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Mathematical aspects of quantum dynamics


Principal Investigator

Prof. Dr. Benjamin Schlein
Hausdorff Center for Mathematics & Institute for Applied Mathematics
Endenicher Allee 60
53115 Bonn



The main goal of this proposal is to reach a better mathematical understanding of the dynamics of quantum mechanical systems. In particular I plan to work on the following three projects along this direction. A. Effective Evolution Equations for Macroscopic Systems. The derivation of effective evolution equations from first principle microscopic theories is a fundamental task of statistical mechanics. I have been involved in several projects related to the derivation of the Hartree and the Gross-Piteavskii equation from many body quantum dynamics. I plan to continue to work on these problems and to use these results to obtain new information on the many body dynamics. B. Spectral Properties of Random Matrices. The correlations among eigenvalues of large random matrices are expected to be independent of the distribution of the entries.

This conjecture, known as universality, is of great importance for random matrix theory. In collaboration with L. Erdos and H.-T. Yau, we established the validity of Wigner's semicircle law on microscopic scales, and we proved the emergence of eigenvalue repulsion. In the future, we plan to continue to study Wigner matrices to prove, on the longer term, universality. C. Locality Estimates in Quantum Dynamics. Anharmonic lattice systems are very important models in non-equilibrium statistical mechanics. With B. Nachtergaele, H. Raz, and R. Sims, we proved Lieb-Robinson type inequalities (giving an upper bound on the speed of propagation of signals), for a certain class of anharmonicity. Next, we plan to extend these results to a larger class of anharmonic potentials, and to apply these bounds to establish other fundamental properties of the dynamics of anharmonic systems, such as the existence of its thermodynamical limit.