Uniqueness of Ground States for Pseudo-Relativistic Hartree Equations

TL;DR

This paper proves the uniqueness of ground states for pseudo-relativistic Hartree equations in three dimensions under a small mass condition, confirming a longstanding conjecture by Lieb and Yau.

Contribution

It establishes the uniqueness of ground states in a pseudo-relativistic setting and introduces a novel proof combining variational methods with a nonrelativistic limit, including a new nondegeneracy result.

Findings

Uniqueness of ground states for small mass in pseudo-relativistic Hartree equations.

Nondegeneracy of the linearized operator for the limiting Hartree equation.

Validation of Lieb and Yau's conjecture under a smallness condition.

Abstract

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.