Association of resonance states with the incomplete spectrum of finite complex-scaled Hamiltonian matrices

The incomplete spectrum of finite complex-scaled Hamiltonian matrices H_η is studied. It is pointed out that the occurrence of an incomplete spectrum of complex-scaled Hamiltonians in the finite-element approximation is neither accidental nor rare, and the existence of a defective eigenvector (orthogonal to itself) of H_η can be associated with a complex-stationary point which represents the resonance state. A physical interpretation of the incomplete spectrum (the eigenvalues of a defective Hamiltonian matrix) is given, supported by numerical results for e^--He⁺ scattering worked out as an example. The numerical procedure suggested here for the purpose of identifying the resonance state with the eigenvalue associated with the defective eigenvector of H_η, may prove to be not very practical. This is so as long as only relatively small basis sets are used. However, in the finite-element approximation, this procedure does yield a better understanding of the behavior of the resonance solution.