Cross between Born and WKB approximations: Variational solutions of nonlinear forms of the Schrödinger equation

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Dashen, taking as his starting point the fact that the Born approximation for the phase shift has a wider range of applicability than would be expected from the usual criterion that the phase shift should be small, investigated a nonlinear form of the "one-dimensional" Schrödinger equation and arrived at an expression which can best be described as "a cross between the Born and WKB approximations." This expression for the phase shift reduces to the Born result even when the phase shift is not small. We investigate this result from a different point of view and establish that the Dashen expression is a variational principle for the nonlinear equation considered by him. This serves to explain the accuracy of the approximation. We also construct similar variational expressions for the more usual nonlinear Riccati equation and contrast it with the WKB series solution and recently proposed alternatives to this series. Contact is made between the techniques of functional analysis that Dashen used in arriving at his result and the unified formulation of variational principles that we adopt. An appendix deals with the principles of "invariant imbedding" in the particular context of such nonlinear versions of the one-dimensional Schrödinger equation. Copyright © 1976 American Institute of Physics.

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Journal of Mathematical Physics

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