A Unified Theoretical Framework for HFB Resonant States: Integration of the Complex-Scaled Jost Function and Autonne-Takagi Normalization

Abstract

We develop a theoretical framework to describe quasiparticle resonance states within the Hartree-Fock-Bogoliubov (HFB) theory by integrating the complex-scaled Jost function method with the Autonne-Takagi factorization. The HFB completeness relation is derived from the analytical properties of the Green's function using contour integration in the complex energy plane, where the complex scaling method (CSM) is shown to be essential for explicitly separating resonance pole contributions from the continuum background. To uniquely define and normalize the resonant wave functions (Gamow states), the Autonne-Takagi factorization is applied to the rank-1 residue matrix of the flux-adjusted S-matrix at the pole energy. This scheme determines the absolute scale and phase of the eigenfunctions without relying on artificial adjustments or phenomenological basis sets. Numerical analysis confirms that physical observables and the defined wave functions remain invariant under the rotation of the complex scaling angle $θ$. Furthermore, the T-matrix residues calculated via the Mittag-Leffler expansion are shown to be in exact numerical agreement with those obtained from the microscopic integrals of the Takagi-normalized Gamow states. Our analysis of the scattering profiles reveals that hole-type quasiparticle resonances can be understood as a manifestation of the Fano process originating from the interference between the discrete poles and the background continuum. The proposed normalization scheme provides a foundation for evaluating the collectivity of various excitation modes in open quantum many-body systems.