The inverse problem of recovery of a potential on a quantum tree graph from Weyl's matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm-Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two-spectra inverse problems on separate edges and reduce them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.

翻译：本文研究了通过给定多个点的Weyl矩阵反演量子树图势能的反问题，并提出了一种数值求解方法。该整体方法基于叶剥离技术与Bessel函数Neumann级数（NSBF）表示下Sturm-Liouville方程解的协同策略。每一步中，反问题的求解可转化为对NSBF系数的处理。叶剥离法使得总体反问题能够局域化为茎叶层上的局部问题，而基于NSBF表示的方法则进一步将局部问题分解为独立边上的双谱反问题，并将其简化为NSBF系数的线性代数方程组。值得注意的是，每条边上的势能可通过首个NSBF系数直接恢复。所提出的方法构建了高效的数值算法，并通过数值试验进行了验证。