Quantum mechanics on Laakso spaces

Christopher J. Kauffman, Johns Hopkins University
Robert M. Kesler, Cornell University
Amanda G. Parshall, York College of Pennsylvania
Evelyn A. Stamey, Ithaca College
Benjamin A. Steinhurst, Cornell University

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 resulting zeta function. 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 some special cases. © 2012 American Institute of Physics.