Variational calculation of bound and metastable-resonance states by finite difference method.

The standard finite difference method (FDM) enables calculation of the bound states of a quantum system, which constitute the real poles of the scattering matrix. However, this method is not applicable for the calculation of the complex poles which are associated with metastable (resonance) states. The origin of this failure can be traced back to the fact that the standard FDM does not satisfy the variational principle. Here we show that a simple change in the selection of the grid points leads to a variational principle and enables calculation of both real and complex poles of the scattering matrix. This approach opens the gate to evaluate the resonances by FDM for atoms and molecules as well as mesoscopic systems. Illustrative numerical examples are given.