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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.

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 mechanical studies of superconductors
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