On the ε → 0 limit of the Lippmann-Schwinger-Low states

Abstract

The Lippmann-Schwinger-Low (LSL) quantum scattering states involve a resolvent operator depending on an infinitesimal adiabatic parameter ε. We reexamine the LSL formalism by taking the ε → +0 limit at the end of the analysis (rather than at the outset). It is found that the LSL state vector |ψkL〉 does not coincide with the Schrödinger eigen vector in Hilbert space as a whole, and the pair |ψn L〉, |ψkL〉 is mutually nonorthogonal if the energy En = Ek, n ≠ k. For this purpose we carefully use a new type of projection operator ηk, a novel nonlinear relation among transition amplitudes, and a separable interaction as illustration. © 2004 NRC Canada.