Sum rules for spectral densities of differential operators and generalized heat kernel expansion

Abstract

The study of the spectral measure for a second-order nonnegative elliptic differential operator acting on smooth fields defined over a noncompact manifold of arbitrary dimension is carried out, using the spectral theorem and a heat type kernel expansion. A set of sum rules concerning the spectral density is given. The singular structure of the resolvent kernel is briefly discussed. Some applications to quantum field theory are pointed out.