  We propose and analyze an extended Fourier pseudospectral (eFP) method for
the spatial discretization of the Gross-Pitaevskii equation (GPE) with low
regularity potential by treating the potential in an extended window for its
discrete Fourier transform. The proposed eFP method maintains optimal
convergence rates with respect to the regularity of the exact solution even if
the potential is of low regularity and enjoys similar computational cost as the
standard Fourier pseudospectral method, and thus it is both efficient and
accurate. Furthermore, similar to the Fourier spectral/pseudospectral methods,
the eFP method can be easily coupled with different popular temporal
integrators including finite difference methods, time-splitting methods and
exponential-type integrators. Numerical results are presented to validate our
optimal error estimates and to demonstrate that they are sharp as well as to
show its efficiency in practical computations.
