  We propose an exponential wave integrator sine pseudospectral (EWI-SP) method
for the nonlinear Schr\"{o}dinger equation (NLS) with wave operator (NLSW), and
carry out rigorous error analysis. The NLSW is NLS perturbed by the wave
operator with strength described by a dimensionless parameter
$\varepsilon\in(0,1]$. As $\varepsilon\to0^+$, the NLSW converges to the NLS
and for the small perturbation, i.e. $0<\varepsilon\ll1$, the solution of the
NLSW differs from that of the NLS with a function oscillating in time with
$O(\varepsilon^2)$-wavelength at $O(\varepsilon^2)$ and $O(\varepsilon^4)$
amplitudes for ill-prepared and well-prepared initial data, respectively. This
rapid oscillation in time brings significant difficulties in designing and
analyzing numerical methods with error bounds uniformly in $\varepsilon$. In
this work, we show that the proposed EWI-SP possesses the optimal uniform error
bounds at $O(\tau^2)$ and $O(\tau)$ in $\tau$ (time step) for well-prepared
initial data and ill-prepared initial data, respectively, and spectral accuracy
in $h$ (mesh size) for the both cases, in the $L^2$ and semi-$H^1$ norms. This
result significantly improves the error bounds of the finite difference methods
for the NLSW. Our approach involves a careful study of the error propagation,
cut-off of the nonlinearity and the energy method. Numerical examples are
provided to confirm our theoretical analysis.
