A Space-Time Wigner Function Approach to Long Time Schrödinger--Poisson Dynamics

Abstract

A long time description of electrostatic Schrodinger--Poisson states, satisfying $ i \partial_t \psi = -\frac{1}{2} \Delta_x \psi + ( \frac{C}{|x|} \ast_x |\psi|^2 + \frac{1}{2} |x|^2 ) \psi, $ is provided in terms of a non-Markovian Wigner formalism through the choice of the simplest charge-preserving scale group, $\psi_{\varepsilon}(t,x) = \psi({\varepsilon}^{-1}t, x)$, in which the position variable $x \in \mathbb R^3$ remains unscaled while time $t \in \mathbb R^+$ is sent to infinity as $\varepsilon \to 0$. Typically, the introduction of this group of scale transformations leads to high frequency, time oscillatory states that may not converge in such a good topology as to deal with the nonlinear term. To overcome this drawback, the sequence of wavefunctions is Wignerized via the action of the so-called $\tau$-convoluted space-time Wigner transform. The main goal of this extended Wigner operator consists in producing an attenuating effect on the temporal oscillations as time grows, which in turn allow...