  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.
