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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the trace of Schrödinger heat kernels and regularity of potentials


Author: Hart Smith
Journal: Trans. Amer. Math. Soc. 371 (2019), 3857-3875
MSC (2010): Primary 58J35; Secondary 35P25
DOI: https://doi.org/10.1090/tran/7486
Published electronically: December 3, 2018
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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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Additional Information

Hart Smith
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350
Email: hfsmith@uw.edu

DOI: https://doi.org/10.1090/tran/7486
Received by editor(s): March 21, 2017
Received by editor(s) in revised form: September 13, 2017
Published electronically: December 3, 2018
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant DMS-1500098
Article copyright: © Copyright 2018 American Mathematical Society