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Bound states of waveguides with two right-angled bends. (English) Zbl 1323.35117

Authors’ abstract: We study waveguides in the plane \({\mathbb R}^2\) with two right-angled bends. These waveguides are in the shape of letter Z or alternatively C. For both cases, we assume the semi-infinite arms of waveguides to be of unit width. These arms are connected to each other by a rectangle with side lengths \(H\) and \(L\). We consider the Dirichlet boundary value problem for the Laplacian and study the spectrum of the corresponding operator. Due to the unboundedness of the domain, the spectrum of the Dirichlet-Laplacian is not discrete. It is shown that the total multiplicity of the discrete part of the spectrum depends on the parameters \(H\) and \(L\). In particular, for the width \(H = 1\), we compare the relation between the eigenvalues of both waveguides and, moreover, we observe that the monotonicity in the height \(L\) of the first eigenvalue of the Z-shaped waveguide is not achieved, while the question of the monotonicity of the second eigenvalue remains open. The eigenvalues in the Z-shaped waveguide are monotone. We construct and justify the asymptotics of the eigenvalues for the cases \(H = 1\), \(L \to 1\), \(H = 1\), \(L \to 1 + 0\), and \(H\), \(L \to \infty\).

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78A50 Antennas, waveguides in optics and electromagnetic theory
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