Resonances near thresholds in slightly twisted waveguides

Abstract

We consider the Dirichlet Laplacian in a straight three dimensional waveguide with non-rotationally invariant cross section, perturbed by a twisting of small amplitude. It is well known that such a perturbation does not create eigenvalues below the essential spectrum. However, around the bottom of the spectrum, we provide a meromorphic extension of the weighted resolvent of the perturbed operator, and show the existence of exactly one resonance near this point. Moreover, we obtain the asymptotic behavior of this resonance as the size of the twisting goes to 0. We also extend the analysis to the upper eigenvalues of the transversal problem, showing that the number of resonances is bounded by the multiplicity of the eigenvalue and obtaining the corresponding asymptotic behavior.

Eigenvalue and Resonance Asymptotics in Perturbed Periodically Twisted Tubes: Twisting Versus Bending

“…Moreover, they are concerned only with the existence of eigenvalues, and the study of the resonances is not treated. We have tackled these questions in the previous article [6] when both β and κ are 0 (i.e. when Ω 0 " R ˆω and the deformation is only given by a twisting effect).…”

“…On the other side, despite the fact that various works on the existence of bound states for deformed tubes are available, the results are mainly concerned with their existence below the bottom of the essential spectrum of H 0 . The spectral effects that the geometric deformations have near the upper thresholds are not well studied (see however [6]). Using our methods we can prove some results concerning this problem.…”

“…As in [2,6], our strategy is to use an explicit decomposition of the resolvent of the free operator. However, unlike these articles, the Fourier multipliers of H 0 are not explicit when β ‰ 0, and so we need to start by studying the resolvent of H 0 .…”

Spectrum of the Dirichlet Laplacian in waveguides with parallel cross-sections

“…Remark 1. We can perform a similar change of coordinates for the quadratic form a ε (ϕ) given (7) in the Introduction. Recall the mapping L ε and the quadratic form b ε (ψ) given by ( 6) and ( 9), respectively.…”

“…Let Ω be an unbounded quantum waveguide in R n , n = 2, 3, and denote by −∆ D Ω the Dirichlet Laplacian operator in Ω. The spectrum of this operator has been extensively studied in the last years [1,2,3,4,5,6,7,8,9,11,12,13,15,17,19,20,21,23,24,25,26,27]. In fact, the subject is non-trivial since the results depend on the geometry of Ω [4,8,12,17,23,24].…”