The heat trace for the drifting Laplacian and Schr\"odinger operators on manifolds

TL;DR

This paper analyzes the small-time asymptotic behavior of the heat trace for drifting Laplacian and Schrödinger operators on compact manifolds, revealing how regularity affects the expansion and eigenvalue distribution.

Contribution

It establishes partial and full asymptotic expansions of the heat trace depending on potential regularity, and explores isospectrality for the drifting Laplacian.

Findings

Existence of a six-term asymptotic expansion for finite regularity potentials.

Full asymptotic expansion of the heat trace for smooth weight functions.

Weyl law remains consistent with the standard Laplace-Beltrami operator.

Abstract

We study the heat trace for both the drifting Laplacian as well as Schr\"odinger operators on compact Riemannian manifolds. In the case of a finite regularity potential or weight function, we prove the existence of a partial (six term) asymptotic expansion of the heat trace for small times as well as a suitable remainder estimate. We also demonstrate that the more precise asymptotic behavior of the remainder is determined by and conversely distinguishes higher (Sobolev) regularity on the potential or weight function. 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.