Minimal gap in the spectrum of the Sierpinski gasket

TL;DR

This paper investigates the eigenvalue spectrum of the Laplace operator on the Sierpinski gasket, revealing that the minimal gap between consecutive eigenvalues equals the spectral gap, with a focus on the Dirichlet case.

Contribution

It establishes that the minimal eigenvalue gap coincides with the spectral gap on the Sierpinski gasket, providing new insights into its spectral structure.

Findings

Minimal eigenvalue gap equals the spectral gap.

Dirichlet spectrum analysis involves complex dynamical system behavior.

Results enhance understanding of fractal Laplacian spectra.

Abstract

This paper studies the size of the minimal gap between any two consecutive eigenvalues in the Dirichlet and in the Neumann spectrum of the standard Laplace operator on the Sierpinski gasket. The main result shows the remarkable fact that this minimal gap is achieved and coincides with the spectral gap. The Dirichlet case is more challenging and requires some key observations in the behavior of the dynamical system that describes the spectrum.