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

TL;DR

This paper provides an exact calculation of the density of states for a quantum particle in a Gaussian-correlated random potential using functional integrals and spectral transformations related to the Korteweg-de Vries hierarchy.

Contribution

It introduces a novel method combining spectral variables and isospectral transformations to solve the density of states problem exactly.

Findings

Exact density of states expression derived

Spectral transformations preserve potential's spectral properties

Reduction of Feynman integrals to solvable quadratures

Abstract

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.