On multiplicity of spectrum for Anderson type operators with higher rank perturbations

TL;DR

This paper investigates the spectral multiplicity of Anderson type operators with higher rank perturbations on infinite graphs, revealing conditions under which the pure point spectrum has non-trivial multiplicity and automorphisms are trivial.

Contribution

It demonstrates the existence of non-trivial spectral multiplicity for certain graph families and characterizes automorphisms fixing the operator as trivial.

Findings

Pure point spectrum can have non-trivial multiplicity on specific graphs.

Automorphisms fixing the operator are trivial in these cases.

Spectrum multiplicity relates to graph symmetries and perturbation structure.

Abstract

Here, we focus on Anderson type operators over infinite graphs where the randomness acts through higher rank perturbations. We show that for special family of graphs, the operator has non-trivial multiplicity for its pure point spectrum. We, also, show that for some family of graphs, any unitary which fixes the random operator, arising from an automorphism of the graph is identity; but that, for these graphs the spectrum of the random operator has non-trivial multiplicity.