This paper discusses a phenomenon known as Klauder's phenomenon, which occurs when a perturbation of a self-adjoint operator does not converge to the original operator in the strong resolvent sense. The paper analyzes a class of pairs of positive self-adjoint operators A and B, where A + λB converges to a different operator A_p as λ approaches zero. The example given involves the harmonic oscillator operator A and a singular perturbation B = |x|^{-3}. The eigenvalues and eigenvectors of A + λB do not converge to those of A but instead to those of A shifted by one level. The paper presents two theorems that characterize the conditions under which Klauder's phenomenon occurs. The first theorem states that Klauder's phenomenon occurs if and only if the intersection of the form domains of A and B is not a form core for A. The second theorem shows that the eigenvalues of A + λB converge to those of A_p as λ approaches zero. The paper also provides several examples, including one where A is the Laplacian and B is a singular potential. It discusses the importance of the form domain in determining the convergence of the perturbed operator. The paper concludes by noting that the form sum can sometimes be unsuitable for describing the behavior of the perturbed operator, and that different boundary conditions can lead to different results. The paper also references several other works that have studied similar phenomena.This paper discusses a phenomenon known as Klauder's phenomenon, which occurs when a perturbation of a self-adjoint operator does not converge to the original operator in the strong resolvent sense. The paper analyzes a class of pairs of positive self-adjoint operators A and B, where A + λB converges to a different operator A_p as λ approaches zero. The example given involves the harmonic oscillator operator A and a singular perturbation B = |x|^{-3}. The eigenvalues and eigenvectors of A + λB do not converge to those of A but instead to those of A shifted by one level. The paper presents two theorems that characterize the conditions under which Klauder's phenomenon occurs. The first theorem states that Klauder's phenomenon occurs if and only if the intersection of the form domains of A and B is not a form core for A. The second theorem shows that the eigenvalues of A + λB converge to those of A_p as λ approaches zero. The paper also provides several examples, including one where A is the Laplacian and B is a singular potential. It discusses the importance of the form domain in determining the convergence of the perturbed operator. The paper concludes by noting that the form sum can sometimes be unsuitable for describing the behavior of the perturbed operator, and that different boundary conditions can lead to different results. The paper also references several other works that have studied similar phenomena.