Existence of global solutions to the nonlocal Schr\"{o}dinger equation on the line

In this paper, we address the existence of global solutions to the Cauchy problem for the integrable nonlocal nonlinear Schr\"{o}dinger (nonlocal NLS) equation with the initial data $q_0(x)\in H^{1,1}(\R)$ with the $L^1(\R)$ small-norm assumption. We rigorously show that the spectral problem for the nonlocal NLS equation admits no eigenvalues or resonances, as well as Zhou vanishing lemma is effective under the $L^1(\R)$ small-norm assumption. With inverse scattering theory and the Riemann-Hilbert approach, we rigorously establish the bijectivity and Lipschitz continuous of the direct and inverse scattering map from the initial data to reflection coefficients.By using reconstruction formula and the Plemelj projection estimates of reflection coefficients,we further obtain the existence of the local solution and the priori estimates, which assure the existence of the global solution to the Cauchy problem for the nonlocal NLS equation.