  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.
