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Generalized Metaplectic Operators and the Schrödinger Equation with a Potential in the Sjöstrand Class

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 Sjöstrand 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 Sjöstrand class consists of generalized metaplectic operators. As a consequence, the Schrödinger equation preserves the phase-space concentration, as measured by modulation space norms.