TY - RPRT
T1 - A class of Hamiltonians for a three-particle fermionic system at unitarity
Y1 - 2015
A1 - Michele Correggi
A1 - Gianfausto Dell'Antonio
A1 - Domenico Finco
A1 - Alessandro Michelangeli
A1 - Alessandro Teta
AB - 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.
UR - http://urania.sissa.it/xmlui/handle/1963/34469
N1 - This SISSA preprint is composed of 29 pages and is recorded in PDF format
U1 - 34644
U2 - Mathematics
U4 - 1
ER -
TY - JOUR
T1 - Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions
JF - Rev. Math. Phys. 24 (2012), 1250017
Y1 - 2012
A1 - Michele Correggi
A1 - Gianfausto Dell'Antonio
A1 - Domenico Finco
A1 - Alessandro Michelangeli
A1 - Alessandro Teta
AB - 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.
PB - World Scientific
UR - http://hdl.handle.net/1963/6069
U1 - 5955
U2 - Mathematics
U3 - Mathematical Physics
U4 - -1
ER -