Variational Principle for Resonant States and its Application to the Spherical Delta Shell Potential.

The self energy for a decaying system is derived from Liouvillian Green's function and the transition energy shift and width are related to the self energy through the second order perturbation expansion. The functional variational technique is applied to the three dimensional spherical delta shell potential in order to find the resonant wave equation, the resonance poles for the decaying states, the corresponding initial state resonant wavefunctions and probability current densities. The time dependent decaying state wavefunctions and current densities evolving from these initial states are obtained by solving the time dependent Schrodinger equations for the decaying system numerically.