Localization for Random Unitary Operators

Abstract

We consider unitary analogs of $1-$dimensional Anderson models on $l^2(\mathbb{Z})$ defined by the product $U_{\omega }=D_{\omega }S$ where $S$ is a deterministic unitary and $D_{\omega }$ 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 $U_{\omega }$ 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 Verblunski coefficients of constant modulus and correlated random phases.

References in corpus (3)

• Spectral Analysis of Unitary Band Matrices
• Density of States and Thouless Formula for Random Unitary Band Matrices
• Fractional Moment Estimates for Random Unitary Operators

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