Non-orthogonality of Lippmann-Schwinger-Low states

Abstract

For the general short-ranged potential the Lippmann-Schwinger-Low (LSL) scattering formalism by taking the limit <img src='/image/spc_char/italics_e.gif' border=0>→+0 at the end of the analysis, with <img src='/image/spc_char/italics_e.gif' border=0> being an infinitesimal adiabatic parameter has been re-examined. It is found that the LSL state |<img src='/image/spc_char/phi.gif' border=0>^L_k)does not strictly satisfy the Schrodinger eigen equation, and the pair |<img src='/image/spc_char/phi.gif' border=0>^L_n),|<img src='/image/spc_char/phi.gif' border=0>^L_k) is mutually non-orthogonal if <i style="">E_n</i> = <i style="">E_k</i>, n ≠ <i>k</i>. For this purpose a new type of projection operator <img src='/image/spc_char/Etta.gif' border=0>⁰_k, a non-linear relation among transition amplitudes, and a separable interaction as illustration have been used.