On the Spectral Asymptotics of Operators on Manifolds with Ends

Abstract

We deal with the asymptotic behaviour, for $\lambda \to +\infty$, of the counting function $N_P(\lambda )$ of certain positive self-adjoint operators P with double order $(m,\mu )$, $m,\mu$ > $0,m\ne \mu$ , defined on a manifold with ends M. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier integral operators associated with weighted symbols globally defined on ${ℝ}^n$. By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae for $N_P(\lambda )$ and show how their behaviour depends on the ratio $m/\mu$ and the dimension of M.