Reformulating the Schrödinger equation as a Shabat–Zakharov system

Abstract

We reformulate the second-order Schrödinger equation as a set of two coupled first-order differential equations, a so-called “Shabat–Zakharov system” (sometimes called a “Zakharov–Shabat” system). There is considerable flexibility in this approach, and we emphasize the utility of introducing an “auxiliary condition” or “gauge condition” that is used to cut down the degrees of freedom. Using this formalism, we derive the explicit (but formal) general solution to the Schrödinger equation. The general solution depends on three arbitrarily chosen functions, and a path-ordered exponential matrix. If one considers path ordering to be an “elementary” process, then this represents complete quadrature, albeit formal, of the second-order linear ordinary differential equation.