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We prove that positive solutions $u\in H^s(\mathbb{R}^N)$ to the equation$(-\Delta )^s u+ u=u^p$ in $\mathbb{R}^N$ are unique up to translations andnondegenerate, for all $s\in (0,1)$, $N\geq 1$ and $p>1$ is strictly smallerthan the critical Sobolev exponent. This generalizes a result of Frank,Lenzmann and Silvestre \cite{FLS}, where the same uniqueness and nondegeneracyis proved for solutions with Morse index 1. Letting $B$ be the unit centeredball and $\lambda_1(B)$ the first Dirichlet eigenvalue of the fractionalLaplacian $(-\Delta )^s$, we also prove that positive solutions to $(-\Delta)^s u+\lambda u=u^p$ in ${B}$ with $u=0$ on $\mathbb{R}^N\setminus B$, areunique and nondegenerate for any $\lambda> -\lambda_1(B)$. This extends thevery recent results in \cite{DIS-1,Azahara-Parini} which provide uniqueness andnondegeneracy of least energy solutions.

Uniqueness and nondegeneracy of solutions to $(-\Delta )^su+ u=u^p$ in $\mathbb{R}^N$ and in balls