Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral Instability of Random Fredholm Operators

Simon Becker, Izak Oltman, Martin Vogel

TL;DR

This work proves that random trace-class perturbations of unbounded Fredholm operators of index $0$ render the spectrum discrete with high probability on each connected component of the Fredholm domain, by a Grushin-problem reduction and probabilistic bounds for the smallest singular value. The authors develop a general framework with quasi-finite-rank Gaussian perturbations $Q_\\omega$ built from finite-rank blocks $S_1,S_2$, derive explicit probability bounds for the invertibility of a finite-dimensional Schur complement, and apply the theory to semiclassical operators. A key application is to twisted bilayer graphene in the chiral limit, where the method shows that generic small random perturbations destroy the non-discrete spectrum associated with magic angles, yielding overwhelming probability of a discrete spectrum. The results bridge non-self-adjoint spectral theory, random perturbations, and numerical perturbations, with matrix-valued symbol quantization on the torus and a detailed Grushin-analysis strategy as the core technical engine.

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.

Spectral Instability of Random Fredholm Operators

TL;DR

This work proves that random trace-class perturbations of unbounded Fredholm operators of index render the spectrum discrete with high probability on each connected component of the Fredholm domain, by a Grushin-problem reduction and probabilistic bounds for the smallest singular value. The authors develop a general framework with quasi-finite-rank Gaussian perturbations built from finite-rank blocks , derive explicit probability bounds for the invertibility of a finite-dimensional Schur complement, and apply the theory to semiclassical operators. A key application is to twisted bilayer graphene in the chiral limit, where the method shows that generic small random perturbations destroy the non-discrete spectrum associated with magic angles, yielding overwhelming probability of a discrete spectrum. The results bridge non-self-adjoint spectral theory, random perturbations, and numerical perturbations, with matrix-valued symbol quantization on the torus and a detailed Grushin-analysis strategy as the core technical engine.

Abstract

If is an unbounded Fredholm operator of index on a Hilbert space with a dense domain , 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.

Paper Structure

This paper contains 8 sections, 13 theorems, 136 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The motivating example: Twisted bilayer graphene
  3. Setup of problem and main result
  4. Random perturbation
  5. Main result
  6. Proof of main result
  7. Applications to twisted bilayer graphene
  8. Quantization of matrix valued functions on C/Γ

Key Result

Theorem 1

Suppose $A - z$ is a Fredholm operator of index $0$ for all $z\in \Omega \subset \mathbb{C}$ (an open set) and $Q_\omega$ is a suitable random trace class perturbation (see random_pertubation), then for sufficiently small $\delta > 0$, the set $\mathop{\mathrm{Spec}}\nolimits(A + \delta Q_\omega) \c

Figures (2)

  • Figure 1: Spectrum of smallest 600 eigenvalues of finite matrix truncation of $D_h(\beta)$, with matrix size 13122, for largest magic $h$ and $\beta=1$ (top left). The accumulation of eigenvalues since $\mathop{\mathrm{Spec}}\nolimits(D_h(\beta))={\mathbb C}$ in the center is clearly visible. Spectrum of finite matrix truncation of $D_h(\beta)$, with same $h,\beta$ and random $\delta =0, 0.01, 10^{-4}, 10^{-7}$ perturbation (clockwise). The accumulation of eigenvalues in the center gets resolved immediately.
  • Figure 2: Counting the number of eigenvalues of a finite rank approximation of $D_h(1)$ (defined in \ref{['eq:first define of Dh']}) for different $h$ in a ball of radius $2$ with random perturbations of size $\delta=0,10^{-5},10^{-1}$ from left to right. Magic angles at $1/h \approx 0.586,2.221$ are clearly visible as spikes in the unperturbed figure (left) but washed out in the perturbed ones (center and right).

Theorems & Definitions (27)

  • Theorem : Heuristic Main Result
  • Theorem 1: Application to TBG
  • Proposition 2.1
  • proof
  • Theorem 2: General Result
  • Theorem 3
  • Lemma 3.1
  • proof : Proof of Theorem \ref{['thm:general result']}
  • Proposition 3.2
  • proof
  • ...and 17 more