Two- and Three-Particle States in a Nonrelativistic Four-Fermion Model in the Fine-Tuning Renormalization Scheme: Goldstone Mode Versus Extension Theory

Abstract

In a nonrelativistic contact four-fermion model we show that simple regularisation prescriptions together with a definite fine-tuning of the cut-off-parameter dependence of "bare" quantities give the exact solutions for the two-particle sector and Goldstone modes. Their correspondence with the self-adjoint extension into Pontryagin space is established leading to self-adjoint semi-bounded Hamiltonians in three-particle sectors as well. Renormalized Faddeev equations for the bound states with Fredholm properties are obtained and analysed.The interaction between all particles in the systems B and C is the same, as in the AA and A A-channels of system A. So it is enough to consider the last one. Hereafter BB means BB, B B, B B, and the same for CC. Let us introduce the two-particle interaction kernels occurring in Eq. (11) and the two-particle energies as: K (QQ ′ ) (s, k) = K P {±} (s, k), for QQ ′ = A A, AA, BB, CC A A respectively,