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On the Schrödinger equation with potential in modulation spaces

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Abstract

This work deals with Schrödinger equations with quadratic and sub-quadratic Hamiltonians perturbed by a potential. In particular we shall focus on bounded, but not necessarily smooth perturbations, following the footsteps of the preceding works (Cordero et al., Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class, 2013; Cordero et al., Propagation of the Gabor wave front set for Schrödinger equations with non-smooth potentials, 2013). To the best of our knowledge these are the pioneering papers which contain the most general results about the time–frequency concentration of the Schrödinger evolution. We shall give a representation of such evolution as the composition of a metaplectic operator and a pseudodifferential operator having symbol in certain classes of modulation spaces. About propagation of singularities, we use a new notion of wave front set, which allows the expression of optimal results of propagation in our context. To support this claim, many comparisons with the existing literature are performed in this work.

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Notes

  1. If \(z\in \mathbb {C}\), with \(\mathrm{Re}\, z\ge 0\), \(z\not =0\), we take as argument of \(z^{1/2}\) that belonging to \([-\pi /4,\pi /4]\). We then define \(z^{k/2}=(z^{1/2})^k\) if \(k\) is an integer.

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Acknowledgments

We would like to thank Professor K. Gröchenig for inspiring this work.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 , Torino, Italy

    Elena Cordero

  2. Dipartimento di Scienze Matematiche, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 , Torino, Italy

    Fabio Nicola

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  1. Elena Cordero
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  2. Fabio Nicola
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Correspondence to Fabio Nicola.

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Cordero, E., Nicola, F. On the Schrödinger equation with potential in modulation spaces. J. Pseudo-Differ. Oper. Appl. 5, 319–341 (2014). https://doi.org/10.1007/s11868-014-0096-2

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