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Asymptotic Behaviour For The 3D SchrödingerPoisson System In The Attractive Case With Positive Energy
, 1999
"... In this paper we study the asymptotic behaviour of solutions to the threedimensional SchrodingerPoisson system in the attractive case with positive energy. In this case, it is proved that for a real initial condition the solutions expand unboundedly as time goes to infinity. The proof of this resu ..."
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In this paper we study the asymptotic behaviour of solutions to the threedimensional SchrodingerPoisson system in the attractive case with positive energy. In this case, it is proved that for a real initial condition the solutions expand unboundedly as time goes to infinity. The proof of this result is based on the derivation of a dispersive equation relating density and linear momentum as well as on optimal bounds for the kinetic energy. Keywords Asymptotic behaviour, SchrodingerPoisson problem. 1 INTRODUCTION The SchrodingerPoisson system (SPS) in (0, #) R 3 associated with a single particle in a vacuum can be written in terms of the wave function (x, t) and the potential V (x, t) as follows i~ # #t =  ~ 2 2m # + V , (x, 0) = #(x), lim x## # = 0, (1.1) where ~ stands for the Planck constant and m for the particle mass, and where the potential is related to the particle density #(x, t) 2 via the Poisson equation #V (x, t) = ##(x, t) 2 , lim x## V =...
HOMOGENIZATION ON LATTICES: SMALL PARAMETER LIMITS, HMEASURES, AND DISCRETE WIGNER MEASURES
"... Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of m ..."
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Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of magnetic dipoles on a Bravais lattice, (letting the lattice parameter tend to zero). In order to describe the smallparameter limit, we use discrete Wigner transforms to transform the storedenergy which is given by the double convolution of a sequence of (dipole) functions on a Bravais lattice with a kernel, homogeneous of degree with N with the cancellation property, as the lattice parameter tends to zero. By rescaling and using Fourier methods, discrete Wigner transforms in particular, to transform the problem to one on the torus, we are able to characterize the smallparameter limit of the energy depending on whether the dipoles oscillate on the scale of the lattice, oscillate on a much longer lengthscale, or converge strongly. In the case where> N, the result is simple and can be characterized by anintegral with respect to the Wigner measure limit on the torus. In the case where = N, oscillations essentially on the scale of the lattice must be separated from oscillations essentially onamuch longer lengthscale in order to characterize the energy in terms of the Wigner measure limit on the torus, an Hmeasure limit, and the limiting magnetization. We show that the classical
Chapter 2 The Wigner–Poisson System
"... Abstract In electrostatic quantum plasmas, the Wigner–Poisson system plays the same rôle as the Vlasov–Poisson system in classical plasmas. This chapter considers the basic properties of the Wigner–Poisson system, including the essentials on the Wigner function method and the derivation of the Wigne ..."
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Abstract In electrostatic quantum plasmas, the Wigner–Poisson system plays the same rôle as the Vlasov–Poisson system in classical plasmas. This chapter considers the basic properties of the Wigner–Poisson system, including the essentials on the Wigner function method and the derivation of the Wigner–Poisson system in the context of a mean field theory. This chapter also contains a discussion on the Schrödinger–Poisson system as well as extensions to include correlation and collisional effects. The Wigner–Poisson system is shown to imply, in the highfrequency limit, the Bohm–Pines dispersion relation for linear waves, which is the quantum analog of the Bohm–Gross dispersion relation for classical plasmas. 2.1 The Wigner Function To maintain the closest resemblance to the familiar methods of classical plasma physics, the Wigner function approach is the natural choice. Indeed, using the Wigner function, one can proceed in almost total analogy with the standard phasespace distribution function method to compute macroscopic quantities like number and current densities. Hence, it is useful to review some of the properties of the Wigner (pseudo) distribution function approach. In addition, the differences between classical and quantum formalisms will be highlighted. The treatment is by no means exhaustive, being intentionally restricted to the bare necessary minimum. More complete reviews on Wigner function methods can be found, for example, in [8, 18, 26, 36]. For simplicity, let us start with a onedimensional, oneparticle pure state quantum system, represented by a wavefunction ψ(x,t). In this case, the Wigner function f = f (x,v,t) is defined [38]as f = m
GLOBAL WELLPOSEDNESS FOR THE L 2 CRITICAL HARTREE EQUATION ON R n, n ≥ 3.
, 806
"... Abstract. We consider the initial value problem for the L 2critical defocusing Hartree equation in R n, n ≥ 3. We show that the problem is globally well posed in H s (R n) when 1> s> 2(n−2). We use the “Imethod ” following [14] combined with a local in time ..."
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Abstract. We consider the initial value problem for the L 2critical defocusing Hartree equation in R n, n ≥ 3. We show that the problem is globally well posed in H s (R n) when 1> s> 2(n−2). We use the “Imethod ” following [14] combined with a local in time
Groups of Operators for Evolution Equations of Quantum ManyParticle Systems
, 2008
"... The aim of this work is to study the properties of groups of operators for evolution equations of quantum manyparticle systems, namely, the von Neumann hierarchy for correlation operators, the BBGKY hierarchy for marginal density operators and the dual BBGKY hierarchy for marginal observables. We s ..."
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The aim of this work is to study the properties of groups of operators for evolution equations of quantum manyparticle systems, namely, the von Neumann hierarchy for correlation operators, the BBGKY hierarchy for marginal density operators and the dual BBGKY hierarchy for marginal observables. We show that the concept of cumulants (semiinvariants) of groups of operators for the von Neumann equations forms the basis of the expansions for oneparametric families of operators for evolution equations of infinitely many particles.
ASYMPTOTIC BEHAVIOR TO THE 3D SCHRÖDINGER/HARTREE POISSON AND WIGNER POISSON SYSTEMS
, 1999
"... Using an appropriate scaling group for the 3D Schrödinger–Poisson equation and the equivalence between the Schrödinger formalism and the Wigner representation of quantum mechanics it is proved that, when time goes to infinity, the limit of the rescaled selfconsistent potential can be identified as ..."
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Using an appropriate scaling group for the 3D Schrödinger–Poisson equation and the equivalence between the Schrödinger formalism and the Wigner representation of quantum mechanics it is proved that, when time goes to infinity, the limit of the rescaled selfconsistent potential can be identified as the Coulomb potential. As a consequence, Schrödinger–Poisson and Wigner–Poisson systems are asymptotically simplified and their longtime behavior is explained through the solutions of the corresponding linear limit problems.
Contents
, 2012
"... Abstract. We consider the classical limit of the quantum evolution, with some rough potential, of wave packets concentrated near singular trajectories of the underlying dynamics. We prove that under appropriate conditions, even in the case of BV vector fields, the correct classical limit can be sele ..."
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Abstract. We consider the classical limit of the quantum evolution, with some rough potential, of wave packets concentrated near singular trajectories of the underlying dynamics. We prove that under appropriate conditions, even in the case of BV vector fields, the correct classical limit can be selected.