Lanczos recursion, continued fractions, Padé approximants, and variational principles in quantum scattering theory

Abstract

Connections are established between the Schwinger and Newton variational principles and recursive generation of the remainder in Born series expansions of individual K-matrix elements. It is shown that Lanczos development of the remainder yields results identical to either of these variational principles, depending upon the starting vectors that are used to initiate the recursion sequence. In all cases, the correction to the Born series is computed from the 1,1 element of the inversion of a small tridiagonal matrix. The relationship to Padé approximants and continued fractions is also noted. Numerical results on the convergence of elastic and inelastic K-matrix elements are presented for a model system.