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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 2013 (Monday, October 7 to Saturday, December 14)


  • Fri 10:05 - 10:55


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


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


Photo of Johar Ashfaque
Johar Ashfaque
Photo of David Beltran Portales
David Beltran Portales
Photo of Michael Neururer
Michael Neururer
Photo of Xiaolong Niu
Xiaolong Niu
Photo of Katie Spalding
Katie Spalding
Photo of Freddy Symons
Freddy Symons


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 will be a single (multi-part) assignment to pass the course, to be completed and submitted by 17 January 2014.

Assessment for Spectral Theory of Ordinary Differential Operators

Files:Exam paper
Released: Friday 13 December 2013 (1346.2 days ago)
Deadline: Sunday 19 January 2014 (1308.2 days ago)

The assessment is based on the questions appearing in the lecture notes at the end of each chapter; observing which chapter each question refers to will help with the solution. 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) 8 (e.g. 8 single-weight questions or 4 double- weight questions or a suitable mixture), of which at least a total weight of 4 should be from Section B (i.e. questions 12-23), 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.


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