On the trace of Schr\"odinger heat kernels and regularity of potentials

For the Schr\"odinger operator $-\Delta_\rm{g}+V$ on a complete Riemannian manifold with real valued potential $V$ of compact support, we establish a sharp equivalence between Sobolev regularity of $V$ and the existence of finite-order asymptotic expansions as $t\rightarrow 0$ of the relative trace of the Schr\"odinger heat kernel. As an application, we generalize a result of S\`a Barreto and Zworski, concerning the existence of resonances on compact metric perturbations of Euclidean space, to the case of bounded measurable potentials.