Title | Dispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3 |

Publication Type | Book Chapter |

Year of Publication | 2017 |

Authors | Iandoli, F, Scandone, R |

Editor | Michelangeli, A, Dell'Antonio, G |

Book Title | Advances in Quantum Mechanics: Contemporary Trends and Open Problems |

Pagination | 187–199 |

Publisher | Springer International Publishing |

City | Cham |

ISBN Number | 978-3-319-58904-6 |

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 $q \in [2,3)$. These estimates are implied by a recent result concerning the Lpboundedness 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 $q \geq 3$. |

URL | https://doi.org/10.1007/978-3-319-58904-6_11 |

DOI | 10.1007/978-3-319-58904-6_11 |

## Dispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3

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