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.
is the study of particles and waves interacting with other objects. When particles are concerned we think of classical scattering and in the case of waves we may be in the setting of electromagnetic, acoustic or quantum scattering. The subject has a long tradition in science and engineering and in mathematics it is part of the theory of dynamical systems and partial differential equations: trajectories of particles describe dynamics and waves are solutions of partial differential equations such as the wave equation or Schroedinger equation. The PI studies scattering mathematically in different settings motivated by physical phenomena. One way that scattered waves are described is in terms of their oscillations and rates of decay. Mathematically the two properties can be described as a complex number, its real part being the rate of oscillation and its imaginary part the rate of decay. These are called resonances: the response of the system is highest when the energy or frequency is closest to the real part of a resonance. One practical setting where resonances are important are MEMS (microelectromechanical systems) which can take a form of resonators: the ratio of real and imaginary part of the resonance is then called a quality factor. Another examples studied extensively by PI's graduate students are quasinormal modes of black holes. They are supposed to describe oscillations and decay of gravitational waves generated by interaction of a black hole with other objects and could in principle describe the structure of a black whole. The PI has discovered many features of the distribution of resonances at the high frequency limit. The key feature is the relation to classical scattering and concepts from the thermodynamic formalism: the topological pressure determines distribution of decay rates and the dimension of the classical repellor a power law for counting resonances. These works have now been studied in the physics literature and led to the PI participation in experimental projects involving microwave billiards. Another example of classical/quantum correspondence studied by the PI is the effective dynamics of solitons moving in external fields. Solitons are non-linear waves exhibiting strong stability properties. They appear in modeling of many physical systems, for instance Bose-Einstein condensates. The presence of external fields is physically natural and the motion of solitons is remarkably well described by suitable classical dynamics in which multiple solitons are treated as interacting particles -- see figure. Earlier work of the PI and his collaborators on interaction of solitons with narrow potentials modeled by delta functions is attracting attention in physics and is related to recent experiments involving Bose-Einstein condensates (total reflection of a soliton by a narrow attractive trap -- the quantum effects allow this counterintuitive phenomenon). The PI has also completed a book on semiclassical analysis which describes mathematical aspects of the particle/wave correspondence in the context of differential equations. To summarize, PIs work uses rigorous mathematical methods to study general phenomana in wave/particle correspondence, seeking, when possible, numerical or experimental verification in specific cases.
