论文标题
广义schrödinger方程和产品歧管上的量子限制的可观察性
Observability for generalized Schrödinger equations and quantum limits on product manifolds
论文作者
论文摘要
Given a closed product Riemannian manifold N = M x M equipped with the product Riemannian metric g = h + h , we explore the observability properties for the generalized Schr{ö}dinger equation i$\partial$ t u = F (g)u, where g is the Laplace-Beltrami operator on N and F : [0, +$\infty$) $\rightarrow$ [0, +$ \ infty $）是一个增加的功能。在本说明中，我们在满足所谓的垂直几何控制条件的任何开放子集$ω$上在有限时间内证明了可观察性，并在其他假设F（g）的频谱满足差距条件的情况下规定了任何垂直测量符合$ω$。第一个结果是，Schr {Ö} dinger方程的$ω$的可观察性比任何球体上的任何产品的常见几何控制条件都严格弱。第二个结果是，沿氮的任何测量沿线的狄拉克度量绝不是量子限制。
Given a closed product Riemannian manifold N = M x M equipped with the product Riemannian metric g = h + h , we explore the observability properties for the generalized Schr{ö}dinger equation i$\partial$ t u = F (g)u, where g is the Laplace-Beltrami operator on N and F : [0, +$\infty$) $\rightarrow$ [0, +$\infty$) is an increasing function. In this note, we prove observability in finite time on any open subset $ω$ satisfying the so-called Vertical Geometric Control Condition, stipulating that any vertical geodesic meets $ω$, under the additional assumption that the spectrum of F (g) satisfies a gap condition. A first consequence is that observability on $ω$ for the Schr{ö}dinger equation is a strictly weaker property than the usual Geometric Control Condition on any product of spheres. A second consequence is that the Dirac measure along any geodesic of N is never a quantum limit.
