On the point-particle (Newtonian) limit of the nonlinear Hartree equation
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 \(\epsilon^{-1}\).
weakly interacting non-relativistic bosons
scientific article; zbMATH DE number 1742843