On the trace of Schrödinger heat kernels and regularity of potentials

Smith, Hart

doi:10.1090/tran/7486

On the trace of Schrödinger heat kernels and regularity of potentials
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Trans. Amer. Math. Soc. 371 (2019), 3857-3875 Request permission

Abstract:

For the Schrödinger operator $-\Delta _{\mathrm {g}}+V$ 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 Schrödinger heat kernel. As an application, we generalize a result of Sà Barreto and Zworski concerning the existence of resonances on compact metric perturbations of Euclidean space to the case of bounded measurable potentials.

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