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274
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.
Uniform Spectral Properties Of OneDimensional Quasicrystals, IV. QuasiSturmian Potentials
 I. Absence of eigenvalues, Commun. Math. Phys
, 2000
"... We consider discrete onedimensional Schrodinger operators with quasiSturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, ..."
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Cited by 51 (32 self)
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We consider discrete onedimensional Schrodinger operators with quasiSturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely ffcontinuous spectrum. All these results hold uniformly on the hull generated by a given potential.
Metalinsulator transition for the almost Mathieu operator
, 1999
"... We prove that for Diophantine ω and almost every θ, the almost Mathieu operator, (Hω,λ,θΨ)(n) = Ψ(n+1)+Ψ(n −1)+λcos2π(ωn+θ)Ψ(n), exhibits localization for λ> 2 and purely absolutely continuous spectrum for λ < 2. This completes the proof of (a correct version of) the AubryAndré conjecture. ..."
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Cited by 47 (5 self)
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We prove that for Diophantine ω and almost every θ, the almost Mathieu operator, (Hω,λ,θΨ)(n) = Ψ(n+1)+Ψ(n −1)+λcos2π(ωn+θ)Ψ(n), exhibits localization for λ> 2 and purely absolutely continuous spectrum for λ < 2. This completes the proof of (a correct version of) the AubryAndré conjecture.
Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs
 J. Phys. A: Math. Gen
"... The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one ..."
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Cited by 45 (5 self)
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The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one to detect that a point λ belongs to the spectrum when a generalized eigenfunction with an subexponential growth integral estimate is available. A theorem on spectral gap opening for “decorated ” quantum graphs is established (its analog is known for the combinatorial case). It is also shown that if a periodic combinatorial or quantum graph has a point spectrum, it is generated by compactly supported eigenfunctions (“scars”). 1
Continuity properties of Schrödinger semigroups with magnetic fields
 Rev. Math. Phys
"... The objects of the present study are oneparameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Katolike conditions. The configuration space ..."
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Cited by 40 (10 self)
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The objects of the present study are oneparameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Katolike conditions. The configuration space is supposed to be an arbitrary open subset of multidimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show localnormcontinuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownianbridge expectation. Altogether, the article is meant to extend some of the results in B. Simon’s landmark paper (Bull. Amer. Math. Soc. (N.S.) 7, 447–526 (1982)) to nonzero vector potentials and more general configuration
Moment Analysis for Localization in Random Schrödinger Operators
, 2005
"... We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the ..."
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Cited by 38 (12 self)
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We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonancediffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the LifshitzKrein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weakL¹ estimate concerning the boundaryvalue distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.
Time dependent resonance theory
, 1995
"... An important class of resonance problems involves the study of perturbations of systems having embedded eigenvalues in their continuous spectrum. Problems with this mathematical structure arise in the study of many physical systems, e.g. the coupling of an atom or molecule to a photonradiation fiel ..."
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Cited by 29 (9 self)
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An important class of resonance problems involves the study of perturbations of systems having embedded eigenvalues in their continuous spectrum. Problems with this mathematical structure arise in the study of many physical systems, e.g. the coupling of an atom or molecule to a photonradiation field, and Auger states of the helium atom, as well as in spectral geometry and number theory. We present a dynamic (timedependent) theory of such quantum resonances. The key hypotheses are (i) a resonance condition which holds generically (nonvanishing of the Fermi golden rule) and (ii) local decay estimates for the unperturbed dynamics with initial data consisting of continuum modes associated with an interval containing the embedded eigenvalue of the unperturbed Hamiltonian. No assumption of dilation analyticity of the potential is made. Our method explicitly demonstrates the flow of energy from the resonant discrete mode to continuum modes due to their coupling. The approach is also applicable to nonautonomous linear problems and to nonlinear problems. We derive the time behavior of the resonant states for intermediate and long times. Examples and applications are presented. Among them is a proof of the instability of an embedded eigenvalue at a threshold energy under suitable hypotheses.
Resonances in One Dimension and Fredholm Determinants
, 2000
"... We discuss resonances for Schrödinger operators in whole and halfline problems. One of our goals is to connect the Fredholm determinant approach of Froese to the Fourier transform approach of Zworski. Another is to prove a result on the number of antibound states namely, in a halfline problem the ..."
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Cited by 24 (1 self)
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We discuss resonances for Schrödinger operators in whole and halfline problems. One of our goals is to connect the Fredholm determinant approach of Froese to the Fourier transform approach of Zworski. Another is to prove a result on the number of antibound states namely, in a halfline problem there are an odd number of antibound states between any two bound states.
Essential selfadjointness for semibounded magnetic Schrödinger operators on noncompact manifolds
, 2001
"... We prove essential selfadjointness for semibounded below magnetic Schrödinger operators on complete Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. Some singularities of the scalar potential are allowed. This is an extension of the Povzner–Wie ..."
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Cited by 23 (3 self)
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We prove essential selfadjointness for semibounded below magnetic Schrödinger operators on complete Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. Some singularities of the scalar potential are allowed. This is an extension of the Povzner–Wienholtz–Simader theorem. The proof uses the scheme of Wienholtz but requires a refined invariant integration by parts technique, as well as a use of a family of cutoff functions which are constructed by a nontrivial smoothing procedure due to Karcher.