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一个韦尔矩阵对无限非自相辅的雅科比矩阵的视角.
Benjamin Eichinger1, Milivoje Lukić2, Giorgio Young3
1School of Mathematical Science, Lancaster University, LA1 4YF Lancaster, UK.
概括
研究人员介绍了一种使用韦尔函数编码非自我附加的雅科比矩阵的新方法. 这种方法简化了证明,并将编码扩展到无限矩阵,与光谱数据建立一个对称.
科学领域:
- 频谱理论是一种光谱理论.
- 运营者理论是运营者的理论.
- 线性代数的线性代数
背景情况:
- 普什尼茨基-斯塔马帕奇方法用光谱测量和相位函数编码局限非自相对应的雅科比矩阵.
- 现有的方法通常依赖于基于时刻的方法,这可能是复杂的.
- 将这些编码技术推广到无边界的 Jacobi 矩阵仍然是一个重大挑战.
研究的目的:
- 开发一种对编码非自我附加的雅科比矩阵的替代视角.
- 为了将普什尼茨基-斯塔马帕奇对应关系推广到无限的情况下.
- 建立一个特定类的雅科比矩阵和光谱数据之间的对称关系.
主要方法:
- 利用韦尔函数作为矩阵编码中光谱时刻的替代品.
- 开发了一种新的方法来处理无边界的非自我附加的雅科比矩阵.
- 使用局部的博格-马琴科定理证明了已建立的映射和注射性的连续性.
主要成果:
- 建立了一个简化和通用的对应对编码非自我附加的雅科比矩阵,适用于无限制的情况.
- 证明了正 Jacobi 矩阵与正非对角元素和特定类的光谱数据之间的对称关系.
- 证明了这个 bijection 的连续性,并建立了一个局部 Borg-Marchenko 定理,用于无边界的非自我附加的 Jacobi 矩阵.
结论:
- 基于韦尔函数的方法为编码非自我附加的雅科比矩阵提供了一个更简单的方法.
- 这项研究成功地将编码框架扩展到无限Jacobi矩阵,扩大了其适用性.
- 开发的局部Borg-Marchenko定理为分析无边界非自我附加的雅科比矩阵提供了一个有价值的工具.
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