A Criterion for the Existence of Decaying Solutions in the Problem on a Resonator with a Cylindrical Waveguide

For the Helmholtz equation $\Delta u+k^2u=0$ in a domain $\Omega$ with a cylindrical outlet $Q_+=\omega\times\mathbb{R}_+$ to infinity, we construct a fictitious scattering operator $\mathfrak{S}$ that is unitary in $L_2(\omega)$ and establish a bijection between the lineal of decaying solutions of the Dirichlet problem in $\Omega$ and the subspace of eigenfunctions of $\mathfrak{S}$ corresponding to the eigenvalue $1$ and orthogonal to the eigenfunctions with eigenvalues $\lambda_n\le k^2$ of the Dirichlet problem for the Laplace operator on the cross-section $\omega$.