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A Class of Hamiltonians for a Three-Particle Fermionic System at Unitarity

We consider a quantum mechanical three-particle system made of two identical fermions of mass one and a different particle of mass $ m $, where each fermion interacts via a zero-range force with the different particle. In particular we study the unitary regime, i.e., the case of infinite two-body scattering length. The Hamiltonians describing the system are, by definition, self-adjoint extensions of the free Hamiltonian restricted on smooth functions vanishing at the two-body coincidence planes, i.e., where the positions of two interacting particles coincide. It is known that for $ m $ larger than a critical value $ m^* \simeq (13.607)^{-1} $ a self-adjoint and lower bounded Hamiltonian $ H_0 $ can be constructed, whose domain is characterized in terms of the standard point-interaction boundary condition at each coincidence plane. Here we prove that for $ m \in( m^*,m^{**}) $, where $ m^{**} \simeq (8.62)^{-1} $, there is a further family of self-adjoint and lower bounded Hamiltonians $ H_{0,β} $, $ β\in \mathbb{R} $, describing the system. Using a quadratic form method, we give a rigorous construction of such Hamiltonians and we show that the elements of their domains satisfy a further boundary condition, characterizing the singular behavior when the positions of all the three particles coincide.