PT-symmetric regularizations in supersymmetric quantum mechanics

TL;DR

This paper explores PT-symmetric regularizations in supersymmetric quantum mechanics, showing how shifting the coordinate axis into the complex plane resolves formal issues and broadens applicability to various models.

Contribution

It introduces a universal PT-SUSY regularization method by complex shifting, connecting it with non-Hermitian Hamiltonians and extending its use to complex and relativistic models.

Findings

Regularization via complex axis shift resolves formal obstacles in SUSY QM.

The method is applicable to non-Hermitian, PT-symmetric, and pseudo-Hermitian Hamiltonians.

Potential extension to complex, superintegrable, and relativistic quantum models.

Abstract

Supersymmetry offers one of the deepest insights in the concept of solvability in quantum mechanics. This insight is, paradoxically, restricted by one of the most serious formal drawbacks of the standard Witten's formulation of supersymmetric quantum mechanics which lies in the Jevicki-Rodrigues' postulate of absence of poles in superpotentials W(x) over all the real axis of coordinates x. In our review we emphasize that this obstacle is artificial and that it disappears immediately after a suitable (say, constant) shift of the axis of x into complex plane. Detailed attention is paid to a close relationship between this common trick and the recent not quite expected increase of interest in non-Hermitian (a. k. a. PT-symmetric or pseudo-Hermitian) Hamiltonians. We show that the resulting PT-SUSY regularization recipe proves both easy and universal. An insight into its mathematics is mediated by the complex harmonic oscillator with a centrifugal-like spike. An exhaustive discussion of the role of the strength of this spike is offered. In addition we recollect the possibility of a re-formulation of the recipe in the second-order SUSY language. Finally we list a few promising directions of applicability of our PT-SUSY regularization prescription to a few more complicated nonrelativistic models (superintegrable Hamiltonians of the Smorodinsky-Winternitz and of the Calogero-Sutherland type) and to the relativistic Klein-Gordon equation (as well as to all of its unphysical higher-order analogues).