Asymptotics of Wave Functions of the Stationary Schrödinger Equation in the Weyl Chamber

Abstract

We study stationary solutions of the Schrödinger equation with a monotonic potential U in a polyhedral angle (Weyl chamber) with the Dirichlet boundary condition. The potential has the form \(U\left( x \right) = \sum _{j = 1}^nV\left( {{x_j}} \right),x = \left( {{x_1}, \ldots ,{x_n}} \right) \in {\mathbb{R}^n}\), with a monotonically increasing function V (y). We construct semiclassical asymptotic formulas for eigenvalues and eigenfunctions in the form of the Slater determinant composed of Airy functions with arguments depending nonlinearly on x_j. We propose a method for implementing the Maslov canonical operator in the form of the Airy function based on canonical transformations.