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Growth estimates for Nevanlinna matrices of order larger than one half

Jakob Reiffenstein

TL;DR

The work investigates growth of Nevanlinna matrices for indeterminate Hamburger moment problems via two-dimensional Hamburger Hamiltonians and their Jacobi counterparts. A central contribution is a new explicit lower bound for max_{|z|=r} log ||W_H(z)|| under a monotone-angle condition, which captures the growth in the large-order regime (ρ > 1/2) and matches existing upper bounds in that regime. By translating Jacobi parameters with two-term power asymptotics into Hamburger Hamiltonians, the author derives exact order results for the Nevanlinna matrix in the simply critical case, showing ρ = 1/[2(β−1)] for 3/2 < β < σ and ρ = 1/β for β > 2 (with ρ = 1/2 at β = 2). The methods hinge on a lower-bound lemma based on Ω_H determinants and regular variation analysis, connecting Hamiltonian rotation, spectral density, and the order problem in a unified canonical-systems framework. Overall, the paper advances explicit growth estimates and order determinations for Nevanlinna matrices in regimes where previous lower bounds were insufficient, with concrete applications to limit-circle Jacobi matrices.

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.

Growth estimates for Nevanlinna matrices of order larger than one half

TL;DR

The work investigates growth of Nevanlinna matrices for indeterminate Hamburger moment problems via two-dimensional Hamburger Hamiltonians and their Jacobi counterparts. A central contribution is a new explicit lower bound for max_{|z|=r} log ||W_H(z)|| under a monotone-angle condition, which captures the growth in the large-order regime (ρ > 1/2) and matches existing upper bounds in that regime. By translating Jacobi parameters with two-term power asymptotics into Hamburger Hamiltonians, the author derives exact order results for the Nevanlinna matrix in the simply critical case, showing ρ = 1/[2(β−1)] for 3/2 < β < σ and ρ = 1/β for β > 2 (with ρ = 1/2 at β = 2). The methods hinge on a lower-bound lemma based on Ω_H determinants and regular variation analysis, connecting Hamiltonian rotation, spectral density, and the order problem in a unified canonical-systems framework. Overall, the paper advances explicit growth estimates and order determinations for Nevanlinna matrices in regimes where previous lower bounds were insufficient, with concrete applications to limit-circle Jacobi matrices.

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.
Paper Structure (11 sections, 13 theorems, 117 equations)

This paper contains 11 sections, 13 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Main results in terms of Hamiltonian parameters
  3. Main results in terms of Jacobi parameters
  4. Regularly varying Hamiltonian parameters and growth of $W_H$
  5. A method of estimating $W_H$ from below
  6. Proof of \ref{['K32']}
  7. The case $\delta_l+\delta_{\phi}=2$
  8. Growth of the Nevanlinna matrix of a Jacobi operator
  9. Rewriting a Jacobi matrix to a Hamburger Hamiltonian
  10. $\gamma$-tempered Jacobi parameters
  11. Jacobi parameters with power asymptotics

Key Result

Theorem 1.1

Consider a Hamburger Hamiltonian $H$ in limit circle case. Let ${\mathscr{d}}_l,{\mathscr{d}}_\phi: [1,\infty) \to (0,\infty)$ be regularly varying with indices $-\delta_l$ and $-\delta_\phi$, respectively. Assume that $\delta_\phi \in (0,1)$, and suppose that angles $(\phi_j)_{j=1}^\infty$ of $H$ c Then for $r$ sufficiently large. In particular, $\rho_H \geq \frac{1-\delta_\phi}{\delta_l-\delta_

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 14 more