  We consider the general Wigner function for a particle confined to a finite
interval and subject to Dirichlet boundary conditions. We derive the boundary
corrections to the "star-genvalue" equation and to the time evolution equation.
These corrections can be cast in the form of a boundary potential contributing
to the total Hamiltonian which together with a subsidiary boundary condition is
responsible for the discretization of the energy levels. We show that a
completely analogous formulation (in terms of boundary potentials) is also
possible in standard operator quantum mechanics and that the Wigner and the
operator formulations are also in one-to-one correspondence in the confined
case. In particular, we extend Baker's converse construction to bounded
systems. Finally, we elaborate on the applications of the formalism to the
subject of Wigner trajectories, namely in the context of collision processes
and quantum systems displaying chaotic behavior in the classical limit.
