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An Analog of the Fourier Transform Associated with a Nonlinear One-Dimensional Schrödinger Equation

Peter E. Zhidkov

出版	SSRN, 2018
URL	http://books.google.com.hk/books?id=AZEAzwEACAAJ&hl=&source=gbs_api

註釋

We consider an eigenvalue problem which includes a nonlinear Schrödinger equation on the half-line [0, ∞) and certain boundary conditions. It is shown that the spectrum of this problem fills a half-line and that to each point of the spectrum there corresponds a unique eigenfunction. The main result of the paper is that an arbitrary infinitely differentiable function () rapidly decaying as → ∞ and satisfying suitable boundary conditions at the point = 0 can be uniquely expanded into an integral over eigenfunctions similar to the representation of functions by the Fourier transform (the latter is obviously associated with a linear self-adjoint eigenvalue problem).