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198
Eigenfunctions, transfer matrices, and absolutely continuous spectrum of onedimensional Schrödinger operators
, 1999
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Spectral Analysis for Systems of Atoms and Molecules coupled to the Quantized Radiation Field
 Commun. Math. Phys
, 1998
"... We consider systems of static nuclei and electrons atoms and molecules coupled to the quantized radiation field. The interactions between electrons and the soft modes of the quantized electromagnetic field are described by minimal coupling, ~ p ! ~ p \Gamma e ~ A(~x), where ~ A(~x) is the elect ..."
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Cited by 106 (17 self)
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We consider systems of static nuclei and electrons atoms and molecules coupled to the quantized radiation field. The interactions between electrons and the soft modes of the quantized electromagnetic field are described by minimal coupling, ~ p ! ~ p \Gamma e ~ A(~x), where ~ A(~x) is the electromagnetic vector potential with an ultraviolet cutoff. If the interactions between the electrons and the quantized radiation field are turned off, the atom or molecule is assumed to have at least one bound state. We prove that, for sufficiently small values of the feinstructure constant ff, the interacting system has a ground state corresponding to the bottom of its energy spectrum and that the excited states of the atom or molecule above the ground state turn into metastable states Heisenberg Fellow of the DFG, supported by SFB 288 of the DFG, the TMRNetwork on "PDE and QM". y Supported by NSERC Grant NA 7901 0 whose lifetimes we estimate. Furthermore the energy spectrum is absolu...
Quantum Dynamics and Decompositions of Singular Continuous Spectra
 J. Funct. Anal
, 1995
"... . We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (wi ..."
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Cited by 99 (13 self)
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. We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (with k/k = 1). The spectral measure ¯/ of / (and H ) is uniquely defined by [24]: h/ ; f(H)/i = Z oe(H) f(x) d¯/ (x) ; (1:1) for any measurable (Borel) function f . The time evolution of the state / , in the Schrodinger picture of quantum mechanics, is given by /(t) = e \GammaiHt / : (1:2) The relations between various properties of the spectral measure ¯/ (with an emphasis on "fractal" properties) and the nature of the time evolution have been the subject of several recent papers [7,13,1518,20,22,33,36,39]. Our purpose in this paper is twofold: First, we use a theory, due to Rogers and Taylor [28,29], of decomposing singular continuous measures with respect to Hausdorff measures to i...
Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations
, 1998
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Spectral Theory Of Elliptic Operators On NonCompact Manifolds
, 1992
"... preliminaries Let H be a complex Hilbert space, A a densely dened linear operator in H (the domain of A will be denoted D(A)). Suppose that A has a closure A or, equivalently, that the adjoint operator A is densely dened (see e.g. [32]). We shall denote by GA the graph of A i.e. the set of pairs ..."
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Cited by 73 (9 self)
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preliminaries Let H be a complex Hilbert space, A a densely dened linear operator in H (the domain of A will be denoted D(A)). Suppose that A has a closure A or, equivalently, that the adjoint operator A is densely dened (see e.g. [32]). We shall denote by GA the graph of A i.e. the set of pairs fu; Aug; u 2 D(A). Then G A = GA , i.e. the graph of A is the closure of the graph of A. Moreover A = A = (A ) . Now let A + be another densely dened linear operator in H. DEFINITION 1.1. A + is called formally adjoint to A if (1:1) (Au; v) = (u; A + v); u 2 D(A); v 2 D(A + ); where (; ) is the scalar product in H. If A = A + then A is called symmetric or formally self{adjoint. Note that since A; A + are densely dened, both A and A + have closures. DEFINITION 1.2. Let A; A + be as in Denition 1.1. Then the minimal and the maximal operator for A are dened as follows: A min = A = A ; A max = (A + ) : Note that both A min and A max are...
Renormalization Group Analysis of Spectral Problems in Quantum Field Theory
, 1998
"... this paper we present a selfcontained and detailed exposition of the new renormalization group technique proposed in [1, 2]. Its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator, the partition function, o ..."
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Cited by 62 (23 self)
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this paper we present a selfcontained and detailed exposition of the new renormalization group technique proposed in [1, 2]. Its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator, the partition function, or correlation functions. We apply this renormalization transformation to a Hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field, and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing. # 1998 Academic Press Key Words: renormalization group; spectrum; resonances; Fock space; QED. I. INTRODUCTION In this paper we give a detailed and mathematically selfcontained presentation of the new renormalization group technique proposed in [1, 2]. The mathematical problems we address in this paper are encountered in the quantum theory of atoms and molecules coupled to the quantized electromagnetic field. As explained in [1, 2], our task is to explore spectral properties of Hamiltonians that generate the time evolution of systems of Article No. AI981733 205 00018708#98 #25.00 Copyright # 1998 by Academic Press All rights of reproduction in any form reserved. * Supported by SFB 288 of the DFG, the TMRNetwork on PDE and QM," and NSF Grant Phy 9010433A02.  Supported by NSERC Grant NA 7901. electrons (particles") bound to static nuclei and interacting with the quantized radiation field (bosons"). In paper [2] we justify studying the following simplified quantum mechanical model system. The Hilbert space of pure state vectors of the system is given by H=H el #F b [L 2 (R d )], (I.1) where H el , the particle Hilbert space", is some separable Hilbert space (for a sin...
Localization for random perturbations of periodic Schrödinger operators
, 1998
"... We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R 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 s ..."
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Cited by 58 (19 self)
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We prove localization for Andersontype random perturbations of periodic Schrödinger operators on R 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 Schrödinger 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.
Magnetic Bottles in Connection With Superconductivity
, 2001
"... Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semiclassical regime of the lowest eigenvalue of the Schrodinger operator with magnetic field have appeared recently. Here ..."
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Cited by 57 (16 self)
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Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semiclassical regime of the lowest eigenvalue of the Schrodinger operator with magnetic field have appeared recently. Here we would like to mention the works by BernoffSternberg, LuPan and Del PinoFelmerSternberg. This recovers partially questions analyzed in a different context by the authors around the question of the so called magnetic bottles. Our aim is to analyze the former results, to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder. In particular, we improve significatively the lower bounds and as a byproduct we solve a conjecture proposed by BernoffSternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields.
Selection of the ground state for nonlinear Schrödinger equations
 REV. MATH. PHYS
, 2008
"... We prove for a class of nonlinear Schrödinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism throu ..."
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Cited by 56 (7 self)
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We prove for a class of nonlinear Schrödinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear Master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of HartreeFock type.
A characterization of the Anderson metalinsulator transport transition
 Duke Math. J
"... We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong... ..."
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Cited by 52 (19 self)
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We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong...