Monodromy-quasifree singular points of the Sturm-Liouville equation of standard form on the complex plane (Q2082815)

Given two simply connected domains \(\Omega_1\subset\Omega_2\) in the complex plane \(\mathbb C\) so that the annular domain \(K=\Omega_2\backslash\Omega_1\) is nonempty, the author considers the standard Sturm-Liouville equation \[ y''(z)+(q(z)-\lambda^2)y(z)=0,\quad z\in K,\tag{1} \] with a potential \(q(z)\) which is holomorphic in \(K\) and has the branching order \(N-1\) for some integer \(N\geq1\). Denoting by \(Y_0(z,\lambda,z_0)\) a matrix of fundamental solutions of Eq. (1) in a neighborhood of a point \(z_0\in K\) for some value of the spectral parameter \(\lambda\), and by \(Y_1(z,\lambda,z_0)\) the analytic continuation of \(Y_0\) along some closed rectifiable curve \(\gamma\subset K\) with start and end at \(z_0\) which goes around \(\Omega_1\) in the positive direction \(N\) times, there exists a constant nonsingular monodromy matrix \(M\) of the domain \(\Omega_1\) such that \[ Y_1(z,\lambda,z_0)=Y_0(z,\lambda,z_0)M(\lambda,z_0). \] In this paper, the author is interested in the existence of potentials with monodromy-quasifree singular points, that is with singular points such that some power of the monodromy matrix \(M\) of the domain \(\Omega_1\) is independent of the spectral parameter \(\lambda\) and equal to \(\pm I\), where \(I\) stands for the identity matrix. In particular, he states necessary and sufficient conditions for the singular points of the potential \(q(z)\) to be monodromy-quasifree and, as an illustration, he develops several examples of potentials with such singular points, including branching points.