Near integrable systems

TL;DR

This paper investigates a two-dimensional quantum billiard with special boundary conditions, revealing that most eigenfunctions are localized and eigenvalues are near those of a Neumann billiard, supported by numerical evidence.

Contribution

It provides a detailed analysis of eigenfunction localization and eigenvalue proximity in a novel quantum billiard system with unusual boundary conditions.

Findings

Eigenfunctions are strongly localized in most cases.

Eigenvalues are close to those of the Neumann boundary condition billiard.

Numerical calculations support the analytical results.

Abstract

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 localized and the corresponding eigenvalues are close to eigenvalues of the circular billiard with Neumann boundary conditions. Deviations from strong localization are also discussed. These results agree well with numerical calculations.