论文标题
与分数Schrödinger操作员有关的可观察性结果
Observability results related to fractional Schrödinger operators
论文作者
论文摘要
我们针对涉及分数schrödinger操作员的各种问题建立可观察性不平等，$（δ）^{α/2}+v $，$α> 0 $，在紧凑的Riemannian歧管上。证明，一旦观察集满足几何控制条件，就证明了与$α> 1 $的相应分数Schrödinger演化方程的可观察性。还表明，当歧管是配备标准度量的$ d $维球体时，这种情况是必要的。这与本征函数的情况形成鲜明对比。我们在两个球体上构建电势，并在球体上存在两个点，以使$-Δ+V $的特征函数从这两个点的任意小社区中均匀观察到。该条件比几何控制条件弱得多，这对于在球体上的自由拉普拉斯式的本征函数均匀观察到必不可少。对于任何$α> 0 $，对于$（-δ）^{α/2}+V $的本征函数也相同。
We establish observability inequalities for various problems involving fractional Schrödinger operators $(-Δ)^{α/2}+V$, $α>0$, on a compact Riemannian manifold. Observability from an open set for the corresponding fractional Schrödinger evolution equation with $α>1$ is proved to hold as soon as the observation set satisfies the Geometric Control Condition; it is also shown that this condition is necessary when the manifold is the $d$-dimensional sphere equipped with the standard metric. This is in stark contrast with the case of eigenfunctions. We construct potentials on the two-sphere with the property that there exist two points on the sphere such that eigenfunctions of $-Δ+V$ are uniformly observable from an arbitrarily small neighborhood of those two points. This condition is much weaker than the Geometric Control Condition, which is necessary for uniform observability of eigenfunctions for the free Laplacian on the sphere. The same result also holds for eigenfunctions of $(-Δ)^{α/2}+V$, for any $α>0$.
