Uniqueness of ground states for pseudorelativistic Hartree equations

We prove uniqueness of ground states Q ∈ H 1/2 (R 3 ) for the pseudo-relativistic Hartree equation, p −� + m 2 Q − ` |x| 1 ∗ |Q| 2 ´ Q = −µQ, in the regime of Q with sufficiently smallL 2 -mass. This result shows that a uniqueness conjecture by Lieb and Yau in (CMP 112 (1987), 147-174) holds true at least for N = R |Q| 2 ≪ 1 except for at most countably many N. Our proof combines variational arguments with a nonrelativistic limit, which leads to a certain Hartree-type 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 nonrelativis- tic Hartree equations.