Spectral analysis on infinite Sierpinski fractafolds

TL;DR

This paper develops a method to explicitly describe the spectral resolution of the Laplacian on noncompact Sierpinski fractafolds, extending previous work on compact and infinite cases, and introduces new formulas including for the 3-regular tree.

Contribution

It combines earlier spectral decimation techniques to provide a comprehensive spectral resolution for noncompact fractafolds, including explicit formulas and applications.

Findings

Explicit spectral resolution for noncompact fractafolds obtained

Plancherel formulas derived for specific fractal examples

New spectral formula established for the graph Laplacian on the 3-regular tree

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.