论文标题
通过定量估计值与分数Shubin操作员相关的演化方程的无零可控性
Null-controllability of evolution equations associated with fractional Shubin operators through quantitative Agmon estimates
论文作者
论文摘要
我们认为各向异性Shubin运算符$（-δ）^m + \ vert x \ vert^{2k} $作用于太空$ l^2（\ Mathbb r^n）$，带有$ k，m \ geq1 $一些正整数。我们在Gelfand-Shilov空间中为这些自我差异差异运算符的特征函数提供了尖锐的定量估计值，即这些功能的指数衰减估计既可以估计这些功能，又在$ l^2（\ Mathbb r^n）中的傅立叶变换。实施的策略是基于获得光谱理论中AGMON估计值的经典方法。通过为各向异性舒宾操作员的特征值使用Weyl定律，我们还描述了这些操作员分数产生的半群的平滑性能，并在短时间内进行了精确的估计。此描述使我们能够从整个空间$ \ Mathbb r^n $上的相关演变方程获得正零可控性结果，这是从相对于密度和任何积极时间较厚的控制支持。我们概括了与分数谐波振荡器相关的演化方程已知的结果。
We consider the anisotropic Shubin operators $(-Δ)^m + \vert x\vert^{2k}$ acting on the space $L^2(\mathbb R^n)$, with $k, m \geq1$ some positive integers. We provide sharp quantitative estimates in Gelfand-Shilov spaces for the eigenfunctions of these selfadjoint differential operators, that is, exponential decay estimates both for these functions and their Fourier transforms in $L^2(\mathbb R^n)$. The strategy implemented is based on the classical approach to obtain Agmon estimates in spectral theory. By using a Weyl law for the eigenvalues of the anisotropic Shubin operators, we also describe the smoothing properties of the semigroups generated by the fractional powers of these operators, with precise estimates in short times. This description allows us to prove positive null-controllability results for the associated evolution equations posed on the whole space $\mathbb R^n$, from control supports which are thick with respect to densities and in any positive time. We generalize in particular results known for the evolution equations associated with fractional harmonic oscillators.
