Finite dimensional global attractor for a semi-discrete nonlinear Schrödinger equation with a point defect

We consider a semi-discrete in time Crank-Nicolson scheme to discretize a weakly-damped forced nonlinear Schrödinger equation with a delta-function impurity in one space dimension. We prove that such a semi-discrete equation provides a discrete infinite dimensional dynamical system in H1(R)H1(R) that possesses a global attractor in H1(R)H1(R). We show also that this global attractor is actually a compact set of H32-ε(R) and has a finite fractal dimension.