PROBLEM OF A QUANTUM PARTICLE IN A RANDOM POTENTIAL ON A LINE REVISITED

The density of states (DOS) and the level statistics for a particle moving in a Gaussian random potential on a line are derived exactly. The degrees of freedom of the potentials in the ensemble are split into the spectral variables and the parameters of isospectral deformations of the potential which are given by the flows of the Korteweg-de Vries (KdV) hierarchy. This allows one to evaluate the functional integral for the ensemble average and to analyze the physical content of possible modifications of the Gaussian ensemble. Contrary to the known results for the finite interval problem, the DOS shows the formation of an impurity band for E < 0, separated from the continuous spectrum.