Nonlinear stationary solutions of the Wigner and Wigner-Poisson equations

TL;DR

This paper derives exact nonlinear stationary solutions for the one-dimensional Wigner and Wigner-Poisson equations, extending classical Bernstein-Greene-Kruskal modes to quantum phase space and analyzing their physical validity.

Contribution

It introduces new exact solutions for the Wigner and Wigner-Poisson equations that depend on both energy and position, adapting classical plasma modes to quantum formalism.

Findings

Solutions for quartic oscillator potential derived

Self-consistent Wigner-Poisson solutions constructed

Conditions for physically meaningful Wigner functions discussed

Abstract

Exact nonlinear stationary solutions of the one-dimensional Wigner and Wigner-Poisson equations in the terms of the Wigner functions that depend not only on the energy but also on position are presented. In this way, the Bernstein-Greene-Kruskal modes of the classical plasma are adapted for the quantum formalism in the phase space. The solutions are constructed for the case of a quartic oscillator potential, as well as for the self-consistent Wigner-Poisson case. Conditions for well-behaved physically meaningful equilibrium Wigner functions are discussed.