JournalsjstVol. 1, No. 3pp. 237–272

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

  • Vincent Bruneau

    Université Bordeaux 1, Talence, France
  • Pablo L. Miranda

    Universidad de Chile, Santiago, Chile
  • Georgi Raikov

    Pontificia Universidad Católica de Chile, Santiago de Chile, Chile
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Abstract

We consider the unperturbed operator H0:=(iA)2+WH_0 : = (-i \nabla - A)^2 + W, self-adjoint in L2(R2)L^2(\mathbb{R}^2). Here AA is a magnetic potential which generates a constant magnetic field b>0b>0, and the edge potential WW is a non-decreasing non constant bounded function depending only on the first coordinate xRx \in \mathbb{R} of (x,y)R2(x,y) \in \mathbb{R}^2. Then the spectrum of H0H_0 has a band structure and is absolutely continuous; moreover, the assumption limx(W(x)W(x))<2b\lim_{x \to \infty}(W(x) - W(-x)) < 2b implies the existence of infinitely many spectral gaps for H0H_0. We consider the perturbed operators H±=H0±VH_{\pm} = H_0 \pm V where the electric potential VL(R2)V \in L^{\infty}(\mathbb{R}^2) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H±H_\pm in the spectral gaps of H0H_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 VV. Further, we restrict our attention on perturbations VV of compact support and constant sign. We establish a geometric condition on the support of VV which guarantees the finiteness of the eigenvalues of H±H_{\pm} in any spectral gap of H0H_0. In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of H+H_+ (resp. HH_-) to the left (resp. right) edge of a given spectral gap, is Gaussian.

Cite this article

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. 237–272

DOI 10.4171/JST/11