Hierarchy of QM SUSYs on a Bounded Domain

TL;DR

This paper develops a systematic hierarchy of isospectral Hamiltonians in one-dimensional supersymmetric quantum mechanics, analyzing boundary conditions on intervals and circles, revealing restrictions and possibilities for constructing such hierarchies.

Contribution

It introduces a structured hierarchy of isospectral Hamiltonians with boundary conditions compatible with supersymmetry, highlighting differences between interval and circle geometries.

Findings

Hierarchy of up to three isospectral Hamiltonians on an interval

Infinite tower of isospectral Hamiltonians on a circle

Boundary conditions severely restrict supersymmetric compatibility

Abstract

We systematically formulate a hierarchy of isospectral Hamiltonians in one-dimensional supersymmetric quantum mechanics on an interval and on a circle, in which two successive Hamiltonians form N=2 supersymmetry. We find that boundary conditions compatible with supersymmetry are severely restricted. In the case of an interval, a hierarchy of, at most, three isospectral Hamiltonians is possible with unique boundary conditions, while in the case of a circle an infinite tower of isospectral Hamiltonians can be constructed with two-parameter family of boundary conditions.