Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials

Abstract

Abstract. We consider the unperturbed operatorHere 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 2 R of .x; y/ 2 R 2 . Then the spectrum of H 0 has a band structure and is absolutely continuous; moreover, the assumption lim x!1 .W .x/ W . x// < 2b implies the existence of infinitely many spectral gaps for H 0 . We consider the perturbed operators H˙D H 0˙V where the electric potential V 2 L 1 .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 C (resp. H ) to the lower (resp. upper) edge of a given spectral gap, is Gaussian.Mathematics Subject Classification (2010). 35P20, 35J10, 47F05, 81Q10.

Eigenvalue Asymptotics for a Schrödinger Operator with Non-Constant Magnetic Field Along One Direction

“…In that context the main asymptotic term is (ln | ln λ|) −1 | ln λ| which goes to infinity faster than the one presented here, | ln λ| 1/2 , implying that the accumulation of the eigenvalues is stronger in our case. Similar results to our was previously obtained in [3], [4], for other magnetic Hamiltonians with compact supported electric potentials (see Remark after Theorem 2.1).…”

“…Remark : Similar results to Theorem 2.1 appear in [3] and [4]. In [3] the discrete spectrum of operators of the form…”

“…Using the Ky-Fan inequalities (3.2) we get (3.22) (see Proposition 3.1 and Theorem 3.2 in [3] for a detailed proof of a similar result). Now we are ready to finish the proof of Theorem 2.1.…”

Threshold singularities of the spectral shift function for a half-plane magnetic Hamiltonian

“…Since our extremal points are only the limits on the band functions, it is necessary to modify the analysis of previous works. The phenomenon of thresholds given by limits of the band functions is also present for some quantum Hall effect models (see [6]) and for some Iwatsuka models (see [26]). In these works, the SSF was studied only in the region where it counts the number of discrete eigenvalues.…”

Spectral asymptotics for two-dimensional Schrödinger operators with strong magnetic fields

“…By applying the Mourre theory, they proved that a part of the absolutely continuous spectrum of P (b, ω) persists. On the other hand, we can consider the model (1.1) as the quantum hall system Hamiltonian with the unbounded edge potential W (x) = ω 2 x 2 (see [4,6,20]). In this work, we give a complete asymptotic expansion in powers of b −1 of the trace of the operators f (P (b, ω)) and f (P (b, ω))F and the stationary techniques developed by M. Dimassi [7] (see also M. Dimassi-J.…”