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We present a barrier potential with bound states that is exactly solvable anddetermine the eigenfunctions and eigenvalues of the Hamiltonian. Theequilibrium density matrix of a particle moving at temperature T in thisnonlinear barrier potential field is determined. The exact density matrix iscompared with the result of the path integral approach in the semiclassicalapproximation. For opaque barriers the simple semiclassical approximation isfound to be sufficient at high temperatures while at low temperatures thefluctuation paths may have a caustic depending on temperature and endpoints.Near the caustics the divergence of the simple semiclassical approximation ofthe density matrix is removed by a nonlinear fluctuation potential. For opaquebarriers the improved semiclassical approximation is again in agreement withthe exact result. In particular, bound states and the form of resonance statesare described accurately by the semiclassical approach.Comment: 32 pages & 11 figures; in press (J. Chem. Phys.

Exact and semiclassical density matrix of a particle moving in a barrier potential with bound states