Heat content and small time asymptotics for Schr\"odinger operators on   $R^d$
  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, $\alpha=2$. 
Luis Acu\~na Valverde and Rodrigo Ba\~nuelos
math.PR math.CA math.SP