  Here we investigate the two-parameter high-frequency localization for the
eigenfunctions of a Schr\"{o}dinger operator with a singular inverse square
potential in high-dimensional balls and spherical shells as the azimuthal
quantum number $l$ and the principal quantum number $k$ tend to infinity
simultaneously, while keeping their ratio as a constant, generalizing the
classical one-parameter localization for Laplacian eigenfunctions [SIAM J.
Appl. Math. 73:780-803, 2013]. We prove that the eigenfunctions in balls are
localized around an intermediate sphere whose radius is increasing with respect
to the $l$-$k$ ratio. The eigenfunctions decay exponentially inside the
localized sphere and decay polynomially outside. Furthermore, we discover a
novel second-order phase transition for the eigenfunctions in spherical shells
as the $l$-$k$ ratio crosses a critical value. In the supercritical case, the
eigenfunctions are localized around a sphere between the inner and outer
boundaries of the spherical shell. In the critical case, the eigenfunctions are
localized around the inner boundary. In the subcritical case, no localization
could be observed, giving rise to localization breaking.
