Mathematical Studies in Quantum Dynamics of Matter and Radiation

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At energy scales common on Earth, the building blocks of matter are electrons and nuclei, with interactions mediating by quantum electro-magnetic field, as well as Coulomb force. The dynamic of the total system - particles and fields - is given by a well-defined evolution equation - the Schroedinger equation. Constructing the mathematical apparatus describing this dynamic is the main thrust of our research in the next several years. We will address a wide gamut of issues, from properties of a single electron, or more precisely the electron surrounded by an induced cloud of photons, to emission and absorption of radiation by atoms and molecules, to behaviour of large aggregates of atoms forming coherent, macroscopic structures (say, localized structures like Bose-Einstein condensates or extended ones like solid crystals), or more generally some continuous medium, with quantized collective oscillations (mechanical or elastic waves), called phonons. The processes we are interested in are "irreversible", accompanied by information loss at infinity. The issues described above underlie the structure and properties of matter as we experience it. Not surprisingly, they have garnered a disproportionate number of Nobel prizes in Physics and Chemistry. Nevertheless the mathematical apparatus dealing with them is only in the initial stages of construction, especially compared with the mathematical techniques developed in the last few centuries to tackle problems of classical physics. However the pace is quickly picking up. We expect that in the next few years, the analysis of evolution equations of the type appearing in this project and the corresponding nano-, meso- and macroscopic approximations will grow into a sizable area of pure and applied mathematics. I am confident that jointly with my collaborators (including my past and present graduate students and postodcs) I will be able to make substantial contributions to this development.