Landau-level splitting by a circular $\delta$ interaction: boundary overlap and the hard-wall crossover

Abstract

A thin circular barrier lifts the angular momentum degeneracy of Landau states according to their overlap with the barrier. We quantify this effect for the two-dimensional Landau Hamiltonian with a circular $\delta$ interaction of radius $a$ and strength $\alpha$. Starting from the known Kummer-function matching equation, we separate the contribution of one Landau level from the remaining radial spectrum. For fixed $n$, $B>0$, $a>0$, and $\alpha\ne0$, we prove $$ E_{n,m}^{(\alpha)}-B(2n+1) =\alpha c_{n,m}\left[1-\frac{\alpha a}{2m}+O(m^{-2})\right] $$ as $m\to+\infty$, where $c_{n,m}$ is an explicit boundary trace weight with factorial decay. The proof gives a uniform estimate of the regular part of the radial resolvent, including at the Landau energy. This estimate also yields an expansion uniform over all repulsive strengths, with crossover parameter $\alpha a/(2m)$, and the exterior Dirichlet limit. Thus a small energy shift does not by itself guarantee relative accuracy of the projected energy shift: the correction depends on the wall strength compared with the local centrifugal scale. The state nevertheless remains close in norm to the chosen Landau subspace, uniformly for repulsion. We also obtain the boundary density from the response to $\alpha$ and distinguish nodal Landau states from coupling-dependent level crossings. High-precision calculations test both the fixed-strength expansion and the crossover to a hard wall.