Matrix superpotentials

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 $W=kQ+\frac{1}{k}R+P$ 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.

The previous version is extended by adding new results conmcerning the dual shape invariance. 27 pages

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• Minimal Realizations of Supersymmetry for Matrix Hamiltonians
• N-fold Supersymmetric Quantum Mechanics with Reflections
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• Supersymmetric Quantum Mechanics, Engineered Hierarchies of Integrable Potentials, and the Generalised Laguerre Polynomials
• Polynomial supersymmetry for matrix Hamiltonians: proofs