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 operators 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.

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

CORDERO, Elena;RODINO, Luigi Giacomo
2014-01-01

Abstract

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 operators 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.
2014
55
081506
1
17
http://arxiv.org/abs/1306.5301
Fourier Integral operators; modulation spaces; metaplectic operator; Short-time Fourier transform; Wiener algebra; Schr\"odinger equation
E. Cordero; K. Gr\"ochenig; F. Nicola; L. Rodino
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/154264
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