Uniqueness and Nondegeneracy of ground states of -Δu + (-Δ)^s u+u = u^{p+1} \quad \hbox{in$\mathbb{R}$^n} when $s$ is close to $0$ and $1$

We are concerned with the mixed local/nonlocal Schrödinger equation \begin{equation} - Δu + (-Δ)^s u+u = u^{p+1} \quad \hbox{in ${\mathbb{R}}^n$,} \end{equation} for arbitrary space dimension $n⩾1$, $s\in (0,1)$, and $p\in (0,2^{\ast }-2)$ with $2^{\ast }$ the critical Sobolev exponent. We provide the existence and several fundamental properties of nonnegative solutions for the above equation. And then, we prove that, if $s$ is close to $0$ and $1$, respectively, such equation then possesses a unique (up to translations) ground state, which is nondegenerate.