THE DENSITY-MATRIX NEAR THE TOP OF A POTENTIAL BARRIER - A PROBLEM OF CAUSTICS IN PATH INTEGRATION

Employing the path integral approach, the real space representation of the equilibrium density matrix of a particle moving in a nonlinear potential field is calculated for coordinates near the top of a potential barrier. As the temperature is decreased below a critical temperature T(c), the harmonic approximation for the potential fails even in the close vicinity of the barrier top and a caustic arises. The divergent semiclassical approximation of the path integral is regularized by taking into account anharmonic terms. This regularization is carried out explicitly for the case of a symmetric barrier. Apart from the behavior of the density matrix near T(c), the crossover to the semiclassical approximations of the path integral for high and low temperatures is discussed.