Finite difference method on flat surfaces with a flat unitary vector bundle
Abstract
We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a halftranslation surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian with von Neumann boundary conditions. As an interesting byproduct of our study, we obtain Harnacktype estimates on "almost harmonic" discrete functions, defined on the graphs, which approximate a given surface. The results of this paper will be later used to relate the asymptotic expansion of the number of spanning trees, spanning forests and weighted cyclerooted spanning forests on the discretizations to the corresponding zetaregularized determinants.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.04862
 Bibcode:
 2020arXiv200104862F
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Functional Analysis;
 Mathematics  Spectral Theory;
 58A99;
 47N30;
 31A05
 EPrint:
 39 pages, 8 figures