Excited bound state logarithmic perturbation theory without nodes

Abstract

The logarithmic perturbation theory is modified slightly in order to deal with excited states. Instead of considering a real wavefunction describing the physical stationary state, the authors consider a complex wavefunction at the same energy, by mixing in the ghost state. For excited bound states, the former has nodes, while the latter is guaranteed not to have any nodes, and can be represented simply as exp(-G), to which the logarithmic perturbation method can be applied in a straightforward manner. The physical entities (the energy corrections) are independent of the amount of mixing of the ghost state. The connection to the Green function method is also shown. The freedom to mix in the ghost state allows the authors to justify an ad hoc approach whereby the simple version of the logarithmic perturbation theory is applied to excited bound states. The formalism is illustrated with simple examples.