Quantum Mechanics on Laakso Spaces

TL;DR

This paper explores the spectral properties of Laplacians and Hamiltonians on Laakso spaces, analyzing their spectra under various potentials, geometric modifications, and boundary conditions, and deriving related physical and mathematical quantities.

Contribution

It provides a comprehensive spectral analysis of Laakso spaces, including modified Hamiltonians, geometric deformations, and the Casimir effect, with explicit formulas and regularizations.

Findings

Spectral spectra of Laplacians on Laakso spaces are characterized.

Explicit spectra for modified Hamiltonians and geometric deformations are derived.

Casimir force calculations are performed with regularized spectral zeta functions.

Abstract

We first review the spectrum of the Laplacian operator on a general Laakso Space before considering modified Hamiltonians for the infinite square well, parabola, and Coulomb potentials. Additionally, we compute the spectrum for the Laplacian and its multiplicities when certain regions of a Laakso space are compressed or stretched and calculate the Casimir force experienced by two uncharged conducting plates by imposing physically relevant boundary conditions and then analytically regularizing the result. Lastly, we derive a general formula for the spectral zeta function and its derivative for Laakso spaces with strict self-similar structure before listing explicit spectral values for cases of interest.