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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 2018 (Monday, October 8 to Friday, December 14)

Hours

  • Live lecture hours: 10
  • Recorded lecture hours: 0
  • Total advised study hours: 40

Timetable

  • Tue 09:05 - 09:55

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


Bibliography


Theory of ordinary differential equationsCoddington and Levinson
Spectral theory of ordinary differential operatorsWeidmann
Introduction to spectral theoryLevitan and Sargsjan
Theory of Linear Operators in Hilbert SpaceN.I. Akhiezer, I.M. Glazman


Note:

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Assessment



The assessment will consist of a number of questions to be selected from a list (related, but not restricted to the practice questions at the end of lecture note chapters). The pass mark will be 50

No assignments have been set for this course.

Files


Files marked L are intended to be displayed on the main screen during lectures.

Week(s)File
stodo2018.pdf


Recorded Lectures


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