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Random Schrödinger operator with singular potentials

Travis Kwan

TL;DR

This work analyzes localization for the Anderson model with singular single-site randomness by treating Hölder-continuous laws and Bernoulli laws within a unified framework. For Hölder laws, it derives a Wegner bound with exponent $\alpha$ and applies multiscale analysis to obtain localization near spectral edges or at large disorder in both lattice and continuum settings. In the Bernoulli case, where spectral averaging fails, it deploys a quantitative unique continuation principle (continuum via Carleman estimates, lattice via discrete UCP) together with a Sperner-type combinatorial argument to obtain a Bernoulli Wegner bound and localization results. The work highlights replacing spectral averaging with propagation-of-smallness and combinatorics, unifying continuum and lattice approaches and extending 3D Li–Zhang results and 2D large-disorder findings under singular randomness. Overall, the paper strengthens universality claims by showing that localization persists under singular randomness through robust analytical and combinatorial mechanisms.

Abstract

We survey the localization theory of random Schrödinger operators with singular single-site distributions, focusing on two regimes: (i) Hölder-continuous laws, where quantitative Wegner estimates enable the classical multiscale analysis (MSA); and (ii) purely atomic (Bernoulli) laws, where the failure of spectral averaging is overcome via quantitative unique continuation principles (UCP). Our discussion covers both lattice and continuum settings and highlights the analytic and combinatorial mechanisms that replace regularity of the single-site measure.

Random Schrödinger operator with singular potentials

TL;DR

This work analyzes localization for the Anderson model with singular single-site randomness by treating Hölder-continuous laws and Bernoulli laws within a unified framework. For Hölder laws, it derives a Wegner bound with exponent and applies multiscale analysis to obtain localization near spectral edges or at large disorder in both lattice and continuum settings. In the Bernoulli case, where spectral averaging fails, it deploys a quantitative unique continuation principle (continuum via Carleman estimates, lattice via discrete UCP) together with a Sperner-type combinatorial argument to obtain a Bernoulli Wegner bound and localization results. The work highlights replacing spectral averaging with propagation-of-smallness and combinatorics, unifying continuum and lattice approaches and extending 3D Li–Zhang results and 2D large-disorder findings under singular randomness. Overall, the paper strengthens universality claims by showing that localization persists under singular randomness through robust analytical and combinatorial mechanisms.

Abstract

We survey the localization theory of random Schrödinger operators with singular single-site distributions, focusing on two regimes: (i) Hölder-continuous laws, where quantitative Wegner estimates enable the classical multiscale analysis (MSA); and (ii) purely atomic (Bernoulli) laws, where the failure of spectral averaging is overcome via quantitative unique continuation principles (UCP). Our discussion covers both lattice and continuum settings and highlights the analytic and combinatorial mechanisms that replace regularity of the single-site measure.

Paper Structure

This paper contains 11 sections, 10 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Hölder--continuous single--site laws: Wegner $\Rightarrow$ MSA
  3. Multiscale framework and initial length scale
  4. Wegner estimate under Hölder continuity
  5. Bernoulli potentials: quantitative unique continuation and antichains
  6. Continuum quantitative unique continuation
  7. Discrete UCP on the lattice and the role of frozen/free sites
  8. A combinatorial antichain bound: Sperner’s theorem and its $\rho$--Sperner variant
  9. How UCP produces $\rho$--Sperner.
  10. Wegner bound for Bernoulli.
  11. Remarks

Key Result

Theorem 2.5

Fix $I\subset\mathbb{R}$ and suppose a Wegner estimate (Definition def:W) holds on $I$ with exponent $\alpha>0$. Let $L_{k+1}=L_k^{1+\eta}$ and assume that for some $m_k\in(0,1)$ and $\kappa>2d$, Then there exist $m_{k+1}\in(0,m_k)$ and $\kappa'>2d$ such that Moreover, $m_{k+1}\to m_\infty>0$ as $k\to\infty$.

Theorems & Definitions (32)

  • Definition 2.1: (E,$m$)-good/nonresonant boxes
  • Remark 2.2: Combes--Thomas control
  • Definition 2.3: Wegner bound
  • Definition 2.4: Initial length scale (ILS)
  • Theorem 2.5: Induction step of MSA
  • proof : Proof sketch
  • Theorem 2.6: Initial length scale
  • Remark 2.7
  • Theorem 2.8: Wegner for the discrete Anderson model with Hölder law
  • proof
  • ...and 22 more