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

Abstract

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 $$S\subset \mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> ensuring that the $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -seminorm associated to S is equivalent to the full $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -norm on $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> 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.