Growth estimates for Nevanlinna matrices of order larger than one half

TL;DR

This paper investigates the growth behavior of Nevanlinna matrices associated with two-dimensional canonical systems, especially for orders greater than one half, providing new bounds and explicit calculations for specific cases.

Contribution

It introduces a new lower bound for the growth of Nevanlinna matrices in the large order case and computes the order for a specific limit circle Jacobi matrix.

Findings

Established an explicit lower bound for growth in large order cases

Computed the order for a Jacobi matrix with two-term power asymptotics

Described the growth behavior for systems with certain monotonicity properties

Abstract

Our objects of study are two-dimensional canonical systems that arise from indeterminate Hamburger moment problems and associated half-line Jacobi operators in limit circle case. The monodromy matrix of such a system coincides, up to a permutation of its entries, with the Nevanlinna matrix of the associated moment problem. Its growth relates to the density of eigenvalues of self-adjoint realisations of the system and the Jacobi operator, respectively. The order of the Nevanlinna matrix is known to be at most 1. In the case of "large" order, meaning order greater than one half, determining the growth of the monodromy matrix is known to be harder than for "small" order, i.e., order less than one half. As our main result, we establish an explicit new lower bound for data featuring a certain kind of monotonicity, which correctly describes the growth in the case of large order. Moreover, we compute the order of the Nevanlinna matrix of a limit circle Jacobi matrix with two-term power asymptotics in a critical case.