Singular Continuous Limiting Eigenvalue Distributions for Schrِdinger Operators on a 2-Sphere

LetH=−Δ+Vbe a Schrödinger operator acting inL2(S), withSthe two-dimensional unit sphere,Δthe spherical Laplacian, andVa continuous potential. As is well known, the eigenvalues ofHin thelth cluster, i.e., those eigenvalues within a radius sup |V| ofl(l+1), thelth eigenvalue of −Δ, have a limiting distribution;l→∞. We provide an alternative self-contained proof of this fact. We then exhibit Hölder continuous potentialsV, both axially- and nonaxially-symmetric, for which the limiting distributions are singular continuous.