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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.

Autumn 2011 (Monday, October 10 to Friday, December 16)

  • Wed 10:05 - 10:55


Karl Michael Schmidt
Phone (029) 20876778
Photo of Karl Michael Schmidt


Photo of Abeer AL-NAHDI
Photo of Christopher Jeavons
Christopher Jeavons
Photo of Benjamin Lang
Benjamin Lang
Photo of Umberto  Lupo
Umberto Lupo
Photo of Adam  Newman
Adam Newman
Photo of Greg Roddick
Greg Roddick
Photo of Magdalena Zajaczkowska
Magdalena Zajaczkowska
Photo of Lu  Zhang
Lu Zhang
Photo of Yiwei Zhang
Yiwei Zhang


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.


  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


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


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.)


There is a single assignment to pass the course. See under Ässignments" and follow the instructions there.


Spectral theory of ordinary differential operators - course assessment

Deadline: Tuesday 3 January 2012 (1025.7 days ago)
Instructions:Instructions for Assessment.
The more substantial questions 3., 7., 10., 16., 17. carry a double weight; all other questions have single weight.
Choose questions to a total weight of (exactly) 10 (e.g. 10 single-weight questions or 5 double-weight questions or a suitable mixture) and submit written solutions to your chosen questions.
Your answers will be marked and contribute according to their weights to the assessment. The pass level is 50%. This is the only assessment for this MAGIC course.
The questions appear in the lecture notes at the end of each chapter; observing which chapter each question refers to will help with the solution.


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