Commutation Relations for Unitary Operators II

TL;DR

This paper investigates the spectral properties of certain unitary operators on multi-dimensional lattices, showing they have finite point spectrum and no singular continuous spectrum away from specific sets, with applications to GGT matrices.

Contribution

It introduces a new approach using positive commutator techniques to analyze spectral properties of unitary operators and GGT matrices with asymptotically constant coefficients.

Findings

Operators have finite point spectrum

No singular continuous spectrum away from specific sets

Propagation estimates are obtained

Abstract

Let $f$ be a regular non-constant symbol defined on the $d$-dimensional torus ${\mathbb T}^d$ with values on the unit circle. Denote respectively by $\kappa$ and $L$, its set of critical points and the associated Laurent operator on $l^2({\mathbb Z}^d)$. Let $U$ be a suitable unitary local perturbation of $L$. We show that the operator $U$ has finite point spectrum and no singular continuous component away from the set $f(\kappa)$. We apply these results and provide a new approach to analyze the spectral properties of GGT matrices with asymptotically constant Verblunsky coefficients. The proofs are based on positive commutator techniques. We also obtain some propagation estimates.