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

TL;DR

This paper studies high-frequency eigenfunction localization of Schrödinger operators with inverse square potentials in high-dimensional balls and shells, revealing a two-parameter phase transition depending on the ratio of quantum numbers.

Contribution

It extends classical one-parameter localization results to a two-parameter setting and uncovers a second-order phase transition in eigenfunction localization in spherical shells.

Findings

Eigenfunctions in balls localize around an intermediate sphere with exponential decay inside and polynomial decay outside.

A second-order phase transition occurs in spherical shells as the ratio crosses a critical value.

Localization behavior varies: around an intermediate sphere, inner boundary, or no localization, depending on the ratio.

Abstract

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.