Related papers: Local limits of high energy eigenfunctions on inte…
The plane wave decomposition method (PWDM) is one of the most popular strategies for numerical solution of the quantum billiard problem. The method is based on the assumption that each eigenstate in a billiard can be approximated by a…
A two-dimensional circular quantum billiard with unusual boundary conditions introduced by Berry and Dennis (\emph{J Phys A} {\bf 41} (2008) 135203) is considered in detail. It is demonstrated that most of its eigenfunctions are strongly…
We investigate the integrability of Kepler billiards-mechanical billiard systems in which a particle moves under the influence of a Keplerian potential and reflects elastically at the boundary of a strictly convex planar domain. Our main…
We study chaotic eigenfunctions in wedge-shaped and rectangular regions using a generalization of Berry's conjecture. An expression for the two-point correlation function is derived and verified numerically.
We use scanning near-field optical microscopy to image hyperbolic phonon polaritons in hexagonal boron nitride (hBN) billiards with integrable and chaotic geometries. In Sinai billiards, we observe irregular mode patterns consistent with…
The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a…
In this work we study the geometrical properties of the high-lying eigenfunctions (200,000 and above) which are deep in the semiclassical regime. The system we are analyzing is the billiard system inside the region defined by the quadratic…
The Random Wave Conjecture of M. V. Berry is the heuristic that eigenfunctions of a classically chaotic system should behave like Gaussian random fields, in the large eigenvalue limit. In this work we collect some definitions and properties…
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…
Non-universal correlations due to localization are observed in statistical properties of experimental eigenfunctions of quantum chaotic and disordered microwave cavities. Varying energy {E} and mean free path {l} enable us to experimentally…
The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove…
In this paper, we introduce a new notion of convergence for the Laplace eigenfunctions in the semiclassical limit, the local weak convergence. This allows us to give a rigorous statement of Berry's random wave conjecture. Using recent…
The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between their centers, as introduced by Heller and Tomsovic in Phys. Today 46 38 (1993). We study two classical and…
We study the statistical properties of the high-lying chaotic eigenstates (200,000 and above) which are deep in the semiclassical regime. The system we are analyzing is the billiard system inside the region defined by the quadratic…
In the present work we explore the concept of solitary wave billiards. I.e., instead of a point particle, we examine a solitary wave in an enclosed region and explore its collision with the boundaries and the resulting trajectories in cases…
Eigenstates and energy levels of a square quantum billiard in a magnetic field, or with an Aharonov-Bohm flux line, are found in quasiclassical approximation, that is, for high enough energy. Explicit formulas for the energy levels and…
The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely, integrability close to the boundary, and prove…
We show that wave functions in planar rational polygonal billiards (all angles rationally related to Pi) can be expanded in a basis of quasi-stationary and spatially regular states. Unlike the energy eigenstates, these states are directly…
The boundary integral method (BIM) is a formulation of Helmholtz equation in the form of an integral equation suitable for numerical discretization to solve the quantum billiard. This paper is an extensive numerical survey of BIM in a…
The methods of the high energy semiclassical quantization in the rational polygon billiards used in our earlier papers are generalized to an arbitrary rational multi-connected polygon billiards i.e. to the billiards which is a rational…