Three-dimensional Matrix Superpotentials

TL;DR

This paper classifies three-dimensional matrix superpotentials of a specific form, enabling the exact solvability of certain Schrödinger equations by explicitly listing all such superpotentials.

Contribution

It provides a complete classification of matrix superpotentials of a particular form in three dimensions, expanding the set of exactly solvable Schrödinger systems.

Findings

Explicit list of three-dimensional matrix superpotentials.

Identification of superpotentials of the form W_k = kQ + P + 1/k R.

Enhanced understanding of solvable quantum systems.

Abstract

This article considers the classification of matrix superpotentials that corresponds to exactly solvable systems of Schrodinger equations. Superpotentials of the following form are considered: $W_k = kQ + P + \frac1kR$, where $k$ --- parameter, $P, Q$ and $R$ --- hermitian matrices, that depend on a variable $x$. The list of three-dimensional matrix superpotentials is presented explicitly.