Derivation of effective evolution equations from many body quantum dynamics

We review some recent results concerning the derivation of effective evolution equations from first principle quantum dynamics. In particular, we discuss the derivation of the Hartree equation for mean field systems and the derivation of the Gross-Pitaevskii equation for the time evolution of Bose-Einstein condensates.Comment: 9 pages. Contribution to the Proceedings of the International Congress on Mathematical Physics held in Prague in August 200

Spectral Properties of Wigner Matrices

In these notes we review recent progress (and, in Section \ref{sec:ados}, we announce a new result) concerning the statistical properties of the spectrum of Wigner random matrices.Comment: 14 pages, contribution to the Proceedings of the Conference QMath 11 held in Hradec Kralove (Czechia) in September 201

Effective Evolution Equations in Quantum Physics

In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand, by analyzing the so called BBGKY hierarchy, describing the time-evolution of reduced density matrices, and, on the other hand, by studying the dynamics of coherent initial states in a Fock-space representation of the many body system.Comment: 17 pages. Contribution to the Proceedings of the Conference "Days on PDEs" held in Biarritz in June 201

Dynamics of Bose-Einstein Condensates

We report on some recent results concerning the dynamics of Bose-Einstein condensates, obtained in a series of joint papers with L. Erdos and H.-T. Yau. Starting from many body quantum dynamics, we present a rigorous derivation of a cubic nonlinear Schroedinger equation known as the Gross-Pitaevskii equation for the time evolution of the condensate wave function.Comment: 20 pages, 2 figures; contribution to the Proceedings of the International Congress on Mathematical Physics held at IMPA in Rio de Janeiro in August 200