Matrix superpotentials

Details Matrix superpotentials Matrix superpotentials

2011-01-21

arxiv.org/abs/1101.4129v2

J.Phys.A44:305204,2011

Physics

Mathematical Physics

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

Scientific paper

10.1088/1751-8113/44/30/305204

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.

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