Efficient Semiclassical Asymptotics with Simple Caustics in a Boundary Value Problem

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In this paper we continue to develop the approach to constructing global uniform asymptotics of solutions of (pseudo)differential problems in terms of special functions based on the theory of the Maslov canonical operator. In particular, we show that if the corresponding Lagrangian manifold has a fold-type singularity, then the canonical operator on it is represented via the Airy function Ai and its derivative of complex arguments. This approach is illustrated by a known problem about construction of asymptotic eigenfunctions of the Laplace operator in an elliptic domain with Dirichlet boundary conditions. DOI 10.1134/S1061920825600072