  We consider the anisotropic Shubin operators $(-\Delta)^m + \vert
x\vert^{2k}$ acting on the space $L^2(\mathbb R^n)$, with $k, m \geq1$ some
positive integers. We provide sharp quantitative estimates in Gelfand-Shilov
spaces for the eigenfunctions of these selfadjoint differential operators, that
is, exponential decay estimates both for these functions and their Fourier
transforms in $L^2(\mathbb R^n)$. The strategy implemented is based on the
classical approach to obtain Agmon estimates in spectral theory. By using a
Weyl law for the eigenvalues of the anisotropic Shubin operators, we also
describe the smoothing properties of the semigroups generated by the fractional
powers of these operators, with precise estimates in short times. This
description allows us to prove positive null-controllability results for the
associated evolution equations posed on the whole space $\mathbb R^n$, from
control supports which are thick with respect to densities and in any positive
time. We generalize in particular results known for the evolution equations
associated with fractional harmonic oscillators.
