Logarithmic Terms in Trace Expansions of Atiyah–Patodi–Singer Problems

Abstract

For Dirac-type operator D on a manifold X with a spectral boundarycondition (defined by a pseudodifferential projection), the associated heatoperator trace has an expansion in integer and half-integer powers and log-powersof t; the interest in the expansion coefficients goes back to the work of Atiyah,Patodi and Singer. In the product case considered by APS, it is known that allthe log-coefficients vanish when dim X is odd, whereas the log-coefficients atinteger powers vanish when dim X is even. We investigate here whether this partialvanishing of logarithms holds more generally. One type of result, shown forgeneral D with well-posed boundary conditions, is that a perturbation of Dby a tangential differential operator vanishing to order k on the boundaryleaves the first k log-power terms invariant (and the nonlocal power termsof the same degree are only locally perturbed). Another type of result is thatfor perturbations of the APS product case by tangential operators commuting withthe tangential part of D, all the logarithmic terms vanish when dim X is odd(whereas they can all be expected to be nonzero when dim X is even). The treatmentis based on earlier joint work with R. Seeley and a recent systematic parameter-dependentpseudodifferential boundary operator calculus, applied to the resolvent.