Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions

Abstract

We discuss some properties of a model Hamiltonian for a system of three bosons interacting via zero-range forces in three dimensions. In order to avoid the well known instability phenomenon, we consider a regularized Hamiltonian with a repulsive three-body force. We review the main result recently obtained in [1] where, starting from a suitable quadratic form Q, the selfadjoint and bounded from below Hamiltonian H is constructed provided that the strength γ of the three-body repulsion is larger than a threshold parameter γ c . We also show that the threshold value γ c found in [1] is optimal, in the sense that the quadratic form Q is unbounded from below if γ < γ c .Finally, we give an alternative and much simpler proof of the result in [1] whenever γ > γ ′ c , with γ ′ c strictly larger than γ c .

On the weakness of short-range interactions in Fermi gases

“…See Proposition 4.1. In order to avoid this effect, suitable many-body forces are required [3,8,9]. This work is organized as follows.…”