THE 1/D EXPANSION FOR THE QUANTUM-MECHANICAL N-BODY PROBLEM - APPLICATION FOR DIRECTED POLYMERS IN A RANDOM MEDIUM

In this paper we give a closed form expression for the 1/d corrections to the leading d --> infinity approximation for the ground-state energy of an n-body quantum mechanical problem in a general potential (where d is the number of spatial dimensions). We then extend the results to the problem of directed polymers in a random medium where the limit n --> 0 has to be taken in the corresponding n-body problem. In this case the effective two-body potential measures the strength of correlations of the quenched disorder. In the n --> 0 limit a replica-symmetry-breaking solution to the saddle-point equations turns out to be the correct solution in the strong-coupling regime, and we obtain the 1/d corrections to the corresponding energy which constitute the gaussian fluctuations about this solution as well as provide information about its basin of stability.