Uniqueness of ground states for pseudorelativistic Hartree equations

Abstract

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

$$\sqrt{-\Delta +m^2}Q-\left({\left|x\right|}^{-1}\ast {\left|Q\right|}^2\right)Q=-\mu Q,$$

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.