Hamiltonian Deformations of Gabor Frames

The aim of this project is to study a class of nonlinear deformations of Gabor frames, and to give applications to time-frequency analysis and quantum mechanics. The action of linear or affine symplectic transformations on Gabor frames is well understood, due to the symplectic/metaplectic covariance property of the Heisenberg operators. On the other hand there are very few non- trivial results when one considers general nonlinear symplectomorphisms (a symplectomorphism is the non-linear generalization of a symplectic transformation: a mapping f : of the phase space on itself is called a symplectomorphism if its Jacobian matrix is symplectic at every point where it is defined. A typical example of a general symplectomorphism is provided by the consideration of Hamiltonian flows, that is of the flows determined by Hamilton`s equations of motion. It turns out that such Hamiltonian symplectomorphisms have been studied very much in detail these last years, particularly in connection with the emerging branch of mathematics known as "symplectic topology", and of which Gromov`s nonsqueezing theorem is one of the pillars. The purpose of this project is to open up a new research line at the crossroad of symplectic geometry, time- frequency analysis, quantum mechanics, and Hamiltonian symplectomorphisms. Recall that time-frequency analysis is basically trying to analyze functions and distributions (signals in an engineering terminology) by representing them over phase space (resp. over the time-frequency plane), using some generalization of the short-time Fourier transform, with respect to a suitably chosen window g (typically the Gauss function). When it comes to understand which transformations of phase space are able to convert one such representation into another (equivalent one) it turns out that one has to preserve the symplectic structure of phase space, i.e. one can only allow symplectic transformations. This approach allows for example to convert Gabor expansions over regular grids (the standard case works with separable grids of the form aZd xbZd into Gabor expansions using alternative, e.g. sheared lattices in the TF-plane. A potential application of these deformed Gabor frames is the study of Schrödinger`s equation by invoking methods from time-frequency analysis. In particular, we have in mind the study of the phase space Schrödinger equation and of its generalization "Landau calculus" closely related to deformation quantization, which we have developed with success in previous work. Parts of this project are consequences and follow-ups of the research and results produced during the previous FWF Project "Symplectic Geometry and Applications to TFA and QM" (FWF P20442-N13), which has produced a large number of publications and unexpected results.

This project, although being of a theoretical nature (the mathematical tools that were used are quite sophisticated), there are obvious practical applications, for instance in engineering and medical imaging. When a subject (patient) is being scanned by an IRM device, or more traditionally by X-ray scanner, it is required that it be motionless to avoid blurring of the images. The deformation techniques developed in this project should avoid to bypass this requirement and allow to produce images of good quality even when the subject is moving (for instance children). Another potential application could be, for similar reasons, the improvement of radar technology, which heavily relies on reconstruction techniques.