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

Generalized Metaplectic Operators and the Schr\"odinger Equation with a Potential in the Sj\"ostrand Class