Analisi tempo-frequenza di operatori integrali di Fourier ed Applicazioni

We present the analysis of a particular class of FIOs, whose representation with respect to Gabor frames is proved to be very efficient in the framework of both pure and applied mathematics. To be more precise, we shall present some boundedness result for FIOs with classical symbols on modulation spaces. Furthermore, using these spaces as symbol classes, we study again continuity results on modulation spaces. In these spaces, functions are estimated in norm with respect to the time and frequency point of view at the meantime. As an application, we will develop a detailed study of the Fourier integral operator which solves the Cauchy problem for the Schr\"odinger equation related to the harmonic oscillator, with Gaussian initial datum. We will also represent the decay of the coefficients of the matrix operator corresponding to the harmonic oscillator with the respect to Gabor frames and the behavior of the solution in the time-frequency plane with a numerical implementation.