Oscillator with singular perturbation

The Rayleign–Schrödinger perturbation theory is formulated for even states of a one-dimensional oscillator with the singular perturbation $\lambda|x|^{-\nu}(1\leq\nu <2)$. It is shown that the matrix elements of the perturbation and the Rayleigh–Schrödinger series evist for $1\leq\nu <3/2$ if the induced point perturbation
$$-2\lambda(\nu-1)^{-1}|x|^{1-\nu}\delta(x) \quad (1<\nu <3/2), \quad 2\lambda\ln |x|\delta(x) \quad (\nu=1).$$
arising as the result of the singular perturbation is taken into account. For $3/2<\nu <2$ the standard perturbation theory cannot be constructed although the energy levels are analytic in $\lambda$.