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We consider the anisotropic Shubin operators $(-\Delta)^m + \vertx\vert^{2k}$ acting on the space $L^2(\mathbb R^n)$, with $k, m \geq1$ somepositive integers. We provide sharp quantitative estimates in Gelfand-Shilovspaces for the eigenfunctions of these selfadjoint differential operators, thatis, exponential decay estimates both for these functions and their Fouriertransforms in $L^2(\mathbb R^n)$. The strategy implemented is based on theclassical approach to obtain Agmon estimates in spectral theory. By using aWeyl law for the eigenvalues of the anisotropic Shubin operators, we alsodescribe the smoothing properties of the semigroups generated by the fractionalpowers of these operators, with precise estimates in short times. Thisdescription allows us to prove positive null-controllability results for theassociated evolution equations posed on the whole space $\mathbb R^n$, fromcontrol supports which are thick with respect to densities and in any positivetime. We generalize in particular results known for the evolution equationsassociated with fractional harmonic oscillators.

Null-controllability of evolution equations associated with fractional Shubin operators through quantitative Agmon estimates