Reformulating the Schroedinger equation as a Shabat-Zakharov system

Journal Article · Mon Feb 15 04:00:00 EST 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3282847· OSTI ID:21335902

Boonserm, Petarpa ^[1]; Visser, Matt ^[2]

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1. Department of Mathematics, Faculty of Science, Chulalongkorn University, Phayathai Rd., Pathumwan, Bangkok 10330 (Thailand)
2. School of Mathematics, Statistics, and Operations Research, Victoria University of Wellington, PO Box 600, Wellington 6140 (New Zealand)

We reformulate the second-order Schroedinger 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 Schroedinger 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.

• OSTI ID: 21335902
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 2 Vol. 51; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
ALGEBRA
DEGREES OF FREEDOM
FUNCTIONS
MATHEMATICAL SOLUTIONS
MATRICES
QUADRATURES
SCHROEDINGER EQUATION