Trace asymptotics for fractional Schrödinger operators

Abstract

This paper proves an analogue of a result of Bañuelos and Sá Barreto [6] on the asymptotic expansion for the trace of Schrödinger operators on R d when the Laplacian ∆, which is the generator of the Brownian motion, is replaced by the non-local integral operator ∆ α/2 , 0 < α < 2, which is the generator of the symmetric stable process of order α. These results also extend recent results of Bañuelos and Yildirim [3] where the first two coefficients for ∆ α/2 are computed. Some extensions to Schrödinger operators arising from relativistic stable and mixed stable processes are obtained.

Heat Content for Stable Processes in Domains of $$\mathbb {R}^d$$

“…It is interesting to notice that the last limit intuitively gives an insight that the surface area of the boundary should be recovered when considering the small time behavior of the function H (1) Ω,Ω c (t)(Cauchy process, α = 1) which is exactly what our next result shows.…”

“…Supported in part by NSF Grant #0603701-DMS, Rodrigo Bañuelos, PI. 1 As for the cases 0 < α < 2, it is a standard fact that the transition densities p (α) t (x, y) can be written in terms of the α/2-subordinator (see [1, p. 522] for further details). That is, p (α) t (x, y) = E p (2) St (x, y) = ∞ 0 ds p (2) s (x, y) η (α/2) t (s).…”

“…The transition densities p (α) t (x, y) are known to have an explicit expression only for α = 2 and α = 1. In fact, for α = 1, the function p (1) t (x, y) is called the Cauchy (or Poisson in analysis) heat kernel and is given by p (1) t (x, y) =…”

“…Here, (−∆) α 2 denotes the fractional Laplacian and the interested reader may consult [1,2,3,6,9] for a detailed treatment and applications to the theory of Schrödinger operators and scattering theory. In other words, the initial value problem (1.10) exactly says that H (α) Ω (t) represents the amount of heat in Ω, if Ω is at initial temperature 1 and if Ω c is at initial temperature 0.…”

Eigenvalue Bounds for the Fractional Laplacian: A Review

“…(8) Heat trace asymptotics for fractional Schrödinger operators and heat content asymptotics. See [17,2,3,4].…”