Geodesic
A Criterion for the Existence of Decaying Solutions in the Problem on a Resonator with a Cylindrical Waveguide
Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 2, pp. 20-32.

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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 $\omega$. 
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S. A. Nazarov. A Criterion for the Existence of Decaying Solutions in the Problem on a Resonator with a Cylindrical Waveguide. Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 2, pp. 20-32. http://geodesic.mathdoc.fr/item/FAA_2006_40_2_a2/

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