We prove uniqueness of ground state solutions Q = Q(|x|)≥0 for the nonlinear equation (−Δ)^sQ + Q − Q^(α+)1 = 0 in R, where 0 < s < 1 and 0 < α < _(4s) ^(1−2s) for s < 1/2 and 0 < α < ∞ for s ≥ 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.,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 ker L_+ = 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
