2014/07/03 by Raikov, Georgi
#35J10 #35P20 #81Q10 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1407.0757
We consider the twisted waveguide Ωθ, i.e. the domain obtained by the rotation of the bounded cross section ω⊂ \mathbb R2 of the straight tube Ω: = ω× \mathbb R at angle θ which depends on the variable along the axis of Ω. We study the spectral properties of the Dirichlet Laplacian in Ωθ, unitarily equivalent under the diffeomorphism Ωθ→ Ω to the operator Hθ', self-adjoint in \rm L2(Ω). We assume that θ' = β- ε where β is a 2π-periodic function, and ε decays at infinity. Then in the spectrum σ(Hβ) of the unperturbed operator Hβ there is a semi-bounded gap (-∞, \mathcal E0+), and, possibly, a number of bounded open gaps (\mathcal Ej-, \mathcal Ej+). Since ε decays at infinity, the essential spectra of Hβ and Hβ- ε coincide. We investigate the asymptotic behaviour of the discrete spectrum of Hβ- ε near an arbitrary fixed spectral edge \mathcal Ej^±. We establish necessary and quite close sufficient conditions which guarantee the finiteness of σ\rm disc(Hβ-ε) in a neighbourhood of \mathcal Ej^±. 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\rm disc(Hβ-ε) near \mathcal Ej^± could be represented as a finite orthogonal sum of operators of the form -μ(d2)/(dx2) - ηε, self-adjoint in \rm L2(\mathbb 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.