Spectral boundary conditions for generalizations of Laplace and Dirac operators

Gerd Grubb

Published 2003-02-24, updated 2003-04-04Version 4

Spectral boundary conditions for Laplace-type operators, of interest in string and brane theory, are partly Dirichlet, partly Neumann-type conditions, partitioned by a pseudodifferential projection. We give sufficient conditions for existence of associated heat trace expansions with power and power-log terms. The first log coefficient is a noncommutative residue, vanishing when the smearing function is 1. For Dirac operators with general well-posed spectral boundary conditions, it follows that the zeta function is regular at 0. In the selfadjoint case, the eta function has a simple pole at zero, and the value of zeta as well as the residue of eta at zero are stable under perturbations of the boundary projection of order at most minus the dimension.