Nonlinear Supersymmetry for Spectral Design in Quantum Mechanics

TL;DR

This paper reviews nonlinear supersymmetry techniques for designing quantum systems with specific spectral properties, including classification of SUSY algebra chains and extensions, providing new insights into quantum spectra and symmetries.

Contribution

It provides a comprehensive classification of SUSY algebra chains in quantum mechanics and explores extensions, including hidden symmetries and spectrum-generating algebras.

Findings

Classification of ladder-reducible and irreducible SUSY chains

Extension of SUSY with hidden symmetries and central charges

Embedding into non-stationary SUSY QM reveals new spectral insights

Abstract

Nonlinear (Polynomial, N-fold) SUSY approach to preparation of quantum systems with pre-planned spectral properties is reviewed. The full classification of ladder-reducible and irreducible chains of SUSY algebras in one-dimensional QM is given. Possible extensions of SUSY in one dimension are described. They include (no more than) ${\cal N} =2$ extended SUSY with two nilpotent SUSY charges which generate the hidden symmetry acting as a central charge. Embedding stationary quantum systems into a non-stationary SUSY QM is shown to yield new insight on quantum orbits and on spectrum generating algebras.