  In this paper, a linearized semi-implicit finite difference scheme is
proposed for solving the two-dimensional (2D) space fractional nonlinear
Schr\"{o}dinger equation (SFNSE).The scheme has the property of mass and energy
conservation on the discrete level, with an unconditional stability and a
second order accuracy for both time and spatial variables. The main
contribution of this paper is an optimal pointwise error estimate for the 2D
SFNSE, which is rigorously established and proved for the first time. Moreover,
a novel technique is proposed for dealing with the nonlinear term in the
equation, which plays an essential role in the error estimation. Finally, the
numerical results confirm well with the theoretical findings.
