Quantum decay rates in chaotic scattering

Quantum decay rates in chaotic scattering

Stéphane Nonnenmacher  and  Maciej Zworski Service de Physique Théorique, CEA/DSM/PhT, Unité de recherche associé CNRS, CEA/Saclay,
91191 Gif-sur-Yvette, France
nonnen@spht.saclay.cea.fr Mathematics Department, University of California
Evans Hall, Berkeley, CA 94720, USA
zworski@math.berkeley.edu

1. Statement of Results

In this article we prove that for a large class of operators, including Schrödinger operators,

(1.1)

with hyperbolic classical flows, the smallness of dimension of the trapped set implies that there is a gap between the resonances and the real axis. In other words, the quantum decay rates are bounded from below if the classical repeller is sufficiently filamentary. The higher dimensional statement is given in terms of the topological pressure and is presented in Theorem 3. Under the same assumptions, we also prove a useful resolvent estimate:

(1.2)

for any compactly supported bounded function - see Theorem 5, and a remark following it for an example of applications.

We refer to §3.2 for the general assumptions on , keeping in mind that they apply to of the form (1.1). The resonances of are defined as poles of the meromorphic continuation of the resolvent:

through the continuous spectrum . More precisely,

is a meromorphic family of operators (here and denote functions which are compactly supported and in , and functions which are locally in ). The poles are called resonances and their set is denoted by — see [4, 46] for introduction and references. Resonances are counted according to their multiplicities (which is generically one [22]).

In the case of (1.1) the classical flow is given by Newton’s equations:

(1.3)

This flow preserves the classical Hamiltonian

and the energy layers of are denoted as follows:

(1.4)

The incoming and outgoing sets at energy are defined as

(1.5)

The trapped set at energy ,

(1.6)

is a compact, locally maximal invariant set, contained inside , for some . That is clear for (1.1) but also follows from the general assumptions of §3.2.

We assume that the flow is hyperbolic on .

The definition of hyperbolicity is recalled in (3.10) – see §3.2 below. We recall that it is a structurally stable property, so that the flow is then also hyperbolic on , for near . Classes of potentials satisfying this assumption at a range of non-zero energies are given in [27], [37, Appendix c], [44], see also Fig.1.

Figure 1. A three bump potential exhibiting a hyperbolic trapped set for a range of energies. When the curve is made of three approximate circles of radii and centers at equilateral distance , the partial dimension in (1.8) is approximately when .

The dimension of the trapped set appears in the fractal upper bounds on the number of resonances. We recall the following result [41] (see [37] for the first result of this type):

Theorem 1.

Let be given by (1.1) and suppose that the flow is hyperbolic on . Then in the semiclassical limit

(1.7)

where

(1.8)

We note that using [33, Theorem 4.1], and in dimension , we strengthened the formulation of the result in [41] by replacing upper Minkowski (or box) dimension by the Hausdorff dimension. We refer to [41, Theorem 3] for the slightly more cumbersome general case.

In this article we address a different question which has been present in the physics literature at least since the seminal paper by Gaspard and Rice [15]. In the same setting of scattering by several convex obstacles, it has also been considered around the same time by Ikawa [19] (see also the careful analysis by Burq [6] and a recent paper by Petkov and Stoyanov [34]).

Question: What properties of the flow , or of alone, imply the existence of a gap such that, for sufficiently small,

In other words, what dynamical conditions guarantee a lower bound on the quantum decay rate?

Numerical investigations in different settings of semiclassical three bump potentials [23, 24], three disk scattering [15, 25, 45], Cantor-like Julia sets for , [42], and quantum maps [31, 35], all indicate that a trapped set of low dimension (a “filamentary” fractal set) guarantees the existence of a resonance gap .

Figure 2. A sample of numerical results of [23]: the plot shows resonances for the potential of Fig. 1 (). For the energies inside the box, the fractal dimension is approximately (see [23, Table 2]), and resonances are separated from the real axis in agreement with Theorem 2.

Some of these works also confirm the fractal Weyl law of Theorem 1. which, unlike Theorem 2 below, was first conjectured in the mathematical works on counting resonances.

Here we provide the following

Theorem 2.

Suppose that the assumptions of Theorem 1 hold and that the dimension defined in (1.8) satisfies

(1.9)

Then there exists , and such that

(1.10)

The statement of the theorem can be made more general and more precise using a more sophisticated dynamical object, namely the topological pressure of the flow on with respect to the unstable Jacobian:

We will give two equivalent definitions of the pressure below, the simplest to formulate (but not to use), given in (3.17).

The main result of this paper is

Theorem 3.

Suppose that satisfies the general assumptions of §3.2 (in particular it can be of the form (1.1) with ), and that the flow is hyperbolic on the trapped set . Suppose that the topological pressure satisfies

Then there exists such that for any satisfying

(1.11)

there exits such that

(1.12)

For , is equivalent to , which shows that Theorem 2 follows from Theorem 3. The connection between and a resonance gap also holds in dimension ; however, for there is generally no simple link between the sign of and the value of (except when the flow is “conformal” in the unstable, respectively stable directions [33]).

The optimality of Theorem 3 is not clear. Except in some very special cases (for instance when consists of one hyperbolic orbit) we do not expect the estimate on the width of the resonance free region in terms of the pressure to be optimal. In fact, in the analogous case of scattering on convex co-compact hyperbolic surfaces the results of Naud (see [29] and references given there) show that the resonance free strip is wider at high energies than the strip predicted by the pressure. That relies on delicate zeta function analysis following the work of Dolgopyat: at zero energy there exists a Patterson-Sullivan resonance with the imaginary part (width) given by the pressure, but all other resonances have more negative imaginary parts. A similar phenomenon occurs in the case of Euclidean obstacle scattering as has recently been shown by Petkov and Stoyanov [34].

Figure 3. The top figure shows the phase portrait for the Hamiltonian , with highlighted. The middle plot shows the resonant state corresponding to the resonance closest to the real axis at , and the bottom plot shows the squared modulus of its FBI tranform. The resonance states were computed by D. Bindel (http://cims.nyu.edu/dbindel/resonant1d) and the FBI transform was provided by L. Demanet. The result of Theorem 4 is visible in the mass of the FBI transform concentrated on , with the exponential growth in the outgoing direction.

The proof of Theorem 3 is based on the ideas developed in the recent work of Anantharaman and the first author [2, 3] on semiclassical defect measures for eigenfuctions of the Laplacian on manifolds with Anosov geodesic flows. Although we do not use semiclassical defect measures in the proof of Theorem 3, the following result provides a connection:

Theorem 4.

Let satisfy the general assumptions of §3.2 (no hyperbolicity assumption here). Consider a sequence of values and a corresponding sequence of resonant states (see (3.19) in §3.2 below) satisfying

(1.13)

where is the trapped set at energy (1.6) and . Suppose that a semiclassical defect measure on is associated with the sequence :

(1.14)

Then

(1.15)

and there exists such that

(1.16)

See Fig. 3 for a numerical result illustrating the theorem. A similar analysis of the phase space distribution for the resonant eigenstates of quantized open chaotic maps (discrete-time models for scattering Hamiltonian flows) has been recently performed in [21, 30]. Connecting this theorem with Theorems 2 and 3, we see that the semiclassical defect measures associated with sequences of resonant states have decay rates bounded from below by , once the dimension of the trapped set is small enough (), or more generally, the pressure at is negative.

Our last result is the precise version of the resolvent estimate (1.2):

Theorem 5.

Suppose that satisfies the general assumptions of §3.2 (in particular it can be of the form (1.1) with ), and that the flow is hyperbolic on the trapped set . If the pressure then for any we have

(1.17)

Notice that the upper bound is the same as in the one obtained in the case of one hyperbolic orbit by Christianson [8]. To see how results of this type imply dynamical estimates see [7, 8]. In the context of Theorem 5, the applications are presented in [9]. Referring to that paper for details and pointers to the literatures we present one application.

Let be the Laplace-Bertrami operator satisfying the assumptions below, for instance on a manifold Euclidean outside of a compact set with the standard metric there. The Schrödinger propagator, , is unitary on any Sobolev space so regularity is not improved in propagation. Remarkably, when , that is, when the metric is nontrapping, the regularity improves when we integrate in time and cut-off in space:

and this much exploited effect is known as local smoothing. As was shown by Doi [12] any trapping (for instance a presence of closed geodesics or more generally ) will destroy local smpoothing. Theorem 5 implies that under the assumptions that the geodesic flow is hyperbolic on the trapped set , and that the pressure is negative at (or, when , that the dimension of is less than ) local smoothing holds with replaced by for any .

Notation. In the paper denotes a constant the value of which may changes from line to line. The constants which matter and have to be balanced against each other will always have a subscript and alike. The notation means that , and the notation means that .

2. Outline of the proof

It this section we present the main ideas whith the precise definitions and references to previous works given in the main body of the paper. The operator to keep in mind is , where , , and the metric is Euclidean outside a compact set. The corresponding classical Hamiltonian is given by . Weaker assumptions, which in particular do not force the compact support of the perturbation, are described in §3.2.

First we outline the proof of Theorem 3 in the simplified case in which resonances are replaced by the eigenvalues of an operator modified by a complex absorbing potential:

where , satisfies the following conditions:

for sufficiently large. In particular, is large enough so that , where is the trapped set given by (1.6). The non-self-adjoint operator has a discrete spectrum in and the analogue of Theorem 3 reads:

Theorem 3.Under the assumptions of Theorem 3, for

(2.1)

there exits such that for ,

(2.2)

This means that the spectrum of near is separated from the real axis by , where is given in terms of the pressure of the square root of the unstable Jacobian, .

This spectral gap is equivalent to the fact that the decay rate of any eigenstate is bounded from below:

This is the physical meaning of the gap between the spectrum (or resonances) and the real axis — a lower bound for the quantum decay rate — and the departing point for the proof. To show (2.2) we will show that for functions, , which are microlocally concentrated near the energy layer (that is, for a supported near ) we have

(2.3)

for any . Taking and applying the estimate to an eigenstate gives (2.2).

To prove (2.3) we decompose the propagator using an open cover of the neighbourhood of the energy surface. That cover is adapted to the definition of the pressure (see §§5.2,5.3) and it leads to a microlocal partition of a neighbourhood of the energy surface:

The definition of the pressure in §5.2 also involves a time , independent of , but depending on the classical cover. Taking

(2.4)

the propagator at time acting of functions microlocalized inside can be written as

(2.5)

Most terms in the sum appearing on the right hand side of (2.5) are negligible. The sequences which are classically forbidden, that is, for which the corresponding sequences of neighbourhoods are not successively connected by classical propagation in time , lead to negligible terms. So do the sequences for which the propagation crosses the region where : the operator is negligible there, due to damping (or “absorption”) by .

As a result, the only terms relevant in the sum on the right hand side of (2.5) come from where indexes the element of the partition intersecting the trapped set , and are the classically allowed sequences — see (6.29). We then need the crucial hyperbolic dispersion estimate proved in §7 after much preliminary work in §§4.3 and 5.1: for , arbitrary, we have for any sequence :

(2.6)

The expression in parenthesis is the coarse-grained unstable Jacobian defined in (5.22), and is a parameter depending on the cover , which can be taken arbitrarily small — see (5.24). From the definition of the pressure in §5.2, summing (2.6) over leads to (2.3), with .

In §9 we show how to use (2.3) to obtain a resolvent estimate for : at an energy for which the flow is hyperbolic on and , we have

(2.7)

To prove Theorem 3, that is the gap between resonances and the real axis, we use the complex scaled operator  : its eigenvalues near the real axis are resonances of . If is a decaying real analytic potential extending to a conic neighbourhood of (for instance a sum of three Gaussian bumps showed in Fig. 1), then we can take , though in this paper we will always use exterior complex scaling reviewed in §3.4, with , where is chosen depending on in (2.4).

To use the same strategy of estimating we need to further modify the operator by conjugation with microlocal exponential weights. That procedure is described in §6. The methods developed there are also used in the proof of Theorem 4 and in showing how the estimate (2.7) implies Theorem 5.

Since we concentrate on the more complicated, and scientifically relevant, case of resonances, the additional needed facts about the study of and its propagator are presented in the Appendix.

3. Preliminaries and Assumptions

In this section we recall basic concepts of semiclassical analysis, state the general assumptions on operators to which the theorems above apply, define hyperbolicity and topological pressure. We also define resonances using complex scaling which is the standard tool in the study of their distribution. Finally, we will review some results about semiclassical Lagrangian states and Fourier integral operators.

3.1. Semiclassical analysis

Let be a manifold which agrees with outside a compact set, or more generally

A weight function on is of the form

where is a distance function on , and any uniform choice of distance in the fibers is allowed — the usual Euclidean distance is taken outside . The typical choice is , for a metric .

The class of symbols associated to the weight is defined as

Most of the time we will use the class with in which case that we drop the subscript. When and , we simply write or for the class of symbols.

We denote by or the corresponding class of pseudodifferential operators. We have surjective quantization and symbol maps:

Multiplication of symbols corresponds to composition of operators, to leading order:

and

is the natural projection map. A finer filtration can be obtained by combining semiclassical calculus with the standard calculus (or in the yet more general framework of the Weyl calculus) — see for instance [40, §3].

The class of operators and the quantization map are defined locally using the definition on :

(3.1)

and we refer to [11, Chapter 7] for a detailed discussion, and to [13, Appendix D.2] for the semiclassical calculus on manifolds.

The semiclassical Sobolev spaces, are defined by choosing a globally elliptic, self-adjoint operator, (that is an operator satisfying everywhere) and putting

When ,

Unless otherwise stated all norms in this paper, , are norms.

For we follow [40] and say that the essential support is equal to a given compact set ,

if and only if

Here denotes the Schwartz class which makes sense since is Euclidean outside a compact. In this article we are only concerned with a purely semiclassical theory and deal only with compact subsets of .

For , , we put

noting that the definition does not depend on the choice of .

We introduce the following condition

(3.2)

and call families, , satisfying (3.2) -tempered. What we need is that for , -tempered, for . That is, applying an operator in the residual class produces a negligible contribution.

For such -tempered families we define the semiclassical -wave front set :

(3.3)

The last condition in the definition can be equivalently replaced with

since we may always take .

Equipped with the notion of semiclassical wave front set, it is useful and natural to consider the operators and their properties microlocally. For that we consider the classe of tempered operators, , defined by the condition

For open sets, , , the operators defined microlocally near are given by the following equivalence classes of tempered operators:

(3.4)

For two such operators we say that microlocally near . If we assumed that, say , where then could be replaced by in the condition. We should stress that “microlocally” is always meant in this semi-classical sense in our paper.

The operators in are bounded on uniformly in . For future reference we also recall the sharp Gårding inequality (see for instance [11, Theorem 7.12]):

(3.5)

and Beals’s characterization of pseudodifferential operators on (see [11, Chapter 8] and [41, Lemma 3.5] for the case) :

(3.6)

Here .

3.2. Assumptions on

We now state the general assumptions on the operator , stressing that the simplest case to keep in mind is

In general we consider

and an energy level , for which

(3.7)

Here the operator near infinity takes the following form on each “infinite branch” of :

with independent of for , uniformly bounded with respect to (here denotes the space of functions with bounded derivatives of all orders), and

(3.8)

We also need the following analyticity assumption in a neighbourhood of infinity: there exist such that the coefficients of extend holomorphically in to

with (3.8) valid also in this larger set of ’s. Here for convenience we chose the same as the one appearing in (3.7), but that is clearly irrelevant.

We note that the analyticity assumption in a conic neighbourhood near infinity automatically strengthens (3.8) through an application of Cauchy inequalities:

(3.9)

where for any the function when .

3.3. Definitions of hyperbolicity and topological pressure

We use the notation

where is the Hamilton vector field of ,

in local coordinates in . The last expression is the Poisson bracket relative to the symplectic form .

We assume and satisfy the assumptions (3.7) and (3.8) of §3.2, and study the flow generated by on . The incoming and outgoing sets, , and the trapped set, , are given by (1.5) and (1.6) respectively.

We say that the flow is hyperbolic on , if for any , the tangent space to at splits into flow, unstable and stable subspaces [20, Def. 17.4.1]:

(3.10)

is a locally maximal hyperbolic set for the flow . The following properties are then satisfied:

(3.11)

The adapted metric can be extended to the whole energy layer, such as to coincide with the standard Euclidean metric outside . We call

(3.12)

the weak unstable and weak stable subspaces at the point respectively. Similarly, we denote by (respectively ) the weak unstable (respectively stable) manifold. The ensemble of all the (un)stable manifolds forms the (un)stable lamination on , and one has

If periodic orbits are dense in , then the flow is said to be Axiom A on [5].

Such a hyperbolic set is structurally stable [20, Theorem 18.2.3], so that

(3.13)

Since the topological pressure plays a crucial rôle in the statement and proof of Theorem 3, we recall its definition in our context (see [20, Definition 20.2.1] or [33, Appendix A]).

Let be the distance function associated with the adapted metric. We say that a set is -separated if for , , we have for some . Obviously, such a set must be finite, but its cardinal may grow exponentially with . The metric induces a volume form on any -dimensional subspace of . Using this volume form, we now define the unstable Jacobian on . For any , the determinant map

can be identified with the real number

(3.14)