On the parametrization of self-adjoint extensions of singular Sturm-Liouville operators

The self-adjoint extensions of a lower semi-bounded minimal Sturm–Liouville operator are parametrized using generalized boundary values defined by fixed choices of special nonoscillatory solutions, called principal and nonprincipal solutions. However, principal and nonprincipal solutions are not unique, and different choices yield different generalized boundary values. Therefore, the parametrization of self-adjoint extensions inherently depends upon the choices of principal and nonprincipal solutions. Using known properties of principal and nonprincipal solutions, Wronskian techniques, and the Pl¨ucker identity, we find the relation between self-adjoint extensions for two different parametrizations.