Asymptotic Eigenfunctions of the Laplace Operator in an Elliptic Domain Corresponding to Birkhoff Billiards with Hyperbolic Caustics

A geometric approach to constructing the asymptotic behavior of the eigenfunctions of the Laplace operator in an elliptic domain with Dirichlet conditions on the boundary is presented for the case in which the caustics are hyperbolas. This approach consists of constructing, from certain caustics, families of noncompact invariant Lagrangian manifolds and generalized asymptotic eigenfunctions of the Laplace operator, defined in the form of Maslov canonical operators on them. Then in the plane we select curves on which the resulting functions vanish, thus serving as solutions to the Dirichlet problem in domains bounded by these curves. In this paper, we present effective formulas for the asymptotic behavior in terms of the Airy function and also discuss the question of choosing suitable coordinates on invariant Lagrangian manifolds, which is important for obtaining constructive asymptotic formulas.