Course details
Semester
 Autumn 2017
 Monday, October 9th to Friday, December 15th
Hours
 Live lecture hours
 10
 Recorded lecture hours
 0
 Total advised study hours
 0
Timetable
 Tuesdays
 12:05  12:55
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 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.
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
The course is planned to be selfcontained and only requires knowledge of
mathematical analysis. 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.
 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 8th January 2018 and is due in by Sunday 21st January 2018 at 23:59.
The assessment will be 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 singleweight questions or 4 doubleweight questions or a suitable mixture), of which at least a total weight of 4 should be from Section B (i.e. questions 1223), 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
Please note that you are not registered for assessment on this course.
Lectures
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