  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.
