  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 finite-rank perturbation
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.
