Exact and quasi-exact solvability of two-dimensional superintegrable quantum systems. I. Euclidean space

TL;DR

This paper demonstrates how separation of variables in 2D Euclidean superintegrable quantum systems leads to exactly and quasi-exactly solvable problems, proposing new definitions based on hypergeometric and polynomial solutions.

Contribution

It introduces new definitions of ES and QES quantum problems and links superintegrability to solvability classifications in quantum mechanics.

Findings

Separation of variables yields ES and QES problems.

New definitions of ES and QES based on solution forms.

Polynomial solutions correspond to QES systems.

Abstract

In this article we show that separation of variables for second-order superintegrable systems in two-dimensional Euclidean space generates both exactly solvable (ES) and quasi-exactly solvable (QES) problems in quantum mechanics. In this article we propose the another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schroedinger equation, can be expressed in terms of hypergeometrical functions $_mF_n$ and is QES if the Schroedinger equation admit polynomial solutions with the coefficients satisfying the three-term or more higher order of recurrence relations