  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.
