Heat content and small time asymptotics for Schr\"odinger operators on $R^d$

TL;DR

This paper derives small time asymptotic expansions for the heat content of Schr"odinger operators involving fractional Laplacians in Euclidean space, revealing new invariants even for classical Laplacians.

Contribution

It introduces novel small time asymptotic formulas and identifies heat content invariants for fractional Schr"odinger operators, extending known results to new cases.

Findings

Asymptotic expansion formulas for heat content

Identification of heat content invariants

Results applicable to classical Laplacian case

Abstract

This paper studies the heat content} for Schr\"odinger operators of the fractional Laplacian $(-\Delta)^{\alpha/2}$, $0<\alpha\leq 2$ in $R^d$, $d\geq 1$. Employing probabilistic and analytic techniques, a small time asymptotic expansion formula is given and the "heat content invariants" are identified. These results are new even in the case of the Laplacian, $\alpha=2$.