Exactly solvable Schr\"odinger operators

TL;DR

This paper classifies 1D Schr"odinger equations solvable via hypergeometric functions, introducing two new classes reducible to the Hermite equation, expanding the set of exactly solvable quantum models.

Contribution

It provides a systematic classification of solvable Schr"odinger equations, including two newly identified classes reducible to Hermite equations.

Findings

Identified all known hypergeometric solvable Schr"odinger equations

Explicitly described two new classes reducible to Hermite equations

Enhanced the understanding of exactly solvable quantum systems

Abstract

We systematically describe and classify 1-dimensional Schr\"odinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe 2 new classes of exactly solvable Schr\"odinger equations that can be reduced to the Hermite equation.