Spectral analysis beyond $\ell^2$ on Sierpinski lattices

TL;DR

This paper investigates the spectrum of the Laplacian on Sierpinski lattices across different $\,p$-spaces, revealing invariance and characterizing spectral points for lattices with boundary.

Contribution

It demonstrates the spectrum's invariance across all $\,p$-spaces and characterizes spectral points for lattices with boundary, extending spectral analysis beyond $\, ext{l}^2$.

Findings

Spectrum remains unchanged across all $\,p$-spaces.

Spectral points are characterized for lattices with boundary.

Provides a comprehensive spectral analysis on fractal lattices.

Abstract

We study the spectrum of the Laplacian on the Sierpinski lattices. First, we show that the spectrum of the Laplacian, as a subset of $\mathbb{C}$, remains the same for any $\ell^p$ spaces. Second, we characterize all the spectral points for the lattices with a boundary point.