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

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 \(\epsilon^{-1}\).


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
Full Text: DOI