Sturm-Lieouville Theory (FSI-9SLT)

Academic year 2022/2023
Supervisor: prof. Aleksandre Lomtatidze, DrSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech or English
Aims of the course unit:
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.
Learning outcomes and competences:
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.
Course contents:
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.
Teaching methods and criteria:
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.
Controlled participation in lessons:
Absence has to be made up by self-study using recommended literature.
Type of course unit:
    Lecture  10 × 2 hrs. optionally                  
Course curriculum:
    Lecture 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.
Literature - fundamental:
1. 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).
2. A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975.
3. E. C.Titchmarsh: Eigenfunction expansions associated with second-order differential equations. Part I. Second Edition Clarendon Press, Oxford 1962 vi+203 pp.
4. A. Zettl, Sturm-Liouville theory: Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005. xii+328 pp. ISBN: 0-8218-3905-5
5. 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.
Literature - recommended:
1. 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).
2. A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975
3. E. C.Titchmarsh: Eigenfunction expansions associated with second-order differential equations. Part I. Second Edition Clarendon Press, Oxford 1962 vi+203 pp.
4. A. Zettl, Sturm-Liouville theory: Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005. xii+328 pp. ISBN: 0-8218-3905-5
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
D-APM-K combined study --- -- DrEx 0 Recommended course 3 1 W
D-APM-P full-time study --- -- DrEx 0 Recommended course 3 1 W