Absolute continuity of the spectrum of coupled identical systems on 1D lattices

TL;DR

This paper proves that the spectrum of a specific 2D discrete Schrödinger operator with a potential depending only on one coordinate is absolutely continuous, indicating extended states in the system.

Contribution

It establishes the absolute continuity of the spectrum for a coupled 2D lattice Schrödinger operator with a one-dimensional potential.

Findings

Spectrum is absolutely continuous for the given operator.

Results imply the presence of extended states in the lattice.

Advances understanding of spectral properties in coupled lattice systems.

Abstract

We prove that the spectrum of the discrete Schr\"odinger operator on $\ell^2(Z^2)$, $(\psi_{n,m})\mapsto -(\psi_{n+1,m} +\psi_{n-1,m} +\psi_{n,m+1} +\psi_{n,m-1})+V_n\psi_{n,m}$ is absolutely continuous.