Spectral Asymptotics for Waveguides with Perturbed Periodic Twisting

We consider the twisted waveguide ${\mathrm{\Omega }}_{\theta }$, i.e. the domain obtained by the rotation of the bounded cross section $\omega \subset {\mathbb{R}}^2$ of the straight tube $\mathrm{\Omega }:=\omega \times \mathbb{R}$ at angle $\theta$ which depends on the variable along the axis of $\mathrm{\Omega }$. We study the spectral properties of the Dirichlet Laplacian in ${\mathrm{\Omega }}_{\theta }$, unitarily equivalent under the diffeomorphism ${\mathrm{\Omega }}_{\theta }\to \mathrm{\Omega }$ to the operator $H_{{\theta }^{\mathrm{\prime }}}$, self-adjoint in ${\mathrm{L}}^2(\mathrm{\Omega })$. We assume that $\theta’ = \beta - \epsilon$where$\beta$isa$2\pi$-periodicfunction,and$\epsilon$decaysatinfinity\mathrm{.}Theninthespectrum$\sigma(H_\beta)$ of the unperturbed operator $H_{\beta }$ there is a semi-bounded gap $(-\mathrm{\infty },{\mathcal{E}}_0^{+})$, and, possibly, a number of bounded open gaps $({\mathcal E}j^-, {\mathcal E}j^+)$\mathrm{.}Since$\epsilon$decaysatinfinity,theessentialspectraof$H\beta$and$H{\beta - \epsilon}$ 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 $\sigma_{\rm disc}(H_{\beta-\epsilon})$inaneighbourhoodof${\mathcal E}j^\pm$. 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 $\sigma{\rm disc}(H_{\beta-\epsilon})$near${\mathcal E}_j^\pm$ could be represented as a finite orthogonal sum of operators of the form $-\mu\frac{d^2}{dx^2} - \eta \epsilon$,self-adjointin${\rm L}^2({\mathbb R})$;here,$\mu > 0$ is a constant related to the so-called effective mass, while $\eta$ is $2\pi$-periodic function depending on $\beta$ and $\omega$.