Abstract

An inverse spectral optimization problem is considered: given a matrix potential \({{Q}_{0}}(x)\) and a value \(\lambda {\kern 1pt} \text{*}\), find a matrix function \(\hat {Q}(x)\) closest to \({{Q}_{0}}(x)\) such that the kth eigenvalue of the Sturm–Liouville matrix operator with potential \(\hat {Q}(x)\) matches \(\lambda {\kern 1pt} \text{*}\). The main result of the paper is the proof of existence and uniqueness theorems. Explicit formulas for the optimal potential are established through solutions to systems of second-order nonlinear differential equations known in mathematical physics as systems of nonlinear Schrödinger equations.

中文翻译：

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向量函数空间中 Sturm-Liouville 算子的优化谱问题

摘要

考虑逆谱优化问题：给定矩阵势\({{Q}_{0}}(x)\)和值\(\lambda {\kern 1pt} \text{*}\)，找到一个矩阵函数\(\hat {Q}(x)\)最接近\({{Q}_{0}}(x)\)，使得 Sturm-Liouville 矩阵算子的第k个特征值具有势\(\ hat {Q}(x)\)匹配\(\lambda {\kern 1pt} \text{*}\)。论文的主要成果是证明了定理的存在性和唯一性。通过解二阶非线性微分方程组（在数学物理学中称为非线性薛定谔方程组），可以建立最佳势的显式公式。