Edge states for the magnetic Laplacian in domains with smooth boundary. (English) Zbl 1481.35350
Let \(\Omega \subset \mathbb{R}^2\) be a bounded \(C^4\)-domain. The authors study the existence of edge states of magnetic Laplacians with strong magnetic fields on \(\Omega\) with Dirichlet boundary conditions. Consider for \(\varepsilon > 0\) in \(L^2(\Omega)\) the operator
\[
H_\varepsilon := -(\nabla + i \varepsilon^{-2} a) (\nabla + i \varepsilon^{-2} a), \quad \mathcal{D}(H_\varepsilon) := H^2(\Omega) \cap H^1_0(\Omega);
\]
the magnetic field is assumed to be \(a := (0,-x_1) b_0\) with \(b_0 \in \mathbb{R}\), however, it is sketched that the main results of the paper also hold for smooth \(a\) that do not oscillate too much. The spectrum \(\sigma(H_\varepsilon)\) is purely discrete. Set \(\sigma_\text{Landau} := \{ b_0 (2n+1): n \in \mathbb{N}_0\}\), which is the spectrum of \(H_\varepsilon\), when \(\Omega=\mathbb{R}^2\) and \(\varepsilon=1\), let \(\lambda_\varepsilon \in \sigma(H_\varepsilon)\) such that \(\varepsilon^2 \lambda_\varepsilon \rightarrow \lambda \notin \sigma_\text{Landau}\), and let \(\Psi_\varepsilon\) be the corresponding eigenfunctions, i.e. \(H_\varepsilon \Psi_\varepsilon = \lambda_\varepsilon \Psi_\varepsilon\). Then, in the main result of the paper the authors prove for sufficiently small \(\varepsilon\) that \(\Psi_\varepsilon\) is mostly supported in an \(\varepsilon\)-neighborhood \(\Omega_\varepsilon\) of \(\partial \Omega\), i.e. \(\| \Psi_\varepsilon \|_{L^2(\Omega_\varepsilon)} \approx \| \Psi_\varepsilon \|_{L^2(\Omega)}\), and that \(|\Psi_\varepsilon|^2\) is distributed homogeneously along \(\partial \Omega\), i.e. that \(\Psi_\varepsilon\) are edge states.
For the proof of the main result, the authors transform the eigenvalue problem for \(H_\varepsilon\) with the help of local coordinates and use knowledge of the spectral properties of the magnetic Laplacian on the half plane. Moreover, techniques similar to the theory of homogenization are employed. It is remarkable that the main result holds for all eigenvalues \(\lambda_\varepsilon\) such that \(\varepsilon^2 \lambda_\varepsilon \rightarrow \lambda \notin \sigma_\text{Landau}\) and not only for the ground state.
For the proof of the main result, the authors transform the eigenvalue problem for \(H_\varepsilon\) with the help of local coordinates and use knowledge of the spectral properties of the magnetic Laplacian on the half plane. Moreover, techniques similar to the theory of homogenization are employed. It is remarkable that the main result holds for all eigenvalues \(\lambda_\varepsilon\) such that \(\varepsilon^2 \lambda_\varepsilon \rightarrow \lambda \notin \sigma_\text{Landau}\) and not only for the ground state.
Reviewer: Markus Holzmann (Graz)
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
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