Siegert State Approach to Quantum Defect Theory

TL;DR

This paper presents a Siegert state approach within quantum collision formalism, deriving a new equation that links Siegert states to quantum defect theory, including multichannel systems and resonance phenomena.

Contribution

It introduces a Siegert state equation based on the R-matrix formalism, extending it to multichannel systems and deriving key quantum defect theory results.

Findings

Siegert states are characterized as poles of the multichannel collision matrix.

The new Siegert state equation relates R-matrix elements to decay channel derivatives.

Derivation of quantum defect theory results such as Seaton's theorem and channel resonances.

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 ${\cal R}_{nn}$. One proves the Siegert state is a pole, $(1 - {\cal R}_{nn} L_{n})^{-1}$, of multichannel collision matrix. The Siegert equation $1 - {\cal R}_{nn} L_{n} = 0$, ($n$ - 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.