Order problem for canonical systems and a conjecture of Valent

TL;DR

This paper derives a precise upper bound for the order of canonical systems based on their Hamiltonian, confirming a conjecture of Valent for a specific class of Jacobi matrices with polynomial coefficients.

Contribution

It provides a sharp upper estimate for the order of canonical systems and proves Valent's conjecture for Jacobi matrices with polynomial coefficients.

Findings

Sharp upper estimate for the order of canonical systems

Equality case for Krein strings

Proof of Valent's conjecture for polynomial-coefficient Jacobi matrices

Abstract

We establish a sharp upper estimate for the order of a canonical system in terms of the Hamiltonian. This upper estimate becomes an equality in the case of Krein strings. As an application we prove a conjecture of Valent about the order of a certain class of Jacobi matrices with polynomial coefficients.