论文标题
各向异性Shubin操作员的定量光谱不平等以及无效控制的应用
Quantitative spectral inequalities for the anisotropic Shubin operators and applications to null-controllability
论文作者
论文摘要
我们证明了整个欧几里得空间上（各向异性）舒宾算子（各向异性）舒宾算子的定量光谱不等式，因此与与有限的能量间隔相关的光谱子空间的函数涉及整个空间的有限l^2 $ norm到合适的子集中的$ l^2 $ -Norm。我们估计值的一个特殊特征是，将这些$ l^2 $ norms相关的常数在整个空间的相应子集的几何参数中非常明确，在无穷大处可能会稀疏，甚至可能具有有限的度量。这扩展了J. Martin最近获得的结果，在谐波振荡器的特殊情况下，A。Dicke，I。Veselić和第二作者获得了结果。我们将结果应用于相关的抛物线方程的无控制性，以及与（脱位）Baouendi-Grushin操作员相关的结果，该操作员作用于$ \ Mathbb r^d \ times \ times \ times \ mathbb t^d $。
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 $L^2$-norm on the whole space to the $L^2$-norm on a suitable subset. A particular feature of our estimates is that the constant relating these $L^2$-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 $\mathbb R^d \times \mathbb T^d$.
