Localization for Random Unitary Operators Eman Hamza

Alain Joye

Description

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

Sujets

• Unitary operator
• Diagonal matrix
• Spectral theory of ordinary differential equations
• Almost surely
• Random variable

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 15
• Langue: English

Extrait

LocalizationforRandomUnitaryOperators EmanHamzaAlainJoye UniversityofAlabamaatBirminghamInstitutFourier DepartmentofMathematicsCH452Universite´deGrenoble,BP74 Birmingham,Al35294-117038402Saint-Martind’He`res U.S.A.France Gu¨nterStolz ∗ UniversityofAlabamaatBirmingham DepartmentofMathematicsCH452 Birmingham,Al35294-1170 U.S.A. Pre´publicationdel’InstitutFouriern 0 673(2005) http://www-fourier.ujf-grenoble.fr/prepublications.html Abstract Weconsiderunitaryanalogsof1 − dimensionalAndersonmodelsonl 2 ( Z )definedby theproductU ω =D ω SwhereSisadeterministicunitaryandD ω isadiagonalmatrix ofi.i.d.randomphases.TheoperatorSisanabsolutelycontinuousbandmatrixwhich dependsonaparametercontrollingthesizeofitsoff-diagonalelements.Weprovethat thespectrumofU ω ispurepointalmostsurelyforallvaluesoftheparameterofS.We providesimilarresultsforunitaryoperatorsdefinedonl 2 ( N )togetherwithanapplication toorthogonalpolynomialsontheunitcircle.Wegetalmostsurelocalizationforpolyno- mialscharacterizedbyVerblunskicoefficientsofconstantmodulusandcorrelatedrandom phases. AMSclassificationnumbers:82B44,42C05,81Q05 Keywords:Localization,randomunitaryoperators,orthogonalpolynomials. ∗ partiallysupportedthroughUS-NSFgrantDMS-0245210 1