Syllabus
* States and observables in quantum mechanics.
A survey of properties of Hilbert spaces and operators on them. Spectral theorem and spectral types of self-adjoint operators, the theory of self-adjoint extensions. Basic postulates of quantum mechanics. Examples of simple quantum systems. Mixed states, superselection rules. Compatibility of observables. Algebraic formulation of the quantum theory.
* Global and local uncertainty relations.
Heisenberg relations. Hilbert space of analytical functions. Coherent states. Local uncertainty relations.
* Canonical commutation relations.
Nelson's example. Weyl relations: Stone - von Neumann theorem about the existence and uniqueness of their representation. Systems with an infinite number of degrees of freedom.
* Time evolution.
The basic dynamical postulate. Time evolution pictures. Wave packet dispersion. Evolution of coherent states. Feynman "integrals". Time evolution of unstable systems. Friedrichs model.
* Schroedinger operators.
Self-adjointness criteria. Discrete spectrum, its cardinality and structure. Essential spectrum, its stability. Systems with a boundary, quantum waveguides.
* Point and contact interactions.
The one-dimensional case: definition of a point interaction, spectral and scattering properties. Kronig-Penney model. Point interactions in dimension two and three. Approximations by scaled potentials. Quantum mechanics on graphs.
Annotation
Advanced parts of quantum theory: operators on Hilbert spaces; postulates of quantum mechanics, states and observables in quantum mechanics; global and local uncertainty relations; canonical commutation relations; time evolution, Schrödinger operators; point and contact interactions. For the 4th and 5th year of the TF and JSF studies and for doctoral students.