Mastery of elementary notions of linear algebra in finite-dimensional spaces and calculus.

The goal of the course is to acquaint the students with spectral methods in the theory of linear differential operators coming both from modern physics and from classical physics, with special emphasis put on geometrically induced spectral properties. We give an overview of classical results and also recent developments in the field, and we always attempt to do this by providing a physical interpretation of the mathematical theorems.

Spectral theory is an extremely rich field, which has found applications in many areas of physics and mathematics. One reason why it is 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.

1. Motivations. Differential equations in physics. Unifying framework of spectral theory. Geometrical aspects.

2. Preliminaries. Glazman's classification of Euclidean domains. Elements of functional analysis. The point and continuous spectra. Self-adjointness. Quadratic forms. Sobolev spaces.

3. Quasi-conical domains. Location of propagating modes. Dimensional features of the Euclidean space. The Hardy inequality. Stability of matter. The crisis of classical physics and the rise of quantum mechanics. Mathematical formulation of quantum theory. Virtual bound states.

4. Quasi-bounded domains. Discrete and essential spectra. Compactness. Vibrational systems. Spectral isoperimetric inequalities. The symmetric rearrangement and the Faber-Krahn inequality for the principal frequency. Optimality of a disk in physics.

5. Quasi-cylindrical domains. Tubes. Elements of differential geometry of curves and its moving frames. Quantum waveguides. Geometrically induced bound states and Hardy-type inequalities. Applications to nanostructure devices and stochastic processes.

Solving specific exercises related to the topic of the course based on the acquired knowledge.

Saturday, November 14, 16:00 – 18:00 Welcome meeting
Sunday, November 15, 15:00 –  18:00 Prague Discovery Game, 18:00 - 21:00 Dinner party
Friday, November 20, 17:00 – 19:00 – Farewell Party