m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

Abstract

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 the theorem of Hochstadt (who proved the result in 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 (δ_n , (H-z)^(-1) δ_n ) in the case N < ∞.