Bound States and the Szego Condition for Jacobi Matrices and Schrodinger Operators

TL;DR

This paper investigates the spectral properties of specific Jacobi matrices and Schrödinger operators, establishing conditions for the Szego condition and the existence of bound states based on parameter decay rates.

Contribution

It provides a new proof of the Szego condition for certain Jacobi matrices and characterizes when bound states are finite or infinite depending on parameters.

Findings

Szego condition holds if gamma > 1/2

Szego condition fails if gamma < 1/2 with alpha=0 and beta≠0

Finite bound states occur when gamma=1 and parameters are sufficiently small

Abstract

For Jacobi matrices with a_n = 1+(-1)^n alpha n^{-gamma}, b_n = (-1)^n beta n^{-gamma}, we study bound states and the SzegHo condition. We provide a new proof of Nevai's result that if gamma > 1/2, the Szego condition holds, which works also if one replaces (-1)^n by cos(mu n). We show that if alpha = 0, beta not equal to 0, and gamma < 1/2, the Szego condition fails. We also show that if gamma = 1, alpha and beta are small enough (beta^2 + 8 alpha^2 < 1/24 will do), then the Jacobi matrix has finitely many bound states (for alpha = 0, beta large, it has infinitely many).