M. Langer, R. Pruckner and H. Woracek,
Canonical systems whose Weyl coefficients have dominating real part,
J. Anal. Math., to appear
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.