Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability

We prove quantitative spectral inequalities for the (anisotropic) Shubin operators on the whole Euclidean space, thus relating for functions from spectral subspaces associated to finite energy intervals their <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -norm on the whole space to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -norm on a suitable subset. A particular feature of our estimates is that the constant relating these <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -norms is very explicit in geometric parameters of the corresponding subset of the whole space, which may become sparse at infinity and may even have finite measure. This extends results obtained recently by J. Martin and, in the particular case of the harmonic oscillator, by A. Dicke, I. Veselić, and the second author. We apply our results towards null-controllability of the associated parabolic equations, as well as to the ones associated to the (degenerate) Baouendi-Grushin operators acting on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>𝕋</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> .