Operator På svenska, tack!
A course for doctor and advance undergraduate students

mf

Spectral analysis

of differential operators

from mathematical physics

P.Kurasov
5 points, Spring 2003

wermeer

The course is devoted to the spectral analysis of differential operators appearing in different problems of mathematical physics and applied mathematics. Most of these operators are unbounded self-adjoint operators and their spectra can have different nature. Our analysis of differential operators will be based on their spectral properties. Different types of the spectrum will be described: discrete, continuous, absolutely continuous, singular continuous. The behaviour of the spectrum under perturbations of the operator will be investigated. This knowledge will be applied to derive explicit solutions to differential equations and to study rigorously operators appearing in problems of mathematical physics. The main goal of the course is to present the spectral theorem for unbounded self-adjoint operators. Elements of scattering theory will be presented as well. Special attention will be given to the spectral theory of the one-dimensional Schrödinger operator describing recent break-through in this area.

This course can be considered as a natural continuation of the standard courses on Ordinary and Partial Differential Equations, which includes elements of Functional Analysis, or the course on Equations of Mathematical Physics (Kontinuerliga System). Knowledge of these courses is not necessary to follow the lectures but will be helpful. The course is oriented towards advanced undergraduate and postgraduate students.

There will be one two-hour lecture every week. The main idea of the course is to introduce students into concrete problems of mathematical physics related to operator theory. Therefore a part of every lecture will be devoted to studies of realistic differential operators appearing in different problems (of modern mathematical physics).

There will be two possibilities to pass the examination: oral discussion on the contents of the course or active participation in the research oriented projects. For the students preferring the second alternative the problems will be distributed in the middle of the semester. The results of these projects will be discussed at the last meeting in May. In case of interest it will be possible to continue successful projects and prepare scientific reports on the obtained results. The aim is that some of these projects will lead to full-size research articles. After a similar course was given at Stockholm University the following two articles were published

http://www.maths.lth.se/matematiklth/personal/kurasov/PDFARCHIV/JMAA02.pdf
http://www.maths.lth.se/matematiklth/personal/kurasov/PDFARCHIV/JPhysA02.pdf

Time and place: The lectures are given during the Spring semester on Mondays, 15.15--17.00 in room MH:333. Please let me know if you are interested to participating and/or cannot come to the first meeting: kurasov@maths.lth.se
Welcome!
Pavel Kurasov

Preliminary plan for lectures and Lecture notes can be downloaded from the following addresses
Plan for lectures
Lecture notes



Literature for the course:


N.I.Akhiezer, I.M.Glazman
The theory of linear operators in Hilbert space
part II (or parts I and II)
(any available edition). 		glazman's photo Buy




Additional reading:

S.Albeverio and P.Kurasov
Singular perturbations of differential operators
London Mathematical Society Lecture Notes 271
Cambridge Univ. Press 2000
(some chapters will be distributed to the participants). 		book's photo Buy


M.S.Birman, M.Z.Solomyak
Spectral theory of self-adjoint operators in Hilbert space
D.Reidel Publishing, Dordrecht, 1987. 		Buy
(check the price)


E.B.Davies
Spectral theory and differential operators
Cambridge Univ. Press, 1995. 		davies's photo Buy


M.Naimark, Linear differential operators, Parts I and II, New York 1967-68 (some chapters will be distributed to the participants).

M.Reed and B.Simon
Methods of modern mathematical physics, vol II
Fourier Analysis, Self-Adjointness
(any edition) 		simon's photo Buy
(check the price)




Preliminary plan for the course

  1. Introduction: linear (unbounded) operators in Hilbert space, domain, elementary differential operators.
  2. Isometric and unitary operators, the Cayley transform, projections.
  3. Unbounded operators: symmetric and self-adjoint.
  4. The spectral theorem for unitary and self-adjoint operators.
  5. Spectral theory for periodic one-dimensional Schrödinger operator.
  6. Perturbation theory for self-adjoint operators.
  7. Extension theory for symmetric operators: defect elements, defect index, von Neumann theory.
  8. Krein's resolvent formula.
  9. Perturbations and extensions: singular perturbations.
  10. Elementary scattering theory.
  11. Scattering theory for singular perturbations.
  12. Schrödinger operators.