An analog of the Fourier transform associated with a nonlinear one-dimensional Schrödinger equation
type: Item
instance of: scholarly article
title: An analog of the Fourier transform associated with a nonlinear one-dimensional Schrödinger equation
zbMATH Open document ID: 1025.34087
DOI: 10.1016/S0362-546X(02)00130-X
author: P. E. Zhidkov
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: 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.
reviewed by: Miloslav Znojil
Mathematics Subject Classification ID: 34L30
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 42C30
Mathematics Subject Classification ID: 47J10
zbMATH DE Number: 1879977
zbMATH Keywords: nonlinear Schrödinger equation on half line
zbMATH Keywords: eigenfunction expanion
zbMATH Keywords: continuous spectrum
zbMATH Keywords: completeness of eigenfunctions
zbMATH Keywords: Fourier transform
MaRDI profile type: Publication
cites work: Q5838785
cites work: Completeness of systems of eigenfunctions for the Sturm-Liouville operator with potential depending on the spectral parameter and for one non-linear problem
cites work: On the completeness of the system of normalized eigenfunctions of a nonlinear Schrödinger-type operator on a segment.
cites work: Riesz basis property of the system of eigenfunctions for a non-linear problem of Sturm-Liouville type
cites work: Q4948679
cites work: On the property of being a basis for a denumerable set of solutions of a nonlinear Schrödinger-type boundary-value problem
cites work: Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory
description: scientific article; zbMATH DE number 1879977