Whispering-gallery modes and resonances of an elliptic cavity

Eigenfunctions of the whispering-gallery type in elliptic cavities are cons idered. Asymptotic expansions for resonances are derived from the uniform a symptotic expansions of Mathieu functions and modified Mathieu functions co nstructed by applying the Langer-Olver method. These asymptotic expansions are improved by including exponentially small terms which lie beyond all or ders of the perturbative series and can be captured by carefully taking int o account Stokes's phenomenon. A classification of resonances along the fou r irreducible representations of C-2v (the symmetry group of the elliptic c avity) is provided, and the splitting up of resonances is then understood i n connection with the breaking of O(2)-symmetry (invariance under any rotat ion).