  In this paper, new spectral inequalities for finite combinations of
eigenfunctions of anisotropic Shubin operators are presented. Given a subset
$\omega$ and an energy level, we provide an explicit control of the ratio of
the L 2 (R d)-norm over the L 2 ($\omega$)-norm with respect to the energy
level. The proofs are based on recent uncertainty principles holding in
Gelfand-Shilov spaces and Bernstein type estimates deduced from quantitative
smoothing effects proved by Paul Alphonse. These spectral inequalities allow to
derive the null-controllability in any positive time from any control subset
with positive Lebesgue measure of evolution equations associated to anisotropic
Shubin operators, except for the harmonic oscillator.
