Localization of Laplacian eigenfunctions in circular, spherical and elliptical domains

TL;DR

This paper investigates high-frequency localization of Laplacian eigenfunctions in circular, spherical, and elliptical domains, revealing mechanisms, inequalities, and examples of localized modes, and contrasting with non-localized behaviors in rectangle-like domains.

Contribution

It provides a detailed analysis of localization mechanisms using special functions and introduces new examples and open problems in polygonal and piecewise smooth convex domains.

Findings

Localization occurs in convex domains like circles, spheres, and ellipses.

High-frequency localized modes exist in elliptical annuli.

Most rectangle-like domains do not exhibit localization.

Abstract

We consider Laplacian eigenfunctions in circular, spherical and elliptical domains in order to discuss three kinds of high-frequency localization: whispering gallery modes, bouncing ball modes, and focusing modes. Although the existence of these modes was known for a class of convex domains, the separation of variables for above domains helps to better understand the "mechanism" of localization, i.e. how an eigenfunction is getting distributed in a small region of the domain, and decays rapidly outside this region. Using the properties of Bessel and Mathieu functions, we derive the inequalities which imply and clearly illustrate localization. Moreover, we provide an example of a non-convex domain (an elliptical annulus) for which the high-frequency localized modes are still present. At the same time, we show that there is no localization in most of rectangle-like domains. This observation leads us to formulating an open problem of localization in polygonal domains and, more generally, in piecewise smooth convex domains.