Fractional Moment Estimates for Random Unitary Operators
We consider unitary analogs of d-dimensional Anderson models on l2( $$\mathbb(z)$$ d ) defined by the product Uω=Dω S where S is a deterministic unitary and Dω is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on parameters controlling the size of its off-diagonal elements. We adapt the method of Aizenman–Molchanov to get exponential estimates on fractional moments of the matrix elements of Uω(Uω −z)−1, provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements of S. Such estimates imply almost sure localization for Uω.
Keywordsfractional moment method unitary operators localization.
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- 1.Aizenman, M., Molchanov, S. 1993Localization at large disorder and at extreme energies: an elementary derivationCommun. Math. Phys.157245278Google Scholar
- 2.Aizenman, M., Graf, G.-M. 1998Localization bounds for an electron gasJ. Phys. A.3167836806Google Scholar
- 5.Bourget, O., Howland, J.S., Joye, A. 2003Special analysis of unitary band matricesCommon. Math3738563880Google Scholar
- 10.Simon, B. (2004). Orthogonal Polynomials on the Unit Circle: Part 1: Classical Theory; Part~2: Spectral Theory, AMS Colloquium Series, Vol 54, American Mathematical Society, Providence RI.Google Scholar
- 11.Simon, B. Aizenman’s theorem for orthogonal polynomials on the unit circle, eprint, math.SP/0411388 and mp-arc:04-386.Google Scholar
- 12.Simon, B., Wolff, T. 1986Singular continuous spectrum under rank one perturbations and localization for random HamiltoniansCommun. Pure Appl. Math.397590Google Scholar
- 13.Stoiciu, M.: The statistical distribution of the zeros of random paraorthonormal polynomials on the unit circle, eprint math-ph/0412025 and mp-arc:04-405.Google Scholar
- 14.Teplyaev, A.V. 1992The pure point spectrum of random polynomials orthogonal on the circleSoviet. Math. Dokl.44407411Google Scholar