Eigenvalue asymptotics for a class of md-elliptic ψdo’s on manifolds with cylindrical exits

Abstract

We obtain a Weyl formula for a class of positive md-elliptic operators on manifolds with finitely many cylindrical exits. Weyl formula is deduced by a classical Tauberian theorem through the asymptotic expansion at t = 0 of the trace of the heat parametrix. The constant of the leading term is expressed invariantly by means of the usual principal symbol and exit symbols.

Complex powers of classical SG-pseudodifferential operators

“…This we shall derive in Section 3, giving also explicit formulas for all homogeneous components of the complex powers. It is worthwhile to point out that the formulas here obtained could be of some interest in connection with the study of the asymptotic behavior of eigenvalues of SGclassical operators in the spirit of [14].…”

Weyl laws on open manifolds

“…Remarkably, some sort of spectral asymptotics for the Laplacian survive for g 0 , as shown by Christiansen and Zworski [8]. Let us note that a similar (though analytically quite different) result was obtained by Maniccia and Panarese [22] for ends isometric to the Euclidean space outside a ball.…”

“…Acknowledgments I am grateful to Andrei Moroianu for several useful discussions, and to Victor Nistor for pointing out the paper [22].…”

A note on the Einstein–Hilbert action and Dirac operators on $${\mathbb{R}^n}$$

“…The asymptotic expansion (3.18) also holds in the case of manifolds with cylindrical ends, since the computations are completely analogous and purely local, see [2]. The evaluation of the first term of the asymptotic expansion can be found in [24]: the expression of the second term then follows, using the properties of generalised Laplacians. In view of our hypotheses, the right hand side of (3.18) is a classical SG-symbol: then, we obtain…”