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

Abstract

We consider the unperturbed operator $H_0:=(-i\nabla -A{)}^2+W$, self-adjoint in $L^2({\mathbb{R}}^2)$. 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 $x\in \mathbb{R}$ of $(x,y)\in {\mathbb{R}}^2$. Then the spectrum of $H_0$ has a band structure and is absolutely continuous; moreover, the assumption ${\lim }_{x\to \infty }(W(x)-W(-x))<2b$ implies the existence of infinitely many spectral gaps for $H_0$. We consider the perturbed operators $H_{\pm }=H_0\pm V$ where the electric potential $V\in L^{\infty }({\mathbb{R}}^2)$ is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of $H_{\pm }$ 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 eigenvalues of $H_{\pm }$ 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 left (resp. right) edge of a given spectral gap, is Gaussian.