Trace Asymptotics for Fractional Schrodinger Operators

TL;DR

This paper extends trace asymptotics results for Schrödinger operators by replacing the Laplacian with fractional and non-local operators, providing new asymptotic expansions and extending previous coefficient computations.

Contribution

It introduces asymptotic trace expansions for fractional Schrödinger operators and extends existing results to relativistic and mixed stable processes.

Findings

Derived asymptotic expansions for fractional Schrödinger operators

Extended coefficient calculations to non-local operators

Applied results to relativistic stable processes

Abstract

This paper proves an analogue of a result of Banuelos and Sa Barreto on the asymptotic expansion for the trace of Schrodinger operators on $\R^d$ 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 Banuelos and Yildirim where the first two coefficients for $\Delta^{\alpha/2}$ are computed. Some extensions to Schrodinger operators arising from relativistic stable and mixed stable processes are obtained.