Browsing by Author "Dell'Antonio, Gianfausto"
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Item A class of Hamiltonians for a three-particle fermionic system at unitarity(2015-05-15) Correggi, Michele; Dell'Antonio, Gianfausto; Finco, Domenico; Michelangeli, Alessandro; Teta, Alessandro; MathematicsWe 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.Item Contact interaction in Quantum Mechanics and Gamma convergence(SISSA, 2020) Dell'Antonio, Gianfausto; MathematicsItem Dynamics on a graph as the limit of the dynamics on a "fat graph"(SISSA, 2014) Dell'Antonio, Gianfausto; Michelangeli, AlessandroWe discuss how the vertex boundary conditions for the dynamics of a quantum particle constrained on a graph emerge in the limit of the dynamics of a particle in a tubular region around the graph (\fat graph") when the transversal section of this region shrinks to zero. We give evidence of the fact that if the limit dynamics exists and is induced by the Laplacian on the graph with certain self-adjoint boundary conditions, such conditions are determined by the possible presence of a zero energy resonance on the fat graph. Pictorially, one may say that in the shrinking limit the resonance acts as a bridge connecting the boundary values at the vertex along the different rays.Item Lp-boundedness of wave operators for the three-dimensional multi-centre point interaction(2017-04-13) Dell'Antonio, Gianfausto; Michelangeli, Alessandro; Scandone, Raffaele; Yajima, Kenji; MathematicsWe prove that, for arbitrary centres and strengths, the wave operators for three dimensional Schrödinger operators with multi-centre local point interactions are bounded in Lp(R3) for 1 < p < 3 and unbounded otherwise.Item Schödinger operators on half-line with shrinking potentials at the origin(SISSA, 2015) Dell'Antonio, Gianfausto; Michelangeli, AlessandroWe discuss the general model of a Schrödinger quantum particle constrained on a straight half-line with given self-adjoint boundary condition at the origin and an interaction potential supported around the origin. We study the limit when the range of the potential scales to zero and its magnitude blows up. We show that in the limit the dynamics is generated by a self-adjoint negative Laplacian on the half-line, with a possible preservation or modification of the boundary condition at the origin, depending on the magnitude of the scaling and of the strength of the potential.