Spectral asymptotics for waveguides with perturbed periodic twisting

Abstract

We consider the twisted waveguide ${\Omega }_{\theta }$, i.e. the domain obtained by the rotation of the bounded cross section $\omega \subset {\mathbb{R}}^2$ of the straight tube $\Omega :=\omega \times \mathbb{R}$ at angle $\theta$ which depends on the variable along the axis of $\Omega$. We study the spectral properties of the Dirichlet Laplacian in ${\Omega }_{\theta }$, unitarily equivalent under the diffeomorphism ${\Omega }_{\theta }\to \Omega$ to the operator $H_{{\theta }^{\prime }}$, self-adjoint in ${\mathrm{L}}^2(\Omega )$. We assume that ${\theta }^{\prime }=\beta -ϵ$ where $\beta$ is a $2\pi$-periodic function, and $ϵ$ decays at infinity. Then in the spectrum $\sigma (H_{\beta })$ of the unperturbed operator $H_{\beta }$ there is a semi-bounded gap $(-\infty ,{\mathcal{E}}_0^{+})$, and, possibly, a number of bounded open gaps $({\mathcal{E}}_j^{-},{\mathcal{E}}_j^{+})$. Since $ϵ$ decays at infinity, the essential spectra of $H_{\beta }$ and $H_{\beta -ϵ}$ coincide. We investigate the asymptotic behaviour of the discrete spectrum of $H_{\beta -ϵ}$ near an arbitrary fixed spectral edge ${\mathcal{E}}_j^{\pm }$. We establish necessary and quite close sufficient conditions which guarantee the finiteness of ${\sigma }_{\mathrm{disc}}(H_{\beta -ϵ})$ in a neighbourhood of ${\mathcal{E}}_j^{\pm }$. 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 ${\sigma }_{\mathrm{disc}}(H_{\beta -ϵ})$ near ${\mathcal{E}}_j^{\pm }$ could be represented as a finite orthogonal sum of operators of the form

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