Global Asymptotics for Functions of Parabolic Cylinder and Solutions of the Schrödinger Equation with a Potential in the Form of a Nonsmooth Double Well

Abstract

In the paper, an approach is discussed that makes it possible to obtain global formulas in terms of Airy functions \({\rm Ai}\) and \({\rm Bi}\) of compound argument for the asymptotics of the functions of parabolic cylinder \(D_{\nu}(z)\) for real \(z\) and large \(\nu\). The parabolic cylinder functions are determined from the Schrödinger equation, with potential in the form of a quadratic parabola, whose asymptotic solution can be constructed using the semiclassical approximation. In this case, the Bohr–Sommerfeld condition singles out the functions with an integer index whose asymptotics is determined only by the function \({\rm Ai}\). For noninteger indices, the function \({\rm Bi}\) also contributes into the asymptotics.