We consider the twisted waveguide , i.e. the domain obtained by the rotation of the bounded cross section of the straight tube at angle which depends on the variable along the axis of . We study the spectral properties of the Dirichlet Laplacian in , unitarily equivalent under the diffeomorphism to the operator , self-adjoint in . We assume that where is a -periodic function, and decays at infinity. Then in the spectrum of the unperturbed operator there is a semi-bounded gap , and, possibly, a number of bounded open gaps . Since decays at infinity, the essential spectra of and coincide. We investigate the asymptotic behaviour of the discrete spectrum of near an arbitrary fixed spectral edge . We establish necessary and quite close sufficient conditions which guarantee the finiteness of in a neighbourhood of . 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 near could be represented as a finite orthogonal sum of operators of the form
self-adjoint in ; here, is a constant related to the so-called effective mass, while is -periodic function depending on and .
Cite this article
Georgi Raikov, Spectral asymptotics for waveguides with perturbed periodic twisting. J. Spectr. Theory 6 (2016), no. 2, pp. 331–372DOI 10.4171/JST/126