Spectral Asymptotics for Waveguides with Perturbed Periodic Twisting

TL;DR

This paper analyzes the spectral properties of the Dirichlet Laplacian in twisted waveguides with perturbed periodic twisting, establishing conditions for the finiteness of discrete spectra and deriving asymptotics near spectral edges.

Contribution

It provides new criteria for the finiteness of discrete spectra and characterizes their asymptotic behavior near spectral edges in waveguides with perturbed periodic twisting.

Findings

Discrete spectrum finiteness depends on specific conditions.

Asymptotic eigenvalue distribution near spectral edges is characterized.

Effective Hamiltonian is a sum of one-dimensional operators.

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 $H_{\theta'}$, self-adjoint in ${\rm L}^2(\Omega)$. We assume that $\theta' = \beta - \epsilon$ where $\beta$ is a $2\pi$-periodic function, and $\epsilon$ 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 $\epsilon$ decays at infinity, the essential spectra of $H_\beta$ and $H_{\beta - \epsilon}$ coincide. We investigate the asymptotic behaviour of the discrete spectrum of $H_{\beta - \epsilon}$ near an arbitrary fixed spectral edge ${\mathcal E}_j^\pm$. We establish necessary and quite close sufficient conditions which guarantee the finiteness of $\sigma_{\rm disc}(H_{\beta-\epsilon})$ 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 $\sigma_{\rm disc}(H_{\beta-\epsilon})$ 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-adjoint in ${\rm L}^2({\mathbb R})$; here, $\mu > 0$ is a constant related to the so-called effective mass, while $\eta$ is $2\pi$-periodic function depending on $\beta$ and $\omega$.