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

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.