  The transparent boundary condition for the free Schr\"{o}dinger equation on a
rectangular computational domain requires implementation of an operator of the
form $\sqrt{\partial_t-i\triangle_{\Gamma}}$ where $\triangle_{\Gamma}$ is the
Laplace-Beltrami operator. It is known that this operator is nonlocal in time
as well as space which poses a significant challenge in developing an efficient
numerical method of solution. The computational complexity of the existing
methods scale with the number of time-steps which can be attributed to the
nonlocal nature of the boundary operator. In this work, we report an
effectively local approximation for the boundary operator such that the
resulting complexity remains independent of number of time-steps. At the heart
of this algorithm is a Pad\'e approximant based rational approximation of
certain fractional operators that handles corners of the domain adequately. For
the spatial discretization, we use a Legendre-Galerkin spectral method with a
new boundary adapted basis which ensures that the resulting linear system is
banded. A compatible boundary-lifting procedure is also presented which
accommodates the segments as well as the corners on the boundary. The proposed
novel scheme can be implemented within the framework of any one-step time
marching schemes. In particular, we demonstrate these ideas for two one-step
methods, namely, the backward-differentiation formula of order 1 (BDF1) and the
trapezoidal rule (TR). For the sake of comparison, we also present a
convolution quadrature based scheme conforming to the one-step methods which is
computationally expensive but serves as a golden standard. Finally, several
numerical tests are presented to demonstrate the effectiveness of our novel
method as well as to verify the order of convergence empirically.
