TY - JOUR

T1 - Abnormal Behavior of Eigenvalues of Mixed Boundary Value Problems for the Laplace Operator in Truncated, but Long Cylinders

AU - Nazarov, S. A.

N1 - Publisher Copyright: © 2020, Springer Science+Business Media, LLC, part of Springer Nature. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/10/1

Y1 - 2020/10/1

N2 - In the spectrum of the mixed boundary value problem for the Laplace operator in a finite, but long cylinder Ωℓ with curved ends, we detect a series of eigenvalues of the strange behavior as the length 2ℓ unlimitedly increases. In addition to usual hard-movable eigenvalues not leaving small neighborhoods of fixed points, we detect eigenvalues gliding down with a large speed along the real axis, but smoothly landing on the lower bound λ† = Λ1 of the continuous spectrum of the limit problem in the half-cylinder Π± obtained by elongating Ωℓ in one of two directions. We give a complete description of the low-frequency range of the spectrum and show that each point in (Λ1, Λ2) is a blinking eigenvalue. Above the second threshold Λ2 of the continuous spectrum of the problem in Π±, the behavior of eigenvalues of the problem in the waveguide Ωℓ as ℓ → + ∞ is chaotic and their asymptotics can be constructed only in particular cases.

AB - In the spectrum of the mixed boundary value problem for the Laplace operator in a finite, but long cylinder Ωℓ with curved ends, we detect a series of eigenvalues of the strange behavior as the length 2ℓ unlimitedly increases. In addition to usual hard-movable eigenvalues not leaving small neighborhoods of fixed points, we detect eigenvalues gliding down with a large speed along the real axis, but smoothly landing on the lower bound λ† = Λ1 of the continuous spectrum of the limit problem in the half-cylinder Π± obtained by elongating Ωℓ in one of two directions. We give a complete description of the low-frequency range of the spectrum and show that each point in (Λ1, Λ2) is a blinking eigenvalue. Above the second threshold Λ2 of the continuous spectrum of the problem in Π±, the behavior of eigenvalues of the problem in the waveguide Ωℓ as ℓ → + ∞ is chaotic and their asymptotics can be constructed only in particular cases.

UR - http://www.scopus.com/inward/record.url?scp=85090378529&partnerID=8YFLogxK

U2 - 10.1007/s10958-020-05020-8

DO - 10.1007/s10958-020-05020-8

M3 - Article

AN - SCOPUS:85090378529

VL - 250

SP - 351

EP - 383

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 2

ER -