Abstract
Our aim in this work is to give some quantitative insight on the dispersive effects exhibited by solutions of a semiclassical Schrödinger-type equation in \(\mathbf{R}^d\). We describe quantitatively the localisation of the energy in a long-time semiclassical limit within this non compact geometry and exhibit conditions under which the energy remains localized on compact sets. We also explain how our results can be applied in a straightforward way to describe obstructions to the validity of smoothing type estimates.
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Notes
- 1.
Think for instance of \(\lambda (\xi ) = \Vert \xi \Vert ^2\), for which (2) corresponds to the standard, non-semiclassical, Schrödinger equation, one of the most studied dispersive equations.
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Acknowledgements
F. Macià has been supported by grants StG-2777778 (U.E.) and MTM2013-41780-P, TRA2013-41096-P (MINECO, Spain). Part of this work was done while V. Chabu was visiting ETSI Navales at Universidad Politécnica de Madrid in the fall of 2015. V. Chabu was partly supported by grant 2017/13865-0, São Paulo Research Foundation (FAPESP)
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Chabu, V., Fermanian-Kammerer, C., Macià, F. (2019). Semiclassical Analysis of Dispersion Phenomena. In: Delgado, J., Ruzhansky, M. (eds) Analysis and Partial Differential Equations: Perspectives from Developing Countries. Springer Proceedings in Mathematics & Statistics, vol 275. Springer, Cham. https://doi.org/10.1007/978-3-030-05657-5_7
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