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Uniqueness and Nondegeneracy of Ground States for $(-Δ)^s Q + Q - Q^{α+1} = 0$ in $\mathbb{R}$

We prove uniqueness of ground state solutions $Q = Q(|x|) \geq 0$ for the nonlinear equation $(-Δ)^s Q + Q - Q^{α+1}= 0$ in $\mathbb{R}$, where $0 < s < 1$ and $0 < α< \frac{4s}{1-2s}$ for $s < 1/2$ and $0 < α< \infty$ for $s \geq 1/2$. Here $(-Δ)^s$ denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $s=1/2$ and $α=1$ in [Acta Math., \textbf{167} (1991), 107--126]. As a technical key result in this paper, we show that the associated linearized operator $L_+ = (-Δ)^s + 1 - (α+1) Q^α$ is nondegenerate; i.\,e., its kernel satisfies $\mathrm{ker}\, L_+ = \mathrm{span}\, \{Q'\}$. This result about $L_+$ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for nonlinear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.