JournalsjstVol. 6, No. 2pp. 331–372

Spectral asymptotics for waveguides with perturbed periodic twisting

  • Georgi Raikov

    Pontificia Universidad Católica de Chile, Santiago de Chile, Chile
Spectral asymptotics for waveguides with perturbed periodic twisting cover
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Abstract

We consider the twisted waveguide Ωθ\Omega_\theta, i.e. the domain obtained by the rotation of the bounded cross section ωR2\omega \subset {\mathbb R}^{2} of the straight tube Ω:=ω×R\Omega : = \omega \times {\mathbb R} at angle θ\theta which depends on the variable along the axis of Ω\Omega. We study the spectral properties of the Dirichlet Laplacian in Ωθ\Omega_\theta, unitarily equivalent under the diffeomorphism ΩθΩ\Omega_\theta \to \Omega to the operator HθH_{\theta'}, self-adjoint in L2(Ω){\rm L}^2(\Omega). We assume that θ=βϵ\theta' = \beta - \epsilon where β\beta is a 2π2\pi-periodic function, and ϵ\epsilon decays at infinity. Then in the spectrum σ(Hβ)\sigma(H_\beta) of the unperturbed operator HβH_\beta there is a semi-bounded gap (,E0+)(-\infty, {\mathcal E}_0^+), and, possibly, a number of bounded open gaps (Ej,Ej+)({\mathcal E}_j^-, {\mathcal E}_j^+). Since ϵ\epsilon decays at infinity, the essential spectra of HβH_\beta and HβϵH_{\beta - \epsilon} coincide. We investigate the asymptotic behaviour of the discrete spectrum of HβϵH_{\beta - \epsilon} near an arbitrary fixed spectral edge Ej±{\mathcal E}_j^\pm. We establish necessary and quite close sufficient conditions which guarantee the finiteness of σdisc(Hβϵ)\sigma_{\rm disc}(H_{\beta-\epsilon}) in a neighbourhood of Ej±{\mathcal E}_j^\pm. In the case where the necessary conditions are violated, we obtain the main asymptotic term of the corresponding eigenvalue counting function. The effective Hamiltonian which governs the the asymptotics of σdisc(Hβϵ)\sigma_{\rm disc}(H_{\beta-\epsilon}) near Ej±{\mathcal E}_j^\pm could be represented as a finite orthogonal sum of operators of the form

μd2dx2ηϵ,-\mu\frac{d^2}{dx^2} - \eta \epsilon,

self-adjoint in L2(R){\rm L}^2({\mathbb R}); here, μ>0\mu > 0 is a constant related to the so-called effective mass, while η\eta is 2π2\pi-periodic function depending on β\beta and ω\omega.

Cite this article

Georgi Raikov, Spectral asymptotics for waveguides with perturbed periodic twisting. J. Spectr. Theory 6 (2016), no. 2, pp. 331–372

DOI 10.4171/JST/126