The exponent of 1D Laplacean (interpreted as the free propagator of
quantum mechanics) is given an approximate matrix (or rather
algorithmic) form of an amendment of the well-known Fourier
pseudospectral approximation where $O(N \log N)$ operations of the
Fast Fourier Transform (FFT) ``diagonalize" the operator. In the new
method the fully discrete Fourier transforms are partially evaluated
explicitly in terms of the error function. In the error analysis the
accuracy of the two approximations are compared and the issues like
stability and conservation are also considered. For linear and
nonlinear Schrodinger equations a few numerical experiments are
presented.

MR2560826 Nash, Patrick L.; Weideman, J. A. C. High accuracy
representation of the free propagator. Appl. Numer. Math. 59 (2009),
no. 12, 2937--2949. 81Q05 (65F30)