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arXiv · 2102.04837

Discrete and zeta-regularized determinants of the Laplacian on polygonal domains with Dirichlet boundary conditions

Rafael Leon Greenblatt
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Abstract

For $Π\subset \mathbb{R}^2$ a connected, open, bounded set whose boundary is a finite union of disjoint polygons whose vertices have integer coordinates, the logarithm of the discrete Laplacian on $LΠ\cap \mathbb{Z}^2$ with Dirichlet boundary conditions has an asymptotic expansion for large $L$ involving the zeta-regularized determinant of the associated continuum Laplacian. When $Π$ is not simply connected, this result extends to Laplacians acting on two-valued functions with a specified monodromy class.

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Rafael Leon Greenblatt. 2023-03-10. Discrete and zeta-regularized determinants of the Laplacian on polygonal domains with Dirichlet boundary conditions. https://doi.org/10.1063/5.0062138

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