Canonical systems whose Weyl coefficients have dominating real part

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 .