Regularized Laplacian determinants of self-similar fractals

TL;DR

This paper explores the spectral zeta functions of Laplacians on self-similar fractals, revealing complex dimensions off the imaginary axis and connecting Laplacian determinants to eigenvalues and graph Laplacians.

Contribution

It demonstrates that certain self-similar fractals have complex dimensions not on the imaginary axis, enabling interpretation of their Laplacian determinant as a regularized eigenvalue product.

Findings

Complex dimensions of some fractals are off the imaginary axis.

Laplacian determinants can be interpreted as regularized eigenvalue products.

A link between discrete graph Laplacian and continuous Laplacian determinants is established.

Abstract

We study the spectral zeta functions of the Laplacian on fractal sets which are locally self-similar fractafolds, in the sense of Strichartz. These functions are known to meromorphically extend to the entire complex plane, and the locations of their poles, sometimes referred to as complex dimensions, are of special interest. We give examples of self-similar sets such that their complex dimensions are not on the imaginary axis, which allows us to interpret their Laplacian determinant as the regularized product of their eigenvalues. We then investigate a connection between the logarithm of the determinant of the discrete graph Laplacian and the regularized one.