Bernstein-Greene-Kruskal approach for the quantum Vlasov equation

TL;DR

This paper extends the Bernstein-Greene-Kruskal method to analyze the one-dimensional stationary quantum Vlasov equation, providing an infinite series solution in the semiclassical regime and testing its accuracy with anharmonic potentials.

Contribution

It introduces a novel application of the BGK method to the quantum Vlasov equation, deriving an infinite series solution in the semiclassical limit with high-order anharmonic potential examples.

Findings

Series expansion is accurate for small quantum diffraction parameters.

Method is effective for smooth, time-independent external potentials.

High-order anharmonic potential solutions demonstrate the approach's robustness.

Abstract

The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables, similarly as in the solution of the Vlasov-Poisson system by means of the Bernstein-Greene-Kruskal method. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed and shown to be immediately integrable up to a recursive chain of quadratures in position space only. { As it stands, the treatment of the self-consistent, Wigner-Poisson system is beyond the scope of the method, which assumes} a given smooth { time-independent} external potential. Accuracy tests for the series expansion are also provided. Examples of anharmonic potentials are worked out up to a high order on the quantum diffraction parameter.