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Random Schrödinger operators and convolution on wreath products

Adam Arras

TL;DR

The paper builds a bridge between random Schrödinger operators on Cayley graphs and deterministic convolution operators on wreath products by establishing a spectral correspondence; the averaged spectral measure of the random operator matches the Plancherel spectral measure of a wreath-product convolution operator, and a direct-integral unitary equivalence is shown when the lamp group is Abelian. This framework yields practical consequences, including a criterion for absolute continuity of spectra on wreath-product Cayley graphs, Lifshitz-tail behavior for density of states on polynomial-growth groups, and an exact, Parseval-based formula for the second moment of the Green function. The approach unifies perspectives from mathematical physics and group theory, providing tools to analyze spectral properties of wreath-product operators and to translate localization/transport questions across the two contexts. Overall, the work broadens the scope of spectral analysis on wreath products and offers new pathways to study AC spectrum, DOS tails, and Green-function statistics via a deterministic-random operator correspondence.

Abstract

We establish a spectral correspondence between random Schrödinger operators and deterministic convolution operators on wreath products, generalizing previous results that relate Lamplighter groups to Schrödinger operators with Bernoulli potentials. Using this correspondence in both directions, we obtain an elementary criterion for the absolute continuity of convolutions on wreath products, Lifschitz tail estimates for Schrödinger operators on Cayley graphs of polynomial growth, and an exact formula for the second moment of the Green function, expressed in terms of the wreath product with an Abelian group of lamps.

Random Schrödinger operators and convolution on wreath products

TL;DR

The paper builds a bridge between random Schrödinger operators on Cayley graphs and deterministic convolution operators on wreath products by establishing a spectral correspondence; the averaged spectral measure of the random operator matches the Plancherel spectral measure of a wreath-product convolution operator, and a direct-integral unitary equivalence is shown when the lamp group is Abelian. This framework yields practical consequences, including a criterion for absolute continuity of spectra on wreath-product Cayley graphs, Lifshitz-tail behavior for density of states on polynomial-growth groups, and an exact, Parseval-based formula for the second moment of the Green function. The approach unifies perspectives from mathematical physics and group theory, providing tools to analyze spectral properties of wreath-product operators and to translate localization/transport questions across the two contexts. Overall, the work broadens the scope of spectral analysis on wreath products and offers new pathways to study AC spectrum, DOS tails, and Green-function statistics via a deterministic-random operator correspondence.

Abstract

We establish a spectral correspondence between random Schrödinger operators and deterministic convolution operators on wreath products, generalizing previous results that relate Lamplighter groups to Schrödinger operators with Bernoulli potentials. Using this correspondence in both directions, we obtain an elementary criterion for the absolute continuity of convolutions on wreath products, Lifschitz tail estimates for Schrödinger operators on Cayley graphs of polynomial growth, and an exact formula for the second moment of the Green function, expressed in terms of the wreath product with an Abelian group of lamps.

Paper Structure

This paper contains 11 sections, 7 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Convolution operators.
  3. Random Schrödinger operator.
  4. Wreath product of group.
  5. Main results.
  6. Applications
  7. Absolutely continuous spectrum on wreath product group
  8. Lifchitz tail on polynomial growth group
  9. An exact formula for the Green’s function second moment
  10. Proofs of main theorems
  11. Direct integral.

Key Result

Theorem 1.1

Let $A=L_{m_\Gamma}$, $B=L_{m_\Lambda}$ be two self-adjoint convolution operators on countable groups $\Gamma,\Lambda$. Consider the random Schrödinger operator on of the form $H(\omega)=A+V$ such that the random potential $(v(x,\omega))_{x\in \Gamma}$ is i.i.d, following the spectral measure $\mu_B

Figures (1)

  • Figure 1: The tree $\mathcal{T}_{\Gamma,\Lambda}$

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • Example 2.2
  • proof : Proof of theorem \ref{['thm:ACspecForWreath']}
  • Theorem 2.3: Lifshitz tail on any polynomial growth group
  • Example 2.4
  • proof
  • Proposition 2.5: Reverse correspondence
  • ...and 10 more