Revisited version of Weyl's limit-point limit-circle criterion for essential self-adjointness

The principal aim of this paper is to present a new proof of Weyl’s criterion in which it is shown that the natural framework for the associated Sturm-Liouville operators is W**(2,1) cap L**2 -i.e.- the intersection of a particular Sobolev space and of the L**2 space. Indeed, we will deal with the special case of the radial operator (-{d^{2}over dx^{2}}+q(x)) on a real line segment (either bounded or unbounded) that often occurs in the study of quantum systems in central potentials.We also derive from first principles the functional behaviour of the coefficients for a general second-order Sturm-Liouville operator by using some extensions of a milestone Carathéodory existence theorem.