N. Burq, M. Zworski
In this note we further develop the idea of using a ``black box'' point of view (see our previous work) to study eigenfunctions for billiards which have rectangular components: they include the Bunimovich billiard, the Sinai billiard, and the recently popular pseudointegrable billiards
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We consider quasimodes on planar domains with a partially rectangular boundary. We prove that for any $ε_0>0$, an $Ø(λ^{-ε_0})$ quasimode must have $L^2$ mass in the "wings" bounded below by $λ^{-2-δ}$ for any $δ>0$. The proof uses the author's recent work on 0-Gevrey smooth domains to approximate quasimodes on $C^{1,1}$ domains. There is an improvement for $C^{2,α}$ domains.
We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called ``pockets''. We prove there are only finitely many immersed periodic tubes missing the pockets and moreover establish a new quantitative estimate for the lengths of such tubes. This extends well-known results in dimension $2$. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results on irrational polyh…
Local limits of high energy eigenfunctions on integrable billiards
本文研究可积台球中高能本征函数的逆局域化性质，发现大多数可积多边形和椭圆的局部极限显著偏离Berry猜想预期的随机波。
This paper is part of a series concerning the isospectral problem for an ellipse. In this paper, we study Cauchy data of eigenfunctions of the ellipse with Dirichlet or Neumann boundary conditions. Using many classical results on ellipse eigenfunctions, we determine the microlocal defect measures of the Cauchy data of the eigenfunctions. The ellipse has integrable billiards, i.e. the boundary phase space is foliated by invariant curves of the billiard map. We prove that, for any invariant curve $C$, there exists a sequence of eigenfunctions whose Cauchy data concentrates on $C$. We use this re…
Suppose that Omega is a bounded, piecewise smooth Euclidean domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on Omega with various boundary conditions are quantum ergodic if the classical billiard map (on the ball bundle of the boundary of Omega) is ergodic. Our proof is based on the classical observation that the boundary value of an interior eigenfunction is an eigenfunction of an operator F_h on the boundary of Omega, where h^{-2} is equal to the eigenvalue. In the case of the Neumann boundary condition, F_h is the boundary integral operator induced…
We study the localization of eigenfunctions produced by a point scatterer on a thin rectangle. We find an explicit set of eigenfunctions localized to part of the rectangle by showing that the one-dimensional Schrödinger operator with a Dirac delta potential asymptotically governs the spectral properties of the two-dimensional point scatterer. We also find the rate of localization in terms of the aspect ratio of the rectangle. In addition, we present numerical results regarding the asymptotic behavior of the localization.
The Laplace-Beltrami eigenfunctions on a compact Riemannian manifold $M$ whose geodesic billiard flow has mixed character have been conjectured by Percival to split into two complementary families, with all semiclassical mass supported in the completely integrable and ergodic regions of phase space respectively. In this paper, we consider the Dirichlet Laplacian on a family of mushroom billiards $M_t$ parametrised by the length $t\in(0,2]$ of their rectangular part. We prove that $M_t$ satisfies Percival's conjecture for almost all $t\in(0,2]$, hence providing the first example of a billiard k…
The paper addresses the the number of nodal domains for eigenfunctions of Schrödinger operators with Dirichlet boundary conditions in bounded domains. In dimension one, the $n$th eigenfunction has $n$ nodal domains. The Courant Theorem claims that in any dimension, the number of nodal domains of the $n$th eigenfunction cannot exceed $n$. However, in dimensions higher than 1 the equality can hold for only finitely many eigenfunctions. Thus, a "nodal deficiency" arises. Examples are known of eigenfunctions with arbitrarily large index $n$ that have just two nodal domains. It was suggested in the…
Consider the eigenfunctions $u$ for a ree rectangular membrane wo that $-Δu=λu$ on $\mathcal R(c,d)=(0,d)\times(0,d)$. In this note we show that if if $u>0$ on $\partial \mathcal R(c,d)) then $u\equiv C$ for some positive constant.
Bouncing-ball modes in ergodic billiards
通过改变翼部并使用参数变化论证，证明水平拉伸的 Bunimovich 台球在几乎每个参数下存在集中于矩形部分的特征函数序列。
We define a random model for the moments of the new eigenfunctions of a point scat-terer on a 2-dimensional rectangular flat torus. In the deterministic setting,Seba conjectured these moments to be asymptotically Gaussian, in the semi-classical limit. This conjecture was disproved by Kurlberg-Uebersch{ä}r on Diophantine tori. In our model, we describe the accumulation points in distribution of the randomized moments, in the semi-classical limit. We prove that asymptotic Gaussianity holds if and only if some function, modeling the multiplicities of the Laplace eigenfunctions, diverges to +$\inf…
We study the effects of perturbing the boundary of a rectangle on the nodal sets of eigenfunctions of the Laplacian. Namely, for a rectangle of a given aspect ratio $N$, we identify the first Dirichlet mode to feature a crossing in its nodal set and perturb one of the sides of the rectangle by a close to flat, smooth curve. Such perturbations will often "open" the crossing in the nodal set, splitting it into two curves, and we study the separation between these curves and their regularity. The main technique used is an approximate separation of variables that allows us to restrict study to the…
We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrodinger operators with mixed assumptions on the Hamiltonian flow.) Specificially, we assume that the billiard flow has an invariant ergodic component, U, and study defect measures, mu, of positivie density subsequences of eigenfunctions (and, more generally, of almost orthogonal quasimodes). We show that any defect measure associated to such a subsequence, when restricted to U, satisfies mu = c mu_L where mu_L is …
This paper is inspired from the nice result of Andrew Hassell on the eigenfunctions in the stadium billiard. From a classical paper of V. Arnol'd, we know that quasi-modes are not always close to exact modes. We show that, for almost all Riemannian metrics on closed surfaces with an elliptic generic closed geodesic C, there exists exact modes located on C. Related problems in the integrable case are discussed in several papers of John Toth and Steve Zelditch.
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