TY - JOUR

T1 - Asymptotics of 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

Y1 - 2019

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 ofthe 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 ofthe trapped modes: the terms of the series have compact supports that expand unboundedly as the number of the term increases.

KW - lattice of thin quantum waveguides

KW - perturbation

KW - essential and discrete spectra

KW - gaps

KW - eigenvalues

KW - asymptotic expansion

UR - https://link.springer.com/content/pdf/10.1134/S0001434619030295.pdf

UR - https://link.springer.com/article/10.1134/S0001434619030295

M3 - Article

VL - 105

SP - 559

EP - 579

JO - Mathematical Notes

JF - Mathematical Notes

SN - 0001-4346

IS - 4

ER -