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Dispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3

  • Felice IandoliEmail author
  • Raffaele Scandone
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 18)

Abstract

The study of dispersive properties of Schrödinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schrödinger equations with singular interactions. In this work we proved that, in the case of one point interaction in \(\mathbb{R}^{3}\), the perturbed Laplacian satisfies the same L p L q estimates of the free Laplacian in the smaller regime q ∈ [2, 3). These estimates are implied by a recent result concerning the L p boundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime q ≥ 3.

Keywords

Dispersive estimates Point interactions Schrödinger operators Weighted Fourier inequalities 

Notes

Acknowledgements

Raffaele Scandone is partially supported by the 2014–2017 MIUR-FIR grant “Cond-Math: Condensed Matter and Mathematical Physics”, code RBFR13WAET and by Gruppo Nazionale per la Fisica Matematica (GNFM-INdAM).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.International School for Advanced Studies SISSATriesteItaly

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