An analog of the Fourier transform associated with a nonlinear one-dimensional Schrödinger equation (Q1863467)

The author summarizes the last five years of his research and ``grey'' as well as standard publication activity on the subject where his attention has been paid to the completeness of the set of solutions to a class of nonlinear Sturm-Liouville problems on the half-axis under certain technical assumptions (not weakened to an extreme yet). Being inspired by the idea that the kernel of the well known Fourier integral transformation (in various versions) satisfies always a linear Sturm-Liouvillean problem, the author emphasizes that the core of his new result lies in the availability of an estimate of the bounds (4) valid for his ``generalized Fourier transform'' of any function from a certain Schwartz space.