DOI

Cylindrical acoustic waveguides with constant cross-section ω are considered, specifically, a straight waveguide Ω = R × ω ⊂ Rd and a locally curved waveguide Ωε that depends on a parameter ε ∈ (0, 1]. For d > 2, in two different settings (ε = 1 and ε ≪ 1), the task is to find an eigenvalue λε that is embedded in the continuous spectrum [0,+∞) of the waveguide Ωε and, hence, is inherently unstable. In other words, a solution of the Neumann problem for the Helmholtz operator Δ + λε arises that vanishes at infinity and implies an eigenfunction in the Sobolev space H1ε). In the first case, it is assumed that the cross-section ω has a double symmetry and an eigenvalue arises for any nontrivial curvature of the axis of the waveguide Ωε. In the second case, under an assumption on the shape of an asymmetric cross-section ω, the eigenvalue λε is formed by scrupulous fitting of the curvature O(ε) for small ε > 0.

Original languageEnglish
Pages (from-to)865-885
Number of pages21
JournalSt. Petersburg Mathematical Journal
Volume31
Issue number5
DOIs
StatePublished - Oct 2020

    Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Algebra and Number Theory

    Research areas

  • Asymptotics, Continuous and point spectra, Curved cylinder, Eigenvalue, Expanded scattering matrix, Neumann problem for the laplace operator, expanded scattering matrix, CONTINUOUS-SPECTRUM, Neumann problem for the Laplace operator, BOUND-STATES, curved cylinder, asymptotics, eigenvalue

ID: 71562148