  It is well known that the matrix of a metaplectic operator with respect to
phase-space shifts is concentrated along the graph of a linear symplectic map.
We show that the algebra generated by metaplectic operators and by
pseudodifferential opertators in a Sj\"ostrand class enjoys the same decay
properties. We study the behavior of these generalized metaplectic operators
and represent them by Fourier integral operators. Our main result shows that
the one-parameter group generated by a Hamiltonian operator with a potential in
the Sj\"ostrand class consists of generalized metaplectic operators. As a
consequence, the Schr\"odinger equation preserves the phase-space
concentration, as measured by modulation space norms.
