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 mo ving in a Gaussian random potential on a line are derived exactly. The degrees of freedom of the potentials in the ensemble are split into t he spectral variables and the parameters of isospectral deformations o f the potential which are given by the flows of the Korteweg-de Vries (KdV) hierarchy. This allows one to evaluate the functional integral f or the ensemble average and to analyze the physical content of possibl e modifications of the Gaussian ensemble. Contrary to the known result s for the finite interval problem, the DOS shows the formation of an i mpurity band for E < 0, separated from the continuous spectrum.