On the epsilon - 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 |ψ kL > does not coincide with the Schrödinger eigen vector in Hilbert space as a whole, and the pair |ψ nL >, |ψ kL > 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. PACS Nos.: 0.3.65.Nk, 0.3.80.+r