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.

Figures (7)

• Figure 1: (Color online) Panel (A): Integration contour $C$ in the complex energy-$E$ plane. The contour $C$ consists of the paths $L = L_1 + L_2$ along the rotated branch cuts (I) and (II), and the outer arcs $D = D_1 + D_2$. The branch cuts are rotated by an angle $2\theta$ from the real axis. The crosses ($\times$) denote the poles of the $S$-matrix (or Green's function), where the poles in the fourth quadrant represent resonance states uncovered by the complex scaling. Panels (B) and (C). Mapping of the integration contours and discrete poles onto the complex momentum planes $k_1^{\theta}(E)$ and $k_2^{\theta}(E)$. The figures illustrate the transformation of the branch cuts and the integration paths from the $E$-plane to the $k$-plane under complex scaling. The rotation angle $\theta$ in the momentum plane corresponds to the $2\theta$ scaling in the energy plane. Crosses ($\times$) denote the discrete eigenvalues, including bound and resonance states, captured within the deformed contours $L$ and $D$. The mapped branch cuts (I) and (II) are shown as blue and green lines, respectively.
• Figure 2: (Color online) Contour plots of $\log |\det \hbox{\boldmath $\mathcal{J}$}_{lj}^{(+)}(E)|$ for the $1d_{3/2}$ state on the first Riemann sheet of the complex energy plane, calculated with the pairing gap $\langle \Delta \rangle = 3.0$ MeV and chemical potential $\lambda = -1.0$ MeV. Panel (a) shows the result without complex scaling ($\theta = 0.0$), while panels (b) and (c) show the results with $\theta = 0.1$ for the original and dual bases, respectively. The dashed lines starting from $E = -\lambda$ represent the branch cuts, which are rotated by $-2\theta$ in (b) and $+2\theta$ in (c). The smooth behavior of the contour lines approaching the branch cuts provides the basis for the analytical connection to the second Riemann sheet shown in Fig. \ref{['fig_mod2-detJ1']}.
• Figure 3: (Color online) Same as Fig. \ref{['fig_mod1-detJ1']}, but on the second Riemann sheet. The resonance pole of the $1d_{3/2}$ state is clearly identified as a distinct zero point (dark region) at $E \approx 4.94 - i0.25$ MeV, which is consistent with the values listed in Tables \ref{['table2']} and \ref{['table3']}. Comparison with Fig. \ref{['fig_mod1-detJ1']} through the contour consistency across the branch cuts demonstrates the seamless analytical connection between the two Riemann sheets. The position of the resonance pole remains strictly identical across panels (a), (b), and (c), numerically confirming the $\theta$-invariance of the resonance poles as analytically described by Eq. (\ref{['invarianceJost']}).
• Figure 4: (Color online) Radial wave functions $\varphi_{1}^\theta(r)$ and $\varphi_{2}^\theta(r)$ for the p-type quasiparticle resonance state of $1f_{7/2}$ (pole No. 2 in Table \ref{['table2']} and \ref{['table3']}). The top panel shows the wave functions in the single-particle limit ($\langle \Delta \rangle = 0.0$ MeV, $\theta = 0.0$). The bottom-left and bottom-right panels show the results with the pairing correlation ($\langle \Delta \rangle = 3.0$ MeV) for the unscaled ($\theta = 0.0$) and complex-scaled ($\theta=0.3$) cases, respectively. The complex scaling regularizes the divergent behavior of the resonance wave function at large distances, ensuring its convergence as $r \to\infty$.
• Figure 5: (Color online) Radial wave functions $\varphi_{1}^\theta(r)$ and $\varphi_{2}^\theta(r)$ for the h-type quasiparticle resonance state of $2s_{1/2}$ (pole No. 4 in Table \ref{['table2']} and \ref{['table3']}). The parameters and panel layout are the same as in Fig.\ref{['fig-wf_f7-2']}. For the h-type resonance, the lower component $\varphi_{2}^\theta$ is dominant, and its non-integrable behavior at $\theta = 0.0$ is effectively regularized to a square-integrable form by the complex scaling at $\theta = 0.3$.