Gauss quadrature generated by diagonalization ofHin finiteL2bases

Abstract

Using the $J$ matrix method, diagonalization of $H_0+V$ in certain finite $L^2$ basis sets is shown to generate a Gauss quadrature of the continuum. Explicit formulas for the corresponding weight function and orthogonal polynomials are given. This leads to a particularly simple expression for the Fredholm determinant and to the equivalence of the $J$ matrix and Fredholm equivalent-quadrature methods. Consideration of the asymptotic behavior of the polynomials for large degree results in a proof of Heller's derivative rule relating the spacing of pseudostate eigenvalues to the relative normalization of pseudostate and continuum matrix elements, and providing an alternative to Stieltjes imaging for the solution of the classical moment problem. Applications include an accurate quadrature of the sum over a complete set of intermediate states in perturbation expansions.