Course details
A specialist MAGIC course
Semester
- Autumn 2020
- Monday, October 5th to Friday, December 11th
Hours
- Live lecture hours
- 10
- Recorded lecture hours
- 0
- Total advised study hours
- 40
Timetable
- Tuesdays
- 12:05 - 12:55 (UK)
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.
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.
Some familiarity with ordinary differential equations and/or linear operator theory will be helpful.
Syllabus
- Regular Sturm-Liouville boundary value problems: Hilbert-Schmidt method, resolvents and Green's function, Stieltjes integrals and the spectral function
- 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
- 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
Follow the link for a book to take you to the relevant Google Book Search page
You may be able to preview the book there and 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.
- Theory of ordinary differential equations (Coddington and Levinson, book)
- Spectral theory of ordinary differential operators (Weidmann, book)
- Introduction to spectral theory (Levitan and Sargsjan, book)
- Theory of Linear Operators in Hilbert Space (N.I. Akhiezer, I.M. Glazman, )
Assessment
The assessment for this course will be released on Monday 11th January 2021 at 00:00 and is due in before Sunday 24th January 2021 at 23:59.
The assessment will consist of a number of questions to be selected from a list (related, but not restricted to the questions at the end of lecture note chapters). The pass mark will be 50%.
Please note that you are not registered for assessment on this course.
Files
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Lectures
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