TY - JOUR

T1 - Embedded Eigenvalues for Water-Waves in a Three-Dimensional Channel with a Thin Screen

AU - Chiadó Piat, Valeria

AU - Nazarov, Sergey A.

AU - Taskinen, Jari

PY - 2018/5/1

Y1 - 2018/5/1

N2 - We construct asymptotic expansions as ϵ→+0 for an eigenvalue embedded into the continuous spectrum of water-wave problem in a cylindrical three-dimensional channel with a thin screen of thickness O(ϵ). The screen may be either submerged or surface-piercing, and its wetted part has a sharp edge. The channel and the screen are mirror symmetric so that imposing the Dirichlet condition in the middle plane creates an artificial positive cut-off-value Λ† of the modified spectrum. Depending on a certain integral characteristic I of the screen profiles, we find two types of asymptotics of eigenvalues, λϵ = Λ† - O(ϵ2) and λϵ = Λ† - O(ϵ4) in the cases I > 0 and I = 0, respectively. We prove that in the case I < 0 there are no embedded eigenvalues in the interval [0,Λ†], while this interval contains exactly one eigenvalue, if I ≥ 0. For the justification of these result, the main tools are a reduction to an abstract spectral equation and the use of the max-min principle.

AB - We construct asymptotic expansions as ϵ→+0 for an eigenvalue embedded into the continuous spectrum of water-wave problem in a cylindrical three-dimensional channel with a thin screen of thickness O(ϵ). The screen may be either submerged or surface-piercing, and its wetted part has a sharp edge. The channel and the screen are mirror symmetric so that imposing the Dirichlet condition in the middle plane creates an artificial positive cut-off-value Λ† of the modified spectrum. Depending on a certain integral characteristic I of the screen profiles, we find two types of asymptotics of eigenvalues, λϵ = Λ† - O(ϵ2) and λϵ = Λ† - O(ϵ4) in the cases I > 0 and I = 0, respectively. We prove that in the case I < 0 there are no embedded eigenvalues in the interval [0,Λ†], while this interval contains exactly one eigenvalue, if I ≥ 0. For the justification of these result, the main tools are a reduction to an abstract spectral equation and the use of the max-min principle.

UR - http://www.scopus.com/inward/record.url?scp=85048644345&partnerID=8YFLogxK

U2 - 10.1093/qjmam/hby002

DO - 10.1093/qjmam/hby002

M3 - Article

AN - SCOPUS:85048644345

VL - 71

SP - 187

EP - 220

JO - Quarterly Journal of Mechanics and Applied Mathematics

JF - Quarterly Journal of Mechanics and Applied Mathematics

SN - 0033-5614

IS - 2

ER -