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

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 Schr\"odinger equation. It is then applied to the E1 excitation of $^6$He in an $\alpha + n + n$ 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.