Reformulating the Schrodinger equation as a Shabat-Zakharov system

Details Reformulating the Schrodinger equation as a Shabat-Zakharov system Reformulating the Schrodinger equation as a Shabat-Zakharov system

2009-10-14

arxiv.org/abs/0910.2600v1

Journal of Mathematical Physics 51 (2010) 022105

Physics

Mathematical Physics

18 pages, plain LaTeX

Scientific paper

10.1063/1.3282847

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.

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