Wegner-type bounds for a two-particle lattice model with a generic "rough" quasi-periodic potential

TL;DR

This paper establishes volume-dependent Wegner-type bounds for two-particle lattice systems with quasi-periodic potentials and short-range interactions, extending previous single-particle results to a more complex two-particle setting.

Contribution

It introduces a method to derive Wegner bounds for two-particle quasi-periodic lattice models with interactions, expanding the understanding of spectral properties in such systems.

Findings

Proved volume-dependent Wegner bounds for two-particle models

Extended single-particle quasi-periodic results to two-particle systems

Demonstrated bounds hold for a class of ergodic deterministic potentials

Abstract

In this paper, we consider a class of two-particle tight-binding Hamiltonians, describing pairs of interacting quantum particles on the lattice $\Z^d$, $d\ge 1$ subject to a common external potential $V(x)$ which we assume quasi-periodic and depending on an auxiliary parameter. Such parametric families of ergodic deterministic potentials ("grand ensembles") have been introduced earlier in \cite{C07}, in the framework of single-particle lattice systems, where it was proved that a non-uniform analog of the Wegner bound holds true for a class of quasi-periodic grand ensembles. Using the approach proposed in \cite{CS08}, we establish volume-dependent Wegner-type bounds for a class of quasi-periodic two-particle lattice systems with a non-random short-range interaction.