A criterion for the existence of decaying solutions in the problem on a resonator with a cylindrical waveguide

Abstract

Listen

Translate article

For the Helmholtz equation Δu + k2u = 0 in a domain Ω with a cylindrical outlet Q+ = ω × ℝ+ to infinity, we construct a fictitious scattering operator \(\mathfrak{S}\) that is unitary in L2(ω) and establish a bijection between the lineal of decaying solutions of the Dirichlet problem in Ω and the subspace of eigenfunctions of \(\mathfrak{S}\) corresponding to the eigenvalue 1 and orthogonal to the eigenfunctions with eigenvalues λn ≤ k2 of the Dirichlet problem for the Laplace operator on the cross-section ω.