
Author(s):
N. Hatano, K. Sasada, H. Nakamura and T. Petrosky 
Title:
Some Properties of the Resonant in Quantum Mechanics and its Computation
Date of publication:
Aug. 2007

Key words:
Resonance, open quantum system, complex eigenvalue, effective potential

Abstract:
The resonant state of the open quantum system is studied from the viewpoint
of the 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 would decay exponentially in a fixed volume of integration.
Moreover, we introduce new numerical methods of treating the resonant state
with the use of the effective potential. We first give a numerical method of
finding a resonance pole in the complex energy plane. The method seeks an
energy eigenvalue iteratively. We found that our method leads to a
superconvergence, the convergence exponential with respect to the iteration
step. The present method is completely independent of commonly used complex
scaling. We also give a numerical trick for computing the time evolution of
the resonant state in a limited spatial area. Since the wave function of the
resonant state is diverging away from the scattering potential, it has been
previously difficult to follow its time evolution numerically in a finite
area.

