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 -
TY - JOUR
T1 - A time-dependent perturbative analysis for a quantum particle in a cloud chamber
JF - Annales Henri Poincare 11 (2010) 539-564
Y1 - 2010
A1 - Gianfausto Dell'Antonio
A1 - Rodolfo Figari
A1 - Alessandro Teta
AB - We consider a simple model of a cloud chamber consisting of a test particle (the alpha-particle) interacting with two other particles (the atoms of the vapour) subject to attractive potentials centered in $a_1, a_2 \\\\in \\\\mathbb{R}^3$. At time zero the alpha-particle is described by an outgoing spherical wave centered in the origin and the atoms are in their ground state. We show that, under suitable assumptions on the physical parameters of the system and up to second order in perturbation theory, the probability that both atoms are ionized is negligible unless $a_2$ lies on the line joining the origin with $a_1$. The work is a fully time-dependent version of the original analysis proposed by Mott in 1929.
PB - Springer
UR - http://hdl.handle.net/1963/3969
U1 - 432
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Blow-up solutions for the SchrÃ¶dinger equation in dimension three with a concentrated nonlinearity
JF - Ann. Inst. H. Poincare Anal. Non Lineaire 21 (2004) 121-137
Y1 - 2004
A1 - Riccardo Adami
A1 - Gianfausto Dell'Antonio
A1 - Rodolfo Figari
A1 - Alessandro Teta
AB - We present some results on the blow-up phenomenon for the Schroedinger equation in dimension three with a nonlinear term supported in a fixed point. We find sufficient conditions for the blow up exploiting the moment of inertia of the solution and the uncertainty principle. In the critical case, we discuss the additional symmetry of the equation and construct a family of explicit blow up solutions.
PB - Elsevier
UR - http://hdl.handle.net/1963/2998
U1 - 1335
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Diffusion of a particle in presence of N moving point sources
JF - Annales Poincare Phys.Theor.69:413-424,1998
Y1 - 1998
A1 - Gianfausto Dell'Antonio
A1 - Rodolfo Figari
A1 - Alessandro Teta
PB - SISSA Library
UR - http://hdl.handle.net/1963/134
U1 - 75
U2 - LISNU
U3 - Interdisciplinary Laboratory for Advanced Studies
ER -
TY - JOUR
T1 - Statistics in space dimension two
JF - Lett. Math. Phys. 40 (1997), no. 3, 235-256
Y1 - 1997
A1 - Gianfausto Dell'Antonio
A1 - Rodolfo Figari
A1 - Alessandro Teta
AB - We construct as a selfadjoint operator the Schroedinger hamiltonian for a system of $N$ identical particles on a plane, obeying the statistics defined by a representation $\\\\pi_1$ of the braid group. We use quadratic forms and potential theory, and give details only for the free case; standard arguments provide the extension of our approach to the case of potentials which are small in the sense of forms with respect to the laplacian. We also comment on the relation between the analysis given here and other approaches to the problem, and also on the connection with the description of a quantum particle on a plane under the influence of a shielded magnetic field (Aharanov-Bohm effect).
PB - SISSA Library
UR - http://hdl.handle.net/1963/130
U1 - 12
U2 - LISNU
U3 - Interdisciplinary Laboratory for Advanced Studies
ER -
TY - JOUR
T1 - Quadratic forms for singular perturbations of the Laplacian
JF - Publ. Res. Inst. Math. Sci. 26 (1990), no. 5, 803--817
Y1 - 1990
A1 - Alessandro Teta
PB - SISSA Library
UR - http://hdl.handle.net/1963/757
U1 - 3034
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - THES
T1 - Singular perturbation of the Laplacian and connections with models of random media
Y1 - 1989
A1 - Alessandro Teta
PB - SISSA
UR - http://hdl.handle.net/1963/6348
U1 - 6281
U2 - Mathematics
U4 - -1
ER -