论文标题
小型谐振器扰动的域上的laplacian频谱
Spectrum of the Laplacian on a domain perturbed by small resonators
论文作者
论文摘要
众所周知，在域的小扰动下，Dirichlet Laplacian的光谱是稳定的，而在Neumann或混合边界条件的情况下，频谱可能会突然改变。在这项工作中，我们讨论了这样一个域扰动的示例。令$ω$为$ \ mathbb {r}^n $中的a（不是{不一定}有限的）域。我们将其驱逐到$ω_\ varepsilon =ω\ setminus \ cup_ {k = 1}^m s_ {k，k，\ varepsilon}，其中$ s_ {k，k，\ varepsilon} $是由小的封闭的表面，具有小的适当缩放的孔（由小规模缩放的孔）（the）curne（ （``谐振器''）连接到外部域。当$ \ varepsilon $变为零时，谐振器会收缩到点。我们证明，在限制$ \ varepsilon \ to to $ω__\ varepsilon $上，在$ s_ {k，\ varepsilon} $上与neumann边界条件在$ω__\ varepsilon $上，以及$ s_ {k，\ varepsilon} $，以及外在边界上的dirichlet $ $ $ $ $ cons $ s of dirichlet laplaclaclacian of dirichlet laplaclac的界限， $ k = 1，\ dots，m $，是$ 1/4 $ $ $ $倍的$ k $ th窗口和$ k $ th谐振器的容量之间的比率。我们获得了相对于Hausdorff型指标的这种收敛速率的估计。此外，还提出了该结果的应用：我们构建了一个无界波导的域，上插入的谐振器，以使Laplacian的特征值在基本频谱阈值以下的该域上与规定的数字相吻合。
It is widely known that the spectrum of the Dirichlet Laplacian is stable under small perturbations of a domain, while in the case of the Neumann or mixed boundary conditions the spectrum may abruptly change. In this work we discuss an example of such a domain perturbation. Let $Ω$ be a (not {necessarily} bounded) domain in $\mathbb{R}^n$. We perturb it to $ Ω_\varepsilon=Ω\setminus \cup_{k=1}^m S_{k,\varepsilon},$ where $S_{k,\varepsilon}$ are closed surfaces with small suitably scaled holes (``windows'') through which the bounded domains enclosed by these surfaces (``resonators'') are connected to the outer domain. When $\varepsilon$ goes to zero, the resonators shrink to points. We prove that in the limit $\varepsilon\to 0$ the spectrum of the Laplacian on $Ω_\varepsilon$ with the Neumann boundary conditions on $S_{k,\varepsilon}$ and the Dirichlet boundary conditions on the outer boundary converges to the union of the spectrum of the Dirichlet Laplacian on $Ω$ and the numbers $γ_k$, $k=1,\dots,m$, being equal $1/4$ times the limit of the ratio between the capacity of the $k$th window and the volume of the $k$th resonator. We obtain an estimate on the rate of this convergence with respect to the Hausdorff-type metrics. Also, an application of this result is presented: we construct an unbounded waveguide-like domain with inserted resonators such that the eigenvalues of the Laplacian on this domain lying below the essential spectrum threshold do coincide with prescribed numbers.
