Energy and Laplacian on Hanoi-type fractal quantum graphs

TL;DR

This paper develops potential theory and spectral analysis for fractal quantum graphs, including Hanoi-type fractals, establishing resistance forms, heat kernel estimates, and eigenvalue bounds that depend on fractal scaling properties.

Contribution

It introduces a framework for resistance forms and spectral analysis on fractal quantum graphs, including Hanoi attractor, with new conditions for resistance form existence and spectral bounds.

Findings

Existence of resistance form on Hanoi attractor

Heat kernel estimates and eigenvalue bounds for Laplacians

Conditions for resistance form existence on general fractal quantum graphs

Abstract

This article studies potential theory and spectral analysis on compact metric spaces, which we refer to as fractal quantum graphs. These spaces can be represented as a (possibly infinite) union of 1-dimensional intervals and a totally disconnected (possibly uncountable) compact set, which roughly speaking represents the set of junction points. Classical quantum graphs and fractal spaces such as the Hanoi attractor are included among them. We begin with proving the existence of a resistance form on the Hanoi attractor, and go on to establish heat kernel estimates and upper and lower bounds on the eigenvalue counting function of Laplacians corresponding to weakly self-similar measures on the Hanoi attractor. These estimates and bounds rely heavily on the relation between the length and volume scaling factors of the fractal. We then state and prove a necessary and sufficient condition for the existence of a resistance form on a general fractal quantum graph. Finally, we extend our spectral results to a large class of weakly self-similar fractal quantum graphs.