标题： 一个带有环的度量图上的反问题 显示英文标题

标题： An inverse problem on a metric graph with cycle

摘要： 考虑一个由环和两个附加边组成的量子图，并假设在内部顶点处满足Kirchhoff-Neumann条件。 与此图相关的是一个Schrödinger型算子$L=-\Delta +q(x)$，其在两个边界节点处具有Dirichlet边界条件。 令$\{ \omega_n^2, \ \varphi_n(x)\}$为特征值及其相关的归一化特征函数。 令$v_1$为一个边界顶点，$v_2$为相邻的内部顶点。 假设我们知道以下数据：$\{ \omega_n^2,\partial_x \varphi_n(v_1),\partial_x\varphi_n(v_2)\}.$。其中$\partial_x\varphi_n(v_2)$指的是在$v_2$处沿另一个内部顶点相连的边的外法向导数。 从这些数据中，我们可以确定以下未知量：边的长度以及每条边上的势函数。 显示英文摘要

摘要： Consider a quantum graph consisting of a ring with two attached edges, and assume Kirchhoff-Neumann conditions hold at the internal vertices. Associated to this graph is a Schr\"{o}dinger type operator $L=-\Delta +q(x)$ with Dirichlet boundary conditions at the two boundary nodes. Let $\{ \omega_n^2, \ \varphi_n(x)\}$ be the eigenvalues and associated normalized eigenfunctions. Let $v_1$ be a boundary vertex, and $v_2$ the adjacent internal vertex. Assume we know the following data: $\{ \omega_n^2,\partial_x \varphi_n(v_1),\partial_x\varphi_n(v_2)\}.$ Here $\partial_x\varphi_n(v_2)$ refers to an outward normal derivative at $v_2$ along one of the edges incident to the other internal vertex. From this data we determine the following unknown quantities: the lengths of edges and the potential functions on each edge.