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, ∞) whose Hamiltonian H is a.e. positive semi-definite, denote by qH its Weyl coefficient. De Branges’ inverse spectral theorem states that the assignment H ↦ qH 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 qH(iy), dominates its Poisson integral Im qH(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.