It remains a difficult task to establish contact between the rigorous quantum theory built on sound foundations by Birkhoff and von Neumann and others and the models usually used to describe resonances. A theorem is proved which gives a set of necessary and sufficient conditions for the existence of solutions of formal perturbation and kindred equations in atomic physics and establishes certain uniqueness properties of solutions which exist. In another theorem it is proved that whenever a state vector belonging to a point eigenvalue of the Hamiltonian is perturbed, one has the phenomenon of spectral concentration and it is possible that in cases where the resulting spectral concentration does not constitute a point eigenvalue of the perturbed Hamiltonian, one has the phenomenon which probably corresponds to a resonance. It is shown that in the autoionising 2s2p^1.3P states of helium, a solution of the formal perturbation equations exists in the underlying Hilbert space only in the zero order, though a formal solution which is outside the underlying Hilbert space exists and belongs to the space of distributions on a subset of the Hilbert space. It is also shown that in cases where a complex eigenvalue is obtained by interfering with a self-adjoint operator, the phenomenon is not related to deficiency indices. The mathematical background of the work of Sharma and Bowtell (1975) has been examined and some progress has been made towards finding the rigorous mathematical framework in which the theory should find a proper formulation.