量子树的若干重构技术
Reconstruction techniques for quantum trees
AI总结：
本文针对量子树图位势反问题，提出结合叶子剥离法与 NSBF 表示的数值方法，将问题局部化并化为线性代数方程组求解，最终实现高效重构。
摘要：
本文考虑了从在若干点处给定的 Weyl 矩阵反演恢复量子树图上位势的反问题，并提出了一种数值求解方法。整体方法基于叶子剥离法与 Sturm-Liouville 方程解的 Neumann 级数 Bessel 函数（NSBF）表示相结合。在每一步中，所出现的反问题的求解都归结为处理 NSBF 系数。叶子剥离法允许将一般反问题局部化为层（sheaves）上的局部问题，而基于 NSBF 表示的方法则可将局部问题分裂为各条独立边上的双谱反问题，并将其归结为关于 NSBF 系数的线性代数方程组。此外，每条边上的位势可由第一个 NSBF 系数直接恢复。所提出的方法导出了一个高效的数值算法，并通过数值试验加以说明。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
