Circulants, symmetry and time-independent wavepacket scattering

Time-independent implementations of wavepacket scattering theory usually require iterative solution of a linear system Ax=b, in which the primary representation is often a spatial grid. We show that for Fourier basis functions, while A is complex-symmetric on a spatial grid, it will generally not be so in the corresponding spectral representation, since the persymmetry properties of the grid-based potential matrix govern the spectral symmetry properties of A. For an illustrative test-case, correct scattering information is yielded on the grid using the symmetric quasi-minimal residual (QMR) method and spectrally via unsymmetric transpose-free QMR.