  We present a barrier potential with bound states that is exactly solvable and
determine the eigenfunctions and eigenvalues of the Hamiltonian. The
equilibrium density matrix of a particle moving at temperature T in this
nonlinear barrier potential field is determined. The exact density matrix is
compared with the result of the path integral approach in the semiclassical
approximation. For opaque barriers the simple semiclassical approximation is
found to be sufficient at high temperatures while at low temperatures the
fluctuation paths may have a caustic depending on temperature and endpoints.
Near the caustics the divergence of the simple semiclassical approximation of
the density matrix is removed by a nonlinear fluctuation potential. For opaque
barriers the improved semiclassical approximation is again in agreement with
the exact result. In particular, bound states and the form of resonance states
are described accurately by the semiclassical approach.
