An analog of the Fourier transform associated with a nonlinear one-dimensional Schrödinger equation

✍ Scribed by Peter E. Zhidkov

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 171 KB
• Volume: 52
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(02)00130-x

✦ Synopsis

We consider an eigenvalue problem which includes a nonlinear Schr odinger equation on the half-line [0; ∞) and certain boundary conditions. It is shown that the spectrum of this problem ÿlls 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 inÿnitely di erentiable function g(x) rapidly decaying as x → ∞ and satisfying suitable boundary conditions at the point x = 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).