UNIQUENESS OF GROUND STATES FOR PSEUDORELATIVISTIC HARTREE EQUATIONS

We prove uniqueness of ground states Q is an element of H-1/2(R-3) for the pseudorelativistic Hartree equation, root-Delta + m(2) Q - (vertical bar x vertical bar(-1) * vertical bar Q vertical bar(2))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 = f vertical bar Q vertical bar(2) << 1 except for at most countably many N. Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartreetype equation (also known as the Choquard-Pekard or Schrodinger-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.