Estimates for the Weyl coefficient of a two-dimensional canonical system

TL;DR

This paper derives bounds for the Weyl coefficient of two-dimensional canonical systems, linking its asymptotic growth to the Hamiltonian near the endpoint, with applications to spectral measures and differential equations.

Contribution

It provides new upper and lower bounds for the Weyl coefficient's growth, relating it to the Hamiltonian's local properties and translating these into spectral measure behavior.

Findings

Bounds depend on Hamiltonian near the endpoint

Growth of |q_H(z)| is independent of off-diagonal entries

Imaginary part of q_H(z) is not fully determined by bounds

Abstract

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on some interval $(a,b)$ whose Hamiltonian $H$ is a.e. positive semi-definite and which is regular at $a$ and in the limit point case at $b$, denote by $q_H$ its Weyl coefficient. De Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. We give upper and lower bounds for $|q_H(z)|$ and $\operatorname{Im} q_H(z)$ when $z$ tends to $i\infty$ non-tangentially. These bounds depend on the Hamiltonian $H$ near the left endpoint $a$ and determine $|q_H(z)|$ up to universal multiplicative constants. We obtain that the growth of $|q_H(z)|$ is independent of the off-diagonal entries of $H$ and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in forthcoming work we shall prove that for "most" Hamiltonians also $\operatorname{Im} q_H(z)$ is fully determined). We translate the asymptotic behaviour of $q_H$ to the behaviour of the spectral measure $\mu_H$ of $H$ by means of Abelian--Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of tails at $\pm\infty$ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm--Liouville equations.