High-energy eigenfunctions of point perturbations of the Laplacian on the spheres $\mathbb{S}^{2}$ and $\mathbb{S}^{3}$

TL;DR

This paper investigates how point scatterers on spheres influence the semiclassical measures of eigenfunctions, revealing conditions under which these measures are invariant or not, thus advancing understanding of quantum limits in perturbed geometries.

Contribution

It demonstrates that adding point scatterers can produce non-invariant semiclassical measures, especially when antipodal points are included, contrasting with the unperturbed case.

Findings

All invariant measures can be realized as semiclassical measures with point scatterers.

Presence of antipodal scatterers allows construction of non-invariant semiclassical measures.

Without antipodal points, invariant and semiclassical measures coincide.

Abstract

We study the set of Quantum Limits, and more generally, of semiclassical measures of sequences of eigenfunctions of perturbations of the Laplacian on the spheres $\mathbb{S}^{2}$ and $\mathbb{S}^{3}$ by point-scatterers. In the unperturbed case, it is known that the set of semiclassical measures coincides with the set of measures that are invariant under the geodesic flow; on the other hand, when the Laplacian is perturbed by a generic smooth potential, the set of semiclassical measures turns out to be strictly contained within that of invariant measures. In this article, we prove that the addition of a perturbation by a finite set of point-scatterers has a different effect: (i) all invariant measures are semiclassical measures for some sequence of eigenstates of the perturbed operator, and (ii) as soon as the set of scatterers contains a pair of antipodal points, it is possible to construct a sequence of eigenfunctions whose semiclassical measure is not invariant under the geodesic flow. We also show that this geometric condition is sharp: if the set of scatterers does not contain a pair of antipodal points, then the sets of invariant and semiclassical measures coincide.