We consider the unperturbed operator $H_0 : = (-i \nabla - A)^2 + W$,
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 $x \in \R$
of $(x,y) \in \R^2$. 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 $V \in L^{\infty}(\R^2)$ 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.