SISSA Open Science

# A class of Hamiltonians for a three-particle fermionic system at unitarity

 dc.contributor.author Correggi, Michele dc.contributor.author Dell'Antonio, Gianfausto dc.contributor.author Finco, Domenico dc.contributor.author Michelangeli, Alessandro dc.contributor.author Teta, Alessandro dc.date.accessioned 2015-05-21T06:42:01Z dc.date.available 2015-05-21T06:42:01Z dc.date.issued 2015-05-15 dc.identifier.uri http://preprints.sissa.it/xmlui/handle/1963/34469 dc.description This SISSA preprint is composed of 29 pages and is recorded in PDF format en_US dc.description.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. en_US 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. dc.description.sponsorship This work was partially supported by the MIUR-FIRB grant 2012 “Dispersive dynamics: Fourier analysis and variational methods”, code RBFR12MXPO (D.F.), the MIUR-FIR grant 2013 “Condensed Matter in Mathematical Physics”, code RBFR13WAET (A.M. and M.C.), a INdAM 2014-2015 Progetto Giovani GNFM grant (A.M.), and a 2013-2014 “CAS-LMU Research in Residence” grant (A.M.). Part of this work has been carried out during a visit of A.M. and M.C. at CIRM (Fondazione Bruno Kessler), Trento, funded by a 2013 Research in Pairs CIRM grant, as well as during a visit of all five authors at the Center for Advanced Studies at LMU Munich, funded by a “CAS-LMU Research in Residence” grant. en_US dc.language.iso en en_US dc.relation.ispartofseries SISSA;22/2015/MATE dc.title A class of Hamiltonians for a three-particle fermionic system at unitarity en_US dc.type Preprint en_US dc.subject.miur en_US dc.miur.area 1 en_US dc.contributor.area Mathematics en_US dc.identifier.sissaPreprint 22/2015/MATE
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