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Level spacing and Poisson statistics for continuum random Schrödinger operators

Adrian Dietlein, Alexander Elgart

Abstract

We prove a probabilistic level-spacing estimate at the bottom of the spectrum for continuum alloy-type random Schrödinger operators, assuming sign-definiteness of a single-site bump function and absolutely continuous randomness. More precisely, given a finite-volume restriction of the random operator onto a box of linear size $L$, we prove that with high probability the eigenvalues below some threshold energy $E_{\rm sp}$ keep a distance of at least $e^{-(\log L)^β}$ for sufficiently large $β>1$. This implies simplicity of the spectrum of the infinite-volume operator below $E_{\rm sp}$. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy $E$.

Level spacing and Poisson statistics for continuum random Schrödinger operators

Abstract

We prove a probabilistic level-spacing estimate at the bottom of the spectrum for continuum alloy-type random Schrödinger operators, assuming sign-definiteness of a single-site bump function and absolutely continuous randomness. More precisely, given a finite-volume restriction of the random operator onto a box of linear size , we prove that with high probability the eigenvalues below some threshold energy keep a distance of at least for sufficiently large . This implies simplicity of the spectrum of the infinite-volume operator below . Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy .

Paper Structure

This paper contains 16 sections, 22 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Model and results
  3. Model
  4. Results
  5. Outline of the proofs
  6. Clusters of eigenvalues
  7. A Feynman-Hellmann type estimate
  8. The local discriminant and a Cartan estimate
  9. Proof of the level spacing estimates
  10. Existence of good configurations
  11. Proof of Theorem \ref{['thm:MainResultv2']}
  12. Proof of Theorem \ref{['thm:MainResult']}
  13. Proof of the Minami-type estimate
  14. Simplicity of spectrum and Poisson statistics
  15. Properties of deformed Schrödinger operators
  16. ...and 1 more sections

Key Result

Theorem 2.1

For a fixed energy $E < \min\left\{ E_{\mathop{\mathrm{sp}}\nolimits},E_{\mathop{\mathrm{loc}}\nolimits} \right\}$ there exist $\mathcal{L}_{\mathop{\mathrm{sp}}\nolimits} =\mathcal{L}_{\mathop{\mathrm{sp}}\nolimits,E}, C_{\mathop{\mathrm{sp}}\nolimits}=C_{\mathop{\mathrm{sp}}\nolimits,E}$ such that holds for $L\geq \mathcal{L}_{\mathop{\mathrm{sp}}\nolimits}$ and $\delta < 1$.

Theorems & Definitions (42)

  • Theorem 2.1: Probabilistic level-spacing estimate, Version 1
  • Theorem 2.2: Probabilistic level-spacing estimate, Version 2
  • Corollary 2.3: Eigenvalue simplicity
  • Theorem 2.4: Minami-type estimate
  • Theorem 2.5: Poisson statistics
  • Lemma 3.1
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['lem:EigCluster']}
  • proof : Proof of Lemma \ref{['lem:TechEst']}
  • Lemma 3.3
  • ...and 32 more