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

Abstract

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 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 problemThere is an enormous literature on inverse spectral problems for − [1,29,[56][57][58][59][60]64] and references therein), but considerably less for their discrete analog, the infinite and semi-infinite Jacobi matrices (see, e.g., [3, 4, 6-8, 13-22, 24, 26-28, 30, 32, 37, 38, 42-44, 50-52, 61-63, 66, 67, 69-71]) and even less for finite Jacobi matrices (where references include, e.g., [9,10,23,25,39,40,41,[45][46][47][48]). Our goal in this paper is to study the last two problems using one of the most powerful tools from the spectral theory of − d 2 dx 2 + V (x), the m-functions of Weyl. Explicitly, we will study finite N × N matrices of the form:

Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum

“…In [11], we consider results related to Theorem 1.4 in that, for Schrödinger operators on (−∞, ∞), "spectral" information plus the potential on one of the half-lines determine the potential on all of (−∞, ∞). In that paper, we consider situations where there are scattering states for some set of energies and the "spectral" data are given by a reflection coefficient on a set of positive Lebesgue measure in the a.c. spectrum of H. The approach is not as close to this paper as is [12], but m-function techniques (see also [10]) are critical in all three papers.…”

“…In [12], we consider, among other topics, analogs of Theorems 1.1 and 1.3 for finite tridiagonal (Jacobi) matrices extending a result in [15]. The approach there is very similar to the current one, except that the somewhat subtle theorems on zeros of entire functions in this paper are replaced by the elementary fact that a polynomial of degree at most N with N + 1 zeros must be identically zero.…”

Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees

“…The fact that the singular continuous part of µ n 1 is singular, with respect to the singular continuous part of µ n 2 , follows, now, from the characterization of the appropriate supports in terms of m-functions (see e.g. [9]) and from the continued fraction expansion of m [3]. (The spaces H ′ n are just P I (H n )).…”

Jacobi Operators and Completely Integrable Nonlinear Lattices

“…Additional results can be found in [88], [89], [92] and, in the case of finite Jacobi matrices, [104].…”