Induced potential and Klauder phenomenon for even confinement potentials with singularity $\lambda x^{-2}$

For the example of a one-dimensional oscillator with singular perturbation the occurrence of the strongly singular potential $A(\lambda)\delta(x)|x|^{-1}$ induced by this perturbation is investigated. The appearance of such a potential can be regarded as a generalization of the Klauder phenomenon ($A=\operatorname{const}$). The existence of induced potentials leads to two physically acceptable sets of even states of a perturbed oscillator, but only one of these goes over continuously into the set of even states of the harmonic oscillator when the perturbation is switched off.