arXiv subjects

Santiago Verdasco

Publications and source records attributed to Santiago Verdasco.

3 recordsLinked to original sources

arXiv2026

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

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.

math.SP↗

arXiv2026

Quantum Limits of the Laplacian perturbed along a geodesic on $\mathbb{S}^{2}$

This article studies the high-frequency behavior of eigenstates of perturbations of the Laplace-Beltrami operator on the two-sphere $\mathbb{S}^{2}$ by a measure supported on an equator. We are interested in understanding to what extent this behavior can be described in terms of the geodesic flow of the sphere. This is done by analyzing quantum limits and semiclassical measures of sequences of high-frequency eigenfunctions, which describe how their $L^2$-masses concentrate in phase space. When the Laplacian on $\mathbb{S}^{2}$ is perturbed by a bounded potential, it is known that the family of all possible semiclassical measures is contained in the set of positive measures on the unit cosphere bundle $S^*\mathbb{S}^{2}$ that are invariant under geodesic flow (with equality in the unperturbed case). In this article, we show that the presence of a singular delta potential on a closed geodesic results in the existence of sequences of eigenfunctions whose semiclassical measure is not invariant under geodesic flow. In particular, one can find a sequence of eigenfunctions whose energy asymptotically concentrates on the hemisphere bounded by the equator on which the potential is concentrated.

math.SP↗

arXiv2026

High-energy eigenfunctions of point-perturbations of the Laplacian

In this paper, we explore the high-frequency properties of eigenfunctions of point perturbations of the Laplacian on a compact Riemannian manifold. These systems cannot be obtained as the quantization of a classical Hamiltonian, as the effect of the perturbation amounts to prescribing certain boundary conditions on a discrete set of points. We are interested in understanding to what extent the high-frequency behavior of eigenfunctions is governed by the global dynamics of the geodesic flow in the manifold (the classical flow corresponding to the unperturbed Laplacian). We prove that as soon as the Laplacian is perturbed by a finite set of point scatterers satisfying a \emph{non-focality} condition, namely, that the family of geodesics starting from this set and coming back to it has zero measure, semiclassical measures corresponding to high-frequency sequences of eigenfunctions are invariant under the geodesic flow. Invariance may fail when the non-focality condition does not hold, as is shown in a companion article [arXiv:2601.19701]. Our results are based on a quasimode construction that requires improved estimates on the spectral function of the Laplacian on the set of scatterers.

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