Wigner measures on the disk

Wigner measures and observability for the Schrödinger equation on the disk

Nalini Anantharaman Université Paris-Sud 11, Mathématiques, Bât. 425, 91405 Orsay Cedex, France Matthieu Léautaud Université Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, Bâtiment Sophie Germain, 75205 Paris Cedex 13 France  and  Fabricio Macià Universidad Politécnica de Madrid. DCAIN, ETSI Navales. Avda. Arco de la Victoria s/n. 28040 MADRID, SPAIN
Abstract.

We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schrödinger equation on the disk, with Dirichlet boundary conditions.

Our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk. We prove a “structure theorem”, expressing the restriction of the Wigner measures on each invariant torus in terms of second-microlocal measures. They are obtained by performing a finer localization in phase space around each of these tori, at the limit of the uncertainty principle, and are shown to propagate according to Heisenberg equations on the circle.

Our construction yields as corollaries (a) that the disintegration of the Wigner measures is absolutely continuous in the angular variable, which is an expression of the dispersive properties of the equation; (b) an observability inequality, saying that the -norm of a solution on any open subset intersecting the boundary (resp. the -norm of the Neumann trace on any nonempty open set of the boundary) controls its full -norm (resp. -norm). These results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary.

Key words and phrases:
Semiclassical measures, microlocal defect measures, Schrödinger equation, disk, observability, completely integrable dynamics

1. Introduction

1.1. Motivation

We consider the unit disk

 D={z=(x,y)∈R2,|z|2=x2+y2<1}⊂R2

and denote by the euclidean Laplacian. We are interested in understanding dynamical properties of the (time-dependent) linear Schrödinger equation

 (1.1) 1i∂u∂t(z,t)=(−12Δ+V(t,z))u(z,t), t∈R,z=(x,y)∈D, (1.2) u⌉t=0=u0∈L2(D)

with Dirichlet boundary condition (we shall write when we want to stress that we are using the Laplacian with that boundary condition). We assume that is a smooth real-valued potential, say . We shall denote by the (unitary) propagator starting at time , such that is the unique solution of (1.1)-(1.2).

This equation is aimed at describing the evolution of a quantum particle trapped in a disk-shaped cavity, being the wave-function at time . The total -mass of the solution is preserved: for all time . Thus, if the initial datum is normalized, , the quantity is, for every fixed , a probability density on ; given , the expression:

 ∫Ω|u(z,t)|2dz

is the probability of finding the particle in the set at time . Having for all solutions of (1.1) means that every quantum particle spends a positive fraction of time of the interval in the set . A major issue in mathematical quantum mechanics is to describe the possible localization – or delocalization – properties of solutions to the Schrödinger equation (1.1), by which we mean the description of the distribution of the probability densities for all solutions . A more tractable problem consists in considering instead of single, fixed solutions, sequences of solutions to (1.1) and describe the asymptotic properties of the associated probability densities or . This point of view still allows to deduce properties of single solutions and their distributions , as we shall see in the sequel.

It is always possible to extract a subsequence that converges weakly:

 ∫D×Rϕ(z,t)|un(z,t)|2dzdt⟶∫¯¯¯D×Rϕ(z,t)ν(dz,dt), % for every ϕ∈Cc(¯¯¯¯D×R),

where is a nonnegative Radon measure on that describes the asymptotic mass distribution of the sequence of solutions . One of the goals of this paper is to understand how the fact that solves (1.1) influences the structure of the associated measure .

As an application, we aim at understanding the observability problem for the Schrödinger equation: given an open set and a time , does there exist a constant such that we have:

 (1.3) ∫T0∫Ω|u(z,t)|2dzdt≥C∥u0∥2L2(D), for all u0∈L2(D) and u associated % solution of~{}(???)?

If such an estimate holds, then every quantum particle must leave a trace on the set during the time interval ; in other words: it is observable from . This question is linked to that of understanding the structure of the limiting measures . Estimate (1.3) is not satisfied if and only if there exists a sequence of data such that and , where is the solution of (1.1) issued from . After the extraction of a subsequence, this holds if and only if the associated limit measure satisfies

 ∫T0∫¯¯¯Dν(dz,dt)=T,∫T0∫Ων(dz,dt)=0.

The question of observability from may hence be reformulated as: can sequences of solution of (1.1) concentrate on sets which do not intersect ? From the point of view of applications, it is of primary interest to understand which sets do observe all quantum particles trapped in a disk. Moreover, the observability of (1.1) is equivalent to the controllability of the Schrödinger equation (see e.g. [Leb:92]), which means that it is possible to drive any initial condition to any final condition at time , with a control (a forcing term in the right-hand side of (1.1)) located within .

It is well-known that the space of position variables does not suffice to describe the propagation properties of solutions to Schrödinger equations (or more generally wave equations) in the high frequency régime. To take the latter into account, one has to add the associated dual variables, (momentum and energy) and lift the measure to the phase space, associated to the variables : this gives rise to the so-called Wigner measures [Wigner32]. We shall hence investigate the regularity and localization properties in position and momentum variables of the Wigner measures associated with sequences of normalized solutions of (1.1). They describe how the solutions are distributed over phase space. We shall develop both the microlocal and semiclassical points of view. These are two slightly different, but closely related, approaches to the problem : the semiclassical approach is more suitable when our initial data possess a well-defined oscillation rate, whereas the microlocal approach describes the singularities of solutions, independently of the choice of a scale of oscillation, at the price of giving slightly less precise results.

Our study fits in the regime of the “quantum-classical correspondence principle” , which asserts that the high-frequency dynamics of the solutions to (1.1) are described in terms of the corresponding classical dynamics; in our context the underlying classical system is the billiard flow on . Wigner measures carry this information, for they are known to be invariant by this flow.

Of course, one may consider similar questions for any bounded domain of or any Riemannian manifold, and not only the disk . As a matter of fact, the answer to these questions depends strongly on the dynamics of the billiard flow (resp. the geodesic flow on a Riemannian manifold), and, to our knowledge, it is known only in few cases (see Section 1.6). For instance, on negatively curved manifolds, the celebrated Quantum Unique Ergodicity conjecture remains to this day open. Two geometries for which the observation problem is well-understood, and the Wigner measures are rather well-described, are the torus (see [JaffardPlaques, Kom:92, MaciaDispersion, BZ:12] and [JakobsonTori97, BourgainLatt, MaciaTorus, AnantharamanMaciaTore]) and the sphere , or more generally, manifolds all of whose geodesics are closed (see [JZ:99, MaciaZoll, MaciaAv, AM:10]), on which the classical dynamics is completely integrable. We shall later on compare these two situations with our results on the disk . We refer to the article [AMSurvey] for a survey of recent results concerning Wigner measures associated to sequences of solutions to the time-dependent Schrödinger equation in various geometries and to the review article [Laurent14] on the observability question.

1.2. Some consequences of our structure theorem

Our central results are Theorems 2.4 and 2.6 below, which provide a detailed structure of the Wigner measures associated to sequences of solutions to the Schrödinger equation, using notions of second-microlocal calculus. As corollaries of these structure Theorems, we obtain:

• Corollary 1.1 (see also Theorem 2.9), which reflects the dispersive character of the Schrödinger equation (1.1);

• Theorem 1.2 (resp. Theorem 1.3), which states the observability/controllability of the equation from any nonempty open set touching the boundary of the disk (resp. from any nonempty open set of the boundary).

Let us first state these corollaries in order to motivate the more technical results of this paper.

Corollary 1.1.

Let be a sequence in , such that for all . Consider the sequence of nonnegative Radon measures on , defined by

 (1.4) νn(dz,dt)=|UV(t)u0n(z)|2dzdt.

Let be any weak- limit of the sequence : then where, for almost every , is a probability measure on , and is absolutely continuous.

This result shows that the weak- accumulation points of the densities (1.4) possess some regularity in the interior of the disk. This result cannot be extended to , since it is easy to exhibit sequences of solutions that concentrate singularly on the boundary (the so-called whispering-gallery modes, see Section 1.3). In Theorem 2.9 below, we present a stronger version of Corollary 1.1 describing (in phase space) the regularity of microlocal lifts of such limit measures . This precise description (as well as all results of this paper) relies on the complete integrability of the billiard flow on the disk. Its statement needs the introduction of action angle coordinates and associated invariant tori, and is postponed to Section 2.7.

The second class of results mentioned above is related to unique continuation-type properties of the Schrödinger equation (1.1). We consider the following condition on an open set , a time and a potential :

 (UCPV,Ω,T) (u0∈L2(D),UV(t)u0⌉(0,T)×Ω=0)⟹u0=0.

As a consequence of Theorem 2.6, we shall also prove the following quantitative version of (UCP).

Theorem 1.2.

Let be an open set such that and . Assume one of the following statements holds:

• the potential , the time , and the open set satisfy (UCP),

• the potential does not depend on .

Then there exists such that:

 (1.5)

for every initial datum .

Roughly speaking, this means that any set touching observes all quantum particles trapped in the disk. As we shall see, these are the only sets satisfying this property (see Section 1.3 and Remark 2.11).

We are also interested in the boundary analogue of (UCP) for a given potential , a time and an open set :

 (UCPV,Γ,T) (u0∈H10(D),∂n(UV(t)u0)⌉(0,T)×Γ=0)⟹u0=0,

where denotes the exterior normal derivative to . As a consequence of Theorem 2.6, we shall also prove the following quantitative version of (UCP).

Theorem 1.3.

Let be any nonempty subset of and . Suppose one of the following holds:

• the potential , the time and the set satisfy (UCP),

• does not depend on .

Then there exists such that:

 (1.6)

for every initial datum .

Note that the unique continuation properties (UCP) and (UCP) are known to hold, for instance, when is analytic in , as a consequence of the Holmgren uniqueness theorem as stated by Hörmander (see e.g. [Hormander:LPDO, Theorem 5.3.1]).

These three results express a delocalization property of the energy of solutions to (1.1). The observation of the -norm restricted to any open set of the disk touching the boundary is sufficient to recover linearly the norm of the data. In particular, the -mass of solutions cannot concentrate on periodic trajectories of the billiard. The observability inequalities (1.5) and (1.6) are especially relevant in control theory (see [LionsSurvey88, BLR:92, Leb:92]): in turn, they imply a controllability result from the set or .

As a consequence of the observability inequality 1.5, we have the following result (where we use the notation of Corollary 1.1).

Corollary 1.4.

For every open set touching the boundary, for every , there exists a constant such that for any initial data and any weak- limit of the sequence as in Corollary 1.1, we have

 ∫T0νt(Ω)dt≥1C(T,Ω).

This translates the fact that any solution has to leave positive mass on any set touching the boundary during the time interval . This may be rephrased by saying that any such set observes all quantum particles trapped in the disk.

1.3. Stationary solutions to (1.1): eigenfunctions on the disk

If the potential does not depend on the time variable , we have as particular solutions of the Schrödinger equation the “stationary solutions”, those with initial data given by eigenfunctions of the elliptic operator involved.

In the absence of potential, i.e. if , these solutions are well understood: the eigenfunctions of on are the functions whose (non-normalized) expression in polar coordinates is

 (1.7) ψ±n,k(reiu)=Jn(αn,kr)e±inu,

where are non-negative integers, is the -th Bessel function, and the are its positive zeros ordered increasingly with respect to . The corresponding eigenvalue is . Putting then gives a time-periodic solution to (1.1)-(1.2). Moreover, the eigenvalues of have multiplicity two. This is a consequence of a celebrated result by Siegel [Siegel29], showing that have no common zeroes for . In particular, the limit measures associated to sequences of eigenfunctions are explicitly computable in terms of the limits of the stationary distributions:

 |ψ±n,k(z)|2∥ψ±n,k∥2L2(D)dz=|Jn(αn,kr)|2∥ψ±n,k∥2L2(D)rdrdu,

as the frequency tends to infinity (this expression has to be slightly modified when considering linear combinations of the two eigenfunctions and , corresponding to the same eigenvalue, with fixed and tending to infinity) . Let us recall some particular cases of this construction. For fixed and for , it is classical  [Lagnese, Lemma 3.1] that

 |ψ±n,k(z)|2∥ψ±n,k∥2L2(D)dz⇀(2π)−1δ∂D,

which corresponds to the so-called whispering gallery modes. On the other hand, letting with being constant, one may obtain for any depending on the ratio  [PTZ:14, Section 4.1]

 |ψ±n,k(z)|2∥ψ±n,k∥2L2(D)dz⇀12π(1−γ2)1/21(|z|2−γ2)1/2\mathds1(γ,1)(|z|)dz.

Except the Dirac measure on the boundary, these measures all belong to for any (hence satisfying Corollary 1.1) and are invariant by rotation and positive on the boundary (hence satisfying Corollary 1.4). These measures in fact enjoy more regularity and symmetry than those asserted by Corollaries 1.1 and 1.4.

The observability question for eigenfunctions can also be simply handled in account of the bounded multiplicity of the spectrum. For any non-empty open set (where is an open subset of , an open interval of ), for any eigenfunction of , one has:

 ∥ψ∥L2(ΩI1,S1)≤C(|I2|)∥ψ∥L2(ΩI1,I2)

where is a positive constant depending only on the size of . On the other hand, if touches the boundary () it automatically satisfies the geometric control condition as defined in [BLR:92, Leb:92]. The results on those references imply that:

 ∥ψ∥L2(D)≤C′(I1)∥ψ∥L2(ΩI1,S1).

Therefore, for such , we have

 ∥ψ∥L2(D)≤C(|I2|)C′(I1)∥ψ∥L2(ΩI1,I2).

It is not known to the authors whether or not any of the results of the present article could be deduced directly from the result for eigenfunctions, even when the potential vanishes identically. This does not seem to appear in the literature. On flat tori, proving observability or regularity of Wigner measures associated to the Schrödinger equation from the explicit expression of the solutions in terms of Fourier series requires a careful analysis of the distribution of lattice points on paraboloids [JakobsonTori97, BourgainLatt] or sophisticated arguments on lacunary Fourier series [JaffardPlaques, Kom:92]. On the disk, and in absence of a potential, one could try to expand the kernel of in terms of Bessel functions:

 eitΔD/2=∑n,k,±e−itα2n,k/2|ψ±n,k⟩⟨ψ±n,k|

and to use some of their known properties. Such an approach would anyway require some very technical work on the spacings between the .

Here, instead, we establish directly the links between the completely integrable nature of the dynamics of the billiard flow and the delocalization and dispersion properties of the solutions to the Schrödinger equation. Note that all results of this paper also hold for eigenfunctions of the operator (as stationary solutions to (1.1)). As a matter of fact, our approach is more general for it applies as well to quasimodes and clusters of eigenfunctions of the operator . The reader is referred to [ALMcras] and Remark 2.5 for more details on this matter.

1.4. The semiclassical viewpoint

In spite of the fact that our statements and proofs are formulated exclusively in terms of the non-semiclassical Schrödinger equation (1.1), our results do have an interpretation in the light of the semiclassical limit for the Schödinger equation. Suppose that solves the semiclassical Schrödinger equation:

 (1.8) hi∂vh∂t(z,t)=(−h22Δ+h2V(ht,z))vh(z,t),vh⌉t=0=u0.

It turns out that is in fact the solution to the (non-semiclassical) Schrödinger equation (1.1) with initial datum . As a consequence, describing properties of solutions to (1.1) on time intervals of size of order amounts to describing properties of solutions to the semiclassical Schrödinger equation (1.8) up to times of order . Our results show that the semiclassical approximation (meaning that the solution to (1.8) should be well-approximated by its initial datum propagated through the billiard flow) breaks down in time . For instance, if we take as initial datum in (1.8) a coherent state localized at ,

 u0n=1hαnρ(z−z0hαn)eihnz⋅ξ0,ρ∈C∞c(D),ρ(0)=1,α∈(0,1),hn→0,

our results imply that the associated solution of (1.8) on the time interval is no longer concentrated on the billiard trajectory issued from . Instead, we show that it spreads on the disk (the associated measure is absolutely continuous) and it leaves a positive mass on any set touching the boundary (even if the trajectory of the billiard issued from avoids this set).

Hence, our analysis goes far beyond the well-understood semiclassical limit for times of order , or even of order (known as the Ehrenfest time, see [BouzouinaRobert]). Such a long time analysis is possible thanks to the complete integrability of the system. In fact, in the paper [AnantharamanKMacia], which deals with the Schrödinger equation (and more general completely integrable systems) on the flat torus, it is shown that the time scale is exactly the one at which the delocalization of solutions takes place; for chaotic systems on the contrary, the semiclassical approximation is expected to break down at the Ehrenfest time [NAQUE, AnRiv, AMSurvey].

1.5. The structure theorem

We would like to stress the fact that all these results are obtained as consequences of our main theorem, Theorem 2.4 or its variant Theorem 2.6, that gives a precise description of the structure of Wigner measures arising from solutions to (1.1). It provides a unified framework from which to derive simultaneously the absolute continuity of projections of semiclassical measures (a fact that is related to dispersive effects) on the one hand, and, on the other hand, the observability estimates (1.5) and (1.6), which are quantitative unique continuation-type properties. Since a precise statement requires the introduction of many other objects, we postpone it to Sections 2.4 and 2.5 (semiclassical and microlocal formulations respectively), and only give a rough idea of the method for the moment.

The standard construction of the Wigner measures, outlined in Section 2.5, allows to lift a measure to a measure on phase space (or in the semiclassical setting): this is the associated microlocal defect measure [GerardMDM91]. The law of propagation of singularities for equation (1.1) implies that is invariant by the billiard flow in the disk, and we want to exploit the complete integrability of this flow.

For this, we use action-angle coordinates to integrate the dynamics of the billiard flow and describe associated invariant tori (Section 3.1). The angular momentum of a point in phase space is preserved by the flow, and so is the Hamiltonian . The actions and are in involution and independent, except at the points of with tangent momentum. The angle that a trajectory makes when bouncing on the boundary is a also preserved quantity (in fact a function of ). The key point of our proof is to analyze in detail the possible concentration of sequences on the sets of all points of phase space sharing a common incidence/reflection angle . To this aim, we perform a second microlocalization on this set, in the spirit of [MaciaTorus, AnantharamanMaciaTore, AnantharamanKMacia]. We decompose a Wigner measure as a sum of measures supported on these invariant sets. The case corresponds to trajectories hitting the boundary in a dense set, and is trivial for us since it supports only one invariant measure. We focus on those for which . Any trajectory of the billiard having this angle is periodic. We wish to “zoom” on this torus to describe the concentration of the associated measure. Assuming that the initial sequence has a typical oscillation scale of order , we perform a second microlocalization at scale , which is the limit of the Heisenberg uncertainty principle. Roughly speaking, the idea is to relocalize in the action variable at scale (i.e. times ), so that the Heisenberg uncertainty principle implies delocalization in the conjugated angle variable. We obtain two limit objects, interpreted as second-microlocal measures. The first one captures the part of our sequence of solutions whose derivatives in directions “transverse to the flow” remain bounded; the second one captures the part of the solution rapidly oscillating in these directions. Understanding the notion of transversality adapted to this problem is achieved by constructing a flow that interpolates between the billiard flow (generated by the Hamiltonian ) and the rotation flow (generated by the Hamiltonian ). The second measure is a usual microlocal/semiclassical measure whereas the first one is a less usual operator-valued measure taking into account non-oscillatory phenomena. We prove that both second-microlocal measures enjoy additional invariance properties: the first one is invariant by the rotation flow, whereas the second one propagates through a Heisenberg equation on the circle. This translates, respectively, into Theorem 2.4 (ii) and (iii).

This program was already completed in [MaciaTorus, AnantharamanMaciaTore, AnantharamanKMacia] for the Schrödinger equation on flat tori, but carrying it out in the disk induces considerable additional difficulties. Our phase space does not directly come equipped with its action-angle coordinates, so that we need first to change variables. This requires in particular to build a Fourier Integral Operator to switch from -variables to action-angle coordinates. These coordinates are very nice to understand the dynamics and are necessary to perform the second microlocalization, but they are extremely nasty to treat the boundary condition, for which the use of polar coordinates is more suitable. It seems that we cannot avoid having to go back and forth between the two sets of coordinates. Our approach to that particular technical aspect is inspired by [GerLeich93]; however, the second-microlocal nature of the problem requires to perform the asymptotic expansions of [GerLeich93] one step further.

1.6. Relations to other works

1.6.1. Regularity of semiclassical measures

This work pertains to the longstanding study of the so-called “quantum-classical correspondence”, which aims at understanding the links between high frequency solutions of the Schrödinger equation and the dynamics of the underlying billiard flow (see for instance the survey article [AMSurvey]).

More precisely, it is concerned with a case of completely integrable billiard flow. This particular dynamical situation has already been addressed in [MaciaTorus] and [AnantharamanMaciaTore] in the case of flat tori, and in [AnantharamanKMacia] for more general integrable systems (without boundary). These three papers use in a central way a “second microlocalization” to understand the concentration of measures on invariant tori. The main tools are second-microlocal semiclassical measures, introduced in the local Euclidean setting in [NierScat, Fermanian2micro, FermanianShocks, FermanianGerardCroisements, MillerThesis, Miller:97], and defined in [MaciaTorus, AnantharamanMaciaTore, AnantharamanKMacia] as global objects.

On the sphere , or more generally, on a manifold with periodic geodesic flow, the situation is radically different. The geodesic flow for this type of geometries is still completely integrable, but it is known [MaciaAv] (see also [JZ:99, MaciaZoll, AM:10] for the special case of eigenfunctions) that every invariant measure is a Wigner measure; those are not necessarily absolutely continuous when projected in the position space. The difference with the previous situation is that the underlying dynamical system, though completely integrable, is degenerate. What was evidenced in [AnantharamanKMacia] is that a sufficient and necessary for the absolute continuity of Wigner measures, is that the hamiltonian be a strictly convex/concave function of the action variables – a condition that is even stronger than non-degeneracy. In the case of the disk, the complete integrability of the billiard flow on degenerates on the boundary. There, both actions coincide, which allows for the concentration of solutions on the invariant torus at the boundary (as was the case with the aforementioned whispering gallery modes).

Note that on the torus and on the disk, it remains an open question to fully characterize the set of Wigner measures associated to sequences of solutions to the time-dependent Schrödinger equation. In the case of flat tori, the papers [JakobsonTori97, AJM] provide additional information about the regularity of the measures.

1.6.2. Observability of the Schrödinger equation

Since the pioneering work of Lebeau [Leb:92], it is known that observability inequalities like (1.5)-(1.6) always hold if all trajectories of the billiard enter the observation region or in finite time. However, since [Har:89plaque, JaffardPlaques], we know that this strong geometric control condition is not necessary: (1.5) holds on the two-torus as soon as ; for different proofs and extensions of this result see [Kom:92, BurqZworski04, MaciaDispersion, BZ:12, BBZ14, AnantharamanMaciaTore]. These properties seem to deeply depend on the global dynamics of the billiard flow.

On manifolds with periodic geodesic flow, it is necessary that meets all geodesics for an observation inequality as (1.5) to hold [MaciaDispersion]. This is due to the strong stability properties of the geodesic flow.

To our knowledge, apart from the case of flat tori, few results are known concerning the observability of the Schrödinger equation in situations where the geometric control condition fails. The paper [AnantharamanKMacia] extends [AnantharamanMaciaTore] to general completely integrable systems under a convexity assumption for the hamiltonian. Note also that the boundary observability (1.6) holds in the square if (and only if) the observation region contains both a horizontal and a vertical nonempty segments [RTTT:05]. Finally, for chaotic systems, the observability inequality (1.5) is also valid on manifolds with negative curvature if the set of uncontrolled trajectories is sufficiently small [NAQUE, AnRiv].

Our Theorems 1.2 and 1.3 provide necessary and sufficient conditions for the observability of the Schrödinger group on the disk. This is clear in the case of boundary observability, and in the case of internal observability, if is such that , the observability inequality (1.5) fails. When this comes from the existence of whispering-gallery modes, see Section 1.3, and this remains true for any , as proved in Remark 2.11.

Let us conclude this introduction with a few more remarks.

Remark 1.5.

In this article, we only treat the case of Dirichlet boundary conditions. The extension of our method to the Neumann or mixed boundary condition deserves further investigation.

Remark 1.6.

Let us comment on the regularity required on the potential . Arguments developed in [AnantharamanMaciaTore] show that all the results of this paper could actually be weakened to or even to the case where is continuous outside a set of zero measure. Corollary 1.1 in fact also holds for any , and in particular for any bounded potentials. See also Remark 2.5 below.

Remark 1.7.

Our results directly yield a polynomial decay rate for the energy of the damped wave equation with Dirichlet Boundary conditions on the disk. More precisely, [AnL, Theorem 2.3] and Theorem 1.2 imply that if is positive on an open set such that , then the norm of solutions decays at rate for data in . This rate is better than the a priori logarithmic decay rate given by the Lebeau theorem [Leb:96]. The latter is however optimal if as a consequence of the whispering gallery phenomenon (see e.g. [LR:97]).

Acknowledgement. We thank Patrick Gérard for his continued encouragement and very helpful discussions.

NA and ML are supported by the Agence Nationale de la Recherche under grant GERASIC ANR-13-BS01-0007-01.

NA acknowledges support by the National Science Foundation under agreement no. DMS 1128155, by the Fernholz foundation, by Agence Nationale de la Recherche under the grant ANR-09-JCJC-0099-01, and by Institut Universitaire de France. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

FM takes part into the visiting faculty program of ICMAT and is partially supported by grants MTM2013-41780-P (MEC) and ERC Starting Grant 277778.

2. The structure Theorem

In this section, we give the main definitions used in the article and state our main structure theorems. We first define microlocal and semiclassical Wigner measures (which are the main objects discussed in the paper) in Section 2.1. We then briefly describe the billiard flow and introduce adapted action-angle coordinates in Section 2.2. This allows us to formulate our main results (Sections 2.4 and 2.5), both in the semiclassical (Theorem 2.4) and in the microlocal (Theorem 2.6) framework. Next, in Section 2.7, we define various measures at the boundary of the disk, that will be useful in the proofs, and explain their links with the Wigner measures in the interior.

2.1. Wigner measures: microlocal versus semiclassical point of view

Let be the cotangent bundle over , and be the cotangent bundle over . We shall denote by (resp. ) the space (resp. time) variable and (resp. ) the associated frequency.

Our main results can be formulated in two different and complementary settings. We first introduce the symbol class needed to formulate their microlocal version, allowing to define microlocal Wigner distributions. We then define semiclassical Wigner distributions and briefly compare these two objects.

Definition 2.1.

Let us call the space of functions , such that

1. is compactly supported in the variables .

2. is homogeneous at infinity in in the following sense: there exists such that

 (2.1) a(z,ξ,t,H)=a(z,λξ,t,λ2H),\quad for |ξ|2+|H|>R0 and λ≥1.

Equivalently, there is satisfying (2.1) for all , such that

Such a homogeneous function is entirely determined by its restriction to the set , which is homeomorphic to a -dimensional sphere . Thus we may (and will, when convenient) identify with a function in the space .

Note that the different homogeneity with respect to the and variables is adapted to the scaling of the Schrödinger operator.

Let be a sequence in , such that for all . For and we denote . In what follows (e.g. in formula (2.2) below), we shall systematically extend the functions , a priori defined on , by the value outside as done in [GerLeich93], where semiclassical Wigner measures for boundary value problems were first considered. The extended sequence now satisfies the equation

 (−12Δ+V−Dt)un=12∂un∂n⊗δ∂D,(z,t)∈R2×R,

where denotes the Laplacian on . Remark that the term has no straightforward meaning at this level of regularity. We shall see below how to give a signification to this equation, both in the semiclassical (see Remark 2.3) and in the microlocal (see Section 2.8.3) settings).

The microlocal Wigner distributions associated to act on symbols by

 (2.2) Wun(a):=⟨un,Op1(a)un⟩L2(R2z×Rt),

where (with the standard notation ) is a pseudodifferential operator defined by the standard quantization procedure. In what follows, will stand for the operator acting on by:

 (2.3) (Opϵ(a)u)(z,t)=1(2πϵ)3∫R2×R∫R2×Reiξ⋅(z−z′)+iH(t−t′)ϵa(z,ξ,t,H)u(z′,t′)dz′dt′dξdH.

Usual estimates on pseudodifferential operators imply that is well defined, and forms a bounded sequence in . The main goal of this article is to understand properties of weak limits of that are valid for any sequence of initial conditions .

The problem also has a semiclassical variant. In this version, one considers , a real parameter , and one defines the semiclassical Wigner distributions at scale by

 (2.4) Whun(a):=⟨un,Op1(a(z,hξ,t,h2H))un⟩L2(R2z×Rt),

where , see (2.3). Note that this scaling relation is the natural one for solutions of (1.1), and its interest will be made clear below. Again is well defined, and forms a bounded sequence in if stays bounded. This formulation is most meaningful if the parameter is chosen in relation with the typical scale of oscillation of our sequence of initial conditions .

Definition 2.2.

Given a bounded sequence in , we shall say that it is -oscillating from above (resp. -oscillating from below) if the sequence extended by zero outside of satisfies:

 limR→∞limsupn→∞∫|ξ|≥R/hn|ˆwn(ξ)|2dξ=0,

(resp.

 limϵ→0limsupn→∞∫|ξ|≤ϵ/hn|ˆwn(ξ)|2dξ=0)

where is the Fourier transform of on .

The property of being -oscillating from above is only relevant if ; if is -oscillating for bounded away from , the (extended) sequence is compact in and the structure of the accumulation points of is trivial. Therefore, we shall always assume that . Note that one can always find tending to zero such that is -oscillating from above (to see that, note that for fixed one may choose such that ). However, the choice of the sequence is by no means unique (-oscillating sequences are also -oscillating as soon as ), although in many cases there is a natural scale given by the problem under consideration.

One can find such that is -oscillating from below if and only if the extended converges to weakly in . It is not always possible to find a common such that is -oscillating both from above and below (see [GerardSobolev] for an example of a sequence with this behavior). However, when it is the case, the semiclassical Wigner distributions contain more information that the microlocal ones (see Section 2.6). On the other hand, if no exists such that is -oscillating from above and below, the accumulation points of may fail to describe completely the asymptotic phase-space distribution of the sequence , either because some mass will escape to or because the fraction of the mass going to infinity at a rate slower that will give a contribution concentrated on . In those cases, the microlocal formulation is still able to describe the asymptotic distribution of the sequence on the reduced phase-space .

This is one of the motivations that has lead us to study both points of view, semiclassical and microlocal.

2.2. The billiard flow

Microlocal or semiclassical analysis provide a connection between the Schrödinger equation and the billiard on the underlying phase space. In this section we first clarify what we mean by “billiard flow” in the disk. The phase space associated with the billiard flow on the disk can be defined as a quotient of (position frequency). We first define the symmetry with respect to the line tangent to the circle at by

 σz(ξ)=ξ−2(z⋅ξ)z,σ(z,ξ)=(z,σz(ξ)),z∈∂D.

Then, we work on the quotient space

 W=¯¯¯¯D×R2/∼where (z,ξ)∼σ(z,ξ) for |z|=1.

We denote by the canonical projection which maps a point to its equivalence class modulo . Note that is one-one on , so that may be seen as a subset of .

A function can be identified with the function satisfying for .

The billiard flow on is the (uniquely defined) action of on such that the map is continuous on , satisfies , and such that

 ϕτ(z,ξ)=(z+τξ,ξ)

whenever and .

In order to understand how the completely integrable dynamics of the flow influences the structure of Wigner measures, we need to introduce coordinates adapted to this dynamics. We denote by

 (2.5) Φ :(s,θ,E,J)↦(x,y,ξx,ξy),

the set of “action-angle” coordinates for the billiard flow (see also Section 3.1), defined by:

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩x=JEcosθ−ssinθ,y=JEsinθ+scosθ,ξx=−Esinθ,ξy=Ecosθ.

These coordinates are illustrated in Figure 1. The inverse map is given by the formulas

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩E=√ξ2x+ξ2y, (velocity)J=xξy−yξx, (angular momentum)θ=−arctan(ξxξy), (angle of ξ % with the vertical)s=−xsinθ+ycosθ, (abscissa of (x,y) along the line (JEcosθ,JEsinθ)+Rξ ).

In other words, we have:

 ⎧⎪ ⎪⎨⎪ ⎪⎩E=|ξ|,J=z⋅ξ⊥,s=z⋅ξ|ξ|,

where , and

 ⎧⎪⎨⎪⎩ξ=(ξx,ξy)=E(−sin(θ),cos(θ)),\parz=(x,y)=s(−sin(θ),cos(θ))+JE(cos(θ),sin(θ))=(z⋅ξ|ξ|)ξ|ξ|+(z⋅ξ⊥|ξ|)ξ⊥|ξ|.

Note that the velocity and the angular momentum are preserved along the free transport flow in , but also along ; the variables and play the role of “angle” coordinates. We call the angle that a billiard trajectory makes with the normal to the circle, when it hits the boundary (see Figure 2). The quantity is preserved by the billiard flow.

Let us denote the level sets of the pair , namely

 (2.6) T(E,J)={(z,ξ)∈¯¯¯¯D×R2:(|ξ|,z⋅ξ⊥)=(E,J)}.

For let us denote the probability measure on that is both invariant under the billiard flow and invariant under rotations. In the coordinates , we have

 λE,J(ds,dθ)=c(E,J)dsdθ,c(E,J)=(∫T(E,J)dsdθ)−1>0.

Note that for and the billiard flow is periodic on whereas corresponds to trajectories that hit the boundary on a dense set. More precisely, if then the billiard flow restricted to has a unique invariant probability measure, namely .

2.3. Standard facts about Wigner measures

We start formulating the question and results in a semiclassical framework: we have a parameter going to , meant to represent the typical scale of oscillation of our sequence of initial conditions .

We simplify the notation by writing , . We will always assume that the functions are normalized in . We define (the reader should be aware that satisfies the classical Schrödinger Equation (1.1); the index only reminds its oscillation scale). Since this is a function on it is natural to do a frequency analysis both in and . Recall that we keep the notation after the extension by zero outside . Recall that the semiclassical Wigner distribution associated to (at scale ) is a distribution on the cotangent bundle , defined by

 (2.7) Whuh:a↦⟨uh,Op1(a(z,hξ,t,h2H))uh⟩L2(R2×R), % for all a∈C∞c(T∗R2×T∗R).

The scaling is performed in order to capture all the information whenever is -oscillating from above (if is not -oscillating from above, the discussion below remains entirely valid but part of the information about is lost when studying ). Under this assumption, if is a function on that depends only on , we have

 (2.8) Whuh(a)=∫Da(z,t)|uh(z,t)|2dzdt.

When no confusion arises, we shall denote for .

By standard estimates on the norm of , it follows that belongs to , and is uniformly bounded in that space as