Spectra of Magnetic Operators on the Diamond Lattice Fractal

TL;DR

This paper extends spectral decimation techniques to magnetic Schrödinger operators on the diamond lattice fractal, bridging historical physics methods with modern functional analysis for spectral computation.

Contribution

It adapts spectral decimation to magnetic operators on fractals, providing a complete method for the diamond lattice fractal and connecting past physics approaches with current mathematical frameworks.

Findings

Successfully computes spectra of magnetic operators on the fractal

Establishes the existence of a limiting fractal operator

Links physics-based techniques with modern analysis methods

Abstract

We adapt the well-known spectral decimation technique for computing spectra of Laplacians on certain symmetric self-similar sets to the case of magnetic Schrodinger operators and work through this method completely for the diamond lattice fractal. This connects results of physicists from the 1980's, who used similar techniques to compute spectra of sequences of magnetic operators on graph approximations to fractals but did not verify existence of a limiting fractal operator, to recent work describing magnetic operators on fractals via functional analytic techniques.