Siegert State Approach to Quantum Defect Theory

Abstract

The Siegert states are approached in framework of Bloch-Lane-Robson formalism for quantum collisions. The Siegert state is not described by a pole of Wigner R- matrix but rather by the equation $1-R_{nn}{L}_n=0$, relating R- matrix element $R_{nn}$ to decay channel logarithmic derivative $L_n$. Extension of Siegert state equation to multichannel system results into replacement of channel R- matrix element $R_{nn}$ by its reduced counterpart ${\mathcal{R}}_{nn}$. One proves the Siegert state is a pole, $(1-{\mathcal{R}}_{nn}{L}_n{)}^{-1}$, of multichannel collision matrix. The Siegert equation $1-{\mathcal{R}}_{nn}{L}_n=0$, ($n$ - Rydberg channel), implies basic results of Quantum Defect Theory as Seaton's theorem, complex quantum defect, channel resonances and threshold continuity of averaged multichannel collision matrix elements.