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

Abstract

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 $m^{\ast }\simeq (13.607{)}^{-1}$ 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 $m\in (m^{\ast },m^{\ast \ast })$, where $m^{\ast \ast }\simeq (8.62{)}^{-1}$, there is a further family of self-adjoint and lower bounded Hamiltonians $H_{0,\beta }$, $\beta \in \mathbb{R}$, 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.

30 pages; pdfLaTeX. Some comments and remarks added

References in corpus (2)

• Energy lower bound for the unitary N+1 fermionic model
• The self-energy of an impurity in an ideal Fermi gas to second order in the interaction strength

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