We prove uniqueness of ground states
for the pseudorelativistic Hartree equation,
in the regime of
with
sufficiently small
-mass.
This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for
except for at most
countably many
.
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.
