Geometrical Aspects of Spectral Theory

Garant předmětu:

David Krejčiřík

Lecturer:

David Krejčiřík

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

1. Motivations. The crisis of classical physics and the rise of quantum mechanics. Mathematical formulation of quantum theory. Spectral problems in classical physics.

2. Elements of functional analysis. The discrete and essential spectra. Sobolev spaces. Quadratic forms. Schrödinger operators.

3. Stability of the essential spectrum. Weyl's theorem. Bound states. Variational and perturbation methods.

4. The role of the dimension of the Euclidean space. Criticality versus subcriticality. The Hardy inequality. Stability of matter.

5. Geometrical aspects. Glazman's classification of Euclidean domains and their basic spectral properties.

6. Vibrational systems. The symmetric rearrangement and the Faber-Krahn inequality for the principal frequency.

7. Quantum waveguides. Elements of differential geometry: curves, surfaces, manifolds. Effective dynamics.

8. Geometrically induced bound states and Hardy-type inequalities in tubes.

Requirements:

Syllabus of lectures:

Syllabus of tutorials:

Study Objective:

Acquired knowledge:

The goal of the lecture is to acquaint the students with spectral methods in the theory of linear differential operators coming both from modern as well as classical physics, with a special emphasis put on geometrically induced spectral properties.

Acquired skills:

Mastering of advanced methods of spectral theory of self-adjoint operators; variational tools, partial differential equations, geometric analysis, Sobolev spaces.

Study materials:

Key references

[1] B. Davies, Spectral theory and differential operators, Cambridge University Press, 1995.

[2] A. Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkhäuser, Basel, 2006.

[3] M. Reed and B. Simon, Methods of modern mathematical physics, I-IV, Academic Press, New York, 1972-1978.

Recommended references:

[1] W. O. Amrein, A. Boutet de Monvel and V. Georgescu, C0 -groups, commutator methods and spectral theory of N-body Hamiltonians, Progress in Math. Ser., vol. 135, Birkhäuser, 1996.

[2] D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford University Press, 1987.

[3] L. C. Evans, Partial Differential Equations, Amer. Math. Soc., 2010.

Note:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (elective course)
• Matematická fyzika (elective course)
• Matematické inženýrství (elective course)