Geometric bounds for the magnetic Neumann eigenvalues in the plane. (English. French summary) Zbl 1531.35193
The authors derive upper and lower bounds for the eigenvalues of the magnetic Laplacian on a bounded domain in \(\mathbb{R}^2\). A constant magnetic field and magnetic Neumann boundary conditions are considered. For the ground state, the authors prove a universal upper bound that is strict and given by the intensity of the magnetic field. A new lower bound for a smooth domain is given as well. Semiclassical estimates on eigenvalue averages which are asymptotically sharp are also obtained. For variable magnetic fields, upper bounds for the ground state eigenvalue are given on a Riemann surface. Finally, several examples are provided, such as the case of domains with small width.
Reviewer: Eric Stachura (Marietta)
MSC:
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
Keywords:
magnetic Laplacian; constant field; Neumann eigenvalues; upper and lower bounds; semiclassical estimatesReferences:
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