Symplectic Geometry for TFA and QM

We propose a doubly interdisciplinary and synergetic project: it aims at building bridges between time-frequency analysis and symplectic geometry on one hand, but also to deepen the connections between time-frequency analysis and quantum mechanics on the other hand. The key issue for this project is an approach to uncertainty principles in time-frequency analysis with methods from symplectic geometry. On the one hand we are heading for a unified and rigorous description of uncertainty principles in time-frequency and Gabor analysis, and on the other hand we want to invoke methods from time- frequency analysis for the study of Schrödinger operators. Uncertainty principles are central research topics both in time-frequency as well as in quantum mechanics, because they are intrinsically related to deeper properties of signals and quantum states respectively. The study of uncertainty principles is closely tied to a variety of notions and theorems in pure mathematics, e.g. Heisenberg groups, the orbit method of Kirillov, spectral theory of unbounded operators, to mention just a few. Recently de Gosson has linked the symplectic structure of phase space with uncertainty principles. Such results highlight a deep links between symplectic geometry and quantum mechanics, especially to Schrödinger equations. The main tools for our investigations will be symplectic geometry, the theory of metaplectic operators, and modulation spaces. Since their invention in 1983 modulation spaces have turned out to be the correct class of function spaces in time-frequency analysis and Gabor analysis. Recently, Gröchenig and his collaborators (and also Cordero and Nicola) have related the properties of Schrödinger operators with modulation spaces. Another goal of our project is to deepen the connections between quantum mechanics and time-frequency analysis. This appears feasible, because from a mathematical point of view they have a very similar structure, but somehow time-frequency analysis so far has been the much less well-known twin of quantum mechanics, gaining only slowly more and more recognition with in the scientific community.

We propose a doubly interdisciplinary and synergetic project: it aims at building bridges between time-frequency analysis and symplectic geometry on one hand, but also to deepen the connections between time-frequency analysis and quantum mechanics on the other hand. The key issue for this project is an approach to uncertainty principles in time-frequency analysis with methods from symplectic geometry. On the one hand we are heading for a unified and rigorous description of uncertainty principles in time-frequency and Gabor analysis, and on the other hand we want to invoke methods from time- frequency analysis for the study of Schrödinger operators. Uncertainty principles are central research topics both in time-frequency as well as in quantum mechanics, because they are intrinsically related to deeper properties of signals and quantum states respectively. The study of uncertainty principles is closely tied to a variety of notions and theorems in pure mathematics, e.g. Heisenberg groups, the orbit method of Kirillov, spectral theory of unbounded operators, to mention just a few. Recently de Gosson has linked the symplectic structure of phase space with uncertainty principles. Such results highlight a deep links between symplectic geometry and quantum mechanics, especially to Schrödinger equations. The main tools for our investigations will be symplectic geometry, the theory of metaplectic operators, and modulation spaces. Since their invention in 1983 modulation spaces have turned out to be the correct class of function spaces in time-frequency analysis and Gabor analysis. Recently, Gröchenig and his collaborators (and also Cordero and Nicola) have related the properties of Schrödinger operators with modulation spaces. Another goal of our project is to deepen the connections between quantum mechanics and time-frequency analysis. This appears feasible, because from a mathematical point of view they have a very similar structure, but somehow time- frequency analysis so far has been the much less well-known twin of quantum mechanics, gaining only slowly more and more recognition with in the scientific community.