Boundedness of Schrödinger type propagators on modulation spaces

It is known that Fourier integral operators arising when solving Schrödinger-type operators are bounded on the modulation spaces M^{p,q}, for 1\leq p=q\leq\infty, provided their symbols belong to the Sjostrand class M^{\infty,1}. However, they generally fail to be bounded on M^{p,q} for p\not=q. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on \cM^{p,q} for p\not=q, and between M^{p,q}\to\cM^{q,p}, 1\leq q<p\leq\infty. We also study similar problems for operators acting on Wiener amalgam spaces, recapturing, in particular, some recent results for metaplectic operators. Our arguments make heavily use of the uncertainty principle.