On indices of the Dirac operator in a non-Fredholm case

TL;DR

This paper investigates the spectral asymmetry of the Dirac operator with Aharonov-Bohm potential, revealing limitations of standard index definitions in non-Fredholm cases with continuous spectra.

Contribution

It demonstrates that traditional index methods fail to capture spectral asymmetry contributions from the continuous spectrum in non-Fredholm operators.

Findings

Standard indices miss continuous spectrum contributions

Spectral asymmetry arises entirely from the continuous spectrum

Regularization methods may give vanishing results in non-Fredholm cases

Abstract

The Dirac Hamiltonian with the Aharonov-Bohm potential provides an example of a non-Fredholm operator for which all spectral asymmetry comes entirely from the continuous spectrum. In this case one finds that the use of standard definitions of the resolvent regularized, the heat kernel regularized, and the Witten indices misses the contribution coming from the continuous spectrum and gives vanishing spectral asymmetry and axial anomaly. This behaviour in the case of the continuous spectrum seems to be general and its origin is discussed.