Canonical systems whose Weyl coefficients have dominating real part

Langer, Matthias and Pruckner, Raphael and Woracek, Harald (2022) Canonical systems whose Weyl coefficients have dominating real part. Journal d'Analyse Mathématique. ISSN 0021-7670 (In Press)

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Abstract

For a two-dimensional canonical system y'(t)=zJH(t)y(t) on the half-line (0, ∞) 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 → 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  → + ∞. 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. 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.

ORCID iDs

Langer, Matthias ORCID logoORCID: https://orcid.org/0000-0001-8813-7914, Pruckner, Raphael and Woracek, Harald;