Geometric scaling laws for quantum reflection at singular corners

This is the authors' abstract. We don't add key points for this paper.

Quantum reflection is usually described as an effectively one-dimensional phenomenon governed by motion along the surface normal. However, when translational symmetry is broken by sharp geometric features, scattering becomes intrinsically multidimensional. Here, we show that singular corners generate new geometric scaling laws for quantum reflection. In a minimal two-dimensional corner model with an attractive quadrant interaction, the reflected scattering width scales as σ refl ∝ E − 2 in the intermediate Born regime, followed at lower energies by a crossover to the unitarity-limited threshold law σ refl ∝ E − 1 / 2 . The E − 2 exponent arises from two geometric ingredients: half-space support of the interaction and specular boundary-layer kinematics. More generally, for corner-supported potentials of the form V ( x , z ) = W ( x , z ) Θ ( x ) Θ ( z ) , we show that the Born exponent is universal under mild regularity assumptions. The leading coefficient is asymptotically determined by a specular boundary layer and depends only on the boundary trace W ( x , 0 ) , while microscopic details away from the boundary enter only through subleading structure. These results identify corners as the simplest surface singularities that support genuinely multidimensional geometric scaling laws, showing that geometry alone can control reflection across different energy regimes beyond the standard one-dimensional framework.