Scattering Theory

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The purpose of the project is the study of quantum/wave mechanics from the mathematical point of view, and of its many manifestations in the theory of partial differential equations and geometry. Specific current interests are the distribution of scattering resonances in physical and geometric settings, dynamical and semiclassical zeta functions, quantum chaos, and scattering of solitons/NLS in external fields.As popular view would have it, resonance is the tendency of a system to oscillate at a maximum amplitude at a certain frequency. Mathematically, it is described by a complex number with the real part being the frequency and the imaginay part, the rate of decay (the resonances "die" as "dying notes of a bell"). These numbers appear as poles of classes of meromorphic operators or functions (such as zeta functions, including the Riemann zeta function).The project focuses on the search for general mathematical principles in the distribution of resonances, and on the detailed study of specific examples motivated by that. The previous work clearly demonstrates this trend: resonances appear in geometry, semi-classical theories, obstacle scattering, open quantum maps.Some results hold universally and some are known in specific cases. Our study of scattering of solitons is also motivated by resonance phenomena, such as the search for the correct concept of resonance transmission in scattering of Bose-Einstein matter waves.The phenomena studied in the project are very general: for instance, microwaves can be used to model quantum scattering and quantum chaos, leading to insights about MEMS (micro-electro-nechanical systems) which are constructed using tiny resonators. Purely mathematical quantum maps (the study of which often has connections to number theory) are used to model nanostructures and transport through quantum dots. Zeros of zeta functions for hyperbolic rational maps can be used as models for resonance distribution in chaotic scattering.