Energy and Laplacian on Hanoi-type 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.

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• Thermodynamics of photons on fractals
• Heat Kernels and Zeta Functions on Fractals
• Laplace Operators on Fractals and Related Functional Equations
• Spatial Log Periodic Oscillations of First-Passage Observables in Fractals
• Metrics and spectral triples for Dirichlet and resistance forms

Cited by in corpus (8)

• Spectral Theory of Infinite Quantum Graphs
• Perfect quantum state transfer on diamond fractal graphs
• Spectra of Perfect State Transfer Hamiltonians on Fractal-Like Graphs
• Power dissipation in fractal Feynman-Sierpinski AC circuits
• Power dissipation in fractal AC circuits
• A note on reflected Dirichlet forms
• Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs
• From non-symmetric particle systems to non-linear PDEs on fractals