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Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities

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Abstract

We study evolution equations associated to time-dependent dissipative non-selfadjoint quadratic operators. We prove that the solution operators to these non-autonomous evolution equations are given by Fourier integral operators whose kernels are Gaussian tempered distributions associated to non-negative complex symplectic linear transformations, and we derive a generalized Mehler formula for their Weyl symbols. Some applications to the study of the propagation of Gabor singularities (characterizing the lack of Schwartz regularity) for the solutions to non-autonomous quadratic evolution equations are given.

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Notes

  1. Even in the case when Hamiltonians actually do not depend on time.

  2. Determined up to its sign.

  3. A set invariant under multiplication with positive reals.

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Authors and Affiliations

  1. IRMAR, CNRS UMR 6625, Université de Rennes 1, Campus de Beaulieu, 263 avenue du Général Leclerc, CS 74205, 35042, Rennes cedex, France

    Karel Pravda-Starov

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  1. Karel Pravda-Starov
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Correspondence to Karel Pravda-Starov.

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Communicated by Y. Giga.

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Pravda-Starov, K. Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities. Math. Ann. 372, 1335–1382 (2018). https://doi.org/10.1007/s00208-018-1667-y

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  • DOI: https://doi.org/10.1007/s00208-018-1667-y

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