Localization for Random Unitary Operators

Abstract

We consider unitary analogs of one-dimensional Anderson models on $$l^2(\mathbb{Z})$$ defined by the product U ω=D ω S where S is a deterministic unitary and D ω is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on a parameter controlling the size of its off-diagonal elements. We prove that the spectrum of U ω is pure point almost surely for all values of the parameter of S. We provide similar results for unitary operators defined on $$l^2(\mathbb{N})$$ together with an application to orthogonal polynomials on the unit circle. We get almost sure localization for polynomials characterized by Verblunsky coefficients of constant modulus and correlated random phases