The Partition Function of the Dirichlet Operator D 2 = ∑ =1 ( − 2 ) on a -Dimensional Rectangle Cavity

Abstract

We study the asymptotic behavior of the free partition function in the → 0 + limit for a diffusion process which consists of -independent, one-dimensional, symmetric, 2 -stable processes in a hyperrectangular cavity ⊂ R with an absorbing boundary. Each term of the partition function for this polyhedron in -dimensions can be represented by a quermassintegral and the geometrical information conveyed by the eigenvalues of the fractional Dirichlet Laplacian for this solvable model is now transparent. We also utilize the intriguing method of images to solve the same problem, in one and two dimensions, and recover identical results to those derived in the previous analysis