Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian. (English) Zbl 1097.47020
The paper deals with the study of eigenvalues near the bottom of the spectrum of the magnetic Schrödinger operator with Neumann boundary conditions in a smooth, bounded domain \(\Omega\):
\[
D(\mathcal{H})\ni u \rightarrowtail \mathcal{H}u = \mathcal{H}_{h,\Omega}u = (-ih\nabla_z-A(z))^2u(z),
\]
where \(A(z)\) is a vector potential generating a constant magnetic field with \(\operatorname{curl} A = 1\) and
\[
D(\mathcal{H}) =\{u\in H^2(\Omega) | \nu \cdot (-ih\nabla_z - A(z))u| _{\partial\Omega}=0\}.
\]
The main result of the paper gives the asymptotic expansion of the lowest eigenvalues of \(\mathcal{H}\). The results are interesting from the point of view of their applications to superconductivity.
Reviewer: Maksim Sokolov (Tashkent)
MSC:
| 47A75 | Eigenvalue problems for linear operators |
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
| 35Q40 | PDEs in connection with quantum mechanics |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
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