On the Spectral Asymptotics of Operators on Manifolds with Ends

We deal with the asymptotic behaviour for λ → +∞ of the counting function N_P(λ) of certain positive selfadjoint operators P with double order (m, μ), m, μ > 0, m different from μ, 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(λ) and show how their behaviour depends on the ratio m/μ and the dimension of M.