Propagation of exponential phase space singularities for Schr\"odinger equations with quadratic Hamiltonians

We study propagation of phase space singularities for the initial value Cauchy problem for a class of Schr\"odinger equations. The Hamiltonian is the Weyl quantization of a quadratic form whose real part is non-negative. The equations are studied in the framework of projective Gelfand--Shilov spaces and their distribution duals. The corresponding notion of singularities is called the Gelfand--Shilov wave front set and means the lack of exponential decay in open cones in phase space. Our main result shows that the propagation is determined by the singular space of the quadratic form, just as in the framework of the Schwartz space, where the notion of singularity is the Gabor wave front set.Comment: 39 pages. To appear in J. Fourier Anal. App

The Weyl product on quasi-Banach modulation spaces

We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.Comment: 29 page

Semigroups for quadratic evolution equations acting on Shubin-Sobolev and Gelfand-Shilov spaces

We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is the Weyl quantization of a homogeneous quadratic form with non-negative definite real part. The solution semigroup is shown to be strongly continuous on several spaces: the Shubin--Sobolev spaces, the Schwartz space, the tempered distributions, the equal index Beurling type Gelfand--Shilov spaces and their dual ultradistribution spaces.Comment: 36 page