Time evolution formalism in the complex scaling method: Application to the E1 response of $^6$He

Abstract

Background: The complex scaling method (CSM) has been successfully used to describe many-body resonances as eigenvalues of the complex-scaled Hamiltonian in an appropriate $L^2$ basis representation. Its scope has subsequently been extended to many-body continuum states, strength functions, and scattering observables. However, a general framework that incorporates time evolution within the same CSM framework has not yet been established. Purpose: We formulate a time-evolution formalism as a natural extension of the CSM based on the extended completeness relation (ECR), and apply it to the electric dipole (E1) excitation of $^6$He in order to clarify how an initially correlated three-body configuration evolves into continuum states. Methods: Time evolution is described by a complex-scaled time-evolution operator represented with the ECR. The formalism is first tested in a simple two-body model through comparison with a direct numerical solution of the time-dependent Schrödinger equation. It is then applied to the E1 excitation of $^6$He in an $α+ n + n$ three-body model, and the density distributions are analyzed in different Jacobi coordinate systems. Results: The present formalism reproduces the wave-packet evolution obtained in the direct time-dependent calculation. In the application to $^6$He, the initial E1-excited state exhibits a correlated configuration and evolves into spatially extended continuum states. The time evolution of the density distributions indicates the coexistence of sequential decay through a core-neutron subsystem and direct breakup. Conclusions: The present formalism extends the scope of the CSM from spectral and scattering observables to real-time continuum dynamics, and provides a unified framework that connects initial-state correlations, continuum structure, and decay dynamics in weakly bound nuclei.

Figures (8)

• Figure 1: Distribution of the complex energy eigenvalues for the model Hamiltonian, obtained with 60 Gaussian basis functions whose range parameters are taken in geometric progression from 0.01 fm to 60 fm. The scaling angle is taken to be $\theta=20$ degrees. The open circles represent the discretized eigenvalues obtained with the $L^2$ basis, and the solid line indicates the $2\theta$-rotated branch cut. The isolated pole corresponds to the resonance at $E_r-i\Gamma/2 = 0.377 - 0.172\,i$ MeV.
• Figure 2: Distribution of the complex energy eigenvalues for the $1^-$ states of $^{6}$He obtained with the scaling angle $\theta = 35$ degrees. The open circles represent the discretized eigenvalues obtained with the $L^2$ basis. The dashed, dotted, and dash-dotted lines indicate the $2\theta$-rotated branch cuts corresponding to the $\alpha+n+n$, $^{5}$He$(3/2^-)+n$, and $^{5}$He$(1/2^-)+n$ channels, respectively. No isolated resonance poles are found for the present $1^-$ calculation.
• Figure 3: Schematic illustration of the Jacobi coordinate systems used in the present work.
• Figure 4: Time evolution of the radial density distribution $\rho(r,t)=|u(r,t)|^2$ for the test calculation. The solid black line indicates the initial wave packet at $ct = 0~\mathrm{fm}$. The results obtained using the present time-evolution formalism are shown by open red circles, closed green circles, and blue squares at $ct = 20~\mathrm{fm}$, $40~\mathrm{fm}$, and $80~\mathrm{fm}$, respectively. These results are compared with those obtained by the Crank--Nicolson (CN) method, shown by the dashed red, dotted green, and dash-dotted blue lines at the corresponding times. The density distribution is given in units of fm$^{-1}$.
• Figure 5: Time dependence of the integrated norm $N(t)$ in the test calculation. The solid red line, open green circles, and blue crosses show the results for $\theta=10$, $15$, and $20$ degrees, respectively, and the dotted black line indicates $N(t)=1$.