Spectral analysis on infinite Sierpinski fractafolds

Abstract

A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpinski gasket, it was shown by the first author how to compute the discrete spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian. A similar problem was solved by the second author for the case of infinite blowups of a Sierpinski gasket, where spectrum is pure point of infinite multiplicity. Both works used the method of spectral decimations to obtain explicit description of the eigenvalues and eigenfunctions. In this paper we combine the ideas from these earlier works to obtain a description of the spectral resolution of the Laplacian for noncompact fractafolds. Our main abstract results enable us to obtain a completely explicit description of the spectral resolution of the fractafold Laplacian. For some specific examples we turn the spectral resolution into a "Plancherel formula". We also present such a formula for the graph Laplacian on the 3-regular tree, which appears to be a new result of independent interest. In the end we discuss periodic fractafolds and fractal fields.

References in corpus (10)

• On the spectra of carbon nano-structures
• Thermodynamics of photons on fractals
• Laplace Operators on Fractals and Related Functional Equations
• Derivations and Dirichlet forms on fractals
• Dirac and magnetic Schrödinger operators on fractals
• Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals
• Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals
• Equilateral quantum graphs and boundary triples
• Extensions and degenerations of spectral triples
• Statistical mechanics of Bose gas in Sierpinski carpets

Cited by in corpus (22)

• Geometry and field theory in multi-fractional spacetime
• Derivations and Dirichlet forms on fractals
• Metrics and spectral triples for Dirichlet and resistance forms
• Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket
• One-dimensional wave equations defined by fractal Laplacians
• Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces
• Power dissipation in fractal AC circuits
• Singularly continuous spectrum of a self-similar Laplacian on the half-line
• Spectral decimation of a self-similar version of almost Mathieu-type operators
• Regularized Laplacian determinants of self-similar fractals
• Geometry of fractional spaces
• Reflected Brownian motion on simple nested fractals
• Spectral Analysis and Dirichlet Forms on Barlow-Evans Fractals
• Sierpiński fractals and the dimension of their Laplacian spectrum
• Heat kernel-based p-energy norms on metric measure spaces
• Eigenvalues of Laplacians on Higher Dimensional Vicsek Set Graphs
• Gaps in the spectrum of the Laplacian on $3N$-Gaskets
• Pseudo-differential Operators on Fractals
• Cyclic Cohomology Groups of Some Self-similar Sets
• Spectra of three-peg Hanoi towers graphs
• BV functions and Besov spaces associated with Dirichlet spaces
• The Mathieu Differential Equation and Generalizations to Infinite Fractafolds