| Abstract: |
| Consider a reference homogeneous and isotropic electromagnetic waveguide with a simply connected cross-section embedded in a perfect conductor. In this setting, when the waveguide is straight, the spectrum of the associated self-adjoint Maxwell operator with a constant twist (which may be zero) is entirely essential, lies on the real line and is symmetric with respect to zero, exhibiting a gap around the origin.
In this talk, we present new results on the effects of the geometric deformations of bending and twisting on the spectrum of the Maxwell operator. More precisely, we provide, on the one hand, sufficient conditions on the asymptotic behaviour of the curvature and twist of a perturbed waveguide ensuring the preservation of the essential spectrum. Our approach relies on a Birman-Schwinger-type principle which has an interest of its own.
On the other hand, we give sufficient conditions, involving in particular the shape of the cross-section, so that the geometrical deformation creates discrete spectrum within the gap, and give some insight into its localization.
Finally, we show some theoretical and numerical results further investigating the sufficient condition involving the geometry of the cross-section. |
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