Isospectral problem for Schrödinger operator: Evolutional viewpoint

Abstract

An isospectral transform of the Schrödinger operator is considered as an evolutional problem. For a transform defined by the McKean–Trubowitz flows associated evolutional equations are derived. It is shown that for one-level and two-level flows these equations can be split into integrable Liouville equations. A relationship between the Liouville equations and the Darboux transforms is discussed; this analysis suggests that the evolutional equations can be split into the Liouville equations in the general case. A Hamiltonian formulation of the isospectral transform defined by the McKean–Trubowitz flows is presented. It is shown that this transform is performed by a canonical change of variables, which is related to the Darboux transform.