Balslev-Combes theorem within the framework of the finite-matrix approximation

The Balslev-Combes theorem states that upon scaling the internal degrees of freedom in the Hamiltonian by a complex factor =exp(i), the resonances are isolated from the other states in the continuum, which are rotated into the lower half complex-energy plane by the angle of 2. On the basis of this theorem, the resonance positions and lifetimes in atomic, molecular, and nuclear systems have been calculated. It is reported here that, due to the finite-matrix approximation, the rotating continuum, Ec(), shows a structure when Im[Ec()] is plotted versus Re[Ec()]. This structure is a fingerprint of the resonances. Moreover, the density of states of the rotating continuum, c, does not show the expected monotonic behavior as Ec is increased but has holes i.e., a drastic decrease of when Ec()=Eres. These holes gradually disappear as and/or the basis set is increased. An illustrating numerical example is given.