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首页> 外文期刊>Annales Henri Poincare >Eigenvalue and Resonance Asymptotics in Perturbed Periodically Twisted Tubes: Twisting Versus Bending
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Eigenvalue and Resonance Asymptotics in Perturbed Periodically Twisted Tubes: Twisting Versus Bending

机译：扰动定期扭曲管中的特征值和共振渐近学：扭曲与弯曲

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摘要

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 delta. 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 d small. We are able to perform the asymptotic expansion of the resonance in d, 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.
机译：我们考虑一种三维波导，其是周期性扭曲管的小变形（特别是直管的情况）。通过耦合恒定的三角形的耦合，通过弯曲和额外的扭曲给出变形。在这种变形的波导中，我们考虑Dirichlet Laplacian。我们在其基本谱的底部扩展其附近的分辨率，我们展示了D小的渐近制度的恰当的一个共鸣。我们能够在D中进行共振的渐近扩展，特别允许我们为存在于基本谱以下的离散特征值存在定量几何标准。在直线管的扰动的情况下，我们能够透露不仅在基本光谱的底部附近的谐振的存在，而是在光谱中的每个阈值附近，特别是弯曲的较高能量的光谱效应是什么。在这种情况下，我们还获得了共振的渐近行为，这与第一种情况不同。

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