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Journal of Mathematics
Volume 2015, Article ID 785720, 7 pages
Research Article

The Partition Function of the Dirichlet Operator on a -Dimensional Rectangle Cavity

Department of Mathematics, School of Sciences, University of Aegean, Samos, 83200 Karlovasi, Greece

Received 9 July 2015; Revised 14 August 2015; Accepted 23 August 2015

Academic Editor: Alfred Peris

Copyright © 2015 Agapitos N. Hatzinikitas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the asymptotic behavior of the free partition function in the limit for a diffusion process which consists of -independent, one-dimensional, symmetric, -stable processes in a hyperrectangular cavity 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.