Summarizing Remarks

The considerations in the preceding text demonstrate that with perturbation theory one can go significantly beyond the examples which are usually treated in elementary texts on quantum mechanics, like the harmonic oscillator, the Coulomb potential and several other simple potentials, and — unlike any other method — one can handle with it such diverse cases as periodic potentials, anharmonic oscillators, screened Coulomb potentials, and even a typical singular potential. In most cases the expansions of interest in physics are asymptotic, in a few cases supplemented by convergent expansions (as in the case of the cosine potential). A crucial point in the formulation of the perturbation method is the full exploitation of the symmetries provided by the parameters of the respective Schrödinger equation. If these are taken into account, the perturbation method can be extended and employed even for the calculation of the behaviour of its large order terms. This is not widely appreciated, and therefore one encounters repeatedly attempts to find a better way with some kind of convergent expansion…