Asymptotics of the whispering gallery-type in the eigenproblem for the Laplacian in a revolutional domain diffeomorphic to a solid torus

TL;DR

This paper develops asymptotic eigenvalues and eigenfunctions of the Laplacian in a solid torus-shaped domain, revealing whispering gallery-type behaviors localized near the boundary and connecting them to classical billiard dynamics.

Contribution

It introduces explicit Airy function-based asymptotics for the Laplacian eigenproblem in revolutional domains, employing adiabatic approximation and linking to billiard dynamics.

Findings

Asymptotic eigenvalues and eigenfunctions constructed

Explicit Airy function representations derived

Connection established between whispering gallery asymptotics and billiard dynamics

Abstract

We consider the eigenproblem for the Laplacian inside a three-dimensional revolutional domain diffeomorphic to a solid torus and construct asymptotic eigenvalues and eigenfunctions (quasimodes) of the whispering gallery-type. The whispering gallery-type asymptotics are localized near the boundary of the domain, and an explicit analytic representations in terms of Airy functions is constructed for such asymptotics. There are several different scales in the problem, which makes it possible to apply the procedure of adiabatic approximation in the form of operator separation of variables to reduce the initial problem to one-dimensional problems up to the small correction. We also discuss the relation between the constructed whispering gallery-type asymptotics and classical billiards in the corresponding domain, in particularly, such asymptotics correspond to almost integrable billiards with proper degeneracy. We illustrate the results in the case when a revolutional domain is obtained by rotation of the triangle with rounded wedges.