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:从Weyl的矩阵中在若干点给出的量子树图中恢复潜力的反面问题得到了考虑。提出了数字解决办法的方法。总体方法的基础是叶子剥皮方法,结合Neumann系列贝塞尔函数(NSBF)的演示,以找到Sturm-Liouville等式的解决方案。在每一步骤中,所产生反面问题的解决方案都减少了对NSBF系数的处理。叶子剥皮方法使得人们能够将一般反向问题与堆积物当地问题区分开来,而基于NSBF的表述方法则导致将本地问题分为不同边缘的两种反向问题,将其减少到NSBF系数的线性代数方程系统。此外,每种边缘的潜力都从NSBFF系数的最初一个系数中收回。拟议的方法导致一种有效的数值算法,用数字测试来说明。