Announcements


There are no announcements

Forum

General


Description

Ordinary differential operators appear naturally in many problems of mathematical physics as well as questions of pure mathematics such as the stability of minimal surfaces. Their spectra often have direct significance, e.g. as sets of vibration frequencies or admissible energies in quantum mechanics. Moreover, ordinary differential operators provide important and sometimes surprising examples in the spectral theory of linear operators.


This course gives a detailed introduction to the spectral theory of boundary value problems for Sturm-Liouville and related ordinary differential operators. The subject is characterised by a combination of methods from linear operator theory, ordinary differential equations and asymptotic analysis. The topics covered include regular boundary value problems, Weyl-Titchmarsh theory of singular boundary value problems, the spectral representation theorem as well as recent developments of oscillation theory as a modern tool of spectral analysis.

Semester

Autumn 2017 (Monday, October 9 to Friday, December 15)

Timetable

  • Tue 12:05 - 12:55

Prerequisites

The course is planned to be self-contained and only requires knowledge of mathematical analysis. Some familiarity with ordinary differential equations and/or linear operator theory will be helpful.

Syllabus

  1. Regular Sturm-Liouville boundary value problems: Hilbert-Schmidt method, resolvents and Green's function, Stieltjes integrals and the spectral function
  2. Singular boundary value problems: Weyl's alternative, Helly's selection and integration theorems, Stieltjes inversion formula, generalised Fourier transform, spectral function, spectral measures and types
  3. Oscillation methods of spectral analysis: Prüfer variables, generalised Sturm comparison and oscillation theorems, uniform subordinacy theory, Kotani's theorem

Lecturer


Karl Michael Schmidt
Email Schmidtkm@cf.ac.uk
Phone (029) 20876778
Photo of Karl Michael Schmidt


Students


Photo of Musawa Almusawa
Musawa Almusawa
(Sussex)
Photo of Nifeen Altaweel
Nifeen Altaweel
(Lancaster)
Photo of Stuart Bond
Stuart Bond
(Sussex)
Photo of Bobby Cheng
Bobby Cheng
(Sussex)
Photo of Maxime Fairon
Maxime Fairon
(Leeds)
Photo of Raffaele Grande
Raffaele Grande
(Cardiff)
Photo of Waleed Hamali
Waleed Hamali
(Sussex)
Photo of Georgina Long
Georgina Long
(Exeter)
Photo of Marianthi Moschou
Marianthi Moschou
(Manchester)
Photo of David Snee
David Snee
(Northumbria)
Photo of Ville Syrjanen
Ville Syrjanen
(Sussex)


Bibliography


Theory of ordinary differential equationsCoddington and Levinson
Spectral theory of ordinary differential operatorsWeidmann
Introduction to spectral theoryLevitan and Sargsjan


Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

Assessment



No assessment information is available yet.

No assignments have been set for this course.

Files



No files have yet been uploaded for this course.

Recorded Lectures


Please log in to view lecture recordings.