Energy of surface states for 3D magnetic Schrödinger operators. (English) Zbl 1344.35116
Let \(\Omega \subset \mathbb{R}^3\) be an interior or exterior domain with compact and smooth boundary \(\partial \Omega \subset \mathbb{R}^3\). Let also \(h>0\) and \(\mathbf{A} \in C^\infty(\bar{\Omega};\mathbb{R}^3)\) be a magnetic vector potential such that \(b:=\inf_{x\in\bar{\Omega}}|\operatorname{curl}\mathbf{A}(x)|>0\). In the Hilbert space \(L^2(\Omega)\), consider the Neumann Schrödinger operator with magnetic field
\[
\mathcal{P}_h = (-\text{i}h\nabla + \mathbf{A})^2.
\]
The main result in the paper under review is a semi-classical formula for the sum of eigenvalues of \(\mathcal{P}_h\). The leading order term is described by means of the eigenvalues of auxiliary operators on the semi-axis and the half-plane. This result can be seen as a generalization of the results for two-dimansional Schrödinger operators from [S. Fournais and A. Kachmar, J. Lond. Math. Soc., II. Ser. 80, No. 1, 233–255 (2009; Zbl 1179.35203)].
Reviewer: Aleksey Kostenko (Donetsk)
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35J10 | Schrödinger operator, Schrödinger equation |
Citations:
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