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

Two-parameter localization and related phase transition for a Schr\"{o}dinger operator in balls and spherical shells