  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.
