Scattering Theory (GQT)

Credits	8 credit points
Instructors	Koelink, H.T. (Radboud Universiteit Nijmegen)
E-mail	e.koelink@math.ru.nl
Aim	Functional analytic aspects of perturbations of unbounded operators on Hilbert spaces, especially applied to Schroedinger operators on the real line. Scattering theory for Schroedinger operators on the real line and the inverse scattering method for the Korteweg-de Vries equation and its soliton solutions.
Description	We start by recalling some aspects of functional analysis, especially compact operators, unbounded operators, the spectral theorem for unbounded self-adjoint operators. Some function spaces, such as Sobolev and Hardy spaces are also recalled, and, whenever possible or when results are not standard, proofs are given. Next, perturbation results, such as Rellich's theorem, are discussed with application to the Schroedinger operator for the general case, as well as to the Schroedinger operator with specific potentials. The scattering operator is defined and studied in a general functional analytic context. Next, the Schroedinger operator on the real line is studied in detail: Jost solutions; transmission and reflection coefficients; Gelfand-Levitan-Marchenko integral equation; reflectionless potentials. Finally, the inverse scattering method is discussed for the Korteweg-de Vries equation using Lax pairs and isospectral deformations of the Schroedinger equation. In particular, the 1- and 2-soliton solutions are obtained in this way.
Organization	Each class there will be three 45 minutes time slots. The last one will be devoted to questions and exercises.
Examination	The final grade is based on the results of the homework assignments (obligatory) and an oral examination. The average of the two results form the final grade. If necessary, the oral examination will be replaced by a written examination.
Literature	Lecture notes (downloadable by the end of January 2008 from the website mentioned below), which also has more references to the literature.
Prerequisites	Operators on Hilbert spaces. Key words: Hilbert spaces, operators, closed graph theorem, compact operators, unbounded operators, symmetric operators, unbounded self-adjoint operators, spectral theorem for unbounded self-adjoint operators, Stone's theorem. NB Not all of these notions occur in a course on functional analysis --especially the last five--, so the precise notions as well as precise statements with some indications of the proof will be given at the beginning of the course. In particular, it is not necessary to have taken the national course on functional analysis beforehand. The entrance level of that course (see its description for details), plus some general analytic maturity, is sufficient.
Remarks	http://www.math.ru.nl/~koelink/edu/ScatteringTheory.html

Last changed: 18-01-2012 10:21