Uniqueness of ground states for pseudorelativistic Hartree equations

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 $N = \int | Q |^{2} ≪ 1$ 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.

Keywords

pseudorelativistic Hartree equation, ground state, uniqueness, boson stars

Mathematical Subject Classification 2000

Primary: 35Q55

Milestones

Received: 25 January 2008

Revised: 17 September 2008

Accepted: 11 January 2009

Published: 1 February 2009

Authors

Enno Lenzmann
Massachusetts Institute of Technology Department of Mathematics Room 2-230 Cambridge, MA 02139 United States

Vol. 2, No. 1, 2009

Enno Lenzmann

Abstract

Keywords

Mathematical Subject Classification 2000

Milestones

Authors