The hammerstein integral equation: a general technique for constructing a rapidly convergent padé‐type approximation to the logarithmic derivative

The radial one‐electron Schrödinger equation can be written as a nonlinear first‐order differential equation by making a suitable logarithmic transformation. The resulting Riccati equation has the equivalent Hammerstein integral representation [1],\documentclass{article}\pagestyle{empty}\begin{document}$$ \beta (r) = \int_{r' = 0}^\infty P(r') N(r,r')dr' \quad 0\buildrel{<}\over{=} r < \infty $$\end{document}where the kernel, N ( r , r ′) is\documentclass{article}\pagestyle{empty}\begin{document}$$ N\left( {r,\,r\prime} \right) = H\left( {r,\,r\prime} \right)\exp \left\{ {\int_{\xi = r\prime}^r {R\left( \xi \right)\beta \left( \xi \right)d\xi } } \right\} $$\end{document}and H ( r , r ′) is the Heaviside unit step function. This kernel is a more general one than that developed in ref. [1]. Both kernels apply in cases where the Riccati equation corresponds to a Sturm–Liouville problem. It is shown that this integral equation can be integrated by parts so that, for any local potential, the integrand decreases as the cyclic folding procedure is applied. During this cyclic folding, the kernel generates an equation that contains only coefficients of β( r ) 0 and β( r ) 1 . Consequently, after truncating at the end of the n th cycle, it is possible to write down a Padé‐type approximation to the logarithmic derivative as a known function of the independent variable. All coefficients in this approximation can be evaluated as simple algebraic formulations of P ( r ), R ( r ), and integrals over P ( r ).