We address the computational spectral theory of Jacobi operators that are compact perturbations of the free Jacobi operator via the asymptotic properties of a connection coefficient matrix. In particular, for Jacobi operators that are finite-rank perturbations we show that the computation of the spectrum can be reduced to a polynomial root finding problem, from a polynomial that is derived explicitly from the entries of a connection coefficient matrix. A formula for the spectral measure of the operator is also derived explicitly from these entries. The analysis is extended to trace-class perturbations. We address issues of computability in the framework of the Solvability Complexity Index, proving that the spectrum of compact perturbations of the free Jacobi operator is computable in finite time with guaranteed error control in the Hausdorff metric on sets.

我们通过连接系数矩阵的渐近性质来解决雅可比算子的计算谱理论，该理论是自由雅可比算子的紧凑扰动。特别地，对于有限秩扰动的Jacobi算子，我们表明，可以将频谱的计算简化为多项式求根问题，这是由明确地从连接系数矩阵的项中导出的多项式得出的。从这些条目还可以明确得出操作员的光谱测量公式。该分析扩展到跟踪类扰动。我们在“可解决性复杂度指数”框架内解决可计算性问题，