On the Spectral Asymptotics of Operators on Manifolds with Ends

Sandro Coriasco,Lidia Maniccia

doi:10.1155/2013/909782

Abstract

We deal with the asymptotic behaviour, forλ→+∞, of the counting functionNP(λ)of certain positive self-adjoint operatorsPwith double order(m,μ),m,μ>0, m≠μ, defined on a manifold with endsM. 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 onℝn. By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae forNP(λ)and show how their behaviour depends on the ratiom/μand the dimension ofM.

Highlights

The aim of this paper is to study the asymptotic behaviour, for λ → +∞, of the counting function NP (λ) = ∑ 1, (1)λ j ≤λ where λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ is the sequence of the eigenvalues, repeated according to their multiplicities, of a positive order, self-adjoint, classical, elliptic SG-pseudodifferential operator P on a manifold with ends
We indicate the subspaces of classical symbols and operators adding the subscript cl to the notation introduced above
We introduce the class of noncompact manifolds with which we will deal

Summary

Introduction

The aim of this paper is to study the asymptotic behaviour, for λ → +∞, of the counting function. We deal with the case of manifolds with ends for P ∈ ELmcl,μ(M), positive and self-adjoint, such that m, μ > 0, m ≠ μ, focusing on the (invariant) meaning of the constants appearing in the corresponding Weyl formulae and on achieving a better estimate of the remainder term. In view of the technique used there, the remainder terms appeared in the form o(λn/ min{m,μ}) and o(λn/m log λ) for m ≠ μ and m = μ, respectively An improvement in this direction for operators on Rn had been achieved by Nicola [16], who, in the case m = μ, proved that. By studying the asymptotic behaviour in λ of the trace of the operator ψλ(−Q), ψλ(t) = ψ(t)e−itλ, ψ ∈ C0∞(R), defined via a Spectral Theorem and approximated in terms of Fourier Integral Operators, we prove the following.

SG-Classical Operators on Manifolds with Ends

Proof of Theorem 1

Solution of Cauchy Problems and SG Fourier Integral Operators