Reformulating the Schrodinger equation as a Shabat-Zakharov 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.

18 pages, plain LaTeX

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• Bounds on variable-length compound jumps
• Greybody factors for Schwarzschild black holes: Path-ordered exponentials and product integrals
• Reflection and transmission resonances and accuracy of the wkb method
• Constraining transmission and reflection probabilities by using the Miller-Good transformation