Complex scaling spectrum using multiple avoided crossings at stabilization graph

This study concerns finite basis set $\{\chi_k\}$ calculations of resonances based on real scaling, $\chi_k(x)\to \chi_k(xe^{-\eta})$. I demonstrate that resonance width is generally influenced by several neighboring quasi-discrete continuum states. Based on this finding I propose a new method to calculate the complex resonance energy together with several states of complex rotated continuum. The theory is introduced for a one-dimensional model, then it is applied for helium doubly excited resonance $2s^2$. The new method requires the real spectrum ("stabilization graph") for a sufficiently large interval of the parameter $\eta$ on which the potential curve of the sought resonance gradually meets several different quasi-continuum states. Diabatic Hamiltonian which comprehends the resonance and the several quasi-continuum states participating at the avoided crossings is constructed. As $\eta$ is taken to complex plane, $\eta\to i\theta$, the corresponding part of the complex scaled spectrum is obtained.