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Random Schrödinger operators with complex decaying potentials

Jean-Claude Cuenin, Konstantin Merz

TL;DR

The paper proves that for continuum Anderson-type Schrödinger operators with complex decaying potentials, eigenvalues can be bounded with high probability in terms of the $L^q$ norm of the potential for all $q\le d+1$, effectively doubling the deterministic exponent from Frank's bound. The authors combine a Born-series framework with sharp local and global operator bounds, employing Bourgain’s entropy method and Stein–Tomás-type restriction estimates to control random perturbations. A detailed local-to-global scheme, including dyadic and sparse decompositions, yields global spectral-radius bounds and almost-sure eigenvalue bounds, with explicit dependence on the randomization scale $h$ and decay/radius parameters. The results demonstrate that randomness can destroy certain deterministic counterexamples and enhance spectral control, offering new insights into almost-sure scattering and the ac spectrum for complex potentials.

Abstract

We prove that the eigenvalues of a continuum random Schrödinger operator $-Δ+ V_ω$ of Anderson type, with complex decaying potential, can be bounded (with high probability) in terms of an $L^q$ norm of the potential for all $q\leq d+1$. This shows that in the random setting, the exponent $q$ can be essentially doubled compared to the deterministic bounds of Frank (Bull. Lond. Math. Soc., 2011). This improvement is based on ideas of Bourgain (Discrete Contin. Dyn. Syst., 2002) related to almost sure scattering for lattice Schrödinger operators.

Random Schrödinger operators with complex decaying potentials

TL;DR

The paper proves that for continuum Anderson-type Schrödinger operators with complex decaying potentials, eigenvalues can be bounded with high probability in terms of the norm of the potential for all , effectively doubling the deterministic exponent from Frank's bound. The authors combine a Born-series framework with sharp local and global operator bounds, employing Bourgain’s entropy method and Stein–Tomás-type restriction estimates to control random perturbations. A detailed local-to-global scheme, including dyadic and sparse decompositions, yields global spectral-radius bounds and almost-sure eigenvalue bounds, with explicit dependence on the randomization scale and decay/radius parameters. The results demonstrate that randomness can destroy certain deterministic counterexamples and enhance spectral control, offering new insights into almost-sure scattering and the ac spectrum for complex potentials.

Abstract

We prove that the eigenvalues of a continuum random Schrödinger operator of Anderson type, with complex decaying potential, can be bounded (with high probability) in terms of an norm of the potential for all . This shows that in the random setting, the exponent can be essentially doubled compared to the deterministic bounds of Frank (Bull. Lond. Math. Soc., 2011). This improvement is based on ideas of Bourgain (Discrete Contin. Dyn. Syst., 2002) related to almost sure scattering for lattice Schrödinger operators.
Paper Structure (25 sections, 25 theorems, 157 equations)

This paper contains 25 sections, 25 theorems, 157 equations.

Table of Contents

  1. Introduction and main result
  2. Born series
  3. Localization and discretization
  4. Localization in momentum space
  5. Localization in position space
  6. Discrete Fourier extension operator
  7. Stein-Tomas theorem
  8. Randomization
  9. Sub-gaussian random variables
  10. Tail bounds
  11. Entropy bound
  12. Local bounds on elementary operators
  13. Local extension bound
  14. Smoothing
  15. Foliation by level sets
  16. ...and 10 more sections

Key Result

Theorem 1

There exist constants $M_0,c>0$ such that the following holds. For any $R,\lambda>0$, $0<h<R$, $|\varepsilon|\ll \lambda$, $q\leq d+1$, for any $V\in L^q(\mathbb{R}^d)$ supported in a ball of radius $R$, and for any $M\geq M_0$, each eigenvalue $z=(\lambda+\mathrm{i}\varepsilon)^2$ of $-\Delta+V_{\o except for $\omega$ in a set of measure at most $\exp(-cM^2)$.

Theorems & Definitions (50)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • proof
  • Remark 5
  • ...and 40 more