Estimates for the Weyl coefficient of a two-dimensional canonical system

For a two-dimensional canonical system y'(t)=zJH(t)y(t) on some interval (a,b) whose Hamiltonian H is a.e. positive semi-definite and which is regular at a and in the limit point case at b, 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.

We give upper and lower bounds for |q_H(z)| and Im q_H(z) when z tends to i∞ non-tangentially. These bounds depend on the Hamiltonian H near the left endpoint a and determine |q_H(z)| up to universal multiplicative constants. We obtain that the growth of |q_H(z)| is independent of the off-diagonal entries of H and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in forthcoming work we shall prove that for "most" Hamiltonians also Im q_H(z) is fully determined).

We translate the asymptotic behaviour of q_H to the behaviour of the spectral measure μ_H of H by means of Abelian–Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of tails at ±∞ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm–Liouville equations.