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Spectral Inequalities For Anisotropic Shubin Operators

In this paper, new spectral inequalities for finite combinations of eigenfunctions of anisotropic Shubin operators are presented. Given a subset $ω$ and an energy level, we provide an explicit control of the ratio of the L 2 (R d)-norm over the L 2 ($ω$)-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.