xv ) is given by (3.25), otherwise, see (3.16) Kv(hv ) (xv ) =

 lhv −1 (x1 , x2 , x3 , x4 ) if v is of type λ, d

x1 , x2 ) hv −1 (

if v is of type z,

(3.29)

Vol. 3, 2002

Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off

1201

where lhv −1 (x1 , x2 , x3 , x4 ) = eix4 (ε1 ω1 pF,σ1 +ε2 ω2 pF,σ2 −ω3 pF,σ3 −ε4 ω4 pF,σ4 ) λhv −1,ω1 ,...,ω4 ((ˆ x1 − x ˆ4 )σ1 , (ˆ x2 − x ˆ4 )σ2 , (ˆ x3 − xˆ4 )σ3 ) dhv −1,ω (x1 , x2 ) = eix2 (ω1 pF,σ1 −ε2 ω2 pF,σ2 ) δhv −1 ((ˆ x1 − x ˆ2 )σ1 ) If v is not an endpoint, (h +1)

v Kv(hi v +1) (xvi ) = RKτi ,P (i) ,Ω(i) (xvi ) ,

(3.30)

where τi is the subtree of τ starting from v and passing through vi (hence with root the vertex immediately preceding v), P(i) and is the restrictions to τi of P. The action of R is defined using the representation (3.9) of the R operation. (3.26) is not the final form of our expansion, since we further decompose V (h) (τ, P), by using the following representation of the truncated expectation in the right-hand side of (3.28). Let us put s = sv , Pi ≡ Pvi \Qvi ; moreover we order in an arbitrary way the sets Pi± ≡ {f ∈ Pi , σ(f ) = ±}, we call fij± their elements and − + we define x(i) = ∪f ∈P − x(f ), y(i) = ∪f ∈P + x(f ), xij = x(fi,j ), yij = x(fi,j ). Note i i s  s − + that i=1 |Pi | = i=1 |Pi | ≡ n, otherwise the truncated expectation vanishes. A couple l ≡ (fij− , fi+
j
) ≡ (fl− , fl+ ) will be called a line joining the fields with labels fij− , fi+
j
and sector indices ωl− = ω(fl− ), ωl+ = ω(fl+ ) and connecting the points xl ≡ xi,j and yl ≡ yi
j
, the endpoints of l. Moreover, we shall put ml = m(fl− ) + m(fl+ ) and, if ωl− = ωl+ , ωl ≡ ωl− = ωl+ . A similar definition is repeated for σ. Then, it is well known (see [Le], [BM], [GM] for example) that, up to a sign, if s > 1, EhT (ψ˜(h) (P1 ), . . . , ψ˜(h) (Ps ))
  (h) = g˜ω− ,σ− (xl − yl )δω− ,ω+ δσ− ,σ+ dPT (t) det Gh,T (t) , (3.31) T

l∈T

l

l

l

l

l

l

where T is a set of lines forming an anchored tree graph between the clusters of points x(i) ∪ y(i) , that is T is a set of lines, which becomes a tree graph if one identifies all the points in the same cluster. Moreover t = {ti,i
∈ [0, 1], 1 ≤ i, i ≤ s}, dPT (t) is a probability measure with support on a set of t such that ti,i
= ui · ui
for some family of vectors ui ∈ Rs of unit norm. Finally Gh,T (t) is a (n − s + 1) × (n − s + 1) matrix, whose elements are given by Gh,T ij,i
j
= −

+

m(f ) m(f ) (h) ti,i
∂ˆσ(f −ij) ∂ˆσ(f +ij) g˜ωl (xij − yi
j
)δω− ,ω+ δσ− ,σ+ with (fij− , fi+
j
) not belonging to T . ij

ij

l

l

l

l

In the following we shall use (3.31) even for s = 1, when T is empty, by interpreting the right-hand side as equal to 1, if |P1 | = 0, otherwise as equal to det Gh = EhT (ψ˜(h) (P1 )).

1202

V. Mastropietro

Ann. Henri Poincar´e

If we apply the expansion (3.31) in each non-trivial vertex of τ , we get an expression of the form 
 (h) V (h) (τ, P, T ) , (3.32) dxv0 ψ˜(≤h) (Pv0 )Wτ,P,T (xv0 ) ≡ V (h) (τ, P) = T ∈T

T ∈T

where T is a special family of graphs on the set of points xv0 , obtained by putting together an anchored tree graph Tv for each non-trivial vertex v. Note that any graph T ∈ T becomes a tree graph on xv0 , if one identifies all the points in the sets xv , for any vertex v which is also an endpoint. We are writing the R operation as acting on the kernels, according to (3.9) and its analogous for n = 1. Such representation for the R operation is however

not suitable to “gain” the convergence factor γ −(h−h ) , or γ −2(h−h ) , for which is much more convenient representation of R in (3.10), (3.14). However if we write simply all the R operations as in (3.10), (3.14) one gets possibly factors (xi −xj )αn with αn = O(n), which when integrated give O(n!) terms. One has to proceed in a more subtle way starting from the vertices of τ closest to the root from which the R operation is non-trivial, and writing R as in (3.10), (3.14) leaving all the other R operations as in (3.9). One distributes the “zero” along a path connecting ˜ 4h = (xi − xj )W ˜ 4h , if xi , xj are a family of end points, and from (3.9) (xi − xj )RW h h ˜ is the term in square brackets in the left-hand ˜ and RW two coordinates of W 4 4 ˜ 2h . There are same technical side of (3.9); an analogous property holds for RW complications in implementing this idea, which are discussed in [BM] (see also [BoM]), §3.2, §3.3 for a different model, but the adapting of such argument to the present case is straightforward. We obtain, in the L → ∞ limit V (h) (τ, P) =

 


dxv0

|Pv0 | Zh Wτ,P,T,α (xv0 )

T ∈T α∈AT

·

  f ∈Pv0

 q (f ) (≤h)σ(f ) [∂ˆjαα(f ) ψ]xα (f ),ω(f ) , (3.33)

where 

·

n  i=1





Wτ,P,T,α (xv0 ) =

Zhv /Zhv −1

v not e.p.

|Pv |/2  

b (v ∗ ) djαα(vi∗ ) (xi , yi )Kvh∗i (xvi∗ ) i i

 v not e.p.

1 sv !


dPTv (tv )

  qα (fl− ) ¯qα (fl+ ) bα (l) ˆml g (h−v ) − + + (xl − yl )] , · det Ghαv ,Tv (tv ) ∂¯j (f − ∂ + [dj (l) (xl , yl )∂ α ) j (f ) σ ,ω ;σ ,ω l∈Tv

α

l

α

l

l

l

l

l

(3.34) where:

Vol. 3, 2002

Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off

1203

1) P is the set of {Pv }; 2) T is the set of the tree graphs on xv0 , obtained by putting together an anchored tree graph Tv for each non-trivial vertex v; 3) AT is a set of indices which allows to distinguish the different terms produced by the non-trivial R operations and the iterative decomposition of the zeros; v1∗ , . . . , vn∗ are the endpoints of τ , fl− and fl+ are the labels of the two fields forming the line l, “e.p.” is an abbreviation of “endpoint”. 4) Ghαv ,Tv (tv ) is obtained from the matrix Ghv ,Tv (tv ), associated with the vertex v and Tv , by substituting + m(f − ) m(f

) (h )

v ,Tv ˆ ij− ∂ˆx i+j g −v − + + (xij − yi
j
) Ghij,i
j
= tv,i,i
∂x ) σ ,ω ;σ ,ω σ(f ) σ(f ij

i
j


l

l

l

l

with −

+

−

ij

ij

m(f + )

) q (f ) m(fij )

(h ) ij ¯ α ij ˆ v ,Tv ¯qα (f− ∂ ∂xσ(f − ) ∂ˆxσ(fi+j ) gσ−v,ω− ;σ+ ,ω+ (xij − yi
j
) . Ghα,ij,i
j
= tv,i,i
∂ j (f ) j (f + ) α

ij

α

i
j


l

l

l

l

(3.35) 5) ∂¯jq , q = 0, 1, 2, are discrete derivatives or operators dimensionally equivalent to derivatives, due to the presence of the lattice and the fact that β is finite, see [BM] §3. Moreover ∂¯j0 denotes the identity and j = 0, +, −. According to q (3.13), (3.15) if σ(f ) = I then in ∂¯j(f ) one has j(f ) = 0, + and if σ(f ) = II then j(f ) = 0, −. 6) d0 (xl − yl ) = πβ sin πβ (x0,l − y0,l ) and di (xl − yl ) = (xi,l − yi,l ), i = ± are the “zeros” produced by the R operation, see (3.13), (3.15). Finally by construction ba (l) ≤ 2. are functions of the coordinates, and such dependence is 7) The factors ZZh−1 h not explicitly written. Of course the coefficients bα and qα are not independent, and, by the definition of R (see the discussion after (3.13)) it holds for any α ∈ AT , the following inequality       γ hα (f )qα (f ) γ −hα (l)bα (l) ≤ γ −z(Pv ) , (3.36) f ∈Iv0

l∈T

v not e.p.

where hα (f ) = hv0 − 1 if f ∈ Pv0 , otherwise it is the scale of the vertex where the field with label f is contracted; hα (l) = hv , if l ∈ Tv and   1 z(Pv ) = 2   0

if |Pv | = 4, if |Pv | = 2; otherwise.

(3.37)

1204

V. Mastropietro

Ann. Henri Poincar´e

It holds

 sv | det Ghαv ,Tv (tv )| ≤ C i=1 |Pvi |−|Pv |−2(sv −1) sv sv hv · γ 2 ( i=1 |Pvi |−|Pv |−2(sv −1)) γ hv i=1 [qα (Pvi \Qvi )+m(Pvi \Qvi )]  −h [q (f + )+qα (fl− )+m(fl+ )+m(fl− )] . (3.38) · γ v l∈Tv α l

This follows from the well-known Gram-Hadamard inequality, see also [Le], [BM], [GM], stating that, if M is a square matrix with elements Mij of the form Mij =< Ai , Bj >, where Ai , Bj are vectors in a Hilbert space with scalar product < ·, · >, then  | det M | ≤ ||Ai || · ||Bi || . (3.39) i

where || · || is the norm induced by the scalar product. In our case it can be shown that −

+

ij ) ¯qα (fij ) ˆm(fij ) ˆm(fi
j
) (hv ) v ,Tv ¯qα (f− Ghα,ij,i gωl ,σl (xij − yi
j
) ∂ ∂
j
= ti,i
∂ + ∂ jα (fij ) jα (fij )   (hv ) (hv ) = ui ⊗ Ax(f − ),ω ,σ , ui
⊗ Bx(f + ),ω ,σ , (3.40) +

−

ij

l

l

l

i
j


l

(h )

where ui ∈ Rs , i = 1, . . . , s, are the vectors such that ti,i
= ui ·ui
, and Ax(fv − ),ω ,σ , (h ) Bx(fv + ),ω ,σ l l i
j


l

ij

l

are such that (in the case q = m = 0 for simplicity):

v) gω(hl ,σ (xij − yi
j
) = l



 (h ) (h ) Ax(fv − ),ω ,σ , Bx(fv + ),ω ,σ l l l l ij i
j

(h ) (hv ) ≡ dyAx(fv − )−y,ω ,σ By−x(f + ij

l

l

),ωl ,σl i
j


(3.41)

For instance A and B can be chosen as:  
(h ) Ax,ωv l = −i dk e−ik x H(a20 sin2 kˆσl )f˜h (k0 , k σl , kˆσl ) k2 +(2 cos kˆ1 sin k
)2 (3.42) σl 0 σl    
(h ) Bx,ωv l ,σl = dk e−ik x H(a20 sin2 kˆσl )f˜h (k0 , kσl , kˆσl ) ik0 + 2ω cos kˆσl sin k σl and from (3.39) we easily get (3.38). By using (3.34) and (3.38) we get, assuming (2.33)
 dxv0 |Wτ,P,T,α (xv0 )| ≤ C n Jτ,P,r,T,α ·C

 sv

· γ hv

i=1

|Pvi |−|Pv |−2(sv −1)

sv

i=1

γ

hv 2

(

sv i=1

v not e.p. |Pvi |−|Pv |−2(sv −1))

[qα (Pvi \Qvi )+m(Pvi \Qvi )] γ −hv



(3.43)  ] ,

qα (fl+ )+qα (fl− )+m(fl+ )+m(fl− ) l∈Tv

[

Vol. 3, 2002

Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off

1205

where
Jτ,P,T,α = ·



n     b (v ∗ ) dxv0  djαα (vi∗ ) (xi , yi )Kvh∗i (xvi∗ )

 v not e.p.

i

i

i=1

 1   ¯qα (fl− ) ¯qα (fl+ ) bα (l)  (h ) ∂j (f − ) ∂j (f + ) [djα (l) (xl , yl )∂ˆml gω−v,σ− ;ω+ ,σ+ (xl − yl )]  . α l α l sv ! l l l l l∈Tv

(3.44) In [BM], [BoM] it is proved that x d(xv0 ) = d¯



drl ,

(3.45)

l∈T

where rl = xl (tl ) − yl (sl ) and xl (tl ), yl (sl ) are interpolated points, see (3.13), ¯ is an arbitrary point of xv0 . By using (3.12), (3.4) we bound dimen(3.15), and x sionally each propagator, each derivative and each zero and we find Jτ,P,T,α ≤ C n

  1 −h b (l)+˜ bα (l) C 2(sv −1) γ v l∈Tv α sv ! v not e.p.   h [q (f + )+qα (fl− )+m(fl+ )+m(fl− )] . (3.46) · γ −hv (sv −1) γ v l∈Tv α l 

We find then
dxv0 |Wτ,P,T,α (xv0 )|





≤ C n L2 β|λ|n γ −hD(Pv0 )

v not e.p.

1 sv |Pvi |−|Pv | −[−2+ |Pv | +z(Pv )] 2 C i=1 γ sv !

 (3.47)

where D(Pv0 ) = −2 + m. The sum over t, P, T, α is standard and we refer to [BM], §3.15; at the end the following theorem is proved. Theorem. Let h > hβ ≥ 0. If (2.33) holds then there exists a constant c0 such that 



P τ ∈Th,n |Pv0 |=2m

 


dxv0 |Wτ,P,T,α (xv0 )| ≤ L2 βγ −hD(Pv0 ) (c0 λ)n , (3.48)

T ∈T α∈AT

where D(Pv0 ) = −2 + m .

(3.49)

1206

V. Mastropietro

Ann. Henri Poincar´e

4 The flow of running coupling functions 4.1

Lemma

It holds that

 η=±



ˆ 2h (η π , ± π , k− ) = W β 2

η=±

ˆ 2h (η π , k+ , ± π ) = 0 . W β 2

ˆ 2h (k) also by a “single scale” integration; in fact W ˆ 2h (k) Proof. We can compute W
+(≤h) −(≤h) ,h is the kernel of the term ψk ψk in V defined by

,h [h,0] ( 0 is a bound for the norm of the second order contribution to λh . Hence sup |λh−1 (k¯σ1 ,ω1 , k¯σ2 ,ω2 , k¯σ3 ,ω3 , k¯σ4 ,ω4 )| ≤ [|λ| + |h|2cλ2 λ2 ] .

(4.13)

{k},{σ}

1 λ

Then sup |λh−1 | ≤ 2|λ| if β < e 2c2 |λ| , as |h| ≤ |hβ |. The same argument can be Z repeated for δh and Zh−1 , and (2.33) holds. h

4.3

Flow of the wave function renormalization

To complete the proof of the main Theorem one has to check that indeed the critical indices η(k+ ) or η(k− ) are non identically vanishing, and this is equivalent  to show that there exists a non-vanishing function a(k¯σ,ω ) > 0 such that e−

¯
a(k σ,ω ) 2

λ2 h

¯


2

 ≤ Zh (k¯σ,ω ) ≤ e−2a(kσ,ω )λ

h

.

(4.14)

∞ h(n)  h(n)   ) = n=2 βξ (k¯σ,ω ) with |βξ (k¯σ,ω )| ≤ cn0 |λ|n , as a From the fact that βξh (k¯σ,ω consequence of (3.49) and (2.33), it is sufficient to find an upper and lower bound h(2) for βξ . From an explicit computation one finds
  ∂ h(2) = 24 [ dk1 dk2 dk3 gσ(≤h) (k1 ) (4.15) 2ω cos kˆσ βξ 1 ,ω1 ∂kσ ω ,ω ,ω σ ,σ ,σ 1

2

3

1

gσ(h) (k2 )gσ(≤h) (k3 )δ(k 2 ,ω2 3 ,ω3

2

3

¯σ , k ¯ 3,σ , k ¯ 1,σ ) + k3 − k1 − k2 )λh (k 3 1 ¯ 1,σ , k ¯ 2,σ , k ¯ σ )]| ¯ . λh (k 1 2 k=kσ,ω

h (≤h) (k) where gσ,ω = k=hβ gσ,ω . As the dependence from the momenta of λh is quite complex, it is convenient to replace in the above integral λh with λ; if the integral −1 so obtained is non-vanishing, the correction will be surely smaller for T ≥ e−(¯c|λ|) for a suitable c¯, as λh = λ + O(λ2 | log β|) from (4.13). We can choose σ = I for definiteness (the analysis for σ = II is identical), and we can distinguish two kind of contributions in the sum over σ1 , σ2 , σ3 ; one in which all the propagators are gI , and the other such that there is at least a propagator gII . The estimate of this second contribution is O(λ2 γ h ), as it can be immediately checked by dimensional considerations and applying the derivative in (4.15) over the gII propagators (one can always do that). We can further simplify the expression we have to compute noting that
H(a20 sin2 k− )f˜h (k0 , k+ ) (h) (h) + g¯I,ω (x − y) , (4.16) gI,ω (x − y) = dke−ikx −ik0 + 2ω sin k+

Vol. 3, 2002

Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off

1209

with (h)

|¯ gI,ω (x − y)| ≤ a−2 0

CN γ h , 1 + [γ h |x0 − y0 | + γ h |x+ − y+ | + |x− − y− |]N

(4.17)

i.e., similar to (3.3) with an extra a−2 0 . We can replace in (4.15) the propagators (h) gI,ω with the first addend in the right-hand side of (4.16); if such term will be given by a non-vanishing constant, the correction will be surely smaller at least for a0 large enough. Hence the dominant contribution to (4.15) is given by 
dk−,1 dk−,3 H(a20 sin2 k−,1 )H(a20 sin2 k−,3 )H(a20 sin2 (−k−,1 +k−,3 +k− ))A, ω1 ,ω2 ,ω3

(4.18) with A=







dk0,1 dk0,3

ω1 ,ω2 ,ω3

dk+,1 dk+,3

f ≤h (k0,1 , k+,1 ) f ≤h (k0,3 , k+,3 ) ∂k −ik0,1 + ω1 2k+,1 −ik0,3 + ω3 2k+,3 +

 f h (−k0,1 + k0,3 + k0 , −k+,1 + k+,3 + k+ )  −i(−k0,1 + k0,3 + k0 ) + 2ω2 (−k+,1 + k+,3 + k+ ) k+ −ωpF =k0 =0

(4.19)

with ω = ω1 + ω2 − ω3 ; it is easy to check that this term is indeed non-vanishing. Note also that A is the first non-trivial contribution to the critical index η of the Schwinger function of a d = 1 systems of interacting fermions.

4.4

Schwinger functions

We will not repeat here the analysis of the Schwinger functions at the temperature scale, as one can proceed as in the d = 1 to obtain an expansion for the Schwinger function once that the expansion for the effective potential is understood; see for instance [GM]. We only remark that the AI and AII in (1.12) and (1.13) are indeed O(λ2 ) as a consequence of

   h  dk− gIh (k0 , k+ + ωpF , k− ) = 0 dk0 dk+ dk− gI (k0 , k+ , k− + ωpF ) = 0 . dk0 dk+ (4.20)

5 Conclusions 5.1

Marginal Fermi liquids and Luttinger liquids

We can compare the behaviour of the half-filled Hubbard model with cut-off with other models. We have found that the wave function renormalization has Z an anomalous flow up to exponentially small temperatures, Zh−1 = 1 + O(λ2 ), see h (4.14); in the case of circular Fermi surfaces one finds instead, see [DR], for |λ| ≤ ε h Zh−1 = 1 + O(ε2 γ 2 ) , Zh

(5.1)

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which means that Zh = 1 + O(λ2 ), up to exponentially small temperatures; the factor γ h/2 in the right-hand side of (5.1) is an improvement with respect to a power counting bound and is found by using a volume improvement based on the geometrical constraints to which the momenta close to the Fermi surface are subjected. An equation similar to (5.1) holds also for any symmetric smooth Fermi surfaces with non-vanishing curvature; a proof can be obtained by combining the results of [BGM] with Appendix 2 of [DR]. The similarity of the equation for Zh with its analogous for one-dimensional systems may suggest that the behaviour of the half-filled Hubbard model with cutoff up to zero temperature is similar to the one of a system of spinless interacting fermions in d = 1 (the so-called Luttinger liquid behaviour). However this is false; in a Luttinger liquid in fact one has that h

λh−1 = λh + O(ε2 γ 2 ) ,

(5.2)

a property known as vanishing of beta function. One can easily check that this cancellation is not present in the half-filled Hubbard model with cut-off; in fact the dominant second order contribution to λh (k¯1,I , k¯−1,I , k¯1,I ) containing only σ = I internal lines is
1 2 2 dk f h (k0 , k+ )f ≥h (k0 , k+ ) 2 2 H(a0 sin k− ) k0 + k+ [H(a20 sin2 (k1,− + k2,− − k− )λh λh − H(a20 sin2 (k3,− − k2,− + k− )λh λh ] ,

(5.3)

where the dependence from k of the λh has not been explicitated. It is clear then that even at the second order the flow of λh is quite complex, and we plan to analyze it in a future work, in order to understand the leading instabilities. Replacing H with 1 and having λh not momentum dependent one recovers the d = 1 situation in which the beta function is vanishing. The theory resembles the theory of d = 1 Fermi systems in which each particle has an extra degree of freedom, the component of the momentum parallel to the flat Fermi surface, playing the role of a “continuous” spin index; and it is known that in d = 1 even a spin- 12 index can destroy the Luttinger liquid behaviour [BoM].

5.2

Marginal Fermi liquid behaviour close to half-filling

A similar analysis can be performed in the case of the Hubbard model with cut-off close to half-filling (µ = −ε with ε small and positive); in such a case the Fermi surface is convex and with finite radius of curvature but still resembles a square with non-flat sides and rounded corners. The propagator has the form χ(k) −ik0 + 2 cos k+ cos k− − ε and it is easy to verify that, if β < C min[ ρ1 ] where ρ is the radius of curvature h of the Fermi surface, the bounds (3.3), (3.4) for the single scale propagator gσ,ω

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still holds; the reason is that, up to temperatures greater than the inverse of the curvature radius, the bounds are insensitive to the fact that the sides of the Fermi surface are not perfectly flat. One can repeat all the analysis of the preceding sections and it is found that the Schwinger functions behave like (1.12), (1.13) for −1 λ small enough and β < C min[min[ ρ1 ], e(¯c|λ|) ]; in other words marginal Fermi liquid behaviour is still found close to half-filling, up to such temperatures. −1 On the other hand at lower temperatures, for min[ ρ1 ] ≤ β ≤ e(¯c|λ|) (of −1

course assuming min[ ρ1 ] ≤ e(¯c|λ|) ) one can apply the results of [BGM] (valid for any convex symmetric and regular Fermi surface) so finding Z = 1+Cρ [λ2 +O(λ3 )] where Cρ is a constant which is very large for small ε (and diverging at half-filling ε = 0). Hence, depending on the values of the parameters, one can have, in the low temperature region and before the critical temperature, two possibilities: the first is to have only marginal Fermi liquid behaviour Z = 1 + O(λ2 log β), and the second is to have marginal Fermi liquid behaviour up to temperatures O(min[ρ−1 ]) and then Fermi liquid behaviour up to the critical temperature.

Acknowledgments This work was partly written during a visit at the Ecole Polytechnique in Paris. I thank J. Magnen and V. Rivasseau for their warm hospitality and useful discussions. I thank also G. Benfatto and G. Gallavotti for important remarks.

References [A]

P.W. Anderson, The theory of superconductivity in high Tc cuprates, Princeton University Press, Princeton (1997).

[AGD]

A.A. Abrikosov, L.P. Gorkov, I. Dzialoshinsky, Methods of quantum field theory in statistical physics, Prentice hall (1963).

[BGM]

G. Benfatto, A. Giuliani, V. Mastropietro, to appear in Annales Henri Poincar´e.

[BM]

G. Benfatto, V. Mastropietro, Renormalization group, hidden symmetries and approximate Ward identities in the XY Z model, Rev. Math. Phys. 13, 1323–143 (2001), no. 11.

[BoM]

F. Bonetto, V. Mastropietro, Beta function and anomaly of the Fermi surface for a d = 1 system of interacting fermions in a periodic potential, Comm. Math. Phys. 172, 57–93 (1995), no. 1.

[DAD]

S. Dusuel, F. De Abreu, B. Doucot, Renormalization Group for 2D fermions with a flat Fermi surface, cond-mat 0107548.

[DR]

M. Disertori, V. Rivasseau, Interacting Fermi liquid in two dimensions, Comm. Math. Phys. 215, 251–290 and 291–341 (2000).

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[FMRT] J. Feldman, J. Magnen, V. Rivasseau, E. Trubowitz, An infinite volume expansion for Many Fermions Green functions, Helvetica Physica Acta 65, 679–721 (1992). [FSW]

A. Ferraz, T. Saikawa, Z.Y. Weng, marginal fermi liquid with a twodimensional Patched Fermi surface, cond-mat9908111.

[FSL]

J.O. Fjaerested, A. Sudbo, A. Luther, Correlation function for a twodimensional electron system with bosonic interactions and a square Fermi surface, Phys. Rev. B. 60, 19, 13361–13370 (1999).

[GM]

G. Gentile, V. Mastropietro, Renormalization group for one-dimensional fermions. A review on mathematical results, Phys. Rep. 352, 273–43 (2001), no. 4–6.

[Le]

A. Lesniewski, Effective action for the Yukawa 2 quantum field Theory, Comm. Math. Phys. 108, 437–467 (1987).

[L]

A. Luther, Interacting electrons on a square Fermi surface, PRB vol.30, n.16 (1994).

[M]

D.C. Mattis, Implications of infrared instability in a two-dimensional electron gas, PRB, vol.36, n.1 (1987).

[Me]

W. Metzner, Renormalization Group analysis of a two-dimensional interacting electron system, Int. Jour. mod. Phys. 16, 11, 1889–1898 (2001).

[R]

V. Rivasseau, The two-dimensional Hubbard model at half-filling. I. Convergent contributions, J. Statist. Phys. 106, no. 3-4, 693–72 (2002).

[S]

Z-X. Shen et al, Science 267, 343 (1995).

[Sh]

R. Shankar, Renormalization Group approach to interacting fermions, Rev. Mod. Phys. 66 (1), 129–192 (1994).

[VZS]

C.M. Varma, Z. Nussinov, W. van Saarloos, Singular Fermi liquids, cond-mat0103393.

[VLSAR] C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams, A.E. Ruckestein, Phenomenology of the normal state of the Cu-O high Tc superconductors, Phys. Rev. Lett. 63, 1996 (1989). [VR]

A. Viroszteck, J. Ruvalds, Nested fermi liquid theory, Phys. Rev. B 42, 4064 (1990).

[ZYD]

A.T. Zheleznyak, V.M. Yakovenko, I.E. Dzyaloshinskii, Parquet solutions for a flat fermi surface, Phys. Rev. B 55, 3200 (1997).

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Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off

Vieri Mastropietro Dipartimento di Matematica Universit` a di Roma “Tor Vergata” Via della Ricerca Scientifica I-00133, Roma Italy email: [email protected] Communicated by Vincent Rivasseau submitted 17/05/02, accepted 14/10/02

To access this journal online: http://www.birkhauser.ch

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auser Verlag, Basel, 2002 1424-0637/02/0601215-18

Annales Henri Poincar´ e

Propagation Properties for Schr¨ odinger Operators Affiliated with Certain C*-Algebras W.O. Amrein, M. M˘ antoiu and R. Purice Abstract. We consider anisotropic Schr¨ odinger operators H = −∆ + V in L2 (Ên ). To certain asymptotic regions F we assign asymptotic Hamiltonians HF such that (a) σ(HF ) ⊂ σess (H), (b) states with energies not belonging to σ(HF ) do not propagate into a neighbourhood of F under the evolution group defined by H. The proof relies on C ∗ -algebra techniques. We can treat in particular potentials that tend asymptotically to different periodic functions in different cones, potentials with oscillation that decays at infinity, as well as some examples considered before by Davies and Simon in [4].

1 Introduction This paper is concerned with propagation properties of scattering states of selfadjoint n-dimensional Schr¨ odinger operators H = −∆+V with potentials V having different asymptotics in different directions. We recall that the scattering states of H are defined by the property that, as the time t tends to ±∞, they propagate away from each bounded region of the configuration space Rn (at least in some time average [2]). In many situations, in particular if V is a bounded function, they can be identified with the states in the continuous spectral subspace of H. If the potential V tends to zero (or to some other constant) sufficiently rapidly at infinity, standard scattering theory provides a description of the behaviour of e−itH f for a scattering state f at large times t. In more complicated situations, in particular if the asymptotic behaviour of V is highly anisotropic, little is known about the propagation of the scattering states. One may expect that certain asymptotic regions of configuration space should be inaccessible to states of certain energies, as illustrated by the following two examples. (1) In one dimension (n = 1), assume that V (x) → V± as x → ±∞, with V+ = V− . If for example V+ > V− , a state in the continuous spectral subspace of H with spectral support in the interval (V− , V+ ) will not propagate to the right. (2) In higher dimension (n ≥ 2) consider a potential V approaching a periodic function V0 as the argument x tends to infinity inside some cone C ⊂ Rn . In addition to the Hamiltonian H = −∆ + V one may introduce the periodic Schr¨ odinger operator H0 = −∆ + V0 . Bearing in mind some hypothetical scattering theory (e−itH0 should furnish a suitable comparison dynamics for the propagation inside C), one could expect that scattering states of H with energy disjoint from the continuous spectrum of H0 will not be able to propagate into C (such states may exist: the

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spectrum of H0 has a band structure, whereas the continuous spectrum of H will depend also on the behaviour of V outside the cone C and thus could intersect some gap in the band spectrum of H0 ). Detailed results about the propagation and non-propagation of states for onedimensional Schr¨ odinger operators with different spatial asymptotics at ±∞ (in particular for the first example above) and for multi-dimensional operators periodic in all but one dimension have been obtained by Davies and Simon [4]. The investigation of these authors includes a careful spectral analysis of the Hamiltonians under study. We propose here a different method for obtaining non-propagation properties, based on a relatively recent approach to spectral theory in the framework of C ∗ -algebras and without invoking any scattering theory. We shall in particular obtain the non-propagation result stated in the second example above. Below we give a brief non-technical description of this method. Typically the potential V to be considered is an element of a C ∗ -algebra A of bounded, continuous functions on Rn . The functions in A are characterized by a specific asymptotic behaviour (for example asymptotic periodicity in certain cones). Then, by invoking the Neumann series for (H − z)−1 (which is convergent for z large enough), one finds that the resolvent of the operator H = −∆ + V belongs to a C ∗ -algebra CA generated by products of elements of A (viewed as multiplication operators in L2 (Rn )) and suitable functions of momentum. We shall say that H is affiliated with CA . A central concept is that of the spectrum of H relative to an ideal K of CA of the form K = CK , where K is an ideal of A and CK is defined similarly to CA (just replace A by K in the definition of CA ). Let us denote this spectrum by σ K (H) and call it the essential spectrum associated with the ideal K (a precise definition is given in Section 2). For K = {0}, σ K (H) is the usual spectrum of H; for K = C0 (Rn ) (the space of continuous functions converging to zero at infinity), K will be the ideal of compact operators and σ K (H) the essential spectrum σess (H) of H. For an ideal K of A such that C0 (Rn ) ⊂ K, σ K (H) is a subset of σess (H). A typical non-propagation result will assert that scattering states of H with spectral support disjoint from σ K (H) will essentially never be localized in certain spatial domains W determined by K. Using the essential spectrum associated with such ideals to characterize some geometric properties for quantum Hamiltonians seems to be new, although in the literature ideals have been used in connection with spectral theory (e.g. in [3], [5]). In these considerations the ideal K will be given in terms of the asymptotic behaviour of the elements of A in some neighbourhood of infinity, and the theory will apply if the spatial domains W associated with K cover or intersect this neighbourhood of infinity. In the second example mentioned at the beginning K could be the set of functions in A that are asymptotically periodic in the cone C and tend to zero in directions not belonging to C, W could be the intersection of C with the complement of a compact set and σ K (H) would be the spectrum of H0 . Our treatment consists in the introduction of a compactification X of Rn related to the algebra A (in fact X will be the character space of A). The ideal K is determined by some closed subset F of the frontier X \ Rn of Rn in X and

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W is a subset of Rn which is sufficiently close to F . In the cited example one may think, intuitively, of F as the part of the compactification attached at infinity to the cone C. In this context the behaviour of various objects under translations is important. The algebra A is assumed to be invariant under translations; then the natural action of the translation group in Rn has a continuous extension to the compactification X of Rn and F must be invariant under this extension in order to allow the application of a result from the theory of crossed products of C ∗ -algebras to determine the spectrum σ K (H). In Section 2 we present some of the C ∗ -algebraic concepts that are useful in the spectral theory of self-adjoint operators. In Section 3 we discuss some algebras A of continuous functions on Rn , the associated compactifications of Rn and the continuous extension of the translation group to these compactifications. Section 4 contains a few remarks on crossed product algebras, and in Section 5 we give the proof of an abstract theorem on non-propagation. An application of this theorem to Schr¨ odinger Hamiltonians that are asymptotically periodic in several cones is studied in detail in Section 6, and in Section 7 we mention other classes of Hamiltonians that can be treated in a similar way. We use the terminology and results of [12] for the theory of C ∗ -algebras. We shall not try to discuss the various applications of C ∗ -algebraic methods in the study of quantum Hamiltonians, in the literature there exist several excellent reviews on these problems. Nevertheless we refer to Chapter 8 of [1] for a presentation of the algebraic approach to spectral theory that we shall use. The algebras CA and CK mentioned above have the structure of a crossed product; Reference [6] contains a description of such algebras that is well adapted to our applications in spectral theory. Finally we point out that various generalizations of our results are possible with almost no extra effort (cf. also [1], [6], [10] and [11]). Local singularities of the potential can easily be taken into account. The kinetic energy −∆ can be replaced by h(P ), an arbitrary continuous function of momentum satisfying |h(p)| → ∞ when |p| → ∞. Instead of the configuration space Rn , one can work with any abelian locally compact group X. The case X = Zn leads to finite difference operators.

2 C*-algebras and generalized essential spectra If H is a self-adjoint operator in a Hilbert space H, the spectral theorem allows one to associate an operator η(H) to a large class of functions η : R → C. We shall be here concerned with the set C0 (R) consisting of all continuous functions η : R → C that vanish at infinity (i.e., satisfying limx→±∞ η(x) = 0). Some parts of the spectrum of H can easily be characterized in terms of these functions: (i) a number λ ∈ R belongs to the spectrum σ(H) of H if η(H) = 0 whenever η ∈ C0 (R)

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and η(λ) = 0, (ii) λ belongs to the essential spectrum σess (H) of H if η(H) is a non-compact operator whenever η ∈ C0 (R) and η(λ) = 0. If C is a C ∗ -algebra of bounded operators in H such that η(H) ∈ C for each η ∈ C0 (R), then H is said to be affiliated with C. A sufficient condition for H to be affiliated with C is the requirement that (H − z)−1 ∈ C for some complex number z∈ / σ(H). The preceding situation can be viewed as a special case of the following abstract definition: Definition 1. (a) An observable affiliated with a C ∗ -algebra C is a ∗- homomorphism from the C ∗ -algebra C0 (R) to C (i.e., a linear mapping Φ : C0 (R) → C satisfying Φ(ξη) = Φ(ξ)Φ(η) and Φ(η)∗ = Φ(η) if ξ, η ∈ C0 (R)). (b) The spectrum σ(Φ) of the observable Φ is defined as the set of real numbers λ such that Φ(η) = 0 whenever η(λ) = 0. σ(Φ) is a closed subset of R. ˆ≡ Now let K be a (closed, self-adjoint, bilateral) ideal in C. We denote by C ∗ C/K the associated quotient C -algebra and by Π the canonical ∗-homomorphism ˆ If Φ is an observable affiliated with C, then clearly Π ◦ Φ determines of C onto C. ˆ an observable affiliated with C. ˆ is called Definition 2. The spectrum σ(Π◦Φ) of the observable Π◦Φ (relative to C) the K-essential spectrum of Φ and will be denoted by σK (Φ): σK (Φ) ≡ σ(Π ◦ Φ). Equivalently, a real number λ belongs to σK (Φ) if and only if Φ(η) ∈ / K whenever η ∈ C0 (R) is such that η(λ) = 0. To motivate the present terminology, let us consider the situation introduced at the beginning, where C is a C ∗ -subalgebra of B(H) and ΦH is the observable determined by a self-adjoint operator H affiliated with C (so ΦH (η) = η(H)). Assume that C contains the ideal K(H) of all compact operators in H. Then σK(H) (ΦH ) is just the essential spectrum σess (H) of the self-adjoint operator H. Now let us observe that, if K1 and K2 are two ideals in C satisfying K1 ⊂ K2 , then σK2 (Φ) ⊂ σK1 (Φ) ⊂ σ(Φ). In particular, if H is a self-adjoint operator affiliated with a C ∗ -subalgebra C of B(H) and if K is an ideal in C with K(H) ⊂ K, then σK (ΦH ) ⊂ σess (H). One of the interesting aspects of the preceding framework in the study of self-adjoint operators in a Hilbert space H is as follows. Let C be a C ∗ -subalgebra of B(H) and consider a class Θ of self-adjoint operators H affiliated with C such that, for some ideal K of C, the ∗-homomorphisms Π ◦ ΦH do not depend on H. So all members H of Θ have the same K-essential spectrum σK . In some situations ˆ will not be it is rather easy to determine σK : although the quotient C ∗ -algebra C identifiable with a subalgebra of B(H), it may be possible to specify a faithful ˆ in a Hilbert space H ˆ (an injective ∗-homomorphism π : C ˆ → representation of C ˆ ˆ ˆ ˆ B(H)) and a simple self-adjoint operator H in H affiliated with π(C) such that

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ˆ for all η ∈ C0 (R) and all H ∈ Θ. Then σK is just the spectrum π [Π(η(H))] = η(H) ˆ Examples will be considered in Sections 6 of the (presumably simple) operator H. and 7. The following result will be used in Section 5: Lemma 1. Let K be an ideal in a C ∗ -algebra C and Φ an observable affiliated with C. If η ∈ C0 (R) is such that η(µ) = 0 for all µ ∈ σK (Φ), then Φ(η) ∈ K. Proof. (i) Let λ ∈ R \ σK (Φ). There are a number ε > 0 and a function θ ∈ C0 (R) such that |θ(µ)| > ε for all µ ∈ (λ − ε, λ + ε) and Φ(θ) ∈ K. Now let ξ ∈ C0 (R) be such that supp ξ ⊂ (λ−ε, λ+ε). Since ξ/θ ∈ C0 (R), we have Φ(ξ) = Φ(θ)Φ(ξ/θ) ∈ K. In conclusion: each λ in R \ σK (Φ) has an open neighbourhood Vλ with the property that Φ(ξ) ∈ K for each ξ ∈ C0 (R) having support in Vλ . (ii) Since K is norm-closed, it is enough to establish the conclusion of the lemma under the additional assumption that η has compact support in R \ σK (Φ). Choose a finite collection of numbers λ1 , . . . , λM ∈ supp η such that supp η ⊂ ∪k Vλk and a corresponding partition of unity on supp η, i.e., a collection of func
M tions ξk in C0 (R) such that supp ξk ⊂ Vλk and k=1 ξk (λ) = 1 for all λ ∈ supp η.
M Since Φ(ξk ) ∈ K by (i), we get Φ(η) = k=1 Φ(η)Φ(ξk ) ∈ K.


3 Some abelian C*-algebra If Y is a locally compact, Hausdorff space, we denote by Cb (Y ) the abelian C ∗ algebra of all bounded, continuous complex functions defined on Y . If G is a closed subset of Y , we set C G (Y ) = {ϕ ∈ Cb (Y ) | ϕ(y) = 0, ∀y ∈ G}. Certain C ∗ -subalgebras of Cb (Y ) will be important further on, in particular the algebras Cbu (Y ) and C0 (Y ) consisting respectively of all bounded, uniformly continuous functions and of all continuous functions vanishing at infinity. In fact C0 (Y ) is an ideal of Cb (Y ). Throughout this paper we set X = Rn . Let Y be as above and assume that X acts on Y as a group of homeomorphisms: so if αx denotes the homeomorphism in Y associated with the element x ∈ X, we have αx ◦ αx
= αx+x
. The mapping X × Y (x, y) → αx (y) ∈ Y is assumed continuous. Then α induces a representation of the group X by ∗-automorphisms of Cb (Y ) as well as of various C ∗ -subalgebras of Cb (Y ): for ϕ ∈ Cb (Y ) and x ∈ X, define ax (ϕ) ∈ Cb (Y ) by [ax (ϕ)](y) = ϕ(αx (y)) (y ∈ Y ). We observe that a C ∗ -subalgebra form C G (Y ) is invariant under  G  of the G this automorphism group (i.e., ax C (Y ) ⊂ C (Y )) if and only if the closed set G is invariant under each αx . Let A be a unital C ∗ -subalgebra of Cb (X) containing C0 (X). We denote its character space Ω(A) by X and we recall that X is a compactification of X, i.e., X is a compact topological space and there is a homeomorphism i from X to a dense subset of X (see e.g. §8.1 of [8]). For x ∈ X, the character i(x) is given by the formula [i(x)](ϕ) = ϕ(x), for ϕ ∈ A. We write Z = X \ i(X) and call it the frontier of X in X . By the Gelfand theorem, A is isomorphic to the C ∗ -algebra C(X ) of

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continuous functions on Ω(A). We shall use the notation G : C[Ω(A)] → A for the inverse of the Gelfand isomorphism. The C ∗ -subalgebra C Z (X ) (consisting of continuous functions on X that vanish on the frontier Z of X ) can be naturally identified with C0 (X), more precisely C0 (X) = GC Z [Ω(A)]. There is a one-to-one correspondence between (self-adjoint, closed) ideals K of A and closed subsets G of X , given by K = GC G (X ) (Theorem 3.4.1. of [9]). In particular each closed subset F of the frontier Z determines an ideal KF in A, viz. KF = GC F (X ). It is clear that such an ideal contains C0 (X). Suppose now that the C ∗ -algebra A considered above is contained in Cbu (X) and invariant under translations, i.e., such that ax A ⊂ A for all x ∈ X, with [ax (ϕ)](y) = ϕ(x + y). Since A ⊂ Cbu (X), the mapping x → ax (ϕ) is norm continuous for each ϕ ∈ A. Furthermore the action of X on itself (given as αx (y) = x+y) induces a continuous representation ρ of X by homeomorphisms of the character space X = Ω(A): for τ ∈ X the character ρx τ is defined as [ρx τ ](ϕ) = τ [ax (ϕ)]. For y ∈ X, set τy = i(y); then ρx τy = τx+y (x ∈ X). We end this section with a result which will be useful in the examples presented further on. Let τ ∈ X be a character of A. A neighbourhood base of τ in X is given by the collection {VF ,ε (τ )}, where ε varies over (0, ∞) and F over all finite families {ϕ1 , . . . , ϕm } of elements of A and where VF ,ε (τ ) = {τ  ∈ X | |τ  (ϕi ) − τ (ϕi )| < ε for each ϕi ∈ F }. Lemma 2. Let A be a unital C ∗ -subalgebra of Cb (X). Let F be a closed subset of Ω(A) and W a neighbourhood of F . Then there exist ε > 0 and a finite family F = {ϕ1 , . . . , ϕm } of elements of A such that F ⊂ ∪τ ∈F VF ,ε (τ ) ⊂ W. Proof. Let τ ∈ F . Then W is a neighbourhood of τ , hence there are a finite family F (τ ) of elements of A and a number ε(τ ) > 0 such that VF (τ ),ε(τ )(τ ) ⊂ W. Since F is compact, there are a finite number of points τ1 , . . . , τM in F such that 1 M F ⊂ ∪M j=1 VF (τj ),ε(τj )/2 (τj ). Let F = ∪j=1 F (τj ) and ε = 2 min{ε(τ1 ), . . . , ε(τM )}. The result of the lemma is true if we can show that, for each τ ∈ F , there is j ∈ {1, . . . , M } such that VF ,ε (τ ) ⊂ VF (τj ),ε(τj ) (τj ). Since clearly VF ,ε (τ ) ⊂ VF (τj ),ε (τ ) ⊂ VF (τj ),ε(τj )/2 (τ ) for each j, it is enough to show that for some j ∈ {1, . . . , M } one has VF (τj ),ε(τj )/2 (τ ) ⊂ VF (τj ),ε(τj ) (τj ). To prove this last inclusion, observe that τ belongs to VF (τj ),ε(τj )/2 (τj ) for at least one value of j. Choose one of these values of j and let τ  ∈ VF (τj ),ε(τj )/2 (τ ). By the triangle inequality one has for each ϕ ∈ A: |τj (ϕ) − τ  (ϕ)| ≤ |τj (ϕ) − τ (ϕ)| + |τ (ϕ) − τ  (ϕ)|. For every ϕ ∈ F(τj ) each term on the right-hand side is less than ε(τj )/2. Hence τ  belongs to VF (τj ),ε(τj ) (τj ).


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4 Some crossed product C*-algebras We consider some C ∗ -subalgebras of the space B(H) of all bounded, linear operators in the Hilbert space H = L2 (X). If ϕ : X → C is a bounded, measurable function, we denote by ϕ(Q) the operator of multiplication by ϕ in H and by ϕ(P ) the operator F∗ ϕ(Q)F (the operator of multiplication by ϕ in momentum space), where F is the Fourier transformation. A C ∗ -subalgebra A of Cbu (X) will be identified with the subalgebra of B(H) consisting of all multiplication operators ϕ(Q) with ϕ ∈ A. If A is a C ∗ -subalgebra of Cub (X), we write CA for the norm closure in B(H) of the set of finite sums of the form ϕ1 (Q)ψ1 (P ) + · · · + ϕN (Q)ψN (P ) with ϕk ∈ A and ψk ∈ C0 (X). We mention the fact that, if A = C0 (X), then CA is the ideal of all compact operators in L2 (X). If A is invariant under translations, then CA is a C ∗ -algebra isomorphic to the crossed product algebra A
X defined in terms of the action ax of X on A. In the proof of Lemma 6 we shall use the following result from the theory of crossed products: If K is an ideal in A that is invariant under translations, then the quotient C ∗ -algebra CA /CK is isomorphic to [A/K]
X. The point is that the general theory allows us to define the crossed product [A/K]
X only by using the continuous action of X by ∗-automorphisms of A/K (the quotient action); the fact that A/K is not a C ∗ -subalgebra of B(H) does not matter. Remark. If V ∈ A, where A is a unital C ∗ -subalgebra of Cbu (X), then the selfadjoint operator H = −∆ + V is affiliated with CA . This is easily seen from the  k
∞ fact that the Neumann series [H − z]−1 = k=0 (P 2 − z)−1 −V (Q)[P 2 − z]−1 converges in the norm of B(H) if z is sufficiently large.

5 A non-propagation theorem For our principal theorem and its corollary we consider the following Framework. A is a unital C ∗ -subalgebra of Cbu (X), invariant under translations and such that C0 (X) ⊂ A, and CA is the associated C ∗ -subalgebra of B(H) introduced in §4 (with H = L2 (X)). X = Ω(A) is the character space of A, F a translation invariant, closed subset of Z = X \ i(X) and C F (X ) the ideal in C(X ) determined by F . We set KF ≡ GC F (X ), which is a translation invariant ideal in A. Then KF ≡ CKF is an ideal in CA that contains all the compact operators in H. We shall work with families W of subsets of X such that their images through i in X are close to F . W will have the structure of a filter base, i.e., a non-void collection of non-void subsets of X such that for any W1 , W2 ∈ W there is a W ∈ W with W ⊂ W1 ∩ W2 . If W is a filter base in X, then the family {i(W ) | W ∈ W } is a filter base in X and we say that W is adjacent to F if all cluster points in X of this family {i(W ) | W ∈ W } belong to F , i.e., if ∩W ∈W W i(W ) ⊂ F , where the closures are taken in X . We observe that the set of these cluster points is

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non-empty since X is a compact space. In the majority of situations considered further on it will suffice to take for W the family {W = i−1 [W ∩ i(X) | W ∈ W]}, where W is a neighbourhood base of F in X (since i(X) is dense in X , each of these sets W is non-void). If W is a filter base adjacent to F and ϕ ∈ C(X ) is such that ϕ|F = 0, then given any ε > 0, there is some W ∈ W such that | ϕ(τ ) |< ε for all τ ∈ i(W ). In the sequel we shall denote by χW the characteristic function of W . Theorem. Let A and F be as in the framework and let W a filter base in X that is adjacent to F . Let H be a self-adjoint operator in H affiliated with CA . Let ε > 0 and η ∈ C0 (R) with supp η ∩ σKF (ΦH ) = ∅. Then there is a W ∈ W such that  χW (Q)η(H) ≤ ε.

(1)

Proof. (i) We use the notation τ for characters in Ω(A) and observe that KF = G{ϕ ∈ C(X ) | ϕ|F = 0} = {ϕ ∈ A | τ (ϕ) = 0 ∀τ ∈ F }. So if ϕ belongs to KF , then for each δ > 0 there exists W ∈ W such that |τ (ϕ)| ≤ δ ∀τ ∈ i(W ). Thus, if ϕ ∈ KF , we have | ϕ(x) |≤ δ for all x ∈ W . (ii) By the hypothesis on the support of η we have η(H) ∈ KF (see Lemma 1). So there are a finite number of functions ϕ1 , . . . , ϕN ∈ KF and ψ1 , . . . , ψN ∈ C0 (X) such that N   η(H) − ϕk (Q)ψk (P ) ≤ ε/2. k=1

We also have  χW (Q)η(H) ≤

N 

 ϕk L∞ (W )  ψk L∞ (X) +

k=1

+  η(H) −

N 

ϕk (Q)ψk (P )  .

(2)

k=1

The first term in the right-hand side of (2) can be made less than ε/2 by using  −1 the result of (i) with δ = N · supk=1,...,N  ψk L∞ (X) · ε/2, so the proof is finished.
Corollary. Let A, F , W and H be as in the theorem. Then for each ε > 0 and each η ∈ C0 (R) with supp η ⊂ R \ σKF (ΦH ), there exists W ∈ W such that  χW (Q)e−itH η(H)f ≤ ε  f  for all t ∈ R and all f ∈ L2 (X).

(3)

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(3) is a straightforward consequence of (1). Note the obvious fact that one may replace in the corollary {e−itH } by any bounded family of bounded operators commuting with H. Remark. The corollary gives the precise meaning, in our framework, of the notion of non-propagation described in the introduction. To be more specific, let us denote by suppH (f ) the spectral support with respect to H of the vector f ∈ H, defined as follows in terms of the spectral measure EH of H: λ∈ / suppH (f ) ⇔ ∃ε > 0 such that EH (λ − ε, λ + ε)f = 0. suppH (f ) is the smallest closed set J ⊂ R such that EH (J)f = f . Then it follows easily that, under the hypotheses of the corollary, for each ε > 0 and each closed subset L of R \ σKF (ΦH ) there exists an element W of W such that  χW (Q)e−itH f ≤ ε  f  for all t ∈ R and all f ∈ L2 (X) with suppH (f ) ⊂ L. In the situation just described, let us take for H a self-adjoint Schr¨ odinger operator affiliated with CA . Then, if f is a unit vector with suppH (f ) ⊂ L, one has  χW (Q)e−itH f ≤ ε for all t ∈ R. In physical terms: the probability of finding f localized in W is less than ε2 at all times. If K is a compact subset of X and if the preceding vector f belongs to the absolutely continuous subspace of H, then there is t0 ∈ R such that  χK (Q)e−itH f ≤ ε for all t > t0 [2]. It follows that  χK c ∩W c (Q)e−itH f 2 ≥ 1−2ε2 for all t > t0 , which (for small ε) essentially means that f describes a state that will propagate into the complement (K ∪ W )c of the set K ∪ W . If f belongs to the singularly continuous subspace of H, a similar conclusion is true, except that some averaging over time may be necessary [2]: t there are 0 < t0 < t1 such that (t − t0 )−1 t0  χK c ∩W c (Q)e−iτ H f 2 dτ ≥ 1 − 3ε2 for all t > t1 .

6 Example: Non-propagation in multicrystalline systems As an application we present in some detail the situation where the potential V of a Schr¨ odinger Hamiltonian becomes asymptotically periodic, with different periodic limit functions in different cones (the more general case in which the limit functions are only almost periodic can be treated analogously). More precisely V will belong to the C ∗ -algebra A introduced below, so that H = −∆ + V will be affiliated with CA . Let S be the unit sphere in X = Rn . For j = 1, . . . , N let Γj be a periodic lattice in X and Σj a non-empty open subset of S, with Σj ∩ Σk = ∅ if j = k. We denote by Cj (X) the C ∗ -algebra Cj (X) = {ϕ ∈ Cbu (X) | ϕ(x + γ) = ϕ(x) ∀x ∈ X, ∀γ ∈ Γj } and we define A as the set of bounded, uniformly continuous complex functions ϕ on X such that for each j ∈ {1, . . . , N } there exists ϕj ∈ Cj (X) such that limr→∞ |ϕ(rω) − ϕj (rω)| = 0 for all ω ∈ Σj , uniformly in ω on each

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compact subset of Σj . The (uniquely determined) collection {ϕj | j = 1, . . . , N } corresponding to ϕ ∈ A will be called the asymptotic functions of ϕ. If Σ is some subset of Σj and R > 0, let WjR (Σ) be the subset {rω | r > R, ω ∈ Σ} of X. The application of the results of Section 5 leads to the following non-propagation property into the cone subtended by Σj : Proposition 1. Let V ∈ A be real and denote its asymptotic functions by {Vj }. Set H = −∆ + V . Fix a number j ∈ {1, . . . , N } and choose η ∈ C0 (R) with supp η disjoint from the spectrum of the periodic Schr¨ odinger operator −∆ + Vj . Then, given a compact subset K of Σj and ε > 0, there is R ∈ (0, ∞) such that for each f ∈ L2 (X) we have sup  χW R (K) (Q)e−itH η(H)f ≤ ε  f  . t∈R

(4)

j

The proof will be given in a series of lemmas. The validity of (4) is obtained by combining the last two lemmas (Lemma 5 and Lemma 6) with the corollary given in Section 5 and with the remark at the end of Section 4. The estimate (4) gives a precise meaning to the statement made in the second example of the introduction that states with spectral support away from certain subsets of R will not propagate into the asymptotic part of the cone Cj subtended by Σj . We shall use the following notations: WjR ≡ WjR (Σj ) = {rω | r > R > 0, ω ∈ Σj } ⊂ X, and Tj = X/Γj (the class of z ∈ X in Tj , denoted by ζ, is given as ζ = {x ∈ X | x = z + γ, γ ∈ Γj }). We observe that Cj (X) is isomorphic to C(Tj ) and that the correspondence ϕ → ϕj defines a ∗-homomorphism Ψj from A to Cj (X). Lemma 3. (a) A is a unital C ∗ -algebra containing C0 (X) and invariant under translations. (b) The ∗-homomorphism Ψj : A → Cj (X) is surjective. Proof. (a) It is clear that A contains C0 (X) and the constants. To see that A is closed, let {ϕ(k) } be a Cauchy sequence in A, denote by ϕ ∈ Cbu (X) its limit (k) and by ϕj (j = 1, . . . , N ) the asymptotic functions of ϕ(k) . Let us show that, for (k)

fixed j, {ϕj (k)

| k ∈ N} is Cauchy in the norm of Cbu (X). We have for any γ ∈ Γj :

(l)

|ϕj (x) − ϕj (x)|

(k)

(l)

(k)

= |ϕj (x + γ) − ϕj (x + γ)| ≤ |ϕj (x + γ) − ϕ(k) (x + γ)| (l)

+ |ϕ(k) (x + γ) − ϕ(l) (x + γ)| + |ϕ(l) (x + γ) − ϕj (x + γ)|. Fix ε > 0. The second term on the right-hand side is less than ε/3 for all x and all γ if k, l > L for some L ∈ N. For fixed k and l the first and the third term are less than ε/3 in the sup norm (with respect to x) since for each fixed x ∈ X and any R > 0 one may find γ ∈ Γj such that x + γ ∈ WjR (K) if K is a compact subset of Σj with non-empty interior.

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(k)

Now define ϕj = limk→∞ ϕj and observe that ϕj ∈ Cj (X). By another ε/3 type argument one then finds that these functions ϕj are asymptotic functions of ϕ. Let us show that A is invariant under translations. If x ∈ X and ϕ ∈ A with asymptotic functions {ϕj }, then the collection {ax (ϕj )} are asymptotic functions of ax (ϕ), hence ax (ϕ) ∈ A: 



x  x    |ax (ϕ)(rω) − ax (ϕj )(rω)| = ϕ r ω + − ϕj r ω +  → 0 as r → ∞, r r uniformly in ω belonging to compact subsets of Σj . (b) Let ψ ∈ Cj (X). Let ϕ = θψ, where θ is a function in Cbu (X) that is homogeneous of degree zero outside the unit ball of X and satisfies θ(x) = 1 on Wj1 and θ(x) = 0 on Wk1 if k = j. Then ϕ ∈ A, with asymptotic functions ϕj = ψ, ϕk = 0 for k = j.
We next show that there is a canonical identification of Tj with a closed subset Tj of Z = X \ i(X), where X = Ω(A) as before. This is a direct consequence of the fact that there is a surjective ∗-homomorphism Φj : C(Z) → C(Tj ), deduced from Ψj : A → Cj (X), which is a surjective ∗-homomorphism with kernel including C0 (X), and from the natural isomorphisms A/C0 (X) ∼ = C(Z) and Cj (X) ∼ = C(Tj ). Below we shall make the construction as explicit as possible. For this we introduce a mapping ij : Tj → X that associates to ζ ∈ Tj (j) (j) the character ij (ζ) ≡ τζ ∈ X given as τζ (ϕ) = ϕj (z), where ϕj is the j-th asymptotic function of ϕ and z is any representative of the class ζ. In other terms (j) τζ = τz ◦ Ψj , where τz is interpreted as a character of C(Tj ). We set Tj = ij (Tj ). Lemma 4. (a) Tj is contained in X \ i(X). (j)

(b) The correspondence ζ → τζ

is injective and continuous.

(c) The set Tj is closed. (d) The set Tj is invariant under all translations. (j)

Proof. (a) If ζ ∈ Tj , then τζ

does not belong to i(X): if x ∈ X, choose a (j)

function ϕ ∈ C0 (X) such that ϕ(x) = 0: then ϕj = 0, so that τζ (ϕ) = 0 = (j)

ϕ(x) = [i(x)](ϕ). Thus τζ = i(x) for each x ∈ X. / Γj . Choose ϕj ∈ Cj (X) such that ϕj (z1 ) = ϕj (z2 ) (b) Assume that z1 − z2 ∈ (j) and let ϕ ∈ A be such that ϕj = Ψj (ϕ) (Lemma 3.(b)). Then τζ1 (ϕ) = ϕj (z1 ) = (j)

(j)

(j)

ϕj (z2 ) = τζ2 (ϕ), so τζ1 = τζ2 . Thus the mapping ij is injective. Its continuity is easy to establish. (c) Tj is the continuous image of the compact space Tj , hence it is a compact subset of Z.

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(d) Let z ∈ X be a representative of ζ ∈ Tj . For x ∈ X, denote the class of z + x in Tj by ζ + ξ. Then (j) (j) (j) ρx (τζ ) (ϕ) = τζ [ax (ϕ)] = [ax (ϕ)]j (z) = [ax (ϕj )](z) = ϕj (z + x) = τζ+ξ (ϕ). (j)

(j)

Hence ρx (τζ ) = τζ+ξ , the character associated with the class of z + x in Tj . We conclude that Tj is a single orbit under the representation ρ of X in X .
We need to find subsets of i−1 [W ∩ i(X)] easy to express in terms of the geometry of X, for arbitrary neighbourhoods W of Tj . Lemma 5. Let W be a neighbourhood of Tj in X and set W = i−1 [W ∩ i(X)]. Given any compact subset K of Σj there is R ∈ (0, ∞) such that WjR (K) ⊂ W . Proof. By Lemma 2 applied to F = Tj , there are ε > 0 and a finite family F = {ϕ1 , . . . , ϕm } of elements of A such that ∪τ ∈Tj VF ,ε (τ ) ⊂ W, in other terms such (j) that ∩i {τ  ∈ X | |τ  (ϕi ) − τζ (ϕi )| < ε} ⊂ W for each ζ ∈ Tj . Upon restricting  to characters τ belonging to i(X) and denoting the j-th asymptotic function of ϕi by ϕi,j , we get immediately that ∩i {x ∈ X | |ϕi (x) − ϕi,j (x)| < ε} ⊂ W . Now for each i there is Ri ∈ (0, ∞) such that |ϕi (x) − ϕi,j (x)| < ε for all x ∈ WjRi (K). Then clearly the assertion of the Lemma holds for R = max{R1 , . . . , Rm }.
We finally specify the KTj -essential spectrum of H. Lemma 6. The set σKTj (ΦH ) coincides with the (band) spectrum of the periodic Schr¨ odinger operator Hj = −∆ + Vj . Proof. Let us denote by Πj : CA → CA /KTj the canonical ∗-homomorphism. By definition, σKTj (ΦH ) is the spectrum of the observable Πj ◦ ΦH affiliated with CA /KTj . It is enough to show that CA /KTj is isomorphic to CCj (X) and that the image of Πj ◦ΦH under this isomorphism is ΦHj ; this will conclude the proof, since isomorphisms of C ∗ -algebras leave the spectra of observables invariant. For any M ∈ N, ϕ1 , . . . , ϕM ∈ A and ψ1 , . . . , ψM ∈ C0 (X) we set 

M M   Θj ϕi (Q)ψi (P ) = ϕi,j (Q)ψi (P ), i=1

i=1

where ϕi,j ∈ Cj (X) is the j-th asymptotic function of ϕi . By the discussion in Section 4, Θj extends to a surjective ∗-homomorphism CA → CCj (X) with kernel KTj = CKTj . A simple argument in terms of the Neumann series shows that Θj [(H− z)−1 ] = (Hj − z)−1 , so that Θj (ΦH ) = ΦHj .
Remark. Since K(H) = CC0 (X) = KZ ⊂ KTj , we have ∪N j=1 σ(Hj ) ⊂ σess (H). The behaviour of the bounded, uniformly continuous function V outside the cones

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{Cj }j=1,...,N is submitted to no constraint and the asymptotic functions Vj are not related. So it is possible to have a large set σess (H) \ ∪N j=1 σ(Hj ) on which the result of the proposition (non-propagation into the asymptotic part of ∪N j=1 Cj ) is relevant and non-trivial. Of course the simplest situation is that where N = 1, and the general case (N > 1) can be reduced to it since Proposition 1 involves only one value of j.

7 Other examples, comments Some other examples will be discussed briefly in the present section. Most of the proofs consist in suitable adaptations of arguments already used above and we shall just sketch them.

7.1

Example 1. Potentials that are asymptotically periodic in a half-space

This is the example that is the most close to that treated in [4]. It is also related to Section 6. Let us write X = R × X  , with X  = Rn−1 . For a periodic lattice Γ+ of X, we denote by C+ (X) the C ∗ -algebra of all complex continuous functions on X that are Γ+ -periodic. We shall consider the unital C ∗ -algebra A+ of all bounded, uniformly continuous functions ϕ : X → C such that there exist a (necessarily unique) element ϕ+ ∈ C+ (X) such that | ϕ(x1 , x ) − ϕ+ (x1 , x ) |→ 0 when x1 → +∞, uniformly in x ∈ X  . As before we call ϕ+ the asymptotic function of ϕ. It is easy to see that A+ is invariant under translations. Since we imposed no conditions on the behaviour of ϕ outside a remote half-space, we cannot determine the character space X of A+ precisely. But it is straightforward to show that the torus T+ = X/Γ+ is a closed invariant subset of its frontier and that any neighbourhood of T+ in X contains {(x1 , x ) ∈ X | x1 > R} for some R > 0 large enough (use Lemma 2). It is also easy to show that the quotient C ∗ -algebra CA+ /KT+ is isomorphic to CC+ (X) in such a way that ϕ(Q)ψ(P ) corresponds to ϕ+ (Q)ψ(P ); here ϕ ∈ A+ , ϕ+ ∈ C+ (X) is its asymptotic function and ψ ∈ C0 (X). By applying the results of Section 5 and the discussion above one gets Proposition 2. Let V ∈ A+ be real and let V+ ∈ C+ (X) be its asymptotic function. Let H = −∆ + V and H+ = −∆ + V+ be the associated self-adjoint operators in H = L2 (X). Let η ∈ C0 (R) with supp η ∩ σ(H+ ) = ∅. Then for each ε > 0 there exist R > 0 such that  χ(Q1 ≥ R)e−itH η(H)f ≤ ε  f  for all f ∈ H and all t ∈ R. Of course, one can also introduce the C ∗ -algebra A− , consisting of all bounded, uniformly continuous functions that become Γ− -periodic (for some other periodic lattice Γ− ) at x1 = −∞, uniformly in the orthogonal variable x . The elements

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of A = A− ∩ A+ are bounded, uniformly continuous functions which have (different) periodic limits at x1 = ±∞; their behaviour in a vertical strip is submitted to no constraint.

7.2

Example 2. Potentials with asymptotic vanishing oscillation

We say that a function ϕ ∈ Cb (X) has asymptotic vanishing oscillation, and we write ϕ ∈ V O(X), if the function x → sup|y|≤1 | ϕ(x+y)−ϕ(x) | is of class C0 (X). We remark that V O(X) contains the C ∗ -algebra C rad (X) of all complex continuous functions on X that have radial limits at infinity, uniformly in all directions. Sums of the form ϕ0 +ϕ1 with ϕ0 ∈ C0 (X) and ϕ1 continuous and homogeneous of degree zero outside a ball are the most general elements of C rad (X). Note that V O(X) is considerably larger than C rad (X). A C 1 -function ϕ with ∂j ϕ ∈ C0 (X) for all j is in V O(X). This includes ϕ(x) = φ [(1 + |x|)p ] for p < 1 and φ, φ continuous and bounded. We point out that Proposition 3 will be particularly simple to interpret for potentials in C rad (X). Let X be the character space of V O(X). By identifying X with its homeomorphic image in X , we can express this character space as the disjoint union X = X  Z. The nice feature is that V O(X) is the largest unital, translation invariant C ∗ -subalgebra of Cbu (X) such that all the elements of Z are fixed points under the extension of the action of the group X. This was used in [11] to show that, for V ∈ V O(X), the essential spectrum of the Schr¨odinger operator H = −∆ + V is [min V (X)asy , ∞), where the asymptotic range of V is defined as V (X)asy = ∩K V (X \ K) with K varying over all compact subsets of X. This result is specific to the class V O(X). The frontier Z is not easy to understand, so we shall consider only closed sets F ⊂ Z which are suitably related to a given potential V . Let Vˆ be the continuous extension of V to X and G a closed subset of R such that its interior Go meets V (X)asy . We set F = Vˆ −1 (G) ∩ Z; it is a closed, non-void subset of Z, and it is automatically invariant under translations (this is the point which makes our analysis possible without extra information on Z). To apply the theorem, one has to find in X a filter base adjacent to F and to calculate the KF -essential spectrum of H. For the latter problem we proceed as in [11], where more details can be found. The set σKF (H) is the spectrum in CV O(X) /KF of the image of the observable ΦH through the canonical ∗-homomorphism CV O(X) → CV O(X) /KF . The quotient C ∗ -algebra CV O(X) /KF is isomorphic to C(F )
X (crossed product constructed in terms of the trivial action of X on C(F )). The latter can be embedded in the direct sum ⊕τ ∈F C
X ∼ = ⊕τ ∈F C0P (X), where C0P (X) is ∗ the C -subalgebra of B(H) of all the operators of the form ψ(P ), with ψ ∈ C0 (X). This leads to a ∗-homomorphism ΠF : CV O(X) → ⊕τ ∈F C0P (X) with kerˆ )ψ(P ))τ ∈F , where ϕˆ is nel KF . This ∗-homomorphism maps ϕ(Q)ψ(P ) to (ϕ(τ the continuous extension of ϕ to X . With the Neumann series  for the resolvent, it follows easily that the observable ΦH is mapped to ΦHτ τ ∈F , where Hτ =

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ˆ −∆ operator in H). It follows that σKF (H) = ∪τ ∈F σ(Hτ ) = + V (τ ) (self-adjoint  ˆ min V (F ), ∞ . By taking into account the definition of the closed set F and the fact that Vˆ (Z) = V (X)asy one easily shows that Vˆ (F ) = G ∩ V (X)asy , thus σKF (H) = [min{G ∩ V (X)asy }, ∞). We next indicate a suitable filter base in X adjacent to F . For any compact subset K of X we set WK = V −1 (G) ∩ K c . The assumption Go ∩ V (X)asy = ∅ implies that WK = ∅. Since WK ∩ WK
= WK∪K
, W = {WK }K is a filter base in X. Then (all closures are taken in X ): ∩K WK = ∩K [V −1 (G) ∩ K c ] ⊂ ∩K V −1 (G) ∩ K c   ⊂ Vˆ −1 (G) ∩ ∩K K c = Vˆ −1 (G) ∩ Z = F. Thus W is adjacent to F . By applying the corollary in Section 5 we obtain Proposition 3. Let V ∈ V O(X) and consider the self-adjoint operator H = −∆ + V in H = L2 (X), which defines an observable affiliated with CV O(X) . Let G ⊂ R be a closed set such that Go ∩V (X)asy = ∅, and let ε > 0. Then for each η ∈ C0 (R) with supp η ⊂ (−∞, min{G ∩ V (X)asy }) there is a compact subset K of X such that  χV −1 (G)\K (Q)e−itH η(H)f ≤ ε  f  for all f ∈ H and all t ∈ R. To illustrate this result, let us take G = [λ, ∞) with min [V (X)asy ] < λ < max [V (X)asy ]. Then, roughly, scattering states at energies situated below λ will not propagate into the asymptotic part of the set {x ∈ X | V (x) ≥ λ}. For a one-dimensional system with a slowly oscillating potential, this corresponds to tunneling through an infinite sequence of more and more widely separated barriers of increasing length. The effective parts of these barriers, for states with energy less than λ, occupy the intervals {x ∈ R \ K | V (x) ≥ λ} for some compact set K ⊂ R (as an example one may consider a potential that is asymptotically of the form cos(|x|β ) with 0 < β < 1; the essential spectrum of the associated Sturm-Liouville operator will often be continuous, cf. Theorem 4 in [7], in particular vectors with spectral support in the interval (−1, λ) will propagate away from each compact set K and thus undergo tunneling of the indicated type). For multi-dimensional systems there are various possibilities: for a spherically symmetric slowly oscillating potential there will be an infinite sequence of spherically symmetric barriers arranged (as a function of the radial variable r) in analogy with the one-dimensional case; for a potential having radial limits (V ∈ C rad (X)) there will be no propagation into the asymptotic part of the cone subtended by {ω ∈ S | limr→∞ V (rω) ≥ λ}; for certain slowly oscillating non-spherically symmetric potentials there may be an infinite collection of inaccessible regions of increasing size towards infinity.

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The above type of behaviour is specific to the class V O(X) and is related to the fact that the action of translations on the frontier Z is trivial. If V would tend at infinity to a periodic function for instance, the connection between the localization in energy and the domains of non-propagation has a different nature, as seen in Section 6.

7.3

Example 3. Potentials with cartesian anisotropy

We shall work here with X = R2 ≡ R1 × R2 . The generalization to arbitrary dimension is straightforward. For j = 1, 2 let us denote by Aj the C ∗ -algebra of all continuous functions ϕ : Rj → C such that the limits c± j = limxj →±∞ ϕ(xj ) exist. Then A = A1 ⊗ A2 is a unital C ∗ -subalgebra of Cbu (X) that is invariant under translations by elements of X. j j = Rj ∪{−∞j , +∞j } be the two-point compactification of Rj . Then R Let R 1 × R 2 is the spectrum of A. The closed invariant is the spectrum of Aj and X = R subsets of the frontier Z = X \ X are as follows: the four corners {(±∞1 , ±∞2 )}, 1 ×{±∞2 } and {±∞1 }× R 2 and all their unions. We shall illustrate the four edges R 1 × {+∞2 }. our theorem for F = {(+∞1 , +∞2 )} and for F = R So let us consider the self-adjoint operator H = −∆ + V in H = L2 (X), for V ∈ A a real function. It is easy to show that V has the following property (which in fact characterizes the elements of A): for j, k ∈ {1, 2} and k = j, the limits Vj± (xj ) = limxk →±∞k V (x) exist uniformly in xj ∈ Rj and define elements of Aj . The values taken by the continuous extension of V to X on the four edges coincide respectively with Vj± , j = 1, 2. Its values at the four corners will be denoted by c++ , c+− , c−+ , c−− . Note that for example c++ = limx1 →+∞1 V1+ (x1 ) = limx2 →+∞2 V2+ (x2 ).   Let us consider the operator H1+ = −∆+V1+ = −∆1 + V1+ ⊗12 +11 ⊗(−∆2 ) + in the representation L2 (R1 ) ⊗ L2 (R2 ). Its spectrum equals [a+ 1 , ∞), where a1 + 1,+ is the infimum of the spectrum of the operator H = −∆1 + V1 acting in of this kind are available and, with obvious H1 = L2 (R1 ). Three other operators   − + − notations, we have σess (H) = min{a+ , a , a , a }, ∞ . This follows quite easily 1 1 2 2 from our arguments and was proved in a greater generality in §3 of [11]. Remark also that the spectrum of H ±± = −∆ + c±± is [c±± , ∞) and that inequalities such −+ ++ as a+ , c } are true. 1 ≤ min{c A neighbourhood base of the point {(+∞1 , +∞2 )} is composed of all the rectangles {(y1 , +∞1 ] × (y2 , +∞2 ] | y1 ∈ R1 , y2 ∈ R2 } and a neighbourhood base 1 × (y2 , +∞2 ] | y2 ∈ R2 }. We get: 1 × {+∞2 } consists of the rectangles {R of R Proposition 4. Let ε > 0. (a) For any η ∈ C0 (R) with supp η ⊂ (−∞, c++ ) there exist y1 ∈ R1 , y2 ∈ R2 such that for all f ∈ H  χ(Q1 > y1 )χ(Q2 > y2 )e−itH η(H)f ≤ ε  f  .

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(b) For any η ∈ C0 (R) with supp η ⊂ (−∞, a+ 1 ) there exists y2 ∈ R2 such that for all f ∈ H  χ(Q2 > y2 )e−itH η(H)f ≤ ε  f  . So, at energies below a+ 1 there is no propagation towards x2 = +∞2 . At ++ this becomes possible, but then the obenergies comprised between a+ 1 and c servable Q1 cannot diverge through positive values. Results of this type are by no means trivial in the sense that, in certain situations, propagation away from any compact subset of X does occur at energies as above (under suitable assumptions on V there will be intervals of purely absolutely continuous spectrum of H in the considered energy range, and associated states must propagate to infinity by the results of [2]).

Acknowledgments We are grateful to G¨ unter Stolz for correspondence and for pointing out to us the paper [7] and to the referee for drawing our attention to Refs. [3] and [5].

References [1] W.O. Amrein, A. Boutet de Monvel and V. Georgescu, C0 -Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians, Birkh¨ auserVerlag, Basel-Boston-Berlin, 1996. [2] W.O. Amrein and V. Georgescu, On the Characterization of Bound States and Scattering States in Quantum Mechanics, Helv. Phys. Acta 46, 635–658 (1973). [3] J. Bellissard, K-Theory of C ∗ -algebras in Solid State Physics, Lecture Notes in Physics 257, 99–156 (1986). [4] E.B. Davies and B. Simon, Scattering Theory for Systems with Different Spatial Asymptotics on the Left and Right, Commun. Math. Phys. 63, 277– 301 (1978). [5] G.A. Elliott, Gaps in the Spectrum of an Almost Periodic Schr¨ odinger Operator, C.R. Math. Rep. Acad. Sci. Canada 4, 255–259 (1982). [6] V. Georgescu and A. Iftimovici, C ∗ -Algebras of Energy Observables: I. General Theory and Bumps Algebras, preprint 00–520 at http://www.ma.utexas.edu/mp arc/. [7] A.Ya. Gordon, Pure Point Spectrum under 1-Parameter Perturbations and Instability of Anderson Localizaton, Commun. Math. Phys. 164, 489–505 (1994).

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[8] A. Guichardet, Special Topics in Topological Algebras, Gordon and Breach, New York (1968). [9] R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, New York (1983). [10] M. M˘ antoiu, On a Class of Anisotropic Schr¨ odinger Operators, preprint 01– 201 at http://www. ma.utexas.edu/mp arc/. [11] M. M˘ antoiu, C ∗ -Algebras, Dynamical Systems at Infinity and the Essential Spectrum of Generalized Schr¨ odinger Operators, preprint 01–298 at http://www.ma.utexas.edu/mp arc/ and to appear in J. Reine Angew. Math. [12] G.J. Murphy, C ∗ -Algebras and Operator Theory, Academic Press, Boston (1990).

W.O. Amrein D´epartement de Physique Th´eorique Universit´e de Gen`eve 1211 Gen`eve 4 Switzerland email: [email protected] M. M˘ antoiu, R. Purice Institute of Mathematics of the Romanian Academy P.O.Box 1-764 Bucharest, RO-70700 Romania email: [email protected] email: [email protected] Communicated by Jean Bellissard submitted 15/05/02, accepted 22/06/02

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