求解量子散射逆问题的一种嬗变算符方法
A transmutation operator method for solving the inverse quantum scattering problem
摘要：
研究了扰动贝塞尔方程的量子逆散射问题。提出了一种直接而实用的求解方法。它允许一个减少反问题的线性代数方程组的系统，和潜在的恢复从系统的解向量的第一个组成部分。该方法是基于一个特殊形式的傅里叶-雅可比级数表示的嬗变算子内核和Gelfand-Levitan方程，用于获得线性代数方程组的系统。证明了该方法的收敛性和稳定性以及截断系统解的存在唯一性。讨论了该方法的数值实现。数值试验结果表明，该方法具有较高的精度和稳定性。
英文摘要：
The inverse quantum scattering problem for the perturbed Bessel equation is considered. A direct and practical method for solving the problem is proposed. It allows one to reduce the inverse problem to a system of linear algebraic equations, and the potential is recovered from the first component of the solution vector of the system. The approach is based on a special form Fourier-Jacobi series representation for the transmutation operator kernel and the Gelfand-Levitan equation which serves for obtaining the system of linear algebraic equations. The convergence and stability of the method are proved as well as the existence and uniqueness of the solution of the truncated system. Numerical realization of the method is discussed. Results of numerical tests are provided revealing a remarkable accuracy and stability of the method.
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