Perturbative approach to some one-dimensional many-body problems. I: Discussion of the Rayleigh-Schrödinger and Brillouin-Wigner series for the Sutherland model

Summary  This is the first of a series of papers in which we investigate some of the more common approximate many-body techniques. A comparison is given of the results provided by the Rayleigh-Schrödinger and the Brillouin-Wigner perturbative approaches to a one-dimensional nonrelativisticN-body problem with those arising from its exact solution. We find that the Rayleigh-Schrödinger series converges to the exact value of the physical quantity considered (the ground-state energy of the system) for a certain range of values of the coupling constant; on the other hand, the Brillouin-Wigner series gives a result which does not agree with its usually assumed behaviour. The origin of this discrepancy is discussed and explained.