INNER PRODUCT FOR RESONANT STATES AND SHELL-MODEL APPLICATIONS

A generalization of the standard inner-product integral is developed, which allows one to treat resonant states on an equal footing with bound states. The resonant and bound states are shown to form an orthonormal set with respect to the generalized inner product. Near a complex resonance energy Es of the Hamiltonian H, the Green function (E+iε-H)-1 is shown to be approximately equal to |s)(s|/(E-Es), where |s) is the normalized (in terms of the new inner product) resonance state. This is exactly the same form as the leading term is the Green function expansion about a bound state pole. This feature can be exploited in a reformulation of the MacDonald or Weidenmüller shell-model theory of nuclear reactions. The new formalism also has possible applications to giant resonance problems, stripping to unstable final state or any other problem where resonances play an important part. © 1968.