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109
Localization for random perturbations of periodic Schrödinger operators
 RANDOM OPER. STOCHASTIC EQUATIONS
, 1996
"... We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R d near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) ..."
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Cited by 59 (20 self)
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We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R d near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A study of the band shift of periodic Schrodinger operators under linearly coupled periodic perturbations. (ii) A proof of the Wegner estimate using properties of the spatial distribution of eigenfunctions of finite box hamiltonians. (iii) An improved multiscale method together with a result of de Branges on the existence of limiting values for resolvents in the upper half plane, allowing for rather weak disorder assumptions on the random potential. (iv) Results from the theory of generalized eigenfunctions and spectral averaging. The paper aims at high accessibility in providing details for all the main steps in the proof.
An invitation to random Schrödinger operators
, 2007
"... This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are m ..."
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Cited by 51 (8 self)
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This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization. Copyright by the author. Copying for academic purposes is permitted.
Internal Lifshits Tails For Random Perturbations Of Periodic Schrödinger Operators
"... . Let H be a \Gammaperiodic Schrodinger operator acting on L 2 (R d ) and consider the random Schrodinger operator H! = H + V! where V! (x) = X fl2\Gamma ! fl V (x \Gamma fl) (here V is a positive potential and (! fl ) fl2\Gamma a collection of positive i.i.d random variables). We prove tha ..."
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Cited by 42 (7 self)
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. Let H be a \Gammaperiodic Schrodinger operator acting on L 2 (R d ) and consider the random Schrodinger operator H! = H + V! where V! (x) = X fl2\Gamma ! fl V (x \Gamma fl) (here V is a positive potential and (! fl ) fl2\Gamma a collection of positive i.i.d random variables). We prove that, at the edge of a gap of H that is not filled in for H! , the integrated density of states of H! has a Lifshits tail behaviour if and only if the integrated density of states of H is nondegenerate. R' esum' e. Soient H un op'erateur de Schrodinger \Gammap'eriodique agissant sur L 2 (R d ), V un potentiel positif et (! fl ) fl2Z d une famille de variables al'eatoires i.i.d positives. Consid'erons l'op'erateur de Schrodinger al'eatoire H! = H + V! o`u V! (x) = X fl2Z d ! fl V (x \Gamma fl). On montre que, au bord d'une lacune spectrale de H qui n'est pas combl'e pour H! la densit'e d"etats int'egr'ee de H! a un comportement asymptotique de Lifshits si et seulement si la densit'e d"...
Localization of Classical Waves I: Acoustic Waves.
 Commun. Math. Phys
, 1996
"... We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existe ..."
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Cited by 35 (0 self)
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We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the selfadjoint operators A = \Gammar \Delta 1 %(x) r on L 2 (R d ). We prove that, in the random medium described by %(x), the random operator A exhibits Anderson localization inside the gap in the spectrum of A 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almost the wh...
Scattering theory for systems with different spatial asymptotics to the left and right
 COMMUN.MATH.PHYS. 63
, 1978
"... We discuss the existence and completeness of scattering for onedimensional systems with different spatial asymptotics at ± oo, for example 2 4 V(x) where V(x) = 0 (resp. sin x) if x < 0 (resp. x> 0). We then extend our results to higher dimensional systems periodic, except for a localised impurit ..."
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Cited by 30 (12 self)
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We discuss the existence and completeness of scattering for onedimensional systems with different spatial asymptotics at ± oo, for example 2 4 V(x) where V(x) = 0 (resp. sin x) if x < 0 (resp. x> 0). We then extend our results to higher dimensional systems periodic, except for a localised impurity, in all but one space dimension. A new method, "the twisting trick", is presented for proving the absence of singular continuous spectrum, and some independent applications of this trick are given in an appendix.
Perturbations of OneDimensional Schrödinger Operators Preserving the Absolutely Continuous Spectrum
, 2001
"... We study the stability of the absolutely continuous spectrum of onedimensional Schrodinger operators [Hu](x) = \Gammau 00 (x) + q(x)u(x) with periodic potentials q(x). Specifically, it is proved that any perturbation of the potential, V 2 L 2 , preserves the essential support (and multiplicity ..."
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Cited by 29 (3 self)
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We study the stability of the absolutely continuous spectrum of onedimensional Schrodinger operators [Hu](x) = \Gammau 00 (x) + q(x)u(x) with periodic potentials q(x). Specifically, it is proved that any perturbation of the potential, V 2 L 2 , preserves the essential support (and multiplicity) of the absolutely continuous spectrum. This is optimal in terms of L p spaces and, for q j 0, it answers in the affirmative a conjecture of Kiselev, Last and Simon. By adding constraints on the Fourier transform of V , it is possible to relax the decay assumptions on V . It is proved that if V 2 L 3 and V is uniformly locally square integrable, then preservation of the a.c. spectrum still holds. If we assume that q j 0, still stronger results follow: if V 2 L 3 and V (k) is square integrable on an interval [k 0 ; k 1 ], then the interval [k 2 0 =4; k 2 1 =4] is contained in the essential support of the absolutely continuous spectrum of the perturbed operator. vi Contents Ackn...
Localization for onedimensional, continuum, BernoulliAnderson models
 Duke Math. J
"... We use scattering theoretic methods to prove strong dynamical and exponential localization for onedimensional, continuum, Andersontype models with singular distributions; in particular, the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two ..."
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Cited by 28 (12 self)
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We use scattering theoretic methods to prove strong dynamical and exponential localization for onedimensional, continuum, Andersontype models with singular distributions; in particular, the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms. Our main tools are the reflection and transmission coefficients for compactly supported singlesite perturbations of a periodic background which we use to verify the necessary hypotheses of multiscale analysis. We show that nonreflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs. 1.
Elliptic algebrogeometric solutions of the KdV and AKNS hierarchiesan analytic approach
 Bul.(New Series) AMS
, 1998
"... Abstract. We provide an overview of elliptic algebrogeometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. Our treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theor ..."
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Cited by 28 (7 self)
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Abstract. We provide an overview of elliptic algebrogeometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. Our treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theory developed by Hermite and Picard. 1. Introduction. The story of J. Scott Russell chasing a soliton for a mile or two along the EdinburghGlasgow channel in 1834 has been told many times. It is the starting point of more than 160 years of an exciting history embracing a variety of deep mathematical ideas ranging from applied mathematics to algebraic geometry, Lie
Perturbations of orthogonal polynomials with periodic recursion coefficients
, 2007
"... We extend the results of Denisov–Rakhmanov, Szegő–Shohat– Nevai, and Killip–Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well ada ..."
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Cited by 26 (15 self)
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We extend the results of Denisov–Rakhmanov, Szegő–Shohat– Nevai, and Killip–Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.