Reformulating the Schrodinger equation as a Shabat-Zakharov system

TL;DR

This paper presents a reformulation of the Schrödinger equation as a coupled first-order system called the Shabat-Zakharov system, enabling a formal general solution involving path-ordered exponentials and arbitrary functions.

Contribution

It introduces a flexible formalism for rewriting the Schrödinger equation as a first-order system with an auxiliary gauge condition, providing an explicit general solution framework.

Findings

Derives the general solution as a path-ordered exponential matrix.

Shows the formal solution depends on three arbitrary functions.

Highlights the utility of gauge conditions in simplifying the system.

Abstract

We reformulate the second-order Schrodinger 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 emphasise 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 Schrodinger 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 ODE.