On one-dimensional discrete variable representations with general basis functions

Abstract

The method of discrete variable representation (DVR) is based on standard orthogonal polynomial bases and the associated Gaussian quadratures. General basis functions correspond either to nonpolynomial expressions or to nonstandard orthogonal polynomials. Although one cannot directly relate any Gaussian quadrature to general basis functions, the DVR-like representation derivable with such basis sets via the transformation (diagonalization) method is, as proved here, of Gaussian quadrature accuracy. The optimal generalized DVR (GDVR) is an alternative to and entirely different from this DVR-like representation. Yet, when built from the same general basis functions and the corresponding quadrature points obtained by the diagonalization method, the two representations are found to give almost identical numerical results. The intricate relationship between the optimal GDVR and the transformation method is discussed.