Jacobi Sum Matrices

Sum Rules for Jacobi Matrices and Their Applications to Spectral Theory

We discuss the proof of and systematic application of Case’s sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J − J0 is Hilbert–Schmidt, and a proof of Nevai’s conjecture that the Szegő condition holds if...

Two modified Jacobi methods for M-matrices

Trace Class Perturbations and the Absence of Absolutely Continuous Spectra

We show that various Hamiltonians and Jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices.

Jacobi Operational Matrix Approach for Solving Systems of Linear and Nonlinear Integro-Differential Equations

‎‎‎‎‎‎‎‎‎‎‎‎‎This paper aims to construct a general formulation for the shifted Jacobi operational matrices of integration and product‎. ‎The main aim is to generalize the Jacobi integral and product operational matrices to the solving system of Fredholm and Volterra integro--differential equations‎ which appear in various fields of science such as physics and engineering. ‎The Operational matr...

A Canonical Factorization for Meromorphic Herglotz Functions on the Unit Disk and Sum Rules for Jacobi Matrices

We prove a general canonical factorization for meromorphic Herglotz functions on the unit disk whose notable elements are that there is no restriction (other than interlacing) on the zeros and poles for their Blaschke product to converge and there is no singular inner function. We use this result to provide a significant simplification in the proof of Killip-Simon [5] of their result characteri...

Sum Rules and the Szegő Condition for Orthogonal Polynomials on the Real Line

We study the Case sum rules, especially C0, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat’s theorem to cases with an infinite point spectrum and a proof that if lim n(an− 1) = α and lim nbn = β exist and 2α < |β|, then the Szegő condition fails.