Matrix superpotentials

TL;DR

This paper introduces a comprehensive set of matrix-valued shape invariant potentials in SUSY quantum mechanics, including known and new potentials, with solutions and spectral properties explicitly derived.

Contribution

It provides a classification of matrix superpotentials of a specific form, recovering known potentials and presenting five new shape invariant potentials with dual invariance properties.

Findings

Recovered all known scalar shape invariant potentials

Presented five new shape invariant potentials

Derived spectra and eigenvectors for the new potentials

Abstract

We present a collection of matrix valued shape invariant potentials which give rise to new exactly solvable problems of SUSY quantum mechanics. It includes all irreducible matrix superpotentials of the generic form $W=kQ+\frac1k R+P$ where $k$ is a variable parameter, $Q$ is the unit matrix multiplied by a real valued function of independent variable $x$, and $P$, $R$ are hermitian matrices depending on $x$. In particular we recover the Pron'ko-Stroganov "matrix Coulomb potential" and all known scalar shape invariant potentials of SUSY quantum mechanics. In addition, five new shape invariant potentials are presented. Three of them admit a dual shape invariance, i.e., the related hamiltonians can be factorized using two non-equivalent superpotentials. We find discrete spectrum and eigenvectors for the corresponding Schroedinger equations and prove that these eigenvectors are normalizable.