%0 Report
%D 2015
%T A class of Hamiltonians for a three-particle fermionic system at unitarity
%A Michele Correggi
%A Gianfausto Dell'Antonio
%A Domenico Finco
%A Alessandro Michelangeli
%A Alessandro Teta
%X 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.
%G en
%U http://urania.sissa.it/xmlui/handle/1963/34469
%1 34644
%2 Mathematics
%4 1
%$ Submitted by Alessandro Michelangeli (alemiche@sissa.it) on 2015-05-21T06:33:20Z
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%0 Journal Article
%J Rev. Math. Phys. 24 (2012), 1250017
%D 2012
%T Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions
%A Michele Correggi
%A Gianfausto Dell'Antonio
%A Domenico Finco
%A Alessandro Michelangeli
%A Alessandro Teta
%X 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.
%B Rev. Math. Phys. 24 (2012), 1250017
%I World Scientific
%G en
%U http://hdl.handle.net/1963/6069
%1 5955
%2 Mathematics
%3 Mathematical Physics
%4 -1
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2012-08-02T07:19:59Z\\nNo. of bitstreams: 1\\n1201.5740v1.pdf: 303047 bytes, checksum: eb2df0afd547514c235422e82f494584 (MD5)
%R 10.1142/S0129055X12500171