Rational fraction representations of the energy: A generalised Rayleigh-Schrodinger perturbation theory

The energy function of a perturbed quantum system is derived directly as a quotient of the form N( lambda )/D( lambda ). The Taylor series coefficients of N( lambda ) and D( lambda ) are found to satisfy certain relations, but there remain many unspecified degrees of freedom which may be freely exploited. Certain choices of coefficients lead to well known Pade and lesser known Levin approximants, but the general formalism includes many other rational function forms. In particular, the present procedure allows information other than the Taylor series coefficients to be included naturally, and it is shown to lead to low-order approximants of high accuracy.