Spectral Instability of Random Fredholm Operators

TL;DR

This paper investigates how random trace-class perturbations affect the spectrum of unbounded Fredholm operators, showing that under certain conditions, the spectrum becomes discrete with high probability, which has implications for physics models like twisted bilayer graphene.

Contribution

It proves that random trace-class perturbations can induce spectral discreteness in unbounded Fredholm operators under specific assumptions.

Findings

Spectrum becomes discrete with high probability after perturbation.

Spectral dichotomy is influenced by random trace-class operators.

Results have implications for understanding spectral properties in physical systems.

Abstract

If $A \colon D(A) \subset \mathcal{H} \to \mathcal{H}$ is an unbounded Fredholm operator of index $0$ on a Hilbert space $\mathcal{H}$ with a dense domain $D(A)$, then its spectrum is either discrete or the entire complex plane. This spectral dichotomy plays a central role in the study of magic angles in twisted bilayer graphene. This paper proves that if such operators (with certain additional assumptions) are perturbed by certain random trace-class operators, their spectrum is discrete with high probability.