  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.
