Steve Zelditch
This is a report for the 2003 Forges Les Eaux PDE conference on recent results with A. Hassell on quantum ergodicity of boundary traces of eigenfunctions on domains with ergodic billiards, and of work in progress with Hassell and Sogge on norms of boundary traces. Related work by Burq, Grieser and Smith-Sogge is also discussed.
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In this paper, the results of Burq and Zworski are further developed to study nonconcentration of eigenfunctions for billiards which have rectangular components: these include the Buminovich billiard, the Sinai billiard, and certain pseudointegrable billiards. The results presented are an application of using a "black box" point of view as presented by the same authors.
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
In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in \cite{M1}. There, the methods developed in Burq-Zworski \cite{BZ3} to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $$ \int_U |u|^2 \geq c \int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$.
The Seba billiard, a rectangular torus with a point scatterer, is a popular model to study the transition between integrability and chaos in quantum systems. Whereas such billiards are classically essentially integrable, they may display features such as quantum ergodicity [KU] which are usually associated with quantum systems whose classical dynamics is chaotic. Seba proposed that the eigenfunctions of toral point scatterers should also satisfy Berry's random wave conjecture, which implies that the semiclassical moments of the eigenfunctions ought to be Gaussian. We prove a conjecture of Keat…
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…
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 consider the Laplacian with a delta potential (a "point scatterer") on an irrational torus, where the square of the side ratio is diophantine. The eigenfunctions fall into two classes ---"old" eigenfunctions (75%) of the Laplacian which vanish at the support of the delta potential, and therefore are not affected, and "new" eigenfunctions (25%) which are affected, and as a result feature a logarithmic singularity at the location of the delta potential. Within a full density subsequence of the new eigenfunctions we determine all semiclassical measures in the weak coupling regime and show that…
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…
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 …
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…
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…
Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface $H$ so that restrictions $ϕ_j |_H$ to $H$ of $Δ$-eigenfunctions of Riemannian manifolds $(M, g)$ with ergodic geodesic flow are quantum ergodic on $H$. We prove two kinds of results: First (i) for any smooth hypersurface $H$, the Cauchy data $(ϕ_j|H, \partial ϕ_j|H)$ is quantum ergodic if the Dirichlet and Neumann data are weighted appropriately. Secondly (ii) we give conditions on $H$ so that the Dirichlet (or Neumann) data is individually quantum ergodic. The condition involves the almost nowhere equalit…
Trace estimates and improved pointwise bounds for joint eigenfunctions
本文改进了量子可积系统中联合本征函数的标准Hölder界，通过建立满足秩k非退化条件的点的sharp界h^{(-n+k+1)/2}。
We prove an analogue of Sogge's local $L^p$ estimates for $L^p$ norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq-Gérard-Tzvetkov, Hu, and Chen-Sogge. The improvements are logarithmic on negatively curved manifolds (without boundary) and by $o(1)$ for manifolds (with or without boundary) with ergodic geodesic flows. In the case of ergodic billiards with piecewise smooth boundary, we get $o(1)$ improvements on $L^\infty$ estimates of Cauchy data away from a shrinking neighborhood of…
Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric optics is ergodic. We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by Gérard, Leichtnam \cite{GeLe93-1} and Hassel, Zelditch \cite{HaZe02} obtained under the additional assumption of the convexity of $M$.
The purpose of this article is to develop a Hadamard-Riesz type parametrix for the wave propagator in bounded planar domains with smooth, strictly convex boundary. This parametrix then allows us to rederive an oscillatory integral representation for the wave trace appearing in \cite{MaMe82} and compute its principal symbol explicitly in terms of geometric data associated to the billiard map. This results in new formulas for the wave invariants. The order of the principal symbol, which appears to be inconsistent in the works of \cite{MaMe82} and \cite{Popov1994}, is also corrected. In those pap…
In this paper, we study the boundary traces of eigenfunctions on the boundary of a smooth and bounded domain. An identity derived by Bäcker, Fürstburger, Schubert, and Steiner, expressing (in some sense) the asymptotic completeness of the set of boundary traces in a frequency window of size O(1), is proved both for Dirichlet and Neumann boundary conditions. We then prove a semiclassical generalization of this identity.
Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus
本文研究d维环面上拉普拉斯特征函数关于任意Borel测度的迹不等式与可观测性不等式，基于测度可积性、正则性及支撑集维数给出充分与必要条件，将Zygmund和Bourgain–Rudnick经典结果推广到高维，并应用于量子极限约束与Schrödinger方程控制理论。
It is an open problem in general to prove that there exists a sequence of $Δ_g$-eigenfunctions $ϕ_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(ϕ_{j_k}) $ of nodal domains tends to infinity with the eigenvalue. Our main result is that $N(ϕ_{j_k}) \to \infty$ along a subsequence of eigenvalues of density $1$ if the $(M, g)$ is a non-positively curved surface with concave boundary, i.e. a generalized Sinai or Lorentz billiard. Unlike the recent closely related work of Ghosh-Reznikov-Sarnak and of the authors on the nodal domain counting problem, the surfaces need not have any …
We consider a spectral optimal design problem involving the Neumann traces of the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset $Ω$ of $\R^n$. The cost functional measures the amount of energy that Dirichlet eigenfunctions concentrate on the boundary and that can be recovered with a bounded density function. We first prove that, assuming a $L^1$ constraint on densities, the so-called {\it Rellich functions} maximize this functional.Motivated by several issues in shape optimization or observation theory where it is relevant to deal with bounded densities, and noticing that …
We prove that the Cauchy data of Dirichlet or Neumann $Δ$- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e. a point at which a positive measure of geodesics leaving the point return to the point. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds.
Characterizations of eigenvalues and eigenfunctions of the Laplacian on a product domain are obtained. When zero Dirichlet, Robin or Neumann boundary conditions are specified on each factor, then the eigenfunctions on the product domain are precisely the products of the eigenfunctions on the individual factors. There is a related result when Steklov boundary conditions are specified on the second factor. These results enable the characterization of certain Hilbert-Sobolev spaces as tensor products and descriptions of some orthogonal bases of the spaces. A different characterization of the trac…
A few sharp estimates of harmonic functions with applications to Steklov eigenfunctions
本文结合微局部分析与频率函数技术，建立了光滑紧流形上调和函数限制的精确下界，并通过构造参数化分析其内部行为，最终应用于 Steklov 特征函数的估计与调和函数的数值逼近。
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