Self-Isospectral Periodic Potentials and Supersymmetric Quantum Mechanics

TL;DR

This paper explores supersymmetric quantum models with periodic potentials that can support zero modes in both isospectral partners, leading to unbroken supersymmetry with a zero Witten index, and provides explicit self-isospectral examples.

Contribution

It introduces the concept of self-isospectral periodic potentials supporting zero modes in both partners, expanding understanding of supersymmetry in periodic quantum systems.

Findings

Both isospectral potentials can support zero modes.

Supersymmetry can be unbroken with a zero Witten index.

Explicit self-isospectral examples are provided.

Abstract

We discuss supersymmetric quantum mechanical models with periodic potentials. The important new feature is that it is possible for both isospectral potentials to support zero modes, in contrast to the standard nonperiodic case where either one or neither (but not both) of the isospectral pair has a zero mode. Thus it is possible to have supersymmetry unbroken and yet also have a vanishing Witten index. We present some explicit exactly soluble examples for which the isospectral potentials have identical band spectra, and which are ``self-isospectral'' in the sense that the potentials have identical shape, but are translated by one half period relative to one another.