The aim of the course is to give an introduction to the mathematical theory of quantum mechanics giving some of the most important classical results and aiming at reaching the edge of research in the field.
In the first part of the course I will introduce some fundamental notions of the theory of operators in Hilbert spaces. Furthermore I will discuss the basic ideas of correspondence between quantum and classical mechanics. From a theoretical poiint of view the main notion will be the discussion of the discrete and essential spectrum of a self adjoint operator. An important result that will be deduced is the classification of states as bound states and decaying states. The corresponding behaviour of the solutions will be discussed.
Subsequently I will discuss some concrete examples allowing to put into evidence some important features of quantum mechanics. In particular the following topics will be discussed: Hydrogen atom, band structure of the spectrum of crystals, stability of atoms and molecules, Aharonov-Bohm effect.
In the second part of the course I will present more advanced topics
1) Scattering theory and existence of wave operators
2) Perturbation theory for discrete spectrum and application to Zeeman effect. Fermi Golden Rule and metastability in quantum mechanics
Finally, depending on the time left, I will present some more topics. Possible topics are
3) KAM theory for the understanding of the dynamics of particle under the effect of an impinging electromagnetic wave.
4) Giustification of Bohr-SOmmerfeld quantization rule for integrable systems. Pseudodifferential calculus.
Prerequisites for admission
Elementary notions of calculus and linear algebra. Elementary notions of mechanics and partial differential equations. Knowledge of elementary theory of functions and operators would be helpful.
The course is based on some chapters of the book
Gustafson, Stephen J.; Sigal, Israel Michael Mathematical concepts of quantum mechanics. Universitext. Springer.
One can find some additional metherial usefull do understand better some mathematical concepts in : Teschl, Gerald: Mathematical methods in quantum mechanics. With applications to Schrödinger operators. Second edition. Graduate Studies in Mathematics, 157. American Mathematical Society, Providence, RI, 2014.