Schr\"odinger type propagators, pseudodifferential operators and modulation spaces

We prove continuity results for Fourier integral operators with symbols in modulation spaces, acting between modulation spaces. The phase functions belong to a class of nondegenerate generalized quadratic forms that includes Schr\"odinger propagators and pseudodifferential operators. As a byproduct we obtain a characterization of all exponents $p,q,r_1,r_2,t_1,t_2 \in [1,\infty]$ of modulation spaces such that a symbol in $M^{p,q}(\mathbb R^{2d})$ gives a pseudodifferential operator that is continuous from $M^{r_1,r_2}(\mathbb R^d)$ into $M^{t_1,t_2}(\mathbb R^d)$.