High-frequency averaging in semi-classical Hartree-type equations (original) (raw)

Semi-Classical Wave Packet Dynamics for Hartree Equations

Reviews in Mathematical Physics, 2011

We study the propagation of wave packets for nonlinear nonlocal Schrödinger equations in the semi-classical limit. When the kernel is smooth, we construct approximate solutions for the wave functions in subcritical, critical and supercritical cases (in terms of the size of the initial data). The validity of the approximation is proved up to Ehrenfest time. For homogeneous kernels, we establish similar results in subcritical and critical cases. Nonlinear superposition principle for two nonlinear wave packets is also considered.

Solitary waves for the Hartree equation with a slowly varying potential

Pacific Journal of Mathematics, 2010

We study the Hartree equation with a slowly varying smooth potential, V (x) = W (hx), and with an initial condition that is ε ≤ √ h away in H 1 from a soliton. We show that up to time |log h|/ h and errors of size ε + h 2 in H 1 , the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian. This result is based on methods of Holmer and Zworski, who prove a similar theorem for the Gross-Pitaevskii equation, and on spectral estimates for the linearized Hartree operator recently obtained by Lenzmann. We also provide an extension of the result of Holmer and Zworski to more general initial conditions.

(Semi)Classical Limit of the Hartree Equation with Harmonic Potential

SIAM Journal on Applied Mathematics, 2005

Nonlinear Schrödinger Equations (NLS) of the Hartree type occur in the modeling of quantum semiconductor devices. Their "semiclassical" limit of vanishing (scaled) Planck constant is both a mathematical challenge and practically relevant when coupling quantum models to classical models. With the aim of describing the semi-classical limit of the 3D Schrödinger-Poisson system with an additional harmonic potential, we study some semi-classical limits of the Hartree equation with harmonic potential in space dimension n ≥ 2. The harmonic potential is confining, and causes focusing periodically in time. We prove asymptotics in several cases, showing different possible nonlinear phenomena according to the interplay of the size of the initial data and the power of the Hartree potential. In the case of the 3D Schrödinger-Poisson system with harmonic potential, we can only give a formal computation since the need of modified scattering operators for this long range scattering case goes beyond current theory. We also deal with the case of an additional "local" nonlinearity given by a power of the local density-a model that is relevant when incorporating the Pauli principle in the simplest model given by the "Schrödinger-Poisson-Xα equation". Further we discuss the connection of our WKB based analysis to the Wigner function approach to semiclassical limits.

On the Point-Particle (Newtonian) Limit¶of the Non-Linear Hartree Equation

Communications in Mathematical Physics, 2002

We consider the nonlinear Hartree equation describing the dynamics of weakly interacting non-relativistic Bosons. We show that a nonlinear Møller wave operator describing the scattering of a soliton and a wave can be defined. We also consider the dynamics of a soliton in a slowly varying background potential W (εx). We prove that the soliton decomposes into a soliton plus a scattering wave (radiation) up to times of order ε −1. To leading order, the center of the soliton follows the trajectory of a classical particle in the potential W (εx).

ASYMPTOTIC BEHAVIOR OF GROUND STATES OF GENERALIZED PSEUDO-RELATIVISTIC HARTREE EQUATION

With appropriate hypotheses on the nonlinearity f , we prove the existence of a ground state solution u for the problem −∆ + m 2 u + V u = (W * F (u)) f (u) in R N , where V is a bounded potential, not necessarily continuous, and F the primitive of f. We also show that any of this problem is a classical solution. Furthermore, we prove that the ground state solution has exponential decay.

Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities

2010

Some years ago, Penrose [26] derived a system of nonlinear equations by coupling the linear Schrödinger equation of quantum mechanics with Newton's gravitational law. Roughly speaking, a point mass interacts with a density of matter described by the square of the wave function that solves the Schrödinger equation. If m is the mass of the point, this interaction leads to the following system in R3:

On the role of quadratic oscillations in nonlinear Schrödinger equations

Journal of Functional Analysis, 2003

We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrödinger equation with harmonic potential.

Interaction of modulated pulses in the nonlinear Schroedinger equation with periodic potential

2007

We consider a cubic nonlinear Schroedinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.