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

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 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).