Spectral Theory, Jacobi Matrices, Continued Fractions and Difference Operators

Measure theory and Perron-Frobenius operators for continued fractions

Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schrödinger Operators

We announce three results in the theory of Jacobi matrices and Schrödinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schrödinger operator − d dx +V (x) on L2(0,∞) with V ∈ L2(0,∞) and u(0) = 0 boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials...

2d Continued Fractions and Positive Matrices

The driving force of this paper is a local symmetry in lattices. The goal is two theorems: a partial converse to the Perron-Frobenius theorem in dimension 3 and a characterization of conjugacy in Sl(Z). In the process we develop a geometric approach to higher dimension continued fractions, HDCF. HDCF is an active area with a long history: see for example Lagarias, [L],[Br]. The algorithm: Let Z...

Trace Formulas and Inverse Spectral Theory for Jacobi Operators

Based on high energy expansions and Herglotz properties of Green and Weyl m-functions we develop a self-contained theory of trace formulas for Jacobi operators. In addition, we consider connections with inverse spectral theory, in particular uniqueness results. As an application we work out a new approach to the inverse spectral problem of a class of reflectionless operators producing explicit ...

A Note on Jacobi Symbols and Continued Fractions

1. INTRODUCTION. It is well known that the continued fraction expansion of a real quadratic irrational is periodic. Here we relate the expansion for √ rs, under the assumption that rX