Asian Journal of Mathematics

Volume 23 (2019)

Number 4

The heat trace for the drifting Laplacian and Schrödinger operators on manifolds

Pages: 539 – 560

DOI: https://dx.doi.org/10.4310/AJM.2019.v23.n4.a1

Authors

Nelia Charalambous (Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus)

Julie Rowlett (Department of Mathematics, Chalmers University and the University of Gothenburg, Sweden)

Abstract

We study the heat trace for both Schrödinger operators as well as the drifting Laplacian on compact Riemannian manifolds. In the case of a finite regularity (bounded and measurable) potential or weight function, we prove the existence of a partial asymptotic expansion of the heat trace for small times as well as a suitable remainder estimate. This expansion is sharp in the following sense: further terms in the expansion exist if and only if the potential or weight function is of higher Sobolev regularity. In the case of a smooth weight function, we determine the full asymptotic expansion of the heat trace for the drifting Laplacian for small times. We then use the heat trace to study the asymptotics of the eigenvalue counting function. In both cases the Weyl law coincides with the Weyl law for the Riemannian manifold with the standard Laplace–Beltrami operator. We conclude by demonstrating isospectrality results for the drifting Laplacian on compact manifolds.

Keywords

heat trace, drifting Laplacian, weighted Laplacian, Schrödinger operator, Weyl law, Weyl asymptotic, eigenvalue asymptotics

2010 Mathematics Subject Classification

Primary 58J35. Secondary 35P20, 58J50.

Full Text (PDF format)

Received 1 February 2017

Accepted 21 March 2018

Published 7 January 2020