Canonical systems whose Weyl coefficients have dominating real part

Matthias Langer, Raphael Pruckner, Harald Woracek

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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.
Original languageEnglish
Number of pages30
JournalJournal d'Analyse Mathématique
Publication statusAccepted/In press - 3 Aug 2022


  • canonical system
  • Weyl coefficient
  • high-energy behaviour
  • singular integral
  • dominating real part


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