A noncanonical transformation which allows perturbative techniques to be applied to the pairing force problem is introduced. The lowest-order eigenvalue equation gives the standard results for both strong and weak coupling.
This paper presents several cases in which the effects of the addition of a delta function potential on bound states can be computed exactly. In the case of the one dimensional Schrödinger equation, a Green’s function technique is used to compute an exact implicit expression that gives the effect of the one−dimensional delta function potential on the bound states eigenvalues. It is also shown that the weak coupling limit of the implicit expression agrees with the perturbative treatment of the delta function. It is pointed out that this technique cannot be applied to obtain an exact treatment of the delta function potential δ3(r) in the three−dimensional Schrödinger equation. Instead, an alternate definition of the delta function in three dimensions as a limit of a sequence of square well potentials is used. Unlike the one dimensional case, the exact effect on the eigenvalue does not agree in the limit of weak coupling with the perturbative results. This indicates that the three−dimensional delta function potential should not be used to approximate strongly coupled short range potentials for which perturbation theory is inadequate.
We compare different methods used for non-perturbative calculations in strongly interacting fermionic systems. Mean field theory shows a basic ambiguity related to the possibility to perform Fierz transformations. The results depend strongly on an unphysical parameter which reflects the choice of the mean field. Renormalization group equations in a partially bosonized setting can overcome this problem if the truncation is chosen appropriately. We argue that the renormalization group approach goes beyond the lowest order Schwinger-Dyson equation and constitutes a very promising basis for the explicit treatment of condensates and spontaneous symmetry breaking.