which obeys the radiation conditions (\[2-5\]) and reasonable boundary condition on a compact surface $S$, identically vanishes. In this case the Laplace operator entering the Helmholtz equation (\[2-3\]) has no eigenfunctions with real eigenvalues.

The physical content of the radiation conditions is very clear. They select only the oscillations with real frequencies driven by external sources which are situated in a compact spatial area. From the mathematical standpoint, these conditions ensure the [*uniqueness*]{} of the solutions to [*inhomogeneous*]{} wave or Helmholtz equations with external sources on the right-hand side, when these solutions are considered in the external region $D_{\text{ex}}$ or in the whole space $D_{\text{in}}+D_{\text{ex}}$ in the case of compound media.

Here the question arises, how to change minimally the conditions in the [*homogeneous*]{} problem at hand in order to get nonzero solutions, i.e., eigenfunctions in unbounded regions. The energy conservation law prompts a simple way to construct nonzero solutions to the homogeneous wave equation (\[2-1\]) or to the Helmholtz equation (\[2-3\]) describing the outgoing waves at spatial infinity, namely, one has to introduce [*complex*]{} frequencies $\omega =
\omega'- i\,\omega '',\quad \omega ''>0$. We may hope that in this case the factor $e^{-\omega ''t}$ will describe the decay of the 