Abstract
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.
Highlights
A Jacobi operator is a selfadjoint operator on 2 = 2({0, 1, 2, . . .}), which with respect to the standard orthonormal basis {e0, e1, e2, . . .} has a tridiagonal matrix representation, ⎛ α0 β0 ⎞ J = ⎜⎝ β0 α1 β1 β1 α2 ⎟⎠ (1.1)where αk and βk are real numbers with βk > 0
Our assumptions on the structure of the operator are sufficient to produce better results in terms the Solvability Complexity Index (SCI) hierarchy: we can compute the discrete spectra and spectral measures of trace-class perturbations or compact perturbations with known decay with error control ( 1 in the notation of [11]), the spectral measure of Jacobi operators that are finite rank perturbations of in finite operations ( 0), and the absolutely continuous spectrum is always [−1, 1]
In this paper we have proven new results about the relationship between the connection coefficients matrix between two different families of orthonormal polynomials, and the spectral theory of their associated Jacobi operators
Summary
Schrödinger operator setting, K is a diagonal potential function which decays to zero at infinity [41] Another reason this class of operators is well studied is because the Jacobi operators for the classical Jacobi polynomials are of this form [36]. Theorem 4.8: The connection coefficient matrix CJ→ can be decomposed into CToe + Cfin where CToe is Toeplitz, upper triangular and has bandwidth 2n − 1, and the entries of Cfin are zero outside the n − 1 × 2n − 1 principal submatrix.
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