New representations for square-integrable spheroidal functions

TL;DR

This paper introduces new mathematical representations for spheroidal functions to solve boundary value problems in quantum mechanics, specifically for oblate spheroidal coordinates, using a generalized Jaffe transformation.

Contribution

It develops a novel approach to construct eigenfunctions for spheroidal boundary value problems where traditional series methods fail, by applying a generalized Jaffe transformation.

Findings

Solution expressed as trigonometric and power series for zero charge parameter

Application to quantum ring spectral problem with spheroidal potential well

Enhanced methods for boundary value problems in spheroidal coordinates

Abstract

We discuss the solution of boundary value problems that arise after the separation of variables in the Schr\"odinger equation in oblate spheroidal coordinates. The specificity of these boundary value problems is that the singular points of the differential equation are outside the region in which the eigenfunctions are considered. This prevents the construction of eigenfunctions as a convergent series. To solve this problem, we generalize and apply the Jaffe transformation. We find the solution of the problem as trigonometric and power series in the particular case when the charge parameter is zero. Application of the obtained results to the spectral problem for the model of a quantum ring in the form of a potential well of a spheroidal shape is discussed with introducing a potential well of a finite depth.