Course details
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 (UK)
Description
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 SturmLiouville 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, WeylTitchmarsh 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
Some familiarity with ordinary differential equations and/or linear operator theory will be helpful.
Syllabus
 Regular SturmLiouville boundary value problems: HilbertSchmidt 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.
 Introduction to Spectral Theory (Boris Moiseevich Levitan and Ishkhan Saribekovich Sargsi͡an, book)
 Spectral Theory of Ordinary Differential Operators (Joachim Weidmann, book)
 Theory of Linear Operators in Hilbert Space (Naum Ilʹich Akhiezer and Izrailʹ Markovich Glazman, book)
 Theory of Ordinary Differential Equations (Earl A. Coddington and Norman Levinson, book)
Assessment
The assessment for this course will be released on Monday 10th January 2022 at 00:00 and is due in before Sunday 23rd January 2022 at 23:59.
Assessment for all MAGIC courses is via takehome 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 uptodate 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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