Singular Hartree equation in fractional perturbed Sobolev spaces

DOI

10.1080/14029251.2018.1503423How to use a DOI?

Keywords

Point interactions; Singular perturbations of the Laplacian; Regular and singular Hartree equation; Fractional singular Sobolev spaces; Strichartz estimates for point interaction Hamiltonians Fractional Leibniz rule; Kato-Ponce commutator estimates

Abstract

We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.

Copyright

© 2018 The Authors. Published by Atlantis and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Alessandro Michelangeli
AU  - Alessandro Olgiati
AU  - Raffaele Scandone
PY  - 2021
DA  - 2021/01/06
TI  - Singular Hartree equation in fractional perturbed Sobolev spaces
JO  - Journal of Nonlinear Mathematical Physics
SP  - 558
EP  - 588
VL  - 25
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1503423
DO  - 10.1080/14029251.2018.1503423
ID  - Michelangeli2021
ER  -