Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials
Résumé
We consider the unperturbed operator H-0 = (-i del - A)(2) + W, self-adjoint in L-2(R-2). Here A is a magnetic potential which generates a constant magnetic field b > 0, and the edge potential W is a non-decreasing non-constant bounded function depending only on the first coordinate x is an element of R of (x, y) is an element of R-2. Then the spectrum of H-0 has a band structure and is absolutely continuous; moreover, the assumption lim(x ->infinity)(W(x) - W(-x)) < 2b implies the existence of infinitely many spectral gaps for H-0. We consider the perturbed operators H-+/- = H-0 +/- V where the electric potential V is an element of L-infinity(R-2) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H-+/- in the spectral gaps of H-0. 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 V. Further, we restrict our attention on perturbations V of compact support and constant sign. We establish a geometric condition on the support of V which guarantees the finiteness of the number of the eigenvalues of H-+/- in any spectral gap of H-0. In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of H+ (resp. H-) to the lower (resp. upper) edge of a given spectral gap, is Gaussian.