Eigenfunction asymptotics and spectral Rigidity of the ellipse
EEIGENFUNCTION ASYMPTOTICS AND SPECTRAL RIGIDITY OF THEELLIPSE
HAMID HEZARI AND STEVE ZELDITCH
Abstract.
Microlocal defect measures for Cauchy data of Dirichlet, resp. Neumann, eigenfunc-tions of an ellipse E are determined. We prove that, for any invariant curve for the billiard map onthe boundary phase space B ∗ E of an ellipse, there exists a sequence of eigenfunctions whose Cauchydata concentrates on the invariant curve. We use this result to give a new proof that ellipses areinfinitesimally spectrally rigid among C ∞ domains with the symmetries of the ellipse. This note is part of a series [HeZe12, HeZe19] on the the inverse spectral problem for ellipticaldomains E ⊂ R . In [HeZe12], it is shown, roughly speaking, that an isospectral deformation ofan ellipse through smooth domains (but not necessarily real analytic) which preserves the Z × Z symmetry is trivial. In [HeZe19] it is shown that ellipses of small eccentricity are uniquelydetermined by their Dirichlet (or, Neumann) spectra among all C ∞ domains, with no analyticityor symmetry assumptions imposed. In both [HeZe12, HeZe19], the main spectral tool is the wavetrace singularity expansion and the special form it takes in the case of ellipses. In this article,we take the dual approach of studying the asymptotic concentration in the phase space B ∗ ∂E ofthe Cauchy data u bj of Dirichlet (or, Neumann) eigenfunctions u j of elliptical domains in the unitcoball bundle of the boundary ∂E . In Theorem 1, we show that, for every regular rotation numberof the billiard map in the ‘twist interval’, there exists a sequence of eigenfunctions whose Cauchydata concentrates on the invariant curve with that rotation number in B ∗ ∂E . The proof uses theclassical separation of variables and one dimensional WKB analysis.Before stating the results we introduce some notation and background. An orthonormal basisof Dirichlet (resp. Neumann) eigenfunctions in a bounded, smooth Euclidean plane domain Ω isdenoted by (∆ + λ j ) ϕ j = 0 , (cid:104) ϕ j , ϕ k (cid:105) := (cid:82) Ω ϕ j ¯ ϕ k dx,ϕ j | ∂ Ω = 0 , (resp . ∂ ν ϕ j | ∂ Ω = 0) , where as usual ∂ ν denotes the inward unit normal. The semi-classcial Cauchy data is denoted by,(1) u bj := ϕ j | ∂ Ω , Neumann λ − j ∂ ν ϕ j | ∂ Ω , Dirichlet . The Cauchy data are eigenfunctions of the semi-classical eigenvalue problem, N ( λ j ) u bj = u bj , where N ( λ ) is a semi-classical Fourier integral operator quantizing the billiard map β : B ∗ ∂ Ω → B ∗ ∂ Ω(see [HaZe04] for the precise statement).We are interested here in the quantum limits of the Cauchy data (1) of an orthonormal basis ofeigenfunctions of an ellipse, i.e. in the asymptotic limits of the matrix elements(2) ρ bj ( Op (cid:126) ( a )) := (cid:104) Op (cid:126) ( a ) u bj , u bj (cid:105)(cid:104) u bj , u bj (cid:105) , ( (cid:126) = (cid:126) j = λ − j ) Date : July 1, 2020.Research of the second author is partially supported by NSF grant DMS-1810747. a r X i v : . [ m a t h . SP ] J un HAMID HEZARI AND STEVE ZELDITCH of zeroth order semi-classical pseudo-differential operators Op (cid:126) ( a ) on ∂E with respect to the L -normalized Cauchy data of eigenfunctions. We note that ρ bj is normalized so that ρ bj ( I ) = 1 andis a positive linear functional, hence all possible weak* limits are probability measures on the unitcoball bundle B ∗ ∂ Ω. Moreover, ρ j ( N ( λ ) ∗ Op (cid:126) ( a ) N ( λ )) = ρ j ( Op (cid:126) ( a )), so that the quantum limitsare quasi-invariant under the billiard map (see [HaZe04] for precise statements). In Theorem 1 wedetermine the quantum limits of sequences in (2) for an ellipse. The proof uses many of the priorresults on WKB formulae for ellipse eigenfunctions, especially those of [KeRu60, WaWiDu97, Sie97].In large part, our interest in matrix elements (2) owes to the fact that the Hadamard variationalformulae for eigenvalues of the Laplacian with Dirichlet boundary condition expresses the eigenvaluevariations as the special matrix elements (2) given by,(3) (cid:90) ∂E ˙ ρ | u bj | ds of the domain variation ˙ ρ (not to be confused with ρ j ) against squares of the Cauchy data (seeSection 5.1). As stated in Corollary 2, the limits of such integrals over all possible subsequencesof eigenfunctions determines the ‘Radon transform’ of ˙ ρ over all possible invariant curves for thebilliard map. Under an infinitesimal isospectral deformation, all of the limits are zero. We use thisresult to give a new proof of the spectral rigidity result in [HeZe12]; see Theorem 4 and Corollary5. The principal motivation for studying the inverse Laplace spectral problems for ellipses stemsfrom the Birkhoff conjecture that ellipses are the only bounded plane domains with completelyintegrable billiards. Strong recent results, due to A. Avila, J. de Simoi, V. Kaloshin, and A.Sorrentino [AvdSKa16, KaSo18] have proved local versions of the Birkhoff conjecture using a weakernotion of integrability known as ‘rational integrability’, i.e. that periodic orbits come in one-parameter families, namely invariant curves of the billiard map with rational rotation number. Inthis article, Bohr-Sommerfeld invariant curves play the principal role rather than curves of periodicorbits.1.1. Statement of results.
The first result pertains to concentration of Cauchy data of sequences ϕ j of Dirichlet (resp. Neumann) eigenfunctions on invariant curves of the billiard map of an ellipse.We denote E by x a + y b ≤
1, 0 ≤ b < a , and choose the elliptical coordinates ( ρ, ϑ ) by( x, y ) = ( c cosh ρ cos ϑ, c sinh ρ sin ϑ ) . Here, c = (cid:112) a − b , ≤ ρ ≤ ρ max = cosh − ( a/c ) , ≤ ϑ ≤ π. We denote the angular Hamiltonian, which we will also call the action, by I = p ϑ /c + cos ϑ. The invariant curves of β are the level sets of I . The range of I is called the action interval. Thereis a natural measure dµ α on each level set I = α called the Leray measure which is invariant under β and the flow of I . We refer to Section 2 for detailed definitions and properties involving thebilliard map of an ellipse, actions, invariant curves, and the Leray measure. Theorem Let E be an ellipse. For any α in the action interval of the billiard map of E ,there exists a sequence of separable (in elliptical coordinates) eigenfunctions { ϕ j } of eigenvalue λ j whose Cauchy data concentrates on the level set { I = α } , in the sense that, for any zeroth ordersemi-classical pseudo-differential operator Op (cid:126) ( a ) on B ∗ ∂E with principal symbol a , (4) (cid:104) Op (cid:126) j ( a ) u bj , u bj (cid:105)(cid:104) u bj , u bj (cid:105) → (cid:82) I = α a dν α (cid:82) I = α dν α , ( h j = λ − j → PECTRAL RIGIDITY OF THE ELLIPSE 3 where (5) dν α = √ c (cosh ρ max − cos ϑ ) dµ α , Dirichlet , (cid:113) c (cosh ρ max − cos ϑ ) dµ α , Neumann . In particular,
Corollary In the special case when the symbol a ( ϑ, p ϑ ) = ˙ ρ ( ϑ ) is only a function of the basevariable ϑ , (cid:82) ∂E ˙ ρ | u bj | ds (cid:82) ∂E | u bj | ds → (cid:82) I = α ˙ ρ dν α (cid:82) I = α dν α , where ds = (cid:113) c (cosh ρ max − cos ϑ ) dϑ is the arclength measure. Remark 3.
If we denote η to be the symplectic dual variable of the arclength s , then our quantumlimit can be expressed as dν α = (cid:112) − | η | dµ α , Dirichlet , √ −| η | dµ α , Neumann . For the proof, see our computation of − | η | in the proof of Corollary 18.The appearance of the (non-invariant) factors (cid:112) − | η | and / (cid:112) − | η | is consistent with theresult of [HaZe04] , where the quantum limits of boundary traces of ergodic billiard tables are studied. To our knowledge, Theorem 1 is the first result on microlocal defect measures of Cauchy dataof eigenfunctions in non-ergodic cases. See Section 1.3 for related results. One of the difficultiesin determining the limits of (2) is that the Cauchy data u bj are not L normalized. It is shown in[HaTa02, Theorem 1.1] that there exists C, c > c ≤ || λ − j ∂ ν ϕ j || L ( ∂ Ω) ≤ C for Dirichlet eigenfunctions of Euclidean plane domains (and more general non-trapping cases).Hence the L normalization in (2) is rather mild. On the other hand, the corresponding inequalitiesdo not hold in general for Neumann eigenfunctions. As pointed out in [HaTa02, Example 7], thereare simple counter-examples to any constant upper bound on the unit disc (whispering gallerymodes). There do exist positive lower bounds for convex Euclidean domains. Hence, in the case ofan ellipse, the L normalization in (2) is necessary to obtain limits.1.2. Spectral rigidity.
Before stating the results, we review the main definitions. An isospectraldeformation of a plane domain Ω is a one-parameter family Ω t of plane domains for which thespectrum of the Euclidean Dirichlet (or Neumann, or Robin) Laplacian ∆ t is constant (includingmultiplicities). The deformation is said to be a C deformation through C ∞ domains if each Ω t isa C ∞ domain and the map t → Ω t is C . We parameterize the boundary ∂ Ω t as the image underthe map(6) x ∈ ∂ Ω → x + ρ t ( x ) ν x , where ρ t ∈ C ([0 , t ] , C ∞ ( ∂ Ω)). The first variation is defined to be ˙ ρ ( x ) := ddt | t =0 ρ t ( x ). Anisospectral deformation is said to be trivial if Ω t = Ω (up to isometry) for sufficiently small t . Adomain Ω is said to be spectrally rigid if all C ∞ isospectral deformations are trivial.In [HeZe12] the authors proved a somewhat weaker form of spectral rigidity for ellipses, with‘flatness’ replacing ‘triviality’. Its main result is the infinitesimal spectral rigidity of ellipses among HAMID HEZARI AND STEVE ZELDITCH C ∞ plane domains with the symmetries of an ellipse. We orient the domains so that the symmetryaxes are the x - y axes. The symmetry assumption is then that ρ t is invariant under ( x, y ) → ( ± x, ± y ). The variation is called infinitesimally spectrally rigid if ˙ ρ = 0.The main result of [HeZe12] is: Theorem Suppose that Ω is an ellipse, and that Ω t is a C Dirichlet (or Neumann) isospectraldeformation of Ω through C ∞ domains with Z × Z symmetry. Let ρ t be as in (6). Then ˙ ρ = 0 . Corollary Suppose that Ω is an ellipse, and that t → Ω t is a C Dirichlet (or Neumann)isospectral deformation through Z × Z symmetric C ∞ domains. Then ρ t must be flat at t = 0 . The proof of Theorem 4 in [HeZe12] used the variation of the wave trace. In the original posting(arXiv:1007.1741) the authors used a more classical Hadamard variational formula for variationsof individual eigenvalues λ j ( t ), which appears in Section 5.1. The authors rejected this approachin favor of the one appearing in [HeZe12] because it was thought that this argument was invalidwhen the eigenvalues were multiple. When a multiple eigenvalue of a 1-parameter family L t ofoperators is perturbed, it splits into a collection of branches which in general are not differentiablein t . Moreover, the authors assumed that the variational formula would express the variation interms of special separable eigenfunctions (see Section 3). This created doubt that one could use thevariational formula for individual eigenvalues. Instead, the authors used the variational formulafor the trace of the wave group or equivalently for spectral projections, which are symmetric sumsover all of the branches into which an eigenvalue splits.However, as we show in this article, the original variational formulae were in fact correct even inthe presence of multiplicities. The first point is that the non-differentiability issue does not arisefor an isospectral deformation since no splitting occurs. Second, the vanishing of the variationof eigenvalues implies that the infinitesimal variation ˙ ρ is orthogonal to squares of all (Dirichlet)eigenfunctions in the eigenspace, and in particular the separable ones. More precisely, we provethat (cid:90) ∂E ˙ ρ | u bj | ds = 0 . Then by Corollary 5, we obtain that for every α in the action interval one has(7) (cid:90) I = α ˙ ρ dν α = 0 . In the final step we calculate the measure dν α and provide two proofs, one via inverting an Abeltransform and another using the Stone-Weierstrass theorem, that (7) implies ˙ ρ = 0. The proof inthe Neumann case is similar and will be provided.1.3. Related results and open problems.
Quantum limits of Cauchy data on manifolds withboundary have been studied in [HaZe04, ChToZe13] in the case where the billiard map β is ergodic.To our knowledge, they have not been studied before in non-ergodic cases. Theorem 1 shows that,as expected, Cauchy data of eigenfunctions localize on invariant curves for the billiard map ratherthan delocalize as in ergodic cases. L norms of Cauchy data of eigenfunctions are studied in [HaTa02] in the Dirichlet case andin [BaHaTa18] in the Neumann case. Further results on the quasi-orthonormality properties ofCauchy are studied in [BFSS02, HHHZ15].The study of eigenfunctions in ellipses has a long literature and we make substantial use of it. Inparticular, we quote several articles in the physics literature, in particular [WaWiDu97, Sie97], andseveral in mathematics [KeRu60, BaBu91], for detailed analyses of eigenfunctions of the quantumellipse. There is also a series of articles of G. Popov and P. Topalov (see e.g. [PoTo03, PT16])on the use of KAM quasi-modes to study Laplace inverse spectral problems. In particular, in PECTRAL RIGIDITY OF THE ELLIPSE 5 [PT16], Popov-Topalov also give a new proof of the rigidity result of [HeZe12] and extend it toother settings. The approach in this article is closely related to theirs, although it does not seemthat the authors directly studied Cauchy data of eigenfunctions of an ellipse.The multiplicity of Laplace eigenvalues of an ellipse appears to be largely an open problem. It is anon-trivial result of C.L. Siegel that the multiplicities are either 1 or 2 in the case of circular billiards;multiplicity 1 occurs for, and only for, rotationally invariant eigenfunctions. The Laplacians of thefamily of ellipses x a + y b = 1 form an analytic family containing the disk Laplacian, and one mighttry to use analytic perturbation theory to prove the following, Conjecture For a generic class of ellipses the multiplicity of each eigenvalue is ≤ . Quantum Birkhoff conjecture.
As mentioned above, ellipses have completely integrablebilliards, and the classical Birkhoff conjecture is that elliptical billiards are the only completelyintegrable Euclidean billiards with convex bounded smooth domains. Despite much recent progress,the Birkhoff conjecture remains open.The eigenvalue problem on a Euclidean domain is often called ‘quantum billiards’ in the physicsliterature (see e.g. [WaWiDu97]). One could formulate quantum analogues of the Birkhoff con-jecture in several related but different ways. The quantum analogue of the Birkhoff conjectureis presumably that ellipses are the only ‘quantum integrable’ billiard tables. A standard notionof quantum integrability is that the Laplacian commutes with a second, independent, (pseudo-differential) operator; we refer to [ToZe03] for background on quantum integrability. In Section3, we explain that the ellipse is quantum integrable in that one may construct two commutingSchr¨odinger operators with the same eigenfunctions and eigenvalues. The symbol of the secondoperator then Poisson commutes with the symbol of the Laplacian, hence the billiard dynamicsand billiard map are integrable. A related version is that one can separate variables in solving theLaplace eigenvalue problem. It is not obvious that these two notions are equivalent; in Section 3we use both separation of variables and existence of commuting operators in studying the ellipse.Classical studies of separation of variables and its relation to integrability go back to C. Jacobi, P.St¨ackel, L. Eisenhart and others, and E.K. Sklyanin has studied the problem more recently. We donot make use of their results here.Quantum integrability is much stronger than classical integrability, and one might guess that itis simpler to prove the quantum Birkhoff conjecture than the classical one. Wave trace techniquesas in [HeZe12, HeZe19] reduce Laplace spectral determination and rigidity problems to dynamicalinverse or rigidity results. The wave trace only ‘sees’ periodic orbits and is therefore well-adaptedto results on rational integrability. The dual approach through eigenfunctions studied in thisarticle gives a different path to the quantum Birkhoff conjecture, in which rational integrabilityand periodic orbits play no role.
Acknowledgment.
We thank Luc Hillairet for a discussion which prompted the revival of thisnote. 2.
Classical billiard dynamics
In this, and the next, section, we review some background definitions and results on the classicaland quantum elliptical billiard. We follow the notation of [Sie97]; see also [BaBu91, WaWiDu97].An ellipse E is a plane domain defined by, x a + y b ≤ , ≤ b < a. Here, a, resp. b , is the length of the semi-major (resp. semi-minor) axis. The ellipse has foci at( ± c,
0) with c = √ a − b and its eccentricity is e = ca . Its area is πab , which is fixed under an HAMID HEZARI AND STEVE ZELDITCH isospectral deformation. We define elliptical coordinates ( ρ, ϑ ) by( x, y ) = ( c cosh ρ cos ϑ, c sinh ρ sin ϑ ) . Here, 0 ≤ ρ ≤ ρ max = cosh − ( a/c ) , ≤ ϑ ≤ π. The coordinates are orthogonal. The lines ρ = constant are confocal ellipses and the lines ϑ = constant are confocal hyperbolas. In the special case of the disc, we have c = 0, but we assumehenceforth that c (cid:54) = 0.2.1. Action variables for the billiard flow.
The billiard flow on the ellipse E is the (broken)geodesic flow of the Hamiltonian H = p x + p y on T ∗ E , which follows straight lines inside E andreflects on ∂E according to equal angle law of reflection.Action-angle variables on T ∗ E are symplectic coordinates in which the billiard flow of the ellipseis given by Kronecker flows on the invariant Lagrangian submanifolds. We refer to [Ar89] for thegeneral principles and to [Sie97] for the special case of the ellipse. Let p ρ and p ϑ be the symplecticdual variables corresponding to the elliptic coordinates ρ and ϑ , respectively. The two conservedquantities of the system are the energy (the Hamiltonian) H and the angular Hamiltonian I (whichwe also call the action), given in the coordinates ( ρ, p ρ , ϑ, p ϑ ), by H = p ρ + p ϑ c (cosh ρ − cos ϑ ) and I = p ϑ cosh ρ + p ρ cos ϑp ρ + p ϑ . In the notation of [Tab97], I = cos θ cosh ρ + sin θ cos ϑ, where θ is the angle between a trajectory of the billiard flow and a tangent vector to the confocalellipse with parameter ρ . Note also that by the notation of [Sie97], I = 1 + L L c H where L L is theproduct of two angular momenta about the two foci. The values of I are restricted to0 ≤ I ≤ a c = cosh ( ρ max ) . The upper limit I = cosh ( ρ max ) corresponds to the motion along the boundary and the lower limit I = 0 corresponds to the motion along the minor axis. Moreover, there are two different kinds ofmotion in the ellipse depending on the sign of I . For 1 < I < cosh ( ρ max ) the trajectories havea caustic in the form of a confocal ellipse. For 0 < I < x -axis between the two focal points. Both kinds of motionsare separated by a separatrix which consists of orbits with I = 1 that go through the focal pointsof the ellipse.In terms of H and I , the canonical momenta, are given by(8) p ρ = c (cosh ρ − I ) H and p ϑ = c ( I − cos ϑ ) H. Therefore, the action variables are I ρ = 12 π (cid:90) p ρ dρ = c √ Hπ (cid:90) cosh ρ ≥ I,ρ ≥ (cid:113) cosh ρ − I dρ, (9) I ϑ = 12 π (cid:90) p ϑ dϑ = c √ Hπ (cid:90) cos ϑ ≤ I, ≤ ϕ ≤ π (cid:112) I − cos ϑ dϑ. (10)In fact these are the actions for the half-ellipse 0 ≤ ϕ ≤ π . The integrals can be calculated in termsof I using elliptic integrals of first and second kind (See [Sie97]). The actions will play a key rolein Section 3.4 in the description of Bohr-Sommerfeld quantization conditions for the eigenvalues ofthe Laplacian. PECTRAL RIGIDITY OF THE ELLIPSE 7
M Sieber
The two conserved quantities of the system are the energy and the product L L of thetwo angular momenta about the two foci L L = p v sinh u p sin v cosh u cos v . (6)It is more convenient to use another conserved quantity, instead of L L , which is energyindependent and is determined by the geometrical properties of a trajectory only. Thisis ↵ = L L /E whose values are restricted to the range b > ↵ > c . The upperlimit ↵ = b corresponds to the motion along the boundary and the lower limit ↵ = c corresponds to the motion along the minor axis.In terms of E and ↵ , the canonical momenta, are given by p = E(c sinh u ↵ )p v = E(c sin v + ↵ ). (7)There are two different kinds of motion in the ellipse depending on the sign of ↵ . For0 < ↵ b the trajectories have a caustic in the form of a confocal ellipse with semiminoraxis b = p ↵ . The motion goes around the two focal points and is composed of a librationin the coordinate u and a rotation in the coordinate v . For c ↵ < b = p ↵ and semitransverse axis a = p c + ↵ . The motion is composed of a libration in the coordinate u and a libration inthe coordinate v , and the trajectories cross the x -axis always between the two focal points.Both kinds of motions are separated by a separatrix which consists of orbits with ↵ = s is the arclength along thebilliard boundary and p is the cosine of the angle between the outgoing trajectory and thetangent to the boundary at the reflection point. The two lines through (s, p) = ( , ) markthe separatrix of the motion. The lines inside the separatrix correspond to the librationalmotion with ↵ < –1.0–0.50.00.51.0 s / L Figure 1.
A Poincar´e section through the motion in an ellipse with ba = . The boundary ofthe billiard is taken as surface of section and the reflections are described in Birkhoff coordinates s and p . L denotes the perimeter of the ellipse. Figure 1.
Invariant curves and caustics.2.2.
Billiard map, invariant curves, Leray measure, and action-angle variables.
Thebilliard map of an ellipse E (or in general any smooth domain) is a cross section to the the billiardflow on S ∗ ∂E E , which we always identify with B ∗ ∂E and call it the phase space of the boundary.To be precise, the billiard map β is defined on B ∗ ∂E as follows: given ( s, η ) ∈ T ∗ ∂E , with s beingthe arc-length variable measured in the counter-clockwise direction from a fixed point say s , and | η | ≤
1, we let ( s, ζ ) ∈ S ∗ E be the unique inward-pointing unit covector at s which projects to ( s, η )under the map T ∗ ∂E E → T ∗ ∂E . Then we follow the geodesic (straight line) determined by ( s, ζ ) tothe first place it intersects the boundary again; let s (cid:48) ∈ ∂E denote this first intersection. (If | η | = 1,then we let s (cid:48) = s .) Denoting the inward unit normal vector at s (cid:48) by ν s (cid:48) , we let ζ (cid:48) = ζ + 2( ζ · ν s (cid:48) ) ν s (cid:48) be the direction of the geodesic after elastic reflection at s (cid:48) , and let η (cid:48) be the projection of ζ (cid:48) to T ∗ s (cid:48) Y . Then we define β ( s, η ) = ( s (cid:48) , η (cid:48) ) . A theorem of Birkhoff asserts that billiard map preserves the natural symplectic form ds ∧ dη on B ∗ ∂E , i.e. β ∗ ( ds ∧ dη ) = ds ∧ dη. In the literature, the coordinates ( s, θ ) are commonly used for phase space of the boundary, where θ ∈ [0 , π ] is the angle that ζ makes with the positive tangent direction at s . In these coordinates, ds ∧ dη = sin θ dθ ∧ ds An invariant set in B ∗ ∂E is a set C such that β ( C ) = C . An invariant curve is a curve (connectedor not) on the phase space that is invariant. The phase space B ∗ ∂E of the ellipse E is in factfoliated with invariant curves. More precisely, Lemma The invariant curves of the billiard map β : B ∗ ∂E → B ∗ ∂E are level sets of I : B ∗ ∂E → R defined by, I = p ϑ c + cos ϑ Proof.
It follows quickly form the second equation of (8) and that H = 1 on S ∗ ∂E . (cid:3) Although I ϑ is the classical angular action on B ∗ ∂E , but we shall call I the action as it ismore convenient and is related to I ϑ via the one-to-one correspondence (10). As is evident from HAMID HEZARI AND STEVE ZELDITCH the Figure 1, the separatrix curve I = 1 divides the phase space into two types of open sets, theexterior corresponding to trajectories with confocal elliptical caustics (1 < I < cosh ρ max ) and theinterior to trajectories with confocal hyperbolic caustics (0 < I < Leray measure.
On each level set I = α of I , there is a natural measure dµ α called the Leraymeasure which in invariant under β and the flow generated by I . In the symplectic coordinates( ϑ, p ϑ ), and on I = α , it is given by dµ α = dϑ ∧ dp ϑ dI . Since dϑ ∧ dI = ∂I∂p ϑ dϑ ∧ dp ϑ , we obtain that(11) dµ α = c p ϑ (cid:12)(cid:12)(cid:12)(cid:12) I = α dϑ = c α − cos ϑ ) − / dϑ. Here, x + = x if x > dµ α is a unique measurethat is invariant under β and the flow of I .2.2.2. Action-angle variables and rotation number.
The billiard map has a Birkhoff normal formaround each invariant curve in B ∗ ∂E . That is, in the symplectically dual angle variable ι to I , thebilliard map has the form, β ( I, ι ) = (
I, ι + r ( I )), where r is often called the rotation number of theinvariant curve. An explicit formula is given for it in [Tab97] (3.5), [CaRa10] (section 4.3 (11)) and[Ko85]. Then, if 0 < I < r ( I ) = π F ( √ I ) F (cid:32) arcsin (cid:32) ρ max ) (cid:112) cosh ρ max − I cosh ρ max − I + I tanh ρ max (cid:33) , √ I (cid:33) , where F ( z, k ) = (cid:90) z dτ (cid:112) − k sin τ , F ( k ) = F (cid:16) π , k (cid:17) . Also, if 1 < I < cosh ( ρ max ) then r ( I ) = π F (1 / √ I ) F (cid:32) arcsin (cid:32) √ I ρ max ) (cid:112) cosh ρ max − I cosh ρ max − I + I tanh ρ max (cid:33) , √ I (cid:33) . Definition:
We define the range of the action variable I as the action interval , i.e. the interval[0 , cosh ( ρ max )], and the range of r ( I ) as the rotation interval .3. Quantum elliptical billiard
The Helmholtz equation in elliptical coordinates takes the form,(12) − (cid:18) ∂ ∂ρ + ∂ ∂ϑ (cid:19) ϕ = λ c (cosh ρ − cos ϑ ) ϕ. The quantum integrability of ∆ owes to the fact that this equation is separable. We put(13) ϕ ( ρ, ϑ ) = F ( ρ ) G ( ϑ ) , and separate variables to get the coupled Mathieu equations,(14) (cid:126) c F (cid:48)(cid:48) ( ρ ) + cosh ρ F ( ρ ) = αF ( ρ ) DBC (resp . NBC) , − (cid:126) c G (cid:48)(cid:48) ( ϑ ) + cos ϑ G ( ϑ ) = αG ( ϑ ) PBC . PECTRAL RIGIDITY OF THE ELLIPSE 9 where (cid:126) = λ − and α is the separation constant. Here, ‘PBC’ stands for ‘periodic boundaryconditions’; DBC (resp. NBC) stands for Dirichlet (resp. Neumann) boundary conditions. Thus,we consider pairs ( (cid:126) , α ) where there exists a smooth solution of the two boundary problems.Each of the angular and radial equations above is an eigenvalue problem for a semiclassicalSchr¨odinger operator with boundary conditions on a finite interval. These commuting operatorsare given by Op (cid:126) ( J ); J = − p ρ /c + cosh ( ρ ) , (15) Op (cid:126) ( I ); I = p ϑ /c + cos ( ϑ ) . (16)The boundary conditions on F take the form,(17) F ( ρ max ) = 0 (Dirichlet) , F (cid:48) ( ρ max ) = 0 (Neumann) . As G ( − ϑ ) is a solution whenever G ( ϑ ) is, we restrict our attention to 2 π -periodic solutions to theangular equation which are either even or odd. One can then see that: Remark 8.
In order to obtain solutions well-defined on the line segment joining the foci, i.e. at ρ = 0 , solutions to the radial equation must satisfy the boundary condition F (cid:48) (0) = 0 in case thesolution G is even and F (0) = 0 in case G is odd. In these cases the solutions F are also respectivelyeven and odd functions. Mathieu and modified Mathieu characteristic numbers.
For each fixed (cid:126) , the angularproblem is a Sturm-Liouville problem and thus there exist real valued sequences { a (cid:48) n ( (cid:126) ) } ∞ n =0 and { b (cid:48) n ( (cid:126) ) } ∞ n =1 so that it has 2 π -periodic non-trivial solutions - even solutions if α = a n ( (cid:126) ) and oddsolutions if α = b n ( (cid:126) ). Here even or odd is with respect to ϑ → − ϑ , or equivalently y → − y .We represent the corresponding solutions by G e n ( ϑ, (cid:126) ) and G o n ( ϑ, (cid:126) ), respectively. The even indicescorrespond to π -periodic solutions, thus they must be invariant under ϑ → π − ϑ , or equivalentlybe even with respect to x → − x . Solutions with odd indices have anti-period π and correspond toodd solutions in the x variable. The sequences a (cid:48) n ( (cid:126) ) and b (cid:48) n ( (cid:126) ) are related to the standard Mathieucharacteristic numbers of integer orders a n ( q ) and b n ( q ) by(18) a (cid:48) n ( (cid:126) ) = 12 + a n ( q )4 q , b (cid:48) n ( (cid:126) ) = 12 + b n ( q )4 q , q = c (cid:126) . Thus using the wellknown properties of a n and b n , for (cid:126) > a (cid:48) ( (cid:126) ) < b (cid:48) ( (cid:126) ) < a (cid:48) ( (cid:126) ) < b (cid:48) ( (cid:126) ) < a (cid:48) ( (cid:126) ) < b (cid:48) ( (cid:126) ) < · · · , (20) b (cid:48) n +1 ( (cid:126) ) − a (cid:48) n ( (cid:126) ) = O n ( e − C/ (cid:126) ) , C > . The sequence (19) is precisely the spectrum of the angular Schr¨odinger operator on the flat circle R / (2 π Z ).Similarly for the radial problem (say with Dirichlet boundary condition F ( ρ max ) = 0), for each (cid:126) ,there exist sequences { A (cid:48) m ( (cid:126) ) } ∞ m =0 and { B (cid:48) m ( (cid:126) ) } ∞ m =1 such that the radial problem has a non-trivialeven solution F e m ( ρ, (cid:126) ) if α = A (cid:48) m ( (cid:126) ), and a odd solution F o m ( ρ, (cid:126) ) if α = B (cid:48) m ( (cid:126) ). The sequencesof A (cid:48) m ( (cid:126) ) and B (cid:48) m ( (cid:126) ) are related to modified Mathieu characteristic numbers A m ( q ) and B m ( q )(See [Ne10]) by the same relations as in (18). They form the spectrum of the radial semiclassicalSchr¨odinger operator on the interval [ − ρ max , ρ max ] with Dirichlet boundary condition and satisfy(21) A (cid:48) ( (cid:126) ) < B (cid:48) ( (cid:126) ) < A (cid:48) ( (cid:126) ) < B (cid:48) ( (cid:126) ) < A (cid:48) ( (cid:126) ) < B (cid:48) ( (cid:126) ) < · · · . Eigevalues of E : Intersection of Mathieu and modified Mathieu curves. In order tofind eigenfunctions of the ellipse E one has to search specific values of (cid:126) such that both radial andangular Sturm-Liouville problems possess non-trivial solutions for the same value of α . By Remark8, we only consider the separable solutions F e m ( ρ, (cid:126) ) G e n ( ϑ, (cid:126) ) and F o m ( ρ, (cid:126) ) G o n ( ϑ, (cid:126) ) . Thus the frequencies of E with Dirichlet boundary condition are of the form λ e mn = 1 (cid:126) e mn and λ o mn = 1 (cid:126) o mn , where (cid:126) e mn and (cid:126) o mn are, respectively, solutions to(22) a (cid:48) n ( (cid:126) ) = A (cid:48) m ( (cid:126) ) and b (cid:48) n ( (cid:126) ) = B (cid:48) m ( (cid:126) ) . The existence of the point of intersection of the curves a (cid:48) n ( (cid:126) ) with A (cid:48) m ( (cid:126) ), and b (cid:48) n ( (cid:126) ) with B (cid:48) m ( (cid:126) )are guaranteed by: Theorem (Neves [Ne10]) . For each ( m, n ) , there is a unique positive solution q to each of theequations a n ( q ) = A m ( q ) and b n ( q ) = B m ( q ) . Hence the same statement holds for the equations (22) by the correspondence (18). The frequen-cies λ j of E are obtained by sorting { λ e mn , λ o mn ; ( m, n ) ∈ N } in increasing order.3.3. Symmetries classes.
The irreducible representations of the Z × Z symmetry group arereal one-dimensional spaces, so that there exists an orthonormal basis of eigenfunctions of the ellipsewhich are even or odd with respect to each Z symmetry, i.e. have one of the four symmetries(even , even) , (even , odd) , (odd , even) , (odd , odd) , where the first and the second entries correspond to symmetries with respect to x → − x and y → − y , respectively. Given the above discussion the symmetric eigenfunctions are:(23) (even , even) : ϕ e m, k = F e m ( ρ, (cid:126) ) G e2 k ( ϑ, (cid:126) ); (cid:126) = (cid:126) e m, k , (even , odd) : ϕ o m, k = F o m ( ρ, (cid:126) ) G o2 k ( ϑ, (cid:126) ); (cid:126) = (cid:126) o m, k , (odd , even) : ϕ e m, k +1 = F e m ( ρ, (cid:126) ) G e2 k +1 ( ϑ, (cid:126) ); (cid:126) = (cid:126) e m, k +1 , (odd , odd) : ϕ o m, k +1 = F o m ( ρ, (cid:126) ) G o2 k +1 ( ϑ, (cid:126) ); (cid:126) = (cid:126) o m, k +1 . Figure 2 shows the symmetries classes of eigenfunctions distinguished by their probability den-sities. It is possible that two symmetric eigenfunctions correspond to the same eigenvalue, or it ispossible that they correspond to different eigenvalues.3.4.
Semiclassical actions and Bohr-Sommerfeld quantization conditions for the ellipse.
Graphs of the one-dimensional classical potentials are given in [WaWiDu97, Figure 1]. The potential − cosh ρ for Op (cid:126) ( J ) in (15) is a potential barrier with a single local maximim which is symmetricaround the vertical line through the local maximum. The classical potential cos ϑ underlying Op (cid:126) ( I ) in (16) is a double-well potential on the circle. Thus, there exists a separatrix curvecorresponding to the two local maxima of the potential, which divides the two-dimensional phasespace into two regions. Inside the phase space curve, the level sets of the potential are ‘circles’paired by the left right symmetry across the vertical line through the local maximum at π . Outsidethe separatrix, the level sets have non-singular projections to the base, i.e. are roughly horizontal. In the Neumann case, A n and B m are different from the ones for the Dirichlet case. PECTRAL RIGIDITY OF THE ELLIPSE 11 even in x , even in y odd in x , even in y even in x , odd in y odd in x , odd in y Figure 2.
Symmetries classes of Dirichlet eigenfunctions corresponding to the firstfour eigenvalues, shown by their probability densities.As will be seen below, the Bohr-Sommerfeld levels inside the separatrix are invariant under theup-down symmetry and have two components exchanged by the left-right symmetry. The levelsoutside the separatrix are invariant under the left-right symmetry and are exchanged under theup-down symmetry.It is more important for our purposes to determine the lattice of semi-classical eigenvalues interms of classical and quantum action variables. The WKB (or EKB) quantization for the actionsare given in [Sie97, (33)] (see also [KeRu60] for the original reference). Up to O ( (cid:126) ) terms theyhave the form:Odd in y I > I ρ = ( m + ) (cid:126) , I ϑ = ( n + 1) (cid:126) , m, n = 0 , , , . . . ,I < I ρ = ( m + 1) (cid:126) , I ϑ = ( n + ) (cid:126) , m, n = 0 , , , . . . , Even in y I > I ρ = ( m + ) (cid:126) , I ϑ = n (cid:126) , m, n = 0 , , , . . . ,I < I ρ = ( m + ) (cid:126) , I ϑ = ( n + ) (cid:126) , m, n = 0 , , , . . . . There is a discontinuity at I = 1 due to the separatrix curve, but it is not important for our problemand we ignore it.3.4.1. Semiclassical action.
In fact for each of the eight Bohr-Sommerfeld quantization conditionabove there is a version which is valid to all orders in (cid:126) which are essentially given by the quantumBirkhoff normal form around each orbit under consideration. To be precise, there exist eight socalled semiclassical actions S e/o, ± ,ρ/ϑ (cid:126) ( α ) , where the choices of e/o corresponds to even or odd in the y (equivalently in the ϑ ) variable, of 1 + or 1 − to I >
I <
1, and ρ or ϑ to actions in the ρ or ϑ variable, respectively. Each of the eightsemiclassical actions has an (cid:126) asymptotic expansion of the form S (cid:126) ( α ) = S ( α ) + (cid:126) S ( α ) + (cid:126) S ( α ) + · · · , where S ( α ) is the corresponding classical action which is I ρ | I = α for S e/o, ± ,ρ (cid:126) and I ϑ | I = α for S e/o, ± ,ϑ (cid:126) . See equations (10) and (9) for formulas for the classical actions in terms of I = α . Then the Bohr-Sommerfeld Quantization Conditions (BSQC) to all orders are given by S e/o, + ,ρ (cid:126) ( α e/o, + ,ρm ( (cid:126) )) = m (cid:126) , valid uniformly for α ∈ [1 + (cid:15), cosh ρ max − (cid:15) ] , (24) S e/o, + ,ϑ (cid:126) ( α e/o, + ,ϑn ( (cid:126) )) = n (cid:126) , valid uniformly for α ∈ [1 + (cid:15), cosh ρ max − (cid:15) ](25) S e/o, − ,ρ (cid:126) ( α e/o, − ,ρm ( (cid:126) )) = m (cid:126) , valid uniformly for α ∈ [ (cid:15), − (cid:15) ] , (26) S e/o, − ,ϑ (cid:126) ( α e/o, − ,ϑn ( (cid:126) )) = n (cid:126) , valid uniformly for α ∈ [ (cid:15), − (cid:15) ] , (27)where (cid:15) > (cid:15) . There are versions of BSQC in the literature that are valid uniformly near the separatrix butwe do not need it here. We also point out that the Maslov indices are not ignored but absorbed inthe corresponding subleading terms S ( α ). Remark 10.
By our notations of Section 3.1 on the Mathieu and modified Mathieu characteristicvalues, away from the separatrix level we have, { α e, ± ,ρm ( (cid:126) ); m = 0 , , , · · · } = { A (cid:48) m ( (cid:126) ) : m = 0 , , · · · } , { α o, ± ,ρm ( (cid:126) ); m = 0 , , , · · · } = { B (cid:48) m ( (cid:126) ) : m = 1 , , · · · } , { α e, ± ,ϑn ( (cid:126) ); n = 0 , , , · · · } = { a (cid:48) n ( (cid:126) ) : n = 0 , , · · · } , { α o, ± ,ϑn ( (cid:126) ); n = 0 , , , · · · } = { b (cid:48) n ( (cid:126) ) : n = 1 , , · · · } . The eigenvalues of E are determined by intersecting the above analytic curves as follows:(28) α e, ± ,ρm ( (cid:126) ) = α e, ± ,ϑn ( (cid:126) ) ,α o, ± ,ρm ( (cid:126) ) = α o, ± ,ϑn ( (cid:126) ) , the solutions of which are precisely (cid:126) emn and (cid:126) omn , respectively, that we introduced in Section 3.2.3.5. Keller-Rubinow algorithm.
In this section we explore the procedure of finding (cid:126) emn corre-sponding to eigenvalues associated to invariant curves outside the separatrix (i.e. 1 + case) whoseeigenfunctions are even in the ϑ variable. All other cases follow a similar procedure and we shalldrop the superscripts for convenience.We are in search of solutions to equation (28) which, in our convenient notation, are given by(29) α ρm ( (cid:126) ) = α ϑn ( (cid:126) ) , where the left and the right hand sides satisfy the BSQC (24) and (25),(30) S ρ (cid:126) ( α ρm ( (cid:126) )) = m (cid:126) , S ϑ (cid:126) ( α ϑn ( (cid:126) )) = n (cid:126) , respectively. Following [KeRu60], we divide these two equations to obtain,(31) A (cid:126) ( α ) := S ρ (cid:126) ( α ) S ϑ (cid:126) ( α ) = I ρ ( α ) − (cid:126) + (cid:80) ∞ k =2 S ρk ( α ) (cid:126) k I ϑ ( α ) + (cid:80) ∞ k =2 S ϑk ( α ) (cid:126) k = mn . The expression A (cid:126) ( α ) has a classical (cid:126) expansion with principal term(32) A ( α ) := I ρ ( α ) I ϑ ( α ) , which is a positive monotonic function on the interval [1 , cosh ρ max ] (See [KeRu60], page 41).Hence, if we choose r in the range of A ( α ) on the domain [1 + 2 (cid:15), cosh ρ max − (cid:15) ], then for (cid:126) PECTRAL RIGIDITY OF THE ELLIPSE 13 sufficiently small there is a unique solution α to the equation A (cid:126) ( α ) = r in the slightly largerinterval [1 + (cid:15), cosh ρ max − (cid:15) ], accepting an (cid:126) expansion of the form:(33) α ( (cid:126) , r ) = ∞ (cid:88) k =0 α ( k ) ( r ) (cid:126) k . It is manifestly the the inverse function of A h ( α ) and its formal power series coefficients α ( k ) ( r ) aresmooth functions of r . The principal term α (0) is the inverse function of A ( α ). By this definition,the solution to (31) is α ( (cid:126) , m/n ) whenever m/n belongs to A [1 + 2 (cid:15), cosh ρ max − (cid:15) ], which isa bounded closed interval in (0 , ∞ ). In particular m/n is bounded above and below by positiveconstants K and K :(34) ( m, n ) ∈ N : K ≤ mn ≤ K . This is the eligible sector of lattice points for our eigenvalue problem outside the separatrix. Plug-ging α ( (cid:126) , m/n ) into the angular BSQC, i.e. the second equation of (30), (the radial one followsimmediately from the angular one and (31)), we arrive at the quantization condition for the eigen-values of E :(35) Q ( (cid:126) , m, n ) := 1 n S ϑ (cid:126) ( α ( (cid:126) , m/n )) = (cid:126) . We claim that for m and n sufficiently large, this equation has a unique solution (cid:126) mn in a sufficientlysmall interval [0 , (cid:126) ], or equivalently the function Q ( · , m, n ) has a unique fixed point. Now, since Q (0 , m, n ) = I ϑ ( α (0) ( m/n )) n , ∂Q∂ (cid:126) (0 , m, n ) = 0 , for (cid:126) sufficiently small, and n sufficiently large Q ( · , m, n ) maps [0 , (cid:126) ] into itself and ∂Q∂ (cid:126) ( (cid:126) , m, n ) < in this interval. The claim follows by the Banach contraction principle. Remark 11.
Since there are many functions α used, it is important to highlight their relationsand differences. If we evaluate α ( (cid:126) , r ) defined in (33) , at (cid:126) = (cid:126) m,n and r = mn , we get the commonvalue of (29) . In short, α (cid:16) (cid:126) mn , mn (cid:17) = α ρm ( (cid:126) mn ) = α ϑn ( (cid:126) mn ) . We also note that the function α (0) ( r ) , with parentheses around , is the principal term of α ( (cid:126) , r ) and should not be confused with α ρ ( (cid:126) ) or α ϑ ( (cid:126) ) . In fact, the above procedure provides an asymptotic for λ mn = 1 / (cid:126) mn and gives a sharper resultthan previously known: Proposition
The frequencies λ e/omn of E associated to invariant curves outside the separatrixcurve, and (cid:15) away from it, correspond to lattice points ( m, n ) ∈ N in the sector min (cid:26) I ρ ( α ) I ϑ ( α ) ; α ∈ [1 + (cid:15), cosh ρ max − (cid:15) ] (cid:27) ≤ mn ≤ max (cid:26) I ρ ( α ) I ϑ ( α ) ; α ∈ [1 + (cid:15), cosh ρ max − (cid:15) ] (cid:27) , and satisfy the asymptotic property, λ e/omn = nI ϑ ( α (0) ( m/n )) + O (cid:18) n (cid:19) . The same asymptotic formula holds for the frequencies λ e/omn associated to invariant curves insidethe separatrix curve, except in this case the sector of lattice points is: min (cid:26) I ρ ( α ) I ϑ ( α ) ; α ∈ [ (cid:15), − (cid:15) ] (cid:27) ≤ mn ≤ max (cid:26) I ρ ( α ) I ϑ ( α ) ; α ∈ [ (cid:15), − (cid:15) ] (cid:27) , The effects of even/odd are only reflected in the remainder term O (1 /n ), which in additiondepends on the distance (cid:15) from the separatix. Note that the explicit formulas for I ϑ and I ρ (hencefor α (0) ) in terms of elliptic integrals are different for the inside and outside the separatrix curve(See for example [Sie97]).4. Localization of boundary values of separable eigenfunctions on invariantcurves. Proof of Theorem 1
In this section, we relate semi-classical asymptotics of eigenfrequencies λ e/o m,n = 1 / (cid:126) e/o m,n and ofthe associated separated eigenfunctions ϕ e/o m,n defined by (23) along ‘ladders’ or ‘rays’ in the actionlattice ( m, n ) ∈ N . In particular, different rays correspond to different invariant Lagrangriansubmanifolds for the billiard flow. It is simpler to use the billiard map and then to relate raysin the joint spectrum to invariant curves for the billiard map. Given an invariant curve, insideor outside the separatrix, we wish to find a ray in the joint spectrum for which the associatedeigenfunctions concentrate on the curve. Since the WKB method is highly developed in dimensionone, it suffices for our purposes to locate the ray in N which corresponds to the invariant curve. Thecorresponding eigenfunctions will then concentrate on the corresponding Lagrangian submanifolds. Proposition
Let ϕ e/om,n ( ρ, ϑ ) be a separable Dirichlet (resp. Neumann) eigenfunction definedin (23) . Then the ‘modified boundary trace’ u e/om,n ( ϑ ) = ϕ e/om,n ( ρ, ϑ ) | ρ = ρ max , Neumann , λ e/omn ∂ϕ e/om,n ( ρ,ϑ ) ∂ρ | ρ = ρ max , Dirichlet . is an eigenfunction of the angular Schr¨odinger operator { Op (cid:126) ( I ) } (cid:126) = (cid:126) e/o m,n , whose eigenvalue α isdetermined by (36) (cid:104) Op (cid:126) ( I ) u e/om,n , u e/om,n (cid:105) L ( ∂E ) (cid:104) u e/om,n , u e/om,n (cid:105) L ( ∂E ) , which is α e/o, + ,ϑn ( (cid:126) ) if it is > and α e/o, − ,ϑn ( (cid:126) ) if it is < .Proof. The proof is obvious by equations (23), (14), and (16). (cid:3)
Remark 14.
It is important to note that although in the Neumann case our modified boundarytrace u e/om,n is the same as the boundary trace (cid:16) u e/om,n (cid:17) b defined by (1) , but they are slightly differentin the Dirichlet case as in this case (cid:16) u e/om,n (cid:17) b = − (cid:113) c (cosh ρ max − cos ϑ ) u e/om,n , which is due to the relation ∂∂ν = − (cid:113) c (cosh ρ max − cos ϑ ) ∂∂ρ (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ max . Our goal is to show that, for any invariant curve I = α , of the billiard map lying inside oroutside the separatrix curve, there exists a ladder of separable eigenfunctions ϕ e/om,n whose Cauchydata (cid:16) u e/om,n (cid:17) b concentrates on the invariant curve in B ∗ ∂E . In order to prove this we first need thefollowing lemma. PECTRAL RIGIDITY OF THE ELLIPSE 15
Lemma
For any α ∈ [0 , cosh ρ max ] , there exists a subsequence of { (cid:126) e/omn : ( m, n ) ∈ N } (foreither Dirichlet or Neumann boundary conditions) along which the eigenvalues of the semiclassicalangular operator { Op (cid:126) ( I ) }| (cid:126) = (cid:126) e/omn converges to α . Here, e/o means that any choice of even or oddcan be selected.Proof. It suffices to prove that(1) For any α ∈ (1 , cosh ρ max ) corresponding to invariant curves outside the separatrix, thereexists a subsequence of { (cid:126) e/omn : ( m, n ) ∈ N } (for either Dirichlet or Neumann boundaryconditions) along which α e/o, + ,ϑn ( (cid:126) e/omn ) → α. (2) For any α ∈ (0 ,
1) corresponding to invariant curves inside the separatrix, there exists asubsequence along which α e/o, − ,ϑn ( (cid:126) e/omn ) → α. A density argument would take care of the levels α = 0, 1 and cosh ρ max .We shall only prove (1), as the proof of (2) is similar. Furthermore, we shall only focus on theeven case because the proof for the odd case is identical. Fix α ∈ (1 , cosh ρ max ). We choose (cid:15) > α ∈ [1 + 2 (cid:15), cosh ρ max − (cid:15) ]. Let (cid:126) mn be the sequence we found in Section 3.5 associatedto the level curves outside the separatrix and to even eigenfunctions (even in the y variable). ByRemark 11, it suffices to show that there is a subsequence ( m j , n j ) along which α (cid:18) (cid:126) m j ,n j , m j n j (cid:19) → α, ( j → ∞ ) . We choose r by α (0) ( r ) = α (recall that α (0) is monotonic) and choose a sequence of lattice points( m j , n j ) ∈ N in the eligible sector (34) such that m j n j → r and | ( m j , n j ) | → ∞ . Since, (cid:12)(cid:12)(cid:12)(cid:12) α (cid:18) (cid:126) m j ,n j , m j n j (cid:19) − α (0) (cid:18) m j n j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) (cid:126) m j n j (cid:1) = O (cid:16) n − j (cid:17) , the lemma follows by letting j → ∞ and using the continuity of α (0) . (cid:3) Quantum limits of Cauchy data and the proof of Theorem 1.
By Proposition 13, themodified boundary traces u e/om,n ( ϑ ) of the separable eigenfunctions ϕ e/om,n ( ρ, ϑ ) of ∆, are eigenfunctionsof the semiclassical angular Schr¨odinger operator { Op (cid:126) ( I ) } (cid:126) = (cid:126) e/omn . It is well-known that eigenfunc-tions of 1 D semi-classical Schr¨odinger operators localize on level sets of the symbol. Thus if we fix α in the action interval and choose a sequence of { (cid:126) e/omn } provided by Lemma 15, then we know thatalong this sequence the quantum limit of | u e/om,n | dϑ is a measure on B ∗ ∂E that is supported on I = α . We also know, by Egorov’s theorem, that this measure must be invariant under the flow of I ,therefore the quantum limit must be the Leray measure dµ α . Since, by Remark 14, in the Dirichletcase the boundary traces (cid:16) u e/om,n (cid:17) b differ from u e/om,n by a factor (cid:0) c (cosh ρ max − cos ϑ ) (cid:1) − / causedby the conformal transformation from Cartesian to elliptical coordinates, and since ds = (cid:113) c (cosh ρ max − cos ϑ ) dϑ, we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) u e/om,n (cid:17) b (cid:12)(cid:12)(cid:12)(cid:12) ds = 1 (cid:113) c (cosh ρ max − cos ϑ ) (cid:12)(cid:12)(cid:12) u e/om,n (cid:12)(cid:12)(cid:12) dϑ → dµ α (cid:113) c (cosh ρ max − cos ϑ ) , which proves Theorem 1 in the Dirichlet case. The Neumann case is essentially the same; we omitthe details.5. Hadamard variational formulae for isospectral deformations
We consider the Dirichlet (resp. Neumann) eigenvalue problems for a one parameter family ofEuclidean plane domain Ω t , where Ω = E is an ellipse:(37) − ∆ ϕ j ( t ) = λ j ( t ) ϕ j ( t ) in Ω t ,ϕ j ( t ) = 0 (resp. ∂ ν t ϕ j ( t ) = 0) on ∂ Ω t . Here, ∂ ν t is the interior unit normal to ∂ Ω t . When λ j (0) is a simple eigenvalue, then under a C deformation the eigenvalue moves in a C curve λ j ( t ). When λ j (0) is a multiple eigenvalue, thenin general the eigenvalue may split into branches. Examples in [Ka95] show that eigenfunctions donot necessarily deform nicely if the deformation is not analytic. Hence we cannot even assume thateigenfunctions are C if the deformation is only C . However, we assume in this section that thedeformation is isospectral. In this case, a multiple eigenvalue does not change multiplicity underthe deformation, and therefore there is no splitting into branches.When an eigenvalue has multiplicity >
1, there exists an orthonormal basis (known as theKato-Rellich basis) of the eigenspace which moves smoothly under the deformation. The multipleeigenvalue splits under a generic perturbation and one can only expect a perturbation formulaalong each path. When we assume that the deformation is isospectral, hence that the eigenvaluedoes not split (or even change) along the deformation, then there exists a Kato-Rellich basis forthe eigenspace.5.1.
Hadamard variational formulae.
As in the introduction, we parameterize the deformationby a function ρ t on ∂E so that ∂ Ω t is the graph of ρ t over ∂ Ω = ∂E in the sense that ∂ Ω t = { x + ρ t ( x ) ν x : x ∈ ∂ Ω } . If ˙ ρ := ddt ρ t | t =0 (cid:54) = 0, then the first order variation of eigenvalues is thesame as for the deformation by x + t ˙ ρ ( x ) ν x . In this section we review the Hadamard variationalformula in the case of simple eigenvalues. We refer to [HeZe12, Section 1] for background on theHadamard variational formula.When λ j (0) is a simple eigenvalue (i.e. of multiplicity one) with L -normalized eigenfunction ϕ j ,then Hadamard’s variational formula for plane domains is that(38) Dirichlet: ( λ j ) · = (cid:90) ∂ Ω ( ∂ ν ϕ j ) ˙ ρ ds, where ds is the induced arc-length measure. Hence, under an infinitesimal isospectral deformationwe have, for every simple eigenvalue,(39) Dirichlet: (cid:90) ∂ Ω ( ∂ ν ϕ j ) ˙ ρ ds = 0 . Hadamard’s variational formula is actually a variational formula for the variation of the Green’sfunctions G ( λ, x, y ) with the given boundary conditions. In the Dirichlet case it states that˙ G ( λ, x, y ) = − (cid:90) ∂ Ω ∂∂ν G ( λ, q, x ) ∂∂ν G ( λ, q, y ) ˙ ρds. The formula (39) follows if we compare the poles of order two on each side. The same comparisonshows that if the eigenvalue λ j (0) is repeated with multiplicity m ( λ j (0)) and if { λ jk ( t ) } m ( λ j (0)) j =1 is PECTRAL RIGIDITY OF THE ELLIPSE 17 the perturbed set of eigenvalues, then ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 m ( λ j (0)) (cid:88) k =1 λ jk ( t ) = m ( λ j (0)) (cid:88) k =1 (cid:90) ∂ Ω ( ∂ ν ϕ j,k ) ˙ ρ ds. Here { ϕ j,k } m ( λ j (0)) j =1 is any ONB of the repeated eigenvalue λ j (0).There exist similar Hadamard variational formulae in the Neumann case. When the eigenvalueis simple, we have (cid:0) λ j (cid:1) · = (cid:90) ∂ Ω (cid:0) |∇ ∂ Ω ( ϕ j ) | − λ j ϕ j (cid:1) ˙ ρ ds, hence(40) Neumann: (cid:90) ∂ Ω (cid:0) |∇ ∂ Ω ( ϕ j ) | − λ j ϕ j (cid:1) ˙ ρ ds = 0 . Hadamard variational formula for an isospectral deformation.
We now assume thatthe deformation is isospectral. As mentioned above, there exists a Kato-Rellich basis which movessmoothly under the deformation. In fact, we show that for an isospectral deformation every eigen-function has a smooth deformation along the path. In the following − ∆ t denotes the Dirichlet(resp. Neumann) Laplacian on Ω t . Lemma
Suppose that Ω t is a C Dirichlet (resp. Neumann) isospectral deformation. Then anyeigenfunction ϕ j (0) of − ∆ on Ω , has a C deformation ϕ j ( t ) of eigenfunctions of − ∆ t on Ω t .Proof. Let λ j (0) be the eigenvalue of ϕ j (0), of multiplicity m j ≥
1, and γ be a circle in C centeredat λ j (0) such that no other eigenvalues of − ∆ are in the interior of γ or on γ . We define P t = − πi (cid:90) γ zR t ( z ) dz, where R t ( z ) = ( − ∆ t − z ) − is the resolvent of − ∆ t . By the Cauchy integral formula, it is clear that P is the orthogonal projector onto the eigenspace of λ j (0). Since the eigenvalues { λ j,k ( t ) } m j k =1 varycontinuously in t , for t small these are the only eigenvalues of − ∆ t in γ . Therefore, in general, P t isthe total projector (the direct sum of projectors) associated with { λ j,k ( t ) } mk =1 . The operator P t is C in t , since the resolvent (hence, Green’s function) is C in t (see [Ka95, Theorem II.5.4]). Nowassume Ω t is an isospectral deformation. Since the spectrum is constant along the deformation, P t projects every function on Ω t onto an eigenfunction of Ω t of eigenvalue λ j (0). Let f t be a C family of smooth diffeomorphisms from Ω t to Ω with f = Id. Then ϕ j ( t ) := P t ( f ∗ t ( ϕ j (0))) , (here , f ∗ t ( ϕ j (0)) = ϕ j (0) ◦ f t )must be an eigenfunction of − ∆ t of eigenvalue λ j (0). (cid:3) We are now in position to prove:
Lemma
Suppose that Ω t is a C isospectral deformation. Then for any eigenfunction ϕ j of Ω , (41) (cid:82) ∂ Ω ˙ ρ | ∂ ν ϕ j | = 0 , Dirichet (cid:82) ∂ Ω (cid:16) |∇ ∂ Ω ( ϕ j ) | − λ j ϕ j (cid:17) ˙ ρ ds = 0 , Neumann
Proof.
Let ϕ j (0) be any eigenfunction of Ω and ϕ j ( t ) be the C deformation of eigenfunction ofΩ t provided by Lemma 16. For t >
0, the eigenvalue problem for the isospectral deformation ispulled back to Ω by a C family diffeomorphisms f t , with f = Id, and has the form,( (cid:101) ∆ t + λ j ) (cid:101) ϕ j ( t ) = 0 , where (cid:101) ∆ t and (cid:101) ϕ j ( t ) are the pullbacks of ∆ t and ϕ j ( t ) to Ω , respectively. Taking the variationgives ˙ (cid:101) ∆ ϕ j (0) + (∆ + λ j ) ˙ (cid:101) ϕ j (0) = 0 . Take the inner product with ϕ k (0) in the same eigenspace. Integration by parts in the second termkills the second term. Thus we get (cid:104) ˙∆ ϕ j (0) , ϕ k (0) (cid:105) = 0 . The variation ˙∆ can be calculated (see for example [HeZe12]) to obtain: (cid:90) ∂ Ω ˙ ρ ( ∂ ν ϕ j )( ∂ ν ϕ k ) ds = 0 , for all ϕ j , ϕ k in the λ j -eigenspace of the Dirichlet problem. A similar proof works for the relevantquadratic form for the Neumann problem. (cid:3) Proof of Theorem 4
Before we prove our main theorem, we need to study the limits of the equations (41) alongsequences of eigenvalues introduced in Theorem 1.
Corollary
Let ˙ ρ be the first variation of a Dirichlet (or Neumann) isospectral deformationof an ellipse E . Then for all ≤ α ≤ cosh ( ρ max ) , (cid:90) I = α ˙ ρ (cid:112) cosh ρ max − cos ϑ dµ α = 0 . Proof.
The Dirichlet case follows immediately from Theorem 1 and Lemma 17. For the Neumanncase, we observe that by Theorem 1 the quantum limit of λ − j |∇ ∂ Ω ( ϕ j ) | − ϕ j , along a sequence of eigenfunctions that concentrates on the invariant curve I = α is( | η | − dµ α . Therefore, in the Neumann case we get(42) (cid:90) I = α ( | η | − (cid:113) c (cosh ρ max − cos ϑ ) ˙ ρ dµ α = 0 . We recall that η is the symplectic dual of the arclength variable s . From the equation ηds = p ϑ dϑ ,we find that in the ( ϑ, p ϑ ) coordinates, η is given by η = p ϑ (cid:113) c (cosh ρ max − cos ϑ ) . Since on I = α , p ϑ = c ( α − cos ϑ ), | η | − α − cosh ρ max cosh ρ max − cos ϑ . PECTRAL RIGIDITY OF THE ELLIPSE 19
The corollary follows in the Neumann case by taking out the constant α − cosh ρ max from theintegral (42). (cid:3) Theorem 4, now reduces to:
Proposition
The only Z × Z invariant function ˙ ρ satisfying the equations of Corollary 18is ˙ ρ = 0 for α ∈ (0 , , i.e. for levels inside the separatrix. Similarly, the same statement holds ifwe only know equations of Corollary 18 for α ∈ (1 , cosh ρ max ) , i.e. levels outside the separatrix.Proof. Since ˙ ρ ( ϑ ) is Z × Z invariant we can put P (cos ϑ ) := ˙ ρ ( ϑ ) (cid:112) cosh ρ max − cos ϑ . By our explicit formula (11) for the Leray measure dµ α , and by the Z × Z symmetry, we have (cid:90) π P (cos ϑ ) (cid:112) ( α − cos ϑ ) + dϑ = 0 , ∀ ≤ α ≤ cosh ρ max . Splitting this equation into α ≤ α ≥ (cid:90) cos − ( √ α )0 P (cos ϑ ) √ α − cos ϑ dϑ = 0 , ∀ ≤ α ≤ . (44) (cid:90) π P (cos ϑ ) √ α − cos ϑ dϑ = 0 , ∀ ≤ α ≤ cosh ρ max . It is sufficient to show that P ≡
0, given (43) or (44).
Proof using invariant curves inside the separatrix.
We change variables to u = cos ϑ andalso set x = √ α . Then the integral (43) becomes(45) (cid:90) x P ( u ) √ x − u du √ − u = 0 , ∀ ≤ x ≤ . Writing f ( u ) = P ( u ) √ − u , this becomes(46) (cid:90) x f ( u ) √ x − u du = 0 , ∀ ≤ x ≤ . The transform A f ( x ) = (cid:90) x f ( u ) √ x − u du is closely related to the Abel transform. We claim that the left inverse Abel transform is given by, A − g ( u ) = 2 π ddu (cid:90) u xg ( x ) √ u − x dx. The key point is the integral identity, I ( u, v ) := (cid:90) uv xdx √ u − x √ x − v = π , ( v ≤ u ) . It follows that if B g ( u ) is the integral in the purported inversion formula, BA f ( u ) = π ddu (cid:82) u x A f ( x ) √ u − x dx = π ddu (cid:82) u (cid:82) x x √ u − x f ( v ) √ x − v dvdx = π ddu (cid:82) u I ( u, v ) f ( v ) dv = ddu (cid:82) u f ( v ) dv = f ( u ) . Since A is left invertible, it follows that ker A = { } . Since f ( u ) = P ( u ) √ − u lies in its kernel, wehave P = 0 and hence ˙ ρ = 0. Proof using invariant curves outside the separatrix.
The proof of the second assertionof Proposition 19 is similar to the final steps in the proofs of spectral rigidity results of [GuMe79],[HeZe12], and [Vi20], for the ellipse in various settings. We need to show that (44) implies P = 0.We change variables by u = cos ϑ and this time we set f ( u ) = P ( u ) √ u (1 − u ) . Then (cid:90) f ( u ) √ α − u du = 0 , ∀ < α ≤ cosh ρ max . Since the left hand side as a function of α is smooth at cosh ρ max , all its Taylor coefficients at thispoint must vanish. Thus (cid:90) f ( u ) (cid:0) cosh ρ max − u (cid:1) − n − du = 0 , ∀ n ∈ N . By the Stone-Weierstrass theorem, f = 0, hence P = 0. (cid:3) Infinitesimal rigidity and flatness.
In Section 3.2 of our earlier paper [HeZe12], we provedthat infinitesimal rigidity implies flatness, which completes the proof of Corollary 5:
References [Ar89] V.I. Arnol’d,
Mathematical methods of classical mechanics . Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989.[AvdSKa16] A. Avila, J. De Simoi and V. Kaloshin,
An integrable deformation of an ellipse of small eccentricity isan ellipse . Ann. of Math. (2) 184 (2016), no. 2, 527–558.[BaBu91] V. M. Babich and V. S. Buldyrev,
Short wavelength Diffraction Theory . Springer Series Wave Phenomena4 (1991), Springer Verlag.[BFSS02] A. B¨acker, S. F¨urstberger, R. Schubert, and F. Steiner,
Behaviour of boundary functions for quantumbilliards . J. Phys. A 35 (2002), no. 48, 10293–10310.[BaHaTa18] A. H. Barnett, A. Hassell, and M. Tacy,
Comparable upper and lower bounds for boundary values ofNeumann eigenfunctions and tight inclusion of eigenvalues . Duke Math. J. 167 (2018), no. 16, 3059–3114.[CaRa10] P. S. Casas and R. Ramirez-Ros,
The frequency map for billiards inside ellipsoids (arXiv:1004.5499, 2010).[ChToZe13] H. Christianson, J. A. Toth, and S. Zelditch,
Quantum ergodic restriction for Cauchy data: interior QUEand restricted QUE . Math. Res. Lett. 20 (2013), no. 3, 465–475.[dSKaWe17] J. de Simoi, V. Kaloshin and Q. Wei,
Dynamical spectral rigidity among Z2-symmetric strictly convexdomains close to a circle . Appendix B coauthored with H. Hezari. Ann. of Math. (2) 186 (2017), no. 1,277–314.[Fo78] D. Fujiwara and S. Ozawa,
The Hadamard variational formula for the Green functions of some normalelliptic boundary value problems . Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), no. 8, 215–220.[Ga98] P. R. Garabedian,
Partial differential equations . AMS Chelsea Publishing, Providence, RI, 1998.[GuMe79] V. Guillemin and R. Melrose,
An inverse spectral result for elliptical regions in R . Adv. Math. 32 (1979),128–148. PECTRAL RIGIDITY OF THE ELLIPSE 21 [HHHZ15] X. Han, A. Hassell, H. Hezari, and S. Zelditch,
Completeness of boundary traces of eigenfunctions . Proc.Lond. Math. Soc. (3) 111 (2015), no. 3, 749–773.[HaTa02] A. Hassell and T. Tao,
Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions . Math.Res. Lett. 9 (2002), no. 2-3, 289–305; Erratum for ”Upper and lower bounds for normal derivatives ofDirichlet eigenfunctions” [MR1909646]. Math. Res. Lett. 17 (2010), no. 4, 793–794.[HaZe04] A. Hassell and S. Zelditch,
Quantum ergodicity of boundary values of eigenfunctions . Comm. Math. Phys.248 (2004), no. 1, 119–168.[HeZe12] H. Hezari and S. Zelditch, C ∞ spectral rigidity of the ellipse . Anal. PDE 5 (2012), no. 5, 1105–1132.(arXiv:1007.1741 ).[HeZe19] H. Hezari and S. Zelditch, One can hear the shape of ellipses of small eccentricity , arXiv:1907.03882.[KaSo18] V. Kaloshin and A. Sorrentino,
On the local Birkhoff conjecture for convex billiards . Ann. of Math. (2) 188(2018), no. 1, 315–380.[KeRu60] J. B. Keller and S. I. Rubinow,
Asymptotic Solution of Eigenvalue Problems , Annals of Physics 9 (1960),24–73.[Ka95] T. Kato,
Perturbation theory for linear operators.
Reprint of the 1980 edition. Classics in Mathematics.Springer-Verlag, Berlin, 1995[Ko85] R. Kolodziej,
The rotation number of some transformations related to billiards in an ellipse , Studia Math81 (1985), 293–302.[Ma1868] E. Mathieu,
M´emoire sur le mouvement vibratoire d’une membrane de forme elliptique . J. Math. PuresAppl. 13 (1868), 137–203.[Ne10] A. Neves,
Eigenmodes and eigenfrequencies of vibrating elliptic membranes: a Klein oscillation theorem andnumerical calculations , Communications on Pure and Applied Analysis 9 (2010), no 3., 611–624.[Pee80] J. Peetre,
On Hadamard’s variational formula . J. Differential Equations 36 (1980), no. 3, 335–346.[PoTo03] G. Popov and P. Topalov,
Liouville billiard tables and an inverse spectral result . Ergodic Theory Dynam.Systems 23 (2003), no. 1, 225–248.[PoTo12] G. Popov and P. Topalov,
Invariants of isospectral deformations and spectral rigidity , Comm. Partial Dif-ferential Equations 37 (2012), no. 3, 369–446 (arXiv:0906.0449).[PT16] G. Popov and P. Topalov,
From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of BilliardTables , arXiv:1602.03155.[Sie97] M. Sieber,
Semiclassical transition from an elliptical to an oval billiard . J. Phys. A 30 (1997), no. 13,4563–4596.[Tab97] M. B. Tabanov,
Separatrices splitting for Birkhoff’s billiard in symmetric convex domain close to an ellipse .Chaos 4 (1994), no. 4, 595–606.[ToZe02] J. A. Toth and S. Zelditch,
Riemannian manifolds with uniformly bounded eigenfunctions . Duke Math. J.111 (2002), no. 1, 97–132.[ToZe03] J. A. Toth and S. Zelditch, L p norms of eigenfunctions in the completely integrable case . Ann. Henri Poincar´e4 (2003), no. 2, 343–368[Vi20] A. Vig, Robin spectral rigidity of the ellipse , to appear in J. Geom. Analysis, 2020 (arXiv:1812.09649).[WaWiDu97] H. Waalkens, J. Wiersig, and H. R. Dullin,
Elliptic quantum billiard . Ann. Physics 260 (1997), no. 1,50–90.
Department of Mathematics, UC Irvine, Irvine, CA 92697, USA
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