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arXiv · 1912.07261

Edge States for the magnetic Laplacian in domains with smooth boundary

Arianna Giunti·Juan J. L. Velázquez
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Abstract

We are interested in the spectral properties of the magnetic Schr\"odinger 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 \textit{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 the boundary $\partial\Omega$, as well as how their mass is distributed along it. From this result, we also infer asymptotic formulas for the eigenvalues of $H_\varepsilon$.

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Arianna Giunti, Juan J. L. Velázquez. 2019-12-16. Edge States for the magnetic Laplacian in domains with smooth boundary. https://arxiv.org/abs/1912.07261

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