Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tail bounds for the Dyson series of random Schrödinger equations

Adam Black, Reuben Drogin, Felipe Hernández

TL;DR

The paper addresses the small-coupling regime for random Schrödinger operators in ${d\ge 2}$ by bounding the Dyson-series terms ${T_j(t)}$ via tail estimates that hold up to times ${t}$ of order ${\lambda^{-2+\varepsilon}}$. The authors employ a random-matrix–theoretic approach, notably the noncommutative Khintchine inequality, together with dispersive bounds for the free evolution to establish a square-root cancellation that propagates from ${T_1}$ to higher ${T_j}$. They derive high-probability approximations of the propagator by truncated Dyson series and deduce spectral/dynamical consequences, including frequency localization and Floquet-state bounds, with minimal use of diagrammatics. The results unify and extend prior work on delocalization and provide robust, time-periodic extensions, with potential applicability to broader background Hamiltonians through similar dispersive and finite-rank localization techniques.

Abstract

We study Schrödinger equations on $\mathbb{Z}^d$ and $\mathbb{R}^d$, $d\geq 2$ with random potentials of strength $λ$. Our main result gives tail bounds for the terms of the Dyson series that are effective at time scales on the order of $λ^{-2+\varepsilon}$. As corollaries, we obtain estimates on the frequency localization and spatial delocalization of approximate eigenfunctions in the spirit of works by Schlag-Shubin-Wolff and T. Chen. These estimates also apply to Floquet states associated to time-periodic potentials. Our proof is elementary in that we use neither sophisticated harmonic analysis nor diagrammatic arguments. Instead, we use only the noncommutative Khintchine inequality from random matrix theory combined with pointwise dispersive estimates for the free Schrödinger equation.

Tail bounds for the Dyson series of random Schrödinger equations

TL;DR

The paper addresses the small-coupling regime for random Schrödinger operators in by bounding the Dyson-series terms via tail estimates that hold up to times of order . The authors employ a random-matrix–theoretic approach, notably the noncommutative Khintchine inequality, together with dispersive bounds for the free evolution to establish a square-root cancellation that propagates from to higher . They derive high-probability approximations of the propagator by truncated Dyson series and deduce spectral/dynamical consequences, including frequency localization and Floquet-state bounds, with minimal use of diagrammatics. The results unify and extend prior work on delocalization and provide robust, time-periodic extensions, with potential applicability to broader background Hamiltonians through similar dispersive and finite-rank localization techniques.

Abstract

We study Schrödinger equations on and , with random potentials of strength . Our main result gives tail bounds for the terms of the Dyson series that are effective at time scales on the order of . As corollaries, we obtain estimates on the frequency localization and spatial delocalization of approximate eigenfunctions in the spirit of works by Schlag-Shubin-Wolff and T. Chen. These estimates also apply to Floquet states associated to time-periodic potentials. Our proof is elementary in that we use neither sophisticated harmonic analysis nor diagrammatic arguments. Instead, we use only the noncommutative Khintchine inequality from random matrix theory combined with pointwise dispersive estimates for the free Schrödinger equation.

Paper Structure

This paper contains 12 sections, 31 theorems, 161 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and Main Results: time-independent potentials
  3. Random Floquet systems
  4. Overview of the proof
  5. Related work
  6. $T_1$ bounds on ${\mathbb{Z}}^d$
  7. $T_1$ bounds on ${\mathbb{R}}^{d}$
  8. From $T_1$ to $T_k$
  9. Modifications for the time-dependent potential
  10. Spectral and dynamical consequences
  11. The noncommutative Khinthine inequality
  12. Finite-rank approximation of a phase-space localizer

Key Result

theorem 1.1

Let $H$ be the random Schrödinger operator defined above on either ${\mathbb{Z}}^d$ or ${\mathbb{R}}^d$ with $d\geq 2$. Then there exists a constant $c>0$ independent of $\lambda$ or $j$ so that for all $K>\sqrt{\log(R)}$ where

Figures (2)

  • Figure 1: The Fourier transform of an eigenfunction with eigenvalue approximately $1$ for the evolution $U(4\pi,0)$ in a periodically driven system with $V(x,t) = \cos(t/2) V(x)$. Here $V(x)$ has independent Bernoulli $\pm1$ entries on a $2048\times 2048$ periodic lattice and $H_0$ is the shifted discrete Laplacian \ref{['eq:HZd']}. The darker pixels represent the Fourier coefficients with larger magnitude inside the square $[-\frac{1}{2},\frac{1}{2}]^2$. Note that the mass is concentrated near the level sets $\cos(2\pi k_x) + \cos(2\pi k_y) \in \frac{1}{2}{\mathbb{Z}}$, as described by Corollary \ref{['corTimeDependent']}.
  • Figure 2: An illustration of the product structure identity for $k=2$. The operator $T_2(t)$ is given by an integral of $V(s_2)V(s_1)$ for $(s_1,s_2)$ in the shaded triangle. In $(s_1,s_2)$ space, translation corresponds to intertwining the free evolution on either side of $V(s_2)$ and $V(s_1)$, since $V(s+s') = e^{is'H_0} V(s) e^{-is'H_0}$. Thus the contribution of the blue square corresponds to a portion of the integral in $T_2(t)$ that has operator norm bounded by $\|T_1(t/2)\|^2$. Similarly the two orange triangles are have operator norms bounded by $\|T_2(t/2)\|$. Iterating this construction allows one to bound $\|T_2(t)\|$ in terms of $\|T_1(2^{-j}t)\|^2$.

Theorems & Definitions (54)

  • theorem 1.1
  • corollary 1.2
  • corollary 1.2
  • theorem 1.3
  • corollary 1.4
  • theorem 1.4: Matrix concentration for structured random matrices
  • proposition 2.1
  • lemma 2.2
  • proof
  • lemma 2.3
  • ...and 44 more