Course detail

Sturm-Lieouville Theory

FSI-9SLTAcad. year: 2022/2023

The course deals with basic topics of the Sturm-Lieouvill theory. The results are applied to solving of certain problems of mathematical analysis and engineering.

Language of instruction

Czech

Mode of study

Not applicable.

Guarantor

prof. Aleksandre Lomtatidze, DrSc.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

Knowledge of basic topics of the spectral theory of second order differential operators and ability to apply this knowledge in practice.

Prerequisites

Differential and integral calculus, ordinary differential equations.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline.

Assesment methods and criteria linked to learning outcomes

Course-unit credit is awarded on condition of having attended the seminars actively and passed the control test.
Examination has a practical and a theoretical part. In the practical part student has to illustrate the given tasks on particular examples.
Theoretical part includes questions related to the subject-matter presented at the lectures.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The aim of the course is to familiarise students with basic topics and procedures of the Sturm-Lieouville theory in other mathematical subjects and applications.

Specification of controlled education, way of implementation and compensation for absences

Absence has to be made up by self-study using recommended literature.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975. (CS)
A. Zettl, Sturm-Liouville theory: Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005. xii+328 pp. ISBN: 0-8218-3905-5 (EN)
E. C.Titchmarsh: Eigenfunction expansions associated with second-order differential equations. Part I. Second Edition Clarendon Press, Oxford 1962 vi+203 pp. (EN)
P. Hartman: Ordinary differential equations. Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA.]. Philadelphia, PA, 2002. xx+612 pp. ISBN: 0-89871-510-5 34-01 (37-01). (EN)
V. A. Marchenko, Sturm-Liouville operators and applications: Revised edition. AMS Chelsea Publishing, Providence, RI, 2011. xiv+396 pp. ISBN: 978-0-8218-5316-0. (EN)

Recommended reading

A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975 (CS)
A. Zettl, Sturm-Liouville theory: Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005. xii+328 pp. ISBN: 0-8218-3905-5 (EN)
E. C.Titchmarsh: Eigenfunction expansions associated with second-order differential equations. Part I. Second Edition Clarendon Press, Oxford 1962 vi+203 pp. (EN)
P. Hartman: Ordinary differential equations. Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA.]. Philadelphia, PA, 2002. xx+612 pp. ISBN: 0-89871-510-5 34-01 (37-01). (EN)

Classification of course in study plans

  • Programme D-APM-K Doctoral 1 year of study, winter semester, recommended course
  • Programme D-APM-P Doctoral 1 year of study, winter semester, recommended course

Type of course unit

 

Lecture

20 hod., optionally

Teacher / Lecturer

prof. Aleksandre Lomtatidze, DrSc.

Syllabus

1. Second order ODE, Sturmian theory.
2. Two-point boundary value problém, Fredholm theorems.
3. Well-possedness of two-point BVP.
4. Eigenvalues and eigenfunctions.
5. Properties of eigenfunctions.
6. Completness of eigenfunctions.
7. Examples and applications.
8. Bessel and hypergeometric functions.
9. Second order equation on half-line, oscillation theory.
10. Spectrum of differential operator.