Geometrical Aspects of Spectral Theory
Code  Completion  Credits  Range 

02SPEC  ZK  2  2+0 
 Lecturer:
 Tutor:
 Supervisor:
 Department of Physics
 Synopsis:

Spectral theory is an extremely rich field which has found its application in many areas of physics and mathematics. One of the reason which makes it so attractive on the formal level is that it provides a unifying framework for problems in various branches of mathematics, for example partial differential equations, calculus of variations, geometry, stochastic analysis, etc.
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. We give an overview of both classical results and recent developments in the field, and we wish to always do it by providing a physical interpretation of the mathematical theorems.
 Requirements:

Equations of mathematical physics, functional analysis
 Syllabus of lectures:

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.
4. The role of the dimension of the Euclidean space. Criticality versus subcriticality. The Hardy inequality. Stability of matter.
5. Bound states. Variational and perturbation methods.
6. Analytic versus asymptotic perturbation theory. The BirmanSchwinger analysis. Asymptotic formulae for weakly coupled eigenvalues.
7. The semiclassical limit. Strongly coupled bound states. Weyltype asymptotics. LiebThirring inequalities.
8. The nature of the essential spectrum. The absolutely and singular continuous spectra, embedded eigenvalues. The limiting absorption principle.
9. Commutator methods and Mourre?s theory.
10. Geometrical aspects. Glazman?s classification of Euclidean domains and their basic spectral properties.
11. Vibrational systems. The symmetric rearrangement and the FaberKrahn inequality for the principal frequency.
12. Quantum waveguides. Elements of differential geometry: curves, surfaces, manifolds. Effective dynamics.
13. Geometrically induced bound states and Hardytype inequalities in tubes.
 Syllabus of tutorials:
 Study Objective:

Knowledge:
gain knowldge on modern spectral theory
Skills:
mastering of advanced methods of spectral theory of operators
 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 Nbody 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:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans:

 Matematické inženýrství (elective course)
 Matematická fyzika (elective course)