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Hearing the zero locus of a magnetic field. (English) Zbl 0827.58076
This is a study of a two-dimensional particle in a magmetic field that vanishes nondegenerately on a closed curve. The author shows that the ground state concentrates on this curve in the limit \(e/\hbar \to \infty\) where \(e\) is the charge and that the ground state energy grows as \((e/\hbar)^{2/3}\). If the gradient of the magnetic field is of constant magnitude \(b_0\) along its zero locus, then every energy level has asymptotes \((e/\hbar)^{2/3} (b_0)^{2/3}E_* + O(1)\) where \(E_* \approx 0.5698\) is the infimum of the ground state energies of the anharmonic oscillator family \(- {d^2\over dy^2} + ({1\over 2} y^2 - \beta)^2\) where \(\beta\) is a parameter.

MSC:
58Z05 Applications of global analysis to the sciences
82D40 Statistical mechanical studies of magnetic materials
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