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 ($C<sup>{(n)}$) and the Collective Phase ($Θ<sup>{(n)}$), which characterize the dynamical phase synchronization and the line-shape orientation, respectively. Furthermore, a unified Total Collectivity Index ($R<sup>{(n)}$) is defined by combining the coherence index with the Normalized Participation Ratio ($η<sup>{(n)}$). Applying this framework to the isoscalar $2^+$, isovector $2^+$, and $E1$ excitations in $<sup>{16}$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.

• The paper introduces a novel framework that formally defines intrinsic collectivity in open quantum systems via Takagi factorization of the Jost-RPA S-matrix residue.
• It quantifies collectivity with intrinsic indices such as coherence, participation ratio, and collective phase, decoupling the intrinsic structure from extrinsic strength functions.
• Numerical analysis in 16O validates the method by revealing hidden collective modes that exhibit high intrinsic synchronization even without prominent observable peaks.

Intrinsic Collectivity in the Continuum: A Formal Definition via Takagi Factorization of the Jost-RPA S-matrix Residue

Introduction

The paper "Formal definition of intrinsic collectivity in the continuum via Takagi factorization of the Jost-RPA S-matrix residue" (2604.02237) addresses a fundamental challenge in the characterization of collective excitations in open quantum systems, particularly atomic nuclei near the drip lines. Classical approaches, relying predominantly on the prominence and symmetry of peaks in strength functions (e.g., energy-weighted sum rules), often fail to distinguish between true collective modes and non-collective configurations that happen to manifest as sharp features due to phase interference. The work presents a structural framework grounded in the microscopic decomposition of resonance residues, decoupling extrinsic line shape from intrinsic collectivity and enabling a rigorous classification of many-body excitations in the continuum.

Methodological Framework

The central innovation lies in the integration of Takagi factorization with the Jost-RPA formalism. The Jost-RPA extension allows for the direct computation of the Green's function and S-matrix in the continuum based on scattering wavefunctions, yielding a complex-symmetric S-matrix whose residue at a resonance pole is rank-1. Takagi factorization is uniquely suited for decomposing such complex-symmetric rank-1 matrices into a single complex vector, associating microscopic transition amplitudes to individual configurations at each resonance pole.

This decomposition formally generalizes the configuration-space analysis of discrete RPA to the continuum, allowing explicit extraction of configuration amplitudes ($F_i^{(n)}$) for each particle-hole (ph) arrangement at a given pole $E_n$. The framework defines several indices:

• Intrinsic Coherence Index ($C^{(n)}$): Quantifies dynamical phase synchronization among configuration amplitudes. Perfect alignment yields $C^{(n)} = 1$.
• Collective Phase ($\Theta^{(n)}$): The argument of the total pole amplitude, governing the orientation and shape of the observed strength function.
• Normalized Participation Ratio ($\eta^{(n)}$): Measures the effective fraction of configurations contributing substantially, capturing the structural scale irrespective of phase.
• Total Collectivity Index ($R^{(n)}$): $L_2$ norm combination of $C^{(n)}$ and $\eta^{(n)}$, providing a unified quantification of collectivity based on both phase coherence and participation scale.

The formalism demonstrates that the strength-function profile—whether symmetric (Breit-Wigner), asymmetric, or a dip—can arise from a variety of intrinsic structures, and that intrinsic collectivity is largely independent of extrinsic manifestation.

Numerical Analysis in $E_n$0O

The methodology was applied to isoscalar and isovector $E_n$1 as well as $E_n$2 excitations in $E_n$3O using a Woods-Saxon potential and residual interaction. Microscopic indices were computed for representative resonance poles, enabling comparison between intrinsic coherence and observable strength-function behavior.

• Isoscalar $E_n$4 Channel: Pole (1) at $E_n$5 MeV displays high collectivity ($E_n$6, $E_n$7), matching the isoscalar giant quadrupole resonance (ISGQR) and manifesting as a symmetric peak. Poles (4) and (5), with negative real residues and minimal coherence ($E_n$8, $E_n$9), correspond to dips with no collective structure. Higher-energy poles (8), (9) are visually symmetric but intrinsically non-collective, driven by fortuitous phase alignment rather than genuine synchronization.
• Isovector $C^{(n)}$0 Channel: Collectivity is distributed across several poles, none reaching high intrinsic coherence, and all exhibiting systematic phase rotation yielding asymmetric line shapes. Pole (9) achieves the highest $C^{(n)}$1 and $C^{(n)}$2 in the channel despite its non-symmetric appearance.
• $C^{(n)}$3 Channel: The observable peak structure anticorrelates with actual collectivity. Pole (4) yields the highest extrinsic strength but lowest $C^{(n)}$4, while pole (6), forming the weakest peak, achieves higher $C^{(n)}$5 and $C^{(n)}$6, typifying a "hidden" collective mode. Pole (7), with the greatest $C^{(n)}$7 and $C^{(n)}$8, further supports the conclusion that extrinsic prominence is unrelated to intrinsic collectivity.

The results emphasize the necessity of intrinsic indices, exposing the inadequacy of traditional strength-based diagnostics and demonstrating the existence of highly collective modes whose dynamical coherence is masked by phase rotation and interference.

Implications and Future Perspectives

The framework formalizes the structural analysis of resonant collectivity, enabling precise identification of highly synchronized modes regardless of their manifestation in the strength function. This approach is particularly salient for open quantum systems and nuclei near the drip lines, where continuum coupling leads to complex interference effects and non-intuitive observable features.

The theoretical decoupling of phase alignment and participation scale allows for the classification of exotic excitations—such as narrow but synchronized or broad but incoherent modes—facilitating nuanced assessment of many-body dynamics beyond conventional sum-rule exhaustion.

Practically, the method is relevant for experimental interpretation of nuclear resonances, the design of continuum-coupling models, and the exploration of non-Hermitian quantum mechanics in extended systems. The Takagi decomposition is broadly applicable to other open-system scenarios, including atomic and molecular physics, quantum optics, and engineered mesoscopic structures.

Future developments could extend the Jost-RPA-Takagi formalism to multi-pole scenarios, non-symmetric residual interactions, and higher-order correlation effects, providing a basis for systematic studies of collectivity in complex many-body continua.

Conclusion

This study establishes a formal, configuration-resolved definition of intrinsic collectivity in the quantum continuum via Takagi factorization of the Jost-RPA S-matrix residue. The resulting indices rigorously quantify dynamical phase synchronization and participation scale, uncoupling intrinsic collectivity from extrinsic strength-function features. The findings confirm that highly collective modes may lack observable prominence, necessitating intrinsic structural analysis for reliable classification. The framework provides a robust foundation for theoretical and computational investigations of collective phenomena in open quantum systems.