Volume 132, Number 2, October 2020
|Number of page(s)||7|
|Published online||22 December 2020|
Bernstein-Greene-Kruskal approach for the quantum Vlasov equation
Instituto de Física, Universidade Federal do Rio Grande do Sul - Av. Bento Gonçalves 9500, 91501-970 Porto Alegre, RS, Brasil
Received: 10 June 2020
Accepted: 11 September 2020
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.
PACS: 05.30.-d – Quantum statistical mechanics / 02.30.Mv – Approximations and expansions / 52.35.Sb – Solitons; BGK modes
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