The Partition Function of the Dirichlet Operator $\mathcal{D}_{2s}=\sum_{i=1}^{d}(-\partial_i^2)^{s}$ on a d-Dimensional Rectangle Cavity

Abstract

In this letter we study the asymptotic behavior of the free partition function in the $t\rightarrow 0^+$ limit for a stochastic process which consists of $d-$independent, one-dimensional, symmetric, $2s-$stable processes in a hyperrectangular cavity $K \subset \mathbb {R}^d$ with an absorbing boundary. Each term of the partition function for this polyhedron in d-dimensions can be represented by a quermassintegral and the geometrical information inherited by the eigenvalues for this solvable model is complete. We also demonstrate the correctness of our result by applying the method of images in one dimension.