Planarizable Supersymmetric Quantum Toboggans

TL;DR

This paper explores the regularization of singularities in supersymmetric quantum mechanics using complex deformations called quantum toboggans, revealing new solvable models with non-singular differential equations.

Contribution

It introduces a method to regularize supersymmetric quantum models via complex deformations, making certain toboggan models reducible to standard Sturm-Schrödinger equations.

Findings

Certain supersymmetric quantum toboggan pairs are shown to be reducible to non-singular differential equations.

Theoretical framework for complex deformations in supersymmetric quantum mechanics is reviewed.

New classes of solvable models with regularized potentials are identified.

Abstract

In supersymmetric quantum mechanics the emergence of a singularity may lead to the breakdown of isospectrality between partner potentials. One of the regularization recipes is based on a topologically nontrivial, multisheeted complex deformations of the line of coordinate $x$ giving the so called quantum toboggan models (QTM). The consistent theoretical background of this recipe is briefly reviewed. Then, certain supersymmetric QTM pairs are shown exceptional and reducible to doublets of non-singular ordinary differential equations a.k.a. Sturm-Schr\"odinger equations containing a weighted energy $E\to EW(x)$ and living in single complex plane.