Inverse problems for quantum trees
Inverse problems for quantum trees
复制标题
DOI：
10.3934/ipi.2008.2.1
复制
发表时间：
2008
影响因子：
1.3
通讯作者：
S. Avdonin;P. Kurasov
中科院分区：
文献类型：
--
作者：
S. Avdonin;P. Kurasov
三个不同的反问题的薛定谔算子的度量树被认为是迄今为止，标准的边界条件的顶点。这些反问题与矩阵Titchmarsh-Weyl函数，响应算子（动态Dirichlet-to-Neumann映射）和散射矩阵有关。我们的方法是基于边界控制（BC）方法，特别是对响应算子的研究。证明了响应算子完全决定了量子树的连通性、边的长度以及边上的势。如果除了一个边界点之外，所有的响应算子都是已知的，以及Titchmarsh-Weyl函数和散射矩阵也是如此。如果图的连通性是已知的，那么边的长度和相应的势由数据的对角项确定。
Three different inverse problems for the Schrodinger operator on a metric tree are considered, so far with standard boundary conditions at the vertices. These inverse problems are connected with the matrix Titchmarsh-Weyl function, response operator (dynamic Dirichlet-to-Neumann map) and scattering matrix. Our approach is based on the boundary control (BC) method and in particular on the study of the response operator. It is proven that the response operator determines the quantum tree completely, i.e. its connectivity, lengths of the edges and potentials on them. The same holds if the response operator is known for all but one boundary points, as well as for the Titchmarsh-Weyl function and scattering matrix. If the connectivity of the graph is known, then the lengths of the edges and the corresponding potentials are determined by just the diagonal terms of the data.