2022/12/21 by Paul Alphonse, Alphonse, Paul, Albrecht Seelmann +1 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Holomorphic and Operator Theory #Optimization and Control (math.OC) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2212.10842
openalex publication_date 2022/12/21 · openalex created_date 2023/01/26 · openalex updated_date 2026/07/28
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 L2-norm on the whole space to the L2-norm on a suitable subset. A particular feature of our estimates is that the constant relating these L2-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 \mathbb Rd × \mathbb Td.