MAGIC057: Spectral Theory of Ordinary Differential Operators

Course details

A specialist MAGIC course

Semester

Autumn 2021
Monday, October 4th to Friday, December 10th

Hours

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

Timetable

Tuesdays
12:05 - 12:55

Course forum

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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. 

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

  • KS

    Professor Karl Michael Schmidt

    University
    Cardiff University

Bibliography

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Assessment

The assessment for this course will be released on Monday 10th January 2022 and is due in by Sunday 23rd January 2022 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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