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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?

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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.

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