Boundary and scattering rigidity problems in the presence of a magnetic field and a potential
Boundary and scattering rigidity problems in the presence of a magnetic field and a potential
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存在磁场和电势时的边界和散射刚度问题
DOI：
10.3934/ipi.2015.9.935
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Hanming Zhou
中科院分区：
文献类型：
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作者：
Yernat M Assylbekov;Hanming Zhou
在本文中，我们考虑一个紧的黎曼流形的边界，赋予一个磁势$\alpha$和一个潜在的$U$。为了简单起见，这种类型的系统被称为$\MP$-系统。在简单$\MP$-系统上，我们同时考虑了边界刚度问题和散射刚度问题，详见引言。我们证明了这两个问题在简单的$\MP$-系统上是等价的。与测地线或磁性系统的情况不同，仅知道一个能级的边界作用函数或散射关系不足以唯一地确定一个简单的$\MP$-系统，即使假设我们知道该系统在边界$\p M$上的限制，我们也提供了一些反例。这些问题只能在$\alpha$和$U$的等距和规范变换下解决。我们证明了刚性结果的度量在一个给定的共形类，简单的真实的分析$\MP$-系统和简单的二维$\MP$-系统。
In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential $\alpha$ and a potential $U$. For brevity, this type of systems are called $\MP$-systems. On simple $\MP$-systems, we consider both the boundary rigidity problem and scattering rigidity problem, see the introduction for details. We show that these two problems are equivalent on simple $\MP$-systems. Unlike the cases of geodesic or magnetic systems, knowing boundary action functions or scattering relations for only one energy level is insufficient to uniquely determine a simple $\MP$-system, even under the assumption that we know the restriction of the system on the boundary $\p M$, and we provide some counterexamples. These problems can only be solved up to an isometry and a gauge transformations of $\alpha$ and $U$. We prove rigidity results for metrics in a given conformal class, for simple real analytic $\MP$-systems and for simple two-dimensional $\MP$-systems.