Two-parameter localization and related phase transition for a Schrödinger operator in balls and spherical shells

Abstract

Here we investigate the two-parameter high-frequency localization for the eigenfunctions of a Schrödinger 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.

Figures (5)

• Figure 1: Eigenfunctions and localized radii of the Laplacian and Schrödinger operators. (a) An eigenfunction of the two-dimensional Laplace operator when $l = 0$ and $k = 2$, which does not collapse at the origin. (b) An eigenfunction of the two-dimensional Schrödinger operator when $c = 1$, $l = 0$, and $k = 2$, which collapses at the origin. (c) The localized radius of Schrödinger eigenfunctions versus the $l$-$k$ ratio $w$ as $l,k\rightarrow\infty$ simultaneously.
• Figure 2: Two-parameter localization for Schrödinger eigenfunctions in the unit disk. As $l,k\rightarrow\infty$ while keeping $l/k\rightarrow w>0$, the eigenfunctions $u_{klm}$ are localized around a sphere with radius $1/h(w)\in(0,1)$. The eigenfunctions are exhibited by both the three-dimensional graphs (upper) and two-dimensional heat maps with colorbars (lower). The eigenfunctions decay exponentially inside the localized circle and decay polynomially outside. (a) $l = 5$ and $k = 1$. (b) $l = 100$ and $k = 20$. (c) $l = 10000$ and $k = 200$. In (a)-(c), we keep the $l$-$k$ ratio as $w = 5$, in which case the localized radius is $1/h(5)\approx 0.5$. The parameter $c = 1$ in all cases and the eigenfunctions are normalized so that the supreme norm is $1$.
• Figure 3: Localized radii, critical values, and localized indices for Schrödinger eigenfunctions in spherical shells. (a) The localized radius $w/g_R(w)$ of the eigenfunctions versus the $l$-$k$ ratio $w$ under different choices of the outer radius $R$ as $l,k\rightarrow\infty$ simultaneously. (b) The critical value $s(R)$ versus the outer radius $R$. (c) The localized index $\gamma_{R,\epsilon}(w)$ versus the $l$-$k$ ratio $w$ in the case of $R = 3$ under different choices of $\epsilon$, which characterizes the width of the neighborhood of the localized sphere.
• Figure 4: Dynamical phase transition for Schrödinger eigenfunctions in an annulus. (a)-(c) Heat maps of the eigenfunctions under different choices of $l$ and $k$ in the supercritical case, where the outer radius is chosen as $R = 3$ and the $l$-$k$ ratio is chosen as $w = 10 > s(R)\approx 1.972$. When $l$ and $k$ are large, the eigenfunctions are localized around the circle with radius $w/g_R(w)\approx 2$. The eigenfunctions decay exponentially inside the localized circle and decay polynomially outside. (a) $k = 1$. (b) $k = 10$. (c) $k = 100$. (d)-(f) Heat maps of the eigenfunctions under different choices of $l$ and $k$ in the subcritical case, where the outer radius is chosen as $R = 3$ and the $l$-$k$ ratio is chosen as $w = 1$. The eigenfunctions fail to be localized in this case. (d) $k = 10$. (e) $k = 20$. (f) $k = 50$. The parameter $c = 1$ in all cases and the eigenfunctions are normalized so that the supreme norm is $1$.
• Figure 5: Whispering gallery modes, critical modes, and the breaking of focusing modes for the Schrödinger operator in spherical shells. (a) Whispering gallery modes, where we take $l = 50$ and $k = 1$. (b) Critical modes, where we take $l = 197500$ and $k = 10000$. (c),(d) Breaking of focusing modes. (c) $l = 0$ and $k = 20$. (d) $l = 1$ and $k = 20$. The parameter $c = 1$ in all cases and the eigenfunctions are normalized so that the supreme norm is $1$.

Theorems & Definitions (30)

• Lemma \oldthetheorem
• proof
• Theorem \oldthetheorem
• proof
• Theorem \oldthetheorem
• proof
• Remark \oldthetheorem: localized radii
• Remark \oldthetheorem: decaying speed
• Corollary \oldthetheorem
• Corollary \oldthetheorem