Robin initial-boundary value problem for nonlinear Schrodinger equation with potential

We study the initial-boundary value problem for the Schrodinger equation with potential, posed on positive half-line t u + ia1. xx u + ia2| u| 2u + a3P 1 2 u = 0, where a1. R, a2. C, a3 > 0, P 1 2 is the fractional derivative operator defined by the modified Risz potential P 1 2 u = 1 2p 8 0 dy sign ( x - y) | x - y| 1 2. yu ( y, t). We are interested in the initial-boundary value problem with small initial data and Robin boundary data given in a suitable weighted Sobolev spaces. We show that problem admits global solutions whose long-time behavior essentially depends on the boundary data.