Curvature-amplified angular localization of radially localized states
We study a continuum Schr\"odinger particle on a surface of constant curvature subject to a radial random potential and a weaker angle-dependent perturbation. Exact angular-momentum decomposition reduces the rotationally symmetric problem to independent one-dimensional random radial channels. A positive radial Lyapunov exponent distinguishes radial Anderson localization from the circular probability profile imposed by symmetry. Projection of a narrow radial mode produces a disordered one-dimensional ring. For angular disorder that is stationary in physical arc length, the ring localization length grows as the inverse square of the perturbation amplitude. Equating this length with the geodesic circumference yields a shell-breaking radius that grows algebraically in flat space but as $2a\ln(1/\varepsilon)+O(1)$ on a hyperbolic surface of curvature radius $a$ and angular-disorder amplitude $\varepsilon$. Transfer-matrix and finite-ring calculations determine the crossover coefficient. Direct polar-grid calculations at the same angular energy as the ring model use unbiased tracking of the degenerate angular-momentum subspace across six independent radial-disorder realizations and quantify the validity of the single-mode projection. The noncommuting weak-disorder and flat-curvature limits establish a curvature-amplified crossover for this deliberately anisotropic disorder ensemble, rather than a generic metal--insulator transition on the hyperbolic plane.