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Schrödinger Flow’s Dispersive Estimates in a regime of Re-scaled Potentials

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Qualitative Properties of Dispersive PDEs (INdAM 2021)

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Abstract

The problem of monitoring the (constants in the estimates that quantify the) dispersive behaviour of the flow generated by a Schrödinger operator is posed in terms of the scaling parameter that expresses the small size of the support of the potential, along the scaling limit towards a Hamiltonian of point interaction. At positive size, dispersive estimates are completely classical, but their dependence on the short range of the potential is not explicit, and the understanding of such a dependence would be crucial in connecting the dispersive behaviour of the short-range Schrödinger operator with the zero-range Hamiltonian. The general set-up of the problem is discussed, together with preliminary answers, open questions, and plausible conjectures, in a ‘propaganda’ spirit for this subject.

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Acknowledgements

This work is partially supported by the Italian National Institute for Higher Mathematics—INdAM (V.G., A.M., R.S.), the project ‘Problemi stazionari e di evoluzione nelle equazioni di campo non-lineari dispersive’ of GNAMPA—Gruppo Nazionale per l’Analisi Matematica (V.G.), the PRIN project no. 2020XB3EFL of the MIUR—Italian Ministry of University and Research (V.G.), the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences (V.G.), the Top Global University Project at Waseda University (V.G.), and the Alexander von Humboldt Foundation, Bonn (A.M.).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Vladimir Georgiev

  2. Faculty of Science and Engineering, Waseda University, Tokyo, Japan

    Vladimir Georgiev

  3. Institute of Mathematics and Informatics at Bulgarian Academy of Sciences, Sofia, Bulgaria

    Vladimir Georgiev

  4. Institute for Applied Mathematics and Hausdorff Centre for Mathematics, University of Bonn, Bonn, Germany

    Alessandro Michelangeli

  5. TQT Trieste Institute for Theoretical Quantum Technologies, Trieste, Italy

    Alessandro Michelangeli

  6. Gran Sasso Science Institute, L’Aquila, Italy

    Raffaele Scandone

Authors
  1. Vladimir Georgiev
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  2. Alessandro Michelangeli
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  3. Raffaele Scandone
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Corresponding author

Correspondence to Raffaele Scandone .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Vladimir Georgiev

  2. Institute for Applied Mathematics and Hausdorff Centre for Mathematics, University of Bonn, Bonn, Germany

    Alessandro Michelangeli

  3. Gran Sasso Science Institute, L’Aquila, Italy

    Raffaele Scandone

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Georgiev, V., Michelangeli, A., Scandone, R. (2022). Schrödinger Flow’s Dispersive Estimates in a regime of Re-scaled Potentials. In: Georgiev, V., Michelangeli, A., Scandone, R. (eds) Qualitative Properties of Dispersive PDEs. INdAM 2021. Springer INdAM Series, vol 52. Springer, Singapore. https://doi.org/10.1007/978-981-19-6434-3_5

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