Some properties of the resonant state in quantum mechanics and its computation

The resonant state of open quantum systems is studied from the viewpoint of the eigen-function with an outgoing momentum flux. We show that the number of particles is conserved for a resonant state if we use an expanding volume of integration in order to take account of the outgoing momentum flux; the number of particles in a fixed volume of integration would decay exponentially. Moreover, we introduce new numerical methods of treating the resonant state with the use of an effective potential. We first present a numerical method for finding a resonance pole in the complex energy plane. This method seeks an energy eigen-value iteratively. We found that it leads to super-convergence, i.e., convergence whose rate is exponential with respect to the iteration step. Also, it is independent of the commonly used complex scaling. We also present a numerical trick for computing the time evolution of the resonant state in a limited spatial area. Because the wave function of the resonant state is diverging away from the scattering potential, it is difficult to follow its time evolution numerically in a finite area using previous methods.