Variational Technique for Scattering Theory

A stationary variational functional for the $T$ matrix that uses trial $T$ matrices rather than trial wave functions is discussed. Taking a trial $T$ matrix expressed as a general linear combination of matrices leads to a Fredholm integral equation of the first kind for the coefficient vector. The special case of a trial function having the same form as the Born series, except with variable coefficients, is treated in detail. Requiring the functional to be stationary with this trial form leads to the previously established result of Padé approximants to the scattering amplitude. Approximate techniques are used to evaluate the high-order Born integrals, and the behavior of the Padé approximants and the Born series is investigated for a Yukawa potential. An upper bound on the series is used to estimate its radius of convergence as a function of energy and potential strength. The variational calculations converge rapidly even for cases where the Born series diverges.