New version of the Rayleigh-Schrodinger perturbation theory

New version of the Rayleigh-Schrodinger perturbation theory based on the linear dependence of the perturbation wavefunctions on the perturbation energies is summarized. It is shown that this method is suitable also for multidimensional problems and the linear dependence can be used at an arbitrary point inside the integration region. The resulting perturbation theory is simple and can be used at large orders. As an example, the method is applied to the Barbanis hamiltonian.