Annales Henri Poincaré - Volume 6

Ann. Henri Poincaré 6 (2005) 1 – 30 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/010001-30 DOI 10.1007/s00023-005-...

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Ann. Henri Poincaré 6 (2005) 1 – 30 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/010001-30 DOI 10.1007/s00023-005-0197-9

Annales Henri Poincar´ e

Quantum Inequalities in Quantum Mechanics Simon P. Eveson, Christopher J. Fewster and Rainer Verch

Abstract. We study a phenomenon occurring in various areas of quantum physics, in which an observable density (such as an energy density) which is classically pointwise non-negative may assume arbitrarily negative expectation values after quantization, even though the spatially integrated density remains non-negative. Two prominent examples which have previously been studied are the energy density (in quantum field theory) and the probability flux of rightwards-moving particles (in quantum mechanics). However, in the quantum field context, it has been shown that the magnitude and space-time extension of negative energy densities are not arbitrary, but restricted by relations which have come to be known as ‘quantum inequalities’. In the present work, we explore the extent to which such quantum inequalities hold for typical quantum mechanical systems. We derive quantum inequalities of two types. The first are ‘kinematical’ quantum inequalities where spatially averaged densities are shown to be bounded below. Specifically, we obtain such kinematical quantum inequalities for the current density in one spatial dimension (imposing constraints on the backflow phenomenon) and for the densities arising in Weyl–Wigner quantization. The latter quantum inequalities are direct consequences of sharp G˚ arding inequalities. The second type are ‘dynamical’ quantum inequalities where one obtains bounds from below on temporally averaged densities. We derive such quantum inequalities in the case of the energy density in general quantum mechanical systems having suitable decay properties on the negative spectral axis of the total energy. Furthermore, we obtain explicit numerical values for the quantum inequalities on the one-dimensional current density, using various spatial averaging weight functions. We also improve the numerical value of the related ‘backflow constant’ previously investigated by Bracken and Melloy. In many cases our numerical results are controlled by rigorous error estimates.

1 Introduction The uncertainty principle lies at the root of many of the counterintuitive features of quantum theory. Consider, for example, a quantum mechanical particle moving in one dimension, whose state is a superposition of right-moving plane waves. Although the expectation value of (any power of) its momentum is positive, nonetheless it is possible for the probability flux at, say, the origin to become negative. Thus the probability of finding the particle in the right-hand half-line can decrease! We will return to this phenomenon, which has come to be known as backflow [3, 7, 30], in Sect. 2.1. Another, related, phenomenon occurs in quantum field theory. Even if one starts with a classical field theory in which energy densities

2

S.P. Eveson, Ch.J. Fewster and R. Verch

Ann. Henri Poincaré

(as measured by all observers) are everywhere positive,1 one finds that the renormalized energy density of the quantized field can assume negative values [11] and (in all models known to date) can even be made arbitrarily negative at a given spacetime point by a suitable choice of state. For example, the energy density between Casimir plates is computed to be negative; a fact indirectly supported by experiment ([9]; see the recent review [5] for an exhaustive list of up to date references). Various authors have suggested employing such effects to sustain exotic spacetime geometries containing wormholes [31] or ‘warp drive’ bubbles [2]. Such suggestions are, however, severely constrained [23, 33] by the existence of bounds, known as quantum inequalities (QIs) or quantum weak energy inequalities (QWEIs) [13, 14, 16, 18, 19, 20, 21, 22, 24, 34] which impose limitations on the magnitude and duration of negative energy densities. To give an example, let ρ(t)ψ be the energy density of the free scalar field2 measured along an inertial worldline in Minkowski space. Then, for any real-valued smooth compactly supported g, the averaged energy density obeys [14, 17] ∞ dt g(t)2 ρ(t)ψ ≥ − du Q(u)| g(u)|2 (1.1) 0

for all physically reasonable (Hadamard) states ψ, where Q is a known function of polynomial growth. The purpose of this paper is to apply techniques developed in the field theoretic setting to quantum mechanical problems. In so doing we wish to draw attention to a circle of ideas – including sharp G˚ arding inequalities, dynamical stability and the QWEIs – which eventually ought to be seen in the wider context of quantization theory. We begin with a discussion of ‘kinematical QIs’ in Sect. 2, taking the probability flux as our main example. We develop bounds on spatially averaged fluxes which share some technical similarity with the QWEI proved by two of us for the Dirac field [19] (see also [15]). An important aspect of our treatment is the numerical analysis of these bounds. To some extent this is motivated by the recent observation of Marecki [29] that QIs may have observational consequences in quantum optics; it is therefore important to know how sharp analytically tractable bounds are. The techniques used here may also be of independent interest, and we give a detailed account in Sect. 5. Before that, in Sect. 3 we establish a very general form of kinematical QIs arising in the Weyl–Wigner approach to quantizing classical systems (see, e.g., [28] as a general reference on that approach). More precisely, we consider the (quantized) configuration space density Rn x → ρF (x)ψ for normalized wave-functions ψ ∈ S (Rn ) which are associated with classical observables F , i.e., functions on phase space Rn × Rn , by n d p ρF (x)ψ = F (x, p)Wψ (x, p) , (1.2) 2π 1 In

general relativity, one would say that the field obeyed the weak energy condition (WEC). statements can be made for the Maxwell, Proca and Dirac fields [32, 16, 15].

2 Similar

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Quantum Inequalities in Quantum Mechanics

3

where Wψ denotes the Wigner function of ψ. Even if F is everywhere non-negative, the density ρF (x)ψ may assume negative values owing to the indefinite sign of the Wigner function; in fact, we show that, under very general conditions on F , this quantity is unbounded above and below for arbitrary given x upon varying ψ. Conversely, if F belongs to a certain class of symbols (in the sense of microlocal analysis [26, 36]) which are of second order (or lower) in the momentum variables, and if F is everywhere non-negative, then we establish a kinematical quantum inequality of the form (1.3) dn x χ(x)ρF (x)ψ ≥ −C with a suitable constant C depending on the non-negative weight-function χ, but not on the (normalized) wave-function ψ. This is a straightforward consequence of the sharp G˚ arding inequality [12, 27, 36]. The general result will be illustrated by a direct derivation of a kinematical QI for the energy density. In Sect. 4 we focus attention on ‘dynamical’ QIs which bound temporal averages of the energy density in general quantum mechanical systems. (In fact our kinematical flux inequality may be regained as the special case, in which the evolution is the group of translations on the line.) These are conceptually much closer to the QIs which have been obtained in quantum field theory; in fact, the method we use to establish these dynamical QIs makes contact with the techniques employed in [19]. Some features of the general result will be illustrated by taking the harmonic oscillator as a concrete example. We summarize our main results in the conclusion, Sect. 6.

2 A motivating example 2.1

Probability backflow

We begin with a simple example: the motion of a quantum mechanical particle in one dimension. Some time ago, Allcock [3] pointed out the existence of rightwards-moving states (in which the velocity is positive with unit probability) but for which the probability of locating the particle in the right-hand half-line is instantaneously decreasing – a phenomenon known as probability backflow. This phenomenon was subsequently studied in much greater detail by Bracken and Melloy [7] (see also [30]). To illustrate the idea, let us suppose the normalized state ψ (as well as being square-integrable itself) has a continuous, square integrable first derivative. Then the corresponding probability flux at position x is given by Re ψ(x)(pψ(x)) , (2.1) m where the momentum operator is, as usual, p = −id/dx and the particle has mass m. Now the spatial integral of the flux is pψ Re ψ | pψ jψ (x) dx = = (2.2) m m jψ (x) =

4



and therefore yields the expected velocity. If ψ is a normalized right-moving wavepacket, it may be written by means of the Fourier transform as a superposition of right-moving plane waves dk ikx e ψ(k) ψ(x) = (2.3) 2π with ψ(k) = 0 for k < 0, so pψ =

0

∞

dk 2 k|ψ(k)| >0 2π

(2.4)

and we see that the spatially integrated flux is positive. However this does not imply that the flux itself is everywhere non-negative. Indeed, suppose that √ (2.5) ψ k0 (k) = N χ[0,k0 ] (k) k 3 − k0 , √ where χΩ denotes the characteristic function of Ω and N = (k03 (2 − 3)/(2π))−1/2 is a normalization constant. One may calculate 1 k02 1 k 2 jψk0 (0) = −√ (2.6) ∼ −0.006 0 , 4πm 2 m 3 which is not merely negative, but can clearly be made as negative as we wish by tuning k0 . Because the probability flux is negative at the origin, the probability of locating the particle in the left-half line is instantaneously increasing, thereby providing an example of the backflow phenomenon mentioned above. Backflow provides a nice illustration of the inadequacy of the phase velocity alone to predict the motion of a wavepacket. The three plots in Figure 1 indicate the time evolution of the position probability density in time under the free Hamiltonian; although the packet moves to the right, the two main peaks are reshaped in such a way that net probability has passed from the right-hand half line to the left. The wavepacket is given by Eq. (2.5) at time t = 0 with k0 = 5, m = 1/2 and = 1. Although these plots were obtained using the free Hamiltonian, we expect qualitatively similar behavior for evolutions generated by a wide class of Schr¨ odinger operators, on sufficiently small timescales, because the flux describes the flow of probability density for all such evolutions and a negative flux at one instant must persist for an interval of time by continuity. Nonetheless the free evolution is of special significance because an initially right-moving state will be purely right-moving at all times: backflow is not a scattering effect.

2.2

A quantum inequality for the flux

As we will see in Sect. 3 the backflow effect may be traced to the uncertainty principle. From this point of view, it is natural to seek bounds on its magnitude

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Figure 1: Evolution of a wavepacket under the free dynamics, illustrating the backflow phenomenon. From left to right, the plots show the position probability density at times t = −0.1, t = 0 and t = 0.1. and extent. Bracken and Melloy [7] studied the probability Pψ (t) of finding the particle in the left-hand half-line at time t, given that its state at t = 0 is ψ. They showed that (2.7) sup(Pψ (t) − Pψ (0)) = λ , ψ

for all t > 0, where the supremum is taken over all right-moving states ψ (i.e., ψ ∈ L2 (R) with ψ supported in R+ ) and the dimensionless constant λ is the largest positive eigenvalue of the equation 1 ∞ sin u2 − v 2 ϕ(v) dv = λϕ(u) (2.8) − π 0 u−v (for ϕ ∈ L2 (R+ )). It is striking that λ is not only independent of t, but also of the particle mass and Planck’s constant: backflow is an example of a purely quantum effect with no dependence on ! Although no analytical solution of Eq. (2.8) is known, Bracken and Melloy presented numerical evidence that λ ∼ 0.04. Using the numerical methods described in Sect. 5, we have recalculated this quantity to a much higher accuracy, although we have been unable to obtain consonant analytical error estimates. It turns out to be convenient to change variables to x = u2 ; we then consider the truncation of the resulting integral kernel to [0, X]. The maximum eigenvalue λ(X) was then calculated for values of X ranging from 6000 to 24000, using X/2 quadrature nodes. This choice was based on calculations using a variety of densities for

6



λ(X) 0.0382 0.03815 0.0381 0.03805 0.038 0

10000

5000

15000

20000

X √ Figure 2: The least squares fit of λ(X) to a + b/ X.

values of X around 2000 for which X/2 nodes provide accuracy to 5 significant figures. By contrast, the largest calculation conducted in [7] corresponds to X = 625, which reflects the increase in available computing power over the past decade. √ The resulting data may be fitted to a remarkable degree by the form λ(X) = a + b/ X (as already noted by Bracken and Melloy for their data). Using a least squares fit to this, we obtain the estimate λ = 0.03845182014 with a maximum percentage residual error under 4 × 10−4 %. Assuming the residual errors would be comparable for larger X, this suggests that λ = 0.038452 to this level of precision. Our data points and the best-fit curve are shown in Fig. 2. One may interpret the Bracken–Melloy bound (2.7) as a demonstration of the transitory nature of backflow: large negative fluxes for right-moving states must be short-lived. Here, we present an apparently new bound, which demonstrates that such fluxes are also of small spatial extent, and whose proof is related to the quantum weak energy inequalities derived by two of us for the Dirac quantum field [19] (see also [15]). We consider spatially smeared quantities of the form jψ (f ) =

jψ (x)f (x) dx ,

(2.9)

which may be regarded as the instantaneous probability flux measured by a spatially extended detector. For any smooth, compactly supported, complex-valued function g, we will show that

jψ (x)|g(x)| dx ≥ − 8πm 2

dx |g (x)|2

(2.10)

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for all normalized states ψ belonging to the class R of right-moving states defined by = 0 for k < 0 and ψ continuous and square-integrable} . R = {ψ ∈ L2 (R) : ψ(k) (2.11) In fact, the conditions on both g and ψ may be weakened slightly.3 Before giving the proof, let us make three observations. 1. First, we note that there is no upper bound on the smeared flux. To see this, choose any normalized ψ ∈ R and let ψλ (x) = eiλx ψ(x). We have ψλ ∈ R for λ ≥ 0; moreover, jψλ (x) = jψ (x) + so

λ |ψ(x)|2 m

(2.12)

jψλ (x)f (x) dx → +∞ as λ → +∞.

2. Second, the scaling behavior of the above bound may be investigated by replacing g by gλ (x) = λ−1/2 g(x/λ), whereupon the right-hand side of inequality (2.10) scales by a factor of λ−2 . The limit λ → 0 corresponds to the unboundedness below of the probability flux at a point, while the limit λ → ∞ is consistent with the fact that pψ ≥ 0 for ψ ∈ R (because the bound vanishes more rapidly than λ−1 ). Roughly speaking, our bound asserts that the magnitude of negative flux times the square of its spatial extent satisfies a state-independent upper bound on R. Thus the extent of backflow is limited both in space and in time. Note also that the bound (2.10) vanishes in both the classical limit → 0 and the limit of large mass. This differs from Bracken and Melloy’s inequality (2.7) in which the dimensionless constant λ is independent of and m. We remark that – again in contrast to [7] – our result is kinematical rather than dynamical: no specific Hamiltonian is invoked. Here, ‘kinematic’ refers to the kinematics of the Schrödinger representation, i.e., the (unique) regular representation of the Heisenberg commutation relations. 3. Finally, on integration by parts, Eq. (2.10) can be reformulated as the assertion that for each normalized ψ ∈ R, the Schr¨ odinger operator 2 d2 + 4πjψ (x) (2.13) 2m dx2 is positive on the space of smooth compactly supported functions g, in the sense that g(x)(Hψ g)(x) ≥ 0 (2.14) Hψ = −

3 In particular, continuity of ψ may be weakened to ψ ∈ AC(R) ∩ L2 (R) with ψ ∈ L2 (R) at the expense of augmenting some statements with the qualification ‘almost everywhere’; by an approximation argument it is easy to see that (2.10) holds for all g belonging to the Sobolev space W 1,2 (R) [1].

8



for all such g. Although the physical significance of this reformulation is not clear, it can provide useful necessary conditions for a given function j(x) to be the flux of a right-moving state. Only if the corresponding Schr¨ odinger operator has no bound states can this be the case. This can be sharpened slightly: as an illustration, suppose jψ (x) is the flux of a state in R with jψ (x) ≤ −M on some open interval I of length a. Then positivity of Hψ on C0∞ (I) implies that the Friedrichs extension HM of the operator −

2 d2 − 4πM 2m dx2

on C0∞ (I) ⊂ L2 (I)

(2.15)

is also positive. Since the Friedrichs extension of this operator corresponds to the imposition of Dirichlet boundary conditions at the boundary of I, HM has spectrum En =

2 n2 π 2 − 4πM 2ma2

(n = 1, 2, 3, . . .)

(2.16)

so we deduce (from E1 ≥ 0) that M ≤ π/(8ma2 ). This provides a more quantitative version of the connection between the magnitude and spatial extent of negative fluxes. Similar ideas have been employed in the context of quantum weak energy inequalities [18] to cast light on the ‘quantum interest conjecture’ of Ford and Roman [25]. We now establish the quantum inequality (2.10). It is sufficient to prove this for the case in which g ∈ C0∞ (R) is real-valued. Setting f (x) = g(x)2 and writing Mf for the multiplication operator (Mf ψ)(x) = f (x)ψ(x), we have 1 jψ (x)f (x) dx = Re ψ | Mf pψ m 1 = Re (ψ | Mg pMg ψ + ψ | Mg [Mg , p]ψ) m 1 ψ | Mg pMg ψ = m dk 2 k Mg ψ(k) , (2.17) = m 2π where we have used the fact that Re ψ | Mg [Mg , p]ψ = Re iψ | Mgg ψ = 0. We therefore may obtain a bound by estimating the portion of this integral arising from k < 0: 2 0 dk 2 ∞ dk k Mg ψ(k) = − k Mg ψ(−k) . (2.18) jψ (x)f (x) dx ≥ m −∞ 2π m 0 2π By the convolution theorem M g ψ(k) =

0

∞

dk ψ(k ) g(k − k ) , 2π

(2.19)

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where the restriction to k ∈ R+ is permissible for ψ ∈ R. Now a straightforward application of Cauchy-Schwarz gives 2 ∞ dk | g (k + k )|2 , ψ(−k) (2.20) M g ≤ 2π 0 where we have also used | g(−k)|2 = | g(k)|2 (since g is real) and ψ = 1. Substituting in (2.18), we now calculate ∞ dk ∞ dk k| g(k + k )|2 jψ (x)f (x) dx ≥ − m 0 2π 0 2π u ∞ du 2 = − | g (u)| dk k m 0 (2π)2 0 ∞ du 2 u | g (u)|2 = − m 0 8π 2 ∞ du 2 = − u | g (u)|2 m −∞ 16π 2 dx |g (x)|2 , = − (2.21) 8πm where we have changed variables from (k, k ) to (u, k) with u = k + k , used evenness of | g(u)| and Parseval’s theorem. This completes the proof of the quantum inequality (2.10). The later stages of this argument may be rephrased as follows. The inequality (2.18) asserts that 2 jψ (x)f (x) dx ≥ − T ψ (2.22) m where the operator T acts on L2 (R+ , dk/(2π)) by dk √ (T ϕ)(k) = k g (−k − k )ϕ(k ) 2π

(2.23)

and is easily seen to be Hilbert–Schmidt. Varying over normalized ψ, the right-hand side of Eq. (2.22) is bounded below by − T 2, where T denotes the operator norm of T . This leads to the bounds (2.24) jψ (x)f (x) dx ≥ − T 2 ≥ − T 2H.S. , m m where the last inequality holds because the Hilbert–Schmidt norm T H.S. dominates the operator norm. The calculation in (2.21) in fact precisely computes this final bound. To summarize, we have seen that, even for a right-moving state ψ ∈ R, the flux jψ need not be pointwise non-negative; moreover by tuning the state, one may

10



arrange the normalized flux jψ (x) to be as negative as one likes at a given fixed x. However, weighted spatial averages of the flux are bounded below in terms of the weight function alone. This condition may be reformulated as asserting the positivity of the Hamiltonian for a particle moving in a potential given (up to constants) by the probability flux of any state in R.

2.3

Numerical results and sharper bounds

We illustrate our bound by reference to four weight functions: a Gaussian, a squared Lorentzian and two compactly supported weights which we call the truncated cosine and the smoothed truncated cosine. (Neither of the compactly supported weights are C ∞ , but they have sufficient smoothness for the above argument to hold; see footnote 3 above.) Our weight functions are summarized in Table 1, along with the corresponding bound arising from Eq. (2.10). In each case, fλ has unit integral, the parameter λ controls the sampling width and gλ (x) = fλ (x). For later reference we have also given the Fourier transforms of fλ and gλ . We wish to compare the above bound with two sharper (but less analytically tractable) bounds: the bound arising from the first inequality in (2.24) and a direct numerical estimate of the infimum of the integrated flux. In the first case, we are required to find the operator norm of T . Our numerical approach proceeds by first truncating the kernel to an interval [0, K] – we are able to estimate the error incurred here by using bounds obtained from the Hilbert–Schmidt norm – and applying a numerical quadrature scheme due originally to Fredholm (see, e.g., Sect. 4.1 of [4] or Chapter 4 of [10]) to the truncated kernel. This leads to a matrix whose eigenvalues approximate those of the truncated kernel and hence the original operator, and which can be computed using standard numerical packages. Full details, including a discussion of error estimates, are given in Sect. 5. This leads to quantum inequalities C , (2.25) jψ (x)fλ (x) dx ≥ − mλ2 where the constant C depends on the particular weight function used and is given: Kernel Gaussian Squared Lorentzian Trunc. cosine Smooth trunc. cosine

C 0.01958128485 (16π)−1 0.08463957004 0.125047838

Accuracy 10 S.F. Exact 2 S.F.? At least 3 S.F.

Improvement 1.6% 0 16% 4.5%

Note that we were only able to obtain fairly weak error bounds in the truncated cosine case. In each case, the improvement on the analytical bound is relatively small. We may interpret these results as showing that T = R+ S where R has rank 1 and the Hilbert–Schmidt norm of S is small relative to that of T . This is most apparent in the squared Lorentzian case, in which T is itself exactly rank 1 and

−/(16πmλ2 )

−/(16πmλ2 ) −0.01989436788

QI bound

[≈ /(mλ2 )×] −0.09817477044

−π/(32mλ2 )

4π 4 sin(λk) λk(k 2 λ2 − 4π 2 )(k 2 λ2 − π 2 )

π 2 sin(λk) λk(π 2 − k 2 λ2 ) √ 4π λ cos(λk) π 2 − 4k 2 λ2

K 2000 2200 2400 2600 2800 3000

Trunc. cosine µ(K) −0.029012801924 −0.029012804495 −0.029012806174 −0.029012807318 −0.029012808114 −0.029012808686

Smoothed trunc. cosine K µ(K) 140 −0.036095566956 160 −0.036095567038 180 −0.036095567056 200 −0.036095567060 220 −0.036095567061 360

Table 2: Numerical estimation of the minimum eigenvalue µ(K) of the truncation of J to [0, K] for various kernels.

Gaussian Squared Lorentzian K µ(K) K µ(K) 10 −0.0048295212087 30 −0.002980544308 20 −0.0048295668511 40 30 −0.0048295668517 50 40 60 50 70 60 80

−0.1308996939

−π/(24mλ2 )

2π 2 sin(λk) √ k 3λ(π 2 − k 2 λ2 )

4ϑ(λ − |x|)/(3λ)cos(xπ/(2λ))4

Smoothed truncated cosine

ϑ(λ − |x|)/λ cos(xπ/(2λ))

Truncated cosine

Table 1: Compendium of sampling functions considered.

−0.01989436788

√ 2λπe−λ|k|

(1 + λ|k|)e−λ|k|

√ 2 2λπ 1/4 e−(uλ) /2

/4

gλ

2

e−(λu)

2λ3 π −1 /(x2 + λ2 )2

√ 2 (λ π)−1 e−(x/λ)

fλ

fλ

Squared Lorentzian

Gaussian

Vol. 6, 2005 Quantum Inequalities in Quantum Mechanics 11

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no improvement is obtained by using the operator norm. It would be interesting to understand the origin of this apparently general phenomenon. Our second numerical calculation aims to compute the infimum of the spectrum of the unbounded integral operator ∞ dk (k + k )fλ (k − k )ϕ(k ) . (Jϕ)(k) = (2.26) 2 0 2π We proceed by truncating the kernel to the interval [0, K], computing the minimum eigenvalue µ(K) using sufficiently many quadrature points to obtain machine precision. We then increase K until convergence of µ(K) is obtained, again to machine precision. Our results are given in Table 2, in which we give µ(K) in units of /(mλ2 ). Blank entries indicate that the computed value was identical to the last printed number in that column. The density of quadrature points used (per unit K) was 5 for the Gaussian, 1 for the truncated cosine, and 5 for the smoothed truncated cosine, although higher densities were also used as a numerical check (40, 2, and 10 respectively). The results for the squared Lorentzian were rather slower to converge as the density increased (perhaps because the kernel fails to be everywhere smooth) and were computed using a density of 60. For K < 80, a density of 70 was used as a check. To summarize, we have seen that a) the limitations of our flux QI do not lie in the estimation of an operator norm by a Hilbert–Schmidt norm, but rather in the earlier stages of the derivation (probably the estimate (2.18)); b) the overall scope for improvement on our flux QI is roughly a factor of between 3 and 7 (in our examples), and it is clear that the sharp bound is not simply a multiple of our bound (2.10) (in contrast to the situation for two-dimensional massless quantum fields [21, 14]).

3 The Wigner function and kinematical quantum inequalities It is worth emphasizing that phenomena similar to those presented above arise naturally in the context of Weyl quantization, in which the phase space aspect of quantum mechanics is brought to the fore. In our discussion we will consider the phase space to be Rn × Rn (see, e.g., [28] for Weyl quantization on manifolds). We recall that the central object in this approach is the Wigner function Wψ defined on phase space by n 2 (3.1) dn y e2ipy/ ψ(x + y)ψ(x − y) , Wψ (x, p) = where ψ ∈ L2 (Rn ) is the corresponding normalized quantum mechanical state vector. The classical analogue of Wψ (x, p) would be a probability distribution on phase space; as is well known, however, Wψ is not itself a probability distribution because it is not guaranteed to be everywhere non-negative. This has important

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consequences for observables obtained via Weyl quantization, which proceeds as follows. Given an observable on the classical phase space, i.e., a smooth function4 F : Rn × Rn → R, Weyl quantization defines an operator Fw whose expectation values are given (for normalized ψ) by n n d xd p F (x, p)Wψ (x, p) . (3.2) Fw ψ = (2π)n The action of this operator may be written in the form n n d yd p F ([x + y]/2, p)ei(x−y)·p/ψ(y) . (Fw ψ)(x) = (2π)n

(3.3)

Let us note that this procedure also yields a natural definition for the quantum mechanical density associated with a classical observable. Namely, setting dn p F (x, p)Wψ (x, p) , (3.4) ρF (x)ψ = (2π)n it is clear that the spatial integral of ρF (x)ψ yields the expectation value Fw ψ for all F and ψ (modulo domain questions5 ). Now, because the Wigner function need not be everywhere positive, we see that the Weyl quantization of a non-negative classical observable may assume negative expectation values. This situation is exacerbated for the densities defined above (see statement (II) below). However, we will show that kinematical quantum inequalities may be derived, under certain conditions. Indeed, these bounds are obtained as applications of the so-called sharp G˚ arding inequalities in the theory of pseudodifferential operators [36, 12, 27]. It is interesting to note that Fefferman and Phong, to whom the most general sharp G˚ arding results are due, were guided by intuition arising from quantum mechanics: in particular, the uncertainty principle. We begin by specifying more precisely the class of classical observables. For m ) is defined to be m ∈ N, the symbol class S m (often denoted more precisely as S1,0 the set of smooth functions F : Rn × Rn → C such that, for each compact K ⊂ Rn and n-dimensional multi-indices α, β, there exists a constant CK,α,β such that |(Dxα Dpβ F )(x, p)| ≤ CK,α,β (1 + |p|)m−|β|

(3.5)

m for all (x, p) ∈ K × Rn (see, e.g., [26, 36] for multi-index notation). By Shom , we m denote the set of F ∈ S admitting a (unique) decomposition F = Fpr + Fsub such that the principal symbol Fpr belongs to S m , is homogeneous of degree m in momentum, i.e., Fpr (x, λp) = λm Fpr (x, p) for all (x, p) ∈ Rn and λ ∈ R+ , and 4 Precise

growth conditions will be specified below. example, this will certainly hold for F of polynomially bounded growth and ψ belonging to the Schwartz class. 5 For

14



is non-zero except at vanishing momentum, while the sub-principal symbol Fsub belongs to S m−1 . For F ∈ S m , the Weyl quantization Fw is a continuous linear map from ∞ C0 (Rn ) to C ∞ (Rn ), so Eq. (3.2) holds for all normalized ψ ∈ C0∞ (Rn ). The density ρF (x)ψ is in fact defined (and indeed smooth in x) for all ψ belonging to the Schwartz class S (Rn ); however, it is only guaranteed to be integrable for ψ ∈ C0∞ (Rn ). Our first result now reads as follows. Theorem 1 m (I) Suppose F ∈ Shom is real, for some m ≥ 1. Then, for each x, the density ρF (x)ψ is unbounded from both above and below as ψ varies in C0∞ (Rn ) with ||ψ||L2 = 1. (II) Suppose F ∈ S 2 is non-negative, F (x, p) ≥ 0 for all (x, p) and let χ ∈ C0∞ (Rn ) be non-negative. Then there exists a constant C ≥ 0, depending on F and χ, such that (3.6) dn x χ(x)ρF (x)ψ ≥ −C

for all ψ ∈ S (Rn ) with ||ψ||L2 = 1. Proof. To establish (I) we may assume without loss of generality that x = 0, for which we have n n n 2 d pd y ρF (0)ψ = F (0, p)e2ipy/ ψ(y)ψ(−y) (3.7) (2π)n for normalized ψ ∈ C0∞ (Rn ). Setting ψλ (x) = λ−n/2 ψ(x/λ), (λ > 0) and making the obvious change of variables, n n n 2 d pd y ρF (0)ψλ = F (0, p/λ)e2ipy/ ψ(y)ψ(−y) (3.8) λ (2π)n so, bearing in mind that |F (0, p/λ) − Fpr (0, p/λ)| ≤ C(1 + |p/λ|)m−1

(3.9)

m and Eq. (3.5), we obtain by definition of Shom

λm+n ρF (0)ψλ −→

n n n 2 d pd y Fpr (0, p)e2ipy/ ψ(y)ψ(−y) (2π)n

(3.10)

as λ → 0+ . It now remains to show that the right-hand side of this expression attains values of both signs as ψ varies in C0∞ (Rn ). To this end, assume (without loss) that Fpr (0, p) depends non-trivially on the first coordinate, p1 , of p. Integrating by parts, the right-hand side of (3.10) in the form Py (ψ(y)ψ(−y))|y=0 , where

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Py is a homogeneous linear partial differential operator (in y) of order m with (possibly complex) constant coefficients cα . We now consider ψ of the form ψ(y) = f (y1 )ei(y2 +···+yn ) χ(y)

(3.11)

where χ ∈ C0∞ (Rn ) is equal to unity in a neighborhood of the origin and f ∈ C0∞ (R). For such ψ we have Py (ψ(y)ψ(−y))|y=0 = Qy1 (f (y1 )f (−y1 ))|y1 =0

(3.12)

for some ordinary differential operator Qy1 =

q

ck (−i)k

k=0

dk dy1k

(3.13)

of order 1 ≤ q ≤ m with constant real coefficients. (That Qy1 is of order at least one is a consequence of our assumption that Fpr (0, p) depends non-trivially on p1 ; reality of the ck holds because the right-hand side of Eq. (3.10) is manifestly real for all ψ ∈ C0∞ (Rn ).) We now choose f so that f (0) = 1 and f (k) (0) = 0 for 1 ≤ k ≤ q − 1. Then by Leibniz’ formula, Py (ψ(y)ψ(−y))|y=0 = cq (−i)q f (q) (0) + (−1)q f (q) (0) + c0 . (3.14) It is now obvious that f may be chosen so that the right-hand side of this expression adopts values of both signs, completing the argument. Statement (II) is straightforward: because χ ∈ C0∞ (Rn ) and F ∈ S 2 , the symbol χF obeys uniform bounds |(Dxα Dpβ χF )(x, p)| ≤ Cα,β (1 + |p|)2−|β|

(3.15)

for all (x, p) ∈ Rn × Rn , so the sharp G˚ arding inequality (Corollary 18.6.11 in [27] with δ = 0, ρ = 1; see also Eq. (18.1.1) therein) entails the existence of a constant C such that (3.16) dn x χ(x)ρF (x)ψ = (χF )w ψ ≥ −C for all normalized ψ ∈ S (Rn ). This is the required kinematical quantum inequality. We now give two examples to illustrate the above ideas. Example 1: Consider a classical Hamiltonian H(x, p) =

p2 + V (x) 2m

(3.17)

on Rn ×Rn with V ∈ C ∞ (Rn ). The Hamiltonian density obtained from Eq. (3.4) is ρH (x)ψ =

2 |∇ψ(x)|2 − Re ψ(x)(ψ)(x) + V (x)|ψ(x)|2 , 4m

(3.18)

16



where = ∇2 is the Laplacian. Clearly ρH (x)ψ may be made arbitrarily negative as ψ varies in C0∞ (Rn ) by arranging that ∇ψ(x) = 0, ψ(x)ψ(x) > 0 and then – as in (I) above – scaling ψ about x, introducing ψλ (y) = λ−n/2 ψ(x + (y − x)/λ)

(3.19)

for which ρH (x)ψλ = −λ−(n+2)

2 Re ψ(x)(ψ)(x) + λ−n V (x)|ψ(x)|2 . 4m

(3.20)

As in the proof of (I), the subprincipal symbol drops out in the limit λ → 0+ , so ρH (x)ψλ → −∞ Since H ∈ S 2 we already know that a kinematical quantum inequality exists. However it is instructive to give a direct argument for this, which also yields an explicit bound. To this end, we note that, for any non-negative χ ∈ C0∞ (Rn ) and normalized ψ ∈ S (Rn ),

dn x χ(x)ρH (x)ψ =

1 1 pi ψ | Mχ pi ψ + ψ | (Mχ p2 + p2 Mχ )ψ + ψ | MχV ψ 4m 8m

(3.21)

where pi = −i∇i and (Mf ψ)(x) = f (x)ψ(x) is the operator of multiplication by f . Now [Mχ , p] = iM∇i χ , so Mχ p2 + p2 Mχ = 2pi Mχ pi + i[M∇i χ , pi ] = 2pi Mχ pi − 2 Mχ and hence

dn x χ(x)ρH (x)ψ =

1 pi ψ | Mχ pi ψ + ψ | ML ψ 2m

(3.22)

(3.23)

where

2 (χ)(x) + V (x)χ(x) . (3.24) 8m Since the first term in (3.23) is non-negative, we obtain the quantum inequality 2 n d x χ(x)ρH (x)ψ ≥ infn − (χ)(x) + V (x)χ(x) , (3.25) x∈R 8m L(x) = −

for all non-negative χ ∈ C0∞ (Rn ) and ψ ∈ S (Rn ). We note as a curiosity the appearance of a Schr¨ odinger operator applied to the weight χ (rather than the state ψ). In the case of a non-negative potential, we may obtain a QI (slightly weaker than that given above) in the form 1 inf (Hχ)(x) . (3.26) dn x χ(x)ρH (x)ψ ≥ 4 x∈Rn

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Example 2: We now show that (II) allows us to deduce the existence of a kinematical flux QI on rightwards moving states. Let f be a non-negative smooth compactly supported function. The averaged probability flux jψ (f ) is easily seen to be the expectation in state ψ of the Weyl quantization j(f ) of f (x)p/m, which is a (first order) element of the symbol class S 2 (but is of course negative for p < 0). Now let η(p) be smooth, vanishing for p < 0 and equal to p for p greater than some p0 , and set F (x, p) = f (x)η(p)/m. Then the quantization Fw differs from j(f ) on R ∩ S (R) only by a bounded operator. Accordingly (II) entails that jψ (f ) is bounded below for normalized ψ ∈ R ∩ S (R). Of course, this argument does not determine the magnitude of the bound, in contrast to the direct approach of Sect. 2.2. We should like to remark that in [6] there appears a result which is complementary to ours; in that reference, the authors consider the one-dimensional case n = 1 and show that there is a ψ-independent bound below (and above) on the integral of the Wigner function over elliptic sub-regions of the phase-plane which is much sharper than that implied by the a priori uniform bounds on the Wigner function. This is again an effect of averaging, this time over a region of finite extension in both x- and p-space. It would be interesting to see if this result can be generalized to higher dimensions through a generalization of (II) to a more general class of symbols; however, it is not at all clear that this can be accomplished as it apparently goes beyond the scope of sharp G˚ arding inequalities.

4 Dynamical quantum inequalities In this section we turn to a different type of QI, which is closer to those studied in quantum field theory. The focus here is on time-averages of the energy density at a fixed spatial point: we will refer to the QI bounds obtained as dynamical quantum inequalities. To keep the discussion fairly general, we assume that configuration space M is a topological space carrying a measure ν, so that the state space is H = L2 (M, dν). The dynamics is assumed to be generated by a self-adjoint Hamiltonian H which is defined on a dense domain in H . Each normalized state ψ belonging to the domain of H then determines both a position probability density ρ(t, x)ψ and a Hamiltonian density h(t, x)ψ by ρ(t, x)ψ

=

|ψt (x)|2

(4.1)

h(t, x)ψ

=

Re ψt (x)(Hψt )(x)

(4.2)

where ψt = e−iHt/ ψ. This definition of the energy density differs from that employed in Sect. 3; note that we are not assuming in this section that H is the quantization of a classical observable, so the above would appear to be the most natural definition. In particular, both the quantities defined are integrable with respect to the measure dν(x) for each t ∈ R, with integrals equal to unity and Hψ respectively. However, we will be interested mainly in time averages of these quantities at some fixed point x ∈ M . In so doing, we immediately encounter the

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problem that it does not generally make sense to speak of the value of an L1 function at a point.6 To avoid this, we introduce the spaces Hk = D((1 + H 2 )k/2 ) and assume that, for some k > 0, each element in Hk should be (almost everywhere equal to) a continuous function and that for each x ∈ M there is a vector ηx ∈ H such that ψ(x) = ηx | pk (H)ψ ∀ψ ∈ Hk . (4.3) Here, ψ(x) means the value at x of the continuous function to which ψ is almost everywhere equal, and we have written pk (E) = (1 + E 2 )k/2 . Therefore, the functional ψ → ηx | pk (H)ψ on Hk coincides with the δ-distribution concentrated at x, so that formally [as it is not an element of H ] pk (H)∗ ηx is the δ-distribution. In practice, these assumptions are fairly mild: in particular, for the case in which H is minus the Laplacian on some manifold they are simply a transcription of the content of Sobolev’s lemma. We remark that the necessary regularity in quantum field theoretic quantum inequalities is obtained by restricting to the class of Hadamard states, which would correspond to H∞ = k∈N Hk in the present context. It now makes sense to define the position and Hamiltonian densities as ρ(t, x)ψ

=

|ηx | pk (H)ψt |2

(4.4)

h(t, x)ψ

=

Re pk (H)ψt | ηx ηx | pk (H)Hψt

(4.5)

for normalized states ψ ∈ Hk+1 . Furthermore, one may easily check (using CauchySchwarz) that these quantities are bounded in t, so the time-averaged quantities ρx (f )ψ and hx (f )ψ given by ρx (f )ψ =

dt f (t)ρ(t, x)ψ

(4.6)

and the analogous equation for hx (f )ψ are well-defined for any smooth compactly supported function f . From now on, we denote the spectral measure of H by dPE . (In the case where H may be diagonalized by a basis of orthogonal eigenvectors φn with simple eigenvalues En ,

ψ | φn φn | ϕf (En ) dψ | PE ϕf (E) = n

=

dE

δ(E − En )ψ | φn φn | ϕf (E);

(4.7)

n

more generally, the projection-valued measure allows for the case of varying – even infinite – multiplicities and for both continuous and discrete spectrum.) While 6 Elements of the space L1 (M, dµ) are really equivalence classes of functions agreeing almost everywhere.

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there is some ambiguity in choosing k and ηx such that Eq. (4.3) holds, the measure on R defined by ηx | dPE ηx pk (E)2 (4.8) µx (∆) = ∆

for bounded Borel sets ∆ has an independent meaning. In fact, µx (∆) is simply the diagonal P∆ (x, x) of the integral kernel of the spectral projection P∆ of H on ∆,7 given by

µx (∆) = |φn (x)|2 , (4.9) n:En ∈∆

if H has purely discrete spectrum. Below, it will occasionally be useful to consider the corresponding measure arising from self-adjoint operators other than H; in (H ) these cases, we will write µx to denote the operator H involved. Finally, since 2 0 ≤ µx (∆) ≤ ηx supE∈∆ pk (E)2 , we see that µx is polynomially bounded. After these preliminaries, we come to the statement of our dynamical quantum inequalities. Theorem 2 (i) Let g be any real-valued, compactly supported function on R and set f = g 2 . Then given real numbers a < b, the inequalities bρx (f )ψ +

du Q+ (u)| g (u)|2 ≥ hx (f )ψ 2π ≥ aρx (f )ψ −

du Q− (u)| g (u)|2 2π

(4.10)

hold for all normalized ψ ∈ P[a,b] H , where Q− (u)

=

Q+ (u)

=

[a,b]

dµx (E){u + a − E}+

[a,b]

dµx (E){u − b + E}+

(4.11)

are non-negative, monotone increasing and polynomially bounded in u, and we have used the notation {λ}+ = max{0, λ}. (Similarly, we will write {λ}− = min{0, λ}.) Moreover, the first (resp., second) inequality in (4.10) also holds for all ψ ∈ P(−∞,b] Hk+1 (resp., P[a,∞) Hk+1 ) provided the integration range in (4.11) is replaced by (−∞, b] (resp., [a, ∞)). 7 Since,

for any ψ ∈ H , we have P∆ ψ ∈ D(pk (H)), it follows that (P∆ ψ)(x) = dν(y) P∆ (x, y)ψ(y) where P∆ (x, y) = (pk (H)P∆ ηx )(y) is continuous in y. This last quantity may easily be expressed as pk (H)P∆ ηx | pk (H)P∆ ηy , so in particular, P∆ (x, x) = pk (H)P∆ ηx 2 = µx (∆).

20



(ii) Suppose R− dµx (E)(1 + |E|) < ∞ and let g be as in (i). Then, for any fixed c ∈ R, the inequality du S(H − c1; u)| g (u)|2 (4.12) hx (f )ψ ≥ cρx (f )ψ − 2π holds for all normalized ψ ∈ Hk+1 , where S(H; u) = dµ(H) x (E) {u − E}+

(4.13)

is non-negative, monotone increasing and polynomially bounded in u. (There

is of course a dual statement, for the case R+ dµx (E)(1 + |E|) < ∞.) Before proceeding to the proof of these statements, we illustrate them by drawing some consequences. The interpretation of (i) is that a state with energy between a and b has an averaged energy density between a and b, suitably weighted by the averaged position probability density, modulo a certain latitude bounded by quantum inequalities. Replacing g by gλ (t) = λ−1/2 g(t/λ), we may consider the two regimes λ → 0+ , representing tightly peaked averages, and λ → ∞, which represents widely spread averages. In the former case, we have µx ([a, b]) ∞ du du Q± (u)| u| g(u)|2 gλ (u)|2 ∼ (4.14) 2π λ 2π 0 provided µx ([a, b]) > 0 (failing which the left-hand side vanishes identically). Thus the latitude afforded by the quantum inequality bound grows as the sampling becomes more tightly peaked. As λ → ∞, the QI latitude tends to zero and one may show that b ≥ lim sup λ→∞

hx (gλ2 )ψ hx (gλ2 )ψ ≥ lim inf 2 2 ) ≥ a λ→∞ ρx (gλ ρx (gλ )ψ ψ

(4.15)

for all ψ ∈ P[a,b] H , provided ρ(t, x)ψ is non-zero for some t. This ergodic result shows that the spatial and temporal averages of energy densities obey related constraints. As a second illustration, consider (ii) in the case where H has a discrete spectrum {En } with corresponding orthonormal eigenfunctions {φn }, and satisfying the integrability condition on µx (for example, H might be semibounded). Then, in the case c = 0,

du 2 | g(u)|2 |φn (x)|2 (u − En ) hx (g )ψ ≥ − 2π En ≤u

αn |φn (x)|2 (4.16) = − n

where αn =

∞

En

du | g(u)|2 (u − En ) .

(4.17)

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These formulae may be used to compare the relative ease of obtaining negative energy densities at different spatial locations. For example, the eigenfunctions φn of the harmonic oscillator H = p2 /(2m) + 12 mω 2 x2 on L2 (R) obey the following bounds (cf. the Appendix to Sect. V.3 in [35]): For any j ∈ N0 there exists cj > 0 and rj ∈ N0 such that sup |(1 + xj )φn (x)| ≤ cj (1 + n)rj

(4.18)

x∈R

for all n ∈ N0 . Thus, for all normalized ψ ∈ S (R) (in fact, for all ψ in a considerably larger domain) we have hx (g 2 )ψ ≥ −

∞

cj αn (1 + n)rj . 1 + |x|j n=0

(4.19)

In this case, it is clear that – for a fixed sampling function g – the αn form a rapidly decaying sequence and so the sum converges for any j. Thus we have shown that the state-independent bound on energy density is itself a rapidly decaying function of x. It is therefore generally easier to maintain negative energy densities near the classical equilibrium point rather than far away. Finally, consider (i) for the case H = −id/dx on H = L2 (R) and a particle of mass m. In this instance, the spaces Hk coincide with the Sobolev spaces W k,2 (R) and Sobolev’s lemma permits us to take k > 1/2. Then the dynamical evolution amounts to spatial translation and the averaged Hamiltonian density is related to the spatially averaged probability flux by hx (f )ψ = m jψ (f˜x )

(4.20)

where f˜x (t) = f (x − t). Moreover, the measure µx is easily seen to be given by µx (∆) = |∆|/(2π), where | · | denotes the usual Lebesgue measure. Then the second inequality in (i) may easily be checked to reproduce the flux inequality (2.10) for all ψ ∈ P[0,∞) W k+1,2 (R). Proof of Theorem 2. The assertions (i) and (ii) are based upon two facts, which will be proved below. First, for any c ∈ R and normalized ψ ∈ Hk+1 , one may show that 2 d hx (f )ψ − cρx (f )ψ = ( − c) ψ | pk (H) g (−1 [1 − H])ηx . 2π

(4.21)

Second, if ∆ is a Borel set then 2 ψ | pk (H) g (−1 [1 − H])ηx ≤ for all normalized ψ ∈ P∆ H .

∆

2

dµx (E) | g([ − E]/)| .

(4.22)

22



Putting these together, we obtain d { − c}− hx (f )ψ − cρx (f )ψ ≥ dµx (E) | g ([ − E]/)|2 2π ∆ du 2 | g (u)| = dµx (E) {u + E − c}− 2π ∆ du | g(u)|2 =− dµx (E) {u + c − E}+ , 2π ∆

(4.23)

for all normalized ψ ∈ P∆ H , where we have made the change of variables u → −u and exploited the fact that | g (u)|2 is even (because g is real-valued). The interchange of variables employed in the first step is justified provided the inner integral in the last line of (4.23) is polynomially bounded in u. To obtain the second inequality in (i), we set c = a and ∆ = [a, b] and observe that the inner integral in Eq. (4.23) reduces to Q− (u) and is polynomially bounded because µx is. The inequality clearly remains true for ψ ∈ P[a,∞) Hk+1 with ∆ = [a, ∞). To obtain (ii), we set ∆ = R and observe that the integrability condition R− dµx (E)(1 + |E|) < ∞ and polynomial boundedness of µx guarantee that S(H − c1, u) exists and is polynomially bounded. Inequality (4.12) follows

(H−c1)

(H) (E) F (E). from the above on observing that dµx (E) F (E − c) = dµx To obtain the first inequality in (i) and the dual statement to (ii), one argues in an analogous fashion from the calculation d 2 { − c}+ dµx (E) | g([ − E]/)| hx (f )ψ − cρx (f )ψ ≤ 2π ∆ du 2 | g(u)| = dµx (E) {u + E − c}+ , (4.24) 2π ∆ which holds for all normalized ψ ∈ P∆ H . It remains to prove the two facts presented as Eqs. (4.21) and (4.22) above. First, observe that for any normalized ψ ∈ Hk , ρx (f )ψ may be expressed as ρx (f )ψ = dt f (t) dψ | PE ηx dηx | PE ψei(E−E )t/ pk (E)pk (E ) (4.25) by the functional calculus. Performing the t integral first (which is legitimate since f is smooth and compactly supported) we obtain ρx (f )ψ = dψ | PE ηx dηx | PE ψf([E − E]/)pk (E)pk (E ) . (4.26) Since f = g 2 , the convolution theorem may be used to write d g([E − ]/) f ([E − E]/) = g([E − ]/) 2π

(4.27)

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using the fact that g(λ) = g(−λ) since g is real-valued. Substituting in (4.26), and again rearranging the order of integration, we obtain ρx (f )ψ

=

d 2π

dψ | PE ηx

dηx | PE ψ

g([E − ]/) g([E − ]/)pk (E)pk (E )

2 d dψ | PE ηx pk (E) = g ([E − ]/) 2π 2 d ψ | pk (H) = g (−1 [1 − H])ηx 2π

(4.28)

To treat hx (f )ψ for normalized ψ ∈ Hk+1 , we write hx (f )ψ =

1 2

dt f (t) dψ | PE ηx dηx | PE ψ ei(E−E

)t/

(E + E )pk (E)pk (E ) , (4.29)

by functional calculus and use the identity (E + E ) f ([E − E]/) = 2

d g ([E − ]/) g([E − ]/) 2π

(4.30)

in place of the convolution theorem. (See [19] and [15] for proofs of this identity.) By a derivation analogous to that used for ρx (f )ψ we then obtain hx (f )ψ =

2 d ψ | pk (H) g (−1 [1 − H])ηx 2π

(4.31)

and Eq. (4.21) follows from this equation and (4.28). The second assertion, Eq. (4.22), is proved by noting that 2 ψ | pk (H) g (−1 [1 − H])ηx ≤ P∆ pk (H) g (−1 [1 − H])ηx 2

(4.32)

using ψ = P∆ ψ and the Cauchy–Schwarz inequality (with ψ = 1). The righthand side may be written ∆

dηx | PE ηx pk (E)2 | g ([ − E]/)|2 =

∆

dµx (E)| g ([ − E]/)|2

(4.33)

which completes the derivation of Eq. (4.22) and hence the proof of Theorem 2.

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5 Numerical details In this section we provide more details on the numerical methods employed in Sect. 2 and discuss rigorous error estimates on the numerical errors. The basic numerical method is easily explained (see, e.g., Sect. 4.1 of [4] or Chapter 4 of [10]). Suppose T is an integral operator on L2 (R+ , dk) with kernel G, i.e., ∞ ) . (T ψ)(k) = dk G(k, k )ψ(k (5.1) 0

To handle this numerically, we first truncate the kernel to [0, K] × [0, K] for some K > 0, which amounts to studying a compression TK of T . Provided that the required properties of T are, for sufficiently large K, well-approximated by the corresponding properties of TK restricted to L2 (0, K), we proceed to approximate this restricted operator by a matrix. To do this, we suppose that (ξj )N j=0 and N (wj )j=0 are the nodes and weights for a suitable quadrature method on [0, K], and define the (N + 1)-square matrix A = (Ajk )N j,k=0 with (j, k) entry Ajk = 1/2

1/2

wj wk Gλ,K (ξj , ξk ). The relevant computations are performed on A and, if N and K are sufficiently large, this will provide a numerical approximation to the required quantity. This technique was applied to the Bracken–Melloy kernel as described in Sect. 2.2. In that case, we were unable to derive useful error estimates. However, the operator norm calculations of Sect. 2.3 are more controlled, as we now describe. The problem is to estimate the squared operator norms of the family of integral operators Tλ (λ > 0) defined in the above fashion8 with kernel 1 √ k gλ (−k − k ) . (5.2) Gλ (k, k ) = 2π It is straightforward to verify that Tλ 2 = λ−2 T1 2 , so we hereafter study only the operator T = T1 with kernel G = G1 and denote by TK its compression onto L2 (0, K). These compressions converge to T in the Hilbert–Schmidt norm, and therefore in operator norm, as K → ∞. That the corresponding N × N matrix approximations have operator norms converging to TK as N → ∞ is a consequence of the convergence of the quadrature formula to the integral for continuous functions. Thus our technique can legitimately be applied to this problem and it remains to control the errors inherent in the scheme for finite N and K. In general we have analytical control of the truncation errors (parametrised by K): in Hilbert–Schmidt norm, this is given by the integral of |G|2 over the region [0, ∞)×[0, ∞)\[0, K]×[0, K] which, by routine manipulations involving symmetry and a change of variables can be written as 2K ∞ 1 1 2 u(u − K)| g (−u)| du + u2 | g (−u)|2 du . (5.3) 4π 2 K 8π 2 2K 8 In Sect. 2.3 the operators were defined on L2 (R+ , dk/(2π)). Here, we absorb the factor of (2π)−1 into the kernel, which leaves the spectral data and operator norms unchanged.

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Although we are not able to control the discretization errors, we are able to observe apparent convergence to machine precision in most cases. We now consider the four functions used in Sect. 2.3. Starting with √ sampling 2 the Gaussian kernel g(k) = 2π 1/4 e−k /2 , the Hilbert–Schmidt norm of T can be found √by substituting K = 0 in (5.3) and evaluating the integral to give T H.S. = 1/(4 π) which is of course an upper bound for the operator norm. For more precise results, we turn to the quadrature method described above. For this kernel, the integrals in (5.3) can be evaluated explicitly to give a relative error in the Hilbert-Schmidt norm of

T − TK H.S. = (1 + erf(2K) − 2 erf(K))1/2 . (5.4)

T H.S. It can be numerically verified that the relative error falls below ε = 0.5×10−10 (for ten-digit precision) at approximately K = 6.756 (this calculation requires about 25-digit precision). The computations were performed in Maple 8 using c-panel repeated Clenshaw-Curtis quadrature (see Section 2.4.4 of [10]) on the interval [0, 6.9] and Maple’s NAG-based SingularValues routine. Using 33, 65 or 129 samples with c = 1, 2 gives in each case the same results for the first largest two singular values: σ1 = 0.1399331442, σ2 = 0.0175697912 to 10 figures. Notice that the second singular value is very much smaller than the first, which means that the matrices, and hence T , can be well approximated by operators of rank 1. This is consistent with the operator norm,√computed here to be .1399331442, being close to the Hilbert-Schmidt norm, 1/(4 π) = .1410473959 . . . . This similarity finally justifies our use of truncation constants based on the relative error in the Hilbert-Schmidt norm: the calculated value of the operator norm is certainly no larger than the true value, since it is the norm of a compression of T , so we have

T − TK

T − TK H.S. T H.S. ≤

T

T H.S.

T .1410473959 = .4912379017 × 10−10 (5.5) ≤ .4873572016 × 10−10 .1399331442 which is still less than the target figure of 0.5 × 10−10 . The next kernel of interest is the squared Lorentzian; however, in this case T is a rank-1 operator so the Hilbert–Schmidt and operator norms coincide and there is no need for numerical investigation. This leaves the two compactly supported kernels. In the truncated cosine case, the same techniques as above lead to an error estimate of the order of 2% relative error in the Hilbert–Schmidt norm for K = 1100. As this is a rather weak estimate, we suppress the details; the numerical estimate of the squared operator norm (for K = 1100, N = 1024) is given in Sect. 2.3. Our last example is the smoothed truncated cosine, defined by √ 2 3π 2 sin(k) . (5.6) g(k) = 3 (π 2 − k 2 )k

26



The relatively slow decay of this function makes the precision obtained in the Gaussian example impractical, but we can obtain results to at least four significant figures. In fact the numerical results appear to be much more precise than would be suggested by this error estimate. Maple is able explicitly to evaluate the integrals in (5.3) to give a rather complicated formula involving the Si and Ci special functions, and from this to give the asymptotic formula 5π/(16K 3) + o(1/K 4 ) for the relative error in the Hilbert-Schmidt norm. Using only the leading term, we can predict that truncation at about K = 732.3 should give a Hilbert-Schmidt norm relative error less than 0.5 × 10−4 ; numerical investigation of the exact formula near this point confirms this value. Proceeding in the same way as for the Gaussian kernel but this time using the faster numerical engine in Matlab 6 to calculate the singular values, we obtain σ1 = 0.3536210388 and σ2 = 0.0733902259 using 513 samples. The same values are obtained if the number of samples is increased to 1025 or two panels are used. Once again, the fact that the second singular value is considerably smaller than the first can be used after the fact to justify the use of relative errors in the Hilbert-Schmidt norm (rather than in the operator norm) in choosing the truncation constant. Although the error analysis only allows us to be confident of the first four figures, it seems likely that this figure for the operator norm is considerably more accurate than that. Doubling the truncation constant and using 2049 points, again with 1 and 2 panels, gives exactly the same results to ten figures as the two 1025point methods above. The last set of calculations reported in Sect. 2.3 concern the unbounded operator J of Eq. (2.26). Here we have not succeeded in obtaining usable estimates of the errors introduced by truncation to [0, K]. However, it is nonetheless true that inf σ(JK ) → inf σ(J) as a consequence of the following arguments. Proposition 3 Suppose k is an absolutely bounded kernel9 on L2 (0, ∞), and let w be a measurable function on (0, ∞) (with respect to the Lebesgue measure). Let D = {f ∈ L2 (0, ∞) : wf ∈ L2 (0, ∞)} .

(5.7)

Suppose (w(x) + w(y))k(x, y) is a Hermitian function of x and y, and define an operator with domain D by ∞ (T f )(x) = (w(x) + w(y))k(x, y)f (y)dy (5.8) 0

and assume that T is bounded below. For K > 0 define the truncated operator K (TK f )(x) = (w(x) + w(y))k(x, y)f (y)dy . (5.9) 0

Then lim inf σ(TK ) = inf σ(T ) .

K→∞ 9 That

is, |k| is the integral kernel of a bounded operator.

(5.10)

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Proof. TK is a compression of T so inf σ(TK ) ≥ inf σ(T ) for all K. For any ε > 0 we can choose f ∈ D with f = 1 and such that f | T f < inf σ(T ) + ε/2. Now ∞ ∞ (w(x) + w(y))k(x, y)f (y)dy f (x)dx (5.11) f | T f = 0

0

and the integrand here is in L2 ((0, ∞) × (0, ∞)) by Lemma 4 (see below). It now follows from Lebesgue’s dominated convergence theorem and Fubini’s Theorem that, provided the above repeated integral can be interpreted as an integral on the measure space (0, ∞) × (0, ∞), K

f | T f = lim

K→∞

K

(w(x) + w(y))k(x, y)f (y)dy 0

f (x)dx .

(5.12)

0

If we let fK (x) = f (x)χ(0,K) (x) then this tells us that fK | TK fK → f | T f ; we also have fK → f = 1 as K → ∞, so fK | TK fK / fK 2 → f | T f as K → ∞. In particular, for sufficiently large K we have ε fK | TK fK < f | T f + < inf σ(T ) + ε

fK 2 2

(5.13)

which implies that inf σ(TK ) < inf σ(T ) + ε. In combination with the earlier in2 equality inf σ(TK ) ≥ inf σ(T ), this establishes the result. It remains to justify the treatment of the repeated integral as an integral on a product measure space. Lemma 4 In the notation of the above theorem, for any f ∈ D, the repeated integral in Eq. (5.11) is absolutely convergent, and so can be interpreted as the integral of a function in L2 ((0, ∞) × (0, ∞)). Proof. We calculate ∞ ∞ |(w(x) + w(y))k(x, y)f (x)f (y)|dy dx 0 0 ∞ ∞ |k(x, y)| |f (y)|dy dx ≤ |f (x)w(x)| 0 0 ∞ ∞ + |k(x, y)| |w(y)f (y)|dy dx . (5.14) |f (x)| 0

0

2

Since both f and wf are L functions and k is an absolutely bounded kernel, both terms on the right-hand side are finite by the Cauchy-Schwarz inequality. The final conclusion now follows from Tonelli’s Theorem.

6 Conclusion The main focus of this paper has been to draw attention to links between the failure of pointwise energy conditions in quantum field theory and a range of similar situations in quantum mechanics. In addition we have seen that there are links at the

28



technical level between the QIs developed in quantum field theory and those obtained here in the quantum mechanical setting. We have also made contact with the ideas and methods of Weyl–Wigner quantization and sharp G˚ arding inequalities. In conclusion, we briefly summarise the new results we have obtained along the way. First, we have seen that the backflow phenomenon is limited in space (as well as in time [7]) as shown by our flux QI (2.10). In particular, the magnitude of the negative flux times the square of its spatial extent is bounded above for all right-moving states in R. We have also provided an improved numerical estimate of Bracken and Melloy’s backflow constant, and also given numerical evidence to support the conjecture that our flux QI is generally within an order of magnitude of the optimal bound (i.e., the infimum of the spectrum of J, given by Eq. (2.26)). Second, we have shown that similar phenomena occur for densities of observables obtained via Weyl quantization. This is a consequence of the indefinite sign of the Wigner function, and therefore an expression of the uncertainty principle. Moreover, for observables which are second order (or less) in momentum, we have seen that sharp G˚ arding inequalities entail the existence of kinematic quantum inequalities. We have also obtained explicit bounds in the case of Schr¨ odinger operators with smooth potentials. Finally, for general quantum mechanical systems describing dynamics on a topological measure space, we have shown that the time-averaged energy density obeys dynamical quantum inequalities (evolution being generated by the spatial integral of the energy density). For the 1-dimensional harmonic oscillator, we saw that the QI bound (for a given sampling function) is a Schwartz-class function: it becomes rapidly much harder to create sustained negative energy densities away from the classical equilibrium point. Moreover, we have seen that a bound on the spectral behaviour of the Hamiltonian on the negative spectral axis, expressed by the integrability condition on µx in (ii) of Thm. 2, already leads to dynamical QIs. This integrability condition can be viewed as a condition on the global dynamical stability of a quantum system, much in the sense of quantum systems in thermal equilibrium, where the spectral weight of the generator of the time-evolution (the Liouvillian) is exponentially suppressed on the negative half-axis (cf. Prop. 5.3.14 in [8]). This indicates again the link between (thermo)dynamical stability and dynamical QIs which was established in [20] and which originally motivated the introduction of QIs in [22]. All these findings corroborate the intimate connection between QIs and the fundamental principles of quantum mechanics: the uncertainty principle and dynamical stability. Acknowledgments. CJF thanks I˜ nigo Egusquiza for bringing ref. [7] to his attention. The work of CJF was assisted by EPSRC Grant GR/R25019/01 to the University of York; RV also thanks the EPSRC for support received under this grant during a visit to York. Numerical calculations were partly conducted using the White Rose Grid node hosted at the University of York. Some of this work was conducted at the Erwin Schr¨ odinger Institute, Vienna, during the programme

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on Quantum Field Theory in Curved Spacetime which took place in July–August 2002; CJF and RV thank the organizers of this programme and the institute for its hospitality. Particular inspiration was drawn from the Hotel Sacher.

References [1] R.A. Adams, Sobolev Spaces, Academic Press, New York 1975. [2] M. Alcubierre, Class. Quantum Grav. 11, L73 (1994). [3] G.R. Allcock, Ann. Phys. 53, 311 (1969). [4] K.E. Atkinson, The numerical solution of integral equations of the second kind, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge 1997. [5] M. Bordag, U. Mohideen, V.M. Mostepanenko, Phys. Rep. 353, 1 (2001). [6] A.J. Bracken, H.-D. Doebner, and J.G. Wood, Phys. Rev. Lett. 83, 3558 (1999). [7] A.J. Bracken and G.F. Melloy, J. Phys. A: Math. Gen. 27, 2197 (1994). [8] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol 2, 2nd Ed., Springer-Verlag, Berlin 1997. [9] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wet. 51, 793 (1948). [10] L.M. Delves and J.L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge 1985. [11] H. Epstein, V. Glaser, and A. Jaffe, Nuovo Cimento 36, 1016 (1965). [12] C. Fefferman and D.H. Phong, Comm. Pure Appl. Math. 34, 285 (1981). [13] C.J. Fewster, Class. Quantum Grav. 17, 1897 (2000). [14] C.J. Fewster and S.P. Eveson, Phys. Rev. D 58, 084010 (1998). [15] C.J. Fewster and B. Mistry, Phys. Rev. D 68, 105010 (2003). [16] C.J. Fewster and M.J. Pfenning, J. Math. Phys. 44, 4480 (2003). [17] C.J. Fewster and E. Teo, Phys. Rev D 59, 104016 (1999). [18] C.J. Fewster and E. Teo, Phys. Rev D 61, 084012 (2000). [19] C.J. Fewster and R. Verch, Commun. Math. Phys. 225, 331 (2002). [20] C. J. Fewster and R. Verch, Commun. Math. Phys. 240, 329 (2003). ´ E. ´ Flanagan, Phys. Rev. D 56, 4922 (1997). [21] E. [22] L.H. Ford, Proc. Roy. Soc. Lond. A364, 227 (1978). [23] L.H. Ford and T.A. Roman, Phys. Rev. D 53, 5496 (1996). [24] L.H. Ford and T.A. Roman, Phys. Rev. D 55, 2082 (1997). [25] L.H. Ford and T.A. Roman, Phys. Rev. D 60, 104018 (1999).

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[26] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer Verlag, Berlin 1983. [27] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer Verlag, Berlin 1994. [28] N. P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, Springer-Verlag, Berlin 1998. [29] P. Marecki, Phys. Rev. A 66, 053801 (2002). [30] G.F. Melloy and A.J. Bracken, Found. Phys. 28, 505 (1998). [31] M.S. Morris, K.S. Thorne, and U. Yurtsever, Phys. Rev. Lett. 61, 1446 (1988). [32] M.J. Pfenning, Phys. Rev. D 65, 024009 (2002). [33] M.J. Pfenning and L.H. Ford, Class. Quantum Grav. 14, 1743 (1997). [34] M.J. Pfenning and L.H. Ford, Phys. Rev. D 57, 3489 (1998). [35] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Academic Press, New York 1975. [36] M.E. Taylor, Pseudodifferential Operators, Princeton University Press, 1981. Simon P. Eveson and Christopher J. Fewster Department of Mathematics University of York Heslington York YO10 5DD United Kingdom email: [email protected] email: [email protected] Rainer Verch Max-Planck-Institut for Mathematics in the Sciences Inselstr. 22 D-04103 Leipzig Germany email: [email protected] Communicated by Yosi Avron submitted 27/01/04, accepted 05/05/04

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Integrated Density of States for the Periodic Schr¨ odinger Operator in Dimension Two Alexander V. Sobolev

1 Introduction The objective of the present paper is to study the high energy asymptotics of the density of states D(λ) for the Schr¨ odinger operator L2 (Rd ), d ≥ 1 with a periodic potential V : H = −∆ + V. (1.1) Here V is a real-valued function periodic with respect to a d-dimensional lattice Γ ⊂ Rd . Below we denote by O ⊂ Rd a standard fundamental domain of the lattice Γ, and by O† the fundamental domain of the dual lattice Γ† . For this operator, as well as for any other elliptic self-adjoint differential operator, the density of states is defined by the formula (R)

N (λ; HD ) . R→∞ Rd

(1.2)

D(λ) = lim (R)

Here HD is the restriction of H to the cube [0, R]d with the Dirichlet boundary (L) conditions, and N (λ; · ) is the counting function of the discrete spectrum of HD . The above limit exists for periodic and almost periodic potentials, see [17], [22]. To be precise, the quantity D(λ) is called the integrated density of states, but for the sake of brevity we call it simply the density of states. Calculation of the density of states D0 (λ) for the unperturbed operator H0 = −∆ is an elementary exercise: one easily proves (see, e.g., Proposition 2.4 below) that D0 (λ) =

d 1 wd λ 2 , λ ≥ 0, d (2π)

d

wd =

π2 , Γ(1 + d/2)

(1.3)

where wd is the volume of the unit ball in Rd . For d = 1 it was shown in [20] (see [19] for the almost periodic case) that the density of states admits a complete asymptotic expansion in the powers of λ−1 , as λ → ∞. On the basis of this result it is natural to conjecture that for general d ≥ 2 the asymptotics of D(λ) exists and has the form N bj λ−j + o(λ−N ) , λ → ∞, ∀N. D(λ) = D0 (λ) 1 + j=1

(1.4)

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A.V. Sobolev


For d = 1 the coefficients bj satisfy simple recursive relations, see [20]. Results for the multidimensional Schr¨ odinger operator are much less advanced: relation (1.4) has been proved only with N = 1 so far. The first formula was found in [21] (see also [22] for an elementary proof) for almost periodic potentials V : D(λ) = D0 (λ) + O(λ 2 −1 ), λ → ∞. d

A two-term asymptotics was first established in [6] for C∞ -smooth periodic potentials V : 3 d D(λ) = D0 (λ) 1 + b1 λ−1 + O(λ− 2 + ) , ∀ > 0, b1 = − V (x)dx. (1.5) 2|O| O The proof in [6] relied based on the powerful methods of microlocal analysis. In [8], via an advanced version of perturbation theory for periodic operators, this asymptotics was generalized to the case of the polyharmonic operator (−∆)l + V , l > 1/2, with an improvement of the remainder estimate: 1 D(λl ) = D0 (λl ) 1 + b1,l λ−l + O(λ 2 −2l ln λ) , b1,l = b1 l−1 .

(1.6)

A more precise result was obtained in [8] for the Schrödinger operator −∆ + V for d = 3: ˆ + O(λ−δ ), (1.7) D(λ) = D0 (λ) 1 + b1 λ−1 + D ˆ =D ˆ V . Recently it was observed with some small δ > 0 and a constant coefficient D ˆ = 0, and hence the above formula is consistent with the conjecture in [11] that D (1.4). Note that the density of states was also studied for the magnetic operator (−i∇ − a)2 + V with a periodic magnetic vector-potential a, see [14] and [8]. In [14] it was shown that D(λ) = D0 (λ) + O(λ(d−2)/2+ ) with an arbitrary > 0. Justification of the hypothesis (1.4) for large N would be a very hard problem. However, if one assumes that (1.4) holds, then the coefficients bj can be calculated relatively easily. They are obtained in [11] by a standard argument from the asymptotics of the heat kernel of the Schr¨ odinger operator. These coefficients are integrals of certain standard polynomials depending on V and derivatives of V . These polynomials are known both in mathematical and physical literature as heat kernel invariants. For recent work on the structure of these polynomials see, e.g., [16], [7]. In particular, this approach gives d(d − 2) V 2 (x)dx, b2 = 8|O| O which implies that b2 = 0 for d = 2. Moreover, for all even d the coefficients bj satisfy bj = 0 if j ≥ d/2 + 1. Our ultimate goal is to justify the conjecture (1.4) with N = 2 for all dimensions d ≥ 2. In this paper this is done for d = 2, while the general case d ≥ 3 will

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be addressed in a subsequent publication. The main result of the present paper is the formula 6 (1.8) D(λ) = D0 (λ) 1 + b1 λ−1 + O(λ− 5 + ), ∀ > 0, for the density of states of the Schrödinger operator with d = 2, see Theorem 2.3. From the technical point of view the paper is a continuation of [25] where the density of states was studied in the case d = 1 for elliptic operators of a more general form than the Schr¨ odinger operator. As in [25], our approach is a variant of the “near-similarity” method, which is usually applied in dimension one (see [18], [1] and [12], [13]). The central idea is to reduce the operator H with the help of a suitable similarity transformation, to an operator with constant coefficients. In the present paper the required similarity is implemented by a unitary operator eiΨ , where Ψ is a bounded self-adjoint PDO with the symbol ψ(x, ξ). In contrast to the one-dimensional case considered in [25], for d ≥ 2 a complete reduction to constant coefficients is not achievable, since instead of a smooth symbol ψ(x, ξ) (which is the case for d = 1) a straightforward application of the method produces a symbol ψ with singularities on a set Λ which is a union of hyper-planes {ξ ∈ Rd : θ(ξ + θ/2) = 0} where θ ∈ Γ† , Γ† being the dual lattice. To avoid the singularity, one studies the neighbourhood of Λ separately from the region away from Λ. Outside the set Λ the symbol ψ is found as in the case d = 1, from a series of commutator equations which emerge as a result of the requirement that the new operator should have constant coefficients. Then the density of states for the new operator is found by an elementary calculation. Near the “singular” set Λ the operator H is reduced to a “one-dimensional” effective operator of the Schrödinger type with a pseudo-differential perturbation. For this operator the density of states is found using the results of [25]. Although the set Λ emerges in a natural way in the context of the PDO calculus, there is also a perturbation-theoretic interpretation. Recall that the eigenvalues of the unperturbed Floquet Hamiltonian H0 (k) are given by λ(ω) (k) = (ω + k)2 , where ω ∈ Γ† are points of the dual lattice and k ∈ O† is the quasi-momentum. The analysis of these eigenvalues under the perturbation V is dramatically different for d = 1 and d ≥ 2. If d = 1, then the standard perturbation theory yields a complete asymptotic expansion of the eigenvalues. On the contrary, for d ≥ 2 the unperturbed eigenvalues split in two groups behaving differently under the perturbation V , which can be described with the help of the set Λ. The eigenvalues / Λ can be more or less completely described by the perturbation λ(ω) (k) with ω ∈ theory, see [4], [9]. The eigenvalues with ω ∈ Λ move by a quantity of order V under the perturbation V . This effect is due to the small divisors arising when the eigenvalues λ(ω) (k) get close together. In the relevant literature these exceptional eigenvalues are sometimes called resonant, unstable or singular, see [5], [9]. It was shown in [5], [9] that their behavior can be described by means of some effective one-dimensional Schr¨ odinger operators. The resonant set presents a major obstacle when studying spectral properties of the periodic Schr¨ odinger operator, and in particular, the asymptotics of the

34

A.V. Sobolev


density of states D(λ). The precision of the asymptotics eventually depends on how well one knows the behavior of the resonant eigenvalues. For instance, estimating their number from above leads to the remainder estimate in (1.6). The more precise result (1.7) requires more thorough study of the set Λ, see [8]. In the present paper the asymptotics (1.8) is also derived via a detailed analysis of the set Λ. Although the study of the density of states is an object of independent interest in its own right, it can be also used to investigate other spectral properties of the Schr¨ odinger operator. One such problem is to justify the Bethe-Sommerfeld conjecture, that is to prove that the number of gaps in the spectrum of H is finite. The conjecture is known to be true for all dimensions d ≥ 2 under the condition that the lattice Γ is rational, see [23]. For general lattices it was justified so far only for dimensions d = 2, 3, 4, see [2], [24], [9], [6], [15] and references therein. This result is derived not directly from (1.5) or (1.6), but from the asymptotics of the same type for the so-called generalized density of states, see, e.g., [6] or [15] for definition. The restriction d ≤ 4 is then dictated by the remainder estimate in this asymptotics. An improved remainder estimate would lead to the inclusion of bigger d’s. Moreover, the justification of the asymptotics with more terms would allow one to increase the dimension even further. The paper is organized as follows. Section 2 contains the precise definitions of objects studied in the paper, and the statement of the main result (see Theorem 2.3). In Section 3 necessary information on the calculus of periodic PDO’s is collected, including their transformations under linear maps. Section 4 describes partitions of PDO’s which are central for their reduction to constant coefficients. Section 5 is devoted to the study of the density of states for the model operator, which is based on its decomposition in the invariant subspaces. Their structure is explicitly described in terms of the resonant set Λ. On each of the invariant subspaces the model operator reduces to a one-dimensional Schr¨ odinger-type operator, which makes possible the application of the asymptotics established in [25]. At the next step, in Section 6, the Schr¨ odinger operator (1.1) is reduced to the model operator with the help of the unitary operator having the form eiΨ with a suitable PDO Ψ. We loosely call this operator a gauge transformation. A further analysis of the model operator leads to the conclusion that its density of states is determined by its constant coefficients part Ao , see Section 7. The proof of the Main Theorem is completed in Section 8 together with the asymptotic formula for the density of states of the operator Ao . The calculation of an integral featuring in Section 8, is done in the Appendix. We emphasize that although the main result concerns the case d = 2, we do the calculations for arbitrary dimension d ≥ 2 whenever possible, indicating the points where the argument requires d = 2.

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2 Main result 2.1

Classes of PDO’s

Before we define the pseudo-differential operators (PDO’s) we introduce first the relevant classes of symbols. Let Γ ∈ Rd be a lattice. Denote by O its fundamental domain. For example, for O one can choose a parallelepiped spanned by a basis of Γ. The dual lattice and its fundamental domain are denoted by Γ† and O† respectively. Sometimes we reflect the dependence on the lattice and write OΓ and O†Γ . In particular in the case Γ = (2πZ)d one has Γ† = Zd and it is natural to take O = [0, 2π)d , O† = [0, 1)d . For any measurable set C ⊂ Rd we denote by |C| or vol(C) its Lebesgue measure (volume). The volume of the fundamental domain does not depend on its choice, it is called the determinant of the lattice Γ and denoted d(Γ) = |O|. By e1 , e2 , . . . , ed we denote the standard orthonormal basis in Rd . For any u ∈ L2 (O) and f ∈ L2 (Rd ) define the Fourier coefficients and Fourier transform respectively: 1 u ˆ(θ) = e−iθ,x u(x)dx, θ ∈ Γ† , d(Γ) O (Ff )(ξ) =

1 (2π)

d 2

Rd

e−iξ,x f (x)dx, ξ ∈ Rd .

Let us now define the periodic symbols and PDO’s associated with them. Let b = b(x, ξ), x, ξ ∈ Rd , be a Γ-periodic complex-valued function, i.e., b(x + γ, ξ) = b(x, ξ), ∀γ ∈ Γ. Let w : Rd → R be a locally bounded function such that w(ξ) ≥ 1, ∀ξ ∈ Rd and w(ξ + η) ≤ Cw(ξ) ηκ , ∀ξ, η ∈ Rd ,

(2.1)

for some κ ≥ 0. We say that the symbol b belongs to the class Sα = Sα (w) = Sα (w, Γ), α ∈ R, if for any l ≥ 0 and any non-negative s ∈ Z the condition b

(α) l,s

= max max sup θp w(ξ)−α+|s| |Dsξˆb(θ, ξ)| < ∞, p≤l |s|≤s ξ,θ

(2.2)

is fulfilled. If necessary, we reflect the dependence of this norm on the weight w and (α) write b l,s;w . Here we have used the standard notation t = 1 + |t|2 , ∀t ∈ Rd . Also, for any s ∈ Zd we denote |s| = s1 + s2 + · · · + sd . We mainly use two types of classes Sα : either with the weight w(ξ) = ξ, which satisfies (2.1) for κ = 1, or with a weight w(ξ) = L where a constant L is chosen in a convenient way. Note that Sα is an increasing function of α, i.e., Sα ⊂ Sβ for α < β. For later

36

A.V. Sobolev


reference write the following convenient bounds that follow from Definition (2.2) and property (2.1): |Dsξ ˆb(θ, ξ)| ≤ b |Dsξ ˆb(θ, ξ

+ η) −

Dsξˆb(θ, ξ)|

≤C b

(α) −l α−s , l,s θ w(ξ)

(α) −l α−s−1

ηκ|α−s−1| |η|, l,s+1 θ w(ξ)

(2.3) s = |s|, (2.4)

with a constant C depending only on α, s. We introduce a separate notation for (α) the set Pα = Pα (w, Γ) ⊃ Sα of symbols b such that b l,0 < ∞ for all l ≥ 0. Periodic functions V ∈ C∞ (Rd ) can be also viewed as symbols from P0 . For such (0) (0) functions V l,s = V l,0 for any s. Now we define the PDO Op(b) in the usual way: 1 b(x, ξ)eiξ,x (Fu)(ξ)dξ, Op(b)u(x) = d (2π) 2 the integral being taken over Rd . Under the condition b ∈ Pα the integral in the r.h.s. is clearly finite for any u from the class B(Rd ) of functions such that their Fourier transforms decay faster than any power of ξ, that is B(Rd ) = {u : sup ξl |(Fu)(ξ)| < ∞, ∀l > 0}. ξ

(2.5)

In particular, Op(b) is well defined for u in the Schwarz class S(Rd ). Moreover, the condition b ∈ P0 guarantees the boundedness of Op(b) in L2 (Rd ), see Proposition 3.1. Unless otherwise stated, from now on S(Rd ) is taken as a natural domain for all PDO’s at hand, although sometimes we need to consider Op(b) on functions from the bigger class B(Rd ) as well. Observe that the operator Op(b) is symmetric if its symbol satisfies the condition ˆb(θ, ξ) = ˆb(−θ, ξ + θ).

(2.6)

We shall call such symbols symmetric. Note that Sα (L) = Sβ (L) for any α, β ∈ R and b

(α) l,s

= Lβ−α b

(β) l,s .

(2.7)

In fact, the introduction of different notation for the same class is done here to reflect the dependence on the parameter L. Throughout the entire paper we adopt the following convention. An estimate (or an assertion) is said to be uniform in a symbol b ∈ Sα (resp. Pα ) if the constants in the estimate (or assertion) at hand depend only on the constants Cl,s (α) (resp. Cl,0 ) in the bounds b l,s ≤ Cl,s . As was indicated in the introduction, our ultimate goal is to study the density of states of the Schr¨ odinger operator H = H0 + V in H = L2 (Rd ) with H0 = −∆

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and a smooth real-valued periodic potential V . However, some general definitions are more natural to give for more general operators. For these purposes it is not even necessary to assume that H0 = −∆, but it would be sufficient to suppose that H0 = Op(h0 ) with h0 (ξ) = |Fξ|m , m > 0, where F is a non-degenerate d × dmatrix. Also, the perturbation is allowed to be an arbitrary PDO of order lower than H0 . Another reason of considering more general operators is methodological: a number of intermediate results requires the use of such PDO’s. Precisely, we consider the operator  H = Op(h), h(x, ξ) = h0 (ξ) + b(x, ξ), (2.8)  h0 (ξ) = |Fξ|m , b ∈ Pα ( ξ), α < m, with a symmetric symbol b. The operator Op(b) is H0 -bounded with an arbitrarily small relative bound. Thus H is self-adjoint on the domain D(H) = D(H0 ) = Hm (Rd ). Sometimes we call symbols (PDO’s) of this type elliptic symbols (PDO’s) of order m. In this paper we do not need to consider more general elliptic symbols. Due to the Γ-periodicity of the symbol b, the operator H commutes with the shifts along the lattice vectors, i.e., HTγ = Tγ H, γ ∈ Γ. with (Tγ u)(x) = u(x + γ). This allows us to use the Floquet decomposition.

2.2

Floquet decomposition

We identify the space H = L2 (Rd ) with the direct integral Hdk, H = L2 (O). G= O†

This identification is implemented by the Gelfand transform 1 e−ik,x e−ik,γ u(x + γ), k ∈ O† , (U u)(x, k) = d(Γ† ) γ∈Γ

(2.9)

which is initially defined on u ∈ S(Rd ) and extends by continuity to a unitary mapping from H onto G. In terms of the Fourier transform the Gelfand transform is defined as follows: (U u)(θ, k) = (Fu)(θ + k), θ ∈ Γ† . The unitary operator U reduces Tγ to the diagonal form: (U Tγ U −1 f )( · , k) = eik,γ f ( · , k), ∀γ ∈ Γ. Let us consider a self-adjoint operator A in H which commutes with Tγ for all γ ∈ Γ, i.e., ATγ = Tγ A. We call such operators Γ-periodic or simply periodic.

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Then A is partially diagonalized by U (see [17]), that is, there exists a measurable family of self-adjoint operators (fibres) A(k) acting in H, such that ∗ U AU = A(k)dk. (2.10) O†

It is easy to show that any periodic symmetric operator T , which is A-bounded with relative bound < 1, can be also decomposed into a measurable set of fibers T (k) in the sense that (U T f )( · , k) = T (k)(U f )( · , k), a.e. k ∈ O† , for all f ∈ D(A). Moreover, the fibers T (k) are A(k)-bounded with the bound , and if T is symmetric, then the operator A(k) + T (k) is self-adjoint on D(A(k)). Suppose that the operator A (and hence A(k)) is bounded from below and that the spectrum of each A(k) is discrete. Denote by λj A(k) , j = 1, 2, . . . , the eigenvalues of A(k) labelled in the ascending order. Using the min-max principle one easily sees that each λj (A( · )) is a measurable function of k. Suppose also that the counting function N λ, A(k) = #{j : λj A(k) ≤ λ}, λ ∈ R, is bounded as a function of k ∈ O† . Then we define the integrated density of states by the formula 1 N λ, A(k) dk. (2.11) D(λ) = D(λ; A) = d (2π) O† This definition makes sense for the operator A + T as well, since N λ, A( · ) + T ( · ) ∈ L∞ (O† ). Sometimes we need to reflect the dependence of the counting function and density of states on the lattice. In this case we use the notation NΓ λ, A(k) , DΓ (λ; A). Let us indicate some elementary general properties of the density of states, following directly from Definition (2.11). Proposition 2.1 Let A, A1 be self-adjoint Γ-periodic operators as defined above. (i) The density of states is monotone in A, that is, if A ≤ A1 , then D(λ; A) ≥ D(λ; A1 ). (ii) The density of states is a unitary invariant. Precisely, for any unitary Γperiodic operator W one has D(λ; W ∗ AW ) = D(λ; A). † Proof. The inequality in (i) follows from the inequality A(k) ≤ A1 (k), a.a.∗ k ∈ O . The unitary invariance follows from the identity N (λ, A(k)) = N λ, W (k)A(k) W (k) , where W (k) are the fibres of the operator W in the decomposition (2.10).

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d If A = Op(a) with a real-valued symbol a ∈ L∞ loc (R ) depending only on ξ, d d then A(k) is a self-adjoint PDO on the torus TΓ = R /Γ defined as follows:

† 1 A(k)u(x) = eiγ ,x a(γ † + k)ˆ u(γ † ). d(Γ) γ † ∈Γ† If a(ξ) → ∞ as |ξ| → ∞, then the spectrum of each A(k) is purely discrete with eigenvalues given by λ(m) (k) = a(m + k), m ∈ Γ† . Consequently, the number of eigenvalues below each λ ∈ R is essentially bounded from above uniformly in k ∈ O† . If T is a periodic symmetric operator which is A-bounded with a bound < 1, then the spectrum of A(k) + T (k) is also purely discrete and the counting function is also bounded uniformly in k. The above applies to the elliptic operator H defined in (2.8), and thus the quantity D(λ; H) is well defined. In fact, if necessary, one can obtain more information on the operators H0 (k) and H(k). Applying the Gelfand transform (2.9) to Op(b), one finds that, similarly to A considered above, the operator H(k) is a PDO on the torus TdΓ of the form † 1 H(k)u(x) = eiγ ,x h(x, γ † + k)ˆ u(γ † ). d(Γ) γ † ∈Γ† Moreover, if the symbol b(x, ξ) is smooth in ξ, then the family H(k) is smooth in k. Note however that for our purposes we need neither this explicit formula, nor the smoothness property. Recall that for periodic elliptic differential operators A there is another, equivalent, definition of the density of states given by (1.2). In the case of a pseudo-differential operator the formula (1.2) is not applicable, and we use (2.11) as definition. It is important however to convince oneself that the formula (2.11) preserves the invariance of D(λ) with respect to the choice of the lattice in the following sense: if the operator H happens to be periodic with respect to the latformula for tices Γ and Λ, then DΓ (λ) = DΛ (λ). To this end we write another D(λ) in terms of the spectral projection E(λ) = χ H; (−∞, λ] for H. Here and everywhere below we denote by χ( · ; C) the characteristic function of a measurable set C ⊂ Rm . Denote T (λ, C) = tr E(λ)χ(x; C) . It is easy to show that this trace is finite for any bounded set C ⊂ Rd . Theorem 2.2 Let the operator H be as defined in (2.8). Let D(λ; H) be its density of states defined by the formula (2.11). Then D(λ; H) = lim

R→∞

where KR = [0, R]d .

1 T (λ, KR ), Rd

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Proof. Let us make a preliminary calculation. Let B be a finite subset of Γ, and let C = ∪γ∈B (γ + O) be the set consisting of O and its translations by the vectors γ ∈ B. Using the invariance of the trace under unitary transformations, rewrite using the Gelfand transform (2.9): T (λ, C) = tr U E(λ)U ∗ U χ( · , C)U ∗ . A straightforward calculation shows that the operator U χU ∗ has the form ∗ M (x; k − k )f (x, k )dk , (U χU f )(x, k) = O†

M (x; t) =

1 −it,x+γ e . d(Γ† ) γ∈B

To find the trace T (λ, C) we use the formula tr K = (Kφn , φn ), n

which gives tr K for any trace-class operator K in a Hilbert space as the sum over an arbitrary orthonormal basis {φn }. Let us take the following orthonormal basis in G: ψγ,ω (x, k) = fγ (k)gω (x), 1 eix,ω , γ ∈ Γ, ω ∈ Γ† . fγ (k) = eik,γ , gω (x) = † d(Γ) d(Γ ) 1

Denote (U E(λ)U ∗ )(k) = E(λ, k). Then (E(λ, k)M ( · ; k − k )gω , gω )fγ (k)fγ (k )dkdk T (λ, C) = =

γ,ω

O†

ω

O†

O†

(E(λ, k)M ( · ; 0)gω , gω )dk.

Note that M (x; 0) =

1 1 1 {γ ∈ B} = |C| = |C|, d(Γ† ) d(Γ† ) d(Γ) (2π)d

where |C| denotes the Lebesgue measure of C. Consequently 1 1 |C| tr E(λ, k)dk = |C| N λ, H(k) dk = |C|D(λ). T (λ, C) = d d (2π) (2π) O† O† (2.12) Now let C> (resp. C< ) be the maximal (resp. minimal) set consisting of the domain O and its translations by the vectors γ ∈ Γ, such that C< ⊂ KR ⊂ C> . Then, clearly, T (λ, C< ) ≤ T (λ, KR ) ≤ T (λ, C> ), and both volumes |C< | and |C> | are Rd + o(Rd ), R → ∞. Applying (2.12) to C< and C> we obtain the required formula.

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Main result

We are now in a position to state the main result of the paper. Let V be a Γ-periodic function, and let H = −∆ + V . In the main Theorem below we assume that d = 2. The multidimensional case will be considered in a subsequent publication. Theorem 2.3 Let d = 2. Assume that V ∈ C∞ (Rd ) is Γ-periodic and that Vˆ (0) = 0. Then there is a number λ0 = λ0 (V ) > 0 such that D(λ; H) =

6 1 λ + Oδ (λ− 5 +δ ), ∀δ > 0, 4π

for all λ ≥ λ0 . The constant λ0 and the remainder estimate are uniform in V in (0) the sense that they depend only on the constants in the bounds V l,0 ≤ Cl . The condition Vˆ (0) = 0 does not restrict generality, since Vˆ (0) can be always incorporated into the spectral parameter λ. This simple argument allows one to deduce the formula (1.8) announced in the introduction, from Theorem 2.3.

2.4

“Partial” density of states

Now we define the density of states for PDO’s on their invariant subspaces. To this end we need to introduce a class of projection operators. Let C ⊂ Rd be a measurable set. Denote by χ(ξ) =χ(ξ; C) the characteristic function of C. Then the operator P(C) = χ(D; C) = Op χ( · ; C) is a projection in H on the subspace H(C) = P(C)H, and the operator P(k) = P(k; C) is a PDO on the torus with the symbol χ(γ † + k). Suppose that H(C) is an invariant subspace of the operator H defined in (2.8), that is PD(H) ⊂ D(H) and HPD(H) ⊂ H(C). Then the subspace H(k) = P(k)H, k ∈ O† , is invariant for H(k). Since the spectrum of H(k) is purely discrete, then so is the spectrum of the restriction of H(k) to the subspace H(k). The counting function of this restriction is denoted by N λ, H(k); C), and the density of states by D(λ; H; C). The same notation can be naturally introduced for any self-adjoint PDO A such that A(k) H(k) has a discrete spectrum. For instance, this is the case for any symbol a ∈ S0 ( ξ, Γ), such that for some bounded C the subspace H(C) is invariant for A. Note that the condition that H(C) is invariant for a bounded C automatically implies that the symbol h is a trigonometric polynomial in x, and that the operator H(k) H(k) is finite-dimensional. If C consists of two disjoint components, i.e., C = C1 ∪ C2 with C1 ∩ C2 = ∅, then obviously, D(λ; H; C) = D(λ; H; C1 ) + D(λ; H; C2 ). Sometimes we indicate which lattice of periodicity is used to compute the partial density of states and write DΓ (λ; H; C). On the other hand, similarly to the “full” density of states (see Theorem 2.2), one can easily show that D(λ; H; C) does not depend on the choice of the lattice of periodicity for the symbol h(x, ξ). The following three statements provide three important examples involving the density of states, in which the answer can be computed either completely or

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partially in terms of the symbol. The first statement provides an explicit formula for D(λ) in the case of a symbol with constant coefficients. Before proceeding we d introduce a convenient notation. Let a ∈ L∞ loc (R ) be a real-valued function. Denote E(λ; a) = {ξ ∈ Rd : a(ξ) ≤ λ}, ∀λ ∈ R.

(2.13)

d Proposition 2.4 Let A = Op(a) with a real-valued symbol a ∈ L∞ loc (R ) such that d a(ξ) → ∞ as |ξ| → ∞. Let C ⊂ R . Then

D(λ; A; C) =

1 vol C ∩ E(λ; a) . d (2π)

Proof. Denote

ϑ(t) =

0, t < 0; 1, t ≥ 0.

Observe that the eigenvalues of A(k)P(k) equal a(µ + k), µ + k ∈ C, µ ∈ Γ† , so that ϑ λ − a(µ + k) dk = dξ, (2π)d D(λ) = µ+k∈C∩Γ†

O†

a(ξ)≤λ, ξ∈C

as required. The next lemma deals with the integral of the density of states.

Lemma 2.5 Let C ⊂ Rd be a bounded set. Suppose that h(x, ξ) is a symbol of the form (2.8) such that the subspace H(C) is invariant for H. If λ ≥ P(C)H, then (2π)d

λ

−∞

D(µ; H; C)dµ = λ|C| −

1 d(Γ)

C

OΓ

h(x, ξ)dxdξ.

Proof. For any self-adjoint operator A in a finite-dimensional Hilbert space E one can write λ N (µ; A)dµ = λ dim E − tr A, ∀λ ≥ A. −∞

Therefore for H(k) = P(D + k, C)H, we have

λ

−∞

N µ; H(k); C dµ = λ#{γ † ∈ Γ† : γ † + k ∈ C} − tr(H(k) H(k)),

if λ ≥ P(C)B. Integrating the first term in k, we obtain λ|C|. To find the second term write the matrix elements of the operator H(k) on the subspace H(k): † † 1 e−ix,ω −γ h(x, γ † + k)dx, ω † , γ † ∈ C − k, Hω † ,γ † = d(Γ) OΓ

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so that tr H(k) H(k) =

Hγ † ,γ † =

γ † :γ † +k∈C

γ † :γ † +k∈C

1 d(Γ)

OΓ

h(x, γ † + k)dx,

After integrating it in k, we obtain the expected integral, thereby completing the proof. In the next example we consider a PDO which admits a partial separation of variables. To illustrate we use the second order elliptic periodic operators, although the argument below can be easily extended to more general operators. Let n, l be two natural numbers such that n + l = d. Let x = (y, t) and ξ = (η, ω) with y, η ∈ Rn and t, ω ∈ Rl . Let b = b(y, η) be a symmetric elliptic symbol of second order on Rn × Rn and periodic in y w.r.t. the lattice Λ ⊂ Rn . Let a(ω) = |Rω|2 , with a non-degenerate matrix R. Then the operator H = I ⊗ Op(a) + Op(b) ⊗ I is self-adjoint on H2 (Rd ). Lemma 2.6 Let the operator H be as above and let C ⊂ Rl , D ⊂ Rn be subsets of Rl and Rn . Then 1 D(λ, H; D × C) = D(λ − a(k), B; D)dk. (2π)l C Proof. The symbol of H is periodic w.r.t. the lattice Λ × (2πZ)l with the fundamental domain M = O × [0, 2π)l . The Floquet representative of the operator H is the operator H(K) = I ⊗ A(k) + B(µ) ⊗ I, K = (µ, k), k ∈ [0, 1)l , µ ∈ O† . Since the symbol a does not depend on x, after separating variables we have N (λ, H(µ, k); D × C) = N (λ, a(m + k) + B(µ); D), ∀µ ∈ O† , m∈Zl , m+k∈C

and thus D(λ, H; D × C) =

1 (2π)l+n

C

which leads to the required formula.

O†

N (λ, a(k) + B(µ); D)dµdk,

3 Properties of periodic PDO’s In this section we collect various properties of periodic PDO’s to be used in what follows.

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A.V. Sobolev

3.1


Some basic results on the calculus of periodic PDO’s

We begin by listing without proofs the results established in [25]. Recall that S(Rd ) is taken as a natural domain of Op(b). Unless otherwise stated, all the symbols are supposed to belong to the class Sα = Sα (w, Γ) with an arbitrary function w satisfying (2.1) and a lattice Γ. The function w and the lattice Γ are usually omitted from the notation. Proposition 3.1 Assume that b ∈ P0 . Then B = Op(b) is bounded in H and (0) B ≤ Cl b l,0 , ∀l > d, with a constant C = Cl independent of b. Remark 3.2 The above proposition automatically implies that PDO’s with symbols (α) b ∈ Sα (L, Γ) are bounded for any α ∈ R and Op(b) ≤ Cl Lα b l,0 . Since Op(b)u ∈ S(Rd ) for any b ∈ Sα and u ∈ S(Rd ), the product Op(b)Op(g), b ∈ Sα ,g ∈ Sβ , is well defined on S(Rd ). A straightforward calculation leads to the following formula for the symbol b ◦ g of the product Op(b) Op(g): (b ◦ g)(x, ξ) =

1 ˆ b(θ, ξ + φ)ˆ g (φ, ξ)ei(θ+φ)x , d(Γ) θ,φ

and hence 1 ˆb(θ, ξ + φ)ˆ (b ◦ g)(χ, ξ) = g (φ, ξ), χ ∈ Γ† , ξ ∈ Rd . d(Γ) θ+φ=χ

(3.1)

Here and below θ, φ ∈ Γ† . Proposition 3.3 Let b ∈ Sα , g ∈ Sβ . Then b ◦ g ∈ Sα+β and b◦g

(α+β) p,s

≤ Cl,s b

(α) l,s

g

(β) l,s ,

for some l = l(p). We are also interested in the estimates for symbols of commutators. For PDO’s A, Ψl , l = 1, 2, . . . , N , denote ad(A; Ψ1 , Ψ2 , . . . , ΨN ) = i ad(A; Ψ1 , Ψ2 , . . . , ΨN −1 ), ΨN , ad(A; Ψ) = i[A, Ψ], adN (A; Ψ) = ad(A; Ψ, Ψ, . . . , Ψ), ad0 (A; Ψ) = A. For the sake of convenience we use the notation ad(a;ψ1 ,ψ2 ,...,ψN ) and adN (a,ψ) for the symbols of multiple commutators. It follows from (3.1) that the Fourier coefficients of the symbol ad(b, g) are given by g)(χ, ξ) = i ˆb(θ, ξ + φ)ˆ g (φ, ξ) − ˆb(θ, ξ)ˆ g (φ, ξ + θ) , ad(b, d(Γ) θ+φ=χ χ ∈ Γ† , ξ ∈ Rd . (3.2)

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Proposition 3.4 Let b ∈ Sα and gj ∈ Sβj , j = 1, 2, . . . , N . Then ad(b; g1 , . . . , gN ) ∈ N Sγ with γ = α + j=1 (βj − 1), and ad(b; g1 , . . . , gN )

(γ) l,s

≤ Cl,s b

(α) p,s+N

N

gj

(βj ) p,s+N −j+1 ,

(3.3)

j=1

with some p = p(l, N, s, α, βj ), and a constant Cl,s independent of b, gj .

3.2

Asymptotics in the case d = 1

An important part is played by the result obtained in [25] for a PDO acting in L2 (R). We state this result in the form convenient for our purposes. Let b ∈ S0 ( ξ, Λ) and A = Op(a0 ) + Op(b) with a0 (ξ) = gξ 2 with a constant g > 0. We assume that Λ is a one-dimensional lattice with period τ > 0, so that d(Λ) = τ . Note that here and everywhere below, in the case d = 1 we use the notation x and ξ instead of the boldface letters x, ξ. Proposition 3.5 Let d = 1. Suppose that b ∈ S0 ( ξ, Λ) is a τ -periodic symmetric symbol. Then the density of states D(λ; A) is given by the formula 1 |λ| A(λ, g) + O( λ−3/2 ), ∀λ ∈ R, 2πD(λ; A) = 2 − g

λg τ 1 b(x, |λ|g −1 ) + b(x, − |λ|g −1 ) dx. A(λ, g) = 2τ 0 This formula is uniform in the symbol b and in the parameters g and τ satisfying the bounds c ≤ g ≤ C, c ≤ τ ≤ C. Remark that the above formula has an asymptotic meaning only for large λ, and for bounded λ we can only claim that the density D(λ; A) is bounded from above uniformly in b. However it is useful to express these two facts in one formula which is valid for all λ ∈ R. In the study of the density of states for the multidimensional case we shall encounter integrals involving densities for lower-dimensional operators. In particular, there is a need to calculate integrals of the density of states for the one-dimensional case, of the type µ p D(t; A)(λ − t) 2 dt, µ < λ, p ∈ R. (3.4) Dp (µ, λ; A) = −∞

In the next lemma we apply Lemma 2.5 and Proposition 3.5 to compare the above integral for the operator A = A0 + Op(b) with that for the “unperturbed” operator Ao = A0 + Op(bo ), where bo = bo (ξ) is the mean value of b(x, ξ): 1 τ bo (ξ) = b(x, ξ)dx. (3.5) τ 0

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Lemma 3.6 Let = [−Lg −1/2 , Lg −1/2 ], L > 0. Suppose that b ∈ S0 ( ξ, τ Z) is a τ -periodic symmetric symbol such that Op(bo ) ≤ L2 , Op(b − bo ) ≤ L2 .

(3.6)

Suppose also that the subspace H() is reducing for the operator Op(b) and that P(R \ ) Op(b) = Op(bo ). Then for λ/2 ≥ µ ≥ 3L2 and p ∈ R one has |Dp (µ, λ; A) − Dp (µ, λ; Ao )| ≤ Cλ

p−2 2

L.

(3.7)

The constant C = C(p) in (3.7) is finite for all p ∈ R. In particular, D0 (µ, λ; A) = D0 (µ, λ; Ao ). The constant C(p) is independent of µ, λ and uniform in the symbol b and in the parameters g and τ satisfying the conditions c ≤ τ ≤ C, c ≤ g ≤ C. Proof. Compare the densities for A and Ao : Dp (µ, λ; A) = Dp (µ, λ; Ao ) µ p p + D(t; A) − D(t; Ao ) (λ − t) 2 − λ 2 dt −∞ µ p + λ2 D(t; A) − D(t; Ao ) dt.

(3.8)

−∞

Let us consider the second integral first. Since P(R \ ) Op(b) = Op(bo ), we have D(t; A) = D(t; A; ) + D(t; Ao ; R \ ), and hence D(t; A) − D(t; Ao ) = D(t; A; ) − D(t; Ao ; ). To find the integrals of the terms in the r.h.s. use Lemma 2.5. In view of (3.6), AP() ≤ L2 + Op(b) ≤ 3L2 . By the conditions of the lemma this does not exceed µ, and therefore, by Lemma 2.5 and (3.5), τ µ 1 −1/2 2π D(t; A; )dt = 2µLg − (gξ 2 + b(x, ξ))dxdξ τ 0 −∞ µ D(t; Ao ; )dt. = 2π −∞

Hence the second integral in (3.8) vanishes. To handle the first integral in (3.8) we note that in view of Proposition 3.5 |D(t; A) − D(t; Ao )| ≤ C|t|−3/2 ,

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for all t = 0 with a constant C uniform in the symbol b. Notice also that in view of the estimates (3.6) and the property that P(R \ ) Op(b) = Op(bo ), we have D(t; A) = D(t; Ao ) for all t ≥ 3L2 , and thus the limits of integration can be replaced by −3L2 and 3L2 . Now, using the straightforward estimate (λ − t) p2 − λ p2 ≤ Cλ p2 −1 |t|, |t| ≤ µ ≤ λ/2, with a constant C = C(p), we conclude that 3L2 p p o 2 − λ 2 dt (D(t; A) − D(t; A )) (λ − t) −3L2 3L2 p−2 ≤ Cλ 2 |t|−1/2 dt −3L2

˜ ≤ Cλ

p−2 2

L.

The last inequality and the formula (3.8) lead to (3.7).

3.3

Linear change of variables

It is an elementary exercise to describe how the symbols of PDO’s and their densities of state transform under a linear change of variables. Since in what follows we heavily use various changes of variables, below we state these elementary formulae in the form of Lemmas. Let a ∈ Pα (w, Γ) with a function w satisfying (2.1). Let M : Rd → Rd be a non-degenerate linear map. Note first of all that the lattice Γ transforms into an˜ = MΓ. Also, Γ ˜† = (MT )−1 Γ† . It is straightforward other lattice which we denote Γ to see that the sets O˜Γ = MOΓ , O†˜Γ = (MT )−1 O†Γ ˜ and Γ ˜ † respectively, and that are fundamental sets of the lattices Γ ˜ = det M d(Γ). d(Γ) Define the unitary operator W = WM : H → H for any u ∈ H as follows: √ (W u)(t) = det M u(Mt).

(3.9)

Lemma 3.7 Let M be a non-degenerate linear map from Rd to Rd , and let a ∈ Pα (w, Γ). Then the symbol b(x, ξ) of the operator B = W ∗ Op(a)W is given by b(x, ξ) = a(M−1 x, MT ξ).

(3.10)

˜ ˜ = MΓ, the Fourier transform ˆb(θ, ξ) of the symbol This symbol is Γ-periodic with Γ b is given by √ ˆb(θ, ξ) = det M a ˜† , ˆ(MT θ, MT ξ), θ ∈ Γ (3.11) T ˜ ˜ ˜ Γ), w(ξ) ˜ = w(M ξ). Moreover, if a ∈ Sα (w, Γ), then b ∈ Sα (w, ˜ Γ). and b ∈ Pα (w,

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A.V. Sobolev


Proof. The formula (3.10) for b follows by a direct elementary calculation. By ˜ To prove (3.11) inspection the symbol b is periodic with respect to the lattice Γ. write, using (3.10): ˆb(θ, ξ) = 1 ˜ d(Γ) det M = ˜ d(Γ)

O˜Γ

OΓ

a(M−1 x, MT ξ)e−iθ,x dx a(x, MT ξ)e−iM

T

θ,x

√ det M a ˆ(MT θ, MT ξ).

dx =

This proves (3.11). ˜ if a ∈ Pα (w, Γ), or, b ∈ Sα (w, ˜ ˜ Γ) ˜ Γ) Now it is immediate to see that b ∈ Pα (w, if a ∈ Sα (w, Γ). Let us find out how the density of states changes under the change of variables. Lemma 3.8 Let M be a non-degenerate linear map from Rd to Rd , and let a be a Γ-periodic elliptic symbol such that H(C), C ⊂ Rd is an invariant subspace for A = Op(a). Then the subspace H((MT )−1 C) is invariant for B = W ∗ AW, W = WM and DΓ (λ; A; C) = det M D˜Γ (λ; W ∗ AW ; (MT )−1 C). In particular, if C = Rd , then DΓ (λ; A) = det M D˜Γ (λ; W ∗ AW ). Proof. If H(C) is invariant for A, then the invariance of H(C ), C = (MT )−1 C, for B follows from the formula P(C ) = W ∗ P(C)W . The representatives in the Floquet decomposition of A and B = W ∗ AW ˜ k ˜ ∈ O† acting in L2 (OΓ ) and L2 (O˜ ) are the operators A(k), k ∈ O†Γ and B(k), ˜ Γ Γ respectively, with the symbols a(x, γ † + k), γ † ∈ Γ† , and ˜ = a(M−1 x, γ † + MT k), ˜† . ˜ γ ˜ † + k) ˜ † = (MT )−1 γ † ∈ Γ b(x, γ ˜ = W ∗ A(MT k)W ˜ As in Lemma 3.7 one checks directly that B(k) , and therefore

(2π) D˜Γ (λ; B; C ) = d

O˜† Γ

˜= ˜ C dk N λ, B(k);

1 = det M

O˜† Γ

˜ C dk ˜ N λ, A(MT k);

(2π)d DΓ (λ; A; C). N λ, A(k); C dk = det M O†Γ

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A special orthogonal transformation

A special role in what follows will be played by an orthogonal change of variables associated with a vector ν ∈ Rd . From now on we use the notation n(ν) = ν|ν|−1 , 0 = ν ∈ Rd . Recall that ej , j = 1, 2, . . . , d denote the vectors of the standard orthonormal basis in Rd . Let M = M(ν) be an orthogonal transformation M : Rd → Rd such that e1 = Mn(ν). Clearly, M(tν) = M(ν) for any real t > 0. Let us find out how this transformation acts on certain domains in Rd . Let L ≥ 0, s ≥ 0 be some numbers. Now define the domains Λν = {ξ ∈ Rd : | ξ, ν| ≤ L|ν|} = {ξ ∈ Rd : | ξ, n(ν)| ≤ L}. 2

2

(3.12) 2

Ων (s) = {ξ ∈ R : |ξ| − | ξ, n(ν)| ≥ s }. d

(3.13)

The number L will be kept the same throughout the paper and thus it is not reflected in the notation. The geometrical meaning of the sets Λν and Ων is simple: in particular, Ων is the set of all vectors ξ such that the distance from ξ to the one-dimensional subspace spanned by ν is greater than s. Clearly, for any t = 0 one has Λtν = Λν and Ωtν = Ων . For the case d = 2 the definition of Ων can be simplified. Namely, let 0 1 J= . (3.14) −1 0 Then for any t ∈ R2 the vector t⊥ = Jt is orthogonal to t. Now Ων can be rewritten as follows: Ων (s) = {ξ ∈ R2 : | n⊥ (ν), ξ| ≥ s}.

(3.15)

The next lemma describes how the sets Λν and Ων (s) transform under M(ν): Lemma 3.9 Let M = M(ν) be the orthogonal transformation defined above. Then MΛν = Λe1 = {ξ1 ∈ R : |ξ1 | ≤ L} × Rd−1 , ˆ ≥ s}. MΩν (s) = Ωe (s) = R × {ξˆ ∈ Rd−1 : |ξ| 1

Proof. As MC = {ξ ∈ Rd : MT ξ ∈ C} for any set C ⊂ Rd , we have MΛν = {ξ ∈ Rd : | MT ξ, n(ν)| ≤ L} = {ξ ∈ Rd : | MT ξ, MT e1 | ≤ L} = Λe1 . Here we have used the property e1 = Mn(ν). Similarly for Ων (s).

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4 Further properties of periodic PDO’s. Partition of symbols In this section we describe two procedures of partitioning the symbols, that will play a crucial role in the study of spectral properties of PDO’s. Roughly speaking, the aim of these partitions is to transform the operators to a form when separation of variables becomes possible.

4.1

Partition I

The first partition is designed to split the symbol into components supported on different parts of the dual space. The general definitions below will be given for symbols b ∈ Pα (w, Γ) with an arbitrary weight w, but later we shall make more restrictive assumptions. Let Υ ∈ C∞ 0 (R) be a non-negative function such that

0 ≤ Υ ≤ 1, Υ(t) =

1, |t| ≤ 1/4; 0, |t| ≥ 1/2.

(4.1)

Assume also for convenience that Υ is even, i.e., Υ(t) = Υ(−t). For a number L ≥ 1 define 

θ, ξ + θ/2  ζθ (ξ; L) = Υ ,  |θ|L (4.2)    ϕθ (ξ; L) = 1 − ζθ (ξ; L). We point out that ϕθ (ξ; L) = ϕ−θ (ξ + θ; L), ζθ (ξ; L) = ζ−θ (ξ + θ; L),

(4.3)

since the function Υ is even. Note that |Ds ϕθ (ξ; L)| + |Ds ζθ (ξ; L)| ≤ Cs L−s , s = |s|.

(4.4)

This inequality shows that the functions ζθ and ϕθ , viewed as symbols, belong to the class S0 (L). Using ϕθ , ζθ , we shall introduce the following linear operations on PDO’s. Fix a positive parameter r and let Θr = Θr (Γ) = {θ ∈ Γ† : 0 < |θ| ≤ r}, Θ0r = Θr ∪ {0}, Ξr = Ξr (Γ) = {θ ∈ Γ† : |θ| > r} = Γ† \ Θ0r (Γ).

(4.5)

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Given a symbol b ∈ Sα (w), define four new symbols bo , b , b , b↑ as follows: 1 ˆb(θ, ξ)eiθx , b↑ (x, ξ) = d(Γ) θ∈Ξr 1 ˆb(θ, ξ)ϕθ (ξ; L)eiθx , b (x, ξ) = d(Γ) θ∈Θr 1 ˆb(θ, ξ)ζθ (ξ; L)eiθx , b (x, ξ) = d(Γ) θ∈Θr 1 ˆ b(0, ξ). bo (x, ξ) = bo (ξ) = d(Γ) By definition of Υ

(4.6) (4.7) (4.8) (4.9)

b = b↑ + bo + b + b .

The symbol b↑ contains only the Fourier coefficients ˆb(θ, ξ) with large θ’s, and later it will be shown not to contribute to the answer. The remaining symbols are trigonometric polynomials in x. The Fourier coefficient ˆb (θ, · ) is supported near the hyperplane θ, ξ + θ/2 = 0, whereas ˆb (θ, · ) lives away from this hyperplane. It is easy to see that for any symbol b ∈ Pα the introduced symbols also belong to the same class. The corresponding operators are denoted by B ↑ = Op(b↑ ), B = Op(b ), B = Op(b ), B o = Op(bo ). We also denote B ,↑ = B + B ↑ . Assume now that b ∈ Sα (w, Γ) with a constant weight w = L ∈ [1, L]. Then the operations introduced above preserve the properties of the symbol b, that is for any b ∈ Sα (L , Γ) the symbols b , b , b↑ , bo belong to the same class and for all l, s and p < l − d

(α) (α) (α) (α) b l,s + b l,s + bo l,s ≤ C b l,s , (4.10) (α) (α) b↑ p,s ≤ Crp−l+d b l,s . The constant C in the above estimates depends on l, s and on the lattice Γ. The first estimate immediately follows from (4.4). The second bound is a consequence of (2.3) and (4.5). In the case d = 1 the operation “” possesses one more useful property: Lemma 4.1 Let d = 1 and let b ∈ Sα (L, Γ), α ≤ 0. Then for r ≤ L the symbol b belongs to Sα ( ξ, Γ) and b

(α) l,s;ξ

≤ Cl,s b

with a constant Cl,s depending only on l, s.

(α) l,s;L

52

A.V. Sobolev


Proof. The proof is similar to the above proof of (4.10) for the classes with a constant weight. One uses the fact that |ξ| ≤ L/2 + r/2 ≤ L on the support of ζθ , and thus |∂ξsˆb(θ, ξ)| ≤ b

−l α−s l,s;L θ L

≤ Cl,s b

−l α−s , l,s;L θ ξ

for all ξ ∈ supp ζθ and θ ∈ Θr . It also follows from (4.4) that |∂ξs ϕθ (ξ; L)| + |∂ξs ζθ (ξ; L)| ≤ Cs ξ−s . The above estimates lead to the proclaimed estimate for the norm b

(α) l,s;ξ .

To check that the introduced operations preserve symmetry, calculate using (4.3) and remembering that θ and −θ do ( or do not ) belong to Θr simultaneously: ˆb (−θ, ξ + θ) = ˆb(−θ, ξ + θ)ζ−θ (ξ + θ; L) = ˆb(θ, ξ)ζθ (ξ; L) = ˆb (θ, ξ). Thus, by (2.6) the operator B is symmetric if so is B. Similarly for B and B ↑ . Let us find out how these symbols transform under an orthogonal change of variables M. Below we use the notation for the transformed objects, introduced in the beginning of Subsect. 3.3. Note first a direct consequence of the Definition (4.2)  ζθ (MT ξ; L) = ζω (ξ; L), ˜† = MΓ† . (4.11) ω = Mθ ∈ Γ  T ϕθ (M ξ; L) = ϕω (ξ; L), Lemma 4.2 Let M : Rd → Rd be an orthogonal transformation. Let a ∈ Pα (w) with some α ∈ R. Denote by any of the symbols ↑, , o or . Then ∗ ∗ WM AWM = WM A WM . ∗ AWM . We consider only the case = . By Definition (4.8) Proof. Let B = WM the symbol of the operator B coincides with

1 ˆb(ω, ξ)ζω (ξ; L)eiω,x . ˜ d(Γ) ω∈Θr (˜Γ) ˜ = Θr (Γ) and by (3.11) the Fourier transform of the On the other hand, MT Θr (Γ) ∗ symbol of the operator WM A WM is given by ˜† . a ˆ(MT ω, MT ξ)ζMT ω (MT ξ; L), ω ∈ Γ In view of (3.11) and (4.11) this coincides with ˆb(ω, ξ)ζω (ξ; L), as required.

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Partition II

Here we describe a way to split symbols into components associated with the socalled primitive vectors ν ∈ Γ† . For a symbol b ∈ Sα (w, Γ) introduce a family of associated symbols constructed as follows. For a vector 0 = ν ∈ Γ† define the subset (4.12) Γ†ν = {nν, 0 = n ∈ Z}. Let us introduce the symbol 1 ˆb(θ, ξ)eiθx = 1 ˆb(nν, ξ)einνx . bν (x, ξ) = d(Γ) d(Γ) † 0 =n∈Z θ∈Γν

Clearly, (bν )ν = bν . Notice that this symbol is symmetric if so is the initial symbol (α) (α) b(x, ξ), see (2.6). Besides, bν ∈ Sα (w, Γ) and bν l,s ≤ b l,s . Let C ⊂ Rd be a set, containing along with each point ξ ∈ C the straight line {ξ + tν, t ∈ R}. Then assuming that all the PDO’s at hand are defined on the set B(Rd ) (see (2.5)), it is straightforward to see H(C) is a reducing subspace of Op(bν ), and in particular, that P(C) Op(bν ) = Op(bν )P(C) (see Subsect. 2.4 for definition of P(C)). For example, the set Ων = Ων (s) defined in (3.13) possesses this property and therefore P(Ων ) Op(bν ) = Op(bν )P(Ων ). Below we decompose any symbol in the sum of symbols of the form bν . To explain how it is done we need to recall the definition of a primitive vector. Definition 4.3 A non-zero vector ν ∈ Λ is said to be a primitive vector of the lattice Λ ⊂ Rd if (i) The first non-zero coordinate of ν = (ν1 , ν2 , . . . , νd ) is positive; (ii) There are no vector η ∈ Λ distinct from ν and no integer n > 0 such that ν = nη. It follows from this definition that for each non-zero vector χ ∈ Λ there exist a uniquely defined integer n and a primitive vector ν ∈ Λ such that χ = nν. Also, every two primitive vectors are linearly independent. The set of all primitive vectors of the lattice Λ is denoted by P Λ. Now we can decompose any symbol b ∈ Sα (w, Γ) into a sum over primitive vectors ν ∈ P Γ† :

b(x, ξ) = bo (ξ) +

bν (x, ξ),

(4.13)

ν∈P Γ†

see (4.9) for definition of bo . From this and Definition (2.2) it follows that max

bo

(α) l,s ,

bη

(α) l,s

≤ b

(α) l,s

≤ max

bo

(α) l,s ,

sup ν∈P Γ†

bν

(α) l,s ,

∀η ∈ Γ† . (4.14)

54

A.V. Sobolev

Denote


r = P Γ† ∩ Θ r . Θ

(4.15)

Let us now combine two types of partition introduced above. Observe first that by definitions (4.6)–(4.8) we always have (b )ν = (bν ) , =↑, , , so that one can use the notation bν without risk of confusion. Moreover, since b (x, ξ) is a trigonometric polynomial, one can rewrite b (x, ξ) = b ν (x, ξ). r ν∈Θ

Due to the presence of a cut-off in the definition, the operators b ν have an additional reducing subspace. Let Λν be the set defined in (3.12). Then the following lemma holds: Lemma 4.4 Let the operator B with the symbol b ∈ Pα (w) be defined on B(Rd ) (see (2.5)). Suppose that r ≤ L. Then BP(Λν ) = Bν P(Λν ) = Bν for any ν ∈ Θr . For a symmetric symbol b this lemma implies that the subspace H(Λν ) is reducing for the operator Bν and that Bν P(R2 \ Λν ) = 0. Proof. It suffices to check that the support of ζθ , θ = nν ∈ Θr is contained in the domain Λν . By Definition (4.1), under the condition r ≤ L we have for each ξ ∈ supp ζθ and n = 0: 1 1 | θ, ξ| ≤ | θ, ξ + θ/2| + |θ|2 ≤ L|θ| + r|θ| ≤ L|θ|. 2 2

It remains to recall (3.12).

Suppose now that the symbol b is symmetric. As was already mentioned, the subspace H(Ων ) is then reducing for Bν . Together with Lemma 4.4 this implies that for the set ˆ ν (s) = Λν ∩ Ων (s) (4.16) Λ ˆ ν ) is reducing for B if r ≤ L. The properties of the operators the subspace H(Λ ν Bν on these reducing subspaces will be a key ingredient in our study of the density of states. In this study the set Λ = ∪ν∈Θr Λν plays the role of the resonant set described in the introduction. ˆν. First of all we need to establish some geometric properties of the sets Λ This will be done for the case d = 2 only.

4.3

Some geometric estimates for d = 2

Recall the notation n(θ) = θ|θ|−1 , and t⊥ = Jt, see (3.14) for definition of J. Lemma 4.5 Let d = 2. Suppose that θ, η ∈ Θr are linearly independent. Then | n(θ), n⊥ (η)| ≥ d(Γ† )r−2 .

(4.17)

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Proof. Let γ 1 , γ 2 be two basis vectors of the lattice Γ† , so that θ = n1 γ 1 + n2 γ 2 and η = m1 γ 1 + m2 γ 2 with some integers n1 , n2 , m1 , m2 . Consequently, ⊥ ⊥

θ, (η)⊥ = n1 m2 γ 1 , γ ⊥ 2 + n2 m1 γ 1 , γ 2 = (n1 m2 − n2 m1 ) γ 1 , γ 2 .

Since θ and η are linearly independent, the integer factor n1 m2 − n2 m1 never vanishes, and thus | θ, η ⊥ | ≥ | Jγ 1 , γ 2 | = d(Γ† ). Since |θ| ≤ r and |η| ≤ r, we obtain (4.17).

Lemma 4.6 Let d = 2. Suppose that ν, µ ∈ Θr are linearly independent. If d(Γ† )ρr−2 L−1 ≥ 4 and ρL−1 1 ≥ 4, then one has ˆ ν (ρ − L1 ). (4.18) | n(µ), ξ| ≥ 2−1 d(Γ† )r−2 ρ, ∀ξ ∈ Λ ˆ ν (ρ − L1 ), Λµ (L) ≥ 4−1 d(Γ† )r−2 ρ. ˆ ν (ρ − L1 ) ∩ Λµ (L) = ∅ and dist Λ Moreover, Λ Proof. Use the short-hand notation nµ = n(µ), nν = n(ν). Decompose the vector nµ into a sum as follows ⊥ nµ = nµ , nν nν + nµ , n⊥ ν nν

= anν + a⊥ n⊥ ν. Then

nµ , ξ = a⊥ n⊥ ν , ξ + a nν , ξ. ˆν = Λ ˆ ν (ρ − L1 ) then | nν , ξ| ≤ L and by (3.15), | n⊥ , ξ| ≥ ρ − L1 , so If ξ ∈ Λ ν that | nµ , ξ| ≥ |a⊥ |(ρ − L1 ) − L. By Lemma 4.5 |a⊥ | ≥ κr−2 with κ = d(Γ† ), and hence | nµ , ξ| ≥ κr−2 ρ − (κr−2 L1 + L) = κr−2 ρ 1 − L1 ρ−1 − κ −1 Lr2 ρ−1 . Since ρr−2 L−1 ≥ 4κ −1 and ρL1 ≥ 4, the r.h.s. is greater than 2−1 κρr−2 , which guarantees (4.18). The r.h.s. of (4.18) is also greater than L, which leads to the ˆ ν and t ∈ Λµ , we have ˆ ν ∩ Λµ = ∅. Also, for any ξ ∈ Λ identity Λ |ξ − t| ≥ | ξ, nµ | − | t, nµ | ≥ 2−1 κρr−2 − L. Again, under the condition imposed on ρ, r, L we have the required lower bound ˆ ν and Λµ . for the distance between Λ

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A.V. Sobolev

4.4


Separation of variables

Observe that the condition d = 2 is crucial for the previous two lemmas. It guarˆ ν and Λµ do not intersect for linearly independent ν and µ. antees that the sets Λ This fact allows one to ”separate variables” when studying the model operator A = Ao + B , Ao = Op(ao ), ao (ξ) = |ξ|2 + bo (ξ).

(4.19)

Lemma 4.7 Let d = 2. Suppose that b ∈ P0 (w) is a symmetric symbol. Let λ0 ≥ 0 be a fixed number. Suppose that 1 ≤ r ≤ L, ρL−1 1 ≥ 4, d(Γ† )ρr−2 L−1 ≥ 4, 2ρL1 ≥ λ0 + 3L2 + L21 , and that B o + B ≤ 2L2 . Then ˆ + ˆν) D(λ; A) = D λ; Ao ; R2 \ Λ D(λ; Ao + Bν ; Λ

(4.20)

r ν∈Θ

ˆν. ˆν = Λ ˆ ν (ρ − L1 ) and Λ ˆ = ∪ν Λ for all λ ≥ ρ2 − λ0 . Here Λ ˆ ν and Proof. Let Cν = Λν \ Λ ˆν. ˆ = ∪ν Λ C = ∪ν Cν , Λ = ∪ν Λν , Λ By Lemma 4.4 the subspace H(Λ) is invariant for the operator A, and AP(R2 \Λ) = ˆ and H(C) are also Ao P(R2 \ Λ). By virtue of Lemma 4.6 the subspaces H(Λ) invariant for A and thus ˆ . N (λ, A(k)) = N λ, Ao (k); R2 \ Λ + N λ, A(k); C + N λ, A(k); Λ (4.21) Furthermore, the third term in the last formula equals ˆ ν ), N λ, Ao (k) + Bν (k); Λ r ν∈Θ

ˆ ν are disjoint for distinct ν’s. Hence it remains to verify that since Λ N λ, A(k); C = N (λ, Ao (k); C . For any ξ ∈ Cν one has: |ξ|2 = | ξ, n(ν)|2 + | ξ, n⊥ (ν)|2 ≤ L2 + (ρ − L1 )2 = ρ2 − 2ρL1 + L21 + L2 . Since ρL−1 ≥ 4 and 2ρL1 ≥ λ0 + 3L2 + L21 , the r.h.s. is bounded from above 1 2 by ρ − λ0 − 2L2 ≤ λ − 2L2 . Consequently, Ao P(C) ≤ λ − 2L2 + B o . By AP(C) virtue of the condition B o + B ≤ 2L2 we have ≤ λ, which implies by an elementary perturbation argument that N λ, A(k); C = N λ, Ao (k); C , as required.

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5 Asymptotics in the invariant subspaces In this section we study the density of states for the model operator (4.19) introduced in the previous section. Our strategy is dictated by Lemma 4.7: since ˆ ν do not intersect for distinct ν ∈ Θ ˜ r , the investigation of each term in the sets Λ the r.h.s. of (4.20) is done independently. Furthermore, the symbol of the operator Ao + Bν depends only on the projection x, n(ν) (see (4.8)), so that in each ˆ ν ) the problem reduces to a one-dimensional one. subspace H(Λ We begin with studying the operator Ao + Bν . Emphasize that the analysis of this operator is carried out without any restrictions on the dimension d ≥ 2. On the contrary, in the closing subsection, dealing with the operator (4.19), we need to assume that d = 2 in order to use Lemma 4.7.

5.1

A new class of symbols

In order to describe the reduction to a one-dimensional problem we need to introduce a new class of symbols that encode the “one-dimensionality”. Let z = z(x, η), x ∈ Rd , η ∈ R be a function and L ≥ 1 be a constant. We say that this function belongs to the class Tα (L, Γ), α ∈ R, if it is Γ-periodic, C∞ -smooth in η and its Fourier coefficients satisfy the condition |∂ηs zˆ(θ, η)| ≤ Cl,s θ−l Lα−s , for all integer l ≥ 0 and s ≥ 0. Similarly to the class Sα (L, Γ) introduce the norm z

(α) l,s

= max max sup sup θp L−α+r |∂ηr zˆ(θ, η)|. r≤s p≤l

θ

η

We are interested in the PDO’s with symbols having the following Fourier coefficients:

ˆb(θ, ξ) = zˆ θ, ξ, n(θ) , θ = 0, (5.1) 0, θ = 0. Clearly, b ∈ Sα (L, Γ) and −1 z Cl,s

(α) l,s

≤ b

(α) l,s

≤ Cl,s z

(α) l,s ,

(5.2)

with some constant Cl,s . We call the symbol z symmetric if zˆ(θ, η) = zˆ(−θ, −η − |θ|).

(5.3)

It is straightforward to check that under this condition the symbol (5.1) is symmetric in the sense of Definition (2.6).

58

5.2

A.V. Sobolev


Reduction to the case d = 1

Assuming, that the symbol b is defined by (5.1) we study the density of states for an operator with the symbol a(x, ξ) = ao (ξ) + bν (x, ξ), ao (ξ) = |ξ|2 + f ξ, n(ν) ,

(5.4)

with some 0 = ν ∈ Γ† , and some real-valued uniformly bounded function f . We are interested in the “partial” density of states D(λ; A; C) with C = Ων ˆ ν . The first step is to perform a change of variables which reduces Ων or C = Λ ˆ ν to Ωe1 and Λ ˆ e1 . Let M = M(ν) be the orthogonal map from Subsect. 3.4, and Λ and let W = WM be the unitary operator defined in (3.9). Then by Lemma 3.7 the symbol of A˜ = W ∗ AW is given by a ˜(x, ξ) = |ξ|2 + f MT ξ, n(ν) + bν (M−1 x, MT ξ). Remembering that Mn(ν) = e1 , we have MT ξ, n(ν) = ξ1 . Using the last relation it is easy to find that bν (M−1 x, MT ξ) = ˜b(x1 , ξ1 ) with the symbol ˜b given by ˜b(x, η) = 1 zˆ lν, l|l|−1 η ei|ν|lx , x ∈ R, η ∈ R. (5.5) d(Γ) l =0 Consequently,

 a ˜(x, ξ) =  o a ˜ (ξ) =

a õ (ξ) + ˜b(x1 , ξ1 ), ˆ 2 + ξ 2 + f ξ1 , ξˆ = (ξ2 , ξ3 , . . . , ξd ). |ξ| 1

(5.6)

Note that the symbol ˜b is periodic in x with the period τ = τ (ν) = 2π

[|ν|] + 1 , |ν|

(5.7)

so that the whole symbol a ˜ is periodic w.r.t. the lattice Λ = (τ Z) × (2πZ)d−1 . Note that τ (ν) is bounded from above and below uniformly in ν ∈ P Γ† . By Lemma 3.8 we have ˜ = MΓ. ˜ MC), Γ DΓ (λ; A; C) = D˜Γ (λ; A; Since the density of states does not depend on the lattice (see Subsection 2.4), we ˜ with Λ defined above, and thus can replace Γ ˜ MC), Λ = (τ Z) × (2πZ)d−1 . DΓ (λ; A; C) = DΛ (λ; A;

(5.8)

In the next three lemmas we show that up to a controllable error the densities of states for the operator A = Op(a) and Ao = Op(ao ) coincide. We begin with a reduction to a one-dimensional operator T with the symbol t(x, η) = η 2 + f (η) + ˜b x, η , x ∈ R, η ∈ R. (5.9)

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The following lemma compares the densities of states for the operators A = Ao +Bν and Ao +Bν with those for the operators T and T o +T respectively, see subsection 4.1 for definitions. Lemma 5.1 Suppose that z ∈ Tα (L, Γ) is a symmetric symbol. Let b be a symbol defined as in (5.1), and let f be a real-valued uniformly bounded function on R. Let ˜b and t be the symbols defined in (5.5) and (5.9) respectively. Then (i) The symbol ˜b belongs to Sα (L, τ Z) with τ specified in (5.7), it is symmetric, (α) (α) and ˜b l,s ≤ Cl,s z l,s with a constant Cl,s depending only on the lattice Γ; (ii) For all λ ∈ R and s ≥ 0 one has  D λ; A; Ων (s) = σDd−3 (λ − s2 , λ, T ), ωd−2 σ= , Dλ; Ao + B ; Ω (s) = σD (λ − s2 , λ; T o + T ), 2(2π)d−1 ν d−3 ν (5.10) where (5.11) ωp = (p + 1) wp+1 , p ≥ 1, is the surface area of a unit sphere in Rp+1 , ω0 = 2, and the quantity Dq is defined in (3.4). Proof. (i) The τ -periodicity of the symbol ˜b with the specified τ has already been observed. The estimate for the norm ˜b follows by inspection. The symmetry of ˜b follows from (5.3). (ii) To prove (5.10) we use Lemma 3.9: ˆ ≥ s}. MΩν (s) = Ωe1 (s) = R × {ξˆ ∈ Rd−1 : |ξ| Thus we can now use (5.8) and Lemma 2.6 to conclude that ˜ Ωe1 (s) (2π)d−1 DΓ λ, A; Ων (s) = (2π)d−1 DΛ λ, A; ˆ 2 ; T )dξˆ = ωd−2 = D(λ − |ξ| D(λ − ξ 2 ; T )ξ d−2 dξ ˆ |ξ|≥s

=

ωd−2 2

ξ≥s λ−s2

−∞

D(µ; T )(λ − µ)

d−3 2

dµ.

By Lemma 4.2 an analogous formula holds for operators Ao + Bν and T o + T . Definition (3.4) leads to the proclaimed formula (5.10). Before calculating the asymptotics of the r.h.s. of (5.10) we need to study the density of states for the operator Ao = Op(ao ) with the symbol ao defined in (5.4). We shall need the notation E(λ; · ) introduced in (2.13).

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Lemma 5.2 Let f be a uniformly bounded function. Suppose that s ≥ 0 and λ ≥ 0 and 0 ≤ λ0 ≤ λ/2 are numbers such that s2 ≤ λ − λ0 − L2 − sup |f (η)|,

(5.12)

η

L2 + sup |f (η)| ≤ η

λ − λ0 . 2

(5.13)

Then for any λ , λ ∈ [λ − λ0 , λ + λ0 ] D(λ , Ao ; Λ ˆ ν (s)) − D(λ , Ao ; Λ ˆ ν (s)) ≤ Cλ d−3 2 |λ − λ |L,

(5.14)

with a constant C independent of the numbers λ, λ , λ , L, s, vector ν and symbol f . Proof. Let M = M(ν) be the orthogonal map from Subsection 3.4. By Definition (4.16) and Lemma 3.9 ˆ ≥ s}. ˆ e1 (s) = {|ξ1 | ≤ L} × {|ξ| ˆ ν (s) = Λ MΛ According to (5.8), formula (5.6) and Proposition 2.4, under the condition (5.12) for any λ ∈ [λ − λ0 , λ + λ0 ] we have the following formula: ˆ e1 (s) ∩ E(λ ; a ˆ ν (s) = (2π)d D λ , Aõ ; Λ ˆ e1 (s) = vol Λ (2π)d D λ , Ao ; Λ õ ) L d−1 = wd−1 λ − η 2 − f (η) 2 dη − 2L wd−1 sd−1 . −L

Here wp is the volume of the unit ball in Rp . Under the conditions (5.12) and (5.13) the above formula yields (5.14). The next lemma yields an important intermediate result – it provides an asymptotic formula for the density of states of the operator Ao + Bν : Lemma 5.3 Let f be a uniformly bounded function, and let b be defined by (5.1) with a symmetric symbol z ∈ T0 (L, Γ). Suppose that r ≤ L and that for some λ0 ∈ [0, λ/2]   Bν ≤ L2 , supη |f (η)| ≤ L2 , (5.15) λ + λ0  3L2 ≤ λ − λ0 − s2 , s2 ≥ . 2 Then for all λ , λ ∈ [λ − λ0 , λ + λ0 ] one has o D λ , A + Bν ; Λ ˆ ν (s) −

1 ˆ ν (s) ∩ E(λ ; ao ) ≤ Cλ d−5 2 L, vol Λ d (2π)

(5.16)

and ˆ ν (s) − D λ ; Ao + Bν ; Λ ˆ ν (s) | |D λ ; Ao + Bν ; Λ ≤ Cλ

d−3 2

L |λ − λ | + λ−1 . (5.17)

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The constants in bounds (5.16) and (5.17) depend on λ0 . They do not depend on λ, s, ν, L, f , and are uniform in the symbol z. Proof. We derive the required formulas from the relation (5.10) with the use of Lemma 3.6. Let us check that its conditions are satisfied. The symbol t = ˜b = ˜b (x, η) (see (5.9) for definition of t) has the form considered in Lemma 3.6 with g = 1. Moreover, since r < L, by Lemma 4.4 we have P() Op(˜b ) = Op(˜b ), where = [−L, L]. Furthermore, by Lemma 4.1 ˜b ∈ S0 ( η, τ Z), and in view of Lemma 5.1(i) we have ˜b (α) ≤ C ˜b (α) ≤ C˜ z (α) , l,s;L l,s l,s;η with some universal constants. And finally, the conditions (5.15) guarantee that Op(f ) ≤ L2 , T ≤ L2 and λ /2 ≥ λ − s2 ≥ 3L2 , for all λ ∈ [λ − λ0 , λ + λ0 ]. Now we can apply Lemma 3.6 with p = d − 3 to the r.h.s. of (5.10), which leads to Dd−3 (λ − s2 , λ ; T o + T ) − Dd−3 (λ − s2 , λ ; T o ) ≤ Cλ d−5 ˜ d−5 2 L ≤ Cλ 2 L, and hence

D(λ , Ao + B ; Ων (s)) − D(λ , Ao ; Ων (s)) ≤ C λ d−5 2 L, ν

(5.18)

ˆ ν (s) note for all λ ∈ [λ − λ0 , λ + λ0 ]. To establish a similar estimate for the set Λ that by Lemma 4.4 ˆ ν (s)) = D(λ ; Ao + B ; Ων (s) − D λ , Ao ; Ων (s) \ Λ ˆ ν (s) , D λ , Ao + Bν ; Λ ν and hence (5.18) yields ˆ ν (s)) − D(λ , Ao ; Λ ˆ ν (s)) ≤ C λ d−5 D(λ , Ao + Bν ; Λ 2 L,

(5.19)

for all λ ∈ [λ − λ0 , λ + λ0 ]. By virtue of Proposition 2.4 the second term in the l.h.s. coincides with 1 ˆ ν (s) ∩ E(λ ; ao ) , vol Ων (s) \ Λ (2π)d which implies (5.16). Note that the conditions (5.15) guarantee (5.12) and (5.13). Now to obtain (5.17) it suffices to use (5.19) for λ , λ , and (5.14).

5.3

Density of states for the model operator (4.19)

Our goal in this subsection is to establish a formula similar to (5.16) for the model operator (4.19). Now it is crucial to assume that d = 2. Let us first specify the symbols that we are working with. Let f (ν) = f (ν) (η), r be a collection of real-valued uniformly bounded functions. Also η ∈ R, ν ∈ Θ suppose that the quantity κν = sup |η|β |f (ν) (η)| |η|≥4L

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A.V. Sobolev


is finite for some β ≥ 0. This implies that |f (ν) (η)| ≤ κν |η|−β , ∀|η| ≥ 4L.

(5.20)

Instead of (5.4) assume that

ao (ξ) = |ξ|2 + f (ξ), f (ξ) =

f (ν) n(ν), ξ .

(5.21)

r ν∈Θ

All the subsequent results will be uniform in the function f in the sense that they depend only on the constant C in the bound supξ |f (ξ)| ≤ C. The perturbation symbol b is chosen in the same way as above, i.e., it is defined by the formula (5.1) for some symmetric z ∈ T0 (L, Γ). Our objective is to compare the density of states for the operator A = Ao + B with that of Ao . We are going to use the notation already exploited in the proof of Lemma 4.7: ˆ ˆ ˆ ˆ Λ = ∪ν∈Θ r Λν , Λν = Λν (ρ − L1 ), Λ = ∪ν∈Θ r Λν . For technical reasons we also need to include a bounded perturbation given by a self-adjoint PDO Q with a symbol q ∈ P0 (w) with an arbitrary weight w. Theorem 5.4 Let d = 2. Let the operator A be as described above with z ∈ T0 (L, Γ), and let q ∈ P0 (w) be a symmetric symbol for some weight w. Denote ˆ ν ), κ = δ = max Q ν P(Λ κν . r ν∈Θ

r ν∈Θ

For a fixed λ0 ≥ 0 denote λ1 = λ0 + δ + κ(4L)−β . Suppose that ρ2 ≥ 16λ1 and that   1 ≤ r ≤ L,    ρ ≥ 4L1 ,     † −2 d(Γ )ρr ≥ 8L, 2ρL1 ≥ 2λ1 + 3L2 + L21 .

(5.22)

(5.23)

(5.24)

Then there exists a constant L0 = L0 (z, q, f ) such that under the condition L ≥ L0 , one has D(λ; Ao + B + Q ) −

(5.25)

1 vol E(λ; ao ) ≤ Cr2 ρ−1 L(κr2β ρ−β + δ) + Cr2 ρ−3 L, 2 (2π)

for all λ ∈ [ρ2 − λ0 , ρ2 + λ0 ]. The constants C, L0 are uniform in the symbols f, q and z, and C may depend on λ0 and λ1 .

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Proof. Since z ∈ T0 (L) and q ∈ P0 (w), the operators B, B , Q, Q are bounded by Proposition 3.1 and Lemma 5.1(i):  Bν + B + B ≤ Cl z (0) l,0 , ∀ν ∈ Γ† , l > 2. Q + Q + Q ≤ C q (0) , l ν l,0 Choosing sufficiently large L0 one ensures that Op(f ) + B + Q ≤ 2L2 . Remembering also (5.23), (5.24), one guarantees that the conditions of Lemma 4.7 are fulfilled. Consequently, in view of (4.20) ˆ + ˆ ν ), (5.26) D(λ; Ao + B + Q ) = D(λ; Ao ; R2 \ Λ) D(λ; Ao + Bν + Q ν ; Λ r ν∈Θ

for all λ ≥ ρ2 − λ1 . By Proposition 2.4 the first term equals ˆ = D(λ; Ao ; R2 \ Λ)

1 ˆ ∩ E(λ; ao ) . vol R2 \ Λ 2 (2π)

(5.27)

r Let us consider each summand in the second term separately. Let us fix a ν ∈ Θ and define y o (ξ) = |ξ|2 + f (ν) n(ν), ξ , Y o = Op(y o ). Note that in view of (4.18) we have | n(µ), ξ| ≥ 2−1 d(Γ† )r−2 ρ ≥ 4L for all ˆ ν for any two ν, µ ∈ Θ r such that ν = µ, so that by (5.20) ξ∈Λ −β 2β −β (µ)

n(µ), ξ ≤ 2β d(Γ† ) max sup f κr ρ ≤ κ(4L)−β . r ξ∈Λ ˆν ν∈Θ

r ν =µ∈Θ

ˆ ν ) one has Consequently, with δν = Q ν P(Λ −β 2β −β κr ρ − δν ≤ Ao + Bν + Q ν Y o + Bν − 2β d(Γ† ) −β 2β −β ≤ Y o + Bν + 2β d(Γ† ) κr ρ + δν . (5.28) Here we assume that all the operators are considered on their invariant subˆ ν ). Choosing L0 sufficiently large we may assume that B ≤ L2 , space H(Λ ν (ν) supη |f (η)| ≤ L2 , so that the first half of the conditions (5.15) are satisfied. Using the bounds ρ2 ≥ 16λ1 and (5.24), under the condition λ ∈ [ρ2 − λ1 , ρ2 + λ1 ] (see (5.22) for definition of λ1 ) one proves that the second half of (5.15) is also satisfied with s = ρ − L1 and λ1 instead of λ0 . Therefore, by (5.16) D(λ; Y o + B ; Λ ˆν) − ν

1 ˆ ν ∩ E(λ; y o ) ≤ Cρ−3 L, vol Λ 2 (2π)

∀λ ∈ [ρ2 − λ1 , ρ2 + λ1 ].

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In view of monotonicity of the density of states (see Proposition 2.1) and the bounds (5.28), (5.17) we have ˆ ν ) − D(λ;Ao + Bν + Q ν ; Λ ˆ ν )| ≤ Cρ−3 L + C ρ−1 L(κr2β ρ−β + δ), |D(λ;Y o + Bν ; Λ ∀λ ∈ [ρ2 − λ0 ,ρ2 + λ0 ]. The last two estimates lead to the bound ˆν) − D(λ; Ao + Bν + Q ν ; Λ

1 ˆ ν ∩ E(λ; y o ) vol Λ 2 (2π)

≤ Cρ−3 L + C ρ−1 L(κr2β ρ−β + δ),

(5.29)

∀λ ∈ [ρ2 − λ0 , ρ2 + λ0 ]. Using this estimate for Q = B = 0, in combination with Proposition 2.4 one also concludes that vol Λ ˆ ν ∩ E(λ; ao ) − vol Λ ˆ ν ∩ E(λ; y o ) ≤ Cρ−3 L + C ρ−1 Lκr2β ρ−β . This shows that in the estimate (5.29) the set E(λ; y o ) can be replaced with ˜ r , taking into account E(λ; ao ). Adding together the formulae (5.29) for all ν ∈ Θ 2 ˜ that card Θr ≤ Cr , we obtain that ˆν) − D(λ; Ao + Bν + Q ν ; Λ ˜r ν∈Θ

1 ˆ ∩ E(λ; ao ) vol Λ 2 (2π) ≤ Cr2 ρ−3 L + C r2 ρ−1 L(κr2β ρ−β + δ).

It remains to combine the obtained formula with (5.27), using (5.26).

6 A “gauge transformation” In this and all the subsequent sections we use the notation Sα for the class Sα (L). For the classes Sα (w) with different weight w we use the full notation to avoid confusion.

6.1

Preparation

Our strategy will be to find a unitary operator which reduces an elliptic PDO H = H0 + Op(b) (see Definition (2.8)) with b ∈ Sα ( ξ), α < m to another PDO, whose symbol, up to some controllable small errors, depends only on ξ. Very soon we shall focus on the operators of second order, but in this subsection the order is irrelevant and it is allowed to be any positive m > 0. The sought unitary operator will be constructed in the form U = eiΨ with a suitable bounded self-adjoint Γperiodic PDO Ψ. This is why we sometimes call it a “gauge transformation”. It is useful to consider eiΨ as an element of the group U (t) = exp{iΨt}, ∀t ∈ R.

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We assume that the operator ad(H0 , Ψ) is bounded, so that U (t)D(H0 ) = D(H0 ). This assumption will be justified later on. Let us express the operator At = U (−t)HU (t) via its (weak) derivative w.r.t. t: t U (−τ ) ad(H; Ψ)U (τ )dτ. At = H + 0

By induction it is easy to show that M 1 j (1) A1 =H + ad (H; Ψ) + RM+1 , (6.1) j! j=1 1 τM τ1 (1) dτ1 dτ2 . . . U (−τM+1 ) adM+1 (H; Ψ)U (τM+1 )dτM+1 . RM+1 = 0

0

0

The operator Ψ is sought in the form Ψ=

N

Ψk , Ψk = Op(ψk ), ψk ∈ Sk(α−m)+1 .

(6.2)

k=1

Substitute this formula in (6.1) and rewrite, regrouping the terms: A1 =H0 + B +

M M 1 j! j=1

ad(H; Ψk1 , Ψk2 , . . . , Ψkj )

l=j k1 +k2 +···+kj =l

(1)

(2)

+ RM+1 + RM+1 , (2)

RM+1 =

M 1 j! j=1

ad(H; Ψk1 , Ψk2 , . . . , Ψkj ).

(6.3)

k1 +k2 +···+kj ≥M+1

Rewrite: A1 = H0 + B +

M

ad(H0 ; Ψl ) +

ad(H0 ; Ψk1 , Ψk2 , . . . , Ψkj )

l=j k1 +k2 +···+kj =l

l=1 M M 1 + j! j=1

M M 1 j! j=2

(1)

(2)

ad(B; Ψk1 , Ψk2 , . . . , Ψkj ) + RM+1 + RM+1 .

l=j k1 +k2 +···+kj =l

Switch the summation signs: A1 = H0 + B +

M l=1

+

l−1 M+1 l=2 j=1

1 j!

l M 1 ad(H0 ; Ψl ) + j! j=2 l=2

k1 +k2 +···+kj =l−1

ad(H0 ; Ψk1 , Ψk2 , . . . , Ψkj )

k1 +k2 +···+kj =l (1)

(2)

ad(B; Ψk1 , Ψk2 , . . . , Ψkj ) + RM+1 + RM+1 .

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A.V. Sobolev


Introduce the notation B1 = B, Bl =

l−1 j=1

Tl =

1 j!

ad(B; Ψk1 , Ψk2 , . . . , Ψkj ), l ≥ 2,

(6.4)

k1 +k2 +···+kj =l−1

l 1 j! j=2

ad(H0 ; Ψk1 , Ψk2 , . . . , Ψkj ), l ≥ 2.

(6.5)

k1 +k2 +···+kj =l

We emphasize that the operators Bl and Tl depend only on Ψ1 , Ψ2 , . . . , Ψl−1 . One more rearrangement: A1 = H0 + B +

M

ad(H0 , Ψl ) +

l=1

RM+1 = BM+1 +

M

Bl +

l=2 (1) RM+1 +

M

Tl + RM+1 ,

l=2 (2) RM+1 .

(6.6)

Now we can specify our algorithm for finding Ψk ’s. The symbols ψk will be found from the following system of commutator equations: ad(H0 ; Ψ1 ) + B1 = 0, ad(H0 ; Ψl ) + and hence

Bl

+

Tl

= 0, l ≥ 2,

 ,↑  A1 = A0 + XM + RM+1 ,    M M XM = l=1 Bl + l=2 Tl ,     M o  o A0 = H0 + M l=1 Bl + l=2 Tl .

(6.7) (6.8)

(6.9)

Below, in Lemma 6.3 we shall prove that all the symbols bl and tl belong to appropriate classes Sβ with some β, and thus by (4.10) the symbols bl , tl possess the same property. This means that Bl and Tl are bounded (see Proposition 3.1) and hence the commutators ad(H0 , Ψl ) are also bounded in view of (6.7), (6.8). This justifies the assumption that ad(H0 , Ψ) is bounded, made in the beginning of the formal calculations in this Section.

6.2

Commutator equations

Since our primary concern is the Schr¨ odinger operator, from now on we assume that m = 2 and F = I in Definition (2.8). Before proceeding to the study of the commutator equations (6.7), (6.8) note that the symbol τθ (ξ) = h0 (ξ + θ) − h0 (ξ) = 2 θ, ξ + θ/2

(6.10)

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satisfies the bound

67

|Dsξ τθ−1 | ≤ Cs |θ|−1 L−s−1 , θ = 0,

(6.11)

for all ξ in the support of the function ϕθ (see (4.2)). This estimate will come in handy in the next lemma. Lemma 6.1 Let A = Op(a) be a symmetric PDO such that a ∈ Sα . Then the PDO Ψ with the Fourier coefficients of the symbol ψ(x, ξ) given by ˆ (θ, ξ) ˆ ξ) = i a , ψ(θ, τθ (ξ)

(6.12)

ad(H0 ; Ψ) + Op(a ) = 0.

(6.13)

solves the equation Moreover, the operator Ψ is bounded and self-adjoint, ψ ∈ Sα−1 and ψ

(α−1) l,s

≤C a

(α) l−1,s .

The constant C is independent of the parameter L ≥ 1 and the symbol a. Proof. Let t be the symbol of ad(H0 ; Ψ). The Fourier transform tˆ(θ, ξ) is easy to find using (3.2): ˆ ξ) = iτθ (ξ)ψ(θ, ˆ ξ). tˆ(θ, ξ) = i h0 (ξ + θ) − h0 (ξ) ψ(θ, Therefore the equation (6.13) amounts to ˆ ξ) = −ˆ a (θ, ξ) = −ˆ a(θ, ξ)ϕθ (ξ; L). iτθ (ξ)ψ(θ, By definition of the function ϕθ , a solution ψˆ exists and is given by (6.12). This symbol satisfies the condition (2.6), so that Ψ is a symmetric operator. Note also that by (4.4) and (6.11) the symbol ψ belongs to Sα−1 and one easily shows that ψ

(α−1) l,s

≤C a

(α) l−1,s .

This estimate for s = 0 and Proposition 3.1 ensure the boundedness of Ψ.

Remark 6.2 Let the symbols a and ψ be as in Lemma 6.1 and consider the commutator Op(a) = ad(Op(g), Ψ) with some symmetric symbol g ∈ Sγ . By (3.2) i ˆ ξ) − gˆ(θ, ξ)ψ(φ, ˆ ˆ gˆ(φ, ξ + θ)ψ(θ, ξ + θ) a(χ, ξ) = d(Γ) θ+φ=χ gˆ(φ, ξ + θ)ˆ a (θ, ξ) gˆ(φ, ξ)ˆ a (θ, ξ + φ) 1 − = − . τθ (ξ) τθ (ξ + φ) d(Γ) θ+φ=χ

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A.V. Sobolev


Analogously, one can easily derive a formula for the commutator symbol aµ,η = ad(gµ , ψη ) with arbitrary µ, η ∈ Γ† , see Subsection 4.2 for definition of these symbols. It is the same formula as above, but with the summation restricted to appropriate subsets of the lattice Γ† : gˆ(φ, ξ + θ)ˆ a (θ, ξ) gˆ(φ, ξ)ˆ a (θ, ξ + φ) 1 ˆ − aµ,η (χ, ξ) = − , τθ (ξ) τθ (ξ + φ) d(Γ) θ+φ=χ θ∈Γ†η ,φ∈Γ†µ

(6.14) see (4.12) for definition of Γ†ν . Recalling that τ−θ (ξ + θ) = −τθ (ξ), and using the property (2.6) we obtain gˆ(−θ, ξ + θ)ˆ a g ˆ (θ, ξ)ˆ a (θ, ξ) (−θ, ξ + θ) 1 − tˆ(0, ξ) = − τθ (ξ) τ−θ (ξ + θ) d(Γ) θ 1 1 gˆ(θ, ξ)ˆ a (θ, ξ) + gˆ(θ, ξ)ˆ a (θ, ξ) . = − τ (ξ) d(Γ) θ θ Let us apply Lemma 6.1 to equations (6.7) and (6.8). Lemma 6.3 Let b ∈ Sα be a symmetric symbol. Then there exists a sequence of selfadjoint bounded PDO’s Ψl , l = 1, 2, . . . with the symbols ψl ∈ Sβl , βl = l(α−2)+1, such that (6.7) and (6.8) hold, and (α) l l) ψl (β (6.15) (i) r,s ≤ C b p,n ) , l ≥ 1; (ii) The symbols bl , tl of the corresponding operators Bl , Tl belong to Sγl with γl = l(α − 2) + 2 and bl

(γl ) r,s

+ tl

bo2 (ξ) + to2 (ξ) = −

(iii)

(γl ) r,s

≤ C( b

(α) l p,n ) ,

l ≥ 2;

1 |ˆb(θ, ξ)|2 1 − ζθ2 (ξ; L) . d(Γ) τθ (ξ)

(6.16) (6.17)

θ∈Θr

The constant C in (6.15) and (6.16) does not depend on b, but depends on l, r, s, α. The integer-valued parameters p, n in (6.15) and (6.16) depend on l, r, s, α. (α)

(iv) If b l,s ≤ Cl,s L2−α for all l and s, then for some positive integer n, p the following bounds hold: ˜ b RM+1 ≤ C( xM

(α) l,s

(α) M+1 (M+1)(α−2)+2 L , p,n )

≤ C˜ b

(α) p,n ,

p = p(M, α), n = n(M, α); (6.18)

p = p(l, s, α, M ), n = n(l, s, α, M ).

(6.19)

The constant C˜ depends only on the constants Cl,s and the parameters M, α.

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Proof. The existence of ψ1 ∈ Sβ1 with required properties follows from Lemma 6.1. Further proof is by induction. To make the calculations less cumbersome, throughout the proof we adopt the following notational convention. If two symbols, φ1 and φ2 satisfy the esti(β) (ω) mate φ1 l,s ≤ C φ2 p,n with some p = p(l, s) and n = n(l, s) we simply write φ1 (β) ≤ C φ2 (ω) . Suppose that ψk with k = 1, 2, . . . , K − 1 satisfy (6.15). In order to conclude that ψK also satisfies (6.15), first we need to check that bK and tK satisfy (6.16). Step I. Estimates for bl . To begin with we prove that all the symbols bl with l ≤ K, satisfy the estimate (6.16). We first obtain a bound for ad(b; ψk1 ψk2 , . . . , ψkj ) with k1 + k2 + · · ·+ kj = l − 1 . To this end we use (6.15) and Proposition 3.4 to conclude that ad(b; ψk1 , ψk2 , . . . , ψkj )

(γ)

≤C b

(α)

j

( b

(α) kn

)

= C( b

(α) l

)

(6.20)

n=1

with γ = α+

j

(βkj − 1) = α +

n=1

j

kj (α − 2) = (l − 1)(α − 2) + α − 2 + 2 = l(α − 2) + 2.

n=1

This implies that bl satisfies (6.16) for all l ≤ K. Step II. Estimates for tl . For the symbols tl the proof is by induction. First of all, note that ad(h0 ; ψ1 , ψ1 ) = − ad(b , ψ1 ), so that, by Proposition 3.4 ad(h0 ; ψ1 , ψ1 )

(2α−2)

≤ C( b

(α) 2

) ,

and thus t2 satisfies (6.16). Suppose that all tk with k ≤ l − 1 ≤ K − 1 satisfy (6.16). Then by Definition (6.8) and (4.10) all ad(h0 ; ψk ), k ≤ l − 1, satisfy the same bound. Remembering that the definition of tl involves only ψk with k ≤ l − 1, and applying Proposition 3.4, we obtain for k1 + k2 + · · · + kj = l, j ≥ 2: ad(h0 ; ψk1 , ψk2 , . . . , ψkj ) (γ) = ad ad(h0 ; ψk1 ); ψk2 , . . . , ψkj (γ) ≤ ( b (α) )l , (6.21) with j γ = k1 (α − 2) + 2 + (kn (α − 2) + 1 − 1) = l(α − 2) + 2. n=2

This leads to (6.16) for all tl , l ≤ K. Step III. To handle ΨK we use the solution Ψ of the equation (6.13) constructed in Lemma 6.1. Then from Definition (6.8) and steps I, II we immediately conclude that ψK ∈ Sγ with γ = β − 1, β = K(α − 2) + 2 and that ψK (β) ≤ C bK (β) + tK (β) ≤ C( b (α) )K , as required.

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Step IV. Proof of (iii). By (6.4) and by (6.5), (6.7) 1 B2 = ad(B; Ψ1 ), T2 = − ad(B ; Ψ1 ). 2 It follows from (6.12) that ˆb (θ, ξ) . ψˆ1 (θ, ξ) = i τθ (ξ)

(6.22)

Remark 6.2 and Definition (4.9) lead to the formulas bo2 (ξ) = −

2 2 |ˆb(θ, ξ)|2 1 |ˆb(θ, ξ)|2 ϕθ (ξ; L), to2 (ξ) = ϕθ (ξ; L) . d(Γ) τθ (ξ) d(Γ) τθ (ξ) θ∈Θr

θ∈Θr

Adding them up and recalling that ϕθ = 1 − ζθ , we get (6.17). Step V. Proof of (iv). The remainder RM+1 (see (6.6)) consists of three components. In view of (6.16), bM+1 ∈ S(M+1)(α−2)+2 , so that by Remark 3.2 the norm of BM+1 is bounded by ( b (α) )M+1 L(M+1)(α−2)+2 as required. M (1) (α) Consider now RM+1 defined in (6.1). Let ψ = ≤ l=1 ψl . Since b 2−α CL , according to (6.15), (2.7) we have ψl

(α−1)

≤ CL(l−1)(α−2)

b

(α) l

≤ C b

(α)

.

Similarly, by Definition (6.9) we have X (α) ≤ C b (α) in view of (6.16), which proves (6.19). It follows from (6.7) and (6.8) that ad(H0 , Ψ) + X = 0. Now, repeating the same argument as on Steps 1 and II, we conclude that adM+1 (H, Ψ) (γ) ≤ (α) M+1 b with γ = (M + 1)(α− 2)+ 2. By Remark 3.2 this leads to the required (1)

estimate for the norm RM+1 .

(2)

In the same way the norm of the error RM+1 defined in (6.3) can be shown to satisfy the same bound. This completes the proof of (6.18).

7 Density of states for operator A1 In this section we apply the transformation constructed in the previous section, to the Schr¨ odinger operator, that is to the operator (2.8) with m = 2 and F = I, b(x, ξ) = V (x), so that α = 0. For the proof of Theorem 2.3 we shall need the representation (6.9) with M = 2. We begin with deriving further consequences from Lemma 6.3. From now on we shall use the notation Vj instead of Bj , j = 1, 2, . . . . All the estimates below are uniform in V . The majority of the results below are obtained for d = 2, although for some intermediate results the condition d ≥ 2 will be sufficient.

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Operators V2 and T2

Let us investigate in more detail the operators V2 and T2 . Recall again that by (6.4), (6.5) and (6.7) 1 V2 = ad(V ; Ψ1 ), T2 = − ad(V ; Ψ1 ). 2 We start by studying the symbols 1 aµ,η = ad Vµ , (ψ1 )η , bµ,η = − ad Vµ , (ψ1 )η 2 with µ, η ∈ P Γ† , see Subsection 4.2 for definition of the symbols bν and of the set of the primitive lattice vectors P Γ† . By (4.14), (4.10) and Lemma 6.3 used with m = 2, α = 0, we have Vµ , Vµ ∈ S0 , (ψ1 )η ∈ S−1 , and Vµ

(0) l,s

+ Vµ

(0) l,s

(0) l,0 ,

≤ Cl,s V

(ψ1 )η

(−1) l,s

≤ Cl,s V

(0) p,0 ,

p = p(l, s).

Consequently by Proposition 3.4 aµ,η , bµ,η ∈ S−2 and aµ,η

(−2) l,s

+ bµ,η

(−2) l,s

≤ Cl,s ( V

(0) 2 p,0 ) ,

p = p(l, s),

(7.1)

uniformly in µ, η ∈ P Γ† . We shall need more detailed properties of these commutators. In particular, let us find bounds for symbols (aµ,η )ν and (bµ,η )ν , ν ∈ P Γ† . In the next lemma and further on we shall need the explicit formulas for these symbols, which follow from (6.14) and (6.22). For brevity we write only the formula for ˆ aµ,η : ϕθ (ξ; L) ϕθ (ξ + φ; L) 1 ˆ ˆ ˆ V (φ)V (θ) − aµ,η (χ, ξ) = − . (7.2) τθ (ξ) τθ (ξ + φ) d(Γ) φ+θ=χ φ∈Γ†µ , θ∈Γ†η ∩Θr

ˆν = In this formula χ ∈ Γ† and τθ is defined in (6.10). Recall the notation Λ ˆ Λν (ρ − L1 ) and (4.15). Lemma 7.1 Let d = 2. Let V be as above and ψ1 be as found in Lemma 6.3. Suppose that   1 ≤ r ≤ L, (7.3) ρ ≥ 4L1 ,   † −2 d(Γ )ρr ≥ 16L. r one has Then for any ν ∈ Θ ˆν) (aµ,η ) ν χ( · ; Λ r µ∈P Γ† ,η∈Θ µ =η

(0) l,0

ˆν) + (bµ,η ) ν χ( · ; Λ

(0) l,0

≤ Cl ( V

(0) 2 4 −2 , p,0 ) r ρ

(7.4)

r . (Here χ( · ; Λ ˆ ν ) denotes the multiplifor all l ≥ 0, p = p(l), uniformly in ν ∈ Θ ˆ cation by the function χ(ξ; Λν ).)

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Proof. Let us estimate each term in the sum (7.4) individually. For the sake of brevity we conduct the proof only for the case of the symbol aµ,η . For this we ˆ ν . Since we are interested in the use (7.2) with χ ∈ Γ†ν , assuming that ξ ∈ Λ operator a µ,η , we may assume that χ ∈ Θr (see Definition (4.8)), and hence we have φ ∈ Θ2r in (7.2). Let us estimate first the terms in the square brackets in (7.2). Since µ = η, the vectors ν and θ, φ in (7.2) are pairwise linearly independent. Consequently, in ˆ ν the bounds view of (4.18) and (7.3) we have for ξ ∈ Λ | n(θ), ξ| ≥ 2−1 d(Γ† )r−2 ρ ≥ 8L, | n(θ), ξ + φ| ≥ 2−1 d(Γ† )r−2 ρ − 2r ≥ 6L, |τθ (ξ)| = 2| θ, ξ + θ/2| 1 ≥ |θ| d(Γ† )r−2 ρ − r ≥ |θ| d(Γ† )r−2 ρ, 2 |τθ (ξ + φ)| = 2| θ, ξ + φ + θ/2| 1 ≥ |θ| d(Γ† )r−2 ρ − 5r ≥ |θ| d(Γ† )r−2 ρ. 2 By Definitions (4.1) and (4.2), in view of the above bounds we have

and

(7.5)

(7.6)

ˆν, ϕθ (ξ) = ϕθ (ξ + φ) = 1, ∀ξ ∈ Λ ˆ ν the symbol âµ,η has the form: and hence for χ ∈ Θr ∩ Γ†ν , ξ ∈ Λ 1 1 1 ˆ Vˆ (φ)Vˆ (θ) aµ,η (χ, ξ) = − − . τθ (ξ) τθ (ξ + φ) d(Γ) φ+θ=χ φ∈Γ†µ , θ∈Γ†η ∩Θr

According to (7.5), (7.6), 1 2| θ, φ| 1 4 −2 τθ (ξ) − τθ (ξ + φ) = τθ (ξ)τθ (ξ + φ) ≤ C|φ|r ρ , and hence ˆν) aµ,η ν χ( · ; Λ

(0) l,0

≤ C max max χl |âµ,η (χ, ξ)| χ∈Θr ξ∈Λ ˆν

≤ Cl r4 ρ−2

φ∈Γ†µ

|φ|l+1 |Vˆ (φ)|

|θ|l |Vˆ (θ)|.

θ∈Γ†η

The r.h.s. is finite since V ∈ S0 . Summing these estimates over µ ∈ P Γ† and η ∈ P Γ† , we bound the r.h.s. by the product (0) 2 |φ|l+1 |Vˆ (φ)| |θ|l |Vˆ (θ)| ≤ Cp V p,0 , φ∈Γ†

θ∈Γ†

for any p > l + 3. Consequently, the estimate (7.4) is fulfilled. The symbol bµ,η can be treated in the same way. These calculations are omitted to avoid repetitions.

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Using (4.13) for V, V and ψ1 , we can now decompose the symbol V2 + t2 as follows:

f=

(aν,ν

V2 + t2 = V2o + to2 + f + g, + bν,ν )ν , g = (aµ,η + bµ,η )ν .

ν∈P Γ†

(7.7)

ν∈P Γ† µ,η∈P Γ† µ =η

Our objective is to show that the symbol f has the form (5.1) and the symbol g has a “small” norm. These properties are proved in the next lemma. Theorem 7.2 Let d = 2. The symbols f, g defined above, satisfy the following properties: (−2)

(i) f, g ∈ S−2 (L) and f l,s + g (ii) For some z ∈ T−2 (L) one has

(−2) l,s

≤ Cl,s ( V

(0) 2 p,0 ) ,

p = p(l, s);

ˆf(χ, ξ) = zˆ(χ, ξ, n(χ)), χ = 0, z

(−2) l,s

≤ Cl,s ( V

(0) 2 p,0 ) ,

(7.8)

p = p(l, s).

(7.9)

(iii) Let the conditions (7.3) be fulfilled. Then ˆ ν ) ≤ Cp ( V Op(g ν )P(Λ

(0) 2 4 −2 , p,0 ) r ρ

ˆν = Λ ˆ ν (ρ − L1 ), Λ

(7.10)

r. with some integer p, uniformly in ν ∈ Θ The constants C in the above inequalities are independent of V and ρ, L, r. (−2)

(0)

Proof. Let us prove (i) first. By Lemma 6.3 we have V2 + t2 l,s ≤ Cl,s ( V p,0 )2 with some p = p(l, s). By (4.14) this guarantees the same estimate for V2o and to2 . Consequently, part (i) will be proved if we establish this estimate for f only. The required bound follows from (7.1) in view of (4.14). Proof of (ii). Use (7.2) with µ = η = ν, and a similar formula for bν,ν . Recalling that ϕθ (ξ; L) = 1 − Υ L−1 ( n(θ), ξ + 2−1 |θ|) , τθ (ξ) = 2 θ, ξ + θ/2, by Definitions (4.2) and (6.10), we conclude that ˆfν = âν,ν + ˆbν,ν has the form (5.1). This means that ˆf also has this property, that is ˆf satisfies (7.8) with some function z. Moreover, in view of (5.2) and part (i), the function z belongs to T−2 and satisfies the bound (7.9). One can write an explicit formula for the function z, but it is too cumbersome and is therefore omitted. Proof of (iii). The estimate (7.10) follows from Definition (7.7) by virtue of (7.4) and Proposition 3.1.

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A.V. Sobolev

7.2


Operator A1

Now we use the established results to study the operator A1 from Section 6 with M = 2. According to (6.9)  A1 = Ao + V + (V2 + T2 ) + X2↑ + R3 , (7.11) Ao = H + V o + T o . 0 2 2 Recall that Vˆ (0) = 0, so that the term V1o drops out. The remainders X2↑ and R3 satisfy the bounds R3 ≤ CL−4 ,

X2↑ ≤ Cp r−p , ∀p > 0.

(7.12)

Indeed, the estimate for R3 follows from (6.18) used with α = 0 and M = 2. Furthermore, the estimate for X2↑ is a consequence of (6.19), (4.10) and Remark 3.2. In this section we establish a suitable asymptotic formula for the density of states of the operator Ao + V + V2 + T2 with the help of Theorem 5.4. Let us verify first that the symbol ao has the required form. Lemma 7.3 Let d ≥ 2. The symbol ao has the form (5.21) with f (ν) (η) = −

1 |Vˆ (lν)|2 ˜ r. 1 − Υ2 (η + l|ν|/2)L−1 , ν ∈ Θ d(Γ) 2l|ν| η + l|ν|/2 0 =l∈Z, lν∈Θr

(7.13) (ν) in (5.21) belongs to S−2 (L) uniformly in V . MoreThe function f = ν∈Θ r f over, under the condition 1 ≤ r ≤ L one has sup |η|2 |f (ν) (η)| ≤ κν ,

|η|≥4L

κ=

r ν∈Θ

κν ≤

1 V 2L2 . d(Γ)

(7.14)

Proof. We need to show that the symbol V2o + to2 has the form f (ξ) as specified in (5.21). To this end rewrite (6.17), replacing the sum over all θ ∈ Θr by the double sum . r θ∈Γ†ν ∩Θr ν∈Θ

Now, denoting θ = lν, 0 = l ∈ Z and η = ξ, n(ν) we can write for each θ ∈ Γ†ν that ζθ (ξ; L) = Υ (η + l|ν|/2)L−1 , τθ (ξ) = 2l|ν| η + l|ν|/2 . Now it is clear that f = V2o + to2 has the form (5.21) with f (ν) as in (7.13). Observe also that according to Lemma 6.3(ii), the function f belongs to S−2 uniformly

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in V . In order to establish (7.14) observe that for r ≤ L and |η| ≥ 4L the function Υ in the Definition (7.13) vanishes (see Definition (4.1)), and hence 1 1 1 (ν) 2 ˆ − |V (lν)| f (η) = − d(Γ)|ν| 2l η + l|ν|/2 2l η − l|ν|/2 0 d/2 M (ρ) + d(d − 2) wd V 22 ≤ Cl ρ−2 r−2l+d + ρ−3 + ρ4−5 ln ρ L (O) 8ρ2 d(Γ) with a constant Cl uniform in V . Proof. It follows from definition of Z(ρ; η0 ) that M =−

|Vˆ (θ)|2 1 Z(ρ; |θ|/2). 8 d(Γ) |θ| θ∈Θr

According to the Lemma 8.2 and Parceval’s identity the leading term of the r.h.s. is given by −

d(d − 2) wd d(d − 2) wd ˆ V 2L2 (O) + O(ρ−2 ) |V (θ)|2 = − |Vˆ (θ)|2 , 2 2 8ρ d(Γ) 8ρ d(Γ) θ∈Θr

θ∈Ξr

see (4.5) for definition of Ξr . The error term does not exceed, up to a multiplicative constant independent of ρ and V , (ρ−3 + ρ4−5 ln ρ)V 2L2 (O) + ρ−4 ∆V 2L2 (O) . In view of (2.3), for any l > d/2 one has (0) |Vˆ (θ)|2 ≤ V l,0 |θ|−2l ≤ Cl V θ∈Ξr

(0) −2l+d . l,0 r

|θ|≥r

This leads to the proclaimed formula.

Proof of Theorem 8.1. The required asymptotics immediately follows from Lemma 8.3 and formula (8.1).

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81

Completion of the proof of Theorem 2.3

In contrast to the previous subsection, where we could allow any dimension d ≥ 2, now we restrict ourselves to d = 2 only. Recall that the unitary operator U = eiΨ constructed in Section 6 is Γperiodic. Consequently, by Proposition 2.1(ii), we have D(λ; H) = D(λ; A1 ), and hence it remains to establish the sought asymptotics for the operator A1 only. To this end we shall use the formula (7.11). By monotonicity of the density of states (see Proposition 2.1(i)) the formulae (7.12) and (7.11) give the estimates D(ρ2 − CL−4 − Cr−p ; Ao + B + B2 + T2 ) ≤ D(ρ2 ; A1 ) ≤ D(ρ2 + CL−4 + Cr−p ; Ao + B + B2 + T2 ). In order to apply Theorem 7.5, assume that L = L1 = ρ /2, r = ρβ with some ∈ (0, 1) and β ∈ (0, min{, (1 − )/2}), so that the conditions (7.3), (5.23), (5.25) are satisfied for all ρ ≥ ρ0 with a sufficiently large ρ0 = ρ0 (V ) which is uniform in V . According to Theorems 7.5 and 8.1 we have D(ρ2 ; A1 ) − w2 ρ2 ≤ Cl ρ−2 ρ−1 + ρ4−3 ln ρ 2 (2π) −2l+d +r + L−4 + ρL−5 + r6 ρ−1 L + C L−4 + Cp r−p , ∀l > d/2, ∀p > 0. Substitute L = ρ /2 and r = ρβ : Cρ−2 r6β ρ−1 + ρ4−3 ln ρ + ρ2−4 + Cp ρ−pβ , ∀p > 0. Optimizing in we get = 3/5. Choose an arbitrarily small β and a suitably large p.

9 Appendix Our aim is to find the value of the integral π d−3 J =i , e−iφ (1 − e2iφ )κ dφ, κ = 2 0 featuring in the proof of Lemma 8.2. Lemma 9.1 The integral J is given by

√ d(d − 2) Γ d+1 wd 2 . J = d(d − 2) = π ωd−2 d − 1 Γ d+2 2

(9.1)

Proof. Recall that (see (1.3), (5.11)) d

wd =

d−1

π2 (d − 1)π 2 d+2 , ωd−2 = (d − 1) wd−1 = , Γ 2 Γ d+1 2

so that we need to prove only that J coincides with the r.h.s. of (9.1).

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Define J(t) = i

0

π


e−iφ (1 − te2iφ )κ dφ, |t| < 1.

Expanding the integrand in the absolutely convergent series, we find J = lim J(t) = lim i t↑1

t↑1

∞ π

tn e(2n−1)iφ (−1)n

0 n=0

∞ 1 κ κ (−1)n dφ = −2 . n 2n − 1 n n=0

(9.2) For d = 2 we have κ = −1/2 and we can use the formula −1/2 1/2 = (1 − 2n), n n so that J =2

∞

(−1)n

n=0

1/2 = 2(1 − 1)1/2 = 0. n

For d ≥ 3 use [3], formula 1.4(2), which implies that the r.h.s. of (9.2) coincides with √ Γ d−1 Γ − 12 Γ κ + 1 2 − = 2 π d−2 Γ κ + 12 Γ 2 In view of the relations d+1 d−1 d−2 d+2 d−1 d(d − 2) Γ Γ Γ = , Γ = , 2 2 2 2 4 2 this leads to (9.1).

Acknowledgments. The author is grateful to L. Parnovski for discussions. The paper was completed during the author’s stay at the Mittag-Leffler Institute in September 2002.

References [1] M.S. Agranovich, Elliptic operators on closed manifolds, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 63, 5–129 (1994); Engl. transl. in: Partial differential equations. VI, Encycl. Math. Sci. 63, 1–130 (1994). [2] B.E.J. Dahlberg, E. Trubowitz, A remark on two-dimensional periodic potentials, Comment. Math. Helvetici 57, 130–134 (1982). [3] A. Erdélyi, Higher transcendental functions, V. I, McGraw-Hill 1953.

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[4] J. Feldman, H. Kn¨ orrer, E. Trubowitz, The perturbatively stable spectrum of a periodic Schr¨ odinger operator, Invent. Math. 100, 259–300 (1990). [5]

, Perturbatively unstable eigenvalues of a periodic Schr¨ odinger operator, Comment. Math. Helvetici 66, 557–579 (1991).

[6] B. Helffer, A. Mohamed, Asymptotics of the density of states for the Schr¨ odinger operator with periodic electric potential, Duke Math. J. 92, 1–60 (1998). [7] M. Hitrik, I. Polterovich, Regularized traces and Taylor expansions for the heat semigroup, J. London Math. Soc.II. Ser. 68, No. 2, 402–418 (2003). [8] Yu. Karpeshina, On the density of states for the periodic Schr¨ odinger operator, Ark. Mat. 38, 111–137 (2000). [9]

, Perturbation theory for the Schr¨ odinger operator with a periodic potential, Lecture Notes in Math. vol 1663, Springer Berlin 1997.

[10] T. Kato, Perturbation theory for linear operators, Springer 1966. [11] E.Korotyaev, A. Pushnitski, On the high energy asymptotics of the integrated density of states, Bull. London Math. Soc. 35, 770–776 (2003). [12] A.S. Milevskij, Similarity transformations and spectral properties of hypoelliptic pseudodifferential operators on a circle, Funk. Anal. Prilozen. 23 no 3, 71–72 (1989); Engl. transl. in Funct. Anal. Appl. 23 no 3, 231–233 (1989). [13]

, A simplification of hypoelliptic pseudodifferential operators on the circle via Fourier integral operators, Moscow 1988, dep. in VINITI 05.09.88, No 6856-B88.

[14] A. Mohamed, Asymptotics of the density of states for the Schrödinger operator with periodic electromagnetic potential, J. Math. Phys. 38, 4023–4051 (1997). [15] L. Parnovski, A.V. Sobolev, Lattice points, perturbation theory and the periodic polyharmonic operator, Ann. H. Poincaré 2, 573–581 (2001). [16] I. Polterovich, Heat invariants of Riemannian manifolds, Israel J. Math. 119, 239–252 (2000). [17] M. Reed, B. Simon, Methods of modern mathematical physics, IV, Academic Press, New York, 1975. [18] G.V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behavior of pseudodifferential operators on the circle, Trudy Moskov. Mat. Obshch. 36, 59–84 (1978) (Russian).

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[19] A.V. Savin, Asymptotic expansion of the density of states for one-dimensional Schr¨ odinger and Dirac operators with almost periodic and random potentials, (in Russian), Sb. Nauchn. Tr. IFTP, 1988, Moscow. [20] D. Shenk and M. Shubin, Asymptotic expansion of the state density and the spectral function of a Hill operator, Math. USSR Sbornik 56 no. 2, 473–490 (1987). [21] M.A. Shubin, Weyl’s theorem for the Schr¨ odinger operator with an almost periodic potential, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 31, no 2, 84–88 (1976)(Russian). Engl. transl.: Moscow Univ. Math. Bull. 31, 133–137 (1976). [22]

, The spectral theory and the index of elliptic operators with almost periodic coefficients, Russian Math. Surveys 34 no 2, 109–157 (1979).

[23] M. Skriganov, Geometrical and arithmetical methods in the spectral theory of the multi-dimensional periodic operators, Proc. Steklov Math. Inst. Vol. 171, 1984. [24]

, The spectrum band structure of the three-dimensional Schr¨ odinger operator with periodic potential, Inv. Math. 80, 107–121 (1985).

[25] A.V. Sobolev, Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one, to appear in Revista Matematica Iberoamericana (2005). Alexander V. Sobolev Department of Mathematics University of Sussex Falmer Brighton BN1 9RH United Kingdom email: [email protected] Communicated by Bernard Helffer submitted 30/04/04, accepted 26/07/04




Some Connections between Dirac-Fock and Electron-Positron Hartree-Fock Jean-Marie Barbaroux, Maria J. Esteban and Eric Séré

Abstract. We study the ground state solutions of the Dirac-Fock model in the case of weak electronic repulsion, using bifurcation theory. They are solutions of a minmax problem. Then we investigate a max-min problem coming from the electronpositron field theory of Bach-Barbaroux-Helffer-Siedentop. We show that given a radially symmetric nuclear charge, the ground state of Dirac-Fock solves this maxmin problem for certain numbers of electrons. But we also exhibit a situation in which the max-min level does not correspond to a solution of the Dirac-Fock equations together with its associated self-consistent projector.

1 Introduction The electrons in heavy atoms experience important relativistic effects. In computational chemistry, the Dirac-Fock (DF) model [1], or the more accurate multiconfiguration Dirac-Fock model [2], take these effects into account. These models are built on a multi-particle Hamiltonian which is in principle not physically meaningful, and whose essential spectrum is the whole real line. But they seem to function very well in practice, since approximate bound state solutions are found and numerical computations are done and yield results in quite good agreement with experimental data (see, e.g., [3]). Rigorous existence results for solutions of the DF equations can be found in [4] and [5]. An important open question is to find a satisfactory physical justification for the DF model. It is well known that the correct theory including quantum and relativistic effects is quantum electrodynamics (QED). However, this theory leads to divergence problems, that are only solved in perturbative situations. But the QED equations in heavy atoms are nonperturbative in nature, and attacking them directly seems a formidable task. Instead, one can try to derive approximate models from QED, that would be adapted to this case. The hope is to show that the Dirac-Fock model, or a refined version of it, is one of them. Several attempts have been made in this direction (see [6, 7, 8, 9] and the references therein). Mittleman [6], in particular, derived the DF equations with “self-consistent projector” from a variational procedure applied to a QED Hamiltonian in Fock space, followed by the standard Hartree-Fock approximation. More precisely, let H c be the free Dirac Hamiltonian, and Ω a perturbation. We denote Λ+ (Ω) = χ(0,∞) (H c + Ω). The electronic space is the range H+ (Ω) of this projector. If one computes the QED energy of Slater determinants of N wave functions in this electronic space, one obtains the DF en-

86

J.-M. Barbaroux, M.J. Esteban and E. Sér´ e


ergy functional restricted to (H+ (Ω))N . Let ΨΩ be a minimizer of the DF energy in the projected space (H+ (Ω))N under normalization constraints. It satisfies the projected DF equations, with projector Λ+ (Ω). Let E(Ω) := E(ΨΩ ). Mittleman showed (by formal arguments) that the stationarity of E(Ω) with respect to Ω implies that Λ+ (Ω) coincides, on the occupied orbitals, with the self-consistent projector associated to the mean-field Hartree-Fock Hamiltonian created by ΨΩ . From this he infers ([6], page 1171) : “Hence, Ω is the Hartree-Fock potential when the Hartree-Fock approximation is made for the wave function”. Recently rigorous mathematical results have been obtained in a series of papers by Bach et al. and Barbaroux et al. [10, 11, 12] on a Hartree-Fock type model involving electrons and positrons. This model (that we will call EP) is related to the works of Chaix-Iracane [9] and Chaix-Iracane-Lions [13]. Note, however, that in [10, 11, 12] the vacuum polarization is neglected, contrary to the Chaix-Iracane approach. In [10], in the case of the vacuum, a max-min procedure in the spirit of Mittelman’s work is introduced. In [12], in the case of N -electron atoms, it is shown that critical pairs (γ, P + ) of the electron-positron Hartree-Fock energy EEP give solutions of the self-consistent DF equations. This result is an important step towards a rigorous justification of Mittleman’s ideas. All this suggests, in the case of N -electrons atoms, to maximize the minimum E(Ω) with respect to Ω. It is natural to expect that this max-min procedure gives solutions of the DF equations, the maximizing projector being the positive projector of the self-consistent Hartree-Fock Hamiltonian. We call this belief (expressed here in rather imprecise terms) “Conjecture M”. In [14] and [15], when analyzing the nonrelativistic limit of the DF equations, Esteban and Séré derived various equivalent variational problems having as solution an “electronic” ground state for the DF equations. Among them, one can find min-max and max-min principles. But these principles are nonlinear, and do not solve Conjecture M. In this paper we try to give a precise formulation of Conjecture M in the spirit of Mittleman’s ideas and to see if it holds true or not, in the limit case of small interactions between electrons. We prove that in this perturbative regime, given a radially symmetric nuclear potential, Conjecture M may hold or not depending on the number of electrons. The type of ions which are covered by our study are those in which the number of electrons is much smaller than the number of protons in the nucleus, with, additionally, c (the speed of light) very large. The paper is organized as follows : in Section 2 we introduce the notations and state our main results (Theorems 9 and 11). Sections 3 and 4 contain the detailed proofs.

2 Notations and main results In the whole paper we choose a system of units in which Planck’s constant, , and the mass of the electron are equal to 1 and Ze2 = 4π0 , where Z is the number

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Some Connections between Dirac-Fock and Electron-Positron Hartree-Fock 87

of protons in the nucleus. In this system of units, the Dirac Hamiltonian can be written as H c = −ic α · ∇ + c2 β, (1) 11 0 where c > 0 is the speed of light , β = , α = (α1 , α2 , α3 ), α = 0 −11 0 σ and the σ ’s are the Pauli matrices. The operator H c acts on 4-spinors, σ 0 i.e., functions from R3 to C4 , and it is self-adjoint in L2 (R3 , C4 ), with domain H 1 (R3 , C4 ) and form-domain H 1/2 (R3 , C4 ). Its spectrum is the set (−∞, −c2 ] ∪ [c2 , +∞). In this paper, the charge density of the nucleuswill be a smooth, radial and compactly supported nonnegative function n, with n = 1, since in our system of units Ze2 = 4π0 . The corresponding Coulomb potential is V := −n ∗ (1/|x|). Then V : R3 → (−∞, 0) is a smooth negative radially symmetric potential such that −

1 ≤ V (x) < 0 (∀x) |x|

,

|x| V (x) −1 for

|x| large enough .

Note that the smoothness condition on V is only used in step 3 of the proof of Proposition 15. Actually we believe that this condition can be removed. It is well known that H c + V is essentially self-adjoint and for c > 1, the spectrum of this operator is as follows: σ(H c + V ) = (−∞, −c2 ] ∪ {λc1 , λc2 , . . . } ∪ [c2 , +∞), with 0 < λc1 < λc2 < · · · and lim λc = c2 . →+∞

Finally define the spectral subspaces Mci = Ker(H c + V − λci 11) and let Nic denote Mci ’s dimension. Since the potential is radial, it is well known that the eigenvalues λci are degenerate (see, e.g., [16]). For completeness, let us explain this in some detail. To any A ∈ SU (2) is associated a unique rotation RA ∈ SO(3) such that ∀x ∈ R3 , (RA x) · σ = A(x · σ)A−1 , where σ = (σ1 , σ2 , σ3 ). This map is a morphism of Lie groups. It is onto, and its kernel is {I, −I}. It leads to a natural unitary representation • of SU (2) in the Hilbert spaces of 2-spinors L2 (S 2 , C2 ) and L2 (R3 , C2 ), given by −1 (A • φ)(x) := A φ(RA x) . (2) Then, on the space of 4-spinors L2 (R3 , C4 ) = L2 (R3 , C2 ) ⊕ L2 (R3 , C2 ), one can define the following unitary representation (denoted again by •) −1 Aφ(RA x) (A • φ)(x) φ (x) := = A• . −1 (A • χ)(x) χ Aχ(RA x)

(3)

88



The radial symmetry of V implies that H c + V commutes with •. The eigenspaces Mci are thus SU (2) invariant. Now, let Jˆ = (Jˆ1 , Jˆ2 , Jˆ3 ) be the total angular momentum operator associated to the representation •. The eigenvalues of Jˆ2 = Jˆ12 + Jˆ22 + Jˆ32 are the numbers (j 2 − 1/4) , where j takes all positive integer values. If φ is an eigenvector of Jˆ2 with eigenvalue (j 2 − 1/4) , then the SU (2) orbit of φ generates an SU (2) invariant complex subspace of dimension 2j ≥ 2. This implies the following fact, which will be used repeatedly in the present paper: Lemma 1. If φ ∈ L2 (R3 , C2 ) is not the zero function, then there is A ∈ SU (2) such that φ and A • φ are two linearly independent functions. Proof of the Lemma. Assume, by contradiction, that C φ is SU (2) invariant. Then φ is an eigenvector of J for = 1, 2, 3, hence it is eigenvector of Jˆ2 . But we have seen that in such a case, the SU (2) orbit of φ must contain at least two independent vectors: this is absurd. As a consequence of the Lemma, the spaces Mci have complex dimension at least 2. The degeneracy is higher in general: for each j ≥ 1 , H c + V has infinitely many eigenvalues of multiplicity at least 2j. Note that in the case of the Coulomb potential, the eigenvalues are even more degenerate (see, e.g., [16]). Now, on the Grassmannian manifold GN (H 1/2 ) := {W subspace of H 1/2 (R3 , C4 ); dimC (W ) = N } we define the Dirac-Fock energy Eκc as follows Eκc (W )

:=

Eκc (Ψ)

κ + 2

:=

N i=1

R3 ×R3

R3

((H c + V )ψi , ψi )dx

ρΨ (x)ρΨ (y) − |RΨ (x, y)|2 dxdy , |x − y|

(4)

where κ > 0 is a small constant, equal to e2 /4π0 in our system of units, {ψ1 , . . . ψN } is any orthonormal basis of W , Ψ denotes the N -uple (ψ1 , . . . , ψN ), ρΨ is a scalar and RΨ is a 4 × 4 complex matrix, given by ρΨ (x) =

N ψ (x), ψ (x) ,

RΨ (x, y) =

=1

N

ψ (x) ⊗ ψ∗ (y) .

(5)

=1

Saying that the basis {ψ1 , . . . , ψN } is orthonormal is equivalent to saying that GramL2 Ψ = 11N . Eκc (W )

(6) Eκc (Ψ).

or The energy can We will use interchangeably the notations be considered as a function of W only, because if u ∈ U (N ) is a unitary matrix,

with the notation (uΨ)k =

Eκc (uΨ) = Eκc (Ψ)

l

ukl ψl .

(7)

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Note that since V is radial, the DF functional is also invariant under the representation • defined above. Its set of critical points will thus be a union of SU (2) orbits. Finally let us introduce a set of projectors as follows: Definition 2. Let P be an orthogonal projector in L2 (R3 , C4 ), whose restriction to 1 1 H 2 (R3 , C4 ) is a bounded operator on H 2 (R3 , C4 ). Given ε > 0, P is said to be 1 c 3 4 2 ε-close to Λ+ c := χ(0,+∞) (H ) if and only if, for all ψ ∈ H (R , C ), 14 P − Λ+ −c2 ∆ + c4 c ψ

14 ≤ ε −c2 ∆ + c4 ψ

L2 (R3 ,C4 )

L2 (R3 ,C4 )

.

In [14] the following result is proved: Theorem 3 ([14]). Take V , N fixed. For c large and 0 , κ small enough, for all P ε0 -close to Λc+ , c(P ) :=

sup

inf

W + ∈GN (P H 1/2 )

W ∈GN (H 1/2 ) P (W )=W +

Eκc (W )

is independent of P and we denote it by Eκc . Moreover, Eκc is achieved by a solution Wκ =span{ψ1 , . . . , ψN } of the Dirac-Fock equations:

c Hκ,Wκ ψi = ci ψic , 0 < ci < 1, (DF) GramL2 Ψ = 1N with c Hκ,W ϕ := (H c + V + κ ρΨ ∗

1 )ϕ − κ |x|

R3

RΨ (x, y)ϕ(y) dy . |x − y|

(MF)

Remark. It is easy to verify that ε0 > 0 given, for c large and κ small enough, c χ(0,∞) (Hκ,W ) is ε0 -close to Λ+ c . κ Corollary 4 ([14]). Take V, N fixed. Choose c large and κ small enough. If we define the projector + c Pκ,W = χ(0,∞) (Hκ,W ) c with Hκ,W given by formula (MF), then

Eκc =

min

W ∈Gn (H 1/2 ) + P W =W κ,W

Eκc (W ) =

min

W ∈GN (H 1/2 ) W solution of (DF)

Eκc (W ) .

(8)

Another variational problem was introduced in the works of Bach et al. and Barbaroux et al. ([10, 11, 12]): define κ = {P + = χ[0,∞) (H c ) ; W ∈ GN (H 1/2 )}, P κ,W κ,W

(9)

90



and N Sκ, W

c {γ ∈ S1 (L2 ) , γ = γ ∗ , Hκ, γ ∈ S1 , W

:=

P + γP − = 0 , −P − ≤ γ ≤ P + , tr γ = N }, κ,W

κ,W

κ,W

κ,W

with the notation P − := 1I − P+ , and S1 being the Banach space of trace-class κ,W

κ,W

N operators on L2 (R3 , C4 ). For all γ ∈ Sκ, , let W

ργ (x)ργ (y) κ |γ(x, y)|2 κ dx dy − dx dy. = tr ((H + V )γ) + 2 |x − y| 2 |x − y| 4 Here, ργ (x) := s=1 γs,s (x, x) = n wn |ψn (x)|2 , with wn the eigenvalues of γ and ψn the eigenspinors of γ, and γ(x, y) = n wn ψn (x) ⊗ ψn (y), i.e., γ(x, y) is the kernel of γ. κ , the infimum of Fκc on In [12] it has been proved that for every P + ∈ P Fκc (γ)

c

κ,W

the set S N is actually equal to the infimum defined in the smaller class of Slater κ,W determinants. More precisely, with the above notations, κ , one has Theorem 5 ([12]). For κ small enough and for all P + ∈ P κ,W

inf

γ∈S N

Fκc (γ)

κ,W

=

inf

W ∈GN (P + H 1/2 )

Eκc (W )

(10)

κ,W

Moreover, the infimum is achieved by a solution of the projected Dirac-Fock equations, namely N

ψi , .ψi γmin = i=1

with

P + ψi κ,W

= ψi (i = 1, . . . , N ), and for Wmin := span(ψ1 , . . . , ψN ) ,

c P + Hκ,W P + ψi = i ψi , 0 < i < 1, min κ,W κ,W GramL2 Ψ = 1N

(11)

Let us now define the following sup-inf: ecκ :=

sup

κ P + ∈P κ,W

inf

W ∈GN (P + H 1/2 )

Eκc (W ) .

κ,W

Then, Theorem 5 has the following consequence: Corollary 6. If κ is small enough, ecκ =

sup

κ P + ∈P κ,W

inf

γ∈S N

κ,W

Fκc (γ).

(12)

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From the above definitions, Theorem 3, Corollary 4 and the remark made after Theorem 3, we clearly see that for all κ small and c large, Eκc ≥ ecκ .

(13)

One can hope more: Conjecture M: The energy levels Eκc and ecκ coincide, and there is a solution Wκc of the DF equations such that Eκc (Wκc ) = ecκ =

inf

+ 1/2 ) V ∈GN (Pκ,W cH

Eκc (V ) .

κ

In other words, the max-min level ecκ is attained by a pair (W, P + ) such that κ,W

= W. W

This paper is devoted to discussing this conjecture, which, if it were true, would allow us to interpret the Dirac-Fock model as a variational approximation of QED. In order to study the different cases that can appear when studying the problems Eκc and ecκ for κ small, we begin by discussing the case κ = 0. Proposition 7. Conjecture M is true in the case κ = 0. Proof. The case κ = 0 is obvious. Indeed, all projectors P + coincide with the 0,W

projector χ[0,∞) (H c + V ). The level E0c , seen as the minimum of Corollary 2, is achieved by any N -dimensional space Wmin spanned by N orthogonal eigenvectors of H c +V whose eigenvalues are the N first positive eigenvalues of H c +V , counted with multiplicity. Then E0c is the sum of these N first positive eigenvalues. Clearly, (Wmin , χ[0,∞) (H c + V )) realizes ec0 . The interesting case is, of course, κ > 0 , when electronic interaction is taken into account. For κ > 0 and small two very different situations occur, depending on the number N of electrons. The first situation (perturbation from the linear closed shell atom) corresponds to I N= Nic , I ∈ Z+ (14) i=1

is treated in detail in Section 3. We recall that Nic is the dimension of the eigenspace Mci = Ker(H c +V −λci 11) already defined. Under assumption (14), for κ = 0, there is a unique solution, W0c , to the variational problems defining E0c and ec0 , W0c =

I i=1

Mci .

92



The “shells” of energy λci , 1 ≤ i ≤ I , are “closed”: each one is occupied by the maximal number of electrons allowed by the Pauli exclusion principle. The subspace W0c is invariant under the representation • of SU (2). We are interested in solutions Wκc of the Dirac-Fock equations lying in a neighborhood Ω ⊂ GN (H 1/2 ) of W0c , for κ small. Using the implicit function theorem, we are going to show that for each κ small, Wκc exists, is unique, and is a smooth function of κ. Information about the properties enjoyed by Wκc is given by Proposition 8. Fix c large enough. Under assumption (14), for κ small enough, Eκc = Eκc (Wκc ) =

inf

+ 1/2 ) W ∈GN (Pκ,W cH

Eκc (W ),

(15)

κ

and Wκc is the unique solution of this minimization problem. This proposition will be proved in Section 3. Our first main result follows from it: Theorem 9. Under assumption (14), for c > 0 fixed and κ small enough, Eκc = ecκ and both variational problems are achieved by the same solution Wκc of the selfκ is P + c . consistent Dirac-Fock equations. For ecκ , the optimal projector in P κ,Wκ Proof. The above proposition implies that for κ small, ecκ ≥

inf

+ 1/2 ) W ∈GN (Pκ,W cH

Eκc (W ) = Eκc (Wκc ) = Eκc .

(16)

κ

Therefore, ecκ = Eκc . Moreover, by Proposition 8, ecκ is achieved by a couple (Wκc , P ) + c such that P = Pκ,W c , Wκ being a solution of the Dirac-Fock equations. This ends κ the proof. The second situation (perturbation from the linear open shell case) occurs when I c Nic + k, I ∈ Z+ , 0 < k < NI+1 . (17) N= i=1

It is treated in detail in Section 4. When (17) holds and when κ = 0, there exists a manifold of solutions, S0 , whose elements are the spaces I

c Mci ⊕ WI+1,k ,

i=1 c c for all WI+1,k ∈ Gk (MI+1 ). These spaces are all the solutions of the variational c problems defining E0 and ec0 . The (I + 1)th “shell” of energy λcI+1 is “open”: it is

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c occupied by k electrons, while the Pauli exclusion principle would allow NI+1 −k more. Note that we use the expression “open shell” in the linear case κ = 0 only: indeed, adapting an idea of Bach et al. [17], one can easily see that for κ positive and small, the solutions to (DF) at the minimal level Eκc have no unfilled shells. For κ > 0 and small we look for solutions of the DF equations near S0 (see Section 4). We could simply quote the existence results of [15], and show the convergence of solutions of (DF) at level Eκc , towards points of S0 , as κ goes to 0. But we prefer to give another existence proof, using tools from bifurcation theory. This approach gives a more precise picture of the set of solutions to (DF) near the level Eκc (Theorem 12). In particular, we obtain in this way all the solutions of (DF) with smallest energy Eκc (Proposition 13). − We now choose one of these minimizers, and we call it Wκc . We have Pκ,W c κ c c (Wκ ) = 0 . Since V is radial, Wκ belongs to an SU (2) orbit of minimizers. We are interested in cases where this orbit is not reduced to a point. Then the mean-field c operator Hκ,W c should not commute with the action • of SU (2), and one expects κ the following property to hold:

(P): Given c large enough, if κ is small, then for any solution Wκc of (DF) at level Eκc , there is a matrix A ∈ SU (2) such that − c Pκ,W c (A • Wκ ) = 0 . κ

(18)

The next proposition shows that whenever (P) holds, Conjecture M does not. This result will imply that Conjecture M is indeed wrong. Proposition 10. If (P) is satisfied, then for c large enough and κ small, given any solution Wκc of the nonlinear Dirac-Fock equations such that Eκc (Wκc ) = Eκ , we have inf Eκc (W ). (19) Eκc = Eκc (Wκc ) > W ∈GN (H 1/2 ) − P c W =0 κ,Wκ

This proposition will be proved in Section 4. Moreover, we verify (see Proposition 15) that (P) holds when I ≥ 1 and k = 1, i.e., when in the linear case there is a single electron in the highest nonempty shell. Our second main result follows directly from Propositions 10 and 15. Theorem 11. Take N=

I

Nic + 1,

I ≥ 1.

i=1

For c large and κ > 0 small, there is no solution W∗ of the nonlinear Dirac-Fock equations with positive Lagrange multipliers, such that the couple + (W∗ , Pκ,W ) ∗

realizes the max-min ecκ . So Conjecture M is wrong.

94



I Remark. Note that the fact that Conjecture M is wrong in the case N = Nic +1,

I ≥ 1, is related to nonuniqueness of the minimizer for the problem inf

W ∈GN (H 1/2 ) − P c W =0 κ,Wκ

i=1

Eκc (W ) .

When such a situation happens, it is well known that one has to be very careful when considering max-min (resp. min-max) problems, since even when solvable, they do not always deliver critical points of the considered functional. A very simple example for this fact is provided by the function f : R2 → R defined by f (x, y) := (1 − x2 )2 + xy. It is easy to verify that sup inf f (x, y) = 0 ,

y∈R x∈R

that the unique maximizer is y = 0 and that there are exactly two minimizers of x → f (x, 0), x± = ±1. But neither (−1, 0) nor (1, 0) are critical points of f .

3 Perturbation from the linear closed shells case Let us recall that we are in the case N=

I

Nic ,

I ∈ Z+ ,

i=1

Nic being the dimension of the eigenspace Mci = Ker(H c + V − λci 11). We want to apply the implicit function theorem in a neighborhood of W0c , for κ small. For this purpose, we need a local chart near W0c . Take an orthonormal basis (ψ1 , . . . , ψN ) of W0c , whose elements are eigenvectors of H c +V , the associated eigenvalues being µ1 ≤ · · · ≤ µN (i.e., λc1 , . . . , λcI counted with multiplicity). Let Z be the orthogonal space of W0c for the L2 scalar product, in H 1/2 (R3 , C4 ). Then Z is a Hilbert space for the H 1/2 scalar product. The map C : χ = (χ1 , . . . , χN ) → span(ψ1 + χ1 , . . . , ψN + χN ) , defined on a small neighborhood O of 0 in Z N , is the desired local chart. Denote Gχ the N × N matrix of scalar products (χl , χ )L2 . Then Eκc ◦ C(χ) = Eκc (I + Gχ )−1/2 (ψ + χ) . The differential of this functional defines a smooth map Fκ : O ⊂ Z N → (Z )N , where Z ⊂ H −1/2 is the topological dual of Z for the H 1/2 topology, identified with the orthogonal space of W0c for the duality product in H −1/2 × H 1/2 . Note that Fκ depends smoothly on the parameter κ. A subspace C(χ) is solution of

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(DF) if and only if Fκ (χ) = 0. To apply the implicit function theorem, we just have to check that the operator L := Dχ F0 (0) is an isomorphism from Z N to its dual (Z )N . This operator is simply the Hessian of the DF energy expressed in our local coordinates: (20) Lχ = (Hc + V − µ1 )χ1 , . . . , (Hc + V − µN )χN . Under assumption (14), the scalars µk , k = 1, . . . , N , are not eigenvalues of the restriction of H c + V to the L2 -orthogonal subspace of W0c . This implies that L is an isomorphism. As a consequence, there exists a neighborhood of W0c × {0} in GN (H 1/2 ) × R, Ω × (−κ0 , κ0 ) and a smooth function hc : (−κ0 , κ0 ) → Ω such that for κ ∈ (−κ0 , κ0 ), Wκc := hc (κ) is the unique solution of the Dirac-Fock equations in Ω. Moreover, for all κ ∈ (−κ0 , κ0 ), the following holds: u(Wκc ) = Wκc , ∀u ∈ SU (2) .

(21)

Indeed, the subset A of parameters κ such that (21) holds is obviously nonempty (it contains 0) and closed in (−κ0 , κ0 ). Now, for κ in a small neighborhood of A, the SU (2) orbit of Wκc stays in Ω. But this orbit consists of solutions of the Dirac-Fock equations, so, by uniqueness in Ω, it is reduced to a point. This shows that A is also open. A is thus the whole interval of parameters (−κ0 , κ0 ). Now we are in the position to prove Proposition 8. + c Proof of Proposition 8. Remember that for κ = 0, P0,W c coincides with χ(0,∞) (H + 0 c c V ). Now, W0 is clearly the unique minimizer of E0 on the Grassmannian sub+ 1/2 manifold G+ ). More precisely, in topological terms, for any 0 := GN (P0,W0c H c 1/2 neighborhood V of W0 in GN (H ), there is a constant δ = δ(V) > 0 such that 1/2 E0c (W ) ≥ E0c (W0c ) + δ , ∀ W ∈ G+ ) \ V) . 0 ∩ (GN (H

(22)

Moreover, looking at formula (20), one easily sees that the Hessian of E0c on G+ 0 is positive definite at W0c . We now take κ > 0 small, and we consider again the chart + 1/2 C constructed above. We define the submanifold G+ ). Then κ := GN (Pκ,Wκc H + + N + c the restriction Cκ of C to (Pκ,Wκc Z) is a local chart of Gκ near Wκ . For κ small enough, there is a neighborhood U of 0 in Z N such that the second derivative of + N . The functional Eκc ◦ Cκ+ is Eκc ◦ Cκ+ is positive definite on Uκ+ := U ∩ (Pκ,W c Z) κ thus strictly convex on Uκ+ . Now, for κ small, there is a unique χκ ∈ Uκ+ such that Cκ+ (χκ ) = Wκc . Then the derivative of Eκc ◦ Cκ+ vanishes at χκ . As a consequence Wκc = Cκ+ (χκ ) is the unique minimizer of Eκc on Vκ+ := Cκ+ (Uκ+ ). Now, we choose, as neighborhood of W0c in GN (H 1/2 ), the set V := C(U), and we consider the constant δ > 0 such that (22) is satisfied. Taking κ > 0 even smaller, we can impose min Eκc + δ/2 ≤ inf Eκc . + Vκ

+ G+ κ \Vκ

Hence, Wκc is the unique solution to the minimization problem (15).

96



4 Bifurcation from the linear open shell case Recall that here we are in the case N=

I

c Nic + k, I ∈ Z+ , 0 < k < NI+1 .

i=1

For κ = 0, there exists a manifold of solutions, S0 , whose elements are the spaces I

c Mci ⊕ WI+1,k ,

i=1 c c for all WI+1,k ∈ G (MI+1 ). These spaces are all the solutions of the variational c problems defining E0 and ec0 . For κ > 0 and small we want to find solutions of the DF equations near S0 , by using tools from bifurcation theory. If λI+1 has only multiplicity 2, then (17) implies k = 1 and by Lemma 1 of §2, S0 is an SU (2) orbit. Then, as in Section 3, one can find, in a neighborhood of S0 , a unique SU (2) orbit Sκ of solutions of (DF). But there are also more degenerate cases in which λI+1 has a higher multiplicity, and S0 contains a continuum of SU (2) orbits. In such situations, κ = 0 is a bifurcation point, and one expects, according to bifurcation theory, that the manifold of solutions S0 will break up for κ = 0, and that there will only remain a finite number of SU (2) orbits of solutions. To find these orbits, one usually starts with a Lyapunov-Schmidt reduction: one builds a suitable manifold Sκ which is diffeomorphic to S0 (see, e.g., [18]). When S0 contains several SU (2) orbits, the points of Sκ are not necessarily solutions of (DF), but Sκ contains all the solutions sufficiently close to S0 . Moreover, all critical points of the restriction of Eκc to Sκ are solutions of (DF). The submanifold Sκ is constructed thanks to the implicit I+1 function theorem. More precisely, we consider the projector Π : L2 → i=1 Mci . To each point z ∈ S0 we associate the submanifold Fz := {w ∈ GN (H 1/2 ) : Πw = z}. For w a point of Fz , let ∆w := Tw Fz ⊂ Tw GN (H 1/2 ). Then the following holds:

Theorem 12. Under the above assumptions, there exist a neighborhood Ω of S0 in GN (H 1/2 ), a small constant κ0 > 0, and a smooth function h : S0 ×(−κ0 , κ0 ) → Ω such that (a) h(z, 0) = z ∀z ∈ S0 (b) Denoting Sκ := h(S0 , κ), Sκ is also the set of all points w in Ω such that

(Eκc ) (w), ξ = 0, (c) h(z, κ) ∈ Fz ,

∀(z, κ) ∈ S0 × (−κ0 , κ0 ).

∀ξ ∈ ∆w

(23)

I+1 Proof. We first fix a point z in S0 . Let N be the orthogonal space of i=1 Mci in H 1/2 for the L2 scalar product. As in Section 3, we can define a local chart

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Cz : O ⊂ (N )N → Fz near z, by the formula C(χ) = span(ψ + χ), where ψ = (ψ1 , . . . , ψN ) is an orthonormal basis of z consisting of eigenvectors of H c + V , with eigenvalues µ1 ≤ · · · ≤ µN (i.e., λc1 , . . . , λcI counted with multiplicity). The Hessian of E0c ◦Cz at χ = 0 is given once again by formula (20). It is an isomorphism between (N )N and its dual. So, arguing as in Section 3, we find, by the implicit function theorem, a small constant κz > 0, a neighborhood ωz of z in Fz and a ˜ z : (−κz , κz ) → Ω z such that: function h ˜ z (0) = z (i) h ˜ z (κ) is the unique point w in Ω z such that (ii) h

(Eκc ) (w), ξ = 0,

∀ξ ∈ ∆w

(24)

Since S0 is compact and Eκc (w) a smooth function of (w, κ), it is possible to choose z such that κ0 := inf z∈S0 κz > 0, with Ω := κz , Ω z∈S0 Ωz a neighborhood of S0 , ˜ and h(z, κ) := hz (κ) a smooth function on S0 × (−κ0 , κ0 ) with values in Ω. This function satisfies (a,b,c). From (b) any critical point of Eκc in Ω must lie on Sκ . From (c) it follows that Sκ is a submanifold diffeomorphic to S0 , and transverse to each fiber Fz in GN (H 1/2 ). If z ∈ S0 is a critical point of Eκc ◦ h(·, κ), then, taking w = h(z, κ), the derivative of Eκc at w vanishes on Tw Sκ . From (b), it also vanishes on the subspace ∆w which is transverse to Tw Sκ in Tz GN (H 1/2 ), hence (Eκc ) (w) = 0. This shows that the set of critical points of Eκc in Ω coincides with the set of critical points of the restriction of Eκc to Sκ . Arguing as in the proof of Proposition 8, one gets more: Proposition 13. For κ > 0 small, the solutions of (DF) of smallest energy Eκc are exactly the minimizers of Eκc on Sκ . We are now ready to prove Proposition 10. Proof of Proposition 10. Since κ is small, for any matrix A ∈ SU (2) the map + + 1/2 ) and Pκ,A•W c induces a diffeomorphism between the submanifolds GN (Pκ,W c H κ κ + 1/2 GN (Pκ, A•Wκc H ) . Now, we fix A ∈ SU (2) such that (18) holds. Then there exists a unique point W + ∈ GN (H 1/2 ) such that − + Pκ,W = 0, cW κ

By (18), we have

+ + Pκ, = A • Wκc A•Wκc W

(25)

W + = A • Wκc .

On the other hand, in [14] it was proved that Eκc (A • Wκc ) =

sup W ∈GN (H 1/2 ) + c P c W =A•Wκ κ,A•Wκ

Eκc (W )

(26)

98



and A • Wκc is the unique solution of this maximization problem. Therefore, Eκc (A • Wκc ) > Eκc (W + ) . But

Eκc (W + ) ≥

inf

+ 1/2 ) W ∈GN (Pκ,W cH

Eκc (W ) ,

κ

hence, by invariance of Eκc

=

Eκc

under the action of SU (2),

Eκc (A

• Wκc ) >

inf

+ 1/2 ) W ∈GN (Pκ,W cH

Eκc (W ) ,

κ

and the proposition is proved.

Since there are no solutions of (DF) under level Eκc , and ecκ ≤ Eκc , Proposition 10 has the following consequence: Corollary 14. If (P) is satisfied, then for c large enough and κ small, there is no solution W∗ of the nonlinear Dirac-Fock equations with positive Lagrange multipliers, such that the couple + ) (W∗ , Pκ,W ∗ realizes the max-min ecκ . So Conjecture M is wrong when (P) holds. We now exhibit a case where (P) holds. Proposition 15. Assume that N =

I

Nic + 1, I ≥ 1. Then (P) is satisfied.

i=1

Proof. Step 0. Fix c large enough and take a sequence of positive parameters (κ )≥0 converging to 0. Let (Wc )≥0 be a sequence in GN (H 1/2 ), with Wc a minimizer of Eκc on Sκ . Let ψc ∈ Wc be an eigenvector of the mean-field Hamiltonian Hκc ,W c , normalized in L2 and corresponding to the highest occupied level. Extracting a subsequence if necessary, we may assume that ψc → ψ c ∈ McI+1 = Ker(H c + V − λcI+1 ). Moreover, from Theorem 12 we have Wc → W0c =

I

Mci ⊕ C ψ c .

i=1

Pκ− ,W c ψc

Step 1. Fix c ≥ 1 . Since = 0, we can write, by a classical result due to Kato, +∞ 1 (27) (Hκc ,Wc − iη)−1−(Hκc ,A•Wc − iη)−1 ψc dη Pκ− ,A•W c ψc = 2π −∞ 1 +∞ c (Hκ ,Wc − iη)−1 (Hκc ,A•Wc −Hκc ,Wc )(Hκc ,A•Wc − iη)−1 ψc dη = 2π −∞ κ +∞ c (H + V − iη)−1 (ΩA•W0c − ΩW0c )(H c + V − iη)−1 ψ c dη + o(κ ) , = 2π −∞

Vol. 6, 2005


c where by ΩW we denote the nonlinear part of Hκ,W : c = H c + V + κ ΩW . Hκ,W I c But note that since the space i=1 Mi is invariant under the action of SU (2), ΩA•W0c − ΩW0c = ΩA•ψc − Ωψc . c , So, we just have to prove that for c sufficiently large and for all ψ c ∈ MI+1 there exists A ∈ SU (2) such that +∞ (H c + V − iη)−1 (ΩA•ψc − Ωψc )(H c + V − iη)−1 ψ c dη = 0 . (28) −∞

Since (H c + V − iη)−1 ψ c =

ψc − iη

λcI+1

and Ωψc ψ c = 0 ,

c , there exists A ∈ SU (2) what we need to prove is that for all nonzero ψ c ∈ MI+1 c c such that L (ΩA•ψc ψ ) = 0, with +∞ dη c . (H c + V − iη)−1 c L := λI+1 − iη −∞

Step 2. We give an asymptotic expression for Lc when c → +∞: 1 1 +∞ 1 η −1 d(η/c2 ) 1 c c (H + V ) − i 2 = 2 Lc + O 2 , L = 2 λcI+1 c −∞ c2 c c c − i η2 2 c

(29)

c

where Lc , in the Fourier domain, is the operator of multiplication by the matrix +∞ ˆ Lc (p) = (−iu + β + (α · p)/c)−1 (−iu + 1)−1 du . (30) −∞

Here, we have used the standard fact that 1 λcI+1 = 1 + O . c2 c2 We have (−iu + β + (α · p)/c)−1 = with ω c (p) :=

1 + |p|2 /c2 ,

1 1 ˆ c (p) + ˆ c (p) Λ Λ −iu + ω c (p) + −iu − ω c (p) − c ˆ c (p) = ω (p) ± (β + (α · p)/c) . Λ ± 2ω c (p)

Hence, by the residues theorem, |p|2 2ˆ (α · p) . Lc (p) = β − 1 + +O π c c2

100



Step 3. It is well known (see [16]) that ψ c can be written as 1 φ c ψ = −i(σ·∇)φ + O 2 , c 2c φ ∈ L2 (R3 , C2 ) being an eigenstate of ( −∆ 2 + V ), with eigenvalue µ = limc→+∞ (λcI+1 − c2 ). Since we have assumed that V is smooth, this asymptotic result holds for the topology of the Schwartz space S(R3 ). So, 1 0 2c2 c i c L (ΩA•ψc ψ ) = +O 2 , π c f (A, φ) c where

x · σ x · σ 2 ∗ f (A, φ) := |A • φ|2 ∗ φ −

A • φ, φ (A • φ) . C |x|3 |x|3

(31)

What remains to prove is: Step 4. For any eigenvector φ of the Schr¨ odinger operator − ∆ 2 + V , there exists an A ∈ SU (2) such that f (A, φ) ≡ 0 . Proof of Step 4. We consider the integral

(x · σ)φ, f (A, φ)C2 (r ω)dω . IA,φ (r) := S2

x Since φ has exponential fall-off at infinity, the electrostatic field |A • φ|2 ∗ |x| 3 x 1 2 when |x| is large. The takes the asymptotic form |x|3 + O |x|3 R3 |A • φ| x same phenomenon holds for the convolution product < A • φ, φ >C2 ∗ |x| 3 . As a consequence, for r large, r IA,φ (r) = |A • φ|2 |φ|2 (r ω) dω R3 S2 −

A • φ, φC2

φ, A • φC2 (r ω) dω R3 S2 1 2 +O |φ| (r ω) dω . r S2

Since • is unitary, the Cauchy-Schwarz inequality gives |φ|2 (r ω) dω = |A • φ|2 (r ω) dω ≥

A • φ, φC2 (r ω) dω . S2

S2

S2

By Lemma 1 of Section 1, we can choose A such that φ and A • φ are not colinear. Then 2 2 |A • φ| = |φ| >

A • φ, φC2 . R3

R3

R3

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So there is a constant δ > 0 such that, for r large enough, 2 |φ| |φ|2 (r ω) dω . |r IA,φ (r)| ≥ δ R3

(32)

S2

Being an eigenvector of the Schr¨ odinger operator − ∆ 2 + V , the function φ cannot have compact support. So the lower estimate (32) implies that the function IA,φ (r) is not identically 0, hence f (A, φ) ≡ 0 . Step 4 is thus proved, and (P) is satisfied. Acknowledgments. The authors wish to thank the referee for useful comments on the first version of this paper. Financial support of the European Union through the IHP networks “Analysis and Quantum” (HPRN-CT-2002- 00277) and “HYKE” (HPRN-CT-200200282), is gratefully acknowledged.

References [1] B. Swirles, The relativistic self-consistent field, Proc. Roy. Soc. A 152 , 625– 649 (1935). [2] I. Lindgren, A. Rosen, Relativistic self-consistent field calculations, Case Stud. At. Phys. 4 , 93–149 (1974). [3] O. Gorceix, P. Indelicato, J.P. Desclaux, Multiconfiguration Dirac-Fock studies of two-electron ions: I. Electron-electron interaction, J. Phys. B: At. Mol. Phys. 20 , 639–649 (1987). [4] M.J. Esteban, E. Séré, Solutions for the Dirac-Fock equations for atoms and molecules, Comm. Math. Phys. 203, 499–530 (1999). [5] E. Paturel, Solutions of the Dirac equations without projector, A.H.P. 1, 1123–1157 (2000). [6] M.H. Mittleman, Theory of relativistic effects on atoms: Configuration-space Hamiltonian, Phys. Rev. A 24(3), 1167–1175 (1981). [7] J. Sucher, Foundations of the relativistic theory of many-particle atoms, Phys. Rev. A 22 (2), 348–362 (1980). [8] J. Sucher, Relativistic many-electron Hamiltonians, Phys. Scrypta 36, 271–281 (1987). [9] P. Chaix, D. Iracane, From quantum electrodynamics to mean-field theory: I. The Bogoliubov-Dirac-Fock formalism, J. Phys. B 22 (23), 3791–3814 (December 1989).

102



[10] V. Bach, J.M. Barbaroux, B. Helffer, H. Siedentop, Stability of matter for the Hartree-Fock functional of the relativistic electron-positron field, Doc. Math. 3, 353–364 (1998). [11] V. Bach, J.M. Barbaroux, B. Helffer, H. Siedentop, On the stability of the relativistic electron-positron field, Comm. Math. Phys. 201(2), 445–460 (1999). [12] J.-M. Barbaroux, W. Farkas, B. Helffer, H. Siedentop, On the Hartree-Fock equations of the electron-positron field, Preprint. [13] P. Chaix, D. Iracane, P.L. Lions, From quantum electrodynamics to mean-field theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation, J. Phys. B 22 (23), 3815–3828 (December 1989). [14] M.J. Esteban, E. Séré, Nonrelativistic limit of the Dirac-Fock equations, A.H.P. 2, 941–961 (2001). [15] M.J. Esteban, E. Séré, A max-min principle for the ground state of the DiracFock functional, Contemp. Mathem. 307, 135–139 (2002). [16] B. Thaller, The Dirac Equation, Springer-Verlag, 1992. [17] V. Bach, E.H. Lieb, M. Loss, J.P. Solovej, There are no unfilled shells in unrestricted Hartree-Fock theory, Phys. Rev. Lett. 72(19), 2981–2983 (1994). [18] A. Ambrosetti, M. Badiale, Homoclinics: Poincaré-Melnikov type results via a variational approach, Annales de l’IHP, Analyse non linéaire 15(2), 233–252 (1998). Jean-Marie Barbaroux CPT-CNRS Luminy Case 907 F-13288 Marseille Cedex 9 France email: [email protected] Maria J. Esteban and Eric Séré Ceremade (UMR CNRS no. 7534) Université Paris IX-Dauphine Place de Lattre de Tassigny F-75775 Paris Cedex 16 France email: [email protected] email: [email protected] Communicated by Bernard Helffer submitted 19/03/04, accepted 30/07/04



Localization for a Family of One-dimensional Quasiperiodic Operators of Magnetic Origin S. Jitomirskaya, D.A. Koslover and M.S. Schulteis

Abstract. We show strong dynamical localization for a family of one-dimensional quasiperiodic Jacobi operators of magnetic origin, throughout the regime of positive Lyapunov exponents.

1 Introduction The study of electrons subjected to a perpendicular magnetic field and twodimensional periodic potentials can be reduced via an appropriate gauge choice to the study of spectral properties of discrete one-dimensional quasiperiodic Jacobi matrices. The simplest and best studied case is the almost Mathieu operator Hθ,λ,ω acting on 2 (Z) by (using a somewhat nonstandard rescaling) (Hθ,λ,ω ψ)(n) = λ(ψ(n + 1) + ψ(n − 1)) + 2 cos 2π(θ + nω).

(1)

This is obtained from the model through a Landau gauge with, in general, anisotropic nearest neighbor couplings (e.g., [15]). For a recent review of the spectral theory of operator (1) see [22]. Two other models that include the almost Mathieu operator for a certain choice of parameters have been proposed in the physics literature [26, 3, 11]. The first model is the case of a square lattice with anisotropic nearest neighbor coupling and isotropic next nearest neighbor coupling. This is also the model that applies if one exposes an ultracold cloud of atoms to a bichromatic standing light wave [11]. The second model has anisotropic coupling to nearest neighbors and next nearest neighbors on a triangular lattice. This paper considers a slightly more general model that includes both cases above. Namely, we study one-dimensional operators corresponding to the case of anisotropic coupling of both nearest neighbors and second nearest neighbors on a two-dimensional lattice. The corresponding Hamiltonian Hθ,λ,ω acting on 2 (Z) is given by (Hθ,λ,ω ψ)(n) = c(n, θ, ω)ψ(n + 1) + c(n − 1, θ, ω)ψ(n − 1) + v(n, θ, ω)ψ(n) (2) where v(n, θ, ω) = v(θ + nω)

c(n, θ, ω) = c(θ + nω)

(3)

The authors were supported in part by BSF grant 2002068 and NSF grant DMS-0300974.

104

S. Jitomirskaya, D.A. Koslover and M.S. Schulteis


and v(θ) c(θ)

= v(θ, λ) = 2 cos 2π(θ) = c(θ, λ) = λ2 + λ3 e

{2πi(θ+ 12 ω)}

(4) + λ1 e

{−2πi(θ+ 12 ω)}

,

the normalized hopping terms λ = (λ1 , λ2 , λ3 ) and θ ∈ T = R/(2πZ). The hopping terms (see Figure 1) are proportional to the probability an electron will hop to a neighboring site. ω is the magnetic flux. We will identify c(n, θ, λ) with c(n) and v(n, θ, λ) with v(n) when it can be done without confusion. The almost Mathieu family of operators corresponds to the choice λ1 = λ3 = 0. λ1 r

rλ r1 * 3 @ I r - rλ2 λ2 r @ @@ Rr r r λ3 λ1 1 Figure 1: Hopping terms. In the case of the almost Mathieu operator, it has been shown that if λ < 1, Hθ,λ,ω has only pure point spectrum with exponentially decaying eigenfunctions. And further, it exhibits strong dynamical localization in that region [13]. If λ = 1, it has purely singular-continuous spectrum and if λ > 1, the spectrum is purely absolutely continuous. (See [16] for a history of results.) For our general family, Hθ,λ,ω , it has been shown that if λ1 + λ3 < 1 and λ2 < 1, then the operator has positive Lyapunov exponents, thus no absolutely continuous spectrum. If λ1 + λ3 < λ2 and 1 < λ2 , or if λ1 or λ3 = 0, and λ1 + λ3 > max(λ2 , 1), there is no pure point spectrum [23] and our preliminary results indicate absolutely continuous spectrum [21]. If λ1 = λ3 and λ1 + λ3 > max(λ2 , 1) there is no absolutely continuous spectrum and we expect to show singular continuous spectrum [21]. (See Figure 2.) In this paper, we will show that if λ1 + λ3 < 1 and λ2 < 1, then under certain arithmetic conditions on (ω, θ), Hθ,λ,ω has only pure point spectrum with exponentially decaying eigenvalues. This is exactly the region of positive Lyapunov exponents, and pure point spectrum is not expected elsewhere. Also, localization does not hold under certain complementary conditions in frequency [9] or in phase [18]. Results on the other regions will follow in a separate paper. As originally shown in [10], spectral localization may not have any dynamical consequences. Dynamical localization, defined as a non-spread of initially localized wave-packets, requires additional arguments. For ergodic families, such as (2)–(4), a stronger statement, so-called strong dynamical localization (see Section 3), is also desirable. This has been achieved for random potentials throughout the regime of multiscale analysis by [14]. Strong dynamical localization also follows naturally

Vol. 6, 2005

Localization for Quasiperiodic Operators of Magnetic Origin

1

+

105

no a.c. spectrum

3

1 s.c. spectrum

p.p. spectrum

s.c. spectrum 1

no p.p. spectrum

2

a.c. spectrum

Figure 2: In the unlabeled region, we expect an interesting dependence of the spectrum on the parameters λ1 and λ3 . In certain cases, it is expected that there is no a.c. spectrum. In other cases, we expect a.c. spectrum. No p.p. spectrum is expected throughout this region. whenever the Aizenman-Mochanov method applies [1]. In the quasiperiodic (and skew-shift) case, it was shown in [6] that dynamical localization follows from the approach of [5]. Strong dynamical localization in the quasiperiodic setting has only been established for the almost Mathieu operator [13]. In this paper, we prove strong dynamical localization throughout the regime of positive Lyapunov exponents. The rest of the paper is organized as follows. In Section 2, we formulate and prove our main result on Anderson localization. We follow the general scheme of [16] highlighting important difference as well as providing more detail. In particular, our proof is designed to work equally well for phases θ = ω/2 or (ω −1)/2 where ∈ Z. That was not the case in [16]. Those cases have recently been shown to be important for the proof of Cantor spectrum [25]. In Section 3, we formulate and prove strong dynamical localization, following the strategy of [13] with a modification as outlined in [7].

2 Anderson localization We say that ω is Diophantine if there exists b(ω) and 1 < r(ω) < ∞ such that | sin πjω| >

b(ω) |j|r(ω)

for all j ∈ Z. We define resonant phases as Θ(ω, α) = {θ : ∃j,

|j| < 3k α ,

−1

| sin 2π(θ + (k/2)ω)| < exp −k (2r(ω))

holds for infinitely many k’s} .

(5)

106



The Lebesgue measures of the complement of {ω : ω is Diophantine} and the set of resonant phases are zero. Theorem 1. Suppose ω is Diophantine, θ ∈ ΘC (ω, α), λ2 < 1, and λ1 +λ3 < 1, then Hθ,λ,ω has only pure point spectrum with exponentially decaying eigenfunctions. Remarks 1. As discussed in the introduction, localization does not hold or is not expected to hold in all other regions of (λ1 , λ2 , λ3 ). 2. For all proofs in this paper, we will assume without loss of generality that 0 ≤ λ2 < 1, 0 ≤ λ1 + λ3 < 1 and at least one of λ1 , λ2 , λ3 > 0 . As usual, a formal solution ψE of the eigenvalue equation Hθ,λ,ω ψE = EψE is called a generalized eigenfunction if |ψE (n)| ≤ C(1 + |n|) for C = C(ψE ) < ∞. We will prove exponential decay of all generalized eigenfunctions, which is sufficient by [4]. The n-step transfer matrix for the eigenvalue equation is 1 1 E − v(i) −c(i − 1) M (θ, n, E) = c(i) 0 c(i) i=n

where

ψ(n + 1) ψ(n)

= M (θ, n, E)

ψ(1) ψ(0)

.

M (θ, n, E) is defined when |c(n, ω, θ, λ)| is strictly greater than zero. Denote Pk (θ, E) = det[(E − Hθ,λ,ω )|[1,k] ], for k ∈ N. Define P−1 (θ, E) = 0 and P0 (θ, E) = 1. Then the n-step transfer matrix can be written as 1 −c(0)Pn−1 (θ + ω, E) Pn (θ, E) M (θ, n, E) = c(1) . . . c(n) c(n)Pn−1 (θ, E) −c(n)c(0)Pn−2 (θ + ω, E) 1 g(θ, n, E). (6) := c(1) . . . c(n) This can be shown, for example, by expanding the determinant of (E −Hθ,λ,ω )|[1,k] in its final row and then using induction in n. 1 Let γn (M (E)) = ln M (θ, n, E). By Kingman’s subadditive ergodic n theorem 1 γ(M (E)) := lim ln M (θ, n, E) n−→∞ n exists for almost every θ ∈ T and fixed E and further 1 1 1 1 γ(M (E)) = lim ln M (θ, n, E)dθ = inf ln M (θ, n, E)dθ. n−→∞ n 0 n n 0 γ(M (E)) is called the Lyapunov exponent. It has been shown that γ(M (E)) is strictly greater than zero in the region we are considering (e.g., [23]).

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107

Note that 0 < γ(M (E))

= :=

1 n−→∞ n

1

lim

0

1 n−→∞ n

ln g(θ, n, E)dθ − lim

0

1

ln |c(1) . . . c(n)|dθ

γ(g(E)) − C(λ),

(7)

where both limits exist by the subadditive ergodic theorem. Further, 1 1 ln g(θ, n, E)dθ. γ(g(E)) = inf n n 0 and γ(g(E)) = lim

n−→∞

(8)

1 ln g(θ, n, E) n

(9)

for any fixed E and almost every θ . Finally, 1 n 1 1 ln |c(k)|dθ = ln |c(0)|dθ. C(λ) = lim n−→∞ n 0 0 k=1

Applying the Poisson-Jensen formula and using (4),  ln λ3 if λ3 ≥ λ1 ≥ 0 and λ1 + λ3 ≥ λ2 ≥ 0     ln λ  1 

if λ1 ≥ λ3 ≥ 0 and λ1 + λ3 ≥ λ2

2λ λ C(λ) =

1 3 ln 

if λ1 + λ3 ≤ λ2 , λ1 , λ3 = 0

  −λ2 + λ22 − 4λ1 λ3

   ln λ2 if λ1 + λ3 ≤ λ2 , λ1 or λ3 = 0 . In the region we are considering, C(λ) < 0. For all n ∈ N, by (7) and (8) 1 1 ln g(θ, n, E)dθ . γ(M (E)) + C(λ) ≤ n 0 So applying (6), en(γ(M(E))+C(λ)) ≤ max θ

Pn (θ, E) −c(0)Pn−1 (θ + ω, E) c(n)Pn−1 (θ, E) −c(n)c(0)Pn−2 (θ + ω, E)

.

Thus, 1 √ en(γ(M(E))+C(λ)) ≤ (10) 4 2 max max [|Pn (θ, E)|, |Pn−1 (θ, E)|, |Pn−1 (θ + ω, E)|, |Pn−2 (θ + ω, E)|] θ

where we used an obvious upper bound for c.

108



1 √ exp[nγ(M (E)) + (n + 2)C(λ)] ≤ 8 2 |Pn (θ, E)|}. From the derivation above, at least one of every {n, n + 1, n + 2} belongs to K. We will use the notation H[x1 , x2 ](θ, λ, ω) for Hθ,λ,ω restricted to the interval [x1 , x2 ] with zero boundary conditions at x1 − 1 and x2 + 1. G[x1 , x2 ](E, θ, x, y) will be the corresponding Green’s function (H[x1 , x2 ](θ, λ, ω) − E)−1 (x, y). Let K = {n ∈ N : ∃θ ∈ [0, 1] :

Definition 1. Let m > 0, E ∈ R, be fixed. A point y ∈ Z will be called (m, k, E, θ)regular if there exists an interval [x1 , x2 ], x2 = x1 + k − 1 containing y such that |G[x1 , x2 ](y, xi )| < exp

−mk 9

and dist(y, xi ) ≥ 19 k for i = 1, 2. Otherwise, y will be called (m, k, E, θ)-singular. The value of any formal solution ψ of the equation Hψ = Eψ, energies E ∈ / σ(H[x1 , x2 ](θ, λ, ω)), at a point x ∈ [x1 , x2 ] ⊂ Z can be reconstructed from the boundary values via ψ(x) = −c(x1 − 1)ψ(x1 − 1)G[x1 , x2 ](x, x1 ) − c(x2 )ψ(x2 + 1)G[x1 , x2 ](x, x2 ). (11) This implies that if ψE is a generalized eigenfunction, then every point y ∈ Z with ψE (y) = 0 is (m, k, E, θ)-singular for k sufficiently large, k > k1 (E, m, θ, y) and m > 0. Define −1 j Θk (y1 , ω, α) = {θ : ∃j, |j| < 3k α , | sin 2π(θ + y1 + ω)| < exp −k (r(ω)) } 2

(12)

and Θ(y1 , ω, α) = =

lim sup Θk (y1 , ω, α) k

−1

{θ : ∃j, |j| < 3k α , | sin 2π(θ + y1 + (k/2)ω)| < exp −k (2r(ω)) for infinitely many k’s}, (13)

Note: Θ(0, ω, α) = Θ(ω, α), as defined in (5). By a Borel-Contelli argument, Θ(y1 , ω, α) has measure zero. Our key technical statement is the following: Lemma 1. Let λ and ω be as in Theorem 1. Suppose θ ∈ Θk (y, ω, α)C . Then if

∈ (0, 12 γ(M (E))) and 1 < α < 2, ∃k2 (θ, ω, y, , α, E) : ∀k > k2 (θ, ω, y, , α, E), if x and y are both (γ(M (E)) − , k, E, θ)-singular and |x − y| > 34 k, then |x − y| > (k − 2)α .

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109

Remarks 1. In Lemma 1, the y dependence of k2 comes entirely through k5 (y, θ, ω). (See the discussion after (20).) 2. The E-dependence of k2 comes entirely through k3 ( , E, ω). (See Lemma 2.) Proof. We start the proof with a series of statements given in Lemmas 2-7. In Lemmas 2-4, we find a uniform upper bound on |Pk (θ, E)|. Applying Lemma 2, we show in Lemma 6 that if y is (γ(M (E)) − , k, E, θ)-singular and x is within a certain range of distance from y, then |Pk (θ + (x− 1)ω)| is unusually small. Finally, we show that if two points y1 and y2 are relatively close together and they are both (γ(M (E)) − , k, E, θ)-singular for some θ, then |Pk (θ)| is uniformly small. Taking this in combination with the lower bound found in (10), for large enough k, we obtain a contradiction. Thus singular points can occur in small clusters, but the clusters must be far apart. First, we wish to bound n1 ln g(θ, n, E) from above uniformly in θ. This bound can actually be made uniform in E as well and that is important for our proof of strong dynamical localization. We will prove an E-independent bound in Section 3. We need to show that 1 for all θ ∈ T. (14) lim sup ln g(θ, k, E) ≤ γ(g(E)) k−→∞ k This result was obtained in [8] for transfer matrices associated with almost periodic Schr¨ odinger operators (for all θ, but a.e. θ is sufficient for our purposes). Examining the proof of [8] shows that it applies to any quasiperiodic cocycle which is analytic in E. Thus (14) holds. Next, in order to bound |Pn (θ, E)|, we will prove Lemma 2. For every E ∈ R, ω irrational, and > 0, there exists k3 ( , E, ω) such that for all n > k3 ( , E, ω), |Pn (θ, E)| ≤ en(γ(g(E))+) = en(γ(M(E))+C(λ)+) for all θ∈T Proof. From (14) for every > 0 and k > K(θ, , E, ω) 1 ln g(θ, k, E) ≤ γ(g(E)) +

k

for a.e. θ ∈ T

and, by (6), |Pk (θ, E)| ≤ g(θ, k, E) ≤ ek(γ(g(E))+)

for a.e. θ .

(15)

To show that K can be chosen uniformly in θ, we will apply the following lemma

k j Lemma 3. ([16]) Let f (z) = j=0 cj z be an arbitrary kth degree polynomial. k Suppose |f (z0 )| ≥ a for some a > 0, z0 ∈ [−1, 1]. Then for any 0 k0 ( ab )) |{θ ∈ (0, 1) : |f (cos 2πθ)| < bk }| ≤ c(a, b) < 1 .

110



To apply Lemma 3, we must show that Pk (θ, E) can be written as a polynomial in cos θ . Lemma 4. 1) Pk (θ, E) is an even function around θ = −(k + 1)/2. 2) It can be written as a kth degree polynomial in cos 2π(θ +

k+1 2 ω).

Proof. Define AS to be the operation which first permutes the rows of A so that they are in reverse order and then similarly permutes the columns of A. This operation requires an even number of permutations, so det AS = det A.

k+1

ω, E = det E − H−θ− k+1 ω,λ,ω

Pk −θ − 2 2 [1,k] S

= det E − H−θ− k+1 ω,λ,ω

2 [1,k]

= det E − Hθ− k+1 ω,λ,ω

2 [1,k] k+1 ω, E = Pk θ − 2 which completes part 1. From part 1, Pk (θ, E) =

∞

aj exp(2πiθ )

j=−∞

where θ = θ + k+1 2 ω and aj = a−j ∈ R. It remains to show that aj = 0 for |j| > k. From (3) and (4), 2n − k − 1 k+1 ω + ω v(n) = 2 cos 2π θ+ 2 2 = b1 (n) exp 2πiθ + b2 (n) exp −2πiθ and c(n) = λ2 + b3 (n) exp 2πiθ + b4 (n) exp −2πiθ where b1 (n), b2 (n) ∈ R and b3 (n), b4 (n) ∈ C. Taking the determinant of

E − Hθ− k+1 ω,λ,ω

, we can write [1,k]

2

Pk (θ, E) =

k

b(n) exp 2πinθ

n=−k

where b(n) ∈ R.

Let k+1 k+1 ω := Q cos 2π θ + ω . (16) am cos 2π θ + Pk (θ, E) = 2 2 m=0 k

m

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From now on, for sets, we will use | · | for Lebesgue measure. Let An = {θ ∈ (0, 1) : ∀k > n, |Q(cos 2πθ)| ≤ exp (γ(g(E)) + 2 )k}. By (15) and (16), |AC n| goes to zero as n goes to infinity. Let N be such that for all k ≥ N , |AC | n < 1 (1 − c(exp[γ(g(E)) +

], exp[γ(g(E)) + ])) 2 2 Assume there exists a θ, such that |Q(cos 2πθ)| > exp k(γ(g(E))+ ) for some k > N . Then by Lemma 3

θ ∈ (0, 1) : |Q(cos 2πθ)| < exp k γ(g(E)) +

2

≤ c exp [γ(g(E)) + ] , exp γ(g(E)) + 0. |AC | > 1 − c exp [γ(g(E)) +

] , exp γ(g(E)) + n 2 This contradiction proves the lemma.

Thus

Remark. As far as more general models are concerned, the proof above uses Lemma 3, and therefore only holds for polynomial v and c. An alternate proof, that can be applied to any continuous v and under very mild restrictions on c, is given in Section 3. However, it requires an exclusion of an additional countable set of frequencies. In order to obtain bounds for the Green’s function, we will need to bound the product c(1) . . . c(n) as given in (6). We will use the following lemma. Lemma 5. ([16]) If f (x) is an analytic function, z ∈ [min f, max f ] and p(f ) the maximum number of times f assumes any value on [0, 1), then for all > 0 there exists a N ( ) such that for n > N ( ) and, if desired, any ∈ [0, . . . , n − 1] n−1

ln |z − f (θ + jω)| ≤ n

j=0 j= n−1

0

ln |z − f (θ + jω)| ≥ n

j=0 j=

+ p(f )D(ω)n

1

1

0

1−r(ω)−1

ln |z − f (θ)|dθ +

(17)

ln |z − f (θ)|dθ −

ln n

min

j=0...n−1 j=

(18)

|z − f (θ + jω)|

.

By (17), for all > 0, there exist k4 ( ) such that if b − a > k4 ( ) ln

b−1 j=a

|c(j)| ≤ (b − a)

0

1

1 ln |c(a)|dθ +

18

= (b − a)(C(λ) +

1

) . 18

(19)

The next lemma shows that an (m, k, E, θ)-singular point “produces” many phases such that, for fixed E, |Pk (θ, E)| is “abnormally” small.

112



Lemma 6. If y ∈ Z is (γ(M (E)) − , k, E, θ)-singular, < 12 γ(M (E)) and k > 1 9 max [k3 ( 18

, E, ω), k4 ( ), 8], then for any x such that y−[ 78 k] ≤ x ≤ y−[ 87 k]+[ 34 k], |Pk (θ + (x − 1)ω)| ≤ exp{kγ(M (E)) + (k − 1)C(λ) −

1 k } . 18

Proof. Let x1 = x and x2 = x + k − 1. If y ∈ [x1 , x2 ] is (γ(M (E)) − , k, E, θ)singular, one of the following is true: 1. G[x1 , x2 ] does not exist or E ∈ σ(H[x1 , x2 ]). Thus Pk (θ + (x1 − 1)ω) = 0 and the lemma holds. 2. y − x1 or x2 − y < 19 k. However, k/9 < k − [7k/8] − 1 < x2 − y and k/9 < [7k/8] − [3k/4] < y − x1 since k > 72. 1 3. |G[x1 , x2 ](y, x1 )| ≥ exp − (γ(M (E)) − ) k 9 1 or |G[x1 , x2 ](y, x2 )| ≥ exp − (γ(M (E)) − )k . 9 By a straightforward computation using Cramer’s rule, if x1 < y < x2 , then

Px2 −y (θ + yω) y−1

|G[x1 , x2 ](y, x1 )| =

|c(j)| and Pk (θ + (x1 − 1)ω) j=x 1

2 −1

Py−x1 (θ + (x1 − 1)ω) x

|c(j)| . |G[x1 , x2 ](y, x2 )| =

Pk (θ + (x1 − 1)ω) j=y Thus, |Px2 −y (θ + yω)| exp

1 (γ(M (E)) − )k 9

|Py−x1 (θ + (x1 − 1)ω)| exp

y−1

|c(j)| ≥ Pk (θ + (x1 − 1)ω) or

j=x1

1 (γ(M (E)) − )k 9

x 2 −1

|c(j)| ≥ Pk (θ + (x1 − 1)ω) .

j=y

Since k > 9k3 ( /18, E, ω), we have y − x1 and x2 − y > k3 ( /18, E, ω), so by Lemma 2, 1 |Px2 −y (θ, E)| ≤ exp (x2 − y) γ(M (E)) + C(λ) +

18 8 1 ≤ exp kγ(M (E)) + (x2 − y)(C(λ) + ) 9 18 and similarly, |Py−x1 (θ, E)| ≤ exp

8 1 kγ(M (E)) + (y − x1 )(C(λ) + ) 9 18

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and since y − x1 and x2 − y > 9k4 , we have by (19) y−1 1 |c(j)| ≤ exp(y − x1 ) C(λ) +

18 j=x 1

x 2 −1 j=y

1 |c(j)| ≤ exp(x2 − y) C(λ) + . 18

7 Next, let y1 and y2 be (γ(M (E)), k, E, θ)-singular, y2 > y1 , ηi = yi − 8 k for i = 1, 2, d = y2 − y1 ≥ 34 k and θ + (η1 + 34 k − 13 k + k−1 0, . .. , [ 13 k] 2 + j)ω for j = θj = (20) 3 k−1 θ + (η2 + 4 k − 2 + j)ω for j = 13 k + 1, . . . , k Combining terms, we get the desired result.

where [x] is the greatest integer less than or equal to x. Since y2 − y1 ≥ 34 k, phases θj , for j = 1, . . . , k, are all distinct. Note that 3/4 here can be improved to any number bigger than 2/3. Additionally, note that cos 2πθm = cos 2πθn for 0 ≤ m, n ≤ k, m = n and k large enough. Indeed, cos 2πθm = cos 2πθn if and only if θm ± θn = 0 or 1. From our definition of θj , θm + θn is of the form 2θ + ω and θm − θn is of the form p ω, where , p ∈ Z. We can eliminate the second 1−ω case by ω ∈ / Q. Thus we should take care of the cases θ = ω 2 and θ = 2 . If ω 1−ω θ ∈ / { 2 , 2 } for ∈ Z, set k5 (y1 , θ, ω) = 1. Otherwise, let k5 (y1 , θ, ω) be the 1 7 −θ 1−2θ smallest k such that η1 + 34 k − 13 k + k+3 2 ≥ y1 + 24 k − 8 > max[ ω , 2ω ]. Then for k > k5 (y1 , θ, ω), we have, by a straightforward computation, that cos θj , for j = 0, . . . , k are distinct. Our choice of θj guarantees that Lemma 6 can be applied. Thus recalling (16)

k + 1

1

ω) < exp kγ(M (E)) + (k − 1)C(λ) − k

|Q (cos 2π (θj ))| = Pk (θj − 2 18 (21) for j = 0, . . . , k. Now we write Qk (z) in the Lagrange interpolation form using points cos 2πθ0 , . . . , cos 2πθk :

k (z − cos 2πθ ) =j

. (22) Qk (cos(2πθj ) |Qk (z)| =

j=0 =j (cos 2πθj − cos 2πθ )

Recalling (12), Lemma 7. Suppose d < k α , for some α ∈ (1, 2), θ ∈ Θk (y1 , ω, α)C , and ω Diophantine. Then for any > 0 there exists k6 ( , ω, α) such that for k > k6 ( , ω, α), for any z ∈ [−1, 1], | =j (z − cos 2πθ )| ≤ exp k . | =j (cos 2πθj − cos 2πθ )|

114



Proof. Applying Lemma 5 for k > N ( 3 ) |I1 | : = ln | (z − cos 2πθ )| =j

≤ = and |I2 | :

= ln |

1

(k + 1) ln |z − cos 2πθ|dθ + 3 0

(k + 1) − ln 2 + 3

(cos 2πθj − cos 2πθ )| =

=j

ln | cos 2πθj − cos 2πθ |

=j

ln | cos 2πθj − cos 2πθ|dθ − 3 0     −1 +2D(ω)(k + 1)1−r(ω) ln(k + 1) min | cos 2πθj − cos 2πθ | =0...n  =j

= (k + 1) − ln 2 − 3     −1 +2D(ω)(k + 1)1−r(ω) ln(k + 1) min | cos 2πθj − cos 2πθ | . =0...n  ≥ (k + 1)

1

=j

Here the integrals were computed using the Poisson-Jensen formula. We want to bound from below min | cos 2πθj − cos 2πθ | = 2 min | sin π(θj + θ )|| sin π(θj − θ )| . We have

 2([ 3 k]−[ 78 k]−[ 13 k])+k−1+j+  θ + y1 + 4 ω  2   2(d+[ 34 k]−[ 78 k])−k+1+j+ θ + y1 + ω 2 θj + θ = 2 2([ 34 k]−[ 78 k])−[ 13 k]+d+j+   ω θ + y1 +  2 

j, ∈ {0, . . . , [ 13 k]}

j, ∈ {[ 13 k] + 1, . . . , k} j ∈ {0, . . . , [ 13 k]}, ∈ {[ 13 k] + 1, . . . , k} .

Since θ ∈ Θ(y1 , ω, α)C , we have for all j, ∈ {0, . . . , k} (2r(ω))−1 3 | sin π(θj + θ )| > exp − 2d + k + 1 . 4 Also, θj − θ =

   

(j − )ω

(−d − [ 13 k] + k − 1 + j − )ω   

j, ∈ {0, . . . , [ 13 k]} or {[ 31 k] + 1, . . . , k} j ∈ {0, . . . , [ 13 k]}, ∈ {[ 31 k] + 1, . . . , k}

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Since ω is Diophantine | sin π(θj − θ )| > b(ω)

1 2 k− 3 3

−r(ω) .

So, min | cos 2πθj − cos 2πθ | ≥ 2b(ω)

=0...n =j

Combining terms, |I1 | |I2 |

≤

=

2 1 k− 3 3

−r(ω)

(2r(ω))−1 3 exp − 2d + k + 1 . 4

−1 2 (k + 1) + 2D(ω)(k + 1)1−r(ω) ln(k + 1) 3 (2r(ω))−1 −r(ω) 2 3 1 × 2d + k + 1 k− − ln 2b(ω) 4 3 3 2 (k + 1) + o(k) ≤ exp k

exp 3

exp

for k > k6 ( , ω, α).

We are now ready to finish the proofs of Lemma 1 and Theorem 1. Proof of Lemma 1. Let θ ∈ Θk (y, ω, α)C . Set kˆ = max[9k3 ( /18, E, ω), 9k4( ), k5 (y1 , θ, ω), k6 ( /36, ω, α, θ), 72]. [kγ(M(E))+(k+2)C(λ)] ˆ ≥ e √ Pick k ∈ K such that k > kˆ and θˆ such that |Pk (θ)| . Let 8 2 α zˆ = cos 2π(θˆ + k+1 2 ω). Assume d < k . Then, by (21) and Lemma 7

e[kγ(M(E))+(k+2)C(λ)] √ z )| ≤ (k + 1)ekγ(M(E))+(k−1)C(λ)−k/36 . ≤ |Qk (ˆ 8 2 There exists a k7 ( ), such that for k > k7 ( ) this statement is contradictory. So ˆ k7 ] and k ∈ K, if x and y are both (γ(M (E)) − , k, E, θ)-singular for k > max[k, and |x − y| > 34 k, then |x − y| > k α . We would like to eliminate the condition ˆ k4 ] + 2 and |x − y| > 3 k > 3 (k − 1) > 3 (k − 2). One k ∈ K. Thus let k > max[k, 4 4 4 ˆ k7 ] + 2 of k − 2, k − 1, k ∈ K. So at worst, |x − y| > (k − 2)α . Letting k2 = max[k, completes the proof. By a Borel-Cantelli argument, we obtain as an immediate corollary Lemma 1a. Let λ and ω be as in Theorem 1. Suppose θ ∈ Θ(y, ω, α)C . Then if

∈ (0, 12 γ(M (E))) and 1 < α < 2, ∃k2 (θ, ω, y, , α, E) : ∀k > k2 (θ, ω, y, , α, E), if x and y are both (γ(M (E)) − , k, E, θ)-singular and |x − y| > 34 k, then |x − y| > (k − 2)α .

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Proof of Theorem 1. Let E(θ) be a generalized eigenvalue of Hθ,λ,ω with ψ(E, x) the corresponding eigenfunction. Assume without loss of generality that ψ(E, 0) = 0. Let 0 < < 12 γ(M (E)), |x| > max[k1 (E,γ(M (E)) − ,θ,0), k2 (θ,ω,0, ,1.5,E),2] and 1.5 . By Lemma k = |x|. Thus 0 is (γ(M (E)) − , k, E, θ)-singular and 3k 4 < |x| < k 1a, x must be (γ(M (E)) − , k)-regular. So there exists an interval [x1 , x2 ] where |G[x1 , x2 ](x, x1 )| |G[x1 , x2 ](x, x2 )| with x − x1 ≥

1 k 9

1 < exp − (γ(M (E)) − )k 9 1 < exp − (γ(M (E)) − )k 9 and

x2 − x ≥

and

1 k. 9

Applying (11) |ψ(x)| ≤ |c(x1 − 1)||ψ(x1 − 1)||G[x1 , x2 ](x, x1 )|

+ |c(x2 )||ψ(x2 + 1)||G[x1 , x2 ](x, x2 )|

≤ C |x1 |e−(γ(M(E))−)k/9 + |x2 |e−(γ(M(E))−)k/9

≤ Ce−{|x|(γ(M(E))−)/10}

for large x.

3 Strong dynamical localization We will use the following Definition 2. An ergodic family H(θ), θ ∈ Θ acting on a Hilbert space H is strongly dynamically localized if, for any q > 0 and initial state ψ ∈ H, ψ = 1, that decays faster than any polynomial, there exists a constant C(q, ψ) < ∞ such that supexp{−iHt}ψ, |X|q exp{−iHt}ψ dP ≤ C(q, ψ) (23) Θ

t

where X is the position operator and P is the probability measure. Remark. Dynamical localization (as opposed to strong dynamical localization) is defined by the same inequality without the integration and holds for almost every θ ∈ Θ. Theorem 2. Assume ω is Diophantine. 1) If λ2 < 1, λ1 + λ3 < 1, λ1 = λ3 , then the family (Hθ,λ,ω )θ∈T is strongly dynamically localized. 2) If λ1 = λ3 < 1/2, λ2 < 1, the same statement is true under the additional condition that −λ2 1 mod 1 S, (24) ωk = cos−1 π 2λ1 for all k ∈ Z\{0}.

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For the proof of strong dynamical localization, we will need to make some of our estimates uniform in energy. For this purpose, we will first establish certain more general continuity results. Given a function c(θ) ∈ C(T) with at most countably many zeros, we will define the countable set Ac ⊂ T by the following rule. Let c0 , . . . , ck , . . . be the zeros of c(θ). We set Ac = {ω : ω = ci − cj (mod 1) for some , i, j ∈ N}. Lemma 8. Let v(θ) and c(θ) in (2) and (3) be in C(T), c(θ) have at most countably many zeros and ω ∈ / Ac . Then ln g(θ, n, E) ∈ C(T). / Ac leads to an additional condition only Remark. For Hθ,λ,ω , the assumption ω ∈ when λ1 = λ3 . The condition is precisely the one given in (24), as can be obtained by a simple computation. Proof. Since ln g(θ, n, E) is a positive continuous function, the proof reduces to showing that g(θ, n, E) = 0 for all θ. For fixed θ and ω, the sequence c(n, θ, ω) can have two or more zeros only if ω ∈ Ac . We will show that g(θ, n, E) = 0 if c(n, θ, ω) has one or no zeros. For fixed θ and ω, we will use c(n), v(n) for c(n, θ, ω), v(n, θ, ω). We will use the following lemma. Lemma 9. 1) If Pn (θ + kω, E) = Pn−1 (θ + kω, E) = 0, then ∃ k + 1 ≤ j < n + k such that c(j)=0 and P (θ + kω, E) = 0 for ≥ j − k. 2) If Pn (θ, E) = Pn−1 (θ + ω, E) = 0, then ∃ 1 ≤ k < n such that c(k)=0 and Pn− (θ + ω, E) = 0 for ≤ k. Proof. Let Pn (θ + kω, E) = Pn−1 (θ + kω, E) = 0. Expanding the determinant Pn (θ + kω, E) in the final row, we get Pn (θ + kω, E) = Pn−1 (θ + kω, E)(E − v(k)) − |c(n + k − 1)|2 Pn−2 (θ + kω, E) . Thus either P (θ + kω, E) = 0 for all ∈ N, or |c(j)| = 0 for some k + 1 < j < n + k and P (θ + kω, E) = 0 for ≥ j − k. If P (θ + kω, E) = 0 for all ∈ N, then P1 (θ + kω, E) = E − v(k + 1) = 0 and P2 (θ, E) = |c(k + 1)|2 = 0. This completes part 1. Now assume Pn (θ, E) = Pn−1 (θ + ω, E) = 0. Expanding Pn (θ, E) in the first row, the argument follows exactly as part 1. We now complete the proof of Lemma 8. Assume g(θ, n) = 0 for some n and θ. Then by (6), Pn (θ, E) = 0 and either Pn−1 (θ, E) = 0 or c(n) = 0. Case 1. If Pn−1 (θ, E) = 0, then by part one of Lemma 9, c(j) = 0 for some 1 ≤ j < n, and Pk (θ, E) = 0 for k ≥ j. By (6) and the fact that there is at most one zero of c(), Pn−1 (θ + ω, E) = Pn−2 (θ + ω, E) = 0. By Lemma 9, part 1, c(˜j) = 0 for some 2 ≤ ˜j < n and P (θ + ω, E) = 0 for ≥ ˜j − 1. Since there is only one zero, 1 < j = ˜j < n − 1. Note particularly that c(1) = 0. We now know

118



Pj (θ, E) = Pj−1 (θ + ω, E) = 0. Thus by Lemma 9, part 2, c(p) = 0 for some p < j which is a contradiction. Case 2. If c(n) = 0, then by (6), Pn−1 (θ + ω, E) = 0. As above this implies c(j) = 0 for some j < n, a contradiction. We will also need another continuity statement. Theorem 3. ([17]) Let v(θ) in (2) and (3) be real analytic and c(θ) in (2) and (3) be analytic such that T |c(θ)| dθ < ∞. Let ω be Diophantine. Then γ(M (E)) = limn→∞ n1 γ(g(E)) − C(λ) is continuous in E. Lemma 8 and Theorem 3 will be an important ingredients in the uniformity results that follow. Lemma 10. Let v, c and ω be as in Lemma 8. For any > 0 and for any interval I there exists L( , ω, I) such that for any k ≥ L( , ω, I), E ∈ I and θ ∈ T, one has 1 ln g(k, θ, E) < γ(g(E)) + . k Remark. In the case of Hθ,λ,ω , we could use the same technique as in Lemma 2 to bound k1 ln g(k, θ, E) uniformly in θ. That proof, however, does not apply to more general c(θ) and v(θ). Proof. We will start with the following theorem. Theorem 4. ([12]) Let {fn } be a continuous subadditive ergodic process on a uniquely ergodic system (X, µ, T ), i.e., fn ∈ C(X) and fn+m (x) ≤ fn (x)+fm (T n x) for all x ∈ X. Then for every x ∈ X and uniformly on X: lim sup n−→∞

1 fn (x) ≤ γ(f ). n

Let T θ = θ + ω. Then (T, µ, T ) is a uniquely ergodic system and ln g(θ, n) is a subadditive process. Thus by Lemma 9 and Theorem 4 , for every θ ∈ T and uniformly on T, lim sup n−→∞

1 ln g(θ, n) ≤ γ(g(E)) . n

(25)

Remark. Clearly, (25) also implies that for all > 0 there exists k( , E, ω) such that for all n ∈ N , n > k( , E, ω) 1 1 ln |Pn (θ, E)| ≤ ln g(θ, n) ≤ γ(g(E)) + . n n Next we show a uniform bound on ln g(θ, n) in E. First note that by subadditivity, the subsequence {supx 21n ln g(θ, 2n )} is monotone decreasing. Let I be

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a compact set containing σ(H). Recalling Theorem 3 and Lemma 8, we can apply Dini’s theorem on I to the subsequence above. Thus given > 0, there exists k0 such that for all k > k0 sup x

1

ln g(θ, 2n ) < γ(g(E)) + . 2n 2

Now let R > 2 be written in binary expansion R = 2k + · · · + 2k0 + M where M < 2k0 . Then, by subadditivity k0

sup x

1 ln g(θ, R) R

For large enough R,

2k 1 1 2k0 sup k ln g(θ, 2k ) + · · · + sup ln g(θ, 2k0 ) R x 2 R x 2k0 1 M ln g(θ, M ) + sup R x M 2k + · · · + 2k0

C < γ(g(E)) + + . R 2 R

≤

C R

6 .1 If y ∈ Z is (γ(M (E))

, ω, I(H)), k4 ( ), 8 , then for any x such that y − [ 87 k] ≤ x ≤ y − [ 87 k]+ 9 max L( 18 [ 34 k], 1 |Pk (θ + (x − 1)ω)| ≤ exp kγ(M (E)) + (k − 1)C(λ) − k . 18

While it is possible to use sets Θk (y, ω, α) as defined in (12) for our proof, we will modify them in order to somewhat simplify the argument. Define for any s > α and k ≥ 1

1 j α

!

Θk (y, ω, α) := θ ∈ T : ∃|j| ≤ 3k , sin 2π θ + y + ω ≤ s . 2 k ! k (y, ω, α)C ⊂ Θk (y, ω, α)C , so Lemma 7 holds on this set. Θ

120


One easily shows

!

Θk (y, ω, α) ≤

C k s−α


.

In our proof of Lemma 1, k2 depended on the choice of singular point y1 . (See remarks after Lemma 1.) In the proof of strong dynamical localization, we wish to eliminate this dependence. We omit the measure zero set 1 − 2θ −2θ or ∈Z . Φ= θ: ω ω This removes the need for constraint k5 (y1 , θ, ω) and the dependence on y1 for our definition of k2 . The bound k2 in Lemma 1 becomes k2 ( , α, I(H)) = max[9L( /18, ω, I(H)), 9k4 ( /18), k6 ( /36, ω, α), k7 ( )] + 2. By Theorem 3, γ(M (E)) is continuous. So on I(H), it has a minimum, γ0 > 0. It follows from definition that for 0 < < 12 γ(M (E)), (γ(M (E)) − , k, E, θ)regularity implies ( 12 γ0 , k, E, θ)-regularity. So we can reformulate Lemma 1 as # "k (y, ω, α)C ΦC , 1 < Lemma 1b. Let λ and ω be as in Theorem 2. Suppose θ ∈ Θ ! ! α < 2, and s > α. Then ∃L(ω, α, s, I(H)) : ∀E ∈ I(H) and ∀k > L(ω, α, s, I(H)), if 34 k < |x − y| ≤ (k − 2)α , then either x or y is ( 12 γ0 ,k)-regular. The rest of the proof closely follows the argument of [13]. By Theorem 1, Hθ,λ,ω has pure point spectrum with exponentially decaying # eigenfunctions for all θ ∈ Θ(y)C ΦC#, a set of full measure. Let ϕ[θ, E] be the orthonormal eigenfunctions on Θ(y)C ΦC with corresponding eigenvalues E[θ]. Define ϕ[θ, E] ϕ[θ, ! E] = (26) Bϕ[θ, E]2 where B is the operator of multiplication by b(x) := (1 + |x|)−δ and δ > 1/2. Notice that uniformly in θ and in energy E[θ] for all x ∈ Z, |ϕ[θ, ! E](x)| ≤ (1 + |x|)δ .

(27)

Clearly |ϕ[θ, ! E](x)| are bounded functions of x. However, what we need here is a uniform bound in (E, θ) as given in (27). ! Lemma 11. Pick k > L(ω, α, s, I(H)) and y ∈ Z.

"k (y,ω,α)C # ΦC , 1) There exists a constant K1 (λ, δ, α, γ0 ) < ∞ such that ∀θ ∈ Θ E = E[θ] and all x ∈ Z such that 34 k < |x − y| < (k − 2)α , we have γ0 k . |ϕ[θ, E](x)||ϕ[θ, E](y)| ≤ K1 (λ, δ, α, γ0 )Bϕ[θ, E]2 (1 + |y|)2δ exp − 20 # 2) There exists a constant K2 (λ, δ, α, γ0 ) < ∞ such that ∀θ ∈ Θk (y)C ΦC γ0 k sup |exp{−iH[θ]t}δx, δy | ≤ K2 (λ, δ, α, γ0 )(1 + |y|)2δ exp − . 20 t

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Proof. By Lemma 1b, either x or y is ( γ20 , E, θ, k)-regular. Let u = x or y be the regular point and v the other one. Let [x1 , x2 ] be the interval such that |G[x1 , x2 ](y, xi )| < exp −γ0 k/18) where i = 1, 2. Note that x2 = x1 + k − 1. Then ϕ[θ, E](u)

=

−c(x1 − 1)ϕ[θ, E](x1 − 1)G[x1 , x2 ](u, x1 ) −c(x2 )ϕ[θ, E](x2 + 1)G[x1 , x2 ](u, x2 ) .

We have using (4), (27) and the definition of regularity |ϕ[θ, ! E](u)| ≤

≤ ≤

|c(x1 − 1)| |ϕ[θ, E](x1 − 1)||G[x1 , x2 ](u, x1 )| Bϕ[θ, E]2 |c(x2 )| + |ϕ[θ, E](x2 + 1)||G[x1 , x2 ](u, x2 )| Bϕ[θ, E]2 $ % γ0 k (λ1 + λ2 + λ3 ) (1 + |x1 − 1|)δ + (1 + |x1 + k|)δ exp − 18 γ0 k δ δ . C(λ, δ)(1 + k) (1 + |u|) exp − 18 −

Therefore, using (27) and the fact that |x − y| < (k − 2)α , we have |ϕ[θ, ! E](x)ϕ[θ, ! E](y)| ≤ ≤

γ0 k C(λ, δ)(1 + k)δ (1 + |x|)δ (1 + |y|)δ exp − 18 k γ 0 K1 (λ, δ, α, γ0 )(1 + |y|)2δ exp − . 20

Substituting (26), we obtain statement 1 of the Lemma. Next, in order to bound the integrand of (23), we find sup |exp{−iH[θ]t}δx, δy | ≤ t

|ϕE [θ, E](x)ϕE [θ, E](y)|

E∈{E[θ]}

≤ K1 (λ, δ, α, γ0 )(1 + |y|)2δ e−

γ0 k 20

Bϕ[θ, E]2

E∈{E[θ]}

≤ K1 (λ, δ, α, γ0 )(1 + |y|)2δ e

γ0 k − 20

2δ −

= K2 (λ, δ, α, γ0 )(1 + |y|) e

γ0 k 20

b2 .

Proof of Theorem 2. Pick α ∈ (1, 2), q > 0 and s > α(q + 1). Define a sequence ! L1 = L(ω, α, s, I(H)), Lj+1 = ( 43 Lj − 2)α . Then if Lj = 34 k, Lj+1 = (k − 2)α . # ! Lj (y, ω, α)C ΦC . By Lemma 11, for all j ≥ 1 and x and y such that Let = Θ

122


Lj < |x − y| ≤ Lj+1

& '

sup e−iH[θ] δx , δy dθ


& '

sup e−iH[θ] δx , δy dθ t∈R

& '

+ sup e−iH[θ] δx , δy dθ

=

T t∈R

C t∈R

≤ ≤ ≤

! Lj (y)| K2 (λ, δ, α, γ0 )(1 + |y|)2δ e−γ0 Lj /15 + |Θ C K2 (λ, δ, α, γ0 )(1 + |y|)2δ e−γ0 Lj /15 + s−α Lj 1 γ0 |x − y| α 2δ K2 (λ, δ, α, γ0 )(1 + |y|) exp − 20 1

+C1 {|x − y| α }−s+α since |x − y| ≤ Lj+1 = ( 43 Lj − 2)α ≤ ( 43 Lj )α . Thus there exists a constant K3 (λ, δ, α, γ0 , s) such that for all x, y ∈ Z

& '

γ 1 s

0 sup e−iH[θ] δx , δy dθ ≤ K3 (1 + |y|)2δ exp − |x − y| α + |x − y|− α +1 . 20 T t∈R (28) The rest of the argument follows exactly as in [13]. Suppose ψ ∈ 2 decays faster than any polynomial and ψ = 1, then ( ) q/2 −iHθ t −iHθ t q −iHθ t sup |X| e ψ dθ = sup e ψ, |X| e ψ, δx δx dθ T t∈R

T t∈R

=

sup

T t∈R

≤

x

≤

y

|x|q

x

|x|q |e−iHθ t ψ, δx |2 dθ

x

sup |e−iHθ t ψ, δx |dθ

T t∈R

|ψ(y)|

x

|x|

q

* +

sup e−iHθ t δy , δx dθ .

T t∈R

Applying equation 28, sup |X|q/2 exp{−iHθ t}ψ dθ T t∈R γ 1 s 0 |ψ(y)| |x|q (1 + |y|)2δ exp − |x − y| α + |x − y|− α +1 . ≤ K3 20 y x Since s > α(q + 1) the sum in x converges and since ψ decays faster than any polynomial, the sum in y converges.

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References [1] M. Aizenman, Localization at weak disorder: some elementary bounds, Rev. Math. Phys. 6, 1163–1182 (1994). [2] J. Avron and B. Simon, Almost periodic Schrodinger operators II. The integrated density of states, Duke Math. J. 50, 369–385 (1983). [3] J. Bellissard, C. Kreft, R. Seiler, Analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods. J. Phys. A 24 , 2329–2353 (1991). [4] Ju.M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Amer. Math. Soc., Providence, RI (1968). [5] J. Bourgain and M. Goldstein, On nonperturbative localization with quasiperiodic potential, Ann. of Math. 152, 825–879 (2000). [6] J. Bourgain and S. Jitomirskaya, Anderson localization for the band model. Geometric aspects of functional analysis, Lecture Notes in Math. 1745, 67–79, Springer, Berlin (2002). [7] J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potentials, J. Stat. Phys. 108, 1203–1218 (2002). [8] W. Craig and B. Simon, Subharmonicity of the Lyaponov index, Duke Math J. 50, 551–560 (1983). [9] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators with Applications to Quantum Mechanics and Global Geometry, SpringerVerlag, New York, (1987). [10] R. Del Rio, S. Jitomirskaya, Y. Last and B. Simon, Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69, 153–200 (1996). [11] K. Drese and M. Holthaus, Phase diagram for a modified Harper model, Phys. Rev. B 55, 693–696 (1997). [12] A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. Henri Poincaré 33, 797–815 (1997). [13] F. Germinet and S. Jitomirskaya, Strong dynamical localization for the almost Mathieu model, Rev. Math. Phys. 13, 755–765 (2001). [14] F. Germinet and A. Klein, Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222, 415–448.

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[15] D. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239–2249 (1976). [16] S. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator, Ann. Math. 150, 1159–1175 (1999). [17] S. Jitomirskaya, D.A. Koslover and M.S. Schulteis, Continuity of the Lyapunov exponent for quasiperiodic Jacobi matrices. Preprint, 2004. [18] S. Jitomirskaya and B. Simon, Operators with singular continuous spectrum III. Almost periodic Schr¨ odinger operators, Comm. Math. Phys. 165, 201–205 (1994). [19] Y. Katznelson, An Introduction to Harmonic Analysis, Dover Publications Inc., New York (1976). [20] P. Koosis, The Logaritmic Integral II, Cambridge University Press, New York (1992). [21] D.A. Koslover, Jacobi Operators with Singular Continuous Spectrum, submitted to Lett. Math. Phys. [22] Y. Last, Spectral theory of Sturm-Liouville operators on infinite intervals: a review of recent developments. Preprint, 2004. [23] V.A. Mandelshtam and S.Ya. Zhitomirskaya, 1D-Quasiperiodic operators. Latent symmetries, Comm. Math. Phys. 139, 589–604 (1991). [24] K. Petersen, Ergodic Theory, Cambridge University Press, New York (1997). [25] J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys. 224, 297–309 (2004). [26] D.J. Thouless, Bandwidths for a quasiperiodic tight-binding model, Phy. Rev. B 28, 4272–4276 (1983). S. Jitomirskaya(1), D.A. Koslover(1) and M.S. Schulteis(1,2) (1) Department of Mathematics University of California, Irvine Irvine, CA 92697, USA email: [email protected] email: [email protected] (2)

Deparment of Mathematics Concordia University, Irvine Irvine, CA 92612, USA email: [email protected] Communicated by Jean Bellissard submitted 7/07/04, accepted 22/07/04



The Aharonov-Bohm Solenoids in a Constant Magnetic Field Takuya Mine Abstract. We study the spectral properties of a two-dimensional magnetic Schr¨ odinger operator HN = ( 1i ∇ + aN )2 . The magnetic field is given by rot aN = B+ N j=1 2παj δ(z − zj ), where B > 0 is a constant, 1 ≤ N ≤ ∞, 0 < αj < 1 (j = 1, . . . , N ) and the points {zj }N j=1 are uniformly separated. We give an upper bound for the number of eigenvalues of HN between two Landau levels or below the lowest Landau level, when N is finite. We prove the spectral localization of HN near the spectrum of the single solenoid operator, when {zj }N j=1 are far from each are the same, and the boundary conditions at zj are other, all the values {αj }N j=1 uniform. We determine the deficiency indices of the minimal operator and give a characterization of self-adjoint extensions of the minimal operator.

1 Introduction 2 Let N = 1, 2, 3, . . . or N = ∞. Let {zj }N j=1 be points in R satisfying

R := inf |zj − zk | > 0 j=k

(the notation X := Y means X is defined to be Y ). Put SN := ∪N j=1 {zj }. Define a differential operator LN on R2 \ SN by 1 LN := ( ∇ + aN )2 , i

√ where i = −1 and ∇ = (∂x , ∂y ). We assume that aN = (aN,x , aN,y ) ∈ C ∞ (R2 \ SN ; R2 ) ∩ L1loc (R2 ; R2 ) and rot aN (z) := (∂x aN,y − ∂y aN,x)(z) = B +

N

2παj δ(z − zj )

(1)

j=1

in D (R2 ) (in the distribution sense), where z = (x, y) ∈ R2 , B, αj are constants satisfying B > 0 and 0 < αj < 1 for every j = 1, . . . , N. We find a proof of the existence of the vector potential aN satisfying above conditions in the paper of Arai [Ar] (see also [Me-Ou-Ro]). Define a linear operator

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LN on L2 (R2 ) by LN u = LN u, u ∈ D(LN ) = C0∞ (R2 \ SN ), where D(L) is the operator domain of a linear operator L and C0∞ (U ) is the space of the compactly supported smooth functions in an open set U . Then, the operator LN is symmetric and positive. Moreover, we will prove that the deficiency indices of LN are (2N, 2N ) (see Proposition 5.7). Thus the operator LN has self-adjoint extensions parametrized by (2N × 2N )-unitary matrices (see [Re-Si, Theorem X.2]). We denote by HN any self-adjoint extension of LN . In particular, we denote by AB the Friedrichs extension of LN , which is called the standard Aharonov-Bohm HN Hamiltonian. The Hamiltonian HN describes the motion of a non-relativistic charged particle moving in the Euclidean plane in the presence of a homogeneous magnetic field B plus a magnetic field concentrated on infinitesimally thin solenoids placed at the points zj with flux 2παj , provided that the mass m = 1/2, the Planck constant (divided by 2π) = 1 and the charge of an electron e = 1. When B = 0, the quantum mechanical system corresponding to HN is known to be a model which explains the Aharonov-Bohm effect ([Ah-Bo]), and is extensively studied from the view point of the scattering theory (e.g., [Rui], [Nam], [It-Ta1], [It-Ta2]), the spectral theory (e.g., [St1], [St2], [St3], [Ta], [Me-Ou-Ro]), and the theory of self-adjoint extensions of symmetric operators (e.g., [Ad-Te], [Da-St]). However, there seem to be few results in the case B > 0. Nambu [Nam] has AB and obtained an integral studied the standard Aharonov-Bohm Hamiltonian HN representation of eigenfunctions corresponding to the Landau level (2n − 1)B, for n = 1, 2, . . .. Particularly in the single solenoid case N = 1, Nambu has obtained all eigenvalues of H1AB and an integral representation of corresponding eigenfunctions. ˇ Exner, St’ov´ ıˇcek and Vytˇras [Ex-St-Vy] have given a complete characterization of the self-adjoint extensions H1 of L1 , written the eigenequation in terms of special functions, and solved it numerically. Their results about σ(H1AB ) (the spectrum of H1AB ) are summarized as follows: σ(H1AB ) =

∞

{(2n − 1)B} ∪ {(2n + 2α − 1)B}

n=1

and mult{(2n − 1)B ; H1AB } = ∞ mult{(2n + 2α − 1)B ; H1AB } = n

for n = 1, 2, . . . , for n = 1, 2, . . . ,

(2)

where mult{λ ; H} := dim Ker(λ − H). In this paper, we consider the following problems: (I) to investigate the spectrum of HN between the gaps of Landau levels, when B > 0 and N ≥ 2, (II) to

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127

give a complete characterization of self-adjoint extensions HN of LN , when B > 0 and N ≥ 2. These problems seem not to be studied yet. For the former problem, we obtain the following two results. First, we consider the finite solenoids case. In the sequel, PI (H) denotes the spectral projection of a self-adjoint operator H corresponding to an interval I. Theorem 1.1. Let 1 ≤ N < ∞. Then, the following holds: (i) For any self-adjoint extension HN of LN , we have mult{(2n − 1)B ; HN } = ∞

for n = 1, 2, 3, . . . .

AB (ii) For the standard Aharonov-Bohm Hamiltonian HN , we have AB ) = dim Ran P(−∞,B) (HN AB ) dim Ran P((2n−1)B,(2n+1)B) (HN

≤

0, nN

for n = 1, 2, 3, . . . .

(3)

(iii) For any self-adjoint extension HN of LN , we have dim Ran P(−∞,B) (HN ) ≤ 2N, dim Ran P((2n−1)B,(2n+1)B) (HN ) ≤ (n + 1)N

for n = 1, 2, 3, . . . .

Next, we consider the case where solenoids are far from each other and the physical situation around all solenoids are the same. To describe this situation rigorously, we prepare some definitions. Definition 1.1. Let w ∈ R2 . Let U be a simply connected open set, and V = U +w = {z + w; z ∈ U }. Let S be at most countable subset of U with no accumulation points in U and T = S + w. Let a ∈ C ∞ (U \ S; R2 ) ∩ L1loc (U ; R2 ) and b ∈ C ∞ (V \ T ; R2 ) ∩ L1loc (V ; R2 ) such that rot a(z) = rot b(z + w) holds in D (U ). By the Poincaré Lemma, we can prove that there exists an operator t−w given by t−w v(z) = Φ(z)v(z + w), Φ(z) ∈ C ∞ (U \ S), |Φ(z)| = 1 and satisfying p(a)t−w v = t−w p(b)v, L(a)t−w v = t−w L(b)v, where 1 ∇ + a, i L(a) = p(a)2 ,

p(a) =

1 ∇ + b, i L(b) = p(b)2 .

p(b) =

We call the operator t−w the magnetic translation operator from V to U intertwining L(b) with L(a). We denote the inverse operator of t−w by tw , that is, tw u(z) = Φ(z − w)u(z − w).

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Remark. The magnetic translation operators are introduced in the classical paper by Zak [Za] and discussed in detail by many authors; see, e.g., Arai [Ar] or Ge˘ıler [Ge]. In the sequel, χ and χj are functions satisfying 0 (|z| > R2 ) χ ∈ C0∞ (R2 ), 0 ≤ χ ≤ 1 on R2 , χ(z) = (4) 1 (|z| < R3 ), χj (z) := χ(z − zj ) for j = 1, . . . , N. Definition 1.2. Let HN be a self-adjoint extension of LN . We say the operator HN has uniform boundary conditions if the following two conditions hold: (i) There exists a constant α with 0 < α < 1 such that αj = α for every j = 1, . . . , N . (ii) There exists a self-adjoint extension H1 of L1 independent of j such that D(HN ) = u ∈ D(LN ∗ ); t−zj (χj u) ∈ D(H1 ) for every j = 1, . . . , N , (5) where t−zj be the magnetic translation operator from {|z − zj | < αj R 2 } intertwining LN with L1 (see (6)).

R 2}

to {|z|
0 and R0 > 0 dependent on B, α, I, H1 (independent of N, R) satisfying the following assertions: (i) If R ≥ R0 , we have σ(HN ) ∩ I ⊂

k

[λl − δ, λl + δ],

l=1 2

where δ = e−uR . (ii) If R ≥ R0 , we have dim Ran PI (HN ) = N dim Ran PI (H1 ). We shall try to give a physical interpretation of our results. In classical mechanics, an electron in a homogeneous magnetic field makes a cyclotron motion. In quantum mechanics, however, the energy of an electron is quantized by the wave property of an electron, and takes a value in Landau levels. If some solenoids are contained in the circle of the cyclotron motion, the Aharonov-Bohm effect causes the phase shift of the electron wave by e/ times the magnetic flux through solenoids in the circle. Thus the energy of the electron

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is obliged to change into a value in a Landau gap, in order to correct the phase shift. For this reason, the number of eigenstates of the single solenoid operator H1AB in the nth Landau gap is considered to be roughly estimated by the possible number of electrons with energy in the nth Landau level, in the circle of the Larmor radius centered at the position of the solenoid. This number can be calculated as follows. Under our normalization of physical constants, the cyclotron radius r of an electron with nth Landau energy (2n − 1)B equals (2n − 1)/B. It is known that the density of states (the number of eigenstates per unit area) for each Landau level is B/2π (see, e.g., [Nak, Proposition 15]). Thus, the number of possible eigenstates in the circle is 1 B =n− . πr2 × 2π 2 The difference between this estimate and the rigorous result (2) is only 12 . AB in the nth Landau gap is roughly Similarly, the number of eigenstates of HN estimated by the number of eigenstates with nth Landau energy in the union set, with respect to j = 1, . . . , N , of the disks of Larmor radius centered at zj . Each disk contains n eigenstates with nth Landau energy. Thus, we conclude that the AB in the nth Landau gap is bounded by nN . Our number of eigenvalues of HN Theorem 1.1 has justified this conclusion. Moreover, if solenoids are far from each other compared with the cyclotron radius, then the interaction between two disks can be ignored. Thus we conclude AB in the nth Landau gap equals to nN . Our that the number of eigenvalues of HN Theorem 1.2 combined with (2) has justified this conclusion. The proof of Theorem 1.1 (section 3.2) depends heavily on a perturbation argument of the canonical commutation relation (CCR) of the annihilation operator and the creation operator (section 3.1; Iwatsuka [Iw] has also used a similar argument). Since our magnetic fields have δ-like singularities, CCR formally holds with a δ-like perturbation. We interpret the δ-like perturbation as a difference of the boundary conditions at zj of two self-adjoint extensions HN appeared in CCR. Determining the operator domain of these operators explicitly (section 5.4) and using a known result by Deift (Lemma 3.2), we obtain Theorem 1.1. Theorem 1.2 is analogous to the result of Cornean and Nenciu ([Co-Ne, Theorem III.1, Corollary III.1]). The proof of Theorem 1.2 (section 4.2) is also analogous to that of their results. However, we need an additional assumption (Definition 1.2) to obtain the result, because of the non-essential self-adjointness of the minimal operator LN . In order to prove Theorem 1.1 and Theorem 1.2, we need a detailed information about the self-adjoint extension of LN . For this reason, we shall give a complete characterization of self-adjoint extensions of LN (Section 5.3). Although there are many results about the self-adjoint extension of Schr¨ odinger operators with singular potentials (e.g., [Al-Ge-Ho-Ho], [Ad-Te], [Da-St], [Ex-St-Vy]), this problem is considered to be rather difficult when N ≥ 2, because of the difficulty

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of determining the deficiency subspaces Ker(LN ∗ ∓ i) explicitly. The main strategy is due to the locality of self-adjoint extensions, which has been pointed out by Bulla and Gesztesy ([Bu-Ge]) in the δ-potential case. Combining their idea with the gauge transformation technique and using the result in the single solenoid case ([Ex-St-Vy]), we obtain our result (Theorem 5.11).

2 Preliminaries We shall prepare some notations used in later sections. We denote the Landau operator (the Schr¨ odinger operator with a constant magnetic field) by L0 , that is, 2 1 B B ∇ + a0 , a0 = − y, x . L0 = i 2 2 It is known that the operator L0 |C0∞ (R2 ) is essentially self-adjoint (see [Ik-Ka], [Le-Si]). We denote by H0 the unique self-adjoint extension of L0 |C0∞ (R2 ) . When we consider the single solenoid operator L1 , we always assume z1 = 0 and take the radial gauge, that is, 2 1 α α B B ∇ + a y − x + = , a (z) = − y, x . (6) L1 = Lα 1 1 1 i 2 |z|2 2 |z|2 The upper suffix α is sometimes used to indicate the value α1 = α explicitly α (similarly, we sometimes denote the self-adjoint extension of Lα 1 by H1 ). ∗ We regard the operator domain D(LN ) as a Hilbert space equipped with the graph inner product and norm (u, v)N = (LN u, LN v) + (u, v), u 2N = (u, u)N , u, v ∈ D(LN ∗ ),

where (u, v) = R2 u¯vdxdy. Notice that the functions LN u and LN v belong to L2 (R2 ) (see (i) of Proposition 5.4 below). We regard D(HN ) and D(LN ) as closed subspaces of D(LN ∗ ). For a Hilbert space H and a closed subspace H of H, we denote H/H the quotient Hilbert space equipped with the norm

[u] = P u H, u ∈ H, where [u] is the equivalence class of u and P is the orthogonal projection onto (H )⊥ .

3 Finite solenoids case 3.1

Perturbation of the canonical commutation relation

Define differential operators ΠN,x , ΠN,y , AN , A†N by ΠN,x := 1i ∂x + aN,x , ΠN,y := 1i ∂y + aN,y , AN := iΠN,x + ΠN,y , A†N := −iΠN,x + ΠN,y .

(7)

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Define linear operators AN , A†N on L2 (R2 ) by AN u := AN u, A†N u := A†N u,

D(AN ) := C0∞ (R2 \ SN ), D(A†N ) := C0∞ (R2 \ SN ).

By the assumption (1), we have the following lemma. Lemma 3.1. The following operator relations hold: (i) A†N AN = LN − B, AN A†N = LN + B. ∗ (ii) A†N ⊂ AN ∗ , AN ⊂ A†N . We review some properties of a pair of operators XX ∗ and X ∗ X. Lemma 3.2. Let X be a densely defined closed linear operator on a Hilbert space H. Then, we have the following: (i) The operators X ∗ X and XX ∗ are self-adjoint. (ii) The operator (XX ∗ )|(Ker XX ∗ )⊥ is unitarily equivalent to the operator (X ∗ X)|(Ker X ∗ X)⊥ .

Proof. (i) See [Re-Si, Theorem X.25]. (ii) See [De, Theorem 3]. Applying Lemma 3.2 to our operators, we have the following. Proposition 3.3. ∗

AB AB (i) We have A†N A†N = HN + B, AN ∗ AN = HN − B.

∗

− − (ii) There exists a self-adjoint extension HN of LN such that A†N A†N = HN − B. AB (iii) The inequality HN ≥ B holds in the form sense. ∗

Proof. (i) By Lemma 3.1, we have A†N A†N ⊃ AN A†N = LN + B. By Lemma 3.2, ∗

we have A†N A†N is self-adjoint. Thus, there is a self-adjoint extension X of LN ∗

AB such that A†N A†N = X + B. To show X = HN , it is sufficient to show that these 2 ∞ operators have a common form core C0 (R \ SN ). This fact follows from the proof of [Re-Si, Theorem X.25] and the definition of the Friedrichs extension. The proof of the second equality is similar.

(ii) The proof is similar to the first part of the proof of (i). AB = AN ∗ AN +B. (iii) This assertion immediately follows from the equality HN

The following lemma is necessary for our proof of Theorem 1.1. − − AB )/(D(HN ) ∩ D(HN )) = N . Lemma 3.4. dim D(HN The proof of Lemma 3.4 will be given in section 5.4. We quote several facts about the spectrum of self-adjoint extensions of a symmetric operator.

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Lemma 3.5. Let L be a symmetric operator on a Hilbert space H. Suppose that the deficiency indices of L are (n, n) and n < ∞. Let A and B be two self-adjoint extensions of L. Then, the following holds: (i) σess (A) = σess (B). (ii) For any open interval I = (c1 , c2 ) such that c1 < c2 and dim Ran PI (A) < ∞, we have dim Ran PI (B) < ∞ and | dim Ran PI (A) − dim Ran PI (B)| ≤ d, where d = dim D(A)/ (D(A) ∩ D(B)) . Proof. (i) See [We, Theorem 8.17]. (ii) This assertion is an immediate corollary of [We, Exercise 8.8].

3.2

Proof of Theorem 1.1

Proof. By (ii) of Lemma 3.2, we have

† ∗ † † † ∗ AN AN AN AN ∗

(Ker A†N A†N )⊥

∗

(Ker A†N A†N )⊥

,

where the notation X Y means that two operators X and Y are unitarily equivalent. By (i) and (ii) of Proposition 3.3, we have − AB + B)|(Ker(HNAB +B))⊥ (HN − B)|(Ker(H − −B))⊥ . (HN N

(8)

Let HN be any self-adjoint extension of LN . By (i) of Lemma 3.5, we have there exists a closed subset S ⊂ R such that − AB ) = σess (HN ) S = σess (HN ) = σess (HN

for any self-adjoint extension HN of LN . In particular we have S ⊂ [B, +∞), by (iii) of Proposition 3.3. Thus {−B} ∈ / S. By (8), we have S \ {B} = S + 2B = {x + 2B; x ∈ S}.

(9)

It is easy to show that the set satisfying (9) is {(2n − 1)B; n = 1, 2, . . .} or the empty set. We show S is not empty. To see this, it is sufficient to construct a Weyl sequence for the spectrum B, that is, an orthonormal sequence {un }∞ n=1 such that (HN − B)un → 0 as n → ∞. Take countable disjoint disks {B(n; wn )}∞ n=1 (B(n; wn ) = {z; |z −wn | < n}) contained in R2 \SN . Let χ be a function satisfying (4). Put z 2 B vn (z) = twn χ( )e− 4 |z| , n

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where twn is the magnetic translation operator from {|z| < n} to {|z − wn | < n} intertwining L0 with LN . It is easy to check that {vn / vn }∞ n=1 is a Weyl sequence by using the equality B

2

B

2

B

LN twn e− 4 |z| = twn L0 e− 4 |z| = Btwn e− 4 |z|

2

in |z − wn | < n

and the Leibniz rule. Thus we have σess (HN ) = {(2n − 1)B; n = 1, 2, . . .}

(10)

for any self-adjoint extension HN of LN . Next, we shall prove (ii). The assertion AB )=0 dim Ran P(−∞,B) (HN

(11)

follows from (iii) of Proposition 3.3. By (8), we have − AB dim Ran P((2n−1)B,(2n+1)B) (HN ) = dim Ran P((2n+1)B,(2n+3)B) (HN )

(12)

for every n = 0, 1, 2, . . .. By Lemma 3.4 and (ii) of Lemma 3.5, we have − AB ) ≤ dim Ran P((2n−1)B,(2n+1)B) (HN ) + N (13) dim Ran P((2n−1)B,(2n+1)B) (HN

for every n = 0, 1, 2, . . .. By (11), (12), (13) and an elementary induction argument, we have (3). Thus (ii) of Theorem 1.1 holds. Since the deficiency indices of LN are (2N, 2N ), we have − ) ≤ 2N (14) dim D(H)/D(H) ∩ D(HN for any self-adjoint extension HN of LN . Thus (iii) of Theorem 1.1 follows from (3), (12), (14) and (ii) of Lemma 3.5. To show (i) of Theorem 1.1, consider the following equality:

=

dim Ran P{(2n−1)B} (HN ) dim Ran P((2n−1)B−,(2n−1)B+) (HN ) − dim Ran P((2n−1)B−,(2n−1)B) (HN ) − dim Ran P((2n−1)B,(2n−1)B+) (HN ).

The first term of the last expression is infinity, by (10). The second term and the third are finite for small > 0, by (iii) of Theorem 1.1. Therefore we have (i) of Theorem 1.1.

4 Large separation case 4.1

Self-adjoint extensions with uniform boundary conditions

We summarize fundamental facts about a self-adjoint extension HN with uniform boundary conditions. The proofs will be given in Section 5.5.

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Lemma 4.1. Let N = 1, 2, . . . or N = ∞. Let 0 < α < 1 and suppose αj = α for every j = 1, 2, . . . , N . Let H1 be any self-adjoint extension of Lα 1 . Then, there exists a unique self-adjoint extension HN of LN satisfying (5). Lemma 4.2. Let HN be a self-adjoint extension of LN which has uniform boundary conditions. Let η ∈ C0∞ ({|z| < R2 }) with η = 1 on some open neighborhood of 0. Let 1 ≤ j ≤ N and put ηj (z) = η(z − zj ). Let t−zj be the magnetic translation operator from {|z − zj | < R2 } to {|z| < R2 } intertwining LN with L1 , and let tzj be the inverse operator. Then, the following holds: (i) For any u ∈ D(HN ), we have ηj u ∈ D(HN ), t−zj (ηj u) ∈ D(H1 ) and HN (ηj u) = tzj H1 t−zj (ηj u).

(15)

(ii) For any v ∈ D(H1 ), we have ηv ∈ D(H1 ), tzj (ηv) ∈ D(HN ) and t−zj HN tzj (ηv) = H1 (ηv). Lemma 4.3. Let HN be a self-adjoint extension of LN which has uniform boundary conditions. Let U be a simply connected open set in R2 \ SN , m ∈ R2 and V = U − m = {z − m; z ∈ U }. Let η ∈ C0∞ (V ) and put ηm = η(z − m). Let t−m be the magnetic translation from U to V intertwining LN with L0 and tm be the inverse operator. Then, the following holds: (i) For any u ∈ D(HN ), we have ηm u ∈ D(HN ), t−m (ηm u) ∈ D(H0 ) and HN (ηm u) = tm H0 t−m (ηm u).

(16)

(ii) For any v ∈ D(H0 ), we have ηv ∈ D(H0 ), tm (ηv) ∈ D(HN ) and t−m HN tm (ηv) = H0 (ηv).

4.2

Proof of Theorem 1.2

Most part of our proof is similar to that of [Co-Ne, Theorem III.1], so we omit the detail of the proof of some lemmas. The main difference between our proof and theirs is that the approximating argument they used in [Co-Ne, Corollary III.1] is not applicable in our case, because the space C0∞ (R2 ) is not a common operator core of HN for N = 1, 2, . . .. Thus we take an approximate eigenfunction ψ instead of an eigenfunction ψ in (17) and prove the statement for H∞ directly. Proof. We shall introduce the notation used in [Co-Ne]. For p = (px , py ) ∈ Z 2 and δ > 0, put K(p, δ) := {z = (x, y) ∈ R2 ; |x − and let m(p) :=

R 10 p.

Since

p∈Z2

R R δ δ px | ≤ , |y − py | ≤ } 10 2 10 2

K(p, δ) = R2

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R R for δ ≥ 10 , we can take pj = (pj,x , pj,y ) ∈ Z 2 such that zj ∈ K(pj , 10 ), for each j = 1, . . . , N . Put K ((pj,x + β, pj,y + γ) , δ) , for δ > 0, Kj (δ) := β,γ∈{−1,0,1} N

FN :=

j=1

Kj (

R ) 10

and let mj := m(pj ). Let Γ0 :=

{m(p) ∈ R2 ; p ∈ Z 2 , K(p,

Γ1 := Γ :=

{mj ; j = 1, . . . , N }, Γ0 ∪ Γ 1 .

R ) ⊂ FN }, 10

For m = m(p) ∈ Γ0 , let t−m be the magnetic translation operator from K(p, R8 ) to K(0, R8 ) intertwining LN with L0 , and tm be the inverse operator. For j = 1, . . . , N , let t−zj be the magnetic translation operator from {|z − zj | < R2 } to {|z| < R2 } α intertwining LN with L1 j , and tzj be the inverse operator. Proof of (i) . Let I = [c, d] be an interval on R satisfying I ∩ {(2n − 1)B; n = 1, 2, . . .} = ∅ and c, d ∈ / σ(H1 ). Take E ∈ σ(HN ) ∩ I. For any > 0, there is some ψ ∈ D(HN ) such that

ψ = 1, ξ < , ξ := (HN − E)ψ .

(17)

Notice that σ(HN ) ∩ I is a finite set when N is finite, by (iii) of Theorem 1.1. In this case, we can take ψ as an eigenfunction corresponding to the eigenvalue E and ξ as 0. We shall prepare three lemmas. Lemma 4.4. Let η ∈ C ∞ (R2 ). Suppose supp |∇η| is a compact set in R2 \ SN and sup |∇η|2 + sup |∇∂x η|2 + sup |∇∂y η|2 ≤ C for some constant C. Then, ηψ ∈ D(HN ) and there exists a constant C5 > 0 dependent only on E, C such that dxdy(|ψ (z)|2 + |ξ (z)|2 ), (18)

[HN , η]ψ 2 ≤ C5 supp |∇η|

where the bracket denotes the commutator, that is, [X, Y ] = XY − Y X. Proof. In the similar way to the proof of [Co-Ne, Lemma II.2]. The fact ηψ ∈ D(HN ) follows from Lemma 4.2.

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Lemma 4.5. Let p ∈ Γ0 . Then, there exist constants C6 > 0 and c > 0 dependent only on B, E such that dxdy|ψ (z)|2 R K(p, 10 )

≤

C6

e

−cR2

2

R K(p, R 8 )\K(p, 9 )

dxdy|ψ (z)| +

K(p, R 8 )

2

dxdy|ξ (z)|

,

if R ≥ 1. Proof. Similar to the proof of [Co-Ne, Lemma III.3], but we use tm (H0 − E)−1 t−m ([HN , ηm ]ψ + ηm ξ ) = ηm ψ instead of (3.28) in [Co-Ne]. The above equality is justified by using Lemma 4.3. Lemma 4.6. There exists a constant C7 > 0 dependent only on B, E such that 2 dxdy|ψ (z)|2 ≤ C7 e−cR , if R ≥ 1, (19) c FN

c

2

where = e− 2 R , c is a constant given in Lemma 4.5. Proof. Similar to the proof of [Co-Ne, Lemma III.4], but we use Lemma 4.4 and Lemma 4.5. Take η0 ∈ C0∞ (R2 ) such that 0 ≤ η0 ≤ 1 and η0 (z) =

1 (z ∈ K0 ( R9 )) 0 (z ∈ / K0 ( R8 )),

sup |∇η0 |2 + sup |∇∂x η0 |2 + sup |∇∂y η0 |2 ≤

C , R2

(20)

where C is a positive constant independent of R and

K0 (δ) =

K((β, γ); δ).

β,γ∈{−1,0,1}

Put ηpj (z) = η0 (z − pj ) for j = 1, . . . , N . By Lemma 4.2, we have ηpj ψ ∈ D(HN ), t−zj ηpj ψ ∈ D(H1 ) and (H1 − E)t−zj ηpj ψ

= t−zj (HN − E)(ηpj ψ ) = t−zj ([HN , ηpj ]ψ + ηpj ξ ).

(21)

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The Aharonov-Bohm Solenoids in a Constant Magnetic Field c

137

2

Let c be the constant given in Lemma 4.5 and put = e− 2 R . Then we have N

(H1 − E)t−zj ηpj ψ 2

j=1

≤

2

N

[HN , ηpj ]ψ 2 + ηpj ξ 2 j=1

≤

2(C5 + 1) C

2

R Kj ( R 8 )\Kj ( 9 )

j=1

≤

N

c FN

dxdy|ψ (z)| +

Kj ( R 8 )

2

dxdy|ψ (z)| +

2

2

≤ Ce−cR ,

2

dxdy|ξ |

if R ≥ 1,

(22)

where C is a constant dependent only on B, E. We used (21) in the first inequality, (18) in the second, (17) in the third and (19) in the last. Moreover, we have N

(H1 − E)t−zj ηpj ψ 2

j=1

≥

dist{E, σ(H1 )}2

N

ηpj ψ 2

j=1

≥

dist{E, σ(H1 )}2

≥

dist{E, σ(H1 )}2 (1 − C7 e−cR ),

FN

dxdy|ψ |2 2

if R ≥ 1,

(23)

2

by (19). Take a large number R0 > 1 such that C7 e−cR0 < 12 . Then we have by (22) and (23) √ c 2 dist{E, σ(H1 )} ≤ 2Ce− 2 R , if R ≥ R0 . Thus (i) of Theorem 1.2 holds. Proof of (ii). Let I be an interval satisfying the assumption of Theorem 1.2. By (i) of Theorem 1.2, there exists a large number R0 > 1 and a small number δ0 > 0 such that I ∩ σ(HN ) ⊂ (E − δ0 , E + δ0 ), if R ≥ R0 , E∈I∩σ(H1 )

(E − δ0 , E + δ0 ) ∩ σ(H1 ) = {E},

for every E ∈ I ∩ σ(H1 ).

Thus it is sufficient to show that there exists R1 > R0 such that dim Ran P(E−δ0 ,E+δ0 ) (HN ) = N k, where k = dim Ker(H1 − E).

if R ≥ R1 ,

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Takuya Mine


First we show that there exists R1 > R0 such that dim Ran P(E−δ0 ,E+δ0 ) (HN ) ≥ N k

(24)

for each eigenvalue E of H1 contained in I, if R > R1 (when N = ∞, this is all to prove). Let Vj = {tzj η0 v ; v ∈ Ker(H1 − E)}, where η0 is the function satisfying (20). We can prove dim Vj = k if R is sufficiently large, with the help of the following inequality: 2 |v|2 dxdy ≤ C7 e−cR , if R ≥ 1 (25) R c K(0, 10 )

N

for v ∈ Ker(H1 − E), which follows from Lemma 4.6. Put V = ⊕ Vj . Then we j=1

have dim V = N k.

(26)

By (ii) of Lemma 4.2, we have V ⊂ D(HN ). Suppose that, for any positive number R3 , there exists a sequence {zj }N j=1 such that R = inf j=j |zj − zj | ≥ R3 and dim Ran P(E−δ0 ,E+δ0 ) (HN ) ≤ N k − 1 (when N = ∞, we replace the latter assumption by ‘dim Ran P(E−δ0 ,E+δ0 ) (H∞ ) < N ∞’). By (26), there exists v = j=1 tzj η0 vj ∈ V , vj ∈ Ker(H1 − E), such that ⊥

v = 1 and v ∈ Ran P(E−δ0 ,E+δ0 ) (HN ) . Then we have

(HN − E)v ≥ δ0 .

(27)

By (ii) of Lemma 4.2, we have (HN − E)v =

N

tzj [H1 , η0 ]vj .

j=1

By Lemma 4.4 and Lemma 4.6, we have 2

tzj [H1 , η0 ]vj 2 ≤ C9 e−cR vj 2 ,

if R ≥ 1,

where C9 and c is a constant dependent only on B, E. Moreover, we have by (25)

vj 2 ≤ 2 η0 vj 2 for sufficiently large R. Thus we have there exists R3 > 0 such that 2

(HN − E)v 2 ≤ Ce−cR ,

if R ≥ R3 ,

where C is a constant dependent only on B, E. This contradicts (27). Thus (24) holds.

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Next, we assume N is finite and prove there exists a constant R2 > R0 dependent only on B, E, δ0 and α such that dim Ran P(E−δ0 ,E+δ0 ) (HN ) ≤ N k,

if R ≥ R2 .

(28)

We prepare three lemmas. Lemma 4.7. Let P be an orthogonal projection and P be a finite rank operator on a Hilbert space. Suppose that

P − P < 1. Then, dim Ran P ≤ dim Ran P .

Proof. See [Co-Ne, Proposition III.1]. Lemma 4.8. There exist smooth functions {ηm }m∈Γ , satisfying 0 ≤ ηm ≤ 1 for m ∈ Γ,

2 ηm = 1 on R2 ,

m∈Γ

R supp ηm ⊂ K(m, ) 9 R supp ηmj ⊂ Kj ( ) 9 C sup(|∇ηm |2 + |∇∂x ηm |2 + |∇∂x ηm |2 ) ≤ 2 R

for m ∈ Γ0 , for mj ∈ Γ1 , for m ∈ Γ, if R ≥ 1,

where C is a constant independent of m, R.

Proof. See e.g. [Cy-Fr-Ki-Si]. (m)

Lemma 4.9. For m ∈ Γ, put η0 AN (w) =

mj ∈Γ1

(mj )

tzj η0

(z) = ηm (z + m). Define an operator AN (w) by

(H1 − w)−1 t−zj ηmj +

(m)

tm η0

(H0 − w)−1 t−m ηm

m∈Γ0

for w ∈ C, |w − E| = δ0 . Then, the sums converge in the strong operator topology of the bounded operators from L2 (R2 ) to D(HN ). Moreover, there exists a constant C10 > 0 dependent on B, E, δ0 , and α such that

AN (w) B(L2 (R2 );D(HN )) ≤ C10 for every w ∈ C with |w − E| = δ0 , if R ≥ 1. Here, B(X, Y ) denotes the space of the bounded operators from X to Y . Proof. Similar to the proof of [Co-Ne, Lemma III.5]. The fact AN (w)u ∈ D(HN ) follows from Lemma 4.2 and Lemma 4.3.

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Takuya Mine

We have (HN − w)AN (w)

=

(mj )

(HN − w)tzj η0


(H1 − w)−1 t−zj ηmj

mj ∈Γ1

+

(m)

(HN − w)tm η0

(H0 − w)−1 t−m ηm

m∈Γ0

=

(mj )

(tzj [H1 , η0

2 ](H1 − w)−1 t−zj ηmj + ηm ) j

mj ∈Γ1

+

(m)

(tm [H0 , η0

2 ](H0 − w)−1 t−m ηm + ηm )

m∈Γ0

=

1 + TN (w),

where TN (w)

=

(mj )

tzj [H1 , η0

](H1 − w)−1 t−zj ηmj

mj ∈Γ1

+

(m)

tm [H0 , η0

](H0 − w)−1 t−m ηm .

m∈Γ0

We used the intertwining property of t−m and the equality (m )

m∈Γ

(29)

2 ηm = 1. Notice

that the operator [H1 , η0 j ](H1 − w)−1 is well defined by Lemma 4.2, and the (m) operator [H0 , η0 ](H0 − w)−1 is well defined by Lemma 4.3. Lemma 4.10. The two sums in the right-hand side of (29) converge in the strong operator topology of the bounded operators from L2 (R2 ) to L2 (R2 ). Moreover, there is a constant C11 > 0 dependent only on B, E, δ0 and α such that

TN (w) B(L2 (R2 );L2 (R2 )) ≤

C11 R

for every w ∈ C with |w − E| = δ0 , if R ≥ 1.

Proof. Similar to the proof of (3.71) in [Co-Ne].

C11 R

1 2.

Take a large number R > R0 such that < By Lemma 4.10, we have

TN (w) ≤ 12 and thus 1 + TN (w) is invertible for w ∈ C with |w − E| = δ0 , if R ≥ R . Then HN − w is also invertible and its inverse is given by (HN − w)−1 = AN (w) − AN (w)TN (w)(1 + TN (w))−1 . Integrating (30) on {w; |w − E| = δ0 }, we have (m ) tzj η0 j P(E−δ0 ,E+δ0 ) (H1 )t−zj ηmj + RN , P(E−δ0 ,E+δ0 ) (HN ) = mj ∈Γ1

where RN =

1 2πi

|w−E|=δ0

AN (w)TN (w)(1 + TN (w))−1 .

(30)

(31)

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By Lemma 4.9 and Lemma 4.10, there exists a constant R2 > R dependent only on B, E, δ0 and α such that

RN < 1, if R ≥ R2 .

(32)

Since the first term of the right-hand side of (31) is a linear operator of at most rank N k, we have (28) by Lemma 4.7 and (32). Thus the proof of Theorem 1.2 is completed.

5 Self-adjoint extensions 5.1

Single solenoid operator

Consider the single solenoid case N = 1. In the sequel, we often identify a vector z = (x, y) ∈ R2 with a complex number z = x + iy ∈ C. Under the gauge (6), the operators defined in (7) is explicitly written as the following: 1 ∂x − i 1 = ∂y + i

Π1,x = Π1,y

B y− 2 B x+ 2

α y, |z|2 α x, |z|2 α B z¯ + , 2 z α B = −2∂z¯ + z + , 2 z¯

A1 = iΠ1,x + Π1,y = 2∂z + A†1 = −iΠ1,x + Π1,y

(33)

where

1 1 (∂x − i∂y ), ∂z¯ = (∂x + i∂y ). 2 2 α α α Define four functions φα −1 , ψ1 , φ0 , ψ0 by ∂z =

B

2

2

B

α −1 − 4 |z| e , ψ1α (z) := |z|−α z¯1 e− 4 |z| , φα −1 (z) := |z| z 2 2 B B α − 4 |z| φα , ψ0α (z) := |z|−α e− 4 |z| . 0 (z) := |z| e

(34)

In the polar coordinate z = reiθ , we have B

2

B

2

α+m imθ − 4 r φα e e , ψnα (z) = r−α+n e−inθ e− 4 r . m (z) = r

(35)

Thus the four functions belong to L2 (R2 ). Lemma 5.1. The following holds: α α α L1 φα −1 = (2α − 1)Bφ−1 , L1 ψ1 = Bψ1 , α α α L1 φ0 = (2α + 1)Bφ0 , L1 ψ0 = Bψ0α .

(36)

Proof. By a simple calculation using (33), (34) and the equality L1 = A†1 A1 + B.

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ˇ We quote a part of the results by Exner, St’ov´ ıˇcek and Vytˇras ([Ex-St-Vy]) for later use. ˇ Proposition 5.2 (Exner-St’ov´ ıˇcek-Vytˇras). (i) We have

2 (R2 \ {0}); L1 u ∈ L2 (R2 ) . D(L1 ∗ ) = u ∈ L2 (R2 ) ∩ Hloc

(ii) The deficiency indices of L1 are (2, 2). (iii) The four linear functionals 2π α 1−α 1 Φ−1 (u) = lim r u(reiθ )eiθ dθ, r→+0 2π 0 2π 1 α −1+α iθ iθ α −1+α u(re )e dθ − Φ−1 (u)r , Ψ1 (u) = lim r r→+0 2π 0 2π α 1 (u) = lim r u(reiθ )dθ, Ψα 0 r→+0 2π 0 2π 1 −α iθ α −α (u) = lim r u(re )dθ − Ψ (u)r Φα 0 0 r→+0 2π 0 are well defined and finite for u ∈ D(L1 ∗ ). (iv) Every u ∈ D(L1 ∗ ) is uniquely decomposed as α α α α α α α u = Φα −1 (u)φ−1 + Ψ1 (u)ψ1 + Φ0 (u)φ0 + Ψ0 (u)ψ0 + ξ,

(37)

where ξ ∈ D(L1 ). In particular, the element u ∈ D(L1 ∗ ) belongs to D(L1 ) if and only if α α α Φα −1 (u) = Ψ1 (u) = Φ0 (u) = Ψ0 (u) = 0. Remark. The decomposition (37) is a paraphrase of [Ex-St-Vy, page 2158, line 8], since the four functions have the asymptotics α−1 −iθ α α −α e , ψ1α ∼ r1−α e−iθ , φα φα −1 ∼ r 0 ∼ r , ψ0 ∼ r

as r → 0. Lemma 5.3. (i) The following equalities hold:

where Γ(z) =

2

φα −1 1

=

π (2α − 1) B + 1

ψ1α 21

=

π(B 2 + 1)

2

2 B

π (2α + 1) B + 1

ψ0α 21

=

π(B 2 + 1)

0

2 B

2 B

α Γ(α),

Γ(2 − α),

=

2−α

2

φα 0 1

∞

2

2

2

1−α

2 B

1+α

Γ(1 − α),

e−t tz−1 dt is the Gamma function.

Γ(1 + α),

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∗ (ii) For u, v ∈ D((Lα 1 ) ), define α [u, v]1 = (Lα 1 u, v) − (u, L1 v).

Then, the following equalities hold: α α α [φα −1 , ψ1 ]1 = 4π(α − 1), [φ0 , ψ0 ]1 = 4πα, α α α α α α α [φα −1 , φ0 ]1 = [φ−1 , ψ0 ]1 = [ψ1 , φ0 ]1 = [ψ1 , ψ0 ]1 = 0, α α α [φα l , φl ]1 = [ψn , ψn ]1 = 0

for l = −1, 0, n = 0, 1.

α α α α ∗ (iii) The operators Φα −1 , Ψ1 , Φ0 , Ψ0 are bounded linear functionals on D((L1 ) ). Moreover, we have

1 1

ψ α 1 , Ψα

φα 1 , 1 ≤ 4π(1 − α) 1 4π(1 − α) −1 1 1

ψ0α 1 ,

φα 1 .

Φα

Ψα 0 ≤ 0 ≤ 4πα 4πα 0

Φα −1 ≤

Proof. (i), (ii) (35) and (36). (iii) Notice that

One can prove these equalities by a short calculation using

[u, v]1 = (L1 ∗ u, v) − (u, L1 v) = 0

for any u ∈ D(L1 ∗ ) and v ∈ D(L1 ). From this equality, (ii) of this lemma and (37), we have 1 1 [ψ1α , u]1 , Ψα [φα , u]1 , 1 (u) = 4π(1 − α) 4π(α − 1) −1 1 1 [ψ α , u]1 , [φα , u]1 . Φα Ψα 0 (u) = 0 (u) = −4πα 0 4πα 0

Φα −1 (u) =

(38)

Moreover, we have by the Schwarz inequality |[u, v]1 | ≤ L1 u v + u L1 v ≤ u 1 v 1 for u, v ∈ D(L1 ∗ ). By (38) and (39), the conclusion holds.

5.2

Deficiency indices

In the sequel, we denote U (r) =

N

{z ∈ R2 ; |z − zj | < r}.

j=1

Proposition 5.4. Let N = 1, 2, . . . , or N = ∞. Then the following holds: (i) We have 2 D(LN ∗ ) = {u ∈ L2 (R2 ) ∩ Hloc (R2 \ SN ); LN u ∈ L2 (R2 )},

where the derivative LN u is interpreted in the distribution sense.

(39)

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Takuya Mine


(ii) Let u ∈ D(LN ∗ ). Suppose that there exists a constant R1 with 0 < R1 < R such that supp u ⊂ R2 \ U (R1 ). Then, u ∈ D(LN ). Proof. (i) This assertion follows from the definition of the adjoint operator and the elliptic inner regularity (see [Ag]). (ii) Take u satisfying the assumption. Since the vector potential aN is smooth in R2 \ U (R1 ), we can approximate u with respect to the graph norm of LN ∗ by a sequence of functions in C0∞ (R2 \ U (R1 )), because of the essential self-adjointness of Schr¨ odinger operators with smooth vector potentials (see [Ik-Ka], [Le-Si]). Since C0∞ (R2 \ U (R1 )) ⊂ D(LN ), we have u ∈ D(LN ). The following lemma shows the continuity of the cut-off map with respect to the graph norm. Lemma 5.5. Let 1 ≤ N < ∞ or N = ∞. Let η ∈ C ∞ (R2 ) and suppose supp |∇η| is a compact set in R2 \ SN . Then, for any u ∈ D(LN ∗ ), we have ηu ∈ D(LN ∗ ) and

[LN , η]u 2 ≤ C0

supp |∇η|

(|LN u|2 + |u|2 )dxdy,

(40)

where C0 = 10 sup(|∇η|2 + |∇(∂x η)|2 + |∇(∂y η)|2 ). Moreover, we have (|LN u|2 + |u|2 )dxdy,

ηu 2N ≤ C1

(41)

supp η

where C1 = 2(sup |η|2 + C0 ). Proof. The proof of (40) is the same as the proof of (2.31) in [Co-Ne], and (41) follows immediately from (40). The fact ηu ∈ D(LN ∗ ) follows from (i) of Proposition 5.4 and the Leibniz rule. Let χ, χj be functions satisfying (4). Define a linear operator T from αj ∗ αj D(LN ∗ )/D(LN ) to ⊕N j=1 D((L1 ) )/D(L1 ) by N

T ([u]) = ⊕ [Tj u]

(42)

j=1

for u ∈ D(LN ∗ ), where α

Tj : D(LN ∗ ) u → t−zj (χj u) ∈ D((L1 j )∗ ) and t−zj is the magnetic translation operator from {|z − zj | < α intertwining LN with L1 j .

(43) R 2}

to {|z|
3, or for the function 0 (n) appearing in Theorem 8.7 below, because some steps of the proof in those dimensions proceed via non-constructive arguments, see Section 7. 2 The

156

R. Beig, P.T. Chru´sciel and R. Schoen


collection be such that K (M ) = {0} .

(1.1)

1. Let p ∈ M and consider the set Vp = {vacuum initial data such that K (U ) = {0} for any neighborhood U of p} . Then Vp is open and dense in a neighborhood of (K0 , g0 ). 2. Define further: V = {vacuum initial data such that K (U ) = {0} for any open subset U of M } . Then V is of second category in a neighborhood of (K0 , g0 ). Identical results hold in the class of initial data with fixed constant trg K, as well as in the class of time symmetric initial data K ≡ 0. (Recall that a set is of second category if it contains a countable intersection of open dense sets; in complete metric or Fréchet spaces such sets are dense.) The C k,α topology in Theorem 1.2, as well as in the remaining results below unless explicitly stated otherwise, can be understood as follows: one chooses some smooth complete Riemannian metric h on M , which is then used to calculate norms of tensors and their h-covariant derivatives. Other choices are possible, and this is discussed in more detail in Appendix A. One expects that for generic initial data the no-global-KIDs condition (1.1) of Theorem 1.2 will be satisfied. Attempts to prove that require analytical tools which impose restrictions on the geometry. We concentrate therefore on three cases which seem to us to be the most important ones from the point of view of applications: compact manifolds without boundary, or asymptotically flat initial data sets, or conformally compactifiable initial data sets. Our next main result, when used in conjunction with Theorem 1.2, establishes Conjecture 1.1 in those cases: Theorem 1.3 Consider the following collections of vacuum initial data sets: 1. Λ = 0 with an asymptotically flat region, or 2. trg K = Λ = 0 with an asymptotically flat region, or 3. K = Λ = 0 with an asymptotically flat region, or 4. with a conformally compactifiable region in which trg K is constant, or

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5. the trace of K is constant and the underlying manifold M is compact, with (trg K)2 ≥

2n Λ, (n − 1)

(1.2)

or 6. K = 0, M is compact, and the curvature scalar R satisfies R = 2Λ ≤ 0, with a C k,α × C k,α (weighted in the non-compact region) topology, with k ≥ k0 (n) for some k0 (n) (k ≥ 6 if n = dim M = 3). For each such collection the subset of vacuum initial data sets without global KIDs is open and dense. The weights in the asymptotic region should be chosen so that the metrics approach the Euclidean one as r−β , for some β ∈ (0, n − 2]. In the conformally compactifiable regions a topology as in [11, Theorem 6.7] with 0 ≤ t < (n + 1)/2 should be used. It would be of interest to have a version of points 4 and 5 without the CMC condition. Since the collection of initial data sets which have no global KIDs is open (see Proposition 4.2 below), for compact manifolds the proof of Theorem 1.3 also provides a large open collection of initial data sets which are close to CMC data and which have no global KIDs. However, the general case remains open. We think that the removal of the CMC condition in point 5 is the most notable problem left open by our paper. Somewhat surprisingly, the above results require a considerable amount of non-trivial work. We first show that generic metrics have no local conformal Killing vectors, or local Killing vectors3 . This is done by reducing the problem to a finite system of linear algebraic equations for the candidate vector, as well as a few of its derivatives, at a given point. While the argument is conceptually straightforward, there is some messy algebra involved when one wishes to show that those algebraic equations lead to the desired conclusion for at least one metric. This result is then used in the proof of Theorem 1.3. A similar argument is used for local KIDs, with an appropriately messier algebra. That would have settled the problem, if not for the fact that we want initial data satisfying the constraint equations. In order to take care of that we first use Taylor expansions to construct approximate solutions of the constraint equations near a point p. The gluing techniques of Corvino-Schoen [14] type, as extended in [10, 11], are then used to go from an approximate solution to a real one, establishing Theorem 1.2. Some comments on the organization of this paper are in order. The heart of our analysis lies in Section 8, where we show how to perturb solutions of the vacuum constraint equations to solutions without KIDs, preserving the constraint 3 The

only related result known to us in the literature is in [15], where it is shown that on a compact boundaryless manifold the set of Riemannian metrics without nontrivial isometries is open and dense. The argument given there does not seem to be useful to get rid of local Killing vector fields, and makes essential use of the fact that M is compact without boundary. Moreover it is not clear how to adapt it to account for conformal Killing vectors, or for KIDs.

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equations. This requires several preliminary results, such as a) perturbing initial data to get rid of KIDs, without necessarily satisfying any constraint equations, and b) perturbing metrics to get rid of conformal Killing vectors. The argument needed for a) is presented in Section 5 in dimension three, and in Section 7 in all dimensions. The advantage of the argument in Section 5 is that it gives an explicit differentiability threshold for the construction, in the physically important case n = 3, while the one in Section 7 leads to some uncontrollable, dimension dependent threshold. In Section 6 we show how to get rid of KIDs in the timesymmetric case, while remaining in the time-symmetric class. In Section 2 we construct functions that control existence, or lack thereof, of conformal Killing vector fields in dimension three. This result is the key for getting rid of KIDs on CMC initial data sets; it also sets the stage for the structure of the argument for KID-removal. As before, the higher-dimensional proof is carried out in Section 7, with some non-explicit differentiability threshold. In Section 3 we construct the corresponding functions for controlling Killing vectors. Here we obtain explicit differentiability thresholds in all dimensions. Perturbations removing conformal Killing vectors do of course remove Killing vectors as well, but the differentiability thresholds we obtain in the Killing case are explicit in all dimensions, and smaller than the corresponding conformal Killing threshold in dimension three. In Section 4 we show how the local perturbation arguments of the previous sections can be translated into category-type statements. This leads immediately to the question of topologies appropriate in our context, this is briefly discussed in Appendix A. Appendix B presents a monodromy-type argument for analytic overdetermined PDE systems, needed in the proofs of Section 7. All the results just described join forces in Section 9, where Theorems 1.2 and 1.3 are established.

2 Metrics without conformal Killing vectors near a point, n=3 We start with some preliminaries. Unless explicitly specified otherwise we assume in this section that dimension equals three. Recall that the Schouten tensor Lij is given by 1 Lij = Rij − gij R , (2.1) 4 where gij is a pseudo-Riemannian metric and Rij and R are respectively its Ricci and scalar curvature. Furthermore we define the Cotton tensor Bijk Bijk = Li[j;k] .

(2.2)

The tensor Bijk has the following algebraic properties Bijk = Bi[jk] ,

B iik = 0 ,

B[ijk] = 0 ,

(2.3)

which makes five degrees of freedom per space point. It also satisfies Bi[jk;l] = 0 .

(2.4)

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Equivalently, we can take Hij = kli Bjkl .

(2.5)

The tensor Hij is symmetric, tracefree and divergence-free. Suppose a metric has a conformal Killing vector X, Di Xj + Dj Xi =

2 ϕgij , 3

(2.6)

where ϕ here is the divergence of the vector field X. Then it has to be the case that (2.7) LX Bijk = 0 . The reason is that the map Cotton sending a metric to its Cotton tensor satisfies Φ∗ Cotton[g] = Cotton[Φ∗ g] for any map Φ of M into itself. One now applies this relation to the case where Φ is a one- parameter family of diffeomorphisms generated by a conformal Killing vector X. Taking the derivative with respect to the parameter and using that the map Cotton is invariant under conformal rescaling of the metric one obtains (2.7). An equivalent form of (2.7) is the relation 1 LX Hij = − ϕHij , 3

(2.8)

Taking cyclic permutations of the equation obtained by differentiating (2.6) one has 2 (2.9) Di Fjk = −Rjki l Xl + ϕ[j gk]i , 3 where we have defined Fij = D[i Xj] and ϕi = Di ϕ. The Lie derivative of the tensor Rij − Rgij /2(n − 1) (here, for future reference, we work in general dimension n) equals R (n − 2) LX Rij − − gij = − Di Dj ϕ . (2.10) 2(n − 1) n In dimension n = 3, to which we return now, this reads Di ϕj = −3LX Lij .

(2.11)

The identities (2.9)–(2.11), together with the relation Di Xj = Fij + ϕgij /3 ,

(2.12)

imply that a conformal Killing vector, for which the quantities (X , Fij , ϕ, ϕi ) are all zero at the point p, has to vanish in a neighborhood of p. Using (2.9), (2.8) takes the form (2.13) X k Dk Hij + 2F (i k Hj)k + ϕHij = 0 .

160



Next we take a derivative of (2.13) with the result that 4 F l k Dk Hij + ϕDl Hij +X k Dl Dk Hij +2(Dl F (i k )Hj)k +2F(i k D|l| Hj)k +ϕl Hij = 0 . 3 (2.14) We are ready now to pass to the proof of the main result of this section: Theorem 2.1 Let (M, g) be a smooth three-dimensional pseudo-Riemannian1 manifold. 1. There exists a non-trivial homogeneous polynomial Q(·, ·, ·) : R6 × R3×6 × R3×3×6 → R such that if

Q(H, DH, D2 H)(p) = 0

(the R6 arises here because H is symmetric), then there exists a neighborhood Op of p on which there are no local conformal Killing vectors. 2. Let Ω be a neighborhood of p ∈ M . For any k ≥ 5 and > 0 there exists a metric g ∈ C ∞ (M ) such that g − g C k (Ω) ¯ < , with g − g supported in Ω, and such that Q(H , DH , D2 H )(p) does not vanish. Remark 2.2 A corresponding result in higher dimensions is proved in Theorem 7.4. Remark 2.3 Recall that a polynomial in the curvature tensor and its derivatives is called invariant if it is independent of the frame used to evaluate its numerical value. Below we arbitrarily choose some orthonormal basis of Tp M to define Q, and it is unlikely that the polynomial Q defined in our proof will be an invariant polynomial if the signature of the metric is Lorentzian; moreover, it is not clear how to modify Q to make it invariant while preserving the claimed properties. Note that one can view Q as a function on the frame bundle. In the Riemannian case ˜ be the integral of Q over those fibers with respect to the Haar measure, we let Q ˜ then Q is a non-trivial invariant polynomial with the properties as above. We note that the polynomial constructed below provides a convenient tool to capture the fact that a certain geometrically defined matrix has rank larger than ten; the latter assertion provides an equivalent invariant statement, regardless of signature. Proof. Before passing to the proof, some auxiliary results will be useful., Let the ˚ij := (Dk Rij )(p). In the superscript “˚” denote “value at the point p”, e.g., Dk R calculations that follow we will assume that the metric has Riemannian signature. The remaining cases require trivial modifications, which we leave to the reader. We start with a Lemma:

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Lemma 2.4 Consider a metric such that ˚ij = 0, Dk R ˚ij = 0 . ˚ gij = δij , R

(2.15)

Furthermore let the second derivatives of the curvature be such that ˚ij = Axk y(i zj) + Byk x(i zj) + Czk x(i yj) , Dk H

(2.16)

where (x, y, z) form an orthonormal basis of Tp M and the three real numbers A, B, C are all non-zero. Then the set of algebraic equations for w := (Xi , Fij := D[i Xj] , ϕ := Dk Xk , ϕi := Di ϕ)(p) obtained from the equations [LX Hij + 13 ϕHij ](p) = 0 , m Hj)m + 13 Dk (ϕHij )](p) = 0 , [LX Dk Hij + 2Ck(i

(2.17) (2.18)

m m Dm Hij + 2(Dl Ck(i )Hj)m + [LX Dl Dk Hij + Ckl m m +2Cl(i D|k| Hj)m + 2Ck(i D|l| Hj)m + 13 Dl Dk (ϕHij )](p) = 0 , i with Cjk defined as i = Cjk

1 i 2ϕ(j δk) − gjk ϕi 3

(2.19)

(2.20)

implies w = 0. Remark 2.5 Equations (2.17)–(2.19) are necessarily satisfied by every conformal Killing vector field X: (2.17) is equivalent to (2.13), while Equations (2.18)–(2.19) are equivalent to the first and second covariant derivatives of (2.13). ˚ij in (2.16) satisfies the necessary algebraic Remark 2.6 It can be seen that Dk H requirements to arise from a metric (i.e., being symmetric in (ij) and trace-free on all index pairs, compare (2.3)–(2.5)); this follows in any case from Proposition 2.7 below. ˚ = 0. Proof. It immediately follows from Equations (2.15), (2.16) and (2.13) that X Let a, b and c be defined as the following components of F in the basis (x, y, z): ˚l k = a(xl y k − yl xk ) + b(zl xk − xl z k ) + c(yl z k − zl y k ). (2.21) F ˚ = 0 we find that Evaluating (2.14) at p, and using X 0 = [b(A + C)zl − a(A + B)yl ]y(i zj) + [a(A + B)xl − c(B + C)zl ]x(i zj) + [c(B + C)yl − b(A + C)xl ]x(i yj) + (aCzl − bByl )xi xj + (cAxl − aCzl )yi yj 4 ˚(Axl y(j zi) + Byl x(j zi) + Czl x(j yi) ). + (bByl − cAxl )zi zj + ϕ (2.22) 3

162



It follows by inspection that a, b, c and ϕ ˚ have all to be zero. Differentiating (2.14) we find that ˚lk )Dk H ˚ij + 4 ϕ ˚ij ˚m Dl H 0 = (Dm F 3 ˚(i|k )Dm| H ˚j) k + 2(Dm F ˚(i|k )Dl| H ˚ j) k + ϕ ˚ij , + 2(Dl F ˚l Dm H (2.23) where we have used the vanishing of X and Di X j at p. Next observe that, by Equations (2.15) and (2.9), there holds ˚jk = Di F

2 ϕ ˚[j gk]i . 3

(2.24)

We now insert (2.24) into (2.23) to find that 0=

8 ˚ij − 1 gml ϕ ˚ij + 2 ϕ ˚j)l ϕ ˚(l Dm) H ˚(i D|m| H ˚k Dk H 3 3 3 2 ˚j)m − 2 ϕ ˚j k − 2 ϕ ˚i k . ˚(i D|l| H ˚k gi(l Dm) H ˚k gj(l Dm) H + ϕ 3 3 3

(2.25)

Direct algebra using (2.16) shows that ϕ ˚i vanishes, which is what had to be established. Let us show now that Proposition 2.7 A metric satisfying (2.15)–(2.16) exists. Proof. We start with two elementary lemmata: Lemma 2.8 Suppose we are given, on a star-shaped domain Ω in (Rn , δij ), a tensor field Bijk satisfying Bijk B[ijk] Bi[jk,l]

= Bi[jk] ,

(2.26)

= 0, = 0.

(2.27) (2.28)

Then there exists a tensor field Lij = L(ij) such that Bijk = Li[j,k] .

(2.29)

If B is a homogeneous polynomial of order p, then L can be chosen to be a homogeneous polynomial of order p + 1. Proof. By Equations (2.26)–(2.28), there exists a tensor field Mij , not necessarily symmetric in i and j, satisfying (2.29) with Lij replaced by Mij . From (2.27) it follows that there exists a covector field Λi with M[ij] = Λ[i,j] . Set Lij = Mij − Λi,j , then Lij = L(ij) and satisfies (2.29) thus proving Lemma 2.8. The fact that solutions can be chosen as polynomials follows from the explicit formula for the primitive of a form used in the proof of the Poincaré Lemma. We will also need the following variation of a result of Pirani [21]:

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Lemma 2.9 Let Ω be as in Lemma 2.8 and on it a tensor field Rijkl having the symmetries of the Riemann tensor and obeying the differential identity Rij[kl,m] = 0 .

(2.30)

Then there exists hij = h(ij) such that Rijlm = 2∂[i hj][l,m] .

(2.31)

If moreover Rijkl is a homogeneous polynomial in the manifestly flat coordinates ξ i of order q, then hij can be chosen as a homogeneous polynomial of order q + 2. Proof. This is proved by inspection of the proof in Pirani [21, pp. 279–280], using the fact that the proof there consists of the repeated use of the Poincaré Lemma. Returning to the proof of Proposition 2.7, let ξ be coordinates on Ω and define 1 Bijk = mjk (∂n Him )ξ n , (2.32) 2 where the constants ∂n Him are given by the right-hand-side of (2.16). The field Bijk defined by (2.32) obviously satisfies (2.26), while (2.27)-(2.28) hold because (2.16) is trace-free in all indices. Now let Lij be the homogenous quadratic polynomial guaranteed to exist by Lemma 2.8. As (2.16) is symmetric in i and j, the field Bijk satisfies the second equation in (2.3). This implies ∂ j Lij = ∂i L,

(2.33)

where L = δ ij Lij . Consider the field Sijkl defined by Sijkl = 2δk[i Lj]l − 2δl[i Lj]k ,

(2.34)

it is a homogeneous quadratic polynomial in ξ which clearly has the symmetries of a Riemann tensor. Equation (2.33) implies that (2.30) holds, hence all the assumptions of Lemma 2.9 are fulfilled. Let hij be the fourth order homogeneous polynomial guaranteed to exist by Lemma 2.9, set gij = δij + hij .

(2.35)

Since h vanishes to order three, both the Riemann tensor and its derivatives vanish at p, which justifies (2.15). Further, the Riemann tensor Rijkl of g coincides with Sijkl up to terms which give zero contribution at p in all the calculations relevant here, so that it is not difficult to show that gij satisfies (2.16), which proves Proposition 2.7.

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We can now pass to the Proof of Theorem 2.1. Consider the linear map L which to w = (Xi , Fij := D[i Xj] , ϕ := Dk Xk , ϕi := Di ϕ)(p) ∈ R10 assigns R10 w → Lw

:=

1 1 m LX Hij + ϕHij , LX Dk Hij + 2Ck(i Hj)m + Dk (ϕHij ) , 3 3 m m LX Dl Dk Hij + Ckl Dm Hij + 2(Dl Ck(i )Hj)m 1 m m +2Cl(i D|k| Hj)m + 2Ck(i D|l| Hj)m + Dl Dk (ϕHij ) (p) 3 ∈ R6 ⊗ R3×6 ⊗ R3×3×6 .

Here the Lie derivative is calculated using the usual formula for the Lie derivative of a tensor, and then the values of X and its derivatives as determined by w are inserted. Further, the second derivatives of ϕ are eliminated using (2.11). It follows from Lemma 2.4 and Proposition 2.7 that the set of metrics for which L is injective is not empty. Standard linear algebra implies that there exists a 10 × 10 matrix, say A, constructed by listing ten appropriately chosen rows of L, which has non-vanishing determinant when H arises from the metric of Proposition 2.7. Let Q be the sum of squares of determinants of all ten-by-ten submatrices of L, then Q ≥ (det A)2 and therefore Q is not identically vanishing by construction. Clearly L is injective whenever Q is non-zero, which proves point 1. To prove point 2, let g be an arbitrary metric, if Q(p), evaluated for the metric g, does not vanish, then the result is true with g = g. Otherwise, define J5

:=

{the set of fifth jets of g in normal coordinates at p as g varies in the set of all Riemannian metrics} .

(2.36)

This a linear space, an explicit parameterisation of which can be found in [23]. Let ei , i = 1, . . . , N , be any basis of J5 , thus every j ∈ J5 can be written as j = i e i , for some numbers i ∈ R. By definition of J5 , for every (i ) ∈ RN there exists some Riemannian metric for which j = i ei . Clearly the map g → (i ) is continuous ¯ ≥ 5, topology on the set of metrics, and a small variation of i can be in a C (Ω), realized by a small variation of g. In a frame such that gij (p) = δij , the map that assigns to the fifth jets of g, at p, the values of the tensors H, DH, and DD2 H at p, is a polynomial on J5 . We want to show that a small variation of g will make Q non-zero. Now, Q is a polynomial in the i ’s. Let i0 be the values of the i ’s corresponding to the metric g, and suppose that we have ∀ i 1 , . . . , in

∂ i1 +···+iN Q (i ) = 0 . ∂ i1 1 . . . ∂ iN N 0

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Then the polynomial Q would identically vanish, contradicting its construction. Hence there exists at least one of the above partial derivatives which does not vanish, and therefore an appropriate, no matter how small, variation of g will lead to a non-vanishing value of Q at p. As the argument depends only upon the jets of g at p, the variation can be made supported in a ball containing p with radius as small as desired.

3 Metrics without Killing vectors near a point Results on non-existence of Killing vectors follow of course immediately from those on non-existence of conformal Killing vectors, as established above. However, for Killing vectors in dimension three the differentiability threshold of Theorem 2.1 can be lowered to three. Further, for Killing vectors a simple proof can be given in all dimensions: Theorem 3.1 Let (M, g) be a n-dimensional pseudo-Riemannian manifold. 1. There exists a non-trivial homogeneous invariant polynomial Pn [g] := (DR, . . . , D2n+1 R) of degree n, where R is the Ricci scalar, such that if Pn (DR, . . . , D2n+1 R)(p) = 0 at a point p ∈ M , then there exists a neighborhood Op of p such that there are no non-trivial Killing vectors on any open subset of Op . In dimension n = 3 there exists such a polynomial Pˆ3 which depends upon Ric and D Ric. 2. Let Ω be a neighborhood of p ∈ M . For any k ≥ 2n + 1 and > 0 there exists a metric g such that g − g C k (Ω) (3.1) ¯ < , with g − g supported in Ω, and such that Pn (DR , . . . , D2n+1 R )(p) does not vanish. In dimension three we can arrange for the non-vanishing of Pˆ3 (Ric , D Ric )(p) using a perturbation supported in Ω and satisfying (3.1) for each arbitrarily chosen k ≥ 3. Remark 3.2 The differentiability required above in dimension n is certainly not optimal, but it allows the simple proof below. Remark 3.3 The polynomial Pn obtained here is completely useless from the point of view of Killing vectors in vacuum space-times, where the Ricci scalar vanishes. In this context it is of interest to have a statement as above with a polynomial depending only upon the Weyl tensor, and we prove existence of such polynomials in Theorem 7.4 below. Further, in Section 8 we will construct small perturbations of initial data which preserve the vacuum constraints.

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Proof. If X is a Killing vector we have LX (∆k R)=0 for all k, where ∆k denotes the kth power of the Laplace operator ∆. At p this gives the linear system of equations Aij X i (p) = 0 ,

Aij = Di (∆j R)(p) ,

j = 0, . . . , n − 1 .

Let Pn = det(Aij ). If Pn (p) does not vanish, then X(q) = 0 for all q in the neighborhood of p defined as {q : Pn (q) = 0}, hence X ≡ 0. It is not too difficult to check, using Taylor expansions of the metric (point 2 of Proposition 5.4 below is useful here), that there exist metrics for which Pn = 0, and the result follows by a repetition of the arguments of the proof of Theorem 2.1. In dimension 3 the number of the derivatives of the metric needed can be improved as follows: Let Gij = Rij − g kl Rkl gij /2, in the notation of Section 2 we assume that ˚ij = λ1 xi xj + λ2 yi yj + λ3 zi zj , G (3.2) with (λ1 − λ2 )(λ2 − λ3 )(λ3 − λ1 ) = 0. We set, as in (2.21), ˚ij = 2(ax[i yj] + bz[i xj] + cy[i zj] ) , F

(3.3)

so that ˚(i k G ˚j)k = b(λ1 − λ3 )x(i zj) + c(λ3 − λ2 )y(i zj) + a(λ2 − λ1 )x(i yj) , F

(3.4)

which has zero components on the diagonal. Finally we assume that ˚ij;k = µ1 (x(i δj)k − 2xk δij ) + µ2 (y(i δj)k − 2yk δij ) + µ3 (z(i δj)k − 2zk δij ) , (3.5) G where µ1 µ2 µ3 = 0. We now set ˚ = αxi + βyi + γzi . X

(3.6)

˚1 + G ˚2 + G ˚3 , we find Writing (3.5) in the form G ij;k ij;k ij;k ˚k ˚1 X G ij;k ˚2ij;k X ˚k G

=

µ1 α(−2yi yj − 2zi zj − xi xj ) + off diagonal terms ,

(3.7)

=

µ2 β(−2xi xj − 2zi zj − yi yj ) + off diagonal terms ,

(3.8)

˚3ij;k X ˚k G

=

µ3 γ(−2xi xj − 2yi yj − zi zj ) + off diagonal terms .

(3.9)

We first consider the relation LX Gij = 0 with i = j. Then (3.4) gives no contribution, while from (3.7) we obtain a linear homogenous system for (α, β, γ) with coefficient matrix ∆ given by   µ1 2µ2 2µ3 ∆ =  2µ1 µ2 2µ3  . 2µ1 2µ2 µ3

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There holds det(∆) = 5µ1 µ2 µ3 = 0. Thus, the equation LX Gij = 0, satisfied by any Killing vector, leads to α = β = γ = 0. The off-diagonal components of LX Gij = 0 imply now, by (3.4), that a = b = c = 0. Since (3.2) is symmetric, and (3.5) satisfies the linearized Bianchi identities, the results in [23] show that there exists a metric gij = δij + hij , with hij = O(ξ 2 ), satisfying (3.2) and (3.5). The proof is completed by the same argument as already given for general n.

4 Generic non-existence of local Killing, or conformal Killing, vector fields In this section we only consider three-dimensional manifolds, the reader will easily formulate an equivalent statement and proof for local Killing vector fields in any dimension using Theorem 3.1, or for local conformal Killing vector fields using Theorem 7.4 below. Theorem 4.1 Let M be a three-dimensional manifold. Then 1. The set of pseudo-Riemannian metrics on M which have no local Killing vector fields is of second category in the C 3 topology. 2. The set of pseudo-Riemannian metrics on M which have no local conformal Killing vector fields is of second category in the C 5 topology. Proof. We start with the following: Proposition 4.2 Let Ω be a domain in M . Then: 1. The set of metrics on Ω which have no Killing vectors on Ω is open in a ¯ topology, k ≥ 2. C k (Ω) 2. The set of metrics on Ω which have no conformal Killing vectors on Ω is ¯ topology, k ≥ 3. open in a C k (Ω) 3. The set of initial data (g, K) on Ω which have no non-trivial KIDs on Ω is ¯ ⊕ C k (Ω) ¯ topology, k ≥ 1. open in a C k+1 (Ω) Remark 4.3 The openness established here holds for any metrisable topology Tk such that convergence in Tk implies uniform convergence in C k norm on compact sets, with k ≥ 2 for Killing vectors, etc; see also Appendix A. Proof. We will show that existence of Killing vectors, or conformal Killing vectors, or KIDs, is a closed property. We start with the slightly simpler case of conditionally compact Ω: Lemma 4.4 Proposition 4.2 holds if Ω has compact closure.

168



Proof. 1. Let γi be a sequence of metrics with non-zero Killing vectors X(i). Rescaling X(i) we can assume that sup γi (X(i), X(i)) = 1 .

(4.1)

p∈Ω

We note that Killing vectors extend by continuity to Ω, we shall use the same symbol to denote that extension. Let pi ∈ Ω be such that the sup is attained, passing to a subsequence if necessary there exists p∗ in Ω such that pi → p∗ . Now, Killing vectors satisfy the system of equations Di Dj Xk = R ijk X ,

(4.2)

which shows that second covariant derivatives of all the X(i)’s are uniformly bounded on Ω. Interpolation [16, Appendix] shows that the sequence X(i) is uniformly bounded in C 2 . The existence of a subsequence converging in C 1 to a non-trivial Killing vector field follows from the Arzela-Ascoli theorem. 2. The argument is essentially identical, with the following modifications: we replace the normalization (4.1) by sup (|X(i)|γi + |DX(i)|γi ) = 1 .

(4.3)

p∈Ω

Equation (4.2) is replaced by the set of equations (2.9)–(2.12). Those equations easily imply boundedness of the sequence X(i) in C 3 , leading to a converging subsequence in C 2 . 3. Let (γi , Ki ) be a sequence of metrics with non-zero KIDs (Y (i), N (i)). We use the normalization sup (|Y (i)|γi + |DY (i)|γi + |N (i)| + |DN (i)|γi ) = 1 .

(4.4)

p∈Ω

From (5.4) and (5.5) one obtains a uniform C 2 bound on (Y (i), N (i)), and one concludes as before. Returning to the proof of point 1 of Proposition 4.2, let Ωj be an increasing sequence of conditionally compact domains such that Ω = ∪Ωj . By Lemma 4.4 we have K (Ωj ) = {0} for all j. The restriction map induces an injection ii,j : K (Ωi ) → K (Ωj ), i ≥ j, so that 1 ≤ dim ii,1 (K (Ωi )) for all i, with ii+1,1 (K (Ωi+1 )) ⊂ ii,1 (K (Ωi )) ⊂ K (Ω1 ). It follows that F := ∩i ii,1 (K (Ωi )) = {0}, and every element of F extends to a globally defined Killing vector field on Ω. Proof of Theorem 4.1. Let pi , i ∈ N be a dense collection of points and let B(pi , 1/j), j ≥ Ni , be a collection of coordinate balls with compact closure. Let Vi,j be the set of metrics such that K (B(pi , 1/j)) = {0}. By Proposition 4.2 the set Vi,j is open, and it is dense by Theorem 3.1. Then any metric in ∩i,j Vi,j has no local Killing vectors. The argument for conformal Killing vector fields is identical, based on Theorem 2.1.

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5 Three-dimensional initial data sets without KIDs near a point We now pass to the construction of initial-data sets without KIDS. C (Kij , gij ) := (Ji , ρ) be the constraints map,

Let

ρ := R + K 2 − Kij K ij − 2Λ ,

(5.1)

Ji := −2Dj (Kij − Kgij ) ,

(5.2)

where Λ ∈ R is the cosmological constant. In this section, and only is this section, the symbol K denotes the trace of Kij ; K stands for the full extrinsic curvature tensor elsewhere in this paper. Let P denote the linearization of C , and let P ∗ be the formal adjoint of P . By definition, a KID (N, X i ) is a solution of the set of equations P ∗ (N, X) = 0; explicitly, in dimension n (cf., e.g., [10]), D(i Xj) = −N Kij , Di Dj N = N (Rij +KKij −2Kil Kj l )−LX Kij +

1 (n − 1)

(5.3)

Jl X l − (ρ + 2Λ) N 2

gij .

(5.4) One checks that any KID (N, X i ) for which X i , Fij = D[i Xj] , N and Ni := Di N all vanish at p has to be zero in a neighborhood of p. This is proved in the usual way from (5.4) together with D Dj Xi = −Rji k Xk − D (N Kij ) − Dj (N Ki ) + Di (N Kj ) .

(5.5)

(Equation (5.5) is obtained by considering cyclic permutations of first derivatives of (5.3).) Since LX Ric(g) = Ric (LX g), the usual formula for Ric leads to LX Rij

= ∆(N Kij ) + Di Dj (N K) − 2D(i Dl (N Kj)l ) l

−2N Rlijm K lm − 2N R(i Kj)l = ∆(N Kij ) + Di Dj (N K) − 2Dl D(i (N Kj)l ) .

(5.6)

From now on we assume that n = 3. By taking the curl of (5.4) one also finds m Rlij k Dk N = −2LX D[l Ki]j − 2Cj[l Ki]m

Jm X m ρ m − + Λ N gi]j , +2D[l N (Ri]j + Ki]j K − 2Ki]m Kj ) + 4 2

(5.7)

where i = −g in [Dj (N Kkn ) + Dk (N Kjn ) − Dn (N Kjk )] . Cjk

(5.8)

We choose some α, β, λi , ai ∈ R and we consider initial data with the following properties at p: ˚jl = 0 , ˚ij = β δij , K ˚ij = α δij , Di K R (5.9) 3 3

170



˚jk = λx xi y(j zk) + (cyclic), Di R

(5.10)

where (cyclic) means cyclic permutations of (x, y, z), and λx λy λz = 0. (This ansatz is general enough to lead to the required result, and simple enough so that the calculations are manageable. We will show shortly that such initial data exist.) We also assume that ˚lm = ax xi xj y(l zm) + (cyclic) , Di Dj K with λx − λy =

a2y a2x − λx λy

(5.11)

(5.12)

and λx + λy = 0 , λx + λz = 0 , λy + λz = 0 .

(5.13)

For further reference we note that, in local coordinates ξ such that p corresponds to ξ = 0, (5.9)–(5.11) imply R + K 2 − Kij K ij = β +

2α2 + O(ξ 2 ) , 3

Dj (Kij − Kgij ) = O(ξ 2 ) .

(5.14)

In particular, if β = 2Λ − 2α2 /3 then ρ = R + K 2 − Kij K ij − 2Λ = O(ξ 2 ) ,

Ji = −2Dj (Kij − Kgij ) = O(ξ 2 ) . (5.15)

Inserting (5.9) into (5.3) and (5.4) we find that ˚j) D(i X

=

˚ = Di Dj N = ˚ = ∆N

α˚ δij , − N 3 ˚ N 3ρ β + α2 − − 3Λ δij 2 3 ˚ Nβ δij , − 6 ˚ Nβ − . 2

(5.16)

(5.17) (5.18)

Evaluating (5.6) at p, it follows that ˚m Dm R ˚ij = N ˚∆K ˚ij , X

(5.19)

˚D[l R ˚i]j = X ˚i]j . ˚m Dm D[l K N

(5.20)

and, from (5.7), that ˚i = αx xi + αy y i + αz z i , that From (5.19) we find, using the expansion X ˚ax , αy λy = N ˚ay , αz λz = N ˚az , αx λx = N

(5.21)

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and from (5.20) ˚(λx −λy ) = αx ax −αy ay , N ˚(λy −λz ) = αy ay −αz az , N ˚(λz −λx ) = αz az −αx ax . N (5.22) Combining (5.22) with (5.21) and using (5.12), it follows that ˚ = 0 = αx = αy = αz . N

(5.23)

Using (5.23) in the first derivative of (5.4) and in (5.5), we infer that ˚j) = − α δij Dl N ˚ , Dl Di Dj N ˚ = − β δij Dl N ˚. Dl D(i X 3 6

(5.24)

We now take a derivative of (5.6) to obtain (recall that Fij is the anti-symmetric part of Di Xj ) ˚ij + 2F ˚j)m = ˚km Dm R ˚(i| m Dk| R F ˚)∆K ˚ij + 2(Dl N ˚)Dk Dl K ˚ij − 2(Dl N ˚)Dk D(i K ˚j) l . (5.25) (Dk N Somewhat surprisingly, all terms involving α and β have dropped out. We have to ˚ij as compute the different terms entering (5.25). Writing F ˚ij = Ax y[i zj] + (cyclic) , F

(5.26)

we obtain

˚ij = 1 λx (Ay zk − Az yk )y(i zj) + (cyclic) , ˚km Dm R F 2 1 ˚j m = (λx xk yj + λy yk xj )(Ax yi − Ay xi ) + (cyclic) . ˚im Dk R F 4 ˚ = ux xi + uy yi + uz zi , we have that Also, decomposing Di N ˚)∆K ˚ij = (ux xk + uy yk + uz zk )(ax y(i zj) + (cyclic)) . (Dk N

(5.27) (5.28)

(5.29)

and ˚)(Dk Dl K ˚ij −Dk D(i K ˚j) l ) = ux (2ax xk −ay yk −az zk )y(i zj) +(cyclic) . (5.30) 2(Dl N We now insert Equations (5.27)–(5.30) into (5.25). Contracting the resulting equation first with xk y i z j and cyclic permutations thereof, one sees that ux , uy , uz have to vanish. Contracting, then, with terms of the form xk xi y j , xk xi z j , y k y i xj , etc., we see that Ax , Ay , Az are also zero, due to (5.13). Thus (N, X i ) is zero near p. We have thus proved: Lemma 5.1 Consider an initial data set (gij , Kij ) satisfying Eqs. (5.9)–(5.11) together with the conditions on the coefficients spelled out above. For any α, β, Λ ∈ R the algebraic equations for r = (Xi , Fij , N, Di N ) obtained from (5.3)–(5.4) by taking derivatives up to order two imply the vanishing of r(p).

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We also have the following KID-analogue of Proposition 2.7: Proposition 5.2 1. A pair (gij , Kij ) satisfying (5.9)–(5.11) exists. 2. Further, one can choose gij = δij + hij and Kij so that, in local coordinates ξ, the tensor fields gij and Kij satisfy the vacuum constraints up to terms which are of O(|ξ|2 ). Proof. By Lemma 2.9 we can find hij of order O(|ξ|2 ), so that (5.9)–(5.10) are satisfied. For Kij we choose Kij =

α 1 ˚ α δij + (K ∂l ∂m˚ hij )ξ l ξ m , ijlm + 3 2 3

(5.31)

where the second term on the right-hand side of (5.31) is given by the right-hand side of (5.11). One checks that (5.11) is valid. Point 2 follows from (5.15). We are ready now to prove: Theorem 5.3 Let α, β ∈ R, p ∈ M , and consider the collection of all threedimensional data sets (M, Kij , gij ) with (Kij , gij ) ∈ C k × C k+1 , k ≥ 3, with the trace K(p) of Kij (p) equal to α, and with R(p) = β. 1. There exists a non-trivial homogeneous invariant polynomial Q[Kij , gij ] := Q(Rij , DRij , D2 Rij , Kij , DKij , D2 Kij , D3 Kij ) such that if Q[Kij , gij ](p) = 0 at a point p ∈ M , then there exists a neighborhood Op of p for which there exist no non-trivial KIDS on any open subset of Op . 2. Let Ω be a domain in M with p ∈ Ω. a) There exists a variation (δKij , δgij ) ∈ (C ∞ × C ∞ )(Ω), compactly supported in Ω, such that Q[Kij + δKij , gij + δgij ](p) = 0 for all small enough. b) The variation can be chosen so that it preserves the value of R(p) and of K(p). One can further arrange for the trace of Kij + δKij to be equal to K throughout Ω when K is a constant. 3. If (Kij , gij ) is vacuum (with perhaps non-zero cosmological constant) with (Kij , gij ) ∈ C k++1 × C k++2 , ≥ 0, then for any p ∈ Ω the variation of point 2 can be chosen to satisfy the linearized constraint equations up to error terms which are o(r ) in a C k (B(p0 , r)) norm, for small r. Proof. The proof of points 1 and 2 follows closely that of points 1 and 2 of Theorem 2.1. Given any constant α, the set J5 of (2.36) is replaced by the {the set of fourth jets of gij and of third jets of Kij at p that one obtains as gij varies in the set of all Riemannian metrics in normal coordinates near p and as Kij varies in the set of all symmetric tensors with trace equal to a prescribed constant α} .

(5.32)

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The intermediate elements of the proof are provided by Lemma 5.1 and the first part of Proposition 5.2. The variations of gij and of the trace-free part of Kij can be chosen to be polynomials multiplied by a smooth cut-off function, and are therefore smooth. One can then adjust the trace part of Kij to achieve K = α. We further note that the non-vanishing of some derivative of Q follows immediately from the fact that Q(p) is a polynomial, when viewed as a function depending upon the jets of gij and Kij in normal coordinates at p. Further details are left to the reader. In order to prove point 3, for r > 0 it is useful to introduce the following set: W+k

= { jets at p of order ( + k + 1, + k + 2) of (Kij , gij ) such that ρ = o(|ξ|+k ) , J = o(|ξ|+k ) in B(0, r)} .

Here ξ are supposed to be geodesic coordinates near p in the metric gij . Equivalently, if (Kij , gij ) ∈ C k++1 × C k++2 has jets in W+k , then we have Dα ρ(p) = 0 , Dα J(p) = 0 ,

0 ≤ |α| ≤ + k ,

(5.33)

where the α = (i1 . . . ij )’s are multi-indices, with |(i1 . . . ij )| = i1 +. . .+ij . Elements of W+k can be uniquely parameterized as follows: Taylor expanding gij and Pij := Kij − Kgij in geodesic coordinates around p, one can write gij = δij + hijα ξ α + O(|ξ|+k+3 ) , with hi(j1 ...jp ) = 0 , (5.34) 2≤|α|≤+k+2

˚ij + Pij = P

Pijα ξ α + O(|ξ|+k+2 )

(5.35)

1≤|α|≤+k+1

(see, e.g., [23] for a justification of the last condition in (5.34)). Then (5.33) can be solved by induction as follows: (5.33) with |α| = 0 gives ˚2 ˚ 2 i,j (hijij − hjjii ) = i,j Pij − ( i Pii ) + 2Λ , i Pijj = 0 . ˚ij ∈ Rm0 := R6 the first equation defines an affine subspace For any given P isomorphic to Rn2 for some n2 , in the vector space of second Taylor coefficients hijkl . The second equation defines a linear subspace isomorphic to Rm1 in the space of Pijk ’s, for some m1 . To understand (5.33) with |α| ≥ 1 we will need the following: Proposition 5.4 Let k ∈ N, and suppose that dim M = n ≥ 2. 1. For every Ji = Jij1 ...jk ξ j1 . . . ξ jk and p = pj1 ...jk+1 ξ j1 . . . ξ jk+1 there exists Pij = Pijj1 ...jk+1 ξ j1 . . . ξ jk+1 , symmetric in i and j, such that ∂j Pij = Ji , Pii = p . i

i

174



2. For every f = fj1 ...jk ξ j1 . . . ξ jk there exists hij = hijj1 ...jk+2 ξ j1 . . . ξ jk+2 , symmetric in i and j, with hi(jj1 ...jk+2 ) = 0, such that (∂j ∂i hij − −∂i ∂i hjj ) = f . i,j

Proof. Consider a system of linear PDEs Pu = I ,

(5.36)

with constant coefficients, of order p, which can be written in the Cauchy-Kowalevska form with respect to a coordinate z. We claim that if I is a polynomial of order l, then there exists a solution of (5.36) which is a polynomial of order l + p. In order to see that, we note that (5.36) determines, at z = 0, the z-derivatives of u of order greater than or equal to p as polynomials in the remaining variables. So choosing zero Cauchy data on {z = 0} one obtains a polynomial solution in z with polynomial coefficients, hence a polynomial. If P is homogeneous of order p, and if I is in addition homogenous of order l, then the above solution is a homogeneous polynomial of order l + p. In order to prove point 1, we make the ansatz

1 p−2 ∂ W δij , Pij = ∂i Wj + ∂j Wi + n

which leads to a homogeneous second order elliptic system for W , and the above argument applies. In order to prove point 2, we first make the ansatz hij = n1 hll δij , solve the resulting Poisson equation in the class of homogeneous polynomials as described above, and introduce a metric gij = δij + hij . In geodesic coordinates y i the metric gij will have an expansion with some new coefficients satisfying the symmetry condition in (5.34) [23]. One has y i = ξ i + O(|ξ|k+3 ), which implies that the polynomial obtained from the y–Taylor coefficients of gij of order k + 2 provides the desired hij . Proposition 5.4 shows that (5.33) can be used to inductively determine higher order Taylor coefficients hijα and Pijβ in terms of lower order ones, as well as in terms of some free P –coefficients in Rm|β| , for some m|β| ∈ N, and some free h-coefficients in Rn|α| , for some n|α| ∈ N. It follows in particular that W+k is diffeomorphic to RN+k , for some N+k ∈ N. For solutions (Kij , gij ) of the constraint equations, the polynomial Q[Kij , gij ](p) can be expressed as a polynomial of (Kij , gij )-jets at p of order (2, 3), call this ˜ Since the W+k ’s are included in each other in the obvious way, polynomial Q. ˜ can actually be viewed as a function defined on W+k which depends only on Q those coefficients which parameterise W1 . The pair (Kij , gij ) constructed in Propo˜ is non-trivial on W1 . It then follows, sition 5.2 has jets in W1 , which shows that Q

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˜ as in the proof of Theorem 2.1, that any jets in W1 can be -perturbed so that Q(p) does not vanish on the perturbed jet, with the jets of the perturbation belonging ˜ with respect to its arguments to W+k ; by analyticity some of the derivatives of Q will not vanish at p. It should be clear from (5.33) that the perturbed solution satisfies the properties described in the statement of point 3 of Theorem 5.3.

6 Riemannian metrics without static KIDs near a point An interesting class of initial data is provided by the time-symmetric ones, K ≡ 0. In this case the KID equations (5.3)–(5.4) decouple, with X in (5.3) being simply a Killing vector field of g. It remains to analyse the equation for N , Di Dj N = N Rij + ∆N gij .

(6.1)

A solution of (6.1) will be called a static KID, and the set of static KIDs on a set Ω will be denoted by N (Ω). (The origin of the adjective “static” will be clarified shortly.) Since time-symmetric initial data are non-generic amongst all initial data, the results of the previous section do not say anything about non-existence of static KIDs, and separate treatment is required. Taking the trace of (6.1) one obtains, in dimension n ∆N = −

1 NR , n−1

(6.2)

so that (6.1) can be rewritten as Di Dj N = N (Rij −

1 gij R) . n−1

(6.3)

Calculating Dj of (6.3) and commuting derivatives one is led to (recall that the Einstein tensor is divergence-free) N Di R = 0 .

(6.4)

Since the zero-set of solutions of (6.1) has no interior except if N ≡ 0, we conclude that existence of non-trivial static KIDs implies that R is constant. It follows that a non-trivial solution of (6.3) does indeed correspond to initial data for a static solution of the vacuum Einstein equations with a cosmological constant. Further, one immediately obtains that generic C 2 metrics have no static KIDs: it suffices to vary the metric so that the scalar curvature is not constant. From now on we assume dim M = 3. In order to prepare the proof, that generic metrics with fixed constant value of scalar curvature have no static KIDs, we consider a metric g with Ricci tensor at p equal to ˚ij = Axi xj + Byi yj + Czi zj , R

(6.5)

176



where we assume that (A − B)(A − C)(B − C) = 0, and we further suppose that ˚j]l = αx[i yj] zl + βz[i xj] yl + γy[i zj] xl − 1 (α + β + γ)ijl , D[i R 6 with (α, β, γ) = 0. We also impose the condition that ˚= 0 . Di R

(6.6)

(6.7)

Taking a curl of (6.1) we infer that (2Rj[l − Rgj[l )Di] N + gj[l Ri]k Dk N = N D[l Ri]j . The left-hand side of (6.8), with ˚ = axi + byi + czi , Di N

(6.8)

(6.9)

takes the form xj x[l [yi] b(A − C) + zi] c(A − B)] +yj y[l [xi] a(B − C) + zi] c(B − A)] +zj z[l [xi] a(C − B) + yi] b(C − A)] .

(6.10)

˚ vanSince no terms with this index structure occur in (6.6) we obtain that Di N ishes, and using (6.8) allows us to finally conclude that ˚ = Di N ˚=0. N (6.11) The arguments of proof of Proposition 2.7 apply and provide existence of a metric gij = δij + hij satisfying (6.5) and (6.6). Clearly A + B + C can be chosen so ˚ has any prescribed value. Now, we can multiply gij by 1 + α, where α is that R ˚ is zero. By conformal a homogeneous third order polynomial chosen so that Di R ˚ invariance this does not change the value of Bijk , hence of (6.6) (compare (2.1)– ˚ either. A repetition of the remaining (2.2)), and does not change the value of R arguments of Section 5, with Kij there set to zero, gives: Theorem 6.1 Let (M, g) be a Riemannian manifold with g ∈ C k , k ≥ 2. A necessary condition for a non-trivial N (Ω) is that the scalar curvature of g be constant on Ω. Further, in dimension three, and for k ≥ 3, the following hold: 1. There exists a non-trivial homogeneous invariant polynomial Q[g] := Q(Ric, D Ric)

such that if

Q(Ric, D Ric)(p) = 0

at a point p ∈ M , for a metric for which the gradient of the scalar curvature vanishes at p, then there exists a neighborhood Op of p for which there exist no non-trivial static KIDS on any open subset of Op . 2. Let Ω be a domain in M , and let p ∈ Ω. There exists a variation δg ∈ C ∞ (Ω), compactly supported in Ω, such that Q[g + δg](p) = 0 for all small enough. If g ∈ C k++2 , ≥ 0, has constant scalar curvature, then the variation above can be chosen to have the same scalar curvature up to error terms which are o(r ) in a C k (B(p0 , r)) norm, for small r.

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7 Results in general dimensions, with non-explicit orders of differentiability The results obtained so far did require rather unpleasant, tedious, and lengthy calculations, and we will present here an argument which avoids those. The draw-back is that one does not obtain an explicit statement on the number of derivatives involved. However, non-genericity of KIDs is obtained in higher dimensions. Further, the proof below generalizes immediately, e.g., to the Einstein-Maxwell equivalent of the KID equations, the details are left to the reader. The starting point of the analysis in this section is the following result (recall that n = dim M ): Lemma 7.1 1. For any n ≥ 2 and for any signature there exists a real analytic compact pseudo-Riemannian manifold (M, g) without local Killing vectors. 2. For any n ≥ 3 there exists a real analytic compact simply connected Riemannian manifold (M, g) without local conformal Killing vectors. 3. For any n ≥ 3, Λ ∈ R, τ ∈ R there exists a real analytic vacuum initial data set (M, g, K), with cosmological constant Λ, with trg K = τ , and without local KIDs. 4. For any n ≥ 3 and Λ ∈ R, there exists a real analytic Riemannian or Lorentzian manifold (M , g), with dim M = n + 1, satisfying the vacuum Einstein equations with cosmological constant Λ and without local Killing vectors. Remark 7.2 The main point of the Lemma is to construct one single example in each category listed. However, our argument makes it clear that there are actually lots of examples. For instance, the proof below shows that in point 1 for any analytic manifold M which is simply connected, compact, with dim M ≥ 2 one can find a g with the required properties. Remark 7.3 If τ2 ≥

2n Λ, (n − 1)

(7.1)

then one can find an M as in point 3 which is compact (without boundary). The proof in the strict inequality case is given below. If the inequality in (1.2) is an equality, in the proof below one should instead choose (M, γ0 ) to be any real analytic compact Riemannian manifold of positive Yamabe class. The monotone iteration scheme for solving the Lichnerowicz equation can then be handled by an argument in [17]. In that last reference only dimension three is considered, but the proof applies in any dimension. Proof. 1: Let M be any simply connected, compact, analytic Riemannian manifold with dim M ≥ 2, let p ∈ M and let g0 be any smooth Riemannian metric on M

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such that the polynomial Pn = Pn [g] of Theorem 3.1 does not vanish at p. We need analytic approximations of g, for example for 0 ≤ t < we can let gt be the family of metrics obtained by evolving g0 using the Ricci flow, then the metrics gt are indeed real analytic for t > 0. By continuity, reducing if necessary, we will have Pn [gt ](p) = 0, hence gt will have no Killing vectors in a neighborhood of p. Now, a theorem4 of Nomizu [20] shows that on a simply connected analytic manifold every locally defined Killing vector extends to a globally defined one. This implies that for 0 < t < the metrics gt have no Killing vectors on any open subset of M . 2: For n = 3 this follows from Theorem 2.1. For any n ≥ 3 one can argue as follows: Let M be any compact real analytic manifold of dimension not less than three. By [18] there exists on M a metric g with strictly negative Ricci curvature. It is well known that such metrics do not have non-trivial conformal Killing vectors, we recall the proof for completeness: from (2.6) with 2/3 replaced by 2/n it follows that 1 Di Dj Xk = −Rjki l Xl + (ϕi gjk + ϕj gik − −ϕk gij ) , (7.2) n hence 2 ∆Xk = −Rk l Xl − (1 − )ϕk . (7.3) n Multiplying by X k and integrating over M one finds (recall that ϕ = divX) 2 |DX|2 − Ric(X, X) + (1 − )ϕ2 = 0 , n M so that X ≡ 0 if Ric < 0. Approximating g by real-analytic metrics gt , gt → g as t → 0, one will have no conformal Killing vectors for gt when t is small enough by Proposition 4.2. It then follows from Theorem B.1, Appendix B, that the gt ’s will have no local conformal Killing vector field either. 3 and 4: We start by noting that in dimension n = 3, an example of initial data as in point 3 can be obtained using vacuum Robinson-Trautman space-times with cosmological constant Λ (cf., e.g., [5]). Because of the parabolic character of the Robinson-Trautman equation, those metrics are always analytic away from the initial data surface. Further, if the initial metric h0 on S 2 used in the RobinsonTrautman equation has no continuous global symmetries, then it follows from Proposition 4.2 that the evolved metrics ht will not have any continuous global symmetries either, at least for t small enough. It is clear that the resulting fourdimensional metric 4 g will then have no globally defined Killing vectors except the zero one. The non-existence of local Killing vectors follows then from Nomizu’s theorem [20]. Finally, the initial data set of point 3 can be obtained as that induced by 4 g on any hypersurface with trg K = τ in M ; such hypersurfaces can be obtained by solving a Dirichlet problem for the CMC equation on the boundary of a sufficiently small spacelike three-ball [4]. 4 We note that in [20] a Riemannian metric is assumed, but the proofs given there apply to any signature.

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In any case, whatever n ≥ 3 one can proceed as follows: consider, first, τ such that the inequality in (7.1) is strict, let (M, γ0 ) be any real analytic compact Riemannian manifold of negative Yamabe class. Let L0 be any non-zero, γ0 -transverse and traceless tensor on M ; such tensors exist by [6]. For t ∈ [0, ) let Lt be a family of analytic symmetric γ0 -trace free tensors converging to L0 . For example, Lt can be obtained from L0 by heat flow using any analytic metric on M , and removing the γ0 trace. Using the conformal5 method [17] with seed fields (γ0 , Lt ) one obtains a family of real analytic vacuum CMC initial data sets (gt , Kt ) with cosmological constant Λ. Since γ0 has no global conformal Killing vectors, gt will have no global Killing vectors. Now, trgt Kt = τ is a constant, which implies (see Remark 9.2 below) that any global KIDs for (gt , Kt ) are of the form (N = 0, Y ), where Y is a Killing vector of gt , therefore none of the (gt , Kt )’s has global KIDs. In the Lorentzian case we let (M , n+1 gt ) be the maximal globally hyperbolic vacuum development of (M, gt , Kt ), then M is diffeomorphic to R × M (hence simply connected), and n+1 g t is analytic by [1]. In the Riemannian case we let M be any simply connected and connected neighborhood Ut of M × {0} in M × (−1, 1), chosen so that there exists a vacuum metric n+1 gt on Ut with Cauchy data (gt , Kt ) on M × {0}, obtained from the Cauchy-Kowalewska theorem. Suppose that there exists an open nonempty subset Ωt ⊂ M such that Kt (Ωt ) = {0}, where Kt denotes the set of KIDs with respect to (gt , Kt ), then a standard argument [13]6 , using the Cauchy-Kowalewska theorem, shows that there exists a non-trivial Killing vector X in a neighborhood of Ωt in M . By Nomizu’s theorem [20] X extends to a globally defined Killing vector on M , hence (M, gt , Kt ) has a globally defined KID, a contradiction. Thus there are no local KIDs on (M, gt , Kt ), and (M , n+1 gt ) is a vacuum metric without local Killing vectors. This proves point 4, as well as Remark 7.3 in the case of a strict inequality there. To prove point 3 for the remaining values of τ one spans [4], within the Lorentzian solution M just constructed, a CMC hypersurface of prescribed τ = trg K on the boundary of a small spacelike ball. The data induced on the resulting CMC hypersurface provide the desired initial data set. We continue with the question of generic non-existence of KIDs, it should be clear that an identical argument applies to conformal Killing vectors (compare [7, Equation (1.15)]), or to Killing vectors. Let (g, K) be any vacuum analytic initial data on a simply connected manifold M which have no global KIDs. As explained above, it follows from a theorem of Nomizu [20], that such an initial data set will not have any local KIDs. Let r(p) ∈ RM be as in Lemma 5.1, for some appropriate M , and for α ∈ Nn , let us write Dα r = Pα r

(7.4)

5 Since the inequality in (1.2) is strict, a small and a large constant provide barriers for the monotone iteration scheme. 6 Compare the proof of [9, Theorem 2.1.1]; the Cauchy-Kowalewska theorem should be invoked for solvability of Eq. (2.1.5) there, or for uniqueness of solutions of Eq. (2.1.7) there.

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for the linear system of equations obtained by calculating the (k + 1)-st derivatives of r by differentiating (5.3)–(5.4) |α| times, and replacing the lower order derivatives that arise in the process by their values already calculated from the previous equations. Let us write Lk r = 0 for the system of equations that arise from first-order integrability conditions of the system (7.4) with |α| ≤ k. Choose some orthonormal frame, then Lk can be identified with a Nk ×M matrix, for some Nk , with entries built out of the extrinsic curvature tensor K, of the Riemann tensor, and of their derivatives. Let Qk denote the sum of squares of determinants of all M × M sub-matrices of Lk . Then the equation Lk r = 0 admits a non-trivial solution if and only if Qk = 0. Suppose that there exists r0 such that Lk r0 = 0 for all k. One can then use (7.4) to calculate all the jets of r with initial value r0 at p so that the Killing equations are satisfied to infinite order at p by a formal solution determined by those jets. To show convergence of the resulting Taylor series one can proceed as follows: let xi ∈ [−, ]n be local analytic coordinates around p = 0, we can solve the linear equation ∂r = P1 r ∂x1 along the path [−, ] x1 → (x1 , 0, . . . , 0), with initial data r0 at the origin, obtaining an analytic solution there. We can use the function so obtained as initial data for the equation ∂r = P2 r ∂x2 to obtain an analytic solution on [−, ]2 × {0} × · · · × {0}. An inductive repetition of this procedure provides an analytic solution on [−, ]n of the equation ∂r = Pn r , ∂xn such that the equation ∂r = Pk r holds on [−, ]k × {0} × · · · × {0}. ∂xk n − k factors

By choice of r0 the analytic functions Lk r have all derivatives vanishing at the origin, hence they vanish on [−, ]n . Standard arguments imply that the function r so obtained provides an analytic solution of the KID equations in a neighborhood of p. This gives a contradiction with the fact that (g, K) has no local KIDs near p. Therefore there exists k such that Qk is non-zero for the initial data set under consideration. This Qk provides the non-trivial polynomial needed in Theorem 5.3. When the metric involved is Riemannian we can integrate Qk , viewed as a function on the frame bundle, over the rotation group to obtain an invariant polynomial.

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We have therefore proved: Theorem 7.4 Theorem 6.1 remains valid in any dimension, with an invariant polynomial that depends upon some dimension-dependent number k of derivatives of g. Similarly Theorem 5.3 remains valid in any dimension, for some polynomial that depends upon k + 1 derivatives of g and k derivatives of K, for some dimensiondependent number k. In dimension n ≥ 4 Theorem 3.1 remains valid with a polynomial which depends upon some dimension-dependent number k of derivatives of the Weyl tensor. Finally, Theorem 2.1 remains valid in any dimension n ≥ 3 with a polynomial that depends upon some dimension-dependent number of derivatives of the Riemann tensor. Proof. The only statement which, at this stage, might require justification is the extension of Theorem 3.1: this result follows from point 4 of Lemma 7.1, as the polynomial obtained in that case by the proof above depends only upon the Weyl tensor.

8 From approximate linearized solutions to small vacuum perturbations The perturbation results of the previous sections can be used to prove nongenericity of KIDs when no restrictions on ρ and J are imposed. They also apply if, e.g., a strict dominant energy condition ρ > |J| is imposed, for then a sufficiently small perturbation of the data will preserve that inequality. However, some more work is needed when vacuum initial data are considered, and this is the issue addressed in this section. Let Ω ⊂ M be open and connected, and let K (Ω) denote the set of KIDs defined on Ω; each K (Ω) is a finite-dimensional, possibly trivial, vector space. If Ω ⊂ Ω we have the natural map iΩ : K (Ω) → K (Ω ) , with iΩ (x) being defined as the restriction to Ω of the KID x ∈ K (Ω). A local KID vanishing on an open subset vanishes throughout the relevant connected component of its domain of definition, which shows that iΩ is injective. We denote by B(p, r) the open geodesic ball of radius r, and for a r1 > 0 there exists 0 < r2 < r1 such that iΓp (r2 ,r) : K (B(p, r)) → K (Γp (r2 , r)) is bijective.

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The proof rests on the following lemma: Lemma 8.2 For every p ∈ M and r1 > 0 there exists σ ∈ (0, 1) such that iσ : K (B(p, r1 )) → K (Γp (σr1 , r1 )) is bijective. Here iσ denotes iΓp (σr1 ,r1 ) . Proof. As already pointed out, injectivity always holds. Suppose that surjectivity fails, then for every σ ∈ (0, 1) there exists a KID xσ ∈ K (Γp (σr1 , r1 )) such that xσ ∈ iσ (K (B(p, r1 ))). Choose any scalar product h on K (Γp (r1 /2, r1 )). For σ < 1/2 without loss of generality we can assume that the restriction xˆσ of xσ to Γp (r1 /2, r1 ) is h-orthogonal to the image of i1/2 , and that h(ˆ xσ , x ˆσ ) = 1. Since K (Γp (r1 /2, r1 )) is finite-dimensional there exists a sequence σi → 0 such ˆ0 , with h(ˆ x0 , x ˆ0 ) = 1. Further x ˆ0 is h-orthogonal to that xˆσi converges to some x i1/2 (K (B(p, r1 ))). It should be clear from (5.4)–(5.5) that for i such that σ > σi , the sequence of KIDs on Γp (σr1 , r1 ) obtained by restricting xσi to Γp (σr1 , r1 ) converges, and defines a non-trivial KID which restricts to x ˆ0 on Γp (r1 /2, r1 ), with the limit being independent of σ in the obvious sense. This shows that there exists a ˆ0 is the restriction of x0 to Γp (r1 /2, r1 ). KID x0 defined on B(p, r1 )\{p1 } such that x But (5.4)–(5.5) further shows that x0 can be extended to a KID defined on B(p, r1 ), still denoted by x0 . It follows that x ˆ0 = i1/2 (x0 ), which contradicts orthogonality of x ˆ0 with the image of i1/2 . Proof of Proposition 8.1: Let r2 = σr1 , with σ given by Lemma 8.2. Every KID on Γp (r2 , r) induces, by restriction, a KID on Γp (r2 , r1 ), therefore dim K (Γp (r2 , r)) ≤ dim K (Γp (r2 , r1 )). By Lemma 8.2 we have dim K (Γp (r2 , r1 )) = dim K (B(p, r)). Again by restriction we have dim K (B(p, r)) ≤ dim K (Γp (r2 , r)), whence the result. Corollary 8.3 Suppose that K (B(p, r)) = {0}. Then for any > 0 there exists > r1 > 0 such that K (Γp (r1 , r)) = {0}. 2 Recall that the constraints map has been defined by the formula:     J 2(−∇j Kij + ∇i trK)   .  (K, g) :=  ρ R(g) − |K|2 + (trK)2 − 2Λ

(8.1)

The following is one of the key steps of the proof: Theorem 8.4 For ∈ N, ≥ 2, α ∈ (0, 1), p ∈ M , r, η > 0, let the symbol P denote the linearization of the constraints operator (8.1) at (K, g) ∈ C +2,α × C +2,α (B(p, r)), and let

xη = (δKη , δgη ) ∈ C +2,α × C +2,α (B(p, r))

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be an “approximate solution” of the linearized constraint equations defined on B(p, r), in the sense that: P xη (C +1,α ×C ,α )(B(p,r)) ≤ η . 1. There exists a constant C such that if iΓp (σr,r) is surjective for some σ ∈ (0, 1/2], then there exists a solution x ∈ C +2,α × C +2,α (B(p, r)) of the linearized constraint equations supported in B(p, r) such that x − xη (C +2,α ×C +2,α )(B(p,σr)) ≤ Cη .

(8.2)

x is smooth if (K, g) and xη are. 2. For ≥ 4, for any (K0 , g0 ) in C +2,α × C +2,α (B(p, r)), and for any r0 such that B(p, r0 ) has smooth boundary, the constant C can be chosen independently of σ ∈ (0,1/2], (K, g), and r satisfying 0 < r ≤ r0 , for all (K, g) sufficiently close in C +2,α × C +2,α (B(p, r0 )) to (K0 , g0 ). Remark 8.5 The restriction σ ≤ 1/2 is arbitrary, the argument applies with any 0 < σ ≤ σ0 ∈ (0, 1), with a constant in (8.2) depending perhaps upon σ0 . Proof. We use the definitions and notation of [11]. In particular if Ω is a domain with smooth boundary, then Hs2 ∩ Csk,α . Λsk,α = ˚ Roughly speaking, functions in that space behave as o(xs ) near the boundary {x = 0}, with derivatives of order j, 0 ≤ j ≤ k, being allowed to behave as o(xs−j ). In particular if s > k + α then functions in the space above are in C k,α (Ω). We will need the following result [11, Proposition 6.5]: Proposition 8.6 Suppose that (K0 , g0 ) ∈ C k+2,α × C k+2,α (M ), k ≥ 2, α ∈ (0, 1), and let Ω ⊂ M be a domain with smooth boundary and compact closure. For all s = (n + 1)/2, (n + 3)/2, the image of the linearization P , at (K0 , g0 ), of −s+2 the constraints map, when defined on Λ−s+1 k+2,α × Λk+2,α (Ω), is

−s (J, ρ) ∈ Λ−s k+1,α × Λk,α such that (J, ρ), (Y, N )(L2 ⊕L2 )(Ω,dµg0 ) = 0 for all (Y, N ) ∈ H1s−n × H2s−n satisfying P ∗ (Y, N ) = 0 . −s+2 Further P −1 (0) ⊂ Λ−s+1 k+2,α × Λk+2,α splits.

The proof of point 1 of Theorem 8.4 will proceed in two steps: Step 1: We set M := B(p, r), k = , (g0 , K0 ) = (g, K), and we use Proposition 8.6 with s = s1 for some s1 < −1. Now, for such s the space K0 ⊂ K(B(p, r)) above is the space of KIDs on B(p, r) which vanish at S(p, r) := ∂B(p, r) together with

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their first derivatives; but Equations (5.4)–(5.5) imply that there are no such nontrivial KIDs. It follows that P is surjective, with the splitting property being −s+2 equivalent to the fact that there exists a closed subspace X ⊂ Λ−s+1 k+2,α × Λk+2,α such that the restriction of P to X is an isomorphism. This shows that there exists −s+2 x ˆη ∈ Λ−s+1 k+2,α × Λk+2,α satisfying ˆ xη Λ−s+1 ×Λ−s+2 ≤ Cη , k+2,α

k+2,α

and P (ˆ xη ) = −P (xη ) ⇔ P (xη + xˆη ) = 0. Step 2: Now, because s = s1 < −1, the correction term xˆη could be blowing-up near S(p, r), while we want a solution which vanishes there to rather high order. To correct that, let ϕ be any smooth non-negative function which is identically one on B(p, 5r/8), and vanishes on Γp (3r/4, r), set yη = ϕ(xη + x ˆη ) . 8.6 Then P (yη ) is supported in Γp (5r/8, 3r/4) ⊂ Γ(σr, r). We now use Proposition ˜η ∈ C +2,α × C +2,α (Γp (σr, r)), once again, with some s = s2 > + 3, to find x which extends by zero both through S(p, σr) and through S(p, r) in a C +2,α × C +2,α manner, such that P (˜ xη + yη ) = 0 ⇐⇒ P (˜ xη ) = −P yη =: zη . This will be possible if and only if zη is orthogonal in L2 (Γp (σr, r)) to K0 (Γp (σr, r)), where now K0 (Γp (σr, r)) coincides with the space of all KIDs on Γp (σr, r). Let, thus, w = (Y, N ) ∈ K0 (Γp (σr, r)), by hypothesis there exists a KID w ˆ defined on ˆ We then have B(p, r) such that w is the restriction to Γp (σr, r) of w. w, P yη = w, ˆ P yη = Pˆ ∗ w, ˆ yη = 0 . Γp (σr,r)

B(p,r)

B(p,r)

Here the first and the second equalities are justified because P yη is supported in Γp (σr, 3r/4), while the last one follows because, by definition of a KID, P ∗ w ˆ = 0. This provides the desired x ˜η . Setting xη = ϕ(x + xˆη ) + x ˜η , point 1 is proved. To prove point 2, we first note that the value of σ does not affect the constant C, as that constant arises from step 1 of the proof of point 1: the perturbation x ˜η from step 2, which could depend upon σ, is supported away from B(p, σr). The result is proved now by the usual contradiction argument: Consider the map ⊥

⊥

g −s −s −s πK⊥g L,x,xs−n/2 : K0 g ∩ (Λ−s k+3,α × Λk+4,α ) −→ K0 ∩ (Λk+1,α × Λk,α ) , 0

(8.3)

with L,x,xs−n/2 being a regularized version, as in [11], of the map Lx,xs−n/2 of [10, Section 5]. Equation (8.2) will fail to hold only if there exists a sequence of radii ⊥ rn and data (Kn , gn ) on B(p, rn ) near (K0 , g0 )|B(p,rn ) , with KIDs (Yn , Nn ) ∈ K0 g such that (Yn , Nn )Λ−s ×Λ−s = 1 and L,x,xs−n/2 (Yn .Nn )|Λ−s ×Λ−s ≤ 1/n . 3,α

4,α

1,α

0,α

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Consider an extracted sequence, still denoted by rn , converging to r∞ . If r∞ > 0, then (K0 , g0 )|B(p,r∞ ) would admit a KID vanishing, together with its first derivatives, at S(p, r∞ ), a contradiction. On the other hand suppose that r∞ = 0, introduce geodesic coordinates for the metrics (Kn , gn ) centred at p; this might lead to a loss of two derivatives of the metric, so we increase the threshold on from ˜ n , gñ ) on B(p, 1) obtained by scaling up the two to four. Consider the sequence (K ˜ ball B(p, rn ) to B(p, 1). Then (Kn , gñ ) converges to (0, δ), where δ is the Euclidean metric on B(p, 1). As before one obtains a contradiction because there are no KIDs vanishing, together with their first derivatives, on S(p, 1) for (K, g) = (0, δ). Smooth solutions can be obtained proceeding as above, but working instead with exponentially-weighted rather than power-weighted spaces. The main result of this section is the following (see footnote 2): Theorem 8.7 Let M be a compact manifold with boundary, suppose that ≥ 0 (n), α ∈ (0, 1) for some 0 (n) ( 0 (3) = 6), and let (M, K, g) be a C ,α × C ,α vacuum initial data set such that K (M ) = {0} . For any p ∈ M \ ∂M and for any > 0 there exists r > 0 and an -small, in a C ,α × C ,α topology, vacuum perturbation (K , g ) of (K, g) such that K (U ) = {0} for all U such that U ∩ B(p, r) = ∅ . Further, (K , g ) can be chosen to coincide with (K, g) in a neighborhood of ∂M . Proof. For definiteness in the proof we will assume n = 3, for n > 3 in the argument below Theorem 5.3 should be replaced by its higher-dimensional generalization provided by Theorem 7.4. If the polynomial Q of point 1 of Theorem 5.3 does not vanish at p, we let r > 0 be small enough so that Q has no zeros on B(p, r). Otherwise, let δx := (δK, δg) be as in point 3 of Theorem 5.3 with = 1 and k = 3. Let > 0, as Q is a polynomial we have Q[x + δx](p) = j (Q(j) [δx])(p) + O(j+1 ) , for some j ≥ 1 such that (Q(j) [δx])(p) = 0. By Proposition 8.1 for any r > 0 there exists σr ∈ (0, 1) such that the conditions of Theorem 8.4 are satisfied. We then have P δx(C 3,α ×C 2,α )(B(p,r)) ≤ P δx(C 4 ×C 3 )(B(p,r)) ≤ C1 r , and Theorem 8.4 provides a solution δ˜ x of the linearized constraint equations supported in B(p, r) such that δx − δ˜ xC 4,α (B(p,σr r)) ≤ CC1 r . Choosing r small enough so that CC1 r ≤ one obtains Q[x + δ˜ x](p) = j (Q(j) [δx + O()])(p) + O(j+1 )) = j (Q(j) [δx])(p) + O(j+1 ) .

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Since δ˜ x satisfies the linearized constraint equations and since K (M ) = {0}, it follows from [10, Theorem 5.6] together with the regularization technique from [11] that for small enough we can find δˆ x(), with δˆ x()C 4 (B(p,r)) ≤ C2 2 , such that x + δ˜ x + δˆ x() satisfies the vacuum constraint equations. Choosing small enough so that C2 ≤ 1/2 we then obtain Q[x + δ˜ x + δˆ x()](p)

x + O(1/2 )])(p) + O(j+1 ) = j (Q(j) [δ˜ = j (Q(j) [δx])(p) + O(j+1/2 ) = 0

for small enough. Replacing r by a smaller number if necessary so that Q[x + δ˜ x + δˆ x()] has no zeros on B(p, r), the resulting data set has no KIDs in any subset of B(p, r) by point 1 of Theorem 5.3. The construction described so far leads to a perturbed initial data set which agrees with the starting one at ∂M to arbitrarily high order. Consider, next, a collar neighborhood Ns = {p ∈ M : d(p, ∂M ) < s} of ∂M . Arguments as in Lemma 8.2 show that K (Ms := M \ Ns ) = {0} for s small enough. Applying the result already established to Ms one obtains a perturbation which vanishes on Ns . An identical proof, based on the results in Section 6, gives: Theorem 8.8 Let (M, g) be a n-dimensional C ,α compact Riemannian manifold with boundary, suppose that ≥ 0 (n), α ∈ (0, 1) for some 0 (n), and suppose that g has constant scalar curvature s. Assume that there are only trivial static KIDs, N (M ) = {0} . For any p ∈ M \ ∂M and for any > 0 there exists r > 0 and an -small, in a C ,α topology, perturbation g of g with scalar curvature s such that N (U ) = {0} for all U such that U ∩ B(p, r) = ∅ . Further, g can be chosen to coincide with g in a neighborhood of ∂M .

9 Proofs of Theorems 1.2 and 1.3 Proof of Theorem 1.2: Let Q be the polynomial of Theorem 5.3, set Vˆp = {vacuum initial data such that Q[K, g](p) = 0} , then Vˆp is open and contained in Vp . To show density, let Mi ⊂ M be a sequence of relatively compact domains with smooth boundary such that M = ∪i Mi . The argument of the proof of Lemma 8.2 shows that K (Mi ) = {0} for i large enough. Point 1 follows then from Theorem 8.7 with M there equal to M i . The timesymmetric case is obtained similarly by Theorem 8.8. Point 2 is established by repeating the argument of the proof of Theorem 4.1.

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Proof of Theorem 1.3: Openness follows from Proposition 4.2, it remains to establish density. We start by showing that for spatially compact CMC initial data KIDs are “purely spacelike”. Somewhat more generally, one has: Proposition 9.1 Consider a vacuum initial data set (M, g, K) with constant τ := trg K = 0, suppose that (1.2) holds, and assume that (M, g) is geodesically complete (perhaps as a manifold with boundary). Let (N, Y ) be a KID on M satisfying lim

sup |N (q)| = 0 ,

r→∞ q∈S (r) p

(9.1)

for some p ∈ M , where Sp (r) is the boundary of the geodesic ball of radius r centred at p. If N ≡ 0, then K is pure trace, M is compact and (M, g) is Einstein. Remark 9.2 A KID satisfying (9.1) will be called asymptotically tangential ; a KID with N ≡ 0 will be called tangential. In the compact boundaryless case we have Sp (r) = ∅ for r large enough, so all KIDs are asymptotically tangential. Proof. We note that if M has a boundary, then Sp (r) ∩ ∂M := ∂Bp (r) ∩ ∂M = ∅ for r large, so that (9.1) implies that N vanishes on ∂M . The KID equations imply 2 2Λ ˜ 2 + (trg K) − 2Λ ∆N = |K|2 − N = |K| N, (n − 1) n (n − 1) ˜ is the trace-free part of K. Equation (1.2) and the maximum principle where K ˜ ≡ 0, with (1.2) being an equality, and N = const, or N ≡ 0. show that either K In the former case the KID equations further imply Ricci flatness of g. The case N ≡ 0 is compatible with (9.1) only if Sp (r) = ∅ for r sufficiently large, which is equivalent to compactness of M . We note the following straightforward consequence of Theorem 2.1 and Proposition 4.2: Proposition 9.3 Consider the collection of Riemannian metrics on a three-dimensional manifold with a C k (weighted, with arbitrary weights, in the non-compact case) topology, k ≥ 5. The set of such Riemannian metrics which have no globally defined conformal Killing vectors is open and dense. Proof. Choose any relatively compact Ω ⊂ M , then by Theorem 2.1 and Proposition 4.2 there exists an open and dense set of metrics which have no conformal Killing vector fields on Ω, then those metrics do not have globally defined conformal Killing vector fields either. k,α defined in Appendix A. The weights ϕ and ψ Our next result uses spaces Cϕ,ψ in our next result have to be chosen in a way compatible with the conformal method in the asymptotically flat regions [8], similarly in the asymptotically hyperbolic regions [2], while ϕ = ψ = 1 in the compact case. The differentiabilities here are different, as compared to Theorem 1.3, because under the CMC restriction the conformal method can be used:

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Corollary 9.4 There exists k1 (n), with k1 (3) = 5, such that for k ≥ k1 (n) and α ∈ k,α k−1,α × Cϕ,ψ -open and dense collection (0, 1) the following holds: There exists a Cϕ,ψ of vacuum CMC initial data sets (M, g, K) which are either 1. asymptotically flat with compact interior (then Λ = 0), or 2. asymptotically hyperbolic as in [2], or 3. defined on a compact M , with Λ satisfying (1.2), and which do not have any asymptotically tangential KIDs. k,α k−1,α × Cϕ,ψ such that g Proof. Let U be the set of vacuum initial data (g, K) ∈ Cϕ,ψ is Einstein. This class of initial data obviously forms a closed set with no interior. k,α k−1,α × Cϕ,ψ initial Let V1 be the complement of U within the set of all vacuum Cϕ,ψ data, then V1 is open and dense. Choose any p ∈ M , and let V2 be the set of initial data in V1 such that K is not pure trace, and such that the polynomial Q[H, DH, D2 H] of Theorem 2.1 does not vanish at p, then V2 is open. Consider any (g, K) which is not in V2 and which is not in U , by Proposition 9.3 for any > 0 there exists a metric g () which is in V2 (and therefore has no conformal 5 Killing vectors) such that g − g ()Cϕ,ψ ≤ . Using K as the seed solution for the extrinsic curvature, the conformal method [2, 8, 17] allows one to solve for a nearby solution (M, g(), K()) of the vacuum constraint equations. Let (N, Y ) be a KID for g(). By Proposition 9.1 the KID (N, Y ) is tangential, N ≡ 0, which implies that Y is a Killing vector field of g(). Now g() is a conformal deformation of g (), therefore Y is a conformal Killing vector field of g (), hence Y = 0. It follows that V2 provides the desired open and dense set.

On compact boundaryless manifolds all KIDs are asymptotically tangential, and Theorem 1.3 is established in this case. Consider, next, the asymptotically flat case, with an r−β weighted topology, β ∈ (0, n − 2). Recall that we want to prove density of metrics without KIDs. For such β the result can be established as follows: consider the set of solutions of the constraint equations on R3 \ B(0, R), which approach (g, K) at S(0, R) exponentially fast as in [11, Theorem 6.6], and which are r−β -asymptotically flat. A straightforward generalization of [11, Corollary 6.3] applies to this space of initial data and shows that this collection forms a manifold. It follows that each linearized solution of the constraint equations constructed as at the beginning of the proof of Theorem 8.7 is tangent to a curve of solutions, which coincide with (g, K) away from the asymptotic region R3 \ B(0, R). This establishes point 1 of Theorem 1.3. We note that the condition trg K = 0 is not necessarily preserved by the perturbation just constructed. However, it follows from the implicit function theorem, or from the results of Bartnik [3], that the deformed initial data set on R3 \ B(0, R) can be deformed in the associated space-time to obtain a data set with vanishing mean extrinsic curvature, proving point 2. Point 3 is established as above using [11] in the K ≡ 0 setting. In the conformally compactifiable case the argument is identical, based on [11, Theorem 6.7].

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In the asymptotically flat case with β = n − 2 some more work is needed. For simplicity we consider only smooth initial data, but the construction works also in the finite differentiability case. The idea is to obtain solutions up to kernel using the techniques of [10, 14], and to show that one can correct for the kernel by changing the metric in the asymptotic region, the argument proceeds as follows. Let Γ(R, 2R) be a coordinate annulus, with inner radius R and outer radius 2R, contained in the asymptotically flat region, let x = (K, g). Let δx = (δK, δg) be a solution of the linearized constraint equations supported in Γ(5R/4, 7R/4), constructed as at the beginning of the proof of Theorem 8.7, so that x = x + δx has no KIDs on Γ(R, 2R) for all positive small enough. By construction x fails to solve the constraint equations by O(2 ). We use the terminology of [10, Sections 8.1 and 8.2]. Let Q0 = (m0 , p 0 , c0 , J0 ) denote the Poincaré charges of x0 = x, and for Q in a neighborhood of Q0 let yQ = (KQ , gQ ) be a reference family of metrics obtained on Rn \B(R) as follows: by scaling, boosting, and space-translating (K, g) one is led to a family of initial data sets with mass m, ADM-momentum p, and centre of mass c covering a neighborhood of (m0 , p0 , c0 ). Choosing R large enough, a construction in [22] can be used to deform each of the solutions obtained so far to initial data sets with arbitrary angular momentum in a neighborhood of J0 .7 One can now glue x with yQ using the techniques described in detail in [10, 14] obtaining, for + |Q − Q0 | small enough, on Γ(R, 2R) a “solution up-to-kernel” z,Q = (K,Q , g,Q ) which smoothly extends across the inner sphere B(0, R) to x, which smoothly extends across the exterior sphere B(0, 2R) to yQ , and which differs from x by terms which are quadratic in and in Q − Q0 . Making and |Q − Q0 | smaller if necessary, the arguments presented in Sections 8.1 and 8.2 of [10] show that one can find Q() so that z,Q() solves the constraints, providing the desired solution without global KIDs.

A

Topologies

In this paper we prove both density and openness results, and there does not seem to be a topology which captures both features in an optimal way. The aim of this appendix is to discuss those issues in some detail. As already pointed out in the introduction, a possible topology for which our results hold is the following: one chooses some smooth complete Riemannian metric h on M , which is then used to calculate norms of tensors and their hcovariant derivatives; we shall denote this topology by T k (h). If M is compact, the resulting topology is h-independent, and all our results in the compact case hold with such topologies, for appropriate k’s. However, when M is not compact, there exist choices of h which will lead to different topologies; nevertheless, for each such 7 The point of the current construction is to obtain a “reference family”, as defined in [10], near the initial data we started with. An alternative way is to first deform the initial data to data which are exactly Kerr outside of a compact set with large radius, and use the Kerr family as the reference family.

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choice Theorems 1.2 holds. Further, all the results, except the perturbations that remove global KIDs in an asymptotically flat or asymptotically hyperbolic region, k,α remain true if, e.g., weighted Cφ,ϕ topologies defined with respect to h are used, as defined in [10], with norm k uC k,α(h) = supx∈M i=0 ϕφi ∇(i) u(x)h φ,ϕ , (i) (i) u(y) h + sup0 =dh (x,y)≤φ(x)/2 ϕ(x)φi+α (x) ∇ u(x)−∇ dα (x,y) h

k,α with any weight functions φ and ϕ; we shall denote such topologies by Tφ,ϕ (h). Finally, all openness and density results established in this paper, including statements involving the field equations, will hold with any choice of h and weight functions except for the following restriction: if (M, g, K) contains an asymptotically flat region, and one wishes to construct a perturbation that gets rid of a globally defined KID while preserving the field equations, then h should be chosen to be, e.g., the Euclidean metric in the asymptotically flat region, with the weights φ = r, ϕ = r−β , for some β ∈ (0, n − 2]. Similarly, in the context of Corollary 9.4 and of point 4 of Theorem 1.3, the weights in the asymptotically hyperbolic region should be chosen in a way compatible with the asymptotic conditions in the conformally compactifiable region as in [2]. While the above topologies seem satisfactory for most purposes, the optimal topology for perturbations that get rid, e.g., of Killing vectors, at a given point p, is that of convergence in the space of kth jets of the metric at p, with k ≥ k0 (n), for some k0 (n) as described above, on the space of metrics which coincide with the starting metric g away from a compact neighborhood of p. However, this space is unnecessarily small for our openness results, which do not hold in such a weak topology in any case; see also Remark 4.3.

B “Local extends to global” in the simply connected analytic setting In this appendix we wish to generalise Nomizu’s theorem [20] concerning Killing vectors to conformal Killing vectors and to KIDs. It should be clear that our argument applies to a large class of similar overdetermined systems with analytic coefficients, such as, e.g., those considered in [7]. In particular the proof given here applies to Killing vector fields in arbitrary signature, and seems to be somewhat simpler than the original one. Theorem B.1 Let (M, g) be a simply connected analytic pseudo-Riemannian manifold. 1. Every locally defined conformal Killing vector extends to a globally defined one. 2. If, moreover, K is also analytic then every locally defined KID extends to a globally defined one.

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Proof. We give the proof for KIDs, the argument for conformal Killing vector fields is identical. Let r, Pα and Lk be as in Section 7. We note the following: Lemma B.2 Consider a KID x defined on an open set Ω, let γ : [0, 1] → M be a differentiable path such that γ : [0, 1) → Ω, with γ(1) ∈ γ([0, 1)). Then there exists a neighborhood U of γ([0, 1]) and a KID xˆ defined on U such that x = xˆ on γ([0, 1)). Proof. Equation (7.4) shows that each covariant derivative Dα r of r satisfies along γ the linear equation D (Dα r ◦ γ) = γ˙ µ ((Dµ Dα r) ◦ γ) = (Pµα r) ◦ γ , ds with the multi-index µα in Pµα defined in the obvious way. It follows that each Fα (s) := (Dα r ◦ γ) (s) extends by continuity to some values, denoted by Fα (1), such that Fα (1) = Pα (γ(1))F (1) , where F (1) = lims→1 r(γ(s)). By continuity the integrability conditions Lk = 0 are satisfied by F (1), and therefore, by the argument given after (7.4), there exists > 0 and a solution of the KID equations defined on B(γ(1), ) for some > 0. We can cover γ([0, 1]) by a finite number of open balls Bi := B(γ(si ), ri ), i = 1, . . . , N , such that s1 = 0, sN = 1, rN ≤ , with the balls pairwise disjoint except for the neighboring ones: Bi ∩ Bj = ∅ if |i − j| > 1. It should be clear that the solution just constructed on B(γ(1), ) coincides with that which exists already on the overlap with B(γ(sN −1 ), rN −1 ). The desired neighborhood is obtained by setting U = ∪i B(γ(si ), ri ). Returning to the proof of Theorem B.1, let q be any point in Ω, let p ∈ M , and let γ : [0, 1] → M be any piecewise differentiable path without self-intersections with γ(0) = q, γ(1) = p. Let I ⊂ [0, 1] be the set of numbers s such that there exists a neighborhood Us of γ|[0,s] and a KID xs defined on Us such that xs = x near p. Then I is open by definition, it is closed by Lemma B.2, therefore I = [0, 1]. We have thus shown: Lemma B.3 For any piecewise differentiable path γ : [0, 1] → M without selfintersections, with γ(0) ∈ Ω, there exist a neighborhood U of γ and a KID xγ defined on U , coinciding with x on U ∩ Ω. Any γ as in Lemma B.3 allows us therefore to extend x to a neighborhood of p. It remains to show that this extension is γ–independent. Let thus γ and

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γˆ be two differentiable paths from q to p without self-intersections, since M is simply connected there exist a homotopy of differentiable paths γt : [0, 1] → M , t ∈ [0, 1], with γt (1) = p, γt (0) = q, γ0 = γ and γ1 = γˆ . If any γt self-intersects at s1 and s2 , with s1 < s2 , we replace it by a new path, still denoted by γt , obtained by staying at γt (s1 ) for s ∈ [s1 , s2 ]; this procedure is repeated until all self-intersections of γt have been eliminated. Let r(t) denote the value of r at p obtained from Lemma B.3 by following γt , then r is a continuous function of t. The set of t’s for which r(t) = r(0) is closed by continuity of r, it is open by Lemma B.3, hence r(0) = r(1), which establishes Theorem B.1. Acknowledgments. We are grateful to L. Andersson, A. Cap, E. Delay, A. Fischer, J. Isenberg, J. Lewandowski, M. McCallum and D. Pollack for comments, discussions and suggestions.

References [1] S. Alinhac and G. Métivier, Propagation de l’analyticité des solutions de systèmes hyperboliques non-linéaires, Invent. Math. 75, 189–204 (1984). [2] L. Andersson and P.T. Chru´sciel, On asymptotic behavior of solutions of the constraint equations in general relativity with “hyperboloidal boundary conditions”, Dissert. Math. 355, 1–100 (1996). [3] R. Bartnik, The existence of maximal hypersurfaces in asymptotically flat space-times, Comm. Math. Phys. 94, 155–175 (1984). [4]

, Regularity of variational maximal surfaces, Acta Math. 161, 145–181 (1988).

[5] J. Biˇcák and J. Podolsk´ y, The global structure of Robinson-Trautman radiative space-times with cosmological constant, Phys. Rev. D 55, 1985–1993 (1996), gr-qc/9901018. [6] J.-P. Bourguignon, D.G. Ebin, and J.E. Marsden, Sur le noyau des opérateurs pseudo-différentiels à symbole surjectif et non injectif, C. R. Acad. Sci. Paris Sér. A-B 282, Aii, A867–A870 (1976). ˇ [7] T. Branson, A. Cap, M. Eastwood, and R. Gover, Prolongations of geometric overdetermined systems, (2004), math.DG/0402100v2. ´ Murchadha, The boost problem in general rela[8] D. Christodoulou and N. O tivity, Comm. Math. Phys. 80, 271–300 (1980). [9] P.T. Chru´sciel, On uniqueness in the large of solutions of Einstein equations (“Strong Cosmic Censorship”), Australian National University Press, Canberra, 1991.

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[10] P.T. Chru´sciel and E. Delay, On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. de France 94, 1–103 (2003), gr-qc/0301073v2. [11]

, Manifold structures for sets of solutions of the general relativistic constraint equations, Jour. Geom Phys. (2004), in press, gr-qc/0309001v2.

[12] P.T. Chru´sciel, J. Isenberg, and D. Pollack, Initial data engineering, (2004), gr-qc/0403066. [13] B. Coll, On the evolution equations for Killing fields, Jour. Math. Phys. 18, 1918–1922 (1977). [14] J. Corvino and R. Schoen, On the asymptotics for the vacuum Einstein constraint equations, gr-qc/0301071, 2003. [15] D.G. Ebin, The manifold of Riemannian metrics, Global Analysis, Proc. Sympos. Pure Math., vol. 15, 1970, pp. 11–40. [16] L. Hörmander, The boundary problems of physical geodesy, Arch. Rat. Mech. Analysis 62, 1–52 (1976). [17] J. Isenberg, Constant mean curvature solutions of the Einstein constraint equations on closed manifolds, Class. Quantum Grav. 12, 2249–2274 (1995). [18] J. Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. (2) 140, 655– 683 (1994). [19] V. Moncrief, Space-time symmetries and linearization stability of the Einstein equations. II, Jour. Math. Phys. 17, 1893–1902 (1976). [20] K. Nomizu, On local and global existence of Killing vector fields, Ann. Math. 72, 105–120 (1960). [21] F.A.E. Pirani, Introduction to gravitational radiation theory, Lectures on general relativity, Brandeis, vol. 1, Prentice Hall, Englewood Cliffs, New Jersey, 1965. [22] R. Schoen, in preparation, (2003). [23] T.Y. Thomas, The differential invariants of generalized spaces, Cambridge University Press, 1934.

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Robert Beig∗ Institut f¨ ur Theoretische Physik Universit¨ at Wien Boltzmanngasse 5 A-1090 Vienna Austria email: [email protected] Piotr T. Chru´sciel∗ Département de Mathématiques Faculté des Sciences Parc de Grandmont F-37200 Tours France email: [email protected] Richard Schoen∗ Department of Mathematics Stanford University Palo Alto USA email: [email protected] Communicated by Sergiu Klainerman submitted 22/02/04, accepted 28/07/04


∗

ESI visiting scientist




A Product Formula Related to Quantum Zeno Dynamics Pavel Exner and Takashi Ichinose Abstract. We prove a product formula which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection P on a separable Hilbert space H, with the convergence in L2loc (R; H). It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace Ran P . The convergence in H is demonstrated in the case of a finite-dimensional P . The main result is illustrated in the example where the projection corresponds to a domain in Rd and the unitary group is the free Schr¨ odinger evolution.

1 Introduction The fact that the decay of an unstable system can be slowed down, or even fully stopped in the ideal case, by frequently repeated measurements checking whether the system is still undecayed was noticed first by Beskow and Nilsson [BN]. It was only decade later, however, when Misra and Sudarshan [MS] caught the imagination of the community by linking the effect to the well-known Zeno aporia about a flying arrow. While at first the subject was rather academical, in recent years the possibility of observing Zeno-type effects experimentally has become real and at present there are scores of physical papers discussing this topic. On the mathematical side, the first discussion of the continuous observation appeared in [Fr]. Two important questions, however, namely the existence of Zeno dynamics and the form of its effective Hamiltonian have been left open both in this paper and later in [MS]. The second problem is particularly important when the subspace into which the state of the system is repeatedly reduced has dimension larger than one. A partial answer was given in [Ex, Sec. 2.4] where it was shown that the results of Chernoff [Ch1, Ch2] allow to determine the generator of the Zeno time evolution naturally through the appropriate quadratic form. Our interest to the problem was rekindled by a recent paper by Facchi et al. [FPS] who studied the important special case when the presence of a particle in a domain of Ω ⊂ Rd is repeatedly ascertained. Using the method of stationary phase the authors showed that the Zeno dynamics describes in this case the free particle confined to Ω, with the hard-wall (Dirichlet) condition at the boundary of the domain. The result cannot be regarded as fully rigorous, because detailed properties of the convergence are not worked out, but the idea is sound without any doubt. In the present paper we combine the results of [Ch1, Ch2] with that of Kato [Ka2] to address this question in a general setting. We show that if the natural

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effective Hamiltonian mentioned above is densely defined – which is a nontrivial assumption – then the Zeno dynamics exists and the said operator is its generator in a topology which includes an averaging over the time variable – cf. Theorem 2.1 for exact statement (a part of the present result given in Corollary 2.3 was announced in [EI]). Our conclusion cannot be thus regarded as fully satisfactory from the mathematical point of view, because the natural topology to be used here is given by the norm of the Hilbert space, and in this respect an important part of the problem remains open. We demonstrate, however, the strong convergence in H for the particular case when the projections involved are finite-dimensional – cf. Theorem 2.4. On the other hand, from the physical point of view the result given in Theorem 2.1 is quite plausible taking into account that any real measurement is burdened with errors – see Remark 2.5 below. We will formulate the theorems together with their corollaries in the next section. Theorem 2.1 will be then proven in Sections 3 and 4, Theorem 2.4 in Section 5. As an example we discuss in the concluding section reduction of a free dynamics to a domain in Rd by permanent observation. We will establish that the Zeno generator mentioned above is in this case the Dirichlet Laplacian, obtaining thus in a different way the result of the paper [FPS].

2 The main result Throughout the paper H will be a nonnegative self-adjoint operator in a separable Hilbert space H, and P will be an orthogonal projection. The nonnegativity assumption is made for convenience; our main result extends easily to any selfadjoint operator H bounded from below as well as one bounded from above, i.e., to each semi-bounded self-adjoint operator in H. Consider the quadratic form u → H 1/2 P u2 with form domain D[H 1/2 P ]. Note that H 1/2 P involved here is a closed operator and HP has the same property. Let HP := (H 1/2 P )∗ (H 1/2 P ) be the self-adjoint operator associated with this quadratic form. In general, HP may not be densely defined in which case it is a self-adjoint operator in a closed subspace of H. More specifically, it is obviously defined and acts nontrivially in a closed subspace Ran P , the closure of the form domain D[H 1/2 P ], while in the orthogonal complement (Ran P )⊥ it acts as zero. The quadratic form u → H 1/2 P u2 defined on D[H 1/2 P ] is a closed extension of the form u → P u, HP u defined on D[HP ], but the former is not in general the closure of the latter. Indeed, if H is unbounded, D[H] is a proper subspace of D[H 1/2 ]. Take u0 ∈ D[H 1/2 ]\D[H] such that the vector H 1/2 u0 is nonzero, and set P to be the orthogonal projection onto the one-dimensional subspace spanned by u0 . Taking into account that D[HP ] = {u ∈ H; P u ∈ D[H]} which u0 = P u0 does not belong to, we find HP u = 0 for u ∈ D[HP ], while H 1/2 P u0 = H 1/2 u0 = 0 by assumption. To describe our results, we denote by L2loc ([0, ∞); H) = L2loc ([0, ∞)) ⊗ H the Fréchet space of the H-valued strongly measurable functions v(·) on [0, ∞)

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such that v(·) is locally square integrable there, with the topology induced by T 1/2 the semi-norms v → 0 v(t)2 dt for a countable set {T }∞ =1 of increasing positive numbers accumulating at infinity, lim→∞ T = ∞. In a similar way one defines the Fréchet space L2loc (R; H) = L2loc (R) ⊗ H. Our main result can be stated as follows: Theorem 2.1 Let H be a nonnegative self-adjoint operator on a separable Hilbert space H and P an orthogonal projection. Let t → P (t) be a strongly continuous function whose values are orthogonal projections in H, defined in some neighborhood of zero, with P (0) =: P . Moreover, suppose that D[H 1/2 P (t)] ⊃ D[H 1/2 P ] and limt→0 H 1/2 P (t)v = H 1/2 P v holds for v ∈ D[H 1/2 P ]. If the operator HP specified above is densely defined in the whole Hilbert space H, then for every f ∈ H and ε = ±1 it holds that [P (1/n) exp(−iεtH/n)P (1/n)]nf −→ exp(−iεtHP ) P f ,

(2.1)

[P (1/n) exp(−iεtH/n)]n f −→ exp(−iεtHP ) P f , [ exp(−iεtH/n) P (1/n)]n f −→ exp(−iεtHP ) P f ,

(2.2) (2.3)

in the topology of L2loc (R; H) as n → ∞. Note that HP differs in general from the operator P HP , which may not be selfadjoint in H, nor even closed, because P H is not necessarily closed, though HP is. HP is a self-adjoint extension of P HP under the requirement of the theorem that HP is densely defined in H, which means nothing else but that the domain D[H 1/2 P ] of the quadratic form in question is dense in H. Note also that for ε = 1, the theorem concerns a nonnegative self-adjoint operator εH = H, while for ε = −1, we get product formulae for the non-positive self-adjoint operator εH = −H. Moreover, the result is preserved when H is replaced with a shifted operator H + cI, i.e., for any semi-bounded self-adjoint operator in a separable Hilbert space. An important particular case, most often met in the applications, concerns the situation when the projection-valued function is constant. Corollary 2.2 Let H be a self-adjoint operator bounded from below in a separable Hilbert space H and P an orthogonal projection. If the operator HP specified above is densely defined, then for every f ∈ H and ε = ±1 we have in the topology of L2loc (R; H) the limiting relation [P exp(−iεtH/n)P ]n f −→ exp(−iεtHP ) P f

(2.4)

for n → ∞ as well as its nonsymmetric counterparts obtained by setting P (1/n) = P in (2.2) and (2.3). From the viewpoint of quantum Zeno effect described in the introduction the optimal result would be a strong convergence on H for a fixed value of the time variable, moreover uniformly on each compact interval in t. Our Theorem 2.1 implies the following weaker result on pointwise convergence.

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Corollary 2.3 Under the same hypotheses as in Theorem 2.1, there exist a set M ⊂ R of Lebesgue measure zero and a strictly increasing sequence {n } of positive integers along which we have

[P (1/n ) exp(−iεtH/n )P (1/n )]n f −→ exp(−iεtHP ) P f , n

[P (1/n ) exp(−iεtH/n )] f −→ exp(−iεtHP ) P f ,

n

[ exp(−iεtH/n ) P (1/n )] f −→ exp(−iεtHP ) P f ,

(2.5) (2.6) (2.7)

for every f ∈ H, strongly in H for all t ∈ R \ M . As we have indicated above, one need not resort to subsequences in the particular case when the projections involved are finite-dimensional. Theorem 2.4 In addition to the hypotheses of Theorem 2.1, assume that the orthogonal projection P as well as P (t) is of finite dimension. Then (i) the formulae (2.1)–(2.3) hold in the norm of H as n → ∞, uniformly on each compact interval of the variable t in R \ {0}, (ii) it also holds for ε = ±1 that as n → ∞, [P (t/n) exp(−iεtH/n)P (t/n)]n −→ exp(−iεtHP ) P , [P (t/n) exp(−iεtH/n)]n −→ exp(−iεtHP ) P , [ exp(−iεtH/n) P (t/n)]n −→ exp(−iεtHP ) P , strongly on H, uniformly on each compact interval in the variable t ∈ R. Before proving Theorems 2.1 and 2.4 and Corollary 2.3 let us comment briefly on several other aspects of the result. Remark 2.5 While the necessity to pick a subsequence makes the pointwise convergence result weaker than desired, let us notice that from the physical point of view the convergence in L2loc (R; H) can be regarded as satisfactory. The point is that any actual measurement, in particular that of time, is burdened with errors. Suppose thus we perform the Zeno experiment on numerous copies of the system. The time value in the results will be characterized by a probability distribution φ : R+ → R+ , which is typically a bounded, compactly supported function – in a precisely posed experiment it is sharply peaked, of course. Corollary 2.2 then gives 2 (2.8) φ(t) [P exp(−iεtH/n)P ]n f − exp(−iεtHP ) P f dt → 0 as n → ∞, in other words, the Zeno dynamics limit is valid after averaging over experimental errors, however small they are.

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Remark 2.6 While the proof of strong convergence in H in Theorem 2.1 and Corollaries 2.2 and 2.3 remains elusive without the finite-dimension assumption, such a claim can be easily established in the orthogonal complement of the subspace P H. Indeed, taking f ∈ QH, where Q := I − P , we have (P (1/n)e−iεtH/n P (1/n))n f = (P (1/n)e−iεtH/n P (1/n))n P (1/n)Qf , (e−iεtH/n P (1/n))n f = (e−iεtH/n P (1/n))n P (1/n)Qf , which converge to zero, uniformly on each compact t-interval in R, as n → ∞, s because P (τ ) → P as τ → 0. This gives the result for (2.5) and (2.7), while for (2.6) one has to employ in addition the relation (3.11) below. Remark 2.7 The fact that the product formulae require HP to be densely defined is nontrivial. Recall the example of [Ex, Rem. 2.4.9] in which H is the multiplication operator, (Hψ)(x) = xψ(x) on L2 (R+ ), and P is the one-dimensional projection onto the subspace spanned by the vector ψ0 : ψ0 (x) = [(π/2)(1+x2 )]−1/2 . In this case obviously HP is the zero operator on the domain D[HP ] = {ψ0 }⊥ . On the other hand, P e−itH P acts on Ran P as multiplication by the function v(t) := e−t −

i −t 2i e Ei(t) − et Ei(−t) = 1 + t ln t + O(t), π π

where Ei (−t) and E i (t) are exponential integrals [AS]; due to the rapid oscillations of the imaginary part as t ↓ 0 a pointwise limit of v(t/n)n for n → ∞ does not exist. Notice also that different limits may be obtained in this example along suitably chosen subsequences {n }. Remark 2.8 In their recent study of Trotter-type formulae involving projections Matolcsi and Shvidkoy [MaS] presented two examples in which expressions of the type [exp(−iH/n)P ]n do not converge strongly. This result does not answer the question, however, whether the product expressions considered here converge in the strong topology of H or not, because our assumptions are not satisfied there. In the first example of [MaS] the analogue of the operator HP is not densely defined, in the second one H is not semi-bounded.

3 Proof of Theorem 2.1 We present the argument for ε = 1, the case ε = −1 can be treated similarly. We first prove (2.1) in (a), and next (2.2), (2.3) in (b). (a) Let us begin with the symmetric product case and prove the formula (2.1) with ε = 1. We will check the convergence in (2.1) on an arbitrary compact t-interval in the closed right half-line [0, ∞). The proof for t-intervals in the closed left half-line (−∞, 0] is analogous, and in addition, it can be included in the case ε = −1 with the convergence in (2.1) on compact t-intervals of the closed right half-line [0, ∞).

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Put Q(t) := I − P (t) and Q := Q(0) = I − P (0) = I − P , where I is the identity operator on H. Since H is nonnegative by assumption, there ∞ exists a spectral measure E(dλ) on the nonnegative real line such that H = 0− λ E(dλ). For ζ ∈ C with Re ζ ≥ 0 and τ > 0, we put F (ζ, τ ) = P (τ ) e−ζτ H P (τ ) ,

(3.1)

which is a contraction, and S(ζ, τ ) = τ −1 [I − F (ζ, τ )] = τ −1 [I − P (τ ) e−ζτ H P (τ )],

(3.2)

which exists as a bounded operator on H with Re f, S(ζ, τ )f ≥ 0 for every f ∈ H. For definiteness we use here and in the following the physicist convention about the inner product supposing that it is antilinear in the first argument. For a non-zero ζ ∈ C with Re ζ ≥ 0, we put also H(ζ) := ζ −1 [I − e−ζH ] .

(3.3)

Each element v(·) in L2loc ([0, ∞); H) is an equivalence class such that any two representatives of it are equal a.e. on [0, ∞). However, at some places we will not avoid an abuse of notation using for a particular representative of such an element the same symbol v(·). At the same time, in the following the convergence of a family of vectors v(·, τ ) to v(·) in the topology of the space L2loc ([0, ∞); H) = L2loc ([0, ∞)) ⊗ H as τ tends to zero will be often written as v(t, τ ) −→ v(t); this will be the case when writing v(·, τ ) −→ v(·) would require to introduce a separate symbol for this v(t, τ ) the meaning of which is clear from the context. The key ingredient of the proof is the following lemma. Lemma 3.1 (I + S(it, τ ))−1 converges to (I + itHP )−1 P as τ → 0 strongly in L2loc ([0, ∞); H), in other words, for all f ∈ H and every finite T > 0 we have 0

T

(I + S(it, τ ))−1 f − (I + itHP )−1 P f 2 dt → 0 ,

τ → 0.

(3.4)

We postpone the proof of Lemma 3.1 to the next section. For the moment we will accept its claim and use it to show that it implies the symmetric case (2.1) of the product formula in Theorem 2.1. To this end, let {mn } be a strictly increasing sequence of positive integers, i.e., a subsequence of the sequence of all positive integers. We have only to show that there exists a subsequence {n } in any such sequence {mn } along which (2.1) holds. Then by a standard argument we can conclude that (2.1) actually holds along the sequence of all positive integers n. For if this were not the case, there would exist a subsequence {n } of strictly increasing positive integers along which

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(2.1) does not converge. However, we see that there is a subsequence {n } of {n } along which the convergence takes place to the same limit, which is a contradiction. Fix {mn } and f ∈ H. Lemma 3.1 holds, in particular, along the sequence {τn } with τn := 1/mn , and since L2 convergence implies pointwise convergence a.e. along a subsequence, there exist a subset Mf of Lebesgue measure zero of the variable t in [0, ∞) and a subsequence {τf,n } of {τn }, both dependent on f , such that (I + S(it, τf,n ))−1 f −→ (I + itHP )−1 P f holds strongly in H for t ∈ [0, ∞) \ Mf . Since H is separable by assumption, we can choose a countable dense subset D = {f }∞ =1 in H. Then we infer that for f1 ∈ D there exist a set M1 := Mf1 of Lebesgue measure zero and a subsequence {τ1,n } of {τn } along which (I + S(it, τ1,n ))−1 f converges to (I + itHP )−1 P f for every t ∈ / M1 . Next, for f2 ∈ D there exist a set M2 := Mf2 of Lebesgue measure zero and a subsequence {τ2,n } of {τ1,n } along which (I + S(it, τ2,n ))−1 f converges / M2 . Proceeding in this way, we associate in the to (I + itHP )−1 P f for every t ∈ th step with f ∈ D a set M := Mf of Lebesgue measure zero and a subsequence {τ,n } of {τ−1,n } along which (I + S(it, τ,n ))−1 f converges to (I + itHP )−1 P f for every t ∈ / M . Now we put τn := τn,n and n := 1/τn , so that {n } is a subsequence of the strictly increasing sequence {mn } of positive integers from which we have started. −1 f} Then it follows that for every t ∈ [0, ∞)\∪∞ =1 M , the sequence {(I +S(it, τn )) −1 converges to (I + itHP ) P f strongly in H as τn → 0 for every f ∈ D, and therefore also in H, because both (I +S(it, τ,n ))−1 and (I +itHP )−1 P are bounded operators on H with the norms not exceeding one. We denote M := ∪∞ =1 M , which is, of course, again a set of Lebesgue measure zero. In this way we have found a subsequence {τn } of {τn = 1/mn } and an exceptional subset M of [0, ∞) such that (3.5) (I + S(it, τn ))−1 f = (I + S(it, 1/n ))−1 f −→ (I + itHP )−1 P f strongly in H as τn → 0 or n → ∞ for every f ∈ H and for each fixed t ∈ / M ; it is important that M is independent of f . Lemma 3.2 For the sequence {n } specified above and every f ∈ H we have

[P (1/n ) exp(−itH/n )P (1/n )]n f −→ e−itHP P f

(3.6)

as n → ∞ strongly in H provided t ∈ / M. Notice that this claim is in fact the “symmetric” part of Corollary 2.3. Proof of Lemma 3.2. We use arguments analogous to those employed in derivation of Chernoff’s theorem – see [Ch2, Theorem 1.1], [Ch1] and [Ka1, Thm IX.3.6]. We divide the proof into two steps referring to f belonging to P H and to its orthogonal complement.

202



Suppose first that f ∈ P H. For t ∈ / M and τ fixed, the operator S(it, τ ) generates a strongly continuous semigroup { e−θS(it,τ ) : θ ≥ 0 } on H, and the resolvent convergence (3.5) implies the convergence of the corresponding semigroups [Ka1, Thm IX.2.16], so we have

e−θS(it,1/n ) f −→ e−iθtHP f s

/ M , uniformly on each compact interval of the variable in H as n → ∞ for t ∈ θ ≥ 0. In particular, choosing θ = 1 we get for each t ∈ [0, ∞) \ M

e−S(it,1/n ) f −→ e−itHP f , s

n → ∞ .

(3.7)

The same equivalence implies for any λ ≥ 0 and t ∈ [0, ∞) \ M that (I + λS(it, 1/n ))−1 f −→ (I + iλtHP )−1 P f , s

in particular, using the diagonal trick we obtain

−1 1 s I + √ S(it, 1/n ) f −→ P f n

as n → ∞,

(3.8)

for every t ∈ [0, ∞) \ M . Next we use [Ch1, Lemma 2] which gives for any g ∈ H the inequality √ F (it, 1/n )n g − e−n (I−F (it,1/n )) g ≤ n (I − F (it, 1/n))g . Choosing g = I +

−1

√1 S(it, 1/n ) n

f we infer that

−1

1 f F (it, 1/n )n − e−S(it,1/n ) 1 + √ S(it, 1/n) n

−1 1 ≤ I + √ S(it, 1/n ) f − f , n where the right-hand side tends to zero as n → ∞ by (3.8). Using (3.8) once again we get (3.9) F (it, 1/n )n f − e−S(it,1/n ) f −→ 0 . The sought relation (3.6) immediately follows from (3.7) and (3.9), since by (3.1) we have F (it, 1/n)n = [P (1/n ) exp(−itH/n )P (1/n )]n . The case f ∈ QH is easier being independent of the arguments preceding Lemma 3.2. We have, along the sequence of all positive integers n, [P (1/n) exp(−itH/n)P (1/n)]n f → 0

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strongly in H and for each t ∈ [0, ∞), since P (1/n)f = P (1/n)Qf converges by assumption to P Qf = 0 as n → ∞, while exp(−itHP )P f = 0.

This yields the sought result because {[P (1/n ) exp(−itH/n )P (1/n )]n } is a bounded sequence for any t ≥ 0 and by Lebesgue dominated-convergence theorem it tends to the expected limit in L2loc ([0, ∞); H). Using the standard “subsequence” trick mentioned above we have thus shown that Lemma 3.1 implies the symmetric product formula (2.1) of Theorem 2.1. (b) Let us turn to the non-symmetric product-formula cases, i.e., to prove that (2.1) implies (2.2) and (2.3). Proof of (2.2). We employ the standard notation, [U, S] = U S − SU , for the commutator of bounded operators U and S. First we observe the following fact. s

Lemma 3.3 It holds that [ e−itτ H, P (τ ) ] −→ 0 as τ → 0, uniformly on each compact t-interval in R. Proof: By (3.3) with ζ = itτ we have [ e−itτ H, P (τ ) ] = i P (τ )tτ H(itτ ) − tτ H(itτ )P (τ ) , and hence for any v ∈ H we can estimate −itτ H [ e , P (τ ) ]v ≤ tτ H(itτ )v + tτ H(itτ )P (τ )v . We rewrite (3.3) with ζ = itτ as iH(itτ ) =

I − cos tτ H sin tτ H +i =: B(tτ ) + iA(tτ ) , tτ tτ

(3.10)

where B(tτ ) and A(tτ ) are obviously bounded self-adjoint operators on H, and B(tτ ) is in addition nonnegative. The definition makes sense if t = 0 but we need not exclude this case because what we really need is the operator tτ H(itτ ). For any w ∈ H we get tτ H(itτ )w2 = [tτ B(tτ ) + itτ A(tτ )]w2 = [(I − cos tτ H) + i sin tτ H]w2 = w, [(I − cos tτ H)2 + sin2 (tτ H)]w = 4 sin(tτ H/2)w2 → 0 , uniformly on compact t-intervals in R. In this way we have proved the claim, noting s that P (τ ) −→ P holds uniformly on each compact t-interval in R as τ → 0. Now we employ the following identity, n n P (1/n) e−itH/n v − P (1/n) e−itH/n P (1/n) v n−1 −itH/n [e , P (1/n) ]v , = − P (1/n)e−itH/n P (1/n)

(3.11)

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the right-hand side of which converges by Lemma 3.3 to zero uniformly on each n−1 compact t-interval for any v ∈ H, because P (1/n)e−itH/n P (1/n) is a contraction on H, and hence also in L2loc ([0, ∞); H). This yields the formula (2.2). Proof of (2.3). In view of the already proven formula (2.1) we have for every f ∈ H and T > 0 the following chain of relations T P f

2

≥ lim sup

T

(e−itH/n P (1/n))n f 2 dt

0

T

= lim sup 0

T

+ ≥ lim sup

T

= 0

e

0

0 T

P (1/n)(e−itH/n P (1/n))n f 2 dt Q(1/n)(e−itH/n P (1/n))n f 2 dt

(P (1/n)e−itH/n P (1/n))n f 2 dt

−itHP

P f 2 dt = T P f 2, s

with the lim sup taken along n → ∞, because I = P (1/n)+Q(1/n) and P (τ ) −→ P T as τ → 0. It follows that 0 Q(1/n)(e−itH/n P (1/n))n f 2 dt −→ 0 as n → ∞. Thus for any v(·) ∈ L2loc ([0, ∞); H) and every T > 0 we have, again by (2.1),

T

0

T

= 0

v(t), (e−itH/n P (1/n))n f dt P (1/n)v(t), (P (1/n)e−itH/n P (1/n))n f dt

T

+ −→

0

0 T

Q(1/n)v(t), Q(1/n)(e−itH/n P (1/n))n f dt

v(t), e−itHP P f dt

as n → ∞. It means that {(e−itH/n P (1/n))n f } converges to e−itHP P f weakly in L2loc ([0, ∞); H) together with all the seminorms, and therefore the convergence is strong in L2loc ([0, ∞); H). This yields the formula (2.3). It remains to prove Lemma 3.1 on which the above arguments were based.

4 Proof of Lemma 3.1 To demonstrate (3.4), we shall use the Vitali theorem – see, e.g., [HP] – for holomorphic functions and employ arguments analogous to those used in Kato’s paper

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[Ka2] for the self-adjoint Trotter product formula with the form sum of a pair of nonnegative self-adjoint operators. We do it in three steps. I. In the first step we will show the following lemma. Lemma 4.1 For a fixed ζ = t > 0, (I + S(t, τ ))−1 −→ (I + tHP )−1 P s

as

τ → 0.

(4.1)

Proof: The argument will be analogous to that in [Ka2], and indeed, validity of the result in the particular case when our projection-valued function is constant is remarked in [Ka2, Eq. (5.2), p. 194]. For ζ = tτ > 0 we have from (3.3) H(tτ ) = (tτ )−1 [I − e−tτ H ], which is a bounded, nonnegative and self-adjoint operator on H. It allows us to rewrite S(t, τ ) = τ −1 [I − P (τ )(I − tτ H(tτ ))P (τ )] = τ −1 Q(τ ) + tP (τ )H(tτ )P (τ ) , which is in this case also a bounded and nonnegative self-adjoint operator. To prove (4.1) take any f ∈ H and put uˆ(t, τ ) := (I + S(t, τ ))−1 f , so that f = (I + S(t, τ ))ˆ u(t, τ ) = [I + τ −1 Q(τ ) + tP (τ )H(tτ )P (τ )]ˆ u (t, τ ) .

(4.2)

Then we have u(t, τ )2 + tH(tτ )1/2 P (τ )ˆ u(t, τ )2 . ˆ u(t, τ ), f = ˆ u(t, τ )2 + τ −1 Q(τ )ˆ

(4.3)

Thus the families {ˆ u(t, τ )}, {τ −1/2 Q(τ )ˆ u(t, τ )} and {t1/2 H(tτ )1/2 P (τ )ˆ u(t, τ )} are all bounded by f for all t > 0, uniformly as τ → 0, and therefore they are weakly compact in H. It follows that for each fixed t > 0 there exists a sequence {τn (t)} with τn (t) → 0 as n → ∞, in general dependent on t, along which these vectors converge weakly in H, w

u ˆ(t, τ ) −→ uˆ(t) ,

τ −1/2 Q(τ )ˆ u(t, τ ) −→ g0 (t) , w

t1/2 H(tτ )1/2 P (τ )ˆ u(t, τ ) −→ h(t) , w

(4.4)

for some vectors u ˆ(t), g0 (t) and h(t) in H. Note that the sequence {τn (t)}∞ n=1 can be chosen the same for all three families. s From this result we see first that Q(τ )ˆ u(t, τ ) −→ 0 uniformly in t > 0 as τ → 0, so that we have Qˆ u(t) = 0 or u ˆ(t) = P u ˆ(t) ∈ P H. For every v ∈ D[H 1/2 ] we have, with the limit taken along {τn (t)}, u(t, τ ) v, h(t) = lim v, t1/2 H(tτ )1/2 P (τ )ˆ = t1/2 lim H(tτ )1/2 v, P (τ )ˆ u(t, τ ) = t1/2 H 1/2 v, P u ˆ(t) , s

because H(tτ )1/2 v −→ H 1/2 v as τ → 0. Hence u ˆ(t) = P u ˆ(t) belongs to D[H 1/2 ] 1/2 1/2 1/2 and h(t) = t H P u ˆ(t) because D[H ] is dense by assumption. Furthermore,

206



multiplying (4.2) by τ 1/2 and taking the weak limit along the sequence {τn (t)} we get g0 (t) = 0. Similarly, multiplying (4.2) by P (τ ) we have for every v ∈ D[H 1/2 P ] v, P (τ )f = v, P (τ )ˆ u(t, τ ) + t1/2 H(tτ )1/2 P (τ )v, t1/2 H(tτ )1/2 P (τ )ˆ u(t, τ ). Then taking the limit along the sequence {τn (t)} we get v, P f = v, P uˆ(t) + t1/2 H 1/2 P v, h(t), because by spectral theorem H(tτ )1/2 (P (τ )−P )v = H(tτ )1/2 (I + H)−1/2 (I + H)1/2 (P (τ )−P )v ≤ (I + H)1/2 (P (τ )−P )v , s

which tends to zero since P (τ ) → P as τ → 0, D[H 1/2 P (τ )] ⊃ D[H 1/2 P ] and H 1/2 P (τ )v → H 1/2 P v for v ∈ D[H 1/2 P ] by assumption1 . Hence H 1/2 P uˆ(t) ∈ D[H 1/2 P ] and Pf

= =

Pu ˆ(t) + t1/2 (H 1/2 P )∗ h(t) = uˆ(t) + t(H 1/2 P )∗ (H 1/2 P )ˆ u(t) ˆ(t) , (4.5) u ˆ(t) + tHP u

because D[H 1/2 P ] is supposed to be dense. Applying once again the standard argument mentioned after Lemma 3.1 to all the three families we conclude that the weak convergence in (4.4) takes place independently of a sequence {τn (t)} chosen. On the other hand, we infer from (4.3) that ˆ u(t), f

≥ lim inf ˆ u(t, τ )2 + lim inf τ −1 Q(τ )ˆ u(t, τ )2 + lim inf t1/2 H(tτ )1/2 P (τ )ˆ u(t, τ )2 ≥ ˆ u(t)2 + g0 (t)2 + h(t)2 = ˆ u(t)2 + t1/2 H 1/2 P u ˆ(t)2 1/2

= ˆ u(t)2 + tHP u ˆ(t)2 with lim inf taken along τ → 0. Since by (4.5) the left-hand side of the above inequality is equal to 1/2

u(t), HP u ˆ(t) = ˆ u(t)2 + tHP u ˆ(t)2 , ˆ u(t), f = ˆ u(t), P f = ˆ u(t)2 + tˆ we see that the norms of these vectors converge to the norms of their limit vectors. It allows us to conclude that the H-valued families in question, {ˆ u(t, τ )}, {τ −1/2 Q(τ )ˆ u(t, τ )} and {t1/2 H(tτ )1/2 P (τ )ˆ u(t, τ )} converge to u ˆ(t), 0 and 1 This part of the proof shows that the hypotheses of Theorem 2.1 can be slightly weakened, because we need in fact only that s − limτ →0 H(tτ )1/2 P (τ )v = H 1/2 P v holds for any v ∈ D[H 1/2 P ].

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t1/2 H 1/2 P uˆ(t) strongly in H, respectively, as τ → 0. In particular, we have shown s u(t) and u ˆ(t, τ ) −→ u ˆ(t) = (I + tHP )−1 P f , or (4.1). This that P f = (I + tHP )ˆ proves Lemma 4.1. II. Next, for a fixed τ > 0, the function ζ → F (ζ, τ ) is holomorphic in the open right half-plane Re ζ > 0 and uniformly bounded in norm by one. This makes it possible to mimic the argument of Feldman [Fe], which is reproduced in Chernoff’s book [Ch2, p. 90], see also [Fr], to conclude by means of the Vitali theorem (see, e.g., [HP, Thm 3.14.1]) that for Re ζ > 0 (I + S(ζ, τ ))−1 −→ (I + ζHP )−1 P s

as τ → 0

(4.6)

holds uniformly on compact subsets of Re ζ > 0. At the boundary Re ζ = 0, or ζ = it with t real, (I + S(ζ, τ ))−1 still converges as τ → 0 but in a weaker sense only. Using the argument of [Fe] based on the Poisson kernel, we can check that for each pair of f, g ∈ H and all φ ∈ L1 (R) the following relation is valid, φ(t)g, (I + S(it, τ ))−1 f dt = φ(t)g, (I + itHP )−1 P f dt . (4.7) s − lim τ →0

R

R

This says that for each pair of f, g ∈ H the family {g, (I + S(it, τ ))−1 f } of functions of t in L∞ (R) converges to g, (I + itHP )−1 P f as τ → 0 weakly∗ , or equivalently, in the weak topology defined by the dual pairing between L∞ (R) and L1 (R) – see, e.g., [K¨ o]. III. Now we shall show the family of the bounded operators {(I + S(it, τ ))−1 } is weakly convergent in L2loc ([0, ∞); H), and in fact, strongly convergent there too. To do so, we will employ an argument analogous to that used in the proof of Lemma 4.1 on the Hilbert space H, however, this time on the Fréchet space L2loc ([0, ∞); H). Using the decomposition (3.10) with t = 0, we find (cf. [Ich]) S(it, τ ) = τ −1 [I − P (τ )(I − itτ H(itτ ))P (τ )] = τ −1 Q(τ ) + tP (τ )(B(tτ ) + iA(tτ ))P (τ ) . To prove (3.4), take any f ∈ H and put u(t, τ ) := (I + S(it, τ ))−1 f. Note that this u(t, τ ) represents an element in L2loc ([0, ∞); H) as well as its unique representative in (0, ∞), because u(t, τ ) is strongly continuous at this interval as a function of t. Then f = (I + S(it, τ ))u(t, τ ) = [I + τ

−1

(4.8)

Q(τ ) + tP (τ )(B(tτ ) + iA(tτ ))P (τ )]u(t, τ ) ,

so we have u(t, τ ), f = u(t, τ ), (I + S(it, τ ))u(t, τ ) = u(t, τ )2 + τ −1 Q(τ )u(t, τ )2 + tB(tτ )1/2 P (τ )u(t, τ )2 +itP (τ )u(t, τ ), A(tτ )P (τ )u(t, τ ) .

(4.9)

208



Observing the real part of (4.9) we see that for τ small enough, each of the H-valued families {u(t, τ )}, {τ −1/2 Q(τ )u(t, τ )} and {t1/2 B(tτ )1/2 P (τ )u(t, τ )} is bounded by f for all t > 0. Moreover, they are strongly continuous in t for fixed τ > 0, and locally bounded as H-valued functions of t in L2loc ([0, ∞); H), uniformly as τ → 0. s Hence we infer first of all that Q(τ )u(t, τ ) −→ 0, uniformly in t ∈ (0, ∞), as 2 τ → 0. Next, since Lloc ([0, ∞); H) is reflexive [GV, Chap. 1, Sec. 3.1, pp. 57-62], any bounded set in it is weakly compact [K¨ o, Sec. 23.5, pp. 302-304]. Consequently, with τ → 0 as n → ∞ along which the above families there is a sequence {τn }∞ n n=1 are weakly convergent in L2loc ([0, ∞); H): w

u(t, τ ) −→ u(t) ,

τ −1/2 Q(τ )u(t, τ ) −→ f0 (t) , w

t1/2 B(t, τ )1/2 P (τ )u(t, τ ) −→ z(t) , w

(4.10)

with some vectors u(·), f0 (·) and z(·) ∈ L2loc ([0, ∞); H). Note that as before the sequence {τn }∞ n=1 can be chosen the same for all three families. Lemma 4.2 These above mentioned vectors have the following properties, u(t) = P u(t) ∈ P H for a.e. t ,

z(·) = 0 ,

f0 (·) = 0 .

Proof. For B(tτ ) and A(tτ ) in (3.10), the spectral theorem gives ∞ 1 − cos tτ λ 1/2 2 E(dλ)H 1/2 v2 → 0 , v ∈ D[H 1/2 ] ; (B(tτ ) v = tτ λ 0− ∞ 1 − cos tτ λ 2 E(dλ)Hv2 → 0 , v ∈ D[H] (4.11) (B(tτ )v2 = tτ λ 0− as τ → 0 by the Lebesgue dominated-convergence theorem. s s Since Q(τ )u(t, τ ) −→ 0 uniformly in t ∈ (0, ∞) and Q(τ ) −→ Q as τ → 0, we have Qu(t) = 0, or in other words u(t) = P u(t) ∈ P H for a.e. t. Moreover, by (4.11) we infer that ∞ ∞ φ(t)v, z(t) dt = lim φ(t)v, t1/2 B(tτ )1/2 P (τ )u(t, τ ) dt 0 0 ∞ 1/2 ¯ B(tτ )1/2 v, P (τ )u(t, τ ) dt φ(t)t = lim 0 ∞ ¯ φ(t)0, P u(t) dt = 0 = 0

holds for every φ ∈ C0∞ ([0, ∞)) and v ∈ D[H 1/2 ], hence z(t) = 0 a.e. because D[H 1/2 ] is dense in H, so that z(·) is the zero element of L2loc ([0, ∞); H). Finally, the relation f0 (·) = 0 follows from (4.8) which implies τ 1/2 Q(τ )f = τ 1/2 (1 + τ −1 )Q(τ )u(t, τ ), yielding the result; this concludes the proof.

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Our next aim is to show that the weak limits in (4.10) do not depend upon a sequence chosen. The u(·, τn ) = (I + S(it, τn ))−1 f converge to u = u(·) weakly in L2loc ([0, ∞); H) as n → ∞. It obviously implies that for all φ ∈ C0∞ ([0, ∞)) and for every g ∈ H we have ∞ ∞ −1 φ(t)g, (I + S(it, τ )) f dt −→ φ(t)g, u(t) dt , 0

0

again along the sequence {τn }. It follows from (4.7) that u(t) = (I + itHP )−1 P f,

for a.e. t in [0, ∞),

(4.12)

¯ because the set of all such φ(·)g is total in L2loc ([0, ∞); H). This shows that for every −1 f ∈ H, (I + S(it, τn )) f converges to (I + itHP )−1 P f weakly in L2loc ([0, ∞); H) as n → ∞. Together with the fact that z(·) = 0, f0 (·) = 0 in view of Lemma 4.2, this yields the desired property, namely that the weak limits of (4.10) are independent of the particular subsequence {τn } chosen. The standard argument sketched below Lemma 3.1 shows that (4.10) holds as τ → 0 without any restriction on subsequences. s

Finally, we are going to check the strong convergence u(·, τ ) −→ u(·) in L2loc ([0, ∞); H) as τ → 0. In fact, we will prove two other limiting relations at the same time. Lemma 4.3 In the topology of L2loc ([0, ∞); H), the family {u(·, τ )} converges to the vector u = u(·) as τ → 0, and moreover, τ −1/2 Q(τ )u(t, τ ) −→ f0 (t) = 0 , t1/2 B(t, τ )1/2 P (τ )u(t, τ ) −→ z(t) = 0 . Proof. In the above reasoning we have already checked the weak convergence in (4.10) as τ → 0. Integrating the real part of (4.9) in t over the interval [0, T ] for any fixed T > 0 and taking lim inf as τ → 0, we get by Lemma 4.2 T T u(t), f dt ≥ lim inf u(t, τ )2 dt Re 0

0

T

τ −1 Q(τ )u(t, τ )2 dt

+ lim inf 0

T

+ lim inf 0

≥ ≥

T 0 T 0

u(t)2 dt + u(t)2 dt .

t1/2 B(tτ )1/2 P (τ )u(t, τ )2 dt

T 0

f0 (t)2 dt +

0

T

z(t)2 dt

210



On the other hand, the left-hand side of the above inequality is by (4.12) equal to Re 0

T

u(t), f dt = Re

0

T

u(t), (I + itHP )u(t) dt =

0

T

u(t)2 dt.

Hence we conclude that all the Fréchet-space semi-norms of the vectors u(t, τ ), τ −1/2 Q(τ )u(t, τ ) and t1/2 B(t, τ )1/2 P (τ )u(t, τ ) converge to the semi-norms of the weak-limit vectors u(t), 0 and 0, respectively, as τ → 0. Thus the convergence is strong with respect to each semi-norm, and since their family induces the topology in L2loc ([0, ∞); H) the lemma is proved. This completes the proof of Lemma 3.1, and by that the verification of our main result, Theorem 2.1.

5 The finite-dimensional case In this section, we will prove Theorem 2.4 in which we assume that P and P (t) are finite-dimensional orthogonal projections. Since the closed operator H 1/2 P is supposed to be densely defined, the domain D[H 1/2 P ] of H 1/2 P becomes the whole space H, for the restriction H 1/2 P |P H of the operator H 1/2 P to the finitedimensional subspace P H is densely defined, so its domain must coincide with P H, and it acts as zero on QH. The same is valid for H 1/2 P (t) when P (t) is of a finite dimension. As a result, H 1/2 P and HP = (H 1/2 P )∗ (H 1/2 P ) as well as H 1/2 P (t) are bounded operators on H by the closed-graph theorem. By the assumptions common with Theorem 2.1, for each fixed f ∈ D[H 1/2 P ] = H the family {H 1/2 P (t)f } converges to H 1/2 P f as t → 0, and hence is uniformly bounded with respect to t near to zero, say, for −1 ≤ t ≤ 1. Then by the uniform boundedness principle we can conclude that sup|t|≤1 H 1/2 P (t) < ∞. To prove the assertions (i) and (ii) of Theorem 2.4 simultaneously, take a fixed a ∈ R and consider instead of F (ζ, τ ), S(ζ, τ ) defined by (3.1) and (3.2), respectively, the following operators Fa (ζ, τ ) := P (aτ ) exp(−ζτ H)P (aτ ),

Sa (ζ, τ ) := τ −1 [I − Fa (ζ, τ )].

In fact, we shall employ Fa (it, τ ), Sa (it, τ ) instead of F (it, τ ), S(it, τ ) in the proof of Lemma 3.2 and Lemma 3.1. Similarly u(t, τ ) used above will be replaced by ua (t, τ ) = (I + Sa (it, τ ))−1 f corresponding to a given f ∈ H. Lemma 5.1 For any t, t ≥ 0 and 0 < τ ≤ 1 we have ua (t, τ ) − ua (t , τ ) ≤ C(a)|t − t | f with a positive C(a) independent of t, t , which is uniformly bounded as a function of a on each compact interval of R.

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Proof. By the resolvent equation we have ua (t, τ ) − ua (t , τ ) = (I + Sa (it, τ ))−1 f − (I + Sa (it , τ ))−1 f

= (I + Sa (it, τ ))−1 P (aτ )τ −1 [e−itτ H − e−it τ H ]P (aτ )(I + Sa (it , τ ))−1 f 1 t d −isτ H = (I + Sa (it, τ ))−1 P (aτ ) ds P (aτ )(I + Sa (it , τ ))−1 f e τ t ds t He−isτ H ds P (aτ )(I + Sa (it , τ ))−1 f = −i(I + Sa (it, τ ))−1 P (aτ ) t

−1

1/2

= −i(I + Sa (it, τ )) (H P (aτ ))∗ t × e−isτ H ds (H 1/2 P (aτ ))(I + Sa (it , τ ))−1 f . t

At the beginning of this section we have argued that the operators H 1/2 P (aτ ) are uniformly bounded on H for 0 < τ ≤ 1. It follows that ua (t, τ ) − ua (t , τ ) ≤ C(a)|t − t | f with C(a) := sup|aτ |≤1 H 1/2 P (aτ )2 . By the argument preceding the lemma the function C(·) is uniformly bounded on each compact a-interval in R; this yields the claim. Proof of Theorem 2.4. It follows from the lemma that the vector family {ua (t, τ )}, continuous in H, is uniformly bounded and equicontinuous. Hence we may infer by the Ascoli–Arzelà theorem that the sequence {τn } used in part III of the proof of Lemma 3.1 can chosen to have an additional property, namely that the sequence {ua (t, τn )} converges strongly to u(t) also pointwise, uniformly on [0, ∞). Then the limit u(t) becomes strongly continuous in t ≥ 0, and coincides with (I + itHP )−1 f for all t ≥ 0. Thus we have instead of Lemma 3.1 the following claim: (I + Sa (it, τ ))−1 −→ (I + itHP )−1 P

(5.1)

as τ → 0, strongly on H and uniformly on each compact interval of the variable t in [0, ∞). Next we will modify the reasoning of Sec. 3 based on [Ch2, Theorem 1.1] with the aim to show the symmetric product case, s

[P (at/n) exp(−itH/n)P (at/n)]n −→ exp(−itHP )P,

n → ∞.

(5.2)

Let f ∈ H. The resolvent convergence (5.1) with t = 1 implies the convergence of the corresponding semigroups, so we have e−θSa (i,τ ) f −→ e−iθHP f s

(5.3)

212



in H as τ → 0, uniformly on each compact interval of the variable θ ≥ 0. Using this equivalence once more we get for any λ ≥ 0 the relation (I + λSa (i, τ ))−1 f −→ (I + iλHP )−1 P f , s

τ → 0.

In particular, taking τ = θ/n and using the diagonal trick, we infer that θ s (I + √ Sa (i, θ/n))−1 f −→ P f , n

n → ∞,

(5.4)

holds uniformly on each compact θ-interval in [0, ∞). Then the mentioned lemma from [Ch1] yields √ Fa (i, θ/n)n g − e−n(I−Fa (i,θ/n)) g ≤ n (I − Fa (i, θ/n))g . Choosing again g = I +

−1

√θ Sa (i, θ/n) n

f we find that

−1

θ n −θSa (i,θ/n) f 1 + √ Sa (i, θ/n) Fa (i, θ/n) − e n

−1 θ ≤ I + √ Sa (i, θ/n) f − f , n where the right-hand side tends to zero as n → ∞ by (5.4). Using the last named convergence once more we get (5.5) Fa (i, θ/n)n f − e−θSa (i,θ/n) f −→ 0 uniformly on each compact θ-interval in [0, ∞). Choosing now θ = t we see that the validity of (5.2) on P H follows immediately from (5.3) and (5.5). Consequently, on the subspace P H the assertion (i) is obtained by taking a = 1/t for any t belonging to a compact interval in R \ {0} and (ii) by choosing simply a = 1. The case f ∈ QH can be treated as in the proof of Lemma 3.2; together this yields the relation (5.2) on H, i.e., the symmetric product case. The non-symmetric product cases can also be checked with the help of Lemma 3.3 – cf. part (b) of the proof of Theorem 2.1 in Section 3. This concludes the proof of Theorem 2.4.

6 An example As we have said, our investigation was motivated by the result by Facchi et al. [FPS] mentioned in the introduction. Let us thus look how the result looks in this case. To see this, consider an open domain Ω ⊂ Rd with a smooth boundary, and denote by P the orthogonal projection on L2 (Rd ) defined as the multiplication operator by the indicator function χΩ of the set Ω. Consider further the free

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quantum Hamiltonian H := −∆, i.e., the Laplacian in Rd which is a nonnegative self-adjoint operator in L2 (Rd ), and the Dirichlet Laplacian −∆Ω in L2 (Ω) defined in the usual way [RS, Sec. XIII.15] as the Friedrichs extension of the appropriate quadratic form. We consider the Zeno dynamics in the subspace L2 (Ω) corresponding to a permanent reduction of the wavefunction to the region Ω, which may be identified with the volume of a detector. In the sense of the L2loc (R; L2 (Rd )) topology, which is physically plausible as explained in Remark 2.5, we then claim that the generator of the dynamics in L2 (Ω) is just the appropriate Dirichlet Laplacian, (P e−it(−∆/n) P )n → e−it(−∆Ω ) P

(6.1)

as n → ∞, or in other words: Proposition 6.1 The self-adjoint operator −∆P = ((−∆)1/2 P )∗ ((−∆)1/2 P )

(6.2)

is densely defined in L2 (Rd ) and its restriction to the subspace L2 (Ω) is nothing but the Dirichlet Laplacian −∆Ω of the region Ω, with the domain D[−∆Ω ] = W01 (Ω) ∩ W 2 (Ω). Proof. Let u ∈ D[−∆P ], so that u and −∆P u belong to L2 (Rd ). We have −∆P u, ϕ = u, −∆ϕ = −∆u, ϕ, for any ϕ ∈ C0∞ (Ω) because ϕ has a compact support in Ω. Thus −∆P u = −∆u holds in Ω in the sense of distributions, which means that ∆u|Ω ∈ L2 (Ω). On the other hand, since (−∆)1/2 P u ∈ L2 (Rd ), we have χΩ u ∈ W 1 (Rd ). Since we have ∇(χΩ u) = ∇((χΩ )2 u) = (∇χΩ )χΩ u(x) + χΩ ∇(χΩ u), in order to belong to L2 (Rd ) the function ∇(χΩ u) must not contain the δ-type singular term, which requires u(·) = 0 on the boundary of Ω. This combined with the fact that u|Ω , ∆u|Ω ∈ L2 (Ω) – see, e.g., [LM, Thm 5.4] – implies that u|Ω belongs to W 2 (Ω) and W01 (Ω). Thus we have shown that u|Ω ∈ D[−∆Ω ] and (−∆P u)|Ω = −∆Ω (u|Ω ) or −∆Ω ⊃ −∆P |L2 (Ω) , but both operators are self-adjoint, so they coincide. In this sense therefore our result given in Theorem 2.1 provides one possible abstract version of the result by Facchi et al. [FPS].

Acknowledgments P.E. and T.I. are respectively grateful for the hospitality extended to them at Kanazawa University and at the Nuclear Physics Institute, AS CR, where parts of this work were done. We thank the referee who spotted a logical gap in the

214



first version of the paper. The research has been partially supported by ASCR and Czech Ministry of Education under the contracts K1010104 and ME482, and by the Grant-in-Aid for Scientific Research (B) No. 13440044 and No. 16340038, Japan Society for the Promotion of Science.

References [AS]

M.S. Abramowitz and I.A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York 1965.

[BN]

J. Beskow and J. Nilsson, The concept of wave function and the irreducible representations of the Poincaré group, II. Unstable systems and the exponential decay law, Arkiv Fys. 34, 561–569 (1967).

[Ch1] P.R. Chernoff, Note on product formulas for operator semigroups, J. Funct. Anal. 2, 238–242 (1968). [Ch2] P.R. Chernoff, Product Formulas, Nonlinear Semigroups, and Addition of Unbounded Operators, Mem. Amer. Math. Soc. 140; Providence, R.I. 1974. [Ex]

P. Exner, Open Quantum Systems and Feynman Integrals, D. Reidel Publ. Co., Dordrecht 1985.

[EI]

P. Exner and T. Ichinose, Product formula for quantum Zeno dynamics, to appear in Proceedings of the XIV International Congress of Mathematical Physics (M ∩ Φ), Lisbon, July 28–Aug 2, 2003.

[FPS] P. Facchi, S. Pascazio, A. Scardicchio, and L.S. Schulman, Zeno dynamics yields ordinary constraints, Phys. Rev. A 65, 012108 (2002). [Fe]

J. Feldman, On the Schr¨ odinger and heat equations for nonnegative potentials, Trans. Amer. Math. Soc. 108, 251–264 (1963).

[Fr]

C. Friedman, Semigroup product formulas, compressions, and continual observations in quantum mechanics, Indiana Math. J. 21, 1001–1011 (1971/72).

[GV] O.M. Gel’fand and N.Y. Vilenkin, Generalized Functions, IV. Applications of Harmonic Analysis, Academic Press, New York 1965. [HP]

E. Hille and R.S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloquium Publ. No. 31, rev. ed., Providence, R. I. 1957.

[Ich]

T. Ichinose, A product formula and its application to the Schr¨ odinger equation, Publ. RIMS Kyoto Univ. 16, 585–600 (1980).

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[Ka1] T. Kato, Perturbation Theory for Linear Operators, Springer, BerlinHeidelberg-New York 1966. [Ka2] T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in Topics in Functional Analysis (I. Gohberg and M. Kac, eds.), Academic Press, New York 1978; pp.185–195. [K¨ o]

G. K¨ othe, Topological Vector Spaces I, Springer, Berlin-Heidelberg-New York 1969.

[LM] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Springer, Berlin-Heidelberg-New York 1972. [MaS] M. Matolcsi and R. Shvidkoy, Trotter’s product formula for projections, Arch. der Math. 81, 309–317 (2003). [MS]

B. Misra and E.C.G. Sudarshan, The Zeno’s paradox in quantum theory, J. Math. Phys. 18, 756–763 (1977).

[RS]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, New York 1978.

Pavel Exner Department of Theoretical Physics Nuclear Physics Institute Academy of Sciences ˇ z 25068 Reˇ Czech Republic and Doppler Institute Czech Technical University Bˇrehov´ a7 11519 Prague Czech Republic email: [email protected] Takashi Ichinose Department of Mathematics Faculty of Science Kanazawa University Kanazawa 920-1192 Japan email: [email protected] Communicated by Gian Michele Graf submitted 21/06/04, accepted 12/10/04



Precise Coupling Terms in Adiabatic Quantum Evolution Volker Betz and Stefan Teufel Abstract. It is known that for multi-level time-dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit. For a family of two-state systems with real-symmetric Hamiltonian we construct such a superadiabatic representation and explicitly determine the asymptotic behavior of the exponentially small coupling term. First order perturbation theory in the superadiabatic representation then allows us to describe the time-development of exponentially small adiabatic transitions. The latter result rigorously confirms the predictions of Sir Michael Berry for our family of Hamiltonians and slightly generalizes a recent mathematical result of George Hagedorn and Alain Joye.

1 Introduction and main result The decoupling of slow and fast degrees of freedom in the adiabatic limit is at the basis of many important approximations in physics, as, e.g., the BornOppenheimer approximation in molecular dynamics and the Peierls substitution in solid state physics. We refer to [BMKNZ, Te] for recent reviews. Generically the decoupling is not exact and a coupling which is exponentially small in the adiabatic parameter remains. However, this small coupling has important physical consequences, as it makes possible, e.g., non-radiative decay to the ground state in molecules. Since Kato’s proof from 1950 [Ka] the adiabatic limit of quantum mechanics was considered also as a mathematical problem, with increased activity during the last 20 years. Some of the landmarks are [Ne1 , ASY, JoPf1 , Ne2 , HaJo]. We consider a two-state time-dependent quantum system described by the Schr¨ odinger equation iε∂t − H(t) ψ(t) = 0 (1) in the adiabatic limit ε → 0. For the moment we take the Hamiltonian H(t) to be the real-symmetric 2 × 2-matrix cos θ(t) sin θ(t) H(t) = ρ(t) . (2) sin θ(t) −cos θ(t) The eigenvalues of H(t) are ±ρ(t) and we assume that the gap between them does not close, i.e., that 2ρ(t) ≥ g > 0 for all t ∈ R. As to be explained, even for this simple but prototypic problem there are open mathematical questions. In order to explain the concern of our work, namely the time-development of the exponentially small adiabatic transitions, let us briefly

218

V. Betz and S. Teufel


recall some important facts about (1). Let U0 (t) be the orthogonal matrix that diagonalizes H(t), i.e., cos(θ(t)/2) sin(θ(t)/2) U0 (t) = . (3) sin(θ(t)/2) − cos(θ(t)/2) Then the Schr¨ odinger equation in the adiabatic representation becomes U0 (t) iε∂t − H(t) U0∗ (t) U0 (t)ψ(t) =: iε∂t − Hεa (t) ψ a (t) = 0

with Hεa (t) =

ρ(t)

iε 2 θ (t)

− iε2 θ (t)

−ρ(t)

and

ψ a (t) = U0 (t)ψ(t) .

Here and henceforth, primes denote time derivatives. First-order perturbation theory in the adiabatic representation (cf. proof of Corollary 1) and integration by parts yields the adiabatic theorem [BoFo, Ka]: The off-diagonal elements of the unitary propagator K a (t, s) in the adiabatic basis, i.e., the solution of iε∂t Kεa (t, s) = Hεa (t)Kεa (t, s) ,

Kεa (s, s) = id ,

vanish in the limit ε → 0. More precisely, let 1 0 0 0 P+ = , P− = , 0 0 0 1

(4)

which project onto the adiabatic subspaces in the adiabatic representation. Then P− Kεa (t, s) P+ = O(ε) .

(5)

Therefore the transitions between the adiabatic subspaces are O(ε). This bound is optimal in the sense that in regions where θ(t) is not constant the leading order term in the asymptotic expansion of P− Kεa (t, s) P+ in powers of ε is proportional to ε. However, if limt→±∞ θ (t) = 0 then in the scattering limit the transitions between the adiabatic subspaces are much smaller: if the derivatives of θ ∈ C ∞ (R) decay sufficiently fast, then for any n ∈ N A(ε) := lim P− Kεa (t, −t) P+ = O(εn ) . t→∞

(6)

If θ is analytic in a suitable neighborhood of the real axis, then transition amplitudes are even exponentially small, A(ε) = O(e−c/ε ) for some constant c depending on the width of the strip of analyticity, see [JoPf1 , Ma]. It is well understood, see [Le, Ga, Ne1 ], how to reconcile the apparent contrariety between the smallness of the final amplitudes in (6) and the optimality of (5): the adiabatic basis is not the optimal basis for monitoring the transition

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process. For any n ∈ N there exist unitary transformations Uεn (t) such that the Hamiltonian in this nth superadiabatic representation takes the form n ρε (t) cnε (t) n Hε (t) = with ρnε (t) = ρ(t) + O(ε2 ) and |cnε (t)| = O(εn+1 ) . cnε (t) −ρnε (t) (7) In the nth superadiabatic basis the off-diagonal components of the propagator and hence also the transitions are of order O(εn ), i.e., there are constants Cn such that P− Kεn (t, s) P+ ≤ Cn εn .

(8)

In the scattering regime, where θ(t) becomes constant, the superadiabatic bases agree with the adiabatic basis, i.e., limt→±∞ Uεn (t) = U0 (t), and therefore the bound in (8) basically yields (6). Typically limn→∞ Cn εn = ∞ for all ε > 0, i.e., choosing n larger while keeping ε fixed does not necessarily decrease the bound in (8). However, one can choose nε = n(ε) in such a way that Cnε εnε is minimal. If θ is analytic, one obtains the improved estimate P− Kεnε (t, s) P+ = O(e−c/ε ) in the optimal superadiabatic basis nε , see [Ne2 , JoPf2 ]. More interesting than bounds on A(ε) is its actual value. Since A(ε) is asymptotically smaller than any power of ε, this question is beyond standard perturbation theory. For the case of analytic coupling θ, asymptotic formulas of the type tc

A(ε) = C e− ε (1 + O(ε))

(9)

have been established, see, e.g., [JKP, Jo], where the constants C and tc depend on the type and location of the complex singularities of θ (t)/ρ(t). These results are obtained by solving (1) along a certain Stokes line in the complex plane except near the singular points, where a comparison equation is solved. As a consequence, the method gives no information at all about the way in which the exponentially small final transition amplitude A(ε) is built up in real time. This question of adiabatic transition histories is the concern of our paper. Berry [Be] and, in a refined way Berry and Lim [BeLi, LiBe], gave an answer on a non-rigorous level and explicitly left a mathematically rigorous treatment as an interesting open problem. Only very recently Hagedorn and Joye [HaJo] succeeded and confirmed Berry’s results rigorously for a specific Hamiltonian. Although our work has been strongly motivated by the findings of Berry, our approach is slightly different. Let us first state our main result before we discuss its relation to the earlier ones. Without loss of generality we assume that ρ(t) ≡ 12 . It was observed in [Be] that this can always be achieved by transforming (1) to the natural time scale t

τ (t) = 2 0

(s) ds .

(10)

220



However, we can only treat a rather special class of Hamiltonians, since we must assume that in the natural time scale the coupling has the form 1 1 γtc θ (t) = iγ − (11) = 2 t + itc t − itc t + t2c with γ ∈ R and tc > 0. In other words we assume t 1 cos θ(t) sin θ(t) H(t) = . (12) with θ(t) = 2 γ arctan sin θ(t) − cos θ(t) 2 tc We shall comment below on the meaning of this special choice and remark here that the Hamiltonian in [HaJo] is (12) with γ = 12 . Our main result is the construction of an optimal superadiabatic basis in which the coupling term in the Hamiltonian is exponentially small and can be computed explicitly at leading order. This optimal basis is given as the nth ε superadiabatic basis where 0 ≤ σε < 2 is such that nε =

tc − 1 + σε ε

is an even integer.

(13)

Theorem 1. Let H(t) be as in (12) and nε as in (13), and let ε0 > 0 be sufficiently small. Then for every ε ∈ (0, ε0 ] one can construct a family of unitary matrices Uεnε (t) ∈ C2×2 , depending smoothly on t ∈ R, such that ε nε Uε (t) − U0 (t) = O (14) 1 + t2 and

n ρε ε (t) cnε ε (t) . Uεnε (t) iε∂t − H(t) Uεnε ∗ (t) = iε∂t − cnε ε (t) −ρnε ε (t)

=: Hεnε (t)

Here ρnε ε (t) = and for every α < cnε ε (t) with

= 2i

1 +O 2

ε2 1 + t2

(15)

,

3 2

πγ tc t2 2ε e− ε e− 2εtc cos sin πtc 2

t t3 σε t − + ε 3εt2c tc

 t t2 α   ε exp − εc 1 + 4t2c φα (ε, t) = 1  tc ln 2  1 + exp − ε 2 1 + t2

+ O (φα (ε, t)) , (16)

if |t| < tc , (17) if |t| ≥ tc .

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√ Remark 1. The explicit term in cnε ε is of order O(e−tc /ε ) only for times |t| = O( ε). For larger times all terms in cnε ε are exponentially small compared to the leading c /ε exponential e−t√ . As a consequence, Taylor expansion of the cosine in cnε ε around t/ε for |t| = O( ε) shows that it can be replaced by cos(t/ε) at the cost of lowering α to α < 1: for every α < 1 πγ tc t2 t 2ε nε e− ε e− 2εtc cos sin cε (t) = 2i + O (φα (ε, t)) . πtc 2 ε Remark 2. The slow time decay of the error in (17) for large times is due to the fact that nε is optimal for t near 0, but not for large t. Remark 3. Taking nε defined in (13) odd instead of even would yield slightly different off-diagonal elements in the effective Hamiltonian Hεnε (t). However, the resulting unitary propagator, cf. Corollary 1, would be the same at leading order. See the end of Section 5 for a discussion of this somewhat surprising fact. Let us shortly explain the idea of the proof of Theorem 1 and at the same time the structure of our paper. First we construct the nth order superadiabatic basis as in (7) in two steps: in Section 2 we construct the projectors on the superadiabatic basis vectors and in Section 3 we construct the unitary basis transformation Uεn (t). We cannot use existing results here, e.g., [Ga, Ne2 ], since we need to keep careful track of the exact form of the off-diagonal terms cnε (t) of the superadiabatic Hamiltonian, and since we aim at a scalar recurrence relation instead of a matrix recurrence relation for the cnε (t)’s. The main mathematical challenge is the asymptotic analysis of the resulting recurrence relation, which is done in Section 4. This is also the only part where we have to assume the special form (11) for θ . Theorem 1 then follows by choosing the order n of the superadiabatic basis as in (13), a choice which minimizes cnε (t) near t = 0. The details of this optimal truncation procedure and the proper proof of Theorem 1 are given in Section 5. Finally in Section 6 we use first order perturbation theory in the optimal superadiabatic basis in order to obtain the following Corollary, in which we abbreviate ∆(t, s) := arctan(t) − arctan(s) . x 2 Also recall that erf: R → (−1, 1) with erf(x) = √2π 0 e−x dx switches smoothly and monotonically from erf(−∞) = −1 to erf(∞) = 1. Corollary 1. The unitary propagator in the optimal superadiabatic basis + kε (t, s) kε (t, s) , Kεnε (t, s) = k ε (t, s) kε− (t, s) i.e., the solution of iε∂t Kεnε (t, s) = Hεnε (t)Kεnε (t, s) ,

Kεnε (s, s) = id ,

222


satisfies kε± (t, s) = e∓

i(t−s) 2ε

and kε (t, s) =

sin

πγ

tc

e− ε e−

i(t+s) 2ε

+ O(ε∆(t, s))

2 √ tc +O εe− ε ∆(t, s) .


erf

t √ 2εtc

(18)

− erf

√

s 2εtc

(19)

Outside the transition region, more precisely for |t| > εβ and |s| > εβ for some tc β < 12 , (19) holds with the error term replaced by O(εα e− ε ∆(t, s)) for every α < 1. Corollary 1 immediately implies the existence of solutions to (1) of the form it √ e− 2ε 1 + O(ε(arctan(t) + π2 )) tc ∗ πγ − tc it εe− ε . (20) ψ(t) = Uε (t) +O t sin 2 e ε e 2ε erf √2εtc + 1 They start at large negative times in the positive energy adiabatic subspace and smoothly and monotonically develop the √ exponentially small component in the negative energy adiabatic subspace in a ε-neighborhood of t = 0. Berry and Lim [Be, BeLi] argue that this behavior is universal: whenever θ has the form θ (t) =

±iγ + O(|t ± itc |α ) t ± itc

for some α > −1

near its singularities ±itc closest to the real axis,√then (20) should hold. For the Landau-Zener Hamiltonian (i.e., (2) with ρ = t2 + δ 2 and θ = arccot(t/δ)), which describes the generic situation, one finds after the transformation (10) that γ = 13 and α = − 31 . Hagedorn and Joye [HaJo] proved (20) for the Hamiltonian (12) with γ = 12 . In the approach of Berry and, slightly modified, of Hagedorn and Joye, the optimal superadiabatic basis vectors are obtained through optimal truncation of an asymptotic expansion of the true solution of (1) in powers of ε. In contrast, in our approach the optimal superadiabatic basis is constructed by approximately diagonalizing the Hamiltonian. The main advantage of “transforming the Hamiltonian” over “expanding the solutions” is that the former approach can be applied, at least heuristically, to more general adiabatic problems, cf. [Te], as for example the Born-Oppenheimer approximation. While we cannot control the asymptotics for the Born-Oppenheimer model rigorously yet, the heuristic application of the idea yields new physical insight into adiabatic transition histories and new expressions for the exponentially small off-diagonal elements of the S-matrix for simple Born-Oppenheimer type models, cf. [BeTe]. Therefore we see the rigorous results obtained in this paper also as a first attempt to justify the application of analogous ideas to more complicated but also more relevant systems. Furthermore, the concept of an adiabatically renormalized Hamiltonian was

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used to derive a criterion for selecting possible transition sequences in multi-level problems [WiMo]. For the specific problem (1) the knowledge of two linearly independent solutions is of course equivalent to the knowledge of the propagator and the effective Hamiltonian in the optimal superadiabatic basis. Therefore we shortly explain which aspects of our result constitute an improvement compared to [HaJo]: Most importantly, our proof does not rely on the a priori knowledge of the scattering amplitude A(ε). Indeed, our result yields for the first time a proof of (9) based on superadiabatic evolution, as expressed in Corollary 2. Moreover, we allow for a slightly larger class of Hamiltonians and obtain more detailed error estimates, which, in particular, give rise to close to optimal error bounds in the expansion of the S-matrix, cf. Corollary 2. Finally, we also get explicitly the next order correction in (19) resp. (20), cf. Section 6. It should be noted, however, that the improved error estimates and the next order corrections could have been obtained also based on the proof in [HaJo]. We finally turn to the scattering limit. Let Kε0 (t, s) denote the propagator in the original basis and define the scattering matrix in the adiabatic basis by 1 iH0 t iH0 t 0 a 0 ∗ 2 ε ε Sε := lim e U0 (t) Kε (t, −t) U0 (−t) e , where H0 = . 0 − 12 t→∞ Since, according to (14), for large negative and positive times the optimal superadiabatic basis agrees with the adiabatic basis, Sεa can be computed with help of the optimal superadiabatic propagator from Corollary 1. Corollary 2. For β < 1 we have πγ − tc ε (1 + O(εβ )) e 1 + O(ε) 2 sin 2 − tc . Sεa = e ε (1 + O(εβ )) 1 + O(ε) 2 sin πγ 2 Proof. According to (14) we have Sεa = lim e t→∞

iH0 t ε

Uεnε (t) Kε0 (t, −t) Uεnε ∗ (−t) e

iH0 t ε

= lim e t→∞

iH0 t ε

Kεnε (t, −t) e

iH0 t ε

.

Now the claim follows from inserting (18) and (19) with the improved error estimate outside of the transition region. From Corollary 2 we conclude that the transition amplitude is given by πγ tc A(ε) = P− Sεa P+ = 2 sin e− ε 1 + O(εβ ) , for any β < 1 , 2 which agrees with the results of [Jo], as explained in [BeLi]. We conclude the introduction with two recommendations for further reading: The numerical results of Berry and Lim [LiBe] beautifully illustrate the idea of optimal superadiabatic bases and universal adiabatic transition histories. The

224



introduction of the paper of Hagedorn and Joye [HaJo] gives a slightly different viewpoint on the problem and, in particular, a short discussion on how exponential asymptotics for the Schrödinger equation (1) fit into the broader field of exponential asymptotics for ordinary differential equations. Acknowledgments. We are grateful to Alain Joye and George Hagedorn for many helpful discussions.

2 Superadiabatic projections For the present and the following section we assume that H(t) has the form (12), but with some arbitrary θ ∈ C ∞ (R). The first aim is to construct time-dependent matrices π (n) ∈ R2×2 with (π (n) )2 − π (n) = O(εn+1 ), iε∂t − H, π (n) = O(εn+1 ).

(21) (22)

Here, [A, B] = AB−BA denotes the commutator two operators A and B. Likewise, we will later use [A, B]+ = AB + BA to denote the anti-commutator of A and B. Equation (21) says that π (n) is a projection up to errors of order εn+1 , while (22) implies that π (n) (t) is approximately equivariant, i.e., Kε0 (t, s) π (n) (s) = π (n) (t) Kε0 (t, s) + O(εn ) . Recall the Kε0 (t, s) is the unitary propagator for (1). Hence π (n) (t) is an almost projector onto an almost equivariant subspace. We construct π (n) inductively starting from the Ansatz π (n) =

n

πk εk .

(23)

k=0

By (12), H has two eigenvalues ±1/2. Let π0 be the projection onto the eigenspace corresponding to +1/2, and π (0) = π0 according to (23). It is easily checked that (21) and (22) are fulfilled for n = 0. In order to construct πn for n > 0, let us write Gn (t) for the term of order εn+1 in (21), i.e., (π (n) )2 − π (n) = εn+1 Gn+1 + O(εn+2 ) . Obviously, Gn+1 =

n

πj πn+1−j .

(24)

(25)

j=1

Proposition 1. Assume that π (n) given by (23) fulfills (21) and (22). Then a unique matrix πn+1 exists such that π (n+1) defined as in (23) fulfills (21) and (22). πn+1 is given by πn+1 = Gn+1 − π0 Gn+1 − Gn+1 π0 − i [πn , π0 ] . (26)

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Furthermore πn+1 is off-diagonal with respect to π0 , i.e., π0 = (1 − π0 )πn+1 (1 − π0 ) = 0, π0 πn+1

(27)

and Gn+1 is diagonal with respect to π0 , i.e., π0 Gn+1 (1 − π0 ) = (1 − π0 )Gn+1 π0 = 0.

(28)

Remark 4. The fact that the superadiabatic projections are unique answers the question raised in [Be] to which extent the superadiabatic basis constructed there is uniquely determined. Remark 5. Our construction can be seen as a special case of the construction in [EmWe], see also [Sj]. It was applied in the same context in [PST, Te]. The role and the importance of the superadiabatic subspaces as opposed to the superadiabatic evolution have been emphasized by Nenciu [Ne2 ]. He constructs the superadiabatic projections for much more general time-dependent Hamiltonians. However, Nenciu’s construction is less suitable for the explicit computations we need to perform. Proof. Let π (n+1) be given by (23) and suppose π (n) fulfills (21) and (22). Let π ñ+1 be an arbitrary matrix, and define π ˜ (n+1) = π (n) + εn+1 π ñ+1 . Then ˜ (n+1) , π ˜ (n+1) = (π (n) )2 − π (n) + εn+1 π ñ+1 − π ñ+1 . (˜ π (n+1) )2 − π +

Using (24), we see that terms of order εn+1 vanish if and only if Gn+1 = π ñ+1 − [π0 , π ñ+1 ]+ = (1 − π0 )˜ πn+1 (1 − π0 ) − π0 π ñ+1 π0 .

(29)

Multiplying (29) with (1 − π0 ) and with π0 on both sides and subtracting the results, we find that π ñ+1 must fulfill πn+1 (1 − π0 ) + π0 π ñ+1 π0 = Gn+1 − [Gn+1 , π0 ]+ . (1 − π0 )˜

(30)

Similarly iε∂t − H, π ˜ (n+1) = iε∂t − H, π (n) + εn+1 [iε∂t − H, π ñ+1 ] . Again terms of order εn+1 vanish if and only if ñ+1 ] . iπn = [H, π

(31)

Since π0 is the projector onto the eigenspace of H, we have π0 H = Hπ0 = Eπ0 , where E = 1/2 is the positive eigenvalue of H, and similarly (1 − π0 )H = H(1 − π0 ) = −E(1 − π0 ). When we multiply (31) first with with π0 from the left and with

226



1 − π0 from the right, then the other way round, and finally subtract the second result from the first, we get 2E(π0 π ñ+1 (1 − π0 ) + (1 − π0 )˜ πn+1 π0 ) = −i [πn , π0 ] .

(32)

Now we divide (32) by 2E and add (30) to find π ñ+1 = Gn+1 − [Gn+1 , π0 ]+ −

i [π , π0 ] . 2E n

(33)

˜ (n+1) should fulfill Thus π ñ+1 is uniquely determined by the requirement that π (21) and (22). On the other hand, H, Gn+1 − [Gn+1 , π0 ]+ = 0 and π0 [πn , π0 ] π0 = (1 − π0 ) [πn , π0 ] (1 − π0 ) = 0 , and thus πn+1 given by the right-hand side of (33) indeed fulfills (30) and (31). This shows existence. (28) and (27) now follow directly from (29) and (31). The calculation of π (n) via the matrix recurrence relation (26) and (25) is now possible in principle, but extremely cumbersome. In order to make more explicit calculations possible, we introduce a special basis of R2×2 . Recall that U0 (t) as defined in (3) is the unitary transformation into the basis consisting of the eigenvectors of H, i.e., the adiabatic basis, and let V0 (t) = θ2(t) U0 (t). With P = P+ as in (4) we then have U02 = V02 = id and P U0 V0 P = P V0 U0 P = 0, and π0 = U0 P U0 . Moreover, since G1 = 0 by (25), (26) implies i π1 = − θ (V0 P U0 − U0 P V0 ). 2

(34)

Motivated by this, we put X = V0 P U0 − U0 P V0 , Z = V0 P U0 + U0 P V0 ,

Y = V0 P V0 − U0 P U0 , W = V0 P V0 + U0 P U0 .

It is immediate that this is a basis of R2×2 for all t, and in fact −1 0 −1 1 0 X= , W = , Y = −2H, Z = Y . 1 0 0 1 θ Our reason for representing X through Z via U0 and V0 is that the following important relations now follow without effort: X = 0, Y = −θ Z, Z = θ Y, [X, Y ]+ = [X, Z]+ = [Y, Z]+ = 0, −X 2 = Y 2 = Z 2 = W,

(35) (36)

[X, π0 ] = Z, [Y, π0 ] = 0, W − [W, π0 ]+ = Y.

(37) (38)

[Z, π0 ] = X,

These relations show that this basis behaves extremely well under the operations involved in the recursion (26). This enables us to obtain

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Proposition 2. For all n ∈ N, πn is of the form πn = xn X + yn Y + zn Z,

(39)

where the functions xn , yn and zn satisfy the differential equations xn yn

zn

= =

izn+1 , −θ zn ,

(40) (41)

=

ixn+1 + θ yn .

(42)

Moreover, i x1 (t) = − θ (t), 2

y1 (t) = z1 (t) = 0.

(43)

Remark 6. Hence, for all even n, xn = 0, while for all odd n, yn = zn = 0. Proof. (43) was already noticed in (34), or alternatively follows from π0 = (W − Y )/2, (35) and (37). Now suppose πn is given by (39). By (36) and (38), Gn+1 − [Gn+1 , π0 ]+ is proportional to Y with a prefactor given through (25), and by (26), (35) and (37), πn+1 =

n (−xj xn+1−j + yj yn+1−j + zj zn+1−j )Y + i(θ yn − zn )X − ixn Z. (44) j=1

Comparing with (39) shows (40) and (42). To show (41), we use (27). This gives 0 = π0 πn π0 = (yn + θ zn )π0 Y π0 + (zn − θ yn )π0 Zπ0 + xn π0 Xπ0 . Since π0 Zπ0 = π0 Xπ0 = 0 and π0 Y π0 = −π0 , the claim follows.

Remark 7. From (40) through (42) we may derive recursions for calculating xn or zn , e.g., d θ (t)zn (t) dt + C . (45) zn+2 (t) = − z (t) + θ (t) dt n The constant of integration C must (and in some cases can) be determined by comparison with (44). In the case where θ is given by (11), this strategy will lead to fairly explicit expressions of the coefficient functions xn , yn and zn , cf. Proposition 5; from these we will extract the asymptotic behavior of xn , yn and zn , cf. Theorem 3. Using (40)–(42), we can give very simple expressions for the quantities appearing in (21) and (22). As for (22), we use (35) and the differential equations to find iε∂t − H, π (n) = iεn+1 πn = iεn+1 (xn X + (yn + θ zn )Y + (zn − θ yn )Z) =

−εn+1 (zn+1 X + xn+1 Z).

(46)

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Now we turn to (π (n) )2 − π (n) , the term by which π (n) fails to be a projector. Let us write n (π (n) )2 − π (n) = εn+k Gn+1,k . (47) k=1

With our earlier convention, Gn+1,1 = Gn+1 . Explicitly, (23) and (47) give Gn+1,k = [πk , πn ]+ + [πk+1 , πn−1 ]+ + . . . =

n−k

πj+k πn−j .

(48)

j=0

Proposition 3. For each n ∈ N, there exist functions gn+1,k , k ≤ n with ((π

) −π

(n) 2

(n)

)(t) =

n

ε

n+k

gn+1,k (t) W.

(49)

k=1

For each k ≤ n, gn+1,k = 2i(xk zn+1 − zk xn+1 ).

Proof. By (36), each Gn+1,k is proportional to W . Using (39) additionally, we find [πk , πm ]+ = 2(−xk xm + yk ym + zk zm )W , and thus (48) yields gn+1,k =

n−k

−xj+k xn−j + yj+k yn−j + zj+k zn−j .

j=0

Thus by using Proposition 2, gn+1,k

=

n−k

i(zj+k+1 xn−j + xj+k zn−j+1 ) − (θ zj+k yn−j + θ yj+k zn−j )

j=0

+θyj+k zn−j + θ yj+k zn−j − i(xj+k+1 zn−j + zj+k xn−j+1 ) =

i

n−k

((zj+k+1 xn−j − zj+k xn−j+1 ) + (xj+k zn−j+1 − xj+k+1 zn−j )

j=0

=

2i(xk zn+1 − zk xn+1 ).

The last equality follows because the sum is a telescopic sum.

Since W = id is independent of t, Proposition 3 gives the derivative of the correction (π (n) )2 − π (n) to a projector. As above, this gives an easy way for estimating the correction itself provided we have some clue how to choose the constant of integration.

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3 Construction of the unitary We now proceed to construct the unitary transformation Uεn into the nth superadiabatic basis. By (23) and (26), π (n) is self-adjoint. Thus it has two orthonormal eigenvectors vn and wn . Let cos(θ/2) sin(θ/2) v0 = , w0 = sin(θ/2) − cos(θ/2) be the eigenvectors of π0 , and write vn = αv0 + βw0 ,

wn = αw0 − βv0

(α, β ∈ C).

(50)

We make this representation unique by requiring 0 ≤ α ∈ R. Let Uεn be the unitary operator taking (vn , wn ) to the standard basis (e1 , e2 ) of R2 , i.e., Uεn = e1 vn∗ + e2 wn∗ ,

(51)

where all vectors are column vectors. Note that the definition (3) of U0 is consistent with (51) for n = 0. Uεn diagonalizes π (n) , thus λ1 0 Uεn π (n) Uεn ∗ = D ≡ , (52) 0 λ2 where λ1,2 are the eigenvalues of π (n) . Although α, β and λ1,2 depend on n, ε and t, we suppress this from the notation. Lemma 1. U0 Uεn ∗ =

α −β β α

,

U0 Uεn ∗ =

and

α + β β − α

α−β α + β

.

Proof. The calculations are straightforward and we only show the second equality. First note that v0 = −w0 and w0 = v0 . Thus

Uεn ∗ = ((α + β)v0 + (β − α)w0 )e∗1 + ((α − β )v0 + (α + β)w0 )e∗2 , and using the orthogonality of v0 and w0 yields the claim,

U0 Uεn ∗ = e1 (α + β)e∗1 + e1 (α − β )e∗2 + e2 (β − α)e∗1 + e2 (α + β)e∗2 . It will turn out that β, α α, and β are small quantities, λ1 , λ2 , and λ2 are even much smaller, while α2 and λ1 are large, i.e., of order 1+O(ε). This motivates the form in which we present the following result.

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Proposition 4. Suppose λ1 = λ2 . Then for each n ∈ N, 2 Uεn (iε∂t −H)Uεn ∗

with R=

1 2

= iε∂t −

α2 εn+1 λ1 −λ2 (−xn+1

εIm(β(2α + β )) + |β|2 εn+1 β 2 λ1 −λ2 (xn+1

− zn+1 )

α εn+1 λ1 −λ2 (xn+1 − 12

− zn+1 ) n+1


− zn+1 )

2

− ελ1 −λβ2 (xn+1 + zn+1 )

−εIm(β(2α + β )) − |β|2

+R,

.

Proof. Let us write Uεn (iε∂t − H)Uεn ∗ = (Mi,j ), i, j ∈ {1, 2}. M1,1 and M2,2 are calculated in a straightforward manner, using Lemma 1 together with the fact 1/2 0 U0 HU0∗ = : 0 −1/2 Uεn (iε∂t − H)Uεn ∗

= =

iε∂t + iεUεn U0∗ U0 Uεn ∗ − Uεn U0∗ U0 HU0∗ U0 Uεn ∗ α β α + β α − β iε∂t + iε −β α β − α α + β 1 1 0 α −β α β − . 0 −1 −β α β α 2

Carrying out the matrix multiplication yields 1 M1,1 = −M2,2 = iε∂t + iε((α(α + β) + β(β − α)) − (α2 − |β|2 ). 2

(53)

We now use α2 + |β|2 = 1 to obtain 0 = 2αα + β β + β β = 2Re(αα + ββ ) and α2 − |β|2 = 1 − 2|β|2 . Plugging these into (53) gives the diagonal coefficients of M . Although we could get expressions for the off-diagonal coefficients by the same method, these would not be useful later on. Instead we use (52), i.e., Uεn ∗ D = π (n) Uεn ∗ together with (46) and obtain Uεn (iε∂t − H)Uεn ∗ D = DUεn (iε∂t − H)Uεn ∗ − εn+1 Uεn (zn+1 X + xn+1 Z)Uεn ∗ . (54) By multiplying (54) with ej e∗j from the left and by ek e∗k from the right (j, k ∈ {1, 2}) and rearranging, we obtain (λk − λj ) ej e∗j Uεn (iε∂t − H) Uεn ∗ ek e∗k

(55)

= −εn+1 ej e∗j Uεn (zn+1 X + xn+1 Z) Uεn ∗ ek e∗k − iδk,j ε λj ej e∗j .

0 1 0 −1 ∗ From the equalities = , U0 ZU0 = and Lemma 1 −1 0 −1 0 we obtain 2 2 α(β − β) α + β Uεn XUεn ∗ = , −(α2 + β 2 ) −α(β − β) U0 XU0∗

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Uεn ZUεn ∗

=

−α(β + β) −(α2 − β 2 )

2

−(α2 − β ) α(β + β)

231

.

The expressions for M1,2 and M2,1 follow by taking k = j in (55).

We now use our results from the previous section to express α, β and λ1,2 in terms of xk , yk and zk , k ≤ n. Let us define n (56) ξ ≡ ξ(n, ε, t) = k=1 εk xk (t), n k (57) η ≡ η(n, ε, t) = k=1 ε yk (t), n k ζ ≡ ζ(n, ε, t) = k=1 ε zk (t). (58) Moreover, let g ≡ g(n, ε, t) =

n

k=1 ε

n+k

gn+1,k (t)

(59)

be the quantity appearing in (49). Lemma 2. The eigenvalues of π (n) solve the quadratic equation λ21,2 − λ1,2 − g = 0. Proof. By (52) and Proposition 3 we obtain 2 g λ1 − λ1 0 n (n) 2 (n) n∗ n n∗ = U ((π ) − π ) U = U gW U = ε ε ε ε 0 λ22 − λ2 0

0 g

.

Lemma 3. α2 (λ1 − λ2 ) = 1 − η − λ2 , Proof. We have

and

αβ(λ1 − λ2 ) = −ξ − ζ.

π (n) = λ1 vn vn∗ + λ2 wn wn∗ .

(60)

Plugging in (50), we obtain π (n) v0

=

λ1 αvn − λ2 βwn = (λ1 α2 + λ2 |β|2 )v0 + (λ1 − λ2 )αβw0

=

(α2 (λ1 − λ2 ) + λ2 )v0 + (λ1 − λ2 )αβw0 .

In the last step, we used |β|2 + α2 = 1. On the other hand, from (23) and (26) we have n π (n) = π0 + εk (xk X + yk Y + zk Z) , (61) k=1

and since Xv0 = Zv0 = −w0 , π0 v0 = v0 and Y v0 = −v0 , we find π (n) v0 = (1 − η)v0 − (ξ + ζ)w0 . Comparing coefficients finishes the proof.

232



Theorem 2. Let ε0 > 0 be sufficiently small. For ε ∈ (0, ε0 ] assume there is a bounded function q on R such that ξ(t), η(t), ζ(t) and their derivatives ξ (t), η (t), ζ (t) are all bounded in norm by εq(t). Then Uεn (iε∂t − H)Uεn ∗ = iε∂t −

1 2

+ O(ε2 q)

εn+1 (−xn+1 − zn+1 ) (1 + O(εq))

εn+1 (xn+1 − zn+1 ) (1 + O(εq)) − 21 + O(ε2 q)

(62) .

Proof. From (61) and our assumptions it follows that π (n) − π0 = O(εq). Thus λ1 = 1 + O(εq) and λ2 = O(εq), and from Lemma 2 we infer g = O(ε) and λ1 = 12 1 + 1 + 4g , λ2 = 12 1 − 1 + 4g . Since λ1 − λ2 = 0, Lemma 3 yields √ 1 + 1 + 4g − 2η √ , α2 = 2 1 + 4g

−ξ − ζ β= √ . 1 + 4gα

Hence α2 = 1 + O(εq), and β, β and αα = (α2 ) /2 are all O(εq). Plugging these into the matrix R in Proposition 4 shows the claim.

4 Solving the recursion: a pair of simple poles In order to make further progress, we need to understand the asymptotic behavior of the off-diagonal elements of the effective Hamiltonian in the nth superadiabatic basis for large n. According to (62) this amounts to the asymptotics of xn and zn as given by the recursion from Proposition 2. It is clear that the function θ alone determines the behavior of this recursion. We will study here the special case θ (t) =

iγ iγ γtc − = 2 . t + itc t − itc t + t2c

(63)

The reason lies in the intuition that the poles of θ closest to the real axis determine the superadiabatic transitions, and that these transitions are of universal form whenever these poles are of order one, see [Be, BeLi] for details. As in [HaJo], we have to restrict to the special case that θ has no contribution besides these poles in order to solve the recursion. We now have two parameters left in θ . The distance tc of the poles from the real axis determines the exponential decay rate in the offdiagonal elements of the Hamiltonian and the strength of the residue γ determines the pre-factor in front of the exponential. As is done in [HaJo], we could get rid of the parameter tc by rescaling time, but we choose not to do so because tc plays a nontrivial role in optimal truncation and the error bounds obtained therein, and keeping this parameter will make things more transparent.

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233

We use (45) in order to determine the asymptotics of zn . From Proposition 2 together with (44) it is clear that yn must be integrable on R. This fixes the constant of integration in (45), and we arrive at the linear two-step recursion zn+2 (t) = −

d dt

zn (t) + θ (t)

t −∞

θ (s)zn (s) ds .

(64)

The fact that the recursion is linear will make its analysis simpler than the one of the nonlinear recursion in [HaJo]. We rewrite θ as θ (t) =

γ (f + f ) tc

with

f (t) =

itc . t + itc

For zn , we will make an Ansatz as a sum of powers of f and f . The reason for the success of this approach is the fact that this representation is stable under differentiation and integration, and also under multiplication with θ through the partial fraction expansion. More explicitly, the following identities hold for the mth power f m of f : Lemma 4. For each m ≥ 1, θ Im(f m ) θ Re(f m ) Im(f m ) Re(f m ) Proof. We have f + f =

m−1 γ −k 2 Im(f m+1−k ), tc k=0 m−1 γ −k m+1−k −m = 2 Re(f )+2 θ , tc k=0 m = − Re(f m+1 ), tc m = Im(f m+1 ). tc

=

2t2c t2 +t2c

f kf =

θf

(66) (67) (68)

= 2f f , and thus

1 k−1 1 f (f + f ) = (f k + f k−1 f ) 2 2

and n−j

(65)

γ = tc

n−j

2

−k n+1−(j+k)

f

+2

−n+j

f

.

(69)

k=0

Taking the complex conjugate of (69) and adding it to resp. subtracting it from (69), we arrive at (65) and (66). To prove (67) and (68), it suffices to use that (f k ) = kf k+1 /(itc ) along with the complex conjugate equation.

234


Ann. Henri Poincaré (n)

Proposition 5. For each even n ∈ N and j = 0, . . . , n − 1, let the numbers aj recursively defined through a0

(2)

=

(n+2)

=

aj

(2)

a1 = 0 , j k 1 (n) n+1−j (n) 2 a (n − j) aj − γ (n + 1) n n − k m=0 m

1,

be

(70) (j < n) , (71)

k=0

a(n+2) n

=

(n+2)

(n+2)

an−1 ,

an+1 = 0 .

Then zn yn

n−1 (n − 1)! −j (n) 2 aj Im(f n−j ) (n even) , tnc j=0 j n−1 1 (n) 2 (n − 1)! −j = γ 2 ak Re(f n−j ) tnc n − j j=0

= −γ

(72)

(n even) ,

(73)

k=0

xn

n−1 n (n+1) (n − 1)! −j aj = iγ 2 Re(f n−j ) tnc n − j j=0

Proof. We proceed by induction. We have x1 = iθ /2 = and (68), i γ z2 = x1 = − 2 Im(f 2 ). tc tc

(n odd) ,

iγ tc Re(f ),

(74)

and thus by (40)

This proves (72) for n = 2. Now suppose that (72) holds for some even n ∈ N. Then by (40) and (68), (74) holds for n − 1. To prove (73) for the given n, we want to use (41). (65) and the induction hypothesis on zn yield

θ zn

−γ

=

2 (n

−γ 2

=

n−1 − 1)!

tn+1 c

j=0 n−1

(n) aj

(n − 1)! tn+1 c m=0

n−j−1

2−(k+j) Im(f n+1−(j+k) )

k=0

  m (n) 2−m  aj  Im(f n+1−m ).

(75)

j=0

Since (75) only contains second or higher order powers of f , it is easy to integrate using (68). Let us write m 1 (n) = a . (76) b(n) m n − m j=0 j Then by (68) we obtain yn = −

t

−∞

θ (s)zn (s) ds = r2

n−1 (n − 1)! −m (n) 2 bm Re(f n−m ), tn−1 c m=0

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proving (73) for n. It remains to prove (72) for n + 2. We want to use (64), and therefore we employ (66) and our above calculations in order to get θ (t)

t

−∞

θ (s)zn (s) ds

n−j+1 n−1 (n − 1)! = −γ 3 n+1 bj 2−(k+j) Re(f n+1−(k+j) ) + 2−n θ tc j=0 k=0    j n−1 n−1 (n − 1)! = −γ 3 n+1  2−j bk Re(f n+1−j ) + 2−n+1 bk Re(f ) . tc j=0 k=0 k=0 By (67), zn = γ

n−1 (n − 1)! −j (n) 2 aj (n − j)Re(f n+1−j ). tn+1 c j=0

Now we sum the last two expressions, differentiate again and obtain  j n−1 (n − 1)!  −j (n) 2 zn+2 = −γ n+2 2 (n + 1 − j) (n − j)aj − γ bk Im(f n+2−j ) tc j=0 k=0 n−1 −2γ 2 2−n bk Im(f 2 ) . k=0

Comparing coefficients, this proves (72) for n + 2.

(n)

We now investigate the behavior of the coefficients aj (n)

Proposition 6. Let aj (n)

(a) a0

=

as n → ∞.

be defined as in Proposition 5.

2 sin(γπ/2) . 1 + O nγ 2 γπ/2

(b) There exists C1 > 0 such that for all n ∈ N (n)

|a1 | ≤ C1

ln n . n−1

(c) For each p > 1 there exists C2 > 0 such that for all n ∈ N sup p−j |aj | ≤ (n)

j≥2

C2 . n−1

236



(2)

Proof. (a) By (70), a0 = 1, and (n+2)

a0

(n)

= a0

γ2 1− 2 . n

Comparing with the product representation of the sine function ([AbSt], 4.3.89) sin(πx) = πx

∞ x2 1− 2 , n n=1

we arrive at (a). (n) (b) Put αn = (n − 1)a1 . Then by (71), 1 1 γ2 (n) 2 + αn+2 = αn 1 − − γ a0 . (n − 1)2 n n−1 thus for n − 1 > γ, we have |αn+2 | ≤ |αn | + γ

2

1 1 + n n−1

(m)

max |a0 |, m∈N

which shows (b). (n) (n) (n) (c) Put cj = (n − 1)p−j aj , and c(n) = maxj≥2 |cj |. We will show that the (n) sequence c is bounded. We have j k 1 −j+m (n) n+1−j (n+2) (n) cj (n − j)cj − γ 2 = p cm n(n − 1) n − k m=2 k=2 j j 1 1 (n) (n) −j 2 + a1 . (77) −(n − 1)p γ a0 n−k n−k k=0

Now

j k=2

k=1

k 1 −j+m (n) 1 p2 p cm ≤ c(n) , n − k m=2 n − j (p − 1)2

and p−j

j k=0

(j + 1)p−j 1 1 ≤ ≤ . n−k n−j (n − j) ln p

We plug these results into (77) and obtain (n + 1 − j)(n − j) (n + 1 − j) γ 2 p2 1 (n+2) |cj | ≤ c(n) + n(n − 1) (n − j)(p − 1)2 n(n − 1) 1 (n + 1 − j) p2 γ 2 (n) (n) (|a | + |a1 |). + n (n − j)(p − 1)2 ln p 0

(78)

(79)

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Precise Coupling Terms in Adiabatic Quantum Evolution (n)

237

(n)

By (a) and (b), a0 and a1 are bounded. Taking the supremum over j ≥ 2 above, we see that there exist constants B1 and B2 with n−2 B1 B2 (n+2) (n) + , ≤c c + n n(n − 1) n hence c(n+2) − c(n) ≤

1 n

B1 −2 + c(n) + B2 . n−1

Now let n − 1 > B1 . Then for c(n) > B2 , the above inequality shows c(n+2) < c(n) , while for c(n) ≤ B2 , c(n+2) ≤ c(n) + B2 /n ≤ B2 (1 + 1/n). Thus c(n) is a bounded sequence. Remark 8. We will make no use of the fact that the logarithmic correction to the (n) 1/n-decay of the higher coefficients occurs only in the coefficient a1 . We chose to include this in the statement of the preceding theorem anyway, because this gives some insight into the nature of the recursion and is not hard to prove. (n)

Remark 9. Numerical calculations of the first few thousand aj suggest that (c) above continues to be true if we choose p = 1, but this seems to be much harder to prove. However, the estimate above is more than good enough for us. Remark 10. The constants appearing in the proof of Proposition 6 (b) and (c) are not optimal, and could be improved by more careful arguments. This is unimportant for our purposes, and for the sake of brevity and readability we chose to use the simple estimates given. (n)

Corollary 3. Let bj such that

be given by (76). Then for each p > 1, there exists C3 > 0 sup p−j bj

(n)

j≥0

≤

C3 . n−1

Proof. For j ≤ n − 1, we have n − 1 ≤ j(n − j), and thus Proposition 6 (c) gives p2 p−j C2 (n) (n) (n) p−j bj (pj−1 − 1) ≤ (a0 + a1 ) + n−j n−1p−1 ≤ p−j

(n)

(n)

p C2 1 C3 j(a0 + a1 ) + ≤ . n−1 p−1n−1 n−1 (n)

Having good control over the coefficients aj , we can now derive relatively sharp estimates on the functions xn , yn and zn . Let us fix α < 1 and define ! " n−2 n 2 1 1 t t c c max , √ . Rnα (t) = (n − 1)α t + itc t + itc 2

238



Obviously, for t ≤ tc the first function in the maximum above dominates, for t > tc the second one does. For families of functions gn (t), Gn (t) we write gn (t) = O(Gn (t)) if there exists C > 0 such that |gn (t)| ≤ C|Gn (t)| for all n ∈ N and all t ∈ R. Theorem 3. For n > 1 and α < 1, we have −n t (n − 1)! 2 sin(γπ/2) α Re 1−i + O(Rn (t)) , xn (t) = i tnc π tc

(80)

(n − 1)! O(Rnα (t)), (81) tnc −n t (n − 1)! 2 sin(γπ/2) Im 1 − i − + O(Rnα (t)) , (82) tnc π tc

yn (t) = zn (t) =

Proof. With the definition of f and Proposition 6 (a) we get −n tc t 2 sin(γπ/2) 1 (n) n Im 1−i +O a0 Im(f ) = 2 πγ tc n t + itc

n

when n is even, and a similar formula for a0 Re(f n ) when n is odd. This covers the j = 0 terms in (74) and (72). For the remaining terms, let ! (n) aj if n is even, (n) cj = (n) naj /(n − j) if n is odd. (n+1)

Now n/(n − j) ≤ j for j < n, and thus by Proposition 6 (b) and (c) for each p > 1 we can find C > 0 such that C (n) cj ≤ jpj (n − 1)α for all j ≥ 1. (For j ≥ 2, we may even choose α = 1, but we will not exploit this.) For |t| ≤ tc , we have |tc /(t + itc )|−j ≤ 2j/2 , so we get   n−j j n−1 n−1 n c(n) C tc p itc j   √ ≤ j . 2j t + itc (n − 1)α j=2 t + itc 2 j=2

√ If we choose p < 2, the sum on the right-hand side is bounded uniformly in n. Combining this with our above calculations, (80) and (82) are proved for |t| < tc . √ For |t| > tc , we have |tc /(t + itc )| ≤ 1/ 2, and thus n−2 j=2

(n)

cj 2j

itc t + itc

n−j

C tc ≤ α (n − 1) t + itc

p j 1 n−2−j √ j . 2 2 j=2

2 n−2

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√ If we√choose again p < 2, the sum on the right-hand side is bounded by ˜ C(1/ 2)n−2 uniformly in n. For the term with j = n − 1, this does not work (n) since then n − 2 − j < 0. But for n even, this term vanishes since then cn−1 = 0, and for n odd, it equals n−2 (n) (n+1) 2 cn−1 nan−1 itc C˜ tc 1 t2c √ Re ≤ . = 2n t + itc 2n t2 + t2c n−1 t + itc 2 This proves (80) and (82) for |t| ≥ tc . The proof of (81) is similar and uses Corollary 3.

5 Optimal truncation By the results of the previous section πk grows like (k − 1)!/tkc . Hence, the sum n (n) π = k=0 εk πk does not converge to an exactly equivariant projection π (∞) as n → ∞. This is the reason why we see exponentially small transitions. The basis in which these transitions develop smoothly is the optimal superadiabatic basis: since we cannot go all the way to infinity with n, we fix ε and choose n = n(ε) such that the off-diagonal elements in (2) become minimal. Using Stirling’s formula and (80) resp. (82), it is easy to see that the place to truncate is at n(ε) = tc /ε. This n(ε) is in general not a natural number, but we will find that a change of n which is of order one does not change the results. Before we go into more details, we need a preliminary result. Lemma 5. Uniformly in x ∈ [0, 1] and for k > 0, we have (1 + x)−k = e−kx + e−kx/2 O k1 . Proof. We start with the equality

(1 + x)−k − e−kx = e−kx ek(x−ln(1+x)) − 1 .

(83)

At first consider x > 1/k. There we use the inequality (x − ln(1 + x)) ≤ x/3, valid for 0 ≤ x ≤ 1, in (83) and obtain |(1 + x)−k − e−kx | ≤ e−kx ekx/3 − 1 = e−kx/2 e−kx/6 − e−kx/2 . For x > 1/k, theterm in the last bracket above is O(1/k), and we are done in this case. For x ≤ 1/k, we use (x − ln(1 + x)) ≤ x2 /2 and rearrange (83) to get 2 ekx/2 ((1 + x)−k − e−kx ) ≤ e−kx/2 ekx /2 − 1 =: f (x, k). To find out where f (x, k) is maximal, we calculate 2 d k f (x, k) = e−kx/2 1 + ekx /2 (2x − 1) . dx 2

240



The derivative is zero exactly at the solutions of the equation ln(1 − 2x)/x2 = −k/2.

(84)

+ R(x), where R(x) is a power series in x, convergent Now ln(1 − 2x)/x2 = −2/x for x < 1/2. Thus for x < 1/k and k sufficiently large, there exists exactly one solution x∗ (k) of (84), and x∗ (k) < C/k uniformly in k for some C > 0. Since d 2 ∗ dx f (x, k) > 0 for x < 1/k , f (x, k) has a maximum at x (k). Thus f (x, k) ≤ f (x∗ (k), k) ≤ e−C/2k − 1 = O(1/k) for x
0 such that |xk (t)| ≤ Cθ (t)(k − 1)!/tkc for each k. The same is true for yn and zn . Using the differential equations (40)–(42), we find that there is C > 0 with |xn (t)| ≤ C θ (t)n!/tn+1 . This means that c

|ξ (t)| ≤ εC θ (t)

n k=1

εk t−k−1 k! εk−1 , c

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241

with similar expressions for the other quantities. Now taking ε = tc /(nε − σε ), we find nε

k t−k c ε (k + 1)! =

k=0

nε (k + 1)! 2 3! = 1 + + + . . . . (nε − σε )k nε − σε (nε − σε )2 k=0

Each of the nε + 1 terms in the sum above is bounded by const/(nε − σε ) except the first which is 1. This shows ε , |ξ (t)| ≤ εθ C 1 + nεn−σ ε and Theorem 2 gives (15) with cnε ε (t) = εnε +1 xnε +1 (t)(1 + O(εθ (t)). Recall that znε +1 (t) = 0 due to our convention. It remains to determine the leading order asymptotics of εnε +1 xnε +1 . For convenience of the reader let us rewrite (80) as εnε +1 xnε +1 (t) = i

εnε +1 nε ! tnc ε +1

# 2 sin(γπ/2) π

Re

1 − i ttc

−(nε +1)

$ + O Rnβε +1 (t) . (87)

Lemma 6. With (86), we have εnε +1 nε ! = tnc ε +1

2πε − tc e ε (1 + O(ε)). tc

Proof. Stirling’s formula for (n + 1)! implies n! =

1 n+1

n+1 e

n+1

√

√ 1 . n + 1 2π 1 + O n+1

Together with (85) this yields ε

nε +1

−(nε +1) 2π σε 1 + O nε1+1 nε ! = 1− nε + 1 nε + 1 2π 1 + O nε1+1 = tnc ε +1 e−(nε +1) eσε nε + 1 tc 2πε (1 + O(ε)). = tnc ε +1 e− ε tc + εσε tnc ε +1 e−(nε +1)

Finally,

2πε = tc + εσε

2πε tc

−1/2 2πε εσε = (1 + O(ε)) . 1+ tc tc

242



Lemma 6 takes care of the first factor in (87). Turning to the terms inside the square brackets in (87), let us first note that for |t| ≥ tc , both terms are O(2−(nε −1)/2 /(1 + t2 )) = O(exp(−tc ln 2/(2ε))/(1 + t2 )) , proving the theorem in this case. For |t| < tc , we investigate the modulus and the phase separately. Let 0 < β < 1. From Lemma 5 it follows that 1 + i ttc

nε +1

= 1+

t2 t2c

(tc /ε+σε )/2

= 1+

t2 t2c

σε /2 t2 t2 e− 2tc ε + O εe− 4tc ε .

For |t| ≥ εβ/2 , exp(−t2 /(2tc ε)) = O(ε exp(−t2 /(4tc ε))). Thus neither the prefactor involving σε above nor the phase play any role in this region. For |t| < εβ/2 , (1 + t2 /t2c )σε /2 = 1 + O(σε εβ ) and therefore 1 + i ttc

nε +1

t2 t2 = e− 2tc ε + O εβ e− 4tc ε .

The same reasoning applies to Rnβε +1 and gives t2 t2 Rnβε +1 (t) ≤ εβ e− 2tc ε + O εβ e− 4tc ε . Turning to the phase in the region |t| < εβ/2 , we find ei(nε +1) arctan(t/tc ) = exp i tεc + σε (t/tc ) − 13 (t/tc )3 + O((t/tc )5 ) σε t 5 3 t3 = exp i εt − 3εt + O(t (t/t ) /ε) + O(σ (t/t ) ) 2 + t c c ε c c c σε t t t3 1 + O(ε5β/2−1 ) + O(ε3β/2 ) . = exp i ε − 3εt2 + tc c

Now we just have to collect all the pieces and add the complex conjugate.

Let us now see what of the above would have changed for nε odd. Then xnε +1 = 0, and (82) together with Lemma 5 and 6 yields cnε ε (t) = =

−εnε +1 znε +1 (t) (1 + O(εθ )) (88) 3 2 tc t t t 2ε πγ σε t − e− ε e− 2εtc sin sin + 2 + O (φα (ε, t)) . πtc 2 ε 3εt2c tc

At first, this looks like an important difference, since now the off-diagonal elements in the transformed Hamiltonian are purely real-valued in leading order, while in the other case they were purely imaginary. However, in the computation of the propagator, another factor of exp(±it/ε) from the dynamical phase appears, cf. (89). At leading order only the resonant term of the Hamiltonian survives, which is the same for odd and even nε .

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6 First-order perturbation in the optimal superadiabatic basis In this section we prove Corollary 1. Since we use standard first-order perturbation theory, we stay sketchy in some parts. After splitting Hεnε (t), see (15), as 1 0 nε 2 + Vε (t) =: H0 + Vε (t) , Hε (t) = 0 − 12 Dyson expansion in the interaction picture (cf. [ReSi], Thm. X.69) yields itH iτ H isH0 i t iτ H0 nε − ε0 − ε0 ε Kε (t, s) = e e Vε (τ ) e dτ e ε id − ε s tc O(ε2 ) O(εe− ε ) ∆(t, s) . + tc O(ε2 ) O(εe− ε ) Thus we only need to evaluate the integral −

i ε

t

e

iτ H0 ε

s

Vε (τ ) e−

iτ H0 ε

dτ

=

iτ i t 0 e ε cnε ε (τ ) dτ iτ ε s e− ε cnε ε (τ ) 0 O(ε) 0 + ∆(t, s) . 0 O(ε)

−

Inserting (16) and using (17) gives −

i ε

t

iτ

e ε cnε ε (τ )dτ s iτ iτ 3 iστ t 3 πγ iτ tc τ2 iτ 2 − + − − iτ + iστ e− ε = sin e ε e− 2εtc e ε 3εt2c tc + e ε 3εt2c tc dτ επtc 2 s α − tεc ∆(t, s) = (∗) , (89) +O ε e ±(

iτ 3

− iστ )

3

iτ iστ for each α < 1. Now we replace the exponentials e 3εt2c tc by 1 ± ( 3εt 2 − t ). c c Using |eiϕ − 1 − iϕ| ≤ ϕ2 , we conclude that the resulting error is bounded by a constant times 6 ∞ tc tc τ2 1 τ τ4 2 + τ e− 2εtc + dτ = O(εe− ε ) . ε − 2 e− ε 2 ε ε −∞

Hence we obtain % (∗) =

2 επtc

sin

πγ 2

tc

e− ε

s α − tεc +O ε e ∆(t, s)

t

τ2

e− 2εtc

1+

iτ 3 3εt2c

−

iστ tc

+e

2iτ ε

1−

iτ 3 3εt2c

+

iστ tc

dτ (90)

244



with α < 1, where the first summand in the integrand gives rise to the explicit term in (19). The remaining terms can be integrated explicitly as well, most conveniently √ tc using Maple or Mathematica. They are all of order O( εe− ε ∆(t, s)) uniformly tc in t and s resp. of order O(εα e− ε ∆(t, s)) for |t| and |s| larger than εβ for some β < 12 . To illustrate the reasoning note that

t s

s2 τ2 t2 e− 2εtc τ dτ = εtc e− 2εtc − e− 2εtc . 2β−1

This is uniformly of order O(ε), but of order O(e−ε ) for |t| and |s| larger than εβ . Finally we emphasize that we could get the next order corrections to (19) explicitly by evaluating (90).

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M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions, 9th printing, Dover, New York, 1972.

[ASY]

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[Be]

M.V. Berry, Histories of adiabatic quantum transitions, Proc. R. Soc. Lond. A 429, 61–72 (1990).

[BeLi]

M.V. Berry and R. Lim, Universal transition prefactors derived by superadiabatic renormalization, J. Phys. A 26, 4737–4747 (1993).

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V. Betz and S. Teufel, Adiabatic transition histories for Born-Oppenheimer type models, in preparation.

[BMKNZ] A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu and J. Zwanziger, The geometric phase in quantum systems, Texts and Monographs in Physics, Springer, Heidelberg, 2003. [BoFo]

M. Born and V. Fock, Beweis des Adiabatensatzes, Zeitschrift f¨ ur Physik 51, 165–169 (1928).

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C. Emmrich and A. Weinstein, Geometry of the transport equation in multicomponent WKB approximations, Commun. Math. Phys. 176, 701–711 (1996).

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L.M. Garrido, Generalized adiabatic invariance, J. Math. Phys. 5, 335 (1964).

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G. Hagedorn and A. Joye, Time development of exponentially small non-adiabatic transitions, Commun. Math. Phys. 250, 393–413 (2004).

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[Jo]

A. Joye, Non-trivial prefactors in adiabatic transition probabilities induced by high order complex degeneracies, J. Phys. A 26, 6517–6540 (1993).

[JKP]

A. Joye, H. Kunz and C.-E. Pfister, Exponential decay and geometric aspect of transition probabilities in the adiabatic limit, Ann. Phys. 208, 299 (1991).

[JoPf1 ]

A. Joye and C.-E. Pfister, Exponentially small adiabatic invariant for the Schr¨ odinger equation, Commun. Math. Phys. 140, 15–41 (1991).

[JoPf2 ]

A. Joye and C.-E. Pfister, Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum, J. Math. Phys. 34, 454–479 (1993).

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T. Kato, On the adiabatic theorem of quantum mechanics, Phys. Soc. Jap. 5, 435–439 (1950).

[Le]

A. Lenard, Adiabatic invariants to all orders, Ann. Phys. 6, 261–276 (1959).

[LiBe]

R. Lim and M.V. Berry, Superadiabatic tracking of quantum evolution, J. Phys. A 24, 3255–3264 (1991).

[Ma]

A. Martinez, Precise exponential estimates in adiabatic theory, J. Math. Phys. 35, 389–391 (1994).

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G. Nenciu, Adiabatic theorem and spectral concentration, Commun. Math. Phys. 82, 121–135 (1981).

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G. Nenciu, Linear adiabatic theory. Exponential estimates, Commun. Math. Phys. 152, 479–496 (1993).

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G. Panati, H. Spohn and S. Teufel, Space-adiabatic perturbation theory, Adv. Theor. Math. Phys. 7, 145–204 (2003).

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M. Reed and B. Simon, Methods of modern mathematical physics II, Academic Press (1975).

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J. Sj¨ ostrand, Projecteurs adiabatiques du point de vue pseudodifférentiel, C. R. Acad. Sci. Paris Sér. I Math. 317, 217–220 (1993).

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Volker Betz Institute for Biomathematics and Biometry GSF Forschungszentrum Postfach 1129 D-85758 Oberschleißheim Germany email: [email protected] Stefan Teufel Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom email: [email protected] Communicated by Yosi Avron submitted 29/06/04, accepted 14/08/04





Existence of the D0–D4 Bound State: a Detailed Proof ∗ Laszlo Erdös, David Hasler and Jan Philip Solovej

Abstract. We consider the supersymmetric quantum mechanical system which is obtained by dimensionally reducing d=6, N =1 supersymmetric gauge theory with gauge group U(1) and a single charged hypermultiplet. Using the deformation method and ideas introduced by Porrati and Rozenberg [1], we present a detailed proof of the existence of a normalizable ground state for this system.

1 Introduction The particular system, which we will consider, belongs to a class of supersymmetric quantum mechanical models. These models appear in the study of quantized membranes [2], D-brane bound states [3], and M-theory [4]. Especially the question of existence respectively absence of normalizable ground states, i.e., zero energy states, is of physical importance. The Hamiltonian of these models is of the form H = −∆ + V + HF . The scalar potential V is polynomial in the bosonic degrees of freedom and admits zero energy valleys extending to infinity while HF is quadratic in the fermionic degrees of freedom and linear in the bosonic degrees of freedom. Moreover, the Hilbert space carries a unitary representation of a gauge group. The physical Hilbert space consists of gauge invariant states. Due to supersymmetric cancellations, the zero energy valleys render the Hamiltonian to have continuous spectrum, which covers the positive real axis. Therefore, the Hamiltonian is non-Fredholm and the question about existence of ground states is subtle. The Witten index IW , i.e., the number of bosonic ground states minus the number of fermionic ground states, can be calculated by means of IW = lim lim Tr((−1)F χR e−βH ) , R→∞ β→∞

where χR denotes the characteristic function of the ball of radius R centered around the origin in configuration space, c.p. [5]. Since there is no gap in the ∗ Work partially supported by NSF grant DMS-0200235, by EU grant HPRN-CT-2002-00277, by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation, and by grants from the Danish research council.

248

L. Erd¨ os, D. Hasler and J.P. Solovej


spectrum one has to deal with a delicate analysis of boundary contributions. As a different approach, Porrati and Rozenberg proposed in [1] a deformation method to detect the existence of normalizable ground states for systems with at least two real supercharges. One deforms the supercharges of the system with a real potential w, D → Dw := e−w Dew ,

† D † → Dw := ew D† e−w ,

† † + Dw Dw such that the spectrum of the deformed Hamiltonian Hw := Dw Dw becomes discrete. This might allow one to show the existence of a ground state Ψw for the deformed problem. Provided that e±w Ψw is normalizable, then, the original problem admits a ground state as well. Using this method the number of ground states for numerous models could be determined, [6]. In this paper, we consider the quantum mechanical system which is obtained by dimensionally reducing N = 1 supersymmetric gauge theory, with gauge group U(1) and with a single charged hypermultiplet from six dimensions. The system appears in the problem of counting H-monopole ground states in the toroidally compactified heterotic string [7]. Moreover, the same system describes the low energy dynamics of a D0-brane in the presence of a D4-brane [8, 9]. String duality arguments predict the existence of exactly one bound state at threshold for this system, c.p. [10]. The existence of such a state provides a check of the correctness of these duality hypotheses. In [5], an analysis was sketched of how to obtain the value one for the Witten index for this system. Combined with vanishing Theorems, [11], such a result implies that the model has a unique ground state. Independently of the work in [5], it was argued in [1] how a deformation method may be used to establish existence of a ground state. In this paper we use this deformation method and follow the main ideas of [1] to present a rigorous proof of the existence of a ground state. In particular, we make the argument in [1] mathematically precise in two important aspects. First we prove the existence of a ground state for the deformed problem: we have to do semiclassical analysis on the space of gauge invariant functions and we have to deal with the fact that HF is unbounded. In a second part we prove a decay estimate for the ground state of the deformed problem. In particular, we show that it decays sufficiently fast implying that the original problem also has a ground state. To obtain this decay property, we use an Agmon [12] estimate and combine it with a symmetry argument. We think this is a clear and direct way to obtain the necessary decay. Alternatively one could also determine the asymptotic form of the ground state by analyzing the effective dynamics along a potential valley. Such an analysis was indicated in [1]. Similarly one could use a supercharge analysis related to the one in [13] (which was used to determine the asymptotic form of the bound state of two D0-branes). Similar considerations have to be taken into account when using the deformation method to study the number of zero energy states for other supersymmetric models of the same type. Moreover, there are results about the structure of the D0-D4 bound state [14].

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Existence of the D0–D4 Bound State: a Detailed Proof

249

The paper is organized as follows. In Section 2, we describe the model. In Section 3, we introduce the deformation method and give an outline of the proof, which is then presented in Section 4.

2 The model The model is obtained by dimensionally reducing N = 1, U(1) supersymmetric gauge theory with a single charged hypermultiplet, from 5 + 1 dimensions to 0 + 1 dimension [8, 14]. The bosonic degrees of freedom are given by q = (qj )j=1,...,4 ∈ R4 ,

and x = (xµ )µ=1,...,5 ∈ R5 ,

and their configuration space is X = R4 × R5 . Let pj , j = 1, . . . , 4, and pµ , µ = 1, . . . , 5, be the associated canonical momenta obeying, [qj , pk ] = iδjk ,

[xµ , pν ] = iδ µν .

The fermionic degrees of freedom are described by the real Clifford generators λa , a = 1, . . . , 8

and ψa , a = 1, . . . , 8 ,

i.e., λ†a = λa , ψa† = ψa , and {λa , λb } = δab ,

{ψa , ψb } = δab ,

{λa , ψb } = 0 .

(Here and below { · , · } stands for the anticommutator.) By F we denote the irreducible representation space of this Clifford algebra. The dimension of F is 28 . We introduce as a preliminary Hilbert space H0 = L2 (X; F ) = L2 (X) ⊗ F . As given in Appendix A, we choose an explicit real irreducible representation µ )a,b=1,...,8 , γ µ = (γab

µ = 1, . . . , 5 ,

of the gamma matrices in 5 dimensions, i.e., {γ µ , γ ν } = 2δ µν . Furthermore we consider the real 8 × 8 matrices si = (siab )a,b=1,...,8 ,

i = 1, . . . , 4 ,

as they are defined in Appendix A. We note that s1 = 1I8×8 and (sl )T = −sl for l = 2, 3, 4 and that each si commutes with the γ-matrices. We define Dab =

1 R 2 R (q s q )ab , 2

250



with qR

=

s1 q1 + s2 q2 + s3 q3 + s4 q4 ,

qR

=

s1 q1 − s2 q2 − s3 q3 − s4 q4 .

We will use the convention of summing over repeated indices. The supercharges are given by Qa = (sj ψ)a pj + (γ µ λ)a pµ + Dab λb + (γ µ sj s2 ψ)a xµ qj ,

a = 1, . . . , 8 .

Note, for any 8 × 8 matrix A we set (Aψ)a = Aab ψb , ψAψ = ψa Aab ψb , and likewise for expressions containing λa . The Hilbert space H0 carries a unitary representation of U(1), called the gauge transformation, defined by the generator i J = W12 + W34 − ψs2 ψ , 2 where Wij = qi pj − qj pi . We set |x| := (xµ xµ )1/2 and |q| := (qi qi )1/2 . The full model satisfies µ µ x J, {Qa , Qb } = δab H + 2γab

(1)

with H

1 = pµ pµ + pi pi + |x|2 |q|2 + |q|4 − ixµ ψγ µ s2 ψ + i2qj λsj s2 ψ 4 = −∆ + V + HF ,

where we have defined 1 V = |x|2 |q|2 + |q|4 , and HF = −ixµ ψγ µ s2 ψ + i2qj λsj s2 ψ . 4 The Hilbert space of the model H is the U(1)-invariant subspace of H0 , i.e., H = {Ψ ∈ H0 | JΨ = 0 } . Note that the supercharges Qa are U(1) invariant and that on H the superalgebra (1) closes, i.e., {Qa , Qb }|H = δab H|H . The Hilbert space H0 carries a natural representation of Spin(5) defined by the infinitesimal generators i µν (λa λb + ψa ψb ) , µ, ν = 1, . . . , 5 , T µν = xµ pν − xν pµ − γab 4 with γ µν = 12 [γ µ , γ ν ]. Under this representation the supercharges Qa transform as spinors and the Hamiltonian H is invariant. The action of Spin(5) commutes with the gauge transformation, and thus leaves the Hilbert space H invariant.

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We introduce the fermionic number operator (−1)F := 28 λ1 λ2 . . . λ8 ψ1 ψ2 . . . ψ8 , which anti-commutes with Qa and commutes with H, and decompose the Hilbert space by means of (−1)F as H± := {Ψ ∈ H | (−1)F Ψ = ±Ψ } , i.e., into bosonic (+) and fermionic (–) sectors. We note that the operators Qa and H are essentially selfadjoint on C0∞ (X; F ). Furthermore their restriction to H is essentially selfadjoint on the space of U(1)-invariant functions in C0∞ (X; F ).

3 Result and outline of the proof The main Theorem is the following: Theorem 1. There exists a state Ψ ∈ H with HΨ = 0. To prove this theorem, we use the deformation method introduced in [1]. We consider the “complex” supercharges D and D† , 1 D = √ (Q1 + iQ2 ) , 2

1 D† = √ (Q1 − iQ2 ) . 2

2

On H, D2 = D† = 0 and

H = {D, D† } .

(2)

We define the U(1)-invariant function wk on X, by wk = k · x1 ,

for k ≥ 0 .

We introduce the deformed supercharges Dk = e−wk Dewk ,

Dk† = ewk D† e−wk .

We have i Dk = D − k √ ((γ 1 λ)1 + i(γ 1 λ)2 ) , 2

i Dk† = D† + k √ ((γ 1 λ)1 − i(γ 1 λ)2 ) . 2

As a little calculation shows, we have on H Hk = {Dk , Dk† } ,

with

Hk := H + k 2 + k(q32 + q42 ) − k(q12 + q22 ) .

We point out that the deformed Hamiltonian is Spin(5) invariant, despite that the function wk = k · x1 is not. The claim of Theorem 1 is an immediate consequence of the following three propositions.

252



Proposition 2. If for some k there exists a state Ψk ∈ H with Hk Ψk = 0 such that e±wk Ψk ∈ H, then HΨ = 0 for some state Ψ ∈ H. Remark. Proposition 2 holds for more general supersymmetric quantum mechanical systems and deformations, c.p. [1]. The proof of Proposition 2, which is presented in Subsection 4.1, makes use of the Hodge decomposition and a cohomology argument. Proposition 3. For k large enough, there exists a unique state Ψ ∈ H with Hk Ψ = 0. Remark. Proposition 3 implies that Hk admits a zero energy ground state for all k > 0. This follows from the stability of the Fredholm index of the continuous family of Fredholm operators, (0, ∞) k → Ak := 2−1/2 (Dk + Dk† )|H− : H− → H+ , where the topology is given by the graph norm with respect to A0 , see for example [15]. However, we will not use this fact to prove Theorem 1. To prove Proposition 3, which is done in Subsection 4.2, we first observe that the set of points, in which the scalar potential of the deformed Hamiltonian, i.e., Vk = V + k 2 − k(q12 + q22 ) + k(q32 + q42 ) , vanishes, is a circle in configuration space X (see, e.g., (5)). Its radius is proportional to k 1/2 . The circle is an orbit of the U (1) action on X. In the direction orthogonal to the circle the Hessian of Vk is non degenerate. Note that up to gauge transformations the scalar potential vanishes exactly in one point. Moreover, at infinity the potential Vk is bounded below by k 2 . Using semiclassical analysis of eigenvalues, as given for example in [16], together with a gauge fixing procedure, we show that there exists only one low lying eigenvalue for k → ∞. In particular, we have to consider the fact that HF is unbounded from below. By supersymmetry this low lying eigenvalue must equal zero for large k. Proposition 4. For k > 0, a state Ψ ∈ H with Hk Ψ = 0 satisfies e±wk Ψ ∈ H. For the proof of Proposition 4, which is given in Subsection 4.3, we need to show that Ψ decays sufficiently fast as |x| → ∞. We write the Hamiltonian as a sum of a free Laplacian in the x-variables and an x-dependent operator, which describes the dynamics in the transverse direction. We show that the latter is bounded below by k 2 − c|x|−2 for some constant c and |x| large. Using an Agmon estimate we then conclude that |x|−1 ek|x| Ψ is square integrable at infinity. As will be shown, this together with the fact that Ψ is invariant under Spin(5) yields e±wk Ψ ∈ H.

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Remark. To be precise, the operators D, D† , Dk , Dk† and Hk are defined in H0 and H as the closure on C0∞ (X; F ) and C0∞ (X; F )∩H, respectively. The domain of D is the set of all Ψ in H0 and H such that DΨ (defined in the sense of distributions) is again in H0 and H, respectively (and analogous for the domains of D† , Dk , Dk† , H, and Hk ). Indeed, D† (resp. Dk† ) is the adjoint of D (resp. Dk ).

4 Proofs 4.1

Proof of Proposition 2

We shall first show the Hodge decomposition H = ker H ⊕ RanD ⊕ RanD† .

(3)

To show the orthogonality, we note that (DΨ, D† Φ) = (D2 Ψ, Φ) = 0 , with Ψ ∈ D(D) and Φ ∈ D(D† ), and Ψ ∈ ker H iff DΨ = 0 and D† Ψ = 0, by (2). To show the completeness, we note that for each Ψ ∈ (ker H)⊥ , Ψ

= = = =

lim P(a,∞) (H)Ψ a↓0

1 1 lim (DD† + D† D) P(a,∞) (H)Ψ a↓0 2 H 1 1 † 1 † 1 D(D P(a,∞) (H)Ψ) + D (D P(a,∞) (H)Ψ) lim a↓0 2 H 2 H 1 1 1 1 lim D(D† P(a,∞) (H)Ψ) + lim D† (D P(a,∞) (H)Ψ) . a↓0 2 a↓0 2 H H

By PΩ (H) we denoted the projection valued measure of H, and the last equality follows since the two terms belong to different orthogonal subspaces. Hence we have shown (3). The equation Hk Ψk = 0 implies Dk Ψk = 0 and Dk† Ψk = 0, and further wk De Ψk = 0 and D† e−wk Ψk = 0. Assume ker H = {0}. Then ewk Ψk ∈ ker D = RanD†

⊥

= RanD

by the Hodge decomposition. It follows that ewk Ψk = limn→∞ DΦn for some Φn , but then (Ψk , Ψk ) = (ewk Ψk , e−wk Ψk ) = =

lim (DΦn , e−wk Ψk )

n→∞

lim (Φn , D† e−wk Ψk ) = 0.

n→∞

This is a contradiction, and hence ker H = {0}.

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4.2



Proof of Proposition 3

We shall first rescale the operators Hk , Dk and Dk† . For Ψ ∈ H and t > 0, we define the unitary operator (U (t)Ψ)(ξ) = t9/2 Ψ(t ξ) , where ξ = (q, x). Furthermore, we define Kt

:=

Ft

:=

Ft†

:=

t2/3 U (t1/3 )Ht2/3 U ∗ (t1/3 ) t1/3 U (t1/3 )Dt2/3 U ∗ (t1/3 )

t1/3 U (t1/3 )Dt†2/3 U ∗ (t1/3 ) .

It follows that on H {Ft , Ft† } = Kt and

,

Ft2 = 0 ,

2

Ft† = 0 ,

Kt = −∆ + t2 V1 + tHF ,

where

1 V1 = |x|2 |q|2 + |q|4 + 1 + (q32 + q42 ) − (q12 + q22 ) . 4 Proposition 3 follows from Lemma 5. Let En (t) denote the nth eigenvalue of Kt counting multiplicity. Then lim E1 (t)/t = 0 and lim inf E2 (t)/t ≥ r > 0 .

t→∞

(4)

t→∞

By supersymmetry, each non-zero eigenvalue of Kt must be two fold degenerate, i.e., occur as the eigenvalue of a pair consisting of a bosonic and a fermionic eigenvector (see Theorem 6.3., [16]). In view of (4), for large t, two fold degeneracy of E1 (t) is not possible. Hence E1 (t) = 0. Moreover, this eigenvalue is nondegenerate. Proof of Lemma 5. Writing the deformed potential V1 as 2 1 1 1 V1 = |x|2 |q|2 + (q12 + q22 ) − 1 + (q32 + q42 ) 1 + (q32 + q42 ) + (q12 + q22 ) (5) 2

4

2

we see that the set of points Γ in which the potential V1 vanishes is given by Γ

:= {(q, x) ∈ X |V1 (q, x) = 0 } = {(q, x) ∈ X | q12 + q22 = 2, q3 = 0, q4 = 0, x = 0 } .

√ The set Γ is a circle in the (q1 , q2 )-plane about the origin with radius 2. The Hessian of V1 at points lying in Γ is   2 0 0 (2qr qs ) ∂ V1 =  , α, β = 1, . . . , 9, (HessV1 )αβ |Γ = 0 41I2×2 0 ∂ξ α ∂ξ β Γ 0 0 41I5×5

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with (ξ 1 , . . . , ξ 9 ) := (q1 , . . . , q4 , x1 , . . . , x5 ) and r, s = 1, 2. At a point p ∈ Γ, the tangent to Γ is the only degenerate direction of the Hessian. To show that there exists only one low lying eigenvalue, we will fix the U(1) gauge. For ω ∈ L1 (X) with (W12 +W34 )ω = 0, we may integrate out the coordinate q1 as follows. We introduce the coordinates Φ : [0, 2π] × [0, ∞) × R2 −→ R4       α q1 cos α − sin α 0  ρ   q2    sin α cos α        v3  −→  q3  =   cos α − sin α 0 v4 q4 sin α cos α

 0 ρ   v3  v4

with α = arctan(q2 /q1 ) and ρ = (q12 + q22 )1/2 . The metric determinant is √ det DΦT DΦ = | det DΦ| = ρ, and R4 ×R5

dq1 . . . dq4 d5 xω(q, x) = 2π

(0,∞)×R2 ×R5

dρdv3 dv4 d5 xρω((0, ρ, v3 , v4 ), x) .

The integration on the right-hand side is reduced to the gauge fixed configuration

:= {0} × (0, ∞) × R2 × R5 ⊂ X. We introduce the Hilbert space space X

:= L2 (X;

F) H

and we denote its canonical scalar product by w.r.t. the Lebesgue measure of X, · , · GF . We define the isometry H Ψ

−→ H

:= 2πρ Ψ| . −→ Ψ X

(6)

we may recover Ψ through By M = − 2i ψs2 ψ we denote the spin part of J. From Ψ Ψ(q, x) = √

1

ρ, q3 cos α − q4 sin α, q4 cos α + q3 sin α, x) . e−iαM Ψ(0, 2πρ

(7)

Under the isometry (6), the corresponding transformation for the operators A ∈

∈ L(H),

is characterized by L(H), i.e., A → A

Ψ

= AΨ . A For f ∈ C0∞ (X), one has

∂ i f = − W12 f , ∂q1 q2

X X

(8)

256



only after the derivatives are performed. where the function f is restricted to X Applying this result to the function ∂f /∂q1 , using the commutation relation [W12 , ∂/∂q1 ] = i∂/∂q2 and again (8), one finds 1 ∂ ∂ ∂ 1 2 f = − 2 W12 f . ∂q1 ∂q1 q2 ∂q2 q2

X X We set L := J − W12 . Then for Ψ ∈ H, W12 Ψ = −LΨ .

= v3 (−i∂/∂v4) − v4 (−i∂/∂v3 ) − i ψs2 ψ. For Ψ ∈ H ∩ C ∞ (X; F ), a Note that L 0 2 straightforward calculation yields 1 2 1 ∂ ∂ 1 ∂ + 2L Ψ Ψ + 2Ψ − Ψ =− ∂q1 ∂q1 ρ ∂ρ 2ρ ρ and

As a result

∂ ∂ ∂ ∂ 1 ∂ 3 1 Ψ. − Ψ+ Ψ− Ψ =− ∂q2 ∂q2 ∂ρ ∂ρ ρ ∂ρ 4 ρ2

2 − 1 Ψ

, (−∆Ψ) = −∆X + ρ−2 L 4

(9)

We will use eq. (9) only for functions in where ∆X is the formal Laplacian on X. ∞ C0 (X; F ). We use the following partition of unity. We define √ j1,t (ξ) = χr (t2/5 ((q12 + q22 )1/2 − 2)) · χa (t2/5 (q3 , q4 , x)) , ξ = (q, x) , where for α = r, a, we have chosen rotation invariant functions χα ∈ C0∞ (Rnα ) with nr = 1, na = 7, 0 ≤ χα ≤ 1, χα (x) = 1 if |x| ≤ 1 and 0 if |x| ≥ 2. Let R ≥ 1 be fixed as t → ∞. We choose j2 ∈ C ∞ (X) with j2 (ξ) = j2 (|ξ|), 0 ≤ j2 ≤ 1, j2 (ξ) = 1 for |ξ| ≥ 2R and j2 (ξ) = 0 for |ξ| < R. Furthermore we set 2 j0,t := (1 − j1,t − j22 )1/2 .

→ X := {0} × R8 and the For technical matters we consider the embedding X 2 9

coordinates √ (0, η , . . . , η ) ∈ X. By η0 we denote the intersection of X with Γ, i.e., η0 = (0, 2, 0, . . . , 0). We define 9 1 V10 (η) = (HessV1 )αβ (η0 )(η α − η0α )(η β − η0β ) . 2 α,β=2

F) and introduce the following operator on L2 (X; Gt = −∆X + t2 V10 + tHF (η0 ) ,

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√ where HF (η0 ) = −i2 2(λ1 ψ1 + ...λ8 ψ8 ) : F → F denotes the evaluation of HF at For χ ∈ L2 (X; F ), we define η0 and −∆X the eight-dimensional Laplacian on X. the unitary transformation (T (t)χ)(η) = t2 χ(t1/2 (η − η0 )) . Then 1 1 T (t)∗ Gt T (t) = −∆X + (HessV1 )αβ (η0 )η α η β + HF (η0 ) . t 2 The eigenvalue problem for this operator can be solved easily. It has purely discrete spectrum and its ground state Φ0 has zero energy and is non degenerate: the sum and has ground of the first two√terms is a harmonic oscillator, which acts on L2 (X) state √ energy 8 2, and HF (η0 ) acts on F and has a unique ground state with energy −8 2, see Appendix B (i). Define

F ) → L2 (X; F) .

t := j1,t T (t)Φ0 ∈ C0∞ (X; Ψ We recall that the corresponding U (1)-invariant wave function in Ψt ∈ H is obtained using (7). Calculating the energy of this state, we find

t

t, K

tΨ Ψt , Kt Ψt = Ψ GF

2 − 1 ) + t2 V 1 + tH

F )Ψ

t = Ψt , (−∆X + ρ−2 (L GF 4 1 −2 2

= Ψt , Gt Ψt GF + Ψt , ρ (L − )Ψt GF (10) 4

t , t(H

t , t2 (V 1 − V10 )Ψ

t

F − H

F (η 0 ))Ψ

t + Ψ . + Ψ GF GF For the first term in (10), we find for t → ∞,

t

t , Gt Ψ Ψ = T (t)Φ0 , j1,t Gt j1,t T (t)Φ0 GF GF 1 2 1 2 = T (t)Φ0 , ( + |∇X j1,t Gt + Gt j1,t j1,t |2 )T (t)Φ0 GF 2

= O(t

4/5

2

),

by ∇ , and we used that Gt T (t)Φ0 = 0 where we denoted the gradient on X X 2 4/5 and ∇X j1,t ∞ = O(t ). By rotation invariance of Φ0 in the v3 , v4 variables, the second term in (10) is an order one term. The estimate 2 2 (V1 − V10 )| ≤ const · t2 |η − η0 |3 ≤ const · t2 · t−6/5 |t2 j1,t j1,t

(11)

t , t2 (V 1 − V 0 )Ψ

t GF = O(t4/5 ). And a similar estimate, yields Ψ 1 2 2

F )| ≤ const · t |t j1,t (HF (η0 ) − H j1,t |η − η0 | ≤ const · t · t−2/5 ,

gives

t , t(H

F − H

F (η0 ))Ψ

t Ψ = O(t3/5 ) , GF

(12)

258



as t → ∞. Collecting terms, we find Ψt , Kt Ψt = O(t4/5 ) , which implies that limt→∞ E1 (t)/t = 0. This shows the first part of (4). To prove the second part, i.e., lim inf E2 (t)/t ≥ r > 0 ,

(13)

t→∞

it suffices to show that there exists an r > 0 such that Kt ≥ (t · r + o(t))1I + Rt ,

(14)

where Rt is a symmetric, rank one operator. To see this, suppose (14) holds. Let ω1,t and ω2,t be the eigenvectors to the eigenvalues E1 (t) and E2 (t) of Kt , respectively. There exists a ωt ∈ Span{ω1,t , ω2,t } in the kernel of Rt . Hence E2 (t)ωt 2 ≥ ωt , Kt ωt ≥ (t · r + o(t))ωt 2 which implies (13). To show (14), we use the IMS localization formula 1

=

Kt

ja,t Kt ja,t + j2 Kt j2 −

a=0

1

|∇ja,t |2 − |∇j2 |2 .

(15)

a=0

Now, supp(j0,t ) ⊂ {ξ ∈ X|dist (ξ, Γ) ≥ t−2/5 }. We have ∇ja,t 2∞ = O(t4/5 ) for a = 0, 1, and ∇j2 2∞ = O(1). We estimate j0,t Kt j0,t

≥

t2 j0,t V1 j0,t + tj0,t HF j0,t

≥

2 (t2 t−4/5 cV − tcF )j0,t ,

≥

t·

2 rj0,t

,

for some cV > 0, cF > 0

for some r > 0 ,

(16)

F) and for t large. By fixing the gauge, we have on L2 (X;

t

j1,t K j1,t

0

1 − V 1 ) = j1,t Gt j1,t t2 (V j1,t j1,t + 1

F − H

F (η0 ))

2 ) + j1,t t(H j1,t + j1,t ρ−2 (− + L j1,t 4

≥ j1,t Gt j1,t + O(t4/5 ) ≥ j1,t t r 1 − |T (t)Φ0 GF · GF T (t)Φ0 | j1,t + O(t4/5 ) ,

2 , the estimates (11, 12), for some r > 0, where we have used the positivity of L and the gap in the spectrum of Gt . On H, this yields 2 − t · r|Ψt · Ψt | + O(t4/5 ) . j1,t Kt j1,t ≥ t · rj1,t

(17)

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To estimate the term j2 Kt j2 , we recall the explicit form of Kt : 1 Kt = pi pi + pµ pµ + t2 |x|2 |q|2 + |q|4 + 1 − (q12 + q22 ) + (q32 + q42 ) 4

+t(−ixµ ψγ µ s2 ψ + 2iqj λsj s2 ψ) . We recall the notation ξ = (q, x). Define a function θ ∈ C ∞ (X) with√θ(ξ) = θ(|q|), 0 ≤ θ ≤ 1, θ(|q|) = 1 if |q| > 4 and θ(|q|) = 0 if |q| < 3. Define θ¯ := 1 − θ2 . Then ¯ 2 )j2 . ¯ t θj ¯ 2 − j2 (|∇θ|2 + |∇θ| j2 Kt j2 = j2 θKt θj2 + j2 θK ¯ 2 = The localization error gives order 1 contributions, i.e., ∇θ2∞ = O(1), ∇θ ∞ O(1). First we consider the case where |q| is large and estimate (see Appendix B (i) for the terms containing fermions) pµ pµ ≥ 0 ,

−ixµ ψγ µ s2 ψ ≥ −4|x| ,

2iqj λsj s2 ψ ≥ −8|q| ,

pi pi + t2 |x|2 |q|2 + t(−ixµ ψγ µ s2 ψ) ≥ pi pi + t2 |x|2 |q|2 − 4t|x| ≥ 0 , where the last inequality follows from the ground state energy of the harmonic oscillator. This yields j2 θKt θj2

1 4

≥

j2 θ(t2 ( |q|4 − |q|2 + 1) − 8t|q|)θj2

≥

t2 · cj22 θ2 ,

for some c > 0 and t sufficiently large. For points ξ = (q, x) ∈ supp j2 , if |q| < 4, then |x| is large for sufficiently large R. We have ¯ t θj ¯2 j2 θK

≥ ≥ ≥

¯ i pi + t2 |x|2 (1 − |x|−2 )|q|2 − 4t|x| + t2 − 8t|q|)j2 θ¯ j2 θ(p ¯ j2 θ(t(4|x|(1 − |x|−2 )1/2 − 4|x|) + t2 − 32t)j2 θ¯ t2 · cj 2 θ¯2 , 2

for some c > 0 and t sufficiently large. Hence there exists an r > 0 such that for large t, j2 Kt j2 ≥ t · rj22 . (18) Now, inserting eqns. (16–18) into (15) yields (14) and therefore (13).

4.3

Proof of Proposition 4.

We decompose the Hilbert space H0 as a constant fiber direct integral [17], with fiber F := L2 (R4 ; F ), H0 =

⊕

R5

F dx ,

260



the isomorphism being Ψ → (x → Ψx := Ψ(·, x)). The Hamiltonian has a direct integral decomposition, Hk = p µ p µ +

⊕

R5

Hk,x dx ,

where the fibers Hk,x , acting on F , are given by 1 Hk,x = Hx0 + |q|4 + 2iqj λsj s2 ψ + k 2 − k(q12 + q22 ) + k(q32 + q42 ) , 4 with

Hx0 := pi pi + |x|2 |q|2 − ixµ ψγ µ s2 ψ .

The scalar product, the norm and operator norm in F will be denoted by (·, ·)F and · F , respectively. Let Px be the projection onto the eigenspace of Hx0 corresponding to its lowest eigenvalue, which is, in fact, zero. We set Px⊥ := 1 − Px , and we define the projection P =

⊕

R5

Px dx ,

(19)

and its complement P ⊥ = 1 − P . As is shown in Appendix B (ii), for x = 0, RanPx = { Ξx · ξ | ξ ∈ F with (uψ)ξ = 0, ∀ u : −iγ µ xµ s2 u = |x|u} , where

1 Ξx (q) := (|x|π)−1 exp(− |x||q|2 ) . 2

Lemma 6. There exists an R > 0 and a constant c > 0 depending on k, such that for |x| > R Hk,x ≥ k 2 − c|x|−2 . Proof. Since all elements in RanPx are spherically symmetric in q it immediately follows that (20) Px Hk,x Px ≥ k 2 Px . We estimate, c.p. Appendix B (i), 2iqj λsj s2 ψ ≥ −8|q| ,

and − k(q12 + q22 ) ≥ −|x|−1 k(|x|−1 pi pi + |x||q|2 ) .

Hence Hk,x ≥ |x|(1 − |x|−1 )(|x|−1 pi pi + |x||q|2 ) + |x|−1 pi pi + |x||q|2 −ixµ ψγ µ s2 ψ − 8|q| − |x|−1 k(|x|−1 pi pi + |x||q|2 ) + k 2 ≥ |x|(1 − |x|−1 − k|x|−2 )(|x|−1 pi pi + |x||q|2 ) − ixµ ψγ µ s2 ψ − 16|x|−1 + k 2 ,

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where we used |x||q|2 − 8|q| ≥ −16|x|−1 in the last inequality. The range of Px⊥ is given by the closure of the set of linear combinations of states which are a product of an eigenstate of pi pi + |x|2 |q|2 and an eigenstate of −ixµ ψγ µ s2 ψ, excluding states which are a product of two ground states. Thus Px⊥ Hk,x Px⊥ ≥ (c0 |x| + k 2 )Px⊥ ,

(21)

for some c0 > 0 and large |x|. Using that |q|α Px F ≤ cα |x|−α/2 from the Gaussian decay of states in RanPx , ⊥ P Hk,x Px = Px⊥ |q|4 /4 − k(q12 + q22 ) + k(q32 + q42 ) + 2iqj λsj s2 ψ Px F x F ≤

c|x|−1/2

for some c > 0 and large |x|. By the selfadjointness of Hk,x also Px Hk,x P ⊥ ≤ c|x|−1/2 . x F

(22)

Let ux ∈ D(Hk,x ) ⊂ F . Then from (20), (21) and (22) (ux , Hk,x ux )F

2

ux 2F

Px ux F Px⊥ ux F

≥ k + Ak,x 2 ≥ k + inf ξ=1 (ξ, Ak,x ξ) ux 2F ,

with

Ak,x :=

0 −c|x|−1/2

−c|x|−1/2 c0 |x|

Px ux F Px⊥ ux F

.

We have inf ξ=1 (ξ, Ak,x ξ) ≥ −c|x|−2 , for some c > 0 and large |x|. Hence the Lemma follows.

Let R ≥ 1 be as in Lemma 6, and let η : R5 → R be a smooth function with η(x) = η(|x|), 0 ≤ η ≤ 1, ∇η∞ ≤ 1, η(x) = 0 for |x| ≤ R and η(x) = 1 for |x| ≥ 3R. The deformed supercharge 1 Q1,k = Q1 + k(γ 1 λ)2 = √ (Dk + Dk† ) 2

(23)

satisfies on H, 2(Q1,k )2 = {Dk , Dk† } = Hk . Hence for Ψ ∈ H, Hk Ψ = 0 iff Q1,k Ψ = 0. Lemma 7. Let Ψ ∈ H with Hk Ψ = 0. Then for any > 0, |x|−1/2− ek|x| ηΨ ∈ H.

262



Proof. It is sufficient to show the claim for arbitrarily small . To prove the lemma, we use an Agmon estimate [12]. Let h : R5 → [0, ∞) be a smooth function such that the set K = {x ∈ R5 | k 2 − c|x|−2 − |∇h(x)|2 < 0 } is compact. Then, as we will show, η 2 Ψx 2F (k 2 − c|x|−2 − |∇h(x)|2 )e2h dx ≤ M0 Ψ2 .

(24)

R5

where M0 :=

sup R≤|x|≤3R

(1 + 2|∇h(x)|)e2h(x) < ∞ .

Define hα := h(1 + αh)−1 . Then, by Lemma 6, (ηehα Ψx , Hk,x ηehα Ψx )F dx (ηehα Ψ, Hk ηehα Ψ) ≥ 5 R ≥ η 2 e2hα (k 2 − c|x|−2 )Ψx 2F dx .

(25)

R5

We estimate hα ηe Ψ, Hk ηehα Ψ = 2 Q1,k ηehα Ψ, Q1,k ηehα Ψ = 2 [Q1,k , ηehα ]Ψ, [Q1,k , ηehα ]Ψ ≤ |∇(ηehα )|2 Ψ, Ψ ≤ (|∇η|2 + 2(∇η)(∇hα )η + |∇hα |2 η 2 )e2hα Ψ, Ψ . Inserting this into inequality (25), we obtain Iα := η 2 e2hα (k 2 − c|x|−2 − |∇hα |2 )Ψx 2F dx R5 ≤ (|∇η|2 + 2|∇η||∇hα |η)e2hα Ψ, Ψ ≤

M0 Ψ2 .

Using Fatou’s Lemma on the set K c and dominated convergence on K yields + η 2 Ψx 2F (k 2 − c|x|−2 − |∇h|2 )e2h dx ≤ lim inf Iα ≤ M0 Ψ2 K

α

Kc

and hence (24). We choose h such that on the support of η, h(x) = k|x| − log |x|. Then k 2 − c|x|−2 − |∇h(x)|2 = 2k|x|−1 − (c + 2 )|x|−2 ≥ k|x|−1 , for large |x|. Hence, by (24) dxη 2 Ψx 2F e2k|x| k|x|−2−1 < ∞ , R5

which proves the Lemma.

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Proof of Proposition 4. We recall that the deformed Hamiltonian commutes with the action of Spin(5). Let k be sufficiently large such that Ψ is the unique zero energy state of Hk . Thus Ψ belongs to a one-dimensional representation of Spin(5), and therefore it is Spin(5) invariant. Let R(S) denote the image of S under the canonical projection Spin(5) → SO(5). By R(S) we denote the spin part of the µν (λa λb + ψa ψb ). Then Spin(5) action, i.e., the representation generated by − 4i γab Ψ(q, R(S)x) = R(S)Ψ(q, x) ,

∀ S ∈ Spin(5) .

This implies that for x, x ∈ R5 with |x| = |x |, |Ψx |F = |Ψx |F . We set ω = x/|x|. Let dΩ denote the surface measure of the unit sphere. Then π 1 e−2k|x|±2kx dΩ(ω) = vol(S 3 ) e−2k|x|(1∓cos θ) sin3 θdθ S4

≤ vol(S 3 )

0

1

−1 −2

≤ const |x|

e−2k|x|(1−cos θ) 2(1 − cos θ) d cos θ .

For the ground state Ψ ∈ H of Hk , we have 1 1 e±2kx |Ψ(q, x)|2 dqdx = e±2kx |Ψx |2F dx 1 1 = (1 − η)e±2kx |Ψx |2F dx + ηe±2kx |Ψx |2F dx 1 ≤ const + ηe±2kx e−2k|x| e2k|x| |Ψx |2F dx ≤ const + const η|x|−2 e2k|x| |Ψx |2F dx 0 for all x ∈ R and all t > 0 no matter whether the nontrivial initial condition u(x, 0) vanishes in some region. Another classical equation that has been used to model diffusion is the wellknown porous medium equation, ut = ∆um with m > 1. This equation also shares several properties with the heat equation but there is a fundamental difference, in this case if the initial data u(·, 0) is compactly supported, then u(·, t) has compact support for all t > 0. In such a case, if the support of the initial condition is a finite interval, one can define the right and left free boundaries of the solution by s+ (t) = sup{x / u(x, t) > 0} and s− (t) = inf{x / u(x, t) > 0} respectively. Properties and the behavior of the free boundary for the porous medium equation have been largely studied over the past years. See for example [1], [7] and the corresponding bibliography. It is worth mentioning that this phenomena also arises in the context of the Stefan problem, see [3] and the references therein. The purpose of this note is to present a simple nonlocal model for diffusion whose solutions, with compactly supported bounded initial data, develop a free boundary. To do this we propose a model where the diffusion at a point depends on the density. The simplest situation we can think of is when the probability distribution of jumping from location y to location x is given by x−y 1 J u(y, t) u(y, t) when u(y, t) > 0 and 0 otherwise. In this case the rate at which individuals are arriving to position x from all other places is x−y J dy u(y, t) R and the rate at which they are leaving location x to travel to all other sites is y−x −u(x, t) = − J dy. u(x, t) R

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271

As before this consideration, in the absence of external sources, leads immediately to the fact that the density u has to satisfy x−y ut (x, t) = J dy − u(x, t). u(y, t) R As for the initial data, although we are mostly interested in functions u(·, 0) ∈ L1 (R) ∩ L∞ (R) it is more convenient, for technical reasons that will become clear later, to consider a slightly more general set of initial conditions. So in this paper we will deal with the problem x−y ut (x, t) = J dy − u(x, t) in R × [0, ∞). u(y, t) R (1.2) u(x, 0) = c + w0 (x) on R, where c ≥ 0, w0 ∈ L1 (R) and w0 ≥ 0. Most of the results contained in this note can be obtained in several dimensions without many changes in the elementary arguments but, we have chosen to treat the one-dimensional case for the sake of simplicity of the exposition. We will address in this paper the questions of existence, uniqueness, comparison principles and some basic facts about the free boundary for solutions of problem (1.2). Several further questions, such as the decay rate of solutions, the speed at which the free boundary moves, the existence of the so-called waiting times for the free boundary and many others, are left open. Also one can consider equations involving a source term and to study, for example, the blow-up phenomena. We hope such questions can be answered by us or by someone else in the near future.

2 Existence and uniqueness The existence and uniqueness result will be a consequence of Banach’s fixed point theorem and it is convenient to give some preliminaries before giving its proof. Fix t0 > 0 and consider the Banach space C([0, t0 ]; L1 ) with the norm |w| = max w(·, t)L1 . 0≤t≤t0

Let

Xt0 = w ∈ C([0, t0 ]; L1 ) / w ≥ 0

which is a closed subset of C([0, t0 ]; L1 ). We will obtain the solution in the form u(x, t) = w(x, t) + c where w is a fixed point of the operator Tw0 : Xt0 → Xt0 defined by t x−y e−(t−s) J Tw0 (w)(x, t) = dy ds w(y, s) + c 0 R +e−t w0 (x) − c(1 − e−t ).

272

C. Cortazar, M. Elgueta and J.D. Rossi


The following lemma is the main ingredient of our proof. Lemma 2.1 Let z0 , w0 be nonnegative functions such that w0 , z0 ∈ L1 (R) and w, z ∈ Xt0 , then |||Tw0 (w) − Tz0 (z)||| ≤ (1 − e−t0 )|||w − z||| + ||w0 − z0 ||L1 (R) . Proof. We have |Tw0 (w)(x, t) − Tz0 (z)(x, t)| dx R t x−y x−y e−(t−s) ≤ J −J dy dx ds w(y, s) + c z(y, s) + c 0 R R −t +e |w0 − z0 |(y) dy. R

Now set

A+ (s) = {y / w(y, s) ≥ z(y, s)}

and

A− (s) = {y / w(y, s) < z(y, s)}.

We have now x−y x−y J − J dy dx w(y, s) + c z(y, s) + c R

R

≤

A+ (s)

R

+ A− (s)

R

J J

x−y w(y, s) + c x−y z(y, s) + c

−J

−J

x−y z(y, s) + c

x−y w(y, s) + c

dy dx dy dx.

Since the integrands are nonnegative we can apply Fubini’s theorem to get x−y x−y J −J dy dx w(y, s) + c z(y, s) + c R A+ (s) = A+ (s)

(w(y, s) − z(y, s))dy

and similarly for the integral over A− (s). Therefore we obtain x−y x−y J − J dy dx w(y, s) + c z(y, s) + c R R |w(y, s) − z(y, s)| dy. ≤ R

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Hence we get |Tw0 (w) − Tz0 (z)| ≤ (1 − e−t0 )|w − z| + ||w0 − z0 ||L1 (R)

as desired. We can state now the main result of this section.

Theorem 2.1 For every nonnegative w0 ∈ L1 and every constant c ≥ 0, there exists a unique solution u, such that (u − c) ∈ C([0, ∞); L1 ), of 1.2. Moreover, the solution verifies u(x, t) ≥ c and preserves the total mass above c, that is (u(y, t) − c) dy = w0 (y) dy for all t ≥ 0. (2.1) R

R

Proof. We check first that Tw0 maps Xt0 into Xt0 . Since w ≥ 0 we have x−y x−y J ≥J w(y, s) + c c and hence Tw0 (w)(x, t) ≥

t

0

e−(t−s)

R

J

x−y c

dy ds (2.2)

+e−t w0 (x) − c(1 − e−t ) = e−t w0 (x) ≥ 0. Taking z0 ≡ 0, z ≡ 0 in Lemma 2.1 we get that Tw0 (w) ∈ C([0, t0 ]; L1 ). Now taking z0 ≡ w0 in Lemma 2.1 we get that Tw0 is a strict contraction in Xt0 and the existence and uniqueness part of the theorem follows from Banach’s fixed point theorem. We finally prove that if u = w + c is the solution, then the integral in x of w is preserved. Since t x−y 0= e−(t−s) J dy ds − c(1 − e−t ), c 0 R we can write w(x, t) =

0

t

e

−(t−s)

(J R

x−y w(y, s) + c

−J

x−y ) dy ds + e−t w0 (x). c

The integrand in the above formula is nonnegative so we can integrate in x and apply Fubini’s theorem to obtain t −(t−s) −t w(x, t)dx = e w(y, s) dy ds + e w0 (x)dx (2.3) R

0

R

R

274


from where it follows that

d dt


R

w(x, t)dx = 0

and the theorem is proved.

We will need in what follows the following lemma which is a direct corollary of the proof of Theorem 2.1 and is a first version of the comparison principle of Section 3 below. Lemma 2.2 With the above notation if 0 ≤ w(x, 0) ≤ M for all x ∈ R, then w(x, t) ≤ M for all (x, t) ∈ R × [0, ∞). Proof. Under the given hypotheses one has that if w(x, t) ≤ M , then Tw0 (w)(x, t) =

t

e

0

−(t−s)

≤

J

x−y w(y, s) + c

R t

0

e

−(t−s)

R

J

x−y M +c

dy ds + e−t w0 (x) − c(1 − e−t )

dy ds + e−t M − c(1 − e−t ) = M.

The lemma follows by the uniqueness of the fixed point for Tw0 .

Lemma 2.1, Theorem 2.1, Lemma 2.2 and their proofs have several immediate consequences that we state as a series of remarks for the sake of future references. Remark 2.1 Solutions of 1.2 depend continuously on the initial condition in the following sense. If u and v are solutions of 1.2, then max u(·, t) − v(·, t)L1 (R) ≤ et0 ||u(·, 0) − v(·, 0)||L1 (R)

0≤t≤t0

for all t0 ≥ 0. Remark 2.2 The function u is a solution of 1.2 if and only if t x−y −(t−s) u(x, t) = e J dy ds + e−t u(x, 0). u(y, s) 0 R Remark 2.3 From the previous remark and Lemma 2.2 we get that if c > 0 and u(·, 0) ∈ C k (R) with 0 ≤ k ≤ ∞, then u(·, t) ∈ C k (R) for all t ≥ 0. Moreover if u(·, 0) is a compactly supported C 1 function, then there exists a constant K depending on c, J and w0 such that ∂u |ut (x, t)| , (x, t) ≤ K. ∂x

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Remark 2.4 A consequence of Remark 2.3 and of (2.1) is that if c > 0 and w0 is a compactly supported C 1 function, then lim u(x, t) = c uniformly on compact intervals [0, T ].

|x|→∞

Remark 2.5 It follows from inequality (2.2) that w(x, t) ≥ e−t w(x, 0). In particular, in the case that u(·, 0) ∈ L1 (R), the support of u(·, t) does not shrink as time increases. By this we understand that if u(x0 , t0 ) > 0, then u(x0 , t) > 0 for all t ≥ t0 .

3 Comparison Principle Comparison principles like the one below have proven to be a very useful tool in studying diffusion problems. Theorem 3.1 Let u and v be continuous solutions of 1.2. If u(x, 0) ≤ v(x, 0) for all x ∈ R, then u(x, t) ≤ v(x, t) for all (x, t) ∈ R × [0, ∞).

(3.1)

Proof. We assume first that u(x, 0) = c + w(x, 0)

and

v(x, 0) = d + z(x, 0)

with 0 < c < d and u(x, 0) < v(x, 0). Moreover we assume for a moment that w(x, 0) and z(x, 0) are compactly supported C 1 functions. In this case there exists δ > 0 such that u(x, 0) + δ < v(x, 0). Assume, for a contradiction that the conclusion does not hold. In view of Remark 2.4 we have that there exists a time t0 > 0 and a point x0 ∈ R such that u(x0 , t0 ) = v(x0 , t0 ) and u(x, t) ≤ v(x, t) for all (x, t) ∈ R × [0, t0 ]. Let us consider the set B = {x ∈ R / u(x, t0 ) = v(x, t0 )}. Clearly B is nonempty and closed. Let x1 ∈ B. We have then x1 − y x1 − y 0 ≤ (u − v)t (x1 , t0 ) = J −J dy ≤ 0 u(y, t0 ) v(y, t0 ) R which implies u(y, t0 ) = v(y, t0 ) for all y ∈ (x1 − c, x1 + c).

276



Hence B is open. It follows that B = R which is the desired contradiction since (u(·, t0 ) − c) ∈ L1 (R). We now get rid of the extra hypothesis that w(x, 0) and z(x, 0) are compactly supported C 1 functions. In order to do this let wn (x, 0) and zn (x, 0) be sequences of compactly supported C 1 functions such that wn (x, 0) → w(x, 0) and zn (x, 0) → z(x, 0) in L1 (R) as n → ∞ and, moreover, un (x, 0) = c + wn (x, 0) < vn (x, 0) = d + zn (x, 0). Let un and vn be the solutions with initial data un (x, 0) and vn (x, 0) respectively. By the previous argument one has un ≤ vn an the result follows by letting n → ∞ in view of Remark 2.1. In order to prove the theorem in the general case pick strictly decreasing sequences an and bn such that 0 < an < bn and bn → 0 as n → ∞. Let un and vn be the solutions with initial conditions un (x, 0) = u(x, 0) + an and vn (x, 0) = v(x, 0) + bn respectively. According to the previous argument one has un ≤ vn . Moreover un+1 ≤ un and vn+1 ≤ vn . By Remark 2.2, after an application of the monotone convergence theorem, it follows that un (x, t) → u(x, t) and vn (x, t) → v(x, t) as n → ∞ and the theorem is proved. An immediate consequence of the comparison principle and Remark 2.4 is the following corollary that extends Remark 2.4 to the case c = 0. Corollary 3.1 If c = 0 and w0 is a compactly supported C 1 function, then lim u(x, t) = 0 uniformly on compact intervals [0, T ].

|x|→∞

4 The free boundary In this section we will prove that solutions of (1.2), with compactly supported continuous initial data, do have a free boundary in the sense that s+ (t) = sup{x / u(x, t) > 0} < +∞ and s− (t) = inf{x / u(x, t) > 0} > −∞ for all t ≥ 0. It follows from Remark 2.5 that s+ and s− are nondecreasing and nonincreasing functions respectively. Moreover we will also prove in this section that the supports of u(·, t) eventually fill at least half a ray of the space, in particular either lim s+ (t) = ∞ or lim s− (t) = −∞. In the case that J is even, that is t→∞ t→∞ the case of an isotropic media, the supports eventually cover the whole of R. The following theorem implies the existence of free boundaries. Theorem 4.1 If u(·, 0) is compactly supported and bounded then u(·, t) is also compactly supported for all t ≥ 0.

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Proof. Due to the scaling invariance of the equation, namely if u(x, t) is a solution then for any λ > 0 the function vλ (x, t) = λu( xλ , t) is also a solution, we can restrict ourselves to initial data supported in [−1, 1] and such that sup u(x, 0) ≤ 1. x∈R

We note first that

ut (x, t) ≤

R

J

x−y u(y, t)

dy.

(4.1)

Therefore, since 0 ≤ u ≤ 1, we get by (4.1) that u(x, t) ≤

1 1 for all t ≤ and all x such that |x| ≥ 1. 2 2

Now if |x| ≥ 2 and t ≤ 12 we have that |x − y| ≤ u(y, t) implies that |y| ≥ 1 and hence u(y, t) ≤ 12 . Therefore, again by (4.1), we have u(x, t) ≤

1 1 for all t ≤ and all x such that |x| ≥ 2. 4 2

We look now at the case |x| ≥ 2 + 12 and t ≤ 12 . In this case |x − y| ≤ u(y, t) implies that |y| ≥ 2 and hence u(y, t) ≤ 14 . Again by (4.1), we have 1 1 1 for all t ≤ and all x such that |x| ≥ 2 + . 8 2 2 Repeating this procedure we obtain by induction that for any integer n ≥ 1 one has u(x, t) ≤

u(x, t) ≤

1 2n+2

n

for all t ≤

1 1 and all x such that |x| ≥ 2 + . 2 2k k=1

It follows that the support of u(·, t) is contained in the interval [−3, 3] for all t ≤ 12 as we wanted to prove. In order to prove our next result we need a preliminary lemma. Lemma 4.1 If u(x, 0) is continuous and not constant, then the function M (t) = max u(x, t) x∈R

is strictly decreasing. Proof. It is clear, by comparison with a constant, that M (t) decreases as t increases. Moreover by Remark 2.5 one has M (t) > c for all t ≥ 0. Fix t0 ≥ 0 and let t1 > t0 . Let us consider the set C = {x / u(x, t1 ) = M (t0 )}.

278



The set C is clearly closed. Since u(x, t) ≤ M (t0 ) for all t ≥ t0 we have that at any point x0 ∈ C one must have x0 − y 0 ≤ ut (x0 , t1 ) = J dy − u(x0 , t1 ) ≤ 0. u(y, t1 ) R This implies that u(x, t1 ) = M (t0 ) for all x in a neighborhood of x0 and hence C is open. Consequently either C = R or C is empty. It is clear that C = R, so C = ∅ and the lemma is proved. We are now in a position to prove that at least one of the free boundaries go to infinity. Theorem 4.2 Let u be the solution of problem 1.2 with c = 0 and w0 = 0. Then either or lim s− (t) = −∞ lim s+ (t) = ∞ t→∞

t→∞

and the supports of u(·, t) eventually cover an infinite half-ray of R. If J is an even function the supports eventually cover the whole of R. Proof. By comparison, and the invariance under translations of the equation, it is enough to prove the theorem under the assumptions that w0 ∈ C 1 , its support is the interval [−A, A] and it is symmetric with respect to the origin. We claim first that the support of u(·, t) is not uniformly bounded. Assume for a contradiction that there exists L > 0 such that u(x, t) = 0 for all x such that |x| ≥ L and all t ≥ 0. Since R u(x, t)dx = R u(x, 0)dx > 0 there exists C > 0 such that lim M (t) = C. t→∞

Let v(x, 0) be a smooth function supported in [−L − 1, L + 1] such that 0 ≤ v(x, 0) ≤ C and v(x, 0) ≡ C if x ∈ [−L, L]. Let us denote by v(x, t) the solution of (1.2) with this initial condition. By Lemma 4.1 we have that max v(x, 1) < C. x∈R

Now for any integer n > 0 let vn (x, 0) be a smooth compactly function supported in [−L − 2, L + 2] such that 0 ≤ v(x, 0) ≤ C + n1 and vn (x, 0) ≡ C + n1 if x ∈ [−L, L]. Assume further that vn+1 (x, 0) ≤ vn (x, 0) and denote by vn (x, t) the solution of (1.2) with initial condition vn (x, 0). By comparison it follows that vn+1 (x, t) ≤ vn (x, t).

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Using Remark 2.2 and the monotone convergence theorem one has vn (x, 1) → v(x, 1) in [−L − 2, L + 2] as t → ∞. Moreover, being the limit continuous the convergence is uniform by Dini’s theorem. Consequently there exists n0 such that max vn0 (x, 1) < C. x∈R

On the other hand there exists t0 such that u(x, t0 ) ≤ vn0 (x, 0). This implies, by comparison, that max u(x, t0 + 1) < C x∈R

a contradiction that proves the claim. We are ready now to prove the statement of the theorem. We claim that if there exists x0 ≥ A such that u(x0 , t) = 0 for all t ≥ 0, then u(x, t) = 0 for all (x, t) ∈ [x0 , ∞) × [0, ∞). Indeed, let d > 0 and we will prove that u(x, t) ≤ d for all x ≥ x0 and all t ≥ 0.

(4.2)

Since u(x0 , t) ≡ 0 one has u(x, t) ≤ |x − x0 | for all x ∈ R and all t ≥ 0. Moreover u(x, 0) = 0 for all x ≥ x0 . So if (4.2) does not hold, using Corollary 3.1, there exists a point x1 ∈ R with x1 ≥ x0 + d and a time t1 > 0 such that u(x1 , t1 ) = d and u(x, t) ≤ d for all (x, t) ∈ R × [0, t1 ]. As in the proof of Theorem 3.1 we consider the set B = {x ≥ x0 + d / u(x, t1 ) = d} which is clearly closed. Also at a point x2 ∈ B one has x2 − y x2 − y J 0 ≤ (d − u)t (x2 , t1 ) = −J dy ≤ 0 d u(y, t0 ) R which implies u(y, t0 ) = d for all y ∈ (x2 − d, x2 + d).

280



It follows that B is open and hence B = [x0 , ∞) which is a contradiction that proves (4.2). Since d > 0 was chosen arbitrarily the claim follows. An analog of the above claim holds for points −x1 < −A such that u(−x1 , t) = 0 for all t ≥ 0. Such a points x0 and x1 can not exist simultaneously because this contradicts the fact that the supports of u(·, t) are not uniformly bounded. This, plus the fact that if J and u(·, 0) are even functions then u(·, t) is even for all t ≥ 0, proves the theorem. Finally we give an example of a nonsymmetric function J such that the supports of solutions u(·, t), with compactly supported bounded initial data, do not eventually cover the whole of R. We will show that for a special choice of J the function 0 if x ≤ 0 u(x) = x+ = x if x ≥ 0 satisfies

0= R

J

x−y u(y)

dy − u(x).

(4.3)

It is immediate that if x ≤ 0, then x−y J dy = 0 u(y) R and hence (4.3) is satisfied. As for the case x > 0 we have that

R

J

=

x−y u(y)

∞ x 2

J

=x

|x−y| y+

≤ 1 implies 0 < x ≤ 2y and hence

dy − u(x) x − 1 dy − x y

dr J(r) −1 . (1 + r)2 −1 1

Now we choose J such that, in addition to the hypotheses already made, satisfies 1 dr J(r) =1 (1 + r)2 −1 and (4.3) also holds. The desired example follows now by a comparison argument, like the one of the proof of Theorem 3.1, using the function x+ , or a translation of it, as a barrier.

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References [1] D.G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems, A. Fasano and M. Primicerio eds. Lecture Notes in Math. 1224, Springer Verlag, (1986). [2] P. Bates, P- Fife, X. Ren and X. Wang, Travelling waves in a convolution model for phase transitions. Arch. Rat. Mech. Anal. 138, 105–136 (1997). [3] J.R. Cannon, Mario Primicerio, A Stefan problem involving the appearance of a phase, SIAM J. Math. Anal. 4, 141–148 (1973). [4] X. Chen, Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations, Adv. Differential Equations 2, 125–160 (1997). [5] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, 153–191, Springer, Berlin, 2003. [6] C. Lederman and N. Wolanski, A free boundary problem from nonlocal combustion, preprint. [7] J.L. Vazquez, An introduction to the mathematical theory of the porous medium equation, in “Shape optimization and free boundaries” (M.C. Delfour ed.), Dordrecht, Boston and Leiden, 347–389, 1992. [8] X. Wang, Metaestability and stability of patterns in a convolution model for phase transitions, preprint. Carmen Cortazar, Manuel Elgueta and Julio D. Rossi Departamento de Matemática Universidad Cat´ olica de Chile Casilla 306, Correo 22 Santiago Chile email: [email protected] email: [email protected] email: [email protected] Communicated by Rafael D. Benguria submitted 29/01/04, accepted 09/09/04


Ann. Henri Poincaré 6 (2005) 283 – 308 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020283-26 DOI 10.1007/s00023-005-0207-y


The Darwin Approximation of the Relativistic Vlasov-Maxwell System Sebastian Bauer and Markus Kunze Abstract. We study the relativistic Vlasov-Maxwell system which describes large systems of particles interacting by means of their collectively generated forces. If the speed of light c is considered as a parameter then it is known that in the Newtonian limit c → ∞ the Vlasov-Poisson system is obtained. In this paper we determine the next order approximate system, which in the case of individual particles usually is called the Darwin approximation.

1 Introduction and main results The relativistic Vlasov-Maxwell system  ∂t f + vˆ · ∇x f + (E + c−1 vˆ × B) · ∇v f     c∇ × B c∇ × E = −∂t B, ∇ · E = 4πρ, ∇·B     ρ := f dv, j

= 0, = =

∂t E + 4πj, 0, := vˆf dv,

(RVMc)

describes the time evolution of a single-species system of particles (with mass and charge normalized to unity) which interact by means of their collectively generated forces. The distribution of the large number of particles in configuration space is modelled through the non-negative density function f (x, v, t), depending on position x ∈ R3 , momentum v ∈ R3 , and time t ∈ R, where vˆ = (1 + c−2 v 2 )−1/2 v ∈ R3

(1.1)

is the relativistic velocity associated to v. The Lorentz force E + c−1 vˆ × B realizes the coupling of the Maxwell fields E(x, t) ∈ R3 and B(x, t) ∈ R3 to the Vlasov equation, and conversely the density function f enters the field equations via the scalar charge density ρ(x, t) and the current density j(x, t) ∈ R3 , which act as source terms for the Maxwell equations. It is supposed that collisions in the system are sufficiently rareso that they can be neglected. The parameter c denotes the speed of light, and always means R3 . At time t = 0, the initial data f (x, v, 0) = f ◦ (x, v),

E(x, 0) = E ◦ (x),

and B(x, 0) = B ◦ (x)

are prescribed. In this work we treat the speed of light as a parameter and study the behavior of the system as c → ∞. Conditions will be established under which

284

S. Bauer and M. Kunze


the solutions of (RVMc) converge to a solution of an effective system. We recall that in [21] it has been shown that as c → ∞ the solutions of (RVMc) approach a solution of the Vlasov-Poisson system with the rate O(c−1 ); see [1, 5] for similar results and [15] for the case of two spatial dimensions. The respective Newtonian limits of other related systems are derived in [20, 4]. It is the goal of this paper to replace the Vlasov-Poisson system by another effective equation to achieve higher order convergence and a more precise approximation. This will lead to an effective system whose solution stays as close as O(c−3 ) to a solution of the full VlasovMaxwell system, if the initial data are matched appropriately. In the context of individual particles, this post-Newtonian order of approximation is usually called the Darwin order, see [23, 13] and the references therein. Let us also mention that weak convergence properties of other kinds of Darwin approximations for the Vlasov-Maxwell system were studied in [6, 2]. In the present paper we mainly view the Darwin approximation as a rigorous intermediate step towards the next order, where in analogy to the case of individual particles [14] radiation effects are expected to play a role for the first time. Since at the radiation order the corresponding dynamics of the Vlasov-Maxwell system most likely will have to be restricted to a center manifold-like domain in the infinite dimensional space of densities (to avoid “run-away”-type solutions [23, 14]), it is clear that several new mathematical difficulties will have to be surmounted in this next step. Then the ultimate goal would be to determine the effective equation for the Vlasov-Maxwell system on the center manifold, which should finally lead to a slightly dissipative Vlasov-like equation, free of “run-away” solutions; see [11, 12] for a model of this equation and more motivation. Compared to systems of coupled individual particles, for the Vlasov-Maxwell system one immediately encounters the problem that so far in general only the existence of local solutions is known. These solutions are global under additional conditions, for instance if a suitable a priori bound on the velocities is available; see the pioneering work [8], and also [10, 3], where this result is reproved by different methods. This means that from the onset we will have to restrict ourselves to solutions of (RVMc) which are defined on some time interval [0, T ] that may be very small. On the other hand, in [21] it has been shown that such a time interval can be found which is uniform in c ≥ 1, so it seems reasonable to accept this restriction. In order to find the desired higher-order effective system, we formally expand all quantities arising in (RVMc) in powers of c−1 : f E

= =

f0 + c−1 f1 + c−2 f2 + · · · , E0 + c−1 E1 + c−2 E2 + · · · ,

B ρ

= =

B0 + c−1 B1 + c−2 B2 + · · · , ρ0 + c−1 ρ1 + c−2 ρ2 + · · · ,

j0 + c−1 j1 + c−2 j2 + · · · , where ρk = fk dv and jk = vfk dv for k = 0, 1, 2, . . .. Moreover, vˆ = v − (c−2 /2)v 2 v + · · · by (1.1), where v 2 = |v|2 . The expansions can be substituted into j

=

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The Darwin Approximation of the Relativistic Vlasov-Maxwell System

285

(RVMc), and comparing coefficients at every order gives a sequence of equations for these coefficients. At zeroth order we obtain ∇ × E0 = 0,

∇ · E0 = 4πρ0 ,

∇ × B0 = 0,

∇ · B0 = 0.

(1.2)

If we set B0 = 0, then the Vlasov-Poisson system  ∂t f0 + v · ∇x f0 + E0 · ∇v f0 = 0,      E0 (x, t) = − |z|−2 z¯ ρ0 (x + z, t) dz,   ρ0 = f0 dv,    f0 (x, v, 0) = f ◦ (x, v),

(VP)

is found, with z¯ = |z|−1 z. Next we consider the equations at first order in c−1 . Here ∇×E1 = −∂t B0 = 0,

∇·E1 = 4πρ1 ,

∇×B1 = ∂t E0 +4πj0 ,

∇·B1 = 0, (1.3)

needs to be satisfied for the fields; also see [12]. Using (1.2), we get ∆B1 = −4π∇× j0 and therefore define (1.4) B1 (x, t) = |x − y|−1 ∇ × j0 (y, t) dy = |z|−2 z¯ × j0 (x + z, t) dz. Regarding the density f1 , we obtain the linear Vlasov equation ∂t f1 + v · ∇x f1 + E1 · ∇v f0 + E0 · ∇v f1 = 0. Hence if we suppose that f1 (x, v, 0) = 0, then we can set f1 = 0 and E1 = 0 consistently. The field equations at the order c−2 are ∇ × E2 = −∂t B1 ,

∇ · E2 = 4πρ2 ,

∇ × B2 = ∂t E1 + 4πj1 = 0,

∇ · B2 = 0.

Therefore we can define B2 = 0. Calculating the equation for the density f2 and taking into account (1.3), we arrive at the following inhomogeneous linearized Vlasov-Poisson system, for which

286



we choose homogeneous initial data:  1 2   ∂t f2 + v · ∇x f2 − 2 v v · ∇x f0 + E0 · ∇v f2 + (E2 + v × B1 ) · ∇v f0 = 0, ∆E2 = ∂t2 E0 + 4π(∇ρ2 + ∂t j0 ),   f2 (x, v, 0) = 0. (LVP) At this point we need to discuss the solvability of the Poisson equation for E2 . Restricting our attention to initial data f ◦ for (RVMc) with compact support, it will turn out below that both ρ2 and j0 have compact support and thus lead to unproblematic sources. The first term ∂t2 E0 has to be examined more closely. Since ρ0 (·, t) has compact support for all t, see (2.1) below, we can calculate the iterated Poisson integrals dy 1 ∂ 2 E0 (y, t) ∆−1 (∂t2 E0 )(x, t) = − 4π |x − y| t 1 dw = |z|−2 z¯ ∂t2 ρ0 (x + w + z, t) dz 4π |w| 1 2 = dy ∂t ρ0 (y, t) du |y − x − u|−1 |u|−3 u 4π 1 1 dy (y − x) ∂t2 ρ0 (y, t) = = z¯ ∂t2 ρ0 (x + z, t) dz (1.5) 2 |y − x| 2 1 z · v)¯ z ) ∂t f0 (x + z, v, t) dz dv, = |z|−1 (v − (¯ 2 where we used (VP), eq. (5.27) from the appendix, and ∂t ρ0 + ∇ · j0 = 0 in conjunction with an integration by parts, the continuity equation itself being a direct consequence of (VP). In view of (1.5) and (LVP) we thus define 1 E2 (x, t) = z¯ ∂t2 ρ0 (x + z, t) dz − |z|−1 ∂t j0 (x + z, t) dz 2 − |z|−2 z¯ ρ2 (x + z, t) dz. (1.6) By (1.6), (VP), and a further integration by parts, we obtain the alternative expression 1 z · v)2 − v 2 ) f0 (x + z, v, t) dz dv E2 (x, t) = |z|−2 z¯ (3(¯ 2 1 |z|−1 (1 + z¯ ⊗ z¯) (E0 ρ0 )(x + z, t) dz − |z|−2 z¯ ρ2 (x + z, t) dz. (1.7) − 2 The first aim of this paper is to show that fD

:=

f0 + c−2 f2 ,

ED

:=

E0 + c−2 E2 ,

B

D

:=

c

−1

B1 ,

(1.8)

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yields a higher-order pointwise approximation of (RVMc) than the Vlasov-Poisson system; we call (1.8) the Darwin approximation. It is clear that for achieving this improved approximation property also the initial data of (RVMc) have to be matched appropriately by the data for the Darwin system. For a prescribed initial density f ◦ , we are able to calculate (f0 , E0 ), B1 , and (f2 , E2 ) according to what has been outlined above. We then consider (RVMc) with initial data   f (x, v, 0) = f ◦ (x, v), (IC) E(x, 0) = E ◦ (x) := E0 (x, 0) + c−2 E2 (x, 0),  B(x, 0) = B ◦ (x) := c−1 B1 (x, 0). Before we formulate our main theorem let us recall that solutions of (RVMc) with initial data (IC) exist at least on some time interval [0, T ] which is independent of c ≥ 1; see [21, Thm. 1], and cf. Proposition 2.2 below for a more precise statement. This time interval [0, T ] is fixed throughout the paper. Theorem 1.1 Assume that f ◦ ∈ C ∞ (R3 × R3 ) is nonnegative and has compact support. From f ◦ calculate (f0 , E0 ), B1 , and (f2 , E2 ), and then define initial data for (RVMc) by (IC). Let (f, E, B) denote the solution of (RVMc) with initial data (IC) and let (f D , E D , B D ) be defined as in (1.8). Then there exists a constant M > 0, and also for every R > 0 there is MR > 0, such that |f (x, v, t) − f D (x, v, t)| D

|E(x, t) − E (x, t)| |B(x, t) − B D (x, t)|

≤ M c−3 −3

≤ MR c ≤ M c−3

(x ∈ R3 ), (|x| ≤ R), (x ∈ R3 ),

(1.9)

for all v ∈ R3 , t ∈ [0, T ], and c ≥ 1. The constants M and MR are independent of c ≥ 1, but do depend on the initial data. Note that if (RVMc) is compared to the Vlasov-Poisson system (VP) only, one obtains the estimate |f (x, v, t) − f0 (x, v, t)| + |E(x, t) − E0 (x, t)| + |B(x, t)| ≤ M c−1 ; see [21, Thm. 2B]. Approximate models have the big advantage that, since by now the VlasovPoisson system is well understood, the existence of (f0 , E0 ), and here also of B1 and (f2 , E2 ), does no longer pose serious problems; note that in (LVP) the equation for f2 is linear. Therefore one can hope to get more information on (RVMc) by studying the approximate equations. As a drawback of the above hierarchy, one has to deal with two densities f0 , f2 and two electric fields E0 , E2 to define f D and E D . Therefore it is natural to look for a model which can be written down using only one density and one field. It turns out that the appropriate (Hamiltonian) system is  ∂t f + (1 − 12 c−2 v 2 )v · ∇x f + (E + c−1 v × B) · ∇v f = 0,     c∇ × E = −∂t B, ∇ · E = 4πρ, (DVMc) c ∆B = −4π∇ × j,     ρ = f dv, j = (1 − 12 c−2 v 2 )v f dv,

288



which we call the Darwin-Vlasov-Maxwell system. We note that (f D , E D , B D ) solves (DVMc) up to an error of the order c−3 . Theorem 1.2 Assume that f ◦ ∈ C ∞ (R3 × R3 ) is nonnegative and has compact support. Then there exist c∗ ≥ 1 and T ∗ > 0 such that the following holds for c ≥ c∗ . (a) If there is a local solution of (DVMc), then the initial data E ◦ and B ◦ of (DVMc) at t = 0 are uniquely determined by the initial density f ◦ . (b) The system (DVMc) has a unique C 2 -solution (f ∗ , E ∗ , B ∗ ) on [0, T ∗] attaining that initial data (f ◦ , E ◦ , B ◦ ) at t = 0. This solution conserves the energy 1 2 1 −2 4 ∗ 1 v − c v f dx dv + |∇φ∗ |2 + |∇ ∧ A∗ |2 dx, H= 2 8 8π where the potentials φ∗ and A∗ are chosen in such a way that B ∗ = ∇ ∧ A∗ , ∇ · A∗ = 0, and −∇φ∗ = E ∗ + c−1 ∂t A∗ . (c) Let (f, E, B) denote the solution of (RVMc) with initial data (f ◦ , E ◦ , B ◦ ). Then there exists a constant M > 0, and also for every R > 0 there is MR > 0, such that |f (x, v, t) − f ∗ (x, v, t)|

≤

M c−3

(x ∈ R3 ),

|E(x, t) − E ∗ (x, t)|

≤

MR c−3

(|x| ≤ R),

∗

|B(x, t) − B (x, t)|

≤

Mc

−3

(x ∈ R3 ),

for all v ∈ R3 , t ∈ [0, min{T, T ∗}], and c ≥ c∗ . Instead of performing the limit c → ∞ in (RVMc) it is possible to reformulate Theorem 1.1 in terms of a suitable dimensionless parameter. Taking this viewpoint means that we consider (RVMc) at a fixed c (say c = 1) by rescaling a prescribed nonnegative initial density f ◦ , for which we suppose that f ◦ ∈ C ∞ (R3 × R3 ) has compact support. To be more precise, let v¯ = vˆf ◦ (x, v) dx dv, where vˆ is taken for c = ε−1/2 ; cf. (1.1). Then v¯ is viewed as an average velocity of the system. Now we introduce f ε,◦ (x, v) = ε3/2 f ◦ (εx, ε−1/2 v) and consider f ε,◦ for c = 1. It follows that √ √ ε ε,◦ v¯ = vˆf (x, v) dx dv = ε wf ˆ ◦ (y, w) dy dw = ε v¯,

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i.e., the system with initial distribution function f ε,◦ has small velocities compared to the system associated to f ◦ . Starting from f ◦ , we next determine (f0 , E0 ), B1 , and (f2 , E2 ), and then the initial data for (RVMc) via (IC) with c = ε−1/2 , as in Theorem 1.1. Next we note that (f, E, B) is a solution of (RVMc) with c = ε−1/2 if and only if f ε (x, v, t)

=

E ε (x, t) = B ε (x, t) =

ε3/2 f (εx, ε−1/2 v, ε3/2 t), ε2 E(εx, ε3/2 t), ε2 B(εx, ε3/2 t),

is a solution of (RVMc) with c = 1. We further introduce f0ε (x, v, t)

E0ε (x, t) B1ε (x, t)

f2ε (x, v, t) E2ε (x, t) ρε0 (x, t) j0ε (x, t) ρε2 (x, t)

= ε3/2 f0 (εx, ε−1/2 v, ε3/2 t), = ε2 E0 (εx, ε3/2 t), = ε5/2 B1 (εx, ε3/2 t), = ε5/2 f2 (εx, ε−1/2 v, ε3/2 t), = ε3 E2 (εx, ε3/2 t), = f0ε (x, v, t) dv = ε3 ρ0 (εx, ε3/2 t), = vf0ε (x, v, t) dv = ε7/2 j0 (εx, ε3/2 t), = f2ε (x, v, t) dv = ε4 ρ2 (εx, ε3/2 t).

Straightforward calculations then confirm the following statements: (a) (f0 , E0 ) is a solution to (VP) with initial data f ◦ if and only if (f0ε , E0ε ) is a solution to (VP) with initial data f ε,◦ , (b) B1 solves ∆B1 = −4π∇ × j0 if and only if B1ε solves ∆B1ε = −4π∇ × j0ε , (c) (f2 , E2 ) is a solution to (LVP) if and only if (f2ε , E2ε ) is a solution to  1 2 ε ε ε ε ε ε ε ε   ∂t f2 + v · ∇x f2 − 2 v v · ∇x f0 + E0 · ∇v f2 + (E2 + v × B1 ) · ∇v f0 = 0, ∆E2ε = ∂t2 E0ε + 4π(∇ρε2 + ∂t j0ε ),   ε f2 (x, v, 0) = 0. Therefore Theorem 1.1 may be reformulated in a way which parallels [13, Thm. 2.2], where the case of individual particles is considered √ which are far apart (of order O(ε−1 )) and have small velocities (of order O( ε)) initially. Note that in this result the Lorentz force is determined up to an error of order O(ε7/2 ), and the dynamics of the full and the effective system can be compared over long times of order O(ε−3/2 ); see [13, p. 448].

290



Theorem 1.3 Assume that f ◦ ∈ C ∞ (R3 × R3 ) is nonnegative and has compact support. From f ◦ calculate (f0 , E0 ), B1 , and (f2 , E2 ), and then define initial data for (RVMc) by (IC) with c = ε−1/2 . Let (f, E, B) denote the solution of (RVMc) on [0, T ] for c = ε−1/2 with initial data (IC). Moreover, let f ε,◦ , f ε , E ε , B ε , f0ε , E0ε , B1ε , f2ε , E2ε , ρε0 , j0ε , and ρε2 be defined as above. Then (f ε , E ε , B ε ) is a solution of (RVMc) on [0, ε−3/2 T ] for c = 1 with initial data (f ε , E ε , B ε )(x, v, 0) = (f ε,◦ (x, v), E0ε (x, 0) + E2ε (x, 0), B1ε (x, 0)). In addition, there exists a constant M > 0, and also for every R > 0 there is MR > 0, such that |f ε (x, v, t) − f0ε (x, v, t) − f2ε (x, v, t)| |E ε (x, t) − E0ε (x, t) − E2ε (x, t)| |B ε (x, t) − B1ε (x, t)|

≤ M ε3 (x ∈ R3 ), 7/2 ≤ MR ε (|x| ≤ ε−1 R), ≤ M ε7/2

(x ∈ R3 ),

for all v ∈ R3 , t ∈ [0, ε−3/2 T ], and ε ≤ 1. The constants are independent of ε. By definition of the rescaled fields, these fields are slowly varying in their space and time variables, which means that we are considering an adiabatic limit. It is clear that also Theorem 1.2 could be restated in an analogous ε-dependent version. The paper is organized as follows. Some facts concerning (VP), (LVP), and (RVMc) are collected in Section 2. The proof of Theorem 1.1 is elaborated in Section 3, whereas Section 4 contains the proof of Theorem 1.2. For the proofs we will mostly rely on suitable representation formulas for the fields (refined versions of those used in [8, 21]), which are derived in the appendix, Section 5. Notation: B(0, R) denotes the closed ball in R3 with center at x = 0 or v = 0 and radius R > 0. The usual L∞ -norm of a function ϕ = ϕ(x) over x ∈ R3 is written as ϕ x , and if ϕ = ϕ(x, v), we modify this to ϕ x,v . For m ∈ N the W m,∞ -norms are denoted by ϕ m,x , etc. If T > 0 is fixed, then we write g(x, v, t, c) = Ocpt (c−m ), if for all R > 0 there is a constant M = MR > 0 such that |g(x, v, t, c)| ≤ M c−m

(1.10)

for |x| ≤ R, v ∈ R3 , t ∈ [0, T ], and c ≥ 1. Similarly, we write g(x, v, t, c) = O(c−m ), if there is a constant M > 0 such that (1.10) holds for all x, v ∈ R3 , t ∈ [0, T ], and c ≥ 1. In general, generic constants are denoted by M .

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2 Some properties of (VP), (LVP), and (RVMc) There is a vast literature on (VP), see, e.g., [7, Sect. 4] or [19] and the references therein. For our purposes we collect a few well-known facts about classical solutions of (VP). Proposition 2.1 Assume that f ◦ ∈ C ∞ (R3 × R3 ) is nonnegative and has compact support. Then there exists a unique global C 1 -solution (f0 , E0 ) of (VP), and there are nondecreasing continuous functions PVP , KVP : [0, ∞[→ R such that

f0 (t) x,v ≤ f ◦ x,v , supp f0 (·, ·, t) ⊂ B(0, PVP (t)) × B(0, PVP (t)),

f0 (t) 1,x,v + E0 (t) 1,x ≤ KVP (t),

(2.1)

for t ∈ [0, ∞[. This result was first established by Pfaffelmoser [18], and simplified versions of the proof were obtained by Schaeffer [22] and Horst [9]; a proof along different lines is due to Lions and Perthame [17]. For our approximation scheme we also need bounds on higher derivatives of the solution. This point was elaborated in [16], where it was shown that if f ◦ ∈ C k (R3 × R3 ), then (f0 , E0 ) possess continuous partial derivatives w.r.t. x and v up to order k. The existence of continuous time-derivatives then follows from the Vlasov equation. Thus (f0 , E0 ) are C ∞ , if f ◦ is C ∞ , and by a redefinition of KVP we can assume that

f0 (t) 3,x,v ≤ KVP (t),

t ∈ [0, ∞[.

(2.2)

The existence of a unique C 1 -solution (f2 , E2 ) of (LVP) follows by a contraction argument, but we omit the details. Furthermore it can be shown that there are nondecreasing continuous functions PLVP , KLVP : [0, ∞[→ R such that supp f2 (·, ·, t) ⊂ B(0, PLVP (t)) × B(0, PLVP (t)),

f2 (t) 1,x,v + E2 (t) 1,x ≤ KLVP (t),

(2.3) (2.4)

for t ∈ [0, ∞[. Concerning solutions of (RVMc), we have from [21, Thm. 1] the following Proposition 2.2 Assume that f ◦ ∈ C ∞ (R3 × R3 ) is nonnegative and has compact support. If E ◦ and B ◦ are defined by (IC), then there exits T > 0 (independent of c) such that for all c ≥ 1 the system (RVMc) with initial data (IC) has a unique C 1 solution (f, E, B) on the time interval [0, T ]. In addition, there are nondecreasing continuous functions (independent of c) PVM , KVM : [0, T ] → R such that f (x, v, t) = 0 if |v| ≥ PVM (t), |E(x, t)| + |B(x, t)| ≤ KVM (t), for all x ∈ R3 , t ∈ [0, T ], and c ≥ 1.

(2.5) (2.6)

292



In fact E ◦ and B ◦ do not depend on c in [21, Thm. 1], but an inspection of the proof shows that the assertions remain valid for initial fields defined by (IC).

3 Proof of Theorem 1.1 In Section 5.1.1 below we will show that the approximate electric field E D from (1.8) admits the following representation: D D D E D = Eext + Eint + Ebd + Ocpt (c−3 ),

(3.1)

with D Eext (x, t) = − |z|−2 z¯ (ρ0 + c−2 ρ2 )(x + z, t) dz |z|>ct 1 |z|−1 ∂t j0 (x + z, t) dz + c−2 z¯ ∂t2 ρ0 (x + z, t) dz, (3.2) −c−2 2 |z|>ct |z|>ct D −2 −2 Eint (x, t) = − |z| z¯ (ρ0 + c ρ2 )(x + z, tˆ(z)) dz |z|≤ct −c−1 z · v)¯ z ) f0 (x + z, v, tˆ(z)) dv dz |z|−2 (v − 2(¯ |z|≤ct −2 z · v)v + v 2 z¯ − 3¯ z (¯ z · v)2 ) f0 (x + z, v, tˆ(z)) dv dz |z|−2 (2(¯ +c |z|≤ct −2 +c |z|−1 (¯ z ⊗ z¯ − 1)E0 ρ0 (x + z, ˆt(z)) dz, (3.3) |z|≤ct D (x, t) = c−1 (ct)−1 (¯ z · v)¯ z f ◦ (x + z, v) dv ds(z) Ebd |z|=ct −2 −1 +c (ct) ((¯ z · v)v − (¯ z · v)2 z¯) f ◦ (x + z, v) dv ds(z), |z|=ct

where the subscripts ‘ext’, ‘int’, and ‘bd’ refer to the exterior, interior, and boundary integration in z. We also recall that z¯ = |z|−1 z and tˆ(z) = t − c−1 |z|. On the other hand, according to Section 5.1.2 below we have E = Eext + Eint + Ebd + O(c−3 ),

(3.4)

with

1 |z|−2 z¯ ρ0 + t∂t ρ0 + t2 ∂t2 ρ0 (x + z, 0) dz 2 |z|>ct 1 −2 + c z¯ ∂t2 ρ0 (x + z, 0) dz − c−2 |z|−1 ∂t j0 (x + z, 0) dz, (3.5) 2 |z|>ct |z|>ct

Eext (x, t) = −

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|z|−2 z¯ρ(x + z, ˆ t(z)) dz |z|−2 (2(¯ z · v)¯ z − v) f (x + z, v, ˆt(z)) dv dz +c−1 |z|≤ct +c−2 |z|−2 (v 2 z¯ + 2(¯ z · v)v − 3(¯ z · v)2 z¯) f (x + z, v, ˆt(z)) dv dz |z|≤ct |z|−1 (¯ z ⊗ z¯ − 1)(Ef )(x + z, v, ˆt(z)) dv dz, (3.6) +c−2

Eint (x, t) = −

|z|≤ct

|z|≤ct

Ebd (x, t) =

D Ebd (x, t).

In order to verify (1.9), we start by comparing the exterior fields. Let x ∈ B(0, R) with R > 0 be fixed. Then we obtain from (3.5) and (3.2), due to |¯ z | = 1, and taking into account ρ2 (x, 0) =

f2 (x, v, 0) dv = 0

by (LVP), as well as (2.1), (2.2), (2.3), and (2.4), D |Eext (x, t) − Eext (x, t)| ≤ |z|−2 ρ0 (x + z, t) − ρ0 (x + z, 0) − t∂t ρ0 (x + z, 0) |z|>ct

−

|z|−1 |v| |∂t f0 (x + z, v, 0) − ∂t f0 (x + z, v, t)| dv dz |z|>ct 1 −2 + c |∂t2 f0 (x + z, v, t) − ∂t2 f0 (x + z, v, 0)| dv dz 2 |z|>ct

t −2 2 3 ≤M |z| (t − s) PVP (s) KVP (s)1B(0,PVP (s)) (x + z) ds dz + c−2

1 2 2 t ∂t ρ0 (x + z, 0) dzρ2 (x + z, 0)| dz 2

|z|>ct

+ M c−2

0

|z|>ct

+ M c−2

|z|>ct

−2

+ Mc 3 ≤ Mt

|z|−1

|z|>ct

|z|>ct

+ Mt c

|z|−2

−2

t 0 t 0

t

PLVP (s)3 KLVP (s)1B(0,PLVP (s)) (x + z) ds dz PVP (s)4 KVP (s)1B(0,PVP (s)) (x + z) ds dz 3

PVP (s) KVP (s)1B(0,PVP (s)) (x + z) ds dz

|z|−2 1B(0,R+M0 ) (z) dz

|z|>ct

≤ MR c−3 ;

0

|z|−1 (|z|−1 + 1 + |z|)1B(0,R+M0 ) (z) dz (3.7)

294



note that here we have used M0 = max PVP (s) + KVP (s) + PLVP (s) + KLVP (s) < ∞, s∈[0,T ]

and for instance t3 |z|−2 1B(0,R+M0 ) (z) dz ≤ (ct)−3 t3 |z|>ct

|z|≤R+M0

To bound |Eint (x, t) −

D Eint (x, t)|,

(3.8)

|z| dz ≤ MR c−3 .

we first recall from [21, Thm. 2B] that

|E(x, t) − E0 (x, t)| = O(c−1 ).

(3.9)

Actually the initial conditions in [21] are different, but we only added terms of order c−2 , so that an inspection of the proof in [21] leads to (3.9). Next we define H(t) = sup {|f (x, v, s) − f D (x, v, s)| : x ∈ R3 , v ∈ R3 , s ∈ [0, t]},

as well as M1 = max

s∈[0,T ]

PVM (s) + PVP (s) + PLVP (s) < ∞.

Then f (x, v, s) = f0 (x, v, s) = f2 (x, v, s) = 0 for x ∈ R3 , |v| ≥ M1 , and s ∈ [0, T ]. Also if R0 > 0 is chosen such that f ◦ (x, v) = 0 for |x| ≥ R0 , introducing the constant M2 = R0 + T M1 + max PVP (s) + PLVP (s) < ∞ s∈[0,T ]

it follows that f (x, v, s) = f0 (x, v, s) = f2 (x, v, s) = 0 for |x| ≥ M2 , v ∈ R3 , and s ∈ [0, T ]. Let x ∈ B(0, R) with R > 0 be fixed. From (3.6), (3.3), (3.8), (3.9), (IC), and 0 ≤ tˆ(z) ≤ t for |z| ≤ ct we obtain D (x, t)| |Eint (x, t) − Eint −2 −2 ˆ ≤ |z| (f − f0 − c f2 )(x + z, v, t(z)) dv dz |z|≤ct −1 −2 −2 ˆ z · v)¯ z − v) (f − f0 − c f2 )(x + z, v, t(z)) dv dz |z| (2(¯ +c |z|≤ct z · v)¯ z − v) f2 (x + z, v, ˆt(z)) dv dz +c−3 |z|−2 (2(¯ |z|≤ct −2 −2 +c |z| (v 2 z¯ + 2(¯ z · v)v − 3(¯ z · v)2 z¯) |z|≤ct −2 ˆ (f − f0 − c f2 )(x + z, v, t(z)) dv dz −4 −2 2 2 ˆ +c |z| (v z¯ + 2(¯ z · v)v − 3(¯ z · v) z¯) f2 (x + z, v, t(z)) dv dz |z|≤ct +c−2 |z|−1 (1 − z¯ ⊗ z¯)([E − E0 ]f )(x + z, v, ˆt(z)) dv dz |z|≤ct

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≤

M (M13 + M14 )H(t) |z|−2 1B(0,M2 ) (x + z) dz |z|≤ct |z|−2 1B(0,M2 ) (x + z) dz +M M14 M0 c−3 |z|≤ct +M M15 H(t) c−2 |z|−2 1B(0,M2 ) (x + z) dz |z|≤ct |z|−2 1B(0,M2 ) (x + z) dz +M M15 M0 c−4 |z|≤ct +M M13 f ◦ x,v c−3 |z|−1 1B(0,M2 ) (x + z) dz

≤

MR (c−3 + H(t)),

|z|≤ct

since for instance |z|≤ct

|z|−2 1B(0,M2 ) (x + z) dz ≤

(3.10) |z|≤R+M2

|z|−2 dz ≤ MR .

D Recalling that the Ebd (x, t) = Ebd (x, t), we can summarize (3.4), (3.1), (3.7), and (3.10) as |E(x, t) − E D (x, t)| ≤ MR (c−3 + H(t)), (3.11)

for |x| ≤ R and t ∈ [0, T ]. Formulas (5.11), (5.13), (5.23), (5.17), (5.18), and (5.19), and an analogous (actually more simple) calculation also leads to |B(x, t) − B D (x, t)| ≤ M (c−3 + H(t)),

(3.12)

for x ∈ R3 and t ∈ [0, T ]. It remains to estimate h = f − f D . Using (RVMc), (1.8), (VP), and (LVP), it is found that ∂t h + vˆ · ∇x h + (E + c−1 vˆ × B) · ∇v h = −∂t f D − vˆ · ∇x f D − (E + c−1 vˆ × B) · ∇v f D 1 = v − c−2 v 2 v − vˆ · ∇x f0 + c−2 (v − vˆ) · ∇x f2 2 +(E D − E) · ∇v f0 + c−2 (E D − E) · ∇v f2 − c−4 E2 · ∇v f2 +c−2 ((v − vˆ) × B1 ) · ∇v f0 + c−1 (ˆ v × (B D − B)) · ∇v f0 − c−3 (ˆ v × B) · ∇v f2 . v | = (1 + c−2 v 2 )−1/2 |v| ≤ |v| ≤ M1 uniformly in c, and If |v| ≤ M1 , then also |ˆ hence 1 v − 1 − c−2 v 2 v ≤ M c−4 . ˆ 2 Next we note the straightforward estimate |B1 (x, t)| ≤ M for |x| ≤ M2 and t ∈ [0, T ], with B1 from (1.4). In view of the bounds (2.1), (2.4), and (2.6), thus by (3.11) and (3.12), |∂t h(x, v, t) + vˆ · ∇x h(x, v, t) + (E(x, t) + c−1 vˆ × B(x, t)) · ∇v h(x, v, t)| ≤ M (c−3 + H(t))

(3.13)

296



for |x| ≤ M2 , |v| ≤ M1 , and t ∈ [0, T ]. But in {(x, v, t) : |x| > M2 } ∪ {(x, v, t) : |v| > M1 } we have h = f − f D = 0 by the above definition of M1 > 0 and M2 > 0. Accordingly, (3.13) is satisfied for all x ∈ R3 , v ∈ R3 , and t ∈ [0, T ]. Since h(x, v, 0) = 0, the argument from [21, p. 416] yields t H(t) ≤ M (c−3 + H(s)) ds, 0 −3

and therefore H(t) ≤ M c for t ∈ [0, T ]. Then due to (3.11) and (3.12), |E(x, t)− E D (x, t)| ≤ MR c−3 for |x| ≤ R and t ∈ [0, T ], as well as |B(x, t) − B D (x, t)| ≤ M c−3 for x ∈ R3 and t ∈ [0, T ]. This completes the proof of Theorem 1.1.

4 Proof of Theorem 1.2 In this section we will be sketchy and omit many details, since the proof is more or less a repetition of what has been said before. First let us assume that there is a C 2 -solution (f ∗ , E ∗ , B ∗ ) of (DVMc), existing on a time interval [0, T ∗ ] for some T ∗ > 0, such that supp f ∗ (·, ·, t) ⊂ R3 × R3 is compact for all t ∈ [0, T ∗ ]. Then ∗ −1 (4.1) |z|−2 z¯ × j ∗ (x + z, t) dz, B (x, t) = c ∆E ∗ (x, t)

= 4π∇ρ∗ (x, t) + c−1 ∂t ∇ × B ∗ (x, t). (4.2) 1 −2 2 ∗ ◦ ∗ ∗ Since f (x, v, 0) = f (x, v) and j (x, 0) = (1 − 2 c v )vf (x, v, 0) dv = (1 − 1 −2 2 v )vf ◦ (x, v) dv, it follows that B ∗ (x, 0) is determined by f ◦ . In order to 2 c compute the Poisson integral for E ∗ , we calculate by means of the transformation y = w − z, dy = dw, and using (5.27) below, c−1 ∆−1 (∂t ∇ × B ∗ )(x, t) 1 dy ∇ × ∂t B ∗ (y, t) = − 4πc |x − y|

1 dy −2 ∗ = − ∇y × |z| z¯ × ∂t j (y + z, t) dz 4πc2 |x − y| 1 = − z · ∇)∂t j ∗ (w, t) dw dz |z|−2 |x − w + z|−1 z¯ ∇ · (∂t j ∗ )(w, t) − (¯ 2 4πc 1 dw ∗ ∗ [x − w] ∇ · (∂ = j )(w, t) − ([x − w] · ∇)∂ j (w, t) t t 2c2 |x − w| dz 1 (1 + z¯ ⊗ z¯)∂t j ∗ (x + z, t). = − 2 2c |z| If we invoke the Vlasov equation for f ∗ and integrate by parts, this can be rewritten as c−1 ∆−1 (∂t ∇ × B ∗ )(x, t) 1 dz 1 1 − c−2 v 2 v ∂t f ∗ (x + z, v, t) dv (1 + z¯ ⊗ z¯) = − 2 2c |z| 2

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297

dz 1 1 −2 2 ∗ 2 2 c v f (x + z, v, t) dv z ¯ (3(¯ z · v) − v ) 1 − 2c2 |z|2 2 1 dz − 2 (1 + z¯ ⊗ z¯) ((E ∗ + c−1 v × B ∗ )f ∗ )(x + z, v, t) dv 2c |z| 1 dz + 4 (1 + z¯ ⊗ z¯) v 2 v ∂t f ∗ (x + z, v, t) dv. 4c |z|

Therefore the solution E ∗ of (4.2) has the representation E ∗ (x, t) = 4π∆−1 (∇ρ∗ )(x, t) + c−1 ∆−1 (∂t ∇ × B ∗ )(x, t) = − |z|−2 z¯ ρ∗ (x + z, t) dz 1 1 z · v)2 − v 2 ) 1 − c−2 v 2 f ∗ (x + z, v, t) dv dz |z|−2 z¯ (3(¯ + 2 2c 2 1 − 2 |z|−1 (1 + z¯ ⊗ z¯) ((E ∗ + c−1 v × B ∗ )f ∗ )(x + z, v, t) dv dz 2c 1 + 4 z · v)¯ z ) ∂t f ∗ (x + z, v, t) dv dz. (4.3) |z|−1 v 2 (v + (¯ 4c Comparison with (VP) and (1.7) reveals the analogy to E D at the relevant orders of c−1 . In particular, if we evaluate this relation at t = 0, the Banach fixed point theorem applied in Cb (R3 ) shows that for c ≥ c∗ sufficiently large the function E ∗ (x, 0) is uniquely determined by f ◦ (x, v) = f ∗ (x, v, 0). Thus f ◦ alone already fixes E ◦ and B ◦ . Concerning the local and uniform (in c) existence of a solution to (DVMc) and the conservation of energy, one can use (4.1) and (4.3) to follow the usual method by setting up an iteration scheme for which convergence can be verified on a small time interval; cf. [7, Sect. 5.8]. Finally, by similar arguments as used in the proof of Theorem 1.1 it can be shown that solutions of (DVMc) approximate solutions of (RVMc) up to an error of order c−3 .

5 Appendix 5.1

Representation Formulas

5.1.1 Representation of the approximation fields E D and B D Here we will derive the representation formula (3.1) for the approximate field E D from (1.8). Since the calculations for the electric and the magnetic field are quite similar, we will only analyze in detail the electric field and simply state the result for its magnetic counterpart. From (1.8) we recall E D = E0 + c−2 E2 , where E0 (x, t) = − |z|−2 z¯ ρ0 (x + z, t) dz, (5.1)

298


E2 (x, t)

=

1 2

−

z¯ ∂t2 ρ0 (x + z, t) dz −


|z|−1 ∂t j0 (x + z, t) dz

|z|−2 z¯ ρ2 (x + z, t) dz,

(5.2)

cf. (VP) and (1.6). We split the domain of integration in {|z| > ct} and {|z| ≤ ct}, and to handle the interior part {|z| ≤ ct} we expand the densities w.r.t. t about the retarded time tˆ(z) := t − c−1 |z|. To begin with, we have −2 |z| z¯ ρ0 (x + z, t) dz = − |z|−2 z¯ ρ0 (x + z, ˆt(z)) dz − |z|≤ct |z|≤ct −1 −1 −2 1 ˆ −c |z| z¯ ∂t ρ0 (x + z, t(z)) dz − c z¯ ∂ 2 ρ0 (x + z, ˆt(z)) dz 2 |z|≤ct t |z|≤ct t 1 − |z|−2 z¯ (t − s)2 ∂t3 ρ0 (x + z, s) ds dz. (5.3) 2 |z|≤ct t(z) Using (2.1) and (2.2), the last term is Ocpt (c−3 ); note that |x| ≤ R for some R > 0 together with the support properties of f0 imply that we only have to integrate in z over a set which is uniformly bounded in c ≥ 1. Since ∂t ρ0 + ∇ · j0 = 0 by (VP), we also find −1 −1 −1 ˆ |z| z¯ ∂t ρ0 (x + z, t(z)) dz = c |z|−1 z¯ ∇x · j0 (x + z, ˆt(z)) dz −c |z|≤ct |z|≤ct −1 = c |z|−1 z¯ v · ∇x f0 (x + z, v, tˆ(z)) dv dz |z|≤ct −1 = c |z|−1 z¯ v · ∇z [f0 (x + z, v, tˆ(z))] + c−1 z¯ ∂t f0 (x + z, v, ˆt(z)) dv dz |z|≤ct

= I + II,

(5.4)

with I

=

|z|−1 z¯ v · ∇z [f0 (x + z, v, tˆ(z))] dv dz

−1 ∇z · |z|−1 z¯i v i=1,2,3 f0 (x + z, v, tˆ(z)) dv dz −c |z|≤ct +c−1 (ct)−1 z¯(¯ z · v)f ◦ (x + z, v) dv ds(z) |z|=ct −c−1 z · v)¯ z )f0 (x + z, v, tˆ(z)) dv dz |z|−2 (v − 2(¯ |z|≤ct z¯(¯ z · v)f ◦ (x + z, v) dv ds(z); +c−1 (ct)−1 c

−1

|z|≤ct

=

=

|z|=ct

(5.5)

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observe that tˆ(z) = 0 for |z| = ct was used for the boundary term. Similarly, by (VP), −2 z · v) ∂t f0 (x + z, v, tˆ(z)) dv dz |z|−1 z¯(¯ II = c |z|≤ct = −c−2 |z|−1 z¯(¯ z · v)(v · ∇x f0 + E0 · ∇v f0 )(x + z, v, tˆ(z)) dv dz |z|≤ct z · v)v)i=1,2,3 f0 (x + z, v, ˆt(z)) dv dz ∇z · (|z|−1 z¯i (¯ = c−2 |z|≤ct −c−2 (ct)−1 z¯(¯ z · v)2 f ◦ (x + z, v) dv ds(z) |z|=ct −c−3 z · v)2 ∂t f0 (x + z, v, tˆ(z)) dv dz |z|−1 z¯(¯ |z|≤ct −2 |z|−1 z¯ z¯ · E0 f0 (x + z, v, tˆ(z)) dv dz +c |z|≤ct −2 = c z · v)v + v 2 z¯ − 3¯ z (¯ z · v)2 ) f0 (x + z, v, tˆ(z)) dv dz |z|−2 ((¯ |z|≤ct −2 +c |z|−1 z¯ z¯ · E0 ρ0 (x + z, ˆt(z)) dz |z|≤ct −2 −1 (5.6) z¯(¯ z · v)2 f ◦ (x + z, v) dv ds(z) + Ocpt (c−3 ). −c (ct) |z|=ct

Next, due to (2.1) and (2.2) we also have 1 1 z¯ ∂t2 ρ0 (x + z, tˆ(z)) dz = −c−2 z¯ ∂ 2 ρ0 (x + z, t) dz + Ocpt (c−3 ). −c−2 2 |z|≤ct 2 |z|≤ct t (5.7) Thus so far by (5.1) and (5.3)–(5.7), −2 |z| z¯ ρ0 (x + z, t) dz − |z|−2 z¯ ρ0 (x + z, t) dz E0 (x, t) = − |z|>ct |z|≤ct = − |z|−2 z¯ ρ0 (x + z, t) dz |z|>ct −2 −2 1 ˆ |z| z¯ ρ0 (x + z, t(z)) dz − c z¯ ∂ 2 ρ0 (x + z, t) dz − 2 |z|≤ct t |z|≤ct −c−1 z · v)¯ z )f0 (x + z, v, ˆt(z)) dv dz |z|−2 (v − 2(¯ |z|≤ct +c−2 z · v)v + v 2 z¯ − 3¯ z(¯ z · v)2 ) f0 (x + z, v, ˆt(z)) dv dz |z|−2 ((¯ |z|≤ct |z|−1 z¯ z¯ · E0 ρ0 (x + z, ˆt(z)) dz +c−2 |z|≤ct

300


+c−1 (ct)−1 −c−2 (ct)−1

|z|=ct

|z|=ct


(¯ z · v)¯ z f ◦ (x + z, v) dv ds(z) (¯ z · v)2 z¯ f ◦ (x + z, v) dv ds(z) + Ocpt (c−3 ).

(5.8)

D −2 Now we turn to E2 , cf. (5.2). Since E2 enters E with the factor c , we first note that c−2 12 |z|≤ct z¯ ∂t2 ρ0 (x + z, t) dz cancels a term on the right-hand side of (5.8). In addition, by analogous arguments, |z|−1 ∂t j0 (x + z, t) dz −c−2 |z|≤ct

t −2 −1 ˆ = −c |z| ∂t j0 (x + z, t(z)) dz − c |z| ∂t2 j0 (x + z, s) ds dz tˆ(z) |z|≤ct |z|≤ct = −c−2 |z|−1 v ∂t f0 (x + z, v, ˆ t(z)) dv dz + Ocpt (c−3 ) |z|≤ct |z|−1 v (v · ∇x f0 + E0 · ∇v f0 )(x + z, v, tˆ(z)) dv dz + Ocpt (c−3 ) = c−2 |z|≤ct = c−2 |z|−1 v v · ∇z [f0 (x + z, v, tˆ(z))] + c−1 z¯ ∂t f0 (x + z, v, ˆt(z)) dv dz |z|≤ct +c−2 |z|−1 v E0 · ∇v f0 (x + z, v, ˆt(z)) dv dz + Ocpt (c−3 ) |z|≤ct z · v)vf0 (x + z, v, tˆ(z)) dv dz |z|−2 (¯ = c−2 |z|≤ct +c−2 (ct)−1 (¯ z · v)vf ◦ (x + z, v) dv ds(z) |z|=ct −2 −c |z|−1 (E0 ρ0 )(x + z, tˆ(z)) dz + Ocpt (c−3 ). (5.9) −2

−1

|z|≤ct

Finally, −c−2

|z|≤ct

|z|−2 z¯ ρ2 (x+z, t) dz = −c−2

|z|≤ct

|z|−2 z¯ ρ2 (x+z, ˆt(z)) dz+Ocpt (c−3 ).

(5.10) Therefore if we write 1 c−2 E2 (x, t) = c−2 z¯ ∂t2 ρ0 (x + z, t) dz − c−2 |z|−1 ∂t j0 (x + z, t) dz 2 |z|>ct |z|>ct 1 −c−2 |z|−2 z¯ ρ2 (x + z, t) dz + c−2 z¯ ∂ 2 ρ0 (x + z, t) dz 2 |z|≤ct t |z|>ct |z|−1 ∂t j0 (x + z, t) dz − c−2 |z|−2 z¯ ρ2 (x + z, t) dz, −c−2 |z|≤ct

|z|≤ct

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use (5.9) and (5.10), and thereafter add the result to (5.8), it turns out that E D = E0 + c−2 E2 can be decomposed as claimed in (3.1). Similar calculations for B D (x, t) = c−1 B1 (x, t) using (1.4) yield B D (x, t) = c−1 |z|−2 z¯ × j0 (x + z, t) dz |z|>ct −1 |z|−2 z¯ × j0 (x + z, tˆ(z)) dz +c |z|≤ct −2 −2c z · v)(¯ z × v) f0 (x + z, v, ˆt(z)) dv dz |z|−2 (¯ |z|≤ct −2 +c |z|−1 z¯ × E0 ρ0 (x + z, ˆt(z)) dz (5.11) |z|≤ct (¯ z · v)(¯ z × v) f ◦ (x + z, v) dv ds(z) + O(c−3 ). −c−2 (ct)−1 |z|=ct

5.1.2 Representation of the Maxwell fields E and B In this section we will verify the representation formula (3.4) for the full Maxwell field E, by expanding the respective expressions from [8, 21] to higher orders. Once again the computation for the corresponding magnetic field B is very similar and therefore omitted. Let (f, E, B) be a C 1 -solution of (RVMc) with initial data (f ◦ , E ◦ , B ◦ ). We recall the following representation from [21, (A13), (A14), (A3)]:

where ED (x, t) EDT (x, t) ET (x, t) ES (x, t)

E

=

ED + EDT + ET + ES ,

(5.12)

B

=

BD + BDT + BT + BS ,

(5.13)

t = ∂t E (x + ctω) dω + ∂t E(x + ctω, 0) dω, 4π |ω|=1 |ω|=1 = −(ct)−1 z , vˆ)f ◦ (x + z, v) dv ds(z), KDT (¯ |z|=ct = − |z|−2 KT (¯ z , vˆ)f (x + z, v, ˆt(z)) dv dz, |z|≤ct = −c−2 |z|−1 KS (¯ z , vˆ)(E + c−1 vˆ × B)f (x + z, v, tˆ(z)) dv dz, t 4π

◦

|z|≤ct

and

t t B ◦ (x + ctω) dω + ∂t B(x + ctω, 0) dω, 4π |ω|=1 4π |ω|=1 (ct)−1 z , vˆ)f ◦ (x + z, v) dv ds(z), LDT (¯

BD (x, t)

=

BDT (x, t)

=

∂t

|z|=ct

302


BT (x, t)

=

BS (x, t)

=

c−1

|z|≤ct

c−2

|z|≤ct

|z|−2 |z|−1


z , vˆ)f (x + z, v, ˆt(z)) dv dz, LT (¯

z , vˆ)(E + c−1 vˆ × B)f (x + z, v, ˆt(z)) dv dz, LS (¯

with z¯ = |z|−1 z and tˆ(z) = t − c−1 |z|. The kernels are given by z , vˆ) = KDT (¯ KT (¯ z , vˆ) = z , vˆ) = KS (¯

(1 + c−1 z¯ · vˆ)−1 (¯ z − c−2 (¯ z · vˆ)ˆ v ), (1 + c−1 z¯ · vˆ)−2 (1 − c−2 vˆ2 )(¯ z + c−1 vˆ), (1 + c−1 z¯ · vˆ)−2 (1 + c−2 v 2 )−1/2 z · vˆ)¯ z − vˆ) ⊗ vˆ − (¯ z + c−1 vˆ) ⊗ z¯ ∈ R3×3 , (1 + c−1 z¯ · vˆ) + c−2 ((¯

and z , vˆ) = (1 + c−1 z¯ · vˆ)−1 (¯ z × c−1 vˆ), LDT (¯ LT (¯ z , vˆ) = (1 + c−1 z¯ · vˆ)−2 (1 − c−2 vˆ2 )(¯ z × vˆ), −1 −2 −2 2 −1/2 (1 + c−1 z¯ · vˆ)¯ LS (¯ z , vˆ) = (1 + c z¯ · vˆ) (1 + c v ) z × (. . .)

−c−2 (¯ z × vˆ) ⊗ (c¯ z + vˆ) ∈ R3×3 .

Next we expand these fields in powers of c−1 . According to (2.5) we can assume that the v-support of f (x, ·, t) is uniformly bounded in x ∈ R3 and t ∈ [0, T ], say f (x, v, t) = 0 for |v| ≥ P := maxt∈[0,T ] PVM (t). Thus we may suppose that |v| ≤ P in each of the v-integrals, and hence also |ˆ v | = (1 + c−2 v 2 )−1/2 |v| ≤ |v| ≤ P uniformly in c. It follows that 1 vˆ = 1 − c−2 v 2 v + O(c−4 ). 2 For instance, for the kernel KDT of EDT this yields z , vˆ) KDT (¯

= (1 + c−1 z¯ · vˆ)−1 (¯ z − c−2 (¯ z · vˆ)ˆ v) −1 −2 2 = 1 − c z¯ · v + c (¯ z · v) + O(c−3 ) z¯ − c−2 (¯ z · v)v + O(c−4 ) z · v)¯ z + c−2 (¯ z · v)2 z¯ − c−2 (¯ z · v)v + O(c−3 ). = z¯ − c−1 (¯

If we choose R0 > 0 such that f ◦ (x, v) = 0 for |x| ≥ R0 , then −1 −(ct) O(c−3 )1B(0,R0 ) (x + z) dv ds(z)

= ct

|z|=ct

|ω|=1

|v|≤P

1B(0,R0 ) (x + ctω) ds(ω) O(c−3 ) = O(c−3 )

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303

by [21, Lemma 1], uniformly in x ∈ R3 , t ∈ [0, T ], and c ≥ 1. Therefore we arrive at −1 EDT (x, t) = −(ct) z · v)¯ z + c−2 [(¯ z · v)2 z¯ − (¯ z · v)v] z¯ − c−1 (¯ |z|=ct ◦

f (x + z, v) dv ds(z) + O(c−3 ).

(5.14)

Concerning ET , we note that f (x, v, t) = 0 for |x| ≥ R0 + T P =: R1 . Since, by distinguishing the cases |x − y| ≥ 1 and |x − y| ≤ 1, |z|−2 1B(0,R1 ) (x + z) dz = |x − y|−2 1B(0,R1 ) (y) dy = O(1) |z|≤ct

|x−y|≤ct

uniformly in x ∈ R3 , t ∈ [0, T ], and c ≥ 1, similar computations as before show that −2 z¯ + c−1 [v − 2(¯ ET (x, t) = − |z| z · v)¯ z ] + c−2 [3(¯ z · v)2 z¯ |z|≤ct

z · v)v] f (x + z, v, ˆt(z)) dv dz + O(c−3 ). −v z¯ − 2(¯ 2

(5.15)

In the same manner, elementary calculations using also (2.6) can be carried out to get ES (x, t) = −c−2 |z|−1 (1 − z¯ ⊗ z¯)(Ef )(x + z, v, tˆ(z)) dv dz +O(c BDT (x, t)

|z|≤ct −3

= (ct)−1

),

|z|=ct

(5.16)

c−1 z¯ × v − c−2 (¯ z · v)¯ z × v f ◦ (x + z, v) dv ds(z)

BT (x, t)

(5.17) +O(c−3 ), = c−1 |z|−2 (¯ z × v − c−1 2v · z¯z¯ × v)f (x + z, v, ˆt(z)) dv dz

BS (x, t)

+O(c−3 ), (5.18) = c−2 |z|−1 z¯ × (Ef )(x + z, v, ˆt(z)) dv dz + O(c−3 ). (5.19)

|z|≤ct

|z|≤ct

Next we consider the data term

t t ◦ ED (x, t) = ∂t E (x + ctω) dω + ∂t E(x + ctω, 0) dω, 4π |ω|=1 4π |ω|=1 =: III + IV. Since f2 (x, v, 0) = 0 by (LVP), we have ρ2 (x, 0) = 0.

(5.20)

304



Thus we get from (IC), (VP), and (1.6), E ◦ (x)

= =

E0 (x, 0) + c−2 E2 (x, 0)

1 − |z|−2 z¯ ρ0 (x + z, 0) dz + c−2 z¯ ∂t2 ρ0 (x + z, 0) dz 2 − |z|−1 ∂t j0 (x + z, 0) dz .

Using the formulas (5.25), (5.26), and (5.24) below, we calculate − |z|−2 z¯ ρ0 (x + ctω + z, 0) dz dω |ω|=1 = − ρ0 (y, 0) |y − x − ctω|−3 (y − x − ctω) dω dy |ω|=1 |z|−2 z¯ ρ0 (x + z, 0) dz, = −4π |z|>ct z¯ ∂t2 ρ0 (x + ctω + z, 0) dz dω |ω|=1 |y − x − ctω|−1 (y − x − ctω) dω dy = ∂t2 ρ0 (y, 0) |ω|=1 1 z¯ − (ct)2 |z|−2 z¯ ∂t2 ρ0 (x + z, 0) dz = 4π 3 |z|>ct 8π + z ∂ 2 ρ0 (x + z, 0) dz, 3ct |z|≤ct t |z|−1 ∂t j0 (x + ctω + z, 0) dz dω − |ω|=1 = − ∂t j0 (y, 0) |y − x − ctω|−1 dω dy |ω|=1 4π −1 = −4π |z| ∂t j0 (x + z, 0) dz − ∂t j0 (x + z, 0) dz. ct |z|≤ct |z|>ct Therefore we get

t III = ∂t E ◦ (x + ctω) dω 4π |ω|=1

t = ∂t − t |z|−2 z¯ ρ0 (x + z, 0) dz + 2 z¯ ∂t2 ρ0 (x + z, 0) dz 2c |z|>ct |z|>ct t3 1 −2 2 − |z| z¯ ∂t ρ0 (x + z, 0) dz + 3 z ∂ 2 ρ0 (x + z, 0) dz 6 |z|>ct 3c |z|≤ct t t 1 − 2 |z|−1 ∂t j0 (x + z, 0) dz − 3 ∂t j0 (x + z, 0) dz c |z|>ct c |z|≤ct

Vol. 6, 2005


=

−

|z|>ct

1 − t2 2

|z|−2 z¯ ρ0 (x + z, 0) dz +

|z|>ct

+(ct)−1

1 −2 c 2

|z|>ct

|z|−2 z¯ ∂t2 ρ0 (x + z, 0) dz − c−2

|z|=ct

305

z¯ ∂t2 ρ0 (x + z, 0) dz

|z|>ct

|z|−1 ∂t j0 (x + z, 0) dz

z¯ ρ0 (x + z, 0) ds(z),

(5.21)

note that several terms have cancelled here. Now we discuss the second part IV of the data term ED , cf. (5.20). To begin with, by (IC), (1.4), and (VP), B(x, 0)

due to j0 = obtain

◦

−1

−1

|z|−2 z¯ × j0 (x + z, 0) dz = B (x) = c B1 (x, 0) = c = c−1 z × v)f0 (x + z, v, 0) dv dz, |z|−2 (¯ vf0 dv. Therefore using (VP) for f0 and integration by parts, we

|z|−2 (¯ z × v)f0 (x + z, v, 0) dv dz ∇ × B(x, 0) = c−1 ∇ × = c−1 z × v) dv dz |z|−2 ∇x f0 × (¯ z − (¯ z · ∇x f0 )v dv dz |z|−2 (v · ∇x f0 )¯ = c−1 −1 = c |z|−2 z¯ (−∂t f0 − E0 · ∇v f ) dv dz −c−1 |z|−2 v z¯ · ∇z [f0 (. . .)] dv dz |z|−2 z¯ ∂t f0 (x + z, v, 0) dv dz + c−1 4π vf0 (x, v, 0) dv, = −c−1 by observing that in general for suitable functions g, invoking the divergence theorem, |z|−2 z¯ · ∇z g(x + z) dz = g(x + εω) dω → 4πg(x), ε → 0. − |z|>ε

|ω|=1

t) = c∇ × B(x, From Maxwell’s equations we have ∂t E(x, t) − 4πj(x, t), and thus in view of j(x, 0) = vˆf (x, v, 0) dv = vˆf ◦ (x, v) dv = vˆf0 (x, v, 0) dv, ∂t E(x, 0) = =

c∇ × B(x, 0) − 4πj(x, 0) −2 − |z| z¯ ∂t f0 (x + z, v, 0) dv dz + 4π (v − vˆ)f0 (x, v, 0) dv.

306



Hence due to (5.25), v − vˆ = O(c−2 ), and by [21, Lemma 1], t ∂t E(x + ctω, 0) dω IV = 4π |ω|=1 t = − |z|−2 z¯ ∂t f0 (x + ctω + z, v, 0) dv dz dω 4π |ω|=1 (v − vˆ)f ◦ (x + ctω, v) dv dω + c−1 (ct) |ω|=1 = −t |z|−2 z¯ ∂t ρ0 (x + z, 0) dz + O(c−3 ).

(5.22)

|z|>ct

If we combine (5.12), (5.20), (5.21), (5.22), (5.14), (5.15), and (5.16), then we see that (3.4) is satisfied. A similar calculation yields −1 |z|−2 z¯ × j0 (x + z, 0) dz BD (x, t) = c |z|>ct +c−1 t |z|−2 z¯ × ∂t j0 (x + z, 0) dz |z|>ct z¯ × j0 (x + z, 0) ds(z), (5.23) −c−1 (ct)−1 |z|=ct

and an analogous decomposition of B into B = Bext + Bint + Bbd + O(c−3 ).

5.2

Some explicit integrals

We point out some formulas that have been used in the previous sections. For z ∈ R3 and r > 0 an elementary calculation yields 4πr−1 : r ≥ |z| |z − rω|−1 dω = . (5.24) 4π|z|−1 : r ≤ |z| |ω|=1 Differentiation w.r.t. z gives −3 |z − rω| (z − rω) dω = |ω|=1

Similarly,

|ω|=1

|z − rω| dω =

4πr + 4π|z| +

and thus by differentiation |z − rω|−1 (z − rω) dω = |ω|=1

0 : r > |z| . 4π|z|−2 z¯ : r < |z|

4π 2 −1 3 z r 4π 2 −1 3 r |z|

4π¯ z−

: r ≥ |z| : r ≤ |z|

8π 3r z 4π 2 −2 z¯ 3 r |z|

(5.25)

,

: r > |z| : r < |z|

.

(5.26)

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Finally, for z ∈ R3 \ {0} also

z |z − v|−1 |v|−3 v dv = 2π¯

307

(5.27)

can be computed. Acknowledgments. The authors are indebted to G. Rein, A. Rendall and H. Spohn for many discussions.

References [1] K. Asano, S. Ukai, On the Vlasov-Poisson limit of the Vlasov-Maxwell equation, in Patterns and Waves, Eds. Nishida T., Mimura M. & Fujii H., Stud. Math. Appl., Vol. 18, North-Holland Publishing Co., Amsterdam 1986, pp. 369–383. [2] S. Benachour, F. Filbet, Ph. Lauren¸cot, E. Sonnendr¨ ucker, Global existence for the Vlasov-Darwin system in R3 for small initial data, Math. Methods Appl. Sci. 26, 297–319 (2003). [3] F. Bouchut, F. Golse, Ch. Pallard, On classical solutions to the 3D relativistic Vlasov-Maxwell system: Glassey-Strauss’ theorem revisited, Arch. Rational Mech. Anal. 170, 1–15 (2003). [4] S. Calogero, H. Lee, The non-relativistic limit of the Nordstr¨ om-Vlasov system, ArXiv preprint math-ph/0309030. [5] P. Degond, Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equation for infinite light velocity, Math. Methods Appl. Sci. 8, 533–558 (1986). [6] P. Degond, P. Raviart, An analysis of the Darwin model of approximation to Maxwell’s equations, Forum Math. 4, 13–44 (1992). [7] R.T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia 1996 [8] R.T. Glassey, W. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal. 92, 59–90 (1986). [9] E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Methods Appl. Sci. 16, 75–85 (1993). [10] S. Klainerman, G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal. 1, 103–125 (2002). [11] M. Kunze, A.D. Rendall, The Vlasov-Poisson system with radiation damping, Ann. H. Poincaré 2, 857–886 (2001).

308



[12] M. Kunze, A.D. Rendall, Simplified models of electromagnetic and gravitational radiation damping, Classical Quantum Gravity 18, 3573–3587 (2001). [13] M. Kunze, H. Spohn, Slow motion of charges interacting through the Maxwell field, Comm. Math. Phys. 212, 437–467 (2000). [14] M. Kunze, H. Spohn, Post-Coulombian dynamics at order c−3 , J. Nonlinear Science 11, 321–396 (2001). [15] H. Lee, The classical limit of the relativistic Vlasov-Maxwell system in two space dimensions, Math. Methods Appl. Sci. 27, 249–287 (2004). at der L¨ osungen des Vlasov-Poisson-Systems par[16] A. Lindner, C k -Regularit¨ tieller Differentialgleichungen, Diplom Thesis, LMU M¨ unchen 1991. [17] P.-L. Lions, B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math. 105, 415–430 (1991). [18] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations 95, 281– 303 (1992). [19] G. Rein, Selfgravitating systems in Newtonian theory – the Vlasov-Poisson system, in Proc. Minisemester on Math. Aspects of Theories of Gravitation 1996, Banach Center Publications 41, part I, 179–194 (1997). [20] A.D. Rendall, The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system, Comm. Math. Phys. 163, 89–112 (1994). [21] J. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Comm. Math. Phys. 104, 403–421 (1986). [22] J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations 16, 1313–1335 (1991). [23] H. Spohn, Dynamics of Charged Particles and their Radiation Field, Cambridge University Press, Cambridge 2004. Sebastian Bauer1 and Markus Kunze Universit¨ at Essen, FB 6 – Mathematik D-45117 Essen Germany email: [email protected] email: [email protected] Communicated by Rafael D. Benguria submitted 07/01/04, accepted 30/07/04 1 Partially

supported by DFG priority research program SPP 1095

Ann. Henri Poincaré 6 (2005) 309 – 326 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020309-18 DOI 10.1007/s00023-005-0208-x


Absence of Continuous Spectral Types for Certain Non-Stationary Random Schr¨ odinger Operators Anne Boutet de Monvel, Peter Stollmann and G¨ unter Stolz In Memory of Robert M. Kauffman Abstract. We consider continuum random Schr¨ odinger operators of the type Hω = −∆ + V0 + Vω with a deterministic background potential V0 . We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of −∆ + V0 . The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (“random tube”) in arbitrary dimension.

1 Introduction In this article we are concerned with spectral properties of certain non-stationary random operators. More specifically, we consider Schrödinger operators of the form Hω = −∆ + V0 + Vω in L2 (Rd ). Here V0 is a deterministic background potential and Vω an Anderson-type random potential which is either sparse near infinity, or concentrated near a lower dimensional surface, or both. This type of models has attracted considerable interest as it allows to study a transition from pure point to continuous spectrum. Here, we are mainly concerned with the former phenomenon. We obtain our results by essentially “deterministic” techniques from [27, 22, 28], establishing conditions on Vω such that Hω has no absolutely continuous spectrum or no continuous spectrum outside the spectrum of −∆ + V0 . This gives us considerable flexibility in the choice of our model. In particular, we are able to avoid some of the typical technical restrictions that come with the usual multiscale analysis or fractional moments proofs of localization. E.g., we can allow for perturbations of changing sign and single site distributions without any continuity. On the other hand, we need decaying randomness in the sense that near infinity the random perturbation is not too effective. That excludes identically distributed random parameters in most cases. An important exception is our result on 1-D “surfaces” (rather tubes) in arbitrary dimensions, see Theorem 4.1 below. The paper is organized in the following way: In Section 2 we present the deterministic techniques we use, recalling the relevant notions and results from [27, 22, 28]; in fact we will need results that are a little stronger than what is explicitly stated in the above cited articles. The common flavor of these methods is that they provide comparison criteria for the absence of continuous and absolutely

310

A. Boutet de Monvel, P. Stollmann and G. Stolz


continuous spectra, respectively. These criteria are formulated in the following way: We consider Schr¨ odinger operators with two potentials that differ only on a set that is “small near infinity in a certain geometrical sense”. Then the spectrum of the first operator has no absolutely continuous component on the resolvent set of the second one. To exclude continuous spectrum one needs a bit more complicated assumptions involving randomization. In Sections 3 and 4 we state and prove our main new results, Theorems 3.1, 4.1 and 4.3. In Section 3 we are dealing with sparse random potentials. The framework we introduce is fairly general and includes as special cases the sparse random models considered in [7], e.g., random scatterers are distributed quite arbitrarily in space and the single site perturbations are assumed to be picked with probabilities that tend to zero near infinity. Then, throughout the resolvent set of the unperturbed operator there is no absolutely continuous spectrum (3.1(a)). Since we can treat quite general unperturbed operators, this includes cases with gaps in the spectrum of the unperturbed operator, a case that is completely new. In the proof we combine elementary combinatorial arguments, Lemma 3.2, with the methods discussed above. In the same fashion, under a bit more incisive conditions concerning the background and at least one random scatterer but with the same condition concerning the decay of probabilities near infinity, we can even deduce absence of continuous spectrum outside the spectrum of the unperturbed operator (3.1(b)). That is, all the new spectrum generated by the random perturbation is pure point. This is quite different from what one can obtain with the usual localization proofs, which require a large disorder condition, or apply to energies near the spectral boundaries of the perturbed operator only (with the exception of the one-dimensional case). In Section 4 we study surface-like structures. This means we consider potentials that are concentrated near a subset of lower dimension. Our strongest result, Theorem 4.1, concerns what we call quasi-1D surfaces. There is quite some literature on surface potentials. Most are dealing with the discrete case [4, 5, 8, 9, 11, 10, 13, 14] while in [3, 7] and the present paper continuum models are treated. Here again, our goal was to be able to exclude absolutely continuous spectrum on all of the unperturbed resolvent set and not just near band edges. Theorem 4.3 deals with absence of absolutely continuous and continuous spectrum, respectively, for m-dimensional surface potentials in Rd under an additional sparseness assumption. In the last section we conclude with a discussion of some possible extensions of our results and a comparison with other works, in particular the results in [10] and [7].

Vol. 6, 2005

Absence of Continuous Spectrum for Certain Random Operators

311

2 Comparison criteria for absence of (absolutely) continuous spectrum In this section we present our methods of proof, essentially taken from [27, 22, 28]. These methods rely on comparison of the spectral properties of Schr¨ odinger operators H1 = −∆ + V1 and H2 = −∆ + V2 whose “difference” is “small” in the sense that the set {V1 = V2 } := {x ∈ Rd | V1 (x) = V2 (x)} is sufficiently sparse. To this end we introduce the following concept, following [27]: Definition. A sequence (Sn )n∈N of compact subsets of Rd with Lebesgue measure |Sn | = 0 (n ∈ N) is called a total decomposition if there exists a family (Ui )i∈I of disjoint, open, bounded sets such that Rd \ Sn = Ui . n∈N

i∈I

A typical example would be Sn = ∂B(0, n), where B(x, r) denotes the closed ball of radius r, centered at x. (Let us stress that the Sn ’s need not be pairwise disjoint.) The sparseness of {V1 = V2 } will be expressed by the existence of a total decomposition (Sn )n∈N with sufficient distance of Sn to {V1 = V2 } compared with the size of Sn . An appropriate notion of size is given by the generalized surface area of a set, a notion introduced in [22] in the following way; here S ⊂ Rd is compact: |{x ∈ Rd | r ≤ dist(x, S) ≤ r + 1}| . σ(S) := sup rd + 1 r≥0 It is easily seen that σ(S) ≤ C ((diam S)d + 1),

(2.1)

i.e., σ(S) is at worst a volume, while for sufficiently regular surfaces it is a surface area measure, for example σ(∂B(x, r)) ≤ C(rd−1 + 1). We cite the following result, essentially taken from [27]: Theorem 2.1 Assume that for each γ > 0 there exists a total decomposition (γ) (Sn )n∈N = (Sn )n∈N such that δn = δn(γ) := dist({V1 = V2 }, Sn ) → ∞ as n → ∞ and

σ(Sn )e−γδn < ∞.

n

Then σac (H1 ) ∩ (H2 ) = ∅.

(2.2) (2.3)

312



The following figure is to help visualizing the geometry one is confronted with in the Theorem.

Sn

δn

Sn−1

Figure 1. {V1 = V2 } must not intersect the shaded region. Here, and in what follows, all potentials V are assumed to be locally uniformly in Lp , where p ≥ 2 if d ≤ 3 and p > d/2 if d > 3, i.e., V pp,unif := sup |V (y)|p dy < ∞. (2.4) x

B(x,1)

Theorem 2.1 is essentially Theorem 4.2 from [27]. We will need the slightly stronger version provided above in which the decomposition Sn may vary with γ. The proof provided in [27] goes through under this weaker assumption. This is roughly seen as follows: It suffices to show that σac (H1 ) ∩ J = ∅

(2.5)

for all compact subsets J of (H2 ). For fixed J the argument in [27] provides a γ > 0 (roughly the exponential decay rate in a Combes-Thomas type bound on the resolvent of H2 for energies in J) such that the validity of (2.2) and (2.3) for a suitable decomposition will imply (2.5). Also, in [27] all potentials are assumed to have locally integrable positive parts and negative parts in the Kato class. Our Lp -type assumptions are a special case. The second result we use is taken from [28] and excludes continuous spectrum. It is clear that a statement of the form of Theorem 2.1 above has to be false, since dense pure point spectrum is extremely unstable and can be destroyed by “tiny” perturbations [26]. The geometry is somewhat similar to what we had above but more restrictive. Namely, consider an increasing sequence (An )n∈N of bounded open sets with n An = Rd . Then Sn := ∂An is a total decomposition. For the arguments in [28] it is not necessary that |∂Sn | = 0, but this will be the case in all our applications.

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313

We assume that δn := min{dist(Sn , {V1 = V2 }), 12 dist(Sn , Sn−1 ∪ Sn+1 )} > 0. d+1

Theorem 2.2 Assume that V1 ∈ Lloc2 (Rd ), W ∈ L∞ with compact support, of fixed sign and such that |W | ≥ cχB(0,s) for suitable c > 0 and s > 0. Moreover, assume that for every γ > 0 there exist An = An (γ) as above such that δn = δn (γ) → ∞ and |An+1 \ An−1 | e−γδn < ∞. (2.6) n

Then for the family Hλ := H1 + λW , λ ∈ R there exists a measurable subset M0 ⊂ R such that |R \ M0 | = 0 and σc (Hλ ) ∩ (H2 ) = ∅ for all λ ∈ M0 . See [28] for the proof which extends to the case of W as specified above. Again, as with Theorem 2.1 above, the possible γ-dependence of the sets An is not explicitly stated in [28], but allowed for by the proof provided there. The requirement that the summability conditions (2.3), (2.6) have to hold for all γ > 0 (and suitable decompositions) comes from the fact that we want to exclude (absolutely) continuous spectrum up to the edges of σ(H2 ). It is possible to quantify and refine the results in a way which says that validity of (2.3), (2.6) for a fixed γ implies absence of (absolutely) continuous spectrum in regions above a certain (γ-dependent) distance from σ(H2 ).

3 Sparse random models In this section we will show how to use the methods from the preceding section to prove absence of continuous or absolutely continuous spectrum for sparse random potentials. As mentioned in the introduction, these models have been set up to study situations in which a transition from singular to absolutely continuous spectrum occurs. This has attracted some interest in the last decade as can be seen in the articles [15, 16, 19, 20, 21, 23, 24] dealing with discrete Schr¨ odinger operators and [7] for the continuum case. We will be concerned mainly with absence of a continuous spectral component away from the spectrum of the unperturbed operator. For this reason we state our results in a generality that does include cases in which no absolutely continuous spectrum survives. As model examples, let us mention two families of models that have been treated in [7].

314



Specific models of sparse random potentials, as considered in [7], are Model I Vω (x) =

ξi (ω)f (x − i),

ω∈Ω

i∈Zd

where f is a compactly supported single site potential and the ξi are independent Bernoulli variables. Set pi := P(ξi = 1). If pi → 0 as |i| → ∞ the random potential will no longer be stationary. In fact, it will be sparse in the sense that almost surely large islands near ∞ will occur where Vω vanishes. For the second model f and the ξi , pi will have the same meaning and, additionally, the qi are i.i.d. nonnegative random variables. Model II Vω (x) =

qi (ω)ξi (ω)f (x − i).

i∈Zd

Again, Vω is sparse in the above sense. Of course, for pi ≡ 1 we would get the usual Anderson model. Hundertmark and Kirsch study in [7] the metal insulator transition for H(ω) = −∆ + Vω in L2 (Rd ) for the case that pi → 0 as |i| → ∞ but not too fast in order to make sure that σess (H(ω)) ∩ (−∞, 0) = ∅. Our Model In the following we consider: (A1 ) V0 : Rd → R which is locally uniformly Lp with p ≥ 2 if d ≤ 3 and p > d/2 if d > 3. (A2 ) Σ ⊂ Rd a set of sites that is uniformly discrete in the sense that inf{|j − i| | j, i ∈ Σ, j = i} =: rΣ > 0. (A3 ) For each i ∈ Σ a single site potential fi ∈ Lp such that, for finite constants ρ and M , supp fi ⊂ B(0, ρ) and fi p ≤ M. (A4 ) Vω (x) =

i∈Σ

ωi fi (x − i)

where ω = (ωi )i∈Σ ∈ (Ω, P) = (RΣ , i∈Σ µi ), i.e., the ωi are independent random variables with distribution µi , and supp µi ⊂ [0, 1] for all i ∈ Σ.

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For our results on absence of continuous spectrum, in order to apply Theorem 2.2, we will also require (d+1)/2

(A5 ) Let V0 , fi ∈ Lloc (Rd ) for all i ∈ Σ. There exists one k ∈ Σ with fk of definite sign, bounded, and such that |fk | ≥ cχB(0,s) for some c > 0 and s > 0. For further reference denote pi (ε) := µi ([ε, 1]) = P{ωi ≥ ε}.

(3.1)

mk := (µk )ac ([0, 1])

(3.2)

Also, denote by the total mass of the absolutely continuous component (µk )ac of µk . We will only use this for the fixed k ∈ Σ given in (A5 ). We consider the self-adjoint random Schr¨ odinger operator H(ω) = H0 + Vω in L2 (Rd )

(3.3)

where H0 = −∆ + V0 . Our assumptions guarantee that the local Lp -bounds (2.4) for V0 + Vω are uniform not only in x, but also in ω. Of course, our model contains Models I and II above as special cases and pi (ε) ≤ pi for any ε > 0 in these cases. We have the following result: Theorem 3.1 Let H(ω) be as above, satisfying (A1 ) to (A4 ), and assume that for all ε > 0, (3.4) pi (ε) = o(|i|−(d−1) ) as |i| → ∞. Then (a) σac (H(ω)) ∩ (H0 ) = ∅ almost surely. (b) Assume, moreover, that (A5 ) holds. Then, with k as in (A5 ), P{σc (H(ω)) ∩ (H0 ) = ∅} ≥ mk .

(3.5)

In particular, σc (H(ω)) ∩ (H0 ) = ∅ holds almost surely if µk is purely absolutely continuous, without any assumption on the distribution at the other sites. In order to apply the results from Section 2 we need to find sufficiently many and sufficiently large regions in which the random potential Vω is small and thus Hω close to H0 . We start by showing that these regions appear with probability one. Definition. Call a set U ε-free for ω if ωi ≤ ε for all i ∈ Σ ∩ U . Denote by Ar,R = B(0, R) \ B(0, r) the annulus with inner radius r and outer radius R.

(3.6)

316



Lemma 3.2 Fix ε > 0 and a > 1. For n ∈ N let an := P Ar,r+n is not ε-free for all r ∈ [an , an+1 − n] . Then

n

(3.7)

an < ∞.

Proof. Choose η > 0 such that a(1 − η) > 1. Using uniform discreteness of Σ we get that for all n ∈ N and r ≥ 1, #(Ar,r+n ∩ Σ) ≤ Cnrd−1 ,

(3.8)

where C depends on d and rΣ . Here # A is the cardinality of a set A. With C from (3.8) choose δ ∈ (0, η/(Cad−1 )). By (3.4), pi (ε) ≤ δ|i|−(d−1) for i sufficiently large. Thus, for sufficiently large n and each r ∈ [an , an+1 − n],

P(Ar,r+n is ε-free) =

(1 − pi (ε))

i∈Ar,r+n ∩Σ

≥ (1 − δ|i|−(d−1) )#(Ar,r+n ∩Σ) ≥ (1 − δa−n(d−1) )Cna

(n+1)(d−1)

≥ (1 − Cδad−1 )n ≥ (1 − η)n .

(3.9)

Aan ,an+1 contains at least n1 (an+1 − an ) − 1 disjoint annuli Aj := Arj ,rj +n of width n. Thus, using independence and (3.9), an ≤ P(no Aj is ε-free) = P(Aj is not ε-free) j −1

≤ (1 − (1 − η)n )n ≤ e−(1−η)

n

(an+1 −an )−1

(n−1 an (a−1)−1)

.

As (1 − η)a > 1, the an are summable. By the Borel-Cantelli lemma we conclude P(Ωε,a ) = 1, where

Ωε,a := ω ∈ Σ : For each sufficiently large n the annulus Aan ,an+1 contains a sub-annulus Arn ,rn +n which is ε-free for ω . Therefore Ωε =

∈N

also has full measure.

Ωε,1+1/

(3.10)

(3.11)

(3.12)

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Based on this we can now complete the Proof of Theorem 3.1. Fix a compact K ⊂ (H0 ). Since (H0 ) can be exhausted by an increasing sequence of compact subsets, it suffices to prove that σac (H(ω)) ∩ K = ∅ almost surely.

(3.13)

It can be shown, using the general theory of uniformly local Lp potentials, e.g., [25], that there is an ε > 0 such that σ(H0 + V ) ∩ K = ∅,

(3.14)

for each V with V p,unif ≤ ε . Thus, by the properties of Σ and fi , there is an ε > 0 such that

δi fi (x − i) ∩ K = ∅ (3.15) σ H0 + i∈Σ

if |δi | ≤ ε for all i ∈ Σ. Fix this ε > 0 and let Ωε be the full measure set found above. For given ˜ i := min{ωi , ε}, i ∈ Σ, and ω ∈ Ωε let ω V2 (x) := ω ˜ i fi (x − i). i∈Σ

By (3.15) we have σ(H0 + V2 ) ∩ K = ∅. Thus, in order to apply Theorem 2.1 and (γ) conclude (3.13), it suffices to find for every γ > 0 a total decomposition (Sn ) of {Vω = V2 } which satisfies (2.2) and (2.3). For given γ > 0 choose an integer > 2(d − 1)/γ. This implies (d − 1) log a < γ/2, where a := 1 + 1/ . As ω ∈ Ωε,a , for each sufficiently large n the annulus Aan ,an+1 contains an ε-free annulus Arn ,rn +n . (γ) Choose Sn := ∂B(0, rn + n2 ). Then δn(γ) := dist({Vω = V2 }, Sn(γ) ) ≥

n −ρ 2 (γ)

since Arn ,rn +n is ε-free (recall that supp fk ⊂ B(0, ρ)). Thus δn → ∞. Also using (γ) that σ(Sn ) ≤ Can(d−1) , we conclude (γ) σ(Sn(γ) )e−γδn ≤ Ceγρ en((d−1) log a−γ/2) < ∞. n

n

This proves part (a) of Theorem 3.1. In order to apply Theorem 2.2 to prove part (b) we slightly modify the above construction, essentially replacing Σ by Σ \ {k}. Let Ω := RΣ\{k} with measure P = ⊗i∈Σ\{k} µi . As the property defining Ωε,a in (3.11) does not depend on the value of ωk , we get that also P (Ωε,a ) = P (Ωε ) = 1, where Ωε,a and Ωε are defined as in (3.11) and (3.12), but as subsets of Ω .

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For compact K ⊂ (H0 ) choose ε > 0 as in the proof For ω ∈ Ωε of part (a). let := min{ωi , ε} (i ∈ Σ \ {k}). Also let Vω (x) = i∈Σ\{k} ωi fi (x − i) and ˜ i fi (x − i). As before, σ(H0 + V2 ) ∩ K = ∅. V2 (x) = i∈Σ\{k} ω For γ > 0 choose > 2d/γ, a = 1 + 1/ . With rn from (3.11), let An = B(0, rn + n2 ) and Sn = ∂An . This yields ω ˜ i

|An+1 \ An−1 | ≤ cd a(n+2)d and

1 n δn = min dist(Sn , {Vω = V2 }), dist(Sn , Sn−1 ∪ Sn+1 ) ≥ − ρ. 2 2 The choice of a guarantees that n |An+1 \An−1 | e−γδn < ∞. By Theorem 2.2 this proves the existence of a measurable subset M0,ω ⊂ R with |R \ M0,ω | = 0 and such that σc (H(λ, ω )) ∩ K ⊂ σc (H(λ, ω )) ∩ (H0 + V2 ) = ∅

for all λ ∈ M0,ω , where H(λ, ω ) = H0 + λfk (x − k) + Vω (x). As µk (M0,ω ) ≥ (µk )ac (M0,ω ) = (µk )ac (R) = mk it follows by Fubini that P{ω ∈ Ω : σc (H(ω)) ∩ K = ∅} ≥ mk . Since this bound is independent of K and we can exhaust (H0 ) by an increasing sequence Kn we arrive at the assertion. This completes the proof of Theorem 3.1. Remarks. (1) While the “volume” term |An+1 \ An−1 | in (2.6) has to be considered larger than the “surface” term σ(Sn ) in (2.3), this did not make a significant difference in the above proof. The same total decomposition Sn can be used to prove absence of absolutely continuous spectrum and absence of continuous spectrum. The difference will become more significant for the quasi-1D surfaces considered in the next section. (2) Crucial for our method to apply is the almost sure appearance of a sequence of ε-free annular regions which must (i) grow in thickness and (ii) not be too far apart, as found in Lemma 3.2. In Theorem 3.1 this was enforced through the assumptions on the distribution of the coupling constants. In Section 4 it will follow from sparseness of the single site set Σ. (3) Note that our methods are sufficiently “soft” to allow for considerable flexibility of the model. The single site potentials fi may depend on the site, do not need to be sign definite, and may include Lp -type singularities. We can deal with quite arbitrary single site distributions. Only for the proof of absence of continuous spectrum we need one of the distributions to be absolutely continuous. These assumptions are weaker than what usually enters into the proof of localization properties through the multiscale analysis or fractional moment methods. (4) The assumption supp µj ⊂ [0, 1] is just a normalization. For our methods to apply, the random potentials have to obey uniform bounds, e.g., in the sense of · p,unif from (2.4).

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Let us finally state the following result for our model which easily follows from the “Almost surely free lunch Theorem” in [7]. For the case V0 = 0 it can be combined with Theorem 3.1 to provide examples with purely singular or pure point (while not discrete) spectrum below zero and an absolutely continuous spectral component above zero. Theorem 3.3 Let µk , fk , V0 be as above, V0 = 0 and assume that, additionally, the fk ∞ are uniformly bounded and that the second moments of the ηk obey 1 2 x2 dµk ≤ C|k|−β E(ηk ) = 0

for some β > 2. Then σac (H(ω)) ⊃ [0, ∞) P -a.s. Proof. The assumptions clearly make sure that 1

W (x) := E(Vω (x)2 ) 2 ≤ C(1 + |x|)−(1+ε) so that we can apply Theorem 2.4 from [7] to see that Cook’s criterion is applicable for P -a.e. ω ∈ Ω. For general V0 the corresponding result, namely that σac (−∆+V0 ) ⊂ σac (Hω ) almost surely, is probably false. It should be true for certain periodic potentials, see [2, 6, 29].

4 Quasi-1D surfaces In Section 3 sparseness of the potential Vω in (A4 ) resulted from an assumption on decaying randomness, e.g., (3.4). In the present section we will modify our methods and results for the case where sparseness of Vω arises directly through sparseness of the deterministic set Σ. By this we mean situations where Σ does not have positive d-dimensional density in Rd , i.e., #(Σ ∩ B(0, R)) = o(Rd ) as R → ∞. A special case would be an m-dimensional sublattice, e.g., Σ = Zm × {0} ⊂ Rm × Rd−m , 0 < m < d, in which case Vω would model a random surface potential. Our most interesting result holds for m = 1, where our methods cover the following more general situation: Definition. A uniformly discrete subset Σ of Rd is called quasi-one-dimensional (quasi-1D) if there exists C < ∞ such that #(Σ ∩ AR,R+1 ) ≤ C

(4.1)

for all R ≥ 0. Theorem 4.1 Let H(ω) = H0 + Vω satisfy (A1 ) to (A4 ). In addition, assume that Σ is quasi-1D and that sup pi (ε) < 1 (4.2) i∈Σ

for every ε > 0. Then σac (H(ω)) ∩ (H0 ) = ∅ almost surely.

320



If Σ is quasi-1D, then by Theorem 4.1, no spatial decay in the randomness of the ηi is required to conclude absence of absolutely continuous spectrum in gaps of σ(H0 ). For example, (4.2) is satisfied for independent, identically distributed random variables ηi such that 0 ∈ supp µ for their common distribution µ. In particular, as every uniformly discrete Σ ⊂ R is quasi-1D, this strengthens Theorem 3.1(a) in the case d = 1, which would require pi (ε) = o(1) as k → ∞. Of course, in the case d = 1 our result is hardly new as (essentially) much stronger results are known for one-dimensional random potentials. More interesting is the case d > 1, where special cases of quasi-1D sets include discrete tubes of the form Σ = Z×S, with S a bounded subset of Zd−1 . Theorem 4.1 shows the absence of absolute continuity in the “surface spectrum” generated by the random (1D) surface potential V (ω). Also, within certain limitations, we can allow for curvature in the tubes Σ, thus covering rather general “random sausages”. Proof. We start with a modification of Lemma 3.2. Lemma 4.2 Fix ε > 0. Let δ = supi pi (ε) < 1, C as in (4.1) and a > the an , as defined in (3.7), are summable.

1 (1−δ)C

. Then

Proof. This follows with the same argument as in the proof of Lemma 3.2, using that now P(Ar,r+n is ε-free) ≥ (1 − δ)Cn . Thus the set Ωε,a , defined as in (3.11), has full P-measure. of Theorem 3.1 to find Fix K ⊂ (H0 ) compact and argue as in the proof ˜ i fi (x − i), ω ˜i = ε > 0 such that σ(H0 + V2 ) ∩ K = ∅, where V2 (x) = i∈Σ ω min{ωi , ε}. Choose a > 1 as in Lemma 4.2 and ω ∈ Ωε,a , i.e., Aan ,an+1 contains ε-free Arn ,rn +n for all sufficiently large n. As before, the spheres Sn = ∂B(0, rn + n2 ) give a total decomposition with dist({Vω = V2 }, Sn ) ≥ n2 − ρ. But, as Lemma 4.2 prevents us from choosing a arbitrarily close to 1, this will not yield convergence of (2.3) for all γ > 0. We will therefore refine our construction by splitting the Sn in two parts. One part is a union of spherical caps for which, due to points of Σ close to Arn ,rn +n , the distance n2 − ρ from {Vω = V2 } can’t be improved. The second part (the remaining “Swiss cheese”) has much bigger distance to {Vω = V2 } and, due to the sparseness of Σ, contains most of Sn . The details of this construction are as follows: Fix α > 1. Let Pn := (Arn −nα ,rn ∪ Arn +n,rn +n+nα ) ∩ Σ α

(4.3)

be the points of Σ in the n -neighborhood of Arn ,rn +n (but outside Arn ,rn +n ). For each j ∈ Pn define the spherical cap

n j Sn,j := Sn ∩ B rn + (4.4) , nα . 2 |j| Also let Sn := Sn \

j

Sn,j .

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j ∈ Pn

Sn

Arn ,rn +n

Sn,j := Sn ∩ B

rn +

n 2

j α |j| , n

Figure 2. The geometry in the proof of Lemma 4.2: the bold face line shows a part of Sn , the shaded region is Arn ,rn +n , the point in the small circle a j ∈ Pn and j the small circle the boundary of B((rn + n2 ) |j| , nα ). Since Sn ∪

j

Sn,j = Sn , we have that {Sn,j : n ∈ N, j ∈ Pn } ∪ {Sn : n ∈ N}

(4.5)

is a total decomposition of Rd . As above, since Arn ,rn +n is ε-free, δn,j := dist({Vω = V2 }, Sn,j ) ≥

n − ρ. 2

(4.6)

If x ∈ Sn and j ∈ Σ ∩ (Arn ,rn +n )c , then, by elementary geometric considerations, dist(x, j) ≥ nα for sufficiently large n. Using this and again that Arn ,rn +n is ε-free, we find (4.7) δn := dist({Vω = V2 }, Sn ) ≥ nα − ρ. From the simple volume bound (2.1) on the generalized surface area one gets σ(Sn,j ) ≤ Cndα ,

(4.8)

σ(Sn ) ≤ Cadn .

(4.9)

Checking (2.3) for the partition (4.5) amounts to proving that σ(Sn ) e−γδn < ∞

(4.10)

n

and

n j∈Pn

σ(Sn,j ) e−γδn,j < ∞

(4.11)

322



for each γ > 0. (4.10) follows from (4.7) and (4.9) since α > 1. (4.11) follows from (4.6) and (4.8), noting that #Pn ≤ 2nα + 2 since Σ is quasi-1D. From Theorem 2.1 we conclude σac (H(ω)) ∩ K ⊂ σac (H(ω)) ∩ (H0 + V2 ) = ∅. Remark. It is possible to prove Theorem 4.1 under a slightly weaker assumption on the set Σ, namely that there exists C < ∞ such that #(Σ ∩ B(0, R)) ≤ CR

(4.12)

for all R ≥ 1. (4.12) is weaker than (4.1) in that it allows the number of points in Σ ∩ AR,R+1 to be unbounded with respect to R. (4.12) is also somewhat more natural as it doesn’t depend on the norm used to define B(0, R) nor on the choice of the center of the ball. A simple counting argument shows that, under the assumption (4.12), for each annulus of the form Aan ,an+1 most sub-annuli AR,R+n satisfy a bound #(Σ ∩ AR,R+n ) ≤ Cn. Here “most” means at least a non-vanishing fraction. One finds sufficiently many disjoint such annuli to construct ε-free regions as before. Moreover, by an additional counting argument, one argues that most of these annuli do not have more than C nα points of Σ in their nα -neighborhoods. Based on this one can construct a partition {Sn , Sn,j } as above and carry through the proof. We skip the somewhat tedious details of this generalization. We are not able to prove a result like Theorem 3.1(b), i.e., absence of continuous spectrum in (H0 ) with positive probability, under the assumptions of Theorem 4.1 (plus (A5 )). For the partition Sn = ∂An , An = B(0, rn + n2 ) the volumes |An+1 \ An−1 | grow too fast to get validity of (2.6) for all γ > 0. A trick like the introduction of {Sn , Sn,j } as above is not applicable here since in Theorem 2.2 the Sn need to arise as boundaries of a growing sequence An . However, if one replaces (4.2) by pi (ε) = o(1) as |i| → ∞ for all ε > 0, then Lemma 4.2 will hold for any a > 1, which allows for an application of Theorem 2.2 with a γ-dependent choice of the Sn , as in the proof of Theorem 3.1(b). Sparseness of the random potential is achieved here through a combination of sparseness of Σ and decaying randomness pi (ε) = o(1), as opposed to Theorem 3.1, where sparseness follows exclusively from stronger decay pi (ε) = o(|i|−(d−1) ). In fact, the correlation between the degree of sparseness of Σ and the rate of decay of pi (ε) can be made more specific. For this, call a uniformly discrete set Σ ⊂ Rd quasi-m-dimensional (1 ≤ m ≤ d, not necessarily integer) if for some C < ∞ and all R ≥ 0, (4.13) #(Σ ∩ AR,R+1 ) ≤ CRm−1 . Then the following result is found with the same methods as above: Theorem 4.3 Let H(ω) satisfy (A1 ) to (A4 ), Σ be quasi-m-dimensional and, for all ε > 0, (4.14) pi (ε) = o(|i|−(m−1) ) as |i| → ∞, then σac (H(ω)) ∩ (H0 ) = ∅ almost surely. If, moreover, (A5 ) holds, then P{σc (H(ω)) ∩ (H0 ) = ∅} ≥ mk .

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5 Concluding remarks Among the known results for discrete surface models, the one most closely related to Theorem 4.1 above is the result of Jakˇsić and Molchanov [10]. They consider the discrete Laplacian on Z × Z+ with random boundary condition ψ(n, −1) = Vω (n)ψ(n, 0), where the Vω (n) are i.i.d. random variables. They show that the spectrum outside [−4, 4], i.e., outside the spectrum of the two-dimensional discrete Laplacian, is almost surely pure point. This is stronger than our continuum analogue in the sense that we can only prove absence of absolute continuity outside the spectrum of the deterministic background operator H0 . The proof in [10] requires a technical tour de force. The two-dimensional problem can be reduced to a one-dimensional problem with long range interactions. Anderson localization for the latter has been proven in [12] with methods based on an approach developed in [17] (which is also behind Theorem 2.2 above). The onedimensional problem depends nonlinearly on the spectral parameter, a difficulty which is resolved by adapting some ideas from the Aizenman-Molchanov fractional moment method [1]. Our methods are comparatively soft. In particular, they work directly in the multi-dimensional PDE setting and do not require a reduction to d = 1. One-dimensionality of the random surface only enters through its probabilistic consequences (Lemma 4.2) for the frequency of the appearance of ε-free regions, which constitute the “potential barriers” required in Theorem 2.1. This makes our methods very flexible. In addition to the extension to continuum models, they allow for rather general quasi-1D surfaces (e.g., curved tubes, unions of tubes), work in arbitrary dimension d and allow for the presence of an additional deterministic background potential V0 . It is possible to adapt our methods to lattice operators and prove absence of absolutely continuous spectrum outside the spectrum of the discrete Laplacian for much more general geometries than the half-plane considered in [10]. Also, our methods can easily be adjusted to work for operators of the type (3.3) on L2 (Ω), Ω = Rd . For example, for H(ω) = −∆ + Vω in L2 ((0, a) × Rd−1 ) with Dirichlet boundary conditions and Vω given through (A2 ) to (A4 ) with i.i.d. coupling constants ωi , we would get that σac (H(ω)) ∩ (−∞, 0) = ∅ almost surely. Of course, for this physically one-dimensional operator (with no bulk space), one would expect the much stronger result that σc (H(ω)) = ∅. But the corresponding result for discrete strips, e.g., [18], does not seem to extend easily to the continuum. Finally, we mention that Hundertmark and Kirsch [7] announce some results on pure point spectrum for continuum models similar to the ones studied here. They will use suitable adaptations of multiscale analysis to show that the negative spectrum of −∆ + Vω is almost surely pure point. Here Vω is either of the type of Model II above or a random potential at the surface of a half space Schr¨ odinger operator. In situations where the multiscale analysis can be carried out, their results should be stronger than ours.

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Acknowledgment Our collaboration has been supported by the University Paris 7 Denis Diderot where part of this work was done, by the DFG in the priority program “Interacting stochastic systems of high complexity” and through the SFB 393, as well as through US-NSF grant no. DMS-0245210.

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[12] V. Jakˇsić and S. Molchanov, Localization for one-dimensional long range random hamiltonians, Rev. Math. Phys. 11, 103–135 (1999). [13] V. Jakˇsić and S. Molchanov, Localization of surface spectra, Commun. Math. Phys. 208, no. 1, 153–172 (1999). [14] V. Jakˇsić, S. Molchanov and L. Pastur, On the propagation properties of surface waves, in “Wave propagation in complex media (Minneapolis, MN, 1994),” IMA Math. Appl., Vol. 96, Springer, New York, 1998, pp. 143–154. [15] W. Kirsch, Scattering theory for sparse random potentials, Random Oper. Stochastic Equations 10, no. 4, 329–334 (2002). [16] W. Kirsch, M. Krishna and J. Obermeit, Anderson model with decaying randomness: Mobility edge, Math. Z., 235, 421–433 (2000). [17] W. Kirsch, S. Molchanov and L. Pastur, One-dimensional Schr¨ odinger operators with high potential barriers, Operator Theory, Adv. Appl. 57, 163–170 (1992). [18] A. Klein, J. Lacroix and A. Speis, Localization for the Anderson model on a strip with singular potentials, J. Funct. Anal. 94, no. 1, 135–155 (1990). [19] M. Krishna, Anderson model with decaying randomness: Existence of extended states, Proc. Indian Acad. Sci. Math. Sci. 100, no. 3, 285–294 (1990). [20] M. Krishna, Absolutely continuous spectrum for sparse potentials, Proc. Indian Acad. Sci. Math. Sci. 103, no. 3, 333–339 (1993). [21] M. Krishna and K.B. Sinha, Spectra of Anderson type models with decaying randomness, Proc. Indian Acad. Sci. Math. Sci. 111, no. 2, 179–201 (2001). [22] I. McGillivray, P. Stollmann and G. Stolz, Absence of absolutely continuous spectra for multidimensional Schr¨ odinger operators with high barriers, Bull. London Math. Soc. 27, no. 2, 162–168 (1995). [23] S. Molchanov, Multiscattering on sparse bumps, In “Advances in differential equations and mathematical physics” (Atlanta, GA, 1997), Contemp. Math. 217, Amer. Math. Soc., Providence, RI, 1998, pp. 157–181. [24] S. Molchanov and B. Vainberg, Multiscattering by sparse scatterers, In “Mathematical and numerical aspects of wave propagation” (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, 2000, pp. 518–522. [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York, 1978. [26] B. Simon, Spectral analysis of rank one perturbations and applications, CRM Proc. Lecture Notes, 8, 109–149 (1995).

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[27] P. Stollmann and G. Stolz, Singular spectrum for multidimensional operators with potential barriers, J. Operator Theory 32, 91–109 (1994). [28] G. Stolz, Localization for Schr¨ odinger operators with effective barriers, J. Funct. Anal. 146, no. 2, 416–429 (1997). [29] D. Yafaev, Eigenfunctions of the continuous spectrum for the N -particle Schr¨ odinger operator, In “Spectral and scattering theory” (Sanda, 1992), 259–286, Lecture Notes in Pure and Appl. Math., 161, Dekker, New York, 1994. Anne Boutet de Monvel Institut de Mathématiques de Jussieu Université Paris 7 2, place Jussieu, case 7012 F-75251 Paris France email: [email protected] Peter Stollmann Fakult¨ at f¨ ur Mathematik Technische Universit¨ at D-09107 Chemnitz Germany email: [email protected] G¨ unter Stolz Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294 USA email: [email protected] Communicated by Jens Marklof submitted 07/04/04, accepted 19/08/04




Spectrum of the Magnetic Schr¨ odinger Operator in a Waveguide with Combined Boundary Conditions Denis Borisov, Tomas Ekholm and Hynek Kovaˇr´ık Abstract. We consider the magnetic Schr¨ odinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann1 . We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.

1 Introduction The existence of bound states of the Laplace operator in the strip with Dirichlet boundary conditions and Neumann window was proven in [1] and independently also in [2]. The so called Neumann window is represented by the segment of the length 2l of the boundary, on which the Dirichlet condition is changed to Neumann. A discrete spectrum of the Laplace operator with Neumann window appears for any nonzero length of the Neumann segment. In particular, for small values of l the eigenvalue emerges from the continuous spectrum proportionally to l4 . The asymptotical estimate for small l was established in [3]. The asymptotics expansion of the emerging eigenvalue for small l was constructed formally in [4], while the rigorous results were obtained in [5]. On the other hand, the results on the discrete spectrum of a magnetic Schr¨ odinger operator in waveguide-type domains are scarce. A planar quantum waveguide with constant magnetic field and a potential well is studied in [6], where it was proved that if the potential well is purely attractive, then at least one bound state will appear for any value of the magnetic field. Stability of the bottom of the spectrum of a magnetic Schrödinger operator was also studied in [7, Sec. 9] In this work we consider the system, where the discrete spectrum in the absence of magnetic field appears due to the perturbation of the boundary of the domain rather than due to the additional potential well. We also assume that the magnetic field is localized in the sense to be specified below. This assumption rules out the case of a constant field. As it has been recently shown in [8] the presence of a suitable magnetic field can prevent the existence of bound states in the Dirichlet strip with a sufficiently small “bump”. Changing the boundary 1 For

the definition of magnetic Neumann boundary conditions see Section 2, Eq. (2.2)

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D. Borisov, T. Ekholm and H. Kovaˇr´ık


conditions to Neumann is however a stronger perturbation in the sense that the existence of a bound state in a waveguide with the bump added to a certain segment of the boundary implies the existence of a bound state in a waveguide with Neumann conditions on the same segment, see [1, Cor. 1.3]. Therefore we cannot mimic the arguments of [8] in the case of the waveguide with Neumann window and a different approach is needed. The main technical tool used in [8] is a modified version of the Hardy inequality for the magnetic Dirichlet quadratic form in the two-dimensional strip. In the present paper we establish a similar inequality in order to prove the absence of a discrete spectrum of the magnetic Schr¨ odinger operator in the straight strip with Neumann window. More exactly speaking, we give sufficient conditions on the magnetic field and the length of the window, under which the discrete spectrum is empty. The above mentioned version of Hardy inequality enables us to reduce the problem to the study of a one-dimensional Laplacian with a purely attractive potential well of a width 2l and a small but fixed positive potential, see Section 4.2 for the details. We then show that for l small enough such a system has no bound state. The main profit of our method is that it gives us an explicit estimate on the critical length of the window, depending on the magnetic field, which guarantees the absence of discrete spectrum. It is of course natural to ask whether a sufficiently large Neumann window will lead to the existence of eigenvalues also in the presence of the magnetic field. In the case of a smooth and compactly supported field we give an answer to this question using a minimax-like argument. The article is organized as follows. In Section 2 we define the mathematical objects that we work with and describe the problem. We also give the statements of the main results separately for the case of a compactly supported bounded magnetic field and for the Aharonov-Bohm field. In Section 3 we show that the essential spectrum of the Dirichlet Laplacian is not affected by the magnetic field, neither by the presence of a Neumann window. Sufficient conditions for the absence of the discrete spectrum are proved in Section 4. Finally, the question of presence of eigenvalues is discussed in Section 5.

2 Statement of the problem and the main results Let x = (x1 , x2 ) be Cartesian coordinates, Ω be the strip {x : 0 < x2 < π}, and γ be the interval {x : |x1 | < l, x2 = 0}. The rest of the boundary will be indicated by Γ, i.e., Γ = ∂Ω \ γ. We denote by B = B(x) a real-valued magnetic field and assume that A is a magnetic vector potential associated with B, i.e., A = A(x) = (a1 (x), a2 (x)) and B = curl A = ∂x1 a2 − ∂x2 a1 . In what follows we will consider two main cases of magnetic fields B. The first case is a smooth compactly supported field. Hereinafter by this we denote the field B belonging to C 1 (Ω) and vanishing in the neighborhood of infinity. The second one is the

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329

Aharonov-Bohm field originated by the potential with components a1 (x) = −

Φ · (x2 − p2 ) , (x1 − p1 )2 + (x2 − p2 )2

a2 (x) =

Φ · (x1 − p1 ) , (x1 − p1 )2 + (x2 − p2 )2

(2.1)

where Φ is a constant and 2πΦ is the flux through the point p = (p1 , p2 ) which is assumed to be inside the strip Ω. We denote by M0 the operator 2

2

(−i∂x1 + a1 ) + (−i∂x2 + a2 )

on the domain D(M0 ) consisting of all functions u ∈ C ∞ (Ω) vanishing in a neighborhood of Γ and in a neighborhood of infinity and satisfying the boundary condition (2.2) (−i∂x2 + a2 )u(x) = 0 on γ. We will call it magnetic Neumann boundary condition. In the case of AharonovBohm field, the functions u ∈ D(M0 ) are assumed to vanish in a neighborhood of the point p. Clearly, the operator M0 is non-negative and symmetric in L2 (Ω) and therefore it can be extended to a self-adjoint non-negative operator by the method of Friedrich. In what follows we will denote this extension by M . The main object of our interest is the spectrum of the operator M . In order to formulate the main results we need to introduce some auxiliary notations. By Ω(α, β) we will indicate the subset of Ω given by {x ∈ Ω : α < x1 < β} and Ω± will be the subsets {x ∈ Ω : x1 > l}, {x ∈ Ω : x1 < −l}, respectively. The symbol Br (q) denotes a ball of radius r centered at a point q in R2 . The flux of the field through the ball Br (q) is given by 1 Φq (r) = B(x) dx. 2π Br (q) Below we give the summary of the main results of the article. Theorem 2.1. The essential spectrum of the operator M coincides with [1, +∞). Theorem 2.2. Assume that the field B is smooth and compactly supported and (1) There exist two balls BR− (p− ) ⊂ Ω− , BR+ (p+ ) ⊂ Ω+ so that at least one of the fluxes Φp± (r) is not identically zero for r ∈ [0, R± ]; (2) The inequality 1 (2.3) (κ− + κ+ ) l≤ 12 holds true, where π κ± := min πc± , , (2.4) 4 ln 2 + π|p± 1 | c± are defined in Lemma 4.1. Then the operator M has empty discrete spectrum.

330



Theorem 2.3. Assume that the field B is the Aharonov-Bohm one with the potential given by (2.1) and (1) The point p is (p1 , p2 ), where p1 < −l; (2) The inequality l< holds true, where

κ := min πc,

κ 6

π 4 ln 2 + π|p1 |

(2.5) ,

(2.6)

c is defined in Lemma 4.2. Then the operator M has empty discrete spectrum. The next theorem provides a condition, that guarantees the existence of discrete eigenvalues in the case of a smooth and compactly supported field. Theorem 2.4. Let the field B be smooth and compactly supported, λ = λ(l) be the lowest eigenvalue of the Laplacian −∆N ,D in the strip Ω subject to the Dirichlet condition on Γ and Neumann condition on γ. Assume that the inequality λ(l) + inf max |A(x)|2 < 1 A

(2.7)

Ω

holds, where infimum is taken over all potentials associated with the field B. Then the operator M has non-empty discrete spectrum. Remark 2.5. In the case of a smooth compactly supported field B we did not define the magnetic potential uniquely. In fact, this is not needed, since the spectrum of the operator M is invariant under the gauge transformation A → A + ∇ϕ, where ϕ is a real-valued function. We will employ this property in section 5 to show that under the hypothesis of this theorem the potential A can be chosen such that |A| is bounded and of compact support. This will imply that the quantity inf max |A(x)|2 A

Ω

in (2.7) is finite. Remark 2.6. The constants κ± and κ in Theorems 2.2 and 2.3 giving the estimates for window length depend on the magnetic field. The constants c± and c in (2.4) and (2.6) are determined by the rational part of the flux and the distance from the support of the field to the boundary (see (4.3) and (4.16)). The important role of the fractional part of the flux is the usual property of the system with magnetic field (see, for instance, [7, Sec. 10], [9, Sec. 6.4]); this is a case in our work too. The distance between the magnetic field and the window is taken into account by π in (2.4) and by the similar term in (2.6). the presence of the terms 4 ln 2+π|p ± | 1

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Throughout the article we will often make use of some notations and it is convenient to introduce them now. The spectrum of an operator T will be indicated by σ(T ) while the essential spectrum will be denoted by σess (T ). We will employ the symbol qT = qT [·, ·] for the sesquilinear form associated with a self-adjoint operator T and D(qT ) will be the domain of the quadratic form produced by the sesquilinear form qT . The Hilbert space we will work in is L2 (Ω); we preserve the notation (·, ·) and · for the inner product and norm in this space. In all other cases the notations of the inner product and norm in a Hilbert space H will be equipped by a subscript H.

3 Proof of Theorem 2.1 To prove the theorem we will need some auxiliary notations and statements. Let H be a Hilbert space and S be a positive definite operator in H whose domain is dense in H. By S1 we indicate the Friedrich’s extension of the operator S and by S2 another self-adjoint positive definite extension of S. By definition, D(qS2 ) is a Hilbert space endowed with the inner product and the norm originated by the quadratic form qS2 . Since S1 is the Friedrich’s extension of S it follows that D(qS1 ) is a subspace of D(qS2 ). Let Q be the orthogonal complement D(qS1 )⊥ in D(qS2 ) in the inner product qS2 [·, ·]. The proof of the theorem is based on the following lemma proven in [10, Lemma 3.1]. Lemma 3.1. If each bounded subset of Q (in the norm · D(qS2 ) ) is compact in H, then the operator T := S2−1 − S1−1 is compact in H. In our case L2 (Ω) plays the role of H and S := (−i∇ + A)2 + 1 with D(S) := C0∞ (Ω). The Friedrich extension S1 of S is in fact the extension of (−i∇ + A)2 + 1 subject to Dirichlet boundary condition. We know from [8] that σess (S1 ) = [2, +∞). We set S2 := M + 1; we naturally can treat M + 1 as an extension of S. If we prove that T := S2−1 − S1−1 is compact, then the essential spectra of the operators S1 and S2 will coincide by the Weyl theorem (see for instance [11, Ch. 9, Sec. 1]). We will prove the compactness of T by Lemma 3.1. First we will establish an auxiliary lemma. By ω we indicate some bounded subdomain of Ω with infinitely differentiable boundary such that dist (γ, Ω \ ω) > 0. In the case of Aharonov-Bohm field we also assume that the point p does not belong to ω. Lemma 3.2. For each function u ∈ Q the inequality u ≤ cuL2 (ω) , holds true, where the constant c is independent on u. Proof. In the proof of the lemma we follow the ideas of the proof of Lemma 3.3 in [10]. The domains D(qS1 ) and D(qS2 ) are completions of C0∞ (Ω) and D(M0 ),

332



respectively, in the norm (−i∇ + A) · 2 + · 2 . In the case of compactly supported field we can choose the vector potential A being from C 1 (Ω) which will make this potential bounded on ω. In the case of Aharonov-Bohm field the potential is in C 1 (ω) as well since the point p does not belong to ω by assumption. Therefore, each element v of D(S2 ) belongs to H 1 (ω) due to the inequality: v2H 1 (ω) = (−i∇ + A)v − Av2L2 (ω) + v2L2 (ω) ≤ 2 (−i∇ + A)v2L2 (ω) + Av2L2 (ω) + v2L2 (ω) ≤ C (−i∇ + A)v2L2 (ω) + v2L2 (ω) = C(S2 v, v),

(3.1)

where the constant C is independent on v. We denote by χ = χ(x) an infinitely differentiable function taking values from [0, 1] and being equal to one in some neighborhood of γ, which is a subdomain of ω, and vanishing outside ω. Since S2 ≥ 1 it follows that S2−1 u ≤ u.

(3.2)

χ)S2−1 u

Let u ∈ Q. Clearly, (1 − ∈ D(qS1 ) ∩ D(S2 ), thus S2 (1 − χ)S2−1 u, u = (1 − χ)S2−1 u, u D(qS

2)

= 0.

Using this equality we deduce

u2 = (u, u) − S2 (1 − χ)S2−1 u, u = (S2 χS2−1 u, u).

(3.3)

Since

S2 χS2−1 u = χu − 2 ∇(S2−1 u), ∇χ R2 − (S2−1 u)∆χ − 2 i (A, ∇χ)R2 S2−1 u

due to (3.1)–(3.3) we have 2 u ≤ χ|u|2 dx + cuL2(ω) S2−1 uH 1 (ω) Ω

−1 ≤ CuL2(ω) u + (S2 u, u) ≤ CuL2 (ω) u, where C is independent on u. This proves the lemma. Let us finish the proof of the Theorem. Given a subset K of Q bounded in the norm · D(qS1 ) , we conclude that it is also bounded in H 1 (ω) due to (3.1). By the well known theorem on compact embedding of H 1 (ω) in L2 (ω) for each bounded domain with smooth boundary (see, for instance, [12, Ch. 1, Sec. 6]) we have that the set K is compact in L2 (ω). Applying now Lemma 3.2, we conclude that K is compact in L2 (Ω). Hence, the assumption of Lemma 3.1 is satisfied and the operator T introduced above is compact. The proof of Theorem 2.1 is complete.

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4 Absence of the discrete spectrum This section is devoted to the proof of Theorems 2.2 and 2.3. By Theorem 2.1 we know that the essential spectrum of the operator M is [1, +∞). Thus, the equivalent formulation of the absence of the discrete spectrum is the following inequality inf σ(M − 1) = inf (−i∇ + A)u2 − u2 ≥ 0. (4.1) u=1 u∈D(qM )

It will be enough to check the infimum for a · D(qM ) -dense subset of D(M ). Hence inf σ(M − 1) = inf (−i∇ + A)u2 − u2 ≥ 0 . (4.2) u=1 u∈D(M0 )

In order to prove this we will need some auxiliary statements which will be established in the next two subsections.

4.1

A Hardy inequality

Here we state a Hardy inequality for the quadratic form of the operator M , which will be one of the crucial tools in the proofs of Theorems 2.2 and 2.3. Let p = (p1 , p2 ) ∈ Ω be some point and the number R be such that BR (p) ⊂ Ω. Given a smooth compactly supported field B, we define the function µ(r) := dist (Φp (r), Z), where we recall that Φp (r) is the flux of the field B through the ball Br (p). We introduce the function  1  , if Φp (r) ≡ 0 as r ∈ [0, R], 16 + c (R)c 1 2 (p, R) c(p, R) = (4.3)  0, if Φp (r) ≡ 0 as r ∈ [0, R], where

64 + 4R2 , R4 2R2 c3 (p2 )c4 (R) + 4c4 (R) + 4R2 c2 (p, R) = , c3 (p2 ) cos2 (|p2 − π2 | + R)

c1 (R) =

c3 (p2 ) = π 2 min{p−2 , (π − p2 )−2 } − 1, 2 µ(r) c4 (p, R) = max , r [0,R] 2 2 c5 (R) = max 2µ0 + 4c5 c6 µ40 , c6 , r02 2R3 − 3R2 r0 + r03 c6 (R) = 4 max 2 , j0,1 6r0

(4.4)

334



and µ0 and r0 are defined by µ0 :=

1 r0 = , max r−1 µ(r) µ(r0 ) [0,R]

j0,1 is a smallest positive root of the Bessel function J0 . It was shown in [8] that the function c(p, R) is well defined. Finally, let us define   1, if |x1 | > l, (4.5) g(x1 ) = 1  , if |x1 | ≤ l. 4 Lemma 4.1. Assume that the field B is smooth and compactly supported and the − condition (1) of Theorem 2.2 is satisfied for the points p− = (p− 1 , p2 ) and p+ = + + (p1 , p2 ), then |(−i∇ + A)u|2 − g(x1 )|u|2 dx, (4.6) ρ(x1 )|u|2 dx ≤ Ω

Ω

holds for all u ∈ D(M0 ), where  c−   − 2,    1 + (x1 − p1 ) 0, ρ(x1 ) =   c  +   , 2 1 + (x1 − p+ 1)

if − ∞ < x1 < p− 1, + if p− 1 < x1 < p1 ,

(4.7)

if p+ 1 < x1 < +∞,

and the constants c± = c(p± , R± ) are given by (4.3). Proof. We start the proof from the estimate |u|2 |(−i∇ + A)u|2 − |u|2 dx, dx ≤ c− − 2 1 + (x1 − p1 ) Ω(−∞,p− Ω(−∞,p− 1 ) 1 )

(4.8)

which is valid for all u ∈ D(M0 ). The proof of this estimate follows from the calculations of [8, Sec. 6], where the similar inequality |u|2 |(−i∇ + A)u|2 − |u|2 dx, (4.9) dx ≤ c − 2 Ω 1 + (x1 − p1 ) Ω is proved for all u ∈ H01 (Ω) with some constant c. The approach employed in [8, Sec. 3] can be applied to prove the inequality (4.8). We will not reproduce all the details of this proof and just note that the only modification needed is to replace the function ϕ defined in [8, Eq. (3.28)] by  R   1 if x1 < p−  1 − √ ,  2  √ − 2(p1 − x1 ) R (4.10) ϕ(x) := − if p−  1 − √ < x1 < p1 ,   R 2    0 elsewhere,

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In the same way the inequality |u|2 2 2 c+ dx ≤ − |u| |(−i∇ + A)u| dx, (4.11) + 1 + (x1 − p1 )2 Ω(p+ Ω(p+ 1 ,+∞) 1 ,+∞) holds for all u ∈ D(M0 ), where c+ = c(p+ , R+ ). We will make use of the diamagnetic inequality (see [13]) |∇|u|(x)| ≤ |(−i∇ + A)u(x)|

(4.12)

which holds pointwise almost everywhere in Ω for each u ∈ D(M0 ). In addition the trivial inequality π π |∂x2 u|2 dx2 ≥ g|u|2 dx2 (4.13) 0

0

holds for each fixed x1 and all u ∈ D(M0 ). The diamagnetic inequality (4.12) and the last estimate lead us to the inequality |(−i∇ + A)u|2 dx ≥ |∇|u||2 dx ≥ g|u|2 dx, Ω(α,β)

Ω(α,β)

Ω(α,β)

which is valid for all α < β. Combining now this inequality with (4.8), (4.11) we arrive at the statement of the lemma. In the case of the Aharonov-Bohm field the similar statement is true. Lemma 4.2. Assume that the field is generated by Aharonov-Bohm potential given by (2.1) and that the condition (1) of Theorem 2.3 is satisfied for the point p = (p1 , p2 ). Then ρ(x1 )|u|2 dx ≤ |(−i∇ + A)u|2 − g(x1 )|u|2 dx, (4.14) Ω

Ω

holds for all u ∈ D(M0 ), where  c  , 1 + (x − p1 )2 1 ρ(x1 ) =  0,

−∞ < x1 < p1 ,

(4.15)

p1 < x1 < +∞,

the constant c = c(p, Φ) is given by R2 µ2 c3 (p2 ) cos2 (|p2 − π2 | + R) , c(p, Φ) = 2 2 8 2µ R c3 (p2 ) + (8µ2 + 8 + c3 (p2 ))(9R2 + 16π 2 )

(4.16)

µ := dist {Φ, Z}, c2 (p2 ) is the same as in (4.4). The proof of this lemma is the same as the one of Lemma 4.8. It is also based on similar calculations of [8, Sec. 7.1], where the inequality (4.9) was proven for Aharonov-Bohm field. Here one also needs to replace the function ϕ in [8, Eq. (3.28)] by the function ϕ defined in (4.10) with p− 1 = p1 .

336

4.2



A one-dimensional model

In this section we will show that the inequality (4.2) holds true if the oned2 2 dimensional Schr¨ odinger operator − dx 2 + V in L (R) with certain potential V 1 is non-negative. We will consider the case of a compactly supported field and the Aharonov-Bohm field simultaneously. In view of Lemmas 4.1 and 4.2 we have 1 (−i∇ + A)u2 − u2 = (−i∇ + A)u2 − (g u, u) 2 1 1 + (−i∇ + A)u2 + ((g − 2) u, u) 2 2 1 1 ≥ (−i∇ + A)u2 + ((ρ + g − 2) u, u) , 2 2 where g is given by (4.5). Here ρ is determined by (4.7) in the case of a compactly supported field and by (4.15) in the case of the Aharonov-Bohm field. Thus, inf (−i∇ + A)u2 − u2 u=1 u∈D(M0 )

≥

1 2

inf

u=1 u∈D(M0 )

(−i∇ + A)u2 + ((ρ + g − 2) u, u) .

By the diamagnetic inequality (4.12) we have inf (−i∇ + A)u2 − u2 u=1 u∈D(M0 )

≥ =

1 2

inf

u=1 u∈D(M0 )

1 2

1 = 2

∇|u|2 + ((ρ + g − 2) u, u)

inf

u=1 u∈D(M0 )

∇u2 + ((ρ + g − 2) u, u)

inf

u=1 u∈D(M0 )

Ω

|∂x1 u|2 + |∂x2 u|2 dx

+ ((ρ + g − 2) u, u) . Using now (4.13) we arrive at inf (−i∇ + A)u2 − u2 u=1 u∈D(M0 )

≥

1 2

inf

u=1 u∈D(M0 )

∂x1 u2 + (ρ u, u) + 2((g − 1) u, u) .

(4.17)

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In order to establish the inequality (4.2) it is therefore enough to show that π |ux1 (x)|2 + ρ(x1 )|u(x)|2 + 2(g(x1 ) − 1)|u(x)|2 dx1 dx2 ≥ 0, 0

R

which is equivalent to the inequality 2 |v | + ρ|v|2 + 2(g − 1)|v|2 dx1 ≥ 0,

(4.18)

R

for all v ∈ C0∞ (R). In other words, to prove Theorems 2.2 and 2.3 it is sufficient to show that the one-dimensional Schr¨ odinger operator −

d2 + ρ + 2(g − 1) dx21

is non-negative in L2 (R). The proof of this fact is the main subject of the next section.

4.3

The proofs of Theorems 2.2 and 2.3

As it has been shown in the previous section to prove the absence of the eigenvalues it is sufficient to check the inequality (4.18). Due to the definition of g it can be rewritten as 3 l |v (t)|2 + ρ(t)|v(t)|2 dt ≥ |v(t)|2 dt. (4.19) 2 −l R Let us show that under the assumptions of Theorems 2.2, respectively 2.3 this inequality holds true. We will show it in detail for the case of compactly supported field only (i.e., for Theorem 2.2); the case of the Aharonov-Bohm field is similar. We introduce a function  π −   c− + arctan(t − p− 1 ) , t < p1 , 2 φ− (t) := πc (4.20)   −, t ≥ p− . 1 2 − We remind that c− and p− 1 are given in Lemma 4.1. Clearly, φ− (t) = ρ(t) for t < p1 − and φ− (t) = 0 if t ≥ p1 . Keeping these properties in mind for each t ∈ (−l, l) we deduce the obvious equality

πc− v(t) = φ− (t)v(t) = 2

t −∞

(φ− (s)v(s)) ds

p− 1

=

t

ρ(s)v(s) ds + −∞

−∞

φ− (s)v (s) ds,

338



where we also employ the fact that by the assumption of Theorem 2.2 we have p− 1 < −l. The equality obtained, definition of φ− and Cauchy-Schwarz inequality give rise to an estimate   2 2 p− t 1 π 2 c2− |v(t)|2 ≤ 2  ρ(s)v(s) ds + φ− (s)v (s) ds  −∞ 4 −∞ − p− t t p1 1 2 2 2 (4.21) ρ(s) ds ρ(s)|v(s)| ds + φ (s) ds |v (s)| ds ≤2 ≤2

−∞

πc− 2

−∞

p− 1

−∞

2

t

ρ(s)|v(s)| ds + −∞

−∞

φ2− (s) ds

−

−∞

l

−∞

2

|v (s)| ds .

Since the function φ− (t) is constant for t > p− 1 it follows that

t

−∞

φ2− (s) ds

p− 1

= −∞

= c2−

− φ2− (s) ds + φ2− (p− 1 )(t − p1 ) 0

−∞

π

2 π 2 c2− + arctan(s) ds + (t − p− 1) 2 4

= c2− π ln 2 +

π 2 c2− (t − p− 1 ). 4

Substituting the last equality into (4.21) and using the expression for φ− (p− 1 ) (see (4.20)) we arrive at p− 1 2 2 ρ(s)|v(s)|2 ds |v(t)| ≤ 2 πc− −∞ (4.22)

l 4 ln 2 − 2 + + (t − p1 ) |v (s)| ds . π −∞ In the case c− = 0 the fraction c1− in this inequality is understood as +∞, so the inequality is valid for all possible values of c− . Integration (4.22) over (−l, l) and using the obvious equality 0 p− 1 2 ρ(s)|v(s)| ds = ρ(s)|v(s)|2 ds −∞

−∞

lead us to the estimate

l l 0 4 ln 2 2 − 2 2 2 − p1 |v(t)| dt ≤ 4l ρ(s)|v(s)| ds + |v (s)| ds πc− −∞ π −l −∞ l 0 4l ≤ ρ(s)|v(s)|2 ds + |v (s)|2 ds , 2 κ− −∞ −∞

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where κ− is given by (2.4). We can rewrite this inequality as l l 0 2 2 2 κ− |v(t)| dt ≤ 4l 2 ρ(s)|v(s)| ds + |v (s)| ds .

(4.23)

−l

−∞

−∞

This inequality is valid also in the case of c− = 0. In the same way one can easily prove similar inequality κ+

l

−l

|v(t)| dt ≤ 4l 2 2

+∞

2

+∞

ρ(s)|v(s)| ds +

0

−l

2

|v (s)| ds ,

(4.24)

where κ+ is given by (2.4). We sum the inequalities (4.23) and (4.24) to get l l 2 2 (κ− + κ+ ) |v(t)| dt ≤ 4l 2 ρ(s)|v(s)| ds + |v (s)|2 ds −l

R

+∞

+ −l

−∞

|v (s)|2 ds .

This implies that

l

−l

|v(t)|2 dt ≤

8l κ

ρ(s)|v(s)|2 ds +

R

R

|v (s)|2 ds ,

where κ = κ− + κ+ . An immediate consequence of the last inequality is that to satisfy (4.19) it is sufficient to set l≤

κ , 12

which coincides with the inequality (2.3). This completes the proof of Theorem 2.2. The proof of Theorem 2.3 is similar. One just needs to use the inequality (4.23) rewritten in a slightly different way: l 0 2 2 |v(t)| dt ≤ 4l ρ(s)|v(s)|2 ds πc− −∞ −l

l 4 ln 2 − 2 + − p1 |v (s)| ds π −∞ l 0 4l 2 2 ≤ ρ(s)|v(s)| ds + |v (s)| ds , κ −∞ −∞ with κ given by (2.6). This inequality will immediately imply the estimate (4.19) if the relation (2.5) is satisfied.

340



5 Presence of eigenvalues In this section we will prove Theorem 2.4. We will use the formula (−i∇ + A)u2 − u2 . inf σ(M − 1) = inf u=1 u∈D(qM )

If we find a test function u ∈ D(qM ) such that (−i∇ + A)u2 − u2 < 0 this will prove the presence of the discrete spectrum due to Theorem 2.1. Clearly, D(qM ) is a subspace of H 1 (Ω) consisting of functions that vanish on Γ. The eigenfunction ψ of −∆N ,D associated with the lowest eigenvalue λ(l) belongs to D(qM ). We can choose this eigenfunction being real-valued and normalized in L2 (Ω). Choosing ψ as a test function we have (−i∇ + A)ψ2 = ∇ψ2 + Aψ2 = λ(l) + Aψ2 ≤ λ(l) + max |A|2 . Ω

(5.1)

Here we used the normalization condition for ψ and an obvious relation λ(l) = ∇ψ2 . The left-hand side of inequality (5.1) is invariant under the gauge transformation of the magnetic potential A. Bearing this fact in mind we take the infimum in (5.1) over all potentials associated with the field B what leads us to (−i∇ + A)ψ2 − ψ2 ≤ λ(l) + inf max |A|2 − 1. A

Ω

By the assumption the right-hand side of the last inequality is less than zero, hence the theorem is proved. In conclusion let us show that the second term on the left-hand side of (2.7) is finite. It is sufficient to show that it is finite for some A. Let A be some potential associated with B. Since B is smooth and compactly supported, the potential A can be chosen in C 1 (Ω). Therefore it is bounded on each bounded subset of Ω. The support of B is a compact set, so there exists number b > 0 such that B = 0 as x ∈ Ω \ Ω(−b, b), i.e., ∂x2 a2 − ∂x1 a1 = 0 as x ∈ Ω \ Ω(−b, b). Since both domains Ω(−∞, −b) and Ω(b, +∞) are simply connected, this immediately implies the existence of functions h− ∈ C 1 (Ω(−∞, −b)), h+ ∈ C 1 (Ω(b, +∞)) such that ∇h− = A as x ∈ Ω(−∞, −b), ∇h+ = A as x ∈ Ω(b, +∞). We introduce the function    h− (x)ζ(x1 ), x1 < −b, −b ≤ x1 ≤ b, h(x) = 0,   h+ (x)ζ(x1 ), x1 > b, where ζ(x1 ) is equal to one as |x1 | > 2b and vanishes as |x1 | ≤ b. By definition h ∈ := A−∇h leads us to a new vector potential A C 1 (Ω). The gauge transformation A is compactly supported associated with the same field B. Moreover the potential A ∈ C 1 (Ω), it follows that max |A| 2 is since ∇h = A if |x1 | is large enough. Since A finite.

Ω

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6 Acknowledgments D.B. has been supported by DAAD (A/03/01031) and partially supported by RFBR and the program “Leading scientific schools” (NSh-1446.2003.1). T.E. has been supported by ESF Program SPECT. The work has also been supported by the DAAD project 313-PPP-SE/05-lk. D.B. and T.E. thank the Stuttgart University, where this work has been done, for the hospitality extended to them. Authors would like to thank T. Weidl for suggesting them the study of the initial problem and for numerous stimulating discussions.

References [1] 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). ˇ [2] P. Exner, P. Seba, M. Tater, D. Vanˇek, Bound states and scattering in quantum waveguide coupled through a boundary window, J. Math. Phys. 37, no. 10, 4867–4887 (1996). [3] P. Exner and S. Vugalter, Asymptotic estimates for bound states in quantum waveguides coupled laterally through a narrow window, Ann. Inst. H. Poincaré: Phys. théor. 65, no. 1, 109–123 (1996). [4] I.Yu. Popov, Asymptotics of bound states for laterally coupled waveguides, Rep. Math. Phys. 43, no. 3, 427–437 (1999). [5] R.R. Gadyl’shin, On regular and singular perturbations of acoustic and quantum waveguides, C.R. Mécanique 332, no. 8, 647–652. [6] P. Duclos, P. Exner, B. Meller, Resonances from perturbed symmetry in open quantum dots. Rep. Math. Phys. 47, no. 2, 253–267 (2001). [7] T. Weidl, Remarks on virtual bound states for semi-bounded operators, Comm. in Part. Diff. Eq. 24, no. 1&2, 25–60 (1999). [8] T. Ekholm and H. Kovaˇr´ık, Stability of the magnetic Schr¨ odinger operator in a waveguide, to appear in Comm. in Part. Diff. Eq., Preprint: arXiv:math-ph/0404069. [9] H.L. Cycon, R.G. Froese, W. Kirsh, B. Simon, Schr¨ odinger operators with application to quantum mechanics and global geometry. Texts and Monographs in Physics. Springer Study Edition. Springer-Verlag, Berlin-New York. 1987. [10] M.S. Birman, Perturbation of the continuous spectrum of a singular elliptic operator under a change of the boundary and the boundary condition, Vestnik Leningradskogo universiteta. 1, 22–55 (1962).

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[11] Michail S. Birman and Michail Z. Solomyak, Spectral theory of self-adjoint operators in Hilbert space, D. Reidel Publishing Company, 1987. [12] O.A. Ladyzhenskaya, The boundary value problems of mathematical physics. Applied Mathematical Sciences, v. 49. Springer-Verlag, New York, 1985. [13] D. Hundertmark and B. Simon, A diamagnetic inequality for semigroup differences, J. Reine Angew. Math. 571, 107–130 (2004).

Denis Borisov Department of Physics and Mathematics Bashkir State Pedagogical University October rev. st., 3a 450000 Ufa Russia email: [email protected]

Tomas Ekholm Department of Mathematics Royal Institute of Technology Lindstedtsv¨ agen, 25 S-100 44 Stockholm Sweden email: [email protected]

Hynek Kovaˇr´ık Faculty of Mathematics and Physics Stuttgart University Pfaffenwaldring, 57 D-70569 Stuttgart Germany email: [email protected] Communicated by Vincent Rivasseau submitted 11/05/04, accepted 21/09/04




The Hopf Algebra of Rooted Trees in Epstein-Glaser Renormalization Christoph Bergbauer and Dirk Kreimer Abstract. We show how the Hopf algebra of rooted trees encodes the combinatorics of Epstein-Glaser renormalization and coordinate space renormalization in general. In particular, we prove that the Epstein-Glaser time-ordered products can be obtained from the Hopf algebra by suitable Feynman rules, mapping trees to operator-valued distributions. Twisting the antipode with a renormalization map formally solves the Epstein-Glaser recursion and provides local counterterms due to the Hochschild 1-closedness of the grafting operator B+ .

Introduction The Epstein-Glaser framework [5, 16] and its modern variants [15, 2, 14] provide a mathematically rigorous approach to perturbation theory and renormalization in coordinate space. Let M = R1,3 denote the Minkowski space. Epstein and Glaser constructed, for a scalar φk field theory say, a sequence of operator-valued distributions Tn on M n respectively, which replace the ill-defined time-ordered products in the standard approach to perturbation theory. The result is a perturbation theory which is a priori finite in each order – no removal of short-distance singularities is needed since all expressions are well defined from the very beginning. The appropriate notion of renormalization in the Epstein-Glaser framework is extension of distributions onto diagonals. Indeed, the objects of interest Tn are a priori determined outside the diagonals by causality. Finite renormalizations correspond to different ways of extending distributions onto diagonals. Moreover, in this approach the S-matrix is local by construction. On the other hand, the combinatorics of momentum space renormalization have been most efficiently described [4, 11] in terms of the Hopf algebra and associated Lie algebra of Feynman graphs. Renormalization and in particular the Bogoliubov recursion boil down to twisting the antipode S of that Hopf algebra by renormalization maps into some target ring of Laurent or formal power series. This is possible due to a coproduct which disentangles 1PI graphs into divergent 1PI subgraphs. There is a universal object behind all Hopf algebras of this kind: the Hopf algebra of rooted trees [9, 3] which encodes nested subdivergences in terms of a tree and their recursive removal in terms of its coproduct and the resulting antipode. We will show how the Hopf algebra of rooted trees works in the realm of Epstein-Glaser renormalization in almost complete analogy to other renormalization programs like BPHZ. In fact it is even easier to understand

344

Ch. Bergbauer and D. Kreimer


its role in Epstein-Glaser renormalization since no regularization is required and overlapping divergences do not exist in the coordinate space language. This paper is organized as follows: In the first section we give a short review of the Epstein-Glaser construction of time-ordered products, emphasizing the point of view of diagonals [2]. The second section recalls the powerful notion of a Hochschild 1-cocycle on a connected graded bialgebra, giving rise to two equivalent presentations of the Hopf algebra of rooted trees. A new convolution-like product is introduced which in cooperation with the antipode allows to recursively generate all terms needed for an Epstein-Glaser time-ordered product, as will be proved using explicit renormalized “Feynman rules” in the final theorem which we already state in a short version: Theorem (Main result) There is a map Φ : H•∗ → F V such that the n-th Epstein-Glaser time-ordered product Tn is given by Φ(SR id)(t) Tn = t∈Tn •∗

where H is a Hopf-algebra of rooted trees, F V something like the tensor algebra of distributions on M , Tn the set of all binary trees with n leaves, SR the twisted antipode of H•∗ and a modified “convolution product” in H•∗ .

1 Some background on Epstein-Glaser renormalization For simplicity we restrict ourselves to a massive neutral scalar field theory with interaction Lagrangian λ LI = φk , (1) k! on the flat Minkowski space-time M := R1,3 . Generalizations to Quantum Electrodynamics and globally hyperbolic space-times have been worked out in [16] and [2], respectively, which though does not affect the combinatorics we are primarily interested in.

1.1

Motivation

As a starting point for the Epstein-Glaser construction of time-ordered products [5] we consider the symbolic Dyson series for the S-matrix S = T ei

LI (x)dx

(2)

which is formally derived from the Schwinger differential equation of motion by transforming it into an iterated integral equation and applying the time-ordering operator T to each summand (i)n T (LI (x1 ) . . . LI (xn ))dx1 . . . dxn n! M n

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345

which has the benefit that we are integrating now over M n rather than over an n-simplex × R3n . Let A, B be operator-valued functions on M. The time-ordering operator T is usually defined by T (A(x1 )B(x2 )) := Θ(x01 − x02 )A(x1 )B(x2 ) + Θ(x02 − x01 )B(x2 )A(x1 )

(3)

where Θ denotes the Heaviside characteristic function of R≥0 . Analogously one defines T on more than two factors. Now S and LI are obviously supposed to be operator-valued distributions, for which (3) does not make sense since distributions can not just be multiplied by noncontinuous functions like Θ. It does make sense though outside the thick diagonal Dn = {x ∈ M n : xi = xj for some i = j} where products of Θ(x0i − x0j ) are continuous. In fact the mathematical origin for the appearance of short-distance singularities in perturbation theory is the ill-defined notion of time-ordering reviewed above. Epstein and Glaser proposed a way to construct well-defined time ordered products Tn , one for each power n of the coupling constant, that satisfy a set of suitable conditions explained below, the most prominent being that of locality or micro-causality. The power series S constructed by (2) using the Epstein-Glaser time-ordered product T is a priori finite in every order, and renormalization corresponds then to stepwise extension of distributions from M n − Dn to M n . In general, distributions can not be extended uniquely onto diagonals. The resulting degrees of freedom are in one-to-one correspondence with the degrees of freedom (finite renormalizations) in momentum space renormalization programs like BPHZ and dimensional regularization. The notion of locality, crucial to the following construction of time-ordered products, can be motivated as follows: Suppose x = (x1 , . . . , xn ) ∈ M n , ∅ I N := {1, . . . , n} and for each i ∈ I, the point xi is not in the past causal shadow of any of the xj for j ∈ N −I. We denote this situation xi xj ∀i ∈ I, j ∈ N −I. Then our time ordered product Tn is supposed to satisfy (in the sense of operator-valued distributions) Tn (x1 , . . . , xn ) = T|I| (xi )i∈I T|N −I|(xj )j∈N −I (4) because we think of the xi to happen after (or at least not before) the xj . If both xi xj and xj xi , ∀i, j, so if all pairs (xi , xj ) are spacelike, we have [T|I| (xi )i∈I , T|N −I| (xj )j∈N −I ] = 0.

1.2

Construction of time-ordered products

In this subsection we give a short review of the mathematical core of EpsteinGlaser renormalization in its modern variant [15, 2, 14] which emphasizes the point of view of nested diagonals. For the proofs, the reader is referred to [2].

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The Minkowski metric on M provides a relation on M as follows: x y − − iff x is not in the past causal shadow of y, that is x ∈ / y + V where V := {z ∈ M : (z)2 ≤ 0, z 0 ≤ 0} is the closed past lightcone. Now, for n ∈ N let N := {1, . . . , n} and ∅ I N. The set CI := {(x1 , . . . , xn ) ∈ M n : xi xj ∀i ∈ I, j ∈ N − I} is obviously a translation invariant open subset of M n . Lemma 1 (Geometric lemma)

CI = M n − ∆n

∅IN

where ∆n = {x ∈ M n : x1 = . . . = xn } is the “thin” diagonal in M n . The proof is an easy induction on n. The geometric lemma tells us that the causality condition (4) determines the time-ordered product Tn everywhere outside the thin diagonal ∆n , once the Tk for k < n are known on whole M k , respectively. It is important to understand that the geometric lemma does not really constitute a specific feature of the Minkowski space. Indeed, the lemma holds if one replaces by any relation such that x y or y x whenever y = x, and such that is “weakly transitive” in the sense that x y and ¬(z y) imply x z. Definition 2 A causal partition of unity {pI,N −I }∅IN is a smooth partition of unity subordinate to the cover {CI }∅IN of M n − ∆n . For simplicity, we will sometimes drop the curly brackets in the subscript, for example p1,2 denotes p{1},{2} . Let D(M ) = C0∞ (M ) denote the space of test functions on M with the usual topology. Let H denote the Hilbert space of the free field theory and D a suitable dense subspace. In principle an Epstein-Glaser time-ordered product is a collection (Tnr )n∈N (r = (r1 , . . . , rn ) an n-multiindex) of operator-valued distributions Tnr : (r ,...,rn ) D(M n ) → End(D), such that Tn 1 replaces the time ordering of the n Wick monomials : φr1 :, . . . , : φrn : . Definition 3 A collection (Tnr ) of operator-valued distributions Tnr : D(M n ) → End(D) is called an (Epstein-Glaser) time-ordered product if (i) T1k (f ) = : φk : (f ) where : φk : (f ) denotes the Wick monomial : φk : smeared with the test function f, (ii) T is symmetric Tnr (f1 ⊗ . . . ⊗ fn ) = Tnr (fπ(1) ⊗ . . . ⊗ fπ(n) )

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when π is a permutation of N := {1, . . . , n}. This allows for the notation T (N ) = Tnr (f1 ⊗ . . . ⊗ fn ) when the fi and ri are clear from the context, (iii) T splits causally: Let ∅ I N. Then T (N ) = T (I)T (N − I)

(5)

for all test functions with support in CI ⊂ M n , (iv) T is translation covariant U (a, 1)T (f1 , . . . , fn )U (a, 1)−1 = T (τa f1 , . . . , τa fn ) where U (·, 1) . . . U (·, 1)−1 is the representation of the translation part of the Poincaré group in D, and τa f (x) = f (x − a) denotes translation by a. (v) The Wick expansion relates time-ordered products corresponding to different Wick-powers Tn(r1 ...rn ) (f1 ⊗ · · · ⊗ fn ) = Ω, Tn(r1 −i1 ,...,rn −in ) (f1 ⊗ · · · ⊗ fn )Ω i1 ,...,in

r1 rn × ··· : φi1 . . . φin : (f1 ⊗ · · · ⊗ fn ) (6) i1 in with Ω the vacuum state in D ⊂ H. Note that by the so-called Theorem 0 in [5] the summands in the right-hand side of (6) as products of translation invariant numerical distributions and Wick monomials are well-defined operator-valued distributions. Once a time-ordered product T = (Tnr ) is given, the S-matrix for the φk -theory is obtained as the formal power series ∞ in S(f ) = T (k,...,k) (f ⊗n ), (7) n!(k!)n n n=0 possibly taking the adiabatic limit f → λ later on, which is a highly nontrivial task we shall not be concerned about in the present work. The S-matrix (7) and the relative S-matrices constructed from T are local. If one imposes additional normalization conditions (Lorentz covariance, Hermiticity etc., see [2]) on T, the S-matrix becomes Lorentz covariant and unitary, etc. Moreover, the interacting field constructed from the relative S-matrices are Lorentz covariant, Hermitean and satisfy the interacting field equation. Theorem 4 Time-ordered products exist.

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A constructive proof is given in [2] and of course, but in a somewhat different notation, in the original paper [5]. The idea is as follows: Provided all (Tm ) for m < n are constructed, the Geometric Lemma 1 ensures that Tn is determined on M n − ∆n by causality (iii). We define TI = T (I)T (N − I) as a distribution on CI . One easily shows that T (I)T (N − I) = T (J)T (N − J) on the intersection CI ∩CJ . Therefore, we can patch the TI together using a causal partition of unity {pI,N −I } 0 T (N ) := pI,N −I T (I)T (N − I) (8) ∅IN

which is a well-defined distribution on M n − ∆n . As usual, 0 T (N ) is independent on the choice of the partition of unity. It remains to extend it to a distribution on M n . Using the Wick expansion (v) and translation invariance, this amounts to an extension problem of numerical distributions 0 tn from M n−1 − {0} to M n−1 . Having quantified the behavior of a numerical distribution at the origin by the Steinmann scaling degree (see [2] for details), a generalization of the degree of homogeneity, one can show that there is a unique extension tn of 0 tn to M n−1 preserving the scaling degree, provided the scaling degree sd(0 tn ) of 0 tn is smaller than the dimension 4(n − 1). Otherwise, if it is larger or equal but still finite, there is a finite dimensional space of extensions obtained as follows: Let f ∈ D(M n−1 ). The distribution

0 α ωα ∂ f (0) (9) tn : f → tn f − α

where the sum goes over all 4(n − 1)-multiindices α such that |α| ≤ sd(0 tn ) − 4(n − 1) and the ωα ∈ D(M n−1 ) such that ∂ β ωα (0) = δα,β , has then scaling degree sd(0 tn ) < 4(n − 1) and is hence uniquely extendible (preserving the scaling degree). There is an ambiguity due to the ωα however, and it is exactly this ambiguity which corresponds to the freedom of finite renormalizations. We call the linear operator id − w on test functions ωα ∂ α f (0) id − w : f → f − α

Taylor subtraction operator and, motivated by the fact that tn = (id − w∗ )0 tn holds on the level of numerical distributions, we write by abuse of notation the extension of 0 T (N ) to the diagonal by ∗ )0 T (N ) T (N ) = (id − W1...n

(10)

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although there is no linear operator W ∗ on the space of operator valued distributions doing this duty. Our abuse of notation is justified though because we are only concerned with the combinatorics with respect to n in the following, and the Wick expansion leaves n obviously unchanged. So we understand W ∗ as the symbolic “operator” which unpacks the operator valued distributions into Wick monomials and numerical distributions, Taylor subtracts the test function for those numerical distributions and produces then a “counterterm” such that (id − W ∗ ) maps a distribution on M n − ∆n to an extension on M n while the possible ambiguity (depending on the scaling degrees) is fixed by a choice of the ωα . The subscript in ∗ W1...n indicating to which coordinates it applies will be useful later on. This constructive proof of Theorem 4 actually proves more than the theorem demands: that in each extension step the scaling degree does not increase. If we make this an additional condition on time-ordered products, we can state Corollary 5 All time-ordered products are uniquely (up to the ωα , more precisely up to the finite set of constants 0 tn (ωα ) in every order n) characterized by equations (8) and (10). Feynman graphs enter the game when one applies Wick’s theorem. It might be instructive to have a look at the examples in [14]. We also note that the usual notions of renormalizable theories, critical dimension etc. can be traced back to the behavior of the scaling degrees as n and the space-time dimension vary. In particular, the scaling degree coincides with the usual power-counting techniques in momentum space.

2 The Hopf algebra of rooted trees in Epstein-Glaser renormalization The combinatorics of renormalization in coordinate space can be most easily described in terms of rooted trees. Given some space-time points, •

•

•

•

•

•

we consider them as leaves of a tree (to be constructed). Whenever some of these points come together on a diagonal in M n , we connect the corresponding vertices to a new vertex such that subdivergences (subdiagonals) correspond to subtrees, for example • @ • @ @ @ @• @• • A A A • A• • A• • A• So a tree represents the (partially ordered) nested or disjoint subdiagonals which are relevant to renormalization. It is now possible to construct a suitable coproduct

350



on the free algebra generated by these trees such that the Bogoliubov recursion is essentially solved by the antipode of the resulting Hopf algebra on trees, as will be made precise in Subsection 2.2. This remarkable property and the fact that local counterterms result [12] are the consequence of the fact that a certain operator on the Hopf algebra is a Hochschild 1-cocycle.

2.1

Hochschild cohomology of bialgebras

All algebras are supposed to be over some field k of characteristic zero, associative and unital, analogously for coalgebras. The unit (and by abuse of notation also the unit map) will be denoted by I, the counit map by . All algebra homomorphisms ∞ are supposed to be unital. A bialgebra (A = i=0 Ai , m, I, ∆, ) is called graded connected if Ai Aj ⊂ Ai+j and ∆(Ai ) ⊂ j+k=i Aj ⊗ Ak , and if ∆(I) = I ⊗ I and ∞ A0 = kI, (I) = I and = 0 on i=1 Ai . We call ker the augmentation ideal of A and denote P the projection A → A onto the augmentation ideal, P = id − I . Let (A, m, I, ∆, ) be a bialgebra. We think of linear maps L : A → A⊗n as n-cochains and define a coboundary map b by bL := (id ⊗ L) ◦ ∆ +

n

(−1)i ∆i ◦ L + (−1)n+1 L ⊗ I

(11)

i=1

where ∆i denotes the coproduct applied to the i-th factor in A⊗n . It is easy to see (using essentially the coassociativity of ∆) that b2 = 0, which gives rise to a cohomology theory called Hochschild cohomology. It is also easy to see that, for A finite dimensional say, the cohomology theory (11) is the dual of the usual Hochschild homology of the dual algebra A∗ . In case n = 1, (11) reduces to, for L : A → A, bL = (id ⊗ L) ◦ ∆ − ∆ ◦ L + L ⊗ I.

(12)

It is known [3] that the category of objects (A, C) consisting of a commutative bialgebra A and a Hochschild 1-cocycle C on A with morphisms bialgebra morphisms commuting with the cocycles has an initial object (H, B+ ), with H the Hopf algebra of (non-planar) rooted trees and the operator B+ which grafts a product of rooted trees together to a new root as described in the next subsection. While the higher (n > 1) Hochschild cohomology of H vanishes [6], the closedness of B+ will turn out to be crucial for what follows. The next lemma will provide a convenient way to construct Hopf algebras out of free or free commutative algebras by choosing linear endomorphisms Ci and demanding that the Ci be Hochschild 1-cocycles. ∞ Lemma 6 Let A = n=0 An be a free or free commutative graded algebra (generated by a graded vector space) such that A0 = kI, and let (Ci )i∈I be a collection

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of injective linear endomorphisms of A such that Ci (A) ∩ Cj (A) = {0} for i = j and such that each free generator y in degree n is the image under some Ci of an x ∈ An−1 for n ≥ 1. Then there is a unique connected graded bialgebra structure (∆, ) on A such that the Ci are Hochschild closed with respect to ∆. In particular, A is a Hopf algebra (with this property) in a unique way. Proof. We will construct ∆ by induction on n. The Hochschild closedness of the Ci demands that ∆ ◦ Ci = (id ⊗ Ci ) ◦ ∆ + Ci ⊗ I.

(13)

∆(I) = I ⊗ I by convention, so ∆ is known on A0 . Now let y be a free generator in An+1 . By assumption there is a unique x ∈ An such that y = Ci x. Assume ∆ is known on x, then by (13) it is also known on y. Hence we can uniquely extend ∆ to an algebra homomorphism on An+1 . By induction, this uniquely defines ∆ as an algebra morphism on A. From (13) it also follows inductively that ∆ respects the grading in all orders: ∆(An ) ⊂

n

Ak ⊗ An−k .

k=0

For the coassociativity (∆ ⊗ id)∆ = (id ⊗ ∆)∆ we note that (∆ ⊗ id)∆Ci

=

(∆ ⊗ id)((id ⊗ Ci )∆ + Ci ⊗ I)

= =

(∆ ⊗ Ci )∆ + ∆Ci ⊗ I (∆ ⊗ Ci )∆ + (id ⊗ Ci ⊗ id)(∆ ⊗ I) + Ci ⊗ I ⊗ I

=

(id ⊗ id ⊗ Ci )(∆ ⊗ id)∆ + (id ⊗ Ci ⊗ id)(∆ ⊗ I) + Ci ⊗ I ⊗ I.

On the other hand, (id ⊗ ∆)∆Ci

=

(id ⊗ ∆)((id ⊗ Ci )∆ + Ci ⊗ I)

= =

(id ⊗ ∆Ci )∆ + Ci ⊗ I ⊗ I id ⊗ ((id ⊗ Ci )∆ + Ci ⊗ I)∆ + Ci ⊗ I ⊗ I

=

(id ⊗ id ⊗ Ci )(id ⊗ ∆)∆ + (id ⊗ Ci ⊗ id)(∆ ⊗ I) + Ci ⊗ I ⊗ I

which proves the coassociativity by induction on the grading. Now setting (I) = I and = 0 elsewhere finishes the proof. Note that any connected graded bialgebra is a Hopf algebra in a unique way.

2.2

The Hopf algebra of rooted trees, relation to previous work

In this section we collect well-known results [3, 4, 9, 12] on Hopf algebra methods in momentum space renormalization which will turn out to be applicable to EpsteinGlaser renormalization as well.

352



A rooted tree is a connected contractible compact graph with a distinguished vertex, the root. A forest is a disjoint union of rooted trees. Isomorphisms of rooted trees or forests are isomorphisms of graphs preserving the distinguished vertex/vertices. Let t be a rooted tree with root o. The choice of o determines an orientation of the edges of t : we draw the root on top and let the rest of the tree “hang down.” Vertices of t having no outgoing edges are called leaves, the other vertices (and the root) are called internal vertices. The set of forests is graded, for instance by the number of vertices a forest has (the weight grading). Let H be the free commutative algebra generated by rooted trees with the weight grading. The commutative product in H will be visualized as the disjoint union of trees, such that monomials in H are scalar multiples of forests. We demand that the linear operator B+ on H, defined by B+ (I)

=

B+ (t1 . . . tn ) =

• • A @ • A@• t1 . . . tn

is a Hochschild 1-cocycle, which makes H a Hopf algebra by virtue of Lemma 6. It is easy to see that the resulting coproduct can be described as follows Pc (t) ⊗ Rc (t) (14) ∆(t) = I ⊗ t + t ⊗ I + adm.c

where the sum goes over all admissible cuts of the tree t. By a cut of t we mean a nonempty set of edges of t that are to be removed. The product of subtrees which “fall down” upon removal of those edges is called the pruned part and denoted Pc (t), the part which remains connected with the root Rc (t). Now a cut c(t) is admissible, if for each leaf l of t it contains at most one edge on the path from l to the root. For instance,   • • • A A A • •   • A • A ∆ = ⊗I+I⊗ • A + AA ⊗ + A A A A A A • • • A A A • • • • • • • • • • • • A • + •⊗ • + 2 • ⊗ • A• + AA • ⊗ • +2 • • ⊗ • + A • • • • A• • • • . + • • ⊗ AA + • • • ⊗ • • • H is obviously not cocommutative. Let V be a unital ring with multiplication mV . Given ring homomorphisms φ, ψ : H → V, one can define their convolution product φ ψ : H → V, x →

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mV (φ ⊗ ψ)∆x, which is a ring homomorphism again. In particular, the antipode S is the inverse of id : H → H with respect to this convolution product. Let Q be the linear endomorphism of H ⊗ H such that Q(I ⊗ I) = −I ⊗ I and Q = id ⊗ P otherwise. So (up to the sign) Q is a projection onto H ⊗ ker ⊕ kI ⊗ kI. The shorthand notation φ Q ψ := mV (φ ⊗ ψ)Q∆ will be useful. Now in any Hopf-algebra approach [9, 12, 3, 4] to perturbative quantum field theory, renormalization boils down to twisting the antipode which, (in any graded Hopf algebra) satisfies the recursive equation S = −m(S ⊗ id)Q∆ = −S Q id, by a homomorphism Φ : H → V, called “Feynman rules”, for example into a ring V of Laurent series (dimensional regularization) or formal power series (BPHZ), and a “renormalization scheme” R : V → V which delivers the counterterm. More explicitly, one considers Φ Φ Φ := −RmV (SR ⊗ Φ)Q∆ = −R(SR Q Φ). SR

(15)

While Φ means application of unrenormalized Feynman rules, the renormalized expression is then given by Φ SR Φ. (16) For details the reader is referred to [3]. In Epstein-Glaser renormalization, essentially the same happens, but in an easier way because no regularization is required. The target ring V is most suitably chosen to be something like the tensor algebra of distributions on M, Φ will then map a given “subdivergence situation” encoded in a rooted tree to the corresponding distribution in V. The meaning of Φ is much easier to understand however if we give a somewhat different presentation of the Hopf algebra and define a modified convolution product.

2.3

The cut product and the Bogoliubov recursion

We enlarge the Hopf algebra H to H•∗ by allowing for two types of vertices: • and ∗. This yields two Hochschild 1-cocycles B+• and B+∗ depending on which type the newly adjoined root has. B+• (I) = • • A@ B+• (t1 . . . tn ) = • A@• t1 . . . tn

B+∗ (I) = ∗

∗ A @ B+∗ (t1 . . . tn ) = • A@• t1 . . . tn

It is easy to see that the coproduct ∆ which we endow H•∗ with using B+• , B+∗ and Lemma 6 has the same form (14) as in H. Now let R be the algebra endomorphism of H•∗ which changes the type of the root to ∗, whatever it was

354



before.  ∗ • A@ @ A   = • A@• . R • A@• t1 . . . tn t1 . . . tn 

R(•) = ∗,

R(∗) = ∗,

Once again we remark that all our algebra endomorphisms are supposed to be unital, so we will not specify their values at I explicitly. Our aim is now to construct a new product called cut product of linear endomorphisms of H•∗ . The usual convolution product (φ, ψ) → φ ψ = m(φ ⊗ ψ)∆ in Endk (H) or Endk (H•∗ ) has the disadvantage that, applied several times with the projection P onto the augmentation ideal, it gets rid of the structure of trees. For example, for any tree t there is an n ∈ N such that P n (t) = (P . . . P )(t) = polynomial in • . Our new product (φ ψ)(t) is supposed to apply φ to Pc (t) and ψ to Rc (t) as well, but reassemble the tree afterwards rather than taking the disjoint union of pruned and root parts using m. For instance, (φ ψ)

• •

:= φ

• •

ψ(I) + φ(I)ψ

• •

+

ψ(•) φ(•)

which should be compared to (φ ψ)

• •

=φ

• •

ψ(I) + φ(I)ψ

• •

+ φ(•)ψ(•).

This is however only possible for a rather small class of φ and ψ which do not change the trees too much. For example, φ is supposed to map trees to trees while ψ is not allowed to kill the vertices where something has been cut. We leave it to the reader to find the most general notion of those maps, because the only ones we need here are B+ and id, P, R, where all this is possible in a rather trivial way. ˜ •∗ be the Hopf algebra of trees as in H•∗ with an additional decoration of Let H ˜ •∗ → H•∗ the vertices by subsets of N. There is an obvious forgetful projection π : H •∗ •∗ ˜ and an inclusion j : H → H decorating all vertices by the empty set. We lift ˜ •∗ → H ˜ •∗ by the any of the maps φ = B+ , id, P, R : H•∗ → H•∗ to a map φ˜ : H prescription that newly created vertices are to be decorated by the empty set while the decorations of the old vertices is to be preserved.

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˜ :H ˜ •∗ → H ˜ •∗ ⊗ H ˜ •∗ which does the same as ∆ in H•∗ We consider the map ∆ but decorates each root in Pc and each vertex in Rc that got separated by a cut by the same integer (by the smallest unused integer say), preserving the existing decoration. For example, ˜ ∆

• • • •1 •2 A = AA ⊗ I + I ⊗ AA + •1 ⊗ + •2 ⊗ + •1 •2 ⊗ •12 . A • • • • • • • •

Here we do not display the empty set and set brackets for simplicity. Note that ˜ is a coproduct. The decoration has the only purpose to we do not contend that ∆ provide “glueing” information. ˜ •∗ → H ˜ •∗ which reconstructs the preimage of ∆ ˜ ˜ •∗ ⊗ H We define a map m ˜ :H by inserting edges between vertices that have been decorated by the same integers ˜ −1 on the image of ∆ ˜ and and discards the used decoration afterwards. So m ˜ =∆ ˜ •∗ . For otherwise, if no decorations match, m ˜ is the free multiplication mH˜ •∗ of H instance, • •1 • m ˜ •1 •2 •3 ⊗ •2 •4 = AA •3 •4 • • • • m ˜ is obviously not an algebra homomorphism. Definition 7 Let φ ∈ {id, P, R} and ψ ∈ {id, P, R, B+ }. Then the linear endomorphism φ ψ of H•∗ , ˜ ∆j ˜ (φ ψ) = π m( ˜ φ˜ ⊗ ψ) is called the cut product of φ and ψ. It is easy to see that if φ and ψ are algebra endomorphisms, so is φ ψ. As a shorthand notation, we will be using ˜Q ˜ ∆j ˜ (φ Q ψ) := π m( ˜ φ˜ ⊗ ψ) ˜ is the obvious lift of Q to (H ˜ •∗ )⊗2 . In analogy to the approach presented in where Q the preceding subsection, we recursively define the twisted antipode by S˜R (I) = I and ˜ m( ˜∆ ˜ = −R ˜ m(− ˜ m(. ˜∆ ˜ ⊗id)Q ˜ ∆. ˜ S˜R := −R ˜ S˜R ⊗ id)Q ˜ R ˜ . . ⊗ id)Q (17) ˜R S

Let SR := π S˜R j. If one is willing to ignore the fact that jπ = id, one can view SR as defined by SR := −R(SR Q id)

356



which might be a helpful motivation when compared to (15). Note that these are ˜∆ ˜ reduces the number of edges and SR (I) = I recursive definitions indeed since Q terminates the recursion. SR will turn out to be the counterterm map in the Epstein-Glaser framework. Remember that R is an idempotent algebra endomorphism, hence in particular a Rota-Baxter operator. Therefore SR and SR id are algebra endomorphism as well by a general inductive argument [10]. Lemma 8 (SR id)B+• = (id − R)B+• (SR id). Proof. We use the Hochschild closedness of B+• , ∆B+• = (id ⊗ B+• )∆ + B+• ⊗ I.

(18)

˜ •∗ )⊗2 in order to apply it to (SR id) : Now we want to lift this equation to (H ˜B ˜+ = C(id ⊗ B ˜+• )∆ ˜ +B ˜+• ⊗ I ∆

(19)

˜ •∗ ⊗ H ˜ •∗ → H ˜ •∗ ⊗ H ˜ •∗ which decorates vertices affected by a where C is a map H cut by the same integer. This is the only adjustment we have to make when going ˜ and j∆ differ only by decoration. This yields from (18) to (19) because ∆j (SR id)B+•

˜ +• = π m( ˜B ˜+• j π m( ˜ S˜R ⊗ id)∆jB ˜ S˜R ⊗ id)∆ ˜ ˜ ˜ ˜ π m( ˜ SR ⊗ id)(C(id ⊗ B+• )∆ + B+• ⊗ I)j ˜+• j ˜+• )∆j ˜ + π S˜R B π m( ˜ S˜R ⊗ id)C(id ⊗ B ˜+• )∆j ˜ − πR ˜ m( ˜∆ ˜B ˜+• j ˜ S˜R ⊗ id)Q π m( ˜ S˜R ⊗ id)C(id ⊗ B

= = = =

= =

˜+• )∆j ˜ − πR ˜ m( ˜+• )∆j ˜ π m( ˜ S˜R ⊗ id)C(id ⊗ B ˜ S˜R ⊗ id)C(id ⊗ B ˜+• )∆j ˜ (id − R)π m( ˜ S˜R ⊗ id)C(id ⊗ B ˜+• )∆j ˜ (id − R)π mC( ˜ S˜R ⊗ id)(id ⊗ B ˜ ˜ +• )(S˜R ⊗ id)∆j (id − R)π mC(id ˜ ⊗B

=

(id − R)B+ (SR id),

= =

where we have used (19), Q(id ⊗ B+• ) = id ⊗ B+• , Q(B+• ⊗ I) = 0 which are obvious, and (S˜R ⊗ id)C = C(S˜R ⊗ id) and mC(id ˜ ⊗ B+• ) = B+• m ˜ which follow from the definition of C. This finishes the proof. Example 9 We illustrate the action of the map ˜∆ ˜ ⊗ id)∆j ˜ SR id = −πRm(−R ˜ m(. ˜ . . ⊗ id)Q on the two trees

• •

• and AA . • •

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•

H•∗

•

˜ ∆j

? ˜ •∗ ⊗ H ˜ •∗ H

• •

⊗I+I⊗

•

+ •1 ⊗ •1

•

˜ Q∆⊗id

? •∗ ⊗3 ˜ (H )

• • + •1 ⊗ •1 ⊗ I − (I ⊗ I) ⊗ + I ⊗ •1 ⊗ •1 I⊗ • •

S˜R ⊗id⊗id

? ˜ •∗ )⊗3 (H

• • − ∗1 ⊗ •1 ⊗ I − (I ⊗ I) ⊗ + I ⊗ •1 ⊗ •1 I⊗ • •

−Rm⊗id ˜

? ˜ •∗ ⊗ H ˜ •∗ H

∗ ∗ • + − ∗1 ⊗ •1 − ⊗I+I⊗ ∗ • •

πm ˜

? H•∗

−

∗ •

+

∗ ∗

+

• •

−

• ∗

.

˜ •∗ Note that we do not need to go into higher than the third tensor power of H because SR (I) = I and hence SR (•) = −∗ terminate the recursion. Now the second, less trivial example:

358


H•∗


• A • A•

˜ ∆j

? ˜ •∗ ⊗ H ˜ •∗ H

• •1 •2 • A ⊗ I + I ⊗ A + •1 ⊗ + •2 ⊗ + •1 •2 ⊗•12 • • • A• • A•

˜ Q∆⊗id

? •1 •2 • • ˜ •∗ )⊗3 I ⊗ A + •1 ⊗ (H + • ⊗ + • • ⊗• ⊗ I − I ⊗ I ⊗ AA 2 1 2 12 A • • • • • • •1 •2 + I ⊗ •2 ⊗ + (I ⊗ •1 •2 + •1 ⊗ •2 + •2 ⊗•1 ) ⊗ •12 +I ⊗ •1 ⊗ • • S˜R ⊗id⊗2

? •1 •2 • • ˜ •∗ )⊗3 I ⊗ A − ∗1 ⊗ (H − ∗ ⊗ + ∗ ∗ ⊗• ⊗ I − I ⊗ I ⊗ AA 2 1 2 12 A • • • • • • •1 •2 +I ⊗ •1 ⊗ + I ⊗ •2 ⊗ + (I ⊗ •1 •2 − ∗1 ⊗ •2 − ∗2 ⊗•1 ) ⊗ •12 • • −Rm⊗id ˜

? ˜ •∗ ⊗ H ˜ •∗ H

• ∗ ∗ ∗ ∗ ⊗ I + I ⊗ AA − AA + AA + AA − AA • • ∗ ∗ • ∗ ∗ • • • •1 •2 − ∗1 ⊗ − ∗2 ⊗ + ∗1 ∗2 ⊗•12 • •

πm ˜

? H•∗

2.4

• • • ∗ ∗ ∗ − AA + 2 AA − AA + AA − 2 AA + AA . ∗ ∗ ∗ • • • ∗ ∗ ∗ • • •

An alternative presentation of the Hopf algebra

In this subsection we give a somewhat different presentation H of H which will turn out to be more instructive for Epstein-Glaser renormalization. The basic idea is as follows: We consider a tree t of the preceding subsections as a trunk and let two more branches, called “hair”, grow out of each leaf and one more branch out

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of each unary vertex of the trunk. This yields a tree t in the presentation H. • A • t= → t = • A A • ◦ A◦ A◦ While the trunk will correspond to an abstract nest of subdivergences, the leaves of the hairy tree actually represent (some unordered set of) space-time points to which that particular subdivergence situation applies. For the reader’s convenience, we visualize hair by ◦ and the trunk vertices by •. This is only to make it easier to distinguish between the bold trees in H and the hairy trees in H, so we are not talking about trees with “two types of vertices” here. Now in order to underline the power of the Hochschild 1-cocycle and to illustrate Lemma 6, we will prescribe the cocycle and see what the coproduct looks like then. Let H be the free commutative algebra generated by rooted trees the leaves of which descend exclusively from binary vertices. In other words each leaf must have one and only one sibling (which is not necessarily a leaf too). For example, the trees • @ • A • @ A , • A , @ @• A A • • ◦ A◦ A A A ◦ A◦ ◦ A A ◦ ◦ ◦ ◦ ◦ A◦ are in H while • A • • A , • •A• , A • •A• • • A• are not. The tree • consisting only of the root is not in H by convention, so the most “primitive” generator is • AA . ◦ ◦ Now we demand B + to act as follows: B + (I)

=

• B + ( AA ) ◦ ◦

=

• A ◦ A◦ • A • A A A ◦ A◦ ◦

B + (t)

=

• • A • A◦ , so t is grafted to a leaf of •AA◦ t

and for a forest, B + (t1 . . . tn )

=

• A@ A@• • t1 . . . tn

in general, for any tree t,

360



Lemma 10 There is a unique Hopf algebra structure (∆, , S) on H such that B + is Hochschild closed. ∆ is given on trees t by

∆(t) = I ⊗ t + t ⊗ I +

P c (t) ⊗ Rc (t)

adm c

where the definition of admissible cuts and P c , Rc is as in the preceding subsections with the following modifications: (i) cuts containing external edges (hair) are not admissible here (ii) if a vertex v of Rc (t) has no more outgoing edges due to cut edges in c, that • vertex v is to be replaced by AA in Rc (t). ◦ ◦ If a vertex v of Rc (t) is left with only one outgoing edge due to cut edges in c, an additional branch is to be adjoined to v in Rc (t). The map β : H → H, given by removing the hair, i.e., all leaves and adjacent edges, is an isomorphism of Hopf algebras. β −1 in turn replaces vertices with fertility 0 or 1 by binary vertices. Sketch of proof. First of all we note that whole H − kI is the iterated image of B + and the multiplication. Moreover, H is graded as an algebra by the number of internal (non-hairy) vertices. The existence and uniqueness of (∆, , S) is then a consequence of Lemma 6. The remaining statements are easy to check using the map β, in particular β

• A = •, ◦ A◦

 • A • . β  • A  = AA A • ◦ ◦ ◦ 

Therefore H is nothing but a somewhat different presentation of H. Using β, we can transfer all notions developed in the preceding subsections to H (which we denote by underlining everything). Note that in H•∗ only internal vertices can ˜ •∗ only internal vertices are decorated etc. From now on, we have type ∗, in H ˜ •∗ . work only in the presentation H, H•∗ , H

2.5

Feynman rules and counterterms. Main result

On the Hopf algebra level, a tree represents a certain subdivergence situation. Internal vertices of type • mean that the unrenormalized Feynman rules have been applied to the respective subdivergence, while ∗ denotes the corresponding counterterm. For example, • A ◦ A◦

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corresponds to the distribution 0 T2 : f1 ⊗ f2 → (p1,2 T1 ⊗ T1 )(f1 ⊗ f2 ) + (p2,1 (T1 ⊗ T1 )(f2 ⊗ f1 ), defined on M 2 − ∆2 . Again we do not display the Wick multiindex r for simplicity. The tree ∗ A ◦ A◦ ∗ 0 represents the counterterm −W12 T2 . We already know that their sum (id − ∗ 0 W12 ) T2 = T2 is the well-defined Epstein-Glaser time-ordered product on whole M 2 . In less trivial cases subtrees represent subdivergences, the root represents the overall divergence. For example

∗ @ • @ @ @ @• @ ∗ ∗ A A A ◦ A◦ ◦ A◦ ◦ A◦ yields ∗ ∗ ∗ W123456 p1234,56 p12,34 W12 p1,2 W34 p3,4 p5,6 T1⊗6 + suitable perm. of indices.

Epstein-Glaser renormalization is essentially a binary operation since in each step only products T (I)T (N − I) of two operator-valued distributions are considered. Indeed, it is impossible to extend a distribution from M n − Dn (for n > 2) onto the thin diagonal in (M n − Dn )∪∆n without extending it to the thicker diagonals, e.g., {xi = xj for some i, j} first. So we will be needing only binary trees here. Now let t be a binary tree in H. All of its internal vertices are of type •. We need a map which changes the types of internal vertices of t in all possible combinations and sums up the resulting trees in order to take care of the Bogoliubov recursion. This is essentially done by S R id, as we have proved in Lemma 8. In order to avoid overcounting, we will have to take care of the symmetry factors which show up whenever the coproduct is applied. For instance, in the second part ∗ of Example 9 we got 2 AA because two cuts, one on the “left”, the other on the ∗ • “right-hand side”, yield the same result. We will compensate that by eventually dividing by symmetry factors. Let T 1 = {I} and for n ≥ 2 let T n be the subset of H of binary trees with n leaves. Furthermore, let F V be the free commutative algebra generated by the graded vector space ∞ V := Dop (M n − Dn ) n=0 (M n − Dop n

Dn ) is the space of collections (Tnr ) of operator-valued distriwhere butions on M − Dn (again r is an n-multiindex referring to the Wick powers under consideration). By Dn we continue to denote the thick diagonal in M n .

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Thus elements of F V are formal free commutative products of operator-valued distributions on the configuration spaces M n − Dn carrying a Wick-multiindex. The free commutative product is supposed to model the analogue of the disjoint union of trees. We don’t actually need it to state the theorem, but it is instructive to keep it in mind. The reader might wish to review the notation for Epstein-Glaser time-ordered products in Subsection 1.2 at this point. Theorem 11 (Main result) Let Φ : H•∗ → F V be the homomorphism of free commutative algebras such that Φ(I) = T1 where T1k =: φk : and n for n ≥ 2, 1 ≤ i ≤ n − 1, ti ∈ T i , tj ∈ T n−i and f1 . . . fn ∈ D(M ) such that i=1 supp fi = ∅, Φ(B+• (ti tj )) is the collection of distributions defined by Φ(B+• (ti tj ))(f1 ⊗ . . . ⊗ fn ) = =

1 S(ti , tj )

pI,N −I Φ(ti )(⊗k∈I fk )Φ(tj )(⊗l∈N −I fl )+

I⊂N,|I|=i

+pN −I,I Φ(tj )(⊗l∈N −I fl )Φ(ti )(⊗k∈I fk ), ∗ = W1...n Φ(B+• (ti tj )).

Φ(B+∗ (ti tj ))

while Φ(t ) = 0 on non-binary trees t . The symmetry factor S(ti , tj ) := 2 if the root of ti has type • and tj = R(ti ), and S(ti , tj ) := 1 otherwise. Using these renormalized Feynman rules Φ, the n-th Epstein-Glaser time-ordered product is (the unique extension onto M n of ) Tn :=

Φ(S R id)(t).

(20)

t∈T n

Note that in an obvious abuse of notation we consider the counterterms as distributions on M n − Dn too. Recall that the extension onto M n is only unique up to the ωα as discussed in Corollary 5. We assume here that for each n, those ωα have been chosen once and forever according to some renormalization scheme. Proof. For n = 1 and n = 2 the statement is obviously true (take t1 = t2 = I). ∗ Now for t ∈ T n it is easy to see that (ΦR)(t) = (W1...n Φ)(t) (note that W ∗ is idempotent as well) and ΦB +• is the very sum of causal partitions times lower order time-ordered products that shows up in the equation ∗ ) Tn = (id − W1...n

∅IN

pI,N −I T (I)T (N − I)

(21)

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which defines the time-ordered product Tn by Corollary 5. Symbolically, the diagrams H•∗

- FV

Φ

H•∗

×

B +•

? H•∗

pI,N −I ...

- FV W∗

R

? H•∗

? - FV

Φ

Φ

Φ

? - FV

commute. This can be seen as follows: T n = B + (T n−1 ) ∪

n−2

B + (T i T n−i ) =

i=2

n−1

B + (T i T n−i )

i=1

where we are overcounting since H is commutative. Using the Hochschild closedness of B+ in the form of Lemma 8 and the fact that S R id is an algebra homomorphism, we get by induction on n, using the symmetry factor S (ti , tj ) := 2 if ti = tj and S (ti , tj ) := 1 otherwise: Tn

=

Φ(S R id)(t)

t∈T n

=

n−1 1 2 i=1

S (ti , tj )Φ(S R id)B + (ti tj )

ti ∈T i tj ∈T n−i

=

=

1 2

n−1

S (ti , tj )Φ(id − R)B +• (S R id)(ti tj )

i=1 ti ∈T i tj ∈T n−i n−1



1 ∗ (id − W1...n ) ΦB +•  2 i=1

ti ∈T i

= =

(S R id)(ti )

 (S R id)(tj ) + C 

tj ∈T n−i

n−1 1 ∗ (id − W1...n ) pI,N −I T (I)T (N − I) + pN −I,I T (N − I)T (I) 2 i=1 I⊂N,|I|=i ∗ (id − W1...n ) pI,N −I T (I)T (N − I) ∅IN

where C is eventually C = t (S R id)(t)(S R id)(t) (for each t such that ti = tj =: t has occurred in the sum above, thus in particular for all t ∈ T n/2 if n is even) which cancels the symmetry factor S(ti , tj ) in the statement of the theorem. This finishes the proof. While the preceding theorem just defines Φ inductively by pushing it forward along B + , which is a perfectly natural way of doing so, one might also work out a

364



non-recursive formula for Φ as follows: Draw the tree, scan it from the top to the bottom and wherever you see an ∗, apply W ∗ . Then symmetrize in all possible ways. Since H is nothing but a different presentation of H, one could also have stated the theorem in terms of trees of H from the very beginning, which would have required a grading on H that is isomorphic to the grading of H by the number of external (hairy) vertices. We encourage the reader to check that one could obtain the same result in complete analogy to momentum space renormalization (BPHZ, dimensional regularisation, etc.) [3, 4, 9, 12] as reviewed in Subsection 2.2 by the following approach: Define the (unrenormalized) Feynman rules Φ : H → H •∗ → F V as in Theorem 11, but let now R : F V → F V be the idempotent algebra endomorphism T → W ∗ T. Note that R is a Rota-Baxter operator. Then replace the cut product by the usual convolution product again, and an adaptation of Theorem 11 yields (S Φ Tn = R Φ)(t) t∈T n

which should be compared to (16). The reason why we preferred the method of letting R act in the Hopf algebra H•∗ and using is that like this we achieved a complete decoupling of the combinatorics (which happen in H•∗ ) and the analysis (which happens in V ), making it easier to see how the essential work is being done on the Hopf algebra side while the renormalized Feynman rules Φ : H•∗ → F V is a rather trivial map translating abstract subdivergence situations into the appropriate operator valued distributions.

3 Conclusions and outlook We have seen how Hopf algebras of rooted trees take care of the combinatorics of Epstein-Glaser renormalization. It is the twisted antipode SR which provides a complete set of counterterms and formally solves the Bogoliubov recursion thanks to the Hochschild closedness of B+ . The statement of Lemma 8 also amounts to the fact that the counterterms are local. Indeed, once the subdivergences are taken care of, it suffices to subtract the superficial divergence, i.e., to extend a distribution onto the thin diagonal. Although we do not claim that the statement of Theorem 11 makes actual calculations easier, it closes the gap between the Epstein-Glaser approach and the Hopf algebra picture in momentum space. Starting from Theorem 11, one rather easily derives Feynman rules Φ for the vaccum expectation values of time-ordered products. One can also try to construct a coproduct on the vacuum expectation values of time-ordered products. Finally, we would like to mention another issue which seems to be intimately related to the above approach to coordinate space renormalization: constructing

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an analogy between extension of distributions from M n − Dn to M n and compactification of the configuration space M n − Dn of n points in M. Indeed, we can already see how this leads to rooted trees if we look at the Fulton-MacPherson compactification of configuration spaces [7, 1, 8] defined as follows: Let M be a smooth manifold. There is an obvious inclusion of the configuration space into a product of blowups, Bl(M |I| , ∆|I| ) (22) M n − Dn → M n × I⊂N,|I|≥2

where Bl(M i , ∆i ) is the (differential-geometric) blowup of M i along ∆i of M i , i.e., M i where the thin diagonal ∆i is replaced by the sphere bundle in the normal bundle over ∆i . For the details, the reader is referred to [1]. The Fulton-MacPherson compactification M [n] of M n − Dn is then the closure of M n − Dn upon this inclusion. Obviously M [n] has only a chance to be compact if M is compact. Now a closer look at what happens in the right-hand side of (22) when a sequence approaches the thin diagonal in M n leads to a nice description of M [n] in terms of nested screens [7, 1]. In particular, it can be shown that there is a stratification of the manifold with corners M [n], M (S) M [n] = S∈S

where S is the set of all nests of subsets of N = {1, . . . , n} with at least 2 elements. Now nested sets are perfectly described by the forests in H. Moreover, if we restrict our attention to M = Rk and replace M n − Dn by the moduli space F˙k (n) := (M n −Dn )/G(k) where G(k) is the subgroup (acting diagonally) of affine transformations of Rk generated by translations by elements of Rk and dilatations by elements of R+ , there is an operad structure behind the Fulton-MacPherson compactifications Fk (n) of the moduli spaces F˙k (n) [8, 13]. The compactifications M [n] of the configuration spaces still furnish a right module over the operad Fk . Operads arise in a natural way when rooted trees are grafted to each other: • @ • • • • @ A , A , A A , → @ A A A A @• • • • • • • • • ••• A A A • A• • A• • A• It seems tempting to explore possible relations between the operad µF M of FultonMacPherson compactified moduli spaces Fk (n), the operad µEG which arises when the trees in H we used for Epstein-Glaser renormalization are grafted to each other, and finally the operad of Feynman graph insertions µF G [13, 11]. The operad µF G is closely related to the pre-Lie structure of Feynman graphs which is dual in a certain sense to the coproduct in H. This might establish a true analogy between the Fulton-MacPherson compactification M [n] of M n −Dn and the renormalization of time-ordered products in the sense of Epstein-Glaser.

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Acknowledgments It is a pleasure to thank Henri Epstein and Ivan Todorov for valuable discussion and helpful comments during a series of talks given on the subject. The first named author would also like to thank the IHES for generous hospitality and the German Academic Exchange Service (DAAD) for financial support.

References [1] S. Axelrod and I.M. Singer, Chern-Simons perturbation theory 2, J. Diff. Geom. 39, 173–213 (1994); hep-th/9304087. [2] R. Brunetti and K. Fredenhagen, Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds, Commun. Math. Phys. 208, 623–661 (2000); math-ph/9903028. [3] A. Connes and D. Kreimer, Hopf algebras, renormalization and non-commutative geometry, Commun. Math. Phys. 199, 203–242 (1998); hep-th/9808042. [4] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210, 249–273 (2000); hep-th/9912092. [5] H. Epstein and V. Glaser, The role of locality in perturbation theory, Ann. Inst. Henri Poincaré, Section A, Vol. XIX, n. 3, 211 (1973). [6] L. Foissy, Les algèbres de Hopf des arbres enracinés décorés, Thesis, 2001, Department of Mathematics, University of Reims. [7] W. Fulton and R. MacPherson, A compactification of configuration spaces, Ann. of Math. 139, 183–225 (1994). [8] E. Getzler and J.D.S. Jones, Operands, homotopy algebra and iterated integrals for double loop spaces, Preprint hep-th/9403055. [9] D. Kreimer, On the Hopf algebra structure of perturbative quantum field theory, Adv. Theor. Math. Phys. 2.2, 303–334 (1998); q-alg/9707029. [10] D. Kreimer, Chen’s Iterated Integral represents the Operator Product Expansion, Adv. Theor. Math. Phys. 3, 3 (2000); Adv. Theor. Math. Phys. 3, 627–670 (1999); hep-th/9901099. [11] D. Kreimer, Combinatorics of (perturbative) quantum field theory, Phys. Rept. 363, 387–424 (2002); hep-th/0010059. [12] D. Kreimer, Factorization in quantum field theory: an exercise in Hopf algebras and local singularities, Contributed to Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry, Les Houches, France, 9–21 Mar 2003; hep-th/0306020.

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[13] M. Markl, S. Shnider and J. Stasheff, Operads is Algebra, Topology and Physics, volume 96 of Mathematical surveys and monographs. Amer. Math. Soc., Providence, RI, 2002. [14] G. Pinter, The Action Principle in Epstein Glaser Renormalization and Renormalization of the S-Matrix in φ4 -Theory, Annalen Phys. 10, 333–363 (2001); hep-th/9911063. [15] G. Popineau and R. Stora, A Pedagogical Remark on the Main Theorem of Perturbative Renormalization Theory, Unpublished preprint. [16] G. Scharf, Finite Quantum Electrodynamics: The Causal Approach, SpringerVerlag, 1995, 2nd edition.

Christoph Bergbauer Freie Universit¨ at Berlin II. Mathematisches Institut Arnimallee 3 D-14195 Berlin Germany email: [email protected] and ´ Institut des Hautes Etudes Scientifiques F-91440 Bures-sur-Yvette France

Dirk Kreimer ´ Institut des Hautes Etudes Scientifiques 35, route de Chartres F-91440 Bures-sur-Yvette France email: [email protected] and Boston University Department of Mathematics and Statistics Center for Mathematical Physics Boston, MA 02215 USA

Communicated by Vincent Rivasseau submitted 29/03/04, accepted 01/06/04




Integrable Renormalization II: The General Case Kurusch Ebrahimi-Fard, Li Guo and Dirk Kreimer Abstract. We extend the results we obtained in an earlier work [1]. The cocommutative case of ladders is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the double Rota-Baxter construction, respectively Atkinson’s theorem. We also outline the extension to the Hopf algebra of Feynman graphs via decorated rooted trees.

1 Introduction The perturbative approach to quantum field theory (QFT) has been spectacularly successful in the past. It is based on a priori formal series expansions of Green functions in orders of a coupling constant, measuring the strength of the corresponding interaction. Terms in these series expansions are indexed by Feynman diagrams, a graphical shorthand for the corresponding Feynman integrals. Physically relevant quantum field theories when treated perturbatively develop short-distance singularities present in all superficially divergent contributions to the perturbative expansion. Renormalization theory [2] allows nevertheless for a consistent way to treat these divergent Feynman integrals in perturbative QFT. The intricate combinatorial, algebraic and analytic structure of renormalization theory within QFT is by now known for almost 70 years. Within the physics community the subject reached its final and satisfying form through the work of Bogoliubov, Parasiuk, Hepp, and Zimmermann. It was very recently that one of us in [3] discovered a unifying scheme in terms of Hopf algebras and its duals underlying the combinatorial and algebraic structure. This Hopf algebraic approach to renormalization theory, as well as the related Lie algebra structures, were exploited in subsequent work [4, 5, 6, 7, 8, 9, 10, 11]. The focus of an earlier work of us [1]1 and this article is on the algebraic Birkhoff decomposition discovered first in [4, 8, 9], and the related Lie algebra of rooted trees, respectively Feynman graphs [10, 11]. The Rota-Baxter algebra structure on the target space of (regularized) Hopf algebra characters showed to be of crucial importance with respect to the Birkhoff decomposition (in [4] this relation appeared under the name multiplicativity constraint). Using a classical r-matrix ansatz, coming simply from the Rota-Baxter map, we were able to derive in (I) the formulae for the factors φ± [8] for the decomposition of a Hopf algebra character φ in the case of the Hopf subalgebra of rooted ladder 1 For

the rest of this work we will cite paper [1] by (I).

370

K. Ebrahimi-Fard, Li Guo and D. Kreimer


¯ trees. Bogoliubov’s R-map finds its natural formulation in terms of a character with values in the double Rota-Baxter algebra of the above target space Rota-Baxter φ algebra. The counterterm SR = φ− and the renormalized character φ+ simply lie ˜ = R − id, respectively, of the in the images of the group homomorphisms R, −R Bogoliubov character. In this work we would like to extend these results to the general case, i.e., the full Hopf algebra of arbitrary rooted trees. The main difference lies in the fact that in the rooted ladder tree case we worked with a cocommutative Hopf algebra, or dually, with the universal enveloping algebra of an Abelian Lie algebra. In the general case the Lie algebra of infinitesimal characters is non-Abelian, correspondingly the Hopf algebra is non-cocommutative, necessitating a more elaborate treatment due to contributions from the Baker-Campbell-Hausdorff (BCH) formula. The modifications coming from these BCH contributions have to be subtracted order by order in the grading of the Hopf algebra. Hence we will define in a recursive manner an infinitesimal character in the Lie algebra which allows for the Birkhoff decomposition of the Feynman rules regarded as an element of the character group of the Hopf algebra. The factors of the derived decomposition φ = φ− give the formulae for the renormalized character φ+ and the counterterm SR ¯ introduced in [3, 4, 8]. Bogoliubov’s R-map becomes a character with values in the double of the target space Rota-Baxter algebra. It should be underlined that the above ansatz in terms of an r-matrix solely depends on the algebraic structure of the Lie algebra of infinitesimal characters, i.e., the dual of the Hopf algebra of (decorated) rooted trees or Feynman graphs, and on the Rota-Baxter structure underlying the target space of characters. When specializing the Rota-Baxter algebra to be the algebra of Laurent series with pole part of finite order, this approach naturally reduces to the minimal subtraction scheme in dimensional regularization, or to the momentum scheme, which are both widely used in perturbative QFT and thoroughly explored in [3, 4, 5, 6], which extend to non-perturbative aspects still using the Hopf algebra [7]. The paper is organized as follows. In the following section, we introduce the notion of Rota-Baxter algebras and recall some related basic algebraic facts, like the relation to the notion of classical Yang-Baxter type identities, the double Rota-Baxter construction and Atkinson’s theorem. After that, we review the notion of a renormalization Hopf algebra by introducing the universal object [8] for such Hopf algebras, the Hopf algebra of rooted trees. Having this at our disposal, generalizations to the Hopf algebra of decorated rooted trees or Feynman graphs are a straightforward generalization which we will outline later on. Its dual, containing the Lie group of Hopf algebra characters, and the related Lie algebra of its generators, i.e., the infinitesimal characters, is introduced without repeating the details which are by now standard [8]. In section four, which contains the main part of this paper, the notion of a regularized character is introduced as a character with values in a Rota-Baxter algebra. Note that Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormaliza-

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tion falls into this class, even though it makes no use of a regulator, but of a Taylor operator on the integrand instead, which provides a Rota-Baxter map similarly. This is immediate upon recognizing that disjoint one-particle irreducible graphs allow for independent Taylor expansions in masses and momenta. This allows us to lift the Lie algebra of infinitesimal characters to a Rota-Baxter Lie algebra, giving the notion of a classical r-matrix on this Lie algebra. We then review briefly the results of (I), i.e., the Birkhoff factorization in the cocommutative case. Motivated by this result for the simple case of rooted ladder trees, we solve here the factorization problem for the non-cocommutative Hopf algebra of rooted trees by defining a BCH-modified infinitesimal character. Section four closes with some calculations using the notion of normal coordinates intended to make the construction of the modified character in terms of the BCH-corrections more explicit, and a remark on decorated, non-planar rooted trees and Feynman graphs.

2 Rota-Baxter algebras: from Baxter to Baxter The Rota-Baxter (RB) relation first appeared in 1960 in the work of the American mathematician Glen Baxter [12]. Later it was explored especially by the mathematicians F.A. Atkinson, G.-C. Rota and P. Cartier [13, 14, 15]. In particular, Rota underlined its importance in various fields of mathematics, especially within combinatorics [16]. But it was very recently that after a period of dormancy it showed to be of considerable interest in several so far somewhat disconnected areas like Loday type algebras [17, 18, 19, 20, 21], q-shuffle and q-analogs of special functions through the Jackson integral [22], differential algebras [23, 24], number theory [25], and the Hopf algebraic approach to renormalization theory in perturbative QFT [1, 4, 9]. In particular, in collaboration with Connes, the connection to Birkhoff decompositions based on Rota-Baxter maps was introduced in [9, 10]. It is the latter aspect on which we will focus in this work. In its Lie algebraic version the RB relation found one of its most important applications within the theory of integrable systems, where it was rediscovered in the 1980s under the name of (operator form of the) classical and modified classical Yang-Baxter2 equation [26, 27, 28]. There some of its main features, already mentioned in [13], and Atkinson’s theorem itself, were analyzed in greater detail. Especially the double Rota-Baxter construction introduced in the work of Semenov-Tian-Shansky [26, 27], and the related factorization theorems [29] will be of interest to us. In the following we will collect a few basic results on Rota-Baxter algebras, some of which we will need later, and some of which we state just to indicate interesting relations of these algebras to other areas of mathematics. Of course, the list is by no means complete, and a more exhaustive treatment needs to be done. 2 Referring

to C.N. Yang and the Australian physicist Rodney Baxter.

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Let us start with the definition of a Rota-Baxter algebra [16, 23, 24]. Suppose K is a field of characteristic 0. A K-algebra neither needs to be associative, nor commutative, nor unital unless stated otherwise. Definition 2.1 Let A be a K-algebra with a K-linear map R : A → A. We call A a Rota-Baxter (RB) K-algebra and R a Rota-Baxter map (of weight θ ∈ K) if the operator R holds the following Rota-Baxter relation of weight θ ∈ K 3 : R(x)R(y) + θR(xy) = R R(x)y + xR(y)), ∀x, y ∈ A. (1) Remark 2.2 1) For θ = 0 a simple scale transformation R → θ−1 R gives the so-called standard form: R(x)R(y) + R(xy) = R R(x)y + xR(y)). (2)

2) 3) 4)

5)

6)

For the rest of the paper we will always assume the Rota-Baxter map to be of weight θ = 1, i.e., to be in standard form. ˜ := id−R fulfills the same Rota-Baxter relation. If R fulfills relation (2) then R The images of R and id − R give subalgebras in A. The free associative, commutative, unital RB algebra is given by the mixable shuffle algebra [23] which is an extension of Hoffman’s quasi-shuffle algebra [30, 31]. The case θ = 0, R(x)R(y) = R R(x)y + xR(y)), naturally translates into the ordinary shuffle relation, and finds its most prominent example in the integration by parts rule for the Riemann integral. A relation of similar form is given by the associative Nijenhuis identity [32]: (3) N (x)N (y) + N 2 (xy) = N N (x)y + xN (y) . Given a RB algebra with an idempotent RB map R, the operator Nγ := ˜ γ ∈ K fulfills relation (3). See [20, 33, 34] for recent results with R − γ R, respect to this relation.

Example 2.3 1) The q-analog of the Riemann integral, or Jackson-integral [16, 22], on a well-chosen function algebra F is given by: x J[f ](x) := f (y)dq y 0 := (1 − q) f (xq n )xq n . (4) n≥0

It may be written in a more algebraic version, using the operator: Pq [f ] := Eqn [f ],

(5)

n>0 3 Some

λ = −θ.

authors denote this relation in the form R(x)R(y) = R R(x)y + xR(y) + λxy . So

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where Eq [f ](x) := f (qx), f ∈ F. Pq and id + Pq =: Pˆq are RB operators of weight −1, 1, respectively. Now let us define a multiplication operator Mf : F → F, f ∈ F, Mf [g](x) := [f g](x) = f (x)g(x) which fulfills the associative Nijenhuis relation (3). The Jackson integral is given in terms of the above operators as: J[f ](x) = (1 − q)Pˆq Mid [f ](x),

(6)

and fulfills the following mixed RB relation J[f ] J[g] + (1 − q)JMid [f g] = J J[f ] g + f J[g] .

(7)

In a forthcoming work two of us (K.E.-F., L.G.) will report some interesting implications of this fact with respect to some recent results on q-analog of multiplezeta-values [31]. 2) A rich class of Rota-Baxter maps is given by certain projectors. Within renormalization theory, dimensional regularization together with the minimal subtraction scheme play an important rˆ ole. Here the RB map RMS is of weight θ = 1 and defined on the algebra of Laurent series C , −1 ] [4] with finite pole part. For −1 ∞ k ] it gives: k=−m ck ∈ C , RMS

∞ k=−m

−1 ck k := ck k .

(8)

k=−m

Of equal importance is the projector which keeps the finite part, closely related to the momentum scheme. We now introduce the modified Rota-Baxter relation. Its Lie algebraic version already appeared in [26, 28]. Definition 2.4 Let A be a Rota-Baxter algebra, R its Rota-Baxter map. Define the operator B : A → A, B := id − 2R to be the modified Rota-Baxter map and call the corresponding relation fulfilled by B: B(x)B(y) = B B(x)y + xB(y) − xy, ∀x, y ∈ A (9) the modified Rota-Baxter relation. Remark 2.5 In the following proposition (2.6), we mention the notion of pre-Lie algebras. Let us state briefly its definition. A (left) pre-Lie K-algebra A is a Kvector space, together with a bilinear pre-Lie product : A × A → A, holding the (left) pre-Lie relation: a (b c) − (a b) c = b (a c) − (b a) c, ∀a, b, c ∈ A. The commutator [a, b] := a b − b a, ∀a, b ∈ A fulfills the Jacobi identity.

374



Proposition 2.6 For the Rota-Baxter algebra A to be either an associative or preLie K-algebra, the (modified) Rota-Baxter relation naturally extends to the Lie algebra LA with commutator bracket [x, y] := xy − yx, ∀x, y ∈ A: [R(x), R(y)] + R([x, y]) = R [R(x), y] + [x, R(y)] [B(x), B(y)] = B [B(x), y] + [x, B(y)] − [x, y].

(10) (11)

The proof is a straightforward calculation. The relations (10) and (11) are well known as the (operator form of the) classical Yang-Baxter and modified YangBaxter equation, respectively. Remark 2.7 1) The same is true for the associative Nijenhuis relation (3). In its Lie algebraic version, identity (3) was investigated in [35, 36]. 2) Let A be an associative K-algebra. We regard A⊗A as an A-bimodule, x⊗y ∈ A⊗A and a(x⊗y)b = (ax⊗y)b = ax⊗yb. A solution r := i s(i) ⊗t(i) ∈ A⊗A of the extended associative classical Yang-Baxter relation: r13 r12 − r12 r23 + r23 r13 = θr13 , θ ∈ K gives a RB map β : A → A of weight θ, defined by β(x) := i s(i) x t(i) . The notation rij means, for instance, r13 := i s(i) ⊗1⊗t(i) . This example implies many more interesting results with respect to unital infinitesimal bialgebras, which will be presented elsewhere. The case θ = 0 was already treated in [21], implying a RB map of weight 0. Atkinson gave in [13] a very nice characterization of general RB K-algebras in terms of a so-called subdirect Birkhoff decomposition: Theorem 2.8 (Atkinson [13]): For a K-algebra A with a linear map R : A → A to be a Rota-Baxter K-algebra, it is necessary and sufficient that A has a subdirect Birkhoff decomposition. The proof of this theorem may be found in [13] and will not be given here. Essentially, the subdirect Birkhoff decomposition in this case means that the Cartesian ˜ product D := (R(A), −R(A)) ⊂ A × A is a subalgebra in A × A and that every ˜ element x ∈ A has a unique decomposition x = R(x) + R(x). This should be compared to the results in the Lie algebra case (10) to be found in [26, 27, 29, 37]. We come now to one of the main facts about RB algebras. In the following we assume every RB algebra A to be either an associative algebra or a pre-Lie or Lie algebra. The RB relation then implies furthermore a possibly infinite hierarchy of the same RB structure in each of the former cases. We call this the double RotaBaxter construction of the RB-hierarchy on the RB algebra A, given as follows.

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Proposition 2.9 Let A be a Rota-Baxter algebra with (modified) Rota-Baxter map R, set B = id − 2R. Equipped with the new product: a ∗R b :=

R(a)b + aR(b) − ab 1 = − B(a)b + aB(b) , 2 A is again a Rota-Baxter algebra of the same type, denoted by AR .

(12) (13)

The proof of this proposition is immediate by the definition of ∗R . Following the terminology in [26, 27], we call this new Rota-Baxter algebra AR the double RB algebra of A. It is also in [26] where already the notion of the double RB structure for associative algebras equipped with a modified Rota-Baxter operator was suggested. Remark 2.10 1) Let A be an associative RB algebra. The composition ab := R(a)b−bR(a)+ ab defines a pre-Lie structure on A. This aspect becomes more apparent in the context of Loday’s dendriform structures, for which associative RB algebras give a rich class of interesting examples, see [17, 18, 19, 20, 21] and references therein. 2) It is obvious that Proposition 2.9 implies a whole, possibly infinite, hierarchy (i) (0) (1) of double RB algebras AR (here, ∗ = ∗R and ∗R = ∗R ): (0)

(1)

(i)

(i)

AR := A, AR := (A, ∗R ), . . . , AR := (A, ∗R ), . . . (i)

a ∗R b :=

1 1 di e− 2 tB (a) e− 2 tB (b), a, b ∈ A. dti |t=0

(i)

Let us call AR the ith double RB algebra of A, or equivalently the double of (i−1) AR . The following diagram serves to visualize the so-called RB-hierarchy: ∗

(1)

(1) ∗

(2)

(2) ∗

(3)

(3)

R R R A −− → AR −− → AR −− → AR → · · ·

3) The RB-hierarchy becomes cyclic of period 2 at level i = 3, for R being an (k) (k+2) idempotent RB map, R2 = R, i.e., the kth double product ∗R = ∗R . (i)

4) The Rota-Baxter map R becomes an K-algebra homomorphism between AR (i−1) and AR , i ∈ N: (i) (i−1) R(b). (14) R(a ∗R b) = R(a) ∗R ˜ := id − R, we have 5) For the Rota-Baxter map R (i−1) ˜ ˜ ∗(i) b) = −R(a) ˜ R(a R(b). ∗R R

We therefore have the following diagram of K-algebra homomorphisms: ˜ R,R

(1)

˜ R,R

(2)

˜ R,R

(3)

A ←−− AR ←−− AR ←−− AR ← · · ·

(15)

376



We introduce the following composition, using the shuffle product notion formally. For an associative K-algebra A and a, b ∈ K, we define a

A b := ab + ba. For fixed ai , bj ∈ A, 1 ≤ i ≤ m, 1 ≤ j ≤ n, define recursively (a1 a2 · · · am )

A (b1 b2 · · · bn ) =

a1 ((a2 · · · am )

A (b1 · · · bn )) +b1 ((a1 · · · am )

A (b2 · · · bn )),

Proposition 2.11 Let A be an associative Rota-Baxter algebra. For n ∈ N, x ∈ A we have ˜ 1) integer powers of R(x) and R(x) can be written explicitly as: (−R(x))n = (−1)n R x∗R n = ˜ n R(x)

= =

n n−1 (−R(x))n−k

A xk −R x +

(16)

k=1 (n−1) ∗R n

˜ (−1) R

˜ xn + R

x

n−1

(−R(x))n−k

A xk .

(17)

k=1

2) for A also being commutative the above formulae simplify to: n

=

˜ n R(x)

=

(−R(x))

n

n n−1 −R x + (−R(x))(n−k) xk , k k=1

n−1 n ˜ xn + R (−R(x))(n−k) xk . k

(18)

(19)

k=1

The proof of this proposition follows by induction on n.

3 The Hopf algebra of rooted trees Rooted trees naturally give a convenient way to denote the hierarchical structure of subdivergences appearing in a Feynman diagram [3], and the structure maps of their Hopf algebras describe the combinatorics of renormalization of local interactions, encapsulating Zimmermann’s forest formula. For a renormalizable theory, the hierarchy of subdivergences can always be resolved into decorated rooted trees (the parenthesized words of [3]) upon resolving overlapping divergences using maximal forests [38] corresponding to Hepp sectors. This amounts to a determination of the closed Hochschild one-cocycles of the Hopf algebra of renormalization for a given quantum field theory. This is always possible as the rooted trees Hopf algebra with its one-cocycle B+ is the universal object [8] of graded commutative Hopf algebras. Hence it suffices to study this universal object, while the details

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of a specific Hopf algebra of renormalization of a chosen quantum field theory only provide additional notational excesses, albeit cumbersome, see [5, 6, 7] for applications. The main ingredient of this universal commutative Hopf algebra of rooted trees is given by a well-suited non-cocommutative coproduct, defined in terms of admissible cuts on these rooted trees. The aforementioned forest formula is then given essentially by the recursively defined antipode of this Hopf algebra, coming for free from mathematical structure.

Figure 1. A rainbow diagram and corresponding rooted tree of weight 8. Having the Hopf algebra of rooted trees, organizing the algebraic and combinatorial aspects of renormalization, the description of the analytical structure in terms of the group of so-called (regularized) Hopf algebra characters takes place within the dual of this Hopf algebra, being an associative algebra with respect to convolution. Let us introduce the Hopf algebra of rooted trees [8, 39, 40], which we will denote as Hrt . The base field K is assumed once and for all to be of characteristic zero. By definition a rooted tree T is made out of vertices and nonintersecting oriented edges, such that all but one vertex have exactly one incoming line. We denote the set of vertices and edges of a rooted tree by V (T ), E(T ) respectively. The root is the only vertex with no incoming line. Each rooted tree is effectively a representative of an isomorphism class, and the set of all isomorphism classes will be denoted by Trt .

···

···

Definition 3.1 The commutative, unital, associative K-algebra of rooted trees Art is the polynomial algebra, generated by the symbols T , each representing an isomorphism class in Trt . The unit is the empty tree, denoted by 1, and the product of rooted trees is denoted by concatenation, i.e., mArt (T, T ) =: T T . We define a grading on the rooted tree algebra Art in terms of the number of vertices of a rooted tree, #(T ) := |V (T )|. This is extended to monomials, i.e., so-called

378



n (n) forests of rooted trees, by #(T1 · · · Tn ) := i=1 #(Ti ), so that Art = n≥0 Art becomes a graded, connected, unital, commutative, associative K-algebra. Let us introduce now the notion of admissible cuts on a rooted tree. A cut cT of a rooted tree is a subset of the set of edges of T , cT ⊂ E(T ). It becomes an admissible cut, if and only if along a path from the root to any of the leaves of the tree T , one meets at most one element of cT . By removing the set cT , E(T ) − cT , each admissible cut cT produces a monomial of pruned trees, denoted by PcT . The rest, which is a rooted tree containing the original root, is denoted by RcT . We exclude the cases, where cT = ∅, such that RcT = T, PcT = ∅ and the full cut, such that RcT = ∅, PcT = T . We extend the rooted tree algebra Art to a bialgebra Hrt by defining the co-unit : Hrt → K: 0 T1 · · · Tn = 1 (T1 · · · Tn ) := (20) 1 else. The coproduct ∆ : Hrt → Hrt ⊗ Hrt is defined in terms of the set of all admissible cuts CT of a rooted tree T : ∆(T ) = T ⊗ 1 + 1 ⊗ T + PcT ⊗ RcT . (21) cT ∈CT

It is obvious, that this coproduct is non-cocommutative. We extend this by definition to an algebra morphism. Definition 3.2 The graded connected Hopf algebra Hrt := (Art , ∆, ) is defined as the algebra Art equipped with the above defined compatible coproduct ∆ : Hrt → Hrt ⊗ Hrt (21), and co-unit : Hrt → K (20). Remark 3.3 1) The coproduct can be written in a recursive way, using the B + operator, which is a Hochschild 1-cocycle [4, 8, 39]: ∆(B + (Ti1 · · · Tin )) = T ⊗ 1 + {id ⊗ B + }∆(Ti1 · · · Tin ).

(22)

B + : Hrt → Hrt is a linear operator, mapping a (forest, i.e., monomial of) rooted tree(s) to a rooted tree, by connecting the root(s) to a new adjoined root: B + (1) = , B + ( ) = , B + ( ) =

, B+(

)=

, B+(

)=

···

It therefore raises the degree by 1. Every rooted tree lies in the image of the B + operator. Its conceptual importance with respect to fundamental notions of physics was illuminated recently in [41]. 2) The Hopf algebra Hrt contains a commutative, cocommutative Hopf subl algebra Hrt , generated by the so-called rooted ladder trees, denoted by the

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symbol tn , n ∈ N and recursively defined in terms of the B + operator, t0 := 1, m tm = B + (1). The coproduct (21) therefore can be written as: ∆(tn ) = tn ⊗ 1 + 1 ⊗ tn +

n−1

ti ⊗ tn−i .

(23)

i=1

The bialgebra Hrt actually is a graded connected Hopf algebra, since due to its grading and connectedness, it comes naturally equipped with an antipode S : Hrt → Hrt , recursively defined by: S(T ) := −T − S(PcT )RcT . (24) cT ∈CT ∗ of the Hopf algebra of rooted trees, i.e., linear We come now to the dual Hrt ∗ , T ∈ maps from Hrt into K. It is convenient to denote f (T ) =: f, T ∈ K, f ∈ Hrt Hrt . Equipped with the convolution product:

f g(T )

:= (21)

=

mK (f ⊗ g)∆(T ), T ∈ Hrt . f (T ) + g(T ) + f (PcT )g(RcT )

(25)

cT ∈CT ∆

f ⊗g

m

K Hrt −→ Hrt ⊗ Hrt −−−→ K ⊗ K −−→ K

it becomes an associative K-algebra. Its unit is given by the co-unit . Remark 3.4 Higher powers of the convolution product are defined as follows: f1 f2 · · · fn := mK (f1 ⊗ f2 ⊗ · · · ⊗ fn )∆(n−1)

(26)

∆(0) := id, ∆(k) := (id ⊗ ∆(k−1) ) ◦ ∆. ∗ contains the set charK Hrt of Hopf algebra characters, i.e., multiplicative Hrt linear maps with values in the field K.

Definition 3.5 A linear map φ : Hrt → K is called a character if φ(T1 T2 ) = φ(T1 )φ(T2 ), Ti ∈ Hrt , i = 1, 2, i.e., φ(1) = 1K . We denote the set of characters by charK Hrt . Proposition 3.6 The set of characters charK Hrt forms a group with respect to the convolution product (25). The inverse of φ ∈ charK Hrt is given in terms of the antipode (24), φ−1 := φ ◦ S. Definition 3.7 A linear map Z : Hrt → K is called derivation, or infinitesimal character if Z(T1 T2 ) = Z(T1 )(T2 )+(T1 )Z(T2 ), Ti ∈ Hrt , i = 1, 2, i.e., Z(1) = 0. The set of infinitesimal characters is denoted by ∂ charK Hrt .

380



Lemma 3.8 For any Z ∈ ∂ charK Hrt and T ∈ Hrt of degree #(T ) = n < ∞, we have for m > n, Z m (T ) = 0. Remark 3.9 The last result implies that the exponential exp∗ (Z)(T ) := k≥0 Z k k! (T ),

Z ∈ ∂ charK Hrt , is a finite sum, ending at k = #(T ).

The above facts culminate into the following important results [8, 40]. Given the explicit base of rooted trees generating Hrt , the set of derivations ∂ charK Hrt is generated by the dually defined infinitesimal characters, indexed by rooted trees: ZT (T ) = ZT , T := δT,T .

(27)

Proposition 3.10 The set ∂ charK Hrt defines a Lie algebra, denoted by LHrt , and equipped with the commutator: ZT ZT − ZT ZT n(T , T ; T ) − n(T , T ; T ) ZT ,

[ZT , ZT ] := =

(28)

T ∈Trt

where the n(T , T ; T ) ∈ N denote so-called section coefficients, which count the number of single simple cuts, |cT | = 1, such that PcT = T and RcT = T . The exponential map exp∗ : LHrt → charK Hrt defined in remark (3.9) is a bijection. Generated by the infinitesimal characters ZT (27), the Lie algebra LHrt carries naturally a grading in terms of the grading of the rooted trees in Hrt , deg(ZT ) := (n) #(T ), and LHrt = n>0 LHrt . The commutator (28) implies then:

(n) (m) (m+n) LHrt , LHrt ⊂ LHrt .

(29)

Let us calculate a few commutators, to get a better feeling for the structure of LHrt : [Z , Z ] = Z + 2Z − Z = 2Z (30) [Z , Z ] = Z + Z

[Z , Z ] =

+ 2Z

−Z =Z

1 [[Z , Z ], Z ] = Z 2

− 3Z

+ 2Z

−Z .

This Lie algebra received more attention recently [11, 42, 43], but needs further structural analysis, since it captures in an essential way the whole of renormalization and the structure of the equations of motion [7] in perturbative QFT. This remark is underlined by the results presented in the next section.

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4 Classical r-Matrix and Birkhoff decomposition For a renormalizable theory, the process of renormalization removes the shortdistance singularities order by order in the coupling constant. For this to work one has to choose a renormalization scheme which determines the remaining finite part. This choice is of analytic nature but also contains an important algebraic combinatorial aspect, which lies at the heart of the Birkhoff factorization, found in [4, 9]. It is the goal of this section to clarify how this algebraic step implies the Birkhoff decomposition in a completely algebraic manner. We derive the corresponding theorem for graded connected Hopf algebras quite generically. The main ingredient is a generalized notion of regularization in terms of a Rota-Baxter structure, which is supposed to underlie the target space of the characters of Hrt . Following the Hopf algebraic approach to renormalization in perturbative QFT, we henceforth introduce the notion of regularized (infinitesimal) characters, maps from Hrt into a commutative, associative, unital Rota-Baxter algebra A. The choice of the Rota-Baxter map is determined by the choice of the renormalization scheme, which can be a BPHZ scheme (Taylor subtractions of the integrand), the before-mentioned minimal subtraction and momentum schemes, and others, which all provide Rota-Baxter maps. Here is not the space to give a complete census of renormalization schemes in use in physics, but we simply assume Feynman rules and a Rota-Baxter map being given. Let us mention that sometimes we write R-matrix, instead of the standard notation r-matrix, to underline its operator form, and origin in the Rota-Baxter relation. ∗ to L(Hrt , A), consisting of K-linear maps from We therefore generalize Hrt Hrt into the Rota-Baxter algebra A, i.e., φ, T ∈ A, φ ∈ L(Hrt , A), T ∈ Hrt . Due to the double RB structure on the Rota-Baxter algebra (12) we naturally get L(Hrt , AR ). We then lift the Rota-Baxter map R : A → A to L(Hrt , A), which is possible since it is linear. Proposition 4.1 Define the linear map R : L(Hrt , A) → L(Hrt , A) by f → R(f ) := R ◦ f : Hrt → R(A). Then L(Hrt , A) becomes an associative, unital Rota-Baxter algebra. The Lie algebra of infinitesimal characters LHrt ⊂ L(Hrt , A) with bracket (28) becomes a Lie Rota-Baxter algebra, i.e., for Z , Z ∈ ∂ charA Hrt , we have the notion of a classical R-matrix respectively classical Yang-Baxter relation: (31) [R(Z ), R(Z )] = R [Z , R(Z )] + R [R(Z ), Z ] − R [Z , Z ] . Notice that we replaced K by A for the target space of the regularized infinitesimal characters. The proof of this proposition was given in (I). Using the double RB construction and Atkinson’s theorem of Section 2 we have the following Lemma 4.2 The Rota-Baxter algebra L(Hrt , A) equipped with the convolution product: f R g = f R(g) + R(f ) g − f g (32)

382



gives a Rota-Baxter algebra structure on the set of linear functionals with values in the double RB algebra AR of A, denoted by L(Hrt , AR ). An analog for LHrt exists, denoted by LHrt R , equipped with the R-bracket: [Z , Z ]R

= [Z , R(Z )] + [R(Z ), Z ] − [Z , Z ] −1 = [Z , B(Z )] + [B(Z ), Z ] . 2

The R map becomes a (Lie) algebra morphism (LHrt R → LHrt ) L(Hrt , AR ) → L(Hrt , A). Remark 4.3 ˜ := id − R ˜ (see Re˜ := id − R, respectively R 1) The above is also true for R mark 2.10). + ˜ 2) We will denote the Lie subalgebras R(LHrt ) by L− Hrt and R(LHrt ) by LHrt . We now apply Atkinson’s theorem to the Lie algebra LHrt of infinitesimal characters, the generators of the group of Hopf algebra characters charA Hrt . Lemma 4.4 Every infinitesimal character Z ∈ LHrt has a unique subdirect Birkhoff ˜ decomposition Z = R(Z) + R(Z). Remark 4.5 1) In the case of an idempotent Rota-Baxter map R we have a direct decompo+ sition A = A− + A+ respectively LHrt = L− Hrt + LHrt . 2) Let Z ∈ LHrt be the infinitesimal character generating the character φ = exp (Z) ∈ charA Hrt . Using the result in Proposition 2.11, we then see that for elements in ker(), the augmentation ideal, we have ˜ ˜ expR (−Z) . = −R exp (−R(Z)) = R expR (−Z) , exp (R(Z))

4.1

Review of the Ladder case

For the Hopf subalgebra of rooted ladder trees, introduced in the last section, we found in (I) the following simple factorization for a regularized character l l φ = exp∗ (Z), Z ∈ ∂ charA Hrt due to the abelianess of LHrt charA Hrt l , and induced by Atkinson’s theorem, i.e., the lifted Rota-Baxter map R:

=

exp∗ (Z) ˜ exp∗ R(Z) + R(Z)

(34)

=

φ−1 −

(35)

φ =

φ+

(33)

where: φ− = exp (−R(Z))

˜ φ+ = exp (R(Z)),

(36)

and such that we arrive at the following formulae [8] for φ± , using Proposition 2.11:

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l Proposition 4.6 In the rooted ladder tree case, Hrt , we find for the factors (36) in the Birkhoff decomposition (35) the following explicit formulae: φ− (tn ) = R expR (−Z) (tn ) (37) n−1 = −R φ(tn ) + φ− (tk )φ(tn−k )

(38)

k=1

˜ expR (−Z) (tn ) φ+ (tn ) = −R ˜ φ(tn ) + = R

n−1

(39)

φ− (tk )φ(tn−k ) .

(40)

k=1

We emphasize that the map: b[φ](tn ) := =

expR (−Z)(tn ) n−1 −φ(tn ) − φ− (tk )φ(tn−k )

(41) (42)

k=1 l → AR , which we will call Bogoliubov character, is a Hopf algebra character Hrt i.e., into the double RB algebra AR of A. This gives a natural algebraic expression ¯ for Bogoliubov’s R-map. In the next section, where we treat the general case, we ˜ i.e., the Lie algebra will formally introduce b[φ]. The Rota-Baxter maps R and −R, ˜ R,−R

homomorphisms, become group homomorphisms charAR Hrt −−−−→ charA Hrt . We will generalize this to arbitrary rooted trees in the following section, using equation (17).

4.2

The general case

As stated above, in (I) we introduced a classical R-matrix coming from the RotaBaxter structure underlying the target space of regularized characters. We saw that in the case of the Hopf subalgebra of rooted ladder trees, the abelianess of the related Lie algebra implies a somehow simple Birkhoff factorization (35, 36) respectively the formulae for the factors φ± (38, 40). The general, i.e., noncocommutative case can be solved due to the graded, connectedness of the Hopf algebra of rooted trees. Suppose we start with an infinitesimal character Z ∈ ∂ charA Hrt generating the regularized Hopf algebra character φ = exp∗ (Z) ∈ charA Hrt . The above mentioned properties of the Hopf algebra allow for a recursive definition of an infinitesimal character χ = χ(Z) ∈ ∂ charA Hrt , defined in terms of Z, using the lower central series of Lie algebra commutators. Setting ∞ (k) χZ (43) χ(Z) = Z + k=1

384



we proceed in the following manner. We first introduce the related series χ(u; Z) = Z +

∞

(k)

u k χZ ,

(44)

k=1 (0)

where we assume that 0 < u < 1 is a real parameter. We also set χZ = Z. We next introduce the Baker-Campbell-Hausdorff (BCH) series

1 1 A, [A, B] − B, [A, B] + · · · . BCH(A, B) := [A, B] + (45) 2 12 See [44] for more details on the BCH formula. Let us write equation (45) in the form ∞ ck K (k) (A, B), (46) BCH(A, B) = k=1 (k)

such that the K are the appropriate nested- or multicommutators of depth k ∈ N, i.e., K (1) = [A, B], K (2) = [A, [A, B]] − [B, [A, B]] and so on. Then, the (k) χZ are defined as the solution of the fix point equation χ(u; Z) = Z −

∞

˜ . ck uk K (k) R(χ(Z)), R(χ(Z))

(47)

k=1

Note that χ(u; Z)(T ) is a polynomial in u of degree m − 2 for any finite tree T of degree #(T ) = m say, i.e., with m vertices and therefore is well defined at u = 1. We hence set χ(Z) ≡ χ(1; Z). Furthermore, χ(Z) is an infinitesimal character as it is a finite linear combination of infinitesimal characters, and thus the above definition on trees implies its action on forests as a derivation in the sense of definition (3.7). (k) It is immediate that χZ vanishes for all k ≥ 1 when applied to cocommutative Hopf algebra elements. Let us work out the cases k = 1, 2 as examples: 1 1 ˜ = − [R(Z), Z] χ(1) = − [R(Z), R(Z)] 2 2 and for k = 2 we have: χ(2)

1 1 ˜ (1) )] ˜ = − [R(χ(1) ), R(Z)] − [R(Z), R(χ 2 2

1 ˜ R(Z), [R(Z), Z] − R(Z), [R(Z), Z] − 12 1 1 ˜ ˜ = + [R([R(Z), Z]), R(Z)] + [R(Z), R([R(Z), Z])] 4 4

1 ˜ − R(Z), [R(Z), Z] − R(Z), [R(Z), Z] 12

(48)

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=

385

1 R([R(Z), Z]), Z 4

1 + R(Z), [R(Z), Z] − [R(Z), Z], Z . 12

(49)

˜ has completely vanished. This nontrivial fact comes partly from R ˜ = where R (k) id − R, and in a moment we show that the χZ solve the simpler recursion: χ(u; Z) = Z +

∞

ck uk K (k) (−R(χ(Z)), Z)) .

(50)

k=1 (k)

Indeed, from the relation (47) and the recursive definition of the χZ we have the ∗ : following factorization for group like elements in Hrt Proposition 4.7 Using the infinitesimal character χ ∈ ∂ charA Hrt defined in (47), we have the following decomposition of a character φ ∈ charA Hrt given in terms of its generating infinitesimal character Z ∈ ∂ charA Hrt : ˜ exp∗ (Z) = exp∗ R(χ(Z)) exp∗ R(χ(Z)) . (51) ˜ is apparThis then implies the simpler recursion (50), in which the vanishing of R ent. Remark 4.8 The above formal derivation of the factorization of Hrt characters using the BCH formula in (47) and (50) to define the infinitesimal character χ(Z) (43) may be summarized in a more suggestive manner by the following two recursive formulae: ˜ χ(Z) = Z − BCH R(χ(Z)), R(χ(Z)) = Z + BCH − R(χ(Z)), Z . Let us define the factors, i.e., characters φ± ∈ charA Hrt : ∗ ˜ R(χ(Z)) , φ+ := exp∗ R(χ(Z)) φ−1 − := exp

(52)

and introduce the Bogoliubov character now in general via the following definition. Definition 4.9 Let exp∗ (Z) = φ ∈ charA Hrt . We define the following character b[φ] ∈ charAR Hrt with values in the double RB algebra of A, and call it Bogoliubov character: (53) b[φ] := exp∗R (−χ(Z)). Remembering the crucial property of exponentiated Rota-Baxter maps, coming from (12), respectively (14, 15): exp∗ − R(Z) = R exp∗R (−Z) (54) ˜ ˜ exp∗R (−Z) , = −R exp∗ R(Z) (55)

386



for elements in the augmentation ideal ker(), we have: φ− := R(b[φ]),

˜ φ+ := −R(b[φ]).

(56)

We use the factorization (51) in proposition (4.7) to derive an explicit formula for the characters b[φ] respectively φ± . Let T ∈ ker(), using the coproduct (21), we get: ˜ −R(b[φ])(T ) = =

exp∗ (−R(χ(Z))) exp∗ (Z)(T )

(57)

∗

exp (Z)(T ) + R(b[φ])(T ) 1 n−1 n

+ R(−χ(Z))(n−j) Z j (T ), n! j=1 j

(58)

n≥0

where (57) again implies the simpler recursion equation (50). Remark 4.10 1) All expressions are well defined since they reduce to finite sums for an element T ∈ ker() of finite order #(T ) = m < ∞. 2) In the last expression in equation (58), the primitive part in ∆(T ) is mapped to zero, since only strictly positive powers of infinitesimal characters appear. Continuing the above calculation, we get the following: ˜ −R(b[φ])(T ) − R(b[φ])(T )

= −b[φ](T ) = exp∗ (Z)(T ) 1 n−1 n

+ R(−χ(Z))(n−j) Z j (T ), n! j=1 j n≥0

and therefore we find the well-known formula: R exp∗R (−χ(Z)) (T ) = +

−R exp∗ (Z)(T ) 1 n−1 n

R(−χ(Z))(n−j) Z j (T ) . n! j=1 j n≥0

Finally, we rederive the results of [3, 4, 8] which gave the counterterm and the renormalized contribution as the image of the Bogoliubov character under the ˜ now derived from the double RB construction group homomorphisms R and −R, for any algebraic Birkhoff decomposition based on a suitable R, i.e., Rota-Baxter type map:

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Theorem 4.11 For T ∈ Hrt , #(T ) = m we have the following formulae for the factors in (51): R(b[φ])(T ) = = =

φ− (T )

m n−1

1 n −R φ(T ) + R(−χ(Z))(n−j) Z j (T ) . n! j=1 j n≥0 φ− (PcT )φ(RcT ) . −R φ(T ) + cT ∈CT

and

˜ b[φ] (T ) = R = =

φ+ (T )

m n−1

1 n ˜ φ(T ) + R R(−χ(Z))(n−j) Z j (T ) . n! j=1 j n≥0 ˜ φ(T ) + φ− (PcT )φ(RcT ) . R cT ∈CT

This should be compared to the general equation (16) including the shuffle: 1 − exp∗R (−χ(Z))(T ) = exp∗ (Z)(T ) + n! n−1

n≥0

n R(−χ(Z))(n−j) Z j (T ). j

j=1 ∗

= exp (χ(Z))(T ) + 1 n−1 R(−χ(Z))(n−j)

χ(Z)j (T ). n! j=1

n≥0

It allows us to define the infinitesimal character χ = χ(Z) to order k > 0 in another (j) way recursively by using the χZ , j < k. We therefore get to order k: (k)

χZ

= −

k−1

(j)

χZ −

j=1

−

k+2 l=1

k+2 n≥0

+

1 n!

1 1 χ(Z) l + Z l l! l!

n−1

k+2

l=1

R(−χ(Z))(n−j)

χ(Z)j

j=1

k+2 n≥0

n−1

1 n R(−χ(Z))(n−j) Z j . n! j=1 j

After these formal arguments based on the general results for Rota-Baxter operators and the structure of the rooted tree Hopf algebra, we end this section and the paper with a remark on calculational aspects.

388



When treating the rooted ladder case, we mentioned at the end of (I) the use of normal coordinates introduced by Chryssomalakos et al. in [43]. Given a regularized character φ ∈ charA Hrt , this provided a very easy way to define the coefficients for its generator Z ∈ ∂ charA Hrt : Z := α(n) Ztn , α(n) ∈ A (59) n>0

such that exp∗ (Z)(tn ) = φ(tn ) ∈ A. Remark 4.12 1) Here and also later, we omit for notational reasons the tensor sign between the α(n) and Ztn , i.e., α(n) Ztn ∈ A ⊗ ∂ charA Hrt , n > 0. 2) The α(n) were given in terms of Schur polynomials, i.e., α(n) := φ(P (t1 , · · · , tn )). And φ− was just exp∗ (−R(Z)). In the general case, i.e., for arbitrary rooted trees, the simple Schur polynomials get replaced by the following set of polynomial equations. Details may be found in [43]. Introducing the symbols xT , indexed by rooted trees, and defining the new coordinates, which are characterized by exp∗ (Z)(xT ) = φ(xT ), φ ∈ charA Hrt , where: T αx ZT ∈ ∂ charA Hrt , (60) Z= T ∈Trt

we arrive at the following infinite set of coupled equations, expressing the coordinates T in terms of the new xT : 1 mHrt (P⊗n+1 )∆(n) (xT ), T ∈ Trt . T = (61) (n + 1)! n≥0

Here, the map P denotes the projector into the augmentation ideal: 0, xT1 · · · xTn = 1 T1 Tn P(x · · · x ) := xT1 · · · xTn , else.

(62)

∆ denotes the coproduct reduced to single simple cuts |cT | = 1, and ∆(n) := this op(id ⊗ ∆(n−1) ) ◦ ∆ , such that ∆(0) := id, ∆(1) = ∆ . One should compare ˆ eration with the formal linear map exp∗ (Z)(T ) on Hrt , where Zˆ := T ∈Trt T ZT , T ZT (T ) = T δT,T , and T1 ZT1 T2 ZT2 := T1 T2 ZT1 ZT2 . The first four equations for the rooted trees , , , =

x

=

1 x + x x 2

=

1 x +x x + x x x , 6

=x

are:

1 +x x + x x x 3

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389

The final step, done in [43], is to invert the above equations, giving the xT ’s in terms of the original rooted trees. In the ladder case we just get the Schur polynomials. In general, we have for example:

x = −

+

1 3

,

x

=

−

+

1 6

.

T

Therefore, the coefficients αx := φ(xT ) ∈ A in (60). Let us briefly dwell on the generalization to decorated non-planar rooted trees, and to Feynman graphs, following [38, 45]. Every Feynman graph provides a number r of maximal forests. The integer r counts the number of terms pi , i = 1, . . . , r in the coproduct which are primitive on the rhs, and in the augmentation ideal on the lhs of the coproduct on graphs. If r > 1, we call the graph overlapping divergent. It is then mapped to a linear combination of r decorated rooted trees, where each of those trees has a root decorated by one of the pi . Iterating this procedure, one obtains a map from Feynman graphs to decorated rooted trees where the decorations are provided by subdivergence free skeleton contributions. Having resolved the overlapping sectors into trees, one then proceeds as before. We close this paper with a study of a simple example on decorated rooted trees using two decorations. The generalization to Feynman graphs including form factor decompositions for theories with spin is somewhat excessive on the notational side, but provides no difficulty for the practitioner of quantum field theory, making full use of the Hopf and Lie algebra of Feynman graphs with external structures. See [5, 38, 45] where examples can be found. We consider the example of vertices with a decoration D by 2 elements {a, b}. Let us denote them by a vertex and a vertex . For the Lie bracket (28) of these two vertices we get: [Z , Z ] = Z − Z .

(63)

Note that though we have here the analog of a simple nesting of one graph in another, this has already a non-vanishing commutator in the Lie algebra. This fact makes it necessary to include the BCH-corrections (50) already at this level. For the above example (63), we have to add the correction (48). Let us do the calculation of the counterterm φ− explicitly for the decorated rooted ladder tree , using the normal coordinates in (60). We have to use χ = Z + χ(1) (43), with χ(1) given in (48). The infinitesimal character Z generating the character φ is given to order 2 in terms of the normal coordinates xT as:

Z = φ(x )Z + φ(x )Z + φ(x )Z + φ(x )Z ,

(64)

390



where: φ(x )

= φ( ),

φ(x )

1 = φ( ) − φ( )φ( ), 2

φ(x ) = φ( ) 1 φ(x ) = φ( ) − φ( )φ( ). 2

(65)

And therefore we have for the infinitesimal character χ(Z) to order k = 1, i.e., including the first correction (48):

χ

=

φ(x )Z + φ(x )Z + φ(x )Z + φ(x )Z −

1 [R(φ(x )Z ), φ(x )Z ] + [R(φ(x )Z ), φ(x )Z ] . 2

So that when the counterterm character φ− = exp∗ (−R(χ)) is applied to get: φ− ( ) = = =

we

exp∗ (−R(χ))( ) 1 −R(χ)( ) + R(χ) R(χ)( ) 2 1 −R φ( )Z ( ) − φ( )φ( )Z ( ) 2 −

1 R(φ( )Z ( )) φ( )Z ( ) 2 −φ( )Z ( ) R(φ( )Z ( ))

=

(66)

(67)

1 + R(φ( )Z ( ))R(φ( )Z ( )) 2 −R φ( ) + R(φ( )) φ( ) ,

(68) (69)

which is the correct result. In line (67) no higher order terms can appear. In the next line we used relations (65). From (68) to the last equality (69) we used the RB relation: 1 1 R φ( )Z ( ) R φ( )Z ( ) + R φ( )Z ( ) φ( )Z ( ) 2 2 1 R R φ( )Z ( ) φ( )Z ( ) = 2 +φ( )Z ( ) R φ( )Z ( ) .

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Remark 4.13 1) Note that now χ(u; Z)(T ), equation (44), will be a polynomial of degree at most m−1 when acting on decorated trees with m vertices, as the tree studied above with two vertices differently decorated is already non-cocommutative under the coproduct. 2) Using the standard example of the QFT Φ36dim , the result (69) should be , which has an compared to the counterterm for the Feynman graph additional factor two reflecting the fact that it is overlapping divergent, r = 2, and it resolves into two identical rooted trees [38].

5 Conclusion and outlook In this work we generalized the results of (I) to arbitrary rooted trees, i.e., we showed how to derive the Birkhoff factorization for characters of the Hopf algebra of rooted trees. Using the Rota-Baxter structure underlying the target space of the characters of a renormalization Hopf algebra, the notion of a classical r-matrix was introduced on the corresponding Lie algebra defined on rooted trees. A couple of simple results for Rota-Baxter algebras were collected which allowed for a straightforward derivation of the twisted antipode formula, defined in [3, 8] concerning the study of the Hopf algebraic approach to perturbative QFT. This gives a firm algebraic basis to any renormalization scheme using an algebraic Birkhoff decomposition together with a suitable double RB construction. We regard this work as a further step towards a more interesting connection to the realm of integrable systems. Sakakibara’s result [46] also points into this direction. This connection was already apparent in [10], in which effectively the grading operator Y served as a Hamiltonian providing the ”scaling evolution” of the coupling constant, and hence the renormalization group flow initiated by scaling transformations, and can and should be worked out for the corresponding flow of many other physical parameters of interest.

Acknowledgments The first author would like to thank the Ev. Studienwerk for financial support. ´ Also the I.H.E.S. and its warm hospitality is greatly acknowledged. We would like to thank Prof. Ivan Todorov, Prof. Olivier Babelon, and Igor Mencattini for valuable discussions, and helpful comments. D.K. is in parts supported by NSF grant DMS-0401262.

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References [1] K. Ebrahimi-Fard, L. Guo, D. Kreimer, Integrable renormalization I: the ladder case, J. Math. Phys. 45, No 10, 3758–3769 (2004). [2] J.C. Collins, Renormalization, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, (1985). [3] D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2, 303 (1998). [4] D. Kreimer, Chen’s iterated integral represents the operator product expansion, Adv. Theor. Math. Phys. 3, no. 3, 627 (1999). [5] D.J. Broadhurst, D. Kreimer, Renormalization automated by Hopf algebra, J. Symb. Comput. 27, 581 (1999). [6] D.J. Broadhurst, D. Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation, Phys. Lett. B 475, 63 (2000). [7] D.J. Broadhurst, D. Kreimer, Exact solutions of Dyson-Schwinger equations for iterated one-loop integrals and propagator-coupling duality, Nucl. Phys. B 600, 403 (2001). [8] A. Connes, D. Kreimer, Hopf algebras, Renormalization and Noncommutative Geometry, Comm. in Math. Phys. 199, 203 (1998). [9] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. in Math. Phys. 210, 249 (2000). [10] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group, Comm. Math. Phys. 216, 215 (2001). [11] I. Mencattini, D. Kreimer, Insertion and elimination Lie algebra: the ladder case, Lett. in Math. Phys. 67, 61–74 (2004). [12] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10, 731 (1960). [13] F.V. Atkinson, Some aspects of Baxter’s functional equation, J. Math. Anal. Appl. 7, 1 (1963). [14] G.-C. Rota, Baxter algebras and combinatorial identities. I, II., Bull. Amer. Math. Soc. 75, 325 (1969); ibid. 75, 330 (1969).

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[15] P. Cartier, On the structure of free Baxter algebras, Advances in Math. 9, 253 (1972). [16] G.-C. Rota, Baxter operators, an introduction, In: “Gian-Carlo Rota on Combinatorics, Introductory papers and commentaries”, Joseph P.S. Kung, Editor, Birkh¨ auser, Boston, (1995). [17] M. Aguiar, J.-L. Loday, Quadri-algebras, J. Pure Applied Algebra 191, 205– 221 (2004). [18] K. Ebrahimi-Fard, Loday-type algebras and the Rota-Baxter relation, Letters in Mathematical Physics 61, no. 2, 139 (2002). [19] Ph. Leroux, Ennea-algebras, Nov 2003, preprint: arXiv:math.QA/0309213. [20] K. Ebrahimi-Fard, L. Guo, On product and Duality of Binary, Quadratic Regular Operads, preprint 2004, J. Pure Applied Algebra, in press. [21] M. Aguiar, Prepoisson algebras, Letters in Mathematical Physics 54, no. 4, 263 (2000). [22] G.-C. Rota, Ten mathematics problems I will never solve, Mitt. Dtsch. Math.Ver., no. 2, 45 (1998). [23] L. Guo, W. Keigher, Baxter algebras and shuffle products, Adv. Math. 150, no. 1, 117 (2000). [24] L. Guo, Baxter algebras and differential algebras, in ”Differential algebra and related topics”, (Newark, NJ, 2000), World Sci. Publishing, River Edge, NJ, 281, (2002). [25] L. Guo, Baxter Algebras, Stirling Numbers, and Partitions, Feb. 2004, preprint, http://newark.rutgers.edu/ liguo/lgpapers.html, to appear in J. Algebra and Its Appl. [26] M.A. Semenov-Tian-Shansky, What is a classical r-matrix?, Funct. Ana. Appl. 17, no.4., 254 (1983). [27] A.G. Reyman, M.A. Semenov-Tian-Shansky, Group theoretical methods in the theory of finite dimensional integrable systems, in: Encyclopedia of mathematical science, v.16: Dynamical Systems VII, Springer, 116, (1994). [28] A. Belavin, V. Drinfeld, Triangle Equations and Simple Lie-Algebras, Classic Reviews in Mathematics and Mathematical Physics, 1. Harwood Academic Publishers, Amsterdam, (1998), viii+91 pp. [29] M.A. Semenov-Tian-Shansky, Integrable Systems and Factorization Problems, Lectures given at the Faro International Summer School on Factorization and Integrable Systems (Sept. 2000), Birkhäuser 2003, Sept. 2002, preprint: arXiv: nlin.SI/0209057.

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[30] M. Hoffman, Quasi-shuffle products, J. Algebraic Combin., 11, no. 1, 49 (2000). [31] K. Ebrahimi-Fard, L. Guo, Rota’s q-shuffle relation for the Jackson integral, work in progress. [32] J. Cari˜ nena, J. Grabowski, G. Marmo, Quantum bi-Hamiltonian systems, Internat. J. Modern Phys. A 15, no. 30, 4797 (2000). [33] K. Ebrahimi-Fard, On the associative Nijenhuis relation, The Electronic Journal of Combinatorics 11 (1), R38 (2004) . [34] Ph. Leroux, Construction of Nijenhuis operators and dendriform trialgebras, Nov. 2003, preprint: arXiv:math.QA/0311132. [35] I.Z. Golubchik, V.V. Sokolov, One more type of classical Yang-Baxter equation, Funct. Anal. Appl. 34, no. 4, 296 (2000). [36] Y. Kosmann-Schwarzbach, F. Magri, Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor. 53, no. 1, 35 (1990). [37] O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, (2003). [38] D. Kreimer, On overlapping divergences, Commun. Math. Phys. 204, 669 (1999). [39] H. Figueroa, J.M. Gracia-Bondia, J.C. Varilly, Elements of Noncommutative Geometry, Birkh¨ auser, (2001). [40] D. Manchon, Hopf algebras, from basics to applications to renormalization, Comptes-rendus des Rencontres mathématiques de Glanon 2001. [41] D. Kreimer, Factorization in quantum field theory: An exercise in Hopf algebras and local singularities, Proceedings From Number Theory to Physics and Geometry, Les Houches March 2003, in press, arXiv:hep-th/0306020. [42] A. Connes, D. Kreimer, Insertion and elimination: the doubly infinite Lie algebra of Feynman graphs, Ann. Henri Poincaré 3, no. 3, 411 (2002). [43] C. Chryssomalakos, H. Quevedo, M. Rosenbaum, J.D. Vergara, Normal coordinates and primitive elements in the Hopf algebra of renormalization, Comm. in Math. Phys. 225, no. 3, 465 (2002). [44] V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Springer-Verlag, (1984).

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[45] D. Kreimer, Structures in Feynman graphs: Hopf algebras and symmetries, arXiv:hep-th/0202110, Proceedings of GRAPHS AND PATTERNS in Mathematics and Theoretical Physics, Department of Mathematics and Institute for Mathematical Sciences Stony Brook, New York June 14–21, 2001. To be published in “Proceedings of Symposia in Pure Mathematics”, AMS. [46] M. Sakakibara, On the Differential equations of the characters for the Renormalization group, Mod. Phys. Lett. A 19, 1453–1456 (2004). Kurusch Ebrahimi-Fard* and Dirk Kreimer ´ Institut des Hautes Etudes Scientifiques 35, Route de Chartres F-91440 Bures-sur-Yvette France and *Universit¨ at Bonn Physikalisches Institut Nussallee 12 D-53115 Bonn Germany email: [email protected] Li Guo Rutgers University Department of Mathematics and Computer Science Newark, NJ 07102 USA email: [email protected] Communicated by Vincent Rivasseau submitted 16/03/04, accepted 09/09/04




Erratum to “Negative Discrete Spectrum of Perturbed Multivortex Aharonov-Bohm Hamiltonians” Ann. Henri Poincaré, 5 (2004) 979–1012 M. Melgaard, E.-M. Ouhabaz and G. Rozenblum

Regrettably, there is an error in the proof of Proposition 4.1. To correct it the following changes are needed. On page 989 : ∂u ∂u The quantity ∂x in the last term of (4.2) has to be replaced by ∂x . 1 2 Line 10 (the last line of the first non-numbered formula for I2 ): the two signs − must be +; the lines 13 and 14 (the last two lines in the second non-numbered formula for I2 ): again the last two − must be +, which gives + instead of − in (4.3), thus (4.3) has to be replaced by

I2 =

Ωn

−

2A1 Im

Im

Ωn

∂u ∂u sign u¯ |v| + 2A2 Im sign u ¯ |v| ∂x1 ∂x2 (A21 + A22 )|u||v| ≥− Ωn

2 2 ∂u ∂u |v| |v| χ{u=0} − χ{u=0} . sign u ¯ sign u ¯ Im ∂x1 |u| ∂x2 |u| Ωn (4.3)

After this, (4.4) follows, and then, on page 990, the rest of the proof, after the first line must be replaced by:

Re hn [u, v] ≥ ln [|u|, |v|] for all u, v ∈

H01 (Ωn )

obeying u · v¯ ≥ 0. This proves (4.1).

We use the occasion to make a comment on the results of the paper. For the case γ ≥ 1/2, alternatively to Theorem 1.2 in the paper, the Lieb-Thirring type estimates can also be obtained by an approximation procedure based on the results of the paper by A. Laptev and T. Weidl ([18] in the reference list, Theorem 3.2) and of the paper by D. Hundertmark, A. Laptev, T. Weidl, New bounds on

398

M. Melgaard, E.-M. Ouhabaz and G. Rozenblum


the Lieb-Thirring constants. Invent. Math. 140 (2000), no. 3, 693–704, Theorem 4.2. This reasoning would give better constants in the estimates, up to 7 times smaller than the ones obtained in our Theorem 1.2. M. Melgaard Department of Mathematics Uppsala University Polacksbacken S-751 06 Uppsala Sweden email: [email protected] E.-M. Ouhabaz Laboratoire Bordelais d’Analyse et Géométrie Université de Bordeaux 1 351, Cours de la Libération F-33405 Talence cedex France email: [email protected] G. Rozenblum Department of Mathematics Chalmers University of Technology and University of Gothenburg Eklandagatan 86 S-412 96 Gothenburg Sweden email: [email protected] Communicated by Bernard Helffer received 03/01/05




Renormalization of the 2-Point Function of the Hubbard Model at Half-Filling Stéphane Afchain, Jacques Magnen and Vincent Rivasseau Abstract. We prove that the Hubbard model at finite temperature T and half-filling is analytic in its coupling constant λ for |λ| ≤ c/| log T |2 , where c is some numerical constant. We also bound the self-energy and prove that the Hubbard model at half-filling is not a Fermi liquid (in the mathematically precise sense of Salmhofer), modulo a simple lower bound on the first non-trivial self-energy graph, which will be published in a companion paper.

I Introduction In [1] we introduced the tools for a multiscale analysis of the two-dimensional Hubbard model at half-filling: momentum slices, sectors and their conservation rules. In this paper we achieve the proof that the correlation functions of the model at finite temperature T are analytic in the coupling constant λ for |λ| ≤ c/| log T |2 , by treating the renormalization of “bipeds” (two-particle subgraphs), that was missing in [1]. This proof requires a new tool which is a constructive two-particle irreducible analysis of the self-energy. This analysis according to the line form of Menger’s theorem ([2]) leads to the explicit construction of three line-disjoint paths for every self-energy contribution, in a way compatible with constructive bounds. On top of that analysis, another one which is scale-dependent is performed: after reduction of some maximal subsets provided by the scale analysis, two vertex-disjoint paths are selected in every self-energy contribution. This requires a second use of Menger’s theorem, now in the vertex form. This construction allows to improve the power counting for two-point subgraphs, exploiting the particle-hole symmetry of the theory at half-filling, and leads to our analyticity result. In the last section we write the upper bounds on the self-energy that follow from our analysis. These upper bounds strongly suggest that the second momentum derivative of the self energy is not uniformly bounded in the region |λ| ≤ c/| log T |2 . A rigorous proof of this last statement follows from a rigorous lower bound of the same type than these upper bounds, but for the smallest nontrivial self-energy graph, so as to rule out any “miraculous cancellation”. This lower bound, which we have now completed, is the tedious but rather straightforward study of a single finite-dimensional integral. Since it is not related to the main analysis in this paper, we postpone it to a separate publication [5].

400

S. Afchain, J. Magnen and V. Rivasseau


Taken all together these bounds prove that the model is not a Fermi liquid in the sense of Salmhofer’s criterion (see [3] and [4]). Indeed to be such a Fermi liquid the second derivative would have to be uniformly bounded in a larger region (of type |λ| ≤ c/| log T |) than the one for which we prove it is unbounded. The scaling properties of the self energy and its derivatives in fact mean that the model is not of Fermi but of Luttinger type, with logarithmic corrections if we compare to the standard one-dimensional Luttinger liquid. Let us state precisely the main result of this paper: Theorem The radius of convergence of the Hubbard model perturbative series at half-filling is at least c/ log2 T , where T is the temperature and c some numerical constant. As T and λ jointly tend to 0 in this domain, the self-energy of the model does not display the properties of a Fermi liquid in the sense of [3], but those of a Luttinger liquid (with logarithmic corrections). Let us also put our paper in perspective and relation with other programs of rigorous mathematical study of interacting Fermi systems. Recall that in dimension 1 there is neither superconductivity nor extended Fermi surface, and Fermion systems have been proved to exhibit Luttinger liquid behavior [6]. The initial goal of the studies in two or three dimensions was to understand the low temperature phase of these systems, and in particular to build a rigorous constructive BCS theory of superconductivity. The mechanism for the formation of Cooper pairs and the main technical tool to use (namely the corresponding 1/N expansion, where N is the number of sectors which proliferate near the Fermi surface at low temperatures) have been identified [8]. But the goal of building a completely rigorous BCS theory ab initio remains elusive because of the technicalities involved with the constructive control of continuous symmetry breaking. So the initial goal was replaced with a more modest one, still important in view of the controversies over the nature of two-dimensional “Fermi liquids” [7], namely the rigorous control of what occurs before pair formation. The last decade has seen excellent progress in this direction. As is well known, sufficiently high magnetic field or temperature are the two different ways to break the Cooper pairs and prevent superconductivity. Accordingly two approaches were devised for the construction of “Fermi liquids”. One is based on the use of non-parity invariant Fermi surfaces to prevent pair formation. These surfaces occur physically when generic magnetic fields are applied to two-dimensional Fermi systems. The other is based on Salmhofer’s criterion [3], in which temperature is the cutoff which prevents pair formation. In a large series of papers [9], the construction of two-dimensional Fermi liquids for a wide class of non-parity invariant Fermi surfaces has been completed in great detail by Feldman, Knörrer and Trubowitz. These papers establish Fermi liquid behavior in the traditional sense of physics textbooks, namely as a jump of the density of states at the Fermi surface at zero temperature, but they do not apply to the simplest Fermi surfaces, such as circles or squares, which are parity invariant.

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Another program in recent years was to explore which models satisfy Salmhofer’s criterion. Of particular interest to us are the three most “canonical” models in more than one dimension namely: • the jellium model in two dimensions, with circular Fermi surface, nicknamed J2 , • the half-filled Hubbard model in two dimensions, with square Fermi surface, nicknamed H2 , • and the jellium model in three dimensions, with spherical Fermi surface, nicknamed J3 . The study of each model has been divided into two main steps of roughly equal difficulty, the control of convergent contributions and the renormalization of the two-point functions. In this sense, five of the six steps of our program are now completed. J2 is a Fermi liquid in the sense of Salmhofer [10]–[11], H2 is not, and is a Luttinger liquid with logarithmic corrections, according to [1], to the present paper, and to [5]. Results similar to [10]–[11] have been also obtained for more general convex curves not necessarily rotation invariant such as those of the Hubbard model at low filling, where the Fermi surface becomes more and more circular, including an improved treatment of the four-point functions leading to better constants [12]. Therefore as the filling factor of the Hubbard model is moved from halffilling to low filling, we conclude that there must be a crossover from Luttinger liquid behavior to Fermi liquid behavior. This solves the controversy [7] over the Luttinger or Fermi nature of two-dimensional many-Fermion systems above their critical temperature. The short answer is that it depends on the shape of the Fermi surface. Up to now only the convergent contributions of J3 , which is almost certainly a Fermi liquid, have been controlled [13]. The renormalization of the two-point functions for J3 , the last sixth of our program, remains still to be done. This last part is difficult since the cutoffs required in [13] do not conserve momentum. This means that the two-point functions that have to be renormalized in this formalism are not automatically one particle irreducible, as is the case both in [11] and in this paper. This complicates their analysis.

II Slices, sectors, propagator decay and momentum conservation We recall here some generalities that were explained in [1], in order to make this paper self-contained. Given a temperature T > 0, the Hubbard model lives on [−β, β[ × Z2 , where β = T1 . Indeed, the real interval [−β, β[ should be thought of as the circle of radius β. A generic element of [−β, β[ × Z2 will be denoted → → x = (x0 , − x ), where x0 ∈ [−β, β[ and − x = (n1 , n2 ) ∈ Z2 .

402



→ Like in every Fermionic model, the propagator C(x0 , − x ) 1 is antiperiodic 1 in the variable x0 , with antiperiod T . Therefore, for the Fourier transform of → ˆ 0, − the propagator C(k k ), the relevant values for k0 are discrete and called the Matsubara frequencies: (2n + 1)π k0 = , n ∈ Z, (II.1) β → − whereas the vector k lives on the two-dimensional torus R2 /(2πZ)2 . At half-filling and finite temperature T , we have: Câ,b (k) = δa,b

1

→ , − ik0 − e( k )

(II.2)

→ − with e( k ) = cos k1 + cos k2 . a and b are spin indices (elements of the set {↑, ↓}), and may sometimes be dropped when they are not essential. Hence the expression of the real space propagator is: π π 1 dk dk2 eik.x Câ,b (k) . (II.3) Ca,b (x) = 1 (2π)2 β −π −π k0

The notation k0 really means the discrete sum over the integer n in (II.1). When T → 0+ (which means β → +∞), k0 becomes a continuous variable, the corresponding discrete sum becomes an integral, and the corresponding propagator C0 (x) becomes singular on the Fermi surface √ defined by k0 = 0 and e(k) = 0. This Fermi surface is a square of side size 2π (in the first Brillouin zone) joining the corners (±π, 0), (0, ±π). We call this square the Fermi square, its faces and corners are called the Fermi faces and corners. Considering the periodic boundary conditions, there are really four Fermi faces, but only two Fermi corners. In the following, to simplify notations, we will write:

1 dk1 dk2 d k ≡ β [−π,π]2 3

,

k0

1 d x ≡ 2

β

3

−β

dx0

.

(II.4)

x∈Z2

The interaction of the Hubbard model is simply SV = λ V

 d3 x 

2 ψ a (x)ψa (x) ,

(II.5)

a∈{↑,↓}

where V := [−β, β[×V and V is an auxiliary volume cutoff in two-dimensional space, that will be sent to infinity eventually. Remark that in (II.1) |k0 | ≥ π/β = 0 1 Indeed, the propagator should be seen as depending on two variables x, y ∈ [−β, β[ × Z2 , but by translational invariance, we have C(x, y) = C(0, y −x) and we shall write in the following simply C(x) instead of C(0, x).

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hence the denominator in C(k) can never be 0 at non-zero temperature. This is why the temperature provides a natural infrared cutoff. We use in this paper the same slices and sectors than in [1] and recall the main points for completeness. Introducing a fixed large number M > 1, we perform a slice analysis according to geometric scales of ratio M . Like in [1] since we have a finite temperature, this analysis should stop for a scale imax (T ) such that M −imax (T ) 1/T . We write simply imax for imax (T ). As in [1] we use the tilted orthogonal basis in momentum space (e+ , e− ), defined by e+ = (1/2)(π, π) and e− = (1/2)(−π, π). In the corresponding coordinates (k+ , k− ) the Fermi surface is given by k+ = ±1 or k− = ±1. This follows from the identity πk− πk+ cos . (II.6) cos k1 + cos k2 = 2 cos 2 2 We also use the convenient notations q± = k± − 1 if k± ≥ 0 ; q± = k± + 1 if k± < 0

(II.7)

so that 0 ≤ |q± | ≤ 1. Picking a Gevrey compact support function u(r) ∈ C0∞ (R) of order α < 1 which satisfies: u(r) = 0

for |r| > 2 ; u(r) = 1 for |r| < 1 ,

(II.8)

we consider the partition of unity:

imax (T )

1=

i=0

with

2 2 πk+ 2 πk− cos ui k0 + 4 cos , 2 2

u0 (r) = 1 − u(r) , ui (r) = u M 2(i−1) r − u M 2i r for i ≥ 1.

(II.9)

(II.10)

The sum over i a priori runs from 0 to +∞ to create a partition of unity, but in fact since k02 is at least of order M −2imax (T ) , the sum over i stops as imax (T ). This is similar to [1]. The i slice propagator Ci (k) = C(k)ui (k) is further sliced into the ± directions exactly as in [1]: Ci (k) = Cσ (k) , (II.11) σ=(i,s+ ,s− )

where Cσ (k) = Ci (k) vs+

πk+ πk− cos2 vs− cos2 2 2

(II.12)

using a second partition of unity 1=

i s=0

vs (r) ,

(II.13)

404

where


  v0 vs   vi (r)


= 1 − u(M 2 r) , = us+1 for 1 ≤ s ≤ i − 1 , = u(M 2i r) .

(II.14)

Like in [1] we need s+ + s− ≥ i − 2 for non-zero Cσ (k), and the depth l(σ) of a sector is defined as l = s+ + s− − i + 2, with 0 ≤ l ≤ i + 2. 2 . We have the scaled decay ([1], Lemma 1):

| Cσ (x, y)| ≤ c.M −i−l e−c [dσ (x,y)]

α

(II.15)

where c, c are some constants and dσ (x, y) = M −i |x0 − y0 | + M −s+ |x+ − y+ | + M −s− |x− − y− | .

(II.16)

Furthermore we recall the momentum conservation rules for the four sectors σj , j = 1, . . . , 4 meeting at any vertex ([1], Lemma 4): Proposition 1: Momentum conservation at a vertex. For M large enough, the two smallest indices among sj,+ , j = 1, . . . , 4 differ by at most one unit, or the smallest one, say s1,+ must coincide with i1 with i1 < ij , j = 1. Exactly the same statement holds independently for the minus direction. We say that the sectors which have smallest indices at a vertex in a direction “collapse” in that direction. We also introduce a new index for each sector, r(σ) = E(i(σ) + l(σ)/2) (where E means the integer part like in [1], section 4) and the corresponding slice propagator Cσ (k) . (II.17) Cr (k) = σ | r(σ)=r

We remark that this slice cutoff respects the symmetries of the theory. It is with respect to this slice index that our main multislice analysis will be performed. The propagator with infrared cutoff r is defined as C≤r (k) =

Cσ (k) .

(II.18)

σ | r(σ)≤r 2 This definition of sectors looks at first sight a bit complicated. It is designed for Proposition 1 to hold, in order to get an analyticity in |λ| ≤ c/ log2 T . With less detailed sectors one could prove analyticity in a radius, e.g., only in |λ| ≤ c/ log4 T . This would be well enough to prove that the Hubbard model is not a Fermi liquid in Salmhofer’s sense. The reader only interested in this result could therefore skip some of the technicalities below by putting everywhere the l index to zero, and skipping Proposition 1. But he should not skip Sections VI–VII because the two particle irreducible analysis and the ring construction are there to fix the correct power counting, not just the logarithmic power counting.

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III Renormalization of the two-point function Let us define S2,≤r (k0 , k) as the connected amputated two-point function with infrared cutoff r, and define also: 1 S2,≤r (k0 , k) + S2,≤r (−k0 , k) . (III.19) G2,≤r (k0 , k) = 2 Consider k such that e(k) = 0. If our cutoff respects the symmetries of the theory, which is the case here, the nesting or particle-hole symmetry forces G2 to vanish for such k. Using the variables q+ and q− defined in (II.7), this is expressed by Lemma III.1 The following equality holds: G2,≤r (k0 , q+ , q− )

q+ =0 or q− =0

=0.

(III.20)

Proof. Using the symmetries of the theory, it is easy to check that for any Feynman two-point function graph G, the Feynman amplitude IG satisfies: IG (k0 , k1 , k2 ) = IG (k0 , k2 , k1 ) ,

(III.21)

IG (k0 , k1 , k2 ) = IG (k0 , −k1 , k2 ) ,

(III.22)

IG (k0 , k1 , k2 ) = −IG (−k0 , k1 + π, k2 + π) .

(III.23)

The last symmetry, the particle-hole symmetry, is the only non-trivial one and it can be checked because it changes all the propagators in momentum space into their opposite with all the momentum conservation laws respected. Since there is an odd number of propagators in a two-point subgraph, (III.23) holds. Now we consider a point k in the first quadrant with 0 ≤ k1 ≤ π and 0 ≤ k2 ≤ π. On the Fermi curve whose equation in this quadrant is k2 = π − k1 , we apply the relation (III.23) and get 0 = IG (k0 ,k1 ,k2 ) + IG (−k0 ,k1 + π,k2 + π) = IG (k0 ,k1 ,k2 ) + IG (−k0 ,2π − k2 ,2π − k1 ). (III.24) By the symmetries (III.21), (III.22) and periodicity 2π we obtain that IG (k0 , k) + IG (−k0 , k) = 0. By symmetry this relation holds also for the other quadrants, hence on all the Fermi square. Summing over all Feynman graphs we obtain the vanishing of G2,≤r (k0 , q+ , q− ) on the Fermi surface whose equation is q+ = 0 or q− = 0. The function being constant on the straight lines of the Fermi square, obviously its partial derivatives to any order along these straight directions also vanish on the Fermi surface. Recall that in [1] analyticity of a simplified Hubbard model at half filling was established in a domain of the expected optimal form |λ| ≤ c/| log T |2 . Indeed

406



and more precisely the result was established only for a model called “bipedfree” in which all two-point subgraphs appearing in the multislice expansion were suppressed. A straightforward extension of the bounds given in [1] is not enough to prove analyticity in the expected domain for the full model. Naive power counting in the style of [1] is indeed not sufficient to sum geometric series made of insertions of a two-point subgraph at a scale r and a propagator at scale s >> r. Consider, e.g., the simplest such sum, made of the chain of Figure 1, where the three internal lines of the biped have main scale r and the external one has main scale s >> r. The naive bound for the contribution of such a chain is M −r−l/2 per propagator at scale r, M −s−l /2 per propagator at scale s, and contains for each irreducible biped one integral over the position of one vertex evaluated through the decay of a propagator of scale s and one evaluated through the decay of a propagator of scale r. Let us neglect the auxiliary “depth indices” l and l which are not essential. The bound is therefore a geometric series with ratio M −3r M −s M 2r M 2s = M s−r .

(III.25)

Figure 1. A simple chain of bipeds. This bad factor M s−r appears always in the naive bounds for any similar twopoint function; it is exponential, not logarithmic in s − r, and certainly prevents a proof of analyticity, not only for |λ| ≤ c/| log T |2 , but for |λ| ≤ c/| log T |q for any integer q as well. As remarked in [1], this is however only a bound, and the true contribution is much smaller due to the particle-hole symmetry of the model at half-filling. To exploit this, and to treat the true model, we must “renormalize” the two-point functions of the theory instead of suppressing them. This is accomplished by a second order Taylor expansion of the two-point function with given cutoff in the style of [11]. In momentum space we change first k0 to the smallest possible values ±πT :

+

1 S2,≤r (k0 , q+ , q− ) − S2,≤r (πT, q+ , q− ) 2 S2,≤r (k0 , q+ , q− ) − S2,≤r (−πT, q+ , q− )

+

G2,≤r (πT, q+ , q− ) .

S2,≤r (k0 , q+ , q− ) =

(III.26)

Then we use (III.20) to write G2,≤r (πT, q+ , q− )

= G2,≤r (πT, q+ , q− ) − G2,≤r (πT, 0, q− ) − G2,≤r (πT, q+ , 0) + G2,≤r (πT, 0, 0) ,

(III.27)

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where the variables (q+ , q− ) are the usual k variables translated, so as to vanish on the Fermi surface. They depend on the patch of coordinates chosen. This patch can be determined by the sector of the external line to which S2 is hooked. For constructive purpose one cannot however work in momentum space and one should write an equivalent dual formula in direct space. In practice a two-point function S2 is integrated in a bigger function against a kernel always made of one external propagator C and a rest called R, which (in momentum space) may be in general a function of the set Pe of external momenta. So in the momentum representation we have to compute not S2 itself but integrals such as I=

dpdq S2 (p)C(q)R(p, q, Pe )

(III.28)

where from momentum conservation R(p, q, Pe ) = δ(p − q)R (p, q, Pe ). To get the corresponding direct space representation we have to pass to the Fourier transform. Using same letters for functions and their Fourier transforms we write (III.29) I = dydz S2 (x, y)C(y, z)R(z, x, Pe ) (this integral being in fact by translation invariance independent of x) where S2 (x, y) = dp S2 (p)eip(x−y) ; C(y, z) = dq C(q)eiq(y−z) ; dpdq R(p, q, Pe )eip(z−x) , (III.30) R(z, x, Pe ) = where the last integral is not really a double integral because of the δ function hidden in R. Any counterterm for I that is expressed in momentum space by an operator τ acting on S2 (p), such as putting S2 to a fixed momentum k, hence τ S2 (p) = S2 (k), can also be represented by a dual operator τ ∗ acting in direct space, but on the external propagator. This τ ∗ is not unique, but a convenient choice is to use x as the reference point for τ ∗ : τ I = dp dq S2 (k)C(q)R(p,q,Pe ) = dy dz S2 (x,y)[eik(x−y) C(x,z)]R(z,x,Pe ), (III.31) hence τ ∗ C(y, z) = eik(x−y) C(x, z).

(III.32)

The dual version of the more complicated expressions (III.26–III.27) is given by (we write the expressions in the patch where q+ = k+ − 1, q− = k− − 1) I = dpdq S2 (p)C(q)R(p, q, Pe ) = I1 + I2

408


I1 =

I2 =


dydz S2 (x, y) C(y, z) − cos πT (x0 − y0 ) C (x0 , y+ , y− ), z R(z, x, Pe )

dydz S2 (x, y) cos πT (x0 − y0 ) C (x0 , y+ , y− ), z R(z, x, Pe ) − dydz S2 (x, y) cos πT (x0 − y0 ) ei(x+ −y+ ) C (x0 , x+ , y− ), z R(z, x, Pe ) − dydz S2 (x, y) cos πT (x0 − y0 ) ei(x− −y− ) C (x0 , y+ , x− ), z R(z, x, Pe ) + dydz S2 (x, y) cos πT (x0 − y0 ) ei[(x+ −y+ )+(x− −y− )] C(x, z)R(z, x, Pe ) (III.33)

where the propagator C is now the natural extension of the propagator to the continuum. Each integral I1 and I2 will be bounded separately. We need to exploit the differences as integrals of derivatives. This means that in I1 we write: C(y, z) − cos πT (x0 − y0 ) C (x0 , y+ , y− ), z 1 d C (ty0 + (1 − t)x0 , y+ , y− ), z cos πT (1 − t)(x0 − y0 ) dt = dt 0 1 1 dt (y0 − x0 ) eiπT (1−t)(x0 −y0 ) (∂0 + iπT )C (ty0 + (1 − t)x0 , y+ , y− ), z = 2 0 + e−iπT (1−t)(x0 −y0 ) (∂0 − iπT )C (ty0 + (1 − t)x0 , y+ , y− ), z (III.34) and in I2 we write C (x0 , y+ , y− ), z − ei(x+ −y+ ) C (x0 , x+ , y− ), z − ei(x− −y− ) C (x0 , y+ , x− ), z + ei[(x+ −y+ )+(x− −y− )] C(x, z) = F (1, 1) − F (0, 1) − F (1, 0) + F (0, 0) (III.35) where

x0 , sy+ +(1−s)x+ , ty− +(1−t)y+ , z ei[(1−s)(x+ −y+ )+(1−t)(x− −y− )] . (III.36) Finally we can use 1 1 d2 F (s, t) . (III.37) F (1, 1) − F (0, 1) − F (1, 0) + F (0, 0) = dsdt dsdt 0 0

F (s, t) = C

to obtain: I2 = dydz S2 (x, y) cos πT (x0 − y0 ) R(z, x, Pe )ei[(1−s)(x+ −y+ )+(1−t)(x− −y− )] (y+ −x+ )(y− −x− )(∂+ +i)(∂− +i)C x0 , sy+ +(1−s)x+ , ty− +(1−t)y+ , z . (III.38)

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IV Multislice expansion We perform a multi-slice expansion, and get a Gallavotti-Nicolò or clustering tree structure as in [1]. In that paper a tree formula was used to express a typical function for the model, namely the pressure, but the analysis applies to any thermodynamic function. Now we would like to focus on the self-energy. A good starting point for this is the connected amputated two-point Schwinger function. We fix here some conventions and notations that have not been introduced in [1]. We will call a “field” (between inverted commas) a five-tuple (x, a, σ, nature, order) where: x ∈ V , a ∈ {↑, ↓} , σ ∈ Sect(T ) , nature ∈ {+, −} , order ∈ {1, 2}. (IV.39) x is the spacetime position of the “field”, a its spin and σ its sector. nature is an element of the set whose elements are denoted + and −; this parameter is introduced in order to distinguish between the fields and the antifields (corresponding respectively to the Grassmann variables ψ and ψ). Thus in the following, it may happen that we use the term field (without inverted commas) to mean a “field” such that nature = + and of course an antifield will be a “field” such that nature = −. At last, the parameter order allows to distinguish between the two copies antifield involved in the expansion of the quartic action: and of each field a∈{↑, ↓} ψ a ψa = a, b ψ a ψa ψ b ψb , in such a way that order = 1 corresponds to the first (anti)field represented by the Grassmann variables ψ a and ψa , while order = 2 corresponds to the second ones, represented by ψ b and ψb . Given an integer n ≥ 1, an n-tuple (x1 , . . . , xn ) of elements of V , two n-tuples (a1 , . . . , an ) and (b1 , . . . , bn ) of elements of {↑, ↓} and four n-tuples of elements of Sect(T ), denoted (σ1j , . . . , σnj ), j ∈ {1, 2, 3, 4}, we define the family of the antifields: AF = (x1 , a1 , σ11 , −, 1), (x1 , b1 , σ12 , −, 2), . . . , (xn , an , σn1 , −, 1), (xn , bn , σn2 , −, 2) . (IV.40) We can imagine it as a 2n-tuple indexed by the set [n] × {1, 2} (where [n] denotes the set {1, . . . , n}), lexicographically ordered: (1, 1) ≺ (1, 2) ≺ (2, 1) ≺ (2, 2) ≺ · · · ≺ (n, 1) ≺ (n, 2) .

(IV.41)

In the same way we introduce the family of the fields: F = (x1 , a1 , σ13 , +, 1), (x1 , b1 , σ14 , +, 2), . . . ,

(xn , an , σn3 , +, 1), (xn , bn , σn4 , +, 2) . (IV.42)

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Observe that AF and F are defined as families and not as sets. Hence the cardinality of AF and F is 2n, whatever may be the values of the parameters {xv }, {av }, {bv } and {σvj }. Given f ∈ AF and g ∈ F, we will simply denote by C(f, g) the propagator: C(f, g) = δa(f ), a(g) δσ(f ), σ(g) C x(f ) − x(g) , (IV.43) where the notations a(f ), a(g), σ(f ), σ(g), x(f ), x(g) have an immediate obvious meaning. With all these notations, we can express the partition function of the model as: ∞ λn d3 x1 . . . d3 xn det C(f, g) . (IV.44) Z(V ) = n! V n (f,g)∈AF ×F j n=0 {av }, {bv } {σv }

AF for the Fermionic determinant (Cayley’s F notation). To write the unnormalized unamputated two-point Schwinger function: 2 3 ψ a (x)ψa (x) , S2 (Y, Z)σ0 = dµC (ψ, ψ) ψ ↑, σ0 (Y )ψ↑, σ0 (Z)exp λ d x Sometimes we shall write simply

V

a

(IV.45) we only need to add the source terms (Y, ↑, σ0 , −) to AF and (Z, ↑, σ0 , +) to F 3 . Since AF and A are indeed totally ordered families, we must specify in which position (y, ↑, σ0 , −) and (z, ↑, σ0 , +) are inserted. Clearly, they must be added in first position, that is, we have: AF = (y, ↑, σ0 , −), (x1 , a1 , σ11 , −, 1),...,(xn , an , σn1 , −, 1), (xn , bn , σn2 , −, 2) (IV.46) and

F = (z, ↑, σ0 , +), (x1 , a1 , σ11 , −, 1),...,(xn , an , σn1 , −, 1), (xn , bn , σn2 , −, 2) . (IV.47) Observe that, with a slight abuse of notation, we denote these two families again by AF and F . With this convention, the expression of the two-point function S2 (y, z)σ0 is exactly the same as the one of Z(V ): ∞ λn C(f, g) . d3 x1 . . . d3 xn det S2 (Y, Z)σ0 = n! V n (f,g)∈AF ×F j n=0 {av }, {bv } {σv }

(IV.48) The main tool to express the connected two-point function is a Taylor jungle formula [14], that is a forest formula which is ordered according to the main slice index, namely r, attached to the propagator, to expand the Fermionic determinant. 3 Note

that these two external “fields” have no order parameter.

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To extract the connected part of the two-point function, namely S2 (Y, Z)c, σ0 = Z −1 S2 (Y, Z)σ0 , we only need to factorize the contributions of the vacuum clusters of the jungle, and we get a tree formula: S2 (Y, Z)c,

σ0 =

∞ λn n=0

∈T

1

dw

0

n!

Vn

d3 x1 ...d3 xn

treesT {av }, {bv }, {σvj } oriented over V

C f (,Ω), g(,Ω)

∈T

det

(f, g)∈AF left ×Fleft

field attributions Ω

C(f, g,{w }) . (IV.49)

The amputated connected two-point function S2 (y, z)c,a is given by a similar formula, in which we should delete the two external sources Y and Z and the two propagators which connect them to two particular external distinguished vertices4 . Let us rename the position of these vertices as y and z, and rename all remaining internal positions as x1 , . . . , xn . So, after integration over positions of these n internal vertices, this amputated function is a function of the positions y and z of the two particular special external vertices. We shall denote V the family of the vertices: V = (y, z, x1 , . . . , xn ). We recall that a tree over V = {y, z, x1 , . . . , xn } is a set of pairs of vertices {v, v } (called the links of the tree), such that the corresponding graph has no loop and connects all the elements of V. As |V| = n + 2; any tree over V has n + 1 links. Once a tree T over V is chosen, a field attribution Ω for T is a family of the form (ω , ω ) where ω is a map from the pair to {1, 2} and ω a one-to-one ∈T

map from to {+, −}. Hence Ω is simply the choice, for each “half-line” of the tree T of a precise “field” of the vertex to which this half-line hooks. We have taken into account the constraint that a field must contract with an antifield by the fact that the maps ω : → {+, −} are one-to-one. Given ∈ T and a field attribution Ω, we denote respectively by f (, Ω) and g(, Ω) the antifield and the field attached to by Ω. AF left and Fleft are the families of the remaining “fields”: AF left = AF \{f (, Ω), ∈ T } and Fleft = F \{g(, Ω), ∈ T } .

(IV.50)

At last we must precise the expression of the entries of the remaining Fermi. onic determinant that depends now on the interpolation parameters w ∈P2 (V)

We recall that (see [14] – [1] for details) the data w allows to define a vector 4 Indeed we can forget the graphs where these two external sources Y and Z connect to the same external vertex, the “generalized tadpoles”, since they are zero by the particle hole symmetry.

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X T {w } whose components are indexed by P2 (V), the set of the (unordered) pairs of vertices. By definition, for {v, v } ∈ P2 (V), X T {w } is the in {v, v }

fimum of the w parametersover the unique path in T from v to v . Then, the expression of C f, g, {w } is simply obtained by multiplying C(f, g) by the component of X T {w } corresponding to the vertices v(f ) and v(g) of f and g. Hence we have: C f, g, {w } = X T {w } C(f, g) . (IV.51) {v(f ), v(g)}

IV.1 The Gallavotti-Nicol` o tree In order to analyze further this sum, it is well known that the main tool is the “Gallavotti-Nicol` o” or clustering tree which represents the inclusion relations of the connected components of “higher scales” (smaller r indices) into those of “lower scales” (bigger r indices) [6]. This tree is also the key tool to identify the components that require some renormalization (here the two-point functions). But before doing this, we want to describe precisely the constraints on the sum over the sectors {σvj }. Indeed, this sum could be let free of constraints, but due to the expression of the propagator: C(f, g) = C (x(f ), a(f ), σ(f )); (x(g), a(g), σ(g)) = δa(f ), a(g) δσ(f ), σ(g) Cσ(f ) (x(f ), x(g)) , (IV.52) we see easily that the sectors and spin indices are conserved along each line of the tree T . Therefore, once T has been fixed, the sum over the σvj ’s can be understood as a sum over the families of sectors indexed by the lines of T , denoted σ ∈T , and the families of sectors indexed by the remaining “fields”, σf f ∈AF ∪F . left left Now let supposewe are given an oriented tree T over V, and an attribution us o tree is defined as of sectors, σ ∈T and σf f ∈AF ∪F . The Gallavotti-Nicol` left left follows: for each index r ∈ [0, rmax (T )], we define a partition Πr . Πr is the set of the connected components of the graph whose set of verticesis V and whose internal tree lines are the lines of T such that r ≤ r. The family r∈[0, rmax ] Πr is partially ordered by the inclusion relation and forms the nodes of the GallavottiNicol` o tree. To visualize better the situation, let us take the example of Figure 2 for an amputated two-point function with external vertices at y and z (the external amputated legs in slice 6 are represented as dotted lines in Figure 2). The total number of vertices is 8, hence there are 7 lines in the tree T represented as bold lines, and 16 internal fields in the determinant represented as thin half-lines. In the attribution of r indices chosen we see that there is a two-point subfunction to renormalize, the one in the dotted box, which is completed at scale 3 with external lines at scale 5.

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5

6

5

5

x1

5 5

x4 2 5

3

3 3

0

y

3

x3

3 3

3

1

x6

x5

3

413

5

5

z

5

6

4 x2

Figure 2. A contribution with eight vertices to the two-point function at scale 6. The corresponding Gallavotti-Nicol` o tree is pictured in Figure 3 (with determinant fields omitted for simplicity). This abstract tree should not be confused with T , whose lines are the bold lines of Figure 3. As in [1], we can now write an expression of S2 (y, z)c,a re-ordered in terms of these “clustering tree structures”, in which all nested sums have to be compatible: S2 (y, z)c,a =

∈T

1

dw 0

∞ λn+2 d3 x1 . . . d3 xn n! n V n=0 × {av }, {bv }

clustering tree treesT field attributions structures C over V Ω

C f (, Ω), g(, Ω)

∈T

{σvj }

det


C(f, g, {w }) . (IV.53)

In the Gallavotti-Nicol` o tree, of particular interest to us are the nodes such as the dotted box of Figure 2 between scales 3 and 5 which correspond to two-point functions. They are the ones that were artificially suppressed in the simplified model [1]. We need to renormalize them to solve the divergent power counting explained in Section III. But we can choose to renormalize only the two-point functions for which external lines have r index bigger than the maximum index of internal lines plus 2, so as to create a gap between internal and external supports5 . Such two-point functions are the dangerous nodes of the GN tree. The gap ensures that all such dangerous two-point functions, which are those that we need to renormalize, are automatically one-particle irreducible by momentum conservation6 . Hence they correspond to the so-called self-energy. 5 The

two-point functions for which the external r index is the maximum r index of internal lines plus 1 don’t really need renormalization, as is obvious from power counting (see (III.25)). 6 Indeed any one particle reducible two-point function would have its external momentum also flowing through any internal one-particle-reducibility line, which is a contradiction with the fact that the internal and external cutoffs have empty intersection.

414


y

x1

x2

x3

x4

x5

x6

z

y

x1

x2

x3

x4

x5

x6

z

y

x1

x2

x3

x4

x5

x6

z

y

x1

x2

x3

x4

x5

x6

z

y

x1

x2

x3

x4

x5

x6

z

x4


x6

y x1

x3

x5

z

x2

Figure 3. The Gallavotti-Nicol` o tree corresponding to Figure 2. We can re-order the expression of S2 (y, z)c,a in terms of these dangerous two-point subgraphs, in the spirit of [1]: S2 (y, z)c,

∈T

0

a

∞ λn+2 = d3 x1 . . . d3 xn n! V n n=0 {av }, {bv }

biped structures external fields clustering tree treesT field attributions B EB structures C over V Ω

1

dw

C f (, Ω), g(, Ω)

∈T

det


{σvj }

C(f, g, {w }) . (IV.54)

V

Main theorem on the self-energy

We have given in the last section an expression for the connected amputated 2point Schwinger function. Now we would like to consider the self-energy Σ(y, z). This quantity can be defined either through its Feynman graph expansion, or through a Legendre transform.

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In the first approach, which we use, Σ(y, z) is given by the same sum (IV.54) than S2 (y, z)c, a but restricted to the contributions which are 1-particle-irreducible in the channel y−z, that is, in which y and z cannot be disconnected by the deletion of a single line. This definition does not look very constructive because in principle we would have to expand out all the remaining determinant in (IV.54) to know which contributions are 1-PI or not. But in the next section we shall see that to extract this information a partial (still constructive) expansion of the determinant is enough. In this section we only formulate our main bound on this connected amputated and one particle irreducible (1-PI) 2-point function or self-energy Σ. Note that, for convenience, we shall simply write in the following “1-PI” to mean: “1particle-irreducibility in the channel y − z”. The sum of all contributions to the self-energy with infrared cutoff r and fixed external positions y and z will be called Σ2 (y, z)≤r . Consider the set Σr of triplets σ ¯ = (i(¯ σ ), s+ (¯ σ ), s− (¯ σ )) with 0 ≤ i ≤ r and 0 ≤ s± ≤ r, also called “generalized sectors”. We can obviously also define the scale distance dσ¯ (y, z) for such triplets as in (II.16), and the index r(¯ σ) = σ ) + s− (¯ σ ))/2 . Then with all the notations of the previous section, the (i(¯ σ ) + s+ (¯ following bound holds: Theorem V.1 There exists a constant K such that: α

|Σ2 (y, z)≤r | ≤ (λ| log T |)2 sup KM −3r(¯σ) e−cdσ¯ (y,z) ,

(V.55)

σ ¯ ∈Σr

α

|y+ − z+ |.|y− − z− |.|Σ2 (y, z)≤r | ≤ (λ| log T |)2 sup KM −2r(¯σ) e−cdσ¯ (y,z) , (V.56) σ ¯ ∈Σr

≤r

|y0 − z0 |.|Σ2 (y, z)

α

| ≤ (λ| log T |) sup KM −2r(¯σ) e−cdσ¯ (y,z) . 2

σ ¯ ∈Σr

(V.57)

For the second equation (V.56), a naive bound would have M −r instead of . So the crucial point is to gain a factor M −r in the bound (V.56). (V.55) M and (V.57) are easy. The next four sections are dedicated to the proof of this theorem. We call a self energy contribution “primitively divergent” if there is no smaller biped in it. The sum of all such “primitively divergent” contributions to the self-energy with infrared cutoff r and fixed external positions y and z is called Σ2,pr (y, z)≤r . We first prove in the next three sections that the bounds (V.56) and (V.57) hold for Σ2,pr (y, z)≤r , then by an inductive argument we extend the bound to the general unrestricted self-energy. The most naive bounds don’t work. Indeed we should optimize power counting and positions integrals separately in the 0 and ± directions in order to bound correctly the effect of the (y − z)± factors in (V.56). But the problem is how to do this constructively. One cannot simultaneously build the three spanning trees that would optimize spatial integrations with respect to the 0 and ± directions, as this −2r

416



may typically develop too many loops out of the determinant. The road to solve this problem is to derive not only a 1-PI, but a 2-PI expansion inside each twopoint contribution to renormalize. This expansion can be controlled constructively; then one can optimize over the 0 and ± multiscale analysis, using only the tree T and the additional loops which the expansion has taken out of the determinant. In this way one obtains a better bound than the one obtained naively by simply exploiting a single tree formula as in [1]. This is the key to our problem of the renormalization of the 2-point function.

VI Multiarch expansion Consider the self-energy of the model. The previous tree expansion insured the connexity of the graphs but not their 1 or 2-particle-irreducibility. We are going now to expand out explicitly some additional lines from the determinant, in order to complete the tree T into a 2-PI graph. Nevertheless it is not trivial to ensure that this additional expansion does not generate “too many” terms, or in other words that it is “constructive”. In the following section, we explain in detail this expansion for an expression of the type F = ∈T Cσ() (f (), g()) detleft,T .

VI.1 1-particle-irreducible arch expansion First, we fix some conventions. We consider the tree T connecting all the vertices y, z, x1 , . . . , xn . We distinguish in T the unique path connecting y and z through T , denoted by P (y, z, T ). Each vertex of this path is numbered by an integer starting with 0 for y and increasing towards z, which is the end of the path (with number p). The set of the remaining 2(n + 2) fields and antifields, denoted by Fleft,T = AF left,T ∪ Fleft,T , is divided into p + 1 disjoint subsets or “packets” F0 , . . . , Fp : by definition, an element f ∈ Fleft,T belongs to Fk if and only if k is the first vertex of P (y, z, T ) met by the unique path in T joining the vertex to which f is hooked to y. Figure 4 allows to visualize better the situation. When f belongs to the packet Fk we also say that the packet index of f is k. F2

F0

Fp 00 11 0 1 0 1 0 1 00 11 00 11 0 1 0 1 0 1 00 11 1 0 0 1 00 11 00 11 1 0 0 1 0 1 0 1 00 11 00 11 1 0 0 00 11 0 1 1 0 00 11 11 00 11 00 11 00 1 0 1 11 1 00 0 00 11 0 00 11 0 1 0 1 1 00 11 00 11 00 11 0 1 0 1 0 1 00 11 0 1 00 11 0 1 11 00 0 1 00 11 00 11 1 0 00 11 0 1 0 1 00 11 0 1 00 11 00 11 00 11 0 1 00 11 00 11 00 11 0 1 1 0 00 11 1 0 00 11 0 1 00 11 00 11 11 00 0 1 00 11 00 11 11 00 00 11 00 11 0 1 00 11 00 11 00 11 0 1 00 11 00 0 1 F1 11 0 1 11 00 0 1 00 11 00 11 00 11 00 11 00 11 00 11 0 1 1 0 0 1 00 11 00 11 00 11 00 11 00 11 00 11 0 1 0 1 00 11 00 11 0 1 11 00 00 11 0 1 0 1 00 11 00 11 00 11 11 00 00 11 00 00 11 0 1 0 1 00 11 00 0 1 00 0 1 00 0 1 0011 11 00 00 00 00 11 00 11 00 11 11 0 11 1 0 11 1 00 11 11 00 11 0 11 1 00 11 0 11 1 111 000 1111 0000

0 y

1

2

3

p−1

p z

Figure 4. The tree T and the “field packets” F0 , . . . , Fp .

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417

In this figure we have represented the external (amputated) propagators by dotted lines, the links of P (y, z, T ) by bold lines, the other links of T by thin lines and at last the remaining fields in the determinant by thinner half-lines. Once the ordered family of subsets of fields F0 , . . . , Fp has been defined, the arch expansion is carried out in the standard way of [11], Appendix B1. Let us recall this expansion here for self-completeness. Among all the possible contraction schemes implicitly contained in detleft, T , we select through a Taylor expansion step with an interpolating parameter s1 those which have a contraction between an element of F0 and ∪pk=1 Fk . Given such a contraction, we call k1 the index of the precise packet joined to F0 by this contraction. Thus we have added to T an explicit line 1 joining F0 to Fk1 . At this stage, the graph obtained is 1-particle-irreducible in the channel y − xk1 (see Figure 5).

1

F2

F0 1 0 0 1 0 1 0 1 0 00 1 0 1 11 00 11 00 11 00 11 00 11 00 11 11 00

0 1 0 1 0000 1111 0 1

0 y

0 1 0 1 1 0 0 1 0 1 0 1 01 1 01 1 0 1 0 0 0 1 0 1 0 1 01 1 0 1 0 1 0 11 0 1 000 1 01 1 0 1 0 0 1 0 0 1 1 0000 1111 0000 1111 1 0 0000 1111 0 1 0000 1111 0 1 0 1 0111 1 0000 1111 01 1 0 111 000 000 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 00001 1111 000 111 0 0 1 0 1 0 0 1 0 1 01 1 0 1 0 1 0 1 0 1 0 1 01 1 0 1 0 1 0 1 0 0 1 01 1 0 0 1 0 01 1

F1

1

2

3

Fp

Fk 1

0 1 1 0 00 11 0 1 011 1 0 1 0 1 0 1 0 1 0 01 1 000 1 11 00 00 11 0 1 0 1 00 11 0 1 11 00 00 00 11 0 1 011 1 0 1 0 1 00 11 11 00 0 1 0 1 0 1 11 0 0011 11 0011 11 111 000 0 1 11 00 0 00 11 00 0 1 0000 1 0 1 0 1 1 0 00 11 00 111 0 1 00 11 0 1 00 11 0011 11 00 11 0 1 00 11 00 11 00 00 11 0011 11 00 11 00 11 00 11 00 00 11 0011 11 00 11 00 11 00 11 00 00 11 00 11 00 11 0011 11 00 11 00 11 00 11 00 11 00 11 00 11 00 0011 11 00 00 11 00 11 0011 11 00

k1

00 11 00 11 00 11 11 00 00 11 00 11 00 11

p−1

0 0 01 1 01 1 0 1 0 1 1 0 0 1 0 0 01 1 01 1 0 1 0 1 00 11 00 0 1 0011 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00 11111 0000000 11 00 11

p z

Figure 5. The tree T completed by a first line from the arch expansion.

Then we continue the procedure, testing whether there is a contraction be1 Fk and one of ∪pk=k1 +1 Fk . If there is not, the line from k1 tween an element of ∪kk=0 to k1 +1 of the path P (x, y, T ) is certainly a line of 1-particle-reducibility (i.e., its deletion would disconnect y and z), and therefore the corresponding contraction schemes do not contribute to the self-energy. But on the contrary, if there exists 1 Fk and ∪pk=k1 +1 Fk , we select it and we have the picture of a line 2 between ∪kk=0 Figure 6. The graph T ∪ {1 , 2 } is clearly 1-particle-irreducible in the channel y − xk2 . Observe that 0 < k1 < k2 therefore, in at most p steps, we shall reach certainly the end vertex z and we shall have a 1-particle-irreducible graph (in the channel y − z). Any final set of m arches derived in this way is called an m-arch system.

418



1

F2

F0 0 0 1 01 1 0 1 0 1

0 0 1 01 1 0 1 0 1 01 1 0 1 1 0 0 1 0 0 0 1 01 1 0 1 01 1 0 1 0 1 0 0 1 0 1 1 01 1 0 0 0 1 1 000 11 0 0 1 0000 1111 11 00 1 0 1 0 1 00 11 0000 1111 1 0 00 11 0000 1111 0 1 00 11 0000 1111 10 0 1 0 0111 1 00 11 0000 1111 1 0 1 111 000 000 00 11 0000 1111 000 111 00 11 0000 1111 11 00 000 111 0000 1111 000 111 0000 1111 000 111 00001 1111 000 111 0 0 1 0 0 1 11 0 0 1 0 1 0 1 0 1 0 1 01 1 0 1 0 1 0 1 0 0 1 01 1 0 0 1 0 01 1

F1

1 0 0 1 0 1

0 y

1

2

3

2

Fp

011 1 0 1 0 1 00 0 1 0 1 0 1 00 0 1 0 1 0 1 01 1 011 1 11 00 0 1 00 11 0 00 11 0 1 11 00 00 00 11 0 1 011 1 0 1 0 1 00 11 00 11 0 1 0 1 1 0 0 1 0 1 001 11 00 0 1 0 1 1 0 0011 11 0011 01 1 111 000 0 0 00 11 0 1 00 11 0011 11 00 11 0 1 00 11 00 11 00 00 11 0011 11 00 00 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 11 00 00 11 00 11 0011 11 00 11 00 00 11 00 11 0011 11 00 11 0011 11 00 00 11 00 11 00

k1

k2

00 11 11 00 00 11 00 11 00 11 00 11 00 11

p−1

0 1 0 1 0 1 0 1 1 0 0 1 0 0 01 1 01 1 0 1 1 0 00 11 00 0 1 0011 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 0011 11 00 0011 11 00 11 00 11 00

p z

Figure 6. The tree T completed by two lines from the arch expansion. We obtain the 1-PI part of the determinant as: !m " m 1 dsr C(fr , gr )(s1 , . . . , sr−1 ) F1−PI = r=1 m−arch systems r=1 0 (f1 ,g1 ),...,(fm ,gm ) with m≤p

∂ m detleft, T {sr } . (VI.58) m r=1 ∂C(fr , gr )

The expansion respects positivity of the interpolated propagator at any stage, because all sr interpolations are always performed between a subset of packets and its complement, hence the final covariance as function of the sr parameters is a convex combination with positive coefficients of block-diagonal covariances. This ensures that the presence of the sr parameters does not alter Gram’s bound on the remaining determinant, which is the same than with all these parameters set to 1 ([10]–[11]). Furthermore it is constructive in the sense that it does not generate any factorial in the bounds for the sum over all derived arches. Here is a subtlety. Once the departure and arrival fields joined by the arches have been fixed (which costs at most 4n ), the arrival fields are determined because their packet indices are strictly growing. But the departure fields are not, and in principle this could create a constructive problem. For example, if the line 1 joins F0 to Fk1 , it is possible for the second one, 2 , to join F0 to Fk2 (see Figure 7). Remark that in this case a posteriori 1 is useless. With three arches, an arch system such as Figure 8 shows the same phenomenon, in the sense that a posteriori 2 is useless. This is not a great disadvantage, because in spite of this lack of minimality, the expansion can indeed be controlled in a constructive way. The reason is that

Vol. 6, 2005


419

2 1 F0

0 0 1 1 0 01 1 00 0 1 0 11 1 1 110 00 00 11 00 11 00 11 00 11 00 11 11 00

1 0 0 1 0 1

0 y

00 11 0 1 1 0 00 11 0 1 0 1 0 1 0 1 00 11 01 0 01 1 0 1 00 11 0 1 00 11 0 001 11 0 1 1 00 11 0 0 001 11 0 1 1 1111 0000 0000 0 1111 0 1 0000 1111 0 1 0000 1111 10 0 1 0 11 00 000 111 0111 1 0000 1111 1 0 1 11 00 000 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 00 11 0 1 00 11 0 1 11 0 00 11 0 0 1 0 1 0 1 0 1 10 0 1 1 0 1 0 00 1 1 0 0 1 1 0 1 0 1

F1

1

2

Fp 00 11 1 0 00 11 1 000 0 0 111 00 11 1 0 11 0 0011 11 00 0 1 00 11 00 0011 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00

3

0011 11 00 11 00 0 1 00 11 00 11 11 00 00 11 0 1 00 11 00 11 000 111 00 11 00 11 00 11 00 11 0 1 00 11 000 111 00 11 00 11 00 11 000 111 00 11 00 11 000 111 00 11 00 11 000 111 00 11 1111 0000 11 00 000 111 00 11 000 111 11 00 000 111 000 111 000 111 000 111 000 111 000 111 00 11 1 0 000 111 00 11 1 0 000 111 00 11

k2

k1

11 00 00 11 0 1 00 11 0 1 00 11 0 1 00 11

p−1

00 11 0 111 00 11 00 0 00 1 11 00 00 11 11 0 1 11 00 00 11 00 11 0 1 00 11 00 11 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00

p z

Figure 7. A pair of arches which is not minimal from the 1-PI point of view. 3

1

2 0 1 0 1 1 0 0 1 0 0 1 01 1 01 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 01 1 01 0 1 0 01 1 1 0 0 1 0 1 0 1 0 1 0000 1111 11 00 1 0 0 0000 1111 0000 1 1111 1111 0 0000 1111 0 1 0 000 111 000 1 0 0000 1111 01 1 0 1 000 111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0 1 0 1 0 1 0 0 1 0 1 01 1 0 1 0 1 0 1 0 1 0 1 00 1 1 0 1 0 1 1 0 1 00 1

1 0 1 0 0 1 0 1 1 0 1 0 0 1 11 00 00 11 00 11 00 11 00 11 00 11 11 00

0 1 0 1 0000 1111 0 1

0 y

1

2

3

0 1 0 1 0 1 0 1 0 1 00 11 0 1 0 1 1 0 0 1 00 0 0 1 00 11 11 0 011 1 11 00 0 1 00 11 0 1 00 11 0 1 11 00 00 00 11 0 1 011 1 1 0 0 1 001 11 00 11 0 1 0 1 0 1 1 0 00 11 00 11 000 11 00 0 0 1 00 11 00 0 1 0111 1 0 1 01 1 0011 11 0011 11 000 111 0 1 0 001 11 0 1 00 11 00 11 00 0 00 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 11 00 00 11 0011 11 00 11 00 11 00 11 00 00 11 00 11 0011 11 00 11 11 00 00 00 11 00 11 0011 11 00 11 00 11 0000 11 00 11

k2

k1

00 11 00 11 00 11 11 00 00 11 00 11 00 11

p−1

0 1 0 1 1 0 0 1 0 1 0 1 01 1 01 1 0 0 0 1 0 1 0 1 0 1 0011 11 00 11 0 1 00 11 00 00 11 00 11 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00 11111 00000 0011 11 00

p z

Figure 8. Another example of a “non-minimal” system of three arches. the arches for which the departure fields indices do not grow are damped by small s interpolation parameters, so that the result is indeed bounded by K n [11]. More precisely the dependence in the sr parameters in front of each arch system is a q m monomial r=1 srr,m−arch , so that we have: m m m q C(fr , gr )(s1 , . . . , sr−1 ) = C(fr , gr ) srr,m−arch . (VI.59) r=1

r=1

r=1

The reader can check that the integer qr,m−arch ≥ 0 is the number of arches which fly entirely over the rth arch, making it useless.

420



Lemma VI.1 There exists some numerical constant K such that (for n ≥ 1): m m p 1 q dsr srr,m−arch ≤ c.K n . (VI.60) 0 m−arch systems m=1 r=1 r=1 (f1 ,g1 ),...,(fm ,gm ) with m≤p

Proof. The proof is identical to [11], Lemma 9. We reproduce it here for completeness. Consider Fkr the arrival packet of the rth arch, which joins the field fr to the field gr ∈ Fkr . The set of possible departure packets to which fr must belong is Er = F0 ∪ F1 ∪ · · · ∪ Fkr−1

(VI.61)

We also define ei = |Ei − Ei−1 | as the number of fields and antifields in Ei and not in Ei−1 . The sum over all m-arch systems which we have to bound is p

m=1 0 0, the “imaginary time”, denoted x0 , belongs to the real interval − T1 , T1 . In the following, we shall denote β = T1 . Indeed this interval should be thought of as a circle of length 2β, that is R/2βZ. Consequently, the momentum space, which is the dual of R/2βZ × Z2 in the sense of the Fourier transform, is πT Z × [R/2πZ]2 . The torus [R/2πZ]2 will be represented by the square [−π, π[2 , with periodic boundary conditions. In Fourier variables, the expression of the propagator at half-filling reads: C(k0 , k1 , k2 ) =

1 ik0 − cos k1 − cos k2

(II.1)

if k0 = (2n + 1)πT for some n ∈ Z. If k0 = 2nπT , C(k0 , k1 , k2 ) = 0 because in the formalism of Fermionic theories at finite temperature, the propagator has an antiperiod β with respect to the x0 variable and therefore each Fourier coefficient of even order vanishes. With a slight abuse of language, we can say that C(k0 , k1 , k2 ) is only defined for k0 = (2n + 1)πT . This set of values is called the Matsubara frequencies. The expression of the propagator in real space is deduced by Fourier transform: eik.x 1 C(x0 , x1 , x2 ) = (II.2) dk dk dk 0 1 2 (2π)3 ik0 − cos k1 − cos k2 where we adopt the notations of [1], namely the integral dk0 really means the discrete sum over the Matsubara frequencies 2πT n∈Z ((2n + 1)πT ) (with k0 = (2n+1)πT ), whereas the integrals over k1 and k2 are “true” integrals, for (k1 , k2 ) ∈ [−π, π[2 . (We do not need any ultraviolet cutoff for the graph studied in this paper, since it is ultraviolet convergent.) For our analysis, it will be convenient to introduce another parametrization of the spaces [−π, π[2 and Z2 . The idea is to “rotate” the Fermi surface of Figure II by an angle of π4 . In the k0 = 0 plane, it is defined by cos k1 + cos k2 = 0, which is equivalent to k2 = π ± k1 or k2 = −π ± k1 . k1 = π2 (k+ + k− ) k1 ±k2 Introducing the variables k± = π ⇐⇒ , the domain k2 = π2 (k+ − k− ) of integration (k1 , k2 ) ∈ [−π, π[2 becomes the set: D = (k+ , k− ) ∈ [−2, 2]2 with −2 ≤ k+ ≤ 0 0 ≤ k+ ≤ 2 . (II.3) or −2 − k+ ≤ k− ≤ 2 + k+ −2 + k+ ≤ k− ≤ 2 − k+ As cos k1 + cos k2 = 2 cos π2 k+ cos π2 k− , the Fermi surface in the variables k± is simply defined by k+ = ±1 , k− = ±1. The new domain of integration, with the Fermi surface is represented on Figure 2.

Vol. 6, 2005

The Hubbard Model at Half-Filling, Part III

π

451

k2

π −π

k1

−π

Figure 1. The square [−π, π[2 and the Fermi surface. k− 2

1 -2

1

-1

2

k+

-1

-2

Figure 2. The domain of integration in (k+ , k− ) and the Fermi surface. In a dual way, we introduce new variables in real space, x+ and x− in such a way that k1 x1 + k2 x2 = k+ x+ + k− x− . We have: x+ = π2 (x1 + x2 ) (II.4) x− = π2 (x1 − x2 ) . Observe that the image of the lattice Z2 by this change of variable is not

π 2 2Z

but

452



π

n , (m, n) ∈ Z2 , m ≡ n[2] . (II.5) 2 2 In other words, the integers m and n must have same πparity.

π 2 k1 k+ 2 2 As the Jacobian of the transformation = π is J = − π2 , π k2 − k − 2 2 we have: ei(k1 x1 +k2 x2 ) ei(k+ x+ +k− x− ) π2 dk1 dk2 = dk+ dk− . ik0 − cos k1 − cos k2 2 D ik0 − 2 cos π2 k+ cos π2 k− [−π,π]2 (II.6) But the domain D is not very convenient for practical computations, and therefore we would like the k+ k− integration domain to factorize. Since the complement set [−2, 2[2 \D is another fundamental domain for the torus R2 /2πZ2 , we have: ei(k+ x+ +k− x− ) ei(k+ x+ +k− x− ) dk+ dk− = dk+ dk− . π π ik0 − 2 cos 2 k+ cos 2 k− ik0 − 2 cos π2 k+ cos π2 k− D [−2,2[2 \D (II.7) Hence: ei(k+ x+ +k− x− ) ei(k+ x+ +k− x− ) 1 dk+ dk− = dk+ dk− . π π ik0 − 2 cos 2 k+ cos 2 k− 2 [−2,2]2 ik0 − 2 cos π2 k+ cos π2 k− D (II.8) Recapitulating, the expression of the propagator that we take as our starting point is: ei(k0 x0 +k+ x+ +k− x− ) C(x0 , x+ , x− ) = d3 k (II.9) ik0 − 2 cos π2 k+ cos π2 k− for x± satisfying the parity condition (II.5). In II.9 the notation d3 k means 1 dk+ dk− , (II.10) dk0 32π [−2,2]2 where we recall that dk0 means 2πT n∈Z η((2n+1)πT )), since k0 = (2n+1)πT . Now, let us consider, in Fourier space, the amplitude of the graph G represented on Figure 3, with an incoming momentum k = (k0 , k+ , k− ). This amplitude ¯ 2 e−ik.x (where is denoted AG (k) and written as AG (k0 , k+ , k− ) = d3 x C(x)C(x) arrows join antifields to fields).

the subset

S=

π

m,

k

k 0

x

Figure 3. The first non-trivial graph contributing to the self-energy.

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453

More precisely, we shall consider the second momentum derivative in the + direction of this quantity, which up to a global inessential minus sign is: 2 ¯ 2 e−ik.x . ∂+ AG (k) = d3 x x2+ C(x)C(x) (II.11) The quantity we are going to study is explicitly written: 2 AG (πT, 1, 0) ∂+

=

3

d x

x2+

d3 k1

eik1 .x ik1,0 − 2 cos π2 k1,+ cos π2 k1,−

d3 k2

eik2 .x −ik2,0 − 2 cos π2 k2,+ cos π2 k2,−

d3 k3

eik3 .x ei(πT x0 +x+ ) , (II.12) −ik3,0 − 2 cos π2 k3,+ cos π2 k3,−

where again d3 x includes the parity condition (II.5). We state now the main result of this paper:

Theorem II.1 There exists some strictly positive constant K such that, for T small enough: 2 ∂+ AG (πT, 1, 0) ≥ K . (II.13) T We recall that this result, joined to the analysis of [2], leads to the result that the self-energy of the model is not uniformly C 2 in the domain |λ| log2 T < K and therefore that the two-dimensional Hubbard model at half-filling is not a Fermi liquid.

III Plan of the proof Theorem (II.1) will be proven thanks to a sequence of lemmas. But before presenting these lemmas, let us give an overview of our strategy. We use the sector decomposition introduced in [1] to write: 2 ∂+ AG (πT, 1, 0) = d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ ) , (III.14) σ1 ,σ2 ,σ3

where a sector σ is a triplet (i, s+ , s− ) with 0 ≤ s± ≤ i and s+ + s− ≥ i. The main idea is that in the sum over sectors of equation (III.14), the leading contribution is given by a restricted sum corresponding to sectors close to the “vertical part” of the Fermi surface, defined by k+ = ±1. To express this more precisely, let Λ be an integer (whose value will be chosen later), which will play the role of a cut-off for the sectors. We want to prove that as soon as one sector is not

454



close to k+ = ±1, then we have a small contribution. Let us denote Λ − {ij },{s+ j },{sj } the sum in which at least one sector is “far” from the vertical sides of the Fermi surface. Precisely, this means that at least one index s+ j is smaller than imax (T )−Λ, where, as in [1], M −imax (T ) ≈ T . This constrained sum can be written explicitly: Λ

inf(i1 ,imax (T )−Λ)

i1 ,s− 1 ,σ2 ,σ3

s+ 1 =0

=

− {ij },{s+ j },{sj }

+


i1

− + i1 ,s− 1 ,i2 ,s2 ,σ3 s1 =imax (T )−Λ

s+ 2 =0

i1

i2


− − i1 ,s− 1 ,i2 ,s2 ,i3 ,s3

s+ 1 =imax (T )−Λ

s+ 2 =imax (T )−Λ

s+ 3 =0

+

.

(III.15)

Defining: AΛ G (πT, 1, 0)

=

Λ

d3 x Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ ) , (III.16)

− {ij },{s+ j },{sj }

we write: 2 2 2 Λ AG (πT, 1, 0) = ∂+ AG,Λ (πT, 1, 0) + ∂+ AG (πT, 1, 0) ∂+

(III.17)

2 2 2 Λ where ∂+ AG,Λ (πT, 1, 0) = ∂+ AG (πT, 1, 0) − ∂+ AG (πT, 1, 0) is expressed as a sum over sectors that are all close to k+ = ±1, i.e., such that each s+ j index is greater than imax (T ) − Λ. 2 Each sector appearing in the sum expressing ∂+ AG,Λ (πT, 1, 0) will be divided into two disjoint subsectors, according to the sign of k+ . We recall that in [1], the sectors were defined as: + − π π π π ik0 − 2 cos k+ cos k− ≈ M −i , cos k+ ≈ M −s , cos k− ≈ M −s . 2 2 2 2 (III.18) We shall call σ r and σ l (“right” and “left”) the subdomains of σ corresponding to k+ > 0 and k+ < 0 respectively. The underlying motivation is that, if a momentum, say k1 , is close to the side k+ = 1, by momentum conservation at each vertex, the other ones are necessarily close to the other side k+ = −1. Let us state precisely this point: 2 AG,Λ (πT, 1, 0), there must be one sector of Lemma III.1 In the sum expressing ∂+ the right type, and two of the left type.

The proof is obvious by momentum conservation in the + direction.

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455

This Lemma implies that: 2 ∂+ AG,Λ (πT, 1, 0) =

+ >imax (T )−Λ j σ1 right

d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ )

{σj },ij ,s

+


+ >imax (T )−Λ j σ2 right

{σj },ij ,s

+

d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ ) .

(III.19)

+ {σj },ij ,s >imax (T )−Λ j σ3 right

Among these three contributions, the last two ones are indeed equal, and we have: 2 AG,Λ (πT, 1, 0) ∂+ =



+2

d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ ) .

(III.20)


In each sum, we replace the cos π2 k+ appearing in the propagators by their Taylor expansions in the neighborhood of +1 in a right sector, and in a neighborhood of −1 in a left sector. We have cos π2 k+ ≈ − π2 (k+ − 1) for k+ in the neighborhood of 1, in which case we put q+ = (k+ − 1) and cos π2 k+ ≈ π2 (k+ + 1) for k+ in the neighborhood of −1, in which case we put q+ = (k+ + 1). This 2 ˜ AG,Λ (πT, 1, 0): replacement gives an expression that we call ∂+ 2 ˜ ∂+ AG,Λ (πT, 1, 0)

=

3

d x

x2+

d3 k1

uΛ (q1,+ )eik1 .x ik1,0 + πq1,+ cos π2 k1,−

uΛ (q2,+ )eik2 .x uΛ (q3,+ )eik3 .x 3 k e−i(πT x0 +x+ ) d 3 −ik2,0 − πq2,+ cos π2 k2,− −ik3,0 − πq3,+ cos π2 k3,− uΛ (q1,+ )eik1 .x + 2 d3 x x2+ d3 k1 ik1,0 − πq1,+ cos π2 k1,− uΛ (q2,+ )eik2 .x uΛ (q3,+ )eik3 .x 3 k e−i(πT x0 +x+ ) , d3 k2 d 3 −ik2,0 + πq2,+ cos π2 k2,− −ik3,0 − πq3,+ cos π2 k3,− (III.21) d3 k2

456



where the uΛ (q+ )’s is now the smooth scaled cutoff function u(M imax (T )−Λ q+ ) which expresses the former sector constraint s+ ≥ imax (T ) − Λ (u is a usual fixed Gevrey function which is 1 on [−1, 1] and 0 out of [−2, 2], see [1]). In (III.21) we can freely change each integral over dk+ which ran over [−2, 2] into an integral on dq+ which runs from [−∞, ∞]. We still denote d3 k the corresponding integrals. We write now for each propagator in (III.21), uΛ (q+ ) = 1 + u1 (q+ ) + uΛ 1 (q+ ) where u1 (q+ ) = u(q+ ) − 1 and uΛ (q ) = u (q ) − u(q ). In this way we generate + Λ + + 1 three terms: • one in which all three functions uΛ (q+ ) are replaced by 1. We call this term 2 ˜ AG (πT, 1, 0) ∂+ 1 • one in which there is at least one factor uΛ 1 (q+ ) and no factor u (q+ ). We 2 Λ call this term ∂+ AG,1 (πT, 1, 0).

• finally one in which there is at least one factor u1 (q+ ). We call this term 2 1 ∂+ AG (πT, 1, 0). At this stage, we recapitulate: 2 2 ˜ 2 Λ 2 1 ∂+ AG (πT, 1, 0) + ∂+ AG (πT, 1, 0) = ∂+ AG,1 (πT, 1, 0) + ∂+ AG (πT, 1, 0)

2 2 ˜ 2 Λ AG,Λ (πT, 1, 0) + ∂+ + ∂+ AG,Λ (πT, 1, 0) − ∂+ AG (πT, 1, 0) . (III.22) 2 This relation shows that the quantity under study, ∂+ AG (πT, 1, 0), is equal to the 2 ˜ approximation ∂+ AG (πT, 1, 0), up to the four error terms 2 Λ 2 Λ AG (πT, 1, 0), ∂+ AG,1 (πT, 1, 0), ∂+

2 2 ˜ 2 1 AG,Λ (πT, 1, 0) , ∂+ ∂+ AG,Λ (πT, 1, 0) − ∂+ AG (πT, 1, 0) .

(III.23)

Now we are going to prove a lower bound similar to the one of Theorem 2 ˜ AG (πT, 1, 0), and establish an upper bound on each II.1, but on the quantity ∂+ 2 ˜ of the four error terms. More precisely, if we have ∂+ AG (πT, 1, 0) > K T for some constant K > 0 and if the modulus of each error term is smaller than K K − 4K , (III.24) T 2 ˜ which shall prove Theorem II.1. The result that ∂+ AG (πT, 1, 0) > K T is really the most difficult to establish, and its proof is the heart of this paper. But the control of the error terms is easier, and each one will correspond to a lemma. We shall begin by these lemmas in the next section, and then turn to the lower bound 2 ˜ AG (πT, 1, 0). on ∂+

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IV The control of the error terms First we state a result that is not necessary for proving Theorem II.1 but whose proof illustrates the way the sector decomposition allows us to establish quite easily upper bounds. Lemma IV.1 There exists some constant K1 > 0 such that: 2 ∂ AG (πT, 1, 0) ≤ K1 . + T

(IV.25)

Proof: We use the decay property of C(i,s+ ,s− ) (x) (see [1], Lemma 1):

α

C(i,s ,s ) (x) ≤ c.M −s+ −s− exp −c dσ (x) , + −

(IV.26)

where α ∈]0, 1[ is a fixed number, c is a constant and dσ (x) = M −i |x0 | + M −s+ |x+ | + M −s− |x− |. We have: 2 + − ∂ AG (k) ≤ c3 .M − 3j=1 sj − 3j=1 sj +   3

α + − M −ij |x0 | + M sj |x+ | + M −sj |x− |  . d3 x x2+ exp −c − {ij },{s+ j },{sj }

j=1

(IV.27) Among the indices i1 , i2 and i3 , we keep the best one, i.e., the smallest one, to perform the integration over x0 . We proceed in an analogous way for the indices + + − − − (s+ 1 , s2 , s3 ) and (s1 , s2 , s3 ) respectively. Thus we have:

2 ∂+ AG (k) ≤ c3

M−

3

j=1

s+ j −

3

j=1

s− j

+

−

M inf{ij } M 3 inf{sj } M inf{sj

}

.

− {ij },{s+ j },{sj }

(IV.28) To carry out our discussion, we introduce several notations. If (a1 , a2 , a3 ) is a family of three (not necessarily distinct) real numbers, we denote as usual inf{aj } the smallest number among the aj ’s, but we define also

inf {aj } = inf {a1 , a2 , a3 }\{inf{a1 , a2 , a3 }} (IV.29) 2

and:

inf {aj } = inf {a1 , a2 , a3 }\{inf{a1 , a2 , a3 }, inf {a1 , a2 , a3 }} . 3

2

(IV.30)

458



Remark that inf 3 {aj } is indeed sup{aj }. Finally in this paragraph we shall write 3 − simply aj instead of j=1 aj , and similarly for the s+ j ’s and the sj ’s. With these notations, it is very easy to check the following identity: inf{aj } =

3 1 1 1 inf {aj } − inf{aj } − inf {aj } − inf{aj } . aj − 3 j=1 3 2 3 3

We introduce the abbreviation: ∆{aj } = inf {aj } − inf{aj } + inf {aj } − inf{aj } , 2

3

(IV.31)

(IV.32)

so that we have:

1 1 (IV.33) aj − ∆{aj } . 3 3 We use this identity to replace inf{ij } and inf{s± j } in formula (IV.28), and we obtain: 2 ∂ AG (k) + + + − + − 1 1 ≤ c3 M − sj − sj M 3 ij − 3 ∆{ij } M sj −∆{sj } M inf{sj } . inf{aj } =

− {ij },{s+ j },{sj }

(IV.34) Since inf{s− j } ≤

1 3

2 ∂+ AG (k) ≤ c3

s− j , we can write: − + 2 1 1 M 3 ij − 3 ∆{ij } M −∆{sj } M − 3 sj .

(IV.35)

− {ij },{s+ j };{sj }

− Now, we use the constraints in the sum to write, for each j ∈ {ij },{s+ j };{sj } {1, 2, 3}: + (IV.36) s− j ≥ i j − sj − 2 . We deduce that: and

1 1 + 1 − sj ≥ ij − sj − 2 3 3 3 1

M−3

s− j

1

≤ M 2M − 3

ij + 13

s+ j

.

(IV.37) (IV.38)

Replacing in equation (IV.35), we get: + − 2 + 1 1 1 ∂+ AG (k) ≤ c3 M 2 M − 3 ∆{ij } M 3 sj −∆{sj } M − 3 sj , (IV.39) − {ij },{s+ j },{sj }

and using relation (IV.33), we have: − 2 + + 2 1 1 ∂+ AG (k) ≤ c3 M 2 M − 3 ∆{ij } M inf{sj }− 3 ∆{sj } M − 3 sj . (IV.40) − {ij },{s+ j },{sj }

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+ At last, let us denote κ the value of the index j such that s+ κ = inf{sj }. We write + inf{sj } = iκ − (iκ − s+ κ ). Finally we obtain: − 2 + + 2 1 1 ∂+ AG (k) ≤ c3 M 2 M iκ M − 3 ∆{ij } M −(iκ −sκ )− 3 ∆{sj } M − 3 sj . − {ij },{s+ j },{sj }

(IV.41) − − Clearly the sums over s− 1 , s2 and s3 can be bounded by K2 = +

2

M . (M 1/3 −1)3

The decay M − 3 ∆{sj } can be used to perform the sums over s+ j for j = κ, also at 1 a cost K2 . In the same way, we use the decay M − 3 ∆{ij } to sum over the values ij , j = κ also at cost K2 per sum. It remains to sum over s+ κ:

+

M −(iκ −sκ ) ≤

0≤s+ κ ≤iκ

M . M −1

(IV.42)

At last, we have: imax (T ) 2 M imax (T )+1 ∂+ AG (k) ≤ K M iκ = K M −1 i =0

(IV.43)

κ

and we have M

imax (T )

∼

1 T

(see [1]), which proves lemma IV.1.

We have then the following lemma, which is a slight refinement of lemma IV.1: Lemma IV.2 2 Λ 2 Λ K1 ∂+ AG (πT, 1, 0) , ∂+ AG,1 (πT, 1, 0) ≤ Λ M T

(IV.44)

where K1 is the constant of Lemma IV.1. 2 Λ Proof: It is similar to the proof of Lemma IV.1. The case of ∂+ AG,1 (πT, 1, 0) can 2 Λ be decomposed into sectors exactly in the same way than ∂+ AG (πT, 1, 0) because away from the singularity and in a bounded domain in k+ , the presence of πq+ instead of cos π2 k+ does not change anything to the bounds on the propagators in sectors. Each step is then similar to the proof of of lemma IV.1 until we arrive at the last sum which we decompose in two pieces. The first piece corresponds to the domain iκ ≤ imax (T ) − Λ and gives imax (T )−Λ

M iκ

=

M imax (T )−Λ+1 − 1 M −1

(IV.45)

≤

K M M imax (T ) . = , M −1 MΛ T.M Λ

(IV.46)

iκ =0

460



and the second piece corresponds to the domain imax (T ) − Λ < iκ ≤ imax (T ). In this case one should improve on equation (IV.42), to get + M −(iκ −sκ ) ≤ K.M −(iκ −(imax (T )−Λ)) , (IV.47) 0≤s+ κ ≤imax (T )−Λ

so that this second piece is bounded by imax (T )

M iκ K.M −(iκ −(imax (T )−Λ)) ≤ K .M imax (T )−Λ ,

(IV.48)

iκ =imax (T )−Λ

hence the bound for this second piece is the same as for the first. This proves the lemma. The following lemma bounds the contributions with at least one large infrared cutoff u1 on one propagator: Lemma IV.3

2 1 ∂+ AG (πT, 1, 0) ≤ K2

(IV.49)

where K2 is some new constant. Proof: The main idea is that a propagator bearing cutoff u1 = 1 − u on q+ decays 2 1 AG is now harmless, and this on a length scale O(1) in x+ , so the factor x2+ in ∂+ prevents the divergence in 1/T of the bound. 2 1 We remark first that in the amplitude ∂+ AG we can change the sum over x+ into a sum over the non zero values of x+ , because of the x2+ integrand. Since a propagator bearing cutoff u1 = 1 − u on q+ is not absolutely integrable at large q+ , we first prepare all such propagators (there are between 1 and 3 of them) using integration by parts. For any such propagator we first split the q+ integration into the two regions ∞ −1 dq + and −∞ dq+ and treat only the first term, the other one being identical. 1 Similarly we can assume that we work on a ‘right’ propagator, so that q+ = k+ −1, the other case being identical. The corresponding object is then:

D(x) = eix+

dk0

ieix+ =− x+

dk0

−2 2

−2

2

dk−

dk−

1

∞

1 ∞

dq+ dq+

[1 − u(q+ )]ei(k0 x0 +k− x− +q+ x+ ) ik0 + πq+ cos π2 k−

[π cos π2 k− ][1 − u(q+ )]ei(k0 x0 +k− x− +q+ x+ ) [ik0 + πq+ cos π2 k− ]2 u (q+ )ei(k0 x0 +k− x− +q+ x+ ) + . (IV.50) ik0 + πq+ cos π2 k−

The last term, having a compact support u is similar to the ones of the previous lemma, and left to the reader. Let us treat the first term.

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We divide it with a partition of unity into new sectors i, s+ , s− according to the size of the denominator ik0 + πq+ cos π2 k− , which is M −i , the size of q+ which is now of order M +s+ , with s+ > 0, and of k− which is of order M −s− = M −i−s+ , with s− = i + s+ . The bounds are: s+

−i

−s−

α

x− ] ≤ K|x+ |−1 M +i M s+ M −2s− e−c[M x0 +M x+ +M 2K −i−s+ −c[M −i x0 +M s+ x+ +M −s− x− ]α M ≤ e , (IV.51) π −1 Hence taking since for non zero x+ , on the tilted lattice |x+ | is bounded by 2/π. into account that the “integral” dx+ is really a discrete sum on π2 Z: −s− −i x− ]α /2 dx+ x2+ |Di,s+ ,s− (x)| ≤ KM −i−3s+ e−[M x0 +M . (IV.52)

|Di,s+ ,s− (x)|

Finally we need to optimize the dx0 and dx− using the best of the three other propagators. This leads to a bound which obviously is uniform in T . For instance if the three propagators have large infrared cutoffs u1 = 1 − u, we get the bound KM − j ij − j s+,j −2 sup s+,j +inf{i}+inf{i+s+ } i1 ,i2 ,i3 s+,1 ,s+,2 ,s+,3

≤

KM −(1/3)

j

ij −(4/3)

j

s+,j

≤ K , (IV.53)

i1 ,i2 ,i3 s+,1 ,s+,2 ,s+,3

and the other cases, when one or two propagators are of ordinary type, are similar and left to the reader. Finally we state the lemma that allows us to control the replacement of cos π2 k− by its Taylor expansion: Lemma IV.4 There exists a constant K3 > 0 such that: 2 2 ˜ AG,Λ (πT, 1, 0) ≤ K3 . ∂+ AG,Λ (πT, 1, 0) − ∂+

(IV.54)

Proof: 2 2 ˜ ∂+ AG,Λ (πT, 1, 0) AG,Λ (πT, 1, 0) − ∂+ d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x) − C˜σ1r (x)C¯˜σ2 C¯˜σ3 (x) = + {σj }, ij ,s >imax (T )−Λ j σ1 right

+2

e−(πT x0 +x+ )

d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x) − C˜σ1 (x)C¯˜σ2r C¯˜σ3r (x) ,

+ {σj }, ij ,s >imax (T )−Λ j σ2 right

(IV.55)

462



where

d3 k

uσr (k)eik.x ik0 + π(k+ − 1) cos π2 k−

(IV.56)

d3 k

uσl (k)eik.x . ik0 − π(k+ + 1) cos π2 k−

(IV.57)

Observing that there exists a constant K4 such that: π cos k+ + (π/2)(k+ − 1) ≤ K4 (k+ − 1)2 2 π cos k+ − (π/2)(k+ + 1) ≤ K4 (k+ + 1)2 2

(IV.58)

C˜σr (x)

=

C˜σ (x)

=

uniformly in k+ , we have: α Cσr() (x) − C˜σr() (x) ≤ c .M −2s+ −s− e−c dσ (x) .

(IV.59)

(IV.60)

Using the relation Cσ1 C σ2 C σ3 − C˜σ1 C˜ σ2 C˜ σ3 = (Cσ1 − C˜σ1 )C σ2 C σ3 + C˜σ1 (C σ2 − C˜ σ2 )Cσ3 + C˜σ1 C˜ σ2 (C σ3 − C˜ σ3 ) , (IV.61) ˜ we gain M −s+ ≤ M −(imax −Λ) in the power to create differences of the type C − C, counting with respect to a single propagator.

V

Main lower bound

Now we can state our main lower bound: Theorem V.1 There exists a constant K5 > 0 such that: K 2 ˜ 5 . ∂+ AG (πT, 1, 0) ≥ T

(V.62)

This theorem with the lemmas of the previous section obviously imply Theorem II.1, hence the remaining of this paper is devoted to the proof of this Theorem V.1.

V.1 Integration over k1,+ , k2,+ and k3,+ We return to equation (III.21), in which all three cutoffs uΛ have been replaced 2 ˜ 2 ˜ 2 ˜ AG (πT, 1, 0) as ∂+ AG,1 + 2∂+ AG,2 and by 1. Let us write in equation (III.21) ∂+ 2 ˜ let us consider the first term ∂+ AG,1 .

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The first propagator (after a change of variable to call the dummy variable q+ again k+ ): eik1 .x eix+ . d3 k1 (V.63) ik1,0 + πk1,+ cos π2 k1,− For cos

π

2 k1,−

+∞

−∞

dk1,+

= 0 we have:

eik1,+ x+ = ik1,0 + πk1,+ cos π2 k1,− +∞ 1 eik1,+ x+ π

. (V.64) dk1,+ ik π cos 2 k1,− −∞ k1,+ + π cos( π1,0k1,− ) 2

The corresponding residue is exp

k1,0 x+ π cos( π 2 k1,− )

. If x+ > 0, then we move the

path of integration upwards. It is oriented in the positive direction, so we get:

k1,0 x+ k1,0 > 0 2iπ exp π χ(x+ > 0) χ − . π cos( π2 k1,− ) π cos 2 k1,−

(V.65)

If x+ < 0, then the path of integration is moved downwards, and we get a minus sign owing to the negative direction. Hence: eik1,+ x+ 2i k1,0 x+ = exp dk1,+ ik1,0 + πk1,+ cos π2 k1,− cos π2 k1,− π cos π2 k1,− −∞ k1,0 k1,0 > 0 − χ(x+ < 0) χ − 0) χ − π cos π2 k1,− π cos π2 k1,− (V.66)

+∞

We treat analogously the integrations over k2,+ and k3,+ . The only difference with the previous case is that these propagators were near the left singularity k+ −1, so there are some sign changes in q2,+ and q3,+ ≈ −1. We obtain:

eik2,+ x+ −2i k2,0 x+ = exp + π π π cos( π2 k2,− ) −ik2,0 − πk2,+ cos 2 k2,− cos 2 k2,− −∞ k2,0 k2,0 < 0 − χ(x+ < 0) χ >0 . χ(x+ > 0) χ π cos π2 k2,− π cos π2 k2,− (V.67)

+∞

dk2,+

464



2 ˜ AG,1 (πT, 1, 0) = −8i d3 x dk1,0 dk1,− dk2,0 dk2,− dk3,0 dk3,− ∂+

k1,0 k2,0 k3,0 + π cos π k + x+ exp π cos( π ( 2 2,− ) π cos( π2 k3,− ) 2 k1,− ) π π x2+ cos 2 k1,− cos 2 k2,− cos( π2 k3,− )

ei(k1,0 +k2,0 +k3,0 +πT )x0 ei(k1,− +k2,− +k3,− )x− k1,0 k2,0 0 χ π cos π2 k3,− π cos π2 k1,− k2,0 k3,0 >0 χ >0 χ . (V.68) π cos π2 k2,− π cos π2 k3,−

V.2 Integration over x0 and k3,0 The calculation is done integrating over x0 , which leads to a delta function in the integrand, denoted with a slight abuse of notation by δ(k1,0 + k2,0 + k3,0 + πT = 0). 1 In fact, compensates the T factor of dk3,0 : remember there is a prefactor T that that dk3,0 means precisely: 2πT k3,0 ∈πT +2πT Z . This yields: 2 ˜ ∂+ AG,1 (πT, 1, 0)

= −8i

x2+

e

dx+ dx−

dk1,0 dk1,− dk2,0 dk2,− dk3,0 dk3,−

k1,0 k k + π cos( 2,0 + π cos( 3,0 πk πk ) ) π cos( π k1,− ) 2 2 2,− 2 3,−

x+

cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )

ei(k1,− +k2,− +k3,− )x− δ(k1,0 + k2,0 + k3,0 + πT = 0)

k2,0 k3,0 k1,0 < 0) χ( < 0)χ( < 0) π cos( π2 k1,− ) π cos( π2 k2,− ) π cos( π2 k3,− ) k2,0 k3,0 k1,0 > 0) χ( > 0)χ( > 0) . − χ(x+ < 0) χ( π cos( π2 k1,− ) π cos( π2 k2,− ) π cos( π2 k3,− ) (V.69) χ(x+ > 0) χ(

At this stage, we can use the delta function to integrate, for instance, over k3,0 : 2 ˜ ∂+ AG,1 (πT, 1, 0) = −8i

x2+

e

dx+ dx−

dk1,0 dk1,− dk2,0 dk2,− dk3,−

k1,0 k k +k2,0 +πT + π cos( 2,0 − π1,0 πk ) π cos( π k1,− ) cos( π k3,− ) 2 2 2,− 2


x+

ei(k1,− +k2,− +k3,− )x−

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k2,0 k1,0 < 0 χ < 0 × χ(x+ > 0) χ π cos( π2 k1,− ) π cos( π2 k2,− )

k1,0 + k2,0 + πT k1,0 χ > 0 − χ(x > 0 < 0) χ + π cos( π2 k3,− ) π cos( π2 k1,− )

k2,0 k1,0 + k2,0 + πT χ > 0 χ < 0 . (V.70) π cos( π2 k2,− ) π cos( π2 k3,− )

V.3 Simplification This rather complicated expression can be slightly simplified. Indeed, if we perform the change of variables:   x+ = −x+ k = −k1,0 (V.71)  1,0 k2,0 = −k2,0 the integral

dx+ dx−

e

dk1,0 dk1,− dk2,0 dk2,− dk3,− x2+

k1,0 k k +k2,0 +πT + π cos( 2,0 − π1,0 πk ) π cos( π k1,− ) cos( π k3,− ) 2 2 2,− 2

k1,0 < 0) χ χ(x > 0 + cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− ) π cos( π2 k1,− )

k1,0 + k2,0 + πT k2,0 > 0 < 0 χ (V.72) χ π cos( π2 k2,− ) π cos( π2 k3,− ) x+

becomes: dx+ dx− dk1,0 dk1,− dk2,0 dk2,− dk3,− x2 +

e

−πT k1,0 k k +k2,0 + π cos( 2,0 − π1,0 πk ) π cos( π k1,− ) cos( π k3,− ) 2 2 2,− 2

k1,0 < 0 cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− ) π cos( π2 k1,− )

k2,0 k1,0 + k2,0 − πT χ 0 . (V.73) π cos( π2 k2,− ) π cos( π2 k3,− ) x+

χ(x+ > 0) χ

2 ˜ Consequently the previous expression of ∂+ AG,1 (πT, 1, 0) can be factorized: 2 ˜ AG,1 (πT, 1, 0) = −8i ∂+

x2+

e

dx+ dx−

dk1,0 dk1,− dk2,0 dk2,− dk3,−

k1,0 k k1,0 +k2,0 + π cos( 2,0 − π cos( πk πk ) ) π cos( π k1,− ) 2 2 2,− 2 3,−


x+

ei(k1,− +k2,− +k3,− )x−

466



k2,0 k1,0 < 0 χ < 0 × χ(x+ > 0) χ π cos( π2 k1,− ) π cos( π2 k2,− )

−T x+ k1,0 + k2,0 + πT π e cos( 2 (k3,− )) χ > 0 π cos( π2 k3,− )

T x+ k1,0 + k2,0 − πT cos( π (k3,− )) 2 −e >0 . (V.74) χ π cos( π2 k3,− )

VI Integration over x− and k3,− We now are going to perform the integration over x− , which will provide a conservation rule for the moments k1,0 , k2,0 and k3,0 , but onlymodulo 2. To understand that, remember that dx+ dx− means more precisely: (x ,x )∈( π Z)2 , where the +

−

2

prime in the sum means that one has to respect a parity condition between x+ and x− . By slight abuse of language, we say that x+ and x−have thesame parity when x+ + x− ∈ πZ. So (x ,x )∈( π Z)2 does not mean: x+ ∈ π Z x− ∈ π Z but + 2 2 − 2 + . Now, π π x+ ∈πZ x− ∈πZ x+ ∈ +πZ x− ∈ +πZ 2

2

ei(k1,− +k2,− +k3,− )x− = δ(k1,− + k2,− + k3,− = 0[2])

(VI.75)

x− ∈πZ

where by δ(k1,− + k2,− + k3,− = 0[2]), we denote: Then it is clear that

n∈Z

δ(k1,− + k2,− + k3,− = 2n).

π

ei(k1,− +k2,− +k3,− )x− = ei 2 (k1,− +k2,− +k3,− ) δ(k1,− + k2,− + k3,− = 0[2]) .

x− ∈ π 2 +πZ

(VI.76) π Indeed, the factor ei 2 (k1,− +k2,− +k3,− ) can take only two values: 1 if k1,− + k2,− + k3,− = 0[4], and −1 if k1,− +k2,− +k3,− = 2[4]. Hence it is convenient to distinguish these two cases and write: δ(k1,− + k2,− + k3,− = 0[2]) = δ(k1,− + k2,− + k3,− = 0[4]) + δ(k1,− + k2,− + k3,− = 2[4]) (VI.77) and π

ei 2 (k1,− +k2,− +k3,− ) δ(k1,− + k2,− + k3,− = 0[2]) = δ(k1,− + k2,− + k3,− = 0[4]) − δ(k1,− + k2,− + k3,− = 2[4]) . (VI.78)

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467

At this stage, we can gather the previous remarks in the following formula : 2 ˜ AG,1 (πT, 1, 0) = −8i ∂+

x2+

e

dk1,0 dk2,0 dk1,− dk2,− dk3,−

∗ x+ ∈ π 2N

k1,0 k k1,0 +k2,0 + π cos( 2,0 − π cos( πk πk ) ) π cos( π k1,− ) 2 2 2,− 2 3,−

x+

δ(k1,− + k2,− + k3,− = 0[4]) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )

k2,0 k1,0 < 0 χ < 0 χ π cos( π2 k1,− ) π cos( π2 k2,− )

−T x+ k1,0 + k2,0 + πT π e cos( 2 (k3,− )) χ > 0 π cos( π2 k3,− )

T x+ k1,0 + k2,0 − πT cos( π (k3,− )) 2 −e >0 χ π cos( π2 k3,− ) − 8i dk1,0 dk2,0 dk1,− dk2,− dk3,−

x2+

e

[χ(x+

∗ x+ ∈ π 2N k1,0 k k1,0 +k2,0 + π cos( 2,0 − π cos( πk πk ) ) π cos( π k1,− ) 2 2 2,− 2 3,−

x+

δ(k1,− + k2,− + k3,− = 2[4]) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )

k2,0 k1,0 0 . (VI.79) π cos( π2 k3,− )

Then we can perform the integration over k3,− . Formally, we only need to replace cos( π2 k3,− ) by cos( π2 (k1,− +k2,− )) for the first piece and with − cos( π2 (k1,− +k2,− )) for the second piece. We obtain: 2 ˜ AG,1 (πT, 1, 0) ∂+

= −8i

dk1,0 dk2,0 dk1,− dk2,−

∗ x+ ∈ π 2N

k1,0 k k +k2,0 + π cos( 2,0 − π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2

k1,0 χ 0 χ π cos( π2 k2,− ) π cos( π2 (k1,− + k2,− ))

T x+ k1,0 + k2,0 − πT π − e cos( 2 (k1,− +k2,− )) χ > 0 π cos( π2 (k1,− + k2,− ))

x2+

e

x+

468


+ 8i

x2+

[χ(x+

e


dk1,0 dk2,0 dk1,− dk2,−

∗ x+ ∈ π 2N k1,0 k k +k2,0 + π cos( 2,0 + π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2

x+

cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− ))

k2,0 k1,0 < 0 χ < 0 even) − χ(x+ odd)]χ π cos( π2 k1,− ) π cos( π2 k2,− )

T x+ k1,0 + k2,0 + πT π e cos( 2 (k1,− +k2,− )) χ < 0 π cos( π2 (k1,− + k2,− ))

−T x+ k1,0 + k2,0 − πT π − e cos( 2 (k1,− +k2,− )) χ < 0 . (VI.80) π cos( π2 (k1,− + k2,− ))

2 ˜ Now it is clear that ∂+ AG,1 (πT, 1, 0) is a purely imaginary number. The first piece gives the leading behavior as T → 0. Indeed the second piece is much smaller, thanks to the compensation in [χ(x+ even) − χ(x+ odd)]. Indeed the sum

∗ x+ ∈ π 2N

x2+ [χ(x+ even) − χ(x+ odd)] . . .

(VI.81)

can be written as a sum of two terms of the type

dke−2A(k)n [(2n)2 − (2n + 1)2 e−A(k) ]B(k)

(VI.82)

n∈N∗

where A and B are independent of n and A(k) > 0. Then we can decompose the remaining integrals dk into two zones, according to whether A(k) ≥ T 1/3 or A(k) ≤ T 1/3 . In the first zone we do not need to exploit the subtraction, but we 1/3 have simply n∈N∗ n2 e−2T n ≤ c.T −2/3 0 . (VI.84) χ π cos( π2 (k1,− + k2,− ))

VII Leading contribution VII.1 Symmetry properties Henceforward, we shall denote the integrand by F (x+ , k1,0 , k2,0 , k1,− , k2,− ) so that: A1 (T ) = −8i

dk1,0 dk2,0 dk1,− dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) .

∗ x+ ∈ π 2N

(VII.85) The couple of variables of integration (k1,− , k2,− ) describes the square [−2, 2]2 . To pursue the calculation, we shall make a partition of [−2, 2]2 , according to the signs of cos( π2 k1,− ), cos( π2 k2,− ) and cos( π2 (k1,− + k2,− )). This partition is represented in Figure 4. The signs of the three cosines determine eight cases we can discuss separately. In fact, it is possible to restrict the domain of integration thanks to symmetries of the integrand involving the variables k1,− and k2,− together with the variables k1,0 and k2,0 , which describe independently the set πT + 2πT Z. It is evident, by the parity of the cosine function, that the integrand is invariant under the replacement k1,− → −k1,− and k2,− → −k2,− , which corresponds to the central symmetry with respect to the origin (0,0). Hence we have: A1 (T ) = −16i

∗ x+ ∈ π 2N

dk1,0 dk2,0

2

−2

dk1,−

0

2

dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ). (VII.86)

470



k−

1 +2 0 0 1 00000000000000 11111111111111 0 1 00000000000000 11111111111111 000000000000001111 11111111111111 (−,−,+) 00000000000000 11111111111111 0(+,−,−) 0000 1 0000 1111 00000000000000 00000000000000 11111111111111 (−,−,+)11111111111111 0 1 0000 1111 00000000000000 11111111111111 00000000000000 11111111111111 0 1 00000000000000 11111111111111 0000 1111 00000000000000 11111111111111 (−,−,−) (+,−,+) 1 0 00000000000000 11111111111111 0000 000000000000001111 11111111111111 00000000000000 11111111111111 0 1 0000 1111 00000000 11111111 00000000000000 11111111111111 0 1 (+,+,−) (−,+,+) 00000000 11111111 00000000000000 11111111111111 0 1 00000000 00000000000000 11111111111111 011111111 1 00000000 (−,+,−) 00000000000000 11111111111111 011111111 1 00000000 11111111 00000000000000 11111111111111 0 1 00000000 11111111 00000000000000 11111111111111 11111111111111111 00000000000000000 (+,+,+) 111k + 000 00000000 11111111 00000000 11111111 00000000000000 11111111111111 0 1 00000000 11111111 −2 +2 00000000000000 11111111111111 0 1 00000000 11111111 00000000000000 11111111111111 00000000 0 1 (−,+,−)11111111 00000000000000 11111111111111 00000000 11111111 0 1 00000000000000 11111111111111 00000000 11111111 0 (+,+,−) 1 (−,+,+) 00000000000000 11111111111111 00000000 11111111 00000000000000 11111111111111 0 1 00000000000000 11111111111111 00000000000000 11111111111111 00000000 11111111 00000000000000 11111111111111 0000 1111 0 1 00000000000000 11111111111111 (+,−,+) (−,−,−) 00000000000000 11111111111111 0000 1111 0 1 00000000000000 11111111111111 00000000000000 (−,−,+) 11111111111111 0000 1111 00000000000000 11111111111111 0 1 00000000000000 11111111111111 (+,−,−) 00000000000000 11111111111111 0000 1111 0 1 00000000000000 11111111111111 (−,−,+) 00000000000000 11111111111111 0000 1111 0 00000000000000 11111111111111 00000000000000 11111111111111 0000 1 1111 0 1 0 1 −2 0 1 0 1 Figure 4. The domain of integration in (k+ , k− ). Symmetry properties of F (x+ , k1,0 , k2,0 , k1,− , k2,− ) can be exploited further. The above integral may be separated into two pieces:  0 dk1,− dk1,0 dk2,0 A1 (T ) = −16i  −2

∗ x+ ∈ π 2N

+

0

2

dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− )

dk1,0 dk2,0

∗ x+ ∈ π 2N

2

0

dk1,−

0

2

 dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) . (VII.87)

For the first integral, one can easily verify that the integrand F (x+ , k1,0 , k2,0 , k1,− , k2,− ) is invariant under the change of variables: k1,0 = k2,0 , k2,0 = k1,0 , k1,− = −k2,− , k2,− = −k1,− .

(VII.88)

We get:

dk1,0 dk2,0 2

0

−2

dk1,−

dk1,0 dk2,0

0

2

dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) =

0 −2

dk1,−

2

−k1,−

dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) .

(VII.89)

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471

We treat analogously the other integral; we set: k1,0 = k2,0 , k2,0 = k1,0 , k1,− = k2,− , k2,− = k1,− .

Hence: dk1,0 dk2,0 2

0

2

dk1,−

dk1,0 dk2,0

0

2

(VII.90)

2

0

dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) =

2

dk1,−

dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) .

(VII.91)

k1,−

Finally, we have established owing to symmetry properties that: A1 (T ) = −32i dk1,0 dk2,0 dk1,− dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) , (VII.92) T

the domain of integration in (k1,− , k2,− ) being the triangle T delimited by the lines k2,− = 2, k2,− = k1,− and k2,− = −k1,− .

VII.2 Discussion of the various cases VII.2.1 The (+, +, +) case As we have said, it is now convenient to carry a discussion about the signs of cos( π2 k1,− ), cos( π2 k2,− ) and cos( π2 (k1,− + k2,− )), which allows us to perform explicitly the summation over k1,0 and k2,0 in each case. We first begin with the case:  > 0 cos( π2 k1,− )  > 0 , cos( π2 k2,− ) (VII.93)  cos( π2 (k1,− + k2,− )) > 0 that we will denote as (+, +, +). The corresponding contribution to A1 (T ) is: (+,+,+) A1 (T ) = −32i dk1,− dk2,− dk1,0 dk2,0 ∗ x+ ∈ π 2N

e x2+

T(+,+,+)


x+

χ(k1,0 < 0) χ(k2,0 < 0) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) −T x+ π e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 > −πT ) T x+ π −e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 > πT ) , (VII.94)

where T(+,+,+) denotes the subset of T where the signs of the cosines are (+, +, +) respectively. Since the conditions k1,0 < 0, k2,0 < 0 and k1,0 + k2,0 > ±πT are (+,+,+) =0. incompatible, A1

472



VII.2.2 The (+, +, −) case Let us consider the case:  

> cos( π2 k1,− ) > cos( π2 k2,− )  cos( π2 (k1,− + k2,− ))
0) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) −T x+ π e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 > −πT ) T x+ cos( π (k1,− +k2,− )) 2 −e χ(k1,0 + k2,0 > πT ) . (VII.100)

Here like in all the other cases, we have to sum geometric sequences whose ratio is explicitly strictly smaller than 1. This facilitates the discussion of the signs of the , corresponding quantities, as we shall see. If we perform the summation over k1,0

we are lead to a geometric sequence whose ratio is e which leads to a factor −2 1−e

T x+ Tx − cos( π k ++k cos( π k1,− ) 2,− ) 2 2 1,−

−2

T x+ Tx − cos( π k ++k cos( π k1,− ) 2,− ) 2 2 1,−

,

!−1

whose sign is not uniform in (k1,− , k2,− ). Consequently we introduce the variable s = k1,0 + k2,0 and replace k2,0 by s − k1,0 . We must compute:

k1,0 s−k + π cos( π1,0 − π cos( π (ks +k k ) π cos( π k1,− ) 1,− 2,− )) 2 2 2,− 2

x+

χ(k1,0 < 0)χ(s > k1,0 ) dk1,0 ds e −T x+ T x+ π π e cos( 2 (k1,− +k2,− )) χ(s > −πT ) − e cos( 2 (k1,− +k2,− )) χ(s > πT ) . (VII.101) The variable s describes the set 2πT Z and the condition χ(s > k1,0 ) can be omitted. Thus the previous expression writes: (2πT )2

+∞

e

−(2n+1)

T x+ Tx − cos( π k+ ) cos( π k1,− ) 2 2 2,−

n=0

e

−T x+ cos( π (k1,− +k2,− )) 2

+∞ p=0

−e

−T x+

cos( π (k1,− +k2,− )) 2

+∞ p=0

e

e

−2p cos( π (k

1 − cos( π1k ) 1,− +k2,− )) 2 2,−

2

−2p cos( π (k 2

1 − cos( π1k ) 1,− +k2,− )) 2 2,−

T x+

T x+

(VII.102)

474



which is equal to: 2

(2πT )

e

−

1 − cos( π1k + cos( π (k 1 +k ) cos( π k1,− ) 1,− 2,− )) 2 2 2,− 2

−2 1−e

1 − cos( π1k ) cos( π k1,− ) 2 2 2,−

−2 1−e

T x+

T x+ 2T x+ π

1 − e cos( 2 k2,− )

1 − cos( π1k ) cos( π (k1,− +k2,− )) 2 2 2,−

T x+

. (VII.103)

This quantity is positive, thus the conclusion follows: (+,−,+)

iA1

(T ) ≤ 0 .

(VII.104)

VII.2.4 The (+, −, −) case Let us examine now the (+, −, −) case. The contribution is: (+,−,−)

A1

(T ) = −32i

dk1,0 dk2,0

∗ x+ ∈ π 2N

e x2+

dk1,− dk2,−

T(+,−,−)


x+

χ(k1,0 < 0)χ(k2,0 > 0) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) −T x+ π e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 < −πT ) T x+ π − e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 < πT ) . (VII.105)

We set k1,0 = s − k2,0 and we compute:

ds dk2,0 e

s−k2,0 k + π cos( 2,0 − π cos( π (k s +k πk ) π cos( π k1,− ) 1,− 2,− )) 2 2 2,− 2

x+

χ(s < k2,0 )χ(k2,0 > 0)

−T x+ T x+ π π e cos( 2 (k1,− +k2,− )) χ(s < −πT ) − e cos( 2 (k1,− +k2,− )) χ(s < πT ) . (VII.106)

The condition χ(s < k2,0 ) may be omitted and we must evaluate: (2πT )2

+∞

e

(2n+1)

1 − cos( π1k ) cos( π k2,− ) 2 2 1,−

n=0

+∞ −T x+ −2p π e cos( 2 (k1,− +k2,− )) e p=1

T x+

1 − cos( π (k 1 +k cos( π k1,− ) 1,− 2,− )) 2 2

T x+

Vol. 6, 2005


−e

T x+ cos( π (k1,− +k2,− )) 2

+∞

e

−2p cos( π1k 2

1,− )

475 1 1,− +k2,− ))

− cos( π (k 2

T x+

. (VII.107)

p=0

We find:

(2πT )2

e

−

1 − cos( π1k − cos( π (k 1 +k ) cos( π k1,− ) 1,− 2,− )) 2 2 2,− 2

1−e

−2 cos( π1k 2

1,− )

2

T x+

− cos( π1k

T x+

2,− )

−2T x+ π e cos( 2 k1,− ) − 1 1−e

−2 cos( π1k

1,−

2

1 1,− +k2,− ))

− cos( π (k ) 2

. (VII.108)

T x+

This is a negative number, therefore (+,−,−)

iA1

(T ) ≤ 0 .

(VII.109)

VII.2.5 The (−, +, +) and (−, +, −) cases There is no discussion to carry out: in fact, for (k1,− , k2,− ) ∈ T , we have never cos( π2 k1,− ) < 0, cos( π2 k2,− ) > 0 and cos( π2 (k1,− + k2,− )) < 0 simultaneously. We also conclude in the same way for the (−, +, −) case. VII.2.6 The (−, −, +) case

(−,−,+)

A1

(T ) = −32i

∗ x+ ∈ π 2N

e x2+

dk1,0 dk2,0

dk1,− dk2,−

T(−,−,+)


x+

χ(k1,0 > 0)χ(k2,0 > 0) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) −T x+ π e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 > −πT ) T x+ π −e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 > πT ) . (VII.110)

We remark that the conditions χ(k1,0 + k2,0 > ±πT ) are superfluous, and that there is no need to introduce the variable s.

476



We have: 2

(2πT )

+∞

e

(2n+1)

1 − cos( π (k 1 +k cos( π k1,− ) 1,− 2,− )) 2 2

n=0 +∞

e

T x+

1 − cos( π (k 1 +k cos( π k2,− ) 1,− 2,− )) 2 2

(2p+1)

T x+

p=0

−T x+ T x+ cos( π (k1,− +k2,− )) cos( π (k1,− +k2,− )) 2 2 −e e =

(2πT )2

e

−

1 − cos( π1k − cos( π1k ) ) cos( π (k1,− +k2,− )) 2 2 1,− 2 2,−

π

T x+

−2T x+

e cos( 2 (k1,− +k2,− )) − 1 1−e 1−e

−2 cos( π (k

−2 cos( π (k

1 − cos( π1k ) 1,− +k2,− )) 2 1,−

2

1 − cos( π1k ) 1,− +k2,− )) 2 2,−

2

T x+

. (VII.111) T x+

This quantity is negative and we conclude that (−,−,+)

iA1

(T ) ≤ 0 .

(VII.112)

VII.2.7 The (−, −, −) case We finally discuss the last case: (−,−,−)

A1

(T ) = −32i

dk1,0 dk2,0

∗ x+ ∈ π 2N

e x2+

dk1,− dk2,−

T(−,−,−)


x+

χ(k1,0 > 0)χ(k2,0 > 0) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) −T x+ π e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 < −πT ) T x+ π −e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 < πT ) . (VII.113)

But it is clear that the conditions k1,0 > 0, k2,0 > 0 and k1,0 + k2,0 < ±πT are incompatible (as in the (+, +, +) case), hence (−,−,−)

A1

(T ) = 0 .

(VII.114)

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477

Lemma VII.1 There exists a constant K > 0 such that: K (+,+,−) (+,−,+) (+,−,−) (−,−,+) . (T ) + A1 (T ) + A1 (T ) + A1 (T ) > A1 T

(VII.115)

Proof: As each one of the quantities are purely imaginary, with non-negative imag(+,+,−) inary part, it is sufficient to prove the inequality |A1 (T )| > KT1 for some constant K1 . We have: (+,+,−)

|A1

(T )| = 32(2πT )2 dk1,− dk2,−

∗ x+ ∈ π 2N

T (+,+,−)

− 2 e x+

1 cos π k1,− 2

1−e

1

cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,−

+ cos π1k − cos π (k 1 +k T x+ )

−2 cos

2,−

2

1 πk 2 1,−

1,−

2

− cos

2,−

1 π (k 1,− +k2,− ) 2

π

+ k2,− ))

T x+

2T x+

1 − e cos 2 (k1,− +k2,− ) 1−e As 1 − e

−2 cos

and 1 − e

− cos

1 πk 2 2,−

1 π (k 1,− +k2,− ) 2

− cos

1 π (k 1,− +k2,− ) 2

(T )| ≤ 32(2πT )2

∗ x+ ∈ π 2N

1 πk 2 2,−

− cos

1 π (k 1,− +k2,− ) 2

. (VII.116)

T x+

1 πk 2 1,−

−2 cos

(+,+,−)

|A1

−2 cos

T x+

T x+

≤1 ≤ 1, we get:

T (+,+,−)

dk1,− dk2,−

1 cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− ))

2T x+ 1 + cos π1k − cos π (k 1 +k T x+ π k1,− ) 2 − cos π 2,− 1,− 2,− 2 2 2 1 − e cos 2 (k1,− +k2,− ) . (VII.117) x+ e As we are seeking a lower bound, we can restrict the integration over the open (+,+,−) ⊂ T (+,+,−) ,where is a strictly domain T (+,+,−) to a compact T positive 1 constant (for example = 10 ), in which we have cos π2 k1,− ≥ , cos π2 k2,− ≥

478



(+,+,−) and cos π2 (k1,− + k2,− ) ≥ . For (k1,− , k2,− ) ∈ T , we have:

0
0) (1 − e−a )3

(VII.121)



 − 3π 2 T

e (T )| ≥ 32π 4 T 2 

≥ 32πT −1

+e

1−e

− 3π T

− 3π 2 T

3 −

e

−( 3π 2 +π )T

−( 3π +2π )T

+e

3 −( 3π 1 − e 2 +π)T

2 + O(T ) 2 + O(T ) − 3 (3/2) (3/2 + 1)3

 

(VII.122)

.

(VII.123)

Hence for T small enough, we obtain the desired result: (+,+,−)

|A1

(T )| ≥ KT −1 .

(VII.124)

for some explicit K and the lemma is proven.

VIII Study of the other configurations We now are going to treat the other configuration, corresponding to:    k1,+ ≈ −1  k1,+ ≈ −1 and k2,+ ≈ 1 k2,+ ≈ −1   k3,+ ≈ −1 k3,+ ≈ 1

(VIII.125)

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479

2 which are equal and form the term called 2∂+ AG,2 (πT, 1, 0). Let us concentrate on the first case. We have to consider the propagator:

+∞

−∞

dk1,+

eik1,+ x+ = ik1,0 − πk1,+ cos( π2 k1,− ) +∞ eik1,+ x+ −1 dk 1,+ ik1,0 π cos( π2 k1,− ) −∞ k1,+ − π

. (VIII.126)

π cos( 2 k1,− )

The pole of the integrand is e

k1,0 x+ − π cos( πk ) 2 1,−

ik1,0 π cos( π 2 k1,− )

and the corresponding residue writes

. Therefore we have:

+∞

k1,0 x+ −2i eik1,+ x+ − π cos( πk ) 2 1,− = e dk1,+ π π ik1,0 − πk1,+ cos( 2 k1,− ) cos( 2 k1,− ) −∞

k1,0 k1,0 > 0 − χ(x < 0 . < 0)χ χ(x+ > 0)χ + π cos( π2 k1,− ) π cos( π2 k1,− ) (VIII.127)

Now, let us consider the integration over k2,+ . We have:

+∞

−∞

dk2,+

eik2,+ x+ = −ik2,0 + πk2,+ cos( π2 k2,− ) +∞ 1 eik2,+ x+ dk2,+ π ik2,0 π cos( 2 k2,− ) −∞ k2,+ − π

. (VIII.128)

π cos( 2 k2,− )

In fact, the only change with the previous case is a global change of sign. We can immediately write:

+∞

k2,0 x+ 2i eik2,+ x+ − π = e π cos( 2 k2,− ) π π −ik + πk cos( k ) cos( k ) 2,0 2,+ 2,− 2,− −∞ 2 2

k2,0 k2,0 > 0 − χ(x+ < 0)χ 0)χ π cos( π2 k2,− ) π cos( π2 k2,− ) (VIII.129)

dk2,+

For the integration over k3,+ , we have cos( π2 k2,+ ) ≈

+∞

−∞

π 2 (k2,+

−1 eik3,+ x+ = dk3,+ π −ik3,0 − πk3,+ cos( 2 k3,− ) π cos( π2 k3,− )

+ 1) and we consider:

+∞

−∞

eik3,+ x+ k3,+ +

ik3,0 π cos( π 2 k3,+ )

.

(VIII.130)

480



k3,0 x+ π

ik

In this case, the pole is − π cos( π3,0k3,− ) and the residue e π cos( 2 k3,− ) . Therefore the 2 above integral writes:

+∞

k3,0 x+ eik3,+ x+ −2i π cos( π k3,− ) 2 = e −ik3,0 − πk3,+ cos( π2 k3,− ) cos( π2 k3,− ) −∞

k3,0 k3,0 χ(x+ > 0)χ < 0)χ < 0 − χ(x > 0 . + π cos( π2 k3,− ) π cos( π2 k3,− ) (VIII.131)

dk3,+

Hence we obtain: 2 ˜ AG,2 (πT, 1, 0) = −8i ∂+

k

k

− π cos( 1,0 πk 2

e

dk1,0 dk1,− dk2,0 dk2,− dk3,0 dk3,− x2+

dx 1,− )

− π cos( 2,0 πk 2

2,− )

k

+ π cos( 3,0 πk 2

3,− )

x+

cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− ) ei(k1,0 +k2,0 +k3,0 +πT )x0 ei(k1,− +k2,− +k3,− )x−

k2,0 k1,0 >0 χ >0 χ(x+ > 0)χ π cos( π2 k1,− ) π cos( π2 k2,− )

k3,0 k1,0 χ < 0 − χ(x+ < 0) χ 0 . (VIII.132) π cos( π2 k2,− ) π cos( π2 k3,− )

Then we integrate over x0 and perform the sum over k3,0 : 2 ˜ AG,2 (πT, 1, 0) = −8i ∂+

x2+

e

k

− π cos( 1,0 πk 2

1,−

dx+ dx− k

− π cos( 2,0 πk ) 2

2,−

− )

dk1,0 dk1,− dk2,0 dk2,− dk3,− k1,0 +k2,0 +πT π cos( π k3,− ) 2

x+

ei(k1,− +k2,− +k3,− )x− cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )

k2,0 k1,0 >0 χ >0 χ(x+ > 0) χ π cos( π2 k1,− ) π cos( π2 k2,− )

k3,0 k1,0 χ < 0 − χ(x+ < 0) χ 0 . (VIII.133) π cos( π2 k2,− ) π cos( π2 k3,− )

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Thanks to the change of variables x+ = −x+ , k1,0 = −k1,0 , k2,0 = −k2,0 , we get: 2 ˜ ∂+ AG,2 (πT, 1, 0) = −8i

x2+

e

−

dx+ dx−

dk1,0 dk1,− dk2,0 dk2,− dk3,−

k1,0 k k1,0 +k2,0 + π cos( 2,0 + π cos( πk πk ) ) π cos( π k1,− ) 2 2 2,− 2 3,−

x+

ei(k1,− +k2,− +k3,− )x− cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )

k2,0 k1,0 >0 χ >0 χ(x+ > 0) χ π cos( π2 k1,− ) π cos( π2 k2,− ) −T x+

k1,0 + k2,0 + πT π e cos( 2 k3,− ) χ > 0 π cos( π2 k3,− )

T x+ k1,0 + k2,0 − πT π −e cos( 2 k3,− ) χ > 0 . (VIII.134) π cos( π2 k3,− )

Then we perform the sum over x− as previously and integrate over k3,− . There is a small contribution with a compensating factor [χ(x+ even)−χ(x+ odd)] that can be bounded as in Section VI, and we have again to study the dominant contribution: A2 (T ) = −8i dk1,0 dk2,0 dk1,− dk2,− x2+

e

−

∗ x+ ∈ π 2N

k1,0 k k +k2,0 + π cos( 2,0 + π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2

x+

k1,0 >0 π cos( π2 k1,− )

χ cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− ))

−T x+ k2,0 k1,0 + k2,0 + πT cos( π (k1,− +k2,− )) 2 χ > 0 > 0 e χ π cos( π2 k2,− ) π cos( π2 (k1,− + k2,− ))

T x+ k1,0 + k2,0 − πT π − e cos( 2 (k1,− +k2,− )) χ > 0 . (VIII.135) π cos( π2 (k1,− + k2,− )) Fortunately, we do not have to carry again a discussion about the signs of the three cosines. In fact, we can remark that A1 (T ) = A2 (T ). To see that, let us perform the following change of variables in A1 (T ): $ k1,− = k1,− +2 , (VIII.136) k2,− = k2,− +2 to obtain: A2 (T ) = −8i

e

∗ x+ ∈ π 2N

dk1,0 dk2,0

dk1,− dk2,− x2+

T

k1,0 k k +k2,0 + π cos( 2,0 − π cos( π1,0 π k ) ) (k +k )) π cos( π k 1,− 2,− 2 1,− 2 2,− 2

x+

) cos( π k ) cos( π (k cos( π2 k1,− 1,− + k2,− )) 2 2,− 2

χ

k1,0 ) < 0 π cos( π2 k1,−

482


×χ

k1,0 + k2,0 + πT e χ )) > 0 π cos( π2 (k1,− + k2,− T x+ + k − πT k 1,0 2,0 )) +k cos( π (k − e 2 1,− 2,− χ , (VIII.137) )) > 0 π cos( π2 (k1,− + k2,−

k2,0 ) π 3, 2. |z(·, λ)| is strictly increasing on (x∗ , ∞), 3. if t∗ =

x∗ √ , λ

then |t∗ |

1 sin π 3

uniformly in λ.

Proof. By Lemma 7.1.4 and (A.4), Γλ intersects the line {z : arg z = − π3 } at least once. By (A.4), the sector S[− π3 − ε, − π3 + ε] in the z-plane for small ε > 0 π 3ε corresponds to the sector S[− π2 − 2ϑ − 3ε 2 , − 2 − 2ϑ − 2 ] in the ξ-plane. By ∂ Lemma 7.2.3, in this sector ∂r Φ(t) > 0. Therefore, by (A.4), ∂x arg z(x, λ) > 0 for π π z ∈ S[− 3 − ε, − 3 + ε]. Hence z∗ is unique and 1 holds. 2. By (A.4), for x ∈ (x∗ , ∞) and t = √xλ the hypothesis of Lemma 7.2.5 is

∂ R(t) > 0 and |z(x, λ)| is strictly increasing in x. fulfilled. Therefore ∂r 3. By (A.4), we have arg ξ(t∗ ) = − π2 − 2ϑ. By Lemma 7.1.2, for t∗ = 1 + sin ϕ ϑ 2π η∗ e−iϕ∗ we have ϕ∗ − ϑ ∈ [ 2π 3 − 3 , 3 ]. Therefore using the relation r = sin(ϕ−ϑ) 1 we obtain |t∗ | sin π . 3

In the next Lemma we analyze the curves Γ± λ , defined in (2.38). Lemma 7.5 Let λ = |λ|e2iϑ ∈ C+ \ {0}. Then 2

2

3 1. if ϑ = 0, then Γλ = [−λ 3 ( 3π 8 ) , ∞), + π 4 π if 0 < ϑ 2 , then Γλ ⊂ S[−π + 43 ϑ, 0), Γ− λ ⊂ S[−π + 3 ϑ, − 3 ], Γλ ⊂ π ϑ S[− 2 − 2 , 0), 2

2. inf z∈Γλ |z| |λ| 3 sin ϑ, 2

2

− 3 3 3. Γ− λ ⊂ {z : |z| C|λ| }, and the length |Γλ | C|λ| ,

4. the function |z(·, λ)| is strictly increasing on [x∗ , ∞).

Vol. 6, 2005

Harmonic Oscillator Perturbed by Bounded Potentials

783

Proof. 1. Direct calculation yields the result for ϑ = 0. For 0 < ϑ π2 the 2 2 assertion on Γλ follows from Lemma 7.1.4 and z = λ 3 ( 32 ξ) 3 . The assertion on Γ± λ for 0 arg λ δ follows from 1 of the present Lemma and Lemma 7.3.1. For δ arg λ π it follows from Lemma 7.4.1. 2 2 2. Follows from Lemma 7.1.3 and z = λ 3 ( 32 ξ) 3 . 3. By Lemma 7.3.4 and Lemma 7.4.3, |t∗ | sin1 π uniformly in λ ∈ C+ \ {0}. 3 Using definition (2.15) we have that k(t) and k (t) for t ∈ [0, t∗ ] are also uniformly 2 2 3 3 bounded. Now the relations |Γ− λ | = |λ| [0,t∗ ] |k (t)||dt| and |z(x, λ)| = |λ| |k(t)|, x t = √λ yield the proof. ∂ |z(x, λ)| > 0 if ∂r |ξ(t)| > 0 for t = re−iϑ = √xλ . Thus the 4. (A.4) gives ∂x result follows from Lemma 7.4.2 (for δ arg λ π) and from Lemma 7.3.2 (for 0 arg λ δ).

Proof of Lemma 2.1. Case 1 (or 2) follows from Lemma 7.3 (or Lemma 7.4). 2iϑ

Re λξ(t)

Proof of Lemma 2.2. Fix λ = |λ|e . By (A.4), we have h(x) = e for t = √xλ = re−iϑ . By Lemma 7.2.2, if either 0 < arg λ π, r 0 or arg λ = 0, r > 1, then ∂r Re (λξ(t)) > 0. It remains to consider the case ϑ = 0 and x ∈ [0, x∗ ); by Lemma 7.2.2, Re ξ(t) = 0 and therefore h = 1. In order to prove Lemma 2.3, for fixed λ = |λ|e2iϑ ∈ C \ {0} define the 3

function Ψ(z) = 23 z|λ|2 . Ψ maps Γλ (z1 , z2 ) onto γλ (u1 , u2 ), uj = Ψ(zj ), j = 1, 2. Similarly, we set γλ (u) = Ψ(Γλ (z)) for u = Ψ(z) and γλ± = Ψ(Γ± λ ). We also set u0 = Ψ(z0 ) and u∗ = Ψ(z∗ ). Consider the integral γλ (u) f (v)|dv|. By (A.4), we have the parametrization 2iϑ −iϑ γλ (u) = v ∈ C : v = v(r) = e ξ(re ), r > ru , where u = Ψ(z(ru |λ|, λ)) . (A.12) Consider the image of the part of Γλ , defined by the hypothesis a) and b) of Lemma 2.3, under the mapping Ψ. Assumption a) transforms into a ) δ arg λ π and u ∈ γλ . Similarly, b) becomes b ) 0 arg λ δ, u ∈ γλ+ . Then by Lemma 7.2.2, Re v(r) is strictly increasing in r. Thus we choose the variable κ = Re v and obtain ∞ |f (v)||dv| = |f (κ + i Im v(κ))| |∂κ v| dκ, |∂κ v| = 1 + (∂κ Re v(κ))2 . γλ (u)

Re u

Let us show that in both cases a ) and b ) the following estimate holds: ∞ |f (v)||dv| C |f (κ + i Im v(κ))| dκ. γλ (u)

Re u

(A.13)

dv . In terms of the parametrization v(r) = e2iϑ ξ(re−iϑ ) we have Let us estimate dκ

∂r Im e2iϑ ξ(t) d Im v(κ) = = tan arg ∂r e2iϑ ξ(t). dκ ∂r Re (e2iϑ ξ(t))

784

M. Klein, E. Korotyaev and A. Pokrovski


Consider the two cases a ) and b ). a ) Let δ arg λ π and u ∈ γλ . For each point v(r) ∈ γλ consider t = re−iϑ π such that v(r) = e2iϑ ξ(t). By Lemma 7.2.1, arg ∂r ξ(t) ∈ [− 2 − ϑ, −2ϑ). Hence arg ∂r e2iϑ ξ(t) ∈ [− π2 + δ2 , 0) and d Imdκv(κ) sin(1 δ ) uniformly for δ arg λ π. 2 dv Therefore dκ is also uniformly bounded. + b ) Let 0 arg λ δ and u ∈ γλ+ = Ψ(Γ+ λ ). For each point v(r) ∈ γλ π −iϑ 2iϑ consider t = re such that v(r) = e ξ(t). Since 0 ϑ 7 , arg ξ(t∗ ) ∈ 3π 11ϑ 3π 3ϑ [− 4 − 4 , − 4 − 4 ], (equivalent to Lemma 7.3.1) implies that −π − ϑ arg ξ(t).

Therefore 7.1.5, for v ∈ γλ+ we have arg e2iϑ ∂r ξ(t) ∈ [− π3 + ϑ6 , ϑ2 ]. by Lemma dv is uniformly bounded. Hence d Imdκv(κ) sin(1 π ) and dκ 3 dv Thus dκ is uniformly bounded in both cases a ) and b ), implying (A.13). Proof of Lemma 2.3. Firstly we prove (2.39) for the case δ arg λ π. Lemma 7.5.2 gives dist(Γλ , {0}) >const uniformly in λ. Thus we replace · by | · |. The change of variables v = Ψ(s) in (2.39) results in the equivalent relation 3 e−2|λ| Re v C e−2|λ| Re u 2 z2 ∈ γλ . |dv| , u = (A.14) 2 (α+ 12 ) |λ| |u| 23 (α+ 12 ) 3 |λ| γλ (u) |v| 3 By (A.13), we have the auxiliary estimate e−2|λ| Re u e−2|λ| Re v |dv| C , |λ| γλ (u)

u ∈ γλ .

(A.15)

We show that (A.14) follows from (A.15). If u ∈ γλ+ , then Lemma 7.5.4 yields |u| = min |v|. Therefore (A.15) gives (A.14). If u ∈ γλ− , then dist(γλ , {0}) >const and

v∈γλ (u)

we have γλ (u)

e−2|λ| Re v 2

1

|v| 3 (α+ 2 )

|dv| C

γλ (u)

e−2|λ| Re v |dv|,

u ∈ γλ− .

(A.16)

By Lemma 7.5.3, γλ− (u) is uniformly bounded. Therefore, |u| is bounded on γλ− , so (A.16) and (A.15) imply (A.14). + b) We prove (2.39) for the case z ∈ Γ+ λ . It suffices to show that for any z ∈ Γλ we have 3 3 4 4 3 3 |e− 3 s 2 | |e− 3 z 2 | − 43 s 2 − 43 z 2 |e | |ds| C|e |, |z| ≤ 1; |ds| C 1 , |z| > 1. |s|α |z|α+ 2 Γλ (z) Γλ (z) (A.17) By the change of variable u = Ψ(z) the first estimate in (A.17) follows from (A.15). For the proof of the second estimate in (A.17) we observe that by Lemma 7.5.4, for z ∈ Γ+ min |v|. Hence, (A.17) follows from λ we have |z| = v∈Γλ (z)

(A.15). This proves (2.39).

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785 3

We shall prove (2.40). The change of variable v = Ψ(s) = 23 s|λ|2 (see (A.12)) yields − 23 3 3λ |ds| |dv| 2 (α−1) 2 z2 ε . = I, I = , ε = < 1, u = 1 2 2 α α 3 3 |λ| Γλ (z) s γλ (u) |v| 3 (ε + |v| 3 ) (A.18) Assume z ∈ Γ+ , 0 < arg λ π and let u = a + ib, a, b ∈ R. Using Lemma 7.3.1, λ Lemma 7.3.3, Lemma 7.1.5 and definition of z∗ , we have b 0. By Lemma 7.2.2, Im v is strictly decreasing on γλ+ , so that |b| = min | Im v|. Thus using (A.13)

we obtain I CJ,

v∈γλ (u)

∞

J= a

dκ 1 3

2

2

|κ| (ε + |b| 3 + |κ| 3 )α

. 2

Consider two cases: a 0 and a < 0. Firstly, let a 0. Then using |u| 3 = ε|z| we have 3 1 3 ∞ dx 2 J= = 2 a 23 (ε + |b| 32 + x)α α − 1 (ε + |b| 32 + a 23 )α−1 C 1−α = Cε1−α z 2 (ε + |u| 3 )α−1 which together with (A.18) proves (2.40). Secondly, let a < 0. Due to Lemma 7.3.1 and Lemma 7.4.1, for u = a + ib ∈ γλ+ and a < 0 we have |a| C|b| uniformly in 2 2 0 arg λ π. Therefore |b| C|u|, and using |u| 3 = ε|z| and x = κ 3 , we obtain ∞ dκ 3 C 1−α = = Cε1−α z . J 2 1 2 2 2 2 α α−1 α−1 3 3 3 3 3 κ (ε + |b| + κ ) (ε + |b| ) (ε + |u| ) 0 (A.19) which together with (A.18) proves (2.40). 3 2

2 z∗ Assume z ∈ Γ− λ and δ arg λ π. Recall that u∗ = 3 |λ| . We have I = I− + I+ , where 3 2 z2 |dv| |dv| I− = , I = , u = . + 1 2 1 2 α + |v| 3 (ε + |v| 3 )α 3 |λ| γλ (u,u∗ ) |v| 3 (ε + |v| 3 ) γλ

3 By Lemma 7.5.2, we have |z| sin δ2 for z ∈ Γλ , so |u| 23 sin δ2 2 for u ∈ γλ . By Lemma 7.5.3, the length of the curve |γλ− | < C. Therefore I− C. By (A.19) for u = u∗ , we have I+ C so that I C. Next, by Lemma 7.5.3, γλ− is uniformly 2 bounded, implying C (ε + |u| 3 )1−α and I |λ|

2 3 (α−1)

C |λ|

2 3 (α−1)

(ε + |u|

which together with (A.18) proves (2.40).

2 3

)α−1

=

C , z α−1

(A.20)

786



Proof of Lemma 2.4. First we prove (2.41). The change of variable u = Ψ(z) yields |ds| |dv| 2 2 α−1 3 ε = I, I = , ε = ( |λ|)− 3 C, (A.21) 1 2 α − s α − 3 2 Γλ γλ |v| 3 (ε + |v| 3 ) where u, v is defined before (A.13). By Lemma 7.2.2, Im v is strictly decreasing on γλ− . Thus we parameterize the last integral by χ = Im v, so that v(χ) = Re v(χ) + iχ. Lemma 7.5.3 gives γλ− ⊂ {u ∈ C : |u| < c} for some c > 0 independent of λ. Therefore c , dχ I2 |∂χ v| 1 , |∂ v| = 1 + (∂χ Re v(χ))2 (A.22) χ 2 χ 3 {ε + χ 3 }α 0 Recall λ = |λ|e2iϑ and t = re−iϑ . In order to estimate d Redχv(χ) we use the parametrization (A.12). Thus we have

∂r Re e2iϑ ξ(t) d Re v(χ) = = cot arg e2iϑ ∂r ξ(t) . dχ ∂r Im (e2iϑ ξ(t)) By Lemma 7.3.3, Lemma 7.1.5 Lemma 7.4.1, and (2.37) , e2iϑ ∂r ξ(t) is in a sector dv d Re v(χ) isolated from the real axis uniformly in λ. Therefore dχ C and dχ C. 2

Substituting this estimate in (A.21) and making the change of variables x = ε−1 χ 3 we obtain C1 /ε 2 dx C 3 I α−1 , ε = ( |λ|)− 3 < 1, (A.23) α ε (1 + x) 2 0 which together with (A.21) yields (2.41). Now we prove (2.42). By the change of variable u = Ψ(z), we have |ds| |dv| I− + I+ = , I± = 4 2 1 4 . 3 3 2 ± 3 3 3 |λ| Γλ |λ| + |s| γλ ( 2 |v|) (1 + ( 2 |v|) 3 ) For I+ we use (A.13), which gives I+ C estimate I− we use the parametrization of

γλ−

∞ 0

1

dκ

4

κ 3 (1+κ 3 )

(A.24)

C. In order to

by χ = Im v. Repeating the ar dv guments used above for the proof of (2.41), we conclude that dχ C and ∞ dχ C. The estimates of I± < C and (A.24) imply I− C 0 1 4 χ 3 (1+χ 3 )

(2.42).

Lemma 7.6 Let λ = |λ|e2iϑ ∈ S[0, δ] and q, q ∈ L∞ (R). Let P± be defined by (4.27). Then for some absolute constant C the following estimates are fulfilled: 3 4 1 |P− (z, z∗ )| C|e− 3 z 2 | q ∞ + q∞ |λ|− 6 ,

z ∈ Γ− λ,

(A.25)

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787

3 4 2 1 |P+ (z1 , z2 )| C|e 3 z2 | q ∞ + q∞ |λ|− 6 ,

x1 x2 x∗ , z1,2 = z(x1,2 , λ) ∈ Γ− λ . (A.26) Proof. Let ξ1,2 = ξ(t1,2 ) and t1,2 = Lemma 2.2 gives |e

2λ(ξ1 −ξ2 )

x1,2 √ . λ

Then, Lemma 7.3.2 yields |ξ1 − ξ2 | 2|ξ1 |.

− 1| 2|λ||ξ1 − ξ2 | and |e2λ(ξ1 −ξ2 ) − 1| C. Therefore

|(e2λ(ξ1 −ξ2 ) − 1)/λξ1 | C(1 + |λ||ξ1 |)−1 . 2

(A.27)

2

Using z1,2 = λ 3 ( 32 ξ1,2 ) 3 we obtain the following estimate uniformly in 0 arg λ δ: 3 3 4

2

|(e 3 (z1

−z22 )

−3

3 ± 43 s 2

Introduce the functions R± (s, z) = e 3 √ 4 ±2 se± 3 s 2 . Integrating by parts, we have P− (z∗ , z) = R− (z, z∗ )f (z) +

3

− 1)z1 2 | Cz1 2 .

Γλ (z,z∗ )

−e

3 ± 43 z 2

(A.28)

. Evidently ∂s R± (s, z) =

R− (s, z∗ )f (s) ds, √ 1 f (s) = ρ(s)ˆ q (s)/(2 sλ 6 ).

(A.29)

1

d qˆ(s)| Cq ∞ |λ|− 6 we obtain Therefore using (2.25), (2.26), (A.28) and | ds 3

4

|P− (z, z∗ )| C %

q∞

|e− 3 z 2 | 1

|λ| 6 +

& |ds|

1

|λ|− 6 q ∞

2

+

|λ|− 3 q∞

q∞

+

'(

3

, (A.30)

which together with (2.40–2.41) gives (A.25). Similarly we have P+ (z1 , z2 ) = −R+ (z1 , z2 )f (z1 ) − R+ (s, z2 )f (s)ds.

(A.31)

1

z 2

1

s 2

Γλ (z,z∗ )

1

s 2

s 2

Γλ (z1 ,z2 )

1

d Again using (2.25), (2.26), (A.28) and | ds qˆ(s)| Cq ∞ |λ|− 6 we have 3 2

4

|P+ (z1 , z2 )| C   

|e 3 z2 | 1

|λ| 6

q∞ (1 + |z2 |)

1 2

+ Γλ (z1 ,z2 )

&

− 16

|λ|

q ∞

s

1 2

which together with (2.40), (2.41) gives (A.26).

+

− 23

|λ|

q∞

s

1 2

+

q∞ s

3 2

'

  |ds| ,

788



2πi

Lemma 7.7 Let q ∈ B. Define F (z) = a2 (ze± 3 )Vq (z). Then uniformly in z ∈ Γ− λ and λ ∈ S[−δ, δ] the following estimates hold for sufficiently large |λ|: 3 3 3 qB qB 4 4 2 2 −s 2 ) (z s 3 e3 F (s) ds C , e F (s) ds C 1 1 . Γλ (z) Γλ (z,z∗ ) |λ| 3 |λ| 3 (A.32) Proof. We show the first estimate in (A.32).Using (2.25), (2.26), (2.34) and (2.39) we obtain 3 3 4 q∞ (z 2 −s 2 ) 3 F (s) ds C (A.33) 1 . +e |λ| 3 Γλ

3 − 43 s 2

4

3 2

−e− 3 z∗ we integrate by parts the integral over Γλ (z, z∗ ). Using R− (s, z∗ ) = e We have 3 3 4 e 3 (z 2 −s 2 ) F (s)ds I= Γλ (z,z∗ )

3 ) 1 4 F (s) e3z2 F (z) = R− (s, z∗ ) √ ds . (A.34) R− (z, z∗ ) √ + 2 z s Γλ (z,z∗ ) 1

d Therefore, using Lemma 2.2, | ds qˆ(s)| C|λ|− 6 q ∞ , (2.25), (2.26), (2.34), (2.35) and (A.28), we obtain from (A.34) ) 1 1 2 q∞ |λ|− 3 q∞ C q∞ |λ|− 6 q ∞ + |I| + + |ds| . (A.35) 1 1 3 1 s s |λ| 3 z 2 s 2 + 2 Γ− λ

By Lemma 2.4, for sufficiently large |λ| this gives 1 1 |I| C|λ|− 3 qB 1 + |λ|− 6 log(|λ| + 1) . Together with (A.33) this proves the first estimate in (A.32). In order to prove 4

3

the second estimate in (A.32) we use similar arguments with R+ (s,z∗ ) = e 3 s 2 − 4

3 2

e 3 z∗ .

Acknowledgments. The authors would like to thank Vladimir Geiler for discussions on the physical models and for the ref. [8] and Horst Hohberger for the Figures 1 and 2.

References [1] M. Abramowitz and A. Stegun, eds. Handbook of Mathematical Functions. N.Y.: Dover Publications Inc. [2] M. Altarelli, G. Platero, Magnetic hole levels in quantum wells in parallel magnetic field, Surf. Sci. 196, 540–544 (1988).

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[3] T. Ando, A. Fauler, F. Stern, Electronic properties of two-dimensional systems, Rev. Modern Phys. 54, 437–672 (1982). [4] D. Chelkak, P. Kargaev, E. Korotyaev, An Inverse Problem for an Harmonic Oscillator Perturbed by Potential: Uniqueness, Lett. Math. Phys. 64 (1), 7–21 (2003). [5] D. Chelkak, P. Kargaev, E. Korotyaev, Inverse problem for harmonic oscillator perturbed by potential, characterization, Comm. Math. Phys. 249, 133–196 (2004). [6] M. De Dios Leyva, V. Galindo, Interband optical absorption in superlattices in an in-plane magnetic field, Phys. Rev. B. 48, 4518–4523 (1993). [7] V.A. Geyler, I.V. Chudaev, Schr¨ odinger operators with moving point perturbations and related solvable models of quantum mechanical systems, Z. Anal. Anwendungen 17, no. 1, 37–55 (1998). [8] C. Hooley, J. Quintanilla, Single-Atom Density of states of an opticallattice, Phys. Rev. Let. 93 no 8, 080404-1-4 (1998). [9] J. Maan, Magneto-optical properties of superlattices and quantum wells, Surf. Sci. 196, 518–532 (1988). [10] F. Olver, Asymptotics and special functions, Academic Press, New YorkLondon, 1974. [11] F. Olver, Two inequalities for parabolic cylinder functions, Proc. Cambridge Philos. Soc. 57, 811–822 (1961) . [12] J. P¨ oschel, E. Trubowitz, Inverse Spectral Theory, Boston: Academic Press, 1987.

Markus Klein Institut f¨ ur Mathematik Universit¨ at Potsdam Germany email: [email protected] Alexis Pokrovski Institute for Physics St.Petersburg State University Russia email: [email protected] Communicated by Bernard Helffer submitted 23/04/04, accepted 26/10/04

Evgeny Korotyaev Institut f¨ ur Mathematik Humboldt Universit¨ at zu Berlin Rudower Chaussee 25 D-12489, Berlin Germany email: [email protected]

Ann. Henri Poincaré 6 (2005) 791 – 799 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/04791-9, Published online 28.07.2005 DOI 10.1007/s00023-005-0223-y


On the Energy and Helicity Conservations for the 2-D Quasi-Geostrophic Equation Yong Zhou Abstract. In this paper, we establish sufficient conditions in Besov space for weak solutions of 2-D quasi-geostrophic equation to guarantee the conservations of energy and helicity. These two conservation laws are of great interest for both mathematical theory and applications in meteorology and oceanography.

1 Introduction A fundamental equation to describe the motion of inviscid Newtonian flow is the following 3-D Euler equations  in Ω × (0, T ),  (∂t + u · ∇)u + ∇P = f, divu = 0, in Ω × (0, T ) (1.1)  in Ω, u(x, 0) = u0 (x), where u(x, t) ∈ R3 is the velocity field, P is pressure, while f is the external force. Onsager [14] conjectured that the energy of weak solutions to 3-D Euler equations is conserved as long as it belongs to a H¨ older space C γ with γ > 13 . This α conjecture was proved by Constantin, E and Titi [5] in the Besov space B3,∞ with 1 older space. Just as stated in [5], the significance of Onsager’s α > 3 instead of H¨ conjecture can be appreciated in the context of Kolmogorov theory of turbulence. However, so far there are many fundamental questions concerning 3-D Euler equations are not clear, cf. [2, 4, 11]. Another way to understand the problems is by constructing and studying models in lower dimensions. A 1-D model was studied by Constantin, Lax and Majda [6] for the 3-D Euler vorticity equation as ωt = H(ω)ω, x where H is the Hilbert transform and the velocity is defined by u = −∞ ω(y)dy. A 2-D model of the quasi-geostrophic equation was studied by Constantin, Majda and Tabak [7]. The 2-D quasi-geostrophic equation is as follows  in Ω × (0, T ),   (∂t + u · ∇)θ = 0, 1 ∂Ψ ∂Ψ ⊥ u = ∇ Ψ = − ∂x2 , ∂x1 , θ = −(−∆) 2 Ψ, in Ω × [0, T ), (1.2)   in Ω, θ(x, 0) = θ0 (x),

792

Y. Zhou


where Ω ⊂ R2 , θ = θ(x, t) ∈ R1 , u(x, t) ∈ R2 is the velocity field, while Ψ(x, t) ∈ R1 is the stream function. The operator (−∆)γ (γ > 0) is defined by [17] γ f (ξ) = |ξ|2γ fˆ, (−∆) where fˆ denotes the Fourier transform of f . In [7] and [15], θ is interpreted as a potential temperature. Just as stated in [9], θ may be regarded as a vorticity (pointed out by R. de la Llave). This important model (1.2) has been intensively studied for its similarity to the 3-D Euler equations and applications in meteorology and oceanography very recently [7, 8, 9, 10, 13, 15]. The purpose of his paper is to find sufficient conditions to guarantee the conservation of energy (Onsager’s conjecture for 2-D quasi-geostrophic equation) and helicity for weak solutions to the quasi-geostrophic equation. To avoid problems involving boundaries, we assume Ω is the whole space R2 2 or Π = [0, 1]2 with periodic boundary conditions. The first main theorem reads α (Ω)) be a weak solution to the Theorem 1.1 Let θ ∈ C(0, T ; L2 (Ω)) ∩ L3 (0, T ; B3,∞ 2-D quasi-geostrophic equation, i.e.,

0

T

Ω

θ(x, t)∂t φ(x, t)dxdt +

Ω

θ0 (x)φ(x, 0)dx

T

+ 0

Ω

u(x, t)θ(x, t) · ∇φ(x, t)dxdt = 0,

for any test function φ(x, t) ∈ C ∞ (Ω × R1+ ) with compact support. If α > 13 , then the energy is conserved, i.e., E(t) = |θ(x, t)|2 dx = |θ0 (x)|2 dx, Ω

Ω

for all t ∈ [0, T ). If we assume f = 0 in (1.1), the vorticity equation for 3-D Euler equations reads (∂t + u · ∇)ω = ∇u · ω,

(1.3)

with ω = ∇ × u. By differentiating the equation, we obtain an equation similar to (1.3) with the gradient perpendicular to θ instead of the vorticity (∂t + u · ∇)∇⊥ θ = ∇u · ∇⊥ θ. For 3-D Euler equations, one of the most important conservation laws is helicity [1, 11, 12] defined by H(t) =

Ω

v(x, t) · ω(x, t)dx.

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793

Very recently, Chae [3] established a regularity condition ω ∈ L3 (0, T ; B α9 ,∞ ) with 5

α > 13 , to show the conservation of helicity for weak solutions to 3-D Euler equations. For the 2-D quasi-geostrophic equation, although θ is a scalar and ∇⊥ θ is a vector, we can define the ‘helicity’ as follows ¯ Hi (t) = θ(x, t)∂i θ(x, t)dx, Ω

for i = 1, 2, where ∂i denotes ∂xi . The second main result of this paper is concerning the conservation of this ‘helicity’. 4

Theorem 1.2 Let ∇θ ∈ C(0, T ; L 3 (Ω)) ∩ L3 (0, T ; B α3 ,∞ (Ω)) be a weak solution to 2

the 2-D quasi-geostrophic equation. If α > 13 , then the ‘helicity’ is conserved, i.e., H¯i (t) = θ(x, t)∂i θ(x, t)dx = θ0 (x)∂i θ0 (x)dx, Ω

Ω

for i = 1, 2, t ∈ [0, T ). Remark 1.1 For smooth solutions to 2-D quasi-geostrophic equation (1.2), the energy and helicity are conserved.

2 Proof of the main theorems For the completeness of this paper, we recall the definition of Besov spaces. We start by recalling the Littlewood-Paley decomposition of temperate distributions. Let S be the class of Schwartz class of rapidly decreasing functions. Given f ∈ S, the Fourier transform is defined by 1 ˆ F (f ) = f = e−ix·ξ f (x)dx. (2π)N/2 RN One can extend F and F −1 to S in the usual way, where S denotes the set of all tempered distributions. Let φ ∈ S satisfying

12 5 ˆ Suppφ ⊂ ξ : ≤ |ξ| ≤ φˆ 2−j ξ = 1, and 6 5 j∈Z

ˆ −j ξ), in other words, φj (x) = 2jN φ(2j x), for any for ξ = 0. Setting φˆj = φ(2 f ∈ S , we define ∆j f = φj ∗ f and Sj f = φk ∗ f. k≤j−1

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Then the homogeneous Besov semi-norm f B˙ s is defined [16, 18] by p,q For −∞ < s < ∞, 0 0, The space B˙ p,q s (p, q) ∈ (1, ∞) × [1, ∞], we define the inhomogeneous Besov space norm f Bp,q of f ∈ S as s f Bp,q = f Lp + f B˙ s . p,q

The inhomogeneous Besov spaces are Banach spaces equipped with the norm s . f Bp,q ∈ C0∞ (R2 ) be the standard mollifier supported in B(0, 1) and denote

φ(x) xLet 1 ε ε ε ε φ ε by φ (x). Let f (x) = (f ∗ φ ) (x). α First, we recall the following inequalities [16, 18] for functions in Bp,∞ with 1 0. α , u(· + y) − u(·)Lp ≤ C|y|α uBp,∞

(2.1)

α ∇uε Lp ≤ Cεα−1 uBp,∞ ,

(2.2)

α . uε − uLp ≤ Cεα uBp,∞

(2.3)

In the above inequalities, C’s are absolute constants. Proof of Theorem 1.1. We follow the idea of Constantin, E and Titi. Due to divergence free of the velocity field u(x, t), we rewrite equation (1.2) as ∂t θ + div(uθ) = 0,

(2.4)

u(x, t) = (−R2 θ, R1 θ),

(2.5)

and

where Ri be the Riesz transform. Now, we do the regularization of the equation (2.4) to obtain ∂t θε + div(uθ)ε = 0, where (uθ)ε = uε θε − (u − uε ) (θ − θε ) + rε (u, θ)

(2.6)

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with rε (u, θ) =

Ω

φε (y) (u(x − y) − u(x)) (θ(x − y) − θ(x)) dy.

Therefore, for (2.6) directly, we have d 2 |θε (x, t)| dx = 2 (uθ)ε (x, t) · ∇θε (x, t)dx dt Ω Ω ε ε = 2 u (x, t)θ (x, t) · ∇θε (x, t)dx Ω −2 (u − uε ) (x, t) (θ − θε ) (x, t) · ∇θε (x, t)dx Ω +2 rε (u, θ)(x, t) · ∇θε (x, t)dx Ω = −2 (u − uε ) (x, t) (θ − θε ) (x, t) · ∇θε (x, t)dx Ω +2 rε (u, θ)(x, t) · ∇θε (x, t)dx Ω ≤ C (u − uε )(t)L3 (θ − θε )(t)L3 + rε (u, θ)(t)

3 L2

∇θε )(t)L3 . (2.7)

From (2.5), and thanks to the boundedness of Riesz transform Ri : Lp → Lp with 1 < p < ∞, cf. [17], we have (u − uε )(t)L3 ≤ C(θ − θε )(t)L3

(2.8)

and rε (u, θ)(t)

3

L2

≤ ≤

φε (y)(θ(· − y) − θ(·))(t)2L3 dy 2 Cθ(t)B3,∞ α |y|2α φε (y)dy ≤ C 2α θ(t)2B3,∞ α , (2.9) Ω

Ω

where we used (2.1). Substituting (2.8) and (2.9) into (2.7), taking (2.2) and (2.3) into account, and integrating with respect to time, then we have t ε |θε (x, t)|2 dx − ≤ Cε3α−1 |θ (x, 0)| dx θ(τ )3B3,∞ α dτ → 0, Ω

Ω

0

α (Ω)) with α > 13 . when ε → 0 as long as θ ∈ C(0, T ; L2 (Ω)) ∩ L3 (0, T ; B3,∞ This finishes the proof for Theorem 1.1.

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Y. Zhou


Proof of Theorem 1.2. Differentiating equation (2.4), after regularization, we have ∂t (∂i θε ) + div(u∂i θ)ε + div(∂i uθ)ε = 0, i = 1, 2,

(2.10)

with (u∂i θ)ε = uε (∂i θ)ε − (u − uε ) (θ − (∂i θ)ε ) + rε (u, θ) and

rε (∂i u, θ) =

Ω

φε (y) (u(x − y) − u(x)) (∂i θ(x − y) − ∂i θ(x)) dy,

similar version for (∂i uθ)ε . By direct computation, we have d ∂i θε (x, t)θε (x, t)dx = − div(uθ)ε (x, t)∂i θε (x, t)dx dt Ω Ω ε ε − div(u∂i θ) (x, t)θ (x, t)dx − div(∂i uθ)ε (x, t)θε (x, t)dx. (2.11) Ω

Ω

Then by integration by parts and due to divergence free of uε and ∂i uε , (2.11) reduces to d ∂i θε (x, t)θε (x, t)dx dt Ω = rε (u, θ)(x, t) · ∇∂i θε (x, t)dx + rε (u, ∂i θ)(x, t) · ∇θε (x, t)dx Ω Ω + rε (∂i u, θ)(x, t) · ∇θε (x, t)dx Ω − (u − uε )(x, t)(θ − θε )(x, t) · ∇∂i θε (x, t)dx Ω − (u − uε )(x, t)(∂i θ − ∂i θε )(x, t) · ∇θε (x, t)dx Ω − (∂i u − ∂i uε )(x, t)(θ − θε )(x, t) · ∇θε (x, t)dx Ω

=

I1 + I2 + I3 + I4 + I5 + I6 .

(2.12)

First, we do estimate for rε (u, θ) as follows. For 1 < p < ∞, we have rε (u, θ)(·, t)Lp ≤ φε (y)u(· − y, t) − u(·, t)L2p θ(· − y, t) − θ(·, t)L2p dy Ω ≤C φε (y)θ(· − y, t) − θ(·, t)2L2p dy Ω ≤C φε (y)∇(θ(· − y, t) − θ(·, t))2 2p dy L p+1 Ω ≤ C∇θ(t)2B α2p |y|2α φε (y) ≤ Cε2α ∇θ(t)2B α2p , (2.13) p+1

,∞

Ω

p+1

,∞

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where the Sobolev embedding is used. By taking p = 3 in (2.13), we have |I1 | ≤ ∇∂i θ(t)

3

L2

rε (u, θ)(t)L3 ≤ Cε3α−1 ∇θ(t)3B α3 2

,∞

,

(2.14)

where we used (2.2). We can do a similar estimate for rε (u, ∂i θ). rε (u, ∂i θ)(·, t)

6

L5

φε (y)u(· − y, t) − u(·, t)L6 ∂i (θ(· − y, t) − θ(·, t)) 32 dy L ≤ C φε (y)θ(· − y, t) − θ(·, t)L6 ∂i (θ(· − y, t) − θ(·, t)) 32 dy L Ω ≤ C φε (y)∇(θ(· − y, t) − θ(·, t))2 3 dy L2 Ω ≤ C∇θ(t)2B α3 |y|2α φε (y) ≤ Cε2α ∇θ(t)2B α3 .

≤

Ω

2

,∞

Ω

2

,∞

Therefore, |I2 | ≤ ∇θε (t)L6 rε (u, ∂i θ)(t) ≤ Cε3α−1 ∇θ(t)3B α3 2

,∞

6

L5

≤ C∇(∇θε (t))

3

L2

ε2α ∇θ(t)2B α3 2

.

,∞

(2.15)

Due to the relation (2.5) between θ and u, the estimate of I3 is similar to I2 , |I3 | ≤ Cε3α−1 ∇θ(t)3B α3 2

,∞

.

(2.16)

I4 can be estimated straightforward as follows. |I4 | ≤ (u − uε )(t)L6 (θ − θε )(t)L6 ∇∂i θε (t) ≤ C∇(θ − θε )(t)2 3 ∇∂i θε (t) L2

≤ Cε

3

L2

3

L2 3α−1

∇θ(t)3B α3

,∞

2

, (2.17)

where (2.2) and (2.3) are used. Now, we turn our attention to I5 . |I5 | ≤ (u − uε )(t)L6 ∂i (θ − θε )(t)

3

L2

≤ C∇(θ − θε )(t)2 3 ∇(∇θε )(t) L2

∇θε (t)L6 3

L2

≤ Cε3α−1 ∇θ(t)3B α3 2

,∞

. (2.18)

I6 is similar to I5 , |I6 | ≤ Cε3α−1 ∇θ(t)3B α3 2

,∞

.

(2.19)

Substituting (2.14)–(2.19) into (2.12) and integrating with respect to time, then we have ε ε θε (x, t)∂i θε (x, t)dx − θ (x, 0)∂i θ (x, 0)dx Ω

Ω

≤ Cε3α−1

0

t

∇θ(τ )3B α3 2

,∞

dτ → 0,

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Y. Zhou


4

when ε → 0 as long as ∇θ ∈ C(0, T ; L 3 (Ω)) ∩ L3 (0, T ; B α3 ,∞ (Ω)) with α > 13 . Here 2 we use θε (x, t)∂i θε (x, t)dx ≤ ∇θ(t) 43 θ(t)L4 ≤ C∇θ(t)2 4 . (2.20) L

Ω

L3

The proof for Theorem 1.2 is complete. Remark 2.1 Under the same conditions of Theorem 1.2, it follows from inequality (2.20) that we have a lower bounds for ∇θ(t) 43 as L

∇θ(t)2 4

L 3 (Ω)

˜ ≥ C θ0 (x)∂i θ0 (x)dx , Ω

˜ for i = 1, 2, t ∈ [0, T ) and some absolute positive constant C.

Acknowledgment The author would like to express sincere gratitude to his supervisor Professor Zhouping Xin for enthusiastic guidance and constant encouragement. Thanks also to Professor Pedlosky for helpful comments. This work is partially supported by Hong Kong RGC Earmarked Grants CUHK-4028-04P and Shanghai Leading Academic Discipline.

References [1] V.I. Arnold, B.A. Khesin, A. Boris, Topological methods in hydrodynamics, Applied Mathematical Sciences, 125, Springer-Verlag, New York (1998). [2] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94, 61–66 (1984). [3] D. Chae, Remarks on the helicity of the 3-D incompressible Euler equations, Comm. Math. Phys. 240, no. 3, 501–507 (2003). [4] J.Y. Chemin, Perfect incompressible fluids, Oxford Lecture Series in Mathematics and its Applications, 14. The Clarendon Press, Oxford University Press, New York, 1998. [5] P. Constantin, W. E, E.S. Titi, Onsager’s conjecture on the energy conservation for solutions of Euler’s equation, Comm. Math. Phys. 165, no. 1, 207–209 (1994). [6] P. Constantin, P.D. Lax, A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38, no. 6, 715– 724 (1985).

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[7] P. Constantin, A. Majda, E. Tabak, Formation of strong fronts in the 2D quasigeostrophic thermal active scalar, Nonlinearity 7, no. 6, 1495–1533 (1994). [8] P. Constantin, Q. Nie, N. Schorghofer, Nonsingular surface quasi-geostrophic flow, Phys. Lett. A 241, no. 3, 168–172 (1998). [9] D. Cordoba, Nonexistence of simple hyperbolic blow-up for the quasigeostrophic equation, Ann. of Math. 2, no. 3, 148, 1135–1152 (1998). [10] D. Cordoba, C. Fefferman, Growth of solutions for QG and 2D Euler equations, J. Amer. Math. Soc. 15, no. 3, 665–670 (2002). [11] A. Majda, A. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. [12] H.K. Moffat, A. Tsinober, Helicity in Laminar and turbulent flow, Ann. Rev. Fluid Mech. 24, 281–312 (1992). [13] K. Ohkitani, M. Yamada, Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow, Phys. Fluids 9, no. 4, 876–882 (1997). [14] L. Onsager, Statistical hydrodynamics, Nuovo Cimento 9, 6 (1949). Supplemento, no. 2 (Convegno Internazionale di Meccanica Statistica), 279–287. [15] J. Pedlosky, Geophysical fluid Dynamics, Springer-Verlag, New York (1987). [16] T. Runst, W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, 3. Walter de Gruyter & Co., Berlin (1996). [17] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. (1970). [18] H. Triebel, Theory of function spaces, II. Monographs in Mathematics, 84, Birkhäuser Verlag, Basel (1992).

Yong Zhou Department of Mathematics East China Normal University Shanghai 200062 China email: [email protected] Communicated by Rafael D. Benguria submitted 17/06/04, accepted 16/12/04

Ann. Henri Poincaré 6 (2005) 801 – 820 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05801-20, Published online 05.10.2005 DOI 10.1007/s00023-005-0224-x


Existence and Stability of Even-dimensional Asymptotically de Sitter Spaces Michael T. Anderson∗

Abstract. A new proof of Friedrich’s theorem on the existence and stability of asymptotically de Sitter spaces in 3+1 dimensions is given, which extends to all even dimensions. In addition we characterize the possible limits of spaces which are globally asymptotically de Sitter, to the past and future.

1 Introduction Consider globally hyperbolic vacuum solutions (M n+1 , g) to the Einstein equations with cosmological constant Λ > 0, so that Ricg −

Rg g + Λg = 0. 2

(1.1)

The simplest solution is (pure) de Sitter space on M n+1 = R × S n , with metric gdS = −dt2 + cosh2 (t)gS n (1) .

(1.2)

More generally, let (N n , gN ) be any compact Riemannian manifold with metric gN satisfying the Einstein equation RicgN = (n − 1)gN . Then the (generalized) de Sitter metric N gdS = −dt2 + cosh2 (t)gN , (1.3) on R × N is also a solution of (1.1), with Λ = n(n − 1)/2. Let dS + be the space of all globally hyperbolic spacetimes (M n+1 , g) satisfying (1.1), with a spatially compact Cauchy surface, which are asymptotically de Sitter (dS) to the future, i.e., future conformally compact in the sense of Penrose; the terminology asymptotically simple is also used in this context. Thus there is a smooth function Ω such that the conformally compactified metric g¯ = Ω2 g,

(1.4)

¯ = M ∪ I + , where I + is a compact nextends to the compactified spacetime M ¯ , with Ω > 0, I + = manifold without boundary. The function Ω is smooth on M −1 + Ω (0) and dΩ = 0 on I . The boundary metric γ = g¯|I + depends on the choice ∗ Partially

supported by NSF Grant DMS 0305865

802

M.T. Anderson


of Ω; however the conformal class [γ] of γ is independent of Ω and is called future conformal infinity. Such spacetimes are geodesically complete to the future of an initial compact Cauchy surface Σ diffeomorphic to I + . There are no restrictions on the class [γ] or the topology of I + , and so such spacetimes are sometimes also called asymptotically locally de Sitter. Changing the time orientation gives the same notion for dS − , with future conformal infinity I + replaced by past conformal ¯ may be measured in H¨ older infinity I − . The smoothness of g¯ in (1.4) up to M m,α spaces C , but we will mostly use Sobolev spaces H s which are more natural in this context. In addition, let dS ± be the space of such globally hyperbolic spacetimes which are in both dS + and dS − ; thus such spacetimes are (completely) global, in the sense that they are geodesically complete and asymptotically simple both to the past and to the future. Mathematically, the most significant result on the structure of such spacetimes is Friedrich’s theorem [8], [9] that in 3 + 1 dimensions, the Cauchy problem with data on I + , (or I − ) is well posed, cf. also [10] for recent discussions. Thus, for arbitrary Cauchy data on I + , there is a unique spacetime (M 4 , g) which realizes this data at future infinity. Moreover, small but arbitrary variations of the Cauchy data give rise to small perturbations of the solution. It follows in particular that the space dS ± of global solutions is open; thus spaces in dS ± , in particular pure de Sitter space (M 4 , gdS ), are stable under small perturbations of the Cauchy data at I + , (or I − ). The same statement holds for perturbations of the data on a compact Cauchy surface Σ for (M 4 , g). The purpose of this paper is to extend Friedrich’s theorem to arbitrary even dimensions. Let I + be any closed n-manifold, n odd, and let γ be any H s+n smooth Riemannian metric on I + , s > n2 + 1. Next, let τ be any H s symmetric bilinear form on I + satisfying the constraints trγ τ = 0, δγ τ = 0,

(1.5)

i.e., τ is transverse-traceless with respect to γ. Define γ1 ∼ γ2 and τ1 ∼ τ2 if these data are conformally related, i.e., there exists λ : I + → R+ such that γ2 = λ2 γ1 and τ2 = f (λ)τ1 , where f is chosen so that (1.5) holds for τ2 , cf. [5] for the exact transformation formula. Let ([γ], [τ ]) be the equivalence class of (γ, τ ). Then Cauchy data for the Einstein equations (1.1) with Λ > 0 consist of triples (I + , [γ], [τ ]). The form τ corresponds to the order n behavior of the metric; roughly for g¯ as in (1.3), τ = (∂Ω )n g¯|I + ; see §2 for further details. Theorem 1.1 The Cauchy problem for the Einstein equations with Cauchy data (I + , [γ], [τ ]) at future conformal infinity is well posed in H s+n × H s , for any s > n 2 + 2. Thus, given any Cauchy data ([γ], [τ ]) ∈ H s+n (I + )×H s (I + ) satisfying (1.5), up to isometry there is a unique Einstein metric (M n+1 , g) ∈ dS + whose conformal compactification as in (1.4) induces the given data ([γ], [τ ]) on I + .

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This result has the following simple consequence: Theorem 1.2 The space dS ± is open with respect to the H s+n ×H s topology on I + , s > n2 + 2. Thus, given any dS solution (M n+1 , g0 ) ∈ dS ± , any H s+n × H s small perturbation of the Cauchy data ([γ], [τ ]) on I + (or I − ) gives rise to complete solution (M n+1 , g) ∈ dS ± globally close to (M n+1 , g0 ). In particular, the evenN in (1.3) are globally stable. dimensional pure de Sitter spaces gdS Here, globally close is taken with respect to a natural H s topology on the ¯ = M ∪ I + ∪ I − , see the proof for details. The conformal compactification M n+1 complete solution (M , g) induces H s+n × H s Cauchy data at both past and future conformal infinity I − , I + . Of course the size of the allowable perturbations in Theorem 1.2 depends on (M n+1 , g0 ). We describe briefly the main ideas in the proof of Theorem 1.1; full details are given in §2. The Einstein equations (1.1) induce a 2nd order system of equations for a compactified metric g¯ in (1.4). However, this system is degenerate at I + = {Ω = 0} and this degeneracy causes severe problems in trying to prove the well-posedness of the system. In 3 + 1 dimensions, Friedrich [8] has developed a larger and more complicated system of evolution equations, the conformal Einstein equations, for the (unphysical) metric g¯ together with other variables. This expanded system is non-degenerate and shown to be symmetric hyperbolic; then standard results on such systems lead to the well-posedness of the conformal field equations. However, it seems very unlikely that this method could succeed in higher dimensions, cf. [10], due at least in part to the special form of the Bianchi equations in 3 + 1 dimensions. The approach taken here is to replace the Einstein equation by a more complicated but conformally invariant higher-order equation for the metric alone, whose solutions include the vacuum Einstein metrics (with Λ term). In 3 + 1 dimensions, this system is the system of 4th order Bach equations, cf. (2.12) below. The Bach equations have been used in a number of contexts in connection with issues related to conformal infinity, cf. [14], [15], [16], for example. In higher even dimensions, in place of the Bach tensor, we use the ambient obstruction tensor H of Fefferman-Graham [6], which agrees with the Bach tensor in 3 + 1 dimensions; this tensor is also characterized as the stress-energy tensor of the conformal anomaly, cf. [5]. The tensor H is a symmetric bilinear form, depending on a given metric g on M n+1 and its derivatives up to order n + 1. The equation H=0 (1.6) is conformally invariant, and includes all Einstein metrics (of arbitrary signature and Λ-term). It is a system of (n + 1)st -order equations in the metric, whose n+1 leading order term in suitable coordinates is of the form 2 , where is the wave operator of the metric g. Conformal invariance implies that the system (1.6) is non-degenerate at I + = {Ω = 0}. Theorem 1.1 is then proved by showing that

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natural gauge choices for the diffeomorphism and conformal invariance of (1.6) lead again to a symmetrizable system of evolution equations. In the context of Theorem 1.2, it is of interest to understand the closure dS ± of the space dS ± , i.e., the structure of spacetimes which are limits of spacetimes in dS ± but not themselves in dS ± . A first step in this direction was taken in [2] in 3+1 dimensions, and Theorem 1.1 allows one to extend this to any even dimension. Let dS ± be the closure of dS ± with respect to the H s+n × H s topology on the Cauchy data on either I + or I − , i.e., the union of the closures with respect to data on I − and I + . Let ∂dS ± = dS ± \ dS ± be the resulting boundary consisting of limits of spaces in dS ± which are not in dS ± . Theorem 1.3 For (n + 1) even, a space in the boundary ∂dS ± of dS ± , is described by one of the following three configurations: I. A pair of solutions (M, g + ) ∈ dS + and (M, g − ) ∈ dS − , each geodesically complete and globally hyperbolic. One has I − = ∅ for (M, g + ) and I + = ∅ for (M, g − ). Both solutions (M, g + ) and (M, g − ) are “infinitely far apart”. II. A single geodesically complete and globally hyperbolic solution (M, g) ∈ dS + , either with a partial compactification at I − , or I − = ∅. III. A single geodesically complete and globally hyperbolic solution (M, g) ∈ dS − , either with a partial compactification at I + , or I + = ∅. Cases II and III have been distinguished here, but these behaviors become identical under a switch of time orientation. One of the main points here is that singularities do not form on spaces within dS ± . One does expect singularities to form “past” the boundary ∂dS ± . The most natural limits are those of type I; this behavior occurs very clearly and explicitly in the family of dS Taub-NUT metrics on R × S n , cf. [2] for further discussion. It would be very interesting to know more about the structure of dS ± ; for instance, is it compact and connected? Theorems 1.1–1.3 are proved in §2, and we close the paper with some remarks on extending these results to vacuum equations with Λ ≤ 0 and to the Einstein equations coupled to matter fields. I would like to thank the referee and Piotr Chru´sciel for very useful comments on the paper.

2 Proofs of the results Throughout the paper, we consider globally hyperbolic vacuum spacetimes (M, g) with Λ > 0 in (n + 1) dimensions. By rescaling if necessary, it is assumed that Λ is normalized to Λ = n(n − 1)/2, so that the Einstein equations read Ricg = ng.

(2.1)

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The simplest solution of (2.1) is (pure) de Sitter space on M = R×S n , with metric (1.2), or its generalization in (1.3). These de Sitter metrics gdS are geodesically complete and globally conformally compact, i.e., in dS ± . In fact, defining s ∈ 1 and letting (− π2 , π2 ) by cosh(t) = cos(s) g¯ = cos2 (s)g,

(2.2)

one has g¯dS = −ds2 + gS n (1) , which is the metric on the Einstein static spacetime in the region s ∈ [− π2 , π2 ]. ¯ = M ∪ I + ∪ I − and the loci The metric g¯dS is real analytic on the closure M π π + − I = {s = 2 } = {t = ∞}, I = {s = − 2 } = {t = −∞} represent future and past conformal infinity. The induced metric on I ± is of course the unit round metric gS n (1) on S n . The same discussion holds for (N, gN ) as in (1.3) in place of gS n (1) . Consider Einstein metrics (M n+1 , g) in dS + , so that there is a compactification (2.3) g¯ = ρ2 g as in (1.4) to future conformal infinity I + , with I + = {ρ = 0}; all of the analysis below works equally well for spaces in dS − . A compactification g¯ = ρ2 g as in (2.3) is called geodesic if ρ(x) = distg¯ + (x, I ). These are often the simplest compactifications to work with for computational purposes. Each choice of boundary metric γ ∈ [γ] on I + determines a unique geodesic defining function ρ, (and vice versa). The Gauss Lemma gives the splitting (2.4) g¯ = −dρ2 + gρ , g = ρ−2 (−dρ2 + gρ ), where gρ is a curve of metrics on I + . The asymptotic behavior of g at I + is thus determined by the behavior of gρ as ρ → 0. For example, the geodesic compactification of the de Sitter metric (1.2) with respect to the unit round metric at I + is ρ g¯dS = −dρ2 + (1 + ( )2 )2 gS n (1) , 2 for ρ ∈ [0, ∞). Now consider a Taylor series type expansion for the curve gρ on I + . This was analyzed in case of asymptotically hyperbolic or AdS metrics with Λ < 0 by Fefferman-Graham [6], and for dS metrics by Starobinsky [19] when n = 3. This idea of course has further antecedents in the Bondi-Sachs expansion and peeling properties of the Weyl tensor when Λ = 0. In any case, the FG expansion holds equally well for metrics in dS + (or dS − ) in place of asymptotically AdS metrics; in fact the two expansions are very closely related, cf. [2], [18] and further references therein. The exact form of the expansion depends on whether n is odd or even. If n is odd, then gρ ∼ g(0) + ρ2 g(2) + · · · + ρn−1 g(n−1) + ρn g(n) + ρn+1 g(n+1) + · · · , with g(0) = γ.

(2.5)

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This expansion is even in powers of ρ up to order n − 1. The coefficients g(2k) , 0 < k < n/2 are locally determined by the boundary metric γ = g(0) ; they are explicitly computable expressions in the curvature of γ and its covariant derivatives. For example for n ≥ 3, g(2) =

1 Rγ (Ricγ − γ), n−2 2(n − 1)

(2.6)

cf. also [5], [2] for formulas for g(k) for k > 2. The term g(n) is transverse-traceless, i.e., trγ g(n) = 0, δγ g(n) = 0,

(2.7)

but is otherwise undetermined by γ and the Einstein equations (2.1); thus, at least formally, it is freely specifiable. For k > n, terms g(k) occur for k both even and odd; the term g(k) depends on two boundary derivatives of g(k−2) . The main point is that all coefficients g(k) are locally computable expressions in g(0) and g(n) . Mathematically, the expansion (2.5) is formal, obtained by compactifiying the Einstein equations and taking iterated Lie derivatives of g¯ at ρ = 0. If the ¯ ), then the expansion holds up to order geodesic compactification g¯ is in C m,α (M m + α, in the sense that gρ = g(0) + ρ2 g(2) + · · · + ρm g(m) + O(ρm+α ).

(2.8)

Suppose instead n is even. Then the expansion reads gρ ∼ g(0) + ρ2 g(2) + · · · + ρn−2 g(n−2) + ρn g(n) + ρn (log ρ)H + · · ·

(2.9)

Again the terms g(2k) up to order n−2 are explicitly computable from the boundary metric γ, as is the coefficient H of the ρn (log ρ) term. The term g(n) satisfies trγ g(n) = a, δγ g(n) = b, where a and b are explicitly determined by the boundary metric γ and its derivatives, but g(n) is otherwise undetermined by γ and the Einstein equations; as before, it is formally freely specifiable. The series (2.9) is even in powers of ρ, (at all orders) and terms of the form ρ2k (log ρ)j appear at order > n. Again the coefficients g(k) and H(k) depend on two derivatives of g(k−2) and H(k−2) . Although the expressions (2.5) and (2.9) are only formal in general, Fefferman-Graham [6] showed that if the undetermined terms (g(0) , g(n) ) are analytic on the boundary I + , then the expansion (2.5) converges, (for n odd), cf. also [2]. Thus gρ is analytic in ρ for ρ small and one has a dS Einstein metric in this region given by (2.4). A similar result has recently been proved by Kichenassamy [13], (cf. also [17]), for n even; in this case the polyhomogeneous expansion (2.9) converges to gρ for ρ small.

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The term H, which appears only when n is even, has a number of important interpretations. First Fefferman-Graham [6] observed that this tensor, locally computable in terms of the boundary metric γ, is a conformal invariant of γ and is (by definition) an obstruction to the existence of a formal power series expansion of the compactified Einstein metric; in fact it is the only obstruction. The tensor H is also important in the (A)dS/CFT correspondence, in that (up a constant) it equals the stress-energy tensor (i.e., the metric variation) of the conformal anomaly of the corresponding CFT, cf. [5]. It also arises as the stress-energy or metric variation of the Q-curvature of the boundary metric γ, cf. [7]. The tensor H is transverse-traceless δH = tr H = 0,

(2.10)

= λ2−n H. and a conformal invariant of weight 2 − n, i.e., if g = λ2 g, then H Further, if g is conformal to an Einstein metric, with any value of Λ, then H = 0.

(2.11)

In addition, as observed in [6], these properties hold for metrics of any signature and so the equation (2.11) can be viewed as a conformally invariant version of the Einstein equations with an arbitrary Λ term and arbitrary signature. We are not aware of any analogue of such a tensor in odd dimensions. We will use the tensor H to study de Sitter type solutions of the Einstein equations (2.1). Although the derivation of the obstruction tensor H arises from the structure at infinity of conformally compactified odd-dimensional Einstein metrics, once it is given, one can use it to study the Einstein equations themselves in even dimensions. Thus, replacing n by n+1, a vacuum solution of the Einstein equations (M n+1 , g) with Λ > 0, (or any Λ), in even dimensions is a solution of (2.11). For (M n+1 , g) ∈ dS + , the equation (2.11), being conformally invariant, also holds for the compactified Einstein metric g¯ in (2.2); moreover it has the important advantage of being a non-degenerate system of equations in g¯. As is well-known [8], the translation of the Einstein equations for (M, g) to the compactified setting g¯ leads to a degenerate system of equations for g¯. When n = 3, so dim M = 4, up to a constant factor H is the Bach tensor B, given by R R (2.12) B = D∗ D(Ric − g) + D2 (tr(Ric − g)) + R, 6 6 where R is a term quadratic in the full curvature of g. (The specific form of R will not be of concern here.) In general, for n ≥ 3 odd, one has, again up to a constant factor, n+1 H = (D∗ D) 2 −2 [D∗ D(P ) + D2 (trP )] + L(Dn γ), (2.13) where P = P (γ) = Ricg −

Rg g, 2n

(2.14)

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cf. [7] for example. This is a system of PDE’s in the metric g, of order n + 1; L(Dn g) denotes lower-order terms involving the metric up to order n. The Bach equation B = 0 was originally developed by Bach as a conformally invariant version of the Einstein equations (with Λ = 0), and has been extensively studied in this context, cf. [14], [15], [16] for some recent work and references therein. It was also used in [3] to study regularity properties of conformally compact Riemannian Einstein metrics. While Einstein metrics, (of any signature and Λ), are solutions of (2.11), of course not all solutions of (2.11) are Einstein. In addition, for Lorentzian metrics, H is not a hyperbolic system of PDE’s in any of the usual senses; the equation (2.11) is invariant under diffeomorphisms and conformal changes of the metric, and so requires at least a choice of diffeomorphism and conformal gauge to obtain a hyperbolic system. To describe these gauge choices, suppose (M, g) ∈ dS + , so that g is an Einstein metric, satisfying (2.1), and so (2.11), which is asymptotically dS to the future. Assume that (M, g) has a geodesic compactification which is at least C n ; then (2.15) g¯ = ρ2 g = −dρ2 + gρ , and gρ has the expansion (2.8), with m = n, α = 0. One has I + = {ρ = 0} and we set γ = g(0) . By the solution to the Yamabe problem, one may assume without loss of generality that the representative γ ∈ [γ] has constant scalar curvature, i.e., Rγ = const,

(2.16)

on I + . However, closer study shows that the operator P in (2.14) is not well behaved in the coordinates adapted to (2.15), i.e., the natural geodesic coordinates (ρ, yi ), where yi are local coordinates on I + extended to coordinate functions on M to be invariant under the flow of ∇ρ. Further, with this choice of conformal ¯ of g¯. gauge, it is difficult to control the scalar curvature R It is simplest and most natural to choose a conformal gauge of constant scalar curvature, (although other choices are possible). Thus, set g = σ 2 g¯,

(2.17)

= const. In this gauge, the equation (2.13) for g where σ is chosen to make R simplifies to n+1 ∗ D) 2 −1 Ric + L(Dn H = (D g) = 0. (2.18) is not important, but it simplifies matters if one The choice of constant for R chooses = R| ¯ I + = − n(n − 2) Rγ ≡ c0 . (2.19) R n−1 The middle equality follows by taking the trace of (2.6), and combining this with the Raychaudhuri equation on g¯ and (2.28) below.

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For the diffeomorphism gauge, we choose, as usual, harmonic coordinates xα with respect to g; α = 0. x It is assumed that the Cauchy data for x0 are such that x0 is a defining function for I + near I + , and we relabel x0 = t so that the coordinates are (t, xi ), i = 1, . . . , n. As usual, Greek letters are used for spacetime indices, while Latin is used for spatial indices. Equivalently, but from a slightly different point of view, given arbitrary local coordinates xα , with x0 a defining function for the boundary, the condition that xα is harmonic with respect to g is α = ∂α gαβ + 1 gαβ gµν ∂α gµν = 0. x 2

(2.20)

In the coordinates xα , the metric g¯ in (2.15) becomes g¯ = g00 (dt2 ) + 2g0i dtdxi + gij dxi dxj ,

(2.21)

g00 = −(∂t ρ)2 , g0i = ∂t ρ∂i ρ, gij = ∂i ρ∂j ρ + (gρ )ij .

(2.22)

where Similarly, for g, one has g0i dtdxi + gij dxi dxj , g= g00 (dt2 ) + 2

(2.23)

with gαβ = σ 2 gαβ . As long as the coordinates are g-harmonic, the Ricci curvature has the form 1 µν gαβ + Qαβ ( g , ∂ g), Ricαβ = − g ∂µ ∂ν 2 ∗ D has the form Similarly at leading order, the Laplacian D g µν ∂µ ∂ν in harmonic coordinates. Thus, with these choices of gauge for the conformal and diffeomor = 0 has the rather simple form phism invariance, the equation H

( g µν ∂µ ∂ν )

n+1 2

gαβ + L(Dn g) = 0.

(2.24)

This is an N × N system of PDE’s for gαβ which is diagonal, i.e., uncoupled, at leading order, N = (n + 1)(n + 2)/2. These choices for the conformal and diffeomorphism gauges are the simplest; however, they are not necessary and other choices, for instance gauges determined by fixed gauge source functions, cf. [12], could also be used. Having discussed the equations for the metric, we have left to determine the equations for ρ in (2.15) and σ in (2.17). The fact that ρ is a geodesic defining ¯ 2 = −1, implies that function for g¯, i.e., |∇ρ| g ¯ ∂t (g αβ ∂α ρ∂β ρ) = 0, or equivalently, g αβ ∂α ρ∂β ρ) = 0. ∂t (σ 2

(2.25)

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To derive the equation for σ, the equation for the Ricci curvature relating g and g¯ is Ric = Ric + (n − 1)

2σ σ D +{ − n|d log σ|2 } g. σ σ

(2.26)

Taking the trace gives the equation relating the scalar curvatures as ¯=R + 2n σ − n(n + 1)|d log σ|2 . σ −2 R g σ

(2.27)

Using the formula analogous to (2.26) relating the Ricci curvature of g and g¯, together with the fact that g satisfies (2.1) and ρ is a geodesic defining function gives ¯ ¯ = −2n ρ = −2nRic(T, T ), R (2.28) ρ where T = ∂ρ = −∇g¯ ρ. (Observe that the middle term in (2.28) is degenerate at I + , since ρ = 0 there; however, the last term in (2.28) is non-degenerate at ρ = 0). Substituting (2.28) in (2.27) and using (2.26) gives then the equation 2σ D + n(n − 1)|d log σ|2 = −2nσ −2 [Ric(T, R (T, T )], T ) + (n − 1) g σ or equivalently, ∇ T T = − T T (σ) − ∇σ,

σ3 1 1 σ Ric(T, T) − c0 − σ|d log σ|2g , (2.29) n−1 2n(n − 1) 2

where we have also used (2.19). The equations (2.24), (2.25), and (2.29) represent a coupled system of evolution equations for the variables ( gαβ , ρ, σ) on a domain U in (Rn+1 )+ with coordinates (t, xi ); the boundary ∂0 U = U ∩ {t = 0} corresponds to a portion of I + . Written out in more detail, these are: ( g µν ∂µ ∂ν )

n+1 2

gαβ = L1 (Dn g)αβ ,

i ρ)(∂i ∂t ρ) = L2 (Dρ, Dσ, D g ), g00 ∂t ∂t ρ + 2(∇ 0 ρ)2 ∂t ∂t σ + 2(∇ 0 ρ)(∇ i ρ)∂i ∂t σ + (∇ i ρ)(∇ j ρ)∂i ∂j σ = L3 (Dσ, D2 g). (∇

(2.30) (2.31) (2.32)

α ρ denotes the Here Dk w denotes derivatives up to order k in the variable w and ∇ αβ α-component of ∇ρ, ∇ρ = g ∂β ρ∂α . The terms Li are lower-order terms. Observe that the system (2.30) for the metric gαβ is a closed sub-system, i.e., it does not involve ρ or σ. Moreover, although the equations (2.31) and (2.32) for ρ and σ are coupled to each other and to (2.30), the system (2.30)–(2.32) is uncoupled at leading order.

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Following common practice, we now reduce the system (2.30)–(2.32) to a system of 1st order equations. There is not a unique way to do this, but we will discuss perhaps the simplest method, which uses pseudodifferential operators. As usual, the domain ∂0 U ⊂ Rn is viewed as a domain in the n-torus T n and the variables ( gαβ , σ, ρ) are extended to functions on I × T n . Recall that a system of 1st order evolution equations ∂t u =

m

Bj (t, x, u)∂j u + c(t, x, u)

(2.33)

j=1

is symmetrizable in the sense of Lax, cf. [20], [21], if there is a smooth matrix valued function R(t, u, x, ξ) on R × Rp × T ∗ (T n ) \ 0, homogeneous of degree 0 in ξ, such that R is a positive definite p × p matrix with R(t, u, x, ξ) Bj (t, u, x)ξj self-adjoint, for each (t, u, x, ξ). It is well known [20], [21] that strictly hyperbolic systems of PDE, diagonal at leading order, are symmetrizable. A symmetrizer R is given by R= Pk Pk∗ , where Pk is the projection onto the k th eigenspace of the symbol Bj (t, u, x)ξj , 1 ≤ k ≤ p. Proposition 2.1 There is a reduction of the system (2.30)–(2.32) to a symmetrizable system of 1st order evolution equations on I × T n . Proof. Consider first the closed system (2.30) for g. This system is not strictly hyperbolic; the leading order symbol is diagonal and has two distinct real eigenvalues, each of multiplicity (n + 1)/2. However, the eigenspaces of the symbol of n+1 n21 is strongly hy2 vary smoothly and do not coalesce. Thus the operator perbolic, cf. [12] and references therein. In these circumstances, it is essentially k is symmetrizable, for any k; for completeness we standard that the operator sketch the proof following [20, §5.3]. Let u = gαβ be the variable in RN , N = (n + 1)(n + 2)/2. Write , (we drop the tilde here and below), in the form (g 00 )−1 = ∂t2 −

1

Aj ( u, Dx )∂tj ,

j=0

where Aj is a differential operator in x, homogeneous of order 2 − j, depending smoothly u . Then 00 −1

[(g )

]

(n+1)/2

=

∂tn+1

−

n

Bj ( u, Dx )∂tj ,

(2.34)

j=0

where Bj are differential operators in x, homogeneous of order n + 1 − j. Set , for j = 0, . . . , n, where Λ = (1 − ∆)1/2 and ∆ is the standard uj = ∂tj Λn−j u Laplacian on T n . Then (2.30) becomes ∂t uj = Λuj+1 , 0 ≤ j < n, ∂t un =

n j=0

Bj (P u, Dx )Λj−n uj + C(P u),

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where P u = Dn u involves u up to n derivatives. More precisely, for β + j ≤ n, ∂xβ ∂tj u = ∂xβ Λj−n uj ; for example Λ−n u0 = u . This is a system of 1st order pseudodifferential equations in the variables u = {uj }, j = 0, . . . , n of the form ∂t u = L(P u, Dx)u + C(P u).

(2.35)

The eigenvalues λν (w, ξ) of the matrix L(w, ξ) are the roots of the characteristic equation τ n+1 − Bj (w, ξ)τ j , (up to an overall factor of i). Hence, from (2.34), one sees that for each (w, ξ), ξ = 0, there are two distinct roots, each of multiplicity (n+ 1)/2. The eigenvalues vary smoothly with (w, ξ) and remain a bounded distance apart on the sphere |ξ| = 1. The same is true of the corresponding eigenspaces. Hence, the system (2.35) has a symmetrizer R constructed in the same way as following (2.33), cf. [20, Prop. 5.2.C] or [21, Prop. 16.2.2]. Next we show that the equation (2.31) is also symmetrizable. Let φi = i ρ/ g 00 . Introducing the vector variable v = (ρ, ρ0 , . . . , ρn ) with ρ0 = ∂t ρ, ρj = −2∇ ∂j ρ, the equation is equivalent to the system ∂t ρ = ρ0 , ∂t ρ0 = φj ∂j ρ0 + c(t, x, v), ∂t ρj = ∂j ρ0 . This has the form ∂t v =

n

Bj (x, t, v)∂j v + c(x, t, v),

(2.36)

j=1

where Bj is an (n + 2) × (n + 2) matrix with φj in the (2, 2) slot, 1 in the (j + 2, 2) slot, and 0 elsewhere. The system (2.36) is coupled at lower order to the equations (2.30) and (2.32) for g and σ respectively, in that Bj depends on g to order 0, while c depends on g and σ to order 1; for the moment, these dependencies are placed in the (x, t) dependence of Bj and c. The matrix Bj ξj has the entry φj ξj in the (2, 2) slot, ξj in the (j + 2, 2) slot for 3 ≤ j ≤ n, and 0 elsewhere. By a direct but uninteresting computation, it is straightforward to see that this matrix is symmetrizable in the sense following (2.33). Essentially the same argument shows that the equation (2.32) for σ is again symmetrizable. Thus let w = (σ, σ0 , . . . , σn ) with σ0 = ∂t σ, σj = ∂j σ. The equation (2.32) is equivalent to the system ∂t σ = σ0 , ∂t σ0 = φj ∂j σ0 + ψij ∂j σi + c(t, x, v), ∂t ρj = ∂j ρ0 , i ρ)/|∇ 0 ρ|, ψij = (∇ i ρ)(∇ j ρ)/|∇ 0 ρ|2 . This system has the form where φj = −2(∇ ∂t w =

n

Bj (x, t, w)∂j w + c(x, t, w),

(2.37)

j=1

where Bj is the (n + 2) × (n + 2) matrix with φj in the (2, 2) slot, 1 in the (j + 2, 2) slot, and ψij in the (i + 2, j + 2) slot. The system (2.37) is again coupled at

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lower order to the equations (2.30) and (2.31) for g and ρ respectively, in that c depends on g to order 2, while Bj depends on g to order 0 and ρ to order 1. Again a straightforward but longer (uninteresting) computation shows that the matrix Bj ξj is symmetrizable in the sense of (2.33). One may then combine the three systems (2.35), (2.36), and (2.37) to a single large system in the variable U = (u, v, w). The resulting system is then a symmetrizable system of 1st order pseudodifferential equations, cf. [20], [22]. Next consider the Cauchy data for the system (2.30)–(2.32). If one is interested in general solutions of this system, then the Cauchy data are essentially ¯ ·) = 0 on I + . However, arbitrary, subject only to the constraint equation H(∇ρ, as will be seen in Proposition 2.3 below, it is the specification of the Cauchy data which determines the class of conformally Einstein metrics among all solutions of (2.30)–(2.32). This is of course closely related to the FG expansion (2.8) of the dS metric g. The Cauchy data for σ are σ = 1, and ∂t σ = 0 at I + ,

(2.38)

while the Cauchy data for ρ are ρ = 0, and ∂t ρ = 1 at I + .

(2.39)

For the metric g, the closed subsystem (2.30) is of order n + 1, so Cauchy data are specified by prescribing (∂t )k gαβ , k = 0, 1, . . . , n at I + . We compute the data inductively. First, the condition (2.38) implies that gij = g¯ij at I + . Thus at order 0, set (2.40) g00 = −1, g0i = 0, gij = γij at I + , since ρi = 0 at I + . At 1st order, (2.38) and (2.39) together with (2.23) show that ∂t gij = ∂t g¯ij + at I , and the FG expansion (2.8) gives ∂t g¯ij = 0 at I + . Thus, set ∂t gij = 0 at I + .

(2.41)

(This condition, and related ones below, are necessary to obtain Einstein metrics.) The first derivatives of the mixed components g0α of g are determined by the requirement that the coordinates xα = (t, xi ) are harmonic at I + with respect to g, i.e., for each β, 1 ∂α gαβ + gαβ g µν ∂α gµν = 0. (2.42) 2 Via (2.40)–(2.41), this determines ∂t g0α at I + . nd At 2 order, the equation (2.27) implies, using the normalization (2.19), that ∂t2 σ = 0 at I + . Also, ∂t2 (ρi ρj ) = 0, and hence, from the FG expansion (2.8), we set ∂t2 gij = 2g(2) at I + . (2.43)

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The 2nd derivatives ∂t g0α are then determined by (2.43), the lower-order Cauchy data, (2.38)–(2.41), and the t-derivative of (2.42) at t = 0. At 3rd order, suppose first n = 3, so that dim M = 4. Then a straightforward computation, using (2.8), the Raychaudhuri equation on g¯ and (2.28), shows that ¯ = 6n tr g(3) = 0 at I + , where the last equality follows from (2.7). Hence, ∂t R (2.27) gives ∂t3 σ = 0. Similarly, (2.39) gives ∂t3 (ρi ρj ) = 0. Thus, set ∂t3 gij = 6g(3) at I + .

(2.44)

This term is free or unconstrained, subject to the transverse-traceless constraint (2.7). As before the mixed term at order 3, ∂t3 g0α is determined by taking two t-derivatives of (2.42) at t = 0, and using (2.44) together with the determination of the lower-order Cauchy data. Suppose instead n > 3 and hence n ≥ 5. Then g(3) = 0 and same arguments as above give (2.45) ∂t3 gij = 0 at I + , with ∂t3 g0α again determined from two t-derivatives of (2.42), (2.45) and lowerorder Cauchy data. At 4th order on I + , (assuming n ≥ 5), by (2.27) and the fact that ∂tk σ = 0, ¯ and again computations as above then k ≤ 3 on I + , one has ∂t4 σ = −2n∂t2 R, ¯ = 24n tr g(4) . Also, taking i-derivatives of (2.31) or (2.25) and using give ∂t2 R (2.38)–(2.39) shows that ∂t4 (ρi ρj ) = 0 at I + . It follows that, at t = 0, ∂t4 gij = 24(g(4) − 2n2 tr g(4) )γ.

(2.46)

Again, ∂t4 g0α is determined by taking three t-derivatives of (2.42) at t = 0, and using (2.46) with the determination of the lower-order Cauchy data. ¯ = c tr g(5) = 0 at I + At 5th order, suppose n = 5. As in the case n = 3, ∂t3 R 5 while ∂t (ρi ρj ) = 0. Hence, as in the case n = 3, ∂t5 gij = (5!)g(5) ,

(2.47)

which is freely specifiable, subject to the transverse-traceless constraint. As before the mixed term at order 5, ∂t5 g0α is determined by taking four t-derivatives of (2.42) at t = 0, and using (2.47) with the determination of the lower-order Cauchy data. If n > 5, then as before, gij = 0, (2.48) ∂t5 with ∂t5 g0α again determined from (2.42). It is clear that one can continue ingαβ up to order n. Since ductively in this way to determine the Cauchy data ∂tk ∂t6 (ρi ρj ) = 0 at t = 0, these and higher derivative terms contribute to the Cauchy data at order 6 and above. However, one sees by differentiations of (2.31) and (2.42) that these terms are all determined by lower-order Cauchy data for g. gαβ , 0 ≤ k ≤ n, are determined In sum, Cauchy data for gαβ , i.e., the data ∂tk by the Cauchy data (2.38), (2.39) for ρ and σ, the equations (2.31)–(2.32), (or

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(2.25), (2.27)) for ρ and σ, the harmonic equation (2.42), and the coefficients g(k) in the FG expansion (2.8). Thus, the Cauchy data are uniquely determined in terms of the free data (2.49) γ = g(0) , and τ = g(n) , which are arbitrary, subject to the transverse-traceless constraint (2.7) on g(n) and the constant scalar curvature constraint (2.16) or (2.19) on the representative γ ∈ [γ]. Abusing notation slightly, we will call (γ, τ ) Cauchy data for gαβ , since this data determines the rest of the Cauchy data ∂tk gαβ , 0 < k < n uniquely. The analysis above then gives: Proposition 2.2 The system (2.30)–(2.32) for ( gαβ , ρ, σ) is well posed in H s+n (I + ) n s + × H (I ), s > 2 + 2. Thus, given Cauchy data (γ, τ ) ∈ (H s+n (I + ), H s (I + )) satisfying (2.38), (2.39) and (2.49), and satisfying the constraints (2.7), (2.19), there is a unique solution ( gαβ , ρ, σ) of (2.30)–(2.32) with ( gαβ , ρ, σ) ∈ C(I, H s+n (I + )) ∩ C n (I, H s (I + )).

(2.50)

Further, if (γs , τs ) is a continuous curve in H s+n (I + )×H s (I + ), then the solutions ( gs , ρs , σs ) vary continuously with s. Proof. In the local coordinates (t, xi ), the system (2.30)–(2.32) is symmetrizable and so for given Cauchy data on T n , it has a unique solution on I × T n satisfying (2.50), with T n in place of I + , cf. [20, §5.2–5.3] or also [21], [22]. The existence of such local solutions holds for s > n2 + 1. By restriction, one thus obtains local solutions on the domain I × U ⊂ I × T n , for U ⊂ I + as preceding (2.30). To prove these local solutions obtained from domains on I×I + patch together to give a unique solution on I × I + , it is necessary and sufficient to prove that the system (2.30)–(2.32) has finite domains of dependence (or equivalently uniqueness in the local Cauchy problem). This is well known to be true for symmetric hyperbolic systems of PDE’s, (cf. [22, §IV.4] for example); however, the reduction of (2.30) is to a symmetric system of first-order pseudodifferential equations, for which finite propagation speed is not true in general. Nevertheless, standard methods show that solutions of (2.30)–(2.32) do have local uniqueness in the Cauchy problem. Thus, consider the closed subsystem (2.30) for g. A standard argument using the mean-value theorem shows that it suffices to prove local uniqueness in the Cauchy problem for the associated linear equation L( g ) = 0, where the coefficients of (2.30) are frozen at a given metric g0 , cf. [12] for instance. The linear operator L is self-adjoint at leading order, and hence by Proposition 2.1, one has existence and uniqueness of solutions to the Cauchy problem on I × T n for the adjoint equation L∗ u = φ. It is then well-known, cf. [22, Thm.IV.4.3], that this suffices to prove local uniqueness in the Cauchy problem for L, and hence for the system (2.30). Given the local uniqueness for g, exactly the same method can be applied to (2.31)–(2.32) to give local uniqueness for ρ and σ.

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It follows that the system (2.30)–(2.32) does have finite propagation speed, and hence the local solutions patch together uniquely to give a unique global solution on I × I + satisfying (2.50). As is well known in the case of Einstein metrics, (cf. [12] for example), the patching of local coordinate charts requires an extra derivative; the same analysis holds here, so that we assume s > n2 + 2. A lower bound for the time of existence I = [0, t0 ) of the solution depends only on an upper bound on the norm of the initial data in H s+n × H s . This implies that the last statement is an immediate consequence of the existence and uniqueness theorem. Given a solution ( gαβ , ρ, σ) of the Cauchy problem, one may then construct the “physical” metric g by setting g = ρ−2 σ −2 g.

(2.51)

Since σ is bounded away from 0 and ∞ near I + and ρ is a geodesic defining function, it is easy to see that the metric g is future geodesically complete, i.e., geodesically complete to the future of some Cauchy surface Σ ⊂ (M n+1 , g). However, it is not so clear that the metric g is Einstein, or equivalently that the metric g is conformally Einstein. For this, one needs to verify first that the gauge condition (2.42) that the coordinates remain harmonic, is preserved for the solution g. If this is so, it then follows that g is a solution of the equation (2.11). Secondly, the equation (2.11) admits solutions which are not conformally Einstein, and so one needs to verify that the constructed solution g actually is conformally Einstein. Both of these conditions on g can be verified by computation; however, the computations will be somewhat long and involved. Instead, we verify both conditions together by using the following simple conceptual technique, based on analyticity. Proposition 2.3 Any solution ( gαβ , ρ, σ) of the Cauchy problem in Proposition 2.2 defines an Einstein metric in dS + via (2.51). Proof. Suppose first that the free data (γ, g(n) ) are analytic on I + (or on a domain in I + ). As noted above, the expansion (2.5) then converges to gρ and gives a solution, denoted gE , to the Einstein equations (1.1); this metric has the form (2.4), with compactification g¯E , in a neighborhood of I + . In particular, both g¯E and ρ are analytic near I + . Moreover, since the coefficients of the equation (2.27) defining σ are then also analytic, and since the Cauchy data (2.38) for σ are analytic, the Cauchy-Kowalewsky theorem ([21, §16.4]) shows that σ and hence gE is a solution of (2.11), and gE = σ 2 g¯E are analytic near I + . Of course the metric hence a solution of (2.30)–(2.32). with Cauchy data determined by (2.38)–(2.39) and (γ, g(n) ). On the other hand, let ( g , ρ, σ) be the solution to the Cauchy problem (2.30)– (2.32) with Cauchy data determined by (2.38)–(2.39) and (γ, g(n) ) given by Proposition 2.2. Since ( gE , ρ, σ) is also a solution of this Cauchy problem, with the same

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initial data, it follows from the uniqueness part of Proposition 2.2 that gE = g in a neighborhood of I + . Moreover, the results in [1] show that gE remains analytic within its globally hyperbolic development, so that gE = g everywhere in the domain of g. Hence g = ρ−2 σ −2 g is an Einstein metric in dS + , realizing the given data on I + . (In particular, this shows that the harmonic gauge (2.42) must be preserved for analytic initial data.) Now analytic data (γ, g(n) ) are dense in H s+n × H s data with respect to the s+n × H s topology on I + . The Cauchy stability given by Proposition 2.2 implies H that if (γi , g(n),i ) is a sequence of analytic Cauchy data converging in H s+n ×H s to gi , ρi , σi ) also converge H s+n ×H s data (γ, g(n) ), then the corresponding solutions ( to the (unique) solution ( g , ρ, σ) with Cauchy data (γ, g(n) ). Hence the metric g in (2.51) is Einstein. Propositions 2.2–2.3 give the existence of an Einstein metric (M n+1 , g) ∈ dS + , with arbitrarily prescribed asymptotic behavior (γ, τ ) on I + , subject to the constraints (2.7) and (2.19). Suppose [γ ] = [γ] and [τ ] = [τ ] as following (1.5), so ¯ →M ¯, that γ = λ2 γ and τ = f (λ)τ . Given λ, there exists a diffeomorphism φ : M with φ|I + = id, such that limρ→0 φ∗ (ρ)/ρ = λ−1 , where ρ is the geodesic defining function determined by γ. Setting g = φ∗ g, one has g¯ = ρ2 φ∗ (g) = φ∗ (¯ g )( φ∗ρ(ρ) )2 . Hence, the boundary metric of g¯ equals γ . Similarly, by the uniqueness, any Einstein metric (M n+1 , g ) with Cauchy data (γ , τ ) satisfying (1.5) differs from (M n+1 , g) by a diffeomorphism φ equal to the identity on I + . This completes the proof of Theorem 1.1. Remark 2.4 Theorem 1.1 is formulated as a global result, in the sense that future conformal infinity I + is a compact smooth manifold. However, Propositions 2.12.3 all hold locally, by the finite propagation speed of the system (2.30)–(2.32). Hence, Theorem 1.1 also holds locally, where I + is an open manifold with a finite number of local charts. Of course the uniqueness statement then holds only within the domain of dependence of the initial data. Many of the standard solutions of the Einstein equations (2.1) have I noncompact; this is the case for instance for the dS Schwarzschild metrics. Proof of Theorem 1.2. Let (M, g0 ) be a dS Einstein metric in dS ± with Cauchy + − data ([γ + ], [g(n) ]) and ([γ − ], [g(n) ]) induced on I + and I − respectively. Thus, there ¯ → R such that g¯0 = Ω2 g0 extends to a exists a smooth defining function Ω : M + − ¯ metric on M = M ∪ I ∪ I I × Σ; here Σ is a Cauchy surface for (M, g0 ) and I is a compact time interval. We will choose Ω to be a geodesic defining function ρ in a neighborhood of I + and I − so that g¯0 is C(I × H s+n (Σ)) ∩ C n (I × H s (Σ)) up to ¯ . The choice of Ω defines representatives (γ + , g + ), (γ − , g − ) in the conformal M (n) (n) + − classes ([γ + ], [g(n) ]), ([γ − ], [g(n) ]). In the following, we work in the H s+n × H s topology. Let U + be an open neighborhood of Cauchy data on I + containing the given + + ). Then for all data (ˆ γ + , gˆ(n) ) ∈ U + , there exists T < ∞, depending data (γ + , g(n)

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on U + , such that the maximal globally hyperbolic dS Einstein metric (M n+1 , gˆ) + having Cauchy data (ˆ γ + , gˆ(n) ) given by Theorem 1.1 is defined on [T, ∞) × Σ; here the time factor is proper time t = − log ρ2 , where ρ is the geodesic defining function. The Cauchy data of such solutions gˆ at Σ = {T } × Σ then forms an open set U T in the space of Cauchy data for the Einstein equations (2.1) on Σ. By passing to an open subset V T ⊂ U T if necessary, the Cauchy stability theorem for the (standard) Einstein equations implies that the maximal globally hyperbolic development of any gˆ with data in V T contains the region [−T, T ]× Σ, and induces again an open set of Cauchy data V −T at {−T } × Σ. Then, as above with I + , there is an open set U − of Cauchy data on I − whose future development gives a non-empty open subset of V −T . Combining these three unique developments gives a global solution (M n+1 , gˆ) ∈ dS ± , which completes the proof of Theorem 1.2. Remark 2.5 The proof above shows that the space dS ± is also stable with respect to perturbations of the Cauchy data (Σ, γ, K), (satisfying the constraint equations), on a compact Cauchy surface Σ ⊂ (M n+1 , g) in the H s+n × H s+n−1 topology on Σ. It is well known, cf. [4] that this is not the case for perturbations of asymptotically flat Cauchy data when Λ = 0, in that smoothness of the resulting space-time at conformal infinity is lost for generic perturbations. Proof of Theorem 1.3. This result is proved for n = 3 in [2], and it is pointed out there that the same proof holds provided one has the Cauchy stability result of Theorem 1.1, (i.e., Friedrich’s result [8] when n = 3). Given then Theorem 1.1, the proof of Theorem 1.3 is exactly the same as that given in [2], to which we refer for details. Remark 2.6 This paper has focused on the de Sitter case Λ > 0 mainly for simplicity, but also because there are no direct analogues of Theorems 1.2 or 1.3 when Λ = 0 or Λ < 0, due to the more complicated nature of conformal infinity. Nevertheless, one expects that the analogues of Theorem 1.1 for Λ = 0 and Λ < 0, as formulated and proved by Friedrich [8] in the case n = 3, hold for all even dimensions. When Λ < 0, future space-like infinity I + is replaced by time-like infinity I, while when Λ = 0, I + is replaced by future null infinity. The case Λ = 0 has recently been worked out with P. Chrusciel, cf. [arXiv: gr-qc/0412020], (to appear in Comm. Math. Phys.). Remark 2.7 We close with a brief remark on the applicability of the methods used above to the Einstein equations coupled to other matter fields. As noted above, up to multiplicative constants, the tensor H is the metric variation, or stressenergy tensor, of the conformal anomaly or of the Q-curvature. The conformal invariance of these functionals corresponds to the conformal invariance of H. For functionals containing the metric coupled to other fields which are conformally invariant, and whose field equations are symmetric hyperbolic, it seems very likely that the methods used above will again lead to a well-posed Cauchy problem at I + , as in Theorem 1.1. In dimensions 3 + 1, this is the case for the Einstein

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equations coupled to gauge fields, i.e., Einstein-Maxwell or Einstein-Yang-Mills fields. Theorem 1.1 has already been proved in this situation by Friedrich [11], and so at best one would have a different method of proof of this result. In higher dimensions, the EM or YM action is not conformally invariant, and it is less clear if the method can be adapted to this situation.

References [1] S. Ahlinhac and G. Métivier, Propagation de l’analyticité des solutions de systèmes hyperboliques non-linéares, Inventiones Math. 75, 189–204 (1984). [2] M. Anderson, On the structure of asymptotically de Sitter and anti-de Sitter spaces, (preprint), [arXiv: hep-th/0407087], to appear in Adv. Theor. Math. Phys. [3] M. Anderson, Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds, Advances in Math. 179, 205–249 (2003). [4] L. Andersson and P.T. Chru´sciel, On “hyperboloidal” Cauchy data for vacuum Einstein equations and obstructions to the smoothness of Scri, Comm. Math. Phys. 161, 533–568 (1994). [5] S. deHaro, K. Skenderis and S.N. Solodukhin, Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence, Comm. Math. Phys. 217, 595-622 (2001), [arXiv: hep-th/0002230]. [6] C. Fefferman and C.R. Graham, Conformal invariants, in Elie Cartan et les Mathématiques d’Aujourd’hui, Astérisque, (1985), numero hors série, Soc. Math. France, Paris, 95–116. [7] C.R. Graham and K. Hirachi, The ambient obstruction tensor and Qcurvature, (preprint), [arXiv: math.DG/0405068]. [8] H. Friedrich, On the existence of n-geodesically complete or future complete solutions of Einstein’s equations with smooth asymptotic structure, Comm. Math. Phys. 107, 587–609 (1986). [9] H. Friedrich, Existence and structure of past asymptotically simple solutions of Einstein’s field equations with postive cosmological constant, J. Geom. Phys. 3, 101–117 (1986). [10] H. Friedrich, Conformal Einstein evolution, in The Conformal Structure of Space-Time, J. Frauendiener and H. Friedrich, Eds., Lecture Notes in Physics, vol. 604, Springer Verlag, Berlin, 1–50 (2002). [11] H. Friedrich, On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations, Jour. Diff. Geom. 34, 275–345 (1991).

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[12] H. Friedrich and A. Rendall, The Cauchy problem for the Einstein equations, in Einstein’s Field Equations and Their Physical Implications, B.G. Schmidt (Ed.), Springer Lecture Notes in Physics, vol. 540, Springer Verlag, Berlin, 127–223 (2000). [13] C.N. Kozameh and E.T. Newman, A new approach to the vacuum Einstein equations, in Asymptoic Behavior of Mass and Space-Time Geometry, Lecture Notes in Physics, Vol. 202, F.J. Flaherty, (Ed.), Springer Verlag, New York, (1984). [14] S. Kichenassamy, On a conjecture of Fefferman-Graham, Advances in Math. 184, 268–288 (2004). [15] C.N. Kozameh, E.T. Newman and K.P. Tod, Conformal Einstein spaces, Gen. Relativ. and Gravit. 17, 343–352 (1985). [16] L.J. Mason, The vacuum and Bach equations in terms of light cone cuts, Jour. Math. Phys. 36(7), 3704–3721 (1995). [17] A. Rendall, Asymptotics of solutions of the Einstein equations with positive cosmological constant, Annales Henri Poincaré 5, 1041–1064 (2004), [arXiv.org: gr-qc/0312020]. [18] K. Skenderis, Lectures notes on holographic renormalization, Class. Quantum Grav. 19, 5849–5876 (2002), [arXiv: hep-th/0209067]. [19] A. Starobinsky, Isotropization of arbitrary cosmological expansion given an effective cosmological constant, JETP Lett 37, 66–69 (1983). [20] M.E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics Series, vol. 100, Birkh¨ auser Verlag, Boston, (1991). [21] M.E. Taylor, Partial Differential Equations III, Applied Math. Sciences Series, vol. 117, Springer Verlag, New York, (1996). [22] M.E. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton, (1981). Michael T. Anderson Department of Mathematics S.U.N.Y. at Stony Brook Stony Brook, NY 11794-3651 USA email: [email protected] Communicated by Sergiu Klainerman submitted 30/08/04, accepted 27/01/05

Ann. Henri Poincaré 6 (2005) 821 – 847 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05821-27, Published online 05.10.2005 DOI 10.1007/s00023-005-0225-9


On Initial Conditions and Global Existence for Accelerating Cosmologies from String Theory Makoto Narita Abstract. We construct a solution satisfying initial conditions for accelerating cosmologies from string/M-theory. Gowdy symmetric spacetimes with a positive potential are considered. Also, a global existence theorem for the spacetimes is shown.

1 Introduction It is expected that the inflation paradigm would be explained within superstring/ M-theory. The theory predicts that spacetime dimension is greater than four. Since observable spacetime dimension is four, it is thought that the extra dimensions would be compactified within Planck scale. Recently, it has been pointed out that it is possible to find cosmological solutions which exhibit a transient phase of accelerated expansion of the universe (like inflation) if the size of the compactified internal hyperbolic space depends on time and/or if they are S(pacelike)-brane solutions. In these models, exponential potential terms like V0 eaψ appear, where ψ denotes the compactification volume or effective dilaton field, a is a coupling constant and V0 is positive number. Explicitly, a typical action for the case is of the form √ 1 S = d4 x −g − 4R + (∇ψ)2 + V0 eaψ . (1) 2 Then, it is explained that if it would be supposed that, in the case of a > 0, the field ψ starts at a large negative value (i.e., the potential term can be neglected) with high kinetic energy (∂t ψ is positive and large enough)1 near cosmological initial singularities, then, the scalar field runs up the exponential potential, turn around and falls back. At the turning point, the potential term becomes dominant, i.e., the universe makes accelerated expansion. Thus, the universe starts out in a decelerated expansion phase (asymptotic past) and enters an accelerating phase (intermediate era), and after these, the expansion becomes deceleration again (asymptotic future). We call this scenario paradigm-A. We would like to investigate this interesting paradigm from viewpoint of mathematical relativity and cosmology. It is important to study rigorously whether or not the paradigm-A is acceptable. In particular, it should be shown that the assumption of the initial conditions for ψ is generic because, as indicated previously [EG], the accelerated expansion of the universe is all the result of the initial 1 In

the case of a < 0, ψ and ∂t ψ start at large positive and negative values, respectively.

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conditions. That is, (Q1): Are there singular solutions satisfying initial conditions in paradigm-A to the Einstein-matter equations in generic? Furthermore, to be complete the scenario of paradigm-A, we should show global existence theorems, i.e., (Q2): Are there global solutions to the Einsteinmatter equations with such exponential potentials? Unlike BKL [BKL] or cosmic no-hair conjectures [WR], which are problems in only asymptotic (local) regions of spacetimes, the paradigm-A is a global (in time) problem as mentioned already. In addition, it is also important as the first step to prove the strong cosmic censorship. For (Q1), to construct solutions satisfying the initial condition of paradigm-A, we will use the Fuchsian algorithm developed by Kichenassamy and Rendall [KR]. It is interpreted that the class of solutions we are looking for here is a subclass of asymptotically velocity-terms dominated (AVTD) singular solutions since potential terms are neglected near the singularities and, in addition, signature of the time derivative of the scalar field is restricted. By using the method, it has been shown that there are AVTD singularities in (non-)vacuum Gowdy, polarized T 2 -, polarized U (1)-symmetric spacetimes and the Einstein-scalar-p-from system without symmetry assumptions [AR, DHRW, IK, IM, NTM]. Also, systems with an exponential potential as given in (1) have been discussed formally in [DHRW, RA00]. Thus, our result is not only an answer for (Q1), but also it complements previous results. For (Q2), we want to analyze Gowdy symmetric spacetimes. Future global existence theorems for spatially compact, locally homogeneous spacetimes [LH03, LH04, RA04] and hyperbolic symmetric spacetimes [TR] with a positive potential (or a positive cosmological constant) have been proved. These spacetimes do not include gravitational waves. Also, although global existence theorems for Gowdy (more generally, T 2 -) symmetric spacetimes with or without matter have been shown [AH, ARW, BCIM, IW, MV, NM02, NM03, WM], it has not been prove the theorems for the spacetimes with a positive potential. Therefore, spacetimes with dynamical degrees of freedom of gravity and with the positive potential should be considered as the next step. As a model, we choose the bosonic action arising in low energy effective superstring (supergravity) theory since we have a similar action with (1) after the toroidal compactification of the extra dimensions. There are anti-symmetric two-form, Bµν , and three-form, Cµνρ fields in the action. It is known that, in general, p-form fields in n-dimensional spacetimes may violate the strong energy condition for p ≥ n − 1 and then, accelerated expansion of the universe would be expected [GG]. Here, we do not consider hyperbolic compactification of the extra higher dimensions, but the only fluxes of four-form field strengths are investigated because these have essentially the same effects to obtain the exponential potential terms as (1) [EG, TP, WMNR]. Then, our purposes are to construct singular solutions satisfying conditions of paradigm-A and to show a global existence theorem for Gowdy symmetric spacetimes with stringy matter fields.

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Global Properties for Accelerating Cosmologies

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Action

The dimensionally reduced effective action in the Einstein frame is given by 1 1 −2λφ 2 1 −2λφ 2 4 √ 4 2 e e H + F , SIIA = d x −g − R + (∇φ) + 2 2 · 3! 2 · 4!

(2)

where g is the determinant of the metric gµν on a four-dimensional spacetime manifold M , 4R is the Ricci scalar of gµν , φ is the dilaton field, H = dB is the three-form field strength, F = dC is the four-form field strength and λ is a coupling constant. If λ = 1, we have the action for the type IIA supergravity in the absence of vector fields and the Chern-Simons term [LWC]. In four dimensions, there is a duality between the three-form field strength and a one-form, which is interpreted as the gradient of a scalar field. Then, we may define the pseudo-scalar axion field σ as follows: H µνρ = µνρκ e2λφ ∇κ σ.

(3)

∇µ e−2λφ F µνρκ = 0,

(4)

∂[α Fµνρκ] = 0,

(5)

Also, the field equation

and the Bianchi identity

for the four-form field strength can be solved by F µνρκ = Qµνρκ e2λφ ,

(6)

where Q is an arbitrary constant. Thus, after taking the dual transformation and solving the field equations for F , we have a reduced effective action for the IIA system of the form 1 4 √ 4 2 2λφ 2 2 2λφ (∇φ) + e (∇σ) + Q e . (7) SIIA∗ = d x −g − R + 2 Hereafter, we assume Q = 0. Thus, we have the action which is the same from with (1).

1.2

Field equations for Gowdy symmetric spacetimes

The Gowdy symmetric spacetimes admit a T 2 isometry group with spacelike orbits and the twists associated to the group vanish [GR]. The topology of spatial section can be accepted S 3 , S 2 × S 1 , T 3 or the lens space [CP]. In this paper, we assume T 3 spacelike topology.

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Now, we will choose a coordinate, which is the areal time one. This means that time t is proportional to the geometric area of the orbits of the isometry group. Explicitly, ds = −e2(η−U) αdt2 + e2(η−U) dθ2 + e2U (dx + Ady)2 + e−2U t2 dy 2

(8)

2

where ∂/∂x and ∂/∂y are Killing vector fields generating the T group action, and η, α, U and A are functions of t ∈ (0, ∞) and θ ∈ S 1 . It is also assumed that functions describing behavior of matter fields are ones of t and θ. Let us show the field equations obtained by varying the action (7) in the areal coordinate (8). Constraint equations η˙ = U˙ 2 t

+ +

e4U αU 2 + 2 (A˙ 2 + αA2 ) 4t 1 ˙2 φ + αφ2 + e2λφ (σ˙ 2 + ασ 2 ) + αQ2 e2λφ+2(η−U) , 4

(9)

η 1 ˙ e4U ˙ α = 2U˙ U + 2 AA + (φφ − + e2λφ σσ ˙ ), t 2t 2tα 2

(10)

α˙ = −tα2 Q2 e2λφ+2(η−U) .

(11)

Evolution equations η¨ − αη = +

η˙ α˙ α2 α η α e4U + − + − U˙ 2 + αU 2 + 2 (A˙ 2 − αA2 ) 2 2α 4α 2 4t 1 ˙2 2 2λφ 2 2 −φ + αφ + e (−σ˙ + ασ ) + αQ2 e2λφ+2(η−U) , (12) 4

4U ˙ ˙ ¨ − αU = − U + α˙ U + α U + e (A˙ 2 − αA2 ) + 1 αQ2 e2λφ+2(η−U) , U t 2α 2 2t2 4

(13)

A˙ α˙ A˙ α A A¨ − αA = + + − 4(A˙ U˙ − αA U ), t 2α 2

(14)

α˙ φ˙ α φ φ˙ φ¨ − αφ = − + + + λe2λφ (σ˙ 2 − ασ 2 ) − λαQ2 e2λφ+2(η−U) , t 2α 2

(15)

α˙ σ˙ α σ σ˙ + + − 2λ(φ˙ σ˙ − αφ σ ). (16) t 2α 2 Hereafter, dot and prime denote derivative with respect to t and θ, respectively. We will call this system of partial differential equations (PDEs) Gowdy symmetric IIA system. Note that these equations are not independent because the wave equation (12) for η can be derived from other equations. Indeed, there are only two dynamical degree of freedom (i.e., U and A) in the Gowdy symmetric spacetimes. σ ¨ − ασ = −

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2 Initial singularities Consider the problem (Q1). To begin with a brief review of the Fuchsian algorithm, which is a method to construct exact singular solutions to a PDE system near a singularity (t = 0). The algorithm is based on the following idea: near the singularity, decompose the singular formal solutions into a singular part, which depends on a number of arbitrary functions, and a regular part u. If the system can be written as a Fuchsian system of the form [D + N (x)] u = tf (t, x, u, ∂x u),

(17)

where D := t∂t and f is a vector-valued regular function, then the following theorem can be applied: Theorem 1 [KR] Assume that N is an analytic matrix near x = x0 such that there is a constant C with ΛN ≤ C for 0 < Λ < 1. In addition, suppose that f is a locally Lipschitz function of u and ∂x u which preserves analyticity in x and continuity in t. Then, the Fuchsian system (17) has a unique solution in a neighborhood of x = x0 and t = 0 which is analytic in x and continuous in t and tend to zero as t → 0. Thus, the regular part goes to zero and the singular part of the formal solution becomes an exact solution to the original PDE system near the singularity. Unlike the vacuum Gowdy case, the evolution equations (13)–(16) do not decouple from the constraint equations (9)–(11), since they contain the function α. Therefore, according to [IK], we take equations (9), (11), (13)–(16) as effective evolution ones and (10) as the only effective constraint equation. This is not a standard setup for the initial-value problem for the Einstein-matter equations (see example [TM]). Therefore, it is not clear whether the initial-value problem for our case away from the singularity at t = 0 has a unique solution or not, unless it is shown that the constraint (10) propagates. Let us show the local existence and uniqueness of our initial-value problem. We can obtain the following first-order system for z from the PDE system (9), (11), (13)–(16): ∂t z = f (t, θ, z , ∂θ z ),

(18)

˙ φ , σ, σ, ˙ A , φ, φ, ˙ σ , α, η). This means that the PDE syswhere z := (U, U˙ , U , A, A, tem is of Cauchy-Kowalewskaya type. Thus, ignoring the constraint equation (10), we have a unique solution to the effective evolution equations by prescribing the analytic initial data for t = t0 > 0 if all functions are analytic. Now, to assure the local existence and uniqueness of the initial-value problem, we must show that the constraint (10) propagates.

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M. Narita


Let us set N := η − 2DU U −

e4U 1 e2λφ α Dφφ Dσσ , DAA − − + 2t2 2 2 2α

(19)

Computing 0 = Dη − (Dη)

e2λφ e4U 1 α Dσσ − = DN + D 2DU U + 2 DAA + Dφφ − (Dη) , 2t 2 2 2α

(20)

we have a linear, homogeneous ordinary differential equation (ODE) for N of the form DN −

Dα N = 0. 2α

(21)

Thus, the uniqueness theorem for ODEs guarantees that N is identically zero for any time t if we set initial data for t = t0 such that N (t0 ) = 0. Thus, the local existence and uniqueness of the initial-value problem for our case has been shown in the analytic case. In appendix, we shall consider the smooth version of the initial-value problem for our non-standard setup of the Gowdy symmetric IIA system.

2.1

Application of the Fuchsian algorithm

Let us construct AVTD singular solutions to the Gowdy symmetric IIA system. First, we will consider the case that a solution has a maximum number of free functions. In this sense, the solution (given in Theorem 2) is generic. Neglecting spatial derivative and potential terms in the effective evolution equations, we have velocity-terms dominated (VTD) equations as follows: Dη = (DU )2 +

e4U 1 e2λφ 2 2 (Dφ) (Dσ)2 , (DA) + + 4t2 4 4 Dα = 0,

(23)

4U

e 1 DU Dα + 2 (DA)2 , 2α 4t 1 2 DADα − 4DU DA, D A = 2DA + 2α 1 D2 φ = DφDα + λe2λφ (Dσ)2 , 2α 1 DσDα − 2λDφDσ. D2 σ = 2α D2 U =

(22)

Solving this system of VTD equations, we have a VTD solution.

(24) (25) (26) (27)

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827

Then, the following formal solution is obtained:

κ(θ)2 η = k(θ)2 + ln t + η0 (θ) + t µ(t, θ), 4

(28)

α = α0 (θ) + t β(t, θ), U = k(θ) ln t + U0 (θ) + t V (t, θ),

(29) (30)

A = h(θ) + t2−4k (A0 (θ) + B(t, θ)) ,

(31)

φ = κ(θ) ln t + φ0 (θ) + t Φ(t, θ),

(32)

σ = ω(θ) + t−2λκ (σ0 (θ) + Σ(t, θ)) ,

(33)

where > 0,

0 < k(θ)
0

(34)

and −1 < λκ(θ) < 0.

(35)

Note that µ, β, V , B, Φ and Σ are regular parts and others are singular parts (=VTD solutions). Inserting this formal solution into the Einstein-matter equations, we obtain the following Fuchsian system: (D + N ) u = tδ f (t, θ, u, ∂θ u), where u := ui = (V , DV , t V , B, DB, t B , Φ, DΦ, i = 1, . . . , 14, f is a vector-valued regular function and  0 −1 0 0 0 0 0 0 0  2 2 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0 0   0 0 0 0 −1 0 0 0 0   0 0 0 0 2 − 4k 0 0 0 0   0 0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 −1 0 N = 2  0 2 0 0 0 0 0 0   0 0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0 0 − κ2 0 −2k −2k 0 0 0 0 − κ

2

(36) t Φ , Σ, DΣ, t Σ , β, µ), 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 −1 −2λκ 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

            .           

(37)

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Note that δ > 0 if the condition (34), (35) and 3 1 κ2 + λκ + > 0 K := (k − )2 + 2 4 4

(38)

holds. To apply Theorem 1 to our Fuchsian system (36), we must verify that the boundedness condition for the matrix N holds. To do this, we have P −1 N P = N0 , where   1 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 2 − 4k 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 1 0 0 0 0 0 0   , (39) N0 =  0 0 0 0 0 0 0 0 0   0 0 0 0   0 0 0 0  0 0 0 0 0 0 0 0 0 0    0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 −2λκ 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0  0 2k 0 0 0 0 0 κ2 0 0 0 0 0 and

            P =           

1 0 − −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 − −1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1

            .           

(40)

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829

Then, ΛN0 =                       

Λ 0 0 0 0 0 0 0 0 0 0 0 0 0

Λ ln Λ Λ 0 0 0 0 0 0 0 0 0 0 0 2kΛ ln Λ

0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0

Λ2−4k 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 Λ 0 0 0 0 0 0 0

0 0 0 0 0 0 Λ ln Λ Λ 0 0 0 0 0 κ Λ ln Λ 2

0 0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0

Λ−2λκ 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 Λ 0

0 0 0 0 0 0 0 0 0 0 0 0 0 Λ

                       (41)

hence P ΛN0 P −1 = ΛN is uniformly bounded for 0 < Λ < 1 if the condition (34) and (35) hold. Thus, there is a unique solution of the Fuchsian system (36) which goes to zero as t → 0, and which is analytic in θ and continuous in t. Note that (U, A, φ, σ, α, η) is a solution of the effective evolution equations of the Einstein-matter equations (9), (11), (13)–(16) if we construct (U, A, φ, σ, α, η) from (28)–(33) with V = u1 , B = u4 , Φ = u7 , Σ = u10 , β = u13 and µ = u14 . This fact follows from equations D(uI+2 − t uI ) = 0, where I = 1, 4, 7, 10. Now, we want to get a constraint condition to ensure that the solution obtained above is a genuine one to the full Einstein-matter equations. Since Dα/α = O(t ), N˙ α˙ = = O(t −1 ), N 2α

(42)

then, the right-hand side of the above equation is integrable. From this result, we can put a function P (t, θ) such that N ∝ exp P (t, θ).

(43)

This means that N is identically zero if we would choose the singular data such that N → 0 as t → 0, and then, the constraint equation (10) is satisfied. Inserting the formal solutions (28)–(33) into the constraint equation (19), we have N = η0 − 2kU0 − e4U0 (1 − 2k)h A0 −

κφ0 α + e2λφ0 κω σ0 + 0 + O(1), 2 2α0

(44)

830

M. Narita


where O(1) is some expression which tends to zero as t → 0. Thus, the constraint holds iff the singular data satisfy η0 − 2kU0 − e4U0 (1 − 2k)h A0 −

κφ0 α + e2λφ0 κω σ0 + 0 = 0. 2 2α0

(45)

To summarize, we have the following theorem: Theorem 2 Choose data such that conditions (34), (35) and (45) are satisfied. Suppose that is a positive constant less than min{4k, 2 − 4k, −2λκ, 2 + 2λκ, 2K}. For any choice of the analytic singular data η0 (θ), α0 (θ), k(θ), U0 (θ), h(θ), A0 (θ), κ(θ), φ0 (θ), ω(θ) and σ0 (θ), the Gowdy symmetric IIA system has a solution of the form (28)–(33), where µ, β, V , B, Φ and Σ tend to zero as t → 0. Although the solution given in Theorem 2 is generic in the sense that the solution has a maximum number of free functions, conditions for paradigm-A does not hold since λκ < 0, i.e., the universe starts with large potential and wrong sign of the time derivative of φ. To verify the validity of the paradigm-A we need to construct a solution allowing a condition λκ > 0. Indeed, this problem can be solved as follows. If an AVTD solution with λκ > 0 are needed, we replace expansion (33) with σ = ω(θ) + t Σ(t, θ).

(46)

In this case, −2λκ and Λ−2λκ sitting the 11th line and the 11th row in the matrices N and ΛN0 are replaced by and Λ , respectively. Also, the constraint condition for the singular data becomes η0 − 2kU0 − e4U0 (1 − 2k)h A0 −

κφ0 α + 0 = 0. 2 2α0

(47)

Thus, we have the following theorem which is consistent with conditions of paradigm-A. Theorem 3 Choose data such that conditions (34), (47) and λκ > −1/2 are satisfied. Suppose that is a positive constant such that max{0,−2λκ} < < min{4k,2− 4k}. For any choice of the analytic singular data η0 (θ), α0 (θ), k(θ), U0 (θ), h(θ), A0 (θ), κ(θ), φ0 (θ) and ω(θ), the Gowdy symmetric IIA system has a solution of the form (28)–(32) and (46), where µ, β, V , B, Φ and Σ tend to zero as t → 0. The positivity of K is automatically satisfied when 0 < k < 1/2 and λκ > −1/2 hold. Then, a solution to the Gowdy symmetric IIA system allowing the initial conditions for paradigm-A has been constructed. Note that we do not have the maximum number of free functions in this case. Thus, the solution given in Theorem 3 is restricted than generic one given in Theorem 2. The reason why we do not have the maximum number is the existence of dilaton coupling with kinetic terms of other fields (the axion field in our case). Generically, all fields arising in superstring/M-theory couple with the dilaton field. Therefore, we may not avoid such restriction for solutions to our problem unless the dilaton coupling is ignored.

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3 Global existence Now, consider the problem (Q2). We will show the following theorem: Theorem 4 Let (M, g, φ, σ) be the maximal Cauchy development of C ∞ initial data for the Gowdy symmetric IIA system. Suppose that the timelike convergence condition (TCC), which is Rµν W µ W ν ≥ 0 for any timelike vector W µ , holds and ¯ such that |λ| ≤ λ ¯ < 1/2. Then, M can be covered there is a positive constant λ by compact Cauchy surfaces of constant areal time t with each value in the range (0, ∞). In the first place, we need a local existence theorem for the Gowdy symmetric IIA system, which is the Einstein-(minimally coupled) scalar system with a positive potential. Fortunately, there is no coupling caused by existence of such matter fields in the principal part of the PDE system. For this reason, since the local existence theorems for vacuum Gowdy (more generically, T 2 -symmetric) spacetimes have been shown [MV, CP], the same theorem for the Gowdy symmetric IIA system can be shown as vacuum case [FR]. Thus, it is enough to verify uniform bounds of functions (η, α, U, A, φ, σ) and their first and second derivatives to prove global existence [MA]. The strategy is similar with the case of T 2 -symmetric Einstein(Vlasov) system [AH, ARW, BCIM, IW, WM]. Let us define γ := η +

1 ln α. 2

(48)

By using γ we can rewrite the constraint equations as follows: Q2 2(γ+λφ−U) γ˙ =E− e , t 4 F γ =√ , t α α˙ = −tαQ2 e2(γ+λφ−U) ,

(49) (50) (51)

where 1 e4U E := U˙ 2 + αU 2 + 2 A˙ 2 + αA2 + φ˙ 2 + αφ2 + e2λφ σ˙ 2 + ασ 2 , 4t 4 and F :=

√ e4U ˙ 1 ˙ φφ + e2λφ σσ α 2U˙ U + 2 AA + ˙ . 2t 2

Define energies for the Gowdy symmetric IIA system 1 1 √ E + αQ2 e2(η+λφ−U) dθ, E(t) := α 4 S1

(52)

(53)

(54)

832

M. Narita

and

˜ := E(t) S1


E √ dθ, α

(55)

In our case, the TCC is as follows: 1 φ˙ 2 + e2λφ σ˙ 2 ≥ αQ2 e2(η+λφ−U) 2

(56)

First, we will show energy decay and energy inequalities (see Lemmas 1 and 3 in [IW]). ¯ < 1/2. Then, E and E ˜ Lemma 1 Suppose the TCC and the condition |λ| ≤ λ decrease monotonically along time t, that is, dE(t) 0, 2

(83)

˜ i ) > 0. From since regular initial data at t = ti are supposed. This means E(t equation (66), we have ˜ C Q2 Cλ α˙ E E dE √ dθ = − λ ≤ (84) te2(γ+λφ−U) √ dθ, dt 2 α α 2 α 1 1 S S where Cλ < 1 is a positive constant depending on only the coupling constant λ. Suppose λ ≥ 0. Integrating this inequality from ti to t (0 < t < ti ), Cλ Q2 ti 2(γ+λφ−U) E ˜ ˜ √ dθ ds E(t) ≥ E(ti ) + se 2 α t S1

ti 2 Cλ Q ˜ ˜ E(s)ds ≥ E(ti ) + s exp 2 min γ + λ min φ − max U 2 S1 S1 S1 t

2 ti ˜ i ) 1 + Cλ Q ≥ E(t s exp 2 min γ + λ min φ − max U ds (85) , 2 S1 S1 S1 t where the monotonicity of E˜ has been used. From Lemma 4, min γ + λ min φ − max U 1 1 1 S

S

S

1/2 ≥ max γ + λ max φ − min U − tE(t) + (C λ − C )E(t) 1 2 S1 S1 S1 1/2 ≥ max γ + λ max φ − min U − t E(T ) + (C λ − C )E(τ ) , i − 1 2 1 1 1 S

S

S

(86)

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where C1 and C2 are positive constants, τ = ti if C1 λ − C2 < 0 and τ = T− if C1 λ − C2 ≥ 0. Thus, we have

ti 2 ˜ ≥ E(t ˜ i ) 1 + Cλ Q e−2(ti E(T− )+(C1 λ−C2 )E(τ )1/2 ) E(t) se2(γ+λφ−U) ds , (87) 2 t and then, ti se2(γ+λφ−U) ds ≤ t

1/2 2 e2(ti E(T− )+(C1 λ−C2 )E(τ ) ) Cλ Q2

˜ −) E(T −1 , ˜ i) E(t

(88)

where the condition (83) has been used. When one consider the case of λ < 0, we have the same results by exchanging maxS 1 φ and minS 1 φ in inequalities (85) and (86). Now, integrating equation (49), we have ti sQ2 2(γ+λφ−U) e sE − ds γ(t, θ) = γ(ti , θ) − 4 t Q2 ti 2(γ+λφ−U) se ds ≤ γ(ti , θ) + 4 t ˜ −) 1 2(ti E(T− )+(C1 λ−C2 )E(τ )1/2 ) E(T ≤ max e γ(ti , θ) + − 1 . (89) ˜ i) S1 Cλ 2 E(t Thus, the boundedness of γ from above has been shown.

Lemma 6 For any numbers a and b, and for n ≤ 12 , αn e2η+aφ−bU is bounded on (T− , ti ] × S 1 . Proof. (cf. Lemma 5 in [WM]). ∂t tk αn e2η+aφ−bU

k nα˙ ˙ ˙ + + 2η˙ + aφ − bU tk αn e2η+aφ−bU = t α

2 b t ˙ a 2 e4U ˙ φ+ = 2t U − + + 2αU 2 + 2 A˙ 2 + αA2 4t 2 t 2t

1 1 − n tαQ2 e2(η+λφ−U) tk αn e2η+aφ−bU + αφ2 + e2λφ (σ˙ 2 + ασ 2 ) + 2 2 ≥ 0, (90) where we have chosen 8k = 4a2 + b2 . Then, we have

k ti α(ti , θ)n e2η(ti ,θ)+aφ(ti ,θ)−bU(ti ,θ) , α(t, θ)n e2η(t,θ)+aφ(t,θ)−bU (t,θ) ≤ T− on (T− , ti ] × S 1 .

(91)

838

M. Narita


Lemma 7 α is bounded on (T− , ti ] × S 1 . Proof. Integrating the constraint equation (11), we have ti ti α˙ − ds = ln α(t) − ln α(ti ) = Q2 se2(γ+λφ−U) ds, α t t

(92)

for t ∈ (T− , ti ]. By using inequality (88), we have boundedness of ln α from above. As a result, 0 < α is also bounded. √ Lemma 8 For any numbers a and b, eγ+aφ−bU (= αeη+aφ−bU ) is bounded on (T− , ti ] × S 1 . Proof. We have already a result that e2η+aφ−bU is bounded on (T− , ti ] × S 1 (Lemma 6). Combining this and Lemma 7, the boundedness of eγ+aφ−bU on (T− , ti ] × S 1 follows directly. Corollary 1 αα ˙ −1 = ∂t (ln α) is bounded on (T− , ti ] × S 1 . Thus, ln α and α˙ are as well. Proof. Boundedness of αe2(η+aφ−bU ) is obtained by Lemma 8. From the constraint equation (51), we have αα ˙ −1 = −tαQ2 e2(λφ+η−U) . If we set a = λ and b = 1, the boundedness of the right-hand side of that equation is obtained. Thus, the conclusion of this lemma is shown. Lemma 9 The functions U , A, φ, σ and their first derivatives are bounded on (T− , ti ] × S 1 . Proof. From Lemma 2 and Corollary 1, we have the boundedness for E on ˙ ˙ 2U ˙ 2U λφ 1 (T− , ti ] × S . Then, U , |U |, φ, |φ |, e /2t A, e /2t A , e σ˙ and λφ e σ are bounded for all t ∈ (T− , T+ ). Once the boundedness on the first derivative of U and φ is obtained, it follows that U and φ are bounded for all t ∈ (T− , T+ ). ˙ A , σ˙ and σ , and consequently on A and σ. Then, we have bounds on A, Lemma 10 The functions α , α˙ and α ¨ are bounded on (T− , ti ] × S 1 . Also, η, η˙ and η are as well. Proof. (cf. Step 3 of Section 6 in [BCIM]). From the constraint equations (49) and (50), we have boundedness for γ˙ and γ directly. Then, γ is controlled. Differentiating both sides of equation (51) with respect to θ, we have α˙ = α −tQ2 e2(γ+λφ−U) − 2tQ2 e2(γ+λφ−U) α (γ + λφ − U ) . (93) Then, we have boundedness for α by integrating this differential equation for α in time since the coefficient of α and the second term in the right-hand side of the equation (93) are controlled. Thus, we have that η, η˙ and η is bounded.

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The boundedness of α˙ is obtained immediately from (93). Now, differentiating both sides of equation (51) with respect to t, we have α ¨ = −Q2 αe2(η+λφ−U) α + 2tα˙ + 2tα η˙ + λφ˙ − U˙ , (94)

which implies that α ¨ is bounded.

Lemma 11 The second derivatives of U , A, φ and σ are bounded on (T− , ti ] × S 1 . Proof. By Lemma 3 we have the boundedness for E˜ on (T− , ti ] × S 1 . Then, we have ¨ φ˙ , σ ¨ , U˙ , A, ¨ A˙ , φ, uniform bounds on U ¨ and σ˙ . Bounds on U , A , φ and σ follows from the wave equations (13)–(16) directly. Lemma 12 α , η¨, η˙ and η are bounded on (T− , ti ] × S 1 . Proof. By taking the time derivative of (49) and (50), we have bounds on γ¨ and γ˙ . Then, bounds on η¨ and η˙ are obtained by the definition of γ. Also, by taking the θ derivative of (50), we have bounds on γ . The boundedness for α follows from the same argument in the proof of Lemma 10. That is, differentiating both sides of equation (93) with respect to θ, we have α˙ = α −tQ2 e2(γ+λφ−U) − 4tQ2 α (γ + λφ − U )e2(γ+λφ−U) (95) 2 − 2tQ2 e2(γ+λφ−U) α γ + λφ − U + 2 (γ + λφ − U ) . Therefore, we have boundedness for α by integrating this differential equation for α in time since the coefficient of α and the second and third terms in the right-hand side of the equation (95) are bounded as shown already. Then, η is bounded by using the wave equation (12).

3.2

Future direction

Now, consider the future direction. We have already a monotonic decreasing property of E(t) along increasing t, dE/dt < 0 (Lemma 1). Therefore, for any t ∈ [ti , T+ ), E(t) ≤ E(ti ).

(96)

Proofs of the following two lemmas are similar with the argument in Step 1 of Section 5 in [AH]. θ Lemma 13 θ12 α−1/2 dθ is bounded on [ti , T+ ). Proof. The constraint equation (11) can be written as ∂t (α−1/2 ) =

t √ 2 2(η+λφ−U) αQ e . 2

(97)

840

M. Narita


Then, −1/2

α

−1/2

(t, θ) − α

Q2 (ti , θ) = 2

t

√ s αe2(η+λφ−U) ds,

(98)

ti

for ant t ∈ [ti , T+ ). Integrating this equation from θ1 to θ2 in S 1 , we have θ2 θ2 θ2 √ 2(η+λφ−U) Q2 t α−1/2 (t, θ)dθ = s αe dθds + α−1/2 (ti , θ)dθ 2 ti θ1 θ1 θ1 t Q2 ≤ E(ti ) sds + 2π sup α−1/2 (ti , θ) 2 S1 ti Q2 E(ti )(t2 − t2i ) + C, 4

≤

(99)

where (96) has been used. Lemma 14 The functions U and φ are bounded on [ti , T+ ) × S 1 . Proof. For any θ1 , θ2 and for each t ∈ [ti , T+ ), θ2 |U (t, θ2 ) − U (t, θ1 )| = U dθ θ1 1/2 θ2

≤

−1/2

α

dθ

1/2 1/2

α

θ1

≤

θ2

2

U dθ

θ1

C(t),

(100)

where the Hölder inequality, energy bound (96) and Lemma 13 have been used. Now, t ˙ U(t, θ)dθds + C U (t, θ)dθ = S1

ti

≤ ≤

t ti

S1

S1

˙ U(t, θ) dθds + |C|

t

α1/2 dθ

1/2

S1

ti

S1

α−1/2 U˙ 2 dθ

1/2 ds + |C| ,(101)

where the Hölder inequality has been used. Since α is monotonically decreasing function along increasing time t, the right-hand side of the above inequality can be bounded. Thus, U (t, θ)dθ ≤ C(t), (102) S1

for some uniformly bounded function C(t).

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Finally, we obtain a uniform bound on U . We have the following identity:

U (t, θ) = U (t, θ)dθ + U (t, θ) − U (t, θ) dθ. (103) 2π max max 1 1 S

S1

S

S1

The right-hand side of this identity is bounded from above since (100) and (102) hold. For minS 1 U (t, θ), one can use the same argument and then, minS 1 U (t, θ) is bounded from below. Thus, U is uniformly bounded on [ti , T+ ) × S 1 . We can obtain uniform boundedness for φ by replacing U with φ in the above argument. Lemma 15 The functions γ is bounded on [ti , T+ ) × S 1 . Proof. (cf. Step 1 of Section 6 in [BCIM]). From the constraint equation (49) for γ, we have two inequalities: γ˙ ≤ te,

(104)

1 γ˙ ≥ − tQ2 e2(γ+λφ−U) . 4

(105)

and

From the inequality (104), we have γ(t, θ)dθ − γ(ti , θ)dθ = S1

S1

t

ti

d ds

γ(s, θ)dθ ds S1

√ ≤ sup α(ti , θ)

S1

≤ C = which controls

S1

S1

t ti

t

sE(s)ds ti

sE(ti )ds

CE(ti ) 2 (t − t2i ), 2

(106)

γ(t, θ)dθ from above. Now, we have the following identity:

γ(t, θ)dθ = 2π max γ+ γ dθ. (107) γ − max 1 1 S

S1

S

By the equation (50) of γ and a basic inequality, we have |γ | dθ ≤ tE(ti ).

(108)

S1

Then,

θ2 θ2 γ dθ ≤ |γ | dθ ≤ |γ | dθ ≤ tE(ti ). |γ(t, θ2 ) − γ(t, θ1 )| = θ1 θ1 S1

(109)

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M. Narita


Therefore, combining (107) and (109), we have the upper bound for γ: max γ ≤ C(t), 1 S

(110)

where C(t) is a bounded function of t ∈ [ti , T+ ). From the inequalities (105) and (110), and Lemma 14, if the coupling constant λ is non-negative,

1 2 2(γ+λφ−U) γ˙ ≥ − tQ e ≥ Ct exp 2 max γ + λ max φ − min U ≥ Ctec(t) ,(111) 4 S1 S1 S1 for some bounded function c(t) of t ∈ [ti , T+ ) and C < 0. If λ is negative, maxS 1 φ is replaced by minS 1 φ. Thus, γ˙ is controlled into the future, so we have upper and lower bounds for γ on [ti , T+ ) × S 1 . Corollary 2 αα ˙ −1 (hence ln α and α), η and α˙ are bounded on [ti , T+ ) × S 1 . Proof. The constraint equation (51) can be written as α˙ = −tQ2 e2(γ+λφ−U) . α

(112)

With boundedness of γ (Lemma 15), φ and U (Lemma 14), αα ˙ −1 = ∂t (ln α) 1 is bounded on [ti , T+ ) × S . As immediate results, ln α and α are bounded on [ti , T+ ) × S 1 . Since η = γ − 12 ln α, η is bounded on [ti , T+ ) × S 1 . Using these results to the constraint equation (51), we have a conclusion that α˙ is bounded on [ti , T+ ) × S 1 . Once the boundedness of αα ˙ −1 has been obtained, the following arguments are similar with ones of the past direction because key lemmas (Lemma 2 and Lemma 3) can be used and the arguments do not depend on time directions. Lemma 16 The functions U , A, φ, σ and their first derivatives are bounded on [ti , T+ ) × S 1 . Proof. From Lemma 2 and Corollary 2, we have the boundedness for E on [ti , T+ )× S 1 . The proof is the same with one of Lemma 9. Lemma 17 The functions η, ˙ η , α , α˙ and α ¨ are bounded on [ti , T+ ) × S 1 . Proof. Since the constraint equation (9) is described in terms of bounded functions and t, we have bounds on η. ˙ From the constraint equation (50), we have bounds ¨ can be obtained by the same arguon γ and then, boundedness for α , α˙ and α ment with the proof of Lemma 10. Combining this result, boundedness of γ and definition of γ, we have that η is bounded. Lemma 18 The second derivatives of U , A, φ and σ are bounded on [ti , T+ ) × S 1 .

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Proof. By Lemma 3 we have the boundedness for E˜ on [ti , T+ ) × S 1 . The proof is the same with one of Lemma 11. Lemma 19 α , η¨, η˙ and η are bounded on [ti , T+ ) × S 1 . Proof. The argument is the same to Lemma 12.

3.3

Proof of Theorem 4

We can continue to obtain bounds on higher derivatives of the fields by repeating the above arguments. Fortunately, to apply the global existence theorem in [MA], it is enough to get C 2 bounds of all functions. Thus, it has been shown that the functions η, α, U , A, φ and σ extend to t → 0 into the past direction and to t → ∞ into the future direction.

4 Comments We should like to comment concerning the TCC and the condition for coupling constant λ. Note that these conditions are needed to prove Theorem 4 into only the past direction. It is expected that the TCC is satisfied near initial singularities because strong focusing effect by gravity is dominant than repulsing one by a positive potential (cosmological constant) there. Note that spacetimes described by our AVTD solutions satisfy the TCC. Also, it is possible to expand in acceleration of the spacetimes into the future direction since the TCC does not hold necessarily there and the positive potential would become dominant. Thus, Theorem 4 does not deny paradigm-A. ¯ < 1/2 admits λ = 0, which means that there is a The condition |λ| ≤ λ positive cosmological constant. Thus, our theorem is applicable to not only theories with dilaton coupling but also ones with a pure cosmological constant. Now, ¯ It is known that there is a critical value λC in n-dimensional let us discuss λ. homogeneous and isotropic spacetimes [TP, WMNR]. In our notation with n = 4, |λC | = 1/2. Here, “critical” means the boundary whether late-time attractor solutions indicate accelerated expansion or not. Roughly speaking, λ describes steepness of the potential. Therefore, for λ2 > λ2C , the dilaton field falls down the potential hill soon and then decelerating expansion solutions with transient accelerating one are obtained, while we have attractor solutions with eternal accelerating expansion if λ2 < λ2C . It is believed that such critical values exist for generic spacetimes, although we do not know λC for spacetimes we considered here, in ¯ particular, our results give us no information about relation between λC and λ. Thus, it is not clear that the solution obtained in Theorem 3 is consistent with paradigm-A at the intermediate- and late-time. To answer this question, we need to analyze future asymptotic behavior (e.g. see [RA04]), which is left for future research.

844

A

M. Narita


Local existence and uniqueness for smooth case

Let us consider the smooth version of the initial-value problem for our nonstandard setup formulated in Section 2. A key idea is to construct a symmetrichyperbolic system by introducing a new variable α := Z14 [IK]. Let us define ˙ φ , σ, σ, ˙ A , φ, φ,

:= Zi = (U, U˙ , U , A, A, ˙ σ , α, α , η). Here, i runs from 1 to 15. Z The system consisting in the effective evolution equations (9), (11), (13)–(16) becomes the following first-order symmetric-hyperbolic one:

= A1 ∂θ Z

+ F (t, θ, Z),

A0 ∂t Z

(113)

A0 = diag(1, 1, α, 1, 1, α, 1, 1, α, 1, 1, α, 1, 1, 1),

(114)

where

and

   A1 =    

0 A2 =  0 0

A2 0 0 0 0

0 A2 0 0 0

0 0 A2 0 0

0 0 0 A2 0

0 0 0 0 A3 

 0 0 0 α  α 0

and

   ,  

0 A3 =  0 0

 0 0 0 0 . 0 0

(115)

(116)

Thus, we have a unique solution to the effective evolution equations by prescribing the smooth initial data for t = t0 > 0 if the constraint equations (10) and α = Z14 hold for any t. Now, as the analytic case, to assure the local existence and uniqueness of the initial-value problem, it is enough to show that the constraints propagate. Let us set N1 := η − 2DU U −

e4U 1 e2λφ Z14 Dσσ + , DAA − Dφφ − 2 2t 2 2 2α

(117)

and N2 := Z14 − α .

(118)

By direct calculation, we have the following linear, homogeneous ODE system:

= 0, (D + B)N

:= (N1 , N2 ) and where N B=

Dα 2α2

α 4α2

−1 −2α

(119) .

(120)

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845

is identically Thus, the uniqueness theorem for ODE systems guarantees that N

zero for any time t if we set initial data for t = t0 such that N (t0 ) = 0. Thus, the local existence and uniqueness of the initial-value problem for our case has been shown in the smooth case.

Acknowledgments I am grateful to Alan Rendall and Yoshio Tsutsumi for commenting on the manuscript.

References [AH]

H. Andreasson, Global foliations of matter spacetimes with Gowdy symmetry, Commun. Math. Phys. 206, 337–365 (1999).

[AR]

L. Andersson, A.D. Rendall, Quiescent cosmological singularities, Commun. Math. Phys. 218, 479–511 (2001).

[ARW]

H. Andreasson, A.D. Rendall and M. Weaver, Existence of CMC and constant areal time foliations in T 2 symmetric spacetimes with Vlasov matter, Commun. Partial Differential Equations 29, 237–262 (2004).

[BCIM]

B.K. Berger, P.T. Chru´sciel, J. Isenberg and V. Moncrief, Global Foliations of Vacuum Spacetimes with T 2 Isometry, Ann. Physics, NY 260, 117–148 (1997).

[BKL]

V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, A general solution of the Einstein equations with a time singularity, Adv. Phys. 13, 639–667 (1982).

[CP]

P.T. Chru´sciel, On space-times with U (1) × U (1) symmetric compact Cauchy surfaces, Ann. Physics, NY 202, 100–150 (1990).

[DHRW] T. Damour, M. Henneaux, A.D. Rendall and M. Weaver, Kasner-like behaviour for subcritical Einstein-matter systems, Ann. Henri Poincaré 3, 1049–1111 (2002). [EG]

R. Emparan and J. Garriga, A note on accelerating cosmologies from compactifications and S-branes, J. High Energy Phys. 05, 028 (2003).

[FR]

H. Friedrich and A.D. Rendall, The Cauchy Problem for the Einstein Equations, Lect. Notes Phys. 540, 127–224 (2000).

[GG]

G.W. Gibbons, Aspects of supergravity theories, in Supersymmetry, supergravity and related topics, edited by F. del Agulis et al., World Scientific, (1985) 123–181.

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[GR]

R.H. Gowdy, Vacuum spacetimes and compact invariant hypersurfaces: Topologies and boundary conditions, Ann. Physics, NY 83, 203–224 (1974).

[IK]

J. Isenberg and S. Kichenassamy, Asymptotic behavior in polarized T 2 symmetric vacuum space-times, J. Math. Phys. 40, 340–352 (1999).

[IM]

J. Isenberg and V. Moncrief, Asymptotic behaviour in polarized and half-polarized U(1) symmetric vacuum spacetimes, Class. Quantum Grav. 19, 5361–5386 (2002).

[IW]

J. Isenberg and M. Weaver, On the area of the symmetry orbits in T 2 symmetric spacetimes, Class. Quantum Grav. 20, 3783–3796 (2003).

[KR]

S. Kichenassamy and A.D. Rendall, Analytic description of singularities in Gowdy spacetimes, Class. Quantum Grav. 15, 1339–1355 (1998).

[LH03]

H. Lee, Asymptotic behaviour of the Einstein-Vlasov system with a positive cosmological constant, gr-qc/0308035.

[LH04]

H. Lee, The Einstein-Vlasov system with a scalar field, gr-qc/0404007.

[LWC]

J.E. Lidsey, D. Wands and E.J. Copeland, Superstring cosmology, Phys. Rep. 337, 343–492 (2000).

[MA]

A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Springer-Verlag, (1984).

[MV]

V. Moncrief, Global properties of Gowdy spacetimes with T 3 × R topology, Ann. Physics, NY 132, 87–107 (1981).

[NM02]

M. Narita, On the existence of global solutions for T 3 -Gowdy spacetimes with stringy matter, Class. Quantum Grav. 19, 6279–6288 (2002).

[NM03]

M. Narita, Global existence problem in T 3 -Gowdy symmetric IIB superstring cosmology, Class. Quantum Grav. 20, 4983–4994 (2003).

[NTM]

M. Narita, T. Torii and K. Maeda, Asymptotic singular behaviour of Gowdy spacetimes in string theory, Class. Quantum Grav. 17, 4597– 4613 (2000).

[RA00]

A.D. Rendall, Blow-up for solutions of hyperbolic PDE and spacetime singularities, gr-qc/0006060.

[RA04]

A.D. Rendall, Accelerated cosmological expansion due to a scalar field whose potential has a positive lower bound, Class. Quantum Grav. 21, 2445–2454 (2004).

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[TM]

M.E. Taylor, Partial Differential Equations III, Nonlinear Equations, Springer-Verlag, (1996).

[TP]

P.T. Townsend, Cosmic Acceleration and M-Theory, hep-th/0308149.

[TR]

S.B. Tchapnda and A.D. Rendall, Global existence and asymptotic behaviour in the future for the Einstein-Vlasov system with positive cosmological constant, Class. Quantum Grav. 20, 3037–3049 (2003).

[WR]

R.M. Wald, Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant, Phys. Rev. D28, 2118–2120 (1983).

[WM]

M. Weaver, On the area of the symmetry orbits in T 2 symmetric spacetimes with Vlasov matter, Class. Quantum Grav. 21, 1079–1097 (2004).

[WMNR] M.N.R. Wohlfarth, Inflationary cosmologies from compactification?, Phys. Rev. D69, 066002 (2004). Makoto Narita Max-Planck-Institut f¨ ur Gravitationsphysik Albert-Einstein-Institut Am M¨ uhlenberg 1 D-14476 Golm Germany email: [email protected] Present address Center for Relativity and Geometric Physics Studies Department of Physics National Central University Jhongli 320 Taiwan email: [email protected] Communicated by Sergiu Klainerman submitted 18/08/04, accepted 18/01/05




Spin, Statistics, and Reflections I. Rotation Invariance Bernd Kuckert Abstract. The universal covering of SO(3) is modelled as a reflection group GR in a representation independent fashion. For relativistic quantum fields, the Unruh effect of vacuum states is known to imply an intrinsic form of reflection symmetry, which is referred to as modular P1 CT-symmetry [1, 2, 11]. This symmetry is used to construct a representation of GR by pairs of modular P1 CT-operators. The representation thus obtained satisfies Pauli’s spin-statistics relation.

1 Introduction A vacuum state of a quantum field theory usually exhibits the Unruh effect, i.e., a uniformly accelerated observer experiences it as a thermal state whose temperature is proportional to his acceleration [27]. This has been shown by Bisognano and Wichmann [1, 2] for finite-component quantum fields (in the Wightman setting). For general quantum fields, it has recently been derived from the mere condition that each vacuum state exhibits passivity to each inertial or uniformly accelerated observer [18], i.e., that in the observer’s rest frame, no engine can extract energy from the state by cyclic processes.1 By the theorem of Bisognano and Wichmann mentioned above, all familiar quantum fields also exhibit an intrinsic form of PCT-symmetry.2 Namely, one can assign to each Rindler wedge W, i.e., the set W1 := {x1 ≥ |x0 |} or its image under some Poincaré transformation, an antiunitary involution JW . This assignment is an intrinsic construction using the vacuum vector and the field operators only. It is also basic to the so-called modular theory due to Tomita and Takesaki, where an operator like JW is called a modular conjugation. JW then implements a P1 CTsymmetry, i.e., a linear reflection in charge and at the edge of W. This property is called modular P1 CT-symmetry. Note as an aside that this symmetry is a typical property of 1 + 2-dimensional quantum fields as well, whereas these fields do not exhibit PCT-symmetry as a whole [23]. Modular P1 CT-symmetry is a consequence of the Unruh effect [11], but the converse implication does not hold: There are examples of P1 CT-symmetric quantum fields that do not exhibit the Unruh property [4]. 1 Two 2 cf.

related uniqueness results can be found in Refs. 16 and 17. also Refs. 11, 16, and 17.

850

B. Kuckert


Guido and Longo have derived Pauli’s spin-statistics relation from the Unruh effect for general quantum fields in 1 + 3 dimensions [11].3 Independently from this, the present author derived the spin-statistics relation making use of modular P1 CT-symmetry only [15]. This symmetry was assumed for the field’s observables only, but since use of a theorem due to Doplicher and Roberts [8] was made later on, the result of Ref. 15 is confined to the massive-particle excitations of the vacuum. In Ref. 11 the Unruh effect was assumed for the whole field on the one hand. On the other hand, no use of the Doplicher-Roberts theorem was made, so a much larger class of fields and states was included; even fields that are covariant with respect to more than one representation of the universal covering group of L↑+ , among which there may be both representations satisfying and violating Pauli’s relation [24]. What one did obtain was a unique representation satisfying the Unruh effect. This representation exhibits Pauli’s spin-statistics connection. All spin-statistics theorems obtained before did not admit this extent of generality. This paper is the first of two that generalize the result of Ref. 15 in this spirit as well. Assuming P1 CT-symmetry with respect to all Rindler wedges whose edges are two-dimensional planes in a given tim-zero plane, a covariant unitary repre˜ of the rotation group’s universal covering is constructed. This repsentation W resentation satisfies Pauli’s spin-statistics relation. The argument does not make use of the Doplicher-Roberts theorem and applies to general relativistic quantum fields. Like its predecessor in Ref. 15, the argument is crucially based on the fact that each rotation in R3 can be implemented by combining two reflections at planes. This is, as such, well known for both SO(3) and L↑+ . A corresponding result for the universal coverings of these groups is, however, less elementary to obtain. of In Section 2, a model GR ∼ = SU (2) of the universal covering group SO(3) SO(3) will be constructed from nothing except pairs of “reflections along normal vectors”, i.e., from the family (ja )a∈S 2 , where ja is the reflection at the plane a⊥ . This representation-independent construction is set up according to the needs of ↑ ∼ SL(2, C) of L the spin-statistics theorem to be proved later on. A model GL = + will be constructed in a forthcoming paper. It is to be expected that the universal coverings of other Lie groups could be constructed the same way. Recently it has been shown by Buchholz, Dreyer, Florig, and Summers that this structure has a representation theoretic consequence: unitary representations of L↑+ can be constructed from a system of reflections satisfying a minimum of covariance conditions, as they are satisfied by the modular conjugations of a quantum field with modular P1 CT-symmetry [4, 9, 5]. This raises the question how to generalize these results to GR and GL , the goal being a considerable generalization of the spin-statistics analysis in Ref. 15. In Section 3, it is shown that this can, indeed, be accomplished for GR ; the group GL will be treated in the forthcoming paper. If a quantum field exhibits 3 cf.

also Refs. 10, 12, and 13.

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851

modular P1 CT-symmetry, then it is elementary to build a distinguished represen˜ of GR from the modular conjugations that implement P1 CT-symmetry. tation W This representation can, eventually, easily be shown to conform with Pauli’s spinstatistics principle. It is well known that not all GR -covariant quantum fields exhibit the spinstatistics relation, and it should be remarked that even for Lorentz covariant fields there are counterexamples [24]. This means that some condition specifying the representation or field under consideration is needed for whatever spin-statistics theorem. In the early spin-statistics theorems, this condition was that the number of internal degrees of freedom is finite, in this paper the condition is that the representation is constructed from modular P1 CT-operators. At the moment, such sufficient conditions are all one has in the relativistic setting; only in the setting of nonrelativistic quantum mechanics, a both sufficient and necessary condition has been established [19, 20].

as a reflection group 2 SO(3) There are many ways to model the universal covering group of the rotation group SO(3) =: R. Among topologists, “the” universal covering group is the group SO(3) of homotopy classes of curves starting at some base point, physicists are more familiar with SU (2), but these are, of course, not the only examples of simply connected covering groups. As a new model, a group GR will be constructed in this section from pairs of “reflections along normal vectors”, i.e., from the family (ja )a∈S 2 , where ja is the reflection at the plane a⊥ . Let MR be the pair groupoid of S 2 , i.e., the set S 2 × S 2 endowed with the concatenation (a, b) ◦ (b, c) := (a, c). Then the map ρ : MR → R defined by ρ(a, b) := ja jb is well known to be surjective. Namely, ρ(a, a) = 1 for all a ∈ S 2 . For σ = 1, choose τ ∈ R such that τ 2 = σ; if a ∈ S 2 is perpendicular to the axis of σ, then ρ(τ a, a) = σ. Call (a, b) and (c, d) equivalent if ρ(a, b) = ρ(c, d) and if there exists a σ ∈ R commuting with ρ(a, b) and satisfying (a, b) = (σc, σd). Let GR be the quotient space MR / ∼ associated with this equivalence relation, and let π : MR → GR denote the corresponding canonical projection. Define ρ˜ : GR → R by ρ˜(π(m)) := ρ(m) for all m ∈ MR . Then the diagram π

MR

−→

ρ↓

ρ˜

GR (1)

R commutes by construction. All maps in this diagram are continuous: π is continuous by definition, and continuity of ρ is elementary to show. The proof for ρ˜ is

852

B. Kuckert


elementary as well: given any open set M ⊂ R, the pre-image ρ˜−1 (M ) is open if and only if π −1 (˜ ρ−1 (M )) is open. This set coincides with ρ−1 (M ), which is open by continuity of ρ. Defining ±1 := π(a, ±a) for arbitrary a ∈ S 2 , and −π(a, b) := π(a, −b) for (a, b) ∈ MR , one verifies that ρ˜−1 (σ) consists of two equivalence classes for each σ ∈ R. Lemma 1 (i) GR is a Hausdorff space. (ii) ρ˜ is a two-sheeted covering map. Before proving this lemma, we introduce some notation. ˙ For each σ ∈ R, ˙ let A(σ) be the rotation Notation. Denote the set R\{1} by R. axis of σ. If a ∈ A(σ) is one of the two unit vectors in A(σ), then there is a unique α ∈ (0, 2π) such that σ is the right-handed rotation around a by the angle α. The vector a and the angle α determine σ, and occasionally we use the notation [a, α] for σ. Note that [a, α] = [−a, 2π − α]. ˙ by M ˙ R . To each (a, b) ∈ M ˙ R , assign the axial unit Denote the set ρ−1 (R) a×b vector a(a, b) := |a×b| , and denote by (a, b) ∈ (0, π) the angle between a and b. ˙ R. Note that ρ(m) = [a(m), 2(m)] for all m ∈ M ˙ by G ˙ R . Since m ∼ n implies Denote the set ρ˜−1 (R) a(m) = a(n) and (m) = (n) one can define ˜ ˜ (π(m)) := a(m) and (π(m)) a := (m). ˙ R. ˜ Note that ρ˜(g) = [˜ a(g), 2(g)] for all g ∈ G Proof of Lemma 1.(i). Define B˙ π := {x ∈ R3 : |x| ∈ (0, π)}, and assign to each x ∈ B˙ π the rotation τ (x) := [x/|x|, |x|]. Choose any x ∈ B˙ π and an a ∈ S 2 ∩x⊥ , and ˙ R . One then obtains ξa (x) = ξb (x) for all b ∈ S 2 ∩ x⊥ , put ξa (x) := π(τ (x)a, a) ∈ G ˙ R is well defined by ξ(x) := ξa (x), where a ∈ S 2 ∩ x⊥ is so a map ξ : B˙ π → G arbitrary. ˙ R → B˙ π defined by η(g) := −(g)˜ ˜ a(g). Namely, ξ is inverse to the map η : G ˙ R , one has since b ⊥ a × b for all (a, b) ∈ M ξ(η(π(a, b))) = ξ ((a, b) · a(a, b)) = π ( τ (−(a, b)a(a, b)) b, b) = π ([−a(a, b), (a, b)] b, b) = π(a, b). So η is continuous, surjective, and has a continuous inverse, so η is a homeomor˙ R is a Hausdorff space. phism, and G

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853

It remains to construct disjoint neighborhoods of two distinct points g, h ∈ GR for the case that g = ±1 and h ∈ GR is arbitrary. ˜ If g = 1, then (h) = 0, so there exist disjoint open neighborhoods X and ˜ ˜ is Y of 0 and (h) in the topological space [0, π], respectively. Since the map ˜ −1 (Y ) are disjoint neighborhoods ˜ −1 (X) and V := continuous, the sets U := of 1 and h. If g = −1, there exist disjoint neighborhoods U and V of −g and −h, so −U and −V are disjoint neighborhoods of g and h, respectively. Proof of (ii). Define ρˆ : B˙ π → R˙ by ρˆ(x) := [x/|x|, 2|x|]. Then the diagram ˙R G ρ˜|G˙ R R˙

ρˆ

←−

↓η

(2)

B˙ π

commutes. ρˆ is a two-sheeted covering map, and η is a homeomorphism, so ρ˜|G˙ R = ρˆ ◦ η is a two-sheeted covering map. In order to prove that ρ˜ as a whole is a covering map, it remains to be shown ˙R ˙ R , but also in ±1. Since GR is Hausdorff, since G that ρ˜ is open not only on G −1 ˙ is a two-sheeted covering space of R, and since ρ˜ (1) = {±1} contains, like all other fibers of ρ˜, precisely two elements, it then follows that ρ˜ has continuous local inverses everywhere. So let (σn )n be any sequence in R˙ converging to 1, then some sequence (gn )n ˙ in GR needs to be found with ρ˜(gn ) = σn for all n and gn → 1; note that (−gn )n then satisfies ρ˜(−gn ) = σn as well and converges to −1. ˙ R , one has (g) ˜ ˜ ≤ π/2 or (−g) ≤ π/2. It follows that for For each g ∈ G −1 ˜ n ) ≤ π/2. Since [0, π/2] is each n some gn ∈ ρ˜ (σn ) can be chosen such that (g ˜ n ))n has at least one accumulation point, and since σn compact, the sequence ((g tends to 1, the only possible accumulation point in the interval [0, π/2] is zero. It ˜ n ) tends to zero and, hence, that gn tends to 1, proving that ρ˜ is follows that (g open. The reason why this proof is nontrivial is that ρ and π are not open. If this were the case, GR would directly inherit the Hausdorff property from MR , and the proof that ρ˜ is a covering map would be elementary. But neither ρ nor π is open. In order to see this, let (σn )n be any sequence of rotations around some fixed a ∈ S 2 , and suppose this sequence to converge to 1. If ρ were open, one would have to find, for each m ∈ π −1 (1) a sequence (mn )n converging to m and satisfying ρ(mn ) = σn for all n. Now choose m = (a, a). Since a ∈ A(σn ) for all n, one knows for all (bn , cn ) ∈ ρ−1 (σn ) that both bn and cn are perpendicular to a. As a consequence, no sequence (mn )n with ρ(mn ) = σn for all n can coverge to m = (a, a). π cannot be open either, since this would, by diagram 1 and the preceding ˙ are open. Lemma, imply that ρ is open. Only the restrictions of ρ and π to ρ−1 (R)

854

B. Kuckert


Theorem 2 (i) GR is simply connected. (ii) There is a unique group product on GR such that the diagram MR × MR

◦

−→ MR

↓π×π GR × GR

↓π

−→

↓ ρ˜ × ρ˜ R×R

GR

(3)

↓ ρ˜ ·

−→

R

commutes. Proof of (i). GR = π(MR ) is pathwise connected because MR = S 2 × S 2 and because π is continuous. Together with Lemma 1, this implies the statement, since the fundamental group of R is Z2 . Proof of (ii). The outer arrows of the diagram commute, so it suffices to prove the existence and uniqueness of a group product conforming with the lower part. ˜ of an arbitrary But it is well known that each simply connected covering space G topological group G can be endowed with a unique group product such that G is a covering group.4

3 Spin and statistics The preceding section has provided the basis of a general spin-statistics theorem, which is the subject of this section. From an intrinsic form of symmetry under a charge conjugation combined with a time inversion and the reflection in one spatial direction, which is referred to as modular P1 CT-symmetry, a strongly continuous ˜ of GR will be constructed using the above and related unitary representation W ˜ exhibits Pauli’s spin-statistics reasoning. It is, then, elementary to show that W relation. In order to make the notion of rotation meaningful, fix a distinguished time direction by choosing a future-directed timelike unit vector e0 . The 2-sphere of 2 unit vectors in the time-zero plane e⊥ 0 will be called S . 1+3 in a Hilbert space H. The Let F be an arbitrary quantum field on R following standard properties of relativistic quantum fields will be used here. (A) Algebra of field operators. Let C be a linear space of arbitrary dimension,5 and denote by D the space C0∞ (R1+3 ) of test functions on R1+3 . The field 4 See,

e.g., Props. 5 and 6 in Sect. I.VIII. in Ref. 7. the “component space”, and its dimension equals the number of components, which may be infinite in what follows. 5 C is

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F is a linear function that assigns to each Φ ∈ C ⊗ D a linear operator F (Φ) in a separable Hilbert space H. (A.1) F is free from redundancies in C, i.e., if c, d ∈ C and if F (c ⊗ ϕ)) = F (d ⊗ ϕ) for all ϕ ∈ D, then c = d. (A.2) Each field operator F (Φ) and its adjoint F (Φ)† are densely defined. There exists a dense subspace D of H contained in the domains of F (Φ) and F (Φ)† and satisfying F (Φ)D ⊂ D and F (Φ)† D ⊂ D for all Φ ∈ C ⊗ D. Denote by F the algebra generated by all F (Φ)|D and all F (Φ)† |D . Defining an involution ∗ on F by A∗ := A† |D , the algebra F is endowed with the structure of a ∗-algebra. For each a ∈ S 2 , denote by Wa := {x ∈ R1+3 : xa > |xe0 |} the Rindler wedge associated with a,6 and let F(a) be the algebra generated by all F (c ⊗ ϕ)|D and all F (c ⊗ ϕ)† |D with supp(ϕ) ⊂ Wa . The algebra F(a) inherits the structure of a ∗-algebra from F by restriction of ∗. (A.3) F(a) is nonabelian for each a, and a = b implies F(a) = F(b). (B) Cyclic vacuum vector. There exists a vector Ω ∈ H that is cyclic with respect to each F(a). (C) Normal commutation relations. There exists a unitary and self-adjoint operator k on H with kΩ = Ω and with kF(a)k = F(a) for all a ∈ S 2 . Define F± := 12 (F ± kF k). If c and d are arbitrary elements of C and if ϕ, ψ ∈ D have spacelike separated supports, then F+ (c ⊗ ϕ)F+ (d ⊗ ψ) = F+ (d ⊗ ψ)F+ (c ⊗ ϕ), F+ (c ⊗ ϕ)F− (d ⊗ ψ) = F− (d ⊗ ψ)F+ (c ⊗ ϕ),

and

F− (c ⊗ ϕ)F− (d ⊗ ψ) = −F− (d ⊗ ψ)F− (c ⊗ ϕ). The involution k is the statistics operator, and F± are the bosonic and fermionic components of F , respectively. Defining κ := (1 + ik)/(1 + i) and F t (d ⊗ ψ) := κF (d ⊗ ψ)κ† , the normal commutation relations read [F (c ⊗ ϕ), F t (d ⊗ ψ)] = 0. This property is referred to as twisted locality. Denote F(a)t := κF(a)κ† . These properties imply that Ω is separating with respect to each algebra F(a), i.e., for each A ∈ F(a), the condition AΩ = 0 implies A = 0.7 6 An observer who is uniformly accelerated in the direction a can interact with precisely the events in Wa . 7 If AΩ = 0 and B, C ∈ F(−a)t , then 0 = BCΩ, AΩ = CΩ, AB ∗ Ω, so A = 0 by cyclicity of Ω.

856

B. Kuckert


As a consequence, an antilinear operator Ra : F(a)Ω → F(a)Ω is defined by Ra AΩ := A∗ Ω. This operator is closable.8 Its closed extension Sa has a unique 1/2 polar decomposition Sa = Ja ∆a into an antiunitary operator Ja , which is called 1/2 the modular conjugation, and a positive operator ∆a , which is called the modular 1/2 operator. Ja is an involution.9 Sa , Ja , and ∆a are the objects of the so-called modular theory developed by Tomita and Takesaki.10 11 For each a ∈ S 2 , let ja be the orthogonal reflection at the plane a⊥ ∩ e⊥ 0, and for each ϕ ∈ D, define the test function ja ϕ ∈ D by ja ϕ(x) := ϕ(ja x).

(D) Modular P1 CT-symmetry. For each a ∈ S 2 , there exists an antilinear involution Ca in C such that for all c ∈ C and ϕ ∈ D, one has Ja F (c ⊗ ϕ)Ja = F t (Ca c ⊗ ja ϕ). The map a → Ja is strongly continuous.12 It will now be shown that pairs of modular P1 CT-reflections give rise to a strongly continuous representation of GR which exhibits Pauli’s spin-statistics connection. Lemma 3 Let K be a unitary or antiunitary operator in H such that KD = D and KΩ = Ω, and suppose there are a, b ∈ S 2 such that KF(a)K † = F(b). Then KJa K † = Jb , and K∆a K † = ∆b . † Proof. If A ∈ F(b), then KSa K † AΩ = KSa K AK Ω = A∗ Ω = Sb AΩ. The state ∈F(a)

ment now follows by the uniqueness of the polar decomposition.

In particular, this lemma implies kJa k = Ja ,

whence

Ja κ = κ† Ja

(4)

by definition of k. Using twisted locality, the lemma also implies κJa κ† = κ† Ja κ = J−a

(5)

8 By twisted locality, the operator κR † † −a κ is formally adjoint to Ra . Namely, if A ∈ κF(−a)κ and B ∈ F(a), then AΩ, Ra BΩ = AΩ, B ∗ Ω = BΩ, A∗ Ω = BΩ, κR−a κ† AΩ. Since κR−a κ† is densely defined, it follows that Ra is closable. 9 R2 = 1 implies S 2 = 1, so J ∆1/2 = S = S −1 = ∆−1/2 J ∗ , i.e., J 2 ∆1/2 = J ∆−1/2 J ∗ . a a a a a a a a a a a a a −1/2 ∗ Ja is positive, one obtains Ja2 = 1 and Ja ∆−1/2 Ja = ∆1/2 from the uniqueness Since Ja ∆a of the polar decomposition [3]. 10 The original work [26] directly applies to von-Neumann algebras, which are normed. But also for the present setting this structure has been applied earlier, e.g., in the classical papers of Bisognano and Wichmann [1, 2]. See, also, Ref. 14 for a monograph on the Tomita-Takesaki theory of unbounded-operator algebras. 11 i.e., the linear reflection with j a = −a, ja e0 = −e0 , and ja x = x for all x ∈ a⊥ ∩ e⊥ a 0 . 12 If one assumes covariance with respect to some strongly continuous representation of G R (which may also violate the spin-statistics connection), this is straightforward to derive; cf. Lemma 3. But covariance, as such, is not needed.

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857

which, in turn, implies Ja Jb Ja = J−ja b = Jja jb b = Jρ(a,b)b

(6)

by modular P1 CT-symmetry. Define a map W from MR into the unitary group of H by W (a, b) := Ja Jb . Lemma 4 (i) m ∼ n implies W (m) = W (n). (ii) W (m) = W (n) implies ρ(m) = ρ(n). Proof of (i). The proof of Lemma 2.4 in Ref. 5 can be taken without any relevant changes. Despite the fact that the Buchholz-Summers paper is confined to bosonic fields, which, in particular, implies Ja = J−a , it is straightforward to translate their proof to the present setting. This will not be spelled out here. The proof makes use of the continuous dependence of Ja from a assumed in Assumption (D). Proof of (ii). ρ(m) = ρ(n) would imply that there is some b ∈ S 2 such that ρ(m)b = ρ(n)b, so F(ρ(m)b) = F(ρ(n)b) by Assumption (A), whence W (m)F(b)W (m)∗ = W (n)F(b)W (n)∗ by Assumption (D), i.e., W (m) = W (n). ˜ : GR → W (MR ) is defined by W ˜ (π(m)) := W (m), By this lemma, a map W and another map ρW : W (MR ) → R is defined by ρW (W (m)) = ρ(m). The diagrams

(A)

π

MR

−→

ρ↓

R

←−

W

ρW

GR ˜ ↓W

and (B)

W (MR )

π

MR

−→

ρ↓

R

←−

ρ˜

ρW

GR ˜ ↓W

(7)

W (MR )

commute. Theorem 5 (i) There is a unique group product W on W (MR ) with the property that the diagram ◦ −→ MR MR × MR ↓π×π GR × GR

↓π

−→

˜ ×W ˜ ↓W W (MR ) × W (MR )

˜ ↓W W

−→ W (MR )

↓ ρW × ρW R×R ˜ is a homomorphism. commutes, i.e., W

GR

↓ ρW ·

−→

R

(8)

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B. Kuckert


(ii) W is the operator product in the algebra B(H) of bounded operators on H, ˜ is a representation. i.e., W ˜ of GR in C such that (iii) There is a representation D ˜ ˜ (g)F (c ⊗ ϕ)W ˜ (g)∗ = F (D(g)c ⊗ ρ˜(g)ϕ) W

for all g, c, ϕ,

(9)

where ρ˜(g)ϕ := ϕ(˜ ρ(g)−1 ·). Proof of (i). The diagram already commutes if the arrow representing W is omitted. For each g ∈ GR and each (a, b) ∈ π −1 (g) one has ˜ (±1)W ˜ (π(a, b)) = W ((±a, a) ◦ (a, b)) = W (±a, b) = W ˜ (±(π(a, b))), W ˜ (±1) ∼ ˜ (±g) ∼ so W = Z2 if and only if W = Z2 . ∼ ˜ ˜ If W (±1) = Z2 , then W is a bijection, so W is defined by ˜ (W ˜ −1 (U ) W ˜ −1 (V )). U W V := W ˜ (±1) ∼ If W = {1}, then ρW is a bijection, so W is defined by U W V := ρ−1 W (ρW (U ) · ρW (V )). ˙ R , the planes ˙ R . Given g, h ∈ G Proof of (ii). The statement is nontrivial only on G ˜ (g)⊥ and a ˜ (h)⊥ intersect in an at least one-dimensional subspace, so one can a choose (a, b) ∈ π −1 (g) and (c, d) ∈ π −1 (h) such that b = c is in this intersection. Then ˜ (π(a, b) π(c, d)) = W ˜ (π((a, b) ◦ (b, d))) W ˜ (π(a, d)) = W (a, d) =W = Ja Jd = Ja Jb Jb Jd ˜ (π(a, b))W ˜ (π(b, d)) = W (a, b)W (b, d) = W ˜ (π(a, b))W ˜ (π(c, d)). =W Proof of (iii). Define a map D from MR into the automorphism group Aut(C) of C by D(a, b) := Ca Cb . If (a, b) ∼ (c, d), then modular P1 CT-symmetry implies F (Ca Cb c ⊗ ja jb ϕ) = W (a, b)F (c ⊗ ϕ)W (a, b)∗ = W (c, d)F (c ⊗ ϕ)W (c, d)∗ = F (Cc Cd c ⊗ jc jd ϕ) = F (Cc Cd c ⊗ ja jb ϕ) for all c and all ϕ. Using Assumption (A.1), one obtains Ca Cb c = Cc Cd c for all c, so ˜ : GR → Aut(C) is defined by D(π(m)) ˜ D(a, b) = D(c, d), and a map D := D(m). ˜ ˜ This map D now inherits the representation property from W .

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Theorem 6 (Spin-statistics connection) F± (c ⊗ ϕ) =

1 ˜ (1 ± F (D(−1)c ⊗ ϕ)) 2

for all c and all ϕ. Proof. For each a ∈ S 2 one has ˜ (−1) = Ja J−a = Ja κJa κ† = Ja2 (κ† )2 = k, W so F (c ⊗ ϕ) = kF (c ⊗ ϕ)k ˜ (−1)F (c ⊗ ϕ)W ˜ (−1) =W ˜ (−1)F (c ⊗ ϕ)W ˜ (−1)† =W ˜ = F (D(−1)c ⊗ ϕ).

˜ is irreducible with spin s, then D(−1) ˜ If, in particular, D = e2πis , so F− = 0 for integer s and F+ = 0 for half-integer s.

4 PCT-symmetry In order to justify the term “modular P1 CT-symmetry”, one should show that this condition yields, at least in 1+3 dimensions, a full PCT-operator in a baseindependent fashion. Theorem 7 (PCT-symmetry) There exists an antiunitary involution Θ with the properties (i) Ja Jb Jc = Θ for each right-handed orthogonal basis (a, b, c) of e⊥ 0. (ii) There exists an antilinear involution C such that ΘF (c ⊗ ϕ)Θ = F (Cc ⊗ ϕ(− ·)). Proof. Let (a , b , c ) be a second right-handed orthonormal base, and define Θ := Ja Jb Jc . Then it follows from modular symmetry that Θ ΘF (c ⊗ ϕ)Ω = Θ ΘF (c ⊗ ϕ)ΘΘ Ω = F (Ca Cb Cc Ca Cb Cc c ⊗ ϕ)Ω ˜ = F (D(1)c ⊗ ϕ)Ω = F (c ⊗ ϕ)Ω. Since Ω is cyclic, this implies the statement.

˜ has to be If (a, b, c) is right-handed and (a , b , c ) is left-handed, then D(1) ˜ replaced by D(−1) in the above computation. Since J−a J−b J−c = κJa Jb Jc κ† , this is no surprise.

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Conclusion Both the classical geometry and the fundamental quantum field theoretic representations of the rotation group SO(3) and its universal covering group are based on reflection symmetries. At the classical level, the universal covering group GR can be constructed from P1 T-reflections. For a quantum field F with SO(3)-symmetry, a class of antiunitary P1 CT-operators exists that are fixed by the intrinsic structure of the respective field. Along precisely the same lines of argument used for the ˜ of GR is constructed. construction of GR , a covariant unitary representation W ˜ W exhibits Pauli’s spin-statistics connection.

Acknowledgments This work has been supported by the Stichting Fundamenteel Onderzoek der Materie and the Emmy-Noether programme of the Deutsche Forschungsgemeinschaft. I would like to thank Professor Arlt, Klaus Fredenhagen, and Reinhard Lorenzen for their critical comments and help concerning this manuscript.

Appendix. SU(2) versus GR can be described The isomorphism between the models SU (2) and GR of SO(3) as follows. 3 First recall the standard representation of SU (2) on 3R . Denote by σ1 , . . . , σ3 the Pauli matrices, and define x ˆ := x σ , x ∈ R . For each ν, the map ν ν ν x ˆ → Ad(±iσν )ˆ x is well known to implement the rotation [eν , π]. Since the parity transformation P is implemented by the map x ˆ → −ˆ x, one finds that for each ν, x implements the reflection jν . The determinants of the the map x ˆ → −Ad(±σν )ˆ Pauli matrices equal −1, and all of them are involutions. Now one can define an isomorphism J from S 2 onto the unitary matrices with determinant −1 by J(a) := a σ . The products of pairs of unitary matrices with determinant −1 yield all of SU (2).

References [1] J.J. Bisognano, E.H. Wichmann, On the Duality Condition for a Hermitean Scalar Field, J. Math. Phys. 16, 985–1007 (1975). [2] J.J. Bisognano, E.H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys. 17, 303 (1976). [3] O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd Edition, Springer 1987 (2nd printing 2002). [4] D. Buchholz, O. Dreyer, M. Florig, S.J. Summers, Geometric Modular Action and Spacetime Symmetry Groups, Rev. Math. Phys. 12, 475–560 (2000).

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[5] D. Buchholz, S.J. Summers, An algebraic characterization of vacuum states in Minkowski space. III. Reflection maps, Commun. Math. Phys. 246, 625–641 (2004). [6] N. Burgoyne, On the Connection of Spin with Statistics, Nuovo Cimento 8, 807 (1958). [7] C. Chevalley, Theory of Lie Groups, Princeton University Press 1946. [8] S. Doplicher, J.E. Roberts, Why There is a Field Algebra with a Compact Gauge Group Describing the Superselection Structure in Particle Physics, Commun. Math. Phys. 131, 51–107 (1990). [9] M. Florig, Geometric Modular Action, PhD-thesis, University of Florida, Gainesville, 1999. [10] J. Fr¨ ohlich, P.A. Marchetti, Spin statistics theorem and scattering in planar quantum field theories with braid statistics, Nucl. Phys. B356, 533–573 (1991). [11] D. Guido, R. Longo, An Algebraic Spin and Statistics Theorem, Commun. Math. Phys. 172, 517–534 (1995). [12] D. Guido, R. Longo, The Conformal Spin and Statistics Theorem, Commun. Math. Phys. 181, 11–36 (1996). [13] D. Guido, R. Longo, J.E. Roberts, R. Verch, Charged Sectors, Spin and Statistics in Quantum Field Theory on Curved, Rev Math. Phys 13, 125–198 (2001). [14] A. Inoue, Tomita-Takesaki Theory in Algebras of Unbounded Operators, Lecture Notes in Mathematics 1699, Springer, 1991. [15] B. Kuckert, A New Approach to Spin & Statistics, Lett. Math. Phys. 35, 319–335 (1995). [16] B. Kuckert, Borchers’ Commutation Relations and Modular Symmetries in Quantum Field Theory, Lett. Math. Phys. 41, 307–320 (1997). [17] B. Kuckert, Two Uniqueness Results on the Unruh Effect and on PCTSymmetry, Commun. Math. Phys. 221, 77–100 (2001). [18] B. Kuckert, Covariant Thermodynamics of Quantum Systems: Passivity, Semipassivity, and the Unruh Effect, Ann. Phys. (N. Y.) 295, 216–229 (2002). [19] B. Kuckert, Spin & Statistics in Nonrelativistic Quantum Mechanics. I, Phys. Lett. A 332, 47–53 (2004). [20] B. Kuckert, J. Mund, Spin & Statistics in Nonrelativistic Quantum Mechanics. II, Ann. Phys. (Leipzig) 14, 309–311 (2005).

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[21] R. Longo, On the spin-statistics relation for topological charges, in: Doplicher, S., Longo, R., Roberts, J.E., Zsido, L. (eds.): Operator Algebras and Quantum Field Theory, proceedings of the conference at the Accademia Nazionale dei Lincei, Rome 1996 (International Press). [22] G. L¨ uders, B. Zumino, Connection between Spin and Statistics, Phys. Rev. 110, 1450 (1958). [23] A.C. Manoharan, in: K.T. Mahanthappa, W.E. Brittin, (eds.) Mathematical Methods in Theoretical Physics, Gordon and Breach 1969 (New York) (conference held in Boulder 1968). [24] R.F. Streater, Local Field with the Wrong Connection Between Spin and Statistics, Commun. Math. Phys. 5, 88–98 (1967). [25] R.F. Streater, A.S. Wightman, PCT, Spin & Statistics, and All That, Benjamin, 1964. [26] M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras and Its Applications, Lecture Notes in Mathematics 128, Springer 1970 (New York). [27] W.G. Unruh, Notes on black-hole evaporation, Phys. Rev. D14, 870–892 (1976).

Bernd Kuckert II. Institut f¨ ur Theoretische Physik Luruper Chaussee 149 D-22761 Hamburg Germany email: [email protected] Communicated by Klaus Fredenhagen submitted 01/12/04, accepted 06/12/04




A Central Limit Theorem for the Spectrum of the Modular Domain∗ Zeév Rudnick

The statistics of the high-lying eigenvalues of the Laplacian on a Riemannian manifold have been intensively studied in the past few years by physicists working on “quantum chaos”. A number of fundamental insights have emerged from these studies, though to date these have yet to be set on rigorous footing. In the case the manifold at hand is of arithmetic origin, these studies are related to some profound number theoretical problems and as such may be more amenable to investigation. In this note I make use of the arithmetic structure of the modular domain to establish Gaussian fluctuations in its spectrum for certain smooth counting functions.

1 Background To set the stage, I start with describing some of what is currently believed to hold for the statistics of the eigenvalues. Given a compact Riemannian surface M , Weyl’s law for the eigenvalues Ej of the Laplacian says that the number of eigenvalues below x grows linearly with x: #{Ej ≤ x} ∼

area(M ) x, 4π

as x → ∞ .

Let n(E, L) be the number of levels in a window around E for which the leading term in Weyl’s law predicts L levels: n(E, L) = #{E −

2π 2π · L < Ej < E + · L} area(M ) area(M )

and more generally for a test function f define nf (E, L) :=

j

∗A

f(

area(M ) (Ej − E) · ) 4π L

(1.1)

version of this paper was presented in the IAS/Park City Mathematics Institute summer session on Automorphic Forms and Applications in July 2002 as part of the author’s mini-course on Arithmetic Quantum Chaos. Supported by a grant from the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities and a Leverhulme Trust Linked Fellowship at Bristol University.

864

Z. Rudnick


4π which counts the levels lying in a “soft” window of length area(M) L about E. In the above L = L(E) depends on the location E. In what follows we will usually write n(L) for n(E, L), the dependence on E implicitly understood. To study the statistical behavior of n(L) we need to consider E as random, drawn from a certain distribution on the line. We denote by · this kind of energy 2E averaging, e.g., F = E1 E F (E )dE . Weyl’s law leads us to expect that the mean value of n(L) is L and likewise that of nf (L) is L · f (x)dx.

1.1

Number variance

The variance of nf (E, L) from its expected value is: Σ2f (E, L) = |nf (L) − nf (L) |2 It is customary to express the number variance by means of an integral kernel KE (τ ), called the “form factor”, so that as E → ∞ ∞ ∞ u Σ2f (E, L) ∼ L · (Lf(Lτ ))2 KE (τ )dτ , f(u)2 KE ( )du = L −∞ −∞ ∞ where f(y) = −∞ f (x)e−2πixy dx is the Fourier transform of f . For “generic” surfaces, Berry [3, 4] argued that as E → ∞, the behavior of Σ2f (E, L) for L in the range1 1 ≺≺ L ≺≺ Lmax =

√ E

(1.2)

is universal, depending only on the coarse dynamical nature of the geodesic flow on the surface, and follows that of one of a small number of random matrix ensembles: If the flow is integrable (as in the case of a flat torus) then Σ2f (E, L) ∼ ∞ L · −∞ f (x)2 dx for L → ∞, as in the Poisson model of uncorrelated levels. If the flow is chaotic (as in the case of negative curvature) then the behavior is as in the Gaussian Orthogonal Ensemble (GOE): For the sharp window (f = 1[−1/2,1/2] ), this is given by Σ2 (E, L) ∼ π22 log L for L → ∞. For sufficiently smooth f , in the ∞ GOE we have Σ2f (E, L) ∼ 2 −∞ f(u)2 |u|du, that is the variance of sufficiently smooth statistics tends to a finite value as L → ∞. The form factors for the random models are K pois (τ ) ≡ 1, and 2|τ | − |τ | log(1 + 2|τ |), |τ | ≤ 1 GOE K (τ ) = . | 2 − |τ | log 1+2|τ |τ | > 1 2|τ |−1 , It is to be emphasized that the √ above behavior is only valid in the universal √ well understood regime 1 ≺≺ L ≺≺ E; for L E the integrable case is fairly √ (at a rigorous level), see the survey [5]: The variance grows as E (a classical result 1 The

symbol f (x) ≺≺ g(x) means that f (x)/g(x) → 0

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865

[12] in the case of the standard flat torus). In the chaotic case it is believed [3, 4] that generically, the number variance continues to be small as in the universal regime.

1.2

Fluctuations

Our main interest here is in the value distribution of the normalized linear statistic nf (E, L) − nf (L) Σ2f (E, L) as E varies. In all the statistical models (Poisson and GOE/GUE), it is a standard Gaussian [22, 14, 13]. √ In the integrable case, when L E, the distribution is known ([17], [5]), and is definitely not Gaussian. Inside the universal regime (1.2), the distribution is believed to be Gaussian in both the integrable [6] and chaotic [1, 9, 25] cases. In the special case of the standard flat torus, this has been proved in a small part √ of the universal regime near E [18].

1.3

The modular domain

√ We start with the upper half-plane H = {x + −1y : y > 0} equipped with the hyperbolic metric ds2 = y −2 (dx2 + dy 2 ), which has constant curvature equal −1. ∂2 ∂2 The Laplace-Beltrami operator for this metric is given by ∆ = y 2 ( ∂x 2 + ∂y 2 ). The orientation-preserving isometries of the metric ds2 are the linear fractional transformations P SL(2, R) = SL(2, R)/{±I}. The modular domain is the Riemann surface obtained by identifying points in the upper half plane which differ by a linear fractional transformation with integer coefficients, that is by elements of the modular group Γ := P SL(2, Z) = SL(2, Z)/{±I} . The resulting surface H/Γ is non-compact and has cone points, but has finite hyperbolic area: area(H/Γ) = π/3. The spectrum of the Laplacian on the modular domain has a continuous component. Nonetheless, Selberg [24] showed that a version of Weyl’s law holds for the discrete spectrum: If we write the eigenvalues of −∆ on L2 (H/Γ) in the form Ej = 1/4 + rj2 , then: #{rj ≤ T } = 2+log

π

area(H/Γ) 2 2 T T − T log T + c1 T + O( ) 4π π log T

(1.3)

2 where c1 = . π In the case of the modular domain, deviations from generic statistics were discovered [10, 7, 11]. Although the geodesic flow is chaotic, the local statistics

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of the spectrum seem Poissonian, and Bogomolny, Leyvraz and Schmit [8] argued that the behavior of the form factor is given by  √ √ c exp(c√2 Eτ ) , √1 ≺≺ τ ≺≺ log√ E 1 E E E √ KE (τ ) ∼ 1, τ log√ E E

for some constants c1 , c2 > 0, that is to say, in the universal regime we have √ √ √ ∞ 2c1 √LE 0 f(u)2 exp(c2 LEu )du, logEE ≺≺ L ≺≺ E 2 √ Σf (E, L) ∼ ∞ 1 ≺≺ L ≺≺ logEE . L · −∞ f(u)2 du, The only rigorous results known concern the closely related case where the modular group is replaced by quaternion groups: In the 1970’s Selberg [15, Chapter 2 2.18] gave a lower bound for the √ variance Σ (E, E) of the sharp counting function n(E) of the form Σ2 (E, E) E/(log E)2 . Luo and Sarnak [20] gave lower bounds for the averaged number variance of the sharp counting function n(E, L): √ √ √ 1 L 2 Σ (E, L )dL E/(log E)2 , E/ log E ≺≺ L ≺≺ E L 0 in the case of arithmetic (co-compact) groups. In the case of the hard window f = 1[−1/2,1/2] no upper bounds are currently available.

2 Formulation of results 2.1

Definition of the smooth counting functions

Let f be an even test function, whose Fourier transform f ∈ Cc∞ (R) is smooth and compactly supported, and normalized by requiring that ∞ f (x)dx = 1, −∞

and that

sup{|ξ| : f(ξ) = 0} = 1 .

The eigenvalues of the Laplacian are parametrized by Ej = 1/4 + rj2 . We define smooth counting functions by Nf,L (τ ) = f (L(rj − τ )) + f (L(−rj − τ )) (2.1) j≥0

This in essence carries the same information as (1.1). The relation between the expected number of levels L and the inverse width L of the momentum window is √ √ E area(H/Γ) E = . L= 2πL 6L

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The leading order behavior of Nf,L is given by ∞ 1 Nf,L (τ ) := {f (L(r − τ )) + f (L(−r − τ ))}M (r)dr 2π −∞ where

Γ Γ 1 area(H/Γ) r tanh(πr) − (1 + ir) − ( + ir) . 2 Γ Γ 2 (In keeping with tradition, I use the symbol Γ for both the modular group and the Gamma function.) The term Nf,L (τ ) is asymptotic to L: log τ 1τ area(H/Γ) ∞ τ )∼ . Nf,L (τ ) ∼ 2 f (x)dx + O( 4π L L 6L −∞ M (r) :=

2.2

The results

We will see that Nf,L − Nf,L has mean zero and show that the variance of Nf,L , when lim sup πL/ log T < 1, is asymptotic to 2κ ∞ 2 πLu 2 := du (2.2) f (u) e σL πL 0 p4 −2p3 +1 where κ = 1015 p=2 (1 + (p2 −1)3 ) = 1.328 . . . . Thus when the expected number 864 of levels L satisfies √ √ E area(H/Γ) < L ≺≺ T = E , log E the form factor KE (τ ) is given by

√ exp c2 Eτ √ KE (τ ) = c1 E

with c1 = 6κ/π, c2 = π/6. Our main result is that the fluctuations of Nf,L are Gaussian: Theorem 2.1 Assume that L → ∞ as T → ∞ but L = o(log T ). Then the limiting value distribution of (Nf,L − Nf,L )/σL is a standard Gaussian, that is x 2 Nf,L (τ ) − Nf,L (τ ) 1 du meas{τ ∈ [T, 2T ] : < x} = e−u /2 √ . lim T →∞ T σL 2π −∞ The reason that we need to assume that L = o(log T ) is that we prove Theorem 2.1 by the method of moments, and in computing the Kth moment we find Gaussian moments for L < cK log T , where cK → 0 as K grows. Plan of the paper: In Sections 3 and 4 we give some results on the hyperbolic conjugacy classes of the modular group. In Section 5 we use Selberg’s trace formula to express Nf,L (τ ) − Nf,L (τ ) as a sum Sf,L (τ ) over hyperbolic conjugacy classes plus a negligible term. The variance of Sf,L is computed in Section 6 and the higher moments in Section 7. We prove Theorem 2.1 in Section 8.

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3 The modular group 3.1

Conjugacy classes

To analyze Nf,L we use the Selberg trace formula, which for a discrete co-finite subgroup Γ ⊂ P SL(2, R) relates a sum over the spectrum of the Laplacian on L2 (H/Γ) with a sum over the conjugacy classes of the group Γ. We review some background material on these classes for the modular group P SL(2, Z). The conjugacy classes are divided into the class which consists of the identity element, hyperbolic, elliptic and parabolic classes. The hyperbolic conjugacy classes in Γ are represented by matrices P which are diagonalizable over the reals

λ and are conjugate to a matrix of the form with λ > 1. The norm of λ−1 2 P is defined as N (P ) = λ . The norm is therefore related to the trace of the corresponding group element by N (P )1/2 + N (P )−1/2 = | tr(P )|. We can write as each such P as P = P0k where P0 is primitive and k ≥ 1, where an element of Γ is primitive if it cannot be written as an essential power of another element. As is well known, primitive hyperbolic conjugacy classes correspond to closed geodesics on the Riemann surface H/Γ. In the case of the modular group Γ = P SL(2, Z), the traces are integers. If P is a hyperbolic class with trace | tr(P )| = n, n > 2 then its norm is √ n + n2 − 4 2 ) . (3.1) N (n) = ( 2 For the modular group, primitive hyperbolic conjugacy classes are parametrized by indefinite binary quadratic forms as follows (cf. [23]): Take a binary quadratic form Qa,b,c (x, y) := ax2 + bxy + cy 2, with a, b, c ∈ Z. The discriminant of Qa,b,c is d := b2 − 4ac. The form Qa,b,c is indefinite iff d > 0. We assume that d is not a perfect square. We say that Qa,b,c is primitive if gcd(a, b, c) = 1. Two binary quadratic forms

Q, Q are equivalent if Q (x, y) = Q(ax+by, cx+dy) for an element a b γ = of SL(2, Z); since the forms are quadratic, they are also equivalent c d under −γ and hence equivalence is over P SL(2, Z). Let h(d) be the number of equivalence classes of primitive binary quadratic forms of discriminant d. The automorphs of Qa,b,c are all of the form

1 (t − bu) −cu ±P (t, u) = ± 2 1 au 2 (t + bu) where (t, u) solve the Pellian equation t2 − du2 = 4

(3.2)

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If u = √ 0 then these are hyperbolic elements of SL(2, Z) with norm N (P ) = (t + u d)2 and trace t. √ Let d = 12 (td + ud d) (td , ud > 0) be the fundamental solution of (3.2). Then the matrix P (td , ud ) is a primitive hyperbolic matrix P0 of trace tr(P0 ) = td and norm N (P0 ) = 2d . It turns out that in this way we get a bijection between equivalence classes of primitive binary quadratic forms and conjugacy classes of primitive hyperbolic matrices in P SL(2, Z). Thus the number of primitive hyperbolic conjugacy classes of norm 2d is precisely the class number h(d).

3.2

The amplitude β(n)

We define, for n > 2, β(n) :=

1 2

tr(P )=n

log N (P0 ) N (P )1/2 − N (P )−1/2

(3.3)

the sum over all conjugacy classes {P } in P SL(2, Z) with | tr(P )| = n, equivalently with norm N (n) given by (3.1). These quantities turn out to be crucial in our analysis. The factor 1/2 in the definition is inserted among other reasons to give numbers with mean value 1 (as can be seen from the Prime Geodesic Theorem): β(n) ∼ N, as N → ∞ . n≤N

As representatives of the conjugacy classes of matrices with trace n > 2 we can take the matrices P0k = P (td , ud )k = P (n, u) where d runs over all discriminants, k ≥ 1 and n2 − du2 = 4, n > 2, u ≥ 1. Thus we see that β(n) =

d,u≥1 n2 −du2 =4

h(d) log d √ . du2

(3.4)

Dirichlet’s class number formula allows us to use (3.4) to express β(n) in terms of Dirichlet L-functions: For a discriminant d on associates the quadratic character χd given by χd (p) = dp for p an odd prime, χd (2) = 1 if d ≡ 1 mod 8, χd (2) = −1 if d ≡ 5 mod 8 and χd (−1) = 1. The associated L-function is L(s, χd ) = −s , Re(s) > 1. Dirichlet’s class number formula is n≥1 χd (n)n h(d) log d =

√ dL(1, χd ) .

Inserting the class number formula into (3.4) we find that β(n) =

d,u≥1:du2 =n2 −4

1 L(1, χd) . u

(3.5)

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As a consequence, one can get an upper bound of β(n) = O((log n)2 ) by using L(1, χd ) log d. What is more useful to us is that, in the mean square, β(n) is constant: Lemma 3.1 (M. Peter [21])

β(n)2 ∼ κN,

N →∞

(3.6)

n≤N

where κ is given by the product over primes κ=

p4 − 2p3 + 1 1015 (1 + ) = 1.328 . . . 864 (p2 − 1)3

(3.7)

p=2

This (complicated) expression was derived heuristically by Bogomolny, Leyvraz and Schmit [8] and proven by Manfred Peter [21], who uses the expression (3.5) for β(n) in terms of L(1, χ) and methods related to work on moments of class numbers [2]. For an extension to the case of congruence groups, see [19].

4 The length spectrum We will need to study alternating sums of the form K

± log N (nj ).

j=1

The first question is to when these alternating sums vanish. We say that a relation K

ηj log N (nj ) = 0,

ηj = ±1

(4.1)

j=1

is non-degenerate if no sub-sum vanishes, that is if there is no proper subset S ⊂ {1, . . . , K} for which j∈S ηj log N (nj ) = 0. The existence of non-degenerate relations (4.1) forces severe constraints. To explain these, recall that N (n) is a √ unit in the real quadratic field Q( n2 − 4). We claim that such such relations can occur only if all these units lie in the same quadratic field. K Lemma 4.1 Let j= ± log N (nj ) = 0 be a non-degenerate relation. Then all the norms N (ni ) lie in the same quadratic field, that is for some common d we have n2i − 4 = dfi2 for all i. Proof. We can write each norm as a power of the fundamental unit of the quadratic field in which it lies. Thus it will suffice to show that if F1 , . . . , FK be distinct real

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quadratic fields, then the fundamental units i of Fi are multiplicatively independent. Let E = F1 ∨ · · · ∨ FK be the compositum of the fields Fi . This is a Galois extension of the rationals with Galois group G = Gal(E/Q) an elementary Abelian 2-group (Z/2Z)s , for some s ≤ K. If we denote by UE the unit group of E, then G acts on UE and hence we get a linear representation on the vector space Q ⊗ UE . χ (σ) We claim that the i are eigenvectors of G, that is σ(i ) = i i for all σ ∈ G, where χi : G → {±1} are distinct characters. This forces them to be multiplicatively independent. Indeed, since we have an Abelian extension, all subfields are Galois and in particular Fi are preserved by G. Since the unit group is also preserved this means that under the action of any element σ ∈ G, i is taken to a unit of Fi which χi (σ) . The is necessarily ±1 i . That is we have a character χi of G with σ(i ) = i characters χi are distinct since the kernel of χi is precisely Gal(E/Fi ). K We next get a lower bound for j=1 ± log N (nj ) in the case it is non-zero. Lemma 4.2 i) If m = n then | log N (m) − log N (n)| ii) Suppose

K j=1

1 . min(m, n)

± log N (nj ) is nonzero. Then |

K j=1

± log N (nj )|

K j=1

1

2K−1 −1/2 N (nj )

Proof. i) Indeed, since log N (n) = 2 log n + O(1/n2 ), if m = n, say m > n, then log N (m) − log N (n) = 2 log 1 n,

n+1 Since log m n ≥ log n

we find

log N (m) − log N (n) ii) Let λj = |α − 1| ≤ 1/2 then

nj +

√ 2

|

n2j −4

K

m 1 + O( 2 ) . n n

1 1 = . n min(m, n)

so that N (nj ) = λ2j , and set α =

K j=1

λ±1 j . If

± log N (nj )| = 2| log α| |α − 1|.

j=1

So it suffices to give a lower bound for |α − 1|, assuming α = 1. This follows from Liouville’s theorem on Diophantine approximation of algebraic numbers by

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√ √ rationals; we give an explicit proof as follows: Let E = Q( n1 , . . . , nK ), which is a Galois extension of the rationals with Galois group G = Gal(E/Q) which is an elementary abelian 2-group of order 2s , for some s ≤ K. Moreover α ∈ E is an algebraic integer, and hence the norm NE/Q (α − 1) is a nonzero rational integer, hence has absolute value at least 1. Thus |NE/Q (α − 1)| = |α − 1| |ασ − 1| ≥ 1. id=σ∈G

for all σ ∈ G, we have Since λσj = λ±1 j K

|ασ − 1| ≤

λj + 1.

j=1

Thus

1 1 |α − 1| ≥ K K . |G|−1 ( j=1 λj + 1) ( j=1 N (nj ))(2K −1)/2

5

An expansion for Nf,L

5.1

The Selberg trace formula

We will transform Nf,L by using the Selberg trace formula [24]: Let g ∈ Cc∞ (R) be an even, smooth and compactly supported function, and let ∞ g(u)eiru du h(r) = −∞

so that g(u) =

1 2π

∞

h(r)e−iru dr .

−∞

The Selberg trace formula for a discrete co-compact sub-group Γ ⊂ P SL(2, R) with no elliptic elements is the identity [24] area(H/Γ) ∞ h(rj ) = h(r)r tanh(πr)dr 4π −∞ j≥0 (5.1) log N (P0 ) + g(log N (P )) N (P )1/2 − N (P )−1/2 {P } hyperbolic

where the sum is over all hyperbolic conjugacy classes of Γ. In the case of the modular group, the hyperbolic terms can be written as β(n)g(log N (n)) (5.2) 2· n>2

where the amplitude β(n) is given by (3.3).

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For groups with elliptic elements, there is an extra contribution to the RHS of (5.1) which is a sum over the finitely many conjugacy classes of elements E of finite order m ≥ 2: m {E} k=1

1 m sin(πk/m)

∞

h(r) −∞

e−2πkr/m dr. 1 + e−2πr

(5.3)

For discrete groups whose fundamental domain is non-compact but of finite volume, that is with cusps, there are extra terms coming from the contribution of the continuous spectrum and parabolic elements. For Γ = P SL(2, Z), these terms are given explicitly by [16]: g(0) log

1 π − 2 2π

∞

h(r) −∞

Γ Γ 1 (1 + ir) + ( + ir) dr Γ Γ 2 ∞ Λ(n) g(2 log n) (5.4) +2 n n=1

where Λ(n) is the von Mangold function.

5.2

Transforming Nf,L

We now apply the trace formula to derive an alternative expression for Nf,L . Taking h(r) = f (L(r − τ )) + f (L(−r − τ )) so that g(u) =

1 u −iτ u f( ) e + eiτ u 2πL 2πL

we find that Nf,L (τ ) = Nf,L (τ ) + Sf,L (τ ) + E

(5.5)

where: • The term Nf,L is given by the contribution of the identity class to (5.1) and part of the parabolic terms in (5.4): ∞ 1 Nf,L (τ ) = {f (L(r − τ )) + f (L(−r − τ ))}M (r)dr 2π −∞ where

Γ Γ 1 area(H/Γ) r tanh(πr) − (1 + ir) − ( + ir) . 2 Γ Γ 2 By Stirling’s formula, we have log τ 1 ∞ τ ). Nf,L (τ ) = f (x)dx + O( 6 −∞ L L M (r) =

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• The term Sf,L (τ ) is the contribution of the hyperbolic classes (5.2): 1 log N (n) −iτ log N (n) ) e Sf,L (τ ) = β(n)f( + eiτ log N (n) . πL n>2 2πL

(5.6)

The sum (5.6) contains only terms with log N (n) ≤ 2πL, that is n < eπL . As we will see below, it is the term Sf,L (τ ) which is responsible for the 2 fluctuations of Nf,L (τ ), and its variance is asymptotic to σL . As we can see from the formula (5.6), since f has compact support we have Sf,L (τ ) ≡ 0 for L 1. • E is the contribution of the elliptic classes (5.3) and the remaining part of the parabolic contribution (5.4), namely ∞ π 1 Λ(n) log n 1 f (0) log + f( )2 cos(2τ log n). πL 2 πL n=1 n πL

(5.7)

E is easily seen to be negligible, that is E = o(σL ). Indeed, the contribution of the elliptic elements is easily seen to be O(e−const.τ /L). As for (5.7), this is bounded as L → ∞ by (say) Mertens’ theorem. Moreover the mean value of (5.7) clearly vanishes as T → ∞. We thus see that the difference between the centered counting function Nf,L (τ ) − Nf,L and the sum Sf,L (τ ) over hyperbolic conjugacy classes is negligible relative to the standard deviation σL of Sf,L (τ ), and thus for our purposes we need only investigate the statistics of Sf,L (τ ).

6 The mean and variance of Sf,L 6.1

The averaging procedure

We define an averaging procedure by taking a non-negative weight ∞ function w ≥ 0, which is smooth and compactly supported in (0, ∞), with −∞ w(x)dx = 1. We then get an averaging operator: 1 ∞ τ F (τ )w( )dτ . F w,T := T −∞ T Let Pw,T be the associated probability measure:

t 1 ∞ 11A (f (t))w Pw,T (f ∈ A) = dt . T −∞ T Note that the requirement w ∈ Cc∞ (0, ∞) implies that the Fourier transform of w decays rapidly: w(x) |x|−A , as |x| → ∞ for all A > 1. In the concluding Section 8 we will relax the restrictions on w to allow other averages, e.g., w = 1[1,2] so that we take t uniformly distributed in [T, 2T ], or w(t) = 2t1[1,√2] when we take the eigenvalue λ = 1/4 + t2 uniformly distributed in [E, 2E], E = 1/4 + T 2 .

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875

The expected value of Sf,L

We will first show that the mean value Sf,L w,T tends to zero as T → ∞ provided L = O(log T ): Averaging (5.6) we find Sf,L w,T =

1 πL

β(n)f(

2 0.

Proof. To compute (Sf,L )2 w,T , use (5.6) to get (Sf,L )2 w,T =

1 (πL)2

β(m)β(n)f(

m,n<eπL

×

1 ,2 =±1

log N (m) log N (n) )f ( ) 2πL 2πL

w(

T (1 log N (m) + 2 log N (n))). 2π

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We now deduce that as T → ∞, the only non-vanishing contribution is from the “diagonal terms” where 1 = −2 and N (m) = N (n) that is m = n. If m = n we may use Lemma 4.2 to get a lower bound | log N (m) ± log N (n)|

1 . min(m, n)

(6.2)

To be included in the sum, we need N (m), N (n) ≤ e2πL , that is m, n ≤ eπL , and so T min(m, n) A eπL A w( (1 log N (m) + 2 log N (n))) 1. This goes to zero if πL ≤ (1 − δ)(log T ) for some δ > 0, which we assume. The diagonal terms m = n give log N (n) 2 1 ) β(n)2 f( (πL)2 n>2 2πL (where we used w(0) = 1). Since there are two such terms (corresponding to 1 = −2 = +1 or −1), we have the total diagonal contribution being 2

1 log N (n) 2 ) . β(n)2 f( 2 (πL) n>2 2πL

This can be evaluated asymptotically as L → ∞ using Peter’s formula (Lemma 3.1) to give 2κ ∞ 2 πLu 2 du =: σL . f (u) e πL 0 Thus we find

if lim sup πL/ log T < 1.

2 (Sf,L )2 w,T ∼ σL

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7 Higher moments We can now show that Sf,L (τ ) has Gaussian moments: Theorem 7.1 For K ≥ 3 the Kth moment of Sf,L /σL converges to that of a normal Gaussian provided that L → ∞ with T but that L = o(log T ):  (2k)!  k!2k ,  Sf,L (τ ) K lim ( ) = T →∞  σL  w,T 0,

7.1

K = 2k even K odd.

Reduction to the pre-diagonal

By (5.6) the Kth moments of Sf,L is given by (Sf,L )K w,T =

1 (πL)K

K

log N (nj ) ) β(nj )f( 2πL πL j=1

n1 ,...,nK <e

×

w(

ηj =±1

T ( ηj log N (nj ))). 2π j

(7.1)

We now show that as T → ∞, the only (possibly) non-vanishing contribution to (7.1) is for terms satisfying: K

ηj log N (nj ) = 0

j=1

that is we have (Sf,L )K w,T =

1 K (πL) η =±1 j

j

K

β(nj )f(

ηj log N (nj )=0 j=1

log N (nj ) ) 2πL + O(

eαK L ) (7.2) T γK

for some αK , γK > 0. Since L = o(log T ) the remainder term vanishes as T → ∞. K To prove this, recall that by Lemma 4.2, if j=1 ηj log N (nj ) = 0 then for some δK > 0 |

K j=1

 ηj log N (nj )| K 

K j=1

−2K−1 +1/2 N (nj )

e−πδK L

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since only terms with N (nj ) < e2πL appear in (7.1). Thus for these terms we have w(

T eπL·δK A ( ) . ηj log N (nj ))) ( 2π j T

Replacing β(n) by log2 n L2 in (7.1) gives that the contribution of the terms with K j=1 ηj log N (nj ) = 0 is dominated by 1 LK

L2K (

n1 ,...,nK <eπL

eπL(K+AδK ) eπL·δK A ) LK . T TA

Since L = o(log T ), this vanishes as T → ∞ (in fact we need only assume that L < cK log T for this, if cK is sufficiently small). This proves (7.2).

7.2

Off-diagonal terms

In (7.2) we consider the sum of non-diagonal terms, that is terms for which there is at least one index j such that nj = ni for all i = j. To handle these, we use Lemma 4.1 which forces the relation K

N (nj )ηj = 1

j=1

to decompose into a union of such relations. Thus there is a decomposition {1, 2 . . . , K} = Sj so that in each subset Sj we have

N (ni )ηi = 1

(7.3)

i∈Sj

andthe norms N (ni ) = (ni + n2i − 4)/2 lie in the same real quadratic field Q( dj ) for all i ∈ Sj . In the diagonal case there are K/2 such sets, e.g., S1 = {1, K/2 + 1}, S2 = {2, K/2 + 2}, . . . and the identities are of the form N (nj ) N (nK/2+j )−1 = 1, j = 1, . . . , K/2. In the off-diagonal case we assume that there is a subset Sj contains at least 3 elements. The number r of subsets is then at most (K − 1)/2, since K=

r

#Sj ≥ 3 + 2(r − 1).

j=1

To count such tuples of ni , we denote for each subset Sj by dj the common value of the square-free kernel of n2i − 4, i ∈ Sj and then write n2i − 4 = dj fi2 ,

i ∈ Sj .

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Let (dj ) be the fundamental unit of the field Q( dj ) and write N (ni ) = (dj )2ki , i ∈ Sj . Since log N (ni ) L we have ki L/ log (dj ), i ∈ Sj and the relation (7.3) implies i∈Sj ±ki = 0. Thus for each subset Sj there are at most O((L/ log (dj ))#Sj −1 ) solutions of (7.3) with log N (ni ) L. Recall that we are summing over log N (n) ≤ 2πL. Using β(n) (log n)2 2 L we find that the off-diagonal contribution is bounded by the sum over all partitions {1, . . . , K} = rj=1 Sj of L2

r

j=1 (dj )≤eπL

(

L )(#Sj −1)/2 LK (#{d fundamental : (d) ≤ eπL })r log (dj ) (7.4)

where r ≤ (K − 1)/2 is the total number of subsets Sj in our partition. Lemma 7.2 The number of fundamental discriminants d > 0 for which (d) < X is O(X 1+δ ) for all δ > 0. Proof. We need to bound the number of√fundamental discriminants d for which the fundamental solution (d) = (xd + dyd )/2 of x2 − dy 2 = 4 is at most X. Since (d) ∼ xd , this is equivalent to bounding the number of fundamental d’s for which xd X. In turn, this number is majored by the number ν(X) of all triples (d, x, y) of positive integers, with d ≡ 0, 1 mod 4, for which x2 − dy 2 = 4 and x < X, which is the sum ν(X) =

#{d, y ≥ 1, d ≡ 0, 1 mod 4 : dy 2 = x2 − 4} .

x<X

Since for x = 2 the number of pairs (d, y) with dy 2 = x2 − 4 is at most the number of divisors τ (x2 − 4) of x2 − 4, we find that ν(X) ≤

τ (x2 − 4)

2<x<X

xδ X 1+δ

2<x<X

for all δ > 0, by virtue of the bound τ (n) nδ for all δ > 0. Note: A more refined argument [23, Lemma 4.2] shows that ν(X) is asymp1+δ ) by O(X). totic to 35 16 X, so that one can replace the bound O(X Thus we find that (7.4) is bounded by LK e(1+δ)πLr LK e(1+δ)πL(K−1)/2 for all δ > 0. Since σL e(1−)πL/2 for all > 0, this shows that the sum of K−1+ the off-diagonal terms is O(σL ), for all > 0. To prove Theorem 7.1 it thus suffices to evaluate the diagonal contributions.

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The diagonal contribution

Assume now that there is the same of + signs as there are − signs. That number is K = 2k is even, and there are 2k k such choices of signs. For simplicity assume the first k are + and the last k are −. Thus we have to evaluate the sum 1 (πL)2k

2k

k

j=1 N (nj )=

2k

j=k+1 N (nj )

log N (nj ) ). β(nj )f( 2πL j=1

(7.5)

There are k! ways to pair off variables from the first k and the last k, such as the pairing nj = nk+j , 1 ≤ j ≤ k. Each such pairing contributes a term 1 (πL)2k

log N (n) 2 ) β(n) f( 2πL n>2 2

k

∼(

2 σL )k . 2

There are overlaps between the different ways of pairing off variables, which correspond to intersection of diagonals such as n1 = n2 = n3 = n4 . The contribution of these was already estimated in the study of the non-diagonal terms, as they correspond to relations (7.3) where some subset has two elements. more than Thus the total contribution of diagonal terms to (Sf,L )2k w,T is asymptotically

2k (2k)! 2k σ2 σ . · k! · ( L )k = k 2 k!2k L This proves Theorem 7.1.

8 Conclusion Since the Gaussian distribution is determined by its moments, Theorem 7.1 implies Theorem 8.1 Assume that L → ∞ as T → ∞ but L = o(log T ). Then x 2 du Nf,L (τ ) − Nf,L (τ ) lim Pw,T ( < x) = e−u /2 √ , T →∞ σL 2π −∞ So far we have assume that the weight function w defining the averages is in Cc∞ (0, ∞). To deduce the results for the standard averages (w = 1[1,2] ) as in Theorem 2.1, one proceeds by approximating 1[1,2] by “admissible” w’s in a standard fashion, see, e.g., [18]. The details are as follows: Fix > 0, and approximate the indicator function 11[1,2] above and below by smooth functions χ± ≥ 0 so that χ− ≤ 11[1,2] ≤ χ+ , where both χ± and their Fourier transforms are smooth and of rapid decay, and so that their total masses are within of unity: | χ± (x)dx − 1| < . Now set ω± := χ± / χ± . Then ω± are “admissible” and for all t, (1 − )ω− (t) ≤ 11[1,2] (t) ≤ (1 + )ω+ (t). (8.1)

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Now ! " ∞

Sf,L (τ ) t Sf,L (τ ) meas t ∈ [T, 2T ] : ∈A = 11A 11[1,2] dt σL σ T L −∞ and since (8.1) holds, we find ! (1 − )Pω− ,T

! " " Sf,L (τ ) Sf,L (τ ) 1 ∈ A ≤ meas t ∈ [T, 2T ] : ∈A σL T σL " ! Sf,L (τ ) ∈A . ≤ (1 + )Pω+ ,T σL

By Theorem 8.1 we find that ! " 2 1 Sf,L (τ ) 1 (1 − ) √ e−x /2 dx ≤ lim inf meas t ∈ [T, 2T ] : ∈A T →∞ T σL 2π A with a similar statement for lim sup; since > 0 is arbitrary this shows that the limit exists and equals ! " 2 Sf,L (τ ) 1 1 lim meas t ∈ [T, 2T ] : ∈A = √ e−x /2 dx T →∞ T σL 2π A which proves Theorem 2.1. The same consideration applies to other positive statistics, such as the number variance.

References [1] R. Aurich, J. Bolte and F. Steiner, Universal signatures of quantum chaos, Phys. Rev. Lett. 73, no. 10, 1356–1359 (1994). [2] M.B. Barban, The “Large Sieve” method and its applications in the theory of numbers, Russian Math. Surveys 21 , 49–103 (1966). [3] M.V. Berry, Semiclassical theory of spectral rigidity, Proc. Roy. Soc. London Ser. A 400, no. 1819, 229–251 (1985). [4] M.V. Berry, Fluctuations in numbers of energy levels, Stochastic processes in classical and quantum systems (Ascona, 1985), 47–53, Lecture Notes in Phys., 262, Springer, Berlin, 1986. [5] P. Bleher, “Trace formula for quantum integrable systems, lattice-point problems and small divisors”, Emerging Applications of Number Theory, D.A. Hejhal, J. Friedman, M.C. Gutzwiller, A.M. Odlyzko, eds. (Springer, 1999) pp. 1–38.

882

Z. Rudnick


[6] P. Bleher and J. Lebowitz, Energy-level statistics of model quantum systems: universality and scaling in a lattice-point problem, J. Statist. Phys. 74, 167– 217 (1994). [7] E. Bogomolny, B. Georgeot, M.-J. Giannoni and C. Schmit, Chaotic billiards generated by arithmetic groups, Phys. Rev. Lett. 69, no. 10, 1477–1480 (1992). [8] E. Bogomolny, F. Leyvraz and C. Schmit, Distribution of eigenvalues for the modular group, Comm. Math. Phys. 176, no. 3, 577–617 (1996). [9] E. Bogomolny and C. Schmit, Semiclassical computations of energy levels, Nonlinearity 6, no. 4, 523–547 (1993). [10] O. Bohigas, M.-J. Giannoni, and C. Schmit, in “Quantum Chaos and Statistical Nuclear Physics”, edited by Thomas H. Seligman and Hidetoshi Nishioka, Lecture Notes in Physics Vol. 263 (Springer-Verlag, Berlin, 1986), p. 18. [11] J. Bolte, G. Steil and F. Steiner, Arithmetic Chaos and Violation of Universality in Energy Level Statistics, Phs. Rev. Lett. 69, no. 15, 2188–2191 (1992). ¨ [12] H. Cramér, Uber zwei Sätze des Herrn G.H. Hardy, Math. Z. 15, 201–210 (1922). [13] O. Costin and J. Lebowitz, Gaussian fluctuations in random matrices, Phys. Rev. Lett 75, no 1, 69–72 (1995). [14] P. Diaconis and M. Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A, 49–62 (1994). [15] D.A. Hejhal, The Selberg Trace Formula for P SL(2, R), Lecture Notes in Mathematics, vol. 548. Berlin, Heidelberg, New York: Springer 1976. [16] D.A. Hejhal, The Selberg Trace Formula for P SL(2, R), Volume 2, Lecture Notes in Mathematics, vol. 1001. Berlin, Heidelberg, New York: Springer 1983. [17] D.R. Heath-Brown, The distribution and moments of the error term in the Dirichlet divisor problem, Acta Arithmetica 60, 389–415 (1992). [18] C.P. Hughes and Z. Rudnick, On the distribution of lattice points in thin annuli, IMRN 13, 637–658 (2004). [19] V. Lukianov, Ph.D. thesis, Tel Aviv university (in preparation). [20] W. Luo and P. Sarnak, Number Variance for Arithmetic Hyperbolic Surfaces, Commun. Math. Phys. 161, 419–432 (1994). [21] M. Peter, The correlation between multiplicities of closed geodesics on the modular surface, Comm. Math. Phys. 225, no. 1, 171–189 (2002).

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[22] H.D. Politzer, Random-matrix description of the distribution of mesoscopic conductance, Phys. Rev. B 40, no. 17, 11917–11919 (1989). [23] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15, no. 2, 229–247 (1982); corrigenda in J. Number Theory 16, no. 2, 284 (1983). [24] A. Selberg, Harmonic Analysis, in Collected Papers. Vol. I, 626–674 SpringerVerlag, Berlin, 1989. [25] F. Steiner, Quantum Chaos, in Universit¨ at Hamburg: Schlaglichter der Forschung zum 75. Jahrestag, edited by R. Ansorge (Reimer, Hamburg 1994), 542–564. Zeév Rudnick Raymond and Beverly Sackler School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israel email: [email protected] Communicated by Jens Marklof submitted 02/08/04, accepted 09/11/04




Long Time Propagation and Control on Scarring for Perturbed Quantized Hyperbolic Toral Automorphisms Jean-Marc Bouclet and Stephan De Bièvre Abstract. We show that on a suitable time scale, logarithmic in , the coherent states on the two-torus, evolved under a quantized perturbed hyperbolic toral automorphism, equidistribute on the torus. We then use this result to obtain control on the possible strong scarring of eigenstates of the perturbed automorphisms by periodic orbits. Our main tool is an adapted Egorov theorem, valid for logarithmically long times.

1 Introduction One of the main results in quantum chaos is the Schnirelman theorem. It states that, if a quantum system has an ergodic classical limit, then almost all sequences of its eigenfunctions converge, in the classical limit, to the Liouville measure on the relevant energy surface [7, 15, 20, 24]. It is natural to wonder if the result holds for all sequences (a statement commonly referred to as “unique quantum ergodicity”). This has been proven to be true for the (Hecke) eigenfunctions of the Laplace-Beltrami operator of a certain class of constant negative curvature surfaces [17] and has been conjectured to be true for all such surfaces [19]. It also has been proven to be wrong for quantized toral automorphisms in [11]. In that case, sequences of eigenfunctions exist with a semiclassical limit having up to half of its weight supported on a periodic orbit of the dynamics. This phenomenon is referred to as (strong) scarring. In [5, 12], it is shown that this last result is optimal: if a measure is obtained as the limit of eigenfunctions then its pure point component can carry at most half of its total weight. Except for the Schnirelman theorem, which holds in very great generality, all cited results are proven by exploiting to various degrees special algebraic or number theoretic properties of the systems studied. It is one of the major challenges in the field to device proofs and obtain results that use only assumptions on the dynamical properties of the underlying classical Hamiltonian system, such as ergodicity, mixing or exponential mixing, the Anosov property, etc. without relying on special algebraic properties. It is argued in [4, 5, 12] for example, that this will require a good control on the quantum dynamics for times that go to infinity (at least) logarithmically as the semiclassical parameter goes to zero: t ≥ k− ln for some constant k− > 0.

886

J.-M. Bouclet and S. De Bièvre


It is well known that such control is in general hard to obtain especially since a good lower bound on k− is needed. In this paper, we concentrate on the quantized perturbed hyperbolic automorphisms of the 2d-torus, which are known to be Anosov systems classically. For those systems, we first prove an Egorov theorem valid for times proportional to ln , with an explicit control on the proportionality constant k− (Theorem 3.1). This result is obtained by adapting the techniques of [8]. We then combine this result with recent sharp estimates on the exponential mixing of the classical dynamics [3] to study the long time evolution of coherent states (Theorem 4.7), showing that on a sufficiently long logarithmic time scale, those evolved coherent states equidistribute on the torus. Roughly, the result is that for all f ∈ C ∞ (T2 ), t a W t a f (x) dx → 0, → 0, (1.1) Q(f, t, ) ≡ U ϕ,κ , Op (f )U ϕ,κ H (κ) −

T2

for times k− |ln | ≤ t ≤ k+ |ln | ,

0 ≤ k− ≤ k+ .

Here U is the unitary quantum dynamical evolution operator, OpW (f ) is the Weyl quantization of f , and ϕa,κ is a coherent state at the point a of the two-torus T2 . For detailed definitions, we refer to the following sections. This result generalizes results obtained in [4] for unperturbed hyperbolic automorphisms. To prove it, we prove an estimate of the type Q(f, t, ) ≤ ϕa,κ , (U−t OpW (f )Ut − OpW (f ◦ Φt ))ϕa,κ H (κ) a W t a + ϕ,κ , Op (f ◦ Φ ))ϕ,κ H (κ) − f (x) dx

≤ 1 (e

γq t

) + 2 (

−1 −γc t

e

T2

).

Here 1 and 2 are functions tending to zero when their argument does. The first term comes from the error term in the Egorov theorem, whereas the second one involves a classical mixing rate γc . It is obvious that this estimate leads to the result only if γq < γc . One therefore needs γc to be large (fast mixing) and γq to be small. Sharp results on the classical mixing rates of Anosov systems are hard to come by, but for some Anosov maps, among which the perturbed toral automorphisms that are the subject of this paper, such results have become available recently [3]. The remaining difficulty resides therefore in controlling the exponent in the error in the Egorov theorem. This is dealt with in the next section. We note that, although we prove the Egorov theorem for systems on the 2dtorus, we only prove the result above in full generality for d = 1. Indeed, denoting for arbitrary d by Γmin and Γmax the smallest and largest Lyapounov exponents of the system, we prove in Section 2 that, essentially, γq = 32 Γmax . On the other hand, the available estimates on the classical mixing rate [3] yield in our context here γc = 2Γmin. Of course, when d = 1, Γmax = Γmin and we have γq < γc as

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needed. This leads to (1.1). For d > 1, on the other hand, our proof of (1.1) still goes through, but only under an artificial “pinching” condition on the Lyapounov exponents of the type 3Γmax < 4Γmin . As an application of the above result, we finally show how to use the information obtained on the evolved coherent states in combination with the basic strategy of [5, 6] to gain some control on the scarring of eigenfunctions (Theorem 4.9, Corollary 4.10). Roughly speaking, we show that if a sequence of eigenfunctions of a quantized perturbed hyperbolic toral automorphism converges to a delta measure on a finite union of periodic orbits, then it must do so slowly. An improvement on this result (basically, on how slowly) has been announced recently in [13]. We do not expect this result to be optimal: indeed, it is expected, as in the case of unperturbed automorphisms, that sequences of eigenfunctions can not concentrate completely on periodic orbits, no matter how slowly. Proving this would involve controlling the quantum dynamics for longer times than we are currently able to do. A result somewhat analogous to our result on the evolution of coherent states was recently obtained for the long time evolution of Lagrangian states on compact Riemannian manifolds of negative curvature [21]. It should however be noted that such a result does not require any control on the proportionality constant preceding ln so that no precise control on either the mixing rate or the exponent in the error term of the Egorov theorem are needed in that case. We suspect that in situations were such control can be obtained, our present strategy will allow to control both coherent state evolution and strong scarring. A related result for the eigenfunctions of Laplace-Beltrami operators on compact, negatively curved Riemannian manifolds is proven using a different strategy in [1]: it is shown there that (under a suitable technical condition that may or may not hold) such eigenfunctions can not concentrate on sets of small topological entropy (and therefore on periodic orbits).

2 Weyl quantization and Egorov Theorem The purpose of this section is to recall (as compactly as possible) some properties of the Weyl quantization on T2d := (R/Z)2d as well as on R2d , for d ≥ 1. More specifically, we want to state a semi-classical version of the Egorov Theorem in the case of T2d . The latter is of course well known for R2d but it requires a proof for T2d all the more so as we need a rather explicit version of this theorem for the applications we have in mind in this paper. The Weyl quantization on R2d can be defined as the linear map f ∈ B(R2d ) → OpW (f ) ∈ L(L2 (Rd )) where OpW (f ) is the operator (belonging a priori to L(S(Rd ), S (Rd ))) with Schwartz kernel q1 + q2 , p dp. Kf (q1 , q2 ) = (2π)−d exp(i(q1 − q2 ) · p) f 2 Rd

888



Here B(R2d ) is the set of smooth functions f on R2d such that ∂ γ f is bounded for all γ ∈ N2d , thus the above integral has to be understood in the sense of oscillatory integrals [16, 22, 18, 14] but it is of course a usual Lebesgue integral if f decays fast enough at infinity. The fact that OpW (f ) can be considered as a bounded operator on L2 (Rd ) follows from the Calder` on-Vaillancourt theorem [16, 22, 18, 14] which states the existence of C > 0 and d¯ > 0 such that W Op (f )ψ 2 d ≤ C sup ||∂ γ f ||L∞ (R2d ) ||ψ||L2 (Rd ) , L (R ) |γ|≤d¯

∀ f ∈ B(R2d ), ψ ∈ S(Rd ). (2.2) It is moreover well known that OpW (f ) maps the Schwartz space S(Rd ) continuously into itself and that OpW (f )∗ = OpW (f ), thus OpW (f ) can be considered as a continuous operator on S (Rd ) too. Note also that OpW (f ) is self-adjoint on L2 (Rd ) when f is real-valued. The Weyl quantization on T2d is obtained by restricting OpW (f ) to certain subspaces of S (Rd ) when f ∈ C ∞ (T2d ) (i.e., is Z2d periodic). The construction is as follows (see [7] for more details). For any ξ = (ξq , ξp ) ∈ R2d , the phase space translation operator U (ξ) is defined by i ξq · ξp U (ξ)ψ(q) = ψ(q − ξq ) exp ξp · q − , ψ ∈ S(Rd ) 2 and is clearly a unitary operator on L2 (Rd ). One easily checks that U (ξ) = OpW (χξ ),

i χξ (q, p) = exp (q · ξp − p · ξq ),

and that the following Weyl-Heisenberg relations hold for all ξ, η ∈ R2d U (ξ)U (η) = exp

i ω(η, ξ)U (ξ + η), 2

(2.3)

with ω the symplectic form defined by ω(ξ, η) = ξq · ηp − ξp · ηq . This relation shows in particular that, if n, m ∈ Z2d , U (n) and U (m) commute if and only if there exists N ∈ N such that 2πN = 1.

(2.4)

Since U (ξ) acts naturally on S (Rd ), we can introduce for any κ ∈ [0, 2π)2d the space H (κ) = {ψ ∈ S (Rd ) | U (n)ψ = eiω(κ,n)+i

nq ·np 2

ψ, ∀ n = (nq , np ) ∈ Z2d }

and it turns out that H (κ) is of dimension N d if (2.4) holds (0 otherwise) with the basis r κq d − , r ∈ {0, . . . , N − 1}d . ψrκ (q) = N − 2 eiκp ·k δ0 q − k − N 2πN 2d k∈Z

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The latter is proven in [7] as well as the existence of a unique scalar product on each H (κ) making the above basis orthonormal and U (n/N ) unitary for all n ∈ Z2d . The Weyl quantization on T2d is then defined by f ∈ C ∞ (T2d ) → OpW (f )|H (κ) . This is indeed a mapping from C ∞ (T2d ) to L(H (κ)), i.e., H (κ) is stable under OpW (f ), since one can easily check that for any Z2d periodic function f OpW (f ) = fn U (n/N ) n∈Z2d

if f (x) = n fn e2iπω(x,n) , x ∈ R2d . Let us emphasize that the spaces H (κ) are very natural in view of the following direct integral decomposition [7] ⊕ H (κ) dκ L2 (Rd ) (2π)−2d [0,2π)2d

in which the operators OpW (f )|H (κ) are the fibers of OpW (f ) for this decomposition. To streamline the discussion we will write both quantizations on R2d and T2d under a single form. From now on, M will denote either R2d or T2d . The Weyl quantization on M can then be defined as the map f ∈ B(M) → OpW (f ) ∈ L(H) where B(M) is either B(R2d ) or C ∞ (T2d ) and H is either L2 (Rd ) or H (κ) (we omit the , κ dependence in the notations). In order to write OpW (f ) in a unified way, we need to introduce the symplectic Fourier transform F on M defined by exp(iω(ξ, x))f (x) dx F f (ξ) = M

where ξ = (ξq , ξp )1 belongs to M∗ = R2d if M = R2d and (2πZ)2d if M = T2d . Then the following inversion formula holds exp iω(x, ξ)F f (ξ) dν(ξ) (2.5) f (x) = M∗

with dν = (2π)−2d × the Lebesgue measure (resp. Z2d δ2πn ) if M∗ = R2d (resp. (2πZ)2d ) and the Weyl quantization can easily be seen to be OpW (f ) = U (ξ)F f (ξ) dν(ξ). (2.6) M∗

1 Throughout this paper, x will denote the running point of M and ξ the one of M∗ , unlike the usual notation of microlocal analysis where (x, ξ) is the running point of R2d .

890



Of course, when M = R2d , all the integrals must be understood in the weak sense d (in (2.6) we use the fact that U (ξ)ψ1 , ψ2 belongs to S(R2d ξ ) if ψ1 , ψ2 ∈ S(R )). ∞ Note also the existence of C, d¯ such that, if ||.||∞ denotes the L norm on M, W Op (f ) ≤ C sup ||∂ γ f ||∞ , ∀ f ∈ B(M). (2.7) H→H |γ|≤d¯

This comes from (2.2) if M = R2d and from the unitarity of U (n/N ) combined with the elementary estimate n |fn | ≤ C sup|γ|≤2d+1 ||∂ γ f ||∞ when M = T2d . This completes the definition of the Weyl quantization on M. Regarding the composition of the corresponding operators, we have the Proposition 2.1 There exists a bilinear map (f, g) → f #g from B(M)2 to B(M) such that OpW (f )OpW (g) = OpW (f #g). The function f #g has a full asymptotic expansion in powers of , meaning that for all integers J f #g = j f #j g + J rJ (f, g) j<J

α β β α −1 where f #j g = = (−1)α α!β! |α+β|=j Γ(α, β)∂q ∂p f ∂q ∂p g, with Γ(α, β) |α+β| 2d and for all γ ∈ N (2i)

γ ∂ rJ (f, g)

J+|γ|

∞

≤

Cd J!

sup

|γ1 |≤J+|γ|+d˜

||∂ γ1 f ||∞

sup

|γ2 |≤J+|γ|+d˜

||∂ γ2 g||∞ , 0 < ≤ 1. (2.8)

for some constants Cd , d˜ depending only on d. Note that (.#j .) is symmetric (resp. skew symmetric) for j even (resp. odd) and that g#1 f − f #1 g

= −i(∇p g · ∇q f − ∇q g · ∇p f ) = −i{g, f }.

(2.9)

Proof. This result is well known if M = R2d (see for instance the appendix of [8] for a simple proof). We briefly sketch the proof in the case M = T2d . Using (2.3) and (2.6), we have OpW (f )OpW (g) = eiω(η,ξ)/2 U ((ξ + η))F f (ξ)F g(η) dν(ξ)dν(η)(2.10) ∗ ∗ M M = U (ξ) eiω(η−ξ,η)/2 F f (ξ − η)F g(η) dν(η) dν(ξ). M∗

M∗

(2.11)

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Expanding eiω(η−ξ,η)/2 by the Taylor formula, we get the expansion of f #g with a remainder rJ = rJ (f, g) defined by its Fourier transform as follows F rJ (ξ) =

(−2i)−J (J − 1)!

1 0

(1 − t)J−1

M∗

eitω(η−ξ,η) ω(η − ξ, η)J F f (ξ − η)F g(η) dν(η) dt. (2.12)

= (γp , γq ) if γ = (γq , γp ) (γq , γp ∈ Nd ), we Since |ξ γ F rJ (ξ)| = |F∂ γ rJ (ξ)|, with γ have to consider γ! J! (η − ξ)β+γ1 η β+γ2 (−1)|βq |+|γ1 | ξ γ ω(η − ξ, η)J = γ !γ ! β! 1 2 γ +γ =γ 1

2

|β|=J

J+|γ|

where β = (βq , βp ) with βq , βp ∈ Nd . The sum contains at most Cd terms and since J! γ! ≤ (2d)J , ≤ 2|γ| β! γ1 !γ2 ! we conclude that (2.8) is now a simple consequence of the fact that α |F∂ α f (ξ)| dν(ξ) ≤ Cd sup ||∂ α1 +α f ||∞ . ξ F f (ξ) dν(ξ) = M∗

|α1 |≤2d+1

M∗

We omit the details.

Remark. The above proof can be repeated verbatim if M = R2d and B(R2d ) is replaced by S(R2d ). We now present a unified version of Egorov Theorem, that is the semiclassiW W cal analysis of eitOp (g)/ OpW (f )e−itOp (g)/ for f, g ∈ B(M), with g real-valued. This result is well known for M = R2d [10, 22, 16, 18] and the purpose of what follows is essentially to prove a similar result for M = T2d , with an explicit remainder term. The result is based on the following simple remark: if A is a bounded self-adjoint operator and B(t) is a strongly C 1 family of bounded operators, then eitA/ B(0)e−itA/ − B(t) i t i(t−s)A/ d = e i B(s) + [A, B(s)] e−i(t−s)A/ . (2.13) 0 ds We shall use this formula with A = OpW (g) and B(t) of the form B(t) = j OpW (fj (t)) j<J

with f0 (t), . . . , fJ−1 (t) ∈ B(M) such that and i

j<J

j fj (0) = f (i.e., B(0) = OpW (f ))

d B(s) + [A, B(s)] = O(J+1 ) ds

(2.14)

892



where O(hJ+1 ) is to be understood the operator norm on H. Expanding [A, B(s)] in powers of by means of Proposition 2.1, (2.14) leads to the following conditions on the functions fj (s) ∂s f0 − {g, f0 }

= 0,

∂s fj − {g, fj }

= 2i

g#k fl ,

f0 (0) = f,

(2.15)

fj (0) = 0 for j ≥ 1

(2.16)

l+k=j+1

where, in the last sum, 3 ≤ k ≤ J − l is odd and l ≤ J − 1, which implies actually that l ≤ j − 2. This system is thus triangular and can be solved using the Hamiltonian flow φs of g, since the solution of ∂s a − {g, a} = b with as=0 = a0 is given by s s a(s, x) = a0 (φ (x)) + b(τ, φs−τ (x)) dτ, x ∈ M. 0

Note that if M = T2d and g is identified with a Z2d periodic function on R2d , the associated Hamiltonian flow φ˜s on R2d is easily seen to satisfy the identity φ˜s (x + n) = φ˜s (x) + n for all x ∈ R2d and n ∈ Z2d . This shows that the formulas for the fj (s) are the same for M = R2d and T2d , if f and g are Z2d periodic. Let us now define the linear operators Lsj on B(M) by Lsj f := fj (s). We have Ls0 f = f ◦ φs ,

Lsj ≡ 0 for j odd

(2.17)

the latter being a consequence of the (skew) symmetry of #k for k (odd) even. For j ≥ 2 even, an induction shows that Lsj =

j/2 1 ··· j (2i) k=1 m1 +···+mk =j/2 |α1 +β1 |=1+2m1 |αk +βk |=1+2mk sk−1 s 1 ··· Ls−s M α1 ,β1 Ls01 −s2 · · · M αk ,βk Ls0k dsk · · · ds1 0 0

(2.18)

0

where m1 ≥ 1, . . . , mk ≥ 1 in the sum and M α,β is the differential operator M α,β =

(−1)|α| β α α β ∂ ∂ g∂ ∂ . α!β! q p q p

(2.19)

Taking the remainders into account, one gets the following result: Theorem 2.2 (Egorov Theorem) For all f, g ∈ B(M) with g real-valued and all J ≥ 1 we have W W eitOp (g)/ OpW (f )e−itOp (g)/ = j OpW (Ltj f ) + J RJt (f, ) j<J

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where the operator RJt (f, ) has the following explicit form i

t

e

i(t−s)OpW (g)/

0

Op

W

(rJ−l+1 (g, Lsl f ))

− Op

W

(rJ−l+1 (Lsl f, g))

l<J

e−i(t−s)Op

W

(g)/

ds.

Note that estimates on ||RJt (f, )||H→H can then be derived from (2.7), (2.8) and estimates on the derivatives of Lsj f . This will be extensively used in the next section.

3 Perturbations of quantized hyperbolic maps In this section, we address the problem of the semi-classical approximation of U−t OpW (f )Ut as ↓ 0 in the Ehrenfest time limit t ≈ | ln |, when U is a unitary operator on H of the form U = e−iOp

W

(g)/

M (A)

with M (A) the quantization of a symplectic matrix with integer entries A ∈ Sp(d, Z). We refer to [7, 4, 5] and [14] for the definition of M (A) by mean of the metaplectic representation of Sp(d, R) and only quote the properties that we need. The operator M (A) is defined, up to a phase, as the unique operator on S (Rd ) such that M (A)−1 OpW (f )M (A) = OpW (f ◦ A),

∀ f ∈ B(R2d ).

(3.1)

If M = R2d , M (A) is unitary on L2 (Rd ), but if M = T2d and H = H (κ) one has to choose special values of κ to ensure that M (A) maps H (κ) into itself, in which case M (A) is unitary (see [7] for more details); from now on, we shall assume that such a choice, which depends on , has been made. Then, (3.1) holds on M = R2d and T2d and this is often expressed by saying that for linear evolutions ‘Egorov is exact’, meaning there is no remainder term. Let us now describe the results of this section. We will denote by φ the Hamiltonian flow associated to a fixed real-valued g ∈ B(M) and consider the discrete group (Φt )t∈Z of symplectomorphisms on M defined by Φ = φ ◦ A. Then, by setting ˜ j f = (L f ) ◦ A L j with the notations of (2.17) and (2.18), we can consider the functions ˜ l1 · · · L ˜ lt f Lt0 f = f ◦ Φt , L Ltj f = l1 +···+lt =j

(3.2)

894



defined for j ≥ 1, t ≥ 0 integers and f ∈ B(M). Note that they depend on but ˜ 0 )t we omit this dependence for notational convenience. Note also that Lt0 = (L t and that Lj ≡ 0 if j is odd. Our goal is to show that U−t OpW (f )Ut ∼

hj OpW (Ltj f ),

↓ 0,

in a scale of times t described in terms of exponents ΓA , Γg that we now define. For the sake of simplicity, we shall assume that A is diagonalizable over R, meaning that there exists an invertible matrix P with real entries such that A = P −1 DP with D diagonal. Note that such a condition is of course satisfied if A is symmetric, e.g., the cat map. At the end of the section, we explain how to cope with general symplectic matrices A ∈ Sp(d, Z). Let us define ΓA ≥ 0 by eΓA = sup |λ|. σ(A)

Of course, this quantity is well defined for any invertible matrix A with real or complex spectrum. For z = (z1 , . . . , z2d ) ∈ C2d , we denote by |z| := (|z1 |2 + · · · + |z2d |2 )1/2 its standard Hermitian norm and set ||z||P := |P z|. The interest of the norm ||.||P is that we have ||Az||P ≤ eΓA ||z||P ,

||Im Az||P ≤ eΓA ||Im z||P

∀ z ∈ C2d ,

(3.3)

which we shall use extensively in the sequel. Then, inspired by [23, 8], we define the open sets Ωδ ⊂ C2d for δ > 0 by Ωδ = {z ∈ C2d | ||Im z||P < δ} and we consider the family of norms ||.||τ,δ defined for τ ∈ (0, 1) by ||f ||τ,δ = sup |f (z)| z∈Ωτ δ

for functions f which are bounded and analytic on Ωδ . We can now set 0 I 2 Γg = sup |||J ∇ g(z)|||P , J = −I 0 z∈Ωδ with ∇2 g the Hessian matrix of g and |||B|||P := supz=0 ||Bz||P /||z||P for B ∈ M2d (C). Note that Γg = 0 unless g is constant which is a trivial situation. We then define Γ = ΓA + Γg and our main result is the following:

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Theorem 3.1 Assume that f, g ∈ B(M), with g real-valued, have bounded and analytic extensions to Ωδ for some δ > 0. Then, for all 0 < ν < 2, there exists J0 > 0 such that for all J > J0 U−t OpW (f )Ut = j OpW Ltj f + J tJ (f, , ) (3.4) j<J

with a remainder such that, for all 0 ≤ ≤ 1, J t (f, , ) J

H→H

→0

as → 0

if

0≤t≤

2−ν | ln |. 3Γ

(3.5)

The reader may wonder what (3.5) means if Γ = 0. In such a case Γg = 0 thus g is constant so (3.4) becomes U−t OpW (f )Ut = OpW (f ◦ At ) by (3.1) which holds for all t ≥ 0. In Section 3, we will anyway be interested in the situation where ΓA > 0 and is small so that Γ > 0. We also emphasize that the analyticity assumption is imposed by our need to control high order derivatives of f and g in order to estimate tJ (f, , ). Similarly to [8], we could probably relax such a condition by considering quasi-analytic functions (e.g., Gevrey functions) which would allow us to consider compactly supported f . The rest of this section is now devoted to the proof of Theorem 3.1. The principle is rather simple and is the following: a straightforward application of Theorem 2.2 shows that ˜ j f ) + J M (A)−1 R (f, )M (A), j OpW (L ∀ J > 0, U−1 OpW (f )U = J j<J

hence an induction on t ≥ 1 shows that (3.4) holds with tJ (f, , ) =

tJ

t j−J OpW (Kj,J f) +

t

Us−t M (A)−1 RJ (EJs−1 f, )

s=1

j=J

M (A)Ut−s

(3.6)

t with the operators Kj,J and EJt defined by t = Kj,J

l1 +···+lt =j l1 <J,··· ,lt <J

˜ lt · · · L ˜ l1 , L

EJt =

l1 <J

···

˜ lt · · · L ˜ l1 . l1 +···+lt L

(3.7)

lt <J

t t Note that Kj,J depends on both j and J unless j < J in which case Kj,J = Ltj . 1 Note moreover that EJt depends on and that we set Kj,J = 0, EJ0 = id. Thus (2.7) reduces the proof of Theorem 3.1 essentially to estimate the deriva˜ l1 f . To that end, we shall use the following extension of a lemma ˜ lt · · · L tives of L of [8].

896



Lemma 3.2 There exists a constant CP depending only on P such that, if a CP ||f ||τ,δ ≤ M , ∀ 0 0. We claim that, in this case, we have the following result: for ˜ A > ΓA there exists an invertible matrix P with real entries such that any Γ ˜

|||P −1 AP ||| ≤ eΓA

(3.14)

where |||.||| is the matrix norm associated to the Hermitian norm |.| on R2 . We can prove this statement as follows. Assume first that the spectrum of A is real and let us choose a basis (e1 , . . . , e2d ) of R2d in which A is in Jordan normal form. If (ej , . . . , ej+p ) corresponds to a Jordan block      J(λ) =    

λ

1

0

0 .. . .. . 0

λ .. .

1 .. . ..

···

. ···

··· .. . .. . .. . 0

 0 ..  .    , 0    1  λ

then by changing (ej , ej+1 , · · · , ej+p ) into (ej , εej+1 , · · · , εp ej+p ) with ε > 0, the above block is changed into the same one with 1 replaced by ε. Proceeding similarly for all the blocks, we obtain the existence of a basis in which A is the sum of a diagonal matrix of norm eΓA and of a nilpotent matrix of norm O(ε). This leads to the statement when the spectrum is real. For non-real eigenvalues λ = ρeiθ , using Jordan normal form over C2d , we have to consider blocks of the form J(λ) 0 . ¯ 0 J(λ) It is then standard that there exists a basis of real vectors in which the endomorphism represented by the above block has a matrix of the form N + ρR(θ) where N is nilpotent and R(θ) is block diagonal matrix of rotations (of dimension 2) of angle θ. Then, by changing this basis as in the case of a real spectrum, we can assume that N is small and we obtain (3.14) in the general case.

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4 Equirepartition of time-evolved localized states 4.1

The example of (generalized) coherent states

In this subsection, we shall prove that the generalized coherent states, defined below, when evolved over sufficiently long times, equidistribute on the torus. To define the states in question, we proceed as follows. Let q (4.1) ϕ (q) = −µ/2 ϕ µ with ϕ ∈ S(Rd ), |ϕ|2 = 1 and µ ∈ (0, 1). Then we set ϕa = U (a)ϕ

(4.2)

which defines a family of states in L2 (Rd ) indexed by a ∈ R2d . These are commonly referred to as (generalized) coherent states. The corresponding states on the torus, i.e., belonging to H (κ), are defined by    ϕa,κ := S (κ)ϕa =  e−iκq ·np U (0, np )  eiκp ·nq U (nq , 0) ϕa (4.3) np ∈Zd

nq ∈Zd

which converges in S (Rd ) (see [7]). The main property of these states that we shall use is a ϕ,κ , OpW (f )ϕb,κ H (κ) = (−1)N nq ·np eiω(κ,n) eiω(n,b)/2 ϕa , OpW (f )ϕb−n L2 (Rd )

(4.4)

n∈Z2d

which is proven in [4]. The best known example of such functions are obtained by 2 choosing µ = 1/2 and ϕ(q) = η(q) := π −d/4 e−q /2 . With this choice one obtains the standard coherent states. If ϕ˜ is another Schwartz function and ϕ˜ is defined similarly to (4.1), the Wigner function W (x) associated to ϕ , ϕ˜ is defined by f (x)W (x) dx ϕ , OpW (f )ϕ˜ L2 (Rd ) = R2d

for all f ∈ B(R2d ). For general ϕ , ϕ˜ in L2 (Rd ), W is a distribution, but for Schwartz functions it is a Schwartz function as well given by −d q, x = (q, p). e−i˜q·p/ ϕ (q − q˜/2)ϕ˜ (q + q˜/2) d˜ W (x) = (2π) With the simple dependence considered in (4.1), it is easy to see that the Wigner (a,b) function W (x) associated to U (a)ϕ and U (b)ϕ˜ takes the following form for

900



any a, b ∈ R2d (a,b)

W

a+b (x) = e−iω(a,b)/2+iω(x,b−a)/ −d W1 Σµ x − 2

(4.5)

where Σµ is the linear map on R2d defined by Σµ (q, p) = (q/µ , p/1−µ ) and W1 the Wigner function of ϕ, ϕ. ˜ Note that, since W1 ∈ L1 (R2d ), (4.5) implies that (a,b) ||W ||L1 = ||W1 ||L1 is independent of . Note also that when ϕ(q) = ϕ(q) ˜ = η(q), one easily checks that W1 (x) = π −d e−x

2

(4.6)

which makes (4.5) completely explicit in this case. Our main result is Theorem 4.7. As explained in the introduction, its proof goes in two steps. use the Egorov theorem to establish that on a suitable time scale t aFirst we U ϕ,κ , OpW (f )Ut ϕa,κ H (κ) is equivalent to ϕa , OpW (f ◦ Φt )ϕa L2 (Rd ) (Propo sition 4.2). Then we use an estimate on the classical evolution (exponential mixing) to control this last term. As a warm up for the first step, we show for a particularly simple class of states how the Egorov expansion (3.4) can be reduced to the first term. Proposition 4.1 Let Ψ ∈ H be a family such that there exists C satisfying Ψ , OpW (f )Ψ ≤ C||f ||∞ , 0 0 j a j+2M−m(2M−|γ|) ||∂ γ Ltj f ||∞ , ≤ Cj |n|−2M ϕ , Opw (Ltj f )ϕa−n L2 (Rd ) |γ|≤2M

where m = max(µ, 1 − µ). We get the result by the simple observation that j+2M−m(2M−|γ|) ||∂ γ Ltj f ||∞ ≤ Cf 2νj+CM → 0,

→0

for |γ| ≤ 2M , with C = 2(1 + ν)/3 if m < (2 − ν)/3 and C = 2(1 − m) otherwise. This follows from (3.11) by distinguishing both cases m ≥ (2 − ν)/3 and m < (2 − ν)/3.

902



This proposition, combined with (4.5), allows us to reduce the study of the matrix elements of evolved coherent states to a problem in classical dynamics. By this, we mean that the main result of this section, Theorem 4.7, is a direct consequence of Proposition 4.2 and of the mixing estimates given in the Appendix A. Note that from now on, we shall be working with d = 1. As explained in the introduction, the reason for this is that, whereas the mixing rate is controlled by the smallest Lyapounov exponent of A, the error in the Egorov theorem is controlled by its largest Lyapounov exponent. As a warm-up, and in order to bring out the main strategy, we first prove a simplified version of the result: Theorem 4.3 Assume that ΓA > 0. Let a in R2 , f ∈ B(T2 ) and 1/3 < µ < 2/3. Then, for all ν > 0 there exists 0 small enough (independent of f ) such that for || < 0 we have t a W t a f (x) dx, → 0, U ϕ,κ , Op (f )U ϕ,κ H (κ) →

T2

provided that m+ν 2−ν | ln | ≤ t ≤ | ln |, Γ 3Γ

m = max(µ, 1 − µ).

(4.10)

Proof. We first remark that, by choosing 0 < Γ < ΓA close enough to ΓA and small enough we have 1>

1 + ν/2 Γ . > Γ 1+ν

(4.11)

Combined with (4.10), this estimate implies that t/| ln | > (m + ν/2)/Γ and thus e−tΓ ≤ m+ 2 . ν

By Proposition 4.2 and (4.5) we only have to study the limit of (a,a) (f ◦ Φt )(x)W (x) dx

(4.12)

(4.13)

R2

(a,a) for which W (x)dx = 1. Choosing a smooth cutoff function χ so that χ = 1 (a,a) near 0 and which is supported close to 0, then setting g (x) := W (x)χ(x − a), (a,a) − g ||L1 = O(h∞ ) thus we have ||W (a,a) (f ◦ Φt )(x)W (x) dx − (f ◦ Φt )(x)g (x) dx → 0, ↓0 R2

R2

uniformly with respect to t ∈ R. The last integral can obviously be interpreted as an integral over T2 since g is supported close to a and consequently we can use Corollary A.2. The result now simply follows from the fact that e−tΓ ||g ||W 1,1 = O(h−m )e−tΓ → 0 by (4.12).

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The above proof is a rather direct application of Proposition 4.2 and Corollary A.2 but it fails if m ≥ 2/3 (i.e., µ ∈ / (1/3, 2/3)) since e−tΓ h−m > 1, for, in that −tΓ 2/3 case, e > . The problem stems from the lower bound in (4.10), which arises (a,a) because ∇W behaves like −m . One expects on intuitive grounds that it should be possible to replace m by m = min(µ, 1−µ) which is of course less than 1/2 which is less than 2/3. We shall prove this is true, but for that purpose we will need to exploit some more detailed knowledge about the Anosov diffeomorphisms we study. The trick consists in applying a well known idea in the theory of Anosov systems: it is possible to replace (4.13) by an expression obtained by performing an integral (a,a) along along the stable foliation. Since the evolution stretches the function W the unstable manifold, this corresponds to smoothening out the fastest oscillations (a,a) in W , replacing the latter by a function that has a derivative controlled by −m . Let us start the proof. By Proposition 4.2, we have to study (4.13) where (a,a) W can be replaced, as in the proof of Theorem 4.3, by g which we can assume to be supported as close to a as we want. This will allow us to use the following result. Theorem 4.4 [3, 2] For all Γ < ΓA , there exists 0 small enough such that for all || < 0 the following holds: there exist σ > 0 and a C 1+σ diffeomorphism x → F (x) = (s(x), u(x)), from a neighborhood of a ∈ T2 to a neighborhood of 0 ∈ R2 such that F (a) = 0 and ∂s f ◦ Φt ◦ F−1 (u, s) ≤ Cf e−Γt (4.14) for all t ≥ 0, all (u, s) in the neighborhood of 0 and all f ∈ C 1 (T2 , R). Here C 1+σ denotes the corresponding H¨ older class. Using this result, we can perform the following change of variables

f ◦ Φt (x)g (x) dx R2 = f ◦ Φt ◦ F−1 (u, s) g ◦ F−1 (u, s)J (u, s) duds

(4.15)

where J ∈ C σ . On the right-hand side of this equation, we eventually want to use Corollary A.2, but the C σ regularity of J (u, s) is not sufficient for that purpose. Fortunately, the term J is essentially irrelevant in view of the following result. Lemma 4.5 i) (g ◦ F−1 )0 1/2, the same result holds if ∂s (q ◦ F−1 )(0, 0) = 0, with 1 − µ instead of µ in (4.17). We are now ready for the proof of the main theorem of this subsection. Theorem 4.7 Assume that 0 < µ ≤ 1/2 (resp. 1/2 ≤ µ < 1) and that the unstable manifold through a is not aligned with the submanifold {q = q(a)} (resp. {p = p(a)}). Assume moreover that ΓA > 0. Then, there exists 0 such that, for || < 0 and all f ∈ C ∞ (T2 ) t a U ϕ,κ , OpW (f )Ut ϕa,κ H

(κ)

→

f (x) dx, T2

→ 0,

provided that m+ν 2−ν | ln | ≤ t ≤ | ln |, Γ 3Γ

m = min(µ, 1 − µ).

(4.18)

Note that, when µ ∈ (1/3, 2/3), Theorem 4.3 holds as well, without the assumption on the stable and unstable manifolds, but with also a smaller time window than in (4.18). Proof. The above discussion shows that we only have to prove that

f ◦ Φt ◦ F−1 (u, 0)k˜ (u) du →

f (x) dx.

(4.19)

T2

Pick a smooth function (s) supported close to 0 such that (s)ds = 1. Then, using Theorem 4.4, the left-hand side of (4.19) takes the form

f ◦ Φt ◦ F−1 (u, s)k˜ (u)(s) duds + O(e−Γ t )

with O(e−Γ t ) uniform with respect to ∈ (0, 1]. This last integral is nothing but T2

f ◦ Φt (x)˜ g (x) dx

(1) where g˜ ◦ F−1 (u, s) = k˜ (u)(s)/J (u, s). Thus g˜ is of the form g˜ g˜(2) with (1) (1) (2) σ g ||L1 + m ||∇˜ g ||L1 = O(1). Note also that g˜ ∈ C independent of and ||˜ g˜ → 1 as → 0. Using Lemma 4.5 again to approach g˜(2) by C 1 functions, T2 (1)

we may assume that g˜ is C 1 and satisfies the same bound as g˜ . We can now repeat the arguments of Theorem 4.3 and the result follows.

906

4.2



Semiclassical behavior of eigenstates

We now come to a more general result having applications in the description of the eigenvectors of U . Assume that Ψ,κ ∈ H (κ) satisfies, for all f ∈ C ∞ (T2d ), ↓ 0. (4.20) Ψ,κ , OpW (f )Ψ,κ H (κ) → f (0),

Rather vaguely, this condition says that Ψ,κ is concentrated at 0. This is confirmed by the following Lemma 4.8 There exists a sequence of positive numbers r → 0 and a family of functions χ ∈ C ∞ (T2d ) supported in a ball of radius r centered at 0 (in T2d ) such that 0 ≤ χ ≤ 1 and a a Ψ,κ − (2π)−d χ (a) η,κ , Ψ,κ H (κ) η,κ da → 0, ↓ 0. (4.21)

T2d

H (κ)

Conversely, if (4.21) holds and ||Ψ,κ ||H (κ) → 1 then (4.20) holds for all f ∈ C ∞ (T2d ). The proof of this lemma is given in Appendix B, where we also recall basic results on the coherent states decomposition over L2 (Rd ) and H (κ). Recall that 2 a η,κ is defined by (4.1), (4.2) and (4.3) with µ = 1/2 and η(q) = π −d/4 e−q /2 . The right-hand side in (4.20)

could of course be replaced by f (a0 ) for some a0 ∈ T2d or more generally by 0≤j≤J αj f (aJ ) for finitely many points a0 , . . . , aJ . Correspondingly, one can then define the concentration on a finite collection of points in a r neighborhood of those points. a , Ψ,κ H (κ) . The above To simplify the notation, we set λ (a) = χ (a) η,κ lemma proves that ψ,κ := (2π)−d

T2d

a λ (a)η,κ da

satisfies (4.20) as well and that Ψ,κ , U−t OpW (f )Ut Ψ,κ H (κ) − ψ,κ , U−t OpW (f )Ut ψ,κ H

(κ)

→ 0,

↓0

uniformly with respect to t ≥ 0. This is the first step of the proof of the next theorem, in which the notations ., . and ||.|| stand for ., .H (κ) and ||.||H (κ) respectively. Theorem 4.9 Assume that ||Ψ,κ || → 1 and that (4.21) holds for some sequence r such that r ≤ 1/2−σ , with σ > 0. Then, as → 0, Ψ,κ , U−t OpW (f )Ut Ψ,κ − (2π)−2d a b dadb → 0 λ (a)λ (b) η,κ , OpW (f ◦ Φt )η,κ T2d

T2d

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907

provided 0 ≤ Γ t ≤

1 +τ 2

| ln |,

1 − 3τ − 4dσ > 0 2

and

τ
0 and τ −5σ > 0, then there exists t → ∞ and (σ, τ ) > 0 such that for all || ≤ (σ, τ ) Ψ,κ , U−t OpW (f )Ut Ψ,κ → f (x) dx, ↓ 0. (4.23) T2

This theorem generalizes a result of [6], Section 5, where only the case = 0 is treated. The proof is then much simpler, since there is then no error term in the Egorov theorem. The theorem says that, if a sequence of states concentrates sufficiently fast on a point a in T2 , then the time evolved states equidistribute on the torus on some logarithmic time scale. Before proving this theorem, we show how it leads to a result on the semiclassical behaviour of the eigenvectors of U . Corollary 4.10 Assume that d = 1 and that ΓA > 0. For any 0 < σ < 1/38, there exists (σ) > 0 such that for all || < (σ), no family Ψ,κ of eigenvectors of U can satisfy simultaneously (4.20) for all f and (4.21) with r ≤ 1/2−σ . We note in passing that a similar result (with a worse value of σ) holds for d > 1 provided we impose a pinching condition on the Lyapounov exponents of A as mentioned in the introduction. Roughly speaking, this corollary shows that, if a family of eigenvectors of U concentrates on a single point in phase space in the semiclassical limit, then it must do so slowly. In other words, no such sequence can ‘live’ in a ball of too small a radius r . In view of the comment after Lemma (4.8), it is clear that this result holds also for a pure point measure supported on a finite number of periodic orbits. Given Theorem 4.9, the proof is very simple and identical to the case = 0 treated in [6], Section 5. We repeat it for completeness. Proof. For any 0 < σ < 1/38, one Furthermore, (4.22). can find τ > 5σ satisfying since Ψ,κ is an eigenfunction, Ψ,κ , U−t OpW (f )Ut Ψ,κ = Ψ,κ , OpW (f )Ψ,κ for all t, thus by choosing t = t and letting ↓ 0 we obtain f (x) dx f (0) = T2

for all f ∈ C ∞ (T2d ), which leads to a contradiction.

Proof of Theorem 4.9. Here again, it is sufficient to assume that f is analytic. Using Theorem 3.1 and Lemma 4.8, it is clear that, if τ < 1/6 and tΓ ≤ (1/2 + τ )| ln |, we have j ↓ 0. ψ,κ , OpW (Ltj f )ψ,κ → 0, Ψ,κ , U−t OpW (f )Ut Ψ,κ − j<J

908



The first part of the theorem will thus be proven if we show that, for any j ≥ 2 (recall that Ltj ≡ 0 if j is odd), we have a b dadb → 0, ↓0 λ (a)λ (b) η,κ , OpW (Ltj f )η,κ (2π)−2d j T2d

T2d

if 1/2 − 3τ − 4dσ > 0. Using (4.4) and (4.5), integrations by parts similar to those of proposition 4.2 show easily that, for all M > 0, a b = (−1)N nq ·np eiω(κ,n)+iω(n,b)/2 η,κ , OpW (Ltj f )η,κ |n|≤C

a η , OpW (Ltj f )ηb−n L2 + O(M )

uniformly with respect to a, b ∈ [0, 1)2d and Γ t ≤ (1/2 + τ )| ln |, with τ < 1/6. The constant C involved in the sum is such that |b − n − a| ≥ C −1 |n| for all a, b ∈ [0, 1)2d and |n| > C. On the other hand, using (4.5) and (3.11), one sees that, for any n ∈ Z2d and any j ≥ 2, −2d (2π) |λ (a)λ (b)| j ηa , OpW (Ltj f )ηb−n dadb ≤ C−2d r4d 1/2−3τ T2d

T2d

since |λ (a)| ≤ ||Ψ,κ || is bounded and λ is supported in a set of volume O(r2d ). The first part of the theorem follows. We now the second part. Since χ can be chosen of the form χ (a) = prove

a+n1 (see Appendix B), it turns out that, for any M , n1 ∈Z2d χ r a b χ (a)χ (b) η,κ , OpW (f ◦ Φt )η,κ

can be written (−1)N nq ·np eiω(κ,n)+iω(n,b)/2 ηa , OpW (f ◦ Φt )ηb−n L2 + O(M ) |b−a−n|=O(r )

uniformly with respect to a, b ∈ [0, 1)2d . Now, if d = 1 and 5σ < τ , using (4.5) and proceeding similarly to the proof of Theorem 4.3, we see that for small enough and Γ < ΓA sufficiently close to ΓA a a b−n W t b−n η , Op (f ◦ Φ )η − η , η f (x)dx = e−tΓ O(−1/2 + r /) L2 L2 T2

uniformly on the set where |b − n − a| = O(r ), a, b ∈ [0, 1)2 . This shows that a 2 b dadb − ||ψ,κ || λ (a)λ (b) η,κ , OpW (f ◦ Φt )η,κ (2π)−2 T2 T2 f (x)dx = O(e−tΓ −1/2−5σ ) T2

and the result follows.

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A mixing theorem for perturbations of hyperbolic maps on T2

A

Let A be a 2 × 2 matrix with integer entries such that |trA| > 2 and detA = 1. For notational convenience, we assume that its eigenvalues are positive and we note them e±ΓA , with ΓA > 0. Let φ be a measure preserving diffeomorphsim on T2 , depending on a parameter , such that in C 3 (T2 ) as → 0.

φ → id

We define the associated Ruelle-Perron-Frobenius operator L as the map L g := g ◦ T−1 ,

T := φ ◦ A.

Using [3] (more precisely (2.1.7), Example 2.2.6 and Theorem 3) one has the following result. Theorem A.1 ([3]) For any Γ < ΓA , one can find 0 > 0 small enough such that the following property holds: for all || ≤ 0 , there exists a Banach space B of distributions of order 1, containing C 1 (T2 ), with norm ||.|| such that

(with ||g||W 1,1

||g|| ≤ C ||g||W 1,1 , ∀ g ∈ C 1 (T2 ) = T2 |g| + T2 |∇g|) and such that L = Π1 + R

with

R Π1 = Π1 R = 0

where Π1 g = g, 11 and R is a bounded operator on B with spectral radius lower than e−Γ . Here ., . is the pairing between distributions of order 1 and C 1 functions. As a direct consequence, we obtain Corollary A.2 For all Γ < ΓA , there exists 0 such that, for all || < 0 , one can find C,Γ satisfying

T2

f Tt (x) g(x)dx −

f

T2

T

g ≤ C,Γ e−tΓ ||f ||C 1 ||g||W 1,1 , 2 for all f, g ∈ C 1 (T2 ), t ≥ 0.

B Generalized coherent states decompositions In this appendix, we briefly recall some results on coherent states decompositions as well as some convenient tools for the proof of Lemma 4.8. As it is for instance proven in [14], it is well known that for any u ∈ S(Rd ) one has −d u = (2π)

ϕa , uL2 ϕa da (B.1) R2d

910



where ϕa is defined by (4.2) with µ = 1/2. This implies in particular that, for any ϕ˜ ∈ S(Rd ), 2 2 2 −d ˜ L2 = (2π) | ϕã , uL2 | da. (B.2) ||u||L2 ||ϕ|| R2d

This decomposition on L2 (Rd ), known as the coherent states decomposition espe2 cially when ϕ(q) = η(q) = π −d/4 e−q /2 , gives rise to a decomposition on H (κ) a ϕ,κ , S (κ)u H (κ) ϕa,κ da, (B.3) S (κ)u = (2π)−d

T2d

with the notation of (4.3). This is proven in [7]. Note the important consequence of that formula: for any ϕ˜ ∈ S(Rd ) 2 2 −d | S (κ)ϕã , S (κ)u| da = Cϕ˜ ||S (κ)u||H (κ) , ∀ u ∈ S(Rd ). (2π) T2d

(B.4) These decompositions are particularly convenient since one knows rather precisely the action of pseudodifferential operators on functions of the form (4.2), as we shall see in Lemma B.1 below. Motivated by Lemma 4.8, we shall consider functions f depending possibly on . Let ε > 0 and assume that r is a sequence such that r ≥ 1/2−ε and let f be a family of functions in B(R2d ) such that γ ∂ f (x) ≤ Cγ r−|γ| ,

x ∈ R2d .

(B.5)

Lemma B.1 There exists a family Pγ of differential operators with polynomial coefficients (independent of ) such that for any f as above and any M > 0, there exists symbols f (,M,γ) satisfying (B.5) as well and differential operators QM γ with polynomial coefficients (independent of too) such that OpW (f )U (a)ϕ =

|γ|/2 ∂ γ f (a)U (a)(Pγ ϕ)

|γ|<M

+ Mε

OpW (f ,M,γ )U (a)(QM γ ϕ) .

|γ|≤2M −d/4 Whenever A = Pγ or QM (Aϕ)(q/1/2 ). γ , we have set (Aϕ) (q) =

Proof. It is essentially standard. Since U (−a)OpW (f )U (a) = OpW (f (. + a)), we are left with the case a = 0. Then, the result simply follows by writing the Taylor expansion of f at 0 and integrating by parts.

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Remark. The operators Pγ can be computed explicitly and in particular P0 = I. Combining this result and (4.4), it is not hard to deduce that for any f ∈ C (T2d ) satisfying (B.5), one has, for all M > 0, W a |γ|/2 γ Op (f )ϕ − ∂ f (a)S (κ)U (a)(P ϕ) ≤ CMε (B.6) γ ,κ |γ|<M ∞

H (κ)

2d

uniformly with respect to a ∈ [0, 1) . We are now ready for the proof of Lemma 4.8. Proof of Lemma 4.8. We only have to show the existence of a sequence r ≥ 1/2−ε for some ε > 0, satisfying r → 0, such that, if 0 ≤ χ ≤ 1 is supported close to 0 and ≡ 1 near 0 then a + n χ χ (a) := r 2d n∈Z

will satisfy the result. Let us fix ε > 0. Then for any sequence r ≥ 1/2−ε , using the Proposition 2.1, one has OpW (1 − χ )2 = j OpW (χj, ) + o(1) j<M

in operator norm, provided M = M (ε) is large enough. The symbols χj, are such −|γ|−2j that ∂ γ χj, = O(r ) and χ0, = (1−χ)2 , thus using (B.3), (B.4) and (B.6), one has a W 2 Op (1 − χ )Ψ,κ 2 = (2π)−d (1 − χ (a))2 η,κ , Ψ,κ da + o(1) T2d

using also the fact that ||Ψ,κ || → 1. By Taylor formula, there exists a function χ ˜ ∈ C ∞ (T2d ), independent of , such that χ(0) ˜ = 0 and (1 − χ (a))2 ≤ χ ˜2 (a)/r2 . Since a 2 −d (2π) χ(a) ˜ 2 η,κ , Ψ,κ da → 0 (B.7) T2d

2

by (4.20) applied to f = χ ˜ , we see that ||OpW (1 − χ )Ψ,κ || → 0 provided r2 → 0 more slowly than the left-hand side of (B.7). Furthermore there is no restriction to choose r ≥ 1/2−ε . Finally, we remark that a a η,κ , Ψ,κ χ (a)η,κ da + o(1) OpW (χ )Ψ,κ = (2π)−d T2d

by (B.3), (B.4) and (B.6) again which completes the proof of (4.21). For the converse, we note that a a Ψ,κ , OpW (f )Ψ,κ − (2π)−d da → 0. χ (a) η,κ , Ψ,κ Ψ,κ , OpW (f )η,κ T2d

The result follows then easily from the dominated convergence theorem using (B.6) and (B.4).

912



References [1] N. Anantharaman, The eigenfunctions of the Laplacian do not concentrate on sets of small topological entropy, preprint june 2004. [2] M. Brin, G. Stuck, Introduction to dynamical systems, Cambridge Univ. Press (2002). [3] M. Blank, G. Keller, C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity 15, no. 6, 1905–1973 (2002). [4] F. Bonecchi, S. De Bièvre, Exponential mixing and | ln | time scales in quantized hyperbolic maps on the torus, Comm. Math. Phys. 211, 659–686 (2000). [5]

, Controlling strong scarring for quantized ergodic toral automorphisms, Duke Math. J 117, No. 3, 571–587 (2003).

[6]

, Controlling strong scarring for quantized ergodic toral automorphisms, Section 5, mp arc 02-81 (2002).

[7] A. Bouzouina, S. De Bièvre, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Comm. Math. Phys. 178, 83–105 (1996). [8] A. Bouzouina, D. Robert, Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J. 111, No. 2, 223–252 (2002). [9] Y. Colin de Verdière , Ergodicité et fonctions propres du Laplacien, Commun. Math. Phys. 102, 497–502 (1985). [10] Y.V. Egorov, On canonical transformations of pseudo-differential operators (in Russian), Uspekhi Mat. Nauk. 24, no. 5, 235–236 (1969). [11] F. Faure, S. Nonnenmacher, S. De Bièvre, Scarred eigenstates for quantum cats of minimal periods, Commun. Math. Phys. 239, 449–492 (2003). [12] F. Faure, S. Nonnenmacher, On the maximal scarring for quantum cat map eigenstates, Commun. Math. Phys. 245, 201–214 (2004). [13] F. Faure, S. Nonnenmacher, contribution at the Workshop on Random Matrix theory and Arithmetic Aspects of Quantum Chaos, Newton Institute, Cambridge, june 2004. [14] G.B. Folland, Harmonic analysis in phase space, Ann. Math. Studies, Princeton Univ. Press 122, (1989). [15] B. Helffer, A. Martinez, D. Robert, Ergodicité et limite semi-classique, Comm. Math. Phys. 109, 313–326 (1987). [16] L. Hörmander, The analysis of linear partial differential operators III, Springer-Verlag (1985).

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[17] E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Annals of Math., to appear. [18] D. Robert, Autour de l’approximation semi-classique, Progress in mathematics 68, Birkh¨ auser (1987). [19] Z. Rudnick, P. Sarnak, The behaviour of eigenstates of hyperbolic arithmetic manifolds, Commun. Math. Phys. 161, 1, 195–213 (1994). [20] A. Schnirelman, Ergodic properties of eigenfunctions, Usp. Math. Nauk. 29, 181–182 (1974). [21] R. Schubert, Semiclassical behaviour of expectation values in time evolved coherent states for large times, preprint january 2004. [22] M. Taylor, Pseudo-differential operators, Princeton Mathematical Series 34, Princeton University Press (1981). [23] F. Trèves, Introduction to pseudo-differential and Fourier integral operators, Vol. 2: Fourier integral operators, Univ. Ser. Math., Plenum, New-York (1980). [24] Zelditch, Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55, 919–941 (1987).

Jean-Marc Bouclet and Stephan De Bièvre Université de Lille 1 UMR CNRS 8524 F-59655 Villeneuve d’Ascq France email: [email protected] email: [email protected] Communicated by Jens Marklof submitted 08/12/04, accepted 11/01/05




Spin-Glass Stochastic Stability: a Rigorous Proof Pierluigi Contucci and Cristian Giardin` a Abstract. We prove the property of stochastic stability previously introduced as a consequence of the (unproved) continuity hypothesis in the temperature of the spinglass quenched state. We show that stochastic stability holds in β-average for both the Sherrington-Kirkpatrick model in terms of the square of the overlap function and for the Edwards-Anderson model in terms of the bond overlap. We show that the volume rate at which the property is reached in the thermodynamic limit is V −1 . As a byproduct we show that the stochastic stability identities coincide with those obtained with a different method by Ghirlanda and Guerra when applied to the thermal fluctuations only.

1 Introduction In a previous paper by Aizenman and Contucci [AC] the property of stochastic stability was introduced as the consequence of a continuity (in term of the inverse temperature β) hypothesis of the quenched state for the Sherrington-Kirkpatrick [SK] model. Stochastic stability says that a suitable class of perturbations of the spin glass Hamiltonian produces very small changes in the quenched equilibrium state and that such a change vanishes in the thermodynamic limit. This property has interesting consequences for the spin glass models: in terms of the overlap distribution it implies that the quenched measure is replica-equivalent [MPV, P] a property originally introduced within the replica symmetry breaking Parisi ansatz. The same property is also used in [FMPP1, FMPP2] to build a bridge between equilibrium and off-equilibrium properties in a spin-glass model being these last the only ones physically accessible to experimental investigation. More recently all and only the constraints that stochastic stability implies for the overlap moments have been completely classified [C, BCK]. In this paper we give a rigorous proof of stochastic stability property in βaverage. This result is achieved in an elementary way by use of the sum law for independent Gaussian variables and works in full generality for both mean-field and finite-dimensional spin glass models. We also derive the explicit form of the stochastic stability identities which first appeared in [AC] and we prove, using integration by parts in the spirit of [CDGG], that they coincide with a subset of the Ghirlanda-Guerra identities [G, GG], namely the part related to the thermal fluctuation bound (see also [T] for a nice set of rigorous results derived from those identities). The proof also provides the rate at which stochastic stability in β-average is reached with the thermodynamic limit which turns out to be V −1 . The paper

916

P. Contucci and C. Giardin` a


is organized with Section 2 containing a list of the definitions and the statement of the two main theorems. Their proof is built in Section 3 while Section 4 shows how to apply the results to both the mean field models, which we illustrate for the Sherrington-Kirkpatrick model [SK], and for the finite-dimensional cases with the Edwards-Anderson model [EA]. Section 5 collects some comments.

2 Definitions and Results We consider a disordered model of Ising configurations σn = ±1, n ∈ Λ ⊂ Zd for some d-parallelepiped Λ of volume |Λ|. We denote ΣΛ the set of all σ = {σn }n∈Λ , and |ΣΛ | = 2|Λ| . In the sequel the following definitions will be used. 1. Hamiltonian. For every Λ ⊂ Zd let {HΛ (σ)}σ∈ΣN be a family of 2|Λ| translation invariant (in distribution) centered Gaussian random variables of volume-size covariance matrix Av (HΛ (σ)HΛ (τ )) = |Λ| QΛ (σ, τ ) ,

(2.1)

QΛ (σ, σ) = 1 .

(2.2)

and By the Schwarz inequality |QΛ (σ, τ )| ≤ 1 for all σ and τ . 2. Random partition function Z(β) :=

e−βHΛ (σ) .

(2.3)

σ∈ ΣΛ

3. Random free energy F (β) −βF (β) := A(β) := ln Z(β) .

(2.4)

4. Quenched free energy F (β) −βF (β) := A(β) := Av (A(β)) .

(2.5)

5. R-product random Gibbs-Boltzmann state Ω(−) :=

σ(1) ,...,σ(R)

(1)

(−)

e−β[HΛ (σ )+···+HΛ (σ [Z(β)]R

(R)

)]

.

(2.6)

6. Quenched equilibrium state − := Av (Ω(−)) .

(2.7)

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Spin-Glass Stochastic Stability: a Rigorous Proof

917

7. Observables. For any smooth bounded function G(QΛ ) (without loss of generality we consider |G| ≤ 1) of the covariance matrix entries we introduce the random (with respect to −) R × R matrix Q = {qk,l } by the formula G(Q) := Av (Ω(G(QΛ ))) .

(2.8)

E.g.: G(QΛ ) = QΛ (σ (1) , σ (2) )QΛ (σ (2) , σ (3) ) q1,2 q2,3 = Av

σ(1) ,σ(2) ,σ(3)

QΛ (σ (1) , σ (2) )QΛ (σ (2) , σ (3) ) e−β[

3 i=1

HΛ ( s(i) )]

. (2.9)

[Z(β)]3

8. Deformed quenched state. For every Λ ⊂ Zd let the {KΛ (σ)}σ∈ΣN be a translation invariant centered Gaussian random family of size one covariance matrix (2.10) Av (KΛ (σ)KΛ (τ )) = QΛ (σ, τ ) , where the families H and K are mutually independent with respect to the joint Gaussian distribution, i.e., Av (HΛ (σ)KΛ (τ )) = 0 . We consider Zλ (β) :=

e−βHΛ (σ)+

√

λKΛ (σ)

(2.11)

,

(2.12)

σ∈ ΣΛ

Aλ (β) := Av (ln Zλ (β)) ,

(2.13)

√

Ωλ (−) :=

λ [KΛ (σ(1) )+···+KΛ (σ(R) )] ) √ (1) )+···+K (σ (R) )] λ [K (σ Λ Λ Ω(e )

Ω((−) e

,

(2.14)

and the deformed quenched state −λ := Av (Ωλ (−)) .

(2.15)

9. Stochastic Stability. The quenched measure is said to be stochastically stable if for every observable G (see Def. 7) the deformed state is stationary in the thermodynamic limit: d Gλ = 0 lim (2.16) d Λ Z dλ It is possible to see (within Theorem 2) that there is a function of the overlap matrix elements: ∆G s.t. ∆Gλ :=

d Gλ . dλ

(2.17)

918



A stochastically stable measure fulfills then the property lim ∆Gλ = 0

Λ Zd

(2.18)

for all the observables G. Our main result state that a spin glass model is stochastically stable β-almost everywhere (Theorem 1), characterizes the functions ∆G (Theorem 2) and establish their coincidence with the quantities obtained with the Ghirlanda-Guerra method when applied only to the thermal fluctuations. Theorem 1 (Stochastic Stability) The spin-glass quenched state is stochastically stable in β-average, i.e., for each interval [β1 , β2 ] and each observable G (as in Def. 7): 2 β2 2 2 . (2.19) ∆Gλ dβ ≤ β12 |Λ| Theorem 2 (Zero average Observables) The explicit form of the zero average quantities is 2∆G =

R

G q l, k − 2RG

k,l=1 k=l

R

q l, R+1 + R(R + 1)G q R+1, R+2 ,

(2.20)

l=1

which coincide with thermal part of the Ghirlanda-Guerra identities.

3 Proof of the results independent from H and K and distributed like Proof of Theorem 1. Since for H H we have, in distribution, that √ λ D HΛ −βHΛ + λKΛ = − β 2 + (3.21) |Λ| from Def. (8) of the deformed quenched state of the function

expecta G, all the λ λ 2 2 tions Gλ turn out to be functions of β + |Λ| : Gλ = g β + |Λ| . From the composite function derivation rule we deduce (the prime denotes derivative w.r.t. the argument): λ d 1 2 Gλ = g β + · (3.22) dλ |Λ| |Λ| and

λ d Gλ = g β 2 + · 2β , dβ |Λ|

(3.23)

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919

from which we have

d 1 d Gλ = Gλ . (3.24) dλ |Λ| dβ Integrating in dβ and using the fundamental theorem of calculus we obtain β22 Gλ (β2 ) − Gλ (β1 ) . (3.25) ∆Gλ dβ 2 = |Λ| β12 2β

Remembering the assumption on boundedness of function G (Def. 7) this complete the proof. Proof of Theorem 2. Let h(σ) = |Λ|−1 HΛ (σ) be the Hamiltonian per particle. From formula (3.24) and a direct computation of the derivative of Gλ with respect to the inverse temperature we have −2β∆Gλ =

R

Av Ωλ (h(σ (l) ) G) − Ωλ (h(σ (l) ))Ωλ (G) .

(3.26)

l=1

For each replica l (1 ≤ l ≤ R), we evaluate separately the two terms in the right side of Eq. (3.26) by using the integration by parts (generalized Wick formula) for correlated Gaussian random variables, x1 , x2 , . . . , xn n ∂ψ(x1 , . . . , xn ) Av (xi ψ(x1 , . . . , xn )) = Av (xi xj ) Av . (3.27) ∂xj j=1 It is convenient to denote by pλ (R) the Gibbs-Boltzmann weight of R copies of the deformed system pλ (R) =

e−β [

R k=1

HΛ (σ(k) ) ] +

√

λ[

[Zλ (β)]R

R k=1

KΛ (σ(k) ) ]

,

(3.28)

so that we have

R 1 dpλ (R) e−β[HΛ (τ )] = pλ (R) . − δσ(k) , τ − R pλ (R) β dHΛ (τ ) [Zλ (β)]

(3.29)

k=1

We obtain

 

1 Av  Av Ωλ (h(σ (l) ) G) = G HΛ (σ (l) ) pλ (R) |Λ| (1) (r) σ ,...,σ   (R) dp λ  = Av  G QΛ (σ (l) , τ ) dH (τ ) Λ (1) (r) τ σ ,...,σ  

R   = −β  + G q l, k λ − RG q l, R+1 λ  G λ   k=1 k=l

(3.30)

(3.31)

(3.32)

920



where in (3.31) we made use of the integration by parts formula and (3.32) is obtained by (3.29). Analogously, the other term reads

Av Ωλ (h(σ (l) )) Ωλ (G)   1 = Av  G HΛ (σ (l) ) pλ (R + 1) (3.33) |Λ| (l) (1) (R) σ τ ,...,τ   (R + 1) dp λ  = Av  (3.34) G QΛ (σ (l) , γ) dHΛ (γ) (l) (1) (R) γ σ

τ

,...,τ

= −β [Gλ + RG q l R+1 λ − (R + 1)G q R+1, R+2 λ ] .

(3.35)

Inserting the (3.32) and (3.35) in Eq. (3.26) we finally obtain Theorem 2. Remark. The proof of the theorems shows that the identities which follow from the stochastic stability property are included in the Ghirlanda-Guerra identities [GG]. Indeed the family of GG identities are obtained from the self-averaging of the internal energy per particle with respect to the full equilibrium quenched measure. This implies, by the use of the Cauchy-Schwartz inequality, the vanishing of the truncated correlation between internal energy per particle and a generic observable G in the thermodynamic limit: hG − hG → 0

as

|Λ| → ∞ .

(3.36)

But clearly the previous fluctuation can be decomposed as a sum of the thermal fluctuation (averaged over the Gaussian disorder) and the fluctuation with respect to the disorder itself, i.e., h G − hG = Av (Ω[h G]) − Av (Ω[h]) Av (Ω[G]) = Av (Ω[h G] − Ω[h]Ω[G])) + Av (Ω[h]Ω[G]) − Av (Ω[h]) Av (Ω[G])

(3.37) (3.38)

By formula (3.26) we see that the thermal fluctuations (Eq.(3.37)) are those controlled by the stochastic stability.

4 Models The results proved in the previous sections hold true in complete generality because they are based on the general property of Gaussian variables. Stochastic stability in particular is fulfilled by both mean field models (like the SherringtonKirkpatrick, its p-spin generalization, the REM and GREM models etc.) and by the finite-dimensional models (like the Edwards-Anderson and Random Field models in general dimension d). The main point to be observed and well stressed is that each one of these models has his own set of observables which describe the

Vol. 6, 2005


921

quenched equilibrium state, namely the Gaussian covariance matrix of their own Hamiltonians, see Eq. (2.1). To be more specific let illustrate the two main cases of the covariance matrix for the Sherrington-Kirkpatrick model and for the EdwardsAnderson. The SK model of Hamiltonian N 1 Ji,j σi σj HN (σ, J) = − √ N i,j=1

(4.39)

with {Jij } identical independent normal Gaussian variables has a covariance matrix given by the standard overlap function between two configurations: (SK) QΛ (σ, τ )

=

N 1 σi τi N i=1

2 .

(4.40)

The Edwards-Anderson Hamiltonian is

HΛ (J, σ) = −

Jn,n σn σn ,

(4.41)

(n,n )∈B(Λ)

where the Jn,n are again independent normal Gaussian variables and the sum runs over all pairs of nearest neighbors sites n, n ∈ Λ ⊂ Zd with |n − n | = 1. Using the standard identification of the space of nearest neighbors with the d-dimensional bond-lattice b ∈ Bd with b = (n, n ) and denoting B(Λ) the d-bond-parallelepiped associated to Λ (|B| = d|V |) we introduce, for two spin configurations σ and τ , the notation σb = σn σn and τb = τn τn . The covariance matrix turns out to be (EA)

QΛ

(σ, τ ) :=

1 Qb (σ, τ ) , |B|

(4.42)

b∈B

where the local bond-overlap Qb (σ, τ ) between σ and τ is Qb (σ, τ ) := σb τb .

(4.43)

The property of stochastic stability for the Edwards-Anderson model in terms of its link-overlap has been originally considered in [C2]. The theorem proved here provides the generalization to the generic observable G.

5 Comments In this paper we have proved that every Gaussian spin glass model is stochastically stable with respect to a suitable class of perturbations. The consequences of such a stability can be expressed as zero average observables in terms of the proper overlap that each model carries: the covariance of its own Hamiltonian. It is finally worth to mention that the identities that we proved for the Edwards-Anderson model

922



are compatible with both the pictures of triviality and those of non-triviality for the overlap distribution at low temperature; for a discussion the reader may see the replica symmetry Breaking theory in [MPV], the Droplet theory in [FH, BM], the chaotic theory in [NS] and the trivial-non-trivial in [PY, KM]. Nevertheless the stochastic stability identities could suggest a test of triviality for the suitable overlap distribution in the same spirit of [MPRRZ]. We plan to return on these questions in a future work. Acknowledgments. We thank F. Guerra for many interesting discussions and in particular for an observation which led to a substantial improvement of this work. We also thanks A. Bovier, A. van Enter, S. Graffi, M. Talagrand and F.L. Toninelli.

References [AC]

M. Aizenman, P. Contucci, On the Stability of the Quenched state in Mean Field Spin Glass Models, J. Stat. Phys. 92, N. 5/6, 765–783 (1998).

[BCK]

A. Bianchi, P. Contucci, A. Knauf, Stochastically Stable Quenched Measures, math-ph/0404002, to appear in J. Stat. Phys. (2004).

[BM]

A.J. Bray and M.A. Moore, in Heidelberg Colloquium on Glassy Dynamics and Optimization, L. Van Hemmen and I. Morgenstern eds. Springer-Verlag, Heidelberg, (1986).

[C]

P. Contucci, Toward a classification theorem for stochastically stable measures, Markov Proc. and Rel. Fields. 9, N. 2, 167–176 (2002).

[C2]

P. Contucci, Replica Equivalence in the Edwards-Anderson Model, J. Phys. A: Math. Gen. 36, 10961–10966 (2003).

[CDGG]

P. Contucci, M. Degli Esposti, C. Giardin` a and S. Graffi, Thermodynamical Limit for Correlated Gaussian Random Energy Models, Commun. Math. Phys. 236, 55–63 (2003).

[EA]

S. Edwards and P.W. Anderson, Theory of spin glasses, J. Phys. F 5, 965–974 (1975).

[FH]

D.S. Fisher and D.A. Huse, Ordered Phase of Short-Range Ising SpinGlasses, Phys. Rev. Lett. 56, 1601–1604 (1986).

[FMPP1] S. Franz, M. Mezard, G. Parisi, L. Peliti, Measuring equilibrium properties in aging systems, Phys. Rev. Lett. 81, 1758 (1998). [FMPP2] S. Franz, M. Mezard, G. Parisi, L. Peliti, The response of glassy systems to random perturbations: A bridge between equilibrium and offequilibrium, J. Stat. Phys. 97, N. 3/4, 459–488 (1999).

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[G]

F. Guerra, About the overlap distribution in a mean field spin glass model, Int. J. Phys. B 10, 1675–1684 (1997).

[GG]

S. Ghirlanda, F. Guerra, General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys. A: Math. Gen. 31, 9149–9155 (1998).

[KM]

F. Krzakala and O.C. Martin, Spin and Link Overlaps in ThreeDimensional Spin Glasses, Phys. Rev. Lett. 85, 3013–3016 (2000).

[MPRRZ] E. Marinari, G. Parisi, F. Ricci-Tersenghi, J. Ruiz-Lorenzo, F. Zuliani, Replica Symmetry Breaking in Short Range Spin Glasses: A Review of the Theoretical Foundations and of the Numerical Evidence, J. Stat. Phys. 98, N. 5, 973–1074 (2000). [MPV]

M. Mezard, G. Parisi, M.A. Virasoro, Spin Glass theory and beyond, World Scientific, Singapore (1987).

[NS]

C.M. Newman and D.L. Stein, Spatial Inhomogeneity and Thermodynamic Chaos, Phys. Rev. Lett. 76, 4821–4824 (1996).

[P]

G. Parisi, On the probabilistic formulation of the replica approach to spin glasses, Int. Jou. Mod. Phys. B 18, 733–744 (2004).

[PY]

M. Palassini, A.P. Young, Nature of the Spin Glass State, Phys. Rev. Lett. 85, 3017–3021 (2000).

[SK]

D. Sherrington and S. Kirkpatrick, Solvable model of a spin-glass, Phys. Rev. Lett. 35, 1792–1796 (1975).

[T]

M. Talagrand, Spin glasses: a challenge for mathematicians, Berlin, Springer (2003).

Pierluigi Contucci and Cristian Giardin` a Dipartimento di Matematica Universit` a di Bologna I-40127 Bologna Italy email: [email protected] email: [email protected] Communicated by Jennifer Chayes submitted 13/10/04, accepted 22/11/04

Ann. Henri Poincaré 6 (2005) 925 – 936 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05925-12, Published online 05.10.2005 DOI 10.1007/s00023-005-0230-z


Bohr-Sommerfeld Rules to All Orders Yves Colin de Verdière

1 Introduction The goal of this paper is to give a rather simple algorithm which computes the Bohr-Sommerfeld quantization rules to all orders in the semi-classical parameter on the real line. The formula gives the highh for a semi-classical Hamiltonian H order terms in the expansion in powers of h of the semi-classical action using only integrals on the energy curves of quantities which are locally computable from the Weyl symbol. The recipe uses only the knowledge of the Moyal formula expressing the star product of Weyl symbols. It is important to note that our method assumes already the existence of Bohr-Sommerfeld rules to any order (which is usually shown using some precise Ansatz for the eigenfunctions, like the WKB-Maslov Ansatz) and the problem we address here is only about ways to compute these corrections. Existence of corrections to any order to Bohr-Sommerfeld rules is well known and can be found for example in [8] and [15] Section 4.5. Our way to get these high-order corrections is inspired by A. Voros’s thesis (1977) [13], [14]. The reference [1], where a very similar method is sketched, was given to us by A. Voros. We use also in an essential way the nice formula of in terms of the resolvent. Helffer-Sj¨ ostrand expressing f (H)

2 The setting and the main result Let us give a smooth classical Hamiltonian H : T R → R, where the symbol H admits the formal expansion H ∼ H0 + hH1 + · · · + hk Hk + · · · ; following [5] p. 101, we will assume that • H belongs to the space of symbols S o (m) for some order function m (for example m = (1 + |ξ|2 )p ) • H + i is elliptic 1 = Op and define H Weyl (H) with

OpWeyl (H)u(x) =

R2

e

i(x−y)ξ/h

dydξ x+y . , ξ)u(y) H( 2 2πh

1 Contrary to the usual notation, we denote by |dx · · · dx | the Lebesgue measure on Rn in n 1 order to avoid confusions related to orientations problems.

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Y. Colin de Verdière


is then essentially self-adjoint on L2 (R) with domain the Schwartz The operator H space S(R). In general, we will denote by σWeyl (A) the Weyl symbol of the operator A. The hypothesis: • We fix some compact interval I = [E− , E+ ] ⊂ R, E− < E+ , and we assume that there exists a topological ring A such that ∂A = A− ∪ A+ with A± a connected component of H0−1 (E± ). • We assume that H0 has no critical point in A • We assume that A− is included in the disk bounded by A+ . If it is not the case, we can always change H to −H. We define the well W as the disk bounded by A+ . Definition 1 Let HW : T R → R be equal to H in W , > E+ outside W and bounded. Then H W = OpWeyl (HW ) is a self-adjoint bounded operator. The semiclassical spectrum associated to the well W , denoted by σW , is defined as follows: σW = Spectrum(H W )∩] − ∞, E+ ] . The previous definition is useful because σW is independent of HW mod O(h∞ ). Moreover, if H0−1 (] − ∞, E + ]) = W1 ∪ · · · ∪ WN (connected components), then Spectrum(H)∩] − ∞, E + ] = ∪σWl + O(h∞ ) . The spectrum σW ∩ [E− , E+ ] is then given mod O(h∞ ) by the following Bohr-Sommerfeld rules Sh (En ) = 2πnh where n ∈ Z is the quantum number and the formal series Sh (E) =

∞

Sj (E)hj

j=0

is called the semi-classical action. Our goal is to give an algorithm for computing the functions Sj (E), E ∈ I. In fact exp(iSh (E)/h) is the holonomy of the WKB-Maslov microlocal soluˆ − E)u = 0 around the trajectory γE = H −1 (E) ∩ A. tions of (H

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H HW E+ I A+

A

A−

A− H = HW K

A+

E−

A

W Figure 1. The phase space.

It is well known that: • S0 (E) = γE ξdx = {H0 ≤E}∩W |dxdξ| is the action integral • S1 (E) = π − γE H1 |dt| includes the Maslov correction and the subprincipal term. Our main result is: Theorem 1 If H satisfies the previous hypothesis, we have: for j ≥ 2, (−1)l−1 d l−2 Sj (E) = Pj,l (x, ξ)|dt| (l − 1)! dE γE 2≤l≤L(j)

where • t is the parametrization of γE by the time evolution dx = (H0 )ξ dt, dξ = −(H0 )x dt

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• The Pj,l ’s are locally (in the phase space) computable quantities: more precisely each Pj,l (x, ξ) is a universal polynomial evaluated on the partial derivatives ∂ α H(x, ξ). The Pj,l ’s are given from the Weyl symbol of the resolvent (see Proposition (1)): L(j) ∞

1 Pj,l ˆ −1 = + hj . σWeyl (z − H) z − H0 j=1 (z − H0 )l l=2

If H = H0 , S2j+1 (E) = 0 for j > 0. In that case, the polynomial Pj,l (∂ α H) is homogeneous of degree l − 1 w.r. to H and the total weight of the derivatives is 2j, so that all monomials in Pj,l are of the form αk Πl−1 H k=1 ∂

with

l−1 k=1

|αk | = 2j and ∀k, |αk | ≥ 1.

Remark 1 We have also the following nice formula

2

(see also [14]): for any l ≥ 2,

hj Pj,l (x0 , ξ0 ) = (H − H0 (x0 , ξ0 ))(l−1) (x0 , ξ0 ) ,

j

where the power (l − 1) is taken w.r. to the star product. Proof. Let us denote h0 = H0 (x0 , ξ0 ). We have ˆ − h0 ) ˆ = (z − h0 ) − (H z−H and ˆ −1 = (z − H)

∞

ˆ − h0 )l−1 (z − h0 )−l (H

l=1

The formula follows then by identification of both expressions of the Weyl symbol of the resolvent at (x0 , ξ0 ). A less formal derivation is given by applying formula (3) to f (E) = (E − h0 )l−1 and computing Weyl symbols at the point (x0 , ξ0 ).

3 Moyal formula Let us define the Moyal product a b of the semi-classical symbols a and b by the rule: OpWeyl (a) ◦ OpWeyl (b) = OpWeyl(a b) 2I

learned this formula from Laurent Charles

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We have the well-known “Moyal formula” (see [5]): ab=

j ∞ 1 h {a, b}j j! 2i j=0

where {a, b}j (z) = [(∂ξ ∂x1 − ∂x ∂ξ1 )j (a(z) ⊗ b(z1 ))]|z1 =z with z = (x, ξ), z1 = (x1 , ξ1 ). In particular {a, b}0 = ab and {a, b}1 is the usual Poisson bracket. From the Moyal formula, we deduce the following (see also [14]): ˆ −1 of H ˆ is Proposition 1 The Weyl symbol j hj Rj (z) of the resolvent (z − H) given by L(j) ∞ ∞ Pj,l 1 j j h Rj (z) = + h (1) z − H0 j=1 (z − H0 )l j=0 l=2

where the Pj,l (x, ξ) are universal polynomials evaluated on the Taylor expansion of H at the point (x, ξ). If H = H0 , only even powers of j occur: R2j = 0. Proof. The proposition follows directly from the evaluation by Moyal formula of the left-hand side of   ∞ (z − H)  hj Rj  = 1 . j=0

The important point is that the poles at z = H are at least of multiplicity 2 for j ≥ 1. Using     ∞ ∞ (z − H)  hj Rj  =  hj Rj  (z − H) = 1 , j=0

j=0

and the fact that {., .}j are symmetric for even j’s and antisymmetric for odd j’s, we can prove the second statement by induction on j.

4 The method ∞ Let f ∈ Co∞ (I) and let us compute the trace D(f ) := Trace(f (H W )) mod O(h ) in 2 different ways:

1. Using the eigenvalues given by the Bohr-Sommerfeld rules we get: f (Sh−1 (2πhn)) + O(h∞ ) Trace(f (H W )) = n∈Z

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and, because f ◦ Sh−1 is a smooth function converging in the Co∞ topology to f ◦ S0−1 we can apply the Poisson summation formula and we get 1 D(f ) = f (Sh−1 (u))|du| + O(h∞ ) 2πh R 1 f (E)Sh (E)|dE| + O(h∞ ) 2πh R or using Schwartz distributions: and

D(f ) =

(a) D =

1 S (E) + O(h∞ ) 2πh h

using Helffer2. On the other hand, we compute the Weyl symbol of f (H) Sj¨ ostrand’s trick (see [5] p. 93): ∂F −1 |dxdy| = −1 (z)(z − H) (2) f (H) π Cz=x+iy ∂ z¯ where F ∈ C0∞ (C) is a quasi-analytic extension of f , i.e., F admits the Taylor expansion ∞ 1 (k) F (x + ζ) = f (x)ζ k k! k=0

at any real x. We start with the Weyl symbol of the resolvent (1). ˆ by putting Equation (1) into (2): We get then the symbol of f (H)   j h ˆ = OpWeyl f (H0 ) + f (l−1) (H0 )Pj,l  . f (H) (l − 1)! j≥1,l≥2

The justification of this formal step is done in [5]. We then compute the trace by using 1 Tr (OpWeyl (a)) = 2πh We get: 1 D(f ) = 2πh

T R

 f (H0 ) +

j≥1,l≥2

T R

a(x, ξ)|dxdξ| .

 1 f (l−1) (H0 )Pj,l  |dxdξ| hj (l − 1)!

We can rewrite using |dtdE| = |dxdξ| and integrating by parts:   l−1 l−1 d (−1) 1  T(E) + (b) D = hj Pj,l |dt| 2πh (l − 1)! dE γE j≥1,l≥2

(3)

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So we get, because l ≥ 2, by identification of (a) and (b), for j ≥ 1: Sj (E) −

(−1)l−1 d l−2 Pj,l |dt| = Cj (l − 1)! dE γE

(4)

l≥2

where the Cj ’s are independent of E. Proposition 2 In the previous formula (4), the Cj ’s are also independent of the operator. Proof. We can assume that (0, 0) is in the disk whose boundary is A− . Let us choose an Hamiltonian K which coincides with HW outside the disk bounded by A− and with the harmonic oscillator ˆ = OpWeyl( 1 (x2 + ξ 2 )) Ω 2 near the origin. We can assume that K has no other critical values than 0. We claim: for all j ≥ 1, ˆ = Cj (Ω) ˆ 1. Cj (K) ˆ = Cj (K) ˆ 2. Cj (H) Both claims come from the following facts: let us give 2 Hamiltonians whose Weyl symbols coincide in some ring B, then (i) The Pj,l are the same for 2 operators in the ring B where both have the same Weyl symbol, because they are locally computed from the symbols which are the same. (ii) The Sj (E)’s are the same for both operators because they have the same eigenvalues in the corresponding well modulo O(h∞ ): both operators have the same WKB-Maslov quasi-modes in B.

5 The case of the harmonic oscillator Proposition 3 For the harmonic oscillator, C1 = π and, for j ≥ 2, Cj = 0. ˆ = OpWeyl ( 1 (x2 + ξ 2 )) is the harmonic oscillator we have: Proof. If Ω 2 Sh (E) = 2πE + πh because En = (n − 12 )h for n = 1, . . . . It remains to compute the Pj,l ’s. Let us put ρ = 12 (x2 + ξ 2 ), and ∞

ˆ −1 = hj Rj σWeyl (z − Ω) j=0

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It is clear that the Rj ’s are functions fj (ρ, z) and from Moyal formula we get: fj+2 = −

1 (f + ρfj ) 4(z − ρ) j

and by induction on j: f2j+1 = 0 and f2j (ρ, z) =

l=3j+1 l=2j+1

al,j ρl−2j−1 , (z − ρ)l

with aj,l ∈ R. The result comes from

d dE

l−2 ρl−2j−1 |dt| = 0 , γE

if l ≥ 2j + 1.

6 The term S2 Let us assume first that H = H0 . From the Moyal formula, we have R2 = − with and

1 1 ∆ Γ {H0 , }2 = − − 3 z − H0 z − H0 4(z − H0 ) 4(z − H0 )4 ∆ = (H0 )xx (H0 )ξξ − ((H0 )xξ )2

Γ = (H0 )xx ((H0 )ξ )2 + (H0 )ξξ ((H0 )x )2 − 2(H0 )xξ (H0 )x (H0 )ξ .

A very similar computation can be found in [9] p. 93, formula (0.13). Using formulae (1) and (4), we get: S2 (E) = − Theorem 2

1 d 8 dE

∆|dt| + γE

1 24

• If H = H0 , we have S2 = −

1 d 24 dE

d dE

2 Γ|dt|.

(5)

γE

∆|dt|. γE

• In the general case, we have: 1 d 1 d S2 = − ∆|dt| − H2 |dt| + H 2 |dt| . 24 dE γE 2 dE γE 1 γE

(6)

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Formula (5) were obtained in [1], formula (3.12), and formula (6) by Robert Littlejohn [10, 2] using completely different methods. Proof. Γdt is the restriction to γE of the 1-form α in R2 with α = ((H0 )xx (H0 )ξ − (H0 )xξ (H0 )x )dx + ((H0 )xξ (H0 )ξ − (H0 )ξξ (H0 )x )dξ . Orienting γE along the Hamiltonian flow, we get using Stokes formula: Γ|dt| = α=− dα γE

γE

DE

where ∂DE = γE and DE is oriented by dx ∧ dξ. We have dα = −2∆dx ∧ dξ and hence:

Γ|dt| = 2 γE

∆|dxdξ| . DE

From |dtdE| = |dxdξ|, we get: d ∆|dxdξ| = ∆|dt| . dE DE γE So that:

d dE

Γ|dt| = 2 γE

∆|dt| γE

from which Theorem 2 follows easily.

7 Quantum numbers Theorem 3 The quantum number “n” in the Bohr-Sommerfeld rules corresponds exactly to the nth eigenvalue in the corresponding well, i.e., the nth eigenvalue of H W. Proof. It is clear that the labelling of the eigenvalues of H W is invariant by homoˆ topies leaving the symbol constant in A. We can then change H W to K for which the result is clear because the quantization rules give then exactly all eigenvalues.

8 Extensions 8.1

2d phase spaces

The method applies to any 2d phase space using only 3 things: • The star product

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• The fact that the trace of operators is given by (1/2πh)× (the integral of their symbols) • An example where you know enough to compute the Cj s The power of our method is that it avoids the use of any Ansatz. Maslov contributions come only from the computation of an explicit example.

8.2

The cylinder T (R/Z)

In that case, we replace the hypothesis by the following: • We fix some compact interval I = [E− , E+ ] ⊂ R, E− < E+ , and we assume there exists a topological ring A, homotopic to the zero section of T (R/Z), such that ∂A = A− ∪ A+ with A± a connected component of H −1 (E± ). • We assume that H has no critical point in A • We assume that A− is “below” A+ (see Figure 2).

A+

A

A− Figure 2. The cylinder. We will use the Weyl quantization for symbols which are of period 1 in x. Then Theorem 1 holds. The only change is S1 which is now 0. The proof is the same except that the reference operator is now hi ∂x instead of the harmonic oscillator.

8.3

Other extensions

It would be nice to extend the previous method to the case of Toeplitz operators on two-dimensional symplectic phase spaces, in the spirit of [3] and [4], and to the case of systems starting from the analysis in [6].

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As remarked by Littlejohn, our method does not obviously extend to semi1 , . . . , H d with d ≥ 2 degrees of freedom. classical completely integrable systems H The reason for that is that, using the same lines, we will get only the jacobian determinant of the d BS actions which is not enough to recover the actions even up to constants.

9 Relations with KdV ˆ = −∂ 2 + q(x) with q(x + 1) = Let us consider the periodic Schrödinger equation H x q(x). Let us denote by λ1 < λ2 ≤ λ3 < λ4 ≤ · · · the eigenvalues of the periodic ˆ Then the partition function problem for H. ∞

Z(t) =

e−tλn

n=1

admits, as t → 0+ , the following asymptotic expansion 1 a0 + a1 t + · · · + aj tj + · · · + O(t∞ ) Z(t) = √ 4πt where the aj ’s are of the following form aj =

0

1

Aj q(x), q (x), . . . , q (l) (x), . . . |dx|

where the Aj ’s are polynomials. The aj ’s are called the Korteweg-de Vries invariants because they are independent of u if qu (x) = Q(x, u) is a solution of the Korteweg-de Vries equation. See [11], [12] and [16]. Let us translate the previous objects in the semi-classical context: we have ˆ and h2 H ˆ is the semi-classical operator of order 0 whose Z(h2 ) = Tr exp(−h2 H) Weyl symbol is ξ 2 +h2 q(x). If we put f (E) = e−E , the partition function is exactly a trace of the form used in our method except that E → e−E is not compactly supported. Nevertheless, the similarity between both situations is rather clear.

References [1] P.N. Argyres, The Bohr-Sommerfeld Quantization Rule and the Weyl Correspondence, Physics 2, 131–199 (1965). [2] M. Cargo, A. Gracia-Saz, R. Littlejohn, M. Reinsch & P. de Rios, Moyal star product approach to the Bohr-Sommerfeld approximation, J. Phys. A: Math and Gen. 38 , 1977–2004 (2005). [3] L. Charles, Berezin-Toeplitz Operators, a Semi-classical Approach, Commun. Math. Phys. 239, 1–28 (2003).

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[4] L. Charles, Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators, Commun. Partial Differential Equations 28, 1527–1566 (2003). [5] M. Dimassi & J. Sj¨ ostrand, Spectral Asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes 268, Cambridge UP (1999). [6] C. Emmrich & A. Weinstein, Geometry of the transport equation in multicomponent WKB approximations, Commun. Math. Phys. 176 No.3, 701–711 (1996). [7] A. Gracia-Saz, The symbol of a function of a pseudo-differential operator, Ann. Inst. Fourier (to appear). [8] B. Helffer & D. Robert, Puits de potentiel généralisés et asymptotique semiclassique, Ann. IHP (physique théorique) 41, 291–331 (1984). [9] B. Helffer & D. Robert, Riesz means of bound states and semiclassical limit connected with a Lieb-Thirring’s conjecture, Asymptotic Analysis 3, 91–103 (1990) . [10] R. Littlejohn, Lie Algebraic Approach to Higher-Order Terms, Preprint 17p. (June 2003). [11] H.P. McKean & P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30, 217–274 (1975). [12] W. Magnus & S. Winkler, Hill’s equation, New York-London-Sydney: Interscience Publishers, a division of John Wiley & Sons. VIII, 127 p. (1966). [13] A. Voros, Développements semi-classiques, Thèse de doctorat (Orsay, 1977). [14] A. Voros, Asymptotic -expansions of stationary quantum states, Ann. Inst. H. Poincaré Sect. A (N.S.) 26, 343–403 (1977). [15] San V˜ u Ngo.c, Sur le spectre des systèmes complètement intégrables semiclassiques avec singularités, Thèse de doctorat (Grenoble, 1998), http://wwwfourier.ujf-grenoble.fr/˜ svungoc/ . [16] V. Zakharov & L. Faddeev, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Funct. Anal. Appl. 5, 280–287 (1977). Yves Colin de Verdière Institut Fourier Unité mixte de recherche CNRS-UJF 5582 BP 74 F-38402 Saint Martin d’Hères Cedex France email: [email protected] Communicated by Bernard Helffer submitted 02/09/04, accepted 29/10/04

Ann. Henri Poincaré 6 (2005) 937 – 990 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05937-54, Published online 05.10.2005 DOI 10.1007/s00023-005-0231-y


Determination of Non-adiabatic Scattering Wave Functions in a Born-Oppenheimer Model George A. Hagedorn∗ and Alain Joye

Abstract. We study non-adiabatic transitions in scattering theory for the timedependent molecular Schr¨ odinger equation in the Born-Oppenheimer limit. We assume the electron Hamiltonian has finitely many levels and consider the propagation of coherent states with high enough total energy. When two of the electronic levels are isolated from the rest of the electron Hamiltonian’s spectrum and display an avoided crossing, we compute the component of the nuclear wave function associated with the non-adiabatic transition that is generated by propagation through the avoided crossing. This component is shown to be exponentially small in the square of the Born-Oppenheimer parameter, due to the Landau-Zener mechanism. It propagates asymptotically as a free Gaussian in the nuclear variables, and its momentum is shifted. The total transition probability for this transition and the momentum shift are both larger than what one would expect from a naive approximation and energy conservation.

1 Introduction We study scattering theory for the time-dependent molecular Schrödinger equation 4 ∂ 2 ∂ − i 2 ψ(x, t, ) = + h(x) ψ(x, t, ) in L2 (R, Cm ), (1.1) ∂t 2 ∂x2 where the electronic hamiltonian h(x) is an m×m self-adjoint matrix that depends on the nuclear position variable x ∈ R. The Born-Oppenheimer parameter > 0 denotes the fourth root of the electron mass divided by the mean nuclear mass. We compute the leading order asymptotics of nuclear wave functions associated with certain non-adiabatic transitions of the electrons. The Landau-Zener mechanism responsible for these makes them exponentially small in 1/2 as → 0. Our most general result can be found in Theorem 5.1. Describing the most general situation requires the development of a significant amount of notation and some technical hypotheses. So, in this introduction, we describe two physically interesting special cases that illustrate the main consequences of our analysis in a simple situation. Theorems 6.1 and 6.2 give precise statements of our results for these special cases. ∗ Partially Supported by National Science Foundation Grants DMS–0071692 and DMS– 0303586.

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Figure 1. A plot of typical electron energy levels involved in an avoided crossing. Energy is plotted vertically. Position is plotted horizontally, and the avoided crossing occurs in the middle of the plot where the difference between the energy levels is a minimum. Suppose h(x) is a real 2 × 2 self-adjoint matrix that depends analytically on x and has limits h(±∞) as x → ±∞ that are approached sufficiently rapidly. Denote the eigenvalues of h(x) by ej (x), and assume that e2 (x) ≥ e1 (x) + δ for all x ∈ R, where δ > 0. √ Near x = 0, assume e1 and e2 have an avoided crossing, i.e., e2 (x) − e1 (x) x2 + δ 2 close to x = 0, with δ small but positive. Such an avoided crossing corresponds to complex crossing points z0 and z0 , where the analytic continuations of e1 and e2 satisfy e1 (z0 ) = e2 (z0 ), and z0 is close to the real axis, with z0 = O(δ). Let φ1 (x) and φ2 (x) denote normalized, real eigenvectors associated with e1 (x) and e2 (x). Among the nuclear wave functions we can accommodate are Gaussian coherent states that are defined by 2

ϕ0 (A, B, , a, η, x) =

1 π 1/4 1/2 A1/2

exp

B (x − a)2 η (x − a) − +i 2 2A 2

,

where the complex numbers A and B satisfy the normalization condition Re BA = 1. These states are localized in position near x = a, and in momentum near p = η. Their position uncertainty is |A| and their momentum uncertainty is |B|. For a thorough discussion of these wave packets, see [9]. Choose E > supx∈R e2 (x). For a state incoming from the left on the upper electronic level, choose η− > 0. We assume η− is large enough so that the classical 2 energy η− /2 + e2 (−∞) > E. There exists a solution to (1.1) whose large negative

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Non-adiabatic Wave Functions in a B.-O. Model

939

t asymptotics are given by 2

2

ei(η− /2−e2 (−∞))t/ ϕ0 (A− + iB− t, B− , 2 , a− + η− t, η− , x) φ2 (x),

(1.2)

where the nuclear part is a free Gaussian. Since the electronic levels are isolated from one another, the large positive t asymptotics of this solution are multiples of φ2 (x), up to exponentially small errors in 1/2 . They have the leading behavior determined by the standard time-dependent Born-Oppenheimer approximation as → 0, see [8]: 2

2

eiθ1 () ei(η1 /2−e2 (∞))t/ ϕ0 (A1 + iB1 t, B1 , 2 , a1 + η1 t, η1 , x) φ2 (x), where eiθ1 () is some explicit phase, and the parameters A1 , B1 , a1 , η1 are determined by the scattering properties of the classical Hamiltonian p2 /2 + e2 (x). Our interest lies with the leading order asymptotics of the non-adiabatic component of the wave function for large positive t and → 0. We prove in Theorem 6.1 that these have the form ∗

c0 e−α

/2

2

eiθ+ () ei(η+ /2−e1 (∞))t/

2

ϕ0 (A+ + iB+ t, B+ , 2 , a+ + η+ t, η+ , x) φ1 (x), and we specify how the phase θ+ (), the -independent amplitude c0 > 0, the exponential decay rate α∗ > 0, and the parameters of the free Gaussian part A+ , B+ , a+ , and η+ > 0 are determined. As a corollary, the leading term of the transition amplitude A() (whose absolute square is the transition probability) is given by the quantity ∗

A() = c0 eiθ+ () e−α

/2

,

as → 0.

(1.3)

Let us describe the main features of this exponentially small transmitted part of the wave function. One may naively expect η+ to be determined by the energy conservation condition 2 η2 η− + e2 (−∞) = + + e1 (∞), 2 2

but this yields the wrong value. The correct value is larger. Intuitively, this is due to the faster parts of the wave function behaving less adiabatically than the slower parts. Because this dependence on the speed appears in an exponent, it leads to an O(1) change in the final momentum η+ . In other words, the higher momentum components of the incoming state are much more likely to make a transition than the lower momentum components. Hence, after the transition, there are more fast pieces of the wave function, and the final average momentum is greater than one would naively expect from an energy conservation calculation based solely on the average incoming momentum.

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This also affects the transition amplitude which is larger than what is naively expected. It is asymptotically composed of an -independent prefactor c0 times an ∗ 2 exponentially small quantity e−α / , whose decay rate α∗ is related to that of the Landau-Zener decay rate for purely adiabatic problems. Actually, α∗ consists of the sum of the imaginary part of some action integral around the complex electronic eigenvalue crossing point z0 and a contribution that depends explicitly on the nuclear part of the initial incoming state (1.2). The action integral depends only on the electronic levels and reads ζ 2(E − e2 (z)dz where ζ is a loop in the complex plane based at the origin encircling z0 . The contribution from the nuclear part of the wave packet depends on the shape of its momentum/energy density. It is that last contribution that makes the obvious candidate given by the imaginary part of the action integral taken at the classical energy E, miss the actual value of the decay rate α∗ . In that sense, (1.3), which we could call a molecular Landau-Zener formula, cannot be determined from the usual adiabatic Landau-Zener formula with just the knowledge of the electronic levels and the classical nuclear momentum close to the avoided crossing. Indeed, our analysis shows that we also need to take into account the details of the incoming wave packet to determine (1.3). This is why we resort to coherent states to get such accurate asymptotics. The way we obtain all our results is by employing a time-independent scattering theory approach that uses generalized eigenfunctions of the full Hamiltonian. We expand the wave function in terms of the generalized eigenfunctions and calculate the large |t| asymptotics. For every incoming momentum k there is classical energy conservation, but a different probability of making the non-adiabatic transition. We obtain the correct α∗ and η+ by computing the averages over k rather than by doing one calculation based on the average incoming momentum η− . Remarks 1. We obtain the analogous results when the incoming state is associated with the lower electronic level e1 , provided that we keep the average total energy above both the levels. 2. There are other components of the scattered wave function. For example, one should expect a reflected wave on the e2 electronic level and also a reflected wave on the e1 level. We prove that if the avoided crossing has a sufficiently small gap, then the other components are exponentially even smaller in 1/2 than the transmitted non-adiabatic term we compute. The second situation we describe in this introduction involves the same setup as above, but with the Gaussian incoming states replaced by more general incoming coherent states. This example illustrates the second key feature that our analysis demonstrates: even if the incoming state is not Gaussian, but is any polynomial times a Gaussian, the outgoing non-adiabatic transition state is Gaussian to leading order in .

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For m = 1, 2, . . . , we define ϕm (A, B, 2 , a, η, x) = 2−m/2 (m!)−1/2 A−m/2 (A)m/2 Hm

x−a |A|

φ0 (A, B, 2 , a, η, x),

(1.4)

where Hm is the mth order Hermite polynomial. We now replace (1.2) by 2

2

ei(η− /2−e2 (−∞))t/ ϕm (A− + iB− t, B− , 2 , a− + η− t, η− , x) φ2 (x).

(1.5)

Again, up to exponentially small errors, the large positive t asymptotics of the solution are multiples of φ2 (x). Their leading behavior is determined by the standard time-dependent Born-Oppenheimer approximation, 2

2

eiθ1 () ei(η1 /2−e2 (∞))t/ ϕm (A1 + iB1 t, B1 , 2 , a1 + η1 t, η1 , x) φ2 (x), where A1 , B1 , a1 , η1 , and θ1 () are the same as in our first example. However, our Theorem 6.2 shows that the leading order asymptotics of the non-adiabatic component of the wave function for large positive t again have the form of a freely propagating Gaussian ∗

cm −m e−α

/2

2

eiθ+ () ei(η+ /2−e1 (∞))t/

2

× ϕ0 (A+ + iB+ t, B+ , 2 , a+ + η+ t, η+ , x) φ1 (x), and display a pre-exponential factor of order −m . The values of α∗ , A+ , B+ , a+ , and η+ are the same as in our first example, and we determine the prefactor cm . The numerics presented below clearly illustrate these features. The presence of the factor of −m can be understood as follows: In momentum space, the Hermite polynomial in the wave function does not get the extra shift that the Gaussian does. For small , the scaling in these two factors causes the Hermite polynomial to behave like a constant times −m pm where the Gaussian is large. Since the Gaussian is highly localized, a zeroth order Taylor approximation can be used to see that the wave function is asymptotically a multiple of the Gaussian times −m . Our most general result, Theorem 5.1, extends these results in several ways. First, we can handle electron Hamiltonians h(x) that are m×m complex hermitian matrices which have two levels of interest that have an avoided crossing. These levels must stay well separated from the rest of the spectrum of h(x). Second, we can handle situations in which several levels display certain patterns of avoided crossings. For example, when two levels have an avoided crossing for one value of x, and one of those levels has another avoided crossing with a third level for some other value of x. However, in such cases, we can only study the non-adiabatic components for certain levels. The ones we can handle depend on the order in

942



which the levels have the avoided crossings. Third, we can consider more general incoming states that do not have the form of the ϕj ’s considered above. They are characterized by an energy (or momentum) distribution which is sharply peaked around some fixed energy, so that a semiclassical analysis can be performed. In such general cases also, the nuclear part of the non-adiabatic wave function is Gaussian and exponentially small, with a decay rate sharing the properties described above. The paper is organized as follows: In the rest of the Introduction, we review the relevant literature and present numerical results for the above examples. They show excellent agreement with our analysis. (For the first example with = 0.2, our formulas give a transition probability of 1.215×10−9 and a momentum expectation after transition of 2.0516. The corresponding values from the numerical simulations are 1.217 × 10−9 and 2.0543. This agreement is remarkable, given how large our is.) In Section 2, we set up the general problem we study. We state most of our hypotheses here and make precise the notion of avoided crossing. In Section 3, we study generalized eigenvectors of the full Hamiltonian. In particular, their WKB-type analysis in the complex plane is performed here. We superimpose the generalized eigenvectors to generate solutions to the time-dependent Schr¨ odinger equation and construct asymptotic scattering states in Section 4. Non-adiabatic transition asymptotics are studied in Section 5, where our most general result is stated as Theorem 5.1. Further properties and estimates on the energy and momentum shifts are provided in Section 5. Section 6 is devoted to the special case of interest where the nuclear part of the incoming state is a Gaussian or a Gaussian times a Hermite polynomial as in (1.5). Finally, Section 7 contains the proofs of several technical results that are stated in the earlier sections. From this outline, one can see that our results depend crucially on the properties of generalized eigenvectors of the full Hamiltonian. We prove these properties by revisiting ideas and results of Joye [14], [15] that provide exponentially accurate WKB-type results in a generic avoided-crossing regime, generalizing earlier two-level adiabatic techniques from [17], [18], [19]. This step is necessary in order to control the dependence of the relevant quantities in the energy parameter. See also [21], [24] for stationary results of the same kind, obtained at fixed energy and, essentially, for two-level systems. That a complex WKB-type analysis plays an important role here should be no surprise. Indeed, in the ODE context of adiabatic-like problems dealt with in the references above, the complex WKB approach proved to be the most efficient method providing a quantitative analysis of the exponentially small leading order term of the Landau-Zener mechanism. See, however, [13] and [2] for a different successful approach of such problems, based on optimal truncation techniques. Nevertheless, the understanding of the stationary scattering process is of course not enough to get control on the time-dependent propagation of states: as our main result demonstrates, even the decay rate of the transition amplitude cannot be determined by the stationary data only. There are mathematical results on the exponentially small size of nonadiabatic transitions in the Born-Oppenheimer approximation, and for related

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problems. See, e.g., [12], [22], [1], [23]. However, to the best of our knowledge, there are no rigorous results on this topic in the literature that actually compute the leading asymptotics of non-adiabatic transitions in our time-dependent PDE setting. We have recently learned that Betz and Teufel, [3], are adapting techniques from [2] to the Born-Oppenheimer setup. They have formal and numerical results for specific electronic hamiltonians in agreement with ours. Also, rigorous results on the propagation of wave packets through avoided crossings, representing first attempts to unravel the molecular Landau-Zener mechanism, are obtained in [10], [11]. (See also [25].) In those papers, the gap δ shrinks to zero with in such a way that the transitions are of order one, so that they can be computed by perturbation theory. This is in contrast to the present situation, in which δ is small but fixed as → 0, and the transitions are exponentially small. Because of the importance of the Landau-Zener mechanism to molecular physics, there are relevant papers in the physics and chemistry literature. See, e.g., [4], [26], [27].

1.1

Numerical simulations for a Gaussian initial state

We now present graphical results of a numerical simulation in which the initial state is a Gaussian function associated with the upper energy level for a two level system. These plots are in very good agreement with the results of our analysis. We have numerically integrated equation (1.1) with = 0.2 for the Hamiltonian function 1 1 tanh(x) h(x) = . −1 2 tanh(x) 1 1 + tanh(x)2 , and there is an avoided crossing at The energy levels are ± 2 x = 0 with a minimum gap of 1. The initial state is the eigenvector associated with the upper energy level times the Gaussian φ0 (A0 +itB0 , B0 , 2 , ηt, η, x), where A0 = B0 = η = 1, with the initial time t = −10. The following two figures show the initial position and momentum probability densities, respectively. In both plots, the probability of being on the lower energy level is zero. The next two plots show the position and momentum probability densities at t = 9 after the wave function has interacted with the avoided crossing. The component associated with the lower energy level has mean momentum 2.05. It is evident from the plot that it is greater than 2. The naive energy conservation calculation predicts the following: The total energy is E = η 2 /2 + 1/2 1 + tanh(−10)2 = 1.2071. After the to the lower surface, the kinetic energy should be this value plus √ transition 2/2, so η12 /2 = 1.9142. This predicts a final momentum after the transition of η1 = 1.9566.

944



0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 −20

−10

−15

−5

5

0

10

20

15

25

Figure 2. Position space plot at time t = −10 of the probability density for being on the upper energy level (solid line), and 3 × 108 times the probability density for being on the lower energy level (dotted line). The dotted line cannot be seen because it coincides with the horizontal axis.

3

2.5

2

1.5

1

0.5

0 −6

−4

−2

0

2

4

6

Figure 3. Momentum space plot at time t = −10 of the probability density for being on the upper energy level (solid line), and 3 × 108 times the probability density for being on the lower energy level (dotted line). As in Figure 2, the dotted line cannot be seen.

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0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 −20

−10

−15

−5

5

0

10

20

15

25

Figure 4. Position space plot at time t = 9 of the probability density for being on the upper energy level (solid line), and 3 × 108 times the probability density for being on the lower energy level (dotted line).

3

2.5

2

1.5

1

0.5

0 −6

−4

−2

0

2

4

6

Figure 5. Momentum space plot at time t = 9 of the probability density for being on the upper energy level (solid line), and 3 × 108 times the probability density for being on the lower energy level (dotted line).

946

1.2



Numerical simulations for more general initial states

We next present in Figures 6 to 9 the results for the same system as above, but where the initial Gaussian φ0 has been replaced by φ3 . See (1.4). Note that the transition amplitude is significantly larger than in the example above, and that the component of the wave function that makes the transition to the lower level is approximately a Gaussian. The value of epsilon = 0.2 is not particularly small, so the component of the final state that does not make a transition is only approximately a φ3 wave packet. We have chosen this relatively large value of epsilon to avoid numerical difficulties in integrating equation (1.1). We should also note that the naive energy conservation calculation again predicts that the component of the wave function on the lower level should have mean momentum 1.9566. Since initial wave function has a greater momentum uncertainty than in the Gaussian example above, we see an even greater discrepancy between this prediction and the correct value. Our simulation yields a value of roughly 2.25. 0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0 −20

−15

−10

−5

0

5

10

15

20

25

Figure 6. Position space plot at time t = −10 of the probability density for being on the upper energy level (solid line), and 107 times the probability density for being on the lower energy level (dotted line). As in Figures 2 and 3, the dotted line cannot be seen.

2 Hypotheses for the electron Hamiltonian We begin with three general assumptions about the electron Hamiltonian h. We then impose two more assumptions that make precise the avoided crossing situations we can handle.

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1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0 −6

−4

−2

0

2

4

6

Figure 7. Momentum space plot at time t = −10 of the probability densityfor being on the upper energy level (solid line), and 107 times the probability density for being on the lower energy level (dotted line). As in Figure 2, 3, and 6, the dotted line cannot be seen.

0.25

0.2

0.15

0.1

0.05

0 −20

−15

−10

−5

0

5

10

15

20

25

Figure 8. Position space plot at time t = 9 of the probability density for being on the upper energy level (solid line), and 107 times the probability density for being on the lower energy level (dotted line).

948



1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0 −6

−4

−2

0

2

4

6

Figure 9. Momentum space plot at time t = 9 of the probability density for being on the upper energy level (solid line), and 107 times the probability density for being on the lower energy level (dotted line). H1: We assume z → h(z) is a m × m matrix-valued analytic function that is analytic in z ∈ ρα = {z = x + iy : |y| ≤ α}, where α > 0. We assume h(z) is self-adjoint for z ∈ R. Since we work in a scattering framework, we further assume: H2: There exist ν > 1/2, c, and two matrices h(±∞), such that for all x ∈ R, sup h(x + iy) − h(±∞) ≤

|y|≤α

c ,

x2+ν

where x denotes (1 + x2 )1/2 . The rate of convergence in this assumption can certainly be weakened. However, general scattering theory is not the main point of the present study. The following assumption is stronger than what one would expect physically to be necessary because it requires conditions on all eigenvalues, not just the ones of involved in the avoided crossings we consider later. However, we need this assumption for our proof. Specifically, this hypothesis for all eigenvalues plays a role in the proof of (7.29) at the end of the proof of Proposition 3.1. H3: We assume the spectrum σ(h(x)) of h(x) consists of m non-degenerate eigenvalues σ(h(x)) = {ej (x)}j=1,...,m , for any x ∈ R ∪ {±∞}.

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We let φj (x), j = 1, . . . , m, denote the corresponding eigenvectors, characterized up to constant phases by the following conditions φj (x) ≡ 1,

and

φj (x), φj (x) ≡ 0,

∀ j = 1, . . . , m,

(2.1)

where the prime denotes the derivative with respect to x. The eigenvectors are analytic in some narrow open strip containing the real axis [20]. By using the Cauchy formula, it is easy to check that our hypotheses imply

and

dn ( ej (x) − ej (±∞) ) = O( x−(2+ν) ) dxn

(2.2)

dn ( φj (x) − φj (±∞) ) = O( x−(2+ν) ), dxn

(2.3)

for any n ∈ N. We now make specific assumptions concerning avoided crossings for h. The idea is to assume h(x) belongs to a smooth family of electron Hamiltonians h(x, δ). When δ = 0, we assume there are actual crossings. When δ = 0, we assume there are no crossings for real values of x. The electron Hamiltonians we actually use have the form h(x, δ) for some small, but fixed value of δ. Our precise assumption is the following: H4: For each fixed δ ∈ [0, d], the matrix h(x, δ) satisfies H1 in a strip ρα independent of δ, and h(z, δ) is C 2 as a function of the two variables (z, δ) ∈ ρα ×[0, d]. Moreover, h(·) satisfies H2 uniformly for δ ∈ [0, d], with limiting values h(±∞, δ) that are C 2 functions of δ ∈ [0, d]. Again, some of our results hold under weaker smoothness assumptions. We can deal with multiple avoided crossings, but cannot deal with all possible patterns of avoided crossings. The following assumption describes the ones we allow. This assumption is complicated, and we recommend the reader look at the picture on page 686 of [15] to get some intuition about its meaning. H5: For each x ∈ R and each δ ∈ [0, d], σ(h(x, δ)) consists of m real eigenvalues σ(h(x, δ)) = {e1 (x, δ), e2 (x, δ), . . . , em (x, δ)} ⊂ R.

(2.4)

When δ > 0 we assume these are distinct for x ∈ [−∞, +∞] and are labeled by e1 (x, δ) < e2 (x, δ) < · · · < em (x, δ). When δ = 0, the eigenvalues are m analytic functions that have finitely many real crossings at x1 ≤ x2 ≤ · · · ≤ xp , with p ≥ 1. We assume the eigenvalues have m distinct limits as x → −∞ and as x → ∞. We label these eigenvalues ej (x, 0) in a way that is discontinuous in δ near δ = 0. This labeling is determined by the following conditions:

950



i) For all x < x1 , e1 (x, 0) < e2 (x, 0) < · · · 0. ∂x

(2.5)

iii) For all j ∈ {1, 2, . . . , n}, the eigenvalue ej (x, 0) crosses eigenvalues whose indices are all superior to j or all inferior to j. Remarks i) The parameter δ can be understood as a coupling constant that controls the strength of the perturbation that lifts the degeneracies of h(x, 0) on the real axis. ii) Because h(x, 0) is self-adjoint, the eigenvalues ej (x, 0) cannot have branch points on the real axis, and they are analytic in a neighborhood of the real axis. iii) The crossings are assumed to be generic in the sense that the derivatives of ej − ek are non-zero at the crossing xr . This ensures that when δ > 0 is small, the generic behavior (3.19) holds at the corresponding complex crossing points. iv) When m = 2, H5 requires that the two eigenvalues have exactly one generic crossings when δ = 0. v) The crossing points {x1 , x2 , . . . , xp } need not be distinct, which is important when the Hamiltonian possesses symmetries. However, for each j = 1, . . . , n, the eigenvalue ej (x, δ) experiences avoided crossings with ej+1 (x, δ) and/or ej−1 (x, δ) at a subset of distinct points {xr1 , . . . , xrj } ⊆ {x1 , x2 , . . . , xp }. For certain results, we also impose the condition that these avoided crossings be generic in the sense of [7] and [14]. This condition essentially says that the low order Taylor series coefficients of certain quantities do not vanish at the crossing when δ = 0. H6: Near an avoided crossing of ej (x, δ) and en (x, δ), there exist a > 0, b > 0, and c ∈ R, such that en (x, δ) − ej (x, δ) = ± ax2 + 2cxδ + b2 δ 2 + R3 (x, δ), (2.6) where c2 < a2 b2 and R3 (x, δ) is a remainder of order 3 in (x, δ) close to (0, 0). Our final hypothesis involves both the electron Hamiltonian and an interval of energies, ∆. We ultimately consider states of the full Hamiltonian whose energy is concentrated in ∆, with ∆ high enough that scattering onto all the electron

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energy levels is possible. An energy range that satisfies this condition can always be chosen for some strip ρα , provided the minimum value in ∆ is large enough. H7: The interval ∆ ∈ R is compact and has non-empty interior. Furthermore, it is chosen so that for all j, inf

E∈∆ z∈ρα δ∈[0,δ]

|E − ej (z, δ)| > 0.

3 Generalized eigenvectors For energies E ∈ ∆, we construct generalized eigenvectors for the full Hamiltonian. For the time being, the parameter δ > 0 is fixed and we drop it in the notation. The generalized eigenvectors are solutions Ψ(x, E, ) ∈ Cm to the time-independent Schr¨ odinger equation 4 ∂ 2 + h(x) Ψ(x, E, ) = E Ψ(x, E, ). (3.1) − 2 ∂x2 For each E ∈ ∆, the set of such solutions is 2m-dimensional, and individual solutions can be characterized by their asymptotics at x = −∞ (or at x = ∞). Let Ψ(x, E, ) Φ(x, E, ) = ∈ C2m . ∂ i 2 ∂x Ψ(x, E, ) Then (3.1) is equivalent to i 2 where

∂ Φ(x, E, ) = H(x, E) Φ(x, E, ), ∂x

H(x, E) =

O

I

2 (E I − h(x))

O

E I − h(x) > 0,

(3.2)

∈ M2m (C)

for all x ∈ R ∪ {±∞}.

and (3.3)

Here, I denotes the identity matrix in Cm . Note that the matrix H(x, E) is not self-adjoint, but satisfies the relation O I ∗ H(x, E) = J H (x, E) J, where J= . (3.4) I O The small asymptotics of solutions to (3.2) are studied in [15]. Of particular importance to us is Section 7 of [15], which is devoted to the computation of exponentially small elements of the related S-matrix that we describe below. We apply the results of [15] to (3.2), keeping track of the dependence on E.

952



By our hypotheses on E and h(x), the spectrum of H(E, x) consists of 2m distinct real eigenvalues =+,− σ(H(x, E)) = { kjτ (x, E) }τj=1,...,m ,

kjτ (x, E) = τ kj (x, E) = τ

with

2 (E − ej (x)) ∈ R.

(3.5)

Note that the kjτ ’s correspond to the classical momenta associated with the classical potentials ej (x). A set of corresponding eigenvectors { χτj (x, E) } is given (in block notation) by φj (x) τ χj (x, E) = (3.6) ∈ C2m . kjτ (x, E) φj (x) From these we produce new eigenvectors 1 ϕτj (x, E) = χτj (x, E) 2 kj (x, E)

(3.7)

that satisfy the normalization convention (3.10) below, that was adopted in [15]. This normalization is motivated by the following: We can write H(x, E) = kjτ (x, E) Pjτ (x, E), j,τ

where {Pjτ (x, E)} denotes a set of non-orthogonal projections onto the eigenspaces of H(x, E). If we define φ (x) j 1 θjτ (x, E) = , (3.8) τ 2 kj (x,E) φj (x) then it is easy to check that Pjτ (x, E) = | χτj (x, E) θjτ (x, E) |,

(3.9)

where we have used the bracket notation relative to the scalar product in C2m . We use the same notation for scalar products in Cm and C2m , since no confusion should arise. We now see that the eigenvectors (3.7) satisfy the normalization conditions τ ∂ Pj (x, E) ∂x ϕτj (x, E) ≡ 0, and (3.10)

ϕτj (0, E), J ϕτj (0, E) ≡ τ ∈ {−1, 1}. We note that H, kjτ , χτj , Pjτ , and ϕτj are analytic functions of x and E when these variables are in a neighborhood of R × ∆. More precisely, if ∆ = [E1 , E2 ], we

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define Dβ = {z ∈ C : dist(z, ∆) < β}, and these functions are analytic in ρα ×Dβ , for α and β small enough. Here α must be chosen small enough so that ej and φj are analytic in ρα , (see [20]), and β must be small enough so that |E − ej (x)| > 0 in ρα × Dβ . We later make use of larger values of α in order to take advantage of the generic multivaluedness of ej and φj as functions of x. From [15], we now see that any solution to (3.2) can be written as x τ 2 Φ(x, E, ) = cτj (x, E, ) e− i 0 kj (y,E) dy/ ϕτj (x, E),

(3.11)

j,τ

where the scalar coefficients cτj ∈ C satisfy the equation x 2 ∂ τ cj (x, E, ) = aτjlσ (x, E) ei 0 (τ kj (y,E)−σkl (y,E)) dy/ cσl (x, E, ), (3.12) ∂x l,σ

with aτjlσ (x, E) = −

∂

ϕτj (x, E), Pjτ (x, E) ∂x ϕσl (x, E) . ϕτj (x, E) 2

We can rewrite (3.12) as an integral equation cτj (x, E, ) = cτj (x0 , E, )

x x 2 aτjlσ (x , E) ei 0 (τ kj (y,E)−σkl (y,E)) dy/ cσl (x , E, ) dx . (3.13) + x0

l,σ

As we shall soon see, our hypotheses imply the existence of the limits lim cτj (x, E, ) = cτj (±∞, E, ),

x±∞

so that with the notation



 cτ1 (x, E, )  τ   c2 (x, E, )   ∈ Cm , cτ (x, E, ) =    ..   . cτm (x, E, )

we can define an associated S-matrix, S ∈ M2m (C), by the identity + + c (+∞, E, ) c (−∞, E, ) = . S(E, ) c− (−∞, E, ) c− (+∞, E, ) This S-matrix naturally takes the block form ++ S (E, ) S +− (E, ) S(E, ) = . S −+ (E, ) S −− (E, )

(3.14)

(3.15)

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Due to the symmetry (3.4), it also satisfies the relation (see [15]), I O −1 ∗ S (E, ) = R S (E, ) R, where R = . O −I Its elements describe transmission and reflection coefficients at fixed energy E which play key roles in our analysis. The off-diagonal elements are exponentially small and their asymptotics are determined in [15]. With this notation, the generalized eigenvectors are given by Ψ(x, E, ) = ×

j

1 φj (x) 2 kj (x, E)

−i c+ j (x, E, ) e

x 0

kj (y,E) dy/2

i + c− j (x, E, ) e

x 0

kj (y,E) dy/2

x Since 0 kj (y, E) dy x kj (±∞, E) = x 2 (E − ej (±∞)) as component of (3.16) that describes a wave traveling from the left labeled by −, and the component that describes a wave traveling to the left is labeled by +. Note also that (3.16) is simply a WKB of the generalized eigenvectors.

. (3.16)

x → ±∞, the to the right is from the right decomposition

We now state some of the general properties of the coefficients cτj (x, E, ) x 2 and of the phases ei 0 kj (y,E) dy/ that allow us to justify the scattering results described above. Lemma 3.1 Our hypotheses on h(x) imply the following, uniformly for E ∈ ∆ and all n ∈ N: ∂n ∂E n kj (x, E) − kjσ (±∞, E))

0 < C1 (n) ≤ ∂n ∂E n

(kjσ (x, E)

≤ C2 (n) < ∞, and −(2+ν)

= O( x

),

as

(3.17) x → ±∞. (3.18)

±∞ Thus, if we define ωjσ (±∞, E) = 0 (kjσ (y, E) − kjσ (±∞, E)) dy, we further have

x kjσ (y, E) dy = x kjσ (±∞, E) + ωjσ (±∞, E) + rjσ (±, x, E) 0

where, uniformly in E and for all n ∈ N, ∂n σ r (±, x, E) = O( x−(1+ν) ), ∂E n j

as

x → ±∞.

Moreover, the limits cσj (±∞, E, ) as x → ±∞ exist, and as |x| → ∞, ∂n σ c (x, E, ) = O(1), ∂E n j

for

n = 0, 1,

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uniformly in E ∈ ∆ and uniformly in ∈ (0, 0 ), where 0 is independent of E. Also, as x → ±∞ and uniformly for E ∈ ∆ and for ∈ (0, 0 ), cσj (x, E, ) − cσj (±∞, E, ) = O( x−(1+ν) ),

and

∂ σ cj (x, E, ) − cσj (±∞, E, ) = O( x−ν ). ∂E Remarks 1. This lemma is proved in Section 7. 2. The error terms in the lemma do not depend on as → 0.

3.1

Complex WKB analysis

All the information about transmissions and transitions among the asymptotic eigenstates of the electronic Hamiltonian is contained in the asymptotic values of the coefficients cσj (x, E, ±∞) defined by (3.13), and hence, in the matrix S(E, ). We extract this information by mimicking the complex WKB method of [15], while keeping track of the E dependence. The complex WKB method requires hypotheses on the behavior in the complex plane of the so-called Stokes lines for the equation (3.2) in order to provide the required asymptotics. These hypotheses are global in nature, and in general, are extremely difficult to check. However, in the physically interesting situation of “avoided crossings,” they can be easily checked. We restrict our attention to these avoided crossing situations that are described below. We consider the coefficients cj that are uniquely defined by the conditions cτj (−∞, E, ) = 1,

and cσk (−∞, E, ) = 0,

for all (k, σ) = (j, τ ).

The key to the complex WKB method lies in the multivaluedness of the eigenvalues and eigenvectors of the analytic generator H(x, E) of (3.2) in the complex x plane. For any fixed E ∈ ∆, H(·, E) is analytic in ρα , and the solutions (3.11) to (3.2) are analytic in x as well. However, the eigenvalues and eigenvectors may have branch points in ρα whose properties are inherited from those of the eigenvalues and eigenvectors of h(·). Analytic perturbation theory as described in [20] states that the eigenvalues and eigenprojections of h(x) for real x are analytic on the real axis and admit analytic multivalued extensions to ρα . The analytic continuations of the eigenvalues have branch points that are located on a set of crossing points Ω = {z0 ∈ ρα \ R : ej (z0 ) = ek (z0 ) for some j, k and some analytic continuations}. Recall that for δ = 0, the eigenvalues are analytic at any crossing points on the real axis. This follows from the self-adjointness of h(·) on the real axis. Note also that Ω = Ω by the Schwarz reflection principle.

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Generically, at a complex crossing point z0 ∈ Ω, we have the following local behavior, where c ∈ C is some constant ej (z) − ek (z) c (z − z0 )1/2 (1 + O(z − z0 )).

(3.19)

The eigenprojectors of h(x) also admit multivalued extensions in ρα \ Ω, but they diverge at generic eigenvalue crossing points. We only have to deal with generic crossing points. To see what happens to a multivalued function f in ρα \ Ω when we turn around a crossing point, we adopt the following convention: We denote by f (z) the analytic continuation of f defined in a neighborhood of the origin along some path from 0 to z. Then we perform the analytic continuation of f (z) along a negatively oriented loop that surrounds only one point z0 ∈ Ω. We denote by f˜(z) the function we get by coming back to the original point z. We define ζ0 to be a negatively oriented loop, based at the origin, that encircles only z0 when Imz0 > 0. When Imz0 < 0, we choose ζ0 to be positively oriented. We now fix z0 with Imz0 > 0. If we analytically continue the set of eigenvalues {ej (z)}m j=1 , along a negatively oriented loop around z0 ∈ Ω, we get the set { ej (z)}m j=1 with ej (z) = eπ0 (j) (z),

for j = 1, . . . , m,

where π0 : {1, 2, . . . , m} → {1, 2, . . . , m}

(3.20)

is a permutation that depends on z0 . As a consequence, the eigenvectors (2.1) possess multivalued analytic extensions in ρα \Ω. The analytic continuation φj (z) of φj (z) along a negatively oriented loop around z0 ∈ Ω, must be proportional to φπ0 (j) (z). Thus, for j = 1, 2, . . . , m, there exists θj (ζ0 ) ∈ C, such that φj (z) = e−iθj (ζ0 ) φπ0 (j) (z).

(3.21)

We now turn from h(x) to H(x, E). From Hypothesis H7, (3.5), and (3.7), we see that the set of crossing points for the eigenvalues ±kj (x, E) of H(x, E) is independent of E and coincides with Ω. Moreover, for j = 1, . . . , m, we have kjτ (z, E) = kπτ 0 (j) (z, E),

ϕ τj (z, E) = e−iθj (ζ0 ) ϕτπ0 (j) (z, E),

where the prefactor e−iθj (ζ0 ) is independent of E. The above implies a key identity for the analytic extensions of the coefficients cτj (z, E, ), z ∈ ρα \Ω. Since the solutions to (3.2) are analytic for all z ∈ ρα , the coefficients cτj must also be multi-valued. In our setting, Lemma 3.1 of [15] implies the following lemma.

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Lemma 3.2 For any j = 1, . . . , m and τ = +, −, we have cτj (z, E, ) e

−i

ζ0

kjτ (u,E) du/2

e− i θj (ζ0 ) = cτπ0 (j) (z, E, )

(3.22)

where ζ0 , θj (ζ0 ) and π0 (j) are defined as above and are independent of E ∈ ∆. Remark. Since Ω is finite, it is straightforward to generalize the study of the analytic continuations around one crossing point to analytic continuations around several crossing points. The loop ζ0 can be rewritten as a concatenation of finitely many individual loops, each encircling only one point of Ω. The permutation π0 is given by the composition of associated permutations. The factors e−iθj (ζ0 ) in (3.21) are given by the product of the factors associated withthe individual loops. The same is true for the factors exp − i ζ0 kjτ (z, E) dz/2 in Lemma 3.2. We now describe how we use the above properties. The details may be found in [15]. The idea is to integrate the integral equation corresponding to (3.13) along paths that go above (or below) one or several crossing points, and then to compare the result with the integration performed along the real axis. As z → −∞ in ρα these paths become parallel to the real axis so that the coefficients take the same asymptotic value cτm (−∞, E, ) along the real axis and the integration paths. Since the solutions to (3.2) are analytic, the results of these integrations must agree as Rez → ∞. Therefore, (3.22) taken at z = ∞ yields the asymptotics of cj (z, E, ) in the complex plane. We argue cτπ0 (j) (∞, E, ), provided we can control below that this can be done in the so-called dissipative domains (See [6], [5]), as proven in [15]. We do not go into the details of these notions because another result of [15] will enable us to get sufficient control on cj (z, E, ) in the avoided crossing situation, to which we restrict our attention. We define

x τ τσ kj (y, E) − kjσ (y, E) dy. ∆jl (x, E) = 0

By explicit computation, using formula (7.3) in (3.13), we check that (3.13) can be extended to ρα \ Ω. We next integrate by parts in (3.13), to see that (3.13) with x0 = −∞ can be rewritten as σ 2 aτml (z, E) τσ ei∆ml (z,E)/ cσl (z, E, ) cτm (z, E, ) = δjm δτ − − i2 σ τ (z, E) − k (z, E) k m l l,σ

z τ σ 2 aml (z , E) ∂ τσ 2 + i ei∆ml (z ,E)/ cσl (z , E, ) dz σ τ ∂z km (z , E) − kl (z , E) −∞ l,σ σ z aτml (z , E) aσθ 2 τθ lp (z , E) i∆ e mp (z ,E)/ + i2 cθp (z , E, ) dz , (3.23) σ τ (z , E) − k (z , E) k −∞ m l l,p,σ,θ as long as the chosen path of integration does not meet Ω. Here, denotes the analytic continuation along the chosen path of integration of the corresponding

958



function defined originally on the real axis. This distinguishes cτm (∞, E, ) from τ cm (∞, E, ) computed along the real axis as x → ∞. These quantities may differ since the integration path may pass above (or below) points of Ω. If the exponentials in (3.23) are all uniformly bounded, as it is the case when the integration path coincides with the real axis, it is straightforward to get bounds of the type (3.24) c˜τm (z, E, ) = δjm δτ − + OE (2 ). In our context, all quantities depend on E ∈ ∆. However, by mimicking the proof of Proposition 4.1 of [15], it is not difficult to check that the estimate (3.24) is uniform for E ∈ ∆. cτm (z, E, ) is For later purposes, we note that by differentiating (3.23), ∂∂E uniformly bounded for 0 < < 0 and E ∈ ∆ for some fixed 0 . See the proof of Lemma 3.1 for this property on the real axis. Again, as is well known, the existence of paths from −∞ to +∞ along which the exponentials do not blow up and which pass above (or below) points in Ω is difficult to check in general. It is linked to the global behavior of the Stokes lines of the problem. See e.g., [6], [5]. This property goes under the name “existence of dissipative domains” in [15]. We avoid these complications by restricting attention to avoided crossing situations where the existence has been proven [15]. When dissipative domains exist, (3.22) and (3.24) imply cτπ0 (j) (∞, E, ) = e

−i

ζ0

kjτ (u,E) du/2

e− i θj (ζ0 ) (1 + OE (2 )),

(3.25)

where the OE (2 ) estimate is uniform for E ∈ ∆. This is the main result of Proposition 4.1 in [15] in our context, under the assumption that dissipative domains exist.

3.2

Avoided crossings

We now explore the avoided crossing situation, alluded to above, that allows us to avoid considerations of the dissipative domains. We now assume that h(x) has the form h(x, δ) and satisfies Hypotheses H4 and H5. We first check that the allowed pattern of avoided crossings for σ(h(x, δ)) can be transfered to the eigenvalues of H(x, E, δ), obtained from h(x, δ) by (3.3). From the explicit formulae (3.5), we see immediately that xc ∈ R is a real crossing point for the eigenvalues ej (x, 0) and el (x, 0) of h(x, 0) if and only if it is a real crossing point for the analytic eigenvalues kjτ (x, E, 0) and klτ (x, E, 0) of H(x, E, 0), for τ = +, −. Moreover, ∂ ∂ τ τ ∂x (el (x, 0) − ej (x, 0)) (kj (x, E) − kl (x, E)) = τ , ∂x 2 (E − ej (x, 0)) x=xc x=xc

so that the real crossings for H(x, E, 0) are also generic, in the sense of (2.5).

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Remark. Our assumption H7 on the parameter E forbids real crossings between eigenvalues kjτ (x, E, 0) and klσ (x, E, 0), with σ = τ . Regarding the ordering of the eigenvalues of H(x, E, δ), if those of h(x, δ) are ordered as in (2.4), we have −k1 (x, E, δ) < · · · < −km (x, E, δ) < 0 < km (x, E, δ) < · · · < k1 (x, E, δ). (3.26) This means that the pattern of the crossings for the group of eigenvalues {−kj (x, E, 0)}j=1,...,m is the same as that for the eigenvalues {ej (x, 0)}j=1,...,m . The pattern of the crossings for the group {kj (x, E, 0)}j=1,...,m is the reflection with respect to the horizontal axis of the one for {ej (x, 0)}j=1,...,m . Therefore, assumptions H5, i), ii), iii) are also satisfied for the eigenvalues of H(x, E, 0), for a relabeling from 1 to 2m of (3.26) with δ = 0, and x close to −∞. To any given pattern of real crossings for the eigenvalues {ej (x, 0)}j=1,...,m of h(x, 0), we associate a permutation π of {1, 2, . . . , m} as follows. Assume the eigenvalues are labeled in ascending order at x = −∞, as in property i) of H5. If ej (∞, 0) is the k th eigenvalue in ascending order at x = ∞, the permutation π is defined by π(j) = k. (3.27) We call π the permutation associated with σ(h(x, 0)). For small δ > 0, the real crossings turn into avoided crossings on the real axis and conjugate complex crossing points appear close to the real axis. Then π corresponds to the permutation π0 (3.20) for a loop ζ0 that surrounds all complex crossing points in the upper half plane that are associated with the avoided crossings. These properties of corresponding patterns of real crossings of the spectra of h(x, δ) and H(x, E, 0) immediately yield the following convenient relation between the permutation π associated with σ(h(x, 0)) and the permutation Π associated with σ(H(x, E, 0)). If we denote Π by the obvious notation Π(j, τ ) = (k, σ), then we have Π(j, τ ) = (π(j), τ ),

for all

(j, τ ) ∈ {1, . . . , m} × {−, +}.

We can now restate the main result of [15] that describes the asymptotics of the coefficients defined in (3.13), adapted to our scattering framework for incoming states entering from the left. (See (3.16).) Intuitively, this result says that for small δ > 0, dissipative domains exist, provided the pattern of real crossings satisfies H5. Therefore, estimates of the type (3.25) are true for certain indices j and n, determined by the permutation (3.27). It is not difficult to see that the permutation π describes the successive exchanges of eigenvalues one gets by following a path in the complex plane that goes above or below all complex crossing points of the eigenvalues ej (x, δ) that are associated with the avoided crossings.

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Theorem 3.1 Let h(x, δ) satisfy H4 and H5. If δ > 0 is small enough, the π(j), j elements of the matrix S −− (E, ) in (3.15), with π(j) defined in (3.27), have small asymptotics for all j = 1, . . . , m, given by −− Sπ(j),j (E, ) =

π(j)∓1

e−iθl (ζl ,δ) e

i

ζl

kl (z,E,δ) dz/2

1 + OE,δ (2 ) ,

π(j)

l=j

>j <j

where, for π(j) > j (respectively π(j) < j), ζl , l = j, . . . , π(j) − 1 (resp. l = j, . . . , π(j) + 1), denotes a negatively (resp. positively) oriented loop based at the origin which encircles the complex crossing point zr only (resp. zr ) corresponding to the avoided crossing between el (x, δ) and el+1 (x, δ) (resp. el−1 (x, δ)) at xr . The ζl kl (z, E, δ) dz denotes the integral along ζl of the analytic continuation of kl (0, E, δ), and θl (ζl ) is the corresponding factor defined by (3.21). Remarks i) Revisiting the proof of this theorem in [15], we see that we can choose δ > 0 small enough so that dissipative domains can be constructed uniformly for E ∈ ∆. This stems from the formula kj (x, E, 0) − kl (x, E, 0) =

2(el (x, 0) − ej (x, 0)) , kj (x, E, 0) + kl (x, E, 0)

whose denominator can be controlled, close to the real axis, uniformly for E ∈ ∆. ii) When there is only one avoided crossing between level j and j + 1 stemming from a real crossing at x = x0 , we have j + 1 = π(j). The theorem says −− S(j+1),j (E, ) = e−iθj (ζj ,δ) e

i

ζj

kj (z,E,δ) dz/2

1 + OE,δ (2 ) ,

where the negatively oriented loop ζj encircles the corresponding complex crossing point z0 , with Imz0 > 0. Similarly, interchanging the roles of j and j + 1, it yields with ζ¯j the conjugate of the loop ζj , ¯

−− Sj(j+1) (E, ) = e−iθj (ζj ,δ) e

i

¯ ζ j

kj (z,E,δ) dz/2

1 + OE,δ (2 ) ,

iii) Since the eigenvalues are continuous at the complex crossings, we have

lim Im kj (z, E, δ) dz = 0, for all j = 1, . . . , p. δ→0

ζj

It is shown in [14] that lim Im θj (ζj , δ) = 0

δ→0

for all

j = 1, . . . , p.

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iv) The OE,δ () errors in Theorem 3.1 depend on δ, but it should be possible to get estimates which are valid as both and δ tend to zero, in the spirit of [14], [21], and [24]. v) This result shows that at least one off-diagonal element per column of the S-matrix can be computed asymptotically. However, it is often possible to get more elements by making use of symmetries of the S-matrix. See [15] and [16]. In our avoided crossings context, transitions of the coefficients between states that correspond to electronic levels that do not display avoided crossings, i.e., that are separated by a gap of order 1 as δ → 0, are expected to be exponentially smaller than the transitions we control by means of Theorem 3.1, as δ shrinks to zero. Since the coefficients in the exponential decay rates given by the theorem vanish in the limit δ → 0, it is enough to show that the decay rates of the exponentially small transitions between well separated levels are independent of δ. That is the meaning of the following proposition, which draws heavily upon [18] and [15] and is proven in Section 7. Proposition 3.1 Let F (x, δ) be a n × n matrix that satisfies H4, except for the condition that F (·, δ) be self-adjoint. Suppose its eigenvalues {fj (x, δ)}j=1,...,m that satisfy H5. Further assume that the eigenvalues can be separated into two groups σ1 (x, δ) and σ2 (x, δ) that display no avoided crossing, i.e., such that inf

δ≥0 x∈ρα ∪{±∞}

dist(σ1 (x, δ), σ2 (x, δ)) ≥ g > 0.

Let P (x, δ) and Q(x, δ) = I − P (x, δ) be the projectors onto the spectral subspaces corresponding to σ1 (x, δ) and σ2 (x, δ) respectively, and let U (x, x0 , δ) be the evolution operator corresponding to the equation i 2

d U (x, x0 , δ) = F (x, δ) U (x, x0 , δ), dx

with

U (x0 , x0 , δ) = I. (3.28)

Then, for any δ > 0, there exists 0 (δ), C(δ) > 0 depending on δ, and Γ > 0 independent of δ, such that for all ≤ 0 (δ), lim

x→∞ x0 →−∞

2

P (x, δ) U (x, x0 , δ) Q(x0 , δ) ≤ C(δ) e−Γ/ .

Remark. This proposition implies that reflections, i.e., the transitions from wave packets traveling to the right to wave packets traveling to the left, on any electronic level, are exponentially smaller than transitions associated with the avoided crossings in which the propagation direction is not changed. This is a consequence of Hypothesis H7 which implies that complex crossings between kj+ and kl− , are far from the real axis for any j, l ∈ {1, . . . , m}. Let us investigate more closely the analytic structure of kj (z, E, δ) in our avoided crossing regime characterized by H4 and H5, in order to deduce the properties of the exponential decay rates Im ζj kj (z, E, δ) dz. We do so for the kj ’s

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that correspond to electronic eigenvalues ej (z, δ) and en (z, δ) that experience only one avoided crossing, i.e., we take n = j ± 1. We can thus drop the index j in ζj in the notation. We follow [14] where a similar analysis is performed, sometimes referring to results proven there. The general case is dealt with by adding the corresponding contributions stemming from each individual avoided crossing. We can assume that the avoided crossing takes place at x = 0, i.e., ej (0, 0) = en (0, 0) ≡ ec , where ec is the electronic eigenvalue at the crossing. We also define the momentum kc (E) at the crossing point by 2 (E − ec ) kc (E) = and the quantity Γ0 (δ) by

Γ0 (δ) = Im ej (z, δ) dz .

(3.29)

ζ

This quantity is the exponential decay rate given by the Landau-Zener Formula for a (time-dependent) adiabatic problem with hamiltonian h(t, δ). See [14]. In Section 7 we prove Lemma 3.3 With the above notation, we have the following as δ → 0, uniformly for E ∈ ∆,

Γ0 (δ) + O(δ 3 ), 2(E − ej (z, δ)) dz = Im k c (E) ζ

∂ Γ0 (δ) Im + O(δ 3 ), 2(E − ej (z, δ)) dz = − 3 and ∂E kc (E) ζ

Γ0 (δ) ∂2 + O(δ 3 ), Im 2(E − ej (z, δ)) dz = 3 5 2 ∂E kc (E) ζ where 0 < Γ0 (δ) = O(δ 2 ). Remarks i) This implies that Im ζ 2(E − ej (z, δ)) dz is a positive, decreasing, convex function of E on ∆. This remains true when the transition is mediated by several avoided crossings. ii) The first relation can be interpreted as saying that in our Born-Oppenheimer context, the (time-dependent adiabatic) Landau-Zener decay rate at fixed energy E has to be modified in order to take into account the classical velocity kc (E) at the crossing. iii) More precise estimates will be derived below, further assuming H6.

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4 Exact solutions to the time-dependent Schr¨ odinger equation We now construct solutions to (1.1) by taking time-dependent superpositions of the generalized eigenvectors Ψ(x, E, ), where E ∈ ∆. These superpositions depend on an energy density Q(E, ) that can be complex and may or may not depend on . We always assume that the following condition holds: C0: The density Q(E, ) is C 1 as a function of E ∈ ∆, for fixed > 0. In this Section, the parameter δ > 0 is kept fixed and plays no role. We therefore drop it from the notation and work under Hypotheses H1, H2, and H3. We define ψ(x, t, )

2

Q(E, ) Ψ(x, E, ) e−itE/ dE ∆

x σ 2 Q(E, ) = φj (x) cσj (x, E, )e−i 0 kj (y,E)dy/ 2 kj (x, E) ∆ j=1,...,m, σ=± =

2

≡

× e−itE/ dE

ψjσ (x, t, ).

(4.1)

j=1,...,m σ=±

Here ψjσ asymptotically describes the piece of the solution that lives on the electronic state φj and propagates in the direction characterized by σ. Since the integrand is smooth and ∆ is compact, ψ(x, t, ) is an exact solution to the timedependent Schrödinger equation (1.1). Note that this solution is not necessarily normalized. The following lemma, whose proof can be found in Section 7, gives a bound that we use to understand the large t behavior of ψjσ (x, t, ). It is a simple corollary that the state (4.1) belongs to L2 (R). Lemma 4.1 Assume H1, H2, H3 and C0. Let K+ =

sup j=1,...,m E∈∆, σ=±

kj (σ∞, E) < ∞

and K− =

inf

j=1,...,m E∈∆, σ=±

kj (σ∞, E) > 0.

Fix α ∈ (0, 1). Then, for either t = 0, or for any x = 0 and t = 0, such that |x/t| > K+ /(1 − α)

or

|x/t| < K− /(1 + α),

we have σ ψ (x, t, ) ≤ C /|x|, j

with C independent of

where the estimate is in the norm on Cm .

t,

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We now introduce freely propagating states ψ(x, t, , ±) ∈ L2 (R, Cm ) that describe the asymptotics of the solutions ψ(x, t, ) as t → ±∞. We use these asymptotic states when we study the scattering matrix for (1.1). We let

2 Q(E, ) ψ(x, t, , ±) = φj (x) e−itE/ cσj (±∞, E, ) 2k (±∞, E) ∆ j j=1,...,m σ=± σ

=

σ

2

× e−i(xkj (±∞,E)+ωj (±∞,E))/ dE ψjσ (x, t, , ±) (4.2)

j=1,...,m σ=±

These states are linear combinations of products of free scalar wave packets in constant scalar potentials times eigenvectors of the electronic Hamiltonian. Their propagation is thus governed by the various channel Hamiltonians. Proposition 4.1 Assume H1, H2, H3 and C0. In L2 (R) norm as t → ±∞, we have (4.3) ψ(x, t, ) − ψ(x, t, , ±) = O (1/|t|). Remarks i) The estimate (4.3) depends on . ii) By a change of variables, we immediately obtain the following corollary. Corollary 4.1 The density of the component of the asymptotic momentum space wave function on the j th electronic level as t → ±∞ is σ 2 k Q(E(k), ) cσj (±∞, E(k), ) e−iωj (±∞,E(k))/ . σ 2 Here E(k) = k 2 /2 + ej (±∞) and σ = −/+ for waves traveling in the positive/negative direction, respectively. iii) Consider a solution ψ(x, t, ) traveling in the positive direction and associated with the electronic eigenstate φj in the remote past. It is characterized by cσk (−∞, E, ) = δk,j δσ,− , and as t → −∞, it is asymptotic to

− 2 2 Q(E,) ψ(x,t,,−) = φj (x) e−itE/ ei(xkj (−∞,E)−ωj (−∞,E))/ dE. 2kj (−∞,E) ∆ (4.4) As t → +∞, the component of this state that has made the transition from state j to state n is asymptotic to the vector ψn− (x, t, , +). It is given in terms of the matrix S by

2 − 2 Q(E,) −− e−itE/ Snj φn (x) (E,)e+i(xkn (+∞,E)−ωn (+∞,E))/ dE. 2kn (+∞,E) ∆ (4.5) iv) Proposition 4.1 is proven in the Section 7.

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5 Non-adiabatic transition asymptotics 5.1

The transition integral

From now on, we assume we are in the avoided crossing situation, but we still do not make explicit the dependence in the variable δ > 0 in the notation. Section 3 gave us the semiclassical asymptotics of the elements of the Smatrix S(E, ). We now compute the small asymptotics of the integrals that describe the asymptotic states ψjσ (x, t, , ±) given by (4.2) as |t| → ∞, for the different channels. We choose our energy density Q(E, ) to be more and more sharply peaked near a specific value E0 ∈ ∆ \ ∂∆ as → 0. As a result, we obtain semiclassical Born-Oppenheimer states that are well localized in phase space. This choice is physically reasonable, and it allows us to relate the quantum scattering process to classical quantities. More precisely we consider, 2

2

Q(E, ) = e− G(E)/ e− i J(E)/ P (E, ),

(5.1)

where C1: The real-valued function G ≥ 0 is in C 3 (∆), and has a unique non-degenerate absolute minimum value of 0 at E0 in the interior of ∆. This implies that G(E) = g (E − E0 )2 /2 + O(E − E0 )3 ,

where

g > 0.

C2: The real-valued function J is in C 3 (∆). C3: The complex-valued function P (E, ) is in C 1 (∆) and satisfies n ∂ ≤ Cn , sup P (E, ) for n = 0, 1. n ∂E E∈∆

(5.2)

≥0

Remark. Typical interesting choices of Q have G = g (E − E0 )2 , J = 0, and P an -dependent multiple of a smooth function with at most polynomial growth in (E − E0 )/. In our avoided crossing situation, we have already proved the following: A wave packet incoming from the left in the remote past produces reflected waves (i.e., components that travel to the left in the remote future) that are exponentially smaller than the components that travel to the right in the remote future. We have also proved that the non-trivial transitions to electronic states that are not involved in the avoided crossing are exponentially smaller than those to electronic states that are involved in the avoided crossing. Thus, the leading non-adiabatic transitions are described by the asymptotics of those coefficients cσl (±∞, E, ) that satisfy cσk (−∞, E, ) =

δj,k δσ,−

c− n (+∞, E, ) =

e−iθj (ζ) ei

(5.3) ζ

kj (z,E)dz/2

(1 + OE (2 )),

(5.4)

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where n = π(j) = j ± 1. We recall that the error term OE (2 ) depends analytically on the energy E in a neighborhood of the compact set ∆. We have already noted in the comments after (3.24) that the term OE (2 ) satisfies (5.2). The form chosen for the energy densities should make it clear that Gaussian wave packets will play a particular role in the asymptotic analysis of (4.5). Therefore we use the specific notation introduced in (1.2) for them. Recall that a normalized free Gaussian state propagating in the constant potential e(+∞) is characterized by the classical quantities A+ (t) B+ (t)

= A+ + i t B+ , = B+ ,

a+ (t) η+ (t)

= a+ + η+ t, = η+ ,

t 2 η+ (s)/2 − e(+∞) ds, =

S+ (t)

and

0

with Re(A+ B+ ) = 1 (see [9]). The associated nuclear wave packet has the form 2

eiS+ (t)/ ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x) 2 2 eit(η /2−e(∞))/ (x − (a+ + η+ t))2 B+ = exp − 22 (A+ + itB+ ) π 1/4 (A+ + itB+ ) η+ (x − (a+ + η+ t)) . (5.5) × exp i 2 We now have everything to state our main result: Theorem 5.1 Let ψ(x, t, ) be a solution of the Schr¨ odinger equation (1.1) with electronic Hamiltonian h(x, δ) that satisfies Hypotheses H4, H5, H7. Assume that the solution is characterized asymptotically in the past by lim

t→−∞

with ψj− (x, t, , −) = φj (x)

∆

ψ(x, t, ) − ψj− (x, t, , −) = 0, − 2 2 Q(E, ) e−itE/ ei(xkj (−∞,E)−ωj (−∞,E))/ dE, 2kj (−∞, E)

where the energy density is supported on the interval ∆, and 2

2

Q(E, ) = e− G(E)/ e− i J(E)/ P (E, ) satisfies C1, C2, and C3. Let n = π(j) be given by (3.27), and let

α(E) = G(E) + Im( kj (z, E) dz), ζ

κ(E) = J(E) − Re( kj (z, E) dz) + ωn− (∞, E). ζ

(5.6) (5.7)

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Assume E ∗ is the unique absolute minimum of α(·) in Int ∆. Then, there exist δ0 > 0, p > 0 arbitrarily close to 3, and a function 0 : (0, δ0 ) → R+ , such that for all δ < δ0 and < 0 (δ), the following asymptotics hold as t → ∞: e− i θj (ζ) 3/2 π 3/4 iS+ (t)/2 ψn− (x, t, , +) = φn (x) 2 1/4 e d ∗ α(E(k))| 2 k dk

√ ∗ 2 × ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x)P (E ∗ , ) k ∗ e−α(E )/ × e−i(κ(E

∗

)−k∗ 2 κ (E ∗ ))/2

+ O(e−α(E

∗

)/2 p

) + O (1/t) ,

where ϕ0 is parametrized by η+ = k ∗ = A+ =

2(E ∗ − en (∞)),

a+ = k ∗ κ (E ∗ ),

d2 d2 ∗ + i α(E(k))| κ(E(k))|k∗ k dk 2 dk 2

B+ =

1 d2 dk2

,

α(E(k))|k∗

d2 α(E(k))|k∗ dk 2

and

S+ (t) = t(k ∗ 2 /2 − en (∞)). All error terms are estimated in the L2 (R) norm, and the estimate O(e−α(E is uniform in t, whereas O (1/t) may depend on .

(5.8) ∗

)/2 p

)

Remarks i) All quantities computed from the electronic Hamiltonian h(x, δ) depend on δ, even though that dependences is not specified in the notation. ii) The function α has a unique absolute minimum if |∆| and δ are small enough. See Proposition 5.1. However, in the case of several absolute minima, one simply adds the contributions associated with each of them. iii) The transitions to states that travel to the left in the future are excluded from our analysis because of the lack of uniformity in E in the semiclassical asymptotics of the relevant elements of the matrix S(E, ). At the price of some more technicalities, it should also be possible to accommodate this situation by our methods. iv) When several avoided-crossings are taken into account and meet the requirements of Theorem 3.1, c− n (∞, E, ) with n = π(j) is given by a product of exponentials of the same form as those in (5.4). The analysis of this situation is essentially identical to the single avoided crossing situation, mutatis mutandis. v) Further properties of ψn (x, t, , +) are given below. In particular, the characteristics of the average momentum k ∗ and its behavior as a function of δ are detailed in Section 5.2. The energy densities corresponding to specific incoming states are studied in Section 6.

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vi) The asymptotics of the incoming wave with the electrons in the state φj in the remote past are described by the same integral, with the replacements  γj → 0,       ωn− (∞, ·) → ωj− (−∞, ·),   2(E − en (∞)) → 2(E − ej (−∞)),     θj → 0.

(5.9)

Proof of Theorem 5.1. Apart from the E-independent factor given by φn (x) − i θj (ζ) √ e , 2 the asymptotics of (4.5) are determined by the integral

P˜ (E, ) (2(E − en (∞)))1/4

T (, x, t) = ∆

×

e

−G(E)/2

2

2

e−i(tE+J(E))/ eiγj (E)/ ei(x

√

− 2(E−en (∞))−ωn (∞,E))/2

dE,

where P˜ (E, ) = P (E, ) (1 + OE (2 )) satisfies (5.2),

γj (E) = kj (z, E) dz, and ζ

ωn− (∞, E)

−

=

∞

0

2(E − en (y)) − 2(E − en (∞)) dy.

The (1 + OE (2 )) factor in P˜ (E, ) comes from Theorem 3.1. Recall that γj and ωj+ (∞, ·) are analytic in a complex neighborhood of ∆, and that Imγj (E) is a positive, decreasing, convex function of E, for δ sufficiently small. In terms of the C 3 functions (5.6) and (5.7) we can write T (, x, t) as

T (, x, t) = ∆

P˜ (E, ) (2(E − e(∞))

2

1/4

2

e−α(E)/ e−i(tE+κ(E))/ ei(x

√

2(E−e(∞))/2

dE,

where we have dropped the index in the asymptotic eigenvalue e(∞) = en (∞). In Section 7 we analyze the small asymptotics of T essentially by Laplace’s method. The result is k2 + e(∞), and 2(E − e(∞)), or equivalently, E(k) = 2 ∗ assume that α(·) has a unique absolute minimum E . For sufficiently small δ, this Lemma 5.1 Let k(E) =

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minimum is non-degenerate and satisfies E ∗ ∈ Int ∆. With k ∗ = k(E ∗ ), there exists p > 0 arbitrarily close to 3, such that as → 0, 2

d ∗ 3 k ∗ dk κ (E ∗ )) 2 α(E(k))|k∗ + i(x + k

T (, x, t) =

2

2

d d 3/2 ( dk 2 α(E(k))|k∗ + i(t + dk2 κ(E(k))|k∗ )) √ P (E ∗ , ) −α(E ∗ )/2 (tE ∗ + κ(E ∗ ) − xk ∗ ) × 2π √ e exp −i 2 k∗ ! (x − k ∗ (t + κ (E ∗ )))2 × exp − d2 d2 22 dk 2 α(E(k))|k∗ + i(t + dk2 κ(E(k))|k∗

+

O(e−α(E

∗

)/2 p

),

(5.10)

where the error estimate is in the L2 (R) norm, uniformly in t. Remarks i) If there are several absolute minima, one simply adds their contributions to get the asymptotics of T . ii) If T is associated with the incoming wave as t → −∞, the formula holds with E0 in place of E ∗ , k0 = 2(E0 − e(−∞)) in place of k ∗ , and the changes (5.9). iii) If P satisfies C3 and P (E ∗ , ) = O(d ) for some d ≥ 32 , then the above analysis yields no information. To relate the integral T to standard Born-Oppenheimer states involving normalized free Gaussian states, we must identify (5.10) with (5.5), making use of (5.8), and taking care of the x and t dependence in the non-Gaussian part of (5.10). That is the content of the next lemma which completes the proof of Theorem 5.1. With the identifications (5.8), we have Lemma 5.2 For small and 0 < p < 3, we have −1/4 d2 P (E ∗ , ) −α(E ∗ )/2 ∗ √ T (, x, t) = 2 π α(E(k))| e k dk 2 k∗ ∗ d2 ∗ 3 ∗ ∗ + i(x + k k α(E(k))| κ (E )) ∗ ∗2 ∗ 2 2 k dk × e−i(κ(E )−k κ (E ))/ d2 d2 ∗ ∗ 2 dk α(E(k))|k + i(t + dk2 κ(E(k))|k ) 3/2

×

1/2

2

3/4

eiS+ (t)/ ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x)

+

O(e−α(E

where the error is estimated in the L2 (R) norm, uniformly in t.

∗

)/2 p

),

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Furthermore, in the L2 norm, for small and large |t|, we have √ ∗ 2 ∗ ∗2 ∗ 2 k ∗ e−α(E )/ e−i(κ(E )−k κ (E ))/ 2 −1/4 d ∗ ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x) α(E(k))| k dk 2

T (, x, t) = 3/2 21/2 π 3/4 P (E ∗ , ) ×

eiS+ (t)/

2

+

O(e−α(E

∗

)/2 p

)

+

O(3/2 /|t|),

where the first error term is uniform in t. Remarks i) We note that the quantities α(·), k ∗ , and B+ depend only on the index j, while κ(·), and hence, A+ depend on both j and n. ii) More detailed computations are carried out in the next section, which is devoted to specific incoming states.

5.2

Energy and momentum shifts

When there is a single avoided crossing, we can be more precise about the energy and momentum shifts revealed by our general analysis. For the rest of this section, we assume h(x, δ) satisfies Hypothesis H6. Under this hypothesis, it is known [14] that the decay rate in the LandauZener formula (3.29) has the form c2 π b2 − 3 + O(δ 3 ) ≡ δ 2 D + O(δ 3 ), Γ0 (δ) = δ 2 4 a a and that Imθj (ζj , δ) = 0(δ). We use these formulas to get more information on E ∗ , the typical energy of the outgoing wave packet, that is determined by the relation α (E ∗ ) = G (E ∗ ) + Imγj (E ∗ ) = 0,

(5.11)

where the primes denote derivatives with respect to E. In the next proposition, we consider two cases: – In the first case, we choose the exponent G(E) in the energy density to be independent of δ. This yields less interesting momentum and energy shifts since they vanish to leading order in δ as δ → 0, in keeping with [10]. – In the second case, we choose G(E) to depend on δ in such a way that the incoming wave packet contains a sufficiently wide spectrum of energies as δ → 0. This implies non-trivial behavior of the relevant quantities to leading order in δ. For obvious reasons, we restore δ in the notation of this discussion.

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Proposition 5.1 Let G(E) = g(E − E0 )2 /2 + O(E − E0 )3 , Imγj (E, δ) =

Γ0 (δ) D δ2 + O(δ 3 ) = + O(δ 3 ), kc (E) kc (E)

and

α(E, δ) = G(E) + Imγj (E, δ), as above. i) Assume G is independent of δ. Then, for E ∗ defined by (5.11), we have E ∗ (δ) = E0 +

Γ0 (δ) + O(δ 3 ), g kc3 (E0 )

as δ → 0. In this case, α(E ∗ (δ)) = α (E ∗ (δ)) =

Γ0 (δ) + O(δ 3 ) < α(E0 ), kc (E0 )

and

g + O(δ 2 ).

If G(E) = g(E − E0 )2 /2 + g1 (E − E0 )3 /6 + O(E − E0 )4 , then α (E ∗ (δ)) = g + g1

Γ0 (δ) Γ0 (δ) +3 5 + O(δ 3 ). g kc3 (E0 ) kc (E0 )

ii) Assume G(E, δ) = L(δ(E − E0 )), for some function L, such that G(E, δ) = g0 δ 2 (E − E0 )2 /2 + O(δ 3 ), for some g0 > 0, uniformly for E ∈ ∆. Then E ∗ (δ) = E1 + O(δ), where 0 < E1 = E1 (D/g0 ) is the unique solution to the equation (E1 − E0 ) = D/(g0 kc3 (E1 )), and is independent of δ. In this case, D ∗ 2 2 + g0 (E1 − E0 ) /2 + O(δ 3 ) α(E (δ)) = δ kc (E1 ) 1/3 = δ 2 D2/3 g0 (E1 − E0 )1/3 + g0 (E1 − E0 )2 /2 + O(δ 3 )

α (E ∗ (δ))

0 and the set ∆, so that η− the interior of ∆, and that the minimum of ∆ lies strictly above the spectrum of h(x) for all x. We choose a smooth cut-off function F (E) whose support is a subset of the interior of ∆, which takes the value 1 on an interval whose interior contains 2 /2 + ej (−∞), and whose length is almost as large as that of ∆. η− From our assumptions on ∆, there is a one-to-one correspondence between E ∈ ∆ and positive k, such that k 2 /2 + ej (−∞) = E. For E ∈ ∆, we make the change of variables from k to E at t = 0 in the (rescaled) Fourier transform of

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the Gaussian in (6.1) (see [9]). Taking into account the normalization (3.7) of the generalized eigenvectors, this leads to the energy density −

2

F (E) ei ωj (E,−∞)/ Q(E, ) = ϕ0 B− , A− , 2 , η− , −a− , k(E) (6.2) π k(E) − 2 F (E) ei ωj (E,−∞)/ 2 = ϕ0 B− , A− , , η− , −a− , 2(E − ej (−∞)) π 2(E − ej (−∞)) that we use in (4.2). 2 Since η− /2 + ej (−∞) is in the interior of the set where F (E) = 1, the wave functions (6.1) and ψ(x, t, , −) defined by (4.2) with the energy density defined 2 by (6.2) differ in L2 (R) norm by an O(e−C/ ) error. To be sure that this error is smaller than the non-adiabatic effect we are studying, we assume any one of the following conditions: 1. Take the avoided crossing gap δ to be small enough that the non-adiabatic effect is larger than the error we make here. 2. Choose |B− | to be sufficiently small. That increases the value of C in this error estimate. 3. Fix the minimum of ∆, but then choose η− large enough so that the cut off is farther out in the tail of the Gaussian in momentum space. This also makes the non-adiabatic effect larger since η− is larger. With Q(E, ) chosen by (6.2), we have in the notation of (5.1), ( 2 (E − ej (−∞)) − η− )2 G(E) = (Re(A− /B− )) 2 ( 2 (E − ej (−∞)) − η− )2 , = |B− |−2 2 J(E)

=

( 2(E − ej (−∞)) − η− )2 2

(Im(A− /B− ))

(6.3)

(6.4)

+ a− ( 2(E − ej (−∞)) − η− ) − ωj− (E, −∞), P (E, ) =

−1/2

π −3/4 −3/2 B−

(2(E − ej (−∞)))−1/4 F (E).

(6.5)

Also, conditions C1, and C2 are satisfied, and provided we remove the trivial normalization factor of −3/2 from P (E, ), then condition C3 is also satisfied. We already know that asymptotically in the past, the interacting wave func2 tion determined by (6.2) agrees with (6.1) up to an O(e−C/ ) error, and we observe that the density Q(E, ) is sharply peaked around the energy η2 E0 = − + ej (−∞) corresponding to η− = 2(E0 − ej (−∞)). 2

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Thus from (6.3), we see that (E − E0 )2 + O((E − E0 )3 ), 2 (η− |B− |)2

G(E) =

i.e.,

g =

1 . (η− |B− |)2

We are not particularly interested in the main component of the wave function for large time that has not made a non-adiabatic transition. However, by a similar analysis, it could be determined by our techniques. Of course, it is what one would expect from the standard time-dependent Born-Oppenheimer approximation. Our focus is on the dominant non-adiabatic component, which is determined to leading order in by Theorem 5.1. From the above calculations and Theorem 5.1, we immediately get our main result for Gaussian incoming states: Theorem 6.1 Assume Hypotheses H4, H5, and H7, and assume ∆, A− , B− , a− , η− , δ, and the levels j and n have been chosen to satisfy the requirements above. Let Ψ(x, , t) be the solution to the Schr¨ odinger equation that is asymptotic as t → −∞ to 2

2

ei(η− /2−ej (−∞))t/ ϕ0 (A− + itB− , B− , 2 , a− + η− t, η− , x) φj (x). The leading non-adiabatic component of Ψ(x, , t) as t → ∞ and → 0 in L2 norm is on electronic level φn (x) and is given by 2

2

Anj () ei(η+ /2−en (∞))t/ ϕ0 (A+ + itB+ , B+ , 2 , a+ + η+ t, η+ , x) φn (x), where the values of A+ , B+ , a+ , η+ = k ∗ are those given by (5.8) as in Theorem 5.1. The amplitude for making this transition from level j to level n is given by " B+ −i(κ(E ∗ )−k∗ a+ )/2 −iθj (ζ) −α(E ∗ )/2 Anj () = e e e . B− In particular

∗

B+ = ((G (E ) + =

Imγj (E ∗ )) k ∗ 2 )−1/2

|B− | η− k∗

+ Imγj (E ∗ )|B− |k ∗

2

=

η− + Imγj (E ∗ ) k ∗ 2 |B− |2 k ∗

−1/2

(6.6)

Remark. Depending on the relative size of |B− | with respect to δ, we can apply Proposition 5.1 to further characterize Anj . We now turn our attention to the situation where the incoming nuclear wave packet is in the state ϕm . The only change from the situation just considered is

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that we must replace the function P (E, ) in (6.5) by −(m+1)/2

P (E, ) = (−i)m 2−m/2 (m!)−1/2 π −3/4 −3/2 (2(E − ej (−∞)))−1/4 B− 2(E − e (−∞)) − η j − F (E). × (B− )m/2 Hm |B|

(6.7)

Again, this satisfies Condition C3 if we take out the trivial factor of −(m+3/2) . Theorem 6.2 Assume the Hypotheses of Theorem 6.1. Let Ψ(x, , t) be the solution to the Schr¨ odinger equation that is asymptotic as t → −∞ to 2

2

ei(η− /2−ej (−∞))t/ ϕm (A− + itB− , B− , 2 , a− + η− t, η− , x) φj (x). The leading non-adiabatic component of Ψ(x, , t) as t → ∞ and → 0 in L2 norm is on electronic level φn (x), and is given by 2

(m)

2

Anj () ei(η+ /2−en (∞))t/ ϕ0 (A+ + itB+ , B+ , 2 , a+ + η+ t, η+ , x) φn (x), where the values of A+ , B+ , a+ , η+ = k ∗ are those given by (5.8), as in Theorem 5.1. The amplitude for making the transition from level j to level n is given by " m/2 ∗ ∗ 2 B+ e−i(κ(E )−k a+ )/ B− (m) −iθj (ζ) −α(E ∗ )/2 Anj () = e e B− B− 2m/2 (m!)1/2 ∗ k − η− × Hm |B− | ∗

=e

−iθj (ζ)

e−α(E )/ m

2

"

∗

∗

B+ e−i(κ(E )−k a+ )/ B− (m!)1/2

2

√ m 2(k ∗ − η− ) B−

× (1 + O()). In particular, B+ is again given by (6.6) and the pre-exponential factor is of order −m .

7 Technicalities Proof of Lemma 3.1. We consider only the limit x → ∞ and the choice σ = +. The other cases are similar. In this proof, cn denotes a finite constant that depends only on n, but may vary from line to line. Explicitly, for any n ∈ N, ∂n ∂E n

2(E − ej (x)) = cn (2(E − ej (x)))1/2−n ,

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uniformly for E ∈ ∆. So, the first assertion is true. Moreover, ∂n ∂E n

2(E − ej (x)) − 2(E − ej (∞)) = cn (2(E − ej (x)))1/2−n − (2(E − ej (∞)))1/2−n .

For n = 0, we have by (2.2), 2 (ej (∞) − ej (x)) 2(E − ej (x)) − 2(E − ej (∞)) = 2(E − ej (x)) + 2(E − ej (∞)) = O(ej (∞) − ej (x)) = O( x−(2+ν) ). (7.1) For n > 0, we can write (2(E − ej (x)))1/2−n − (2(E − ej (∞)))1/2−n =

+

n−1 k=0

(2(E − ej (x)))1/2 − (2(E − ej (∞)))1/2 (2(E − ej (x)))n

(2(E − ej (∞)))1/2 (2(E − ej (x)))k+1 (2(E − ej (∞)))n−k

(2(ej (x) − ej (∞)),

(7.2)

to which the estimate (7.1) applies. The second assertion follows. By definition,

∞ + (2(E − ej (y)))1/2 − (2(E − ej (∞)))1/2 dy, rj (+, x, E) = − x

n

+ ∂ so that (7.2) implies the estimates on ∂E n rj (+, x, E). We now study the properties of the cτj ’s. Again, we shall consider x → +∞; the other case is similar. We first compute 1 1 τσ ajl (x, E) = −

φj (x), φl (x)(kj (x, E) + τ σkl (x, E)) 2 kj (x, E)kl (x, E) kj φj (x), φl (x) ∂ kl (x, E) + στ − kl 2 ∂x 1 1 = −

φj (x), φl (x)(kj (x, E) + τ σkl (x, E)) 2 kj (x, E)kl (x, E) kj φj (x), φl (x) el (x) . + στ − (7.3) kl 2 kl (x, E)

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The presence of the factors φj (x), φl (x) and el (x), which are independent of E and decay as 1/ x2+ν , implies together with (3.17) that ∂ n+m aτ σ (x, E) = O( x−(2+ν) ). ∂xm ∂E n jl

(7.4)

We denote the coefficients cτj collectively by + c (x, E, ) ∈ C2m , c(x, E, ) = c− (x, E, ) and the generator of equation (3.12) by the 2m × 2m block matrix M(x, E, ) x x 2 2 ei 0 (kj (y,E)+kl (y,E))dy/ a+− ei 0 (kj (y,E)−kl (y,E))dy/ a++ jl (x, E) jl (x, E) , = x 2 i 0x (−kj (y,E)+kl (y,E))dy/2 −− ei 0 (−kj (y,E)−kl (y,E))dy/ a−+ ajl (x, E) jl (x, E) e so that (3.12) can be rewritten as ∂ c(x, E, ) = M(x, E, ) c(x, E, ). ∂x Expressing the solutions as Dyson series, we obtain c(x, E, ) =

∞

n=0

0

x

0

x1

···

xn−1 0

× M(x1 , E, )M(x2 , E, ) · · · M(xn , E, )dx1 dx2 · · · dxn c(0, E, ). (7.5) ∞ Because of (7.4), 0 M(y, E, ) dy < ∞, uniformly for E ∈ ∆ and for → 0, and we get the usual bound c(x, E, ) ≤ e

∞ 0

M(y,E,) dy

c(0, E, ) .

Thus, by the Lebesgue Dominated Convergence Theorem, we get from (7.5) that, as x → ∞ and uniformly for E ∈ ∆ and → 0, c(x, E, ) = O(1). Next we show that c(x, E, ) − c(y, E, ) is arbitrarily small for large x and y, so that limx→∞ c(x, E, ) = c(∞, E, ) exists. It is enough to consider

y M(z, E, ) c(z, E, ) dz c(x, E, ) − c(y, E, ) = − x

and (7.4). The expression above with y = ±∞, and the properties of M, c, just proven yield c(x, E, ) − c(±∞, E, ) = O( x−(1+ν) ), uniformly in E ∈ ∆ and → 0, as x → ±∞.

978



To get similar bounds on the derivatives of c with respect to E which are uniform in , we resort to integration by parts in (3.13) with x0 = −∞

σ τσ 2 aτml (x, E) ei∆ml (x,E)/ cσl (x, E, ) τ (x, E) − k σ (x, E) km l l,σ

x τ σ τσ 2 ∂ aml (z , E) 2 + i ei∆ml (z ,E)/ cσl (z , E, ) dz σ τ ∂z km (z , E) − kl (z , E) l,σ −∞

σ x aτml (z , E) aσθ θ lp (z , E) i∆τmp (z ,E)/2 θ e + i2 cp (z , E, ) dz , (7.6) σ τ (z , E) − k (z , E) k −∞ m l

cτm (x, E, ) = δjm δτ − − i2

l,p,σ,θ

where

∆τjlσ (x, E)

=

x 0

(kjτ (y, E) − klσ (y, E)) dy.

Taking care of the derivatives of the phases with respect to E ∂ i x (τ kj (y,E)+σkl (y,E))dy/2 e 0 = O( x/2 ), ∂E making use of the properties of c, M and kj , of the estimates x > 1, and choosing ≤ 1, we get by differentiation of (7.6) 2 1 ∂E c(x, E, ) ≤ K0 ∂E c(x, E, ) + 2+ν

x

x1+ν

x

x 1 ∂E c(y, E, ) 2 dy + dy , (7.7) 1+ν

y2+ν −∞ y −∞ where K0 is a constant independent of and E. By choosing small enough so that K0 2 < 1/2, (7.7) yields

x ∂E c(y, E, ) ∂E c(x, E, ) ≤ K1 1 + 2 dy , (7.8)

y2+ν −∞ with another constant K1 uniform in E and . Hence, by Gronwall’s Lemma, ∂E c(x, E, ) ≤ K1 eK1

∞

−∞

y −(2+ν ) dy

= O(1),

(7.9)

if < 1/(2K0 ), uniformly in E. Similar manipulations on the difference ∂E c(x, E, ) − ∂E c(±∞, E, ) yield the estimate ∂E (c(x, E, ) − c(±∞, E, ))

±∞ 1 ∂E (c(y, E, ) − c(±∞, E, )) ≤ K2 ± dy ,

xν

y2+ν x for smaller than a constant uniform in E, and for all x ≥ 0, respectively x ≤ 0. Gronwall’s Lemma implies in that case ∂E (c(x, E, ) − c(±∞, E, )) = O( x−ν ), as x → ±∞, uniformly in E ∈ ∆ and in → 0.

(7.10)

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Proof of Lemma 4.1. We assume t = 0 and rewrite the exponential factors in (4.1) as e

−i(

x 0

kjσ (y,E) dy + tE)/2

= i

x

e−i( 0 x t+ 0

∂ 2 ∂E

kjσ (y,E) dy + tE)/2 ∂ ∂E

kjσ (y, E) dy

.

(7.11)

Then, for each integral in (4.1), we have

∆

Q(E, ) cσj (x, E, ) −i( x 0 e kj (x, E) = i 2 e−i(

x

dE

E2 Q(E, ) cσj (x, E, ) x ∂ σ kj (x, E) t + 0 ∂E kj (y, E) dy E 1 ! σ Q(E, ) cj (x, E, ) ∂ x ∂ σ ∂E kj (x, E) t + 0 ∂E kj (y, E) dy


0

−


∆

× i 2 e−i(

x 0


dE.

(7.12)

The quantities cσj (x, E, ), kj (x, E), and their derivatives with respect to E are uniformly bounded in x and E. Also,

x ∂kjσ (±∞, E) ∂kjσ (y, E) dy = x + O(1) ∂E ∂E 0 σx = + O(1), kj (±∞, E) uniformly for E ∈ ∆ as x → ±∞. Thus, the boundary terms in (7.12) satisfy 2 −i(

i e

x 0

=O

kjσ (y,E)dy+tE)2

E2 Q(E, ) cσj (x, E, ) x ∂ σ kj (x, E) t + 0 ∂E kj (y, E)dy E 1

1 1 k σ (±∞, E1 )t + x + O(1) + k σ (±∞, E2 )t + x + O(1) j j

. (7.13)

We now apply the restrictions on x/t in the statement of the Lemma. For any choice of j and σ, they ensure that the denominators on the right hand side of (7.13) can be estimated, uniformly in E and for large |x|, by σ kj (±∞, E)t + x + O(1) = |x| 1 + kjσ (±∞, E)t/x + O(1/x) ≥

|x| (α + O(1/|x|)),

where α is the number that appears in the statement of the lemma. From this, we see that the boundary terms in (7.12) are O(1/|x|).

980



We estimate the integral term in (7.12) in a similar way. Under the restrictions on x/t in the lemma, we obtain

2

i e

−i(

x 0

∂ ∂E

kjσ (y,E) dy +tE)/2

∆

(Q(E, ) (kjσ (x, E))−1/2 cσj (x, E, )) x ∂ σ t + 0 ∂E kj (y, E) dy

!

x ∂2 σ Q(E, ) (kjσ (x, E))−1/2 cσj (x, E, ) 0 ∂E 2 kj (y, E) dy − dE = O(1/|x|). 2 x ∂ σ t + 0 ∂E kj (y, E) dy This implies the lemma for t = 0. When t = 0, the estimate (7.13) with t = 0 yields the result in a more direct way. Proof of Proposition 4.1. We can write ψ(x, t, ) − ψ(x, t, , ±) =

φj (x)

j=1,2 σ=±

x σ 2 2 Q(E, ) dE e−itE/ (cσj (x, E, ) − cσj (±∞, E, )) e−i 0 kj (y,E)dy/ 2kj (±∞, E) ∆

x σ 2 2 Q(E, ) dE + e−itE/ cσj (±∞, E, ) × e−i 0 kj (y,E)dy/ 2kj (±∞, E) ∆ Q(E, )cσ (x, E, )dE 2 j −i(xkjσ (±∞,E)+ωjσ (±∞,E))/2 + −e e−itE/ kj (±∞, E)kj (x, E) ∆ ! kj (±∞, E) − kj (x, E) −i 0x kjσ (y,E)dy/2 . (7.14) × e 2kj (±∞, E) + 2kj (x, E)

×

The first step of the proof consists of integrating by parts to get a factor of 1/t according to

2

f (x, E, ) e−itE/ dE =

∆

E2 2 i 2 f (x, E, ) e−itE/ t E1 −

i 2 t

∆

2 ∂ f (x, E, ) e−itE/ dE. ∂E

(7.15)

We then bound the L2 (Rx ) norm of each term that arises from these integrations by parts, with bounds that are uniform in t. From the estimates in Lemma 3.1, we see that all the boundary terms in (7.15) coming from (7.14) are of order x−(1+ν) . Thus, their L2 norms are bounded, uniformly in t. The integral terms in (7.15) coming from (7.14) all have the form

x σ 2 gj (x, E, ) e−i( 0 kj (y,E)dy+tE)/ dE, j = 1, 2, 3, (7.16) ∆

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where the first integral from (7.14) contains the function g1 (x, E, ) =

∂ ∂E

Q(E, ) (cσj (x, E, ) − cσj (±∞, E, )) kj (±∞, E)

!

Q(E, ) −i (cσj (x, E, ) − cσj (±∞, E, )) kj (±∞, E)

x ∂ σ kj (y, E) dy. (7.17) × 0 ∂E With the notation of Lemma 3.1, the second integral in (7.14) contains the function

Q(E, ) σ i(rjσ (±,x,E))/2 cj (±∞, E, ) 1 − e kj (±∞, E) σ 2 Q(E, ) cσj (±∞, E, ) 1 − ei(rj (±,x,E))/ −i kj (±∞, E)

x ∂ σ kj (y, E) dy. (7.18) × ∂E 0

∂ g2 (x, E, ) = ∂E

The third integral contains ∂ g3 (x, E, ) = ∂E

Q(E, ) cσj (x, E, ) kj (±∞, E) − kj (x, E) kj (±∞, E)kj (x, E) 2kj (±∞, E) + 2kj (x, E)

Q(E, )cσj (x, E, ) kj (±∞, E) − kj (x, E) −i kj (±∞, E)kj (x, E) 2kj (±∞, E) + 2kj (x, E)

x ∂ σ kj (y, E) dy. (7.19) × 0 ∂E By Lemma 3.1, and the condition ν > 1/2, each of these functions gj (x, E, ) satisfies the following bound, uniformly in E, gj (x, E, ) = O( x−ν ) ∈ L2 (R). Therefore, we can estimate the L2 norm of the corresponding expression (7.16) by

R

∆

gj (x, E, ) e

−i(

x 0

kjσ (y,E)dy+tE)/2

2 dE dx ≤ C1 (),

where C1 () is a finite constant that is independent of t. This finishes the proof.

982



Proof of Proposition 3.1. Since the argument is virtually identical to the one presented in [18] and [15], we will be rather sketchy and mainly point out the effects of the parameter δ and of the non self-adjointness of the generator F (x, δ). Expressing the projector P (x, δ) as a integral of the resolvent (F (x, δ) − z)−1 along a loop L (or a finite number of such loops) around the set σ1 (x, δ) by means of the Riesz formula, # 1 P (x, δ) = − (F (x, δ) − z)−1 dz, (7.20) 2πi L we get a bound, uniform in δ > 0 and x ∈ ρα , P (x, δ) ≤ c. Indeed, for each x ∈ ρα , we can choose the path L uniformly in δ by hypothesis. The existence of the limits F (±∞, δ) allows us actually to consider only a finite number of distinct loops a finite distance g/4 away from spectrum of F (x, δ), for all (x, δ) . Also, uniformly in δ > 0, (F (x, δ) − z)−1 ≤ c,

(7.21)

for z on the corresponding loop L, since | det(F (x, δ) − z)| ≥ (g/4)n and F (x, δ) is uniformly bounded. By a similar argument, using Hypothesis H4, we get, uniformly in δ c , P (x, δ) − P (±∞, δ) ≤

x2+ν ∂ , we get from (7.20), as x → ±∞ in ρα . With the notation for ∂x # 1 P (x, δ) = (F (x, δ) − z)−1 F (x, δ) (F (x, δ) − z)−1 dz. 2πi L

Thus, Hypothesis H4 yields, uniformly in δ, P (x, δ) ≤

c ,

x2+ν

and a similar uniform estimate for K(x, δ) = [P (x, δ), P (x, δ)], K(x, δ) ≤

c .

x2+ν

(7.22)

The operator K is the generator of the intertwining operator W defined by W (x, x0 , δ) = K(x, δ) W (x, x0 , δ),

with

W (x0 , x0 , δ) = I.

It satisfies W (x, x0 , δ) P (x0 , δ) = P (x, δ) W (x, x0 , δ), for all (x, δ) (including x = ±∞).

(7.23)

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983

Following [19], we construct a hierarchy of generators. Let F0 (x, δ) = F (x, δ), P0 (x, δ) = P (x, δ), and K0 (x, δ) = K(x, δ). For q ∈ N∗ , we inductively define Fq (x, δ, ) = F (x, δ) − 2 Kq−1 (x, δ, ), assuming is small enough so that the spectrum of Fq is separated into two disjoint parts corresponding to those of F . We define Pq (x, δ, ) to be the spectral projector for Fq corresponding to P (x, δ) as → 0 by perturbation theory. Then, Kq (x, δ, ) = [Pq (x, δ, ), Pq (x, δ, )]. Sections II.A and II.B of [19] and (7.21) and (7.22) show that there exist constants ∗ > 0, r > 0, Γ > 0 and c > 0, all independent of δ > 0, such that for all < ∗ , all x ∈ R, and q = q ∗ = [r/2 ], Kq∗ −1 (x, δ, ) ≤

c ,

x2+ν

(7.24) 2

e−Γ/ Kq∗ (x, δ, ) − Kq∗ −1 (x, δ, ) ≤ c .

x2+ν We define F∗ (x, δ, ) = Fq∗ (x, δ, ) = F (x, δ) − Kq∗ −1 (x, δ, ), P∗ (x, δ, ) = Pq∗ (x, δ, ),

(7.25)

K∗ (x, δ, ) = Kq∗ (x, δ, ), and the evolution operators W∗ and Ξ∗ by W∗ (x, x0 , δ, ) = K∗ (x, δ, ) W∗ (x, x0 , δ, ), and

with

W (x0 , x0 , δ, ) = I,

i 2 Ξ∗ (x, x0 , δ, ) = W∗ (x0 , x, δ, ) F∗ (x, δ, ) W∗ (x, x0 , δ, ) Ξ∗ (x, x0 , δ, ), with Ξ∗ (x0 , x0 , δ, ) = I .

(7.26)

The intertwining property (7.23) still holds with the ∗ indices. Therefore, Ξ∗ satisfies [Ξ∗ (x, x0 , δ, ), P∗ (x0 , δ, )] ≡ 0, for all x ∈ R. It follows from the definitions that the operator V∗ (x, x0 , δ, ) = W∗ (x, x0 , δ, ) Ξ∗ (x, x0 , δ, ) satisfies i 2 V∗ (x, x0 , δ, ) = (F∗ (x, δ, ) + i 2 K∗ (x, δ, )) V∗ (x, x0 , δ, ) and V∗ (x, x0 , δ, ) P∗ (x0 , δ, ) = P∗ (x, δ, ) V∗ (x, x0 , δ, ).

(7.27)

984



Moreover, U (x, x0 , δ) − V∗ (x, x0 , δ, ) (7.28)

x V∗ (x0 , y, δ, ) (Kq∗ (y, δ, ) − Kq∗ −1 (y, δ, )) U (y, x0 , δ) dy. = i x0

The proposition will follow from 2

U (x, x0 , δ) − V∗ (x, x0 , δ, ) = O(e−Γ/ ),

(7.29)

(7.27) and lim

x→±∞

P∗ (x, δ, ) = P (±∞, δ),

due to (7.24) and (7.25). To prove (7.29), we first prove that V∗ is uniformly bounded in x, x0 , and . The analysis leading to Lemma 3.1 implies that U is uniformly bounded in x, x0 , and . This property is a consequence on the fact that the eigenvalues of F are simple and real, so that the decomposition (3.11) holds and the singular exponential factors are phases. Note that the lack of orthogonality of the eigenprojectors of F (x, δ) makes the bound on U dependent on δ. Choose B(δ) > 0, such that U (x, x0 , δ) ≤ B(δ). From (7.28) we get the inequality

x 2 C e−Γ/ sup V∗ (y, y0 , δ, ) dx, V∗ (x, x0 , δ, ) ≤ B(δ) 1 +

x 2+ν x0 y0 ,y the quantity v(, δ) = sup V∗ (x, x0 , δ, ) for some C. This implies that for some C, x0 ,x

satisfies v(, δ) ≤ B(δ)

v(, δ) e−Γ/ 1+C

2

.

This implies v(, δ) ≤

B(δ) ≤ v(δ), 1 − C B(δ) e−Γ/2

where v(δ) is uniformly bounded for sufficiently small . We now use (7.28) again to see that

U (x, x0 , δ) − V∗ (x, x0 , δ, ) ≤ v(δ)

R

2

B(δ) C e−Γ/ dx

x 2+ν

1 e−Γ/2 . ≤ C This proves (7.29) and completes the proof of the proposition.

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985

Proof of Lemma 3.3. Degenerate perturbation theory for self-adjoint matrices and Hypothesis H5 (see [14]) show that there exist f (z, δ) and ρ(z, δ), analytic in z for fixed δ, and C 1 as functions of (z, δ), such that 1 ρ(z, δ) (7.30) ej (z, δ) = f (z, δ) − 2 1 en (z, δ) = f (z, δ) + ρ(z, δ). 2 where, as (z, δ) → (0, 0), f (z, δ) = f (0, 0) + O(|z| + δ) = ec + O(|z| + δ),

and ρ(z, δ) = O(|z|2 + δ 2 ).

Moreover, ρ(z, δ) has two simple zeros, the complex crossing points, z0 (δ) and z¯0 (δ) that have z0 (δ) = O(δ). For concreteness, we arbitrarily choose ej < en on the real axis, although this is irrelevant for the analysis. Thus, by H7, we can write 1/2 ρ(z, δ) 2 (E − ej (z, δ)) = 2 (E − f (z, δ)) 1 + , 2 (E − f (z, δ)) where (E − f (z, δ)) and its inverse are analytic in ρα , uniformly in E ∈ ∆. Moreover, 1/2 ρ(z, δ) ρ(z, δ) 1 + O(|z|2 + δ 2 ). 1+ = 1 + 2 (E − f (z, δ)) 2 2 (E − ec ) Therefore, since 2 (E − f (z, δ)) is analytic, and we can choose the loop ζ encircling z0 (δ) or z¯0 (δ) to satisfy |ζ| = O(δ), we see that

ρ(z, δ) 1 2(E − ej (z, δ)) dz = dz + O(δ 3 ), 2 2(E − ec ) ζ ζ

ρ(z, δ) dz = O(δ 2 ). and 2 ζ

ρ(z, δ) In these two expressions, dz = ej (z, δ) dz due to the analyticity 2 ζ ζ of f in (7.30). Taking the imaginary part yields the first statement of the lemma. Note that we do not

have to worry about sign issues because Theorem 3.1 ensures 2(E − ej (z, δ)) dz, is positive. the decay rate, Im ζ

The two other statements follow from similar considerations for the integrals

1 ∂ Im 2 (E − ej (z, δ)) dz = Im dz and ∂E 2 (E − ej (z, δ)) ζ ζ

∂2 1 Im 2 (E − ej (z, δ)) dz = − Im dz. 3/2 ∂E 2 ζ ζ (2 (E − ej (z, δ)))

986



Proof of Lemma 5.1. Consider first the minimization of the negative of the real part of the exponent. Since γj (E) tends to zero with δ (absent in the notation), if δ is small enough, we must look for minima in a neighborhood of E0 that satisfy the equation α (E) = g (E − E0 ) + Imγj (E) + O(E − E0 )2 = 0. We consider the absolute minimum E ∗ of α and assume it is unique. By Lemma 3.3, Imγj (E) < 0, so E ∗ > E0 . Note that E ∗ does not depend on x or t. Also, (n)

Imγj (E) = o(δ), n = 0, 1, 2, uniformly in E. So, we can assume E ∗ is nondegenerate since α (E ∗ ) = g + Imγj (E ∗ ) + O(E ∗ − E0 ) > 0. In terms of the variable k ∈ [k1 , k2 ], we view T as the (scaled) inverse Fourier transform of the function √ √ 2 2 R(k, t, ) = 2π2 e−α(E(k))/ P˜ (E(k), ) k e−iκ(E(k))/ e−it(k

2

/2+e(∞))/2

χ[k1 ,k2 ] (k),

where χS (·) is the characteristic function of the set S. That is T (x, t, ) = (F−1 R(·, t, ))(x), where F is defined by 1 (F g)(x) = √ 2π2

2

g(k) e−ikx/ dk.

R

With the variable k ∈ [k1 , k2 ] we have ∂2 α(E(k)) = k ∗ 2 α (E ∗ ), ∂k 2 k∗ and expanding around k ∗ , ∗

2

T (, x, t) = e−α(E )/

∂ 2 α(E(k))| √ k∗ ∂k2 (k−k∗ )2 O((k−k∗ )3 )/2 −iβ(k,x,t)/2 22 × k P˜ (E(k), ) e− e e dk, [k1 ,k2 ]

where the negative of the imaginary part of the exponent is denoted by 2 k + e2 (∞) + κ(E(k)) − x k. β(k, x, t) = t 2 We now introduce µ() = s > 0, with 2/3 < s < 1. It goes to zero in such a way that µ()/ 1 and µ()3 /2 1.

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987

Because E ∗ is a unique absolute minimum, the behavior of α(E) close to E ∗ , and the assumption (5.2) on P , we can reduce the integration range in T to [k ∗ − µ(), k ∗ + µ()] at the expense of a relative error whose L2 norm is of order O(∞ ), uniformly t. More precisely, T (x, t, ) = ((F−1 (R1 + R2 ))(·, t, ))(x), where R1 (k, t, ) = χ[k∗ −µ(),k∗ +µ()] (k) R(k, t, ),

and

R2 (k, t, ) = χ[k∗ −µ(),k∗ +µ()]C (k) R(k, t, ). For some a∗ > 0 and r > 0, |R2 (k, t, )| ≤ r e−α(E

∗

)/2

∗

e−a

(µ()/)2

√ | k P˜ (E(k), )|.

Hence, by the Parseval identity, uniformly in t, we have F−1 (R2 )(·, t, ))

!1/2 2

= [k∗ −µ(),k∗ +µ()]C

|R2 (k, t, )| dk = O(e−α(E

∗

)/2 ∞

).

In the remaining integral containing R1 , we further estimate eO(k−k and

∗ 3

) /2

= 1 + O(µ()3 /2 ) = 1 + O(3s−2 ),

(7.31)

√ √ √ k P˜ (E(k), ) = k ∗ P˜ (E ∗ , ) + O(µ()) = k ∗ P (E ∗ , ) + O(s + 2 ).

The contribution of order 2 comes from the error in the computation of the coefficient c− n . Using the Parseval identity again with uniform bounds on the exponential factors of R1 , we see that the contribution to T coming from the error term O(s ) ∗ 2 is bounded uniformly in t in the L2 (Rx ) norm by O(e−α(E )/ 1+2s ). Similarly, 2 the error term stemming from (7.31) yields an error in the L (Rx ) norm of order ∗ 2 O(e−α(E )/ 4s−1 ). To compute the leading term, we expand β(·, x, t) around k ∗ as β(k, x, t)

= t E ∗ + κ(E ∗ ) − x k ∗ ∂ ∗ ∗ κ(E(k)) − x + (k − k ) k t + ∂k ∗ k (k − k ∗ )2 ∂2 + κ(E(k)) t+ 2 ∂k 2 k∗ ∗ 3 3 ∂ (k − k ) κ(E(k)) , + 6 ∂k 3 ˜ k

(7.32)

988



where k˜ lies between k and k ∗ , and the third derivative is independent of t and x. The last term in (7.32) gives rise to a contribution which is of order ∗ 2 O(e−α(E )/ 4s−1 ) in the L2 (Rx ) norm, uniformly in t, as above. Therefore, in the L2 sense, T (, x, t) = e−α(E

×

∗

)/2

e−i(tE

∗

+κ(E ∗ )−xk∗ )/2

√ (k−k∗ ) ∗ ∂ k ∗ P (E ∗ , ) e−i 2 (k t+ ∂k κ(E(k))|k∗ )−x)

[k∗ −µ(),k∗ +µ()]

×e

∗ 2

2

2

) ∂ ∂ − (k−k ( ∂k 2 α(E(k))|k∗ +i(t+ ∂k2 κ(E(k))|k∗ )) 22

dk + O( ) + O( ) , p

∞

where p = min(1 + 2s, 4s − 1) ∈ (0, 3) can be chosen arbitrarily close to 3. Again, ∗ 2 at the cost of an error whose L2 norm is O(e−α(E )/ ∞ ), uniformly in t, we can extend the interval of integration to the whole real line and compute the Gaussian integral explicitly according to the formula (for ReM > 0)

∞√ N2 ∗ 2 ∗ 2 2π − −(M(k−k ) /2+iN (k−k ))/ ∗ (k − iN/M ) . k∗ e dk = √ e 22 M M k∗ −∞ We then get the result with ∂2 ∂2 ∗ ∗ α(E(k))| + i t + κ(E(k))| , k k ∂k 2 ∂k 2 ∂ κ(E(k))|k∗ − x. N = k∗ t + ∂k

M=

and

Proof of Lemma 5.2. The first assertion is straightforward. The second follows from the identity ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x) x = ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x) (x − a+ (t)) + ϕ0 (A+ (t), B+ (t), a+ (t), 2 , η+ (t), x, ) a+ (t). The first term is O() in L2 (R) by scaling, and the second is of order a+ (t) = k ∗ t(1 + O(1/|t|)) for |t| large. We insert this in the first part of the lemma to obtain the second part as t → ±∞.

Acknowledgments George Hagedorn wishes to thank the Institut Fourier and the City of Grenoble for their kind hospitality and support during 2003 and 2004 when this research was conducted.

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References [1] M. Benchaou and A. Martinez, Estimations Exponentielles en Théorie de la Diffusion des Opérateurs de Schrödinger Matriciels, Ann. Inst. H. Poincaré Sect. A 71, 561–594 (1999). [2] V. Betz and S. Teufel, Precise coupling terms in adiabatic quantum evolution, Preprint mp− arc 04–102, and Ann. H. Poincaré (to appear). [3] V. Betz and S. Teufel, Precise coupling terms in adiabatic quantum evolution: The generic case, Preprint mp− arc 04–400. [4] D.F. Coker and L. Xiao, Methods for Molecular-Dynamics with Nonadiabtic Transitions, J. Chem. Phys. 102, 496–510 (1995). [5] M. Fedoriuk, Méthodes Asymptotiques pour les Equations Différentielles Ordinaires Linéaires, Mir, Moscou, 1987. [6] M. Fedoriuk, Analysis I, in Encyclopaedia of Mathematical Sciences, Vol 13, R.V. Gamkrelidze, ed. Springer-Verlag Berlin Heidelberg New York, 1989. [7] G.A. Hagedorn, Proof of the Landau-Zener Formula in an Adiabatic Limit with Small Eigenvalue Gaps, Commun. Math. Phys. 136, 433–449 (1991). [8] G.A. Hagedorn, Molecular Propagation Through Electronic Eigenvalue Crossings, Memoirs Amer. Math. Soc. 111 (536), (1994). [9] G.A. Hagedorn, Raising and lowering operators for semiclassical wave packets, Ann. Phys. 269, 77–104 (1998). [10] G.A. Hagedorn and A. Joye, Landau-Zener Transitions Through Small Electronic Eigenvalue Gaps in the Born-Oppenheimer Approximation. Ann. Inst. H. Poincaré, Phys. Théor. 68, 85–134 (1998). [11] G.A. Hagedorn and A. Joye, Molecular Propagation Through Small Avoided Crossings of Electron Energy Levels. Rev. Math. Phys. 11, 41–101 (1999). [12] G.A. Hagedorn and A. Joye, A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates, Commun. Math. Phys. 223, 583–626 (2001). [13] G.A. Hagedorn and A. Joye, Time Development of Exponentially Small NonAdiabatic Transitions, Commun. Math. Phys., 250, 393–413 (2004). [14] A.Joye, Proof of the Landau-Zener Formula, Asymptotic Analysis 9, 209–258 (1994). [15] A. Joye, Exponential asymptotics in a singular limit for n-level scattering systems, SIAM J. Math. Anal. 28, 669–703 (1997). [16] A.Joye, C.-E. Pfister, Complex WKB Method for 3-Level Scattering Systems. Asymptotic Anal. 23, 91–109 (2000). [17] A. Joye, H. Kunz, C.-E. Pfister, Exponential Decay and Geometric Aspect of Transition Probabilities in the Adiabatic Limit, Ann. Phys. 208, 299–332 (1991). [18] A. Joye, C.-E. Pfister, Semi-Classical Asymptotics beyond All Orders for Simple Scattering Systems, SIAM J. Math. Anal. 26, 944–977 (1995).

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[19] A. Joye, C.-E. Pfister, Superadiabatic Evolution and Adiabatic Transition Probability between Two Non-degenerate Levels Isolated in the Spectrum, J. Math. Phys. 34, 454–479 (1993). [20] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag Berlin Heidelberg New York 1980. [21] Ph.-A. Martin and G. Nenciu, Semiclassical Inelastic S-Matrix for OneDimensional N -States Systems, Rev. Math. Phys. 7, 193–242 (1995). [22] A. Martinez and V. Sordoni, A general reduction scheme for the timedependent Born-Oppenheimer approximation, C.R.A.S. 334, 185–188 (2002). [23] G. Nenciu and V. Sordoni, Semiclassical limit for multistate Klein-Gordon systems: almost invariant subspaces and scattering theory, Preprint mp− arc 01–36. [24] T. Ramond, Semiclassical Study of Quantum Scattering on the Line, Commun. Math. Phys. 177, 221–254 (1996). [25] V. Rousse, Landau-Zener Transitions for Eigenvalue Avoided Crossings in the Adiabatic and Born-Oppenheimer Approximations, Asymptotic Analysis 37, 293–328 (2004). [26] J.C. Tully, Molecular Dynamics with Electronic Transitions, J. Chem. Phys. 93, 1061–1071 (1990). [27] F. Webster, P.J. Rossky and R.A. Friesner, Nonadiabatic Processes in Condensed Matter: Semi-Classical Theory and Implementation, Comp. Phys. Commun. 63, 494–522 (1991).

George A. Hagedorn Department of Mathematics and Center for Statistical Mechanics and Mathematical Physics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0123 USA email: [email protected] Alain Joye Institut Fourier Unité Mixte de Recherche CNRS-UJF 5582 Université de Grenoble I BP 74 F-38402 Saint Martin d’Hères Cedex France email: [email protected] Communicated by Yosi Avron submitted 14/10/04, accepted 18/01/05

Ann. Henri Poincaré 6 (2005) 991 – 1023 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/060991-33, Published online 15.11.2005 DOI 10.1007/s00023-005-0232-x


Heat-Kernel Approach to UV/IR Mixing on Isospectral Deformation Manifolds Victor Gayral Abstract. We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions of Rl . Our framework, incorporating background field methods and tools of QFT in curved spaces, allows to deal both with compact and non-compact spaces, as well as with periodic and non-periodic deformations, essentially in the same way. We compute the quantum effective action up to one loop for a scalar theory, showing the different UV/IR mixing phenomena for different kinds of isospectral deformations. The presence and behavior of the non-planar parts of the Green functions is understood simply in terms of off-diagonal heat kernel contributions. For periodic deformations, a Diophantine condition on the noncommutivity parameters is found to play a role in the analytical nature of the non-planar part of the one-loop reduced effective action. Existence of fixed points for the action may give rise to a new kind of UV/IR mixing.

1 Introduction Noncommutative geometry (NCG), specially in Connes’ algebraic and operatorial formulation [4], is an attempt to free oneself from the classical differential structure framework in modeling and understanding space-time, while keeping in algebraic form geometry’s tools such as metric and spin structures, vector bundles and connection theory. The NCG framework is well adapted to deal with quantum field theory over ‘quantum’ space-time (NCQFT) [34]. However, there is a lack of computable examples crucially needed to progress in this direction. Here we present a large class of models, the isospectral deformation manifolds, in which we show the intrinsic nature of UV/IR mixing through the analysis of a scalar theory. In [6, 7] Connes, Landi and Dubois-Violette gave a method to generate noncommutative spaces based on the noncommutative torus paradigm. For any closed Riemannian spin (this last condition could be relaxed for our purpose) manifold with isometry group of rank l ≥ 2, one can build a family of noncommutative spaces, called isospectral deformations by the authors. The terminology comes from the fact that the underlying spectral triple, that is, the dual object / ) encoding all the topological, differential, metric and spin (C ∞ (MΘ ), L2 (M, S), D structures of the original manifold, and so defining the ‘quantum Riemannian’ space [5], has the same space of spinors and the same Dirac operator as the unde/ ); only the algebra is modified. formed one (C ∞ (M ), L2 (M, S), D

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More precisely, the noncommutative algebra C ∞ (MΘ ) can be defined as a fixed point algebra under a group action [7]: α⊗τ −1 lΘ C ∞ (MΘ ) := C ∞ (M )⊗T ,

(1.1)

where TlΘ is a l-dimensional NC torus(-algebra) with deformation matrix Θ ∈ Ml (R), Θt = −Θ; α is the action of Tl on M given by an Abelian part of its is a suitable tenisometry group, τ is the standard action of Tl on TlΘ and ⊗ sor product completion. By the Myers-Steenrod Theorem [26], which asserts that Isom(M, g) ⊂ SO(n) for any n-dimensional compact Riemannian manifold (M, g), one can see that the class of such manifolds whose isometry group has rank greater or equal to two is far from small. V´ arilly [33] and Sitarz [31] independently remarked that this construction fits into Rieffel’s theory of deformation quantization for actions of Rl [28]. Given a Fréchet algebra A with seminorms {pi }i∈I and a strongly continuous isometric (with respect to each seminorms) action of Rl , one can deform the product of the subalgebra A∞ , consisting of smooth elements of A with respect to the generators X k , k ∈ {1, . . . , l} of the action α. The algebra A∞ can be canonically endowed with a new set of seminorms {˜ pi,m }i∈I,m∈N given by p˜i,m (.) := supj≤i |β|≤m pj (X β .), β ∈ Nl . Those seminorms have the property of being compatible with the deformed product defined by the A∞ -valued oscillatory integral: −l dl y dl z e−i α 1 Θy (a)α−z (b), a, b ∈ A∞ . aΘ b := (2π) R2l

2

Here Θ is the (real, skewsymmetric) deformation l×l matrix, < y, z >= li=1 y i z i , ∞ and if we denote by A∞ Θ the algebra (A , Θ ), the deformation process verifies ∞ ∞ ∞ (AΘ )Θ = AΘ+Θ , and hence is reversible. In [16], we investigate the equivalent of (1.1) in the non-periodic case and extend the construction of isospectral non-periodic deformations (called also θ-deformations to distinguish them from q-deformations) to non-compact manifolds within Rieffel’s framework, whose paradigms are now the Moyal planes [12]. Although we will not use directly the fixed point characterization (1.1), we want to insist on its crucial importance to understand the situation. Indeed, such a characterization means that we are transferring the noncommutative structure of the NC torus or of the Moyal plane inside the commutative algebra of smooth functions, in a way compatible with the Riemannian structure. The first studied examples of NCQFT were the NC tori and the Moyal planes, in pioneer works like [3, 11, 21, 23, 24, 34] (see also [10] and [32] for reviews). In those flat space situations, the main novelty in regard to renormalization aspects is that two kinds of Feynman diagrams coexist, respectively called planar and nonplanar. The first one yields ordinary UV divergences, while the non-planar graphs, characterized by vertices which depend on external momenta through a phase, are finite except for some values of the incoming momenta. That happens in particular

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for the zero mode in λϕΘ 4 theory on the NC torus and in the limit pµ → 0 for the same theory on the Moyal plane. This is the famous UV/IR entanglement phenomenon, which gives rise to difficulties for any renormalization scheme. In this paper, we show that for any (in general non-flat) isospectral deformation, UV/IR mixing in (Euclidean) NCQFT exists as in the (flat) paradigmatic examples of the NC torus and the Moyal planes. In the next section, isospectral deformations are constructed and their basic NCG properties are reviewed. The third section is devoted to the study of the λϕΘ 4 theory. One derives a field expansion from a (modified) heat kernel asymptotics to compute the effective action up to one loop. This construction gives a simple algebraic meaning to the presence and behavior of planar and non-planar sectors in those theories. In sections 4 and 5, using off-diagonal heat-kernel estimates, we prove the inherent generic character of the divergent structures for all kinds of isospectral deformations. Fixed points for the Rl action potentially yield a new kind of UV/IR mixing.

2 Isospectral deformations As explained in the Introduction, isospectral deformations are curved noncommutative spaces generalizing Moyal planes and noncommutative torus. To construct those NC Riemannian spaces (spectral triples), we use an approach developed in [16]. Advantages of this twisted product approach ` a la Rieffel are that it allows to treat on the same footing compact and non-compact cases (unital and nonunital algebras) as well as periodic and non-periodic deformations, and that it is well adapted for Hilbertian analysis. Let (M, g) be a locally compact, complete, connected, oriented Riemannian n-dimensional manifold without boundary, and let α be a smooth isometric action of Rl , 2 ≤ l ≤ n α : Rl −→ Isom(M, g) ⊂ Diff(M ), where l is less or equal to the rank of the isometry group of (M, g). We can then define a deformed or twisted product. The isometric action α yields a group of automorphisms on C ∞ (M ) that we will again denote by α: for all z ∈ Rl αz f (p) := f (α−z (p)). For brevity we will often write z.p ≡ αz (p) to designate the action of a group element on a point of the manifold. Obviously, the group action property reads z1 .(z2 .p) = (z1 + z2 ).p The infinitesimal generators of this action ∂ α (.) , Xj (.) := z j ∂z z=0

and 0.p = p.

j = 1, . . . , l,

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are ordinary smooth vector fields, so they leave Cc∞ (M ) invariant. Hence, given a real skewsymmetric l × l matrix Θ, one defines the deformed product of any f, h ∈ Cc∞ (M ) as a bilinear product on Cc∞ (M ) with values in C ∞ (M ) ∩ L∞ (M, µg ) by the oscillatory integral −l dl y dl z e−i α 1 Θy (f )α−z (h), (2.1) f Θ h := (2π) 2

R2l

l

where < y, z >:= j=i y j z j can be viewed as the pairing between Rl and its dual group. In spite of appearances this formula is symmetric, even with a degenerate Θ matrix (see the discussion near the end of this section), as one can rewrite the deformed product: dl y dl z ei α−y (f )α 1 (h). f Θ h := (2π)−l 2 Θz

R2l

The non-locality of this product generates a non-preservation of supports. In particular, the twisted product of two functions with disjoint support turns out to be non-zero a priori. Whereas in the periodic case (ker α Zl ) the fixed point characterization gives rise to a reasonable locally convex topology on the α⊗τ −1 invariant sub-algebra of the algebraic tensor product C ∞ (M ) ⊗ TlΘ or α⊗τ −1 depending whether M is compact or not, to obtain a smooth Cc∞ (M )⊗TlΘ algebra structure in the non-periodic case one has to complete Cc∞ (M ) to a Fréchet algebra with seminorms defined through the measure associated to the Riemannian volume form, so that the action becomes strongly continuous and isometric with respect to each seminorm. This feature is investigated in [16]. In the sequel, as we mainly work at the linear level, Cc∞ (M ) will be deemed “large enough”. The associativity of the product (2.1) can be easily checked. The ordinary integral with Riemannian volume form µg is a trace (a proof is provided in [16]): µg f Θ h = µg f h = µg hΘ f ; (2.2) M

M

M

α is still an automorphism for the deformed product: αz (f )Θ αz (h) = αz (f Θ h);

(2.3)

the complex conjugation is an involution: (f Θ h)∗ = h∗ Θ f ∗ ;

(2.4)

and the Leibniz rule is satisfied for the generators of the action X k (f Θ h) = X k (f )Θ h + f Θ X k (h), k = 1, . . . , l.

(2.5)

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In fact, the Leibniz rule is satisfied for any order one differential operator which commutes with the action α, thus for the Dirac operator when the manifold has a spin structure. We have basically two distinct situations. When the group action is effective (ker α = {0}), i.e., for a non-periodic deformation, it is seen that the good topological assumption on α in order to avoid serious difficulties is properness. That is, we assume the map (z, p) ∈ Rl × M → (p, αz (p)) ∈ M × M to be proper. Recall that a map between topological spaces is proper if the preimage of any compact set is compact as well. On the other hand, for periodic deformations the action factors through a torus action α ˜ : Rl /Zl → Isom(M, g), and the factorized action α ˜ is automatically proper. When M is compact, α must be periodic to be proper, while in the noncompact case both situations appear. We point out that the (non-compact) nonperiodic case is the most difficult one. First, when the manifold is not compact, the essential spectrum of the Laplacian is non-empty, so its negative powers are no longer compact operators. Furthermore, for periodic deformations (of compact manifolds or not) we have a spectral subspace decomposition, indexed by the dual group of Tl , which does simplify proofs and computations. We do not explicitly treat the mixed case α : Rd × Tl−d → Isom(M, g), but its general features will be clear from what follows. The hypothesis of geodesically completeness of M guarantees selfadjointness of the (closure of the) Laplace-Beltrami operator ∆ restricted to (the dense subset Cc∞ (M ) of) L2 (M, µg ), the separable Hilbert space of squared integrable functions with respect to the measure space (M, µg ). In our convention, ∆ = (d + δ)2 is positive, and reduced to 0-forms ∆ = δd = ∗H d ∗H d where ∗H is the Hodge star. Completeness (plus boundedness from below of the Ricci curvature) is needed to have conservation of probability [2, 9]: µg (p) Kt (p, p ) = 1, M

where Kt := Ke−t∆ is the heat kernel of the manifold. Recall that Kt (p, p ) for t > 0 is a smooth strictly positive symmetric function on M × M . The restriction to manifolds without boundary is required to have a simple (with vanishing of the odd terms [17]) on-diagonal expansion of the heat kernel Kt (p, p) (4πt)−n/2 tl a2l (p), t → 0, (2.6) l∈N

where al (p) are the so called Seeley-De Witt coefficients. It is proved in [16] that for non-compact non-periodic deformations (the stateΘ ment being immediate in the periodic case) Lf ≡ LΘ f (resp. Rf ≡ Rf ), the operator of left (resp. right) twisted multiplication by f , defined by Lf ψ = f Θ ψ (resp.

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Rf ψ = ψΘ f ), for ψ ∈ H := L2 (M, µg ), is bounded for any f ∈ Cc∞ (M ). This will be also true for smooth functions decreasing fast enough at infinity. Denote by Vz the induced action of Rl on L2 (M, µg ) by unitary operators Vz ψ(p) := ψ(−z.p); then one can alternatively define Lf and Rf by an operator valued integral Lf = (2π)−l dl y dl z e−i V 1 Mf V−z , (2.7) 2 Θy

R2l

Rf = (2π)−l

R2l

dl y dl z e−i V−z Mf V 1

2 Θy

,

(2.8)

where Mf denotes the operator of pointwise multiplication by f . Such integrals do not define B¨ ochner integrals in the vector space L(H). Indeed, the operatorial norm of the integrands in (2.7) and (2.8) are not integrable functions on R2l , since they depend on y and z only through unitary operators. Actually, the latter must be understood as L(H)-valued oscillatory integrals [28]. Formulas (2.7) and (2.8) can be easily derived from (2.1) using Vz Mf V−z = Mαz (f ) and the translation z → z − 12 Θy which leaves invariant the phase due to the skewsymmetry of the deformation matrix. Note that they can be used to define (left and right) ‘Moyal multiplications’ of any bounded operator on H, taking the place of Mf in the formulas. Within this presentation, it is straightforward to check that L and R are two commuting representations (in fact R is an antirepresentation): [Lf , Rh ] = 0, ∀f, h ∈ Cc∞ (M ). Thus formulas (2.7) and (2.8) provide an other way to check the associativity of the twisted product, which is equivalent to the commutativity of the left and right regular representations. Using the trace property (2.2), one can also prove that the adjoint of the left (resp. right) twisted multiplication by f equals the left (resp. right) twisted multiplication by the complex conjugate of f : (Lf )∗ = Lf ∗ , (Rf )∗ = Rf ∗ . Again, this fact can be directly checked using formulas (2.7) and (2.8). For Lf it reads ∗ −l (Lf ) = (2π) dl y dl z ei Vz Mf ∗ V 1 − 2 Θy R2l = (2π)−l dl y dl z e−i V 1 Θz Mf ∗ V−y , R2l

2

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where the changes of variable z → 12 Θz, y → 2Θ−1 y and the relation < Θ−1 y, Θz > = − < y, z > have been used. The primary example of such a space is the n-dimensional Moyal plane RnΘ . In this case, the manifold is the flat Euclidean space Rn , l = n, and Rn acts on itself by translation. Another interesting non-compact space which carries a smooth action of Rn−1 by isometry is the n-dimensional hyperbolic space Hn , that we can make into noncommutative HnΘ by the previous prescription. For periodic actions, there is a lattice L = βZl , β ∈ Ml (Z) in the kernel of α which factors through a torus Tlβ := Rl /βZl . This quotient is a compact space if and only if the rank of β equals l. In this case, we have a spectral subspace (PeterWeyl) decomposition (see [6,28,33] for details): for any bounded operator A which is α-norm smooth (the map z ∈ Tlβ → Vz AV−z is smooth for the norm topology of L(H)), one can define a l-grading by declaring A of l-degree r = (r1 , . . . , rl ) ∈ βZl when Vz AV−z = e−i(r1 z1 +···+rl zl ) A, ∀z ∈ Tlβ . Then, any α-norm smooth operator can be uniquely written as a norm convergent sum A= Ar , r∈βZl

where each Ar is of l-degree (r1 , . . . , rl ). This is in particular the case for the operator of pointwise multiplication by any function f ∈ Cc∞ (M ), since Mf lies inside the smooth domains of the derivations δj (.) := [Xj , .]. This assertion is obtained iterating the relation

[Xj , Mf ] = MXj (f ) = Xj (f ) ∞ , which is finite since f ∈ Cc∞ (M ) and because the Xj are ordinary smooth vector fields. Writing the spectral subspace decomposition of suchoperator, we find the PeterWeyl decomposition of any f ∈ Cc∞ (M ), as f = r∈βZl fr , where fr satisfies αz (fr ) = e−i(r1 z1 +···+rl zl ) fr . The twisted product of homogeneous components satisfies the noncommutative torus relation: i

fr Θ hs = e− 2 fr hs .

(2.9)

Noncommutative tori TnΘ , odd and even Connes-Landi spheres Sθ2n+1 , S2n θ [7] are examples of such compact noncommutative spaces; and the ambient space of Sn−1 θ is a non-compact periodic deformation. In summary, it is clear that the noncommutative structures of isospectral deformations are inherited from the NC tori or Moyal planes one’s, depending whether the deformation is periodic or not. When Θ is not invertible, the deformed product reduces to another twisted product associated with the restricted action σ := α|V ⊥ , where V is the null space

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of Θ – see for example [28]. Hence, one can handle non-invertible deformation matrices without any trouble. But of course, the “effective” deformation is always of even rank. Finally, in the non-periodic case only, properness of α implies that it is also free. To see that, recall that properness of any G-action is equivalent to {g ∈ G|g.X ∩ Y = ∅} is compact for any X, Y compact subset of M – see [25]. So, taking X = Y = {p0 } for any p0 ∈ M , its isotropy group Hp0 = {z ∈ Rl |z.p0 = p0 } = {z ∈ Rl |z.{p0 } ∩ {p0 } = ∅} is compact as well. But the only compact subgroup of Rl is {0}, hence the action is automatically free. This implies that the quotient map π : M → M/Rl defines a Rl -principal bundle projection. In the periodic case, the action is no longer automatically free, and the set Msing of points with non-trivial isotropy groups can give rise to additional divergences in the effective action. This will be shown to constitute a new feature of the UV/IR mixing on isospectral deformation manifolds.

3 ϕΘ 4 theory on 4-d isospectral deformations 3.1

The effective action at one-loop

For the sake of simplicity, we now restrict to the four-dimensional case; n = dim(M ) = 4. It will be clear, nevertheless, that our techniques apply to higher dimensions without essential modifications. We consider the classical functional action for a real scalar field ϕ: λ (3.1) µg 12 (∇µ ϕ)Θ (∇µ ϕ) + 12 m2 ϕΘ ϕ + ϕΘ 4 . S[ϕ] := 4! M We could add a coupling with gravitation of the type ξR(ϕΘ ϕ) (or even ξRΘ ϕΘ ϕ), where R is the scalar curvature and ξ a coupling constant, without change in our conclusions. Indeed, this term is not modified by the deformation: due to the α-invariance of the scalar curvature, we have RΘ f = R.f for any f ∈ Cc∞ (M ), thus µg R.(ϕΘ ϕ) = µg RΘ ϕΘ ϕ = µg (RΘ ϕ).ϕ = µg R .ϕ .ϕ. M

M

M

M

Similarly, thanks to the trace property (2.2), S[ϕ] can be rewritten as λ µg 12 ϕ∆ϕ + 12 m2 ϕ ϕ + (ϕΘ ϕ) (ϕΘ ϕ) , S[ϕ] = 4! M

(3.2)

so that, as in the falt cases, the kinetic part is not affected by the deformation. Recall that in our conventions the Laplacian is positive: ∆ = −∇µ ∇µ . We aim to compute the divergent part of the effective action Γ1l [ϕ] associated to S[ϕ] at one loop. This is formally given by 12 ln(det H), where H is the effective potential. In our case (as in the commutative one) it will be seen that H = ∆+m2 +

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B, where B is positive and bounded; so that when the manifold is not compact H has a non empty essential spectrum (typically the whole interval [m2 , +∞[). In order to deal with operators having pure-point spectrum (discret with finite multiplicity), we need first (independently of any regularization scheme) to redefine formally the one-loop effective action as:

Γ1l [ϕ] := 12 ln det HH0−1 , where H0−1 := (∆ + m2 )−1 is the free propagator. We are “not so far” from having a well-defined determinant since: HH0−1 = (H0 + B)H0−1 = 1 + BH0−1 , and BH0−1 is ‘small’: not trace-class in general, but compact; more precisely BH0−1 lies inside the p-th Schatten-class for all p > 2 (see below for the concrete expression of B and [16] for a proof of this claim). Physically, to replace H by HH0−1 corresponds to remove the vacuum-to-vacuum amplitudes. We then define the logarithm of the determinant by the Schwinger “proper time” representation:

1

1 ∞ dt −1 Tr e−tH − e−tH0 . (3.3) Γ1l [ϕ] = ln det(HH0 ) := − 2 2 0 t Before giving a precise meaning to the previous expression, that is to choose a regularization scheme, we go through the computation of the effective potential H. For that, the following definition will be useful. Definition 3.1. Let (X, dµ) a measure space. A kernel operator on E, a functions space on X, is a linear map A : E → E which can be written as

Af (p) = dµ(q) KA (p, q) f (q), f ∈ E, p, q ∈ X, X

where KA is the kernel of A. This definition leads to the following rules for the product of two kernel operators and for the kernel of the adjoint: dµ(u) KA (p, u) KB (u, q), and KA∗ (p, q) = KA (q, p)∗ . (3.4) KAB (p, q) = X

In our case, (X, dµ) ≡ (M, µg ) as a measure space, E ≡ Cc∞ (M ) and we will only be interested on distributional kernels, that is those KA lying on Cc∞ (M ×M ) , the space of distributions on M × M . Recall that the effective potential (see for example [36]) is the operator whose distributional kernel is given by the second functional derivative of the classical action: δ 2 S[ϕ] δ 2 S[ϕ] , K (p, p ) := , KH (p, p ) := H0 δϕ(p)δϕ(p ) δϕ(p)δϕ(p ) λ=0

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with functional derivatives defined as usual in the weak sense

δS[ϕ] dS[ϕ + tψ] , ψ := , δϕ dt t=0 where the coupling is given by the integral with Riemannian volume form f, h = µ f h. g M Using the trace property (2.2) we find out:

dS[ϕ + tψ] λ Θ 3 2 ϕ = ∆ϕ + m ϕ + , ψ . dt 3! t=0 Hence, δS[ϕ] λ S˜p [ϕ] := = ∆ϕ(p) + m2 ϕ(p) + ϕΘ 3 (p). δϕ(p) 3! The second functional derivative reads

2 δ S[ϕ] dS˜p [ϕ + tψ] , ψ := δϕ(p)δϕ dt t=0

λ 2 = ∆ + m + (LϕΘ ϕ + RϕΘ ϕ + Rϕ Lϕ ) δpg , ψ , 3! where δpg is the distribution defined by δqg , φ = M µg (p)δqg (p)φ(p) = φ(q), for any test function φ ∈ Cc∞ (M ). In conclusion, the explicit form of the operator H is: H = ∆ + m2 +

λ (LϕΘ ϕ + RϕΘ ϕ + Rϕ Lϕ ). 3!

Because ϕ is real, the operators Lϕ and Rϕ are self-adjoint, and we can check directly the strict positivity of H: ∗ LϕΘ ϕ + RϕΘ ϕ + Lϕ Rϕ = 12 (Lϕ + Rϕ )∗ (Lϕ + Rϕ ) + 12 L∗ϕ Lϕ + 12 Rϕ Rϕ .

We are come to an important point: the existence of UV/IR mixing for field theory on isospectral deformations comes from the simultaneous presence of left and right twisted multiplications in the effective potential. Precisely, we wish to illustrate the smearing nature of the product of left and right twisted multiplication operator Lf Rh . The crucial consequence, employed in subsection 2 3.3, is that the trace of Lf Rh e−t(∆+m ) is regular when t goes to zero, contrary 2 2 2 to Tr(Lf e−t(∆+m ) ), Tr(Rf e−t(∆+m ) ), Tr(Mf e−t(∆+m ) ), which in n dimensions −n/2 behave as t when t → 0 (In fact the three latter traces are identical). Remark 3.2. For a potential reads:

λ Θ 3 3! ϕ

theory on a six dimensional manifold, the effective

H = ∆ + m2 +

λ (Lϕ + Rϕ ). 2!

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Even in the lack of the ‘mixed’ term Rϕ Lϕ , those theories have a non-planar sector, but which will be present only at the level of the two-point function; the tadpole is not affected by the mixing. Consider the non-degenerate (n = 2N, Θ invertible) Moyal plane case. The operator Lf Rh turns out to be trace-class whenever f, h ∈ S(R2N ), say. This fact is known to the experts, but rarely mentioned – to the knowledge of the author, its first mention in writing is in [1]. We do a little disgression to see how it comes about. Recall [12] that there is an orthonormal basis for L2 (R2N , d2N x), the harmonic oscillator eigentransitions (2πθ)−N/2 {fmn }m,n∈NN , θ := (det Θ)1/2N which are matrix units for the Moyal product: fmn Θ fkl = δnk fml . Expanding f, h ∈ S(R2N ) in this basis: f = m,n cmn fmn , h = m,n dmn fmn , we obtain:

Tr Lf Rh = (2πθ)−N ckl dst fmn , fkl Θ fmn Θ fst m,n,k,l,s,t

−N

= (2πθ) =

ckm dnt fmn , fkt

m,n,k,t

cmm dnn

m,n

= (2πθ)−N

d2N x f (x)

d2N y h(y) < ∞.

Then in this case one can factorize HH0−1 and extract a finite part in the effective action. We have λ 1 H0 H −1 = 1 − (Lϕθ ϕ + Rϕθ ϕ ) λ 3! ∆ + m2 + 3! (Lϕθ ϕ + Rϕθ ϕ ) λ 1 × 1 − Lϕ Rϕ . (3.5) λ 3! ∆ + m2 + 3! (Lϕθ ϕ + Rϕθ ϕ + Lϕ Rϕ ) Now, 1−

λ Lϕ Rϕ 3! ∆ + M2 +

1 λ 3! (Lϕθ ϕ

+ Rϕθ ϕ + Lϕ Rϕ )

∈ 1 + L1 (H),

so that its determinant is well defined. Thus only the determinant of the first piece of (3.5) needs to be regularized. The determinant of the second piece of (3.5) contains the whole non-planar contribution to the two-point function, while for the four-point function the finite non-planar part lies in both pieces. The structure of the effective potential, i.e., the presence of mixed products of left and right twisted multiplication operators, and thus the existence of two distinct sectors in the theory is fairly general: for noncommutative scalar field theories

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whose classical field counterparts are regarded as elements of a noncommutative algebra, and a classical action built from a trace on the algebra, the effective potential will contain in general sums and mixed products of left and right regular representation operators. Let us go back to the computation of Γ1l [ϕ]. The t-integral in (3.3) is divergent because of the small-t behavior of the heat kernel on the diagonal. We thus define a one-loop regularized effective action by:

1 ∞ dt Tr e−tH − e−tH0 . Γ1l [ϕ] := − (3.6) 2 t One can invoke less rough regularization schemes, for example a ζ-function regularization

1 ∞ dt 2 σ σ,µ tµ (3.7) Tr e−tH − e−tH0 , Γ1l [ϕ] := − 2 0 t akin to dimensional regularization. However, for the purposes of this article (3.6) will do. One can think of as of the inverse square of Λ, with Λ a momentum space cutoff. To show that the expressions (3.6) and (3.7) are now well defined, we have to prove that e−tH − e−tH0 is trace-class for all t > 0. Note that for t → ∞ 2 convergence is ensured by the global e−tm factor, and that when the spectrum of the Laplacian has a strictly positive lower bound one can construct massless, IR divergence-free NCQFT. That is the case for the twisted hyperbolic planes HnΘ since the L2 -spectrum of ∆ on Hn is the whole half line [n2 /4, ∞[. Lemma 3.3. The semigroup difference e−tH − e−tH0 is trace-class for all t > 0. Proof. Using positivity of H and H0 , the semigroup property and the holomorphic functional calculus with a path γ surrounding both the spectrum sp(H) ⊂ R+ and sp(H0 ) ⊂ R+ , we have 1 e−tH − e−tH0 = (2iπ)2 dz1 dz2 e−t(z1 +z2 )/2 (RH (z1 )RH (z2 ) − RH0 (z1 )RH0 (z2 )) , γ×γ

where RA (z) = (z − A)−1 denotes the resolvent of A. But H = H0 + B where B is bounded. Using next RH (z) = RH0 (z)(1 + BRH (z)), we find RH (z1 )RH (z2 ) − RH0 (z1 )RH0 (z2 ) = RH0 (z1 )RH0 (z2 )BRH (z2 ) + RH0 (z1 )BRH (z1 )RH0 (z2 ) + RH0 (z1 )BRH (z1 )RH0 (z2 )BRH (z2 ). The first resolvent equation and the fact that Lf (z − ∆)−k , Rf (z − ∆)−k ∈ Lp (H), for p > 2/k, f ∈ Cc∞ (M ) [16], together with the H¨ older inequality for Schatten

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classes, yield

dz1 dz2 e−t(z1 +z2 )/2 RH0 (z1 )RH0 (z2 )BRH (z2 )

γ×γ

is absolutely convergent for the trace norm. Similarly for the other terms. So e−tH − e−tH0 is trace-class as required.

3.2

Field expansion

We now tackle the -behavior of Γ1l [ϕ] to describe the divergences. We will then show that, as for the Moyal planes and noncommutative tori, there exist for general isospectral deformations two kind of contributions to the Green functions, the planar one giving rise to ordinary singularities and the non-planar one exhibiting the UV/IR mixing phenomenon. Note that, since we are in a curved background, we can no longer work with Feynman diagrams in momentum space. However, by abuse of language we continue to speak about planar and non-planar contributions, because there is a splitting at the operator level which coincides with the splitting of planar and non-planar Feynman graphs in the known flat cases. This point will become clearer in subsequent subsections. As we are only interested in the -behavior of Γ1l [ϕ] (we only consider the potentially divergent part of the regularized effective action), we need a small t

expansion for Tr e−tH − e−tH0 . This expansion will be managed in the same vein as the ones obtained in [13,35]. The Baker-Campbell-Hausdorff formula is written: t2

t3

t3

e−tH = e−tB+ 2 [∆,B]− 6 [∆,[∆,B]]− 12 [B,[∆,B]]+··· e−tH0 .

(3.8)

We now expand the first exponential up to factors which, after taking the trace, give terms of order less or equal to zero in t. Only a few terms will be important: We have first to take into account that (in n dimensions) Tr(Lf ∆k e−t∆ ) t−n/2−k , t → 0, Tr(Rf ∆k e−t∆ ) t−n/2−k , t → 0.

(3.9)

Indeed, for the “left” case (the right one being similar) since, as proved in [16], one has Lf (1 + ∆)−k ∈ Lp (H) for any p > n/2k and any f ∈ Cc∞ (M ), we conclude for all > 0:

Lf ∆k e−t∆ 1 ≤ Lf (1 + ∆)−n/2− 1

∆k

(1 + ∆)n/2+k+ e−t∆ (1 + ∆)k

≤ C()t−(n/2+k+) . The last estimate follows from functional calculus. Therefore, in the field expansion we need to correct the power in t by the order of the differential operator appearing when we expand the first exponential in the equation (3.8).

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Secondly, we have to notice that the commutators [∆, Lf ], [∆, Rf ] (and also [∆, Rf Lh ]) reduce by one the order of the differential operator (cf. equation (3.10) below). To see this, we compute the commutators [∆, Lf ], [∆, Rf ] and [∆, Rf Lh ]. The simplest way is to use the formulas (2.7) and (2.8). By [Vz , ∆] = 0 for all z ∈ Rl (from the isometry property of α) and choosing a local coordinate system {xµ }, one obtains −l

[∆, Lf ] = (2π)

−l

R2l

dl y dl z e−i V 1 Θy [∆, Mf ] V− 1 Θy−z 2

l

= (2π)

l

d yd z e

−i

R2l

2

V 1 Θy (M∆f − 2M∇µ f ∇µ ) V− 1 Θy−z 2

2

= L∆f − 2L∇µ f ∇µ ,

(3.10)

and similarly, [∆, Rf ] = R∆f − 2R∇µ f ∇µ ,

(3.11)

[∆, Rf Lh ] = Rf [∆, Lh ] + [∆, Rf ]Lh

= Rf L∆(h) + R∆(f ) Lh − 2R∇µ f L∇µ h − 2 Rf L∇µ h + R∇µ f Lh ∇µ . (3.12)

The local coordinate system used must be compatible with the deformation, that is, defined on some α-invariant open neighborhood U ⊂ M . To obtain one such, I }i∈I by letting Rl act on choose any open covering {UI }i∈I of M and define {U l it: Ui := R .Ui . This implies that in n dimensions, one only needs to use the BCH formula up to order n − 2 to capture the divergent structure of the effective action. Moreover, that the commutators decrease the degree of the differential operator is a necessary condition to make the BCH expansion meaningfull: In [15], we consider a field theory on a noncommutative 4-plane with an (associative) position-dependant Moyal product (coming from a rank-2 Poisson structure on R4 ). It turns out that the commutators [∆, Lf ] and [∆, Rf ] contain now a term with an order two differential operator. This makes the BCH development useless since the k-times iterated commutator [t∆, [· · · , [t∆, tLf ] · · · ]] contains a term which gives after the exponential expansion a contribution of order t−n/2+1 , independently of k, the number of commutators involved. Thus, in this case the whole BCH serie will be needed to capture the divergences. Putting all together, we finally obtain: e−tH =

t3 t2 t2 1 − tB + [∆, B] − [∆, [∆, B]] + B 2 e−tH0 + O(t); 2 6 2

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we mean by this estimate that we have a small-t expansion:

Tr e−tH − e−tH0 =

t2 t3 t2 2 −tH0 − tB + [∆, B] − [∆, [∆, B]] + B e Tr + O(t). 2 6 2

(3.13)

We now show that in fact, the commutators in the expression (3.13) give no contribution to the effective action. Indeed, if each terms C∆e−t∆ and ∆Ce−t∆ are trace-class, with C = B or C = [∆, B], then by the cyclicity of the trace and the fact that the Laplacian commutes with the heat semigroup, one gets

Tr ∆ C e−t∆ − C ∆ e−t∆ = Tr C ∆ e−t∆ − C ∆ e−t∆ = 0 (3.14) That C∆e−t∆ is trace-class is obvious from functional calculus and using the same arguments than those used to obtain the estimate (3.9). For ∆Ce−t∆ , it is a little bit less immediate since the latter appears as a product of a trace-class operator (Ce−t∆ ) times an unbounded one (∆). Actually, using the tautological relation ∆ C e−t∆ = C ∆ e−t∆ + [∆, C] e−t∆ , and the equations (3.10) and (3.11) (iterated once more when C = [∆, B]), one sees that this term appears also as a sum of trace-class operators. Hence (3.14) is proved and we are left with

2 λ Tr e−tH − e−tH0 = − t Tr LϕΘ ϕ + RϕΘ ϕ + Rϕ Lϕ e−t(∆+m ) 3!

t2 λ2 Tr LϕΘ4 + RϕΘ 4 + 3RϕΘ ϕ LϕΘ ϕ + 2 (3!)2 2 + 2Rϕ LϕΘ3 + 2RϕΘ 3 Lϕ e−t(∆+m ) + O(t).

3.3

Planar and non-planar contributions

We split the previous expansion in two parts. In the first one, we only keep terms like Lf e−t∆ and Rf e−t∆ . Those belong to the “planar part”, since they give commutative-like contributions as easily seen from equation (3.15) below. The second contribution, corresponding to the “non-planar part”, consists of crossed terms like Lf Rh e−t∆ . The planar contribution to the effective action is λ

2 1 ∞ Tr LϕΘ ϕ + RϕΘ ϕ e−t∆ Γ1l,P [ϕ] := dt e−tm 2 3!

t λ2 + O(0 ). − Tr LϕΘ4 + RϕΘ 4 e−t∆ 2 2 (3!)

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To compute those traces, let us show that first the trace is a dequantizer for the deformed product

(3.15) Tr Lf e−t∆ = Tr Rf e−t∆ = Tr Mf e−t∆ , whenever Mf e−t∆ is trace-class. Here Mf still denotes the operator of pointwise multiplication by f . We only treat the Lf case, since for the Rf case the arguments are similar. From the definition 2.1 and the product rule (3.4) for kernel operators, a little calculation gives the following expression for the Schwartz kernel of Lf e−t∆ : −l dl y dl z e−i f (− 21 Θy.p) Kt (z.p, p ). KLf e−t∆ (p, p ) = (2π) R2l

Then

Tr Lf e−t∆ =

µg (p) KLf e−t∆ (p, p) = (2π)−l µg (p) dl y dl z e−i f (− 21 Θy.p) Kt (z.p, p). M

M

R2l

Using next the invariance of the volume form under the isometry p → 12 Θy.p and the fact that [e−t∆ , Vz ] = 0, translated in terms of invariance of its kernel Kt (z.p, z.p ) = Kt (p, p ),

(3.16)

the claim follows after a plane waves integration:

−t∆ −l = (2π) Tr Lf e µg (p) dl y dl z e−i f (p) Kt (z.p, p) M R2l = µg (p) f (p) dl z δ(z) Kt (z.p, p) l R M

= µg (p) f (p) Kt (p, p) = Tr Mf e−t∆ . M

Hence, the planar part of the one loop effective action reads: ∞ λ t λ2 2 −t∆ Θ 4 e Tr MϕΘ ϕ e−t∆ − Γ1l,P [ϕ] = dt e−tm Tr M + O(0 ). ϕ 3! 2 (3!)2 Using the on-diagonal heat kernel expansion up to order one

t Kt (x, x) = (4πt)−2 1 − R(x) + O(t0 ), 6 where R is the scalar curvature, together with the relation KMf e−t∆ (x, x) = f (x)Kt (x, x),

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one obtains at 0 order: ∞ 2 λ 1 λ dt e−tm 1 λ2 Θ 4 . Γ1l,P [ϕ] = ϕΘ ϕ − t (ϕΘ ϕ)R + µg ϕ 2 (4πt) 3! 6 3! 2 (3!)2 M (3.17) The planar part thus yields ordinary 1 and | ln | divergences. They can be substracted adding local counter-terms to the original action. The contribution for the non-planar part is λ 2 1 ∞ Γ1l,N P [ϕ] := Tr Rϕ Lϕ e−t∆ dt e−tm 2 3! 2

t λ − + O(0 ). Tr 3RϕΘ ϕ LϕΘ ϕ + 2Rϕ LϕΘ3 + 2RϕΘ 3 Lϕ e−t∆ 2 2 (3!) We now simplify this expression. By the definition of the twisted product (2.1) g and using the identity ψ(z.p) = M µg (p ) δz.p (p ) ψ(p ), one can easily derive the Schwartz kernel of the left and right twisted multiplication operators: −l g KLf (p, p ) = (2π) dl y dl z e−i f (− 21 Θy.p) δz.p (p ), R2l

and

−l

KRf (p, p ) = (2π)

dl y dl z e−i f (z.p) δ g

(p 1 − 2 Θy.p

R2l

).

By the kernel composition rule (3.4), we obtain after few changes of variables and a plane waves integration, the kernel of Lf Rh e−t∆ in term of the heat kernel Kt : KLf Rh e−t∆ (p, p ) = dl y dl z e−i f ((− 12 Θy − z).p) h(z.p) Kt(− 12 Θy.p, p ). (2π)−l R2l

Hence, the trace of Lf Rh e−t∆ reads (with a few changes of variable):

Tr Lf Rh e−t∆ = (2π)−l µg (p) dl y dl z e−i f (p) h(z.p) Kt (−Θy.p, p) 2l M R

(3.18) = Tr Rf Lh e−t∆ . To obtain the last equality, we used the fact that Kt is symmetric, its invariance under α and the isometry p → −z.p. Invoking formula (3.18), we obtain for Γ1l,N P [ϕ]: Γ1l,N P [ϕ] = (2π)−l −

1 2

∞

dt e−tm

2

µg (p) M

R2l

dl y dl z e−i

λ 3!

ϕ(p)ϕ(z.p)

t λ2 3ϕΘ ϕ(p)ϕΘ ϕ(z.p) + 4ϕ(p)ϕΘ 3 (z.p) Kt (−Θy.p, p) + O(0 ) . 2 2 (3!)

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We shall see that the better -behavior of the non-planar part and the UR/IV entanglement phenomenon come from the presence of the off-diagonal heat kernel in the previous expression. Depending on the precise geometric setup, the non-planar contributions could still be divergent. In the unfavorable cases, the divergences are non-local as shown is the next subsections. This makes the renormalization problematic.

4 Non-periodic deformations 4.1

NCQFT on the Moyal plane in configuration space

When M = R4 with the flat metric, l = 4 and R4 acting on itself by translation, isospectral deformation gives R4Θ . In this case, the heat kernel is exactly given by Kt (x, y) = (4πt)−2 e−

|x−y|2 4t

,

so we can explicitly compute Γ1l,P (ϕ) and Γ1l,N P (ϕ). For the planar part, we obtain from (3.17) ∞ 2 λ e−tm t λ2 4 2 2 ϕ dt d x (x) − (ϕ ϕ) (x) + O(0 ), Γ1l,P [ϕ] = Θ (4πt)2 R4 3! 2 (3!)2 that will give the ordinary −1 and | ln | divergences for the respectively planar two- and four-point functions. The non-planar part is given by: Γ1l,N P [ϕ] = (2π)−4

2

∞

dt

e−tm (4πt)2

d4 x d4 y d4 z e−i e−

|Θy|2 4t

R12

1 λ λ2 t

Θ 3 × ϕ(x)ϕ ϕ(x+z)+4ϕ(x)ϕ (x+z) +O(0 ). ϕ(x)ϕ(x+z)− 3ϕ Θ Θ 2 3! (3!)2 4 The Gaussian y-integration can be performed to obtain: ∞ −1 2 −4 −tm2 Γ1l,N P [ϕ] = (2πθ) dt e d4 x d4 z e−t|Θ (z−x)| R8

1 λ λ2 t

ϕ(x)ϕ(z) − × 3ϕΘ ϕ(x)ϕΘ ϕ(z) + 4ϕ(x)ϕΘ 3 (z) + O(0 ), 2 2 3! (3!) 4 where θ := (det Θ)1/4 . Finally, the t-integration gives

2

−1

2

e−(m +|Θ (z−x)| ) m2 + |Θ−1 (z − x)|2 R8 λ λ2 3ϕΘ ϕ(x)ϕΘ ϕ(z) + 4ϕ(x)ϕΘ 3 (z) ϕ(x)ϕ(z) − + O(0 ). × 2.3! (3!)2 4(m2 + |Θ−1 (z − x)|2 )

Γ1l,N P [ϕ] = (2πθ)−4

d4 x d4 z

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This expression is regular when goes to zero – we are now in the full noncommutative picture. From the previous formula one reads off the associated (non-planar) two- and four-point functions in configuration space in the limit → 0: 1 λ , 96 m2 + |Θ−1 (x − y)|2 −1 λ2 3 e2i δ(x−y+z−u) d4 v 2 G41l,N P (x, y, z, u) = −(πθ)−8 24 2 (m + |Θ−1 (z − v − x)|2 )2 −1 e2i<x−y,Θ (z−y)> . + (m2 + |Θ−1 (x − y + z − u)|2 )2 G21l,N P (x, y) = (πθ)−4

We see that the UV/IR mixing in configuration space manifests itself in the long-range behavior of the correlation functions. The slow decreasing at infinity of the two- and four-point functions is equivalent to a IR singularity in momentum space, as shown by a Fourier transform: 2 1l,N P (ξ, η) ∝ m K1 (m|Θξ|)δ(ξ + η). G θ|ξ| Here Kn (z) denotes the n-th modified Bessel function. We retrieve the known UV/IR mixing (se for example [29]): m K1 (m|Θξ|) ∼ (θ|ξ|)−2 , |ξ| → 0. θ|ξ| This last result at one loop in the Moyal (translation-invariant) context is usually obtained by means of Feynman diagrams in momentum space – see for example [29]. We just checked that the Fourier transform for the two-point function coincides with the standard calculation’s result. However, this is not the end of the story. The behavior of the amplitudes as θ ↓ 0 presents interesting differences in configuration and momentum spaces. Assume that Θ has been put in the canonical form   θ  −θ  Θ=  ,  θ −θ and choose θ = θ for simplicity. In effect, developing the two-point expression in terms of θ, we find 1 θ 4 m4 θ 2 m2 1 = + − · · · . 1 − θ4 m2 + θ2 |x|2 θ2 |x|2 |x|2 |x|4 First of all, we remark that the logarithmic dependence on θ of the UV/IR mixing in momentum space (in addition to its quadratic divergence) found in [29] is apparently absent here. Now, with the sole exception of the first term, the previous

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series is made of functions that are not tempered distributions, and so they have no Fourier transform. In other words, the passage to the “commutative limit” does not commute with taking Fourier transforms. The question is subtler, though. We can ask ourselves to which kind of divergences the terms of the last development are associated to. The answer is that first term is infrared divergent in configuration space; the second one is both ultraviolet and infrared divergent, and the following are all ultraviolet divergent. It is perhaps surprising that there is a way to recover the exact result from that nearly nonsensical infinite series; this involves precisely the correction to the indicated UV divergences. Indeed we can “renormalize” (in the sense of Epstein and Glaser) the 1/|x|2k+4 functions, with the result that the redefined distributions [1/|x|2k+4 ]R are tempered. Those [.]R distributions depend on a mass scale parameter. Their 2k+4 ] (making a long history short) have been calculated Fourier transforms [1/|x| R as well [18, 30], with the result (−)k+1 |ξ|2k |ξ| 2k+4 − Ψ(k + 1) − Ψ(k + 2) , ]R (ξ) = k+1 [1/|x| 2 ln 4 k!(k + 1)! 2µ Now, a natural mass scale parameter in our context is 1/θm. This is where ln θ can sneak back in. Upon substituting this for µ in the previous formula, and summing the series of Fourier transforms, we recover on the nose the exact result: ∞ 1 m m2 θ2n m2n |ξ|2n θm|ξ| − Ψ(n + 1) − Ψ(n + 2) K1 (θm|ξ|). = + ln θ2 |ξ|2 2 n=0 4n n!(n + 1)! 2 θ|ξ| For the four-point function, again in the θ ↓ 0 limit no dependence on ln(θ) is apparent in configuration space. The resulting expression is however (UV- and) IR-divergent, and its redefinition ` a la Epstein and Glaser allows one to reintroduce the ln θ. The effect of the rank of Θ becomes clearer in position space. Indeed, for a generic n-dimensional Moyal plane with a deformation matrix of rank l ≤ n, the two-point function in momentum space is always finite and behaves as |Θξ|−n+2 , when ξ → 0. However, since Θξ ∈ Im(Θ) = Rl , the IR singularity is not locally integrable if l ≤ n − 2. It follows that the two-point Green function does not have a Fourier transform since it is not a temperate distribution. Thus in the fourdimensional case, the non-planar contribution to the tadpole in position space remains infinite if l = 2! The four-point function has a Fourier transform, its IR singularity in momentum space being of the ln type, and the Green function in position space is finite whenever l = 0. For example, had we treated R2θ ×R2 instead of R4Θ , we would have found that the four-point part of Γ1l,N P [ϕ] is convergent, while the two-point part diverges as ln . This point is discussed in details in [14], where we use the ζ-regularization scheme and the Duhamel asymptotic expansion (instead of the BCH one), in order to compare our results with those present in the literature.

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These features of the UV/IR mixing phenomenon on position space reappear in the general non-periodic case, where the effective action will still be divergent for l = 2. This is shown in the next subsection.

4.2

The divergences of the general non-periodic case

Assume ϕ ∈ Cc∞ (M ). We have also to make some more precise assumptions on the behavior of the geometry at infinity in order to control the heat kernel. In [2, 9], it is proved that if M is non-compact, complete, with Ricci curvature bounded from below (plus either uniform boundness of the inverse of the volume or of the inverse of the isoperimetric constant of the Riemannian ball for some fixed radius), then the heat kernel satisfies 2

(4πt)−2 e−dg (p,p )/4t ≤ Kt (p, p ) 2

≤ C(4πt)−2 e−dg (p,p )/4(1+c)t ,

(4.1)

where dg is the Riemannian distance and C, c are strictly positive constants. In the general periodic case, we have shown that Γ1l,N P [ϕ] is given by: Γ1l,N P [ϕ] ×

λ 3!

1 = 2(2π)l

ϕ(p)ϕ(z.p) −

∞

dt e

−tm2

µg (p) M

R2l

dl y dl z e−i Kt (−Θy.p, p)

t λ2 Θ 3 3ϕ ϕ(p)ϕ ϕ(z.p) + 4ϕ(p)ϕ (z.p) + O(0 ). Θ Θ 2 (3!)2

We now show that this expression cannot produce more important divergences than the planar contribution. Again, the regularity of those integrals depends only on l (that we may call the effective noncommutative dimension), and on the metric through the Riemannian distance function. Before estimating the two-point part of Γ1l,N P [ϕ], which is our main purpose in this section, we make the following remark: in our present setting, the two-point non-planar Green function reads ∞ 2 λ l l −i g G1l,N P,2P (p, p ) = d y d z e dt e−tm Kt (−Θy.p, p) δz.p (p ). 6(2π)l R2l Now, one can qualitatively see in this distributional expression the UV/IR entanglement phenomenon: thanks to the estimate (4.1), we have 0

∞

2

2

∞

e−tm −d2g (Θy.p,p)/4(1+c)t e (4πt)2 0 √

m dg (Θy.p, p) C 4m 1 + c √ K1 = 2 16π dg (Θy.p, p) 1+c −2 ∼ C dg (Θy.p, p), y → 0,

dt e−tm Kt (Θy.p, p) ≤ C

dt

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and the reverse inequality also holds ∞ 2 dt e−tm Kt (Θy.p, p) ≥ C d−2 g (Θy.p, p) 0

l has to be interpreted as which points precisely to the UV/IR mixing, since y ∈ R a momentum. For the two-point part of Γ1l,N P [ϕ] we have l Γ1l,N P,2P [ϕ] ≤ C λ sup d z |ϕ(z.p)| 12(2π)l p∈M Rl ∞ 2 2 e−tm × dt µ (p) |ϕ(p)| dl y e−dg (−Θy.p,p)/4(1+c)t g 2 (4πt) M Rl Cλ ≤ sup dl z |ϕ(z.p)| ϕ 1 12(2π)l p∈M Rl 2 ∞ 2 e−tm × sup dt dl y e−dg (−Θy.p,p)/4(1+c)t . 2 (4πt) Rl p∈supp(ϕ) By the properness of α, Rl dl z |ϕ(z.p)| is finite for all p ∈ M since {z ∈ Rl : z.p ∈ supp(ϕ)} is compact for each p ∈ M because ϕ has compact support. Thus, ϕ(p) ˜ := Rl dl z |ϕ(z.p)| is constant and finite on each orbit of α, and if we denote π : M → M/Rl the projection on the orbit space, then ϕ˜ factors through π to give a map ϕ¯ defined by ϕ(π(p)) ¯ := ϕ(p). ˜ Finally, ϕ¯ ∈ Cc∞ (M/Rl ) l because if p ∈ / R . supp(ϕ), so that π(p) is! not in the compact set π(supp(ϕ)), " then ϕ(π(p)) ¯ = 0. This proves that supp∈M Rl dl z |ϕ(z.p)| < ∞. Furthermore, since α acts isometrically the induced metric g˜ on the orbits (which are closed submanifolds since the action is proper [25]) is constant, so d2g (y.p, p) =

l

g˜ij (p)y i y j .

i,j=1

Here, g˜ij (p) (which depend only on the the orbit of p) are strictly positive continuous functions since in the non-periodic case the action is free, and then {(0, p) ∈ Rl × M } is the only set for which F (y, p) := dg (y.p, p) vanish. Note that we can use a global coordinate system (on one orbit) given by a suitable basis of Rl in such a way that g˜ij (p) is diagonal. Thus, with θ := (det Θ)1/l , we have: l/2 4π(1 + c)t l −d2g (−Θy.p,p)/4(1+c)t d ye = (det g˜(p))−1/2 . θ2 Rl Hence, one obtains Γ1l,N P,2P [ϕ] ≤ λ C(l, g˜, ϕ, ϕ) θ−l 6

∞

2

dt tl/2−2 e−tm ,

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where C(l, g˜, ϕ1 , ϕ2 ) := C(4π)l/2−2 (1 + c)l/2

ϕ1 1 sup 2(2π)l p∈M

d z |ϕ2 (z.p)| l

Rl

sup

(det g˜(p))−1/2 .

p∈supp(ϕ1 )

Four the four-point part, similar estimates read: Γ

≤

1l,N P,4P [ϕ] λ2 −l

72

θ

3C(l, g˜, ϕΘ ϕ, ϕΘ ϕ) + 4C(l, g˜, ϕ, ϕΘ ϕΘ ϕ)

∞

2

dt tl/2−1 e−tm .

We then have proved the following: Theorem 4.1. When M is non-compact, satisfying all assumptions on the behavior of the geometry at infinity displayed above and endowed with a smooth proper isometric action of Rl , then for ϕ ∈ Cc∞ (M ) we have: for l = 4, Γ1l,N P,2P [ϕ] ≤ C1 (ϕ, Θ) i) C2 (ϕ, Θ)| ln | for l = 2, ii)

Γ1l,N P,4P [ϕ] ≤ C3 (ϕ, Θ) for l = 4 or l = 2.

The possible remaining divergence for l = 2 refers to the fact that the IR singularity might be not integrable, as illustrated previously. In this case, the two-point non-planar Green function does not define a distribution and the theory is not renormalizable by addition of local counter-terms, already in its one-loop approximation order.

5 Periodic deformations Periodic deformations (when the kernel of α is an integer lattice) behave rather differently from non-periodic ones. In the following, we consider ker α = βZl with β a l × l integer matrix of rank l, so that Rl /βZl =: Tlβ is compact. For the sake of simplicity, we will often suppress the subscript β. Momentum space (the dual group of Tlβ ) being discrete, IR problems only occur for some values of the momentum. In favorable cases one can extract the divergent field configurations in the non-planar part (which are often finite in number when (2π)−1 Θ has irrational entries) and renormalize them like the planar contributions; then there is no really UV/IR mixing. When (2π)−1 Θ has rational entries, the theory is equivalent to the undeformed one, in the sense that there are infinitely many divergent field configurations.

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Although in all periodic cases we have a Peter-Weyl decomposition for fields, only in the compact manifold case shall we be able to describe the individual behavior of non-planar “Feynman graphs”, defined through that isotypic decomposition. Both in the compact and in the non-compact case, by means of the off-diagonal heat kernel estimate (4.1), we show in the second subsection how, for periodic deformations, the arithmetical nature of the entries of Θ, more precisely, the existence or nonexistence of a Diophantine condition on Θ, plays a role in determining the analytical nature of Γ1l,N P [ϕ].

5.1

Periodic compact case and the individual behavior of non-planar graphs

Because everything is explicit, we look first at the flat compact case. Let M = T4 with the flat metric, let R4 act on it by rotation (so and l = 4 and we are in i the ‘fully noncommutative picture’). With the orthonormal basis e (2π)2 of L2 (T4 , d4 x) the heat kernel is written 2 Kt (x, y) = (2π)−4 e−t|k| ei ,

k∈Z4

k∈Z4

and we have

i

ei Θ ei = e− 2 Θ(k,q) ei ,

with Θ(k, q) := k, Θq. Expanding the background field ϕ in Fourier modes ϕ = i , with {ck }k∈Z4 ∈ S(Z4 ) whenever ϕ ∈ C ∞ (T4 ), we obtain: k∈Z4 ck e 1 1 e−(m +|k| ) λ λ2 iΘ(k,r) c c e − r −r 2 2 2 2 2 m + |k| 3! r 2(3!) m + |k|2 k i + O(0 ). × cr cs cu−s c−r−u e− 2 Θ(r+s,u) 3 eiΘ(k,r+s) + 4 eiΘ(k,r+u) 2

2

ΓN P [ϕ] =

r,s,u

We can now analyze the individual behavior of non-planar Feynman diagrams. One sees that, thanks to the phase factors, the sum over k is finite when goes to zero, whenever (2π)−1 Θ has irrational entries and r = 0 for the two-point part, or r +s = 0 and r +u = 0 for the four-point part. In effect, returning to the Schwinger parametrization (which exchanges large momentum divergences with small-t ones) and applying the Poisson summation formula with respect to the sum over k we get: ∞ e−tm2 eiΘ(k,r) 2 = dt e−|2πk−Θr| /4t . 2 2 2 m + |k| (4πt) 0 4 4 k∈Z

k∈Z

Hence, the t-integral is finite whenever r = 0 and conclusion holds for the four-point part.

Θr 2π

∈ / Ql . Essentially the same

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We now go to the general periodic compact case. In order to be able to calculate, we make explicit use of the invariance of the heat kernel under α. Let us decompose H = L2 (M, µg ) in spectral subspaces with respect to the group action: # H= Hk . k∈Zl

Each Hk is stable under Vz (recall that Vz denotes the induced action on H) for all z ∈ Rl ; and furthermore all ψ ∈ Hk satisfy Vz ψ = e−i ψ. Note that if ψ ∈ Hk then |ψ| ∈ H0 . Let Pk be the orthogonal projection on Hk . Because the Laplacian −t∆ commutes with Vz , the heat operator also commutes with is block $ Pk ; hence e diagonalizable with respect to the decomposition H = k∈Zl Hk : Pk e−t∆ Pk . e−t∆ = k∈Zl

In each Hk the operator 0 ≤ Pk e−t∆ Pk is trace-class, so it can be written as Pk e−t∆ Pk = e−tλk,n |ψk,n ψk,n |, n∈N

where {ψk,n }n∈N is an orthonormal basis of Hk consisting of eigenvectors of Pk ∆Pk with eigenvalue λk,n . The heat semigroup being Hilbert-Schmidt, its kernel can be written as a (L2 (M × M, µg × µg )-convergent) sum: Kt (p, p ) = e−tλk,n ψk,n (p)ψk,n (p ). (5.1) k∈Zl n∈N

Because each ψk,n (p) lies in Hk , the invariance property (3.16) Kt (z.p, z.p) = Kt (p, p ) is explicit. Any ϕ ∈ C ∞ (M ) has a Fourier decomposition ϕ = r∈Zl ϕr , such that { ϕr ∞ } ∈ S(Zl ) and αz (ϕr ) = e−i ϕr . Furthermore, this decomposition provides a notion of Feynman diagrams, that is of amplitude associated with a fixed field configuration. The non-planar one-loop regularized effective action reads: Γ1l,N P [ϕ] = r,s∈Zl

1 2

µg (p) M

ϕr (p) ϕs (p) e

λ e−(m2 +λk,n ) 2 |ψ | (p) k,n m2 + λk,n 3! l

k∈Z n∈N λ2 −iΘ(k,s)

−

2(3!)2

m2

1 + λk,n

ϕr (p) ϕs (p) ϕu (p) ϕv (p)

r,s,u,v∈Zl

i i + O(0 ). × 3 e− 2 (Θ(r,s)+Θ(u,v)) e−iΘ(k,u+v) + 4 e− 2 Θ(r+s,u+s) e−iΘ(k,v)

Although we do not know the explicit form of the ψk,n , we can by momentum conservation reduce the sums exactly as in the NC torus case, as shown in the following lemma.

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Lemma 5.1 (Momentum conservation). Let ψi ∈ Hki ∩ Lq (M, µg ) for i = 1, . . . , q. Then: µg ψ1 . . . ψq = C(ψ1 , . . . , ψq ) δk1 +···+kq ,0 . M

Proof. By the α-invariance of µg and with the relation αz (ψi ) = e−i ψi we have µg ψ1 . . . ψq = ei µg ψ1 . . . ψq , M

M

for all z ∈ Rl ; the result follows. Because |ψk,n |2 (p) is constant on the orbits of α and ϕr ∈ C ∞ (M ) ⊂ L (M, µg ) for all q ≥ 1, Lemma 5.1 gives q

Γ1l,N P [ϕ]

1 = 2

λ e−(m2 +λk,n ) 2 µg (p) |ψ | (p) k,n m2 + λk,n 3! M l l k∈Z n∈N 2

r∈Z

1 λ ϕr (p) ϕ−r (p) ei − ϕr (p) ϕs (p) ϕu−s (p) ϕ−r−u (p) 2 2 2(3!) m + λk,n r,s,u∈Zl i + O(0 ). (5.2) × e− 2 Θ(r+s,u) 3 eiΘ(k,r+s) + 4 eiΘ(k,r+u) To analyze the divergences when → 0 for a fixed field configuration, note that if we re-index λk,n in a standard way (λ0 ≤ · · · ≤ λn ≤ · · · ), Weyl’s estimate asserts that λn ∼ n1/2 , hence k∈Zl n∈N

|ψn (p)|2 |ψk,n |2 (p) = = K(m2 +∆)−N (p, p), (m2 + λk,n )N (m2 + λn )N n∈N

is finite if and only if N > 2. We see that the sum over n and k in (5.2) diverges in the limit → 0 for certain values of the momenta (r = 0 for the two-point part, r + s = 0 and r + u = 0 for the four-point part) if (2π)−1 Θ has irrational entries. When the entries (2π)−1 Θ are rational, there are infinitely many divergent field l configurations since e−i = 1 for infinitely many k whenever Θr 2π ∈ Q . For other configurations, convergence is guaranteed by the estimate (4.1), as shown in the next subsection. In summary, we have shown that the behavior of an individual field configuration in the non-planar sector for any periodic compact deformation reproduces the main features of the noncommutative torus. In the next paragraph, the arithmetic nature of the entries of Θ gets into the act; also we show there that the possible existence of fixed points for the action may give rise to additional divergences.

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1017

General periodic case and the Diophantine condition

Assume now that α periodic, but M can be compact or not (within the hypothesis of section 4.2 when M is not compact). In this general setup, the Peter-Weyl decomposition still exists, but the heat operator, not being a priori compact, cannot be written as (5.1). Thus we return to the off-diagonal heat kernel estimate. In this case, using Lemma 5.1 and the α-invariance of Kt , we obtain: λ 1 ∞ −tm2 Γ1l,N P [ϕ] = dt e µg (p) Kt (Θr.p, p)ϕr (p) ϕ−r (p) 2 3! M l r∈Z

t λ2 i − ϕr (p) ϕs (p) ϕu−s (p) ϕ−r−u (p) e− 2 Θ(r+s,u) 2(3!)2 r,s,u∈Zl × 3 Kt (Θ(r + s).p, p) + 4 Kt (Θ(r + u).p, p) + O(0 ). We consider only the case (2π)−1 Θ has irrational entries, from now on. Then divergences appear when r = 0 for the two-point function and r + s = 0, r + u = 0 for the four-point functions. This leads us to introduce a reduced non-planar oneloop effective action Γ,red 1l,N P [ϕ] by subtracting the divergent field configurations; for renormalization purposes, they have to be treated together with the planar sector. λ 1 ∞ −tm2 Γ,red [ϕ] := dt e µ (p) Kt (Θr.p, p)ϕr (p) ϕ−r (p) g 1l,N P 2 3! M t λ2 i − ϕr (p) ϕs (p) ϕu−s (p) ϕ−r−u (p)e− 2 Θ(r+s,u) 2(3!)2 × 3 Kt (Θ(r + s).p, p) + 4 Kt (Θ(r + u).p, p) .

is the notation for r∈Zl , r=0 in the two-point part, r,s,u∈Zl , r+s=0 and r,s,u∈Zl , r+u=0 in respectively the first and second piece of the four-point part. Using now the estimate (4.1) and performing the t-integration, we obtain: √ λ C 4m 1 + c ≤ lim Γ,red [ϕ] µ (p) |ϕ (p)| |ϕ (p)| g r −r 1l,N P →0 32π 2 M 3! dg (Θr.p, p) 2

m dg (Θr.p, p) λ √ K1 |ϕr (p)| |ϕs (p)| |ϕu−s (p)| |ϕ−r−u (p)| + 2(3!)2 1+c

m dg (Θ(r + s).p, p)

m dg (Θ(r + u).p, p) √ √ + 4K0 . (5.3) × 3K0 1+c 1+c Here

Definition 5.2. θ ∈ Rl \ Ql satisfies a Diophantine condition if there exists C > 0, β ≥ 0 such that for all n ∈ Zl\{0} :

nθ Tl := inf |nθ + k| ≥ C|n|−(l+β) . k∈Zl

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Diophantine conditions constitute a way to characterize and classify irrational numbers which are “far from the rationals” in the sense of being badly approximated by rationals. The set of numbers satisfying a Diophantine condition is ‘big’ (of full Lebesgue measure) in the sense of measure theory, but ‘small’ (of first category) in the sense of category theory [27]. Again because the metric is constant on the orbits we have:   l g˜ij (p)(y i + k i )(y j + k j ) . d2g (y.p, p) = inf  k∈Zl

i,j=1

Recall also that the modified Bessel functions have the following behavior near the origin K1 (x) =

1 + O(x0 ), x

K0 (x) = −γ + ln(2) − ln(x) + O(x),

where γ is the Euler constant. Thus, in view of { ϕr ∞ } ∈ S(Zl ), and provided the integral over the manifold with the measure µg can be carried out, in (5.3) we have l convergence if and only if d−2 g (Θr.p, p) ∈ S (Z ), that is, if and only if the entries of Θ satisfy a Diophantine condition. This result seems to be new, although the pertinence of Diophantine conditions in NCQFT had been conjectured by Connes long ago. Recently, these conditions have been found to play a role in Melvin models with irrational twist parameter in conformal field theory [22]. We said above: “provided the integral over the manifold with the measure l µg can be carried out”. This because d−2 g (αy (.), .) for a non-zero y ∈ T might not be locally integrable with respect to the measure given by the Riemannian volume form. Problems may appear on a neighborhood of the set of points with non-trivial isotropy groups. In fact, by simple dimensional analysis, we expect serious trouble when the isotropy group is one dimensional. For p ∈ M let Hp its isotropy group and let Msing := {p ∈ M : Hp = {0}}. Recall that Msing is closed and of zero-measure in M since the action is proper (see [25]), and note that for a non-zero y ∈ Tl , dg (y.p, p) = 0 if and only if p ∈ Msing and y ∈ Hp . On Mreg := M \ Msing (the set of principal orbit type), since the action is free, one can define normal coordinates on a tubular neighborhood of an orbit Tl .p. Let (ˆ xµ , x ˜i ), µ = 1, . . . , n − l, i = 1, . . . , l be respectively the transverse and the torus coordinates of a point p ∈ Mreg . Because the action is isometric, in this coordinate system the metric takes the form h(ˆ x) l(ˆ x) g(ˆ x, x ˜) = , l(ˆ x) g˜(ˆ x) where g˜ is the induced (constant) metric on the orbit. Such coordinate system is xµ , x˜i ) singular with singularities located at each point of Msing , and when x ≡ (ˆ approach p0 ∈ Msing , g˜(ˆ x) collapses to a l − dim(Hp0 ) rank matrix. Since in this

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coordinate system µg (p) d−2 g (y.p, p) equals l

% det g(ˆ x)

˜ij (ˆ x)y i y j i,j=1 g

dl x ˜ dn−l x ˆ,

when dim(Hp0 ) = 1 the singularity of d−2 g (y.p, p) for p → p0 cannot be can√ celled by det g. This is a new feature of the UV/IR mixing for generic periodic isospectral deformations which needs to be investigated in detail in each model; it occurs, for instance, for the Connes-Landi spheres and their ambient spaces. Let us summarize: Theorem 5.3. For M compact or not (within the assumptions displayed in section 4.2 in the non-compact case), endowed with a smooth isometric action of the compact group Tl , l = 2 or l = 4 and with a deformation matrix whose entries satisfy a Diophantine condition, then for any external field ϕ ∈ Cc∞ (M, ) vanishing in a neighborhood Msing the one-loop non-planar reduced effective action is finite. In other words, if the Diophantine condition is not satisfied or if d−2 g (αy (.), .) ∈ / then the reduced non-planar two-point function does not define a distribution and the theory is not renormalizable, already at one-loop, by addition of local counter-terms. L1loc (M, µg )

6 Summary and perspectives We have shown the existence of the UV/IR mixing for isospectral deformations of curved spaces. For periodic deformations the entanglement only concerns (at the level of the two-point function) the 0-th component of the field in the spectral subspace decomposition induced by the torus action. In this case, the UV/IR mixing does not generate much trouble since one can treat it for renormalization purposes together with the planar sector. In the non-periodic situation, we obtain non-planar Green functions which present the mixing in a similar form to the Moyal plane paradigms. Our approach gives an algebraic way to understand the presence of the nonplanar sector for those theories: it comes from the product of left and right regular representation operators. As a byproduct of our trace computations, we obtain that the better behavior of the non-planar sector is due to the presence of the off-diagonal heat kernel in the integrals. However, its regularizing character depends highly on the geometric data. For non-periodic deformations, the conclusion is that when the noncommutative rank is equal to two, the non-planar 1PI two-point Green function does not define a distribution and the associated effective action remains divergent [14]. Only the group action of rank four gives rise to a UV divergent-free non-planar sector in the 4-dimensional manifold case. When the action is periodic, we have shown

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that it is necessary that the entries of (2π)−1 Θ satisfy a Diophantine condition to ensure finiteness of the reduced non-planar effective action, i.e., in order that the reduced non-planar 1PI two-point Green function define a distribution. Additional divergences may exist due to the possible fixed points structure of the action α. Our treatment of the generic UV/IR behavior, can be generalized to higher dimensional isospectral deformations and/or to gauge theories. Also, we have restricted ourselves to the 4-dimensional case, for the sake of simplicity and physical interest, but it is clear that the heat kernel techniques employed here apply to higher dimensional scalar theories. For gauge theory on (any dimensional) isospectral deformations manifolds, there is an intrinsic way to define noncommutative actions of the Yang-Mills type. For any ω ∈ Ωp (M ), η ∈ Ωq (M ) (say compactly supported and smooth with respect to α) one can set ω ∧Θ η := (2π)−l dl y dl z e−i (α∗ 1 ω) ∧ (α∗z η), − 2 Θy

R2l

where α∗z is the pull-back of αz on forms. Given now an associated vector bundle π : E → M with compact structure group G ⊂ U (N ), and a connection A ∈ Ω1 (M, Lie(G)) we define the NC analogue of the YM action tr(FΘ ∧Θ ∗H FΘ ), SY M (A) := M

where FΘ := dA + A ∧Θ A. In this context, one can prove a trace property, namely: ω ∧Θ ∗H η = ω ∧ ∗H η, ∀ω, η ∈ Ωp (M ). M

M

Hence SY M (A) equals M tr(FΘ ∧ ∗H FΘ ). To manage the quantization, one can once again use the background field method in the background gauge, and if we ignore the Gribov ambiguity, the one-loop effective action reduce to the computation of determinants of operators (quadratic part in A of SY M +Sgf and Faddeev-Popov determinant) which can be locally expressed as (∇µ + LAµ − RAµ )(∇µ + LAµ − RAµ ) + B, where B is bounded and contains left, right and a product of left and right twisted multiplication operators. It is then clear that UV/IR mixing will appears in the same form as in the flat situations (see [21, 23, 24]). A further interesting task is be to look at what happens for a Grosse-Wulkenhaar like model for the non-compact case. In [20] it is proved that if we add a confining potential (harmonic oscillator in their work) in the usual λϕΘ 4 theory on the four dimensional Moyal plane, i.e., the Grosse-Wulkenhaar action Ω2 SGW [ϕ] := d4 x 12 (∂µ ϕθ ∂ µ ϕ)(x) + 2 2 (xµ ϕ)θ (xµ ϕ) θ m2 λ ϕθ ϕ(x) + ϕθ ϕθ ϕθ ϕ(x) , + 2 4!

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then the theory is perturbatively renormalizable to all orders in λ. The deep meaning of this result is not yet fully understood, but some explanations can be mentioned. First, to add a confining potential is in some sense equivalent to a compactification of the Moyal plane and in the second hand, the particular choice of the potential corresponds to a Moyal-deformation of both the configuration and the momentum space. This can be seen by the invariance (up to a rescaling) of this ↔ (πθ)2 ϕ(x). This point needs to be clariaction under pµ ↔ 2(θ−1 )µν xν , ϕ(p) fied. It would be good to know whether their renormalizability conclusion (UV/IR decoupling) holds in the general context when one adds a coupling with a confining potential in the scalar theory. Last, but not least, it remains to see whether the UV/IR entanglement concerns only θ-deformations or not. Connes-Dubois-Violette [7] 3-spheres and 4planes, whose defining algebras are related to Sklyanin algebras, are good candidates to test this point.

Acknowledgments I am very grateful to J. M. Gracia-Bond´ıa, J. C. V´ arilly and my advisor B. Iochum for their help. I also would like to thank M. Grasseau, T. Krajewski, F. Ruiz Ruiz and R. Zentner for fruitful discussions and/or suggestions. Special thanks are also due to the Departamento de F´ısica Teórica I of the Universidad Complutense de Madrid for its hospitality during the final stages of this work. I finally would like to thank the Referee for his enlightened remarks.

References [1] G. Braunss, On the regular Hilbert space representation of a Moyal quantization, J. Math. Phys. 35, 2045–2056 (1994). [2] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, London and San Diego, 1984. [3] I. Chepelev and R. Roiban, Renormalization of Quantum Fields Theories on Noncommutative Rd . I. Scalar, J. High Energy Phys. 5, 137–168 (2000). [4] A. Connes, Noncommutative Geometry, Academic Press, London and San Diego, 1994. [5] A. Connes, Gravity coupled with matter and the foundation of noncommutative geometry, Commun. Math. Phys. 182, 155–176 (1996). [6] A. Connes and G. Landi, Noncommutative manifolds, the instanton algebra and isospectral deformations, Commun. Math. Phys. 221, 141–159 (2001). [7] A. Connes and M. Dubois-Violette, Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples, Commun. Math. Phys. 230), 539–579 (2002.

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[8] T. Coulhon, E. Russ and V. Tardivel-Nachef, Sobolev algebras on Lie groups and Riemannian manifolds, Amer. J. Math. 123, 283–342 (2001). [9] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. [10] M.R. Douglas and N.A. Nekrasov, Noncommutative Fields Theory, Rev. Modern Phys. 73, 977–1024 (2001). [11] T. Filk, Divergences in a Field Theory on Quantum Space, Phys. Lett. B 376, 53–58 (1996). [12] V. Gayral, J.M. Gracia-Bond´ıa, B. Iochum, T. Sch¨ ucker and J.C. V´ arilly, Moyal planes are spectral triples, Commun. Math. Phys. 246, 569–623 (2004). [13] V. Gayral and B. Iochum, The spectral action for Moyal planes, J. Math. Phys. 46, 043503 (2005). [14] V. Gayral, J.M. Gracia-Bond´ıa and F. Ruiz Ruiz, Trouble with space and nonconstant noncommutativity field theory, Phys. Lett. B 610, 141–146 (2005). [15] V. Gayral, J.M. Gracia-Bond´ıa and F. Ruiz Ruiz, Position-dependent noncommutative products: classical construction and field theory, hep-th/0504022. [16] V. Gayral, B. Iochum and J.C. V´ arilly, Dixmier trace on non-compact isospectral deformations, in preparation. [17] P.B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem, 2nd edition, CRC Press, Boca Raton, FL, 1995. [18] J.M. Gracia-Bond´ıa, Improved Epstein-Glaser renormalization in coordinate space I. Euclidean framework, Math. Phys. Analysis Geom. 6, 59–88 (2003). [19] J.M. Gracia-Bond´ıa, J.C. V´ arilly and H. Figueroa, Elements of Noncommutative Geometry, Birkh¨ auser Advanced Texts, Birkh¨ auser, Boston, 2001. [20] H. Grosse and R. Wulkenhaar, Renormalisation of φ4 -Theory on Noncommutative R4 to all order, hep-th 0403232. [21] T. Krajewski and R. Wulkenhaar, Perturbative Quamtum Gauge Fields on the Noncommutative Torus, Int. J. Mod. Phys. A 15, 1011–1030 (2000). [22] D. Kurasov, J. Marklof and G.W. Moore, Melvin models and Diophantine approximation, hep-th 0407150. [23] C.P. Martin and D. Sanchez-Ruiz, The One Loop UV Divergent stucture of U (1) Yang-Mills Theory on Noncommutative R4 , Phys. Rev. Lett. 83, 476–479 (1999).

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[24] S. Minwalla, M.V. Raamsdonk and N. Seiberg, Noncommutative Perturbative Dynamics, J. High Energy Phys. 2, 20–31 (2000). [25] P.W. Michor, Isometric actions of Lie groups and invariants, Notes of a lecture course at the University of Vienna, July 1997. [26] S.B. Myers and N. Steenrod, On the group of isometries of a Riemannian manifold, Ann. Math. 40, 406–416 (1939). [27] J.C. Oxtoby, Measure and Category, Springer, Berlin, 1972. [28] M.A. Rieffel, Deformation Quantization for Actions of Rd , Memoirs Amer. Math. Soc. 506, Providence, RI, 1993. [29] F. Ruiz Ruiz, UV/IR mixing and the Goldstone theorem in noncommutative field theory, Nucl. Phys. B637, 143–167 (2002). [30] O. Schnetz, Natural renormalization, J. Math. Phys. 38, 738–758 (1997). [31] A. Sitarz, Rieffel’s deformation quantization and isospectral deformations, Int. J. Theor. Phys. 40, 1693–1696 (2001). [32] R.J. Szabo, Quantum Fields Theory on Noncommutative space, Phys. Rep. 37, 207–299 (2003). [33] J.C. V´ arilly, Quantum symmetry groups of noncommutative spheres, Commun. Math. Phys. 221, 511–523 (2001). [34] J.C. V´ arilly and J.M. Gracia-Bond´ıa, On the ultraviolet behavior of quantum fields over noncommutative manifolds, Int. J. Mod. Phys. A14, 1305–1323 (1999). [35] D.V. Vassilevich, Non-commutative heat kernel, Lett. Math. Phys. 67, 185– 194 (2004). [36] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, fourth edition, Clarendon Press, Oxford, 2002. Victor Gayral CPT-CNRS UMR 6207 Luminy Case 907 F-13288 Marseille Cedex 9 France email: [email protected] Communicated by Vincent Rivasseau submitted 22/12/04, accepted 22/03/05



Non-Amenability and Spontaneous Symmetry Breaking – The Hyperbolic Spin-Chain – Max Niedermaier and Erhard Seiler

Abstract. The hyperbolic spin chain is used to elucidate the notion of spontaneous symmetry breaking for a non-amenable internal symmetry group, here SO(1, 2). The noncompact symmetry is shown to be spontaneously broken – something which would be forbidden for a compact group by the Mermin-Wagner theorem. Expectation functionals are defined through the L → ∞ limit of a chain of length L; the functional measure is found to have its weight mostly on configurations boosted by an amount increasing at least powerlike with L. This entails that despite the nonamenability a certain subclass of noninvariant functions is averaged to an SO(1, 2) invariant result. Outside this class symmetry breaking is generic. Performing an Osterwalder-Schrader reconstruction based on the infinite volume averages one finds that the reconstructed quantum theory is different from the original one. The reconstructed Hilbert space is nonseparable and contains a separable subspace of ground states of the reconstructed transfer operator on which SO(1, 2) acts in a continuous, unitary, and irreducible way.

1 Introduction Spontaneous symmetry breaking is typically discussed for compact internal or for Abelian translational symmetries, see, e.g., [1, 2, 3]. Both share the property of being amenable [4]; we recall the definition below but mention already that all semisimple nonabelian noncompact Lie groups are non-amenable. The goal of this note is to elucidate the notion of spontaneous symmetry breaking for a non-amenable internal group. This is motivated by the ubiquitous appearance of noncompact internal symmetries in a gravitational context, specifically in the dimensional reduction of gravitational theories [5], further in integrable sectors of QCD [6], or in ghost- or θ-sectors of gauge theories, and also in condensed matter physics [7, 8, 9, 10, 11]. The very fact that the group is non-amenable turns out to entail a number of surprising new features. In particular spontaneous symmetry breaking becomes possible in low dimensions where it is forbidden by the Mermin-Wagner theorem [1, 12, 13] in the case of compact internal symmetries. In order to have a concrete computational framework at hand we consider a definite lattice statistical system, the hyperbolic spin chain. This is a spin chain where the dynamical variables take values in a hyperbolic (Riemannian) space of constant negative curvature and the interaction is through nearest neighbors only. The lattice formulation was chosen in order to have control over the thermodynamic limit and in preparation to the quantum field theoretical case. Indeed we expect

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M. Niedermaier and E. Seiler


that many of the qualitative results generalize to generic statistical systems as well as to quantum field theories. In an accompanying paper [14] we study the nonlinear sigma-model with a hyperbolic target space in 2 or more dimensions. The systems treated always can be regarded in two different ways: either as a system of classical statistical mechanics, or as a quantum system in imaginary time. We mostly use the former interpretation, but discuss in some detail the reconstruction of the associated quantum system. Following [3], in the quantum interpretation we consider dynamical systems (C, τ ) consisting of a ∗-algebra C (“the observables”) and a one-parameter group of automorphisms (“the time evolution”), which we take to be discrete here τ x , x ∈ Z. In addition a group of automorphisms ρ(g), g ∈ G, (“the symmetry group”) is supposed to act on C and to commute with the time evolution, τ ◦ ρ = ρ ◦ τ . A state ω (positive linear functional over C) is said to be τ -invariant if ω ◦ τ = ω and extremal τ -invariant if it is not a convex combination of different invariant states. The symmetry ρ is said to be spontaneously broken (see, e.g., [1, 2, 3]) by an (extremal) τ -invariant state ω if ω ◦ ρ = ω. In the classical statistical mechanics interpretation C is a commutative C ∗ algebra (though there may be reasons to relax this condition) and the ‘time evolution’ really plays the role of space translations. A symmetry is again given by a group of automorphisms ρ(g), g ∈ G, acting on C and leaving the Hamiltonian (or action) invariant, except for possible symmetry violating boundary condition (a very precise definition of the notion of symmetry and its spontaneous breaking can be found in [15]). The definitions of states and their invariance or noninvariance are as in the quantum interpretation. Spontaneous symmetry breaking is then said to occur if there is an infinite volume Gibbs state (for instance obtained as a limit of finite volume Gibbs states) that is noninvariant. We shall be interested in the above situation when the symmetry group is a non-amenable Lie group. A Lie group G is called amenable if there exists an (left) invariant state (“a mean”) on the space Cb (G) of all continuous bounded functions on G equipped with the sup-norm. Conversely, G is called non-amenable if no such invariant mean over Cb (G) exists. All non-compact semisimple nonabelian Lie groups are known to be non-amenable. The notion of amenability has also been extended from Lie groups to homogeneous spaces (see for instance [16, 17]). Note that if in the above definition C was taken to be Cb (G), spontaneous symmetry breaking would be automatic for all non-amenable symmetries. We shall find however that the non-amenability also forces one to consider smaller algebras of observables (e.g., C ∗ -subalgebras of Cb (G)) so that the issue becomes non-trivial again. As a guideline it may be helpful to contrast the peculiar features we find in the hyperbolic spin chain with those in the corresponding compact model. Here the expectations of an observable refer to the thermodynamic limit of the chain where the number of sites goes to infinity while the lattice spacing is still finite. Moreover we require that the expectations are defined through a thermodynamic limit that does not involve the selection of ‘fine-tuned’ subsequences. This defines a subclass of ‘regular’ observables to which we mostly limit the discussion.

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Hyperbolic Spin-Chain

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quantity

spherical spin-chain

hyperbolic spin chain

ground state(s)

unique, normalizable SO(3) invariant SO(3) invariant independent of bc SO(3) invariant independent of bc

∞ set, non-normalizable not SO(1, 2) invariant SO(1,2) invariant depend on bc bc selects ground state SO(1,2) non-invariant depend on bc

reproduces original one

different from original one

expectations of selected SO(2)-invariant observables expectations of generic non-invariant observables reconstructed quantum theory

These regular observables (later called “asymptotically translation invariant”) presumably include all bounded ones, but an explicit formula for their expectations can be derived regardless of boundedness. For the hyperbolic chain it turns out that one has to impose boundary conditions (bc) at the end(s) of the chain which keep at least one spin fixed. Remarkably, we find that even the expectations of invariant observables may depend on the choice of bc, even though in the limit the ends are separated by an infinite number of sites from those where the observable is supported! The bc we are using also single out a preferred subgroup SO(2) ⊂ SO(1, 2) and the expectation functionals turn out to project any observable onto its SO(2) invariant part. Since this averaging over SO(2) does not commute with the action of the full SO(1, 2) group generic non-invariant observables will signal spontaneous symmetry breaking, i.e., their expectations are not SO(1, 2) invariant. This is accompanied by an infinite family of nonnormalizable ‘ground states’ transforming under an irreducible representation of SO(1, 2). This representation becomes unitary under a suitable change of the scalar product; such a scalar product will be produced by the Osterwalder-Schrader reconstruction described in Section 5. Somewhat different indications of spontaneous symmetry breaking in this context have been obtained in [18, 19]. In a situation of conventional symmetry breaking (say, of a compact Lie group symmetry in higher dimensions) one can always switch to invariant expectation functionals by performing a group average over the original noninvariant ones, at the expense of making the clustering properties worse. Here, due to the non-amenability of SO(1, 2) this cannot be done; the symmetry breaking is more severe, and in this respect resembles somewhat the ‘spontaneous collapse’ of supersymmetry in a spatially homogeneous state at finite temperature [20]. It is therefore remarkable that there exists a class of ‘selected’ SO(2) but not SO(1, 2) invariant observables (later called “SO(2) and asymptotically invariant”) which get averaged to yield a SO(1, 2) invariant result. One sees that the impact of the non-amenability is quite subtle: an invariant mean for all bounded

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(let alone unbounded) observables cannot exist, however an invariant mean on a subalgebra does exist and can be constructed explicitly as a thermodynamic limit of probability measures. Schematically, the mechanism behind this is that for a finite chain of length L the functional measure has support mostly at configurations which are boosted with a parameter depending on and increasing with L. So provided a limit exists at all it will be SO(1, 2) invariant as all non-invariant contributions die out. This can be paraphrased by saying that the thermodynamic limit provides a partial invariant mean, that is a mean which is invariant only on the before-mentioned class of ‘selected’ noninvariant observables. Finally we consider the counterpart of the Osterwalder-Schrader reconstruction in this context; here it is important not only to consider the regular observables but the full algebra Cb . For the compact chain one recovers (a lattice analogue of) the quantum mechanics of a particle moving on a sphere, as expected. In the hyperbolic case, however, the reconstructed quantum theory is different from that of a particle moving on H: whereas the former has purely continuous spectrum, the latter has at least some point spectrum. The reconstructed Hilbert space turns out to be nonseparable and the reconstructed quantum theory can be viewed as an interacting (though quantum mechanical) version of the “polymer representations” of the Weyl algebra studied in other contexts [21, 22, 23]. Consistent with these results we find that the symmetry breaking disappears in the limit of a flat target space, when the symmetry group R2 becomes amenable again. The rest of the article is organized as follows. In the next section we introduce the (iterated) transfer matrix and use its asymptotics in the limit of large separations to identify the ground states. Expectation values for a chain of finite length with various bc are studied in Section 3. The thermodynamic limits for the algebras of observables outlined are constructed in Section 4. Finally these infinite volume expectations are used as the basis for the Osterwalder-Schrader reconstruction.

2 The transfer matrix The hyperbolic spin chain can be regarded as a dynamical system in the sense outlined above, with the observables being operators on a Hilbert space. On the other hand, in the classical statistical interpretation the algebra of observables is a suitable algebra of functions over (direct products of) H which we detail in Section 3. We represent H as the hyperboloid H = {n ∈ R1,2 | n · n = 1 , n0 > 0}, where a · b = a0 b0 − a1 b1 − a2 b2 is the bilinear form on R1,2 . The time evolution of the spin chain is governed by the transfer matrix Tx , x ∈ N, which we study first. The symmetry group G is SO0 (1, 2) which acts unitarily via the (left) quasiregular representation ρ on L2 (H), i.e., ρ(A)ψ(n) = ψ(A−1 n), A ∈ SO0 (1, 2). Since we use the identity component exclusively we write SO(1, 2) for SO0 (1, 2). The time evolution commutes with the group action Tx ◦ ρ = ρ ◦ Tx ,

x ∈ N,

(2.1)

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as required. In the following we analyze the spectrum, the eigenfunctions, and the large x limit of Tx , x ∈ N, in terms of its integral kernel Tβ (n · n ; x). Some results from the harmonic analysis on H are needed which we have collected in Appendix A and use freely in the following.

2.1

Spectrum and integral kernel of Tx

The basic (1-step) transfer matrix acts on L2 (H) and is defined by β β(1−n·n ) (Tψ)(n) = dΩ(n ) 2π e ψ(n ) .

(2.2)

From (A.9), (A.20), one infers that the functions ω,k and ω,l defined in (A.8) and (A.10) are exact generalized eigenfunctions of T with eigenvalues 2β β (2.3) λβ (ω) = e Kiω (β) < 1 . π The eigenvalues are even functions of ω with a unique maximum at ω = 0 (but only ω ≥ will appear in the spectral resolution). In particular it follows that the operator T has absolutely continuous spectrum given by the generalized eigenvalues λβ (ω); the spectrum covers an interval [−q, λβ (0)] with 0 < q < 1 and is infinitely degenerate. It is interesting to note that, although real and bounded above by 1, the generalized eigenvalues are positive only for 0 < ω < ω+ (β), where ω+ (β) increases with β like ω+ (β) ∼ β + const β 1/3 . For ω > ω+ (β) the behavior of λβ (ω) is oscillatory with exponentially decaying amplitude λβ (ω) ∼

2ω π β − π ω+β e 2 + ω ln −1 2 sin ω 4 β

as ω → ∞ .

(2.4)

The fact that some of the spectrum of the transfer operator is negative means that there is no reflection positivity under reflections between the lattice points. However positivity of the eigenvalues is restored in the continuum limit: introducing momentarily the lattice spacing a, physical distances xphys = xa, as well as a coupling g 2 = 1/(βa) one has lim [λ

a→0

1 g2 a

(ω)]

xphys a

= exp

− xphys

g2 1 + ω2 . 2 4

(2.5) 2

These ‘eigenvalues’ are readily recognized as those of the heat kernel exp(− g2 Cxphys ), see (A.8); 1/g could be removed by rescaling the n-fields; g then parameterizes the curvature of the hyperboloid. Besides (2.4) another feature distinguishing the non-compact spin chain from the compact ones is that the iterated transfer matrix is bounded but, having continuous spectrum, is not trace class. Heuristically this is because due to the

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invariance (2.1) the infinite volume of SO(1, 2) gets “overcounted” in any trace operation. More precisely we have the following: Lemma 2.1 Let K be a self-adjoint operator on L2 (H) commuting with the unitary representation ρ. Then K has only essential spectrum, implying that K cannot be compact. In particular K cannot be trace class. Proof. Assume that K has an eigenvalue λ. The corresponding eigenspace Hλ ⊂ L2 (H) then is invariant under the action of ρ and therefore the representation ρ can be restricted to a unitary subrepresentation ρλ . Since SO(1, 2) is noncompact, ρλ is either infinite dimensional or it is a direct sum of copies of the trivial representation. But the trivial representation cannot be a subrepresentation of ρ since the only functions carrying the trivial representation are constants, and thus are not square integrable. Remark 1. There is a stronger version of the last statement in the proof: the trivial representation also is not even weakly contained in the direct integral decomposition of L2 (H) because SO(1, 2) is not amenable [4]. Remark 2. As noted above, Tx has only absolutely continuous spectrum. Since Tx is not trace class, correlators cannot be defined by the usual expressions involving traces. The obvious remedy is gauge-fixing. This could be done by introducing a damping factor at one site and by adopting twisted boundary conditions. Then analytic computations are still feasible but are not much different from those in the simpler gauge fixing approach in which one completely freezes one spin. This is the procedure we use in Section 3. Also the iterated transfer matrix acts as an integral operator on L2 (H) with kernel dΩ(n )Tβ (n · n ; x)ψ(n ) , x = 1, 2, 3, . . . , (Tx ψ)(n) =

Tβ (n · n ; x) =

0

∞

dω ω tanh πω P−1/2+iω (n · n ) [λβ (ω)]x , 2π

where the kernels have the semigroup property dΩ(n )Tβ (n · n ; x)Tβ (n · n ; y) = Tβ (n · n ; x + y) .

(2.6)

(2.7)

Manifestly the naive expression for the trace, i.e., the dΩ(n) integral over Tβ (1; x) does not exist due to the infinite volume of H. In passing we note that in terms of the Legendre functions (2.7) amounts to the following identity (“projection property”) 2πδ(ω − ω ) P−1/2+iω (n · n ) , dΩ(n ) P−1/2+iω (n · n )P−1/2+iω (n · n ) = ω tanh πω (2.8)

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which can also be verified directly from (A.12). Integral kernels of spectral projections in the proper sense are easily obtained by integrating over intervals I ω: dω ω tanh πω P−1/2+iω (n · n ) . (2.9) PI (n · n ) := ω∈I 2π Using Eq. (2.8) one easily verifies for two intervals I , J dΩ(n )PI (n · n )PJ (n · n ) = PI∩J (n · n ) ,

(2.10)

showing that the operators PI are spectral projections for an interval in ω and hence for a corresponding spectral interval for T. Absolute continuity of the spectrum follows from the completeness relation of the generalized eigenfunctions given in Appendix A. Before proceeding let us note the continuum limit of the iterated transfer matrix. Using the notation of (2.5) one has x phys Tc (ξ; g 2 xphys ) := lim T 21 ξ; a→0 g a a ∞ dω g2 1 ω tanh πω P−1/2+iω (ξ) exp − xphys + ω 2 , (2.11) = 2 4 0 2π where the limit is understood in the strong sense. With t = −ixphys this is the correct result for the Feynman kernel evolving a wave function for time t, see, e.g., [24, 25] and [26] for the propagators on other homogeneous spaces.. Most of the discussion on the large x limit of Tβ (ξ; x) below transfers directly to the large xphys limit of Tc (ξ, g 2 xphys ). Clearly for the further analysis the properties of the transfer matrix (2.6) will be crucial. By (2.2) and by iteration of the convolution property ξ → Tβ (ξ; x) is a positive function for all x ∈ N and β > 0. For small x it can be evaluated explicitly Tβ (ξ; 0) =

1 δ(ξ − 1) , 2π

β β −βξ e e , 2π √ β 2β e−β 2(1+ξ) e , Tβ (ξ; 2) = 2π 2(1 + ξ)

Tβ (ξ; 1) =

(2.12)

with ξ = n · n ≥ 1. The fact that Tβ (ξ; 2) can be given in closed form could be used to define a coarse grained action corresponding to decimation of half of the spins. Note also the strictly monotonic decay in ξ, stronger than any power, which is masked by the rapidly oscillating integrand in (2.6). Numerical evaluation of some x ≥ 3 transfer matrices suggests that these are generic features, see Fig. 1.

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0.05 0.04 0.03 0.02 0.01 0.5 1 1.5 2 2.5 3 3.5 4

lnΞ

Figure 1. x-step transfer matrix Tβ=1 (ξ; x) for x = 3, 6, 10, in order of decreasing slope. Note the non-uniformity: Tβ (ξ; x + 1) is smaller/larger than Tβ (ξ; x) for ξ smaller/larger than an intersection point ξx . By (2.26b) the enclosed area is always the same; the value at ξ = 1 is the x-site partition function. We proceed to prove these and some further properties of the kernels of Tx : Lemma 2.2 For fixed x the kernel Tβ (ξ; x) has the following properties: (i) For any integer p ≥ 0 ∞ 1 tβ (p; x) := dξ ξ p Tβ (ξ; x) < ∞ . 2π 1

(2.13)

(ii) Tβ (ξ; x) is strictly decreasing in ξ and vanishes for ξ → ∞. (iii) Tβ (ξ; x) ≤ Tβ (1; x) P−1/2 (ξ) for all ξ ≥ 1. (iv) Let f : [1, ∞) → R+ be a strictly positive locally integrable function satisfying sup n·n↑ >K

f (n · n ) ≤ C(n · n↑ )p , f (n · n↑ )

for some constants p ≥ 0 and C, K > 0. Then

1 ↑

)T (n · n ; x)f (n · n ) dΩ(n

f (n · n↑ )

≤C ,

(2.14)

(2.15)

with some constant C . Remark. Condition (2.14) holds for any function f with power-like growth or decay at ∞. This follows from the fact that the geodesic distance between two points n, n behaves aymptotically like ln(n · n ) and the globally valid triangle inequality for the geodesic distance on H.

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Proof. (i) The proof proceeds by induction in x. Note that naively exchanging the order of integrations in (2.6) would suggest a divergent answer already for the zero-th moment. The point to observe is that the convolution property (2.7) implies the recursion relation ∞ du jβ (ξ, u) Tβ (u; x) , Tβ (ξ; x + 1) = 1

jβ (ξ, u) :=

βeβ(1−ξ u) I0 (β

u2 − 1 ξ 2 − 1) ,

(2.16)

where I0 (u) is a modified Bessel function. The kernel jβ (ξ, u) has the following properties: its integral wrt to either variable equals 1; for fixed (not too small) ξ it √ is a bell-shaped function of u decaying like exp{−β(ξ − ξ 2 − 1)u}/ u for large u, and with a single maximum whose position grows linearly in ξ and whose value decays like 1/ξ, for large ξ. In particular the troublesome rapidly oscillating integrand of (2.6) is gone. So in the expression defining the ξ-moments the interchange of the ξ and u integrations is legitimate. The ξ-integral can be done by repeated differentiation with respect to α of the formula ([28], p.722) √ ∞ − β 2 +2uαβ+α2 −αξ β e dξe jβ (ξ, u) = βe =: Fβ (α, u) . (2.17) β 2 + 2uαβ + α2 1 This will be used below to obtain explicit expressions for the low moments. Note that both √ sides of the equation are holomorphic functions of α for |α| < r0 = β(u − u2 − 1), so that we may freely differentiate at the origin. By Cauchy’s estimate

p

− ∂ Fβ (α, u)

≤ p! r−p M (r, u) , (2.18)

∂α α=0 where M (r, u) is the maximum of |F | on the circle α = r, r < r0 . With the choice r1 = β(u − u2 − 1/2) it is not hard to see that the maximum is attained for α = −r – this follows from the fact that the zeros of the quadratic form Q(α) := β + 2uαβ + α2 are both real and negative, so both the real part and the modulus of Q(reiφ ) take on their minimal value β 2 /2 for φ = π. One concludes from (2.18)

p √ √

− ∂ Fβ (α, u)

≤ p! [β(u − u2 − 1/2]−p 2 eβ(1−1/ 2) . (2.19)

∂α α=0 If we finally use the fact that u − u2 − 1/2 ≥ constu−1 , and insert into the integral defining tβ (p; x + 1) the convolution formula (2.16) we obtain √

tβ (p; x + 1) ≤ p! constp tβ (p; x) eβ(1−1/

2)

.

(2.20)

Since for x = 1 all moments exist trivially, this inequality shows the existence of all moments for all x and (i) is proven.

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(ii) For x = 1 this is manifest from (2.12). For x > 1 we again proceed by induction. Assuming that Tβ (ξ; x) is already known to be strictly decreasing, we want to show ∞ ! ∂ξ Tβ (ξ; x + 1) = du ∂ξ jβ (ξ, u) Tβ (u; x) < 0 . (2.21) 1

This follows from the properties of the kernel jβ , namely

∞ < 0 for u < u0 (ξ) , du ∂ξ jβ (ξ, u) = 0 . and ∂ξ jβ (ξ, u) > 0 for u > u0 (ξ) , 1 Using (2.22) one gets for the rhs of (2.21) u0 (ξ) du ∂ξ jβ (ξ, u)Tβ (u; x) + 1

< 1

∞

u0 (ξ)

∞

(2.22)

du ∂ξ jβ (ξ, u)Tβ (u; x)

du ∂ξ jβ (ξ, u) Tβ (u0 (ξ); x) = 0 ,

(2.23)

where in the first integral Tβ (u; x) was replaced by its minimum and in the second one by its maximum over the range of integration, using the induction hypothesis. Thus ξ → Tβ (ξ; x) is strictly decreasing for all x. The vanishing for ξ → ∞ follows from (iii). (iii) This is obtained from (2.6) and the estimate |P−1/2+iω (ξ)| ≤ P−1/2 (ξ), which is manifest from (A.11). (iv) The proof is an elementary consequence of (i). Remark 1. Iteration of Eq. (2.16) provides an efficient way to compute numerically Tβ (ξ; x) for moderately large x. This was used to produce Figs. 1 and 2. Remark 2. Explicit expressions for the low moments are obtained by differentiating (2.17) and inserting the result in the recursion (2.16). This gives p=0: p=1: p=2:

tβ (0; x + 1) = tβ (0; x) , 1 tβ (1; x + 1) = 1 + tβ (1; x) , (2.24) β 1 1 tβ (2; x + 1) = − 2 (1 + β 2 ) tβ (0; x) + 2 (3 + 3β + β 2 )tβ (1; x) , β β

etc. Since for x = 1 all moments are known tβ (p; 1) = βeβ (−∂β )p (βeβ )−1 ,

(2.25)

(which is basically a Laguerre polynomial in β) the x-recursions can be solved successively for p = 0, 1, 2, . . .. The solution of the first two is trivial and gives 1 x tβ (1; x) = 1 + , ∀x ∈ N. (2.26) tβ (0; x) = 1 , β The higher ones won’t be needed explicitly.

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In summary, the qualitative properties of all the Tβ (ξ; x), x ∈ N, are very much like the ones exemplified in Fig. 1. The rate of decrease becomes softer with increasing x but remains faster than any power. The overall scale is set by the maximum Tβ (1; x), which turns out to decrease like x−3/2 for large x. (This is to be contrasted with the flat case of the Euclidean plane R2 , where the decay is only like x−1 ).

2.2

Large x asymptotics of Tβ (ξ; x)

We next determine the large x asymptotics of Tβ (ξ, x). This is of interest because in this limit the iterated transfer matrix normally becomes a (generalized) projector onto the ground state(s), which can in particular be used to identify the latter. In a compact spin chain the kernel of the iterated transfer matrix (normalized such that the largest eigenvalue is 1) tends to a constant for x → ∞, which is indeed the ground state (unique eigenstate to the highest eigenvalue) of the transfer matrix. This is a reflection of the Mermin-Wagner theorem, i.e., of the absence of spontaneous symmetry breaking. The decay in x is exponential due to the gap in the spectrum. In our noncompact model, since the spectrum is gapless, one expects the limit of large separations x to be approached power-like rather than exponentially. This is correct, but concerning the structure of the limit we are in for a surprise: the large x behavior is not invariant under the symmetry group SO(1, 2). Instead one finds lim

x→∞

Tβ (ξ; x) = P−1/2 (ξ) , Tβ (1; x)

(2.27)

as will be shown below. So in some sense P−1/2 (ξ) plays the role of a ground state, but unlike the compact case, where there is a unique, invariant and normalizable ground state, in our case we have a whole family of generalized non-normalizable ground states ψn0 (n) = P−1/2 (n0 · n), spanning a representation space of SO(1, 2). We shall explore the consequences of (2.27) in more detail below. However already at this point it is clear that in this 1D noncompact model the Mermin-Wagner theorem cannot hold in the usual sense. For later use we also introduce the SO↑(2) averaged versions of the iterated transfer matrix and the corresponding bounds. The former is given by π 1 2 T β (ξ, ξ ; x) = dϕ Tβ ξ ξ − (ξ 2 − 1)1/2 (ξ − 1)1/2 cos(ϕ − ϕ ); x , (2.28) 2π −π where n = (ξ, ξ 2 − 1 sin ϕ, ξ 2 − 1 cos ϕ), etc.. Note that T β (ξ, ξ ; 1) = jβ (ξ, ξ ) is the convolution kernel in (2.16). We collect our results on the asymptotics of the kernels Tβ (ξ; x) and T β (ξ, ξ ; x), which contain (2.27) as a special case, in the following

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Proposition 2.3 The large x asymptotics of Tβ (ξ; x) is governed by the relations: (i) P−1/2 (ξ) Tβ (ξ; x − y) = λβ (0)−y . x→∞ Tβ (ξ ; x) P−1/2 (ξ )

(2.29)

Tβ (ξ; x)

c(β) 2

Tβ (1; x) − P−1/2 (ξ) ≤ Const (ln ξ) P−1/2 (ξ) x ,

(2.30)

lim

(ii)

where c(β) is given below in (2.33) (iii) T β (ξ, ξ ; x) = P−1/2 (ξ)P−1/2 (ξ ) , x→∞ Tβ (1; x)

(2.31)

T β (ξ, ξ ; x)

≤ [ln2 ξ + ln2 ξ ] O(x−1 ) . Tβ (1; x)P−1/2 (ξ )P−1/2 (ξ)

(2.32)

lim

(iv)

1 −

The main ingredient in the proof of this proposition is contained in Lemma 2.4 Let f be an even function of ω ∈ R, which is at least twice differentiable at 0 and grows at most polynomially as ω → ∞. Then 0

∞

˜ β (ω)]x ∼ dω ω shπωf (ω) [λ

with

˜β (ω) = λβ (ω) λ λβ (0)

π [c(β)x]3/2 π f (0) + 2f (0)[c(β)x]−1/2 + O(x−1 ) × 2 ∞

and

dt t2 exp(−βcht) c(β) = 0 ∞ . dt exp(−βcht) 0

(2.33)

Proof of Lemma 2.4. The principle behind this is that the contributions of all ˜ β (ω)| for ω > 0 get exponentially suppressed, because they are less than 1, so |λ only the ω = 0 contribution survives for x → ∞. The leading power x−3/2 arises ˜ β (ω), which from the double zero of the integrand at ω = 0 and the structure of λ has a unique maximum at ω = 0. In more detail (2.33) one applies the Laplace ˜ β (ω) is expansion (see, e.g., [29]) to the kernel exp(−xhβ (ω)), where hβ (ω) = − ln λ strictly increasing in 0 < ω < ω+ (β) with hβ (0) = hβ (0) = 0 and hβ (0) = c(β) > 0. Here ω+ (β) is the position of the first zero of λβ (ω) described after Eq. (2.2). The ˜ β (ω)| < 1 also fact that λβ (ω) changes sign at ω+ (β) is inconsequential because |λ for ω ≥ ω+ (β) and the contribution of this region to the integral is exponentially suppressed.

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Proof of Proposition 2.3. We first prove (ii). To this end set ∞ dω 1+p ˜ β (ω)]x , p = 0, 1, . . . , Dp := ω tanh πω [λ 0 2π ∞ dω ˜ β (ω)]x [P−1/2+iω (ξ) − P−1/2 (ξ)] . ω tanh πω [λ N := 0 2π This is chosen such that Tβ (ξ; x)/Tβ (1; x) − P−1/2 (ξ) = N/D0 , as is manifest from (2.6) and P−1/2+iω (1) = 1. On the other hand, using the integral representation (A.11) given in Appendix A one obtains the bound

Thus

|P−1/2+iω (ξ) − P−1/2 (ξ)| ≤ Const ω 2 P−1/2 (ξ) (ln ξ)2 .

(2.34)

Tβ (ξ; x) D2 2

Tβ (1; x) − P−1/2 (ξ) ≤ Const (ln ξ) P−1/2 (ξ) D0 .

(2.35)

Using again Laplace’s theorem, one finds D2 /D0 = O((c(β)x)−1 ), which establishes (ii) (and therefore also (2.27)). (i): We apply Lemma 2.4 to D0 = Tβ (1; x) to obtain √ π Tβ (1; x) ∼ λβ (0)x + · · · . (2.36) [2c(β) x]3/2 Combining (2.27) with Tβ (ξ; x − y) Tβ (1; x − y) Tβ (1; x) Tβ (ξ; x − y) = , Tβ (ξ ; x) Tβ (1; x − y) Tβ (1; x) Tβ (ξ ; x)

(2.37)

and (2.36) gives (i). (iii): This follows from averaging (2.29) over SO↑(2) and using (A.12c). (iv): This is proven similarly as (2.35) starting from the spectral representation for the kernels T β , which is obtained from (2.6) by averaging over the angles using (A.12c). This concludes the proof of Proposition 2.3. We want to mention a stronger bound for which we do not have a complete proof. Conjecture 2.5 The following global bound holds for all x ∈ N and for all ξ ≥ 1: ln ξ (2.38) Tβ (ξ; x) ≤ Tβ (1; x) P−1/2 (ξ) E √ , x for some function E : R+ → R+ having finite moments of all orders. c1 2 t + Remark. The asymptotics of (2.35) for large x suggests that E(t) = 1 − c(β) 3 O(t ), with a constant c1 of order unity and c(β) as in (2.33). The proposal E(t) =

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c1 2 exp(− c(β) t ) is thus plausible. In the continuum limit (2.38) then reduces to a known global bound on the heat kernel (see, e.g., [27]), noting that the geodesic distance from the origin is arccoshξ ∼ ln ξ (for large ξ) and c(β) ∼ 1/β for large β. Given (2.38) a similar global bound on T β (ξ, ξ ; x) can be obtained from (2.38) 2 2 by using ξξ − (ξ 2 − 1)1/2 (ξ − 1)1/2 cos(ϕ − ϕ ) ≥ ξ[ξ − (ξ − 1)1/2 ], and then averaging over SO↑ (2).

Let us now explore the consequences of (2.27) in more detail. Consider the map ψ → P ψ (2.39) (P ψ)(n) := dΩ(n ) P−1/2 (n · n )ψ(n ) . As map from L2 (H) to itself this would have only the null vector in its domain, because it maps even strongly decreasing functions ψ into functions with a decrease so slow that they are not square integrable. But it may be regarded for instance as a map from the test function space S into its dual space S (see Appendix A). However, the range of P does not intersect its domain of definition, so the map (2.39) cannot even be iterated. This is in strong contrast to the situation in a compact model, where the corresponding operator is a well defined projection onto the 1-dimensional subspace of constant functions. Here, on the other hand, the image (P ψ)(n) is in general not even invariant under some SO(2) subgroup. The Fourier transform of P ψ can be defined nevertheless in a distributional sense. l), consistent with Using Eq. (A.14) and one finds ω tanh πω P ψ(ω, l) = 2πδ(ω)ψ(0, the picture that the limit (2.27) lowers the ‘energy’ as much as possible. There are two further important properties that encode the ground state property of P−1/2 (n · n ). The first one is Lemma 2.6 Let K be an SO(1, 2) invariant integral operator with kernel κ(n · n ), κ ∈ L1 (ξ −1/2 ln ξdξ). Then P ψ is a generalized eigenfunction of K with eigenvalue κ (0); explicitly κ(0)P−1/2 (n · n ) . (2.40) dΩ(n )κ(n · n )P−1/2 (n · n ) = Proof. Applying the Mehler-Fock transformation (A.16) for κ and the convolution formula for the Legendre functions the left-hand side becomes ∞ dω ω tanh(πω) κ(ω)P−1/2+iω (n · n )P−1/2 (n · n ) dΩ(n ) 2π 0 ∞ = dωδ(ω) κ ˆ(ω)P−1/2+iω (n · n ) = κ (0) P−1/2 (n · n ) , (2.41) 0

(where the integral and the δ function have to be interpreted suitably to include from 1 to ∞; this ω = 0). The L1 condition ensures that κP−1/2 is integrable √ follows from the global bound P−1/2 (ξ) ≤ (1 + ln ξ)/ ξ, valid for all ξ ≥ 1.

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Taking for K the iterated transfer matrix one has in particular Tx P ψ = λβ (0)x P ψ. For x = 1 this gives explicitly ∞ du jβ (ξ, u) P−1/2 (u) = λβ (0) P−1/2 (ξ) , (2.42) 1

using (A.21) and the fact that jβ is the SO↑(2) average of Tβ (n · n; 1). Thus P−1/2 is also an eigenfunction of the recursion relation (2.16) with the correct eigenvalue. The second property is the ‘cyclicity’ of the function ψ↑ (n) := P−1/2 (n · n↑ ) for the SO↑(2) invariant subspace of L2 (H) under the action of SO↑(2) invariant operators. See Appendix A for an explicit description of this subspace and the SO↑(2) invariant operators acting on it. We repeat that the SO↑(2) denotes the stability subgroup of n↑ = (1, 0, 0). The cyclicity of ψ↑ then follows trivially from the fact that P−1/2 does not vanish anywhere, so any SO↑(2) invariant element ψ ∈ L2 (H) can be obtained by acting on it with a multiplication operator. On the other hand ψ↑ has no nice properties with respect to operators that are not SO↑(2) invariant. Defining P as the integral operator with kernel P−1/2 (n↑ ·n)P−1/2 (n↑ ·n ) one has (P ψ)(n) = P−1/2 (n↑ ·n) Cψ for some constant Cψ . As with P one needs sufficiently strong falloff of ψ for this to be well defined and the image is again not an element of L2 (H). As expected, P automatically projects out the part of a wave function lying in the orthogonal complement of the SO↑(2) invariant subspace.

3 Expectation functionals for finite length Since the transfer operator is not trace class the overall SO(1, 2) invariance has to be (‘gauge-’) fixed already for a chain of finite length. We do this by keeping the spin at one end of the chain fixed and impose various boundary conditions at the other end. Expectation functionals (mapping observables, i.e., functions of the spins into complex numbers) then are always well defined. However in the limit of infinite length interesting statements can only be made about certain subalgebras of observables which we also introduce here. As before, SO↑(2) ⊂ SO(1, 2) denotes the stability group of the vector n↑ .

3.1

Boundary conditions and algebras of observables

We consider chains of length 2L + 1, with sites x = −L, L + 1, . . . , L − 1, L, and spins nx on them, in order to make the boundary go to infinity as L → ∞, so as to obtain an infinite volume Gibbs state for the chosen action. As discussed earlier, some ‘gauge fixing’ is needed, which is accomplished conveniently by fixing the spin at the left boundary of our chain: n−L = n↑ = (1, 0, 0) ∈ H. At the other end we consider the following choices: fixed (Dirichlet) bc nL = An−L , with A ∈ SO(1, 2), or free bc (integrating with the invariant measure of H over nL ). We refer to Dirichlet bc with A = 1 as ‘periodic bc’ and with A = 1 as ‘twisted bc’. The fixing of the spin n−L avoids the overcounting of the infinite

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volume of H induced by the invariance (2.1); since the associated Faddeev-Popov determinant is just 1, it is justified to refer to fixed bc with A = 1 as ‘periodic bc’. In some cases a nontrivial twist matrix would explicitly break the otherwise manifest SO↑(2) invariance. In those cases we shall average nL over an SO↑(2) orbit, thereby maintaining the SO↑(2) invariance of the bc. We consider several classes of observables, all of which consist of functions of finitely many spins. They form algebras with addition and multiplication defined pointwise. Of particular interest is the algebra Cb of bounded continuous functions on direct products of H. Equipped with the sup-norm and completed with respect to this norm, this is a commutative C ∗ -algebra, and the expectation functionals constructed later fit the usual concept of a ‘state’ ω as a normalized positive (and therefore bounded) functional on the observable algebra, see, e.g., [1]. More generally we consider the ∗-algebra Cp of polynomially bounded functions. For the construction of expectation functionals we introduce a system of subsets of Cp , closed under a suitable norm and designed such that explicit results for thermodynamic limit can be obtained. It turns out that the expectations of a multilocal observables O ∈ Cp can always be expressed in terms of a kernel K O associated with O as follows: Definition 3.1 For O ∈ Cp and ≥ 2 set −1 dΩ(ni )O(n1 , . . . , n ) Tβ (ni−1 · ni ; xi − xi−1 ) , K O (n, n ) := i=2

(3.1)

i=2

where n1 = n and n = n . For observables O depending only on one spin set K O (n, n ) := O(n) δ(n, n ),

(3.2)

where δ(n, n ) is the delta-distribution (point measure) concentrated at n = n , defined with respect to the measure dΩ. Lemma 3.2 The assignment O → K O mapping observables O ∈ Cp into integral operators K O on L2 (H) with kernel (3.1) has the following properties: (i) let A, B ∈ Cp be two observables of ordered non-overlapping ‘support’, i.e., A depends on nx1 , . . . , nxk and B on nxk+1 , . . . , nx with xk+1 ≥ xk ; then K AB = K A Txk+1 −xk K B , (3.3) K AB (n1 , n ) = dΩ(n)dΩ(n ) K A (n1 , n) Tβ (n · n ; xk+1 − xk )K B (n , n ) , where (AB)(nx1 , . . . , nx ) = A(nx1 , . . . , nxk ) B(nxk+1 , . . . , nx ), k, − k ≥ 2. If xk+1 = xk , the transfer matrix Tβ (n · n ; 0) is interpreted as δ(n, n ). (ii) The action of SO(1, 2) on Cp , i.e., ρ(A)O(nx1 , . . . , nxl ) = O(A−1 nx1 , . . . , A−1 nx ) induces an action on the kernels K ρ(A)O (n1 , n ) = K O (A−1 n1 , A−1 n ) . (iii) For the unit 1 ∈ Cp one has: K 1 (n1 , n ) = Tβ (n1 · n ; x − x1 ).

(3.4)

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Proof. This is a straightforward computation. Remark 1. The last property also implies that the correspondence O → K O is unique only for the equivalence classes obtained by inserting into a given K O extra powers of T. For example taking in (i) for A = 1 one obtains K 1B = Txk+1 −x1 K B . In the multipoint functions this just means that not all of the ‘unobstructed’ integrations have been performed. We shall therefore usually work with a reduced representative, i.e., one which cannot written in the form Ty1 K A1 Ty2 K A2 . . . with some smaller y1 , y2 , . . . ≥ 0. Remark 2. For observables depending only on one spin neither (3.1) nor (3.3) are directly applicable. However the assignment K O (n1 , n2 ) = O(n1 ) δ(n1 , n2 ) is compatible with the formulas for the 1-point functions (3.13), (3.17) and the convolution (3.3), provided we associate n1 and n2 with the same lattice point. We now introduce various classes of observables, where the SO↑ (2) average of an observable O is denoted by O. Definition 3.3 (i) An observable O = O ∈ Cp is called invariant if [K O , ρ] = 0 ,

(3.5)

i.e., O(An1 , . . . , An ) = O(n1 , . . . , n ) for all A ∈ SO(1, 2). The set of these observables is denoted by Cinv . (ii) An observable O ∈ Cp is called asymptotically invariant if lim ρ(A)[K O , ρ] = 0 .

A→∞

(3.6)

The set of these observables is denoted by Cainv . (iii) An observable O ∈ Cp is called translation invariant if [K O , T] = 0 .

(3.7)

The set of these observables is denoted by CT inv . (iv) An observable O ∈ Cp is called asymptotically translation invariant if lim ρ(A)[K O , P ] = 0 .

A→∞

(3.8)

The set of these observables is denoted by CT ainv . In (3.8) P is the integral operator (2.39). Both in (3.6) and (3.8) A → ∞ refers to a sequence of SO(1, 2) transformations such that A → ∞, and the commutator has to obey some decay condition detailed in the next section (Definitions 4.2 and 4.5).

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These subsets of Cp are related as follows: CT inv ∪ Cinv

⊂ CT ainv ∪ ⊂ Cainv

(3.9)

where all inclusions are proper. ↑ , CT↑ inv and CT↑ ainv are defined as the SO↑(2) invariant Definition 3.4 The sets Cainv subsets of Cainv , CT inv and CT ainv , respectively.

Of course the inclusion relations are preserved and the counterpart of the diagram (3.9) remains valid for the SO↑(2) invariant subsets.

3.2

Expectation functionals

The expectation functionals for finite L are defined by explicitly given measures and for the largest class of observables Cp . For states over Cb it follows from the general though not very constructive Banach-Alaoglu theorem [30] that thermodynamic limits always exist. The system of algebras (3.9) is designed to make useful and explicit statements about the limit, even for unbounded observables. Sometimes we refer to the expectation values as ‘correlators’ by a common abuse of language. With twisted bc the finite volume average of an observable O({n}) is then defined as L−1 1 β β(1−nx ·nx+1 ) dΩ(nx ) 2π e O({n}) δ(n−L , n↑ ) , (3.10) OL,β,α = Zβ,α (2L) x=−L

Here we anticipate that in the cases of interest the dependence on the twist matrix A is only through the scalar product n↑ · nL or equivalently the “twist parameter” α := arcosh n↑·nL ≥ 0. Zβ,α (2L) is the partition function normalizing the averages, 1L,β,α = 1. The technique to evaluate expressions like (3.10) is well known from the compact models: one uses the semigroup property (2.7) to perform all integrations not ‘obstructed’ by the variables in O({n}). For the partition function there are no obstructions and one readily finds Zβ,α (2L) = Tβ (chα; 2L) .

(3.11)

For the expectation value of some multilocal observable O one has Proposition 3.5 (twisted bc): For ≥ 2 O(nx1 , . . . , nx )L,β,α (3.12) 1 = dΩ(n1 )dΩ(n ) Tβ (n↑ · n1 ; L + x1 )K O (n1 , n ) Tβ (n · nL ; L − x ) , Zβ,α (2L)

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where x1 < · · · < x . For = 1 we have 1 dΩ(n) O(n)Tβ (n↑ · n; L + x)Tβ (n · nL ; L − x) . (3.13) O(nx )L,β,α = Zβ,α (2L) Proof. This is a simple consequence of (3.10) and the definition of K O .

Remark. As will become clear later for a SO↑(2) noninvariant field O one should average nL over SO↑(2), which amounts to replacing Tβ (n · nL ; L − x ) by T β (n↑ · n , n↑ · nL ; L − x ) defined in (2.28). For a field O which is SO↑(2) invariant the replacement is an identity. Since the expectation value is taken with a positive probability measure, for observables O ∈ Cb we have |O| ≤ ||O|| where O is the supremum norm, and for nonnegative O the expectation value is nonnegative. Observe also that due to the gauge fixing the functions (3.12) are in general not translation invariant; we shall later find a simple supplementary condition which restores translation invariance even at finite L. For free boundary conditions at x = L the situation is similar: First note that the partition function with free bc at L is Zβ,free (2L) = 1. (3.14) This follows from the normalization dΩ(n ) Tβ (n · n ; 2L) = 1 and the semigroup property of Tβ (n · n ; x), see Eqs. (2.12) and (2.7). Thus the expectation of an observable O({n}) with free bc at L is simply OL,β,free =

L x=−L

dΩ(nx )

L−1

β β(1−nx ·nx+1 ) 2π e

O({n}) δ(n−L , n↑ ) . (3.15)

x=−L

Again these expectation values can be rewritten similarly as in Proposition 3.5: Proposition 3.6 (free bc): For ≥ 2 O(nx1 , . . . , nx )L,β,free = dΩ(n1 )dΩ(n ) Tβ (n↑ ·n1 ; L + x1 ) K O (n1 , n ) , (3.16) where again x1 < · · · < x and K O is as in (3.1). For = 1 O(nx )L,β,free = dΩ(n) O(n)Tβ (n↑ · n; L + x) . Proof. Again a simple consequence of (3.15) and the definition of K O .

(3.17)

Remark 1. By comparing Eqs (3.16) and (3.12) one sees that the expectation values of observables with free and twisted bc are related by dΩ(nL ) Tβ (n↑ · nL ; 2L) OL,β,α = OL,β,free . (3.18)

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In other words for finite L the free expectation is some kind of weighted average over the twisted expectations. In the thermodynamic limit this is no longer true, as we will find below. Remark 2. Due to (3.4) a SO(1, 2) transformation on the observable can always be compensated by a change in the bc

. (3.19) ρ(A)OL,β,bc = OL,β,bc ↑ −1 ↑ −1 n →A

n , nL →A

nL

Of course our interest will be in the invariance or noninvariance of the expectations when the bc are kept fixed as L → ∞. For translation invariant observables the expectation values can be simplified. Recall that for O ∈ CT inv [K O , T] = 0 ⇐⇒ ∀n, n ∈ H (3.20) dΩ(n ) K O (n, n ) Tβ (n · n ; 1) = dΩ(n ) Tβ (n · n ; 1) K O (n , n ) . For these expressions to make sense, one has to impose some technical conditions; it suffices to demand that K O is a bounded operator. Using the convolution property (2.7) it is then easy to show that for translation invariant observables the expressions (3.12) and (3.16) simplify to Proposition 3.7 (translation invariant observables): For O ∈ CT inv 1 dΩ(n) K O (n↑ , n)Tβ (n · nL ; 2L + x1 − x ) , O(nx1 , . . . , nx )L,β,α = Zβ,α (2L) O(nx1 , . . . , nx )β,free = dΩ(n) K O (n↑ , n) . (3.21) Remark 1. For twisted bc also the equivalent form of the integrand K O (n, nL ) × Tβ (n · n↑ ; 2L + x1 − x ) could be used. Observe that these expectations are translation invariant already for finite L. Moreover for free bc they are L independent altogether, so that taking the thermodynamic limit becomes trivial. Remark 2. For observables whose kernels admit a Fourier expansion (A.18) a necessary and sufficient condition for (3.20) to hold is that expansion takes the form ∞ dω ω tanh πω κ l,l (ω) ω,−l (n)ω,−l (n ) . (−)l+l (3.22) K O (n, n ) = 2π 0 l,l ∈Z

It differs from the most general one in (A.18) only by the fact that it is diagonal in the energy parameter ω, as expected. An important special case is when

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the spectral weight is (up to a sign factor) independent of l1 , l2 . Due to the addition theorem (A.12c) the kernel becomes a function of n1 · n only. In this case the corresponding observables O can be characterized directly as being SO(1, 2) invariant. Remark 3. As already seen in Section 2 the ‘vacuum structure’ can be explored by taking the thermodynamic limit of the discrete system. Equivalently one can first take the continuum limit and then consider its behavior for large Euclidean times. The continuum limit of the correlators in Propositions 3.5, 3.6, and 3.7 is obtained by substituting (xi )phys = axi ,

Lphys = aL ,

β=

1 , ag 2

(3.23)

and taking the limit a → 0. In view of (2.11) this basically amounts to replacing Tβ by Tc everywhere, with the rescaled arguments. This procedure yields the additional bonus of restoring reflection positivity.

3.3

Projection onto SO↑(2) invariant observables

For SO(1, 2) non-invariant observables we did not assume special symmetry properties. It turns out, however, that one needs to consider only SO↑(2) invariant observables (bounded or unbounded) since with our gauge fixing SO↑(2) noninvariant ones are effectively projected onto SO↑(2) invariant ones. In order to see this let us apply an SO↑(2) rotation A(ϕ), A(ϕ)n↑ = n↑ (with ϕ the rotation angle) to an SO↑(2) noninvariant observable O. Using (3.4) one finds for ≥ 2 O(A(ϕ)nx1 , . . . , A(ϕ)nx )L,β,α 1 = dΩ(n1 )dΩ(n ) Tβ (n↑ · n1 ; L + x1 )K O (n1 , n ) Zβ,α (2L)

(3.24a)

Tβ (n · A(ϕ)nL ; L − x ) , O(A(ϕ)nx1 , . . . , A(ϕ)nx )L,β,free = dΩ(n1 )dΩ(n ) Tβ (n↑ · n1 ; L + x1 ) K O (n1 , n ) ,

(3.24b)

and similarly for = 1. For free bc one sees that the dependence on the rotation angle drops out, so that the expectations with these bc are SO↑(2) invariant even if the observable is not. Equivalently SO↑(2) noninvariant observables have the same expectations as their SO↑(2) averages. For twisted periodic bc this is not quite true. However the SO↑(2) noninvariance of (3.24) is evidently caused by the noninvariance of the bc. To retain the SO↑(2) invariance of the bc one can average nL over an SO↑(2) orbit. Then Tβ is replaced with T β in Eq. (2.28) and the situation is the same as with free bc. In summary, the expectations (3.10)

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(when nL is averaged over an SO↑(2) orbit) and (3.15) for finite L are already SO↑(2) invariant and hence we need not distinguish between SO↑(2) noninvariant and SO↑(2) invariant observables. In terms of the algebras introduced in Section 3.1 a projection Cp → Cp↑ , takes place upon insertion into the expectation functionals. In terms of the kernels K O the projection amounts to the replacement π 1 O dϕ K O (A(ϕ)n1 , A(ϕ)n ) =: K (n↑ · n1 , n↑ · n ) . (3.25) K O (n1 , n ) −→ 2π −π For later reference let us issue the warning K

ρ(A)O

O

(n↑ ·n1 , n↑ ·n ) = K (n↑ ·A−1 n1 , n↑ ·A−1 n ) ,

(3.26)

that is, SO↑(2) averaging does not commute with the SO(1, 2) action. In compact sigma models, where there is no need for gauge fixing, one can choose invariant bc, so that the expectation of any noninvariant observable is equal to that of its group average. By the Mermin-Wagner theorem in dimensions 1 and 2 this remains also true in the thermodynamic limit, irrespective of the bc used. Here we find an analogous situation only with respect to the maximal compact subgroup SO↑(2), singled out by the gauge fixing. In contrast, for the full SO(1, 2) group the expectations of noninvariant and of invariant observables cannot be related by group averaging. This is because – due to the amenability of SO(1, 2), such averages (invariant means) do not exist [4]. Heuristically this can be understood by viewing the group averaging as a projector onto the trivial subrepresentation in the direct integral decomposition of tensor products of L2 (H) functions. By the nonamenability the trivial representation does not occur, though. This lack of amenability is the source of many peculiarities in the vacuum structure of the noncompact model.

4 The thermodynamic limit as a partial invariant mean By the non-amenability of SO(1,2) an invariant mean on Cb cannot exist; a fortiori this holds for the unbounded functions Cp . It is known, however, that there are subspaces of the space of bounded continuous functions on any group, such as the spaces of almost periodic or weakly almost periodic functions on which a unique invariant mean exists [4]. These spaces are defined rather abstractly by relative compactness resp. weak compactness of their orbits under the group action. In ↑ ⊂ Cp for which there is the following we will introduce concretely a subspace Cainv a unique, invariant, and explicitly computable thermodynamic limit. The infinite volume averages therefore define a ‘partial invariant mean’. We presume that the ↑ ↑ ∩ Cb of our class Cainv (viewed as functions on SO(1,2)) bounded subalgebra Cainv consists of weakly almost periodic functions, but not of almost periodic functions (the latter set contains only the constant functions [31]). For the construction of the thermodynamic limit we proceed in several steps, where we first construct the

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thermodynamic limit for the algebras in the top row of the diagram (3.9). The limit is shown to be explicitly computable and unique (but different) for free and for twisted bc. The construction does not require the selection of subsequences, i.e., works without recourse to the Banach-Alaoglu theorem. In each case we then proceed to show that this limit is SO(1, 2) invariant for the described subalgebras ↑ . in the bottom row of the diagram, trivially for Cinv and nontrivially for Cainv

4.1

Thermodynamic limit for translation invariant observables

We begin by studying the thermodynamic limit of translation invariant observables. The distinction between the bounded observables and the polynomially bounded observables turns out to be inessential and we assume O ∈ CT inv throughout. With free bc, as seen in Eq. (3.21), there is no L dependence left – so no limit has to be taken. For CT inv expectations defined with twisted bc the existence of an L → ∞ limit needs to be established. First there is a slight complication that needs to be taken care of: twisted bc nL = An−L , n−L = n↑ , with A = 1 explicitly break SO↑(2) invariance. Since in this study we are interested in the spontaneous symmetry breaking for the nonamenable SO(1, 2), we restore the SO↑(2) invariance of the bc by performing an average of nL = An↑ over SO↑(2). For finite length L the expectations will then still depend on the ‘height’ n0L = chα. In a slight abuse of terminology we shall keep referring to these bc as ‘twisted’ ones and also keep the original notation . L,β,α . Only when a confusion is possible we emphasize the additional averaging by denoting the corresponding expectations by . L,β,α,av . Proposition 4.1 For O ∈ CT inv and twisted bc the thermodynamic limit is given by the equivalent expressions: ∞ O O(nx1 , . . . , nx )∞,β,α = λβ (0)x1 −x 2π dξ K (ξ, 1) P−1/2 (ξ) . (4.1) 1

O(nx1 , . . . , nx )∞,β,α = λβ (0)x1 −x 2π = λβ (0)x1 −x 2π

∞

1

O

dξ K (1, ξ) P−1/2 (ξ) O

∞

dξ 1

(4.2)

K (ξ, n↑ ·nL ) P−1/2 (ξ) . P−1/2 (n↑ ·nL )

Proof. For Eq. (4.1) we use the SO↑(2) invariance of the bc to replace the transfer matrix Tβ by Tβ (see (2.28))and then K O by its SO↑(2) average (see (3.25)). Since by assumption the integral dΩ(n)|K O (n, n )| exists one can in the first equation of (3.21) take the L → ∞ limit inside the integral. To obtain Eq. (4.2) one uses the fact that for translation invariant observables O the integral operators K O commute with P in (2.39). Remark 1. There are no elements of CT inv depending only on one spin, except constants. For free bc no thermodynamic limit has to be taken, see Proposition 3.6.

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Remark 2. In particular Proposition 4.1 is valid for SO(1, 2) invariant observables O ∈ Cinv ⊂ CT inv where the kernel K O (n1 , n ) depends only on the invariant distance n1 ·n . The thermodynamic limit (4.1) is then independent of the twist n−L · nL = n↑ · nL = cosh α, i.e., O(nx1 , . . . , nx )∞,β,α = O(nx1 , . . . , nx )∞,β,0 ,

O ∈ Cinv .

(4.3)

This can be verified directly using the ground state property (2.40). Indeed, if one does not take the thermodynamic limit in (3.21) with the SO↑(2) averaged transfer matrix one obtains initially an alternative version of the second Eq. in (4.2) λβ (0)x1 −x O O(nx1 , . . . , nx )∞,β,α = dΩ(n) K (n↑ ·n) P−1/2 (n·nL ) , O ∈ Cinv . P−1/2 (chα) (4.4) Averaging over nL and use of the addition theorem (A.12c) shows that the dependence on α drops out. Alternatively one can use (2.40) to verify (4.3). Remark 3. For generic translation invariant observables the infinite volume expectations are in general not SO(1, 2) invariant. Rather one finds from (3.19) the following induced action on the kernels by O → ρ(A−1 )O: O

O

K (1, ξ) −→ K (n↑ ·An↑ , n↑ ·An) , O

K (1, ξ) −→ O

P−1/2 (n↑ ·AnL ) O ↑ K (n ·An↑ , ξ) P−1/2 (n↑ ·nL ) O

K (ξ, n↑ ·nL ) −→ P−1/2 (n↑ ·An↑ ) K (ξ, n↑ ·AnL ) ,

(4.5a) (4.5b) (4.5c)

where for free bc only (4.5a) applies while for twisted bc all three (equivalent) expressions are applicable. We shall return to these formulae later but note already here that observables in CT inv \ Cinv will in general show spontaneous symmetry breaking: ρ(A)O∞,β,bc = O∞,β,bc . For the rest of this subsection we now focus on the special case of SO(1, 2) invariant observables. Then symmetry breaking is not an issue, nevertheless the result (4.1) is surprising. Besides the mere existence of a thermodynamic limit one would of course expect that the effect of the different bc is washed out. While we have found that the dependence on the twist chα actually does disappear, free bc in general give a different thermodynamic limit. In other words, even invariant observables show a dependence on the boundary conditions, even after the boundary is removed to infinity! To illustrate this consider specifically the usual ‘spin-spin’ two-point functions with the various bc. For twisted bc the thermodynamic limit is obtained from (4.1) and (3.1) (for = 2 with O(n1 , n2 ) = n1 · n2 ) as ∞ 2π dξ ξ Tβ (ξ; x)P−1/2 (ξ) . lim n0 · nx L,β,α = lim n0 · nx L,β,0 = L→∞ L→∞ λβ (0)x 1 (4.6)

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The independence of the twist angle has been seen before to be a general feature. However the same expectation with free bc at the right end of the chain gives a different result. One finds ∞ 1 x dξ ξ Tβ (ξ; x) = 1 + , (4.7) n0 · nx β,free = 2π β 1 using Eq. (2.26b) in the second step. As seen generally in Eq. (3.21) the correlator is L-independent and thus coincides with its thermodynamic limit. But this thermodynamic limit is now different from the previous one. To make sure that the analytical expressions (4.6) and (4.7) really define different functions we evaluated them numerically; the results are shown in Figure 2 below. For periodic bc also the approach to the thermodynamic limit is shown, which turns out to be nonuniform and extremely slow.

n0 nx 500

100 50

10 5

1

0

2

4

6

8

10

x

Figure 2. Spin two-point function for β = 1: for periodic bc and L = 8, 16, 32, 64, ∞, and for free bc, in order of increasing values at fixed x. For the ‘internal energy’ Eβ,bc := limL→∞ n0 · n1 L,β,bc the discrepancy can be seen immediately: Eβ,bc = 1 +

1 ∂ 1 − lim ln Zβ,bc(2L) β L→∞ 2L ∂β

 1 ∂   ln λβ (0) for twisted bc,  1+ − β ∂β = 1    1+ for free bc, β using Eq. (2.36) and (3.14), respectively.

(4.8)

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Technically the discrepancy can be traced back to the fact that in Eq. (3.18) the operations ‘averaging’ and ‘taking the thermodynamic limit’ do not commute, schematically: limL→∞ dΩ(nL )(. . .) = dΩ(nL ) limL→∞ (. . .). Indeed, the lhs is L-independent and equals Oβ,free while the integrand and hence the integral on the rhs vanishes pointwise. In fact the integrand on the right-hand side behaves very nonuniformly for L → ∞: for instance the two-point function with twisted bc is unbounded as a function of α and the convergence as L → ∞ takes place more and more slowly as α increases. These features are in sharp contrast to those of the compact O(N ) spin chains where it is well known that all boundary conditions yield the same thermodynamic limit for the correlators of invariant as well as noninvariant quantities; see for instance [32]. In the compact models no gauge fixing is required, but one could fix a spin at the boundary just as we did here, and the thermodynamic limit would be insensitive to it. This is a consequence of the Mermin-Wagner theorem, which holds in this case. One might suspect that this ‘long range order’ in the non-compact model reflects the poor choice of observables, i.e., that the kernel O(n, n ) = n · n does not define an operator on L2 (H) (as explained after Eq. (A.13)). However the situation is the same for invariant kernels O(n, n ) = κ(n · n ) which obey (A.17) and which therefore do define integral operators on L2 (H). The thermodynamic limit of the corresponding two-point functions is obtained simply by replacing ξTβ (ξ; x) with κ(ξ)Tβ (ξ; x) in Eqs. (4.6) and (4.7). These two-point functions will be conventional, decreasing functions of x. Nevertheless they will in general be different for free and for periodic bc. Another potential problem could be the lack of clustering. However for SO(1, 2) invariant observables the situation turns out to be peculiar – there is perfect clustering even at finite distance. Consider two invariant observables, A(nx1 , . . . , nx )

and

B(ny1 , . . . , nyk )

such that x1 < · · · < x ≤ y1 < · · · < yk . We claim that for all bc AB∞,β,bc = A∞,β,bc B∞,β,bc .

(4.9)

For twisted periodic bc the derivation proceeds along the lines leading to Eq. (4.3) via (4.4): we define kernels K A and K B as above and use the ground state property Eq. (2.40) of P−1/2 . It turns out that both sides of Eq. (4.9) are equal to the same multiple of κ A (0) κB (0) (and thus in particular are independent of the twist parameter). For free bc the expectations of invariant observables are already Lindependent; the asserted factorization can be seen in a way similar to the step from (3.16) to (3.21). This ‘hyperclustering’ property is unpleasant, because it means that from the correlators of invariant fields one can only reconstruct a one-dimensional Hilbert space. The latter is suggested by the fact that all vectors obtained by applying

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invariant kernels to the ground state will by (2.40) be proportional to it. Technically it follows from the Osterwalder-Schrader reconstruction of the Hilbert space, as detailed in Section 5. On the other hand this feature is a peculiarity present likewise for other one-dimensional spin models, like the compact O(N ) chains or the harmonic chains. In these models, since they are based on amenable symmetries, there exists a unique thermodynamic limit also for noninvariant correlators and therefore one obtains by the reconstruction a nontrivial infinite dimensional Hilbert space. We now show that in the noncompact models the situation encountered for invariant observables persists for a class of noninvariant ones: for all observables in CT ainv for fixed bc a unique thermodynamic limit exists but is in general different for periodic and for free bc. The hyperclustering, however, does not carry over to those observables, as we will see.

4.2

TD limit for asymptotically translation invariant observables

We now relax the condition of translation invariance to “asymptotic translation invariance”. It suffices to consider SO↑(2) invariant bc (such as free, periodic, or SO↑(2) averaged twisted bc). As explained in Section 3.3 this allows one to restrict attention to SO↑(2) invariant observables. As before we denote by K O and P the integral operators with kernels K O (n, n ) in (3.1) and P−1/2 (n · n ), respectively. Similarly [K O , P ](n, n ) is the kernel of the commutator of K O with P . We give now the precise version of Definition 3.3 (iv): Definition 4.2 O ∈ Cp is called asymptotically translation invariant iff its SO↑(2) average satisfies

O

p(ξ) ∼ ξ −1/2 (ln ξ)−3 , (4.10)

[K , P ](n, n ) ≤ p(n↑ · n) p(n↑ · n ) , for some fixed n and all n ∈ H or vice versa. For observables O(n) depending on a single spin only we define asymptotic translation invariance by the condition that their SO↑(2) average O(n) has a limit as n → ∞. The function p(ξ) needs to be bounded but it is mainly the large ξ asymptotics that matters; for definiteness we take p(ξ) = p1 ξ −1/2 (1 + ln ξ)−3 , for some p1 = p(1) > 0. To motivate the terminology “asymptotically translation invariant” recall from Section 2 that P can be viewed as a weak limit of transfer operators TL /Tβ (1; L) for L → ∞. Further, for O ∈ CT ainv one has lim ρ(A)[K O , P ] = 0 .

A→∞

(4.11)

Here we assumed that the commutator acts on L1 wave functions so that β ρ(A)([K O , T]ψ)(n) can be bounded by 2π p1 p(An↑ · n)ψ1 . The thermodynamic limit for asymptotically translation invariant multi-spin observables and twisted bc is given by the same expressions as for translation invariant observables:

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Proposition 4.3 (i) Let O ∈ CT ainv be a 1-point observable, i.e., any function of one spin such that its SO↑(2) average has a limit O(∞). Then: lim O(nx )L,β,bc = O(∞) .

L→∞

(4.12)

(ii) Let O ∈ CT ainv be a multi-point observable, ≥ 2. If Conjecture 2.5 holds then: ∞ O x1 −x 2π dξ K (ξ, 1) P−1/2 (ξ) . (4.13) O(nx1 , . . . , nx )∞,β,α = λβ (0) 1

Proof. (i) By definition the SO↑ (2) average of O(n) has a limit O(∞) for n → ∞ (and therefore is a bounded function). We write (4.14) O(nx )L,β,bc = O(∞) + dµL,β,bc(n; x)[O(n) − O(∞)] . Decomposing the second term into an integral over n↑ ·n ∈ [1, Λ] and n↑ ·n ∈ [Λ, ∞[, given choose Λ so large that sup|O(∞) − O(n)| < , with the supremum over n↑ · n ∈ [Λ, ∞[. In Lemma 4.7 below is shown that the 1-spin measure of any bounded set in H goes to 0 as L → ∞, so sending L → ∞ the first integral vanishes. This shows that the total integral goes to 0 for L → ∞ and one obtains (4.12). (ii) The proof is based on a reduction to the case (i) of a one-spin observable. It is convenient to write (AB)(n, n ) for the kernel of AB, for any pair of integral operators A, B. With this notation one starts from T β (n↑ ·n , n↑ ·nL ; L − x ) OL,β,α = dΩ(n ) (TL+x1 K O )(n↑ , n ) . (4.15) Tβ (n↑ ·nL ; 2L) These multipoint averages can be written as one-point averages as follows OL,β,α = O0,L,α L,β,α , with T β (n↑ ·n, n↑ ·nL ; L − x ) . (4.16) O0,L,α (n0 ) := dΩ(n) (Tx1 K O )(n0 , n) T β (n↑ ·n0 , n↑ ·nL ; L) For the time being (4.16) is just an identity (Fubini’s theorem); later we shall put it in the context of the Osterwalder-Schrader reconstruction. Next we observe that O0,L,α (n0 ) has a L → ∞ limit, pointwise for all n0 ∈ H, which is independent of the twist parameter α defining nL modulo SO↑(2) rotations: lim O0,L,α (n0 ) = O0,∞ (n0 ) ,

L→∞

O0,∞ (n0 ) := Here we used Eqs. (2.31).

with

dΩ(n) (Tx1 K O )(n0 , n)

P−1/2 (n↑ · n) λβ (0)−x . (4.17) P−1/2 (n↑ · n0 )

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The crucial identity now is Lemma 4.4 Assume that Conjecture 2.5 holds. Then lim O0,L,α L,β,α = lim O0,∞ L,β,α ,

L→∞

L→∞

for all O ∈ Cb .

(4.18)

Proof of Lemma 4.4. We start with the bound

(4.19)

(O0,L,α − O0,∞ )L,β,α

T (n · n↑ ; L)T (n↑ ·n , n↑ ·n ; L)

β 0 β 0 L ≤ dΩ(n0 ) O0,L,α (n0 ) − O0,∞ (n0 )

, Tβ (n↑ ·nL ; 2L) To examine the difference |O0,L,α (n0 ) − O0,∞ (n0 )| we write O0,L,α (n0 ) − O0,∞ (n0 ) = D1 (n0 ) + D2 (n0 )

(4.20)

with

and

E(n0 ) :=

D1 (n0 ) :=

O0,L,α (n0 ) − E(n0 )

D2 (n0 ) :=

E(n0 ) − O0,∞ (n0 )

dΩ(n)(Tx1 K O )(n0 , n)

(4.21)

T β (n↑ ·n; n↑ ·nL ; L − x ) . P−1/2 (n↑ · n0 )P−1/2 (n↑ ·nL )Tβ (1; L) (4.22)

Using the bound

x1 O

(T K )(n0 , n) ≤ O T (n0 · n; x )

and the convolution property of the transfer matrices, a bound for D1 is

T β (n↑ ·n0 , n↑ ·nL ; L)

, |D1 (n0 )| ≤ O

1 − ↑ ↑ P−1/2 (n · n0 )P−1/2 (n ·nL )Tβ (1; L)

while for D2 one obtains simply Tβ (n0 · n; x ) |D2 (n0 )| ≤ O dΩ(n) P−1/2 (n↑ · n0 )

T β (n↑ ·n, n↑ ·nL ; L − x )

↑ −x

− × P−1/2 (n · n)λβ (0) . P (n↑ ·nL )Tβ (1; L)

−1/2

According to (4.19) we have to estimate Tβ (n0 · n↑ ; L)T β (n↑ ·n0 , n↑ ·nL ; L) , d1,2 := dΩ(n0 ) |D1,2 (n0 )| Tβ (n↑ ·nL ; 2L)

(4.23)

(4.24)

(4.25)

(4.26)

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In a first step we use T β (ξ0 , ξL ; x) ≤ Tβ (1; x)P−1/2 (ξ0 )P−1/2 (ξL ) and the fact that D1,2 (n0 ) depend on ξ0 = n↑ ·n0 only to write P−1/2 (ξL )Tβ (1; L) ∞ d1,2 ≤ 2π dξ0 |D1,2 (ξ0 )|Tβ (ξ0 , L)P−1/2 (ξ0 ) . (4.27) Tβ (ξL ; 2L) 1 Next we claim

|D1,2 (ξ0 )| ≤ [ln2 ξ0 + ln2 ξL ]O(1/L) .

(4.28)

For D1 (ξ0 ) this follows directly from (4.24) and Proposition 2.3(iv). For D2 (ξ0 ) we likewise use Proposition 2.3(iv) and then apply Lemma 2.2(iv) with the function f (ξ) = P−1/2 (ξ)[ln2 ξ + ln2 ξL ] (see the remark after Lemma 2.2). This gives ∞ P−1/2 (ξ) 2 [ln ξ + ln2 ξL ] dξ T β (ξ0 , ξ; x ) |D2 (n0 )| ≤ O(1/L) P−1/2 (ξ0 ) 1 ≤ O(1/L)[ln2 ξ0 + ln2 ξL ] ,

(4.29)

as asserted. On account of (4.28) the integrand I(ξ0 ) in (4.27) vanishes pointwise for L → ∞. Using (4.28), assuming Conjecture 2.5 and recalling that P−1/2 (ξ0 ) ≤ √ √ ξ0 / L) (1 + ln ξ0 )/ ξ0 we can bound I(ξ0 ) by O(1/L)Tβ (1; L)ξ0−1 (1 + ln ξ0 )2 E(ln √ [ln2 ξ0 + ln2 ξL ]. Changing now the integration variable to t := (ln ξ0 )/ L the new integrand is bounded by 1 ln2 ξL )E(t) (4.30) F (t) := const (t + √ )2 (t2 + L L and the right-hand side is bounded uniformly in L by an integrable function. By the dominated convergence theorem we can interchange the limit with the integration and conclude that d1,2 → 0 for L → ∞, completing the proof of Lemma 4.4. This lemma, combined with (4.16), reduces the computation of the thermodynamic limit for multipoint functions to that of one-point functions: from Eqs. (4.12), (4.16) it follows that the thermodynamic limit of a multipoint observable can be computed as lim OL,β,α = lim O 0,∞ (n0 ) ,

L→∞

n0 →∞

(4.31)

whenever the limit exists. We claim that for all O ∈ CT ainv the limit does exist and is given by the rhs of Eq. (4.13). To see this we return to (4.17) and swap the order of K O and P : λβ (0)−x (Tx1 K O P )(n0 , n↑ ) O0,∞ (n0 ) = P−1/2 (n↑ · n0 ) λβ (0)x1 −x = dΩ(n)P−1/2 (n0 · n)K O (n, n↑ ) P−1/2 (n↑ · n0 ) λβ (0)−x dΩ(n) Tβ (n0 · n; x1 )[K O , P ](n, n↑ ) . + (4.32) P−1/2 (n↑ · n0 )

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We now take the SO↑(2) average wrt n0 . In the first term the n0 dependence then drops out by (A.12c) and produces the announced result. For the second term we use the defining bound (4.10) and distinguish between x1 = 0 and x1 = 0. In the first case a bound on the second term is λβ (0)−x p1 p(ξ0 )/P−1/2 (ξ0 ), which vanishes for ξ0 → ∞. For x1 = 0 we bound the integral by 2πp1 dξ T β (ξ0 , ξ; x)p(ξ), using the definition (4.10). To this integral we apply Lemma 2.2(iv) to get constp(ξ0 ), which again vanishes for ξ0 → ∞. This completes the proof of Proposition 4.3(ii). Let us add a number of comments on (4.13), (4.12). First one should note that the thermodynamic limit can be computed explicitly for all of CT ainv without having to select ‘fine-tuned’ subsequences, i.e., without recourse to the BanachAlaoglu theorem. Second one observes that translation invariance is restored in the thermodynamic limit even though for O ∈ CT ainv the finite volume expectations are not translation invariant. Third, just as for translation invariant observables the expectations (4.13), (4.12) will in general not be SO(1, 2) invariant. An exception ↑ ⊂ CT ainv to be discussed below. are observables in a subclass Cainv Just as CT inv contained the SO(1, 2) invariant observables Cinv as special cases, here there is a subspace Cainv of observables which decay sufficiently fast to an SO(1, 2) invariant one after averaging over SO↑(2). We denote the limiting observable by (4.33) O∞ (n1 , . . . , n ) := lim O(An1 , . . . , An ) , A→∞

and specify the rate of approach to the limit below. Provided the limit exists it will automatically be SO(1, 2) invariant. For example one can build a large class of Cp↑ observables satisfying (4.33) by replacing in a function of ni · nj each ni · nj with ni ·nj f (ni , nj ) or with ni ·nj + f (ni , nj ), for some SO↑(2) – but not SO(1, 2) – invariant function f that goes to a constant in the limit. Note that the dependence on the invariant part may correspond to an unbounded function. In addition any dependence on the n0i is allowed, constrained only by the requirement that the limit (4.33) exists. Of course Cainv contains the SO(1, 2) invariant observables Cinv as a proper subset. For observables depending only on one spin ( = 1) asymptotic translation invariance just reduces to the existence of the limit in (4.33), as it did for CT ainv observables with = 1. For > 1 we specify the rate of approach in which the limit in (4.33) is reached in terms of the kernels K O as follows, thereby giving a technically precise version of Definition 3.3(ii): Definition 4.5 O ∈ Cp is called asymptotically invariant, O ∈ Cainv , iff after SO↑(2) averaging the associated kernel obeys

O

K (n, n ) − K O∞ (n · n ) ≤ p(n↑ ·n) p(n↑ ·n ) , with K O∞ (n1 · n ) := lim K ρ(A)O (n1 , n ) = lim K O (A−1 n1 , A−1 n ) , A→∞

where p(ξ) is as in (4.10).

A→∞

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Note that in analogy with (4.11) this implies limA→∞ ρ(A)[K O , ρ] = 0, for O ∈ Cainv , which was used as the defining property in 3.3(ii). Further Cainv ⊂ CT ainv .

(4.34)

To see this one writes [K O , TL ] = [(K O − K O∞ ), TL ] + [K O∞ , TL ]. The second commutator vanishes because SO(1, 2) invariant observables are translation invariant. The kernel of the first commutator is bounded in modulus by P−1/2 (n↑ · n) p(n↑ · n )Tβ (1; L). It follows that [K O , P ] satisfies (4.10), which verifies (4.34). It follows that the formulae (4.13), (4.12) are valid also for observables in O Cainv . Moreover the kernel K can in fact be replaced with the invariant limiting kernel K O∞ . Proposition 4.6 (i) For a multi-point observable O ∈ Cainv , ≥ 2: ∞ O(nx1 , . . . , nx )∞,β,α = λβ (0)x1 −x 2π dξ K O∞ (ξ) P−1/2 (ξ) .

(4.35)

1

↑ (ii) On Cainv the expectation functional O → O∞,β,bc is an invariant mean.

Proof. (i) We decompose K O again as K O = K O∞ +(K O −K O∞ ). For the invariant limiting kernel the manipulations proceed as for the translation invariant case and yield Eq. (4.13) with the indicated replacement. The average of the remainder by Q(1; L + x1 )Q(n↑ ·nL ; L − x )/Tβ (n↑ ·nL ; 2L) with K O − K O∞ can ∞be bounded Q(ξ; x) := 2π 1 dξT β (ξ, ξ ; x)p(ξ ), for x ∈ N. The bound vanishes in the limit L → ∞. (ii) This is a direct consequence of (4.35). Remark 1. For later reference we note again that this reasoning remains valid if the lower boundary in the Q integrals was replaced with an arbitrarily large constant Λ 1. Remark 2. The reason why the expectation functionals do no not provide an invariant mean for all of Cainv is that the SO↑(2) averaging effected by the expectations does not commute with the SO(1, 2) action. As a consequence observables ↑ in Cainv \ Cainv will typically signal spontaneous symmetry breaking. See (3.26) and the examples in Section 4.3. Likewise the hyperclustering (4.9) for SO(1, 2) invari↑ ant observables trivially generalizes to the class Cainv but fails in general for Cainv : ↑ have support as in the premise of (4.9) the limit of the products if A, B ∈ Cainv equals the product of the limits, i.e., (AB)∞ = A∞ B∞ , and to the latter (4.9) applies. A counterexample to hyperclustering in Cainv will be given in Section 4.3. We can summarize these results by saying that the thermodynamic limit effectively projects CT ainv onto CT inv and Cainv onto Cinv , i.e., the top row in the diagram (3.9) is projected onto the bottom row. In the first case translation invariance emerges but SO(1, 2) invariance is in general still absent, while in the second

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case, given SO↑(2) invariance as a ‘seed’, both properties emerge. The second result is more interesting because SO(1, 2) is not amenable, so one could not ‘by hand’ switch to invariant states by group averaging of noninvariant ones. (Presumably this is still true if one adopts a distributional group averaging as in [34, 35].) Rather the thermodynamic limit itself defines a partial invariant mean, that is ↑ gets averaged to the subclass of (bounded as well as unbounded) observables Cainv yield a SO(1, 2) invariant result. An invariant mean in the proper sense would do the same for all continuous bounded observables Cb↑ , but it cannot exist on general grounds.

4.3

The support of the functional measures

Here we discuss the results (4.12) and (4.35) in more detail. Both properties express a partial symmetry restoration and are due to a remarkable concentration (actually rather dilution) property of the underlying functional measures. Roughly speaking the measures have their support concentrated at configurations that are boosted from the origin by an amount growing at least powerlike with the number of sites; the measure of any bounded set of configurations goes to zero in the thermodynamic limit. The derivation of (4.12) given below explicitly makes use of this concentration property; the previous derivation of (4.35) did for technical reasons not explicitly rely on it. We shall explain later why the underlying concentration property is nevertheless visible in the derivation. In Section 5.2 we shall also describe an alternative proof of (4.35) which links it explicitly to the concentration property of the 1-spin measures instrumental for (4.12). This concentration property is due to the large fluctuations present in D = 1, which in compact models are the ‘enforcers’ of the Mermin-Wagner theorem, but which are here insufficient to restore the symmetry. We begin with re-evaluating the thermodynamic limit for asymptotically translation invariant observables depending only on a single spin. Take some O(nx ) ∈ CT ainv (which for = 1 coincides with C ainv by definition) depending on a single spin at site x only. By definition its SO↑(2) average has a limit as nx → ∞. We claim that this limiting value coincides with the thermodynamic limits of the O(nx ) expectation. The mechanism behind this is that the relevant measures have support ‘mostly at infinity’. To make this precise, recall that in view of (3.13) and (3.17) the spin n := nx is for finite L distributed according to the probability measures dµL,β,α,av (n; x) =

Tβ (n↑ · n; L + x)T β (n↑ · n, chα; L − x) dΩ(n) , Tβ (chα; 2L) for twisted bc

dµL,β,free(n; x) = Tβ (n↑ · n; L + x) dΩ(n) , for free bc .

(4.36)

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In the first case T β is defined as in (2.28). From Eqs. (2.27), (2.36) and (2.31) one sees that the densities multiplying dΩ behave for large L as 2 L−3/2 P−1/2 (n↑ · n) , for twisted bc, λβ (0)L+x L−3/2 P−1/2 (n↑ · n) ,

for free bc ;

(4.37)

(the approach to this asymptotic form is, however, very nonuniform in n0 , as can be seen from Eqs. (2.35),(2.38)). In particular the dependence on chα drops out in the first case. Both expressions in (4.36),(4.37) vanish pointwise in the limit but are not integrable. This implies Lemma 4.7 For any bounded subset M ⊂ H and for twisted as well as free bc dµL,bc = 0 , (4.38) lim L→∞

M

where dµL,bc stands for either of the measures in (4.36). As a consequence these measures do not have a limit as L → ∞; they ‘spread out’ over H (though not evenly); Lemma 4.7 may be interpreted as saying that the measure is getting concentrated more and more near infinity. The measures dµL,β,bc form a sequence of bounded, normalized linear functionals (‘states’) on the space Cb . By the theorem of Banach-Alaoglu [30] there is therefore a subsequence convergent to such a functional – a so-called ‘mean’; see, e.g., [4]. Because SO(1, 2) is not amenable, this mean cannot be invariant. We will give below explicit examples of elements of Cb that show this non-invariance, i.e., spontaneous symmetry breaking. However 1-spin observables invariant under SO↑(2) still have a unique thermodynamic limit, which is independent of x and β, see Proposition 4.3(i). In view of Lemma 4.7 this expresses the fact that the thermodynamic limit effectively projects a one spin observable onto the ‘boundary at infinity’ of the hyperbolic plane; for SO↑(2) invariant functions we may use the one-point compactification of H so that there is only one such boundary point at infinity. It is also instructive to estimate the size of the ‘cup’ in the hyperboloid whose contribution to the functional integral is negligible. We integrate the observable under consideration with the pointwise vanishing density in (4.37) over the compact domain {n ∈ H| n↑ ·n ≤ Λ(L)}. Demanding that the contribution of this domain still vanishes in the limit L → ∞ constrains the permitted growth of Λ(L) with Λ(L) Λ(L) dξP−1/2 (ξ)2 ∼ d ln ξ(ln ξ)2 ∼ L. For twisted bc the relevant integral is 3 (ln Λ(L)) , using (4.37) and the asymptotics in (A.13). Thus any Λ(L) satisfying ln Λ(L) = o(L1/2 ) will still give a contribution vanishing in the limit L → ∞. For Λ(L) dξP−1/2 (ξ) ∼ Λ(L) ln Λ(L). Thus any growth free bc the relevant integral is 2 3/2 Λ(L) = o(L / ln L) is allowed. To conclude our discussion of 1-point functions let us consider some examples. A simple example of a bounded SO↑(2) invariant observable is O(n) = tanh(n↑ ·

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n). Then (4.12) gives 1 for the thermodynamic limit of its expectation, which in particular is SO(1, 2) invariant. The spin field nax itself is neither bounded nor SO↑(2) invariant. However by the SO↑(2) invariance of the measures (4.36) one has nax L,β,bc = δ a0 n−L · nx L,β,bc ,

(4.39)

leaving only the SO↑(2) invariant part O(nx ) = n↑ · nx = n0x to study. Slightly generalizing the above discussion it follows that the n0x expectation with both twisted periodic and free bc diverges for L → ∞: for any constant Λ, the measure of the (compact) subset of H where |n0x | ≤ Λ goes to 0; since the total weight of the measure is always 1, the expectation value will eventually be larger than Λ(1 − ) for any > 0. For the rhs in (4.39) this is also illustrated by the numerical results for the 2-point functions shown in Fig. 2. In both of these examples the limit is SO(1, 2) invariant and does not signal spontaneous symmetry breaking. As seen before the computation of the thermodynamic limit for multi-point observables can be reduced to that of 1-point functions. Nevertheless it is instructive to outline the origin of the concentration property also for the multi-point measures. For simplicity we restrict attention to twisted bc. The counterpart of the normalized measures (4.36) for > 1 are (after integrating out nx2 , . . . , nx−1 ) dµL,β,α (n1 , n ; x1 , x ) =

(4.40)

Tβ (n↑ · n1 ; L + x1 )Tβ (n1 · n ; x − x1 )T β (chα, n↑ · n ; L − x ) dΩ(n1 )dΩ(n ) . Tβ (chα; 2L) The finite volume expectation of some observable O can be written in terms of these measures as K O (n1 , n ) . (4.41) OL,β,bc = dµL,β,bc(n1 , n ) Tβ (n1 · n ; x − x1 ) The asymptotics of the density in (4.40) is λβ (0)x1 −x P−1/2 (n↑ · n1 ) Tβ (n1 · n ; x − x1 )P−1/2 (n↑ · n ) L−3/2 .

(4.42)

This density vanishes pointwise as L → ∞ and is integrable wrt one but not wrt both variables. As before the limit of the measures therefore only exists as a mean. The concentration property ensued by (4.42) is however more subtle than for the 1-point measures. This is because invariant combinations like n1 · n contribute even for highly boosted individual n1 and n . Conditions like (asymptotic) translation invariance or (asymptotic) SO(1, 2) invariance allow one to isolate the invariant contribution by swapping the order of K O and TL+x1 while implying that the commutator does not contribute to the invariant part. In order to illustrate the mechanism we set k(n↑ ·n1 ) := sup n

O

K (n↑ · n1 , n↑ · n ) . Tβ (n1 · n ; x − x1 )

(4.43)

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Clearly |k(ξ1 )| ≤ O. If O and hence K O is SO(1, 2) invariant, k(ξ1 ) equals a constant. If K O does not contain a SO(1, 2) part the function k(ξ1 ) vanishes for ξ1 → ∞. Then dµL,β,α (n1 , n ; x1 , x )

n↑ ·n1 1/2, for which the integral in (4.44) approaches a finite but nonzero constant as L → ∞. The upshot is that in the original (n1 , n ) integral over H × H only the region n↑ ·n1 ≥ Λ1 (L), with Λ1 (L) as in (4.45), contributes significantly to the result for the average as L becomes large. For free bc the analysis is similar, except that the change in the rate of decay also involves powers of λβ (0). We omit the details and simply state that one can likewise allow the cutoff Λ1 to grow at least powerlike in L, without affecting the limit formulas.

4.4

Examples

We begin with some examples where a finite thermodynamic limit does not necessarily exist, like for the components of the spin field or of the Noether current.

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The individual components of the energy observable a EL,β,bc := nax nax+1 L,β,bc,

a = 0, 1, 2,

can be shown to diverge for L → ∞ by an argument similar to the one used in Section 4.2. On the other hand the invariant combination (E 0 − 2E 1 )L,β,bc has a finite limit given by (4.8). Next consider the Noether current Jxa = β(nx × nx+1 )a , where n× n denotes the SO(1, 2) invariant vector product of n, n ∈ H. (Explicitly (n × m)a = η aa a bc nb mc , with abc totally antisymmetric and 012 = 1.) For the current two-point function one finds Jx0 Jy0 L,β,bc = 0 ,

for x < y ,

1 Jx1 Jy1 L,β,bc = Jx2 Jy2 L,β,bc = − Jx · Jy L,β,bc , 2

for x < y , (4.46)

so that all components have a finite L → ∞ limit. The first equation is a special case of the more general result Jx01 O(nx2 , . . . , nx )L,β,bc = 0 ,

for x1 < x2 < · · · < x ,

(4.47)

which is obtained by specializing the general formulas (3.12), (3.16) and then using

where nx · nx+1

∂ Tβ (nx · nx+1 ; 1) = −Jx0 Tβ (nx · nx+1 ; 1) , (4.48) ∂ϕx 2 = ξx ξx+1 − ξx2 − 1 ξx+1 − 1 cos(ϕx − ϕx+1 ). Since Jx0 is es-

sentially the Noether charge generating infinitesimal SO↑(2) rotations (see below) Eq. (4.47) expresses the SO↑(2) invariance of the ‘ground states’ 1 and ψ↑ (n), respectively. Conversely, the fact that correlators involving Jx1 , Jx2 are non-zero is yet another manifestation of the SO(1, 2) symmetry breaking. Ward identities expressing the invariance of the measure and of the action can be derived along the familiar lines. For example one has a ) nby L,β,bc + δx,y (ta ny )b L,β,bc , (Jxa − Jx+1

(4.49)

with (ta )dc = −η aa η dd a d c . Replacing nby with a generic (non-invariant) observa able O(nx1 , . . . , nx ) a similar identity arises where the correlator with Jxa − Jx+1 produces a sum of contact terms. As is clear from (4.49) these linear Ward identities will in general not have a non-boring thermodynamic limit. In particular no conflict, even in spirit, with Coleman’s theorem [36] arises. Ward identities where the current enters nonlinearly can likewise be derived but are hampered by the fact that the ‘response’ are in general functions which fail to be translation invariant. In 2 or more dimensions a useful quadratic Ward identity can be derived which relates the components of the longitudinal part of the current-current correlator to the energies E a ; see [37]. In one dimension only

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the longitudinal part exists and only the SO(1, 2) invariant – and hence translation invariant – combination of these component Ward identities is useful. It reads Jp · J−p L,β,bc + 2β(E 0 − 2E 1 )L,β,bc = 0 ,

∀p = 0 ,

(4.50)

where Jpa = x e−ipx Jxa , with p = 2πn/(2L + 1), n = 0, . . . , 2L. Next we consider some examples of asymptotically translation invariant observables. They also serve to highlight the significance of the SO↑ (2) averaging in the definition of the algebras in (3.9). Recall that Cainv = {O ∈ Cp | SO↑ (2) average ↑ lies in Cainv }. The point here is that in general O and ρ(A)O, A ∈ SO(1, 2), will ↑ ↑ have different SO↑ (2) invariant images in Cainv . The elements of Cainv \ Cainv will therefore typically signal spontaneous symmetry breaking although by Section 3.3 ↑ they get effectively projected back into Cainv . An instructive example of such a ‘symmetry breaking observable’ in Cainv arises as follows: given a spacelike unit vector e = ( q 2 − 1, q sin γ, q cos γ) we define Te (n)

:=

T q (ξ)

:=

tanh(n · e) ∈ Cainv , π 1 ↑ dϕ tanh ξ q 2 − 1 − q ξ 2 − 1 cos ϕ ∈ Cainv . (4.51) 2π −π

The observable Te (n) indeed enjoys the property (4.34): after SO↑(2) averaging it has a unique limit T q (∞), which can be obtained by acting with a sequence of SO(1, 2) transformations going to infinity. This limit does not depend on n any more, so in a trivial sense it is an invariant function of the spins. It does, however, depend on e or rather on the scalar product n↑ · e and is therefore not invariant under the action of SO(1, 2) on the original observable. Spontaneous symmetry breaking is shown by the following Proposition 4.8 For all bc considered Te (nx )∞,β,bc = T q (∞) = 1 −

2 arccos 1 − q −2 ; π

(4.52)

this expectation value is manifestly not invariant under SO(1, 2): for a general A ∈ SO(1, 2) one has Te (nx )∞,β,bc = Te (Anx )∞,β,bc . Proof. We use the fact that in finite volume expectations we may replace Te (nx ) by it average over SO↑(2) rotations T q (n0x ). The argument of the tanh, i.e., αξ (ϕ) := ξ( q 2 − 1 − q 1 − ξ −2 cos ϕ), then has its minimum at ϕ = 0 and its maximum at ϕ = ±π. Because e is spacelike, there is a ξ0 (q) such that for all ξ > ξ0 (q) the minimum αξ (0) is negative, the maximum αξ (±π) is positive and there are two zeros at ϕ = ± arccos[(1 − q −2 )/(1 − ξ −2 )]1/2 whose modulus converges to

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ϕ0 := arccos(1 − q −2 )1/2 . This implies that   |ϕ| > ϕ0 1, lim tanh αξ (ϕ) = −1, |ϕ| < ϕ0  ξ→∞  0, |ϕ| = ϕ0 .

1063

(4.53)

By the dominated convergence theorem we can pull the limit ξ → ∞ under the integral for the ϕ averaging and obtain 2 lim T q (ξ) = 1 − arccos 1 − q −2 . (4.54) ξ→∞ π The result then follows from Eq. (4.12).

A large class of observables in Cainv can now be built by algebraic operations. Of course sums and products of Te (n) at the same or different sites will lie in Cainv , but so will be algebraic combinations built from elements of Cinv . The crucial ↑ is that hyperclustering and even ordinary clustering will now difference to Cainv fail in general. This is because ‘SO↑(2) averaging’ and ‘taking the A → ∞ limit’ in (4.33) are noncommuting operations in general. A simple example is given by the product of two tanh-observables (4.51), where 1 = Te (n)2 ∞,β,bc = Te (n)2∞,β,bc = T q (∞)2 ,

(4.55)

from (4.53) and (4.54). So we have so far found observables that show hyperclustering and others that do not cluster at all. Observables showing ordinary (exponential or powerlike) clustering presumably also exist in the large space Cb , but it is more difficult to find explicit examples.

5 Reconstruction of a Hilbert space and transfer operator The Osterwalder-Schrader type reconstruction allows to reconstruct a Hilbert space and a transfer matrix from expectation values satisfying reflection positivity as well as translation invariance. The original expectation values are recovered as expectations in a genuine, i.e., normalizable ground state vector. This construction is well documented in the literature [38, 39, 40], but in our case there are peculiarities and surprises. For this reason we describe in some detail how the construction works here. First there is a rather harmless complication: reflection positivity for reflections both in lattice sites and in midpoints between lattice points is equivalent to positivity of the transfer operator; as we found in the beginning, however, this does not hold in our case. But we still have reflection positivity for reflection in lattice points, at least if we take the thermodynamic limit with periodic bc, and this is enough for the reconstruction of the Hilbert space and a positive two-step transfer matrix.

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There is a much more serious complication: as stated above, the reconstruction produces a ground state in the proper sense, whereas we know that the original transfer matrix T on L2 (H) does not have such a ground state. So it is unavoidable that there is some discrepancy between the reconstructed quantum mechanics and the one we started from. This mismatch is also related to the fact that our expectation functional in the infinite volume is not given by a measure, but only a mean on the configuration space. In this section we consider periodic bc exclusively and denote the expectation functional (the state) . L,β,0 in Eq. (3.12) by ωL ( . ). A reconstruction in the usual sense won’t work for twisted or free bc because the x ≥ 0 and the x ≤ 0 halves of the chain have to enter symmetrically. For the algebra we take Cb in order to have the usual concept of a state available. For Cb ∩ CT ainv we saw before that the thermodynamic limit is explicitly computable and translation invariant. For the rest of Cb a thermodynamic limit exists likewise, though it may be necessary to select subsequences and to average over translations in order to have it translation invariant. We denote such a weak limiting state by ω∞ ( . ) = w − limL→∞ ωL ( . ). We denote by C+ (C− ) the subalgebra of bounded observables Cb depending only on the spins nx with x ≥ 0, (x ≤ 0) and C0 = C+ ∩ C− . Our chain admits a reflection x → −x and we introduce an antilinear time reflection ϑ acting on on Cb by replacing any function O by the same function of the reflected arguments and taking the complex conjugate: (ϑO)(n−x−1 , . . . , n−x0 ) = O(n−x0 , . . . , n−x−1 )∗ ,

x0 < · · · < x −1 ,

(5.1)

where the asterisk denotes complex conjugation. To interpret this formula correctly note that on the lhs we have written the observable ϑO in the customary form as a function of the spins on which it actually depends, in the order of increasing indices. On the rhs O is to be read as a function of spins, with the displayed arguments now appearing in the order of decreasing indices. For example O = n1 ·n3 +c n↑ ·n3 gives ϑO = n−1 · n−3 + c∗ n↑ · n−3 . We discuss the reconstruction first for a finite and then for an infinite chain.

5.1

Finite chains

Recall that we adopt untwisted periodic bc, n−L = nL = n↑ , and consider a chain of total length 2L + 1. We begin by assigning to each O ∈ C+ an element O0,L ∈ C0 with the same expectation value via dΩ(ni )O(n1 , . . . , n ) Tβ (ni−1 · ni ; xi − xi−1 ) O0,L (n0 ) := i=1

=

i=1

Tβ (n · n↑ ; L − x ) × Tβ (n0 · n↑ ; L) dΩ(n) (Tx1 K O )(n0 , n)

Tβ (n↑ · n; L − x ) , Tβ (n↑ · n0 ; L)

(5.2)

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where x0 = 0 and the first transfer matrix is to be interpreted as the identity operator if x1 = 0. Note the properties |O0,L (n)| ≤ O ,

(1)0,L (n) = 1 ,

(ρ(A)O)0,L (n) = O0,L (A−1 n)

↑ −1 n →A

n↑

(5.3)

where we denote by 1 the unit element of C+ . Further we set Tβ (n · n↑ ; L) ψ O (n) := O0,L (n) . Tβ (1; 2L)

(5.4)

The expressions (5.2) and (5.4) are designed such that 1 ωL ([ϑO] O) = dΩ(n) |O0,L (n)|2 Tβ (n · n↑ ; L)2 = dΩ(n) |ψ O (n)|2 , Tβ (1; 2L) (5.5) holds, as one can verify from (3.12). In particular reflection positivity ωL ([ϑO]O) ≥ 0 ,

∀ O ∈ C+ ,

(5.6)

is manifest. With these preparations at hand the reconstruction of the Hilbert space HL for a finite chain works as usual: a positive semidefinite scalar product is introduced on C+ by (A, B)L := ωL ([ϑB] A) ; (5.7) there will be a nontrivial null space N of elements with ωL ([ϑO] O) = 0 . The Hilbert space HL is then the completion of the quotient space C+ /N with respect to the norm induced by ωL . The necessity to divide out N becomes clear if one notices that for any O ∈ C+ one can find a unique element O0,L ∈ C0 such that O − O0,L ∈ N , namely just the one given in (5.2). The uniqueness follows from (5.5), which implies C0 ∩ N = {0}. Note that the OS norm for O coincides with the L2 -norm for ψ O . The above construction makes it manifest that for a finite chain there is a natural isometry between the reconstructed Hilbert space HL and the original L2 (H): HL turned out to be the completion of C0 with respect to the norm induced by (5.7), i.e., HL = C 0 . Note that although C0 is the universal L-independent space of bounded continuous functions on H, its completion with respect to ( , )L depends on L. Of course the L-dependence is of a rather trivial nature in that by (5.4) the map VL : HL −→ L2 (H) ,

Tβ (n · n↑ ; L) (VL ψ)(n) = ψ(n) , Tβ (1; 2L)

(5.8)

defines an isometry between Hilbert spaces. Alternatively HL could be regarded as the preimage of L2 (H) with respect to VL . It is worth noting that HL is by itself

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a commutative C ∗ -algebra, so the reconstruction of the Hilbert space can be considered as an instance of the well-known Gel’fand-Na˘ımark-Segal reconstruction, see, e.g., [1, 41]. To sum up, for a finite chain the original Hilbert space L2 (H) and the reconstructed one HL can really be identified. Unsurprisingly, for a finite chain HL also carries a unitary representation ρL of SO(2, 1) (spontaneous symmetry breaking can only arise in the thermodynamic limit); it is obtained simply by conjugating the representation ρ with VL : ρL = VL−1 ρVL .

(5.9)

Explicitly for A ∈ SO(2, 1) and ψ ∈ C0 this gives (ρL (A)ψ)(n) = ψ(A−1 n)

Tβ (A−1 n · n↑ ; L) . Tβ (n · n↑ ; L)

(5.10)

For O ∈ C+ we define (ρL (A)O)(nx0 , . . . , nx−1 ) = O(A−1 nx0 , . . . , A−1 nx−1 )

Tβ (A−1 nx−1 · n↑ ; L − x −1 ) , (5.11) Tβ (nx−1 · n↑ ; L − x −1 )

which is compatible with (5.10) and induces it via (5.2) in that (ρL (A)O)0,L (n) = (ρL (A)O0,L )(n), for all A ∈ SO(1, 2). This also ensures that ρL maps elements O − O0,L of N onto other elements of zero norm. The rhs of (5.10), (5.11) is in general no longer a bounded function of n because the asymptotics of Tβ (A−1 n · n↑ ; L) and Tβ (n · n↑ ; L) do not match, but it is of course still an element of HL with the same norm as ψ. Likewise by (5.10), (5.11) the completion NL of N wrt ωL is mapped onto itself under ρL . In preparation of the thermodynamic limit let us consider the action of ρL on the function 1 (an approximate ground state for large L): (ρL (A)1)(n) = The scalar product (ρL (A)1, ρL (B)1)L

1 = Tβ (1; 2L) =

Tβ (A−1 n · n↑ ; L) . Tβ (n · n↑ ; L)

(5.12)

dΩ(n)Tβ (n · An↑ ; L)Tβ (n · Bn↑ ; L)

Tβ (An↑ · Bn↑ ; 2L) , Tβ (1; 2L)

(5.13)

then has the finite and nonzero limit P−1/2 (An↑ ·Bn↑ ), as L → ∞. For finite L the state ωL ( · ) will not be translation invariant outside the subalgebra CT inv ∩ C+ . As a consequence there is no reconstructed transfer matrix for finite L. Conversely this provides an intrinsic reason to consider the reconstruction based on the expectations of the infinite chain.

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Thermodynamic limit

Let us thus turn to the thermodynamic limit ω∞ = w − limL→∞ ωL . Reflection positivity remains true in this limit; so one can still define a scalar product and a null space N as for the finite chain. As before a Hilbert space HOS can be constructed as the completion of C+ /N = C0 /(N ∩ C0 ). In order not to clutter the notation we continue to use the same symbols for the algebra of observables and the spaces N etc., however one should keep in mind that the spaces C+ etc. for a finite and for the infinite chain cannot be identified. In particular equation (5.4) loses its meaning in the limit: the left-hand side goes to zero pointwise, even though its norm in HOS in general does not. For observables in C+ ∩ CT ainv the explicit formula (4.13) can be used to compute the inner products ( , )OS . Outside this class in general the original definition ( , )OS = limL→∞ ( , )L has to be used. On the other hand (5.2) always has a sensible limit: Proposition 5.1 (i) The limit limL→∞ O0,L (n0 ) =: O0,∞ (n0 ) exists and obeys P−1/2 (n↑ · n) λβ (0)−x . (5.14) O0,∞ (n0 ) := dΩ(n) (Tx1 K O )(n0 , n) P−1/2 (n↑ · n0 ) (ii) 10,∞ = 1 and |O0,∞ (n0 )| ≤ O, where · denotes the sup norm. (iii) If Conjecture 2.5 holds, O − O0,∞ ∈ N ,

(5.15)

with respect to ( , )OS . Proof. (i) and (ii) are straightforward. (iii), while very plausible, requires nevertheless a proof; the one given here relies on the validity of Conjecture 2.5. It suffices to show that for any A ∈ C− lim ωL (A(O − O0,∞ )) = 0 .

L→∞

(5.16)

In order to show this, we write ωL (A(O − O0,∞ )) (5.17) = ωL (A(O − O0,L )) + ωL (A(O0,L − O0,∞ )) + (ω∞ − ωL )(A(O − O0,∞ )) . The first term vanishes by construction of O0,L , the third term goes to zero as L → ∞ by definition of ω∞ , whereas the second term requires a closer look. In view of

T (n · n↑ ; L)2

β 0

,

ωL (A(O0,L − O0,∞ )) ≤ A dΩ(n0 ) O0,∞ (n0 ) − O0,L (n0 )

Tβ (1; 2L) (5.18) the difference |O0,∞ (n0 ) − O0,L (n0 )| needs to be examined. This however has been done in Section 4.2, and the proof that the right-hand side of (4.19) vanishes for L → ∞ carries over. This completes the proof of (iii).

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The relation (5.15) has several important consequences, which we discuss consecutively. For bounded observables in CT ainv a crucial consistency condition arises from (5.15) and (4.13). Since an explicit formula for the state ω∞ = ∞ is known for these observables it must come out that AO∞ = AO0,∞ ∞

∀A ∈ C− , C+ ∈ O ,

(5.19)

using directly the limiting formulae (4.13) and (5.14). This is indeed the case: a computation shows that both sides of (5.19) reduce to A O O λβ (0)x< −x> dΩ(n)dΩ(n )K A (n↑ , n)Tβ (n · n ; xA > − x< ) ×

dΩ(n )K O (n · n )P−1/2 (n↑ · n ) .

(5.20)

O Here we wrote xO < for the leftmost and x> for the rightmost site where an observable O ∈ C is supported. The consistency condition for O ∈ CT ainv is therefore satisfied. On the other hand (5.15) is valid for all bounded observables and the computation leading to (5.20) does not seem to leave much room for expressions other than (4.13) having the same property (5.19). This suggests that (4.13) is actually valid for all bounded observables though our proof is not. Next let us consider asymptotically invariant observables. For them the result (5.22) yields an alternative derivation of (4.35). Since it is based on (5.22) this derivation highlights that the origin of the result (4.35) lies in the concentration property of the measures described in Section 4.3. To this end we write K O as K O∞ + (K O − K O∞ ) and insert into the definition of O0,∞ (n0 ). Since (5.23) is trivially satisfied for the integral operators coming from SO(1, 2) invariant operators the first term gives (4.13) with K O replaced by K O∞ , which is the asserted result. Using (4.34) the modulus of the second term can be bounded by

λβ (0)−x

Q(ξ0 , x1 ) , P−1/2 (ξ0 )

(5.21)

where ξ0 = n↑·n0 and Q(ξ0 ; x1 ) is defined in after Eq. (4.35). According to (5.22) we have to analyze the limit n0 → ∞ of this expression. To this end we split the region of integration in Q into a bounded part ξ ∈ [1, Λ] and a remainder ξ ∈ [Λ, ∞[. For the unbounded part we use Tβ (ξ0 , ξ ; x1 ) ≤ P−1/2 (ξ0 )P−1/2 (ξ ) Tβ (1; x1 ) to get a n0 independent bound p1 /Λ on it. In the bounded part we use the fact that Tβ (ξ0 , ξ ; x1 ) vanishes faster than any power in ξ0 , and does so uniformly for all ξ ∈ [1, Λ]. For large enough ξ0 the supremum sup[Tβ (ξ0 , ξ ; x1 )/P−1/2 (ξ0 )] over ξ ∈ [1, Λ] can be therefore be made smaller than 1/Λ2 . The upshot is that (5.21) can be made smaller than any prescribed quantity. This completes the derivation of (4.35) based on (5.22).

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A simple consequence of (5.15) is that in contrast to the finite volume case C0 now also intersects the null space N : for instance all functions going to zero for n → ∞ (i.e., n↑ ·n → ∞) will be mapped into the null vector of HOS , according to the previous section. The same is true for all functions that go to zero for n → ∞ after averaging over SO↑(2). In fact, according to the discussion in Section 4.2, this exhausts the intersection N ∩ C0 . Likewise products of the form c(n)ψ(n) ∈ C0 , where c(n) → c for n → ∞, differ from cψ(n) only by an element of N , since their difference goes to zero as n → ∞. This means that in linear combinations of vectors constant coefficients can always be replaced with coefficients satisfying this decay condition without changing the equivalence class mod N . Setting A = 1 in (5.17) and using the results of Section 4.3 one infers ω∞ (O) = ω∞ (O0,∞ ) = w − lim O0,∞ (An↑ ) A→∞

for all O ∈ Cb .

(5.22)

The weak limit arises because for a 1-point observable only the behavior at infinity, defined through some unbounded sequence of A’s, is relevant. This limit does not necessarily exist, however as the O0,∞ (An↑ ) form a bounded sequence in R one can always select a convergent subsequence. As shown in Section 4.2 the limit does exist for all O ∈ CT ainv without taking subsequences and is given by (4.13). For the following discussion it is convenient to introduce a somewhat smaller class of observables which we call P -invariant: O ∈ CP inv

iff P K O = K O P .

(5.23)

One has CT inv ⊂ CP inv ⊂ CT ainv .

(5.24)

The second inclusion is trivial; the first inclusion follows by taking the x → ∞ limit Tβ (1; x)−1 [K O , Tx ] = 0. The condition (5.23) is chosen such that O0,∞ (n0 ) is independent of n0 , so that by (4.32) the value of O0,∞ directly coincides with the thermodynamic limit of O ∈ CP . By a computation similar to the one in (4.32) one shows from (5.22) that for separately SO↑(2) invariant A, B ∈ CP↑ inv one has the ‘hyperclustering’ relation ω∞ (A B) = ω∞ (A) ω∞ (B) .

(5.25)

Since in general AB = A B this does not extend to all of CP inv . The above properties of CP↑ inv observables render them at the same time uninteresting from the viewpoint of the OS reconstruction. More generally we have Proposition 5.2 Observables O ∈ C+ are mapped onto multiples of the ‘canonical’ ground state ψ0 in HOS if and only if the following ‘hyperclustering relation’ holds. ω∞ ([ϑO] O) = ω∞ (ϑO) ω∞ (O) .

(5.26)

Sufficient conditions for (5.26) to hold are: (i) O 0,∞ (An↑ ) in (5.22) has a unique ↑ ∪ CP↑ inv . (and hence invariant ) limit as A → ∞. (ii) O ∈ Cainv

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Proof. The relation (5.26) is equivalent to O−ω∞ (O) ∈ N being a null vector. This in turn is equivalent to O and ω∞ (O) giving rise to the same vector in HOS . But the latter is a multiple of the ground state, as asserted. The condition (i) is sufficient because the Cauchy-Schwarz inequality then implies ω∞ ([O−ω∞ (O)]A) = 0 for all A ∈ C+ , which for A = ϑ[O −ω∞ (O)] amounts to (5.26). The fact that observables ↑ in Cainv or in CP↑ inv have hyperclustering expectations has been seen before.

5.3

The action of SO(1, 2) on HOS .

Next let us consider the action of SO(1, 2) on the reconstructed Hilbert space. Both (5.10) and (5.11) have well-defined limits for L → ∞ given by (ρ∞ (A)ψ)(n) = ψ(A−1 n)

P−1/2 (A−1 n · n↑ ) , P−1/2 (n · n↑ )

(ρ∞ (A)O)(nx0 , . . . , nx−1 ) = O(A−1 nx0 , . . . , A−1 nx−1 )

ψ ∈ C0 ,

(5.27)

P−1/2 (A−1 nx−1 · n↑ ) , P−1/2 (nx−1 · n↑ ) O ∈ C+ .

Here ρ∞ (A) is a well-defined bounded linear map from C+ onto itself because the quotient P−1/2 (A−1 n · n↑ )/P−1/2 (n · n↑ ) is a bounded continuous function with a bounded inverse. One readily verifies the representation property ρ∞ (A) (ρ∞ (B)ψ)(n) = (ρ∞ (AB)ψ)(n). Further the action on C+ is again compatible with that on C0 and induces it via (5.14), namely: (ρ∞ (A)O)0 (n) = (ρ∞ (A)O0 )(n), for all A ∈ SO(1, 2). In particular this ensures that the null space N and its completion are mapped onto itself under ρ∞ . For clarity’s sake let us add the reminder that for O ∈ C+ the assignment of x −1 ≥ 0 as the index of the last argument on which O actually depends is ambiguous since one may always consider a constant dependence on further arguments; see the comment after Eq. (3.4). Proposition 5.3 (i) The representation ρ∞ of SO(1, 2) on HOS is uniformly bounded and measurable. (ii) It does not act unitarily on all of HOS . Remark 1. Uniform boundedness means that supA ρ∞ (A)ψOS < ∞, measurability of the representation means that the functions A → (ψ1 , ρ∞ (A)ψ2 )OS and A → (ρ∞ (A)ψ1 , ψ2 )OS are measurable wrt the Haar measure on SO(1, 2). Remark 2. The fact (ii) may be surprising at first sight, upon second thought it is not: the inner product ( , )OS is constructed in terms of the limiting expectation functional w−limL→∞ ωL = ω∞ , and we already know that this functional is not ρ invariant for all O ∈ Cb . Of course ρ∞ is different from ρ but it seems ‘unlikely’ that the universal ratio P−1/2 (A−1 n·n↑ )/P−1/2 (n·n↑ ) by which they differ could ‘undo’ the symmetry breaking for all of the relevant observables at the same time. As a consequence the ‘square root’ of a bounded observable signaling the ρ symmetry breaking is likely to give rise to a wave function in C0 on which the unitarity of ρ∞ is violated.

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Proof of Proposition 5.3. (i) By (C.1) in fact ρ∞ (A)ψOS ≤ |ψ|2 ∞ using (C.1). Measurability follows from the fact that for each L the functions A → ωL (ψ1∗ ρ∞ (A)ψ2 ) and A → ωL ((ρ∞ (A)ψ1∗ ) ψ2 ) are in L∞ (SO(1, 2)). By construction of the state ω∞ ( . ) = w − limL→∞ ωL ( . ) the L → ∞ limit of the above functions exists pointwise for almost all A ∈ SO(1, 2) wrt the Haar measure. On general grounds the limiting functions are therefore measurable. As a warning we should add that for generic ψ1 , ψ2 continuity in A may be lost in the limit, as we shall see later. (ii) It suffices to give examples. One class is provided by wave functions only depending on the SO↑(2) phases. Consider ψl (n) := eilϕ , with l ∈ Z and ϕ = arctan(n1 /n2 ). Then 0 = (ψl , ψl )OS = (ρ∞ (A)ψl , ρ∞ (A)ψl )OS ,

l = l .

(5.28)

An example for the square root construction mentioned in Remark 2 is Se (n) := [Te (n)]1/2 ,

(5.29)

where Te (n) is the symmetry breaking observable of Eq. (4.51) and the principal branch of the square root is taken. Then T q (∞) = (Se , Se )OS = (ρ∞ (A)Se , ρ∞ (A)Se )OS 2 P−1/2 (n·An↑ ) = lim Te (A−1 n) , P−1/2 (n·n↑ ) n↑ ·n→∞

(5.30)

for A ∈ SO(1, 2)/SO↑(2) and with the overbar referring to the SO↑(2) average. Of course one could also extend the action of ρ from L2 (H) to C0 and thereby to HOS in the obvious way. It acts, however, uninterestingly: first of all ψ0 is ↑ are mapped onto a multiple of mapped onto itself, likewise all elements of Cainv ψ0 . Thus ρ acts ‘unitarily’ on multiples of ψ0 by not acting at all, and since ↑ symmetry breaking is generic, ρ cannot be expected to outside of the class Cainv act unitarily on sizeable subspaces of HOS . On which subspaces of HOS does ρ∞ act unitarily? Let us introduce the 0 be the closed linear subspace generated by following subsets of HOS : first let HOS the ‘ground state orbit’ {ψ ∈ HOS | ψ = ρ∞ (A)ψ0 , A ∈ SO(2, 1)} ,

(5.31)

α be the closed linear subspace generated by and HOS

{ψ ∈ HOS | ψ = ρ∞ (A)ψα , A ∈ SO(2, 1)} ,

α ∈ R \ {0} ,

(5.32)

0 with ψα (n) = exp(iα n↑ ·n). HOS does not change if we allow the coefficients to be ↑ 0 α from Cainv . HOS and HOS are by construction invariant subspaces of HOS under the action of the representation ρ∞ . It is convenient to introduce the notation

ψn0 ,α (n) :=

P−1/2 (n0 · n) iαn0 ·n e , P−1/2 (n↑ · n)

n0 ∈ H , α = 0 ,

(5.33)

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for the basis vectors; then ρ∞ acts simply by ‘rotating’ n0 , i.e., ψn0 ,α → ψAn0 ,α , A ∈ SO(1, 2). Note that this action inherits the properties of the action of SO(1, 2) on H. As such it is transitive and effective but not free. It is not free because An = n for some fixed n ∈ H implies only that A is in stability group of n. The action is manifestly transitive and also effective in that An = n for all n ∈ H implies p (p being mnemonic for ‘phase’ or ‘polymer’) as the closed A = 1. We define HOS subspace generated by all the vectors ψn0 ,α , α = 0, in (5.33). p 0 and HOS : We now describe how ρ∞ acts on HOS p 0 Theorem 5.4 HOS and HOS are orthogonal subspaces of HOS . ρ∞ acts unitarily 0 on both of these subspaces; the action is continuous on HOS , but discontinuous on p 0 HOS . Furthermore, on HOS one has (ρ∞ (A)ψ0 , ρ∞ (B)ψ0 )OS = P−1/2 (An↑ · Bn↑ ) ,

(5.34)

p whereas on HOS one has

(ψn1 ,α1 , ψn2 ,α2 )OS = δn1 ,n2 δα1 ,α2 ,

∀ n1 , n2 ∈ H , α1 , α2 ∈ R , α1 α2 = 0 . (5.35)

Proof. The derivation of Eqs (5.34), (5.35) as well as the proof of the orthogonality of the two subspaces is somewhat technical and is deferred to Appendix C. 0 From (5.34) it is easy to see that ρ∞ acts continuously on HOS : By (5.34) the scalar product of any two elements in the orbit of ψ0 is a continuous function of the group elements, and this continuity trivially extends to finite linear combinations of elements of this orbit. Denote this linear space by D. This implies that for any φ ∈ D we have limA→0 ρ∞ (A)φ − φOS = 0. Now for any element 0 and any > 0 there is a φ ∈ D such that φ − ψOS < . By the ψ ∈ HOS triangle inequality therefore limA→0 ρ∞ (A)ψ − ψOS ≤ , and since was arbitrary, limA→0 ρ∞ (A)ψ − ψOS = 0 follows. The group

structure of SO(1,2) yields (strong) continuity of the whole representation ρ∞ H0 . The discontinuity of the p is obvious from (5.35). action of ρ∞ on HOS

OS

Corollary 5.5 HOS is nonseparable. Proof. The vectors (5.33) provide an explicit nondenumerable orthonormal family. Remark. The representation ρ∞ of SO(1, 2) acts as a kind of ‘nondenumerable discrete permutation group’, ρ∞ (A)ψn0 ,α = ψAn0 ,α , on the orthonormal family (5.33) (see Appendix C). The above result should be viewed in the context of an alternative described by Segal and Kunze ([46], p. 274) which characterizes measurable unitary representations π of some locally compact group G on a nonseparable Hilbert space H. Namely let Hs be the subspace of all vectors ψs in H such that for all ϕ ∈ L1 (G)

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dµ(A) ϕ(A)(ψ, π(A)ψs ) = 0 .

(5.36)

G

Then H is the direct sum of two invariant subspaces H = Hc ⊕ Hs . The restriction of π to Hc is continuous while the restriction to Hs is singular, in the sense that (ψs , π(A)ψs ) = 0 for almost all A ∈ G and all ψs ∈ Hs . If H is separable Hs is absent, as follows from a theorem of von Neumann (see [30], Theorem VIII.9). u In our case we denote by HOS the maximal closed subspace of HOS on u which ρ∞ is unitary and measurable. The above alternative entails that HOS u c s c decomposes into HOS = HOS ⊕ HOS , where the restriction of ρ∞ to HOS and s HOS is continuous and singular, respectively. Our results amount to the explicit construction of subspaces 0 c ⊂ HOS , HOS

p s HOS ⊂ HOS ,

(5.37)

together with a formula for the inner products. In particular the singular subspace is non-empty (for which the nonseparability of the Hilbert space is a necessary but p s ⊂ HOS not a sufficient condition). The assumption (5.36) is satisfied for ψ ∈ HOS because (5.35) projects onto a 1-dimensional submanifold of the group which has zero measure wrt the full Haar measure. The restriction of ρ∞ to Hp ⊂ Hs is indeed singular; in fact by (5.35) (ψn0 ,α , ρ∞ (A)ψn0 ,α )OS = 0 holds for all A = 1. The simple explicit action ρ∞ (A)ψn0 ,α = ψAn0 ,α as a permutation group (acting transitively and effectively for fixed α) is somewhat surprising. The continuous 0 will later be identified as the ground state sector of the reconstructed subspace HOS transfer operator. As outlined in Appendix C there are other nondenumerable orthonormal p 0 and HOS . We did not explore families in HOS which are orthogonal to both HOS the action of ρ∞ on them, but it may well be that HOS contains other invariant subspaces on which ρ∞ acts unitarily. In this case they would likewise be subject to the above continuous-discontinuous alternative and render the inclusions in (5.37) proper. We proceed with the construction of a transfer operator TOS on HOS , which 0 in particular will justify the term ‘ground state sector’ for HOS (see Proposition 5.6 below). To this end let τ be the map from C+ to C+ that shifts all variables by 1 unit to the right, i.e., τ O(nx1 , . . . , nx ) = O(nx1 +1 , . . . , nx +1 ). τ satisfies the relation τ ϑτ = ϑ; it maps N into itself as can be seen by using the Cauchy-Schwarz inequality and translation invariance ω∞ (ϑ[τ O]τ O) = ω∞ ([ϑO] [τ 2 O]) ≤ ω∞ ([ϑO] [τ 4 O])1/2 ω∞ ([ϑO] O)1/2 .

(5.38)

τ therefore induces a well-defined operator TOS on the equivalence classes modulo N , and hence on HOS . By translation invariance TOS is symmetric. Once known to be bounded it extends to a unique selfadjoint operator on HOS . The boundedness

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follows by iterating the Cauchy-Schwarz inequality (using a classic argument of Osterwalder and Schrader) ω∞ ([ϑO] τ 2 O) ≤ ω∞ ([ϑO] τ 4 O)1/2 ω∞ ([ϑO] O)1/2 ≤ · · · n+1

≤ ω∞ ([ϑO] τ 2

−n

O)2

−n

ω∞ ([ϑO] O)1−2

.

(5.39)

−n+1

The first factor is bounded by O2 , which goes to 1 as n → ∞; the second factor goes to ω∞ ([ϑO] O), which proves that T2OS ≤ 1 and thus also TOS ≤ 1. Importantly, the vector ψ0 corresponding to O = 1 is an eigenvector (of norm 1) of the reconstructed transfer operator TOS with eigenvalue 1 = TOS OS , i.e., ψ0 is a ground state of the system. Already the mere existence of at least one normalizable ground state indicates that the reconstructed quantum mechanics given by (HOS , TOS ) is very different from the original one given by (L2 (H), T). This mismatch can be traced back to the purely continuous spectrum of the original system, which in turn stems from the noncompactness of the target space H. A further drastic discrepancy is the nonseparability of HOS . Similar surprising features arise already in the much simpler model with flat target space R, on which R also acts as an amenable symmetry. This example is also instructive because it shows that in the limit of an amenable symmetry the symmetry breaking disappears. We therefore discuss this example briefly in Appendix B. Returning to the hyperbolic model, we summarize the properties of TOS : Proposition 5.6 TOS is a self-adjoint operator on HOS with following properties: (i) ||TOS || = 1 . (ii) ρ∞ ◦ TOS = T OS ◦ ρ∞ . (iii) TOS H0 = 1 H0 . OS

OS

0 (iv) HOS = {ψ ∈ HOS | TOS ψ = ψ}. (v) TOS acts on C+ /N , i.e., on the representatives (5.14) as x −x ↑ −1 (TOS ψ)(n) = λβ (0) P−1/2 (n·n ) dΩ(n )Tβ (n·n ; x)ψ(n )P−1/2 (n ·n↑ ) ,

(5.40) up to an element of N . Remark. (ii) and (iv) show that TOS , in contrast with T, has at least some point spectrum. Despite the concrete expression in (v), it seems difficult to say more 0 . about the spectrum of TOS outside the vacuum space HOS Proof. (i) has already been shown; it is a general feature of the OsterwalderSchrader reconstruction. (ii) Recall that TOS is defined in terms of the shift τ on C+ . As τ trivially commutes with the ρ∞ action (5.27) of SO(1, 2) on C+ and both τ and ρ∞ preserve the nullspace N , the same will be true for TOS induced on the equivalence classes. This gives (ii).

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0 (iii) A simple consequence of (ii) is that TOS acts like the identity on HOS , because τ acts like the identity on the constants, in particular on the unit observable O = 1, corresponding to the ‘canonical’ vacuum ψ0 . By (ii) the same must 0 . Equivalently, the eigenspace of TOS of eigenvalue 1 hold for all elements of HOS 0 contains HOS . (iv) Let O ∈ C+ be such τ O − O ∈ N . Then also τ O0 − O0 ∈ N , because τ O − O = τ O0 − O0 + τ (O − O0 ) − (O − O0 ), and the last two terms on the rhs are in N . By the remark after (5.14) therefore (τ O0 )0 − O0 ∈ N ∩ C0 , and it suffices to consider the exact identity (τ O0 )0 = O0 . From the definition of the map (5.14) one sees that all solutions ψ ∈ C0 of (τ ψ)0 = ψ are such that ψ(n)P−1/2 (n · n↑ ) is an eigenfunction of Tβ of eigenvalue λβ (0). From (2.42) one infers that the solutions lie in the closed subspace of C0 spanned by ratios of the form (5.33) with α = 0. (v) Since (τ O0 )0 − (τ O)0 ∈ N , for all O ∈ C+ , one can use (5.14) to compute the action of τ on the representative O0 as before. One finds (5.40), first for x = 1 and then by iteration for all x ∈ N. 0 . This follows In addition to acting unitarily, ρ∞ also acts irreducibly on HOS directly from the definition (5.31). Alternatively one can use the addition theorem (A.12c) to replace the generating set ρ∞ (A)ψ0 , A ∈ SO(1, 2), by the alternative l generating set P−1/2 (ξ)/P−1/2 (ξ), l ∈ Z. These functions are known to span an irreducible and unitary representation of SO(1, 2); in the Bargmann classification it corresponds to the limit of the discrete series. To sum up, we have found that the space HOS is nonseparable and that it carries a representation ρ∞ of the symmetry group. This representation acts unitarily and discontinuously on a nonseparable proper subspace of HOS , and 0 unitarily and continuously on the separable subspace of ground states HOS of the reconstructed transfer operator TOS . This ground state sector is irreducible and can be described explicitly as

l P−1/2 (n↑ ·n)

P−1/2 (n↑ ·A−1 n)

0 HOS

A ∈ SO(1, 2)

l ∈ Z ⊂ HOS , P−1/2 (n↑ ·n)

P−1/2 (n↑ ·n)

(5.41) where the symbol ‘’ denotes equality of the span of the lhs and rhs for the 0 is given by (ρ∞ (A)ψ0 , equivalence classes modulo N . The inner product on HOS ↑ ↑ ρ∞ (B)ψ0 )OS = P−1/2 (An ·Bn ).

6 Conclusions and outlook We have found that the concept of spontaneous symmetry breaking for a nonamenable continuous internal symmetry group differs in some crucial ways from the familiar situation of an amenable symmetry group: • Symmetry breaking is unavoidable, even in dimensions 1 and 2, where it is forbidden for an amenable continuous symmetry. In one dimension the (improper) ground state in the quantum mechanical interpretation is infinitely

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degenerate; in the statistical mechanics interpretation invariant states over a ‘large’ algebra cannot be defined by group averaging. These features have been worked out in some detail for an analytically solvable model, the hyperbolic spin chain with symmetry group SO(1, 2). • In this 1-dimensional model, however, there is still some vestige of the large fluctuations that are responsible for the symmetry restoration in the compact and abelian models: the sequence of functional measures defined through the thermodynamic limit becomes concentrated at configurations ‘at infinity’ of the hyperbolic plane. As a consequence a certain subclass of non-invariant observables gets averaged to yield an invariant result. The limit of the functional measures provides an invariant mean for this subclass of observables, while outside this class symmetry breaking is generic. • While the quantum mechanics described by our model can be simply interpreted as motion of a particle on the hyperbolic plane, with absolutely continuous spectrum of the transfer matrix, the Osterwalder-Schrader reconstruction based on the infinite volume expectation values yields some surprises: the reconstructed transfer matrix has at least some point spectrum, in particular it has normalizable ground states, and the full reconstructed Hilbert space is nonseparable. These features are, however, due to the noncompact nature of the symmetry group, not its nonamenability, as can be seen from the ‘flat’ analogue discussed in Appendix B. • The Osterwalder-Schrader reconstructed Hilbert space has a nonseparable proper subspace on which a unitary representation of SO(1, 2) acts discontinuously as a kind of ‘nondenumerable discrete permutation group’, not unlike the way the spatial diffeomorphism group acts on the embedded graphs in the framework of [22, 23]. In contrast, the space of ground states of the reconstructed transfer operator is separable and a nontrivial unitary representation of SO(1, 2) acts on it continuously and irreducibly. These features are specific to the case of a nonamenable symmetry and are not present in the ‘flat’ case. In a follow-up paper we study these issues in the D-dimensional (D ≥ 2) version of the model, i.e., the nonlinear sigma-model with a hyperbolic targetspace; see, e.g., [18, 42, 19] for earlier investigations. There we use a combination of analytical techniques and of numerical simulations [14]. We also expect that there will still be a marked difference between dimensions D ≤ 2 and D ≥ 3: whereas in the low dimensional case there is, as stated above, dominance of highly boosted configurations, we expect that in D ≥ 3 spontaneous symmetry breaking in the usual sense takes place, showing normal, approximately Gaussian fluctuations around a fully ordered state, in which for instance unbounded observables like n00 have finite expectation values. Some time after the first version of this paper was posted on the web, a paper by Spencer and Zirnbauer [47] appeared, which showed that indeed in dimensions D ≥ 3 at low temperature the suitable defined spin fluctuations have finite moments.

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It would be interesting to elucidate the physical meaning of the unavoidable spontaneous symmetry breaking in the context of Anderson localization, in which such nonlinear sigma models were studied for instance in [7, 8, 9, 10, 11]. In order not to blur the discussion with (further) technicalities we contrasted here only the simplest compact and noncompact symmetric spaces. However the situation would be similar if the sphere S 2 SO(3)/SO(2) and H SO(1, 2)/SO(2) were replaced with any other dual pair of compact and noncompact Riemannian symmetric spaces (see [26] for the propagators). A further generalization would be to consider a similar dynamical system where the variables take values in an arbitrary Riemannian manifold. In particular this would allow one to examine the interplay between invariant dynamics and non-invariant states for the diffeomorphism group of the target manifold. Finally we cannot resist mentioning a potential application to quantum gravity. Supposing that in a suitable topology an appropriate version of the diffeomorphism group is nonamenable, variants of the above concepts become applicable. This would suggest a scenario in which there is no diffeomorphism invariant ground state, yet a family of selected observables has invariant expectations in each of an infinite set of ground states, while outside this family spontaneous collapse of diffeomorphism invariance is generic.

Appendix A: Harmonic analysis on H Let a · b = a0 b0 − a1 b1 − a2 b2 be the bilinear form of R1,2 and let SO0 (1, 2) =: SO(1, 2) be the component of its symmetry group connected to the identity. Consider the hyperboloid H = {n ∈ R1,2 | n · n = 1 , n0 > 0}. It is isometric to the symmetric space SO(1, 2)/SO(2) and can be parameterized either by points (∆, B), ∆ > 0, B ∈ R, in the Poincaré upper half plane, or by geodetic polar coordinates (ξ, ϕ), ξ ≥ 1, −π ≤ ϕ < π, via 1 + ∆2 + B 2 B = ξ, n1 = − = ξ 2 − 1 sin ϕ , 2∆ ∆ 2 2 −1 + ∆ + B = ξ 2 − 1 cos ϕ . = 2∆

n0 = n2

(A.1)

The (ξ, ϕ) parameterization is adapted to a preferred SO(2) subgroup of SO(1, 2) ↑ which leaves n↑ = (1, 0, 0) invariant and which we denote by SO (2). We also note the relations ∆−1 = ξ − ξ 2 − 1 cos ϕ, B = 1 − ξ −2 sin ϕ/( 1 − ξ −2 cos ϕ − 1). For the invariant distance n · n ≥ 1 of two points n, n ∈ H, one has n · n =

2

∆2 + ∆ + (B − B )2 2 = ξξ − (ξ 2 − 1)1/2 (ξ − 1)1/2 cos(ϕ − ϕ ) . (A.2) 2∆∆

Function spaces on H come naturally equipped with the inner product (ψ1 , ψ2 ) = dΩ(n) ψ1 (n)∗ ψ2 (n) ,

(A.3)

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induced by the invariant measure dΩ(n) := 2d3 n δ(n2 − 1) θ(n0 ), which translates into dΩ(∆, B) = dBd∆∆−2 and dΩ(ξ, ϕ) = dξdϕ, respectively. As indicated we shall freely switch back and forth between the different parameterizations. The Schwartz space S(H) is defined as the space of smooth functions on H decaying faster than any power of B and ∆. The space of tempered distributions S (H) on it together with L2 (H) form a Gel’fand space triple S(H) ⊂ L2 (H) ⊂ S (H) .

(A.4)

The SO(1, 2) rotations of the ‘spins’ n induce a unitary representation ρ on S(H) via ρ(A)ψ(n) = ψ(A−1 n), A ∈ SO(1, 2). On integral operators K with kernel κ(n, n ) it acts as K → ρ(A)−1 Kρ(A) and thus as κ(n, n ) → κ(An, An ) on the kernels. Invariant operators have kernels depending on the inner product n · n only. Similarly operators invariant under ρ restricted to the SO↑(2) subgroup have kernels depending on ξ, ξ and the relative angle ϕ − ϕ only. In general the representation ρ will not be irreducible. Generic functions in S(H) can be expanded into a generalized Fourier integral whose basis functions form unitary irreps of SO(1, 2). Moreover these basis functions comprise S (H) eigenfunctions of the Laplace-Beltrami operator. To make this concrete consider the Killing vectors of H, which generate the Lie algebra sl2 e = ∂B ,

h = 2(B∂B + ∆∂∆ ) ,

[h, e] = −2e , [h, f ] = 2f ,

f = (∆2 − B 2 )∂B − 2B∆∂∆ , [f , e] = h ,

(A.5)

and are anti-hermitian wrt ( , ). Up to a sign the quadratic Casimir coincides with the Laplace-Beltrami operator −C :=

1 ∂2 ∂ 2 ∂ 1 2 1 2 2 h + (ef + fe) = ∆2 (∂∆ (ξ − 1) + 2 + ∂B )= . (A.6) 4 2 ∂ξ ∂ξ ξ − 1 ∂ϕ2

If one just blindly lets the differential operators e, h, f act on the spins (A.1) (which are not elements of L2 (H)) one sees that they act as 3 × 3 matrices t(e), t(h), t(f ) with Casimir C = −213 ; the matrices are however not (anti)hermitian even though the original differential operators (multiplied by i) are essentially self-adjoint on S(H) ⊂ L2 (H). The exponentiated differential operators therefore extend to the unitary action of SO(1, 2) on L2 (H) ρ(e−st(x) )ψ(n) = esx ψ(n) = ψ(est(x) n) ,

x = e, h, f ;

s ∈ R.

(A.7)

A more explicit description of the exponentiated differential operators is possible on irreducible representations. Simultaneous eigenstates of C and e are given by ω,k (n) := ω,k (∆, B) = ∆1/2 Kiω (|k|∆) eikB , with ω,0 (n) := ω,0 (∆, B) = ∆iω+1/2 , 1 + ω 2 ω,k , e ω,k = ik ω,k , C ω,k = 4

k = 0 .

ω > 0,

(A.8)

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where Kν (x) is a modified Bessel function defined, e.g., by Kν (β) = cosh(νt)dt. The Fourier inversion on S(H) takes the form ∞ dω k) ω,k (n) ψ(n) = ω sinh πω dk ψ(ω, π3 0 R k) = ψ(ω, dΩ(n) ψ(n)ω,k (n)∗ .

∞ 0

e−β

cosht

(A.9)

Simultaneous eigenstates of C and e − f , i.e., of the SO↑(2) rotations are given by l (ξ) , l ∈ Z, ω > 0, ω,l (n) := ω,l (ξ, ϕ) = eilϕ P−1/2+iω 1 + ω 2 ω,l , C ω,l = (e − f ) ω,l = il ω,l , 4

(A.10)

where Psl (ξ) are Legendre functions, defined, e.g., by Psl (ξ) =

Γ(s + l + 1) 2πΓ(s + 1)

2π

du eilu [ξ +

0

ξ 2 − 1 cos u]s ,

ξ ≥ 1.

(A.11)

We further note the following properties Γ(s + 1 + l) −l P (ξ) , Γ(s + 1 − l) s

(A.12a)

(−)l δ(ω − ω ) , ω tanh πω

(A.12b)

l (ξ) = Psl (ξ) = P−s−1

∞

1

−l l dξ P−1/2+iω (ξ)P−1/2+iω (ξ) =

2 Ps ξξ − (ξ 2 − 1)1/2 (ξ − 1)1/2 cos ϕ = (−)l eilϕ Ps−l (ξ) Psl (ξ ) ,

(A.12c)

l∈Z

as well as the asymptotics for ξ → ∞ Γ(iω) (2ξ)−1/2+iω + c.c. , √ πΓ( 12 + iω − l)

l P−1/2+iω (ξ) ∼

ω > 0,

ln ξ 2 √ . √ 1 πΓ( 2 − l) 2ξ

l P−1/2 (ξ) ∼

(A.13)

The Fourier inversion in the basis (A.14) takes the form ∞ dω l l) ω,−l (n) , ω tanh πω ψ(ω, (−) ψ(n) = 0 2π l∈Z

l) = ψ(ω,

dΩ(n) ψ(n)ω,l (n) .

(A.14)

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In group theoretical terms the expansions (A.9), (A.14) correspond to the decomposition of the unitary representation ρ on L2 (H) into a direct integral of unitary irreducible representations, namely those of the type 0 principal series in the Bargmann classification, see, e.g., [43]. In terms of the representation spaces ⊕ 2 dµ(ω) Cω (H) , (A.15) L (H) = with the spectral weight dµ(ω) = dω 2π ω tanh ω. We shall frequently encounter SO↑(2) invariant functions ψ = ψ(ξ), for which (A.14) reduces to the Mehler-Fock transform ∞ dω ω tanh(πω) P−1/2+iω (ξ) ψ(ω) , ψ(ξ) = 0 2π ∞ 0) . ψ(ω) = 2π dξ P−1/2+iω (ξ) ψ(ξ) = ψ(ω, (A.16) 1

It holds in the classical sense provided ∞ 2 dξ|ψ(ξ)| < ∞ ⇐⇒ 1

∞ 0

2 |ψ(ω)| ω tanh πω < ∞ ,

(A.17)

see, e.g., [44]. It is possible, however, to interpret the Mehler-Fock transform in the distributional sense and therefore give it a wider range of applicability. The Fourier decomposition of a kernel κ(n, n ) defining an integral operator K makes some of its properties manifest. Subject to suitable regularity conditions the generic form of the expansion wrt the basis (A.14) is

κ(n, n ) =

l1 +l2

∞

(−)

l1 ,l2 ∈Z

0

dω1 dω2 ω1 thπω1 ω2 thπω2 κ l1 .l2 (ω1 , ω2 ) 2π 2π ω1 ,−l1 (n) ω2 ,−l2 (n ) , (A.18)

Depending on the properties of the spectral weight κ l1 ,l2 (ω1 , ω2 ) = (ω1 ,l1 , Kω2 ,l2 ) the corresponding integral operator K will enjoy certain bonus properties: κ l1 ,l2 (ω1 , ω2 ) =

2π κl1 ,l2 (ω1 ) δ(ω1 − ω2 ) ω1 tanh πω1

κ l1 ,l2 (ω1 , ω2 ) = δl1 +l2 ,0 κ l1 (ω1 , ω2 ) κ l1 ,l2 (ω1 , ω2 ) = δl1 +l2 ,0

2π κ(ω1 ) δ(ω1 − ω2 ) ω1 tanh πω1

translation inv. (A.19a) SO↑(2) inv.

(A.19b)

SO(1, 2) inv.

(A.19c)

For K itself these properties amount to a vanishing commutator with T, ρ|SO↑(2) and ρ, respectively. The fact that the spectral weights (A.19c) lead to SO(1, 2)

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invariant operators follows from (A.12c); the kernels κ(n, n ) of these operators depend on the invariant distance n · n only. As an example of a spectral decomposition consider the transfer matrix itself where the weights are just the eigenvalues (2.3). Using, e.g., [28], p.804 and the completeness relation for the Legendre functions one verifies ∞ dω β exp{β(1 − n · n )} = ω tanh πω P−1/2+iω (n · n ) λβ (ω) . (A.20) 2π 2π 0 In representation theoretical terms this expresses the exponential of a singlet wrt the non-unitary vector irrep as a superposition of singlets wrt the unitary irrep (A.15). We note that the inverse Mehler-Fock transform (A.16) gives ∞ λβ (ω) = β dξeβ(1−ξ) P−1/2+iω (ξ). (A.21) 1

Clearly the integral kernels κ(n · n ) that give rise to well-defined operators on L2 (H) must have suitable regularity and decay properties. The asymptotics in (A.13) suggests that the kernels κ(ξ) should also decay at least like ξ −1/2 . Some decay stronger than ξ −1/2 is also necessary in order for κ to be the integral kernel of a densely defined operator from L2 (H) to L2 (H). A sufficient condition seems to be more difficult to obtain, but in any case kernels like n · n do not correspond to densely defined operators on L2 (H) (they give rise only to densely defined quadratic forms). The integrands of the Legendre functions (A.11) likewise provide eigenfunctions of the Laplace-Beltrami operator (A.6). Explicitly (A.22) Eω,u (n) := Eω,u (ξ, ϕ) = [ξ − ξ 2 − 1 cos(u − ϕ)]−1/2−iω , are bounded complex solutions for all |ϕ − u(mod2π)| > > 0), decaying like ξ −1/2+iω for ξ → ∞. √ The upper bound will diverge as → 0 because for φ = u one has |Eω,u (ξ, u)| ∼ 2ξ, for ξ → ∞. The Legendre functions (A.11) are recovered as the Fourier modes of (A.22) and vice versa. The orthogonality and completeness relations take the form (2π)2 δ(ω − ω )δ(θ − θ ) , (Eω,θ , Eω ,θ ) = ω tanh πω 2π ∞ 1 dω ω tanh ω dθ Eω,θ (n)∗ Eω,θ (n ) = δ(n, n ) . (A.23) (2π)2 0 0 The main virtue of these solutions is their simple transformation law under SO(1, 2), see, e.g., [45]. For a boost A−1 = A(θ, α)−1 mapping ξ = n0 into ξchθ − shθ cos(ϕ − α) one has Eω,u (A−1 n) = [chθ + cos(ϕ − α)shθ]−1/2−iω Eω,u (n) ,

(A.24)

for some angle u = u (θ, α). This is also a convenient starting point to show that the Fourier decomposition (A.14) indeed has the representation theoretical significance (A.15), see, e.g., [43].

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Appendix B: Flat noncompact spin chain In order to elucidate the relation between the original Hilbert space and the one obtained by Osterwalder-Schrader reconstruction it is useful to consider the simplest noncompact spin chain where the target space is R. The symmetry group in this case is also R, which in contrast to SO(1,2) is amenable. Some of the unusual aspects of this model have been analyzed already in [21]. Of course all results generalize trivially to target spaces Rn , n > 1. We consider the Hilbert space L2 (R) and take as the one-step transfer matrix simply the heat kernel exp[β −1 ∆](u, v), so that ∞ x (T ψ)(u) = dv Tβ (u − v; x) ψ(v) , x ∈ N , −∞

Tβ (u; x) =

β β exp − u2 . 2πx 2x

(B.1)

The transfer operator trivially commutes with the action of R on the wave functions, i.e., T ◦ ρ = ρ ◦ T, with ρ(a)ψ(u) = ψ(u − a). It is well known that the spectrum of T is continuous and covers the interval [0, 1]; the generalized eigenfunctions are imaginary exponentials. As in the hyperbolic case, a gauge fix is necessary; we simply fix the leftmost ‘spin’ u−L to 0, which is analogous to fixing n−L = n↑ . For the purpose of the Osterwalder-Schrader reconstruction we choose again in addition the bc uL = 0, i.e., we choose 0 Dirichlet conditions. As observable algebra we take C = Cb , the algebra of continuous bounded functions of finitely many variables ux1 , . . . , ux , and we introduce the subalgebras C+ , C− and C0 = C+ ∩ C− as in Section 5. For a finite chain the reconstruction of the Hilbert space HL proceeds as in Section 5; we define for each O ∈ C+ O0,L (u0 ) =

dui O(u1 , . . . , u )

i=1

and

Tβ (ui − ui−1 ; xi − xi−1 )

i=1

Tβ (u ; L − x ) , Tβ (u0 ; L) (B.2)

Tβ (u; L) ψ O (u) = O0,L (u) . Tβ (0; 2L)

(B.3)

The reconstructed Hilbert space HL is the completion of C0 wrt ωL in (B.5). It can be identified with the original L2 (R) by the isometry VL : HL −→ L2 (R) ,

Tβ (u; L) (VL ψ)(n) = ψ(u) . Tβ (0; 2L)

(B.4)

Equivalently HL can be viewed as the preimage of L2 (R) wrt VL . T (u;L) The thermodynamic limit can be readily understood here. The ratio Tβ(0;L) approaches a constant for L → ∞, signaling a unique ground state. On elements

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O0,∞ ∈ C0 the expectation functionals becomes an invariant mean, which exists in this case. There is a subspace HAP of almost periodic functions on which this mean is unique, see [4]; this subspace consists of the completion (in the Hilbert space norm defined by the mean) of the space of trigonometric polynomials. A brief account of the theory of almost periodic functions on R, which is due to H. Bohr, can be found in [48]. For ψ ∈ HAP the mean is u2 1 ψ(u) =: lim ωL (ψ) . (B.5) ω(ψ) = lim √ du exp − L→∞ L→∞ 2L 2πL A better known expression of the invariant mean on HAP is L 1 du ψ(u) , ω(ψ) = lim L→∞ 2L −L

(B.6)

see for instance [48]. By the uniqueness these two expressions have to be the same for an almost periodic ψ and it is straightforward to verify this equivalence for the dense subspace of trigonometric polynomials. The scalar product induced by this invariant mean can be written as L 1 (ψ , ψ)OS = lim du ψ(u)∗ ψ(u) , (B.7) L→∞ 2L −L and the unitarity of ρ on HAP is manifest. It might be surprising that the Hilbert space obtained by the OS reconstruction from C0 is nonseparable; but it is well known that already the space HAP is nonseparable [48]: there is an uncountable set of mutually orthonormal functions, namely the set (B.8) {ψα (u) = eiαu | α ∈ R} . One can introduce a shift automorphism τ like the one used in the hyperbolic case. From this one obtains a reconstructed transfer operator TOS acting on HOS ; in this case it is nonnegative and has again norm 1. TOS acts on C0 simply by Eq. (B.1). This shows that the functions (B.8) are eigenvectors (in the proper 1 2 α ). sense) of TOS with eigenvalue exp(− 2β The relation between the original system (L2 (R), T) and the reconstructed one (HOS , TOS ) turns out to be simply that the spectrum as a set remains the same, namely the interval [0, 1]. However there is now pure point spectrum on every point of the spectral interval and the generalized eigenfunctions become normalizable eigenstates. With respect to the representation of symmetry group R the original L2 (R) is a direct integral of the one-dimensional irreducible representations on the imaginary exponentials (B.8), whereas HAP is (and hence HOS contains) a direct sum over the continuous parameters α: Hα , (B.9) HOS ⊃ HAP = α∈R

where Hα is the one-dimensional Hilbert space spanned by eiαu .

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Let us end this appendix with the remark that the space HAP , huge as it is, is still only a small subspace of the full space HOS . It turns out that there are uncountably many more functions orthogonal to the exponentials discussed so far, for instance the functions pα (u) = |u|iα . Using distributional Fourier transformation one can show that (pα , ψα )OS = 0 ,

∀α = 0 , α ∈ R.

(B.10)

Presumably these functions belong to the continuous spectrum overlaying the point spectrum we have found. p 0 Appendix C: Inner products on HOS and HOS 0 Here we derive the formulas (5.34) and (5.35) for the inner products on HOS and p ↑ HOS . We begin with (5.34), i.e., (ρ∞ (A)ψ0 , ρ∞ (B)ψ0 )OS = P−1/2 (An ·Bn↑ ). By (4.12) this is equivalent to

lim

n↑ ·n→∞

fA,B (n↑ · n) ↑

=

P−1/2 (An↑ ·Bn↑ ) ,

fA,B (n · n) :=

P−1/2 (n·An↑ ) P−1/2 (n·n↑ )

with P−1/2 (n·Bn↑ ) P−1/2 (n·n↑ )

,

(C.1)

↑ ↑ where the bar as before denotes the average over SO (2). Writing ξ = n · n and ↑ 2 2 momentarily n · An = ξξA − ξ − 1 ξA − 1 cos(ϕ − ϕA ), and similarly for n · Bn↑ , the SO↑(2) average evaluates by means of (A.12c) to

fA,B (ξ) =

l∈Z

l −l P−1/2 (ξ)P−1/2 (ξ) −l l e−il(ϕA −ϕB ) P−1/2 (ξA )P−1/2 (ξB ) . P−1/2 (ξ)2

(C.2)

The series converges uniformly in ξ: using the Cauchy-Schwarz inequality, the geometric-arithmetic mean inequality and the bound −l l (ξ)P−1/2 (ξ)| ≤ P−1/2 (ξ)2 |P−1/2

the rhs is bounded by 1 and likewise the tail of the sum can be bounded uniformly in ξ. Taking now the limit ξ → ∞ under the sum, which is permitted because of the uniform convergence of the series, one obtains −l l eil(ϕA −ϕB ) (−)l P−1/2 (ξA )P−1/2 (ξB ) = P−1/2 (An↑ ·Bn↑ ) , lim fA,B (ξ) = ξ→∞

l∈Z

(C.3) using (A.13) and (A.12c). This gives (5.34); note that the result coincides with the one obtained from the ‘correlated’ limit in (5.13).

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The derivation of (5.35) we break up in several steps. Recall the notation ψα (n) = exp(iαn↑ · n), α ∈ R \ {0}. We first show that these functions form an orthonormal system (ψα , ψα )OS = 1 ,

for α = α .

(ψα , ψα )OS = 0 ,

The normalization is clear. For the orthogonality consider for α = 0 ∞ Tβ (ξ; L)2 . Iα (L) := dξ eiαξ Tβ (1; 2L) 1 To analyze this expression we integrate by parts and obtain 2 ∞ T ∂ (ξ; L) Tβ (1; L)2 β Iα (L) = − dξeiαξ . eiα + iαTβ (1; 2L) ∂ξ Tβ (1; L) 1

(C.4)

(C.5)

(C.6)

The first term is O(L−3/2 ) by (2.36); the modulus of the second term can be bounded, using the monotonicity of Tβ (ξ; L) by −

Tβ (1; L)2 αTβ (1; 2L)

∞

dξ 1

∂ ∂ξ

Tβ (ξ; L) Tβ (1; L)

2 =

Tβ (1; L)2 , αTβ (1; 2L)

(C.7)

which is also O(L−3/2 ). Together, limL→∞ Iα (L) = 0 and (C.4) is proven. We remark that this construction readily generalizes to all wave functions oscillating ‘sufficiently fast’ as ξ → ∞. Consider ψp (n) = exp i

ξ

du p(u)

with

1

(ln ξ)2 = 0. ξ→∞ ξp(ξ) lim

(C.8)

Then every pair of wave functions ψp1 (n), ψp2 (n), where the difference p1 (ξ)−p2 (ξ) is strictly monotonous for sufficiently large ξ and obeys the decay condition in (C.8) is orthogonal: (ψp1 , ψp2 )OS = 0, using Lemma 2.2 (iii) to get bounds uniform in L for the ξ → ∞ limits. For example exp{iα(ln ξ)4 }, α ∈ R, provides another nondenumerable orthonormal family, each member of which is orthogonal to each of the plain exponentials in (C.4). Here we shall only pursue the plain exponentials ψα , α ∈ R, further. Repeating the above computations with the transformed exponentials ρ∞ (A)ψα one readily shows that they remain orthogonal if they were initially. For the computation of the norms the phases are irrelevant, so they remain unity if (ρ∞ (A)ψ0 , ρ∞ (A)ψ0 )OS = (ψ0 , ψ0 )OS = 1. This however is a special case of (5.34). Thus (ρ∞ (A)ψα , ρ∞ (A)ψα )OS = (ψα , ψα )OS ,

α, α ∈ R .

(C.9)

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In a last step we show ∀ A ∈ SO(1, 2) , α , α ∈ R , αα = 0 .

(ρ∞ (A)ψα , ψα )OS = 0 ,

(C.10)

By definition one has (ρ∞ (A)ψα , ψα )OS

=

lim 2π

L→∞

Jα (ξ, ξA ) :=

0

2π

1

∞

dξ eiα ξ Jα (ξ, ξA )

Tβ (ξ, L)2 , (C.11) Tβ (1, 2L)

dϕ −iαAn↑ ·n P−1/2 (An↑ · n) e , 2π P−1/2 (ξ)

2 where we view An↑ ·n = ξξA − (ξ 2 − 1)1/2 (ξA − 1)1/2 cos(ϕ − ϕA ) as a function of ξ, ξA and ϕ − ϕA . As anticipated by the notation Jα (ξ, ξA ) is independent of ϕA . Clearly Jα (ξ, 1) = e−iαξ and |Jα (ξ, ξA )| ≤ P−1/2 (ξA ) by the addition theorem (A.12c). We take now ξA > 1 and by (C.9) we may also assume that α = 0 and wlog α > 0 (while α ∈ R may be zero). By the argument familiar from Section 4.2 only the behavior of Jα (ξ, ξA ) for large ξ will be relevant for the inner product (C.11). We claim that 1 as ξ → ∞ , Jα (ξ, ξA ) ∼ √ Q+ (ξA )e−iαp+ (ξA )ξ + Q− (ξA )e−iαp− (ξA )ξ αξ 2 − 1, with p± (ξA ) = ξA ± ξA (C.12)

and some complex constants Q± (ξA ) nowhere zero for ξA > 1. Note that 1 < ξA < p+ (ξA ) and 0 < p− (ξA ) < 1. We first show that the rhs of (C.12) is the leading term in an asymptotic expansion of Jα (ξ, ξA ) for large ξ. The point to observe is that from (A.13) we have 1 2 − 1)1/2 cos ϕ P−1/2 (ξ)−1 ∼ , P−1/2 ξξA − (ξ 2 − 1)1/2 (ξA 2 [ξA − ξA − 1 cos ϕ]1/2 (C.13) with additive corrections of O(1/ ln ξ). Asymptotically the integral becomes √2 2π eiαξ ξA −1 cos ϕ dϕ −iαξξA Jα (ξ, ξA ) ∼ e . (C.14) 2 − 1 cos ϕ]1/2 2π [ξA − ξA 0 For large ξ this integral can now be evaluated by the method of stationary phase (see, e.g., [29]) with the result (C.12). The constants Q± (ξA ) come out as −1/2 2 −1 ξ ± 2 −1 Q± (ξA ) = 2±1/2 e±iπ/4 2π ξA ξA . (C.15) A Subleading terms in the asymptotic expansion of Jα (ξ, ξA ) could be worked out similarly, but are not needed. The properties relevant in the following are that

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|Jα (ξ, ξA )| vanishes for ξ → ∞, and that the phases are linear in ξ with the given frequencies. To make sure that these are properties of Jα (ξ, ξA ) and not just of its asymptotic expansion, we verified them numerically. With (C.12) at our disposal, the rest of the derivation of (C.10) is straightforward. Substituting (C.12) into (C.11) one shows for generic ξA the vanishing of the L → ∞ limit along the lines of (C.5)–(C.7). If both α and α are nonzero one of the ξ-dependent phases might cancel for the special boost parameter ξA = 12 ( αα + αα ). The modulus of this term in the asymptotics of Jα (ξ, ξA ) then is proportional to ↑ (αξ)−1/2 which is an element of Cainv , and the L → ∞ limit vanishes on account of (4.12). This establishes (C.10). The result (5.35) then follows by combining (C.4), (C.9), and (C.10).

Acknowledgments We like to thank A. Duncan for the enjoyable collaboration in [14]. M.N. also wishes to thank M. Lashkevich for contributing to another aspect of this project, and A. Ashtekar for asking about the reconstructed state space. E.S. would like to thank S. Ruijsenaars for helpful discussions. This work was supported by the EU under contract EUCLID HPRN-CT-2002-00325.

References [1] D. Ruelle, Statistical Mechanics, W.A. Benjamin, Reading, Mass. 1969. [2] G. Sewell, Quantum mechanics and its emergent macrophysics, Princeton UP, 2002. [3] H. Narnhofer and W. Thirring, Spontaneously broken symmetries, Ann. Inst. Henri Poincaré 70, 1 (1999) . [4] A. Paterson, Amenability, American Mathematical Society, Providence, R.I. 1988. [5] M. Niedermaier, Dimensionally reduced gravity theories are asymptotically safe, Nucl. Phys. B 673, 131 (2003); M. Niedermaier and H. Samtleben, An algebraic bootstrap for dimensionally reduced gravity, Nucl. Phys. B 579, (2000). [6] L. Faddeev and G. Korchemsky, High energy QCD as a completely integrable system, Phys. Lett. B 342, 311 (1995); S. Derkachov, G. Korchemsky, and A. Manashov, Noncompact Heisenberg spin chains from high energy QCD, Nucl. Phys. B 617, 375 (2001); Nucl. Phys. B 661, 533 (2003). [7] F. Wegner, The mobility edge problem: continuous symmetry and a conjecture, Z. Phys. B 35, 207 (1979).

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[8] A. Houghten, A. Jevicki, R. Kenway, and A. Pruisken, Noncompact sigmamodels and the existence of a mobility edge in disordered electronic systems near two dimensions, Phys. Rev. Lett. 45, 394 (1980). [9] S. Hikami, Anderson localization in a nonlinear sigma-model representation, Phys. Rev. B 24, 2671 (1981). [10] K.B. Efetov, Supersymmetry and theory of disordered metals, Adv. Phys. 32, 53 (83). [11] K.B. Efetov, Supersymmetry in Disorder and Chaos, Cambridge University Press, Cambridge, U.K. 1997. [12] D. Mermin and H. Wagner, Absence of ferromagnetism or anti-ferromagnetism in one or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17, 1133 (1966). [13] R.L. Dobrushin and S.B. Shlosman, Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics, Comm. Math. Phys. 42, 31 (1975). [14] T. Duncan, M. Niedermaier, and E. Seiler, Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma-models, hep-th/0405143. [15] H.O. Georgii, Gibbs measures and phase transitions, de Gruyter, Berlin and New York 1988. [16] F.P. Greenleaf, Amenable actions of locally compact groups, J. Funct. Anal. 4, 295 (1969). [17] P. Eymard, Moyénnes invariantes et représentations unitaires, Lecture Notes in Mathematics 300, Springer-Verlag, Berlin-New York 1972. [18] D. Amit and A. Davies, Symmetry breaking in the non-compact sigma model, Nucl. Phys. B 225, 221 (1983). [19] J.W. van Holten, Quantum noncompact sigma models, J. Math. Phys. 28, 1420 (1987). [20] D. Buchholz and I. Ojima, Spontaneous collapse of supersymmetry, Nucl. Phys. B 498, 228 (1997). [21] J. Löffelholz, G. Morchio and F. Strocchi, Spectral stochastic processes arising in quantum mechanical models with a non-L2 ground state, Lett. Math. Phys. 35, 251 (1995). [22] A. Ashtekar, J. Lewandowski, and H. Sahlmann, Polymer and Fock representations for a scalar field, Class. Quant. Grav. 20, L1 (2003). [23] A. Ashtekar, S. Fairhurst, and J. Willis, Quantum gravity, shadow states, and quantum mechanics, Class. Quant. Grav. 20, 1031 (2003). [24] C. Grosche and F. Steiner, The path integral on the pseudosphere, Ann. Phys. 182, 120 (1988).

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[25] J. Schaefer, Covariant path integral on hyperbolic surfaces, J. Math. Phys. 38, 11 (1997). [26] R. Camporesi, Harmonic analysis and propagators on homgeneous spaces, Phys. Repts. 196, 1 (1990). [27] J.P. Anker and P. Ostellari, The heat kernel on noncompact symmetric spaces, in: Lie groups and symmetric spaces, pp. 27–46, Amer. Math. Soc. Transl. Ser.2, 210, AMS. Providence, RI 2003. [28] I. Gradshteyn and I. Ryzhik, Table of integrals and products, Academic Press, New York and London 1980. [29] F. Olver, Introduction to asymptotics and special functions, Academic Press, New York and London 1978. [30] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 1, Academic Press, New York and London 1972. [31] J. Dixmier, C ∗ algebras, North Holland, Amsterdam 1977. [32] E. Seiler and K. Yildirim, Critical behavior in a quasi D-dimensional spin model, J. Statist. Phys. 112, 457 (2003); [hep-lat/0209166]. [33] K. Ziegler, Divergencies in a Vector Model with Hyperbolic Symmetry on a Chain, Z. Phys. B 43, 275 (1981). [34] D. Giulini and D. Marolf, A uniqueness theorem for constraint quantization, Class. Quant. Grav. 16, 2489 (1999); [gr-qc/9902045]. [35] A. Gomberoff and D. Marolf, On group averaging for SO(n, 1), Int. J. Mod. Phys. D8 (1999); [gr-gc/9902069]. [36] S. Coleman, There are no Goldstone bosons in two dimensions, Comm. Math. Phys. 31, 259 (1973). [37] A. Patrascioiu and E. Seiler, Continuum limit of 2D spin models with continuous symmetry and conformal field theory, Phys. Rev. E 57, 111 (1998); Does conformal quantum field theory describe the continuum limits of 2D spin models with continuous symmetry? Phys. Lett. B 417, 123 (1998). [38] K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, Comm. Math. Phys. 31, 83 (1973); Axioms for Euclidean Green’s functions 2, Comm. Math. Phys. 42, 281 (1975). [39] J. Glimm and A. Jaffe, Quantum Physics, Springer-Verlag, New York etc. 1987. [40] E. Seiler, Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Lecture Notes in Physics vol. 159, SpringerVerlag Berlin etc. 1982. [41] R. Haag, Local Quantum Physics, Springer-Verlag Berlin etc. 1992.

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[42] Y. Cohen and E. Rabinovici, A study of the non-compact non-linear sigmamodel: A search for dynamical realizations of non-compact symmetries, Phys. Lett. B124, 371 (1983). [43] N. Vilenkin and A. Klimyk, Representations of Lie groups and special functions, Kluwer, Dordrecht 1993. [44] H. Dym and H.P. McKean, Fourier Series and Integrals, Academic Press, New York and London 1972. [45] N. Balazs and A. Voros, Chaos on the pseudosphere, Phys. Repts. 143, 109 (1986). [46] I. Segal and R. Kunze, Integrals and operators, Springer-Verlag, Berlin – New York 1978. [47] T. Spencer and M.R. Zirnbauer, Spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions, Comm. Math. Phys. 252, 167 (2004), [arXiv:math-phys/0410032]. [48] N.I. Akhiezer and I.M. Glazman, Theory of linear operators in Hilbert space, Dover, New York 1993. Max Niedermaier Laboratoire de Mathématiques et Physique Théorique CNRS/UMR 6083 Université de Tours Parc de Grandmont F-37200 Tours France email: [email protected] Erhard Seiler Max-Planck-Institut f¨ ur Physik Werner-Heisenberg-Institut F¨ ohringer Ring 6 D-80805 M¨ unchen Germany email: [email protected] Communicated by Joel Feldman submitted 17/12/04, accepted 20/04/05



The Translation Invariant Massive Nelson Model: I. The Bottom of the Spectrum Jacob Schach Møller Abstract. In this paper we analyze the bottom of the energy-momentum spectrum of the translation invariant Nelson model, describing one electron linearly coupled to a second quantized massive scalar field. Our results are valid for all values of the coupling constant and include an HVZ theorem, non-degeneracy of ground states, existence of isolated groundstates in dimensions 1 and 2, non-existence of ground states embedded in the bottom of the essential spectrum in dimensions 3 and 4, (i.e., at total momenta where no isolated groundstate eigenvalue exists), and we study regularity and monotonicity properties of the bottom of the essential spectrum, as a function of total momentum.

1 Introduction and results In this section we introduce the Nelson model and formulate our main results. The notation we use is standard, but for the sake of completeness we give the basic constructions in Subsection 2.1.

1.1

Non-relativistic QED: An overview

In the last decade there has been a surge of interest in non-relativistic QED, sparked by a string of papers by H¨ ubner and Spohn, and by Bach, Fr¨ ohlich, and Sigal. See, e.g., [5, 4, 39, 38]. The purpose of this subsection is to give an overview over different aspects of the problem and place the model we study, as well as the results derived, into context. The fundamental Hamiltonian in non-relativistic QED, describing one charged particle, with mass M > 0 and charge e, coupled to a radiation field, is the minimally coupled one Hmin := 1l ⊗ dΓ(|k|) +

2 1 p ⊗ 1l − e A(x) , on L2 (R3x ) ⊗ Γ(L2 (R3k )) . (1.1) 2M

Here dΓ(|k|) is the kinetic energy of the radiation field, p = i∇x is the particle momentum operator, and A is the second quantized (massless) Maxwell field in the Coulomb gauge, i.e., ∇x · A = 0. The Hilbert space Γ(L2 (R3k )) is the bosonic Fock-space. See [34] and [5, 42]. In order to make sense of this operator (a priori as a form) one must introduce an ultraviolet cutoff into A. We recall that the model is translation invariant, in the sense that it commutes with the operator

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of total momentum P := p ⊗ 1l + 1l ⊗ dΓ(k). We remark that often the second quantized Pauli operator is taken as a starting point instead of (1.1). It is defined by replacing (p − eA)2 by (σ · (p − eA))2 , where σ is the vector of Pauli matrices. This operator differs from (1.1) by a magnetic term σ · (∇x × A) (and with L2 (R3x ) replaced by L2 (R3x ) ⊗ C2 , thus taking into account the spin of the particle). The study of Hmin is a natural starting point in non-relativistic QED. In particular in the context of scattering theory, where the dynamics of Hmin is a natural choice for ”free” dynamics. Unfortunately there are not many rigorous results established for the minimally coupled model, as it is formulated in (1.1), which are valid for all values of e (viewed as a coupling constant). We refer the reader to [35, 41]. Most results obtained in the literature are for Hmin perturbed by an electric potential, and results then pertain to existence and properties of ground states for the perturbed model, or localization in L2 (R3x ) of states below an ionization threshold. See [29, 30, 42, 43]. For a recent textbook treatment of the minimally coupled model and its classical counterpart, the Abraham model, see [56]. There are a number of different ways to obtain simpler problems. Some involve passing to phenomenological Hamiltonians, which are simpler to analyze than (1.1). We list some choices typically considered in the literature: S1) Consider the problem in the weak coupling regime, i.e., for |e| small. S2) Replace the massless photons by massive photons, which amounts to√replacing the massless dispersion relation k → |k| by a massive one k → k 2 + m2 , m > 0. This removes the infrared problem. S2 ) Set the interaction between soft photons (photons with small momenta) equal to zero. S3) Replace the minimal coupling with a linear coupling to a scalar field, i.e., replace Hmin by H = 1l ⊗ dΓ(|k|) +

1 2 p ⊗ 1l + gΦ(v), 2M

where Φ(v) is a field operator and g is a coupling constant. S4) Place the system in a confining external electric potential V , that is lim V (x) = ∞.

|x|→∞

This breaks the translation invariance of the problem. An extreme version of this are the spin-boson and Wigner-Weisskopf models. S4 ) Place the system in an external potential V such that p2 + V has isolated eigenvalues below the essential spectrum. Then consider Hmin + V ⊗ 1l in a low energy regime where states are isolated bound states of p2 + V dressed with photons. S5) A combination of the above.

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In this paper we consider the massive translation invariant linearly coupled model in any dimension, which (in dimension 3) can be viewed as a simplification of the minimally coupled model, by applying S2) and S3) as mentioned above. This model was considered by Nelson in [47], and it is distinguished by being renormalizable in a Hamiltonian setting, cf. also [10, 32, 53]. This model is often referred to as the Nelson model, a convention also adopted here. The models discussed in this introduction is part of a body of models sometimes referred to as Pauli-Fierz models. In this paper we do not consider renormalized operators. In addition we √ note that we work with more general dispersion relations ω and Ω than k 2 + m2 and p2 /2M respectively. We emphasize that we are interested in results which hold for all values of the coupling constant g. See Subsection 1.2 below for a more detailed description of the model. We remark that one can formulate the model and the simplifications discussed above for multiple particles coupled to a radiation field. For confined versions of the model, cf. S4) and S4 ) above, this makes no difference. However, for translation invariant models, not much is known. We pause to remind the reader that translation invariance, the fact that [H, P ] = 0, gives a direct integral representation H(ξ)dξ of the Hamiltonian. What we study in this paper is the bottom of the spectrum and essential spectrum of H(ξ) as functions of total momentum ξ. The former function is also called the ground state mass shell, or simply the mass shell. We note that in the massive case isolated excited states could exist and would give rise to excited mass shells. We are mainly inspired by works of Fr¨ ohlich [19, 20], Spohn [55], and one of Derezi´ nski and Gérard [14]. The results proved in these papers hold for all values of the coupling constant. Fr¨ ohlich considered properties of the ground state mass shell for the massless translation invariant Nelson model. Most of his results hold (suitably translated) also for massive photons. Derezi´ nski and Gérard were concerned with confined, in the sense of S3) above, massive linearly coupled models. They give a geometric proof of a HVZ theorem, thus locating the essential spectrum. (They furthermore apply Mourre theory and time-dependent scattering theory to the model.) Spohn proved a HVZ theorem for the translation invariant model, using in part ideas of Glimm and Jaffe (via a reference to [20]). He furthermore showed, in dimension 1 and 2, that the Hamiltonian at fixed total momentum admits an isolated groundstate. The results of Spohn are for a class of massive and subadditive dispersion relations ω. The result on existence of groundstates requires an additional assumption which excludes the dispersion relation √ k 2 + m2 , m > 0. In this paper we prove the following results for the structure of the bottom of the spectrum of the massive translation invariant Nelson model: A HVZ theorem, Theorem 1.2 (valid for ω which are not necessarily subadditive). The ground state mass shell is non-degenerate, Theorem 1.3, using a Perron-Frobenius argument of [19]. Existence of an isolated groundstate for all total momenta, Theorem 1.5√i) (dimensions 1 and 2), thus extending the result of Spohn to the case ω(k) = k 2 + m2 . Non-existence of a ground state embedded in the essential

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spectrum, Theorem 1.5 ii) (dimensions 3 and 4). Analyticity of the bottom of the essential spectrum, away from a closed countable set, Theorem 1.9. Maximality of the spectral gap and analyticity at local minima for the bottom of the essential spectrum, Theorem 1.10. See Subsection 1.3 for a precise formulation of the main results. In Subsection 4.2 we discuss how to extend the results to the model with a cutoff in the photon number operator. The models considered in this paper only fails to include the so-called (optical mode) polaron model of an electron in a ionic crystal by the requirement that ω(k) → ∞, |k| → ∞. This requirement is a consequence of our use of geometric methods to prove the HVZ theorem, and an adoption of the Glimm-Jaffe approach, as used in [20], might remedy this. However, the geometric approach is important for future work on Mourre and scattering theory. For mathematical work on the polaron model see [32, 44, 53, 54, 55], and for a textbook discussion see [18]. We remark that there are not many results on the translation invariant Nelson model, other than what we have already mentioned above, which are valid for all values of g. See however [31], Lemma 4.1 in this paper. In [54] upper and lower bounds on the effective mass are obtained (the effective mass is the inverse of the Hessian of the ground state mass shell at zero total momentum). There are more complete results available if one imposes a cutoff at small photon number, cf. [23] (the massless case with at most one photon). In the case of weak coupling there are more results, cf. [11, 22, 48]. See also [33, 36, 37]. (We remark however, that although the photon dispersion relation in [22] is massless, the interaction is of the type mentioned in S2 ) above, and the model thus retains massive features.) Finally we recall that for confined massive models, cf. S4) and S4 ) above, quite strong results, valid for all coupling constants, are available. See, apart from [14] mentioned above, the papers [1, 3, 21]. As for the massless confined model we refer the reader to [7, 9, 24, 26, 38]. We end this section with an overview of the paper. The rest of this chapter is devoted to a formal definition of the model and a presentation of assumptions and our main results. In Chapter 2 we present the second quantization formalism, extended objects pertaining to partitions of unity in Fock-space, basic estimates, and the pull-through formula. The core of the paper is Chapter 3, where all the main theorems are proved. In Chapter 4 we have assembled some miscellaneous results, and formulated the main theorems in the setting of the Nelson model with a cutoff in boson number. Finally in Appendix A, we present two mathematical tools, namely the calculus of almost analytic extensions, for a vector of commuting self-adjoint operators, and an abstract Perron-Frobenius result of Faris.

1.2

The translation invariant Nelson model

We consider a particle moving in Rν and interacting with a scalar radiation field. We write x and p = −i∇x for the particle position and momentum respectively.

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The particle Hilbert space is K := L2 (Rνx ) , and the Hamiltonian for a free particle is taken to be Ω(p), where Ω : Rν → R is a smooth dispersion relation. We are primarily interested in the standard nonp2 and Ω(p) = p2 + M 2 . Here relativistic and relativistic choices, i.e., Ω(p) = 2M M > 0 is the mass of the particle. The photon coordinates will be denoted by x = i∇k and k respectively and the one-photon space is hph := L2 (Rνk ) . The Hilbert space for the radiation field is the bosonic Fock-space F ≡ Γ(hph ) :=

∞

sn F (n) , where F (n) ≡ Γ(n) (hph ) := h⊗ ph .

(1.2)

n=0

We write Ω = (1, 0, 0, . . . ) for the vacuum. The creation and annihilation operators, a∗ (k) and a(k) satisfy the canonical commutation relations (CCR for short) [a∗ (k), a∗ (k )] = [a(k), a(k )] = 0 , [a(k), a∗ (k )] = δ(k − k ) ,

(1.3)

and a(k)Ω = 0. The free photon energy is the second quantization of the onephoton dispersion relation ω ω(k) a∗ (k) a(k) dk , where ω(k) := k 2 + m2 . (1.4) dΓ(ω) := Rν

Here m > 0 is the mass of the scalar photon. Our methods do not extend to the case of massless photons, m = 0. The full Hilbert space of the combined system is H := K ⊗ F . We will make the following identification H ≡ L2 (Rνx ; F ). The interaction considered here is linear in the field operator and is given by e−ik·x v(k) 1lK ⊗ a∗ (k) + eik·x v(k) 1lK ⊗ a(k) dk , V := (1.5) Rν

where the physical form of the interaction is v(k) = χ(k)/ ω(k) and χ is an ultraviolet cutoff, which ensures that v ∈ hph = L2 (Rνk ). The free and coupled Hamiltonians for the combined system are H := H0 + V , where H0 := Ω(p) ⊗ 1lF + 1lK ⊗ dΓ(ω) .

(1.6)

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The total momentum for the combined system is given by P := p ⊗ 1lF + 1lK ⊗ dΓ(k) .

(1.7)

The Hamiltonian H is translation invariant. That is, the energy momentum vector (P, H) has mutually commuting coordinates. Similarly for H0 . Translation invariance implies that H0 and H are fibered operators. We introduce a unitary transformation Ifib := F Γ(e−ik·x ) : H → L2 (Rνξ ; F ) , (1.8) where F is the Fourier transform F : L2 (Rνx ; F ) → L2 (Rνξ ; F ) and Γ(e−ik·x ) restricted to K ⊗ F (n) is multiplication by e−i(k1 +···+kn )·x . We have ∗ ∗ = H0 (ξ) dξ and Ifib H Ifib = H(ξ) dξ . (1.9) Ifib H0 Ifib Rν

Rν

The fiber operators H0 (ξ) and H(ξ), ξ ∈ Rν , are operators on F given by H(ξ) = H0 (ξ) + Φ(v) where H0 (ξ) = dΓ(ω) + Ω(ξ − dΓ(k)) and the interaction is

Φ(v) = Rν

v(k) a∗ (k) + v(k) a(k) dk .

(1.10)

(1.11)

We will in general use the notation v ∈ hph to denote a form-factor. In this paper we study the properties of the bottom of the joint spectrum of the vector (P, H).

1.3

Main results

In this subsection we will formulate precise conditions and state our main results. Proofs will be given in Section 3. The first condition is on the particle dispersion relation. We use the standard notation t := (1 + t2 )1/2 . Condition 1.1 (The particle dispersion relation) Let Ω ∈ C ∞ (Rν ). There exists sΩ ∈ {0, 1, 2}, a constant C, and for any multi-index α, with |α| ≥ 1, constants Cα , such that Ω(η) ≥ C −1 η sΩ − C and ∀α : |∂ α Ω(η)| ≤ Cα η sΩ −|α| . 2

p We note that the standard choices Ω(p) = 2M and Ω(p) = this condition with sΩ = 2 and sΩ = 1 respectively.

p2 + M 2 satisfy

Condition 1.2 (The photon dispersion relation) Let ω ∈ C ∞ (Rν ) satisfy i) There exists m > 0, the photon mass, such that inf k∈Rν ω(k) = ω(0) = m. ii) ω(k) → ∞, in the limit |k| → ∞.

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iii) There exists sω ≥ 0, a constant Cω , and for any multi-index α, with |α| ≥ 1, constants Cα , such that: ω(k) ≥ Cω−1 k sω − Cω and ∀α : |∂kα ω(k)| ≤ Cα k sω −|α| . The condition iii) is used in connection with pseudo differential calculus. The physical choice of ω used in (1.4) satisfies this condition (with sω = 1), and so does ω(k) = k 2 + m (with sω = 2). We introduce a space of test functions C0∞ := Γfin (C0∞ (Rν )) .

(1.12)

Note that since H0 (ξ) is a bounded from below multiplication operator on each n-particle sector, we find that it is essentially self-adjoint on C0∞ . We recall the following result, cf. [47], [19], and [20]. For completeness we give a proof in the beginning of Section 3. Proposition 1.1 Let v ∈ L2 (Rν ). Assume Ω and ω, satisfy Conditions 1.1 and 1.2 i) respectively. Then i) D(H0 (ξ)) is independent of ξ and we denote it by D. ii) Φ(v) is H0 (ξ)-bounded with relative bound 0. In particular H(ξ) is bounded from below, self-adjoint on D(H(ξ)) = D(H0 (ξ)), and essentially self-adjoint on C0∞ . iii) The bottom of the spectrum of the fiber Hamiltonians, ξ → Σ0 (ξ) := inf σ(H(ξ)), is Lipschitz continuous. We introduce some notation. First the bottom of the spectrum of the full operator: Σ0 := infν Σ0 (ξ) > −∞ . ξ∈R

For n ≥ 1 and k = (k1 , . . . , kn ) ∈ Rnν we often write k (n) = k1 + · · · + kn . We now introduce the bottom of the spectrum for a composite system at total momentum ξ, consisting of an interacting system at total momentum ξ − k (n) and n non-interacting photons with momenta k: n (n) ω(kj ) . Σ0 (ξ; k) := Σ0 ξ − k (n) +

(1.13)

j=1

The following functions are thresholds due to ground states dressed by n photons, at critical momenta: (n) (n) Σ0 (ξ) := infnν Σ0 (ξ; k) . (1.14) k∈R

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The bottom of the essential spectrum (see Theorem 1.2 below) (n)

Σess (ξ) := inf Σ0 (ξ) .

(1.15)

n≥1

We have the following elementary properties of the functions introduced above. Namely 0 ≤ Σess (ξ) − Σ0 (ξ) ≤ m Σ0 (ξ) = Σ0 ⇒ Σess (ξ) = Σ0 (ξ) + m lim Σ0 (ξ) =

|ξ|→∞

lim Σess (ξ) =

|ξ|→∞

(n)

lim Σ0 (ξ) = ∞

|ξ|→∞

(n)

lim Σ0 (ξ) = ∞ .

n→∞

(1.16) (1.17) (1.18) (1.19)

Our first result is Theorem 1.2 (HVZ) Let v ∈ L2 (Rν ). Assume Conditions 1.1, and 1.2. Then i) Eigenvalues of H(ξ) below Σess (ξ) have finite multiplicity and can only accumulate at Σess (ξ). ii) σess (H(ξ)) = [Σess (ξ), ∞). The method of proof for the HVZ theorem is geometric and follows ideas of [14], cf. Subsection 3.2. See also [1, 2, 9, 15, 24]. The name ”HVZ” (Hunziker–van Winter–Zhislin) is used because the geometric idea of the proof is quite similar to that employed in the proof of the standard HVZ theorem for N -body Schr¨ odinger operators, cf. [13, Theorem 6.2.2]. We recall that there is another method, due to Glimm and Jaffe [28], one can employ to obtain an HVZ theorem. See [55, Section 4], for the case of subadditive dispersion relations ω, and in addition [8, 20]. In the following we will impose either v ∈ L2 (Rν ) , v is real-valued, and v = 0 a.e.

(1.20)

or v ∈ L2 (Rν ) , v is real-valued, and ∀R > 0 : essinf |v(k)| > 0 .

(1.21)

k:|k|≤R

We have the following result on non-degeneracy of groundstates. This type of result is not new, cf. [31, Section 6] and [20, Section 2]. Theorem 1.3 (Non-degeneracy of ground states) Let v satisfy (1.20) and assume Conditions 1.1 and 1.2. Then, if Σ0 (ξ) is an eigenvalue for H(ξ), it is nondegenerate. We note that the result of Gross [31] is for zero total momentum only, and assumed that p → exp(−tΩ(p)) is a positive definite function for all t > 0. Gross pass to the Schr¨ odinger representation of the Fock-space, where H0 (ξ) is positivity improving if and only if ξ = 0.

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For the remaining results we will need either i) or i ) in the condition below. Condition 1.3 ω ∈ C ∞ (Rν ) satisfies i) Subadditivity: For k1 , k2 ∈ Rν we have ω(k1 + k2 ) ≤ ω(k1 ) + ω(k2 ). i ) Strict subadditivity: For k1 , k2 ∈ Rν we have ω(k1 + k2 ) < ω(k1 ) + ω(k2 ). √ The standard dispersion relation ω(k) = k 2 + m2 satisfies Condition 1.3 i ), but ω(k) = k 2 + m does not. We remark that if ω is convex and satisfies: ∀k ∈ Rν : ω(k) − k · ∇ω(k)

(>) ≥

0,

(1.22)

then ω is (strictly) subadditive. If ω is (strictly) subadditive we find, for all ξ ∈ Rν , (n)

Σ0 (ξ)

(n Σ0 (ξ), then k ∈ I0 (ξ). The following lemma allows us to restrict the analysis to one dimension. Lemma 1.7 Assume Conditions 1.1, 1.2 i), and 1.4 i), ii). Let ξ ∈ Rν \{0} and (n) (n) n ∈ N. Any local minimum k ∈ I0 (ξ) of k → Σ0 (ξ; k) is of the form k1 = · · · = kn = θξ, for some θ ∈ R. Let u be a unit vector in Rν . We write σ(t) = Σ0 (tu), for t ∈ R. By rota(n) tion invariance, σ is independent of u. Similarly we write σ (n) (t) := Σ0 (tu) and σess (t) := Σess (tu). With a slight abuse of notation we write ω(s) = ω(su) and I0 (n) to denote the set of t’s such that tu ∈ I0 . We furthermore use the symbol I0 (t), n > 0 (not necessarily integer), to denote the set {s ∈ R : t − ns ∈ I0 }. In light of the previous lemma, we introduce now, for n > 0 and not necessarily integer, the following functions σ (n) (t; s) = σ(t − ns) + n ω(s) and σ (n) (t) = inf σ (n) (t; s) . s∈R

(1.26)

(n)

Note that by Lemma 1.7 we have, for integer n, Σ0 (ξ) = σ (n) (|ξ|), and in particular Σess (ξ) = σ (1) (|ξ|). In this connection we mention that a local minimum (n) for Σ0 (ξ; ·) induces a local minimum for σ (n) (|ξ|; ·). Conversely however, a local minimum for σ (n) (t; ·), which is not a global minimum, could be associated with (n) a saddle point for Σ0 (tu; ·). We have, cf. also [19, Lemma 1.6], Proposition 1.8 Let v satisfy (1.20) and assume Conditions 1.1, 1.2, and 1.4. Let λ < Σ0 . The family of self adjoint operators t → (H(tu) − λ)−1 is analytic of type A. Furthermore, the map I0 t → σ(t) is analytic.

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See [40, Chapter VII] for analytic perturbation theory. The following regularity result is proved by keeping track of global minima of s → σ (n) (t; s), as functions of t. Theorem 1.9 Let v satisfy (1.20) and assume Conditions 1.1, 1.2, 1.3 i), and 1.4. Let n > 0. There exists a closed countable set T (n) ⊂ R, and an analytic map (n) R\T (n) t → Θ(n) (t) ∈ I0 (t) with the property that the maps s → σ (n) (t; s), t ∈ R\T (n) , has a unique global minimum at s = Θ(n) (t) which is non-degenerate, i.e., ∂s2 σ (n) (t; Θ(n) (t)) > 0. In particular R\T (n) t → σ (n) (t) is analytic and d (n) σ (t) = ∂ω Θ(n) (t) , for t ∈ R \ T (n) . (1.27) dt Our final main result is concerned with the structure of the spectrum near local minima of the essential spectrum Theorem 1.10 Let v satisfy (1.20) and assume Conditions 1.1, 1.2, 1.3 i), and 1.4. Let t0 be a local minimum of t → σess (t). Then the spectral gap at t0 is maximal, i.e., σess (t0 ) − σ(t0 ) = m, the map t → σ(t) has a local minimum at t0 , the map t → σess (t) is analytic near t0 , and d2 ∂ 2 ω(0) ∂ 2 σ(t0 ) . σ (t ) = ess 0 dt2 ∂ 2 ω(0) + ∂ 2 σ(t0 )

(1.28)

2 Notation and preliminaries In this section we recall known facts. The reader is urged to consult in particular [14], where most of the results pertaining to second quantization can be found.

2.1

The second quantization functor Γ

Let h be a complex Hilbert space with inner product ·, · , which is conjugate linear in the first variable and linear in the second. We use the standard notation Γ(h) for the associated bosonic Fock-space, see (1.2). For a (not necessarily dense) subspace C ⊂ h, we write Γfin (C) for the subspace of Γ(h) consisting of finite linear combinations of elements of the algebraic tensor products C ⊗s n , n ≥ 0. If C is dense in h, then Γfin (C) is dense in Γ(h). operators. We write a∗ (f ) and a(f ), f ∈ h, for the creation and annihilation √ Recall that for u ∈ Γ(n) (h) := h⊗s n , the n-particle sector; a∗ (f )u = n + 1Sn+1 f ⊗ u ∈ Γ(n+1) (h). Here Sk is the symmetrization operator on h⊗k . We furthermore recall that a∗ (f ) and a(f ) are closed and densely defined, and that D(a(f )) = D(a∗ (f )). They satisfy the CCR: [a∗ (f ), a∗ (g)] = [a(f ), a(g)] = 0 , [a(f ), a∗ (g)] = f, g

(2.1)

and a(f )Ω = 0, for f ∈ h. The field operator Φ(f ) := a∗ (f ) + a(f )

(2.2)

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is self-adjoint on D(a∗ (f )) = D(a(f )) and essentially self-adjoint on Γfin (h). In

the case h = hph we have the relation with (1.3): a∗ (f ) = Rν f (k)a∗ (k)dk and

a(f ) = Rν f (k)a(k)dk. In particular (2.2) and (1.11) coincide. We frequently write a# (k) to denote either a(k) or a∗ (k). Similarly for a# (f ). Recall that a(k) is well defined on C0∞ = Γfin (C0∞ (Rν )), but it is not closable. The domain of its adjoint (a(k))∗ equals {0}. The ”operator” a∗ (k) should be understood as a form. See the monograph by Berezin [6]. Let b be a bounded operator between Hilbert spaces h1 and h2 . We define Γ(b) : Γ(h1 ) → Γ(h2 ) by its restriction to Γ(n) (h1 ) n times

Γ(b)|Γ(n) (h1 )

:= b ⊗ · · · ⊗ b .

In particular we have Γ(b)Ω = Ω. Recall that Γ(b) is bounded if and only if bB(h1 ;h2 ) ≤ 1. We introduce dΓ(a) for operators a : h → h with domain D(a) by dΓ(a)|Γ(n) (h) := a ⊗ 1lh ⊗ · · · ⊗ 1lh + · · · + 1lh ⊗ · · · ⊗ 1lh ⊗ a ,

(2.3)

a priori on the domain Γfin (D(a)). In particular; dΓ(a)Ω = 0. The operators Γ(b) and dΓ(a) are related through the formula Γ(ea ) = edΓ(a) (suitably interpreted). It is easy to see that if a is closed (or closable) on D(a) then dΓ(a) is closable on Γfin (D(a)). See [24, Section 3.2] for a simple proof, which applies also to similar situations below. In addition, if a is self-adjoint, then dΓ(a) is essentially selfadjoint on Γfin (D(a)), cf. [51, Subsection VIII.10, Theorem VIII.33 and Example 2]. For closed a we will by dΓ(a) understand the closure of (2.3). Otherwise dΓ(a) denotes the operator in (2.3) with the a priori domain Γfin (D(a)). For a quadratic form a with form-domain Q(a) we also write dΓ(a) for the quadratic form defined on Γfin (Q(a)) by (2.3). An important operator is the number operator N := dΓ(1lh ) ,

(2.4)

which in the case h = hph can be written as N = Rν a∗ (k)a(k)dk. See also (1.4). Let a and b be densely defined operators on h and v ∈ D(a). We have the following commutation properties, which should be interpreted as forms on Γfin (D(a∗ ) ∩ D(b∗ )) × Γfin (D(a) ∩ D(b)) and Γfin (D(a∗ )) × Γfin (D(a)) respectively. i[dΓ(a), dΓ(b)] = dΓ(i[a, b]) , ∗

[a (v), dΓ(a)] = −a∗ (av) , [a(v), dΓ(a)] = a(av) , and i[Φ(v), dΓ(a)] = − Φ(iav) .

(2.5)

Let b : h1 → h2 be a contraction and a : h1 → h2 with domain D(a). We define dΓ(b, a) : Γ(h1 ) → Γ(h2 ) on Γfin (D(a)) by dΓ(b, a)|Γ(n) (h1 ) := a ⊗ b ⊗ · · · ⊗ b + · · · + b ⊗ · · · ⊗ b ⊗ a .

(2.6)

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In particular (in the case h1 = h2 = h) dΓ(1lh , a) = dΓ(a); cf. (2.3). If a is closed (or closable) we find, as above, that dΓ(b, a) is closable on Γfin (D(a)). As for dΓ(a) we use the notation dΓ(b, a) also in the case where a is a form on h2 × h1 . Let b : h1 → h2 be a contraction, a1 : h1 → h1 and a2 : h2 → h2 be densely defined. As a form on Γfin (D(a∗2 )) × Γfin (D(a1 )) we have (Γ(b) dΓ(a1 ) − dΓ(a2 ) Γ(b)) = dΓ(b, (ba1 − a2 b)) .

2.2

(2.7)

Basic estimates involving Γ

We have the following lemma, cf. [14, Lemma 2.1], Lemma 2.1 For f ∈ h and s ≥ 0, we have a# (f ) : D(N s+1/2 ) → D(N s ) and the following holds true i) Let f1 , . . . , fn ∈ h and k ≥ 0. Then (N + 1)k a# (f1 ) · · · a# (fn ) (N + 1)− n2 −k ≤ Ck,n f1 · · · fn . ii) The following map is norm-continuous n

hn (f1 , . . . , fn ) → (N + 1)k a# (f1 ) · · · a# (fn ) (N + 1)− 2 −k ∈ B(Γ(h)) . iii) Let {f1, }∈N , . . . , {fn, }∈N be uniformly bounded sequences, converging weakly to zero in h. Then n

s − lim (N + 1)k a(f1, ) · · · a(fn, ) (N + 1)− 2 −k = 0 . →∞

˜ and a2 : h2 → ˜h. Define Suppose b ∈ B(h1 ; h2 ) is a contraction, a1 : h1 → h a as a form on D(a2 ) × D(a1 ) by (f, ag) := (a2 f, a1 g). Then, for v ∈ Γfin (D(a1 )) and u ∈ Γfin (D(a2 )), 1

1

|u, dΓ(b, a)v | ≤ u, dΓ(a∗2 a2 )u 2 v, dΓ(a∗1 a1 )v 2 .

(2.8)

˜ = h2 , a2 = 1lh2 , Here a∗# a# denote the obvious forms on h# . Taking in particular h and a1 = a we get, for v ∈ Γfin (D(a)), 1

1

(N + 1)− 2 dΓ(b, a)v ≤ v, dΓ(a∗ a)v 2 .

(2.9)

In connection with this bound we also use the easy property a ≤ b

=⇒

dΓ(a) ≤ dΓ(b) ,

(2.10)

where a and b are self-adjoint operators (or symmetric forms) on h. We also make use of the following estimate, cf. [27, Lemma A.2]. Let k ∈ N and let a and b be self-adjoint operators on h. If 0 ≤ a ≤ b for all 1 ≤ ≤ k, with ∈ N. Then (dΓ(a))k ≤ (dΓ(b))k .

(2.11)

We note that there are several bounds involving powers of second quantized operators, cf., e.g., [15, Lemma 3.2] and [24, Section 3.2] for a selection.

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ˇ The extended space and Γ

Let h0 and h∞ be two Hilbert spaces. We will use the standard unitary identification U : Γ(h0 ⊕ h∞ ) → Γ(h0 ) ⊗ Γ(h∞ ), which is determined uniquely by linearity and the two properties UΩ = U a ((f, g)) = ∗

Ω⊗Ω ∗ a (f ) ⊗ 1lΓ(h∞ ) + 1lΓ(h0 ) ⊗ a∗ (g) U .

Let a0 : h0 → h0 and a∞ : h∞ → h∞ . We have the intertwining property U dΓ(a0 ⊕ a∞ ) = dΓ(a0 ) ⊗ 1lΓ(h∞ ) + 1lΓ(h0 ) ⊗ dΓ(a∞ ) U ,

(2.12) (2.13)

(2.14)

as an identity on Γfin (D(a0 ) ⊕ D(a∞ )). Let h, h0 and h∞ be Hilbert spaces and let b = (b0 , b∞ ), where b0 ∈ B(h; h0 ) and b∞ ∈ B(h; h∞ ). We view b as an element of B(h; h0 ⊕ h∞ ) and define the ˇ associated operator Γ(b) by ˇ Γ(b) := U Γ(b) : Γ(h) → Γ(h0 ) ⊗ Γ(h∞ ) .

(2.15)

In this paper we always require b∗0 b0 +b∗∞ b∞ = 1lh , which implies bB(h;h0 ⊕h∞ ) = 1 ˇ and Γ(b) is an isometry: ˇ ∗ Γ(b) ˇ Γ(b) = 1lΓ(h) . (2.16) ˇ We interpret Γ(b) as a partition of unity. Let b = (b0 , b∞ ) be as above, and let a = (a0 , a∞ ) be an operator from h to h1 ⊕ h2 , with domain D(a) = D(a0 ) ∩ D(a∞ ). We introduce the operator ˇ a) : Γfin (D(a)) → Γ(h0 ) ⊗ Γ(h∞ ) by dΓ(b, ˇ a) := U dΓ(b, a) . dΓ(b,

(2.17)

We use the same notation for forms a = (a0 , a∞ ), where a# are forms on h# × h. Let r : h → h, q0 : h0 → h0 and q∞ : h∞ → h∞ , be densely defined operators. We have the following intertwining relation, viewed as an identity between forms ∗ ))} × Γfin (D(r)): on {Γfin (D(q0∗ )) ⊗ Γfin (D(q∞ ˇ ˇ ˇ a) , (2.18) Γ(b)dΓ(r) − dΓ(q0 ) ⊗ 1lΓ(h∞ ) + 1lΓ(h0 ) ⊗ dΓ(q∞ ) Γ(b) = dΓ(b, ∗ where a = (b0 r − q0 b0 , b∞ r − q∞ b∞ ) has form-domain {D(q0∗ ) ⊕ D(q∞ )} × D(r).

2.4

ˇ Basic estimates involving Γ

˜# and a#,2 : h# → h ˜# , where Let b = (b0 , b∞ ) be as in (2.17). Let a#,1 : h → h ˜ h# are auxiliary Hilbert spaces. Here # denotes 0 and ∞. We define a form a = (a0 , a∞ ) on {D(a0,2 ) ⊕ D(a∞,2 )} × {D(a0,1 ) ∩ D(a∞,1 )} by prescribing the forms a0 and a∞ as follows: (f, a# g) := (a#,2 f, a#,1 g) on D(a#,2 ) × D(a#,1 ).

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Let u0 ∈ Γfin (D(a0,2 )), u∞ ∈ Γfin (D(a∞,2 )), v ∈ Γfin (D(a0,1 ) ∩ D(a∞,1 )). The following key estimate follows from (2.14) and (2.8) ˇ a)v | |u0 ⊗ u∞ , dΓ(b,

1 1 ≤ u0 , dΓ(a0,2 a∗0,2 )u0 2 u∞ + u0 u∞ , dΓ(a∞,2 a∗∞,2 )u∞ 2 1

×v, dΓ(a∗0,1 a0,1 + a∗∞,1 a∞,1 )v 2 .

(2.19)

Again a∗#,2 a#,2 denote the obvious forms on D(a#,2 ), and a∗0,1 a0,1 + a∗∞,1 a∞,1 is a form on D(a0,1 ) ∩ D(a∞,1 ). ˜# = h# , a#,2 = 1lh , and a#,1 = a# ) As for (2.9) this implies (here h # 1 ˇ a)v ≤ v, dΓ(a∗ a0,1 + a∗ a∞,1 )v 12 . (N0 + N∞ )− 2 dΓ(b, 0,1 ∞,1

(2.20)

Here and in the following we use the notation (cf. (2.4)) N0 = dΓ(1lh0 ) ⊗ 1lΓ(h∞ ) and N∞ = 1lΓ(h0 ) ⊗ dΓ(1lh∞ ) .

2.5

(2.21)

Auxiliary spaces and operators

In this subsection we introduce some notation which will be used in the proof of the HVZ theorem in Subsection 3.2. We introduce auxiliary Hilbert spaces for an interacting system accompanied by a fixed number ≥ 1 of auxiliary photons H() := F ⊗ F () ≡ L2sym (Rν ; F ) . Here the subscript sym indicates that functions are symmetric under permutation, i.e., f (kτ (1) , . . . , kτ () ) = f (k1 , . . . , k ) a.e., for any τ ∈ S( ) the group of permutations of the set {1, . . . , }. For ∈ N we extend the notation for second quantization as follows dΓ() (a) =

dΓ(a) ⊗ 1lF () + 1lF ⊗ dΓ(a)|F () ,

for operators a on hph . Again dΓ(a) defined on Γfin (D(a))⊗ D(a)⊗s is closable (essentially self-adjoint) if a is closable (essentially self-adjoint). For the Hamiltonian we write () (2.22) H () (ξ) := H0 (ξ) + Φ(v) ⊗ 1lF () , where We note that

() H0 (ξ) := dΓ() (ω) + Ω ξ − dΓ() (k) . () H0 (ξ)

(2.23)

is essentially self-adjoint on ∞()

C0 ()

:= C0∞ ⊗ Γ() (C0∞ (Rν )) .

(2.24)

and write D() = D(H0 (ξ)), which is independent of ξ. Observe that there is no interaction between the auxiliary photons, nor are they coupled with the

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interacting system (apart from the coupling coming from the dispersive structure). () Note that as for Proposition 1.1, Φ(v) ⊗ 1lF () is H0 (ξ)-bounded with relative ∞() and self-adjoint on D() . bound 0, so H () (ξ) is essentially self-adjoint on C0 Using a direct integral representation we can write the auxiliary Hamiltonian for each total momentum ξ as () H (ξ) = H () (ξ; k) dν k , (2.25) Rν

where H

()

(ξ; k) := H(ξ − k) +

ω(kj ) 1lF .

(2.26)

j=1 ()

Here dν k = Πj=1 dν kj . We have a similar fibration of H0 (ξ). The fiber operators, being spectral translates of a Hamiltonian at a different total momentum, are clearly self-adjoint on D and essentially self-adjoint on C0∞ . We note the following important observations

() Σ0 (ξ; k) = inf σ H () (ξ; k) , (2.27)

() () Σ0 (ξ) = inf σ H (ξ) . (2.28)

2.6

Geometric partition of unity and extended operators

In the analysis of the many-body problem, a central tool is a geometric partition of unity in the configuration space; cf. [13]. Here we will need a similar notion, made complicated by the fact that we have to partition an infinite number of particles. The type of partition of unity used here was introduced in [14] and subsequently used by many authors, cf. [1, 2, 15, 21, 24, 27]. Here h = h0 = h∞ = hph . Let j0 , j∞ ∈ C ∞ (Rν ) be non-negative functions 2 = 1. satisfying: j0 = 1 on {k : |k| ≤ 1}, j0 = 0 on {k : |k| > 2}, and finally j02 + j∞ R R By j , R > 1, we understand the operator j = (j0 (x/R), j∞ (x/R)). Recall that x = i∇k is a differential operator. We view j R as a map from hph into hph ⊕ hph ˇ R ) is an isometry, see (2.16), and the operator Γ(j ˇ R ) : F → F ext := F ⊗ F and Γ(j ˇ R ) = 1lF . ˇ R )∗ Γ(j Γ(j

(2.29)

The partition of unity is used to decouple photons at infinity from photons near the electron. In fact the reader should think of the first component as the Fock-space for interacting photons and the second component as the Fock-space for non-interacting photons at infinity. As in the previous section we extend the notation for second quantization to these extended spaces. We will in general call operators constructed this way, extended operators. The simplest extended operator is the extended number operator, already encountered in Subsection 2.4 N ext := N0 + N∞ .

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This is a particular case of the following notation, which will be used for operators a on hph , dΓext (a) = dΓ(a) ⊗ 1lF + 1lF ⊗ dΓ(a) . (2.30) As in the previous section dΓext (a) is closable (essentially self-adjoint) if a is closable (essentially self-adjoint). Using this notation we introduce the extended Hamiltonian as (2.31) H ext (ξ) := H0ext (ξ) + Φ(v) ⊗ 1lF , where

H0ext (ξ) := dΓext (ω) + Ω ξ − dΓext (k) .

(2.32) C0∞

C0∞

The free extended Hamiltonian (2.32) is essentially self-adjoint on ⊗ and we write Dext = D(H0ext (ξ)), which is independent of ξ. Note that as for Proposition 1.1, Φ(v) ⊗ 1lF is H0ext (ξ)-bounded with relative bound 0, so H ext (ξ) is essentially self-adjoint on C0∞ ⊗ C0∞ and self-adjoint on Dext . Using the notation introduced in the previous subsection we have ∞ (2.33) F ext = F ⊕ H() , =1

and H ext (ξ) = H(ξ) ⊕

∞

H () (ξ) .

(2.34)

=1

2.7

The pull-through formula

In the following we use that a(k) makes sense as an operator on C 0 = Γfin (hph ∩ C 0 (Rν )). Here C 0 (Rν ) denotes the space of continuous functions on Rν . Note that a(k) : C 0 → C 0 , a(k) : C0∞ → C0∞ , and under the assumption v ∈ L2 (Rν ) ∩ C 0 (Rν ), we have H(ξ) : C0∞ → C 0 . Note that v need not be real-valued. For the definition of C0∞ , see (1.12). The type of formula presented here has been used previously in the study of ground states of translation invariant models, cf. [19], and confined models, see, e.g., [5, 24, 26]. Proposition 2.2 Suppose v ∈ L2 (Rν ) ∩ C 0 (Rν ). Let ξ ∈ Rν , n ≥ 1, k ∈ Rnν , and z ∈ C. For ψ ∈ C0∞ we have the identity a(k1 ) · · · a(kn ) (H(ξ) − z) ψ n = H(ξ − k (n) ) + ω(ki ) − z a(k1 ) · · · a(kn ) ψ i=1

+

n

v(ki ) a(k1 ) · · · a(k i ) · · · a(kn ) ψ ,

i=1

where k

(n)

= k1 + · · · + kn .

The notation a(k i ) indicates that the term a(ki ) is omitted from the product.

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For n = 1 we formulate another pull through formula. Note that for ψ ∈ 1 D(N 2 ), the map k → a(k)ψ is in L2 (Rν ; F ). In general, for ψ ∈ F we have 1 k → a(k)ψ in L2 (Rν ; D(N 2 )∗ ). The following proposition can be proved directly as in [26, Proposition 3.4], or by using Proposition 2.2 and an approximation argument. Proposition 2.3 Suppose v ∈ L2 (Rν ). Let ξ ∈ Rν and z ∈ C, Imz = 0. For ψ ∈ D, we have the L2 (Rν ; F )-identity −1 H(ξ − k) + ω(k) − z a(k) (H(ξ) − z) ψ −1 ψ. = a(k) ψ + v(k) H(ξ − k) + ω(k) − z

3 Spectral theory We start this section by giving a proof of Proposition 1.1. First some simple observations. Since 1 ≤ m− ω(k) for any ≥ 0, we obtain from (2.11) that N k ≤ −k m dΓ(ω)k , for k ∈ N. Since 0 ≤ dΓ(ω) ≤ H0 (ξ) and they commute, we find that dΓ(ω)k ≤ H0 (ξ)k for any k ∈ N. We thus get N k ≤ m−k H0 (ξ)k , for k ∈ N. This estimate in particular shows that for k ∈ N k

k

k

k

N 2 is H0 (ξ) 2 − bounded and N ext 2 is H0ext (ξ) 2 − bounded .

(3.1)

Proof of Proposition 1.1. We begin by showing that D(H0 (ξ)) is independent of

1 ξ. We compute on C0∞ as an operator identity H(ξ) − H(0) = ξ · 0 ∇Ω(tξ − dΓ(k))dt. By Condition 1.1 and the estimate ab ≤ aq + bp , q −1 + p−1 = 1 we obtain (H(ξ) − H(0))ψ ≤ Ω(dΓ(k))ψ + C(, ξ)ψ, for any > 0 and ψ ∈ C0∞ . That the domain is independent of ξ now follows from the Kato-Rellich theorem [49, Theorem X.12]. As for ii), the observation (3.1) (applied with k = 1), together with the N 1/2 -boundedness of Φ(v), cf. Lemma 2.1 i), implies the result. The last part follows from the variational principle and an argument similar to the one given for i). We leave it to the reader. Clearly Proposition 1.1 also holds with {H0 (ξ), H(ξ)} replaced by either of () the pairs {H0ext(ξ), H ext (ξ)} or {H0 (ξ), H () (ξ)}. We note the following consequence, for k ∈ {1, 2}, k

k

k

k

N 2 is H(ξ) 2 − bounded , N ext 2 is H ext (ξ) 2 − bounded , N () Here N () := dΓ() (1lhph ).

k 2

k

is H () (ξ) 2 − bounded .

(3.2) (3.3)

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Localization errors

ˇ R) In this subsection we show that localization errors arising when we apply Γ(j are small for large R. Lemma 3.1 Let s ∈ N0 ∩ [0, sΩ ] and f ∈ C ∞ (Rν ) satisfy the bound |(∂ α f )(η)| ≤ Cα η s−|α| , for any multi-index α. Let t = 1, if s = 0, and t = (1 + sΩ − s)/2 if s ≥ 1. We have as a form on F ext × F, R ˇ R ) (H0 (ξ) − i)−1 ˇ ) f (ξ − dΓ(k)) − f (ξ − dΓext (k)) Γ(j (H0ext (ξ) − i)−1 Γ(j = (H0ext (ξ) − i)−t B1 (R) = B2 (R)(H0 (ξ) − i)−t , where B1 and B2 are families of bounded operators which satisfy B1 (R) + B2 (R) = O(R−1/2 ), as R → ∞, locally uniformly in ξ. Proof. As a first step we compute as a form on (C0∞ ⊗ C0∞ ) × C0∞ , for 1 ≤ p ≤ ν, ˇ R ) dΓ(k;p ) − dΓext (k;p ) Γ(j ˇ R ) = dΓ(j ˇ R , sR Γ(j p),

(3.4)

R R R sR p = ([j0 , k;p ], [j∞ , k;p ]). Clearly sp are bounded operators and R , k;p ] = O(R−1 ) , as R → ∞ . [j#

(3.5)

Here we used the notation k;p to denote the p’th coordinate of a vector k ∈ Rν . (This notation should not be confused with the labeling kj of a family of vectors kj ∈ Rν .) We consider first the case s = 0. Let f˜ ∈ C ∞ (Cν ) denote an almost analytic extension of f . Let χ ∈ C0∞ (Rν ) be equal to 1 near 0. Write χn (η) = χ(η/n). Then fn = χn f has almost analytic extensions fñ satisfying that, for all z ∈ Cν : ∂¯fñ (z) → ∂¯f˜(z), and the estimates |∂¯fñ (z)| ≤ C z −1− |Imz|

(3.6)

hold uniformly in n, cf. (A.4). If we take for example the Borel construction (A.2), for f˜ and the fñ ’s, then this property is easy to verify. This well-known approximation technique has been used by many authors (in the case ν = 1), see, e.g., [52, Section 5] and [46, Section 4]. We use (3.4) to compute as a form on (C0∞ ⊗ C0∞ ) × C0∞ , for Imz = 0,

=

ˇ R) ˇ R ) |ξ − dΓ(k) − z|2 − |ξ − dΓext (k) − z|2 Γ(j T (z; R) := Γ(j ν ext ˇ R , sR ˇ R , sR dΓ(j (k;p ) + z;p ) dΓ(j p ) (ξ;p − dΓ(k;p ) − z;p ) + (ξ;p − dΓ p) . p=1

˜# = h# = hph and a#,1 = [j R , k;p ]), and (3.5), we Using (2.10), (2.20) (with h = h # conclude the following estimate 1

1

(N ext + 1)− 2 |ξ − dΓext (k) − z|−1 T (z; R) |ξ − dΓ(k) − z|−1 (N + 1)− 2 (3.7) = O |Imz|−1 R−1 . The estimate is valid uniformly in ξ and Rez = {Rez1 , . . . , Rezν }.

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We proceed to compute

=

ˇ R) ˇ R ) |ξ − dΓ(k) − z|−2ν − |ξ − dΓext (k) − z|−2ν Γ(j Γ(j ˇ R ) |ξ − dΓ(k) − z|2ν −|ξ − dΓ(k) − z|−2ν Γ(j ˇ R ) |ξ − dΓ(k) − z|−2ν − |ξ − dΓext (k) − z|2ν Γ(j

=

−

ν−1

|ξ − dΓ(k) − z|−2(ν−j) T (z; R) |ξ − dΓ(k) − z|−2(j+1) .

(3.8)

j=0

Combining this identity with (3.7), we obtain the estimate 1 ˇ R ) |ξ − dΓ(k) − z|−2ν (N ext + 1)− 2 Γ(j ˇ R ) (N + 1)− 12 − |ξ − dΓext (k) − z|−2ν Γ(j = |ξ − dΓext (k) − z|−1 O |Imz|−2ν R−1 = O |Imz|−2ν R−1 |ξ − dΓ(k) − z|−1 .

(3.9)

A small calculation using (3.4) (and again the estimates (2.10), (2.20), and (3.5)) in conjunction with (3.6) and (3.9) gives the following estimate for all 1 ≤ p ≤ ν and ≥ 0 1 ˇ R ) (ξ;p − dΓ(k;p ) + z;p ) |ξ − dΓ(k) − z|−2ν ∂¯p fñ (z)(N ext + 1)− 2 Γ(j ˇ R ) (N + 1)− 12 − (ξ;p − dΓext (k;p ) + z;p ) |ξ − dΓext (k) − z|−2ν Γ(j (3.10) = O z −−1 |Imz|−2ν R−1 . By choosing = 2ν, in order to dampen the singularity at the real axis, we get an integrable weight factor z −2ν−1 , uniformly in n. We can now invoke the Lebesgue theorem on dominated convergence, and remove the cutoff by taking n → ∞ in the representation formula (A.6). This gives finally 1 ˇ R ) f (ξ − dΓ(k)) − f (ξ − dΓext (k)) Γ(j ˇ R ) (N + 1)− 12 (N ext + 1)− 2 Γ(j = O(R−1 ) . Note that the term in the brackets above is a bounded operator with norm bounded uniformly in R and ξ. We thus get by interpolation (and since powers of N can be moved around as we please) for 0 ≤ ρ ≤ 1/2. 1 ˇ R ) f (ξ − dΓ(k)) − f (ξ − dΓext (k)) Γ(j ˇ R ) (N + 1)−ρ (N ext + 1)ρ− 2 Γ(j 1

= O(R− 2 ) . By (3.1), this concludes the proof for the case s = 0.

(3.11)

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1111

Next we consider the case s = 1 (and hence sΩ ∈ {1, 2}). Use Taylor’s formula

1 to write f (η) = f (0) + η · F0 (η), where F0 (η) = 0 (∇f )(tη)dt. It is easy to verify that F0 ’s coordinate functions satisfy the assumption of the lemma with s = 0. From (3.4) (again combined with (2.10), (2.20), and (3.5)) and (3.11) we get, as a form estimate on (C0∞ ⊗ C0∞ ) × C0∞ , 1 ˇ R ) f (ξ − dΓ(k)) − f (ξ − dΓext (k)) Γ(j ˇ R ) (N + 1)−ρ (N ext + 1)ρ− 2 Γ(j 1

ν

1

p=1 ν

= O(R− 2 ) + = O(R− 2 ) +

1

(ξ;p − dΓext (k;p )) O(R− 2 ) 1

O(R− 2 ) (ξ;p − dΓ(k;p )) .

(3.12)

p=1

Note that if sΩ = 1 then dΓ(k) is H0 (ξ)-bounded, and if sΩ = 2 then dΓ(k) is H0 (ξ)1/2 -bounded. Corresponding relative bounds for the extended operators hold as well. This implies the lemma for s = 1. In the remaining case s = 2 (and hence sΩ = 2). We proceed in a similar fashion, writing f (η) = f (0) + η · F1 (η), where F1 ’s coordinate functions satisfy the assumptions of the lemma with s = 1. Since in this case dΓ(k) and F1 (ξ − dΓ(k)) are H0 (ξ)1/2 -bounded, the result follows (by a similar argument) from the s = 1 case. Lemma 3.2 We have as a form on F ext × F,

R ˇ ˇ R ) (H0 (ξ) − i)−1 ) H(ξ) − H ext (ξ) Γ(j (H0ext (ξ) − i)−1 Γ(j 1

1

= (H0ext (ξ) − i)− 2 B1 (R) = B2 (R)(H0 (ξ) − i)− 2 , where B1 and B2 are families of bounded operators satisfying B1 (R)+B2(R) = o(1), as R → ∞, locally uniformly in ξ. Proof. By Lemma 3.1, applied with f = Ω and s = sΩ , we only need to prove the lemma with H(ξ) replaced by dΓ(ω) and Φ(v), and H ext (ξ) replaced by dΓext (ω) and Φ(v) ⊗ 1lF respectively. We begin by computing as a form on Dext × D ˇ R ) dΓ(ω) − dΓext (ω) Γ(j ˇ R ) = dΓ(j ˇ R , rR ) , Γ(j R where rR = ([j0R , ω], [j∞ , ω]). By Condition 1.2 iii) and pseudo differential calculus, 1 R the components of r satisfies, as operators on D(ω 2 )∗ , 1

1

R , ω] ω − 2 = O(R−1 ) , for R → ∞ . ω − 2 [j#

(Alternatively one could also use here the calculus of almost analytic extensions.) The contribution to B1 and B2 coming from dΓ(ω) thus satisfies the required

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1 bounds by (2.10), (2.19), and (3.1). Here we choose h = h# = hph , ˜h# = D(ω 2 )∗ , 1 1 1 1 R a#,2 = ω 2 , and a#,1 = {ω − 2 [j# , ω]ω − 2 }ω 2 , when applying (2.19). It remains to treat the contribution from the perturbation. We compute as a form on Dext × D, using [14, Lemma 2.14 (iii)]

ˇ R ) Φ(v) − Φ(v) ⊗ 1lF Γ(j ˇ R) Γ(j 1 ∗ R ˇ R) a ((1 − j0R )v) ⊗ 1lF + 1lF ⊗ a∗ (j∞ v) Γ(j = −√ 2 ˇ R ) a((1 − j R )v) . + Γ(j 0

R Eq. (3.1) and Lemma 2.1 ii) now yield the result, since s − limR→∞ j∞ = s− R 2 ν limR→∞ (1 − j0 ) = 0 and v ∈ L (R ).

We immediately get the following two corollaries. Corollary 3.3 We have for any R > 1 ˇ R ) : D → Dext and Γ(j ˇ R )∗ : Dext → D1/2 , Γ(j 1/2 ext where D1/2 = D(H0 (ξ)1/2 ) and D1/2 = D(H0ext (ξ)1/2 ) are independent of ξ.

The first part of the following corollary follows from Lemma 3.2 while the second part follows from the first part and the calculus of almost analytic extensions (with ν = 1), as presented in Subsection A.1. Corollary 3.4 We have, in the limit R → ∞, i) The following estimate holds true locally uniformly in ξ and z ∈ C with Imz = 0 ˇ R ) (H(ξ) − z)−1 − (H ext (ξ) − z)−1 Γ(j ˇ R ) = |Imz|−2 o(1) . Γ(j

ii) For f ∈ C0∞ (R), we have uniformly in ξ ˇ R ) f (H(ξ)) − f (H ext (ξ)) Γ(j ˇ R ) = o(1) . Γ(j

3.2

The HVZ-Theorem

In this subsection we prove Theorem 1.2. Recall the abbreviations k = (k1 , . . . , kn ) ∈ Rnν and k (n) = k1 + · · · + kn . We start by establishing three lemmas (n) Proof of Lemma 1.6. Suppose to the contrary that k ∈ I0 (ξ), that is Σ0 (ξ − k (n) ) ≥ Σess (ξ − k (n) ), cf. (1.25). Then there exist ≥ 1 and kn+1 , . . . , kn+ ,

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cf. (1.15), such that (writing k (n+) = (n)

Σ0 (ξ; k) =

n+ i=1

1113

ki )

n Σ0 ξ − k (n) + ω(ki ) i=1

≥

n+ (n+) Σ0 ξ − k + ω(ki )

≥

Σ0

i=1 (n+)

(n)

(ξ) > Σ0 (ξ; k) ,

which is a contradiction. This proves the lemma.

Lemma 3.5 Let n ≥ 1, and B ∈ L2sym (Rnν ; B(F )). Suppose B(k) commute with N for almost all k ∈ Rnν . Define for ψ ∈ C0∞ the map a(B) ψ := B(k) a(k1 ) · · · a(kn ) ψ dnν k . Rnν

Then (N + 1)−n/2 a(B) extends from C0∞ to a bounded operator on F and there exists C = C(n) such that C −1 (N + 1)−n/2 a(B)B(F ) ≤ B :=

Rnν

B(k)2B(F ) dnν k

12

.

(3.13)

Proof. Let ψ ∈ C0∞ and ϕ ∈ F, with ϕ = 1. We estimate ϕ, (N + n + 1)−n/2 a(B) ψ ϕ, (N + n + 1)− n2 B(k) a(k1 ) · · · a(kn ) ψ dnν k ≤ nν R ϕ, B(k) a(k1 ) · · · a(kn ) (N + 1)− n2 ψ dnν k = nν R n ≤ B(k)B(F ) a(k1 ) · · · a(kn ) (N + 1)− 2 ψ dnν k Rnν 1 a(k1 ) · · · a(kn ) (N + 1)− n2 ψ 2 dnν k 2 ≤ B ≤

Rnν

B ψ .

Here we used the representation N = Rν a∗ (k)a(k)dν k repeatedly in the last step. This estimate yields the lemma (with C = ((n + 1)/2)n/2 ). Lemma 3.6 Let χ ∈ C0∞ (R) and ξ ∈ Rν . Then, for all k, ≥ 0, the form N k χ(H(ξ))N extends from C0∞ to a bounded form on D∗ . Remark. We employ the standard triple: D ⊂ F ⊂ D∗ continuously and densely.

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Proof. Recall from [14, Lemma 3.2] that N k χ(H(ξ))N extends to a bounded form on F . It remains to prove that it extends further by continuity to D∗ . It is sufficient to verify that H(ξ)N k χ(H(ξ)), viewed as a form on C0∞ × F, extends to a bounded form on F ⊗ F. Let ψ ∈ C0∞ and ϕ ∈ F. We compute for k ≥ 1, H(ξ) ψ, (N + 1)k χ(H(ξ)) ϕ = Φ(v) ψ, (N + 1)k χ(H(ξ)) ϕ + (N + 1)k ψ, H0 (ξ) χ(H(ξ)) ϕ 1 1 = (N + 1)− 2 Φ(v) ψ, (N + 1)k+ 2 χ(H(ξ)) ϕ + (N + 1)k ψ, H(ξ) χ(H(ξ)) ϕ 1 1 − (N + 1)−k− 2 Φ(v) (N + 1)k ψ, (N + 1)k+ 2 χ(H(ξ))ϕ .

An application of Lemma 2.1 i) now yields the result.

Proof of Theorem 1.2. We begin with i). Let ξ ∈ Rν and let f ∈ C0∞ (R) be such that supp f ⊂ (−∞, Σess (ξ)). By definition of Σess (ξ) (see (1.13–1.15)), (2.21), (2.33), (2.34), and (2.28), we observe that H

ext

(ξ) 1l(N∞ ≥ 1) =

∞

H () (ξ)

=1

≥

∞

()

Σ0 (ξ) 1lH() ≥ Σess (ξ) 1l(N∞ ≥ 1) .

=1

Here we used the identification 1lH() = 1l(N∞ = ). The lower bound above, together with (2.29) and Corollary 3.4 ii), yields f (H(ξ)) = = =

ˇ R ) + o(1) ˇ R )∗ f (H ext (ξ)) Γ(j Γ(j Γ(j0R ) f (H(ξ)) Γ(j0R ) ˇ R )∗ f (H ext (ξ)) 1l(N∞ ≥ 1) Γ(j ˇ R ) + o(1) + Γ(j Γ(j0R ) f (H(ξ)) Γ(j0R ) + o(1) , for R → ∞ .

The first term on the right-hand side is compact, by a standard argument using Condition 1.2 ii). This implies that f (H(ξ)) is a compact operator, and hence; that the spectrum of H(ξ) below Σess (ξ) is locally finite. As for ii), fix ξ ∈ Rν and λ ≥ Σess (ξ). We wish to show that there exists n0 ≥ 1 and η = (η1 , . . . , ηn0 ) ∈ Rn0 ν such that λ = Σ0 (ξ − η (n0 ) ) +

n0 i=1

where η (n0 ) =

n0 i=1

ηi .

(n0 )

ω(ηi ) and η ∈ I0

(ξ) ,

(3.14)

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1115 (n )

Let n0 be given by n0 + 1 = min{n : λ < minn ≥n Σ0 (ξ)}. The minima exist, and n0 ≥ 1, due to (1.15) and (1.19). There exists k = (k1 , . . . , kn0 ) such n0 (n ) that Σ0 0 (ξ) = Σ0 (ξ − k (n0 ) ) + i=1 ω(ki ) ≤ λ, where k (n0 ) = k1 + · · · + kn0 . By Condition 1.2 ii), (1.18), and continuity of Σ0 (ξ), cf. Proposition 1.1, we can find η such that the first part of (3.14) is fulfilled. The choice of n0 and Lemma 1.6 implies the last part. By i); Σ0 (ξ − η (n0 ) ), given by (3.14), is an eigenvalue for H(ξ − η (n0 ) ). We write ϕ0 for a corresponding ground state; H(ξ − η (n0 ) )ϕ0 = Σ0 (ξ − η (n0 ) )ϕ0 . Let f ∈ C0∞ (Rν ) with f ≥ 0 and f (0) = 1. Write fi, (k) = ν/2 f ( (k − ηi )). Then {f1, }∈N , . . . , {fn0 , }∈N is a family of uniformly bounded sequences in hph , which all converge weakly to 0. Let ψ = a∗ (fn0 , ) · · · a∗ (f1, )ϕ0 . The rest of the proof is concerned with showing that ψ is a Weyl sequence for the energy λ. Note that by Lemma 3.6 and Lemma 2.1 i), we have ϕ0 ∈ D(a∗ (fn0 , ) · · · a∗ (f1, )). Lemma 2.1 iii) furthermore implies that {ψ }∈N converges weakly to zero in F . For ψ to be a Weyl sequence it must satisfy ψ > 0 uniformly in . Let S(n) denote the group of permutations of n elements, and write (σk)j = kσ(j) , for σ ∈ S(n) and k ∈ Rnν . (n) Let n be such that ϕ0 = 0. Pick a compact set (of non-zero measure) nν K ⊂ R with the following properties: (K1) If k ∈ K then σk ∈ K, for all σ ∈ S(n). (K2) For k ∈ K we have ki = ηj , 1 ≤ i ≤ n and 1 ≤ j ≤ n0 . (K3) (n) 1l(k ∈ K)ϕ0 = 0. (n ) (n) (n) Let ψK be defined by ψK := 0, for n = n, and ψK := 1l(k ∈ K)ϕ0 . By property (K2), there exists 0 such that a(fj, )ψK = 0, for any 1 ≤ j ≤ n0 , and

≥ 0 . By the CCR (2.1) we thus get, for ≥ 0 , ∗ 0 fj, , fσ(j), ψK , ϕ0 = Πnj=1 a (fn0 , ) · · · a∗ (f1, ) ψK , ψ σ∈S(n0 )

=

(n) (n) 0 Πnj=1 fj , fσ(j) ϕ0 , 1l(k ∈ K) ϕ0

σ∈S(n0 ) (n)

≥ f 2n0 1l(k ∈ K) ϕ0 2 . This estimate and property (K3), implies ψ > 0 uniformly in ≥ 0 . It remains to prove that (H(ξ) − λ)ψ → 0 as → ∞. for the fiber Hamiltonian with the Let v˜ ∈ L2 (Rν ) ∩ C(Rν ). Write H(ξ) interaction Φ(v) replaced by Φ(˜ v ). Compute, as an identity on D, − k (n0 ) ) − H(ξ − η (n0 ) ) H(ξ = (k (n0 ) − η (n0 ) ) · (∇Ω)(ξ − η (n0 ) − dΓ(k)) (3.15) (n0 ) (n0 ) (n0 ) (n0 ) (n0 ) (n0 ) −η ), T (k ,η ) (k −η ) + Φ(˜ v − v) , + (k

1 where T (ζ1 , ζ2 ) = 0 (1 − t)(∇2 Ω)(ξ − tζ1 − (1 − t)ζ2 − dΓ(k))dt.

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Note that this operator is continuous and bounded uniformly in ζ1 (and ζ2 ) and commutes with the number operator. Abbreviate n0 ω Σ (k, η) := ω(kj ) − ω(ηj ) . j=1

By (3.14), (3.15), and the pull-through formula, Proposition 2.2, we get for ψ ∈ C0∞ ϕ0 , a(k1 ) · · · a(kn0 ) (H(ξ) − λ) ψ

(n0 ) = H(ξ − k ) − H(ξ − η (n0 ) ) + ω Σ (k, η) ϕ0 , a(k1 ) · · · a(kn0 ) ψ +

n0

v˜(ki ) ϕ0 , a(k1 ) · · · a(k i ) · · · a(kn0 ) ψ

i=1

=

Φ(˜ v − v)ϕ0 , a(k1 ) · · · a(kn ) ψ + ω Σ (k, η) ϕ0 , a(k1 ) · · · a(kn ) ψ − (k (n0 ) − η (n0 ) ) · (∇Ω)(ξ − η (n0 ) − dΓ(k)) ϕ0 , a(k1 ) · · · a(kn0 ) ψ + (k (n0 ) − η (n0 ) ), T (k (n0 ) , η (n0 ) )(k (n0 ) − η (n0 ) ) ϕ0 , a(k1 ) · · · a(kn ) ψ +

n0

v˜(ki ) ϕ0 , a(k1 ) · · · a(k i ) · · · a(kn0 ) ψ .

i=1

Abbreviate B1 (k) := 2 (k) := Bp,

B3 (k) :=

0 ω Σ (k, η) Πnj=1 fj, (kj ) 1lF ,

(n0 ) (n0 ) 0 (k;p − η;p ) Πnj=1 fj, (kj ) 1lF , (n0 ) 0 (n0 ) (n0 ) (n0 ) (k −η ), T (k ,η )(k (n0 ) − η (n0 ) ) Πnj=1 fj, (kj ) .

By construction of the fj, ’s we find (see (3.13) for the definition of the norm) B1 +

ν

2 Bp, + B3 → 0 , for → ∞ .

p=1

Using the notation introduced in Lemma 3.5, we can now compute ψ , (H(ξ) − λ)ψ = ϕ0 , Φ(˜ v − v)a(f1, ) · · · a(fn0 , )ψ 1 + ϕ0 , a(B ) ψ + ϕ0 , a(B3 ) ψ ν 2 + ∂p Ω(ξ − η (n0 ) − dΓ(k)) ϕ0 , a(B,p )ψ p=1 n0 ∗ (f ) · · · a∗ (f )ϕ , ψ fi, , v˜ a∗ (fn0 , ) · · · a + i, 1, 0 i=1

(3.16)

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1117

By Lemma 2.1 ii) we can take the limit v˜ → v in L2 (Rν ). This amounts to replacing v˜ by v and H(ξ) by H(ξ) in the equation above. The resulting identity together with Condition 1.1, Lemma 2.1 i), and Lemma 3.6 implies that ψ ∈ D and n (H(ξ) − λ) ψ ≤ C (N + 1) 20 ϕ0 B1 + B3 ν n ∂p Ω(ξ − η (n0 ) − dΓ(k)) (N + 1) 20 ϕ0 B 2 +C ,p p=1

+ C0,n0 −1

n0 n0 −1 max |fj, , v | (N + 1) 2 ϕ0 Πk =i fk, .

1≤j≤ν

i=1

By (3.16) and the fact that w − lim→∞ fj, = 0, we thus find (H(ξ) − λ)ψ → 0 as → ∞, and hence; ψ is a Weyl-sequence. This concludes the proof.

3.3

Uniqueness, existence, and non-existence of ground states

We begin by applying the Perron-Frobenius theorem of Faris, which is presented in Appendix A.2. See also Fr¨ ohlich [19, 20]. We write hph = hph R ⊕ ihph R , where hph R is the real Hilbert space consisting ⊗s n of the real valued functions in hph . We define HR := ⊕∞ n=0 hph R , which is also a real Hilbert space. We define the cone (n) C := ×∞ , n=0 C (n)

C

:= { f ∈

⊗s n hph R

n−times : (−1) v ⊗ · · · ⊗ v f ≥ 0 a.e. } . n

(3.17)

In this section we assume (1.20), i.e., that the coupling function v ∈ L2 (Rν ) is real valued and non-zero a.e., which implies that C is a Hilbert cone in the sense of Definition A.1. Clearly f (H0 (ξ)) is positivity preserving in the sense of Definition A.2 ii), for any bounded non-negative Borel function f . For µ > 0 sufficiently large, the Neumann series −1

(H(ξ) + µ)

=

∞

k (H0 (ξ) + µ)−1 (−Φ(v)) (H0 (ξ) + µ)−1

(3.18)

k=0 1

converge. Note that Φ(v)(H0 (ξ) + µ)−1 ≤ Cµ− 2 ; cf. Lemma 2.1 i) and (3.1). We find from this formula that for any real-valued v ∈ L2 (Rν ), that the resolvent (H(ξ)+µ)−1 is positivity preserving. In fact, we find from (3.18) that, the resolvent (H(ξ) + µ)−1 is a sum of terms of the form

k (H0 (ξ) + µ)−1 − a# (v) (H0 (ξ) + µ)−1 , where all powers k and combinations of a∗ (v) and a(v) occur. Furthermore each of these terms are positivity preserving.

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Let u ∈ C\{0}. There exists n ≥ 0 such that un ∈ hph c⊗s n , the projection onto the n-particle sector, is non-vanishing; un = 0. We wish to prove that, under the assumption (1.20) on v, (H(ξ) + µ)−1 u is strictly positive in the sense of Definition A.2 i). Let w ∈ C\{0}. There exists n ≥ 0 such that wn ∈ hph c⊗s n is non-zero; wn = 0. We estimate (H(ξ) + µ)−1 u, w ≥ (H(ξ) + µ)−1 un , wn

n

n ≥ − a(v)(H0 (ξ) + µ)−1 un , − (H0 (ξ) + µ)−1 a(v) (H0 (ξ) + µ)−1 wn ≥ µ−n−n −1 (−1)n v(k1 ) · · · v(kn )un (k)dnν k νn R × (−1)n v(k1 ) · · · v(kn )wn (k)dn ν k . Rνn

The right-hand side is strictly positive and hence; (H(ξ) + µ)−1 u is strictly positive. Since u ∈ C\{0} was arbitrary we conclude that (H(ξ) + µ)−1 is positivity improving in the sense of Definition A.2 iii). The abstract result of Faris, Theorem A.3 now implies that a ground state, if it exists, is unique and strictly positive in the sense of Definition A.2 i). This proves Theorem 1.3. We now embark on: Proof of Theorem 1.5 ii). Let ξ be such that Σ0 (ξ) = Σess (ξ). Assume Σ0 (ξ) is an eigenvalue. By Theorem 1.3, the eigenvalue is non-degenerate and we can choose an eigenfunction ψξ ∈ C which is strictly positive. (1) Recall from Corollary 1.4 that Σess (ξ) = Σ0 (ξ), under Condition 1.3. Let (1) (1) M := {k ∈ Rν : Σ0 (ξ; k) = Σ0 (ξ)} be the set of minimizers. By (1.23) and (1) Lemma 1.6, M is a compact subset of the open set I0 (ξ). There exists k0 ∈ ∂M, ν a unit vector u ∈ R , and a number r > 0, with the following property: For any δ > 0 we have (1)

Ωrδ := {k ∈ Rν : k − k0 ≤ r and (k − k0 ) · u ≥ δ } ⊂ I0 (ξ) \ M . We also use this notation with δ = 0. For any δ > 0 there exists C(δ) such that inf Σ0 (ξ − k) + ω(k) − Σ0 (ξ − k0 ) ≥ C(δ)−1 .

k∈Ωrδ

(3.19)

Recall that Σ0 (ξ − k), k ∈ Ωr0 , are isolated eigenvalues and, again by Theorem 1.3, they are non-degenerate and we can choose eigenfunctions ψξ−k ∈ C (1) which are strictly positive. Since I0 (ξ) k → ψξ−k is continuous, we find inf ψξ−k , ψξ > 0 .

k∈Ωr0

(3.20)

Let Nδ := dΓ(1l(k ∈ Ωrδ )) = Ωr a∗ (k)a(k)dν k. Note that 0 ≤ Nδ ≤ N , δ and hence ψξ ∈ D(Nδ ) with Nδ ψξ ≤ N ψξ < ∞ uniformly in δ > 0. Using

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1119

Proposition 2.3, (3.19), and the Lebesgue theorem on dominated convergence (to replace z, Imz = 0, by z = Σ0 (ξ)), we get ψξ , Nδ ψξ −1 2 v(k)2 H(ξ − k) + ω(k) − Σ0 (ξ) ψξ dk ≥ Ωrδ

≥ ≥

Ωrδ

−2 v(k)2 Σ0 (ξ − k) + ω(k) − Σ0 (ξ) |ψξ−k , ψξ |2 dk

inf r {|ψξ−k , ψξ |2 v(k)2 }

k∈Ω0

Ωrδ

(3.21)

−2 Σ0 (ξ − k) + ω(k) − Σ0 (ξ) dk . (1)

Since Σ0 (ξ − k) is a smooth function of k in I0 (ξ) and k0 is a global minimum of the function k → Σ0 (ξ − k) + ω(k), we find that there exists C > 0 such that 0 ≤ Σ0 (ξ − k) + ω(k) − Σ0 (ξ) ≤ C |k − k0 |2 , for k ∈ Ωr0 . This estimate together with (3.20), (3.21), and the assumption 3 ≤ ν ≤ 4 implies that |ψξ , Nδ ψξ | → ∞, as δ → 0. This contradicts ψξ ∈ D(N ), and hence; Σ0 (ξ) is not an eigenvalue. The first step in the proof of Theorem 1.5 i) is the following Lemma. Lemma 3.7 Let ξ ∈ Rν and z < Σ0 (ξ). Then Ω(ξ) − z − v(k)2 Ω, (H(ξ − k) + ω(k) − z)−1 Ω dk > 0 . Rν

Proof. Let PΩ := |Ω Ω|, and P Ω := 1lF − PΩ . Using the Feshbach projection method, cf., e.g., [4, Section II], we find −1 . (3.22) Ω, (H(ξ) − z)−1 Ω = Ω(ξ) − z − v, (H(ξ) − z)−1 v Ran P Ω

Here H(ξ) = P Ω H(ξ)P Ω as an operator on Ran P Ω , and v is viewed as an element of the one-particle space which is contained in Ran P Ω . By the spectral theorem the left-hand side of (3.22) is strictly positive and hence (3.23) Ω(ξ) − z − v, (H(ξ) − z)−1 v Ran P Ω > 0 . Viewing (H(ξ) − z)−1 v as an element of F we write −1 v, (H(ξ) − z) v Ran P Ω = v(k) Ω, a(k)(H(ξ) − z)−1 v dk .

(3.24)

Rν

Applying the pull-through formula, Theorem 2.3, with ψ = (H(ξ) − z)−1 v ∈ D, yields as an L2 (Rν ; F ) identity a(k) (H(ξ) − z)−1 v = (H(ξ − k) + ω(k) − z)−1 a(k) (H(ξ) − z) (H(ξ) − z)−1 v − v(k) (H(ξ − k) + ω(k) − z)−1 (H(ξ) − z)−1 v .

(3.25)

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We now make two observations. The first is the identity a(k) (H(ξ) − z) (H(ξ) − z)−1 v = a(k) v = v(k) Ω .

(3.26)

The second observation is that (H(ξ) − z)−1 is positivity preserving, with respect to the cone C introduced in (3.3) (after extending it by zero to the vacuum sector). This follows by a Neumann expansion, as for (H(ξ) + µ)−1 in (3.18), and Lemma A.4. Since (H(ξ − k) + ω(k) − z)−1 is also positivity preserving we find that, for a.e. k ∈ Rν , (3.27) Ω, (H(ξ − k) + ω(k) − z)−1 (H(ξ) − z)−1 v ≤ 0 . Combining (3.25)–(3.27) we get the following estimate a.e. v(k) Ω, a(k) (H(ξ) − z)−1 v ≥ v(k)2 Ω, (H(ξ − k) + ω(k) − z)−1 Ω . This estimate in conjunction with (3.23) and (3.24) concludes the proof.

Proof of Theorem 1.5 i). Assume that the statement is false at ξ, i.e., Σ0 (ξ) = Σess (ξ). The aim is to show that the equation v(k)2 Ω, (H(ξ − k) + ω(k) − z)−1 Ω dk (3.28) Ω(ξ) − z = Rν

has a solution z < Σess (ξ), which would by Lemma 3.7 provide a contradiction. In the limit z → −∞ the left-hand side dominates the right-hand side. A solution to (3.28) exists (and is necessarily unique by monotonicity) if we can show that the right-hand side diverges as z approaches Σess (ξ) from below. As in the proof of Theorem 1.5 ii) we choose a minimizer k0 ∈ Rν satisfying (1) (1) (1) Σ0 (ξ; k0 ) = Σ0 (ξ) = Σess (ξ). Then, by (1.23) and Lemma 1.6, k0 ∈ I0 (ξ) and (1) there exists a neighbourhood O ⊂ I0 (ξ) of k0 satisfying inf k∈O ψξ−k , Ω > 0. Here ψξ−k ∈ C, k ∈ O, are the strictly positive ground state eigenfunctions of H(ξ − k). We thus get v(k)2 Ω, (H(ξ − k) + ω(k) − z)−1 Ω dk Rν 2 2 ≥ inf {ψξ−k , Ω v(k) } (Σ0 (ξ − k) + ω(k) − z)−1 dk . k∈O

O

Since the right-hand side diverges in dimension 1 and 2, as z → Σess (ξ) from below, we conclude the result.

3.4

Regularity of t → σess (t)

We begin with (n) (n) Proof of Lemma 1.7. Let k be a local minimum of I0 (ξ) k → Σ0 (ξ; k). That the kj ’s must be equal follows from strict convexity of ω: Assume n ≥ 2. Let

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kj,s = (1−s)kj +s 21 (k1 +k2 ), j = 1, 2 and 0 ≤ s ≤ 1. Note that k1,s +k2,s = k1 +k2 , (n) so that substituting k1,s , k2,s for k1 , k2 only changes the contribution to Σ0 (ξ; k) coming from ω. We compute

1 d

ω(k1,s ) + ω(k2,s ) = (k2 − k1 ) ∇ω(k1,s ) − ∇ω(k2,s )} . ds 2

1 2 Since ∇ω(k1 ) − ∇ω(k2 ) = ( 0 ∇ ω(tk1 + (1 − t)k2 )dt(k1 − k2 ), we find that the derivative is strictly negative at s = 0, unless k1 = k2 . Write k1 = · · · = kn = Θ. We proceed to argue that Θ is a multiple of ξ. A local minimum is in particular a critical point, i.e., it satisfies ∇j Σ(n) (ξ; k) = −∇Σ(ξ − k (n) ) + ∇ω(kj ) = 0, 1 ≤ j ≤ n. By rotation invariance, this implies that ξ − nΘ is a multiple of Θ. This completes the proof. We introduce an index for a local minimum of s → σ (n) (t; s). (n)

Definition 3.8 Let n > 0, t ∈ R and s ∈ I0 (t). Assume s is a local minimum. We define the index to be Ind(n) (t; s) = min{ ∈ N : ∂s2 σ (n) (t; s) > 0}, with the convention that the index is ∞ if ∂s2 σ (n) (t; s) = 0 for all . For simplicity we (n) define Ind(n) (t; s) = 0 if s ∈ I0 (t) is not a local minimum for s → ∂s σ (n) (t; s ). Note that index 1 means that the local minimum is non-degenerate. (n)

Proposition 3.9 Let n > 0, t ∈ R and s ∈ I0 (t) be such that Ind(n) (t; s) ≥ 1. (n) There exist neighbourhoods Ot t and Os s, with Os ⊂ ∪t ∈Ot I0 (t ), such that the following holds 1) If Ind(n) (t; s) = 1, then there exists an analytic map Θ : Ot → Os , such that: Ind(n) (t ; Θ(t )) = 1 and Ind(n) (t ; s ) = 0, if s = Θ(t ). 2) If Ind(n) (t; s) = 2, then: For t ∈ Ot , s → σ (n) (t ; s ) has either one or two local minima in Os . For t = t, they have index 1. 3) If Ind(n) (t; s) = ∈ [3, ∞), then there exists a countable set K ⊂ Ot \{t}, with K ∪ {t} closed, such that: For t ∈ Ot , s → σ (n) (t ; s ) has between 1 and local minima in Os . For t ∈ Ot \(K ∪ {t}), they all have index 1. For t ∈ K all local minima s ∈ Os satisfies Ind(n) (t ; s ) ≤ − 1. 4) If Ind(n) (t; s) = ∞, then for t ∈ Ot \{t}, we have Ind(n) (t ; s ) = 0, for all s ∈ Os . Proof. 1) follows by analyticity in t and s of ∂s2 σ (n) (t; s), and the implicit function theorem. As for 2) and 3), we write = Ind(n) (t; s). We again invoke the implicit function theorem to construct an analytic function Θ from a neighbourhood Ot of t, into a neighbourhood Os of s, with the property that ∂s2−1 σ (n) (t ; Θ(t )) = 0, t ∈ Ot . Note that by choosing Ot small enough we have t − nΘ(t ) ∈ I0 . We begin by showing that near t no local minima can disappear to the same order as at t. We note that near t we may have at most local minima, but there is

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at least one. Let Ot tj → t and Os sj → s be such that sj is a local minimum of r → σ (n) (tj , r). Assume ∂sk σ (n) (tj , sj ) = 0 for k ≤ 2 − 1. Then necessarily, we must have sj = Θ(tj ). For 1 ≤ k ≤ 2 − 2, the function t → ∂sk σ (n) (t , Θ(t )) is analytic in Ot and vanishes on the sequence {tj }, hence it is identically zero in Ot . We can now compute 0 =

d 2−2 (n) ∂s σ (t ; Θ(t )) = n2−2 ∂ 2−1 σ(t − nΘ(t )) . dt

This implies that ∂ 2−1 ω(Θ(t )) = 0. The function ∂ 2−1 ω(s) has only isolated zeroes, since it is a analytic (and not identically zero). Hence Θ(t ) = Θ0 is a constant function on Ot . Since t → σ(t − nΘ0 ) + nω(Θ0 ) is thus linear near t, we find that σ is linear near t − ns. This implies in particular that ∂s2 σ (n) (t; s) = n∂ 2 ω(s) = 0. Recalling that ω is strictly convex we arrive at a contradiction. The statement 2) is now proved. The statement 3) follows from an induction argument in , starting with = 2. As for 4) we note that we must have σ (n) (t; s ) = C, for some constant C. In other words: σ(t − ns ) = C − nω(s ), for s near s. Compute σ(t − ns ) + nω(s ) = σ(t − n(s + (t − t )/n)) + nω(s ) = C + n{ω(s ) − ω(s + (t − t )/n)}. This gives ∂s σ (n) (t ; s ) = n{∇ω(s ) − ∇ω(s + (t − t )/n)}. This expression can only vanish if t = t . Proof of Theorem 1.9. We argue first that for a given t, the set M of global minima (n) of s → σ (n) (t; s) is finite. Note that by Lemma 1.6 we have M ⊂ I0 (t). Suppose to the contrary that M is infinite. Then either M contains a connected component (n) (n) of I0 (t) or there is a sequence in M converging to ∂I0 (t). In either case, this (n) is a contradiction since M is closed and I0 (t) is bounded and open. We remark that this also implies that a global minimum has finite index. By Proposition 3.9 2)–3), we find that the set T0 of t for which at least one of the global minima for the map s → σ (n) (t; s) have index strictly larger than 1, is closed and countable. It remains to show that the set of t for which there is more than one global minimum, all with index 1, is countable and can accumulate only at T0 . Suppose t is such that the map s → σ (n) (t; s) has global minima all with index 1. Note that for t near t these minima will persist at least as local minima, and any global minima will be found amongst these. There exists analytic maps t → Θj (t ), which parameterize these local minima. they are all defined in a neighbourhood of t, and satisfies Ind(n) (t ; Θj (t )) = 1. We estimate the rate of change of the global minima near t, using twice the critical equation (∂s σ (n) )(t ; Θj (t )) = 0, d (n) σ (t ; Θj (t )) = ∂σ(t − nΘj (t )) = ∂ω(Θj (t )) . dt

(3.29)

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Since ∂ω is monotonically increasing we find that that there exists a neighbourhood Ot of t such that for t ∈ Ot \{t}, the map s → σ (n) (t ; s) has a unique global minimum, with index 1. A compactness argument now concludes the proof. Note that (1.27) is implied by (3.29) since σ (n) (t) = σ (n) (t; Θ(n) (t)), for t ∈ T (n) .

3.5

Local minima of t → σess (t)

This subsection is devoted to the following proof. Proof of Theorem 1.10. Let t0 be a local minimum of t → σess (t) and let U t0 be an open set such that σess (t) ≥ σess (t0 ), t ∈ U. (1) The function Rν s → σ (1) (t0 ; s) has finitely many global minima Θ1 (t0 ) < (1) · · · < Θ (t0 ), all in I (1) (t0 ) and with finite index, cf. the proof of Theorem 1.9. (1)

Assume there exists 1 ≤ j ≤ such that s0 := Θj (t0 ) > 0. By Proposition 3.9 there exist Ot0 , Os0 , and K, with t0 ∈ Ot0 ⊂ U, s0 ∈ Os0 ⊂ (0, ∞) ∩ (1) (∪t∈Ot0 I0 (t)), and K ⊂ U is countable with K ∪ {t0 } closed, such that: For t ∈ Ot0 \(K ∪ {t0 }) all local minima of Os0 s → σ (1) (t; s) have index 1 (and at least one such local minimum exist). Furthermore, the set Ot0 \(K ∪{t0 }) can be written as a countable union of disjoint open intervals Iλ . On each of these intervals we get from the Implicit Function Theorem, that the number of local minima λ ≥ 1, is independent of t ∈ Iλ , and the local minima, Θλ,j (t), 1 ≤ j ≤ λ , are analytic in Iλ . As for (3.29) we compute ∂t σ (1) (t; Θλ,j (t)) = ∂ω(Θλ,j (t)) , for t ∈ Iλ .

(3.30)

Let τ (1) (t) := inf s∈Os0 σ (1) (t; s). Note that τ (1) is continuous on Ot0 and on any Iλ we have τ (1) (t) = min1≤j≤λ σ (1) (t; Θλ,j (t)). Since Θλ,j (t) > 0 we conclude from (3.30) that τ (1) is monotonely strictly increasing on any Iλ and hence by continuity on Ot0 . We now arrive at a contradiction with the assumption that t0 is local minimum for σess = σ (1) as follows. Estimate for t ∈ (−∞, t0 ) ∩ Ot0 : σ (1) (t) ≤ τ (1) (t) < τ (1) (t0 ) = σ (1) (t0 ). (1) We conclude from the argument above that any global minimum Θj (t0 ) (1)

must be less than or equal to zero. Similarly one can show that Θj (t0 ) ≥ 0, thus (1)

leaving only the possibility: = 1 and Θ(1) (t0 ) ≡ Θ1 (t0 ) = 0. This implies the first part of the theorem, namely that σess (t0 ) = σ (1) (t0 ; 0) = σ(t0 ) + m. Since the gap is m at t0 , and σess has a local minimum at t0 , we find from (1.16) that σ also has a local minimum at t0 . In particular σ has a critical point at t0 , with ∂ 2 σ(t0 ) ≥ 0, which yields the bound ∂s2 σ (1) (t0 ; s)|s=0 ≥ ∂ 2 ω(0). Hence Ind(1) (t0 ; 0) = 1. By Proposition 3.9 1), this implies that σess is analytic near t0 and ∂σess (t) = ∂ω(Θ(1) (t)) near t0 , cf. (3.29). Computing 0 = ∂t (∂s σ (1) (·; Θ(1) (·))),

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near t0 , yields the formula d (1) ∂ 2 σ(t − Θ(1) (t)) Θ (t) = 2 (1) . dt ∂s σ (t; Θ(1) (t))

(3.31)

The equation (1.28) for ∂ 2 σess (t0 ) now follows by (1.27), (3.31), and the computation dd d d2 ω(Θ(1) (t)) = ∂ 2 ω(Θ(1) (t)) Θ(1) (t) . σess (t) = 2 dt dt dt dt Recall that Θ(1) (t0 ) = 0.

We end this section with a comment on jump discontinuities of the bounded function ∂σess (t) = ∂ω(Θ(1) (t)). When t increases (away from 0), global minima are a priori not monotone, but when they jump, they jump from large s to smaller s. Passing to larger s, can only happen analytically (where ∂ 2 σ(t − s) ≥ 0, and hence a local minimum has index 1). This implies that Jump discontinuities of ∂σess always decrease the derivative.

(3.32)

4 Additional results In this section we collect some additional results, most of which have appeared elsewhere in some form. They serve to give a more complete picture of the bottom of the joint energy momentum spectrum. In addition we explain how to extend the results described in this paper to models with a number cutoff in the interaction.

4.1

Complimentary results

In this section we recall some known and partly known related results on the structure of the ground state mass shell. The first is due to Gross [31, (6.30)], cf. also [56, (15.26)]. Lemma 4.1 (Gross) Let √ v ∈ L2 (Rν ) be real-valued and symmetric, i.e., v(k) = v(−k) a.e., and ω(k) = k 2 + m2 , m ≥ 0. Assume Condition 1.1 and that, for any t > 0, the map p → e−tΩ(p) is positive definite. Then for all ξ ∈ Rν Σ0 (ξ) ≥ Σ0 (0) . Gross proved this statement for m > 0, but as remarked in [20] this extends by a limiting argument to m = 0. The second result we mention is an extension of a result of Hiroshima and Spohn. See [36, Lemma 3.1] and its proof. See also [56, (15.34)].

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Lemma 4.2 Let v satisfy (1.20), and assume Conditions 1.1 and 1.2. Let ξ ∈ I0 , write ψξ for a normalized ground state eigenfunction, and P ξ := 1lF − |ψξ ψξ |. Then {∇2 Σ0 (ξ)}ij = ψξ , ∂i ∂j Ω(ξ − dΓ(k))ψξ − P ξ ∂i Ω(ξ − dΓ(k)) ψξ , (H(ξ) − Σ0 (ξ))−1 P ξ ∂j Ω(ξ − dΓ(k)) ψξ . In particular ∇2 Σ0 (ξ) ≤ supp σ(∇2 Ω(p)) 1lRν . Note that by Theorem 1.3, H(ξ) − Σ0 (ξ) is bounded invertible on the range of P ξ . If ξ ∈ I0 is a critical point for ξ → Σ0 (ξ), then ∂j Σ0 (ξ) = ψξ , ∂j Ω(ξ − dΓ(k))ψξ = 0, 1 ≤ j ≤ ν, and hence the P ξ in the formula above for the Hessian is superfluous. This is the case considered in [36] (see also [54]). We leave the proof to the reader. In the case Ω(p) = p2 /2M , Lemma 4.2 implies a lower bound Meff ≥ M −1 := ∂ 2 σ(0) (assuming rotation invariance). See on the effective mass, where Meff [56, Section 15.2] for a discussion of effective mass. In [54] an upper bound for the effective mass is derived, implying in particular that ∂ 2 σ(0) > 0. This is still an open problem for Ω(p) = p2 /2M . We note that similarly one can prove the following statement: Replace v by gv, where g ∈ R is a coupling constant. Let g and ξ be such that ξ ∈ I0 , which is a g-dependent set. Then Σ0 (ξ) is an analytic function of the coupling constant in d a neighbourhood of g, dg Σ0 (ξ) = ψξ , Φ(v)ψξ , and d2 Σ0 (ξ) = − P ξ Φ(v) ψξ , (H(ξ) − Σ0 (ξ))−1 P ξ Φ(v) ψξ . 2 d g

(4.1)

In particular, the function g → Σ0 (ξ) is concave in the set {g : ξ ∈ I0 }. Thirdly we formulate a result, which follows from the proof of [20, Theorem 3.2]. We give a short proof of the statement here because Fr¨ ohlich concentrated on the massless case, and the proof simplifies for massive bosons. We remark that the infrared cutoff σ > 0 in [20] can be viewed as a mass. Theorem 4.3 Let v ∈ L2 (Rν ). Assume Conditions 1.1, 1.2, and that the following bounds hold for all p, k ∈ Rν |∇Ω(p)| ≤ 1 and ω(k) − |k| > 0 .

(4.2)

Then I0 = Rν . Remark. This theorem the case of relativistic elec implies in particular that in √ trons, i.e., Ω(p) = p2 + M 2 (M > 0), and ω(k) = k 2 + m2 (m > 0), we have an isolated ground state mass shell for all total momenta. This type of result was an important ingredient in [22]. Proof. Suppose I0 = Rν , and let ξ ∈ R\I0 .

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Define, for ξ, k ∈ Rν with k = 0,

F (ξ, k) := |k|−1 Ω(ξ − dΓ(k)) − Ω(ξ − k − dΓ(k) . This self adjoint operator extends from C0∞ to a bounded operator on F , and by (4.2) it satisfies the bound

Let

F (ξ, k)B(F ) ≤ 1 .

(4.3)

(n ) n := max n ≥ 1 : Σ0 (ξ) = Σess (ξ) .

(4.4) (n)

By Theorem 1.2 and (1.19) this choice of n is well defined. For k ∈ I0 (ξ), we write ψξ−k(n) ∈ D for a (normalized) ground state eigenfunction at total momentum ξ − k (n) . Note that k (n) = 0. For such k we use (4.3) and the Rayleigh-Ritz variational principle to estimate Σ0 (ξ)

≤ ψξ−k(n) , H(ξ) ψξ−k(n) = Σ0 (ξ − k (n) ) + |k (n) | ψξ−k(n) , F (ξ, k (n) ) ψξ−k(n)

(4.5)

≤ Σ0 (ξ − k (n) ) + |k (n) | . (n)

Let U := I0 (ξ) ∩ {η ∈ Rnν : Σ0 (ξ − η (n) ) ≤ Σ0 (ξ)}. The bound (4.5), Lemma 1.6, and the choice (4.4) of n, implies (n)

Σ0 (ξ) = ≥

n

(n) infnν Σ0 (ξ; k) = inf Σ0 (ξ − k (n) ) + ω(kj )

k∈R

k∈U

Σ0 (ξ) + inf

k∈U

n

j=1

ω(kj ) − |k (n) | .

j=1

By (1.18) there exists CU > 0, independent of n, such that |k (n) | ≤ CU , k ∈ U. Now choose R such that ω(k) ≥ CU + 1 for |k| ≥ R. Since |k (n) | ≤ |k1 | + · · · + |kn |, we arrive at the following estimate, cf. (4.4),

(n) Σ0 (ξ) = Σess (ξ) = Σ0 (ξ) ≥ Σ0 (ξ) + min 1 , By (4.2) this is a contradiction.

inf (ω(k) − |k|) .

k:|k|≤R

In addition to Theorem 1.9 we have a complimentary result which is concerned with the regularity of σ (n) (t) as a function of n. We leave the proof, which follows closely the proof of Theorem 1.9, to the reader Proposition 4.4 Let v satisfy (1.20). Assume Conditions 1.1, 1.2, 1.3 i), and 1.4. Let t ∈ R. There exists a closed countable set T (t) ⊂ (0, ∞), and an analytic (n) map (0, ∞)\T (t) n → Θ(n) (t) ∈ I0 (t), with the property that the maps s →

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σ (n) (t; s), n ∈ (0, ∞)\T (t), has a non-degenerate global minimum at s = Θ(n) (t), i.e., ∂s2 σ (n) (t; Θ(n) (t)) > 0. Let (a, b) ⊂ (0, ∞)\T (t). The global minimum is either unique for all n ∈ (a, b), or it is accompanied by another global minimum sitting at s = −Θ(n) (t), for all n ∈ (a, b). The case of two global minima can occur if and only if σ(t−r) = σ(t+r) for r in a neighbourhood of nΘ(n) (t). We furthermore have d (n) σ (t) = ω Θ(n) (t) − ∂ω Θ(n) (t) Θ(n) (t) , for n ∈ (0, ∞) \ T (t) . (4.6) dn The function x → ω(x) − x∂ω(x) appearing on the right-hand side of (4.6), is (n+1) (n) the one from (1.22). The identity (4.6) can be used to estimate Σ0 (ξ)−Σ0 (ξ).

4.2

Interactions with a number cutoff

In this subsection and the next we consider models of the form, cf. (1.6), HN := H0 + 1lK ⊗ 1l(N ≤ N ) V 1lK ⊗ 1l(N ≤ N ) . Here N ∈ Z is the cutoff parameter. Clearly these operators also commute with the total momentum and The corresponding fiber Hamiltonians are, cf. (1.8)–(1.10), HN (ξ) := H0 (ξ) + ΦN (v), where ΦN (v) := 1l(N ≤ N ) Φ(v) 1l(N ≤ N ) . Note that the notation is consistent since Φ0 (v) = 0. For N < 0 we clearly also have HN (ξ) = H0 (ξ). We remark that for N = 1 a complete picture can be obtained, cf. [23], (mass zero case). We note that the spin-boson model has been studied in the weak coupling regime for N = 2 in [45]. See also [25, 38, 39]. We now formulate our main results from Subsection 1.3 in the context of the cutoff models. We impose for brevity of exposition (1.21), Conditions 1.1, 1.2, 1.3 i), and 1.4 throughout this subsection. Let N ≥ 1. We introduce some notation. First the bottom of the spectrum of the full operator: ΣN ,0 := infν ΣN ,0 (ξ) , where ΣN ,0 (ξ) := inf σ(HN (ξ)) . ξ∈R

For n ≥ 1 and k = (k1 , . . . , kn ) ∈ Rnν we introduce n (n) ΣN ,0 (ξ; k) := ΣN −n,0 ξ − k (n) + ω(kj )

(4.7)

j=1

and

(n)

ΣN ,0 (ξ) :=

(n)

inf ΣN ,0 (ξ; k) .

k∈Rnν

(4.8)

The bottom of the essential spectrum is (1)

(1)

Σess,N (ξ) := ΣN ,0 (ξ) = infν ΣN ,0 (ξ; k) . k∈R

(4.9)

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We furthermore write ξ ∈ Rν : ΣN ,0 (ξ) < Σess,N (ξ) ,

(n) IN ,0 (ξ) := k ∈ Rnν : ξ − k (n) ∈ IN −n,0 . IN ,0 :=

(n)

The energies ΣN ,0 (ξ), n ≥ 1, are bottoms of branches of essential spectrum corresponding to having stripped off n photons to infinity, and having the interacting systems in a groundstate. Subadditivity of ω, (1.21), the fact groundstates lie in the cone (3.3), and the Rayleigh-Ritz variational principle ensures that the thresholds are ordered: (n) (n ) (4.10) ΣN ,0 (ξ) > ΣN ,0 (ξ) , for all n > n ≥ 1. Here the assumption v = 0 a.e. comes in. It ensures that the thresholds appear in an ordered fashion as in the full model. Note that the properties (1.16) and (1.17) do not hold for the cutoff model. The gap Σess,N (ξ) − ΣN ,0 (ξ) may exceed m. However, we do have that Σess,N (ξ) − ΣN −1,0 (ξ) ≤ m (it may be negative). We introduce, as in Subsection 1.3, the following notation. Let u be a unit vector in Rν . We write σN (t) = Σ0,N (tu), for t ∈ R. By rotation invariance, σN is inde(n) pendent of u. Similarly we write, for n ∈ N, σN (t; s) := σN −n ((t−ns)u)+nω(su), (n) (n) σN (t) := Σ0,N (tu), and σess,N (t) := Σess (tu). With a slight abuse of notation, we use the same symbol I0,N to denote the (n) set of t’s such that tu ∈ I0,N . We furthermore use the symbol I0,N (t), n ∈ N, to denote the set {s ∈ R : t − ns ∈ I0,N }. We now list a number of results, which we do not prove here. See however the following subsection. In each case the reader can readily mimic the proofs, given in Section 3, of the corresponding results for the full model. I For each N ≥ 1 and ξ ∈ Rν , ΦN (v) is H0 (ξ) bounded with relative bound zero. In particular HN (ξ) is essentially self-adjoint on C0∞ , and D(HN (ξ)) is independent of ξ. II (HVZ) The bottom of the essential spectrum of HN (ξ) is Σess,N (ξ). Eigenvalues below Σess,N (ξ) have finite multiplicity and can only accumulate at Σess,N (ξ). See also [25, 38] for the cutoff spin-boson model. III The ground state is non-degenerate, and in addition: If 1 ≤ ν ≤ 2 then IN ,0 = Rν . If 3 ≤ ν ≤ 4 then the bottom of the spectrum ΣN ,0 (ξ) is an eigenvalue if and only if ξ ∈ IN ,0 . As a consequence of the non-degeneracy, the map IN ,0 t → σN (t) is analytic. (n)

IV Let n ∈ N. There exists a closed countable set TN ⊂ R, and an analytic (n) (n) (n) map R\TN t → ΘN (t) ∈ IN ,0 (t) with the property that the maps s → (n)

(n)

(n)

σN (t; s), t ∈ R\TN , has a unique global minimum at the point s = ΘN (t),

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1129 (n)

(n)

with index Ind(n) (t; ΘN (t)) = 1. In particular R\TN t → σN (t) is ana(n) (n) (1) (1) d (n) lytic and dt σN (t) = ∂ω(ΘN (t)), for t ∈ R\TN . Recall σN (t) = σess,N (t). V Let t0 be a local minimum of t → σess,N (t). Then the ’spectral gap’ at t0 is maximal, i.e., σess,N (t0 ) − σN −1 (t0 ) = m, the map t → σN −1 (t) has a local minimum at t0 , the map t → σess,N (t) is analytic near t0 , and ∂ 2 σess,N (t0 ) =

4.3

∂ 2 ω(0) ∂ 2 σN −1 (t0 ) . ∂ 2 ω(0) + ∂ 2 σN −1 (t0 )

Comments on proofs

The key difference between the cutoff models and the full model, lies in the selfsimilarity of the full model. By self-similarity we mean that after removing a number of bosons to infinity, the remaining interacting system has the same Hamiltonian as the original system, albeit at a different total momentum. For the cutoff model the interacting system, after removing bosons to infinity, has a different cutoff. This is manifested in two instances, in the extended Hamiltonian and in the pull-through formula. For the cutoff model(s) one should replace the extended Hamiltonian, cf. (2.31) and (2.34), by ext (ξ) HN

:= HN (ξ) ⊕

∞

() HN (ξ) ,

(4.11)

=1 ()

where HN (ξ) =

Rν

()

HN (ξ; k)dν k and ()

HN (ξ; k) = HN − (ξ − k () ) +

ω(kj ) .

(4.12)

j=1

With this choice of extended Hamiltonian, the localization estimates derived in Subsection 3.1 applies. This is one of the inputs to the HVZ theorem. The second manifestation of the lack of self-similarity is in the pull-through formula which should be replaced by a(k) (HN (ξ) − z) ψ = HN −1 (ξ − k) + ω(k) − z a(k) ψ + v(k) 1l(N ≤ N − 1) ψ .

(4.13)

It is now left as an exercise to the reader to verify that the proofs go through. We just remark that when applying the Perron Frobenius argument, as in Subsec(j) tion 3.3, one should work only in the sub Hilbert space ⊕N j=0 Γ (hph ) of F . Any eigenfunction will vanish in n-particle sectors with n > N , which is reflected in the fact that the cutoff resolvents, (HN (ξ) + µ)−1 , are not positivity improving in the full Hilbert cone.

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Mathematical tools

A.1 Almost analytic extension In this subsect. we briefly recall the functional calculus provided by almost analytic extensions. In particular we will use a version which handles functions of a vector of commuting operators. See the monographs by Davies [12] and Dimassi and Sj¨ ostrand [16] for details. Below α will denote multi-indices. Let s ∈ R and f ∈ C ∞ (Rν ) satisfy ∀α : ∃Cα such that |∂ α f (x)| ≤ Cα x s−|α| .

(A.1)

We define an almost analytic extension f˜ ∈ C ∞ (Cν ) of f , through a Borel construction. Fix a function χ ∈ C0∞ (R) to be equal to 1 in a neighbourhood of 0, and a sequence {λk }k∈N0 , going sufficiently fast to infinity. The following choice will do: λk := max{max|α|=k Cα , λk−1 + 1}, for k ≥ 1, and λ0 = C0 . Here the constants Cα are coming from (A.1). Then, writing z = u + iv ∈ Rν ⊕ iRν , f˜(z) :=

∂ α f (u) α!

α

(iv)α

ν λ|α| vj . χ u j=1

(A.2)

Note that there exists C > 0 such that supp(f˜) ⊂ {u + iv : u ∈ supp(f ), |v| ≤ Cu } .

(A.3)

We furthermore have the property that ∀ ≥ 0 : ∃C such that |∂¯f˜(z)| ≤ C z s−−1 |Imz| . Here ∂¯ = (∂¯1 , . . . , ∂¯ν ), ∂¯j := ∂uj + i∂vj , and Imz = (v1 , . . . , vν ). If s < 0 we have the following representation, (x + z) 2ν 2ν−1 −1 ∂¯f˜(z), d z, f (x) = 2 |S | |x − z|2ν ν C

(A.4)

(A.5)

where d2ν z = Πνj=1 duj dvj is the Lebesgue measure on Cν , and |S 2ν−1 | is the volume of the unit ball in R2ν . (Note that for s < 0 the integral is absolutely convergent.) For a vector of pairwise commuting self-adjoint operators A = (A1 , . . . , Aν ), and a function f satisfying (A.1) with s < 0, the almost analytic extension thus provides a functional calculus via the formula ν 2ν−1 −1 | (A.6) ∂¯j f˜(z) (Aj + zj ) |A − z|−2ν d2ν z . f (A) = 2 |S j=1

Cν

In the case ν = 1 this reduces to 1 f (A) = ∂¯f˜(z) (A − z)−1 du dv . π C

(A.7)

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A.2 Invariant cones In this subsect. we recall a result of Faris, cf. [17], which will be used to show nondegeneracy of the ground state. It is an abstract version of the Perron-Frobenius Theorem in L2 -spaces, cf. [50, Theorem XIII.43], which together with the Q-space representation of Fock-space, has been used frequently to show non-degeneracy of the ground state, cf. [5, 28, 31]. Definition A.1 Let HR be a real Hilbert space. We say C ⊂ HR , C = {0}, is a Hilbert cone if: i) u, v ∈ C implies u + v ∈ C. ii) u ∈ C, λ ≥ 0 implies λu ∈ C. iii) C ∩ (−C) = {0}. iv) C is closed. v) u, v ∈ C implies u, v ≥ 0. vi) For all w ∈ HR there exists u, v ∈ C s. t. w = u − v and u, v = 0. An important example of a Hilbert cone is, as mentioned above, the subset of real non-negative functions in L2 (Q, dµ), where Q is a measure space. Definition A.2 Let HR be a real Hilbert space, C ⊂ HR a Hilbert cone and A a bounded operator on HR . i) We say u ∈ C is strictly positive if u, v > 0 for any v ∈ C\{0}. ii) A is positive preserving if AC ⊂ C. iii) A is positivity improving if Au is strictly positive for all u ∈ C\{0}. iv) A is ergodic if for any u, v ∈ C\{0} there exists n ≥ 0 s. t. An u, v > 0. Note that a positivity improving operator is in particular ergodic. The following theorem is due to Faris Theorem A.3 (Faris) Let HR be a real Hilbert space, C ⊂ HR a Hilbert cone and A a bounded positive self-adjoint operator on HR . Suppose furthermore that A is positivity preserving and that A is an eigenvalue for A. Then A is ergodic if and only if A is an eigenvalue of multiplicity one and there exists a strictly positive u ∈ C with Au = Au. s

−1

The lemma below follows from the identities e−s = limn→∞ ( ns + 1)−n and

∞ = 0 e−ts ds, for s > 0, in conjunction with the first resolvent formula.

Lemma A.4 Let A be a bounded from below self-adjoint operator on a real Hilbert space. Assume that there exists a λ0 < inf σ(A) such that (A − λ)−1 is positivity preserving (improving) for all λ < λ0 . Then (A − λ)−1 is positivity preserving (improving) for all λ < inf σ(A).

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Acknowledgments The author thanks Z. Ammari, V. Bach, J. Fr¨ ohlich, and C. Gérard, for useful discussions, and Dokuz Eyl¨ ul University for hospitality. This work was supported in parts by Carlsbergfondet and by a Marie-Curie individual fellowship from the European Union.

References [1] Z. Ammari, Asymptotic completeness for a renormalized nonrelativistic Hamiltonian in quantum field theory: The Nelson model, Math. Phys. Anal. Geom. 3, 217–285 (2000). [2]

, Scattering theory for a class of Fermionic Pauli-Fierz models, J. Funct. Anal. 208, 302–359 (2004).

[3] A. Arai and M. Hirokawa, Stability of ground states in sectors and its application to the Wigner-Weisskopf model, Rev. Math. Phys. 13, 513–528 (2001). [4] V. Bach, J. Fr¨ ohlich, and I.M. Sigal, Quantum electrodynamics of confined non-relativistic particles, Adv. in Math. 137, 299–395 (1998). [5]

, Renormalization group analysis of spectral problems in quantum field theory, Adv. in Math. 137, 205–298 (1998).

[6] F.A. Berezin, The method of second quantization, 1 ed., Academic Press, New York, San Francisco, London, 1966. [7] V. Betz, F. Hiroshima, J. L¨ orinczi, R.A. Minlos, and H. Spohn, Ground state properties of the Nelson Hamiltonian - A Gibbs measure-based approach, Rev. Math. Phys. 14, 173–198 (2002). [8] L. Bruneau, The ground state problem for a quantum Hamiltonian model describing friction, C. R. Math. Acad. Sci. Paris 339, 151–156 (2004). [9] L. Bruneau and J. Derezi´ nski, Pauli-Fierz Hamiltonians defined as quadratic forms, Rep. Math. Phys. 54, 169–199 (2004). [10] J.T. Cannon, Quantum field theoretic properties of a model of Nelson: Domain and eigenvector stability for perturbed linear operators, J. Funct. Anal. 8, 101–152 (1971). [11] T. Chen, Operator-theoretic infrared renomalization and construction of dressed 1-particle states in non-relativistic QED, Ph.D. thesis, ETH Z¨ urich, 2002, ETH-Diss 14203. [12] E.B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge. [13] J. Derezi´ nski and C. Gérard, Scattering theory of classical and quantum N -particle systems, Text and Monographs in Physics, Springer, Berlin, 1997.

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[14]

, Asymptotic completeness in quantum field theory. massive PauliFierz Hamiltonians, Rev. Math. Phys. 11, 383–450 (1999).

[15]

, Spectral and scattering theory of spatially cut-off P (φ)2 Hamiltonians, Comm. Math. Phys. 213, 39–125 (2000).

[16] M. Dimassi and J. Sj¨ ostrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, 1999. [17] W.G. Faris, Invariant cones and uniqueness of the ground state for fermion systems, J. Math. Phys. 13, 1285–1290 (1972). [18] R.P. Feynman, Statistical mechanics. A set of lectures, Frontiers in physics, W. A. Benjamin, Inc., Reading, Massechusets, 1972. [19] J. Fr¨ ohlich, On the infrared problem in a model of scalar electrons and massless scalar bosons, Ann. Inst. Henri Poincaré 19, 1–103 (1973). [20]

, Existence of dressed one-electron states in a class of persistent models, Fortschr. Phys. 22, 159–198 (1974).

[21] J. Fr¨ ohlich, M. Griesemer, and B. Schlein, Asymptotic completeness for Rayleigh scattering, Ann. Henri Poincaré 3, 107–170 (2002). [22]

, Asymptotic completeness for Compton scattering, Comm. Math. Phys. 252, 415–476 (2004).

[23] A. Galtbayar, A. Jensen, and K. Yajima, The Nelson model with less than two photons, Ann. Henri Poincaré 4, 239–273 (2003). [24] V. Georgescu, C. Gérard, and J. S. Møller, Spectral theory of massless Pauli Fierz models, Comm. Math. Phys. 249, 29–78 (2004). [25] C. Gérard, Asymptotic completeness for the spin-boson model with a particle number cutoff, Rev. Math. Phys. 8, 549–589 (1996). [26]

, On the existence of ground states for massless Pauli-Fierz Hamiltonians, Ann. Henri Poincaré 1, 443–445 (2000).

[27]

, On the scattering theory of massless Nelson models, Rev. Math. Phys. 14, 1165–1280 (2002).

[28] J. Glimm and A. Jaffe, The λ(ϕ4 )2 quantum field theory without cutoffs: II. The field operators and the approximate vacuum, Ann. Math. 91, 362–401 (1970). [29] M. Griesemer, Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics, J. Funct. Anal. 210, 321–340 (2004). [30] M. Griesemer, E. Lieb, and M. Loss, Ground states in nonrelativistic quantum electrodynamics, Invent. math. 145, 557–595 (2001).

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[31] L. Gross, Existence and uniqueness of physical ground states, J. Funct. Anal. 10, 52–109 (1972). [32]

, The relativistic polaron without cutoffs, Comm. Math. Phys. 31, 25–73 (1973).

[33] C. Hainzl, One non-relativistic particle coupled to a photon field, Ann. Henri Poincaré 4, 217–237 (2003). [34] W. Heitler, The quantum theory of radiation, 3 ed., Dover Publications, 1983. [35] F. Hiroshima, Essential self-adjointness of translation-invariant quantum field models for arbitrary coupling constant, Comm. Math. Phys. 211, 585–613 (2000). [36] F. Hiroshima and H. Spohn, Ground state degeneracy of the Pauli-Fierz Hamiltonian including spin, Adv. Theor. Math. Phys. 5, 1091–1104 (2001). [37]

, Mass renormalization in nonrelativistic QED, mp arc (2003).

[38] M. H¨ ubner and H. Spohn, Radiative decay: nonperturbative approaches, Rev. Math. Phys. 7, 363–387 (1995). [39]

, Spectral properties of the spin-boson Hamiltonian, Ann. Inst. Henri Poincaré 62, 289–323 (1995).

[40] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin Heidelberg, 1995 (reprint of the 1980 2nd edition). [41] E.H. Lieb and M. Loss, Self-energy of electrons in non-perturbative QED, Differential equations and mathematical physics (Providence, RI) (R. Weikard, ed.), AMS/IP Stud. Adv. Math., vol. 16, American Mathematical Society, 2000, pp. 279–293. [42]

, A bound on binding energies and mass renormalization in models of quantum electrodynamics, J. Stat. Phys. 108, 1057–1069 (2002).

[43]

, Existence of atoms and molecules in non-relativistic quantum electrodynamics, Adv. Theor. Math. Phys. 7, 667–710 (2003).

[44] E.H. Lieb and L.E. Thomas, Exact ground state energy of the strong-coupling polaron, Comm. Math. Phys. 183, 511–519 (1997), Erratum 188, 499–500 (1997). [45] R. Minlos and H. Spohn, The three-body problem in radioactive decay: The case of one atom and at most two photons., Topics in statistical and theoretical physics. F. A. Berezin memorial volume. (Providence, RI) (R.L. Dobrushin, ed.), Transl., Ser. 2, Am. Math. Soc., vol. 177, American Mathematical Society, 1996, pp. 159–193. [46] J.S. Møller, An abstract radiation condition and applications to N -body systems, Rev. Math. Phys. 12, 767–803 (2000).

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[47] E. Nelson, Interaction of non-relativistic particles with a quantized scalar field, J. Math. Phys. 5, 1190–1197 (1964). [48] A. Pizzo, One-particle (improper) states in Nelson’s massless model, Ann. Henri Poincaré 4, 439–486 (2003). [49] M. Reed and B. Simon, Methods of modern mathematical physics: II. Fourier analysis and self-adjointness, 1 ed., Academic Press, San Diego, 1975. [50]

, Methods of modern mathematical physics: IV. Analysis of operators, 1 ed., Academic Press, San Diego, 1978.

[51]

, Methods of modern mathematical physics: I. Functional analysis, Academic Press, San Diego, 1980, Revised and Enlarged Edition.

[52] E. Skibsted, Smoothness of N-body scattering amplitudes, Rev. Math. Phys. 4, 619–658 (1992). [53] A.D. Sloan, The polaron without cutoffs in two space dimensions, J. Math. Phys. 15, 190–201 (1974). [54] H. Spohn, Effective mass of the polaron: A functional integral approach, Ann. Phys. 175, 278–318 (1987). [55]

, The polaron at large total momentum, J. Phys. A 21, 1199–1211 (1988).

[56]

, Dynamics of charged particles and their radiation field, Cambridge University Press, Cambridge, 2004.

Jacob Schach Møller Johannes Gutenberg Universit¨ at FB Mathematik (17) D-55099 Mainz Germany email: [email protected] Communicated by Joel Feldman submitted 04/11/04, accepted 17/02/05




An Extension Principle for the Einstein-Vlasov System in Spherical Symmetry Mihalis Dafermos and Alan D. Rendall

Abstract. We prove that “first singularities” in the non-trapped region of the maximal development of spherically symmetric asymptotically flat data for the EinsteinVlasov system must necessarily emanate from the center. The notion of “first” depends only on the causal structure and can be described in the language of terminal indecomposable pasts (TIPs). This result suggests a local approach to proving weak cosmic censorship for this system. It can also be used to give the first proof of the formation of black holes by the collapse of collisionless matter from regular initial configurations.

1 Introduction A fundamental problem in mathematical relativity is to resolve the so-called weak cosmic censorship conjecture, the statement that for “reasonable” Einstein-matter systems, generic asymptotically flat data do not lead to singularities visible from infinity. The notion of “reasonable” above is of course not a precise one, and depends very much on the context one has in mind. A natural matter source for models is provided by kinetic theory. The simplest example is then a self-gravitating collisionless gas. The study of the equations describing such a gas, the Einstein-Vlasov system, was initiated by Choquet-Bruhat in [1], where the existence of a unique maximal development was proven for the Cauchy problem. The problem of weak cosmic censorship concerns the global behaviour of the maximal development for asymptotically flat initial data. Given the current state of the art in nonlinear evolution equations, symmetry must be imposed on initial data for there to be any hope of making progress. The global study of the initial value problem for the Einstein-Vlasov equations for spherically symmetric asymptotically flat initial data was begun in [7], where, in particular, it was proven that for sufficiently small initial data, the maximal development was future causally geodesically complete. The analysis took place in so-called Schwarzschild coordinates. In [8], an extension principle was proven, again in these coordinates, saying in particular that if the solution stopped existing after finite coordinate time t, there was necessarily a singularity at the center. These results were meant to provide a first step for a global existence theorem in Schwarzschild coordinates. If this coordinate system could then be shown to cover the domain of outer communi-

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M. Dafermos and A.D. Rendall


cations, and if null infinity could moreover be shown to be complete, this would then imply a proof of weak cosmic censorship for this system. There is another approach to the problem of weak cosmic censorship, due to Christodoulou [3], for the problem of a self-gravitating spherically symmetric scalar field. Christodoulou showed that initial data leading to a naked singularity was codimension 1 in the space of all initial data. This was shown by embedding such exceptional data in a one-dimensional subset of the space of initial data, such that all other initial data in this subset evolved to a spacetime with the following property, which can be expressed in the language of causal sets [6]. Given a terminal indecomposable past (TIP) with compact intersection with the Cauchy surface, then the domain of dependence of any open set containing this intersection contains a trapped surface. The statement that this latter property is true for generic initial data can be termed the trapped surface conjecture. From this property, the completeness of null infinity was then inferred, proving weak cosmic censorship. It turns out that the relation between the existence of trapped surfaces and the completeness of null infinity is quite general. Specifically, in [12], it was proven that a weaker version of the trapped surface conjecture is sufficient to prove weak cosmic censorship for a wide variety of matter in spherical symmetry. In particular, the completeness of null infinity follows from the existence of a single trapped or marginally trapped surface in the maximal development. The only really restrictive hypothesis on the matter is that “first” singularities necessarily emanate from the center. Here, the notion of “first” is tied to the causal structure and can be formulated in terms of TIPs. The goal of this paper is to prove that the above mentioned hypothesis of [12] is indeed satisfied by the Einstein-Vlasov system. As noted before, extension principles similar in spirit to this one have been proven before (cf. [8, 10]). These earlier results, however, concern the portion of the development of the Einstein-Vlasov system covered by particular coordinate systems. Thus, these previous results, as far as they concern the maximal development itself, are weaker than the results presented here, and in particular, are not sufficient to deduce the assumptions of [12].1 Finally, we make the following remark: In view of [9], there do exist spherically symmetric asymptotically flat initial data for the Einstein-Vlasov system possessing a trapped surface. Thus, the results of this paper provide in particular the first proof of the existence of solutions for collisionless matter representing the formation of a black hole.

2 Initial data Initial data in this paper are always given as follows: 1. We have a C ∞ Riemannian manifold (Σ, g¯), together with an additional symmetric 2-tensor Kab , such that there do not exist closed antitrapped surfaces 1 Of course, the results of [8, 10] also say something about the behaviour of the coordinate system to which they apply, something not addressed here.

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in the data, and a compactly supported function f0 defined on the tangent bundle of Σ, such that these satisfy √ ab 2 ¯ R − Kab K + (trK) = 16π f0 (pa )pa pa /(1 + pa pa )1/2 g¯dp1 dp2 dp3 , √ ∇a K a b − ∇b (trK) = 8π f0 (pa )pa g¯dp1 dp2 dp3 . Here the metric g¯ is used to move indices and to define the trace and covari¯ is the scalar curvature of g¯ and √g¯ the square root of its ant derivative. R determinant. 2. A smooth SO(3) action on Σ such that g¯, Kab , f0 are preserved, and such that Σ/SO(3) inherits naturally the structure of a 1-dimensional manifold. Here and throughout this paper physical units are chosen so that the gravitational constant has the numerical value unity. We recall the definition of a closed antitrapped surface. Let S be a surface in Σ which is closed, i.e., compact without boundary. Suppose that there is a preferred choice na of an outward normal to this surface and let σab be the second fundamental form of S in Σ corresponding to the outward normal. Then S is said to be antitrapped if trσ < −trK + Kab na nb .

3 The maximal development The theorem of Choquet-Bruhat [1], applied to the data considered here, together with a standard argument on preservation of symmetry, yields Proposition 1. There exists a unique C ∞ collection (M, g, f ) such that 1. 2. 3. 4.

g and f satisfy the Einstein-Vlasov equations (M, g) is globally hyperbolic, (M, g, f ) induces the initial data (Σ, g¯, K, f0 ) and Σ is a Cauchy surface Any other collection (M, g, f ) with these properties 1–3 can be embedded in the given one.

Moreover, SO(3) acts smoothly by isometry on M and preserves f , and Q = M/SO(3) inherits the structure of a time-oriented 2-dimensional Lorentzian manifold, with timelike boundary Γ, the center. Let π : M → Q denote the natural projection. On Q we can define the so-called area-radius function r(p) = Area(π −1 (p))/4π . We have r(p) = 0 iff p ∈ Γ. We can always choose global future directed null coordinates on Q, i.e., such that the metric takes the from −Ω2 dudv. The metric of M then takes the form: (1) −Ω2 dudv + r2 γ

1140



where γ = γAB dxA dxB is the standard metric on S 2 and xA , A = 2, 3, are local ∂ ∂ coordinates on S 2 . Let u and v be chosen so that ∂u points “inwards” and ∂v “outwards”. Such definitions are meaningful in view of the assumption of asymptotic flatness. We define ν = ∂u r , λ = ∂v r . The assumption of no antitrapped surfaces initially means by definition that ν 0, the regular region, and denote it R. We call the region where λ = 0 the marginally trapped region, and denote it by A, and finally, we shall can the region where λ < 0 the trapped region, and denote it by T.

4 The extension theorem The extension principle proven in this paper will apply to a region D ⊂ Q with Penrose diagram:

?

Q

D

(i.e., a subset D = [u1 , u2 ] × [v1 , v2 ] \ (u2 , v2 )) such that D ⊂R∪A . Let Cin and Cout be the parts of the boundary of D defined by v = v1 and u = u1 respectively. One can think of D as the “top” of a non-trapped non-central indecomposable past (IP) corresponding to a candidate “first” singularity. In this language, the result of this paper is that such an IP cannot be a TIP, i.e., Theorem 1. If D ⊂ Q, then D ⊂ J − (q) for a q ∈ Q. The theorem thus says that there is no singularity of this form after all! As one might expect, the proof of Theorem 1 proceeds by obtaining a priori estimates in D and then applying an appropriate local existence result. The a priori estimates make use of a certain energy flux along null hypersurfaces. This fact, together with the fact that regular null coordinates can always be chosen, makes it natural to stick to these. We give the form of the equations in local

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null coordinates in the next two sections. Then, in Section 7, we formulate a local existence theorem (Proposition 2) for a double characteristic initial value problem. The “time” of existence, in the sense of null coordinates, will depend only on the C 2 norm of the metric and the C 1 norm (and the support) of f . We obtain energy estimates in Section 8, and use these, together with the structure of the Vlasov equation, to derive in Sections 9–10 a priori estimates for the norm of Proposition 2. The proof of Theorem 1 will follow immediately in Section 11. Finally, in Section 12, we state two applications of our results, discussed already in the Introduction. The above theorem depends on having a well-behaved matter model and the analogous result must be expected to fail for dust. This is illustrated by the Penrose diagram Fig. 1 in [13].

5 The Einstein equations in null coordinates The reader should consult [2] for general facts about the initial value problem in spherical symmetry. When specialized to this case, the Einstein equations are: ∂u ∂v r = −

1 Ω2 − λν + 4πrTuv , 4r r

∂u ∂v log Ω = −4πTuv +

Ω2 1 πΩ2 + 2 λν − 2 γ AB TAB , 2 4r r r

∂v (Ω−2 ∂v r) = −4πrTvv Ω−2 , ∂u (Ω

−2

∂u r) = −4πrTuu Ω

−2

.

(3) (4) (5) (6)

The former two equations can be viewed as wave equations for r and Ω, while the latter two equations can be viewed as constraint equations on null hypersurfaces. A specific choice of matter model, such as a collisionless gas, leads to expressions for the components of the energy-momentum tensor.

6 The Vlasov equation To describe the Vlasov equation in local coordinates, we need a coordinate system on T M. Let pu , pv , and pA denote the functions on T M, defined by writing an arbitrary X ∈ T M as X = pu

∂ ∂ ∂ + pv + pA A . ∂u ∂v ∂x

Together with the pull-back of the coordinates on spacetime these functions define a local coordinate system on T M. Let P ⊂ T M be defined by P = {g(X, X) = −1} ,

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where X ranges over future-pointing vectors. We call P the mass shell. It follows that −Ω2 pu pv + r2 γAB pA pB = −1 . (7) We use pu , pA and the pull-back of the coordinates on spacetime to define coordinates on P and pv is regarded as a function of these coordinates defined by the relation (7). The Vlasov equation is an equation for a non-negative function f :P →R which, in the case that f is spherically symmetric, is given by pu

∂f ∂f + pv ∂u ∂v

= (∂u (log Ω2 )(pu )2 + 2Ω−2 rλγAB pA pB ) + 2r−1 (νpu + λpv )pA

∂f . ∂pA

∂f ∂pu (8)

In deriving this we have used the expressions for the Christoffel symbols given in Appendix A and the fact that a spherically symmetric function f on the mass shell is a function of the variables u, v, pu , and γAB pA pB . This implies the identity pA

∂f B C ∂f = ΓA BC p p ∂xA ∂pA

which has been used to simplify the Vlasov equation. Note that both the expressions γAB pA pB and pA ∂p∂A have a meaning independent of the particular choice of coordinates xA on S 2 . Finally, to close the system, we must define the energy-momentum tensor. We first note that for any point q ∈ M, it follows that Pq , as a spacelike hypersurface in Tq M, inherits a volume form from the Lorentzian metric. In local √ coordinates this volume form can be written r2 (pu )−1 dpu γdpA dpB or alterna√ 2 v −1 v√ A B tively r (p ) dp γdp dp , where γ is the square root of the determinant of γAB . We then have ∞ ∞ ∞ √ Tab = r2 pa pb f (pu )−1 γdpu dpA dpB , (9) 0

−∞

−∞

where pa = gab pb . It follows immediately that this matter model satisfies the energy conditions: (10) Tuv ≥ 0, Tvv ≥ 0, Tuu ≥ 0 .

7 A local existence theorem To prove our extension theorem, we will certainly need to appeal to some sort of local existence theorem. In particular, it is the norm in this theorem that will tell us what quantities we must bound a priori in D. In principle, one could try to

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prove estimates so as to apply the local existence result of [1]. For various reasons, however, the following local existence theorem for a characteristic initial value problem will be more convenient: Proposition 2. Let k ≥ 2. Let Ω, r be positive C k -functions defined on [0, d] × {0} ∪ {0} × [0, d], and let f be a non-negative C k−1 function defined on the part of the mass shell over [0, d] × {0} ∪ {0} × [0, d]. Suppose that equations (5), (6) hold on {0} × [0, d] and [0, d] × {0} respectively, where Tuu and Tvv are defined by (9), and suppose in addition that the C k compatibility condition holds at (0, 0). Define the norm: Nu

=

sup {|Ω|, |Ω−1 |, |∂u Ω|, |∂u2 Ω|, |r|, |r|−1 , |∂u r|, |∂u2 r| ,

[0,d]×{0}

S, |f |, |∂u f |, |∂pu f |, |∂pA f |γ } , Nv

=

sup {|Ω|, |Ω−1 |, |∂v Ω|, |∂v2 Ω|, |r|, |r|−1 , |∂v r|, |∂v2 r| ,

{0}×[0,d]

S, |f |, |∂v f |, |∂pu f |, |∂pA f |γ } , N = sup{Nu , Nv } , were S denotes the supremum of (pu )2 + (pv )2 + γAB pA pB on the support of f and |vA |γ = (γ AB vA vB )1/2 . Then there exists a δ, depending only on N , and C k functions (unique among C 2 functions) r, Ω and a C k−1 function (unique among C 1 functions) f , satisfying equations (3), (4), (5), (6), (8) in [0, δ ∗ ] × [0, δ ∗ ], where δ ∗ = min{d, δ}, such that the restriction of these functions to [0, d]×{0}∪{0}×[0, d] is as prescribed. Proof. See Appendix B. The compatibility conditions referred to in the statement of the proposition are as follows. The data includes the values of the function f on the part of the mass shell over [0, d] × {0}. All derivatives of f tangential to this manifold can be calculated by direct differentiation. By using the field equations transverse derivatives (and thus all derivatives) of f can be computed up to order k − 1. In a similar way, all derivatives up to order k − 1 can be computed on {0} × [0, d]. The condition that derivatives determined in these two different ways agree at (0, 0) is what is referred to above as the C k compatibility condition. Let us add the remark that, defining g on M by (1), the above gives rise to a solution of the Einstein-Vlasov equations upstairs, with the obvious relation to characteristic data, interpreted upstairs.

8 Energy estimates A fundamental fact about the analysis of spherically symmetric Einstein matter systems in the non-trapped region is the existence of energy estimates. To describe these, let us first settle for a particular null-coordinate description of the set D. We normalize our u-coordinate such that ν = −1 along Cin . For the

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v coordinate, we first define the quantity 1 κ = − Ω2 ν −1 . 4 and then define v such that κ = 1 along Cout . D is thus given by [0, U ] × [0, V ] \ {(U, V )}. The concept of energy in spherical symmetry is given by the so-called Hawking mass, given by: m=

r r r (1 − ∂ a r∂a r) = (1 − 2g uv ∂u r∂v r) = (1 + 4Ω−2 λν) . 2 2 2

We will also introduce the so-called mass-aspect function µ=

2m . r

Note that κ(1 − µ) = λ .

(11)

From (3)–(6), we compute the identities: ∂u m

= =

∂v m

= =

8πr2 Ω−2 (Tuv ν − Tuu λ) 1−µ 2 r Tuu , −2πκ−1 r2 Tuv + 2π ν 8πr2 Ω−2 (Tuv λ − Tvv ν) 1−µ 2 r Tuv + 2πκ−1 r2 Tvv . −2π ν

(12)

(13)

The first point to note is that the signs of (12) and (13), together with the signs of λ and ν, give a priori bounds for both r and m. Indeed, set m0 = m(U, 0) ≥ 0 , M = m(0, V ) ,

r0 = r(U, 0) > 0 , R = r(0, V ) .

By (2) and the fact that D ⊂ R ∪ A, we have that r0 ≤ r ≤ R

(14)

throughout D. On the other hand, (12), (13) and (10) give ∂u m ≤ 0, ∂v m ≥ 0, and thus m0 ≤ m ≤ M . (15) Now we make a trivial observation. In view of the fact that we have the a priori bounds (15), if we reexamine the equations (12), (13), keeping in mind that both terms on the right-hand side have the same sign, we obtain the bounds: v2 2π(1 − µ) 2 r Tuv (u, v)dv ≤ M − m0 , (16) −ν v1

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v2

2πκ−1 r2 Tvv (u, v)dv ≤ M − m0 ,

(17)

2πκ−1 r2 Tuv (u, v)du ≤ M − m0 ,

(18)

2π(1 − µ) 2 r Tuu (u, v)du ≤ M − m0 . −ν

(19)

v1 u2

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u1 u2 u1

These will be our energy estimates. As we shall see, our use of the above estimates will not quite be symmetric for u and v. The reason is this: The “constraint” equation (6) can be seen to be equivalent to the following equation for κ: ∂u κ = 4πrν −1 Tuu κ .

(20)

From (2), (20) and (10), we see immediately 0 2Σ for some 0 ≤ v(s) ≤ V , and let s be the last previous time s > s > 0 such that pv (s ) ≥ 2Σ, i.e., we have pv (s∗ ) ≥ 2Σ on [s∗ , s]. By (30), (27), the angular momentum bound (25), and (41), we have ˜ −1 eV G V (2Σ)−1 , pv (s) ≤ 2ΣeV G + r0−3 X Nc

i.e., pv (s) ≤ C˜ . We can now easily estimate Tuu pointwise: ∞ dpv √ r2 (pu )2 f v γdpA dpB Tuu = p 0 |γAB pA pB |≤Xr −4 ∞ dpv √ = (guv )2 r2 (pv )2 f v γdpA dpB p 0 |γAB pA pB |≤Xr −4 ∞ v √ dp = 4ν 2 κ2 r2 (pv )2 f v γdpA dpB p 0 |γAB pA pB |≤Xr −4

(31)

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=

4ν 2 F κ2

˜ C

0

≤ ≤

√ r2 (pv )dpv γdpA dpB

|γAB pA pB |≤Xr −4

r0−2 ν 2 F C˜ 2 X 2

16π ˜ 2E , N


= ν2E

in view of (31), (24), (21), (14) and the angular momentum bound (25). (Note that Tuu ν −2 ≤ E is a coordinate invariant2 bound.) Integrating (20), we obtain now Tuu ˜ κ ≥ e− 4πr ν 2 νdu ≥ e−4πRE N U . (Actually, we have in fact already estimated κ from below since κ−1 = 4(−ν)Ω−2 .) From the inequality pu pv ≤

1 2 (p + p2v ) , 2 u

we have

Tuv ≤

1 (Tuu + Tvv ) . 2

This allows us to estimate ∂u log Ω2 = Γuuu : |Γuuu | ≤ G + 8πTuv dv − 8πκνr−2 γ AB TAB dv − 4κmr−3 νdv ˜ 2 EV + 8πTvv dv − 4κmr−3 νdv ≤ C¯ . ≤ G + 8π N We can easily obtain an estimate now for Tvv . λ can be bounded by integrating (3).

10 C 2 estimates for the metric In this section, we derive C 2 estimates on the metric and C 1 estimates for f . The ideas of this section originate in [7]. It has already been shown that the following quantities are bounded: r, r−1 , m, m−1 , κ, κ−1 , ν, ν −1 , λ, Ω, Ω−1 , all first order derivatives of Ω, all components of the energy-momentum tensor, and all Christoffel symbols in (36)–(41). From these estimates and (22) and (23), it follows that ∂v ν and ∂u λ are bounded, from (12) and (13) it follows that ∂u m and ∂v m are bounded, and from (26), it follows that ∂u ∂v Ω is bounded. Writing ν = − 41 Ω2 κ−1 and differentiating in u, we see from (20) that ∂u ν is bounded, while writing κ = − 41 Ω2 ν −1 and differentiating in v, we see that ∂v κ is bounded, and thus, from (11), we see that ∂v λ is bounded. These estimates and the formulas (36)–(41) allow us to control all first order derivatives of the Christoffel symbols, except ∂u Γuuu and ∂v Γvvv . Since the components of the curvature tensor can be expressed in terms of those derivatives of the Christoffel symbols which have already been estimated, we obtain bounds for all components of the curvature tensor in our coordinates. The above estimates allow us to estimate the first derivatives of the exponential map 2 i.e.,

it does not depend on the normalization of u

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on the tangent bundle. This, in turn allows one to estimate the derivatives of f in terms of initial data. We can, however, argue more directly as follows. Let us abbreviate the Vlasov equation (8) by X(f ) = 0 where X is the Vlasov operator written in these coordinates. Note that pv is to be thought of as expressed in terms of pu and pA via the mass shell condition (7). Define f1 = ∂u f − pu ∂u log Ω2 ∂pu f . Differentiating the Vlasov equation with respect to v, pu and pA gives the following equations: X(∂v f ) = −(∂v pv )∂v f + (∂u ∂v log Ω2 (pu )2 + ∂v (−2Ω−2 rλ)γAB pA pB )∂pu f + 2(∂v (νr−1 )pu + ∂v (λr−1 )pv + λr−1 ∂v pv )pA ∂pA f ,

(32)

2 u

X(∂pu f ) = −∂u f − (∂pu p )∂v f + 2∂u log Ω p ∂pu f v

+ 2(νr−1 + λr−1 ∂pu pv )pA ∂pA f , X(p ∂pD f ) = −p (∂pD p )∂v f − 4Ω D

D

+ 2r

−1

v

D

−2

(33)

A B

rλγAB p p ∂pu f

v A

λp ∂pD p p ∂pA f .

(34)

Differentiating the Vlasov equation with respect to u gives the following equation for f1 : X(f1 ) =

−pu ∂u (log Ω2 )X(∂pu f ) − ∂u pv ∂v f

+ −

(−pu pv ∂u ∂v log Ω2 − ∂u log Ω2 (∂u log Ω2 (pu )2 + 2Ω−2 rλγAB pA pB ) 2∂u (Ω−2 rλ)γAB pA pB )∂pu f

+

2(∂u (νr−1 )pu + ∂u (λr−1 )pv + ∂u pv λr−1 )pA ∂pA f .

(35)

The quantity X(∂pu f ) can be substituted for by one of the previous equations and ∂u f may be eliminated from the equations in favour of f1 . The result is a linear system of equations for the evolution of (f1 , ∂v f, ∂pu f, pA ∂pA f ) along the characteristics of the Vlasov equation. The coefficients are known to be bounded and so we can conclude that ∂u f , ∂v f , ∂pu f and pA ∂pA f are also bounded. (Note that since pu and pv are bounded the derivative with respect to X is uniformly equivalent to a derivative along the characteristic with respect to u or v as parameter.) From this, we immediately estimate ∂u Tab and ∂v Tab pointwise. We now estimate ∂u Γuuu by differentiating (26) in u and integrating in v, and similarly, ∂v Γvvv by differentiating in v and integrating in u. Note that |∂pA |γ can also be bounded. This can be seen by passing from polar to Cartesian coordinates and noting that the resulting metric components are C 2 . As a consequence f is C 1 .

11 The Proof of Theorem 1 Let N/2 denote the sup of the norm defined in Proposition 2, where the sup is taken now in all of D. By the estimates of the previous section, we have that N/2 < ∞. Let δ be the constant of Proposition 2 corresponding to N . Consider

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the point (U − δ/2, V − δ/2). Translate the coordinates so that this point is (0, 0). Since Q is by definition open, by continuity, there exists a δ > δ ∗ > δ/2 such that {0} × [0, δ ∗ ] ∪ [0, δ ∗ ] × {0} ⊂ Q and the assumptions of Proposition 2 hold on {0} × [0, δ ∗ ] ∪ [0, δ ∗ ] × {0}, with N and δ ∗ as already defined. It follows that there exists a unique solution of in E = [0, δ ∗ ] × [0, δ ∗ ] .

q

?

E

Q D

Thus the solution coincides in E ∩ Q by uniqueness. One sees that E ∪ Q is clearly the quotient of a development of initial data. By maximality of M, we must have E ∪ Q ⊂ Q. Thus, in particular, in the old coordinates we have (U, V ) ∈ Q, and the theorem holds with q = (U, V ).

12 Applications We will say that a spherically symmetric maximal development has a black hole, if I + is complete in the sense of [4],3 and if J − (I + ) has a non-empty complement. We have shown that the results of [12] apply to our matter model. In particular, the fact that the complement of J − (I + ) is non-empty implies the completeness of null infinity. That this set is non-empty can be inferred in turn from the existence of a single trapped or marginally trapped surface. Asymptotically flat spherically symmetric solutions of the Einstein-Vlasov system possesing a trapped surface were constructed in [9]. Thus we have Corollary 1. There exist solutions of the Einstein-Vlasov system which develop from regular initial data and contain black holes. The fundamental open question in gravitational collapse is to show that generically, either the solution is future geodesically complete or a black hole forms. In view of [12] and the results of this paper we have Corollary 2. Suppose that for generic initial data, the maximal development either contains a trapped surface or marginally trapped surface, or is future causally geodesically complete. Then weak cosmic censorship is true. Thus, weak cosmic censorship can be reduced to a slightly weaker version of Christodoulou’s trapped surfaces conjecture. As remarked in the Introduction, this suggests a local approach to its proof (cf. [3]). 3 See

[12] for a definition of I + in this context.

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A


1151

The Christoffel symbols

Note: 1 guv = − Ω2 , 2 g uv = −2Ω−2 , Ω2 = −4κν . The nonvanishing Christoffel symbols are given by: ΓuAB = −g uv rλγAB ,

(36)

= −g rνγAB ,

(37)

ΓvAB ΓA Bv A ΓBu Γuuu Γvvv

uv

= =

A λr−1 δB A νr−1 δB

,

(38)

,

(39)

2

= ∂u log Ω ,

(40)

2

= ∂v log Ω .

(41)

In fact the Christoffel symbols ΓC AB , which depend on a choice of coordinates on the spheres of symmetry need not vanish but the expressions for them are not needed in this paper.

B Proof of Proposition 2 The proof of local existence follows from simpler considerations than the proof of the estimates of Sections 8–10. In particular, one does not need to consider energy estimates, for one can recover naive pointwise estimates using the smallness parameter. As in Section 10, the idea of [7] again makes its appearance, to show C 1 bounds on f directly from C 0 bounds on the curvature, before bounding the C 2 norm of the metric. Since all these methods have appeared before, we will only sketch the details here. Let initial data be fixed. Define the space A ⊂ C 2 ([0, δ] × [0, δ]) × C 1 ([0, δ] × [0, δ]) , for δ to be determined later, consisting of all twice continuously differentiable nonnegative functions r, continuously differentiable nonnegative functions Ω, extending the prescribed values, such that N −1 /2 ≤ r ≤ 2N , N

−1

(42)

/2 ≤ Ω ≤ 2N ,

sup{|∂u r|, |∂v r|, |∂u2 r|, |∂v2 r|}

(43) ≤ 2N ,

sup{|∂u Ω|, |∂v Ω|} ≤ 2N .

(44) (45)

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Consider the subset B ⊂ A, consisting of those (r, Ω) for which Ω is C 2 , and for which sup{|∂u2 Ω|, |∂v2 Ω|, |∂u ∂v Ω|} ≤ 2N . (46) Note that the closure of B in A, denoted B, consists of (r, Ω) such that ∂u Ω, ∂v Ω, are Lipschitz, with Lipschitz constants given by the above. We shall define in the next few paragraphs a continuous map Φ : B → A ˜ taking (r, Ω) to (˜ r , Ω). Given r, Ω, first, let f be defined to solve the Vlasov equations on the metric defined by r and Ω, with given initial conditions. Note that since the Christoffel symbols of this metric are Lipschitz, it follows that geodesics can be defined, and thus f can be defined by the requirement that it is preserved by geodesic motion. It follows immediately that 0≤f ≤N , (47) and, after appropriately restricting to sufficiently small δ, it follows easily by integration of the geodesic equations that S ≤ 2N .

(48)

In the case where (r, Ω) ∈ B, we have that f is in fact C 1 , since the exponential map is differentiable. If δ is chosen sufficiently small, it is clear from (42)–(46) that, in this case, we can arrange for sup{|∂v f |, |∂u f |, |∂pu f |, |∂pA f |γ } ≤ 2N .

(49)

Given now f , we can define T uv , T vv , T uu in the standard way. In view of (42)–(45), (47), and (48), these terms can be estimated. Now, set ν = ∂u r, λ = ∂v r. We define r˜ by u v 1 1 − r−2 Ω2 − λν+4πrΩ4 T uv dudv . (50) r˜(u, v) = r(u, 0)+r(0, v)−r(0, 0)+ 4 r 0 0 By appropriate differentiation of (50), it is clear from our bounds thus far that ˜ = ∂v r˜, and ∂u ∂v r˜. We can retrieve the bound we can define and estimate ν˜ = ∂u r˜, λ ¯ (42) for r˜ by integration of the ν˜, after restricting to small δ. For (r, Ω) ∈ B ⊂ B, 2 2 it is clear we can also define and estimate ∂u r˜, ∂v r˜, by differentiating (50) twice in u or twice in v, in view of the fact that all other derivatives, including ∂u T uv , ∂u ν, etc., are clearly defined and bounded, in view of (49), and since these derivatives are defined initially. By appropriate choice of δ, we can clearly arrange–for (r, Ω) ∈ B– so as to retrieve the bound (44). ˜ > 0 by the relation Define now Ω ˜2 log Ω

= log Ω2 (u, 0) + log Ω2 (0, v) − log Ω2 (0, 0) (51) u v 1 2 −2 ˜ ν − 2πΩ2 r˜−2 γ AB TAB )dudv . + (−8πTuv + Ω r˜ + 2˜ r−2 λ˜ 2 0 0

˜ to satisfy (43). Again, for small enough δ, it is clear that one can arrange for Ω

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˜ it Differentiating (51) appropriately, in view of the initial conditions for Ω, 1 ˜ follows that, for (r, Ω) ∈ B, Ω is C , and for δ small enough satsfies (45), while for ˜ is C 2 , and for δ small enough, satisfies (46). (r, Ω) ∈ B, Ω Thus, we have shown that after judicious choice of δ, Φ maps B to itself. By continuity, it maps B to itself. The map Φ can easily be shown to be a contraction in B for the norm of A, i.e., we can show that ˜ 1 ), (˜ ˜ 2 )) ≤ dA ((r1 , Ω1 ), (r2 , Ω2 )) , r1 , Ω r2 , Ω dA ((˜

(52)

for an < 1 and all (ri , Ωi ) ∈ B. To see this, define first fi , corresponding to (ri , Ωi ). Let Γi denote an arbitrary Christoffel symbol for (ri , Ωi ). We clearly have |Γ1 − Γ2 | ≤ CdA ((r1 , Ω1 ), (r2 , Ω2 )) . We easily obtain |f1 − f2 | ≤ Cδ sup |Γ1 − Γ2 | sup (|∂fi | + |fi |) . Γ

i=1,2

Clearly we can also bound sup |T1uv − T2uv | ≤ C sup |f1 − f2 |. One bounds (ν1 − ν2 ) by expressing ∂v (˜ ν1 − ν˜2 ) as a linear combination of Ω1 − Ω2 , r1 − r2 , ν1 − ν2 , λ1 − λ2 and (T1uv − T2uv ) with bounded coefficients. One immediately obtains a similar bound for sup |˜ r1 − r˜2 |. The terms sup |∂u r˜1 − ∂u r˜2 |, sup |∂v r˜1 − ∂v r˜2 |, and sup |∂u ∂v r˜1 − ∂u ∂v r˜2 |, can be handled in the same way. One then obtains a bound ˜ 2 − ∂v log Ω ˜ 2 |, and similarly for sup |∂u log Ω ˜2 − of the above form for sup |∂v log Ω 1 2 1 2 2 2 ˜ ˜ ˜ ∂u log Ω2 |. Either of these bounds of course implies a bound for sup |Ω1 − Ω2 |. To bound sup |∂u2 r˜1 − ∂u2 r˜2 |, we compute

v 1 −2 2 2 2 −1 ∂u r˜ = ∂u r˜|v=0 + ∂u − r Ω − r λν 4 0 =

=

+ 4π∂u (rΩ4 )T uv + 4πrΩ4 ∂u T uv dv

v 1 −2 2 2 −1 ∂u − r Ω − r λν + 4π∂u (rΩ4 )T uv ∂u r˜|v=0 + 4 0 − 4πrΩ4 ∂v T vv + 4πrΩ4 ( T · Γ)dv

∂u2 r˜|v=0 − 4πrΩ4 T vv (u, v) + 4πrΩ4 T uv (u, 0)

v 1 ∂u − r−2 Ω2 − r−1 λν + 4π∂u (rΩ4 )T uv + 4 0 4 + 4π∂v (rΩ )T vv + 4πrΩ4 ( T · Γ)dv .

(53)

(54)

Here we have used the equation ∇a T ab = 0, which follows from the Vlasov equation, and we have integrated by parts. It is now clear that estimates for differences follow as before. We argue in an entirely analogous way for sup |∂v2 r˜1 − ∂v2 r˜2 |.

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After restricting to sufficiently small δ, all constants in the above bounds can be made small. We thus have indeed shown (52). It follows by continuity that Φ is also a contraction on B ⊂ A, and thus, since B is closed, has a fixed point in B. Given such a fixed point (r, Ω), define f as before. To show that (r, Ω, f ) corresponds to a solution of the equations, we have basically only to show that f and ∂u Ω, ∂v Ω, which a priori are Lipschitz, are in fact C 1 . (In particular, from this it will follow that the constraint equations (5)–(6) are also satisfied.) But, in view of the fact that f is initially C 1 , it follows that f is C 1 if the exponential map is C 1 . (The C 2 compatibility condition is used at the point.) But this latter fact follows from the continuity of the curvature, as shown in Exercise 6.2 of Chapter V of [5] 4 . That the curvature is continuous follows by computation, since r is C 2 , Ω is C 1 and ∂u ∂v Ω is C 0 , and ∂u2 Ω and ∂v2 Ω do not appear in the expressions for curvature. From the C 1 property of f , the C 2 property of Ω follows immediately. Similarly, higher regularity follows immediately if it is assumed.

Acknowledgment We gratefully acknowledge the support of the Erwin Schr¨ odinger Institute, Vienna, where an important part of this research was carried out.

References [1] Y. Choquet-Bruhat, Problème de Cauchy pour le système intégro-différentiel d’Einstein-Liouville, Ann. Inst. Fourier 21, 181–201 (1971). [2] D. Christodoulou, Self-gravitating relativistic fluids: a two-phase model, Arch. Rat. Mech. Anal. 130, 343–400 (1995). [3] D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. Math. 149, 183–217 (1999). [4] D. Christodoulou, On the global initial value problem and the issue of singularities, Class. Quantum Grav. 16, A23–A35 (1999). [5] P. Hartman (1992) Ordinary differential equations. Birkhäuser, Basel. [6] R.P. Geroch, E.H. Kronheimer and R. Penrose, Ideal points in spacetime, Proc. R. Soc. Lond. A 327, 545–567 (1972). 4 If the reader does want to apply to this fact, then one can argue as follows: in view of the computations above, in the space B, we have that curvature is in fact C 1 with estimates; since derivatives of the exponential map are computed by integrating curvature on geodesics, and geodesics certainly depend C 1 on their initial conditions, in view of the fact that the Christoffel symbols are C 1 with bounds in B, it follows that we have C 2 estimates for the exponential map in B, and thus by an easy compactness argument, the exponential map of the fixed point must be C 1 . There is only one catch with this argument: r and Ω2 have to be assumed to be initially C 3 to differentiate (51) and (53) three times.

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[7] G. Rein and A.D. Rendall, Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data, Commun. Math. Phys. 150, 561–583 (1992). (Erratum: Commun. Math. Phys. 176, 475–478 (1996).) [8] G. Rein, A.D. Rendall, and J. Schaeffer, A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system, Commun. Math. Phys. 168, 467–478 (1995). [9] A.D. Rendall, Cosmic censorship and the Vlasov equation, Class. Quantum Grav. 9, L99–L104 (1992). [10] A.D. Rendall, An introduction to the Einstein-Vlasov system. Banach Center Publications 41, 35–68 (1997). [11] A.D. Rendall. The Einstein-Vlasov system. In: Chru´sciel, P.T. and Friedrich, H. (eds.) (2004) The Einstein equations and the large scale behavior of gravitational fields. Birkh¨ auser, Basel. [12] M. Dafermos, Spherically symmetric spacetimes with a trapped surface, Class. Quantum Grav. 22, 2221–2232 (2005). [13] P. Yodzis, H.-J. Seifert and H. M¨ uller zum Hagen, On the occurrence of naked singularities in general relativity, Commun. Math. Phys. 34, 135–148 (1973). Mihalis Dafermos University of Cambridge Department of Pure Mathematics and Mathematical Statistics Wilberforce Road Cambridge CB3 0WB United Kingdom email: [email protected] Alan D. Rendall Max Planck Institute for Gravitational Physics Albert Einstein Institute Am Muehlenberg 1 D-14476 Golm Germany email: [email protected] Communicated by Sergiu Klainerman submitted 15/11/04, accepted 17/02/05



Stability of Standing Waves for Nonlinear Schr¨ odinger Equations with Inhomogeneous Nonlinearities Anne De Bouard and Reika Fukuizumi

Abstract. The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves eiωt φω (x) for a nonlinear Schr¨ odinger equation with an inhomogeneous nonlinearity V (x)|u|p−1 u, where V (x) is proportional to the electron density. Here, ω > 0 and φω (x) is a ground state of the stationary problem. When V (x) behaves like |x|−b at infinity, where 0 0. The main point of this paper is to analyze the linearized operator at standing wave solution for the case of V (x) = |x|−b . Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method.

1 Introduction The nonlinear Schr¨ odinger equations i∂t u = −∆u − g(x, |u|2 )u,

(t, x) ∈ R1+n

(1.1)

arise in various physical contexts such as nonlinear optics and plasma physics. When g(x, |u|2 ) = V (x)|u|p−1 , equation (1.1) can model beam propagation in an inhomogeneous medium where V (x) is proportional to the electron density. L. Bergé [2] studied formally the stability condition for soliton solutions of the above type of equations, depending on the shape of g(x, |u|2 ). The real function g(x, |u|2 ) is a potential which can either stand for corrections to the nonlinear power-law response, or for some inhomogeneities in the medium. In addition, Towers and Malomed [29] recently observed by means of variational approximation and direct simulations that a certain type of time-dependent nonlinear medium gives rise to completely stable beams. Akhmediev [1], Jones [17] and Grillakis, Shatah and Strauss [13] studied the existence and stability of solitary waves of (1.1) when g(x, |u|2 ) describes three layered media where the outside two are nonlinear and the sandwiched one is linear. Also, Merle [23] investigated the existence and nonexistence of blowup solutions of (1.1) for inhomogeneities of the form g(x, |u|2 ) = V (x)|u|4/n . In this paper, we will not exactly deal with the same nonlinearity as those in [2, 29], we consider the case g(x, |u|2 ) = V (x)|u|p−1 with V (x) satisfying the following assumptions (V1) and (V2) with n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/(n − 2).

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(V1) (V2)

A. De Bouard and R. Fukuizumi

V (x) ≥ 0,

V (x) ≡ 0,

V (x) ∈ C(Rn \ {0}, R),

Ann. Henri Poincaré ∗

V (x) ∈ Lθ (|x| ≤ 1),

where θ∗ = 2n/{(n + 2) − (n − 2)p}. There exist C > 0 and a > {(n + 2) − (n − 2)p}/2 > b such that V (x) − 1 ≤ C |x|b |x|a

for all x with |x| ≥ 1. The main purpose in this paper is to show that under the above assumptions on V (x), the standing wave solution of (1.1) is stable for p < 1 + (4 − 2b)/n and sufficiently small frequency. As an example satisfying (V1) and (V2), we keep V (x) = (1 + |x|2 )−b/2 in mind. By a standing wave, we mean a solution of (1.1) of the form uω (t, x) = eiωt φω (x), where ω > 0 and φω (x) is a ground state of the following stationary problem x ∈ Rn , −∆φ + ωφ − V (x)|φ|p−1 φ = 0, (1.2) 1 n φ ∈ H (R ), φ ≡ 0. We recall previous results. Several authors have been studying the problem of stability and instability of standing waves for (1.1) (see, e.g., [3, 6, 7, 9, 11, 13, 22, 25, 30, 32]). First, we consider the case V (x) ≡ 1, namely, i∂t u = −∆u − |u|p−1 u,

(t, x) ∈ R1+n ,

where 1 0, there exists a unique positive radial solution ψω (x) of −∆ψ + ωψ − |ψ|p−1 ψ = 0, x ∈ Rn , 1 n ψ ≡ 0. ψ ∈ H (R ),

(1.3)

(1.4)

(See Strauss [26] and Berestycki and Lions [4] for the existence, and Kwong [19] for the uniqueness). It is known that a positive solution of (1.4) is a ground state. In [6] Cazenave and Lions proved that if p < 1 + 4/n then the standing wave solution eiωt ψω (x) is stable for any ω > 0. On the other hand, it is shown that if p ≥ 1+4/n then the standing wave solution eiωt ψω (x) is unstable for any ω > 0 (see Berestycki and Cazenave [3] for p > 1 + 4/n, and Weinstein [30] for p = 1 + 4/n). The aim of the paper is to study, in the case where V (x) satisfies (V1) and (V2), what happens in the complementary case of the result in [11], where instability of standing waves was shown for p > 1 + (4 − 2b)/n and sufficiently small ω > 0. We define the energy functional E and the charge Q on H 1 (Rn ) by 1 1 1 2 E(v) := ∇v2 − V (x)|v(x)|p+1 dx, Q(v) := v22 . 2 p + 1 Rn 2

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We remark that by the assumptions (V1) and (V2), the functional E is well defined on H 1 (Rn ) if p < 1 + (4 − 2b)/(n − 2). The time local well-posedness for the Cauchy problem to (1.1) with g(x, |u|2 ) = V (x)|u|p−1 in H 1 (Rn ) and the conservation of energy and charge hold (see, e.g., Theorem 4.4.6 of Cazenave [5]). Exactly, we have the following proposition. Proposition 1 Let n ≥ 3 and 1 0 and a unique solution u(t) ∈ C([0, T ], H 1 (Rn )) of (1.1) with u(0) = u0 satisfying E(u(t)) = E(u0 ),

Q(u(t)) = Q(u0 ),

t ∈ [0, T ].

Before we state our theorem, we give some precise definitions. Definition 1 For ω > 0, we define two functionals on H 1 (Rn ): Sω (v) := E(v) + ωQ(v)

(action), Iω (v) := ∇v22 + ωv22 − V (x)|v(x)|p+1 dx. Rn

Let Gω be the set of all non-negative minimizers for inf{Sω (v) : v ∈ H 1 (Rn ) \ {0}, Iω (v) = 0}.

(1.5)

The existence of non-negative minimizers for (1.5) was proved by the standard variational argument since V (x) vanishes as |x| → ∞ (see [26, 11]). Namely, we have Lemma 1.1 Let n ≥ 3 and 1 0. |x|→∞

Remark 1.1 (i) We note that

Iω (v) = ∂λ Sω (λv)|λ=1 = Sω (v), v.

(ii) Let φω ∈ Gω . Then, there exists a Lagrange multiplier Λ ∈ R such that Sω (φω ) = ΛIω (φω ). Thus, we have Sω (φω ), φω = Λ Iω (φω ), φω . Since

Sω (φω ), φω = Iω (φω ) = 0 and

Iω (φω ), φω = −(p − 1)

V (x)|φω |p+1 < 0,

we have Λ = 0. Namely, φω satisfies (1.2). Moreover, for any v ∈ H 1 (Rn )\{0} satisfying Sω (v) = 0, we have Iω (v) = 0. Thus, by the definition of Gω , we have Sω (φω ) ≤ Sω (v). Namely, φω ∈ Gω is a ground state (minimal action solution) of (1.2) in H 1 (Rn ). It is easy to see that a ground state of (1.2) in H 1 (Rn ) is a minimizer of (1.5).

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The stability and instability in this paper is defined as follows. Definition 2 For φω ∈ Gω and δ > 0, we put 1 n iθ Uδ (φω ) := v ∈ H (R ) : inf v − e φω H 1 < δ . θ∈R

We say that a standing wave solution eiωt φω (x) of (1.1) is stable in H 1 (Rn ) if for any ε > 0 there exists δ > 0 such that for any u0 ∈ Uδ (φω ), the solution u(t) of (1.1) with u(0) = u0 satisfies u(t) ∈ Uε (φω ) for any t ≥ 0. Otherwise, eiωt φω (x) is said to be unstable in H 1 (Rn ). The following theorem is our main result in this paper. Theorem 1 Let n ≥ 3 and 1 0 such that eiωt φω (x) is stable in H 1 (Rn ) for any ω ∈ (0, ω∗ ). In particular, we can take ω∗ = ∞ in the case where V (x) = |x|−b with 0 < b < 2. Remark 1.2 We make use of Hardy’s type inequality to control the degree of nonlinearity in the space H 1 (Rn ). That is why the restriction on the spatial dimensions, i.e., n ≥ 3 appears in the assumption of Theorem 1. Grillakis, Shatah and Strauss [13, 14] gave an almost sufficient and necessary condition for the stability and instability of stationary states for the Hamiltonian systems under certain assumptions. By the abstract theory in Grillakis, Shatah and Strauss [13, 14], under some assumptions on the spectrum of linearized operators, eiω0 t φω0 (x) is stable (resp. unstable) if the function φω 22 is strictly increasing (resp. decreasing) at ω = ω0 . In the papers of Shatah [24], Shatah and Strauss [25], the authors used the variational characterization of ground states instead of assumptions on the spectrum√of linearized operators. In the case V (x) ≡ 1, by the scaling ψω (x) = ω 1/(p−1) ψ1 ( ωx), it is easy to check the increase and decrease of ψω 22 . However, it seems difficult to check this property of φω 22 for V (x) ≡ 1 since we do not have the scaling invariance in general. To avoid such difficulty, we apply another sufficient condition for stability. Proposition 2 Let n ≥ 3 and 1 0 such that

Sω (φω )v, v ≥ δv2H 1

(1.6)

for any v ∈ H 1 (Rn ) satisfying Re(φω , v)L2 = 0 and Re(iφω , v)L2 = 0, then the standing wave solution eiωt φω (x) of (1.1) is stable in H 1 (Rn ). Remark 1.3 In Proposition 2, the condition Re(φω , v)L2 = 0 is related to the conservation of charge Q. In fact, we have Q (φω ), v = Re(φω , v)L2 . Moreover, since it follows from Sω (eiθ φω ) = 0 for θ ∈ R that Sω (φω )iφω = 0, (1.6) does not hold if we do not restrict v ∈ H 1 (Rn ) to satisfy Re(iφω , v)L2 = 0.

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To check this sufficient condition (1.6) for the case of V (x) satisfying (V1) and (V2), we first consider the case where V (x) = |x|−b with 0 < b < 2 as a limiting problem since the stability results are already known in the case V (x) = |x|−b , which simply follow from the arguments by [24] and [25]. Indeed, in [11], the authors investigated the rescaling limit of φω (x) as ω → 0. It was shown in [11] that as ω → 0, the rescaled function φ˜ω (x) defined by √ φω (x) = ω (2−b)/2(p−1) φ˜ω ( ωx),

ω>0

(1.7)

tends to the unique positive radial solution ψ1,b (x) of (1.2) with ω = 1 and V (x) = |x|−b . Using this convergence, they proved in [11] that eiωt φω (x) is unstable for p > 1 + (4 − 2b)/n and sufficiently small ω > 0. Due to the inhomogeneous medium, the standing wave solution tends to be more unstable for small ω > 0 since 1+(4−2b)/n < p < 1+4/n is the stability region in the case where V (x) ≡ 1. From known stability properties of ψ1,b (x) (see Section 2 of [11]), we would be able to prove (1.6) in the limit. However, to our knowledge, there is no verification of (1.6) even in the case V (x) = |x|−b . For that reason, in Section 2, we first study the properties of the linearized operator at standing wave solution for the case where V (x) = |x|−b in (1.1). In Section 3, we continue analyzing the linearized operator, in particular, we observe that the kernel of real part of the linearized operator is only zero, following the method of Kabeya and Tanaka [18]. We remark that their idea could not be applied directly to our case. We need to modify their perturbed functional in order that the singularity of |x|−b at the origin does not affect the linear part of the equation (1.2). The crucial part is Section 3 because uniqueness and nondegeneracy of a solution of semilinear elliptic equations often plays an essential role in stability problems. In Section 4, we check the condition (1.6) for V (x) satisfying (V1) and (V2), following Esteban and Strauss [8] (see also [10]) and we prove Theorem 1. We remark that Fibich and Wang [9] and Liu, Wang and Wang [22] treated the stability and instability problems of standing waves for (1.1) with g(x, |u|2 ) = V (εx)|u|4/n in a radial space, where ε is a small parameter. Their ways of proof are also a sort of perturbation method. However, they use (1.4) with p = 1 + 4/n as a limiting equation, their assumptions for V (x) are different from those in this paper and it is not clear whether there exists a simple relation between ε and ω.

2 The case V (x) = |x|−b We consider the stability of standing waves for i∂t u = −∆u −

1 |u|p−1 u, |x|b

(t, x) ∈ R1+n ,

where n ≥ 3, 0 0 there exists a unique positive radial solution ψω,b ∈ H 1 (Rn ) of −∆ψ + ωψ −

1 |ψ|p−1 ψ = 0, |x|b

x ∈ Rn .

(2.2)

See Stuart [27] and Remark 3.1 of [11] for existence. The positivity of solutions follows from the maximum principle. Radial symmetry of solutions was showed by Gidas, Ni and Nirenberg [12] and Li [20] (see also Li and Ni [21]), and Yanagida [33] proved the uniqueness. Moreover ψω,b is in C 2 (Rn ) and vanishes as |x| → ∞, particularly decays exponentially (see [4, 5]). This unique solution is a minimizer of db (ω) := inf{Sω,b (v) : v ∈ H 1 (Rn ) \ {0}, Iω,b (v) = 0}, where

1 1 ω 1 ∇v22 + v22 − |v(x)|p+1 dx, 2 2 p + 1 Rn |x|b 1 Iω,b (v) = ∇v22 + ωv22 − |v(x)|p+1 dx. b |x| n R

Sω,b (v) =

In this section, we note the following fact as a special case of Theorem 1. Proposition 3 Let n ≥ 3, 0 0. Actually, this fact can be proved simply by applying the method of [24, 25] to the present case. Using the variational characterization db (ω), we may check the sufficient condition for stability db (ω) > 0 in [24] and instability db (ω) < 0 in [25]. Since ψω,b (x) is a solution of Sω,b (v) = 0, we have db (ω) = Q(ψω,b ). In √ (2−b)/2(p−1) ψ1,b ( ωx), we have 2Q(ψω,b ) = this case, by the scaling ψω,b (x) = ω ψω,b 22 = ω {(2−b)/(p−1)}−n/2 ψ1,b 22 . Therefore, for any ω > 0, the standing wave solution is stable if 1 < p < 1 + (4 − 2b)/n, and unstable if 1 + (4 − 2b)/n < p < 1+(4−2b)/(n−2). We have also blow-up instability for the case p ≥ 1+(4−2b)/n, following Weinstein [30] and Berestycki and Cazenave [3]. However, stability of standing wave solution does not always seem to imply (1.6) immediately. The constraints in (1.6) depend on the negative and zero eigenvalues of the linearized operator at ψω,b . Therefore, the main aim in this section is to show the following proposition. Proposition 4 Assume n ≥ 3, 0 0 such that (ψ1,b )v, v ≥ δv2H 1

S1,b for any v ∈ H 1 (Rn ) satisfying Re(ψ1,b , v)L2 = 0 and Re(iψ1,b , v)L2 = 0.

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Remark 2.1 By combining this proposition with Proposition 2, it follows that the standing wave solution eit ψ1,b (x) of (2.1) is stable in H 1 (Rn ), that is, Proposition 3 holds. We define two self-adjoint operators L1,b and L2,b on L2 (Rn ) by L1,b = −∆ + 1 − p

1 p−1 ψ (x), |x|b 1,b

L2,b = −∆ + 1 −

1 p−1 ψ (x) |x|b 1,b

p−1 v ∈ L2 (Rn )} for j = 1, 2. We with domain D(Lj,b ) = {v ∈ H 2 (Rn , R) : |x|−b ψ1,b 1 n remark that for v ∈ H (R ) with v1 (x) = Re v(x) and v2 (x) = Im v(x), (ψ1,b )v, v = L1,b v1 , v1 + L2,b v2 , v2 ,

S1,b 1 p−1 2

L1,b v1 , v1 = v1 H 1 − p ψ (x)|v1 (x)|2 dx, b 1,b |x| n R 1 p−1 ψ (x)|v2 (x)|2 dx,

L2,b v2 , v2 = v2 2H 1 − b 1,b |x| n R

and Re(ψ1,b , v)L2 = (ψ1,b , v1 )L2 ,

Re(iψ1,b , v)L2 = (ψ1,b , v2 )L2 .

Thus it suffices to show the following. Lemma 2.1 Assume n ≥ 3, 0 0 such that

L1,b v, v ≥ δ1 v2H 1 for any v ∈ H 1 (Rn , R) satisfying (v, ψ1,b )L2 = 0. (ii) There exists δ2 > 0 such that

L2,b v, v ≥ δ2 v2L2 for any v ∈ H 1 (Rn , R) satisfying (v, ψ1,b )L2 = 0. The part (ii) of Lemma 2.1 is obtained since L2,b ψ1,b = 0 and ψ1,b (x) > 0 for x ∈ Rn . Namely, ψ1,b is the first eigenfunction of L2,b corresponding to the eigenvalue 0. Moreover, by Weyl’s theorem, the essential spectrum of L2,b are in [1, ∞), since ψ1,b tends to zero at infinity. These conclude (ii). Therefore, we prove the part (i) of Lemma 2.1. For that purpose, we need to show the following two propositions. Proposition 5 Assume n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/(n − 2). If v ∈ H 1 (Rn , R) satisfies L1,b v = 0, then v ≡ 0.

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Proposition 6 Assume n ≥ 3, 0 < b < 2 and 1 < p ≤ 1 + (4 − 2b)/n. Then we have inf{ L1,b v, v : v ∈ H 1 (Rn , R), (v, ψ1,b )L2 = 0} = 0.

(2.3)

We shall prove Proposition 5 in the next section. As to Proposition 6, we give a proof in the same way as Proposition 2.7 in Weinstein [31]. First, we show the following lemma. Lemma 2.2 Assume n ≥ 3, 0 0 and γ = {n + 2 − (n − 2)p − 2b}/2 > 0. Then, α := inf{J(v) : v ∈ H 1 (Rn )} is attained at a positive radial function ψ ∗ (x) ∈ H 1 (Rn ) ∩ C ∞ (Rn ) such that ψ∗ (x) =

γ 1−b/2 θb/2 α(p + 1)

1/(p−1)

ψ1,b (γ 1/2 θ−1/2 x).

Proof. We follow the proof of Theorem B of [30]. Since J(v) ≥ 0, there exists a minimizing sequence {vν } ⊂ H 1 (Rn ), that is, limν→∞ J(vν ) = α. We can assume that vν is positive since ∇|v|2 ≤ ∇v2 . Now, let v λ,µ (x) = λv(µx) for λ, µ > 0. Then we have J(v λ,µ ) = J(v), ∇v λ,µ 22 = λ2 µ2−n ∇v22 , v λ,µ 22 = λ2 µ−n v22 , 1 λ,µ p+1 1 p+1 −n+b |v | = λ µ |v|p+1 . |x|b |x|b n/2−1

We choose µν = vν 2 /∇vν 2 and λν = vν 2 v λν ,µν has the following properties.

ψν (x) ∈ H 1 (Rn ), ψν (x) ≥ 0, ψν 2 = 1, ∇ψν 22 = 1,

n/2

/∇vν 2

so that ψν :=

x ∈ Rn ,

J(ψν ) → α as ν → ∞. Namely {ψν } is bounded in H 1 (Rn ). Thus there exists a subsequence {ψν } and a limit ψ∗ (x) ∈ H 1 (Rn ) such that ψν converges to ψ∗ weakly in H 1 (Rn ). It follows

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from the Sobolev embedding on a bounded domain and the smallness of |x|−b for large |x| that 1 p+1 1 p+1 ψ (x)dx → ψ (x)dx as ν → ∞ b ν b ∗ |x| |x| n n R R for 1 < p < 1 + (4 − 2b)/(n − 2) (see the argument in [27], Lemma 1.1 and Remark 3.1 of [11]). By weak convergence, ψ∗ 2 ≤ 1 and ∇ψ∗ 2 ≤ 1. Furthermore, ∇ψν θ2 ψν γ2 ∇ψ∗ θ2 ψ∗ γ2 α ≤ J(ψ∗ ) = ≤ lim inf = lim inf J(ψν ) 1 1 ν→∞ ν→∞ p+1 p+1 |ψ | |ψ | ∗ ν b b |x| |x| 1 = α. = lim inf 1 ν→∞ p+1 |ψν | |x|b It follows that ∇ψ∗ θ2 ψ∗ γ2 = 1 and therefore ∇ψ∗ 2 = ψ∗ 2 = 1, which implies that ψν → ψ∗ strongly in H 1 (Rn ). This minimizing function ψ∗ satisfies the Euler-Lagrange equation: d J(ψ∗ + εη) = 0 for any η ∈ C0∞ (Rn ). dε ε=0 Taking into account that ∇ψ∗ 2 = ψ∗ 2 = 1 and that |x|−b |ψ∗ |p+1 = 1/α, we have 1 −θ∆ψ∗ + γψ∗ − α(p + 1) b ψ∗p = 0. |x| The smoothness of ψ∗ follows from the same method as Section 8 of Cazenave [5]. 1−b/2 b/2 1/(p−1) θ ψ(γ 1/2 θ−1/2 x) makes ψ(x) be a posThe scaling ψ∗ (x) = γ α(p+1) itive solution of (2.2) with ω = 1. By the results in [12] and [20], ψ(x) is radial. Accordingly, ψ(x) is the unique solution ψ1,b (r). Proof of Proposition 6. We remark that the infimum of (2.3) is nonpositive because the value L1,b v, v is zero for v = 0. Since J(v) attains its minimum at ψ1,b , d2 J(ψ + εη) ≥0 1,b 2 dε ε=0 for all η ∈ C0∞ (Rn ). A simple calculation concludes 2θ θ

L1,b v, v ≥ 1− (∇ψ1,b , ∇v)2L2 α 2

(2.4)

for any v ∈ H 1 (Rn , R) with (v, ψ1,b )L2 = 0, where α and θ have been defined in Lemma 2.2. The result follows since the right-hand side of (2.4) is nonnegative for p ≤ 1 + (4 − 2b)/n.

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Now we are ready to give a proof of part (i) of Lemma 2.1. Proof of Lemma 2.1 (i). Let τ := inf{ L1,b v, v : v ∈ H 1 (Rn , R), (v, ψ1,b )L2 = 0, vH 1 = 1} and suppose τ = 0 under the condition 1 < p < 1 + (4 − 2b)/n. Let {vj } ⊂ H 1 (Rn ) be a minimizing sequence, that is, lim L1,b vj , vj = 0,

j→∞

vj H 1 = 1,

(vj , ψ1,b )L2 = 0.

Since {vj } is bounded in H 1 (Rn ), there exists a subsequence still denoted by {vj } ⊂ H 1 (Rn , R) which converges weakly to some f∗ ∈ H 1 . By weak convergence, f∗ satisfies (f∗ , ψ1,b )L2 = 0. We also have 1 p−1 2 1 p−1 2 ψ v → ψ f (2.5) |x|b 1,b j |x|b 1,b ∗ as j → ∞ for 1 < p < 1 + (4 − 2b)/(n − 2). Indeed, we note that vj2 converges p−1 weakly to f∗2 in Ln/(n−2) (Rn ) by the Sobolev embedding, and that |x|−b ψ1,b (x) ∈ Ln/2 (Rn ) since |x|−b vanishes at infinity and ψ1,b (x) decays exponentially for |x| ≥ p−1 C with some C > 0. For |x| ≤ C, we know that |x|−b ψ1,b (x) ∈ Ln/2 (|x| ≤ C) if p < 1 + (4 − 2b)/(n − 2). Thus, we have 0 = lim L1,b vj , vj j→∞ 1 p−1 2 = 1 − p lim ψ vj j→∞ Rn |x|b 1,b 1 p−1 2 =1−p ψ f b 1,b ∗ Rn |x| and then, f∗ ≡ 0. Moreover, by weak convergence, f∗ H 1 ≤ 1 and 0 ≤ L1,b f∗ , f∗ ≤ lim L1,b vj , vj = 0, j→∞

where the first inequality follows from Proposition 6. We define g∗ := f∗ /f∗ H 1 and then g∗ satisfies g∗ ∈ H 1 (Rn ), g∗ H 1 = 1, (g∗ , ψ1,b )L2 = 0, g∗ ≡ 0 and

L1,b g∗ , g∗ = 0. Since the minimum is attained at an admissible function g∗ ≡ 0, there exists (g∗ , λ, β) solution of the Lagrange multiplier problem L1,b g∗ = λ(−∆g∗ + g∗ ) + βψ1,b ,

λ, β ∈ R,

(2.6)

g∗ H 1 = 1,

(2.7)

(g∗ , ψ1,b )L2 = 0.

(2.8)

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By (2.6), (2.7) and (2.8), λ = L1,b g∗ , g∗ . Thus, λ = 0 since we have assumed τ = 0. Therefore, L1,b g∗ = βψ1,b . On the other hand, let g :=

b−2 2

1 1 ψ1,b + x · ∇ψ1,b . p−1 2−b

Then we have L1,b g = ψ1,b . Accordingly, L1,b (g∗ − βg) = 0. It follows from Proposition 5 that g∗ = βg. If β = 0, then g∗ = 0, which is a contradiction. Thus β = 0. Here, β 2−b n − (g∗ , ψ1,b )L2 = (βg, ψ1,b )L2 = − ψ1,b 22 , 2 p−1 2 which violates (2.8) when p < 1 + (4 − 2b)/n. Thus, g∗ ≡ 0, a contradiction. We now conclude that τ > 0 if p < 1 + (4 − 2b)/n.

3 Nondegeneracy of unique positive radial solution for (2.2) In this section, we give a proof of Proposition 5, following Kabeya and Tanaka [18]. We always assume that n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/(n − 2). Let ψ1,b (r) ∈ H 1 (Rn ) be the unique positive radial solution of (2.2). ψ1,b (r) decays exponentially and can be characterized as a critical point of the C 2 functional 1 1 1 1 p+1 v dx, S1,b,+ (v) = ∇v22 + v22 − 2 2 p + 1 Rn |x|b + where v+ = max{v, 0}. Remark 3.1 We briefly explain why S1,b,+ (v) is C 2 on H 1 (Rn ) when 1 < p < 1 + (4 − 2b)/(n − 2). For v ∈ H 1 (Rn ), let s 1 1 p+1 N (v) = v dx, M (s) = m(x, τ )dτ, p + 1 Rn |x|b + 0 p where m(x, τ ) = |x|−b τ+ . For v, h ∈ H 1 (Rn ) and t ∈ (−1, 1) \ {0}, we have M (v + th) − M (v) ≤ C|x|−b (|v+ + th+ |p + |v+ |p )|h| (3.1) t p+1 is a C 2 function on R if p > 1. The right-hand side of since the function y → y+ 1 n (3.1) belongs to L (R ) if 1 < p < 1 + (4 − 2b)/(n − 2). Therefore, by Lebesgue’s convergence theorem, N (v + th) − N (v) M (v + th) − M (v) = dx lim lim t→0 t→0 t t n R t 1 p = lim m(x, v + th)hdt dx = v hdx. b + t→0 |x| n n 0 R R

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p We conclude N (v) ∈ C 1 (H 1 (Rn ), R) and N (v)h = Rn |x|−b v+ hdx, for v, h ∈ 1 n 2 H (R ). C regularity follows from the same argument. Any non-zero critical point of S1,b,+ (v) is a positive solution by the maximum principle. On the other hand, as we mentioned in Section 2, radial symmetry of a positive solution and the uniqueness of positive radial solutions follow from [12, 20] and [33]. Thus it is ψ1,b (r). For δ > 0 small, we consider the following perturbed functional: 1 1 p+1 p−1 2 v+ dx − ψ1,b v dx . Sδ (v) = S1,b,+ (v) − δ p + 1 Rn 2 Rn Critical points v(x) of Sδ (v) satisfy −∆v + (1 +

p−1 δψ1,b )v

=

1 p + δ v+ , |x|b

x ∈ Rn .

By the maximum principle, non-zero solutions are positive. Furthermore, positive solutions are radial for small δ > 0 (see [12, 20]). Thus they satisfy 1 p−1 −∆v + (1 + δψ1,b )v = + δ v p , x ∈ Rn , |x|b v(x) > 0, v(x) = v(|x|), x ∈ Rn , v ∈ H 1 (Rn ).

such

(3.2) (3.3) (3.4)

By Yanagida [33], we see that (3.2)–(3.4) has a unique positive radial solution for small δ > 0 (see Appendix). Since ψ1,b (r) satisfies (3.2)–(3.4), the unique solution of (3.2)–(3.4) is ψ1,b (r). For δ ≥ 0, we define the Morse index index Sδ (ψ1,b ) = max{dim H : H ⊂ H 1 (Rn ) is a subspace such that

Sδ (ψ1,b )h, h < 0 for all h ∈ H \ {0}}.

ψ1,b (r) has the following properties. Lemma 3.1 (i) For sufficiently small δ ≥ 0, ψ1,b is a mountain pass critical point of Sδ (v), i.e., Sδ (ψ1,b ) = inf max Sδ (γ(s)), γ∈Γ s∈[0,1]

where Γ = {γ(s) ∈ C([0, 1], H 1 (Rn )) : γ(0) = 0, γ(1) = e0 }. Here, e0 ∈ H 1 (Rn ) satisfies Sδ (e0 ) < 0. (ii) The Morse index at ψ1,b is equal to 1 for small δ ≥ 0, i.e., index Sδ (ψ1,b ) = 1.

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For the proof of Lemma 3.1, we recall Hofer’s result in [15] (see also Tanaka [28] as a related reference). Proposition 7 ([15]) Let F be a real Hilbert space and U ⊂ F be a nonempty open subset. Assume that I ∈ C 2 (U, R) satisfies Palais-Smale condition and the gradient I has the form identity−K, where K is compact. Define A, c, d by A = {a ∈ C([0, 1], F ) : a(i) = ei , i = 0, 1}, d = inf sup I(a[0, 1]), a∈A

c = max{I(e0 ), I(e1 )} and assume d > c. Let u0 ∈ U is an isolated critical point of I at the level d. Then the Morse index at u0 is at most 1. Proof of Lemma 3.1. (i) For some ρ0 > 0 and e0 ∈ H 1 (Rn ), we have inf

vH 1 =ρ0

Sδ (v) > 0,

e0 H 1 ≥ ρ0

and

Sδ (e0 ) < 0.

Therefore Sδ (v) has mountain pass geometry. Since the embedding H 1 ⊂ L2 is compact on a bounded domain and |x|−b vanishes at infinity, Sδ (v) satisfies the Palais-Smale compactness condition if p < 1 + (4 − 2b)/(n − 2) (see Lemma 1.1 and Remark 3.1 of [11]) and small δ ≥ 0. Therefore we can apply the mountain pass theorem. Since ψ1,b is the unique non-zero critical point of Sδ (v) for sufficiently small δ ≥ 0, ψ1,b is the mountain pass critical point. (ii) By Proposition 7, the Morse index is at most one at the mountain pass critical point, i.e., index Sδ (ψ1,b ) ≤ 1. Indeed, Sδ (v) satisfies the conditions in Proposition 7. For v, h ∈ H 1 (Rn ), let Sδ (v)h = v − K(v), hH 1 , where K(v) = p−1 K1 (v) + K2 (v) : H 1 (Rn ) → H 1 (Rn ) defined by K1 (v), hH 1 = Rn δψ1,b hdx, p hdx. We see that K1 is compact and that K2 is

K2 (v), hH 1 = Rn (|x|−b + δ)v+ compact for sufficiently small δ ≥ 0. Furthermore, ψ1,b is the unique mountain pass critical point for sufficiently small δ ≥ 0. On the other hand, 1 p−1 p−1 2

Sδ (ψ1,b )h, h = ∇h22 + (1 + δψ1,b )|h|2 − p + δ ψ1,b h dx. |x|b Rn Rn Setting h = ψ1,b and using Sδ (ψ1,b ), ψ1,b = 0, we have 1 p+1

Sδ (ψ1,b )ψ1,b , ψ1,b = −(p − 1) + δ ψ1,b dx < 0. b |x| n R Thus we get index Sδ (ψ1,b ) = 1.

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Using Lemma 3.1, we verify Proposition 5. Proof of Proposition 5. Suppose that there exists a non-zero solution w0 ∈ H 1 (Rn ) of L1,b w0 = 0. It satisfies (ψ1,b )w0 , ξ = 0

S1,b,+

for all ξ ∈ H 1 (Rn ).

By Lemma 3.1 (ii) with δ = 0, we may also find a w1 ∈ H 1 (Rn ) such that (ψ1,b )w1 , w1 < 0.

S1,b,+

We define a 2-dimensional subspace H of H 1 (Rn ) by H = span{w0 , w1 }. Then we have

S1,b,+ (ψ1,b )h, h ≤ 0 for all h ∈ H. On the other hand, we have for all δ > 0, p−1 2 (ψ1,b )h, h − δ(p − 1) ψ1,b h dx

Sδ (ψ1,b )h, h = S1,b,+ Rn p−1 2 ≤ −δ(p − 1) ψ1,b h dx for all h ∈ H. Rn

n

We remark that ψ1,b (x) > 0 in R and we get

Sδ (ψ1,b )h, h < 0

for all h ∈ H \ {0}.

It means that for all δ > 0, index Sδ (ψ1,b ) ≥ 2, which is a contradiction to Lemma 3.1 (ii) with sufficiently small δ ≥ 0.

4 Proof of Theorem 1 In this section, we prove the following Lemma 4.1 to show Theorem 1. For ω > 0, we define (v, w)H 1 (ω) = Re(∇v, ∇w)L2 + ω Re(v, w)L2 , 1/2

vH 1 (ω) = (v, v)H 1 (ω) ,

v, w ∈ H 1 (Rn ).

(4.1)

Then, we see that · H 1 (ω) is an equivalent norm on H 1 (Rn ) to · H 1 . We remark that for v ∈ H 1 (Rn ) with v1 (x) = Re v(x) and v2 (x) = Im v(x), we have

Sω (φω )v, v = L1,ω v1 , v1 + L2,ω v2 , v2 , 2 2

L1,ω v1 , v1 = v1 H 1 (ω) − p V (x)φp−1 ω (x)|v1 (x)| dx, Rn 2

L2,ω v2 , v2 = v2 2H 1 (ω) − V (x)φp−1 ω (x)|v2 (x)| dx,

(4.2)

Re(φω , v)L2 = (φω , v1 )L2 ,

(4.5)

(4.3) (4.4)

Rn

under the assumptions in Proposition 2.

Re(iφω , v)L2 = (φω , v2 )L2 ,

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Lemma 4.1 Let n ≥ 3, 0 0 with the following property: for any ω ∈ (0, ω1 ), there exists δ1 > 0 such that

L1,ω v, v ≥ δ1 v2H 1 (ω) for any v ∈ H 1 (Rn , R) satisfying (v, φω )L2 = 0. (ii) For any ω ∈ (0, ∞), there exists δ2 > 0 such that

L2,ω v, v ≥ δ2 v2H 1 (ω) for any v ∈ H 1 (Rn , R) satisfying (v, φω )L2 = 0. Proof of Theorem 1. Since · H 1 (ω) is equivalent to · H 1 , by (4.2) and Lemma 4.1, there exists δ > 0 such that (1.6) holds for any v ∈ H 1 (Rn ) satisfying Re(φω , v)L2 = 0 and Re(iφω , v)L2 = 0. Hence, Theorem 1 follows from Proposition 2. In order to show Lemma 4.1, we use the rescaled function φ˜ω defined by (1.7). ˜ 2,ω by ˜ 1,ω and L For ω > 0, we define the rescaled operators L x 2 −b/2 2 ˜ V √

L1,ω v, v = vH 1 − pω φ˜p−1 ω (x)|v(x)| dx, ω Rn x 2 −b/2 2 ˜ φ˜p−1

L2,ω v, v = vH 1 − ω V √ ω (x)|v(x)| dx. ω Rn √ Then, for v(x) = ω (2−b)/2(p−1) v˜( ωx), we have v 2H 1 , v2H 1 (ω) = ω 1+(2−b)/(p−1)−n/2 ˜ (φω , v)L2 = ω (2−b)/(p−1)−n/2 (φ˜ω , v˜)L2 , ˜ j,ω v˜, v˜,

Lj,ω v, v = ω 1+(2−b)/(p−1)−n/2 L

j = 1, 2

(see (4.1), (4.3) and (4.4)). Proof of Lemma 4.1. We show (i) by contradiction. Suppose that (i) were false. Then, there would exist {ωj } and {vj } ⊂ H 1 (Rn , R) such that ωj → 0, ˜ 1,ωj vj , vj ≤ 0, lim L

(4.6)

vj 2H 1 = 1,

(4.7)

j→∞

(vj , φ˜ωj )L2 = 0.

Since {vj } is bounded in H 1 (Rn ), there exists a subsequence of {vj } (still denoted by {vj }) and v0 ∈ H 1 (Rn , R) such that vj → v0 weakly in H 1 (Rn , R). Therefore, |vj |2 → |v0 |2 weakly in Ln/(n−2) (Rn ). Further, by Proposition 3 of

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[11], we see that φ˜ωj → ψ1 strongly in H 1 (Rn ), so that φ˜p−1 → ψ1p−1 strongly ωj in L2n/{(n−2)(p−1)} (Rn ) ∩ L(p+1)/(p−1) (Rn ). Moreover, by (V1) and (V2) if p < 1 + (4 − 2b)/(n − 2),

−b/2 x 1

lim ωj V √ =0 − b

j→∞

ωj |x| θ∗ follows from Lemma 4.2 of [11]. Thus, we have x 1 p−1 2 lim ωj −b/2 V √ (x)|v (x)| dx = ψ (x)|v0 (x)|2 dx. φ˜p−1 j ωj b 1,b j→∞ ωj Rn Rn |x| (4.8) Indeed, x 1 p−1 2 p−1 2 −b/2 ˜ V √ ωj φωj vj − b ψ1,b v0 dx ωj |x| Rn 1 p−1 2 1 ˜p−1 p−1 2 2 ψ (v − v )dx + (φωj − ψ1,b )vj dx = j 0 1,b b b Rn |x| Rn |x| x 1 2 + ωj −b/2 V √ − b φ˜p−1 ωj vj dx. ωj |x| Rn p−1 The first term converges to 0 as j → ∞ since |x|−b ψ1,b ∈ Ln/2 (Rn ) (see Proof of Lemma 2.1 (i)). The two remaining terms are estimated as follows: For some R > 0 such that |x|−b ≤ ε if |x| ≥ R, 1 ˜p−1 p−1 2 (φωj − ψ1,b )vj dx b |x| n R φ˜p−1 − ψ p−1 2n/{(n−2)(p−1)} ≤ |x|−b θ∗ L

(|x|≤R)

ωj

vj 22n/(n−2)

1,b

p−1 2 + εφ˜p−1 ωj − ψ1,b (p+1)/(p−1) vj p+1 ,

1 x 2 ωj −b/2 V √ − b φ˜p−1 ωj vj dx ω |x| n j R

−b/2 x 1

2

≤ ωj V √ φ˜ωj p−1 − b

2n/(n−2) vj 2n/(n−2) , ωj |x| θ∗

which conclude (4.8). Therefore, by (4.6), (4.7) and (4.8), we have 0 ≥ = =

˜ 1,ωj vj , vj lim inf L j→∞ x 2 lim inf vj 2H 1 − pωj −b/2 V √ (x)|v (x)| dx φ˜p−1 j ωj j→∞ ωj Rn 1 p−1 1−p ψ (x)|v0 (x)|2 dx. (4.9) b 1,b |x| n R

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Again, by (4.6), (4.8), we have 0 ≥ = ≥

˜ 1,ωj vj , vj lim inf L j→∞ x 2 p−1 2 −b/2 ˜ V √ lim inf vj H 1 − pωj φωj (x)|vj (x)| dx j→∞ ωj Rn 1 p−1 v0 2H 1 − p ψ (x)|v0 (x)|2 dx = L1,b v0 , v0 . b 1,b |x| n R

Moreover, by (4.7), we have (v0 , ψ1,b )L2 = 0. Therefore, by Lemma 2.1 (i), we have v0 ≡ 0. However, this contradicts (4.9). Hence, we conclude (i). By an analogous argument as (ii) of Lemma 2.1, we can also prove (ii).

5 Appendix 5.1

Uniqueness for (3.2)–(3.4)

We have cited the uniqueness result by Yanagida [33]. Here, we briefly check the conditions to prove the uniqueness of a solution (3.2)–(3.4). The condition appeared as (C1)–(C6) in Theorem 2.2 of [33]. In the paper [33], the following type of semilinear elliptic equations was treated: u (r) +

n−1 u (r) + g(r)u(r) + h(r)u(r)p = 0, r

r > 0,

n ≥ 3,

where we denote d/dr by . p−1 As an application to our present case, we consider g(r) = −(1 + δψ1,b ) and h(r) = r−b + δ, where δ ≥ 0, n ≥ 3, 0 < b < 2 and ψ1,b (r) is the unique positive radial solution of (2.2) with ω = 1. We remark that ψ1,b (r) ∈ C 2 (Rn ) decays exponentially as r → ∞ by the standard argument for radial solutions of elliptic equations (see, for example, Berestycki and Lions [4]) and ψ1,b (r) is monotone (r) < 0 for r > 0. First, decreasing with respect to r > 0 from [12, 20, 21], i.e., ψ1,b we know that two conditions (A1) g(r) and h(r) are in C 1 ((0, ∞)), (A2) r2−σ g(r) → 0 and r2−σ h(r) → 0 as r → +0 for some σ > 0, are satisfied. Now let m ∈ [0, n − 2] be a parameter and define G(r; m)

:=

H(r; m) :=

p−2 p−1 −δ(p − 1)rm+2 ψ1,b (r)ψ1,b (r) + 2(n − 3 − m)rm+1 (1 + δψ1,b (r))

+m(n − 2 − m)(n − 2 − m/2)rm−1 , 2b − 2(m + 2) − 2(n − 2) − m + rm−b+1 p+1 2(m + 2) − 2(n − 2) − m − δrm+1 . p+1

These are related to Pohozaev identity (see Yanagida [33] for details).

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Required conditions in [33, Theorem 2.2] are following: (C1) h(r) ≥ 0 for all r ∈ (0, ∞) and h(r) > 0 for some r ∈ (0, ∞). (C2) G(r; n − 2) ≤ 0 for all r ∈ (0, ∞). (C3) For each m ∈ [0, n − 2), there exists an α(m) ∈ [0, ∞] such that G(r; m) ≥ 0 for r ∈ (0, α(m)) and G(r; m) ≤ 0 for r ∈ (α(m), ∞). (C4) H(r; 0) ≤ 0 for all r ∈ (0, ∞). (C5) For each m ∈ (0, n − 2], there exists a β(m) ∈ [0, ∞] such that H(r; m) ≥ 0 for r ∈ (0, β(m)) and H(r; m) ≤ 0 for r ∈ (β(m), ∞). (C6) When g(r) ≡ 0 for all r ≥ 0, h(r) satisfies h(r) ≡ C0 rq , where C0 > 0 is an n−2 n+2 arbitrary constant and q := p− . 2 n−2 The condition (C6) is excluded in the present case. It is clear that (C1), (C4) and (C5) hold since 2 2b −b+1 r −2 n−2− H(r; 0) = − r(r−b + δ). p+1 p+1 Also, since p−1 p−2 (r) + δr(p − 1)ψ1,b (r)ψ1,b (r)), G(r; n − 2) = {r2 g(r)} = −r(2 + 2δψ1,b

taking δ so small that the right-hand side is nonpositive for all r ≥ 0, we can conclude (C2) for sufficiently small δ ≥ 0. The condition (C3) follows for small δ ≥ 0, too. Indeed, if 0 ≤ n − 3 − m, then we have G(r, m) > 0 for all r > 0, therefore we may take α(m) = ∞. If −1 < n − 3 − m < 0 and m ≥ 1, we have that G(r, m) → −∞ as r → ∞ and that G(r, m) tends to a nonnegative constant as r → 0. For the case where −1 < n − 3 − m < 0 and m < 1, we see that G(r,m) → −∞ as r → ∞ and G(r, m) → ∞ as r → 0. Moreover, in both cases, d G (r, m) < 0 for r > 0 and sufficiently small δ ≥ 0. Thus, there exists α(m) dr rm−2 satisfying (C3) (see a similar investigation in [18, Lemma 1.3]).

5.2

Orbital stability

Next, we remark on the proof of Proposition 2. Proposition 2 implies the following lemma: Lemma 5.1 Under the assumptions in Proposition 2, there exist C > 0 and ε > 0 such that E(u) − E(φω ) ≥ C inf u − eiθ φω 2H 1 θ∈R

for u ∈ Uε (φω ) with Q(u) = Q(φω ). We can prove this lemma following Grillakis, Shatah and Strauss [13, Theorem 3.4] (see also [16, Proposition 1], Section 2 of [10]). Theorem 1 follows from Lemma 5.1 and the proof of Theorem 3.5 of [13].

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Acknowledgment This study started while one of us (R. F) stayed in Université de Paris-Sud, Orsay. R. F is grateful to the staff of Laboratoire d’Analyse Numérique for their warm hospitality. Also, the authors wish to express their sincere appreciation to Professor Kazunaga Tanaka for his helpful advice about Section 3.

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[28] K. Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Comm. Partial Differential Equations 14, 99– 128 (1989). [29] I. Towers and B.A. Malomed, Stable (2 + 1)−dimensional solutions in a layered medium with sign-alternating Kerr nonlinearity, J. Opt. Soc. Am. B 19, 537–543 (2002). [30] M.I. Weinstein, Nonlinear Schr¨ odinger equations and sharp interpolation estimates, Comm. Math. Phys. 87, 567–576 (1983). [31] M.I. Weinstein, Modulational stability of ground states of nonlinear Schr¨ odinger equations, Siam J. Math. Anal. 16, 472–491 (1985). [32] M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39, 51–68 (1986). [33] E. Yanagida, Uniqueness of positive radial solutions of ∆u+g(r)u+h(r)up = 0 in Rn , Arch. Rat. Mech. Anal. 115, 257–274 (1991). Anne De Bouard Laboratoire de Mathématiques Université de Paris-Sud F-91405 Orsay France email: [email protected] Reika Fukuizumi Department of Mathematics Hokkaido University Sapporo 060-0810 Japan email: [email protected] Communicated by Bernard Helffer submitted 14/07/04, accepted 28/02/05



Dispersive Estimates of Solutions to the Schr¨ odinger Equation Georgi Vodev Abstract. We prove time decay L1 → L∞ estimates for the Schr¨ odinger group eit(−∆+V ) for real-valued potentials V ∈ L∞ (R3 ) satisfying V (x) = O |x|−δ , |x| 1, with δ > 5/2.

1 Introduction and statement of results Let V ∈ L∞ (R3 ) be a real-valued function satisfying |V (x)| ≤ Cx−δ ,

∀x ∈ R3 ,

(1.1)

with constants C > 0 and δ > 5/2, where x = (1 + |x|2 )1/2 . Denote by G0 and G the self-adjoint realizations of the operators −∆ and −∆ + V (x) on L2 (R3 ). By Kato’s theorem the operator G has no strictly positive eigenvalues. This implies that G has no strictly positive resonances neither. Indeed, it is possible to show that, under the assumption (1.1) with δ > 2, such a resonance is in fact an eigenvalue (e.g., see [1], [7]). It is well known that the free Schr¨ odinger group satisfies the following dispersive estimate itG e 0 1 ∞ ≤ C|t|−3/2 , t = 0. (1.2) L →L Hereafter, given 1 ≤ p ≤ +∞, Lp denotes the space Lp (R3 ). Given any a > 0 denote by χa ∈ C ∞ (R) a function supported in the interval [a, +∞), χa = 1 on [a + 1, +∞). Let also χ denote the characteristic function of the interval [0, +∞) (the absolutely continuous spectrum of G). For potentials satisfying (1.1) with δ > 7 as well as an extra technical assumption, the following analogue of (1.2) was proved in [5]: itG e χ(G) 1 ∞ ≤ C|t|−3/2 , (1.3) L →L provided that zero is neither eigenvalue nor resonance of G (i.e., 0 is a regular point for G). Note that without the assumption that 0 is a regular point, χ in (1.3) should be replaced by χa . Note also that in [5] an analogue of (1.3) in all space dimensions n ≥ 3 is proved. Later on (1.3) was proved in [8] for potentials satisfying (1.1) with δ > 5 via the properties of the wave operators. Recently, (1.3) is proved in [6] for a class of small potentials, while in [3] (1.3) is proved for potentials satisfying (1.1) with δ > 3, provided 0 is a regular point for G. The

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purpose of this work is to extend the three dimensional result in [3] to the larger class of potentials satisfying (1.1) with δ > 5/2. Our main result is the following Theorem 1.1 Assume (1.1) fulfilled. Then, for every a > 0 there exists a constant C > 0 so that the following estimate holds itG e χa (G) 1 ∞ ≤ C|t|−3/2 , |t| ≥ 1. (1.4) L →L It is worth noticing that the estimate (1.4) with O |t|−3/2+ , ∀0 < 1, in the RHS is obtained in [6] for a class of potentials including those satisfying (1.1) with δ > 2, provided the parameter a is taken big enough (independent of t). Note also that an weaker version of (1.4) is proved in [4] for potentials satisfying (1.1) with δ > 3 as well as an extra technical assumption. Our approach is quite different from those developed in the papers mentioned above. It consists of reducing the estimate (1.4) to the following semi-classical estimate on weighted L2 spaces. Theorem 1.2 Let ϕ ∈ C0∞ ((0, +∞)) and assume (1.1) fulfilled. Then, for 0 < 1, 0 ≤ s ≤ 3/2, we have −s− itG x e ϕ(h2 G)x−s− L2 →L2 ≤ Chs |t|−s , t = 0, 0 < h ≤ 1, (1.5) with a constant C > 0 independent of t and h. The fact that supp ϕ is disjoint from zero plays an essential role in our proof of (1.5). However, if the estimate (1.5) holds with h = 1 and ϕ replaced by (1 − χa )χ, works then (1.4) holds with χa (G) replaced by χ(G). Note also that our method in all space dimensions n ≥ 3 and gives an analogue of (1.4) with O |t|−n/2 in the RHS for potentials (1.1) with δ > n2 + 1, while for n = 2 it leads −1 satisfying to (1.4) with O |t| log |t| in the RHS for potentials satisfying (1.1) with δ > 2. In the general case (1.5) holds for 0 ≤ s ≤ n/2. The only thing concerning the free operator we need is the explicit formula for the kernel of the operator f (G0 ) for suitable functions f ∈ L1loc (R+ ). In the three dimensional case it is given by a very simple formula (see (2.16) below), while in the general case this kernel can be expressed in terms of the Bessel function J(n−2)/2 . It has been pointed out by the referee that (1.4) was proved in [2] for a class of potentials including those satisfying (1.1) with δ > 2 by using different methods. It seems, however, that the fact that the dimension is three plays an important role in the method developed in [2].

2 Proof of Theorem 1.1 In this section we will derive Theorem 1.1 from Theorem 1.2. Without loss of generality we may suppose that t ≥ 1. Given a parameter 0 < h ≤ 1 and a function ϕ ∈ C0∞ ((0, +∞)), denote Φ(t; h) = eitG ϕ(h2 G) − eitG0 ϕ(h2 G0 ).

Vol. 6, 2005

Dispersive Estimates of Solutions to the Schr¨ odinger Equation

We also set

F (t) = i

t

1181

ei(t−τ )G0 V eiτ G0 dτ.

0

Theorem 1.1 follows easily from the following Theorem 2.1 Under the assumption (1.1), for t ≥ 1, 0 < h ≤ 1, we have Φ(t; h) + F (t)ϕ(h2 G0 ) 1 ∞ ≤ Chβ t−3/2 , L →L

(2.1)

with constants C, β > 0 independent of t and h. Indeed, writing the function χa as χa (σ) =

1

ϕ(σθ) 0

dθ , θ

where ϕ(σ) = σχa (σ) ∈ C0∞ ((0, +∞)), we obtain itG e χa (G) − eitG0 χa (G0 ) + F (t)χa (G0 ) 1 ∞ L →L 1 √ dθ ≤ Φ(t; θ) + F (t)ϕ(θG0 ) 1 ∞ θ L →L 0 1 θ−1+β/2 dθ ≤ C t−3/2 . (2.2) ≤ Ct−3/2 0

On the other hand, it is proved in Section 6 of [6] (where in fact much more is proved) that F (t)L1 →L∞ ≤ Ct−3/2 , (2.3) provided the potential V satisfies sup x∈R3

R3

|V (y)| dy < +∞, |x − y|

which in turn is fulfilled for potentials satisfying (1.1). Now (1.4) follows from combining (1.2), (2.2), (2.3) and the fact that the operator χa (G0 ) is bounded on Lp , 1 ≤ p ≤ +∞. Proof of Theorem 2.1. We will first prove the following Lemma 2.2 For 1 ≤ p ≤ +∞, 0 ≤ s ≤ 2, 0 < h ≤ 1, we have ϕ(h2 G) − ϕ(h2 G0 ) p p ≤ Ch2 , L →L

ϕ(h2 G) − ϕ(h2 G0 ) x2 2 2 ≤ Ch2 , L →L ϕ(h2 G0 )(1 + G0 )s/2 p p ≤ Ch−s , L →L

(2.4) (2.5) (2.6)

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ϕ(h2 G)(1 + G0 )s/2 p p ≤ Ch−s , L →L −s x ϕ(h2 G0 )xs 2 2 ≤ C, L →L −s x ϕ(h2 G)xs 2 2 ≤ C, L →L

(2.7) (2.8) (2.9)

with a constant C > 0 independent of h. Proof. We will take advantage of the Helffer-Sj¨ ostrand formula ∂ϕ 1 ϕ(h2 G) = (ζ)(h2 G − ζ)−1 L(dζ), π C ∂ζ

(2.10)

where L(dζ) denotes the Lebesgue measure on C, and ϕ ∈ C0∞ (C) is an almost analytic continuation of ϕ supported in a small complex neighborhood of supp ϕ and satisfying ∂ϕ (ζ) ≤ CN |Im ζ|N , ∀N ≥ 1. (2.11) ∂ζ Thus we have

ϕ(h2 G) − ϕ(h2 G0 ) p p L →L ∂ ϕ (ζ) (h2 G − ζ)−1 V (h2 G0 − ζ)−1 p p L(dζ) ≤ O(h2 ) ∂ζ L →L C N ≤ ON (h2 ) |Im ζ| (h2 G0 − ζ)−1 V (h2 G0 − ζ)−1 p p L →L

Cϕ

+h2 (h2 G0 − ζ)−1 V (h2 G − ζ)−1 V (h2 G0 − ζ)−1 Lp →Lp L(dζ) 2

≤ ON (h )

Cϕ

|Im ζ|

N

−1

1 + h2 |Im ζ|

2 (h G0 − ζ)−1 2 p (h2 G0 − ζ)−1 p 2 L(dζ), L →L L →L

Hence (2.4) follows from this and the following well-known where Cϕ = supp ϕ. estimate 2 (h G0 − ζ)−1 2 p ≤ C|Im ζ|−1−q , ζ, ζ ∈ Cϕ , Im ζ = 0, (2.12) L →L for every 1 ≤ p ≤ +∞, where q = 3 12 − p1 , with a constant C > 0 independent of h. In the same way, to prove (2.5) it suffices to show that, with 0 ≤ s ≤ 2, V (h2 G0 − ζ)−1 xs 2 2 ≤ C|Im ζ|−2 , ζ ∈ Cϕ , Im ζ = 0, (2.13) L →L with a constant C > 0 independent of h. Using the identity V (h2 G0 − ζ)−1 xs = V xs (h2 G0 − ζ)−1 + h2 V (h2 G0 − ζ)−1 [∆, xs ](h2 G0 − ζ)−1 ,

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we get


1183

V (h2 G0 − ζ)−1 xs 2 2 ≤ C1 (h2 G0 − ζ)−1 2 2 L →L L →L

+ C1 h2 V (h2 G0 − ζ)−1 xs−1 L2 →L2 ∇x (h2 G0 − ζ)−1 L2 →L2 + (h2 G0 − ζ)−1

L2 →L2

≤ C2 |Im ζ|−1 1 + h V (h2 G0 − ζ)−1 xs−1 L2 →L2 . Repeating this once again leads to (2.13). To prove (2.6) observe that the function ψh (ζ) = (ζ +h2 )s/2 is holomorphic in Cϕ and satisfies there the bound |ψh (ζ)| ≤ C with a constant C > 0 independent of h. By the formula (2.10) we have 1 ∂ϕ 2 ψh (ζ) (ζ)(h2 G0 − ζ)−1 L(dζ), (ϕψh )(h G0 ) = π C ∂ζ so we get (ϕψh )(h2 G0 ) p p ≤ CN L →L

Cϕ

N |Im ζ| (h2 G0 − ζ)−1 Lp →Lp L(dζ).

Therefore, using that 2 (h G0 − ζ)−1 p p ≤ C|Im ζ|−q0 , L →L

ζ ∈ Cϕ , Im ζ = 0,

for every 1 ≤ p ≤ +∞, with constants C, q0 > 0 independent of h, we deduce (ϕψh )(h2 G0 ) p p ≤ Const, L →L which is clearly equivalent to (2.6). To prove (2.7) we will make use of the fact that the operator (G0 + 1)−1+s/2 , 0 ≤ s ≤ 2, is bounded on Lp , 1 ≤ p ≤ +∞. Therefore, using (2.4) and (2.6) with s = 0, we obtain ϕ(h2 G) − ϕ(h2 G0 ) (1 + G0 )s/2 p p L →L 2 ≤ C1 ϕ(h G) − ϕ(h2 G0 ) (1 + G0 )Lp →Lp ≤ C2 h−2 ϕ1 (h2 G) − ϕ1 (h2 G0 )Lp →Lp + C2 ϕ(h2 G) − ϕ(h2 G0 )Lp →Lp + C2 ϕ(h2 G0 )Lp →Lp ≤ Const, where ϕ1 (σ) = σϕ(σ), which together with (2.6) imply (2.7). The estimate (2.8) can be proved in the same way as (2.6) with s = 0, using (2.13) with V replaced by x−s . The estimate (2.9) follows from (2.8) and (2.5).

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G. Vodev


Using the above lemma we will prove the following Lemma 2.3 For 0 < 1, 0 ≤ s ≤ 1, 0 < h ≤ 1, t = 0, we have ≤ Ch−s |t|−3/2+s , ϕ(h2 G0 )eitG0 x−3/2+s− L2 →L∞

ϕ(h2 G)eitG0 x−3/2+s−

L2 →L∞

≤ Ch−s− |t|−3/2+s ,

(2.14) (2.15)

with a constant C > 0 independent of t and h. Proof. Without loss of generality we may suppose that t > 0. To prove (2.14) we will make use of the fact that the kernel of operator f (G0 ) is given by the formula √ ∞ 1 sin( λ|x − y|) dλ. (2.16) [f (G0 )](x, y) = f (λ) (2π)2 0 |x − y| Thus, the kernel of the operator ϕ(h2 G0 )eitG0 is of the form Kh (|x − y|; t), where ∞ 2 1 sin(σλ) λdλ. Kh (σ; t) = eitλ ϕ(h2 λ2 ) 2 2π 0 σ Let us see that |Kh (σ; t)| ≤ Ch−m t−3/2+m σ −1 (1 + σ)1−m ,

(2.17)

for all 0 ≤ m ≤ 1. Clearly, it suffices to show (2.17) only for m = 0 and m = 1. It is easy to see that for m = 1, (2.17) follows from the bound ∞ itλ2 +iaλ ≤ Ct−1/2 , e φ(λ)dλ (2.18) −∞

for every t > 0, a ∈ R, with a constant C > 0 independent of t and a, where φ ∈ C0∞ (R) is independent of t and a. Note that (2.18) is proved in the proof of Lemma 2.4 of [6]. On the other hand, integrating by parts allows to write the function Kh in the form σ −1 t−1 ∞ itλ2 ϕ(h2 λ2 )σ cos(σλ) + 2h2 λϕ (h2 λ2 ) sin(σλ) dλ, e Kh (σ; t) = 2 i(2π) 0 so (2.17) with m = 0 follows again from (2.18). Applying (2.17) with m = s, we obtain 2 |Kh (|x − y|; t)|2 y−3+2s−2 dy ϕ(h2 G0 )eitG0 x−3/2+s− 2 ∞ ≤ sup L →L

≤ Ch−2s t−3+2s sup

x∈R3

R3

x∈R3

2−2s

(1 + |x − y|) |x − y|2

R3

y−3+2s−2 dy ≤ C h−2s t−3+2s .

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1185

The estimate (2.15) follows from (2.7) and the following estimate ≤ Ct−3/2+s . (G0 + 1)−s/2−/2 eitG0 x−3/2+s− L2 →L∞

(2.19)

We will derive (2.19) from (2.14) and the easy observation that the estimate (2.6) holds in fact for all real s. Thus we have (2.20) (G0 + 1)−s/2−/2 ϕ(h2 G0 )eitG0 x−3/2+s− 2 ∞ ≤ Ch t−3/2+s . L →L

Let η ∈ C0∞ (R) be an even function such that η(σ) = 1 for −1 ≤ σ ≤ 1. Writing the function 1 − η(σ) for σ > 0 in the form 1 − η(σ) =

1

ϕ(σθ) 0

dθ , θ

where ϕ(σ) = −ση (σ), we can use (2.20) to get (G0 + 1)−s/2−/2 (1 − η)(G0 )eitG0 x−3/2+s− 2 ∞ L →L 1 ≤ (G0 + 1)−s/2−/2 ϕ(θG0 )eitG0 x−3/2+s− 0

≤ C t−3/2+s

0

L2 →L∞

1

dθ θ

θ−1+/2 dθ ≤ Ct−3/2+s . (2.21)

On the other hand, it is easy to see that (2.14) still holds with h = 1 and ϕ replaced by the function η(σ)(σ 2 + 1)−s/2− . This observation together with (2.21) imply (2.19). Proposition 2.4 For 0 < 1, 0 ≤ s ≤ 1/2 + /2, 0 < h ≤ 1, t > 0, we have −3/2+s− Φ(t; h) ≤ Ch1−s− t−3/2+s , (2.22) x L1 →L2

with a constant C > 0 independent of t and h. Proof. Let ϕ1 ∈ C0∞ ((0, +∞)) be such that ϕ1 ϕ ≡ ϕ. By Duhamel’s formula we have 3 Φj (t; h), (2.23) Φ(t; h) = j=1

where Φ1 (t; h) = ϕ(h2 G)eitG0 ϕ1 (h2 G) − ϕ1 (h2 G0 ) + ϕ(h2 G) − ϕ(h2 G0 ) eitG0 ϕ1 (h2 G),

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G. Vodev

Φ2 (t; h) = i

t/2

0

Φ3 (t; h) = i

t

t/2


ϕ(h2 G)eiτ G V ei(t−τ )G0 ϕ1 (h2 G)dτ,

ϕ(h2 G)eiτ G V ei(t−τ )G0 ϕ1 (h2 G)dτ.

Using (2.4), (2.5) and Lemma 2.3, we get −3/2+s− Φ1 (t; h) x

L1 →L2

≤ Ch2−s− t−3/2+s .

(2.24)

By (1.1), Lemma 2.3 and (1.5) used with s = 1 − /4 and s = 1 + /4, we have −3/2+s− Φ2 (t; h) x L1 →L2 t/2 −1−/2 ≤ C1 ϕ(h2 G)eiτ G x−1−/2 x L2 →L2 0 −3/2−/2 i(t−τ )G0 × x e ϕ1 (h2 G) 1 2 dτ L →L t/2 ≤ C2 h1−s− t−3/2+s τ −1+/4 τ −/2 dτ ≤ Ch1−s− t−3/2+s . (2.25) 0

Similarly, using (1.5) with s replaced by 3/2 − s and Lemma 2.3 with s = 1 − /2 and s = 1 + /2, we have −3/2+s− Φ3 (t; h) x L1 →L2 t/2 −3/2+s− ≤ C1 ϕ(h2 G)ei(t−τ )G x−3/2− x L2 →L2 0 −1− iτ G × x e 0 ϕ1 (h2 G)L1 →L2 dτ t/2 1−s− −3/2+s ≤ C2 h t τ −1+/2 τ − dτ ≤ Ch1−s− t−3/2+s . (2.26) 0

Now (2.22) follows from (2.23)–(2.26). We are ready now to prove (2.1). By Duhamel’s formula we have Φ(t; h) + F (t)ϕ(h2 G0 ) =

4

Qj (t; h),

j=1

where

Q1 (t; h) = ϕ1 (h2 G)eitG0 ϕ(h2 G) − ϕ(h2 G0 ) + ϕ1 (h2 G) − ϕ1 (h2 G0 ) eitG0 − F (t) ϕ(h2 G0 ), t/2 ϕ1 (h2 G)ei(t−τ )G0 V Φ(τ ; h)dτ, Q2 (t; h) = −i 0

(2.27)

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Q3 (t; h) = −i

t

t/2

1187

ϕ1 (h2 G)ei(t−τ )G0 V Φ(τ ; h)dτ,

Q4 (t; h) = (1 − ϕ1 )(h2 G0 )F (t)ϕ(h2 G0 ). By (1.2), (2.3), (2.4), (2.6) and (2.7) with s = 0, we get Q1 (t; h)L1 →L∞ ≤ Ch2 t−3/2 .

(2.28)

By (1.1), (1.2) and (2.22) used with s = 1/2 + /2 and s = 1/2 − /2, we obtain Q2 (t; h)L1 →L∞ t/2 i(t−τ )G0 −3/2− ≤ C1 x e 0

≤ C2 h1/2−3/2 t−3/2

L2 →L

t/2

0

−1−3/2 Φ(τ ; h) x ∞

L1 →L2

dτ

τ −1+/2 τ − dτ ≤ Ch1/2−3/2 t−3/2 . (2.29)

By (2.15) used with s = 1/2 + /2 and s = 1/2 − /2 and (2.22) used with s = 0, we obtain Q3 (t; h)L1 →L∞ t/2 −3/2− ϕ1 (h2 G)eiτ G0 x−1− 2 ∞ ≤ C1 Φ(t − τ ; h) x L →L 0

≤ C2 h

1/2−2 −3/2

t/2

t

0

L1 →L2

dτ

τ −1+/2 τ − dτ ≤ Ch1/2−2 t−3/2 . (2.30)

To estimate the norm of the operator Q4 observe that its kernel, Kh (x, y; t), is of the form U h−1 |x − x |, h−1 |x − y|; h−2 t V (x )dx , Kh (x, y; t) = ch−2 R3

where c is some constant and √ ∞ ∞ √ itλ itµ sin(σ1 λ) sin(σ2 µ) ψ(λ, µ) e − e dλdµ, U (σ1 , σ2 ; t) = σ1 σ2 0 0 where ψ(λ, µ) = ϕ(µ)

ϕ1 (λ) − ϕ1 (µ) , λ−µ

λ = µ,

extends to a smooth function on R2 , compactly supported in µ and satisfying α α ∂ 1 ∂ 2 ψ(λ, µ) ≤ Cα λ−1−α1 , ∀(λ, µ), (2.31) µ λ for every multi-index α = (α1 , α2 ). Write the function U as U = U1 − U2 , where √ ∞ sin(σ1 λ) itλ U1 (σ1 , σ2 ; t) = e b1 (λ, σ2 ) dλ, σ1 0

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G. Vodev

U2 (σ1 , σ2 ; t) =

∞

0

eitµ b2 (µ, σ1 )


√ sin(σ2 µ) dµ, σ2

√ sin(σ2 µ) b1 (λ, σ2 ) = ψ(λ, µ) dµ, σ2 0 √ ∞ sin(σ1 λ) b2 (µ, σ1 ) = ψ(λ, µ) dλ. σ1 0

where

∞

In view of (2.31), integrating by parts easily yields |∂λα b1 (λ, σ2 )| ≤ Cα λ−1−α σ2−1 (1 + σ2 )−1 ,

∀α.

(2.32)

Let us see that a similar bound holds for the function b2 , namely α ∂µ b2 (µ, σ1 ) ≤ Cα σ −1 (1 + σ1 )−1 , ∀α. 1

(2.33)

Integrating by parts we obtain ∂µα b2 (µ, σ1 ) = 2σ1−1 = 2σ1−2

∞

0

0

∞

∂µα ψ(λ2 , µ) sin(σ1 λ)λdλ

α ∂µ ψ(λ2 , µ) + 2λ2 ∂λ ∂µα ψ(λ2 , µ) cos(σ1 λ)dλ,

where the last integral is absolutely convergent in view of (2.31), which implies (2.33) for σ1 ≥ 1. To prove (2.33) for 0 < σ1 ≤ 1, observe that there exists a constant λ0 1 so that for λ ≥ λ0 the function ∂µα ψ can be written in the form ∂µα ψ(λ, µ) = λ−1 ∂µα ϕ(µ) + ψα (λ, µ), with a smooth function ψα = O(λ−2 ). Therefore, we have λ0 √ −1 α ∂µ b2 (µ, σ1 ) = σ1 ∂µα ψ(λ, µ) sin(σ1 λ)dλ 0

+σ1−1

∞

λ0

√ ψα (λ, µ) sin(σ1 λ)dλ + σ1−1 ∂µα ϕ(µ)

∞

λ0

√ sin(σ1 λ) dλ. λ

The first two integrals are absolutely convergent, while the other one is equal to 2

∞

√ λ0

sin(σ1 λ) dλ = 2 λ

0

∞

sin(σ1 λ) dλ − 2 λ

0

√

λ0

sin(σ1 λ) dλ = Const +O(σ1 ). λ

Thus we conclude that (2.33) holds for 0 < σ1 ≤ 1, too. For j = 1, 2, set x 2 2 kj (x; σj , t) = 2 eitλ sin(σj λ)λdλ = (it)−1 eitx sin(σj x) − σj (it)−1 kj0 (x; σj , t), 0

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1189

where kj0 (x; σj , t)

2

x

=

e 0

itλ2

e−iσj /4t cos(σj λ)dλ = 2t1/2

× k(t1/2 x + σj t−1/2 ) + k(t1/2 x − σj t−1/2 ) − k(σj t−1/2 ) − k(−σj t−1/2 ) ,

where the function k(a) =

a

2

eiλ dλ

0

is known to satisfy the bound |k(a)| ≤ C,

∀a ∈ R,

with a constant C > 0 independent of a. Hence 0 kj (x; σj , t) ≤ Ct−1/2 ,

(2.34)

with a constant C > 0 independent of x, σj and t. Integrating by parts, we can write the function U1 in the form ∞ b1 (λ2 ; σ2 )dk1 (λ; σ1 , t) U1 (σ1 , σ2 ; t) = σ1−1 0 ∞ ∂b1 2 (λ ; σ2 )k1 (λ; σ1 , t)dλ λ = −2σ1−1 ∂λ 0 ∞ 2 ∂b1 2 (λ ; σ2 )eitλ sin(σ1 λ)dλ λ = −2σ1−1 (it)−1 ∂λ ∞ 0 ∂b1 2 −1 (λ ; σ2 )k10 (λ; σ1 , t)dλ +2(it) λ ∂λ 0 ∞ 2 ∂ 2 b1 −1 −2 = 2σ1 (it) λ 2 (λ2 ; σ2 )eitλ sin(σ1 λ)dλ ∂λ 0 ∞ ∂ 2 b1 −2 λ 2 (λ2 ; σ2 )k10 (λ; σ1 , t)dλ −2(it) ∂λ 0 ∞ ∂b1 2 (λ ; σ2 )k10 (λ; σ1 , t)dλ. λ +2(it)−1 ∂λ 0 By (2.32) and (2.34), since t ≥ 1, we conclude |U1 (σ1 , σ2 ; t)| ≤ Ct−3/2 σ1−2 + σ2−2 .

(2.35)

Similarly, using (2.33) instead of (2.32) and the fact that the function b2 (µ, σ1 ) is compactly supported in µ, we get |U2 (σ1 , σ2 ; t)| ≤ Ct−3/2 σ1−2 + σ2−2 . (2.36)

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G. Vodev


By (2.35) and (2.36), we have |Kh (x, y; t)| ≤ Cht−3/2 B(x, y),

where B(x, y) =

R3

|V (x )|dx + |x − x |2

R3

|V (x )|dx ≤ Const . |y − x |2

Clearly, the above estimates imply Q4 (t; h)L1 →L∞ ≤ Cht−3/2 .

(2.37)

Now (2.1) follows from (2.27)–(2.30) and (2.37).

3 Proof of Theorem 1.2 Without loss of generality we may suppose that t > 0. Since (1.5) is trivial for t/h ≤ 1, we may suppose that t/h ≥ 1. We will first show that (1.5) with s = 3/2 implies (1.5) for all 0 ≤ s ≤ 3/2. The function g(z) = (t/h)z x−z− ϕ(h2 G)eitG x−z− is analytic for z ∈ C, Re z ≥ 0, with values in L(L2 ) and satisfies the trivial bounds g(z)L2→L2 ≤ C, g(z)L2 →L2 ≤ C(t/h)Re z ,

Re z = 0, 0 ≤ Re z ≤ 3/2,

(3.1) (3.2)

with a constant C > 0 independent of z, h and t. Moreover, supposing that (1.5) holds with s = 3/2 means that (3.1) holds for Re z = 3/2. Thus, in view of (3.1) and (3.2), by Phragmèn-Lindel¨ of principle we conclude g(z)L2→L2 ≤ C,

0 ≤ Re z ≤ 3/2,

which clearly implies (1.5) for 0 ≤ s ≤ 3/2. We will now show that (1.5) with s = 3/2 follows from the estimate −s− ϕ(h2 G)eitG x−3/2− ≤ Chs t−s , x 2 2 L →L

(3.3)

(3.4)

for 0 ≤ s < 1. Denote by r = |x| the radial variable and set −∆ = −r∆r−1 = −∂r2 + r−2 ∆S 2 , where ∆S 2 denotes the (positive) Laplace-Beltrami operator on S 2 = {x ∈ R3 : |x| = 1}. We have the identity −2∆ + [r∂r , ∆ ] = 0.

(3.5)

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1191

Denote by G the self-adjoint realization of the operator −∆ + V on the Hilbert space H = L2 (R+ × S 2 , drdw). Clearly, the operator G is unitary equivalent to G, so we have −s −s1 2 itG −s2 x 1 ϕ(h2 G)eitG x−s2 2 2 = ϕ(h G )e r , (3.6) r L →L H→H

for all s1 , s2 ≥ 0. We will show that −s1 r b(r)∂r ϕ(h2 G )eitG r−s2 H→H −1 −s1 ≤ Ch r ϕ(h2 G )eitG r−s2 H→H + Ch−1 r−s1 ϕ(h 2 G )eitG r−s2

H→H

, (3.7)

where b(r) = r−1 r, ϕ(σ) = σϕ(σ). Clearly, (3.7) follows from the estimate −s −s r b(r)∂r u ≤ Ch r G u + Ch−1 r−s u , ∀u ∈ D(G ). (3.8) H H H To prove (3.8), set v = r−s G u and observe that the function r−s b(r)u satisfies the equation 2 −∂r + r−2 ∆S 2 + V r−s b(r)u = v + [−∂r2 , r−s b(r)]u. Integrating by parts leads to the estimates −s ∂r r b(r)u 2 H 2

≤ C r−s b(r)uH + v + [−∂r2 , r−s b(r)]u, r−s b(r)u H 2 2 ≤ C r−s b(r)uH + h2 v2H + h−2 r−s b(r)uH 2 2 +γ 2 b(r)[−∂r2 , r−s b(r)]uH + γ −2 r−s uH 2 2 ≤ O h−2 r−s uH + O h2 v2H + O γ 2 r−s b(r)∂r uH , ∀ 0 < γ 1 independent of h. Hence, −s r b(r)∂r u ≤ ∂r r−s b(r)u + C r−s u H H H −1 −s r u H + O (h) vH + O (γ) r−s b(r)∂r uH , ≤O h which implies (3.8) by taking γ > 0 small enough. By Duhamel’s formula and (3.5), we obtain the identity

t r∂r , eitG = ei(t−τ )G r∂r , G eiτ G dτ 0 t ei(t−τ )G V eiτ G dτ = 2tG eitG − 2 0 t t ei(t−τ )G r∂r V eiτ G dτ − ei(t−τ )G V r∂r eiτ G dτ. + 0

0

1192

G. Vodev


Let ϕ1 ∈ C0∞ ((0, +∞)) be such that ϕ1 ϕ ≡ ϕ, and denote ϕ2 (σ) = σ −1 ϕ(σ). By the above identity we obtain 2th−2 ϕ(h2 G )eitG

= + + −

ϕ1 (h2 G )r∂r eitG ϕ2 (h2 G ) − ϕ1 (h2 G )eitG r∂r ϕ2 (h2 G ) t ϕ1 (h2 G )ei(t−τ )G V eiτ G ϕ2 (h2 G )dτ 2 0 t ϕ1 (h2 G )ei(t−τ )G r∂r V eiτ G ϕ2 (h2 G )dτ 0 t ϕ1 (h2 G )ei(t−τ )G V r∂r eiτ G ϕ2 (h2 G )dτ. (3.9) 0

Using (2.9), (3.6) and (3.9), we get th−2 r−1−s ϕ(h2 G )eitG r−1−s H→H −s 2 itG ≤ C r b(r)∂r ϕ2 (h G )e r−1−s H→H −1−s itG +C r e ϕ1 (h2 G )∂r b(r)r−s H→H t −1−s + C ϕ1 (h2 G )ei(t−τ )G r−δ1 r H→H 0 −δ2 iτ G ϕ2 (h2 G )r−1−s dτ r e H→H t −1−s i(t−τ )G + C e ϕ1 (h2 G )b(r)∂r r−δ1 r H→H 0 −δ2 iτ G 2 ϕ2 (h G )r−1−s dτ r e H→H t −1−s + C ϕ1 (h2 G )ei(t−τ )G r−δ1 r H→H 0 −δ2 dτ, r b(r)∂r ϕ2 (h2 G )eiτ G r−1−s H→H

(3.10) for 0 ≤ s ≤ 1, δ1 , δ2 > 0 such that δ1 + δ2 = δ − 1 > 3/2. Applying (3.10) with s = 1/2 + , δ1 = δ2 and assuming that (3.4) holds, we obtain in view of (3.6) and (3.7), th−1 r−3/2− ϕ(h2 G )eitG r−3/2− H→H t ≤ C(t/h)−1/2 + Ch τ −3/4 (t − τ )−3/4 dτ ≤ C (t/h)−1/2 . (3.11) 0

Vol. 6, 2005


1193

To prove (3.4) we will use the fact that the free operator satisfies the following well-known estimate −s x ϕ(h2 G0 )eitG0 x−s 2 2 ≤ Chs t−s , (3.12) L →L for all s ≥ 0. We also need the following Proposition 3.1 For 0 < 1, 0 ≤ s ≤ 1, 0 < h ≤ 1, t > 0, we have −3/2− ϕ(h2 G)eitG x−3/2− ≤ Ct−s . x 2 2 L →L

(3.13)

By Duhamel’s formula we can write ϕ(h2 G)eitG = ϕ1 (h2 G0 )eitG0 ϕ(h2 G) + ϕ1 (h2 G) − ϕ1 (h2 G0 ) eitG ϕ(h2 G) t −i ϕ1 (h2 G0 )ei(t−τ )G0 V eiτ G ϕ(h2 G)dτ. 0

Using this identity together with (2.5), (2.9), (3.12) and (3.13) with s = 1 − /2, we arrive at −s− ϕ(h2 G)eitG x−3/2− 2 2 x L →L −s− 2 itG0 ≤ C x ϕ1 (h G0 )e x−3/2− 2 2 L →L 2 −3/2− +Ch x ϕ(h2 G)eitG x−3/2− L2 →L2 t −s− + C ϕ1 (h2 G0 )ei(t−τ )G0 x−1− 2 2 x L →L 0 −3/2− 2 iτ G ϕ(h G)e x−3/2− dτ x L2 →L2 t ≤ C(t/h)−s + C(t/h)−1+/2 + Chs (t − τ )−s−/2 τ −1+/2 dτ ≤ C (t/h)−s , 0

if 0 ≤ s < 1 and > 0 is taken so that s + /2 < 1, which is the desired result.

4 Proof of Proposition 3.1 Clearly, (3.13) is trivial for s = 0, so it suffices to prove it for s = 1. We will make use of the formula ∞ 2 1 −3/2− 2 itG −3/2− x ϕ(h G)e x = eizt/h ϕ(z)T (z; h)dz, (4.1) 2πi 0 where

T (z; h) = T + (z; h) − T − (z; h),

1194

G. Vodev


T ± (z; h) = x−3/2− (h2 G − z ± i0)−1 x−3/2− = lim x−3/2− (h2 G − z ± iε)−1 x−3/2− . ε→0

Note that the limit exists as an operator in L(L2 ) in view of the limiting absorption principle. Integrating by parts leads to the identity

∞

2

eizt/h ϕ(z)T ± (z; h)dz

0 2 −1

= (it/h )

∞

e 0

izt/h2

dT ± ± (z; h) dz. ϕ (z)T (z; h) + ϕ(z) dz

Therefore, (3.13) with s = 1 follows from the following Lemma 4.1 Let I ⊂ (0, +∞) be a compact interval. Then, the operator-valued functions T ± (z; h) satisfy the estimates (for z ∈ I) T ± (z; h)L2 →L2 ≤ Ch−1 , ± dT −2 dz (z; h) 2 2 ≤ Ch , L →L

(4.2) (4.3)

with a constant C > 0 independent of h and z. Proof. Clearly, (4.2) follows from the estimate −s x 1 (G − λ ± iε)−1 x−s2 2 2 ≤ Cλ−1/2 , L →L

λ ≥ λ0 ,

(4.4)

for all s1 , s2 > 1/2, 0 ≤ ε ≤ 1, ∀λ0 > 0, with a constant C = C(λ0 ) > 0 independent of λ and ε. It is well known that (4.4) holds with G replaced by G0 . To prove (4.4) for G we will use the identity x−s1 (G − λ ± iε)−1 x−s2 (1 + K(λ ∓ iε)) = x−s1 (G0 − λ ± iε)−1 x−s2 , (4.5) where the operator K(λ ∓ iε) = xs2 V (G0 − λ ± iε)−1 x−s2 takes values in the compact operators in L(L2 ). Clearly, it suffices to prove (4.4) for 0 < s2 − 1/2 1. For such s2 there exists λ1 > λ0 so that K(λ ∓ iε)L2 →L2 ≤ 1/2,

λ ≥ λ1 ,

(4.6)

and hence (4.4) follows for λ ≥ λ1 from (4.5) and the fact that (4.4) holds for G0 . Furthermore, since G has no strictly positive resonances, the operator 1+K(λ∓iε) is invertible for λ0 ≤ λ ≤ λ1 , 0 ≤ ε ≤ 1, and satisfies (1 + K(λ ∓ iε))−1 2 2 ≤ C, (4.7) L →L

Vol. 6, 2005


1195

with a constant C > 0 independent of ε. Therefore, (4.4) for λ0 ≤ λ ≤ λ1 follows again from (4.5) combined with (4.7). Clearly, (4.3) follows from the following estimate −3/2− 2 (h G − z ± iε)−2 x−3/2− 2 2 ≤ Ch−2 , z ∈ I, 0 < ε 1. (4.8) x L →L

It suffices to prove (4.8) with G and L2 replaced respectively by G and H introduced in the previous section. Using the identity (3.5) we can write 2(z ∓ iε)r−3/2− (h2 G − z ± iε)−2 r−3/2− =

2r−3/2− (h2 G − z ± iε)−1 r−3/2− −r−3/2− (h2 G − z ± iε)−1 r∂r r−3/2− +r−3/2− r∂r (h2 G − z ± iε)−1 r−3/2− −2h2 r−3/2− (h2 G − z ± iε)−1 V (h2 G − z ± iε)−1 r−3/2− +h2 r−3/2− (h2 G − z ± iε)−1 r∂r V (h2 G − z ± iε)−1 r−3/2− −h2 r−3/2− (h2 G − z ± iε)−1 V r∂r (h2 G − z ± iε)−1 r−3/2− .

Note that in view of (3.8) we have −s r 1 b(r)∂r (h2 G − z ± iε)−1 f ≤ Ch−1 r−s1 f H H + Ch−1 r−s1 (h2 G − z ± iε)−1 f H ,

∀f ∈ H, (4.9)

for all s1 ≥ 0. Taking into account (3.6) and using (4.4) with λ = z/h2 and (4.9), we obtain from the above identity −3/2− 2 (h G − z ± iε)−2 r−3/2− ≤ Ch−2 , r H→H

which clearly implies (4.8).

References [1] V. Georgiev and N. Visciglia, Decay estimates for the wave equation with potential, Commun. Partial Differential Equations 28, 1325–1369 (2003). [2] M. Goldberg, Dispersive bounds for the three-dimensional Schr¨ odinger equation with almost critical potentials, GAFA, to appear. [3] M. Goldberg and W. Schlag, Dispersive estimates for Schr¨ odinger operators in dimensions one and three, Commun. Math. Phys. 251, 157–178 (2004). odin[4] A. Jensen and S. Nakamura, Lp -mapping properties of functions of Schr¨ ger operators and their applications to scattering theory, J. Math. Soc. Japan 47, 253–273 (1995).

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G. Vodev


[5] J.-L. Journé, A. Soffer and C. Sogge, Decay estimates for Schrödinger operators, Commun. Pure Appl. Math. 44, 573–604 (1991). [6] I. Rodnianski and W. Schlag, Time decay for solutions of Schr¨ odinger equations with rough and time-dependent potentials, Invent. Math. 155, 451–513 (2004). [7] G. Vodev, Local energy decay of solutions to the wave equation for short-range potentials, Asympt. Anal. 37, 175–187 (2004). odinger operators, [8] K. Yajima, The W k,p -continuity of wave operators for Schr¨ J. Math. Soc. Japan 47, 551–581 (1995).

Georgi Vodev Université de Nantes Département de Mathématiques UMR 6629 du CNRS 2, rue de la Houssinière, BP 92208 F-44332 Nantes Cedex 03 France email: [email protected] Communicated by Bernard Helffer submitted 27/11/04, accepted 29/04/05


Ann. Henri Poincaré 6 (2005) 1197 – 1199 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/061197-3 DOI 10.1007/s00023-005-0238-4


Erratum to “Determination of Non–Adiabatic Scattering Wave Functions in a Born–Oppenheimer Model” G.A. Hagedorn∗ and A. Joye

We correct some typographical errors and mistakes in the published paper [1]. We freely use the paper’s notation and equation numbering. In Section 4, on the scattering properties of exact solutions to the molecular Schr¨ odinger equation, we used too rough a definition of asymptotic states. We need to consider ψ σ (x, t, , ±) =

φj (x)

j=1,··· ,m

∆

2 Q(E, ) e−itE/ cσj (±∞, E, ) 2kj (±∞, E) σ

σ

2

× e−i(xkj (±∞,E)+ωj (±∞,E))/ dE, instead of ψ(x, t, , ±) = should be replaced by

σ=±

ψ σ (x, t, , ±). With this definition, Proposition 4.1

Proposition 4.1 Assume H1, H2, H3 and C0. Then, for any 0 < β < 1/2, we have the following L2 (IR)–norm estimate as t → ±∞, ψ(x, t, ) − (ψ − (x, t, , ±) + ψ + (x, t, , ∓)) = O (1/|t|β ). In eqn. (4.4), “ψ(x, t, , −) =” should be replaced by “ψ − (x, t, , −) ,” and the qualification “for negative x’s” should be added. Similarly, in the next sentence, the stipulation “for positive x’s” should be added. The proof of Proposition 4.1 should be amended as follows: Equation (7.14) should begin with ψ(x, t, ) − ψ σ (x, t, , ±) =

φ(x)×

j=1,2

instead of “ψ(x, t, ) − ψ(x, t, , ±) = j=1,2 σ=± φ(x)×,” and every L2 (IR) should be replaced by L2 (IR± ), where IR± = {x ∈ IR : ±x > 0}. ∗ Partially supported by National Science Foundation Grants DMS–0071692 and DMS– 0303586.

1198



At the end of the proof, one should add the sentence: If x ∈ IR∓ and sign(t) = ∓σ, one integrates by parts as in (7.12), and one uses x σ t + kj (y, E) dy ≥ c (|t| + |x|) ≥ c |t|β |x|1−β , 0

for all 0 < β < 1, to bound the corresponding L2 (IR∓ ) norms by constants times |t|−β , with 0 < β < 1/2. A

Consequently, the statement of Theorem 5.1 should be amended as follows: The first equation should read lim

t→−∞

ψ(x, t, ) − ψj− (x, t, , −) L2 (R− ) = 0,

instead of “limt→−∞ ψ(x, t, ) − ψj− (x, t, , −) = 0,” and it should be followed by the qualification “for negative x’s.” B The sentence before equation (5.8) should begin: Then, there exist δ0 > 0, p > 0 arbitrarily close to 5/2, and a function 0 : (0, δ0 ) → IR+ , such that for all 0 < β < 1/2, δ < δ0 , and < 0 (δ), the following asymptotics hold as t → ∞: C Finally, each occurrence of O (1/t) should be replaced by O (1/tβ ). As indicated in B, p is arbitrary close to 5/2 instead of 3, as previously erroneously stated. This comes from a missing square root in the computation of the L2 (IR) norm of the error terms in the last paragraph of page 987. They should ∗ ∗ ∗ read O(e−α(E ) 1+3s/2 ) and O(e−α(E ) 7s/2−1 ) instead of O(e−α(E ) 1+2s ) and −α(E ∗ ) 4s−1 ). Keeping track of the consequences of this correction, one sees O(e that p is arbitrarily close to 5/2. This change should also be made in Lemma 5.1 and Lemma 5.2. Finally, due to a miscalculation, formula (5.10) in Lemma 5.1 must be simplified to √ ∗ 2 2πk ∗ P (E ∗ , ) e−α(E )/ ∗ ∗ )−xk∗ ) exp −i (tE +κ(E 2 × d2 d2 1/2 ( dk2 α(E(k))|k∗ + i(t + dk 2 κ(E(k))|k∗ )) ∗ 2 (x − k ∗ (t + κ (E ∗ )))2 × exp − d2

+ O(e−α(E )/ p ). d2 2 ∗ ∗ 2 dk2 α(E(k))|k + i(t + dk2 κ(E(k))|k

T (, x, t) =

This leads to a simplification of Lemma 5.2: Only the second statement should be kept, and the error term O(3/2 /|t|) should be removed. This mistake came from the incorrect computation integral at the ∞ of the Gaussian ∗ 2 ∗ 2 end of the proof of Lemma 5.1. It should read, −∞ e−(M(k−k ) /2+iN (k−k ))/ dk

2 − 2N2 M (for Re M > 0). = 2π M e

Vol. 6, 2005

Erratum to “Non-Adiabatic Wave Functions in a B.-O. Model”

1199

References [1] G.A. Hagedorn and A. Joye, Determination of Non–Adiabatic Scattering Wave Functions in a Born–Oppenheimer Model, Ann. Henri Poincaré 6 (2005), 5, 937–990.

George A. Hagedorn Department of Mathematics and Center for Statistical Mechanics and Mathematical Physics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0123 USA Alain Joye Institut Fourier Unité Mixte de Recherche CNRS-UJF 5582 Université de Grenoble I BP 74 F–38402 Saint Martin d’Hères Cedex France Communicated by Yosi Avron received 14/10/04

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