  We apply both the theory of boundary triples and perturbation theory to the
setting of semi-bounded Sturm-Liouville operators with two limit-circle
endpoints. For general boundary conditions we obtain refined and new results
about their eigenvalues and eigenfunctions.
  In the boundary triple setup, we obtain simple criteria for identifying which
self-adjoint extensions possess double eigenvalues when the parameter is a
matrix. We also identify further spectral properties of the Friedrichs
extension and (when the operator is positive) the von Neumann-Krein extension.
  Motivated by some recent scalar Aronszajn-Donoghue type results, we find that
real numbers can only be eigenvalues for two extensions of Sturm-Liouville
operator when the boundary conditions are restricted to corresponding to affine
lines in the space from which the perturbation parameter is taken. Furthermore,
we determine much of the spectral representation of those Sturm-Liouville
operators that can be reached by perturbation theory.
