Vol. 2, No. 1, 2009

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Uniqueness of ground states for pseudorelativistic Hartree equations

Enno Lenzmann

Vol. 2 (2009), No. 1, 1–27
Abstract

We prove uniqueness of ground states $Q\in {H}^{1∕2}\left({ℝ}^{3}\right)$ for the pseudorelativistic Hartree equation,

$\sqrt{-\Delta +{m}^{2}}\phantom{\rule{0.3em}{0ex}}Q-\left({\left|x\right|}^{-1}\ast {\left|Q\right|}^{2}\right)Q=-\mu Q,$

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 $N=\int |Q{|}^{2}\ll 1$ 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
Primary: 35Q55