Phase space analysis and applications to quantum mechanics
Several problems in partial differential equations, such as that of the propagation of singularities, suggest an analysis by a decomposition of functions and operators in elementary wave packets - the so-called Gabor atoms in time-frequency analysis or coherent states in quantum mechanics - and the study of the evolution of each packet (cf. the propagation of the Gabor wave front set). According to this paradigm, a fundamental role is played by the uncertainty principle, which limits the maximum possible resolution in phase space, and which manifests itself in several functional inequalities, such as the Sobolev embedding theorems. Often, it also represents the ultimate reason for the existence of a solution (e.g. of the ground state for certain linear and nonlinear Schroedinger equations). It is therefore interesting to find, in several contexts, the functions which are maximally concentrated in phase space. Such isoperimetric optimization problems require techniques from geometric measure theory, harmonic and complex analysis, nonlinear analysis and calculus of variations.