Bound states of waveguides with two right-angled bends

Abstract

We study waveguides with two right-angled bends. These waveguides are in 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 Laplacian and study the spectrum of the corresponding operator. It is shown that the total multiplicity of the discrete 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 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 C-shaped waveguide are monotone. We construct and justify the asymptotics of the eigenvalues for the cases H = 1, L → ∞, H = 1, L → 1 + 0, and H, L → ∞.