M-functions and Inverse Spectral Analysis for Finite and Semi-infinite Jacobi Matrices

We study inverse spectral analysis for finite and semi-infinite Jacobi matrices H. Our results include a new proof of the central result of the inverse theory (that the spectral measure determines H). We prove an extension of Hochstadt’s theorem (who proved the result in the case n = N) that n eigenvalues of an N ×N Jacobi matrix, H, can replace the first n matrix elements in determining H uniquely. We completely solve the inverse problem for (δn, (H − z)−1δn) in the N < ∞ case. §