Upper and lower bounds in nonrelativistic scattering theory

• Published in: Journal of mathematical physics 1980-09, Vol.21 (9), p.2398-2405
• Main Authors: Darewych, J. W., Pooran, R.
• Format: Article
• Language: English
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.524677

We consider the problem of determining rigorous upper and lower bounds to the difference between the exact and approximate scattering phase shift, for the case of central potential scattering. The present work is based on the Kato identities and the phase‐amplitude formalism of potential scattering developed by Calogero. For nonstationary approximations, a new first‐order (in small quantities) bound is established which is particularly useful for partial waves other than s waves. Similar, but second‐order, bounds are established for approximations which are stationary. Some previous results, based on the use of the Lippman–Schwinger equation are generalized, and some new bounds are established. These are illustrated, and compared to previous results, by a simple example. We discuss the advantages and disadvantages of the present results in comparison to those derived previously. Finally, we present the generalization of some of the present formalism to the case of many‐channel scattering involving many‐particle systems, and discuss some of the difficulties of their practical implementation.