Trace Asymptotics for Fractional Schrodinger Operators

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.