Completeness of states, shadow states, Heisenberg condition, and poles of the S-matrix

A study of the connection between poles of the S-matrix and states of the Hamiltonian of non-relativistic quantum mechanical systems is made with a view to elucidate the concept of shadow states which have been used by one of the authors for the elimination of divergences in quantum field theory with the aid of an indefinite metric. By specific examples we demonstrate that there exist non-relativistic systems for which the S-matrix has poles which correspond to shadow (redunant) states which are not needed in the completeness relation. Systems with such states do not fulfill a condition on S-matrix which was derived by Heisenberg. It is further shown that there exist phase equivalent systems in which these very poles of the S-matrix correspond to genuine bound states which are absolutely necessary to complete the set of states of the system, and these discrete states may have positive or negative norm depending on the choice of the S-matrix