  We introduce a numerical method for the solution of the time-dependent
Schrodinger equation with a smooth potential, based on its reformulation as a
Volterra integral equation. We present versions of the method both for periodic
boundary conditions, and for free space problems with compactly supported
initial data and potential. A spatially uniform electric field may be included,
making the solver applicable to simulations of light-matter interaction.
  The primary computational challenge in using the Volterra formulation is the
application of a space-time history dependent integral operator. This may be
accomplished by projecting the solution onto a set of Fourier modes, and
updating their coefficients from one time step to the next by a simple
recurrence. In the periodic case, the modes are those of the usual Fourier
series, and the fast Fourier transform (FFT) is used to alternate between
physical and frequency domain grids. In the free space case, the oscillatory
behavior of the spectral Green's function leads us to use a set of
complex-frequency Fourier modes obtained by discretizing a contour deformation
of the inverse Fourier transform, and we develop a corresponding fast transform
based on the FFT.
  Our approach is related to pseudo-spectral methods, but applied to an
integral rather than the usual differential formulation. This has several
advantages: it avoids the need for artificial boundary conditions, admits
simple, inexpensive high-order implicit time marching schemes, and naturally
includes time-dependent potentials. We present examples in one and two
dimensions showing spectral accuracy in space and eighth-order accuracy in time
for both periodic and free space problems.
