### Abstract

The spectrum of truncated cross-shaped waveguides is studied under the Dirichlet conditions on the lateral surface and various boundary conditions on the ends of the column and the cross bar. The monotonicity and asymptotics of the eigenvalues are discussed in dependence on the size of a cross whose section may be fairly arbitrary. In the case of a round section, the estimates found for the second eigenvalue agree with the asymptotic formulas obtained. Such information is needed for the spectral analysis of thin periodic lattices of quantum waveguides.

Original language | English |
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Pages (from-to) | 423-437 |

Number of pages | 15 |

Journal | St. Petersburg Mathematical Journal |

Volume | 29 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

Externally published | Yes |

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### Scopus subject areas

- Analysis
- Algebra and Number Theory
- Applied Mathematics

### Cite this

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*St. Petersburg Mathematical Journal*, vol. 29, no. 3, pp. 423-437. https://doi.org/10.1090/spmj/1500

**Rectangular lattices of cylindrical quantum waveguides. I. spectral problems on a finite cross.** / Bakharev, F. L.; Matveenko, S. G.; Nazarov, S. A.

Research output

TY - JOUR

T1 - Rectangular lattices of cylindrical quantum waveguides. I. spectral problems on a finite cross

AU - Bakharev, F. L.

AU - Matveenko, S. G.

AU - Nazarov, S. A.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - The spectrum of truncated cross-shaped waveguides is studied under the Dirichlet conditions on the lateral surface and various boundary conditions on the ends of the column and the cross bar. The monotonicity and asymptotics of the eigenvalues are discussed in dependence on the size of a cross whose section may be fairly arbitrary. In the case of a round section, the estimates found for the second eigenvalue agree with the asymptotic formulas obtained. Such information is needed for the spectral analysis of thin periodic lattices of quantum waveguides.

AB - The spectrum of truncated cross-shaped waveguides is studied under the Dirichlet conditions on the lateral surface and various boundary conditions on the ends of the column and the cross bar. The monotonicity and asymptotics of the eigenvalues are discussed in dependence on the size of a cross whose section may be fairly arbitrary. In the case of a round section, the estimates found for the second eigenvalue agree with the asymptotic formulas obtained. Such information is needed for the spectral analysis of thin periodic lattices of quantum waveguides.

KW - Asymptotics

KW - Discrete spectrum

KW - Infinite and truncated cross-shaped quantum waveguides

KW - Stable and decaying solution at the threshold of the continuous spectrum

UR - http://www.scopus.com/inward/record.url?scp=85045539527&partnerID=8YFLogxK

U2 - 10.1090/spmj/1500

DO - 10.1090/spmj/1500

M3 - Article

AN - SCOPUS:85045539527

VL - 29

SP - 423

EP - 437

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 3

ER -