Canonical systems whose Weyl coefficients have dominating real part

Abstract

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on the half-line $(0,\infty)$ whose Hamiltonian $H$ is a.e. positive semi-definite, 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. The main result of the paper is a criterion when the singular integral of the spectral measure, i.e. Re $q_H(iy)$, dominates its Poisson integral Im $q_H(iy)$ for $y\to+\infty$. Two equivalent conditions characterising this situation are provided. The first one is analytic in nature, very simple, and explicit in terms of the primitive $M$ of $H$. It merely depends on the relative size of the off-diagonal entries of $M$ compared with the diagonal entries. The second condition is of geometric nature and technically more complicated, but explicit in terms of $H$ itself. It involves the relative size of the off-diagonal entries of $H$, a measurement for oscillations of the diagonal of $H$, and a condition on the speed and smoothness of the rotation of $H$.

Theorems & Definitions (40)

• Theorem 1.1
• Remark 1.3
• Theorem 1.4
• Example 1.6
• Example 1.7
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4