一维高对比度缺陷周期问题本征函数的极端局部化
Extreme localisation of eigenfunctions to one-dimensional high-contrast periodic problems with a defect
摘要：
继最近的一些研究的预解式和频谱收敛的非均匀椭圆族的微分算子描述的行为周期性复合介质的高对比度，我们研究了相应的一维版本，包括一个“缺陷”：包含固定大小与一组给定的材料参数。已知的是，当周期$\varepsilon$变为零时，没有缺陷及其极限的纯周期情况的谱具有带隙结构。我们考虑一个序列的特征值$λ_\varepsilon$的缺陷引起的，并收敛到一个点$ł_0$位于间隙的极限谱的周期性的情况下。我们表明，相应的本征函数是“非常”本地化的缺陷，在这个意义上说，本地化指数的行为作为$\exp(-ν/\varepsilon),$ $ν>0,$，这在现有的文献中还没有观察到。因此，我们认为，$ł_0$是一个特定的极限算子上定义的缺陷只的本征值。在二维和三维的配置，其一维的横截面所描述的设置考虑，这意味着存在的传播波是本地化的缺陷。我们还证明了未扰动算子在实轴上的范数预解式近似于退化算子，并给出了明确的描述。
英文摘要：
Following a number of recent studies of resolvent and spectral convergence of non-uniformly elliptic families of differential operators describing the behaviour of periodic composite media with high contrast, we study the corresponding one-dimensional version that includes a "defect": an inclusion of fixed size with a given set of material parameters. It is known that the spectrum of the purely periodic case without the defect and its limit, as the period $\varepsilon$ goes to zero, has a band-gap structure. We consider a sequence of eigenvalues $λ_\varepsilon$ that are induced by the defect and converge to a point $ł_0$ located in a gap of the limit spectrum for the periodic case. We show that the corresponding eigenfunctions are "extremely" localised to the defect, in the sense that the localisation exponent behaves as $\exp(-ν/\varepsilon),$ $ν>0,$ which has not been observed in the existing literature. As a consequence, we argue that $ł_0$ is an eigenvalue of a certain limit operator defined on the defect only. In two- and three-dimensional configurations, whose one-dimensional cross-sections are described by the setting considered, this implies the existence of propagating waves that are localised to the defect. We also show that the unperturbed operators are norm-resolvent close to a degenerate operator on the real axis, which is described explicitly.
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