Abstract
We investigate the ground state of a two-dimensional quantum particle in a magnetic field where the field vanishes nondegenerately along a closed curve. We show that the ground state concentrates on this curve ase/h tends to infinity, wheree is the charge, and that the ground state energy grows like (e/h)2/3. These statements are true for any energy level, the level being fixed as the charge tends to infinity. If the magnitude of the gradient of the magnetic field is a constantb 0 along its zero locus, then we get the precise asymptotics(e/h) 2/3(b 0)2/3 E * +O(1) for every energy level. The constantE * ≃ .5698 is the infimum of the ground state energiesE(β) of the anharmonic oscillator family\( - \frac{{d^2 }}{{dy^2 }} + \left( {\frac{1}{2}y^2 - \beta } \right)^2\).
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Montgomery, R. Hearing the zero locus of a magnetic field. Commun.Math. Phys. 168, 651–675 (1995). https://doi.org/10.1007/BF02101848
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DOI: https://doi.org/10.1007/BF02101848