Canonical systems whose Weyl coefficients have regularly varying asymptotics

Abstract

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on an interval $(0,L)$ with $0<L\le\infty$ whose Hamiltonian $H$ is a.e.\ positive semidefinite, denote by $q_H$ its Weyl coefficient. De~Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between trace-normalised Hamiltonians and Nevanlinna functions. We prove that $q_H$ has an asymptotics towards $i\infty$ whose leading term is some (complex) multiple of a regularly varying function if and only if the primitive $M$ of $H$ is regularly or rapidly varying at $0$ and its off-diagonal entries do not oscillate too much. The leading term in the asymptotics of $q_H$ towards $i\infty$ is related to the behaviour of $M$ towards $0$ by explicit formulae. The speed of growth in absolute value depends only on the diagonal entries of $M$, while the argument of the leading coefficient corresponds to the relative size of the off-diagonal entries. Translated to the spectral measure $μ_H$ and the Hamiltonian $H$, this means that the diagonal of $H$ determines the growth of the symmetrised distribution function of $μ_H$, and the relative size and sign distribution of its off-diagonal is a measure for the asymmetry of $μ_H$. The results are applied to Sturm--Liouville equations, Krein strings and generalised indefinite strings to prove similar characterisations for the asymptotics of the corresponding Weyl coefficients.