Abstract

We consider the twisted waveguide Ωθ, i.e. the domain obtained by the rotation of the bounded cross section ω⊂ ℝ2 of the straight tube Ω:= ω × ℝ at angle θ which depends on the variable along the axis of Ω. We study the spectral properties of the Dirichlet Laplacian in Ωθ, unitarily equivalent under the ditfeomorphism Ωθ → Ω to the operator Hθ, self-adjoint in L2(Ω). We assume that θ' = - β where (β is a 2π-periodic function, and e decays at infinity. Then in the spectrum σ(Hβ) of the unperturbed operator Hβ there is a semi-bounded gap (-∞,ϵ+0), and- possibly, a number of bounded gaps (ϵ j-,ϵj+). Since ϵ decays at infinity, the essential spectra of Hβ and Hβ-ϵ coincide. We investigate the asymptotic behaviour of the discrete spectrum of Hβ-ϵ near an arbitrary fixed spectral edge β. We establish necessary and quite close sulficient conditions which guarantee the finiteness of (σdic(Hβ-ϵ) in a neighbourhood of ϵ j+ In the case where the necessary conditions are violated, we obtain the main asymptotic term of the corresponding eigenvalue counting function. Ihe effective Hamiltonian which governs the the asymptotics of σdisc(Hβ-ϵ) near ϵ j +could be represented as a finite orthogonal sum of operators of the form d2/dx2 -μ - ν ϵself-adjoint in L2(ℝ); here, μ > 0 is a constant related to the so-called effective mass, while ν is 2π-periodic function depending on β and ω. © European Mathematical Society.