Complex resonance as a phenomenon described by a self-consistent adjoint boundary value problem

A correspondence is established between the properties of a complex resonance and the singularities of non-self-adjoint self-consistent adjoint boundary-value problems. A phenomenon classified as a «complex resonance» takes place in electrodynamic structures described by non-self-adjoint boundary-value problems. It is shown that in the case of pair excitation of complex waves having complex conjugation in wave numbers and amplitudes, a vibration «attached» to the source is formed, which is described by the adjoint Helmholtz equation, that is, the equation on the right side of which is the solution of the homogeneous boundary value problem. A feature of complex resonance is that it exists in the entire range of complex waves with the obligatory presence of a source.