Generic matrix superpotentials

TL;DR

This paper provides a comprehensive classification of matrix superpotentials of various dimensions, enabling the construction of new integrable coupled Schrödinger systems and multidimensional models through shape invariance.

Contribution

It offers an explicit, algorithmic classification of matrix superpotentials of dimensions 2x2, 3x3, and arbitrary sizes, expanding the set of integrable quantum systems.

Findings

Complete list of 2x2 matrix superpotentials.

Explicit description of special 3x3 superpotentials.

Construction of multidimensional integrable models.

Abstract

A simple and algorithmic description of matrix shape invariant potentials is presented. The complete lists of generic matrix superpotentials of dimension $2\times2$ and of special superpotentials of dimension $3\times3$ are given explicitly. In addition, a constructive description of superpotentials realized by matrices of arbitrary dimension is presented. In this way an extended class of integrable systems of coupled Schr\"odinger equation is classified. Examples of such systems are considered in detail. New integrable multidimensional models which are reduced to shape invariant systems via separation of variables are presented also.