Localization for Random Unitary Operators Eman Hamza

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Niveau: Supérieur, Licence, Bac+2
Localization for Random Unitary Operators Eman Hamza University of Alabama at Birmingham Department of Mathematics CH 452 Birmingham, Al 35294-1170 U.S.A. Alain Joye Institut Fourier Universite de Grenoble, BP 74 38402 Saint-Martin d'Heres France Gunter Stolz? University of Alabama at Birmingham Department of Mathematics CH 452 Birmingham, Al 35294-1170 U.S.A. Prepublication de l'Institut Fourier n0 673 (2005) Abstract We consider unitary analogs of 1?dimensional Anderson models on l2(Z) defined by the product U? = D?S where S is a deterministic unitary and D? 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? is pure point almost surely for all values of the parameter of S. We provide similar results for unitary operators defined on l2(N) together with an application to orthogonal polynomials on the unit circle. We get almost sure localization for polyno- mials characterized by Verblunski coefficients of constant modulus and correlated random phases. AMS classification numbers: 82B44, 42C05, 81Q05 Keywords: Localization, random unitary operators, orthogonal polynomials. ?partially supported through US-NSF grant DMS-0245210 1

model alluded

unitary operators

diagonal matrix

spectral measure

adjoint anderson model

almost surely

introduction unitary operators

random variable

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Publié par

profil-vieg-2012

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LocalizationforRandomUnitaryOperatorsEmanHamzaAlainJoyeUniversityofAlabamaatBirminghamInstitutFourierDepartmentofMathematicsCH452Universite´deGrenoble,BP74Birmingham,Al35294-117038402Saint-Martind’He`resU.S.A.FranceGu¨nterStolz∗UniversityofAlabamaatBirminghamDepartmentofMathematicsCH452Birmingham,Al35294-1170U.S.A.Pre´publicationdel’InstitutFouriern0673(2005)http://www-fourier.ujf-grenoble.fr/prepublications.htmlAbstractWeconsiderunitaryanalogsof1−dimensionalAndersonmodelsonl2(Z)deﬁnedbytheproductUω=DωSwhereSisadeterministicunitaryandDωisadiagonalmatrixofi.i.d.randomphases.TheoperatorSisanabsolutelycontinuousbandmatrixwhichdependsonaparametercontrollingthesizeofitsoﬀ-diagonalelements.WeprovethatthespectrumofUωispurepointalmostsurelyforallvaluesoftheparameterofS.Weprovidesimilarresultsforunitaryoperatorsdeﬁnedonl2(N)togetherwithanapplicationtoorthogonalpolynomialsontheunitcircle.Wegetalmostsurelocalizationforpolyno-mialscharacterizedbyVerblunskicoeﬃcientsofconstantmodulusandcorrelatedrandomphases.AMSclassiﬁcationnumbers:82B44,42C05,81Q05Keywords:Localization,randomunitaryoperators,orthogonalpolynomials.∗partiallysupportedthroughUS-NSFgrantDMS-02452101

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