# Positive and negative powers of small parameter in perturbation problem

I'd like to perturbatively handle an eigenvalue problem similar to this: $$\lambda f = (\hat{H} + (1/\epsilon^2) \hat{V} + \epsilon {W}) f,$$ where $$f$$ is a function, $$\lambda$$ is an eigenvalue, $$\epsilon$$ is a small parameter, and the rest are linear operators. The problem is, that if one writes a series for the eigenvalue and the eigenfunction, $$f = f_0 + \epsilon f_1 + \epsilon^2 f_2 + ...\\ \lambda = \lambda_0 + \epsilon \lambda_1 + \epsilon^2 \lambda_2 + ...,$$ one will get e.g. $$\lambda_0 f_0 = \hat{H} f_0 + \hat{V} f_2\\ \lambda_1 f_0 + \lambda_0 f_1 = \hat{H} f_1 + \hat{W} f_0 + \hat{V} f_3\\ ...$$ i.e. the different orders of the series start to mix. Is there a way to develop a systematic perturbation theory for this case?

A perturbation approach requires the additional perturbing Hamiltonian to be a weak disturbance to the unperturbed system. However in your statement you have the small parameter $$\epsilon$$ at the denominator. It is an inconsistency, that is the smaller the parameter the higher the disturbance! The system posted in the question is unstable.