LOCALIZATION OF LAPLACIAN EIGENFUNCTIONS IN CIRCULAR, SPHERICAL, AND ELLIPTICAL DOMAINS

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 has been known for a class of convex domains, the separation of variables for circular, spherical, and elliptical domains helps us 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 inequalities which imply and clearly illustrate localization. Moreover, we provide an example of a nonconvex 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 rectangle-like domains. This observation leads us to formulate an open problem of localization in polygonal domains and, more generally, in piecewise smooth convex domains.