### Abstract

The spectrum of the Dirichlet problem on the planar square lattice of thin quantum waveguides has a band-gap structure with short spectral bands separated by wide spectral gaps. The curving of at least one of the ligaments of the lattice generates points of the discrete spectrum inside gaps. A complete asymptotic series for the eigenvalues and eigenfunctions are constructed and justified; those for the eigenfunctions exhibit a remarkable behavior imitating the rapid decay of the trapped modes: the terms of the series have compact supports that expand unboundedly as the number of the term increases.

Original language | English |
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Pages (from-to) | 559-579 |

Journal | Mathematical Notes |

Volume | 105 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 1 Mar 2019 |

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### Scopus subject areas

- Mathematics(all)

### Cite this

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*Mathematical Notes*, vol. 105, no. 3-4, pp. 559-579. https://doi.org/10.1134/S0001434619030295

**Asymptotics of the Eigenvalues and Eigenfunctions of a Thin Square Dirichlet Lattice with a Curved Ligament.** / Nazarov, S. A.

Research output

TY - JOUR

T1 - Asymptotics of the Eigenvalues and Eigenfunctions of a Thin Square Dirichlet Lattice with a Curved Ligament

AU - Nazarov, S. A.

N1 - Nazarov, S.A. Math Notes (2019) 105: 559. https://doi.org/10.1134/S0001434619030295

PY - 2019/3/1

Y1 - 2019/3/1

N2 - The spectrum of the Dirichlet problem on the planar square lattice of thin quantum waveguides has a band-gap structure with short spectral bands separated by wide spectral gaps. The curving of at least one of the ligaments of the lattice generates points of the discrete spectrum inside gaps. A complete asymptotic series for the eigenvalues and eigenfunctions are constructed and justified; those for the eigenfunctions exhibit a remarkable behavior imitating the rapid decay of the trapped modes: the terms of the series have compact supports that expand unboundedly as the number of the term increases.

AB - The spectrum of the Dirichlet problem on the planar square lattice of thin quantum waveguides has a band-gap structure with short spectral bands separated by wide spectral gaps. The curving of at least one of the ligaments of the lattice generates points of the discrete spectrum inside gaps. A complete asymptotic series for the eigenvalues and eigenfunctions are constructed and justified; those for the eigenfunctions exhibit a remarkable behavior imitating the rapid decay of the trapped modes: the terms of the series have compact supports that expand unboundedly as the number of the term increases.

KW - asymptotic expansion

KW - eigenvalues

KW - essential and discrete spectra

KW - gaps

KW - lattice of thin quantum waveguides

KW - perturbation

UR - http://www.scopus.com/inward/record.url?scp=85065501881&partnerID=8YFLogxK

U2 - 10.1134/S0001434619030295

DO - 10.1134/S0001434619030295

M3 - Article

AN - SCOPUS:85065501881

VL - 105

SP - 559

EP - 579

JO - Mathematical Notes

JF - Mathematical Notes

SN - 0001-4346

IS - 3-4

ER -