We 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.

1 aDell'Antonio, Gianfausto1 aMichelangeli, Alessandro1 aScandone, Raffaele1 aYajima, Kenji uhttps://doi.org/10.1007/s00023-017-0628-401416nas a2200169 4500008004100000020002200041245008400063210007000147260004400217300001400261520082100275100002001096700002301116700002901139700002901168856004901197 2017 eng d a978-3-319-58904-600aDispersive Estimates for Schrödinger Operators with Point Interactions in ℝ30 aDispersive Estimates for Schrödinger Operators with Point Intera aChambSpringer International Publishing a187–1993 aThe study of dispersive properties of Schrödinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schrödinger equations with singular interactions. In this work we proved that, in the case of one point interaction in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p$−$L^q$ estimates of the free Laplacian in the smaller regime $q \in [2,3)$. These estimates are implied by a recent result concerning the Lpboundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime $q \geq 3$.

1 aIandoli, Felice1 aScandone, Raffaele1 aMichelangeli, Alessandro1 aDell'Antonio, Gianfausto uhttps://doi.org/10.1007/978-3-319-58904-6_1101139nas a2200157 4500008004100000020002200041245007400063210006900137260004400206300001400250520058600264100002400850700002900874700002900903856004900932 2017 eng d a978-3-319-58904-600aEffective Non-linear Dynamics of Binary Condensates and Open Problems0 aEffective Nonlinear Dynamics of Binary Condensates and Open Prob aChambSpringer International Publishing a239–2563 aWe report on a recent result concerning the effective dynamics for a mixture of Bose-Einstein condensates, a class of systems much studied in physics and receiving a large amount of attention in the recent literature in mathematical physics; for such models, the effective dynamics is described by a coupled system of non-linear Schödinger equations. After reviewing and commenting our proof in the mean-field regime from a previous paper, we collect the main details needed to obtain the rigorous derivation of the effective dynamics in the Gross-Pitaevskii scaling limit.

1 aOlgiati, Alessandro1 aMichelangeli, Alessandro1 aDell'Antonio, Gianfausto uhttps://doi.org/10.1007/978-3-319-58904-6_1400917nas a2200157 4500008004100000020002200041245008300063210006900146260004400215300001400259520035500273100002400628700002900652700002900681856004900710 2017 eng d a978-3-319-58904-600aRemarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian0 aRemarks on the Derivation of GrossPitaevskii Equation with Magne aChambSpringer International Publishing a257–2663 aThe effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the ``projection counting'' scheme.

1 aOlgiati, Alessandro1 aMichelangeli, Alessandro1 aDell'Antonio, Gianfausto uhttps://doi.org/10.1007/978-3-319-58904-6_1501837nas a2200145 4500008004100000245007900041210006900120520133000189100002201519700002901541700002001570700002901590700002101619856005101640 2015 en d00aA class of Hamiltonians for a three-particle fermionic system at unitarity0 aclass of Hamiltonians for a threeparticle fermionic system at un3 aWe 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.1 aCorreggi, Michele1 aDell'Antonio, Gianfausto1 aFinco, Domenico1 aMichelangeli, Alessandro1 aTeta, Alessandro uhttp://urania.sissa.it/xmlui/handle/1963/3446901018nas a2200121 4500008004100000245007900041210007000120260001000190520058700200100002900787700002900816856005100845 2015 en d00aSchödinger operators on half-line with shrinking potentials at the origin0 aSchödinger operators on halfline with shrinking potentials at th bSISSA3 aWe 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.1 aDell'Antonio, Gianfausto1 aMichelangeli, Alessandro uhttp://urania.sissa.it/xmlui/handle/1963/3443901107nas a2200121 4500008004300000245007000043210006800113260001000181520068600191100002900877700002900906856005000935 2014 en_Ud 00aDynamics on a graph as the limit of the dynamics on a "fat graph"0 aDynamics on a graph as the limit of the dynamics on a fat graph bSISSA3 aWe 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.1 aDell'Antonio, Gianfausto1 aMichelangeli, Alessandro uhttp://urania.sissa.it/xmlui/handle/1963/748500568nas a2200121 4500008004100000245003800041210003800079260003200117520021000149653002200359100002900381856003600410 2012 en d00aSome remarks on quantum mechanics0 aSome remarks on quantum mechanics bWorld Scientific Publishing3 aWe discuss the similarities and differences between the formalism of Hamiltonian Classical Mechanics and of Quantum Mechanics and exemplify the differences through an analysis of tracks in a cloud chamber.10aQuantum mechanics1 aDell'Antonio, Gianfausto uhttp://hdl.handle.net/1963/701801607nas a2200157 4500008004100000245009600041210006900137260002100206520106500227100002201292700002901314700002001343700002901363700002101392856003601413 2012 en d00aStability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions0 aStability for a System of N Fermions Plus a Different Particle w bWorld Scientific3 aWe 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.1 aCorreggi, Michele1 aDell'Antonio, Gianfausto1 aFinco, Domenico1 aMichelangeli, Alessandro1 aTeta, Alessandro uhttp://hdl.handle.net/1963/606900467nas a2200121 4500008004100000245010400041210006900145260001900214100002900233700002600262700002100288856003600309 2011 en d00aOn the number of eigenvalues of a model operator related to a system of three particles on lattices0 anumber of eigenvalues of a model operator related to a system of bIOP Publishing1 aDell'Antonio, Gianfausto1 aMuminov, Zahriddin I.1 aShermatova, Y.M. uhttp://hdl.handle.net/1963/549600702nas a2200121 4500008004300000245011200043210007000155260001900225520025100244100002900495700002000524856003600544 2010 en_Ud 00aEffective Schroedinger dynamics on $ ε$-thin Dirichlet waveguides via Quantum Graphs I: star-shaped graphs0 aEffective Schroedinger dynamics on εthin Dirichlet waveguides vi bIOP Publishing3 aWe describe the boundary conditions at the vertex that one must choose to obtain a dynamical system that best describes the low-energy part of the evolution of a quantum system confined to a very small neighbourhood of a star-shaped metric graph.1 aDell'Antonio, Gianfausto1 aCosta, Emanuele uhttp://hdl.handle.net/1963/410601175nas a2200133 4500008004300000245008500043210006900128260001300197520072500210100002900935700002000964700002100984856003601005 2010 en_Ud 00aA time-dependent perturbative analysis for a quantum particle in a cloud chamber0 atimedependent perturbative analysis for a quantum particle in a bSpringer3 aWe 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 1929.1 aDell'Antonio, Gianfausto1 aFigari, Rodolfo1 aTeta, Alessandro uhttp://hdl.handle.net/1963/396901687nas a2200121 4500008004300000245008300043210007000126520125600196100002201452700002901474700002601503856003601529 2007 en_Ud 00aThe number of eigenvalues of three-particle Schrödinger operators on lattices0 anumber of eigenvalues of threeparticle Schrödinger operators on 3 aWe consider the Hamiltonian of a system of three quantum mechanical particles (two identical fermions and boson)on the three-dimensional lattice $\\\\Z^3$ and interacting by means of zero-range attractive potentials. We describe the location and structure of the essential spectrum of the three-particle discrete Schr\\\\\\\"{o}dinger operator $H_{\\\\gamma}(K),$ $K$ being the total quasi-momentum and $\\\\gamma>0$ the ratio of the mass of fermion and boson.\\nWe choose for $\\\\gamma>0$ the interaction $v(\\\\gamma)$ in such a way the system consisting of one fermion and one boson has a zero energy resonance.\\nWe prove for any $\\\\gamma> 0$ the existence infinitely many eigenvalues of the operator $H_{\\\\gamma}(0).$ We establish for the number $N(0,\\\\gamma; z;)$ of eigenvalues lying below $z<0$ the following asymptotics $$ \\\\lim_{z\\\\to 0-}\\\\frac{N(0,\\\\gamma;z)}{\\\\mid \\\\log \\\\mid z\\\\mid \\\\mid}={U} (\\\\gamma) .$$ Moreover, for all nonzero values of the quasi-momentum $K \\\\in T^3 $ we establish the finiteness of the number $ N(K,\\\\gamma;\\\\tau_{ess}(K))$ of eigenvalues of $H(K)$ below the bottom of the essential spectrum and we give an asymptotics for the number $N(K,\\\\gamma;0)$ of eigenvalues below zero.1 aAlbeverio, Sergio1 aDell'Antonio, Gianfausto1 aLakaev, Saidakhmat N. uhttp://hdl.handle.net/1963/257600818nas a2200109 4500008004300000245007400043210006900117520043500186100002200621700002900643856003600672 2005 en_Ud 00aDecay of a bound state under a time-periodic perturbation: a toy case0 aDecay of a bound state under a timeperiodic perturbation a toy c3 aWe study the time evolution of a three dimensional quantum particle, initially in a bound state, under the action of a time-periodic zero range interaction with ``strength\\\'\\\' (\\\\alpha(t)). Under very weak generic conditions on the Fourier coefficients of (\\\\alpha(t)), we prove complete ionization as (t \\\\to \\\\infty). We prove also that, under the same conditions, all the states of the system are scattering states.1 aCorreggi, Michele1 aDell'Antonio, Gianfausto uhttp://hdl.handle.net/1963/229801006nas a2200133 4500008004300000245007100043210006900114520056200183100002200745700002900767700002000796700002000816856003600836 2005 en_Ud 00aIonization for Three Dimensional Time-dependent Point Interactions0 aIonization for Three Dimensional Timedependent Point Interaction3 aWe study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the ``strength\\\'\\\' of the interaction (\\\\alpha(t)) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of (\\\\alpha(t)), we prove that there is complete ionization as (t \\\\to \\\\infty), starting from a bound state at time (t = 0). Moreover we prove also that, under the same conditions, all the states of the system are scattering states.1 aCorreggi, Michele1 aDell'Antonio, Gianfausto1 aFigari, Rodolfo1 aMantile, Andrea uhttp://hdl.handle.net/1963/229700908nas a2200145 4500008004300000245010400043210007000147260001300217520040600230100002000636700002900656700002000685700002100705856003600726 2004 en_Ud 00aBlow-up solutions for the Schrödinger equation in dimension three with a concentrated nonlinearity0 aBlowup solutions for the Schrödinger equation in dimension three bElsevier3 aWe 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.1 aAdami, Riccardo1 aDell'Antonio, Gianfausto1 aFigari, Rodolfo1 aTeta, Alessandro uhttp://hdl.handle.net/1963/299801012nas a2200121 4500008004300000245005300043210005300096260001300149520064100162100002200803700002900825856003600854 2004 en_Ud 00aRotating Singular Perturbations of the Laplacian0 aRotating Singular Perturbations of the Laplacian bSpringer3 aWe study a system of a quantum particle interacting with a singular time-dependent uniformly rotating potential in 2 and 3 dimensions: in particular we consider an interaction with support on a point (rotating point interaction) and on a set of codimension 1 (rotating blade). We prove the existence of the Hamiltonians of such systems as suitable self-adjoint operators and we give an explicit expression for their unitary semigroups. Moreover we analyze the asymptotic limit of large angular velocity and we prove strong convergence of the time-dependent propagator to some one-parameter unitary group as (\\\\omega \\\\to \\\\infty).1 aCorreggi, Michele1 aDell'Antonio, Gianfausto uhttp://hdl.handle.net/1963/294500849nas a2200121 4500008004300000245005800043210005800101260001900159520046100178100002900639700002300668856003600691 2004 en_Ud 00aSemiclassical analysis of constrained quantum systems0 aSemiclassical analysis of constrained quantum systems bIOP Publishing3 aWe study the dynamics of a quantum particle in R^(n+m) constrained by a strong potential force to stay within a distance of order hbar (in suitable units) from a smooth n-dimensional submanifold M. We prove that in the semiclassical limit the evolution of the wave function is approximated in norm, up to terms of order hbar^(1/2), by the evolution of a semiclassical wave packet centred on the trajectory of the corresponding classical constrained system.1 aDell'Antonio, Gianfausto1 aTenuta, Lucattilio uhttp://hdl.handle.net/1963/299700418nas a2200121 4500008004100000245006600041210006600107260001800173100002900191700002000220700002100240856003500261 1998 en d00aDiffusion of a particle in presence of N moving point sources0 aDiffusion of a particle in presence of N moving point sources bSISSA Library1 aDell'Antonio, Gianfausto1 aFigari, Rodolfo1 aTeta, Alessandro uhttp://hdl.handle.net/1963/13401073nas a2200133 4500008004100000245003800041210003800079260001800117520069900135100002900834700002000863700002100883856003500904 1997 en d00aStatistics in space dimension two0 aStatistics in space dimension two bSISSA Library3 aWe 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).1 aDell'Antonio, Gianfausto1 aFigari, Rodolfo1 aTeta, Alessandro uhttp://hdl.handle.net/1963/13000328nas a2200097 4500008004100000245005400041210005300095260001800148100002900166856003500195 1995 en d00aClassical solutions for a perturbed N-body system0 aClassical solutions for a perturbed Nbody system bSISSA Library1 aDell'Antonio, Gianfausto uhttp://hdl.handle.net/1963/12600347nas a2200097 4500008004100000245006500041210006200106260001800168100002900186856003400215 1993 en d00aWorkshop on point interactions, Trieste, 21-23 December 19920 aWorkshop on point interactions Trieste 2123 December 1992 bSISSA Library1 aDell'Antonio, Gianfausto uhttp://hdl.handle.net/1963/7100455nas a2200109 4500008004100000245012600041210006900167260001800236100002900254700002700283856003500310 1989 en d00aOn the number of families of periodic solutions of a Hamiltonian system near equilibrium. II. (English. Italian summary)0 anumber of families of periodic solutions of a Hamiltonian system bSISSA Library1 aDell'Antonio, Gianfausto1 aD'Onofrio, Biancamaria uhttp://hdl.handle.net/1963/60900412nas a2200097 4500008004100000245012200041210006900163260001800232100002900250856003500279 1988 en d00aMethods of stochastic stability and properties of the Gribov horizon in the stochastic quantization of gauge theories0 aMethods of stochastic stability and properties of the Gribov hor bSISSA Library1 aDell'Antonio, Gianfausto uhttp://hdl.handle.net/1963/817