TY - JOUR

T1 - Embedded eigenvalues of the Neumann problem in a strip with a box-shaped perturbation

AU - Cardone, G.

AU - Durante, T.

AU - Nazarov, S. A.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide Π l ε formed by the union of an infinite strip and a narrow box-shaped perturbation of size 2l×ε, where ε>0 is a small parameter. We prove the existence of the length parameter l k ε =πk+O(ε) with any k=1,2,3,. such that the waveguide Π l k ε ε supports a trapped mode with an eigenvalue λ k ε =π 2 −4π 4 l 2 ε 2 +O(ε 3 ) embedded into the continuous spectrum. This eigenvalue is unique in the segment [0,π 2 ], and it is absent in the case l≠l k ε . The detection of this embedded eigenvalue is based on a criterion for trapped modes involving an artificial object, the augmented scattering matrix. The main difficulty is caused by the rather specific shape of the perturbed wall ∂Π l ε , namely a narrow rectangular bulge with corner points, and we discuss available generalizations for other piecewise smooth boundaries.

AB - We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide Π l ε formed by the union of an infinite strip and a narrow box-shaped perturbation of size 2l×ε, where ε>0 is a small parameter. We prove the existence of the length parameter l k ε =πk+O(ε) with any k=1,2,3,. such that the waveguide Π l k ε ε supports a trapped mode with an eigenvalue λ k ε =π 2 −4π 4 l 2 ε 2 +O(ε 3 ) embedded into the continuous spectrum. This eigenvalue is unique in the segment [0,π 2 ], and it is absent in the case l≠l k ε . The detection of this embedded eigenvalue is based on a criterion for trapped modes involving an artificial object, the augmented scattering matrix. The main difficulty is caused by the rather specific shape of the perturbed wall ∂Π l ε , namely a narrow rectangular bulge with corner points, and we discuss available generalizations for other piecewise smooth boundaries.

KW - Acoustic waveguide

KW - Asymptotics

KW - Box-shaped perturbation

KW - Continuous spectrum

KW - Embedded eigenvalues

KW - Neumann problem

UR - http://www.scopus.com/inward/record.url?scp=85041631138&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2018.01.002

DO - 10.1016/j.matpur.2018.01.002

M3 - Article

AN - SCOPUS:85041631138

VL - 112

SP - 1

EP - 40

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

ER -