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


  • Tue 12:05 - 12:55


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
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Photo of Musawa Almusawa
Musawa Almusawa
Photo of Nifeen Altaweel
Nifeen Altaweel
Photo of Stuart Bond
Stuart Bond
Photo of Bobby Cheng
Bobby Cheng
Photo of Maxime Fairon
Maxime Fairon
Photo of Raffaele Grande
Raffaele Grande
Photo of Waleed Hamali
Waleed Hamali
Photo of Georgina Long
Georgina Long
Photo of Marianthi Moschou
Marianthi Moschou
Photo of David Snee
David Snee
Photo of Ville Syrjanen
Ville Syrjanen


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


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