In particular, this includes many operators which are not bounded from below. Nevertheless, we use the rank one perturbation approach, which is more general in other respects. Note that this result does not say if the singular spectrum is point or singular continuous. In this section, we prove our main new result that the occurrence of any specific spectral type for a set of positive measures of boundary conditions for a Sturm-Liouville operator is invariant under local perturbations of potential. Follows from Corollary 2.8 and Theorem 5.1.
Similar results hold for point spectrum and singular spectrum.
In this case the singular spectrum guaranteed by Corollary 4.3 is just discrete eigenvalues. Note that this does not exclude the occurrence of singular spectrum in addition.