Logarithmic terms in trace expansions of Atiyah-Patodi-Singer problems

TL;DR

This paper investigates how perturbations of Dirac-type operators with spectral boundary conditions affect the logarithmic terms in the heat trace expansion, revealing stability properties and conditions for vanishing coefficients.

Contribution

It introduces a pseudodifferential calculus approach to analyze the stability of log-terms under boundary-vanishing perturbations and identifies conditions for their vanishing in odd dimensions.

Findings

First k log-terms are stable under boundary-vanishing perturbations.

Log-coefficients vanish for certain perturbations in odd dimensions.

Nonlocal power coefficients are only locally affected by perturbations.

Abstract

For a Dirac-type operator D with a spectral boundary condition, the associated heat operator trace has an expansion in powers and log-powers of t. Some of the log-coefficients vanish in the Atiyah-Patodi-Singer product case. We here investigate the effect of perturbations of D, by use of a pseudodifferential parameter-dependent calculus for boundary problems. It is shown that the first k log-terms are stable under perturbations of D vanishing to order k at the boundary (and the nonlocal power coefficients behind them are only locally perturbed). For perturbations of D from the APS product case by tangential operators commuting with the tangential part A, all the log-coefficients vanish if the dimension is odd.