[Paper Review] Anderson localization in a two-particle continuous model with an alloy-type external potential

Anne Boutet de Monvel, Victor Chulaevsky+2 more|ArXiv.org|Jul 9, 2009

Spectral Theory in Mathematical Physics18 references3 citations

TL;DR

This paper establishes exponential localization for a two-particle continuous Anderson model in $\mathbb{R}^d$ ($d \geq 1$) with a short-range interaction and an alloy-type random potential. By reducing the continuous multi-scale analysis to an auxiliary lattice problem and applying geometric resolvent inequalities and partial decoupling techniques, the authors prove that all eigenfunctions near the spectral edge decay exponentially in the $L^2$-norm, confirming pure point spectrum with localized eigenstates.

ABSTRACT

We establish exponential localization for a two-particle Anderson model in a Euclidean space ${\mathbb R}^{d}$, $d\ge 1$, in presence of a non-trivial short-range interaction and a random external potential of the alloy type. Specifically, we prove that all eigenfunctions with eigenvalues near the lower edge of the spectrum decay exponentially in $L^2$-norm.

Motivation & Objective

• To establish exponential localization for a two-particle quantum system in continuous space with a random alloy-type potential.
• To extend the multi-scale analysis (MSA) framework from discrete lattices to continuous Euclidean space by reducing the problem to an auxiliary lattice model.
• To combine geometric resolvent inequalities and partial decoupling techniques to handle the interaction and disorder in the two-particle system.
• To prove that eigenfunctions near the lower spectral edge decay exponentially in $L^2$-norm, confirming Anderson localization in the continuous setting.

Proposed method

• Reduces the continuous two-particle MSA to an auxiliary lattice problem via a geometric embedding and rescaling, enabling application of established discrete MSA techniques.
• Employs the Geometric Resolvent Inequality (GRI) to control the decay of Green's functions across disjoint regions in configuration space.
• Applies partial decoupling of particle cubes to isolate localized regions and bound tunneling effects between distant clusters.
• Uses iterative resolvent estimates and exponential bounds derived from the lattice MSA to establish uniform decay of eigenfunctions.
• Implements a recursive multi-scale scheme with dynamically growing length scales $L_k$, ensuring exponential decay at each scale.
• Relies on Wegner-type bounds for alloy-type potentials and spectral averaging to control the probability of resonant states.

Experimental results

Research questions

• RQ1 Can exponential localization be rigorously established in a two-particle continuous model with short-range interaction and alloy-type disorder?
• RQ2 To what extent can the continuous MSA be reduced to a discrete lattice problem without loss of generality?
• RQ3 How do geometric resolvent inequalities and partial decoupling techniques facilitate the proof of localization in continuous systems?
• RQ4 Does the spectrum of the two-particle Hamiltonian remain pure point with exponentially decaying eigenfunctions near the spectral edge?

Key findings

• All eigenfunctions corresponding to eigenvalues near the lower edge of the spectrum decay exponentially in the $L^2$-norm, confirming Anderson localization in the continuous two-particle model.
• The reduction of the continuous MSA to a lattice problem allows direct application of established discrete MSA techniques, significantly simplifying the analysis.
• Exponential decay of eigenfunctions is established via iterative application of the Geometric Resolvent Inequality and partial decoupling, yielding bounds of the form $\|\mathbf{1}_{C_1(\mathbf{x})}\mathbf{f}\| \leq e^{-\rho\rho' m \|\mathbf{x}-\mathbf{u}\|}$.
• The spectrum of the Hamiltonian is proven to be pure point, with all eigenfunctions in $L^2(\mathbb{R}^{2d})$, due to exponential decay and separability of the Hilbert space.
• The result holds for arbitrary dimension $d \geq 1$ and extends to models with singular or continuous random potentials, including Gaussian fields, as shown in follow-up work.