Zero modes and self-isospectral potentials in periodic supersymmetric quantum mechanics

Abstract

In standard discussions of nonperiodic supersymmetric quantum mechanics either one or neither (but not both) of the isospectral pair of potentials has a zero mode. In contrast, in supersymmetric quantum mechanical models with periodic potentials it is possible for both isospectral potentials to support zero modes. 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.