Bounds on Energy Levels and Lifetimes of Auto-Ionizing States

Upper and lower bounds on the energy level of an auto-ionizing state are derived by a method which is a generalization of the Stevenson-Crawford formula for the lower bound on the ground state of a system. These bounds are nonvariational in character, although the lower bound can be used together with a variational upper bound to obtain a narrower gap between upper and lower bound. In addition, an upper bound on the transition rate for the decay of the auto-ionizing state is derived. Although primarily designed to yield information for more than two-electron systems, for which the Hahn, O'Malley, and Spruch version of the Feshbach projection operators cannot explicitly be written down, the methods are here applied to auto-ionizing states in helium below the $n=2$ threshold. These bounds are found to be a far more sensitive test of the adequacy of an approximate state than is the conventional energy criterion used in conjunction with the variational approach.