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 eigenfunction 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 eigenvalue 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. (author)