Uncertainty Principle for Hermite Functions and Null-Controllability with Sensor Sets of Decaying Density

Abstract

We establish a family of uncertainty principles for finite linear combinations of Hermite functions. More precisely, we give a geometric criterion on a subset $$S\subset \mathbb {R}^d$$ S ⊂ R d ensuring that the $$L^2$$ L 2 -seminorm associated to S is equivalent to the full $$L^2$$ L 2 -norm on $$\mathbb {R}^d$$ R d when restricted to the space of Hermite functions up to a given degree. We give precise estimates how the equivalence constant depends on this degree and on geometric parameters of S. From these estimates we deduce that the parabolic equation whose generator is the harmonic oscillator is null-controllable from S. In all our results, the set S may have sub-exponentially decaying density and, in particular, finite volume. We also show that bounded sets are not efficient in this context.