Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals

Abstract

We investigate the existence of the meromorphic extension of the spectral zeta function of the Laplacian on self-similar fractals using the classical results of Kigami and Lapidus (based on the renewal theory) and new results of Hambly and Kajino based on the heat kernel estimates and other probabilistic techniques. We also formulate conjectures which hold true in the examples that have been analyzed in the existing literature.

References in corpus (3)

• Thermodynamics of photons on fractals
• Laplace Operators on Fractals and Related Functional Equations
• Eigenmodes of a Laplacian on Some Laakso Spaces

Cited by in corpus (12)

• Thermodynamics of photons on fractals
• Laplace Operators on Fractals and Related Functional Equations
• Spectral analysis on infinite Sierpinski fractafolds
• Dirac and magnetic Schrödinger operators on fractals
• Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces
• Fractal AC circuits and propagating waves on fractals
• Wave equation on one-dimensional fractals with spectral decimation and the complex dynamics of polynomials
• Regularized Laplacian determinants of self-similar fractals
• Disconnected Julia sets and gaps in the spectrum of Laplacians on symmetric finitely ramified fractals
• Gaps in the spectrum of the Laplacian on $3N$-Gaskets
• Measurable operators and the asymptotics of heat kernels and zeta functions
• Random walks on barycentric subdivisions and the Strichartz hexacarpet