Mathematical Quantum Mechanics

Various different topics of Quantum Mechanics are being studied, having as a starting point the Schrödinger and Dirac equations. The aim is to obtain a rigorous understanding of specific problems.

Many-body Quantum Mechanics

The aim is to study models in computational quantum chemistry, e.g., Hartree, Hartree-Fock, Thomas-Fermi, and multiconfiguration models. We will investigate issues such as the existence of ground states (both for the electronic structure and the configuration of nuclei), foundations of the models of the crystalline phase, and the macroscopic limits. Relations between the physical modelling, the numerical concerns and the mathematical analysis of the problems are emphasized.

Resonances and Scattering Theory

The so-called ``scattering theory'' allows to analyze the phenomenon of collision between two or more quantum particles. Functional analytic methods provides a mathematical description of these collision processes, in terms of the ``Asymptotic Completeness'' property for 2 or N body systems; they lead to recent extensions to the case of ``Coulombic'' interactions (charged particles) and more generally to ``long range'' pair interactions. The resonance notion plays a crucial role in the physics of collision processes. A growing number of scientists is involved in their study, in order to better understand how they manifest themselves in spectral and scattering problems (including the nonlinear case). Determining the location of resonances is a subtle and promising problem, often linked to geometric and dynamical properties of the system.

Spectral Theory and Mesoscopic Systems

Quantum Operator Theory concerns the analytic properties of mathematical models of quantum systems. Its achievements are among the deepest and most interesting in quantum theory, e.g., the calculation of the energy levels of atoms and molecules which lies at the core of computational quantum chemistry. A new and very exciting source of problems has recently appeared in quantum operator theory motivated by todays experimental techniques of mesoscopic physics which permits fabrication of tiny semiconductor structures (nanostructures) of various shapes (dots, wires, wells etc). Although devised in the laboratory, these systems are small enough to exhibit quantum effects, like interference, and they are expected to become the building blocks of the next-generation electronics. A pressing challenge is to investigate spectral and scattering properties of such systems (Localized states, transport properties etc.).

Other (related) areas

Inverse problems and wave propagation in perturbed stratified media.