Magnetic Laplacians of locally exact forms on the Sierpinski Gasket

TL;DR

This paper rigorously constructs a magnetic Schr"odinger operator on the Sierpinski Gasket with flux through holes, analyzing its spectrum and eigenfunctions, and showing its eigenvalue distribution matches that of the Laplacian.

Contribution

It introduces a rigorous method to define magnetic Schr"odinger operators on fractals with finitely many holes, including spectral analysis and eigenfunction computation techniques.

Findings

Operator has discrete spectrum with eigenvalues accumulating at infinity

Eigenvalue distribution asymptotically matches the Laplacian

Eigenfunctions can be computed via gauge transformations

Abstract

We give a mathematically rigorous construction of a magnetic Schr\"odinger operator corresponding to a field with flux through finitely many holes of the Sierpinski Gasket. The operator is shown to have discrete spectrum accumulating at $\infty$, and it is shown that the asymptotic distribution of eigenvalues is the same as that for the Laplacian. Most eigenfunctions may be computed using gauge transformations corresponding to the magnetic field and the remainder of the spectrum may be approximated to arbitrary precision by using a sequence of approximations by magnetic operators on finite graphs.