  We consider the scattering problem for the nonlinear Schr\"{o}dinger equation
with a potential in two space dimensions. Appropriate resolvent estimates are
proved and applied to estimate the operator $A(s)$ appearing in commutator
relations. The equivalence between the operators
$\left(-\Delta_{V}\right)^{\frac{s}{2}}$ and $\left(-\Delta
\right)^{\frac{s}{2}}$ in $L^{2}$ norm sense for $0\leq s <1$ is investigated
by using free resolvent estimates and Gaussian estimates for the heat kernel of
the Schr\"{o}dinger operator $-\Delta_{V}$. Our main result guarantees the
global existence of solutions and time decay of the solutions assuming initial
data have small weighted Sobolev norms. Moreover, the global solutions obtained
in the main result scatter.
