Uniqueness of Ground States for Pseudo-Relativistic Hartree Equations

Details Uniqueness of Ground States for Pseudo-Relativistic Hartree Equations Uniqueness of Ground States for Pseudo-Relativistic Hartree Equations

2008-01-25

arxiv.org/abs/0801.3976v2

Anal. PDE 2 (2009), no. 1, 1-27

Mathematics

Analysis of PDEs

27 pages. Revised version. Statement of Theorem 2 changed

Scientific paper

10.2140/apde.2009.2.1

We prove uniqueness of ground states $Q$ in $H^{1/2}$ for pseudo-relativistic Hartree equations in three dimensions, provided that $Q$ has sufficiently small $L^2$-mass. This result shows that a uniqueness conjecture by Lieb and Yau in [CMP 112 (1987),147--174] holds true at least under a smallness condition. Our proof combines variational arguments with a nonrelativistic limit, which leads to a certain Hartree-type equation (also known as the Choquard-Pekard or Schroedinger-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.

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