Uniqueness of ground states for pseudorelativistic Hartree equations

We prove uniqueness of ground states Q ∈ H^(1∕2)(ℝ^3) for the pseudorelativistic Hartree equation,
sqrt(−Δ + m^2) * Q -( x^-1 ∗ |Q|^2) * Q = −μQ,
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 = ∫ |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.