On the Spectral Asymptotics of Operators on Manifolds with Ends

We deal with the asymptotic behaviour for $λ\to+\infty$ of the counting function $N_P(λ)$ of certain positive selfadjoint operators $P$ with double order $(m,μ)$, $m,μ>0$, $m\not=μ$, 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 $\mathbb{R}^n$. 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 $\frac{m}μ$ and the dimension of $M$.

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