### Abstract

Original language | English |
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Pages (from-to) | None |

Journal | Journal of Mathematical Physics |

Issue number | 2 |

DOIs | |

Publication status | Published - 2015 |

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### Cite this

*Journal of Mathematical Physics*, (2), None. https://doi.org/10.1063/1.4907559

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*Journal of Mathematical Physics*, no. 2, pp. None. https://doi.org/10.1063/1.4907559

**Bound states of waveguides with two right-angled bends.** / Nazarov, S.A.; Ruotsalainen, K.; Uusitalo, P.

Research output ›

TY - JOUR

T1 - Bound states of waveguides with two right-angled bends

AU - Nazarov, S.A.

AU - Ruotsalainen, K.

AU - Uusitalo, P.

PY - 2015

Y1 - 2015

N2 - © 2015 AIP Publishing LLC.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 → ∞.

AB - © 2015 AIP Publishing LLC.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 → ∞.

U2 - 10.1063/1.4907559

DO - 10.1063/1.4907559

M3 - Article

SP - None

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 2

ER -