The Witten Index of massless $(d+1)$-Dirac-Schr\"odinger Operators

TL;DR

This paper computes the Witten index for a class of non-Fredholm Dirac-Schr"odinger operators in odd dimensions, extending previous results and providing explicit formulas for certain potentials, revealing the index can be any real number.

Contribution

It generalizes the calculation of the Witten index for higher odd-dimensional Dirac-Schr"odinger operators and provides explicit formulas for specific potentials.

Findings

Witten index can be any real number for certain operators

Extended known results from 1D to higher odd dimensions

Provided explicit index formulas for specific potentials

Abstract

We calculate the Witten index of a class of (non-Fredholm) Dirac-Schr\"odinger operators over $\mathbb{R}^{d+1}$ for $d\geq 3$ odd, and thus generalize known results for the case $d=1$. For a concrete example of the potential, we give a more explicit index formula, showing that the Witten index assumes any real number on this class of operators.