@article {2015,
title = {A class of Hamiltonians for a three-particle fermionic system at unitarity},
year = {2015},
note = {This SISSA preprint is composed of 29 pages and is recorded in PDF format},
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 $m^* \simeq (13.607)^{-1}$ 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 $H_{0,\beta}$, $\beta \in \mathbb{R}$, 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.},
url = {http://urania.sissa.it/xmlui/handle/1963/34469},
author = {Michele Correggi and Gianfausto Dell{\textquoteright}Antonio and Domenico Finco and Alessandro Michelangeli and Alessandro Teta}
}
@article {2012,
title = {Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions},
journal = {Rev. Math. Phys. 24 (2012), 1250017},
number = {arXiv:1201.5740;},
year = {2012},
publisher = {World Scientific},
abstract = {We study the stability problem for a non-relativistic quantum system in\\r\\ndimension three composed by $ N \\\\geq 2 $ identical fermions, with unit mass,\\r\\ninteracting with a different particle, with mass $ m $, via a zero-range\\r\\ninteraction of strength $ \\\\alpha \\\\in \\\\R $. We construct the corresponding\\r\\nrenormalised quadratic (or energy) form $ \\\\form $ and the so-called\\r\\nSkornyakov-Ter-Martirosyan symmetric extension $ H_{\\\\alpha} $, which is the\\r\\nnatural candidate as Hamiltonian of the system. We find a value of the mass $\\r\\nm^*(N) $ such that for $ m > m^*(N)$ the form $ \\\\form $ is closed and bounded from below. As a consequence, $ \\\\form $ defines a unique self-adjoint and bounded from below extension of $ H_{\\\\alpha}$ and therefore the system is stable. On the other hand, we also show that the form $ \\\\form $ is unbounded from below for $ m < m^*(2)$. In analogy with the well-known bosonic case, this suggests that the system is unstable for $ m < m^*(2)$ and the so-called Thomas effect occurs.},
doi = {10.1142/S0129055X12500171},
url = {http://hdl.handle.net/1963/6069},
author = {Michele Correggi and Gianfausto Dell{\textquoteright}Antonio and Domenico Finco and Alessandro Michelangeli and Alessandro Teta}
}