ABSTRACT

We consider semi-infinite Jacobi matrices with discrete spectrum. We prove that a Jacobi operator can be uniquely recovered from one spectrum and subsets of another spectrum and norming constants corresponding to the first spectrum. As a corollary, we obtain semi-infinite Jacobi analog of Marchenko's inverse spectral theorem for Schödinger operators, i.e. a Jacobi operator can be uniquely recovered from the Weyl m-function (or the spectral measure). We also solve our Borg–Marchenko-type problem under some conditions on two spectra, when missing part of the second spectrum and known norming constants have different index sets.

抽象的

我们考虑具有离散频谱的半无限Jacobi矩阵。我们证明，可以从一个光谱和另一个光谱的子集以及对应于第一个光谱的范数常数中唯一地恢复Jacobi算子。作为推论，我们为Schödinger算子获得了Marchenko逆谱定理的半无限Jacobi模拟，即，可以从Weyl m函数（或谱测度）唯一地恢复Jacobi算子。当第二个谱图的缺失部分和已知的规范常数具有不同的索引集时，我们还可以在某些条件下在两个谱图上解决Borg–Marchenko型问题。