Formal definition of intrinsic collectivity in the continuum via Takagi factorization of the Jost-RPA S-matrix residue

Abstract

A formal and systematic framework is proposed to quantify the intrinsic collectivity of resonance states in the continuum, independent of their extrinsic manifestation in the strength function. By integrating Takagi factorization into the Jost-RPA framework, we utilize the rank-1 property of the S-matrix residue at a resonance pole to uniquely decompose it into microscopic transition amplitudes for each configuration. To evaluate the nature of these modes, we introduce the Intrinsic Coherence Index ($C^{(n)}$) and the Collective Phase ($Θ^{(n)}$), which characterize the dynamical phase synchronization and the line-shape orientation, respectively. Furthermore, a unified Total Collectivity Index ($R^{(n)}$) is defined by combining the coherence index with the Normalized Participation Ratio ($η^{(n)}$). Applying this framework to the isoscalar $2^+$, isovector $2^+$, and $E1$ excitations in $^{16}$O, we demonstrate that the intrinsic collectivity is decoupled from the observable line shape. Our analysis identifies "hidden" collective modes -- states with high internal synchronization that do not appear as prominent peaks -- and clarifies that distorted structures or dips can either be highly collective or non-collective depending on their microscopic phase alignment. This approach provides a well-defined structural basis for investigating many-body excitations in open quantum systems and nuclei near the drip lines.