Criterion for discreteness of the spectrum of a singular canonical system (Q1916677)

The discreteness of the spectrum of the system \[ J{d\over dt} x= z{\mathcal K}(t)x, x(0)=\begin{pmatrix} 1\\ 0\end{pmatrix},\text{ where }{\mathcal K}=\begin{pmatrix} a(t) b(t)\\ c(t) d(t)\end{pmatrix}, J=\begin{pmatrix} 0 &-1\\ 1 & 0\end{pmatrix}, t\in I\subset\mathbb{R},\tag{\(*\)} \] with \(I\) interval, \({\mathcal K}(t)\geq 0\) for any \(t\in I\) and \(z\) is a complex parameter is studied. After some definitions involving the properties of the interval \(I\) and some kind of functions, one theorem is given for the discreteness of the spectrum of the system \((*)\). Another theorem gives necessary and sufficient conditions between the regularity of the solution of \((*)\) and the discreteness of the spectrum of the system.