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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.

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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