Finite dimensional global attractor for a fractional nonlinear Schrodinger equation

We consider a weakly damped forced nonlinear fractional Schrodinger equation u(t) - i(-Delta)(alpha)u + i vertical bar u vertical bar(2)u + gamma u = f for a given alpha is an element of(1/2, 1) considered in the whole space R. We prove that this equation provides an infinite dimensional dynamical system in H (alpha) (R) that possesses a global attractor in H (alpha) (R) that is regular. We show also that if the external force is in a suitable weighted space then this global attractor has finite fractal dimension.