The origin of an artifact known as the appearance of ghost states in mapped Fourier grid methods is investigated. It was found that the ghost states can be attributed to under sampling of the high momentum components which are folded from the inner to the outer region of the potential to create the ghosts. The effect was corrected by addition of a complex potential at the outer region. The exterior complex potential was shown to shift the ghost states to the continuum part of the spectrum in a controllable way. The various methods to improve the mapped grid method are discussed in this context, and the use of zero boundary conditions is shown to be not essential. (c) 2006 Elsevier B.V. All rights reserved.