Problem of a quantum particle in a random potential on a line revisited

The density of states for a particle moving in a random potential with a Gaussian correlator is calculated exactly using the functional integral technique. It is achieved by expressing the functional degrees of freedom in terms of the spectral variables and the parameters of isospectral transformations of the potential. These transformations are given explicitly by the flows of the Korteweg-de Vries hierarchy which deform the potential leaving all its spectral properties invariant. Making use of conservation laws reduces the initial Feynman integral to a combination of quadratures which can be readily calculated. Different formulations of the problem are analyzed.