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

We consider a three-dimensional waveguide that is a small deformation of a periodically twisted tube (including in particular the case of a straight tube). The deformation is given by a bending and an additional twisting of the tube, both parametrized by a coupling constant δ. In this deformed waveguide, we consider the Dirichlet Laplacian. We expand its resolvent near the bottom of its essential spectrum, and we show the existence of exactly one resonance, in the asymptotic regime of δ small. We are able to perform the asymptotic expansion of the resonance in δ, which in particular permits us to give a quantitative geometric criterion for the existence of a discrete eigenvalue below the essential spectrum. In the case of perturbations of straight tubes, we are able to show the existence of resonances not only near the bottom of the essential spectrum but near each threshold in the spectrum, showing in particular what are the spectral effects of the bending for higher energies. We also obtain the asymptotic behavior of the resonances in this situation, which is generically different from the first case. © 2019, Springer Nature Switzerland AG.