论文标题
从Weyl的矩阵上恢复量子星图上的电势
Recovery of a potential on a quantum star graph from Weyl's matrix
论文作者
论文摘要
考虑了从有限数量点给出的Weyl矩阵中量子星图上恢复电位的问题。提出了一种近似解决方案的方法。它包括将问题减少到每个边缘上的两个逆晶体逆变液问题。总体方法基于neumann系列的Bessel函数（NSBF）表示Sturm-Liouville方程解决方案的表示，实际上，量子图上的逆问题的解决方案还原为处理NSBF系数。 NSBF表示允许对级数剩余的估计值，这些估计值与光谱参数的平方根的实际部分无关。此功能使它们对于解决需要在光谱参数中大部分时间计算解决方案的直接和反问题特别有用。此外，仅NSBF表示的第一个系数就足以恢复电位。 Weyl基质在一组点上的知识使一个人可以在每个边缘的终点计算许多NSBF系数，从而导致两个Sturm-Liouville问题的特征函数近似，并允许一个人计算Dirichlet-Dirichlet和Neumant-Dirichlet和Neumann-dirichlet and neumann-dirichlet在每个边缘上。反过来，为了求解这个两谱逆的sturm-liouville问题，用于计算第一个NSBF系数的线性代数方程系统，从而恢复电势。提出的方法导致了一种有效的数值算法，该算法通过许多数值测试来说明。
The problem of recovery of a potential on a quantum star graph from Weyl's matrix given at a finite number of points is considered. A method for its approximate solution is proposed. It consists in reducing the problem to a two-spectra inverse Sturm-Liouville problem on each edge with its posterior solution. The overall approach is based on Neumann series of Bessel functions (NSBF) representations for solutions of Sturm-Liouville equations, and, in fact, the solution of the inverse problem on the quantum graph reduces to dealing with the NSBF coefficients. The NSBF representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the NSBF representation alone is sufficient for the recovery of the potential. The knowledge of the Weyl matrix at a set of points allows one to calculate a number of the NSBF coefficients at the end point of each edge, which leads to approximation of characteristic functions of two Sturm-Liouville problems and allows one to compute the Dirichlet-Dirichlet and Neumann-Dirichlet spectra on each edge. In turn, for solving this two-spectra inverse Sturm-Liouville problem a system of linear algebraic equations is derived for computing the first NSBF coefficient and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests.
