A simple example of a singular perturbation expansion

For the Hamiltonian?=?? _x ² -αδ(x)+λx ² the bound-(λ>0) and resonance(λ<0) states are discussed. The perturbation expansion of the ground-state? ₀(λ) in powers ofλ is divergent for allλ. The nature of the singularity atλ=0 is investigated and it turns out, that the perturbation series is asymptotic and therefore useful even forλ<0.