  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.
