A new proof of uniqueness of ground state for pseudo-relativistic Hartree equation

We present a new proof of the uniqueness of ground states for the pseudo-relativistic Hartree equation in the non-relativistic regime. The proof treats the Euler-Lagrange equation and the mass constraint as a single augmented system, with the Lagrange multiplier as an additional unknown. The key point is the invertibility of the augmented linearization at the non-relativistic ground state.