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Localization for Random Unitary Operators

Eman Hamza·Alain Joye·Gunter Stolz
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Abstract

We consider unitary analogs of $1-$dimensional Anderson models on $l^2(\Z)$ defined by the product $U_ω=D_ωS$ 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 $l^2(\N)$ 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.

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Eman Hamza, Alain Joye, Gunter Stolz. 2005-04-25. Localization for Random Unitary Operators. https://doi.org/10.1007/s11005-005-0044-4

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