Logarithmic terms in asymptotic expansions of heat operator traces

Abstract

Let P be an elliptic selfadjoint positive classical pseudodifferential operator of order d on a compact m-dimensional manifold without boundary. The heat trace of P has an asymptotic expansion inand tk log t for l=0,1,2,... and k=1,2,... We show that the coefficients of all terms in this expansion are non-trivial for a dense set of P. We show that the coefficient of the term is not locally computable when is a positive integer; the ramaining coefficients are known to be locally computable. —Let PB be an operator of Dirac type on a compact n-dimentional commanifold with smooth boundary such that the structures are product near the boundary; here a spectral boundary condition is imposed. LetIf n is even, the heat trace ofhas and asymptotic expansion inand k=0,1,2,...; if n is odd, there is an expansion without theterms. We show that all coefficients (all but one if n is odd) are nontrivial for a dense set of operators.