Spectral and dynamical analysis of delocalization for random Schrödinger operators

• Applicant: Dr. Leander Geisinger
• Subject Area: Mathematics
• Term: from 2011 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 211408066

The spectral analysis of partial differential operators, in particular the analysis of random Schrödingeroperators, plays a crucial role in mathematical physics. On the one hand randomness is of relevance for physical models, where it carries significant implications, for example on conduction properties of a material. On the other hand the study of random operators raises deep mathematical problems and various aspects of the subject have inspired the development of modern mathematical techniques.An important example is the Anderson model: Motivated by the quest for a theory of quantum transport in disordered media, the physicist P. W. Anderson came up with a model for a quantum mechanical electron in a crystal with impurities. The electron is described by a Schrödinger operator with a random potential incorporating the interaction of the electron with randomly placed impurities. The spectral properties of the Anderson operator determine the dynamics of the electron and the properties of electronic motion and conductance.The main objective of my research project is to contribute to new developments concerning delocalization for random Schrödinger operators. Delocalized, extended states, corresponding to absolutely continuous spectrum, enable electronic transport and conductance. However, only very few rigorous mathematical results are known that imply the existence of such states. Recently, a new approach to this problem was found based on the analysis of rare resonances in the Anderson model on metric trees.Starting from these new insights I am looking for extensions of this mechanism to other models of random Schrödinger operators and for implications on the dynamics generated by the Anderson operator. This might contribute to the understanding of longstanding questions about the mathematics of the Anderson model and reveal interesting relations to other problems in the field of probability, statistical mechanics and spectral analysis.

• DFG Programme: Research Fellowships
• International Connection: USA
• Host: Professor Dr. Michael Aizenman