# Spectral asymptotics for waveguides with perturbed periodic twisting

### Georgi Raikov

Pontificia Universidad Católica de Chile, Santiago de Chile, Chile

## Abstract

We consider the twisted waveguide $Ω_{θ}$, i.e. the domain obtained by the rotation of the bounded cross section $ω⊂R_{2}$ 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 $Ω_{θ}$, unitarily equivalent under the diffeomorphism $Ω_{θ}→Ω$ to the operator $H_{θ_{′}}$, self-adjoint in $L_{2}(Ω)$. We assume that $θ_{′}=β−ϵ$ where $β$ is a $2π$-periodic function, and $ϵ$ decays at infinity. Then in the spectrum $σ(H_{β})$ of the unperturbed operator $H_{β}$ there is a semi-bounded gap $(−∞,E_{0})$, and, possibly, a number of bounded open gaps $(E_{j},E_{j})$. Since $ϵ$ decays at infinity, the essential spectra of $H_{β}$ and $H_{β−ϵ}$ coincide. We investigate the asymptotic behaviour of the discrete spectrum of $H_{β−ϵ}$ near an arbitrary fixed spectral edge $E_{j}$. We establish necessary and quite close sufficient conditions which guarantee the finiteness of $σ_{disc}(H_{β−ϵ})$ in a neighbourhood of $E_{j}$. 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 $σ_{disc}(H_{β−ϵ})$ near $E_{j}$ could be represented as a finite orthogonal sum of operators of the form

self-adjoint in $L_{2}(R)$; here, $μ>0$ is a constant related to the so-called *effective mass*, while $η$ is $2π$-periodic function depending on $β$ and $ω$.

## 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