On the point-particle (Newtonian) limit of the nonlinear Hartree equation. (English) Zbl 1025.81015

Fröhlich, Jürg; Tsai, Tai-Peng; Yau, Horng-Tzer

Commun. Math. Phys. 225, No. 2, 223-274 (2002).

The authors explore a classical regime for a class of systems of identical, non-relativistic bosons, with very weak two-body interactions described by the potential \(-k\Phi\) of van der Waals or Newtonian type satisfying certain regularity properties. They consider the nonlinear Hartree equation \[ i\partial_t\psi_t=-\frac{1}{2}\Delta\psi_t+\lambda V\psi_t-\nu(\Phi*|\psi_t|^2)\psi_t, \] here \(\psi_t(x)=\psi(t,x)\), \(t\in\mathbb{R}, x\in\mathbb{R}^n\) is taken from the Sobolev space \(H^1(\mathbb{R}^n)\), \(\lambda\in\mathbb{R}\), \(V\) is smooth, bounded, positive function on \(\mathbb{R}^n\), \(-\Phi\) is a radially symmetric two-body potential, with \(\Phi\in L^p(\mathbb{R}^n,d^x)+L^{\infty}(\mathbb R^n)\), \(p\geq n/2\) and \(*\) denotes the usual convolution. The authors describe the dynamics of the given Hartree equation with given initial condition, under some imposed conditions on the Hamiltonian function. It is shown that a nonlinear Møller wave operator describing the scattering of a soliton and a wave can be defined. The authors also consider the dynamics of a soliton in slowly varying background potential \(W(\epsilon x)\). It is proved that the soliton decomposes into a soliton plus a scattering wave up to times of order \(\epsilon^{-1}\).

Reviewer: Farruh Mukhamedov (Tashkent)

MSC:

81Q15	Perturbation theories for operators and differential equations in quantum theory
35Q51	Soliton equations
35Q40	PDEs in connection with quantum mechanics
82C10	Quantum dynamics and nonequilibrium statistical mechanics (general)
49N60	Regularity of solutions in optimal control
81U20	\(S\)-matrix theory, etc. in quantum theory

Keywords:

nonlinear Hartree equation; weakly interacting non-relativistic bosons; Møller wave operator; soliton; Newton equation