Lenzmann, Enno.
(2009)
* Uniqueness of ground states for pseudorelativistic Hartree equations.*
Analysis & PDE, 2 (1).
pp. 1-27.

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Official URL: http://edoc.unibas.ch/49582/

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## Abstract

We prove uniqueness of ground states Q ∈ H^(1∕2)(ℝ^3) for the pseudorelativistic Hartree equation,

sqrt(−Δ + m^2) * Q -( x^-1 ∗ |Q|^2) * Q = −μQ,

in the regime of Q with sufficiently small L^2-mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for N = ∫ |Q|^2 ≪ 1 except for at most countably many N.

Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartree-type equation (also known as the Choquard–Pekard or Schrödinger–Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the so-called nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations.

sqrt(−Δ + m^2) * Q -( x^-1 ∗ |Q|^2) * Q = −μQ,

in the regime of Q with sufficiently small L^2-mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for N = ∫ |Q|^2 ≪ 1 except for at most countably many N.

Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartree-type equation (also known as the Choquard–Pekard or Schrödinger–Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the so-called nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations.

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Lenzmann) |
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UniBasel Contributors: | Lenzmann, Enno |

Item Type: | Article, refereed |

Article Subtype: | Research Article |

Publisher: | Mathematical Sciences Publishers |

ISSN: | 2157-5045 |

e-ISSN: | 1948-206X |

Note: | Publication type according to Uni Basel Research Database: Journal article |

Identification Number: | |

Last Modified: | 29 Nov 2017 08:07 |

Deposited On: | 29 Nov 2017 08:07 |

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