Variational principle for quasibound states

A minimum-variance principle for quasibound states is defined and utilized to investigate the corresponding energy resonances. The variational principle yields simultaneously a resonance energy E and a square-integrable wave function φ, such that minimum variance is obtained for arbitrary variations of a restricted class of square-integrable functions as well as with respect to variations of E. It is further shown that the optimum energy E₀ obtained from this method can simultaneously be written as an expectation value of the actual Hamiltonian with respect to φ=φ(E₀). The example of the Stark effect in the hydrogen atom is studied. It is shown that the variationally obtained resonance energy coincides with the real part of the complex pole of the m function of Weyl, related to the Green's function of the system under consideration. It is also shown that the corresponding numerical application of the Rayleigh-Ritz variational method only gives meaningful results for field intensities below 0.06 a.u., as compared with the "exact" results of Hehenberger, McIntosh, and Br\"andas.