Jakob Reiffenstein: Quantitative formulae for the Weyl coefficient of a canonical system

A two-dimensional canonical system is a differential equation of the form
\[\begin{align} y'(t)=z \Bigl(\begin{smallmatrix} \hspace*{-0.1ex} 0 \hspace*{0.1ex} & \hspace*{0.1ex} -1 \hspace*{-0.1ex} \\[0.5ex] \hspace*{-0.1ex} 1 \hspace*{0.1ex} & \hspace*{0.1ex} 0 \hspace*{-0.1ex} \end{smallmatrix}\Bigr) H(t)y(t) \end{align}\]
on an interval \([0,L)\), where \(H\) is a locally integrable \(\mathbb R^{2 \times 2}\)-valued function with \(H(t) \geq 0\) almost everywhere. Together with suitable boundary conditions, equation (1) is the eigenvalue equation of a self-adjoint operator. The spectrum of this operator is encoded in an analytic function \(q_H\) called Weyl coefficient. We will discuss two-sided bounds for absolute value and imaginary part of \(q_H(ir)\) and some consequences for the spectral theory of the equation.