Uniqueness and nondegeneracy of ground states for $(-\mathrm{\Delta }{)}^{s}u+u=2(I_2\star u^2)u$ in ${\mathbb{R}}^N$ when $s$ is close to 1

In this article, we study the uniqueness and nondegeneracy of ground states to a fractional Choquard equation of the form: $(-\mathrm{\Delta }{)}^{s}u+u=2(I_2\star u^2)u$ where $s\in (0,1)$ is sufficiently close to $1$. Our method is to make a continuation argument with respect to the power $s\in (0,1)$ appearing in $(-\mathrm{\Delta }{)}^s$. This approach is based on [M. M. Fall and E. Valdinoci, Comm. Math. Phys., 329 (2014) 383-404].