  The split-operator pseudo-spectral method based on the fast Fourier transform
(SO-FFT) is a fast and accurate method for the numerical solution of the
time-dependent Schr\"odinger-like equations (TDSE). As well as other grid-based
approaches, SO-FFT encounters a problem of the unphysical reflection of the
wave function from the grid boundaries. Exterior complex scaling (ECS) is an
effective method widely applied for the suppression of the unphysical
reflection. However, SO-FFT and ECS have not been used together heretofore
because of the kinetic energy operator coordinate dependence that appears in
ECS applying. We propose an approach for the combining the ECS with SO-FFT for
the purpose of the solution of TDSE with outgoing-wave boundary conditions.
Also, we propose an effective ECS-friendly FFT-based preconditioner for the
solution of the stationary Schr\"odinger equation by means of the
preconditioned conjugate gradients method.
