We consider the unperturbed operator , self-adjoint in . Here is a magnetic potential which generates a constant magnetic field , and the edge potential is a non-decreasing non constant bounded function depending only on the first coordinate of . Then the spectrum of has a band structure and is absolutely continuous; moreover, the assumption implies the existence of infinitely many spectral gaps for . We consider the perturbed operators where the electric potential is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of in the spectral gaps of . We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian involves a pseudo-differential operator with generalized anti-Wick symbol equal to . Further, we restrict our attention on perturbations of compact support and constant sign. We establish a geometric condition on the support of which guarantees the finiteness of the eigenvalues of in any spectral gap of . In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of (resp. ) to the left (resp. right) edge of a given spectral gap, is Gaussian.
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Vincent Bruneau, Pablo L. Miranda, Georgi Raikov, Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials. J. Spectr. Theory 1 (2011), no. 3, pp. 237–272DOI 10.4171/JST/11