Logarithmic perturbation theory for a spiked oscillator and sum rules

We show that logarithmic perturbation theory nicely yields the wavefunction correction terms in closed forms for the spiked perturbation λ/ x 2 on the first excited state of the harmonic oscillator, where the conventional Rayleigh‐Schrödinger scheme is known to encounter serious problems. The study also provides a direct route to obtain several sum rules, some of which appear to be new.