SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 3602-3643, January 2021.
We are interested in the spectral properties of the magnetic Schrödinger operator $H_\varepsilon$ in a domain $\Omega \subset \mathbb{R}^2$ with compact boundary and with magnetic field of intensity $\varepsilon^{-2}$. We impose Dirichlet boundary conditions on $\partial\Omega$. Our main focus is the existence and description of the so-called edge states, namely eigenfunctions for $H_{\varepsilon}$ whose mass is localized at scale $\varepsilon$ along the boundary $\partial\Omega$. When the intensity of the magnetic field is large (i.e., $\varepsilon <<1$), we show that such edge states exist. Furthermore, we give a detailed description of their localization close to $\partial\Omega$, as well as how their mass is distributed along it.

中文翻译：

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光滑边界域中磁性拉普拉斯算子的边缘状态

SIAM 数学分析杂志，第 53 卷，第 3 期，第 3602-3643 页，2021 年 1 月。
我们对域 $\Omega\subset\mathbb{R} 中磁性薛定谔算子 $H_\varepsilon$ 的光谱特性感兴趣^2$ 具有紧凑的边界和强度为 $\varepsilon^{-2}$ 的磁场。我们对 $\partial\Omega$ 强加 Dirichlet 边界条件。我们的主要焦点是所谓的边缘状态的存在和描述，即 $H_{\varepsilon}$ 的特征函数，其质量沿边界 $\partial\Omega$ 定位在尺度 $\varepsilon$。当磁场强度很大时（即 $\varepsilon <<1$），我们证明存在这样的边缘状态。此外，我们详细描述了它们在 $\partial\Omega$ 附近的定位，以及它们的质量如何沿着它分布。