Variational matrix Padé approximations in potential scattering and low-energy Lagrangian field theory

Abstract

In the first part, we develop in potential scattering theory the method of operator Padé approximations (OPA), starting from the variational principle for phase shifts. We show that the lowest approximation, constructed from the first two terms of the perturbative expansion of the Green function, is, for the Schrödinger equation, the exact solution of the problem. However the practical calculation of the OPA requires the inversion of an operator. This inversion is achieved by discretizing the Hilbert space: the OPA is replaced by an MPA (matrix Padé approximation). The “best choice” of the discrete basis of the Hilbert space is given, at each energy, angular momentum, parity, and internal quantum numbers, by the variational principle for phase shifts. The quality of the results is such, that, for potentials giving rise to phase-shifts varying over hundreds of degrees, it is possible, using only the first two terms of the Green function perturbative expansion, to reproduce the exact phase shifts within one part in one thousand, at any energy; this result is achieved with a discrete basis involving no more than two to four vectors. New theorems are derived which ensure the reliability of this method. In the second part, we show how this formalism extends naturally to Lagrangian field theory in the low-energy region. As a first application we give preliminary results for nucleon-nucleon scattering. The information coming from the Green function is essential to obtain the existence of the deuteron and dineutron from the Yukawa Lagrangian, without any adjustable parameter or hard core. All the partial waves L ⩾ 2 are correctly reproduced. The zeroes of the S and P wave phase shifts are still missing: this is very likely due to the fact that, in this preliminary calculation, only the spin part of the amplitude is taken off-shell. This effect happens in potential scattering when the momenta are kept on-shell. The complete analysis involving the introduction of the off-shell momenta is in progress. In the conclusion, we show, by giving a simple example, that the variational matrix Padé approximation is no longer a rational fraction in the coupling constant (expansion parameter), but may have the full analyticity structure of the exact solution. It then defines what could be called a hyper-Padé approximant.