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Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians

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  • Karel Pravda†Starov
  • Luigi Rodino
  • Patrik Wahlberg

Abstract

We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with nonnegative real part. We point out that the singular space associated with the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated with the imaginary part of the quadratic form. As an application we obtain for the heat equation a sufficient condition on the Gabor wave front set of the initial datum tempered distribution that implies regularization to Schwartz regularity for positive times.

Suggested Citation

  • Karel Pravda†Starov & Luigi Rodino & Patrik Wahlberg, 2018. "Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians," Mathematische Nachrichten, Wiley Blackwell, vol. 291(1), pages 128-159, January.
    • Handle: RePEc:bla:mathna:v:291:y:2018:i:1:p:128-159
    DOI: 10.1002/mana.201600410
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    Cited by:

    1. Alden Waters, 2023. "Observability for Schrödinger equations with quadratic Hamiltonians," Partial Differential Equations and Applications, Springer, vol. 4(2), pages 1-33, April.

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