Mathematical methods in many-particle quantum dynamics

For about a century the principles of quantum physics have provided the scientifically recognised method for the description of the very small: atoms, molecules, nano structures, their interaction with light as well as chemical reactions. Given that the fundamental equation of quantum physics, Schrödinger`s equation, describes every single particle in its relation to all other particles, accurate computations demand extensive calculating and storage capacities. Even for small molecules these exceed by far today`s possibilities. Among numerous ways to restrict complexity while preserving precise results through approximations, density functional theory has a privileged status because of its widespread use in chemical physics and material sciences. If interactions of particles with electro-magnetic fields, thus including light, are to be described, a time-dependent version of density functional theory is employed. To make sure in the first place that such approximations yield sensible results that agree with those of Schrödinger`s equation, a mathematical proof is required. This proof was already devised by Erich Runge and E.K.U. Gross in 1984 but only under very specific constraints and not with full mathematical rigour. The proposed project ought to pay substantial attention to the further development of formal structures for a rigorous mathematical treatment. This will also have implications on present, practically oriented research in density functional theory that as of late is applied to the description of light--matter interaction in quantum physics as well. Since such methods are employed in wide areas of physics, chemistry, biology, and material sciences, it is imperative that alongside its continually widening scope of practical applications, also its mathematical foundations are scrutinised and further developed.

This project, conducted at the Max Planck Institute for the Structure and Dynamics of Matter in Hamburg, was devoted to the theoretical study of new methods for calculating properties of molecules and solid matter according to the principles of quantum mechanics. It included the particular treatment of special physical settings, like the presence of a magnetic field or an optical resonator, that lead to the alteration of material properties or are even capable to create entirely new exotic states of matter. Since the fundamental equation of quantum mechanics, Schrödinger's equation, at high particle numbers is far too complex to be directly solvable, special approximation techniques are applied for the calculation of material properties that have to be adopted to the respective physical setting. In this project, in order to achieve this, the necessary mathematical structures were accurately studied and a strict mathematical formulation was sought on the one hand, while on the other completely new methods were developed and tested with concrete numerical examples. In the process, numerous interesting discoveries could be made that relate to vastly different areas of mathematics: One of the most important approximation techniques in quantum chemistry, density-functional theory, uses an iterative procedure for which it was previously unknown if it always yields a solution. With the help of tools from convex analysis we were able to show that this is indeed the case for systems with a finite number of particle states if the procedure is additionally subjected to a particular regularization. In another subproject, we considered electrons in a regular atomic grid under the influence of a magnetic field that take energies that collectively form a fractal, the so-called Hofstadter butterfly. We studied how this fractal changes under the influence of an optical resonator and moreover found that a further fractal is formed by varying the coupling strength to the resonator. A further research topic was the reformulation of the usual method for calculating material properties based on a description of the energies of the system into a description using the acting forces. This leads to new efficient techniques that were compared to previous methods and form the basis for entirely new developments in this field. The above mentioned physical situations, the presence of magnetic fields and optical resonators, are also examined in current experiments, with the aim of finding new and useful material properties. The theoretical research advanced during this project is meant to help building a theoretical foundation for these experiments and to better understand their findings.