Regularized Laplacian determinants of self-similar fractals

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.

16 pages, 5 figures

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• Spanning trees on the Sierpinski gasket
• Thermodynamics of photons on fractals
• Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces
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Cited by in corpus (3)

• Explicit formulas for heat kernels on diamond fractals
• Discrete and zeta-regularized determinants of the Laplacian on polygonal domains with Dirichlet boundary conditions
• Spectral decimation of the magnetic Laplacian on the Sierpinski gasket: Solving the Hofstadter-Sierpinski butterfly