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This work is devoted to the study of uncertainty principles for finitecombinations of Hermite functions. We establish some spectral inequalities forcontrol subsets that are thick with respect to some unbounded densities growingalmost linearly at infinity, and provide quantitative estimates, with respectto the energy level of the Hermite functions seen as eigenfunctions of theharmonic oscillator, for the constants appearing in these spectral estimates.These spectral inequalities allow to derive the null-controllability in anypositive time for evolution equations enjoying specific regularizing effects.More precisely, for a given index $\frac{1}{2} \leq \mu <1$, we deducesufficient geometric conditions on control subsets to ensure thenull-controllability of evolution equations enjoying regularizing effects inthe symmetric Gelfand-Shilov space $S^{\mu}_{\mu}(\mathbb{R}^n)$. These resultsapply in particular to derive the null-controllability in any positive time forevolution equations associated to certain classes of hypoellipticnon-selfadjoint quadratic operators, or to fractional harmonic oscillators.

Spectral inequalities for combinations of Hermite functions and  null-controllability for evolution equations enjoying Gelfand-Shilov  smoothing effects