arXiv subjects

Roman Vanlaere

Publications and source records attributed to Roman Vanlaere.

3 recordsLinked to original sources

arXiv2026

Observability properties of the singular Grushin equation

We study the observability properties of the Grushin equation with an inverse square potential, whose singularity occurs at the boundary of two-dimensional rectangular domains or in the interior of the domain in higher dimensions. In some specific configurations of the observation set, we obtain the exact minimal time of observability. The analysis we present relies on recent Carleman estimates obtained by K. Beauchard, J. Dard\'e, and S. Ervedoza. As a byproduct of these results, we observe, for the heat equation associated to the Laplace-Beltrami operator on almost-Riemannian manifolds, a dependence of the minimal time of observability on the dimension of the singularity.

math.AP↗

arXiv2025

Spectral asymptotics for a class of singular Sturm-Liouville operators with applications to magnetic Laplacian and a-zeros of Kummer functions

We provide a precise description of the bottom of the spectrum in the semiclassical limit of a harmonic-type Schr\"odinger operator with an inverse square potential. By exploiting the connection between the eigenfunctions of these operators and the Kummer and Whittaker functions, we derive accurate localization results for the non-asymptotic zeros of these functions with respect to their first parameter, uniformly with respect to the argument taken large and real. Moreover, our operators are linked to the magnetic Dirichlet Laplacian in the presence of both a constant magnetic field and an Aharonov-Bohm flux line, so that our results describe its spectrum in the strong magnetic field limit. Our spectral analysis relies on a WKB-type approach.

math.SP↗

arXiv2025

Non null-controllability properties of the Grushin-like heat equation on 2D-manifolds

We study the internal non null-controllability properties of the heat equation on 2-dimensional almost-Riemannian manifolds with an interior singularity, and under the assumption that the closure of the control zone does not contain the whole singularity. We show that if locally, around the singularity, the sub-Riemannian metric can be written in a Grushin form, or equivalently the sub-Laplacian writes as a generalized Grushin operator, then, achieving null-controllability requires at least a minimal amount of time. As locally the manifold looks like a rectangular domain, we consequently focus ourselves on the non null-controllability properties of the generalized Grushin-like heat equation on various Euclidean domains.

math.OC↗