The Sturm-Liouville Hierarchy of evolution equations and the Weyl functions of related spectral problems

The goal of the present paper is that of defining the so-called Sturm-Liouville hierarchy of evolution equations, firstly by using the zero-curvature formalism, then by using the asymptotic properties of the Weyl $m$-functions for certain classes of pairs $(q,y)$ of the Sturm-Liouville eigenvalue equation $$-{\phi }^{\mathrm{\prime }\mathrm{\prime }}+q\phi =\lambda y\phi ,$$ in the space $L^2(\mathbb{R},ydx)$. Since the Weyl $m$-functions are known to contain all the information concerning the spectral properties of the above equation, the determination of the evolution of some spectral characteristics will allow to determine solutions of the hierarchies of evolution equations.