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Trace asymptotics for fractional Schrödinger operators. (English) Zbl 1290.35301
Summary: This paper proves an analogue of a result of R. Bañuelos and A. Sá Barreto [Commun. Partial Differ. Equations 20, No.  11–12, 2153–2164 (1995; Zbl 0843.35016)] on the asymptotic expansion for the trace of Schrödinger operators on \(\mathbb R^d\) when the Laplacian \(-{\Delta}\), which is the generator of the Brownian motion, is replaced by the non-local integral operator \((-{\Delta})^{{\alpha}/2}\), \(0 < {\alpha} < 2\), which is the generator of the symmetric stable process of order \({\alpha}\). These results also extend recent results of R. Bañuelos and S. Y. Yolcu [J. Lond. Math. Soc., II. Ser. 87, No. 1, 304–318 (2013; Zbl 1270.35142)] where the first two coefficients for \((-{\Delta})^{{\alpha}/2}\) are computed. Some extensions to Schrödinger operators arising from relativistic stable and mixed-stable processes are obtained.

MSC:
35R11 Fractional partial differential equations
35J10 Schrödinger operator, Schrödinger equation
35K08 Heat kernel
60G52 Stable stochastic processes
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