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

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