Perturbation theory for nonlinear time-independent Schrödinger equations

A special nondegenerate perturbation theory is developed for solving the "nonlinear Schrödinger equation" of the type $[\hat{H}+\hat{V}(\Psi \left)\right]\Psi =E\Psi$, which arises, e.g., from the study of the environmental effects on molecules. The interaction operator is taken as $\hat{V}\left(\Psi \right)=\hat{A}〈\Psi \left|\hat{B}\right|\Psi 〉$. The in vacuo problem $\hat{H}\Phi =\epsilon \Phi$ is considered the zeroth-order description. For the perturbed energies and wave functions recursive formulas are derived at any order of the perturbation. The formulas obtained are compared to the usual Rayleigh-Schrödinger perturbation theory where $\hat{V}$ is supposed to be independent of $\Psi$. The formal results are applied to the case of the Onsager cavity model.