Magnetic bottles in connection with superconductivity. (English) Zbl 1078.81023
Let \(\Omega\subset{\mathbb R}^2\) be an open set, and let \(P_{h,A,\Omega}=(hD_{x_1}-A_1)^2+ (hD_{x_2}-A_2)^2,\) where \(h>0\) is a small parameter. In this paper, the authors treat in a systematic way and sharpen some results present in the literature on the lowest eigenvalue of \(P_{h,A,\Omega}\) in the Dirichlet and Neumann realizations, respectively, and give accurate estimates on the localization of the ground-state in the case of the Neumann realization. In particular, they prove the following result conjectured by A. Bernoff and P. Sternberg [J. Math. Phys. 39, 1272–1284 (1998; Zbl 1056.82523)].
Theorem: Suppose that the magnetic field \(B=\partial_{x_1}A_2-\partial_{x_2}A_1\) is a non-zero constant. Then any normalized ground-state of the Neumann realization of \(P_{h,A,\Omega}\) is exponentially localized as \(h\to 0\) in the neighborhood of the points of the boundary \(\partial\Omega\) with maximal curvature.
In the final section of the paper, they indicate a number of interesting open problems.
In the final section of the paper, they indicate a number of interesting open problems.
Reviewer: Alberto Parmeggiani (Bologna)
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35Q40 | PDEs in connection with quantum mechanics |
| 82D55 | Statistical mechanics of superconductors |
Keywords:
Schrödinger operators; semiclassical analysis; lowest eigenvalue; localization of eigenfunctionsCitations:
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