Uniqueness of ground states for pseudorelativistic Hartree equations

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

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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.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Lenzmann)
UniBasel Contributors:Lenzmann, Enno
Item Type:Article, refereed
Article Subtype:Research Article
Publisher:Mathematical Sciences Publishers
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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