Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density
arXiv:2201.11703 · doi:10.1007/s00041-022-09989-5
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 $S\subset \RR^d$ ensuring that the -seminorm associated to is equivalent to the full -norm on $\RR^d$ 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 . From these estimates we deduce that the parabolic equation whose generator is the harmonic oscillator is null-controllable from . In all our results, the set may have sub-exponentially decaying density and, in particular, finite volume. We also show that bounded sets are not efficient in this context.
Changes compared to previous version: Minor typos corrected, minor editorial changes, two references added, one removed. Manuscript to appear in slightly different form in Journal of Fourier Analysis and Applications with DOI 10.1007/s00041-022-09989-5. Changes compared to manuscript in publication process: Minor editorial changes, several references added
References in corpus (1)
Cited by in corpus (7)
- Observability of the Schr{ö}dinger equation with subquadratic confining potential in the Euclidean space
- Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density
- Spectral inequality with sensor sets of decaying density for Schrödinger operators with power growth potentials
- Uncertainty principles with error term in Gelfand-Shilov spaces
- Unique continuation estimates for Baouendi--Grushin equations on cylinders
- Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs
- Null-controllability of the Generalized Baouendi-Grushin heat like equations