Abstract
We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order (1, 1), on an asymptotically Euclidean manifold. We first prove a two-term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator Q=(1+|x|^2)(1-varDelta ) on mathbb {R}^d.
Highlights
IntroductionX belongs to a class of compact manifolds with boundary, whose interior is equipped with a Riemannian metric g which assumes a specific form close to the boundary ∂X (see Definition 29 in Section A.1 of “Appendix”)
Let (X, g) be a d-dimensional asymptotically Euclidean manifold
In the case of SG-operators on manifolds with cylindrical ends, the leading order of the Weyl asymptotics was found by Maniccia and Panarese [22]
Summary
X belongs to a class of compact manifolds with boundary, whose interior is equipped with a Riemannian metric g which assumes a specific form close to the boundary ∂X (see Definition 29 in Section A.1 of “Appendix”). Hormander [18] proved, for a positive elliptic self-adjoint classical pseudodifferential operator of order m > 0 on a compact manifold, the Weyl law. In the case of SG-operators on manifolds with cylindrical ends (see Definition 41 and the relationship with asymptotically Euclidean manifolds at the end of Section A.4 of “Appendix”), the leading order of the Weyl asymptotics was found by Maniccia and Panarese [22]. Let P ∈ Op SGm cl ,m(X), m > 0, be a self-adjoint, positive, elliptic SG-classical pseudodifferential operator on an asymptotically Euclidean manifold X, and N (λ) its associated counting function. We give a short summary of the various trace operators and SG-Fourier integral operators
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