On Functional Determinants of Laplacians in Polygons and Simplices

1993/04/08 by Erik Aurell, Per Salomonson · 1 citation
Mathematics · Physics and Astronomy · #hep-th #math.DG

paper · pdf · doi:10.1007/bf02099770

published as Commun.Math.Phys. 165 (1994) 233-260 · 40 pages

arxiv created 1993/04/08 · arxiv updated 2009/11/30

Abstract

The functional determinant of an elliptic operator with positive, discrete spectrum may be defined as e^-Z'(0), where Z(s), the zeta function, is the sum ∑_n^∞ λ_n^-s analytically continued to s around the origin. In this paper Z'(0) is calculated for the Laplace operator with Dirichlet boundary conditions inside polygons and simplices with the topology of a disc in the Euclidean plane. The domains we consider are hence piece--wise flat with corners on the boundary and in the interior. Our results are complementary to earlier investigations of the determinants on smooth surfaces with smooth boundaries. We have explicit closed integrated expressions for triangles and regular polygons.

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