Virtual states in the coupled-channel problems with an improved complex scaling method

We improve the complex scaling method (CSM) to obtain virtual states, which were previously challenging in the conventional CSM. Our approach solves the Schr\"odinger equation in the momentum space as an eigenvalue problem by choosing the flexible contours. It proves to be highly effective in identifying the poles across the different Riemann sheets in the multichannel scatterings. It is more straightforward and efficient than searching for the zeros of the Fredholm determinant of the Lippmann-Schwinger equation using the root-finding algorithms. This advancement significantly extends the capabilities of the CSM in accurately characterizing the resonances and virtual states in quantum systems.