Results 1  10
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445
Time decay for solutions of Schrödinger equations with rough and timedependent potentials.
, 2001
"... In this paper we establish dispersive estimates for solutions to the linear SchrSdinger equation in three dimension (0.1) 1.0tO  A0 + Vb = 0, O(s) = f where V(t, x) is a timedependent potential that satisfies the conditions suPllV(t,.)llL(R) + sup f f IV(*'x)l drdy < Co. ..."
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Cited by 72 (14 self)
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In this paper we establish dispersive estimates for solutions to the linear SchrSdinger equation in three dimension (0.1) 1.0tO  A0 + Vb = 0, O(s) = f where V(t, x) is a timedependent potential that satisfies the conditions suPllV(t,.)llL(R) + sup f f IV(*'x)l drdy < Co.
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
A Kreinlike formula for singular perturbations of selfadjoint operators and applications
"... Given a selfadjoint operator A: D(A) ⊆ H → H and a continuous linear operator τ: D(A) → X with Range τ ′ ∩ H ′ = {0}, X a Banach space, we explicitly construct a family A τ Θ of selfadjoint operators such that any Aτ Θ coincides with the original A on the kernel of τ. Such a family is obtained ..."
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Cited by 39 (11 self)
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Given a selfadjoint operator A: D(A) ⊆ H → H and a continuous linear operator τ: D(A) → X with Range τ ′ ∩ H ′ = {0}, X a Banach space, we explicitly construct a family A τ Θ of selfadjoint operators such that any Aτ Θ coincides with the original A on the kernel of τ. Such a family is obtained by giving a Kreĭnlike formula where the role of the deficiency spaces is played by the dual pair (X, X ′); the parameter Θ belongs to the space of symmetric operators from X ′ to X. When X = C one recovers the “ H−2construction” of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which H = L 2 (R n) and τ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudodifferential operators, thus unifying and extending previously known results. 1.
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.
Solitary wave dynamics in an external potential
 Comm. Math. Phys
"... We study the behavior of solitarywave solutions of some generalized nonlinear Schrödinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We construct soluti ..."
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Cited by 36 (7 self)
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We study the behavior of solitarywave solutions of some generalized nonlinear Schrödinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We construct solutions of the equations with a nonvanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton’s equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping. 1
Vishik: Determinants of elliptic pseudo–differential operators
, 1994
"... Abstract. Determinants of invertible pseudodifferential operators (PDOs) close to positive selfadjoint ones are defined through the zetafunction regularization. We define a multiplicative anomaly as the ratio det(AB)/(det(A)det(B)) considered as a function on pairs of elliptic PDOs. We obtained a ..."
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Cited by 34 (1 self)
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Abstract. Determinants of invertible pseudodifferential operators (PDOs) close to positive selfadjoint ones are defined through the zetafunction regularization. We define a multiplicative anomaly as the ratio det(AB)/(det(A)det(B)) considered as a function on pairs of elliptic PDOs. We obtained an explicit formula for the multiplicative anomaly in terms of symbols of operators. For a certain natural class of PDOs on odddimensional manifolds generalizing the class of elliptic differential operators, the multiplicative anomaly is identically 1. For elliptic PDOs from this class a holomorphic determinant and a determinant for zero orders PDOs are introduced. Using various algebraic, analytic, and topological tools we study local and global properties of the multiplicative anomaly and of the determinant Lie group closely related with it. The Lie algebra for the determinant Lie group has a description in terms of symbols only. Our main discovery is that there is a quadratic nonlinearity hidden in the definition of determinants of PDOs through zetafunctions. The natural explanation of this nonlinearity follows from complexanalytic properties of a new trace functional TR on PDOs of noninteger orders. Using TR we easily reproduce known facts about noncommutative residues of PDOs and obtain several new results. In particular, we describe a structure of derivatives of zetafunctions at zero as of functions on logarithms of elliptic PDOs. We propose several definitions extending zetaregularized determinants to general elliptic PDOs. For elliptic PDOs of nonzero complex orders we introduce a canonical determinant in its natural domain of definition. Contents
Notes on infinite determinants of Hilbert space operators
 Adv. Math
, 1977
"... We present a novel approach to obtaining the basic facts (including Lidskii's theorem on the equality of the matrix and spectral traces) about determinants and traces of trace class operators on a separable Hilbert space. We also discuss Fredholm theory, "regularized " determinants and Fredholm theo ..."
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Cited by 34 (2 self)
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We present a novel approach to obtaining the basic facts (including Lidskii's theorem on the equality of the matrix and spectral traces) about determinants and traces of trace class operators on a separable Hilbert space. We also discuss Fredholm theory, "regularized " determinants and Fredholm theory on the trace ideals, c#~(p < oo). 1.
Spectral properties of the laplacian on bondpercolation graphs
"... Abstract Bondpercolation graphs are random subgraphs of the ddimensional integer lattice generated by a standard bondpercolation process. The associated graph Laplacians, subject to Dirichlet or Neumann conditions at cluster boundaries, represent bounded, selfadjoint, ergodic random operators wi ..."
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Cited by 33 (9 self)
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Abstract Bondpercolation graphs are random subgraphs of the ddimensional integer lattice generated by a standard bondpercolation process. The associated graph Laplacians, subject to Dirichlet or Neumann conditions at cluster boundaries, represent bounded, selfadjoint, ergodic random operators with offdiagonal disorder. They possess almost surely the nonrandom spectrum [0, 4d] and a selfaveraging integrated density of states. The integrated density of states is shown to exhibit Lifshits tails at both spectral edges in the nonpercolating phase. While the characteristic exponent of the Lifshits tail for the Dirichlet (Neumann) Laplacian at the lower (upper) spectral edge equals d/2, and thus depends on the spatial dimension, this is not the case at the upper (lower) spectral edge, where the exponent equals 1/2.
Lowtemperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitelymany ground states
"... Starting from classical lattice systems in d 2 dimensions with a regular zerotemperature phase diagram, involving a finite number of periodic ground states, we prove that the addition of a small quantum perturbation and/or increasing the temperature produce only smooth deformations of their phase ..."
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Cited by 31 (13 self)
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Starting from classical lattice systems in d 2 dimensions with a regular zerotemperature phase diagram, involving a finite number of periodic ground states, we prove that the addition of a small quantum perturbation and/or increasing the temperature produce only smooth deformations of their phase diagrams. The quantum perturbations can involve bosons or fermions and Present address: Facultad de Matem'atica, Astronom'ia y F'isica, Universidad Nacional de C'ordoba, Ciudad Universitaria, 5000 C'ordoba, Argentina. Email: fernande@fis.uncor.edu can be of infinite range but decaying exponentially fast with the size of the bonds. For fermions, the interactions must be given by monomials of even degree in creation and annihilation operators. Our methods can be applied to some anyonic systems as well. Our analysis is based on an extension of PirogovSinai theory to contour expansions in d + 1 dimensions obtained by iteration of the Duhamel formula. Keywords: Phase diagrams; quantum latti...