  We construct numerical basis function sets on a lattice, whose spatial
extension is scalable from single lattice sites to the continuum limit. They
allow us to compute small and large bound states with comparable, moderate
effort. Adopting concepts of discrete variable representations, a diagonal form
of the potential term is achieved through a unitary transformation to Gaussian
quadrature points. Thereby the computational effort in three dimensions scales
as the fourth instead of the sixth power of the number of basis functions along
each axis, such that it is reduced by two orders of magnitude in realistic
examples. As an improvement over standard discrete variable representations,
our construction preserves the variational principle. It allows for the
calculation of binding energies, wave functions, and excitation spectra. We use
this technique to study central-cell corrections for excitons beyond the
continuum approximation. A discussion of the mass and spectrum of the yellow
exciton series in the cuprous oxide, which does not follow the hydrogenic
Rydberg series of Mott-Wannier excitons, is given on the basis of a simple
lattice model.
