Spectral properties of singular Sturm–Liouville operators via boundary triples and perturbation theory

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