Uniform Asymptotic Solution in the Form of an Airy Function for Semiclassical Bound States in One-Dimensional and Radially Symmetric Problems

Abstract

We consider stationary scalar and vector problems for differential and pseudodifferential operators leading to the appearance of asymptotic solutions of one-dimensional problems localized in a neighborhood of intervals (“bound states”). Based on the semiclassical approximation and the Maslov canonical operator, we develop a constructive algorithm that allows writing an asymptotic solution globally under certain conditions using an Airy function of complex argument.