Gaussian flexibility with Fourier accuracy: the periodic von Neumann basis set

We propose a new method for solving quantum mechanical problems, which combines the flexibility of Gaussian basis set methods with the numerical accuracy of the Fourier method. The method is based on the incorporation of periodic boundary conditions into the von Neumann basis of phase space Gaussians [F. Dimler et al., New J. Phys. 11, 105052 (2009)]. In this paper we focus on the Time-independent Schr\"odinger Equation and show results for the harmonic, Morse and Coulomb potentials that demonstrate that the periodic von Neumann method or pvN is significantly more accurate than the usual vN method. Formally, we are able to show an exact equivalence between the pvN and the Fourier Grid Hamiltonian (FGH) methods. Moreover, due to the locality of the pvN functions we are able to remove Gaussian basis functions without loss of accuracy, and obtain significantly better efficiency than that of the FGH. We show that in the classical limit the method has the remarkable efficiency of 1 basis function per 1 eigenstate.