We 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 Sch{\"o}dinger 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.

}, isbn = {978-3-319-58904-6}, doi = {10.1007/978-3-319-58904-6_14}, url = {https://doi.org/10.1007/978-3-319-58904-6_14}, author = {Alessandro Olgiati}, editor = {Alessandro Michelangeli and Gianfausto Dell{\textquoteright}Antonio} } @inbook {Olgiati2017, title = {Remarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian}, booktitle = {Advances in Quantum Mechanics: Contemporary Trends and Open Problems}, year = {2017}, pages = {257{\textendash}266}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {The 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 {\textquoteleft}{\textquoteleft}projection counting{\textquoteright}{\textquoteright} scheme.

}, isbn = {978-3-319-58904-6}, doi = {10.1007/978-3-319-58904-6_15}, url = {https://doi.org/10.1007/978-3-319-58904-6_15}, author = {Alessandro Olgiati}, editor = {Alessandro Michelangeli and Gianfausto Dell{\textquoteright}Antonio} } @article {2015, title = {A class of Hamiltonians for a three-particle fermionic system at unitarity}, year = {2015}, note = {This SISSA preprint is composed of 29 pages and is recorded in PDF format}, 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. 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.}, url = {http://urania.sissa.it/xmlui/handle/1963/34469}, author = {Michele Correggi and Gianfausto Dell{\textquoteright}Antonio and Domenico Finco and Alessandro Michelangeli and Alessandro Teta} } @article {2015, title = {Sch{\"o}dinger operators on half-line with shrinking potentials at the origin}, number = {SISSA;06/2015/MATE}, year = {2015}, institution = {SISSA}, abstract = {We discuss the general model of a Schr{\"o}dinger 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.}, url = {http://urania.sissa.it/xmlui/handle/1963/34439}, author = {Gianfausto Dell{\textquoteright}Antonio and Alessandro Michelangeli} } @article {2014, title = {Dynamics on a graph as the limit of the dynamics on a "fat graph"}, number = {SISSA;69/2014/MATE}, year = {2014}, institution = {SISSA}, abstract = {We 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.}, url = {http://urania.sissa.it/xmlui/handle/1963/7485}, author = {Gianfausto Dell{\textquoteright}Antonio and Alessandro Michelangeli} } @article {2012, title = {Some remarks on quantum mechanics}, journal = {International Journal of Geometric Methods in Modern Physics, Volume 9, Issue 2, March 2012, Article number1260018}, year = {2012}, publisher = {World Scientific Publishing}, abstract = {We 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.}, keywords = {Quantum mechanics}, doi = {10.1142/S0219887812600183}, url = {http://hdl.handle.net/1963/7018}, author = {Gianfausto Dell{\textquoteright}Antonio} } @article {2012, title = {Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions}, journal = {Rev. Math. Phys. 24 (2012), 1250017}, number = {arXiv:1201.5740;}, year = {2012}, publisher = {World Scientific}, abstract = {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.}, doi = {10.1142/S0129055X12500171}, url = {http://hdl.handle.net/1963/6069}, author = {Michele Correggi and Gianfausto Dell{\textquoteright}Antonio and Domenico Finco and Alessandro Michelangeli and Alessandro Teta} } @article {2011, title = {On the number of eigenvalues of a model operator related to a system of three particles on lattices}, journal = {J. Phys. A 44 (2011) 315302}, year = {2011}, publisher = {IOP Publishing}, doi = {10.1088/1751-8113/44/31/315302}, url = {http://hdl.handle.net/1963/5496}, author = {Gianfausto Dell{\textquoteright}Antonio and Zahriddin I. Muminov and Y.M. Shermatova} } @article {2010, title = {Effective Schroedinger dynamics on $ ε$-thin Dirichlet waveguides via Quantum Graphs I: star-shaped graphs}, journal = {J. Phys. A 43 (2010) 474014}, number = {arXiv.org;1004.4750v2}, year = {2010}, publisher = {IOP Publishing}, abstract = {We 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.}, doi = {10.1088/1751-8113/43/47/474014}, url = {http://hdl.handle.net/1963/4106}, author = {Gianfausto Dell{\textquoteright}Antonio and Emanuele Costa} } @article {2010, title = {A time-dependent perturbative analysis for a quantum particle in a cloud chamber}, journal = {Annales Henri Poincare 11 (2010) 539-564}, number = {arXiv.org;0907.5503v1}, year = {2010}, publisher = {Springer}, abstract = {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.}, doi = {10.1007/s00023-010-0037-4}, url = {http://hdl.handle.net/1963/3969}, author = {Gianfausto Dell{\textquoteright}Antonio and Rodolfo Figari and Alessandro Teta} } @article {2007, title = {The number of eigenvalues of three-particle Schr{\"o}dinger operators on lattices}, journal = {J. Phys. A 40 (2007) 14819-14842}, number = {arXiv.org;math/0703191v1}, year = {2007}, abstract = {We 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.}, doi = {10.1088/1751-8113/40/49/015}, url = {http://hdl.handle.net/1963/2576}, author = {Sergio Albeverio and Gianfausto Dell{\textquoteright}Antonio and Saidakhmat N. Lakaev} } @article {2005, title = {Decay of a bound state under a time-periodic perturbation: a toy case}, journal = {J. Phys. A 38 (2005) 4769-4781}, number = {SISSA;54/2004/FM}, year = {2005}, abstract = {We 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 {\textquoteleft}{\textquoteleft}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.}, doi = {10.1088/0305-4470/38/22/002}, url = {http://hdl.handle.net/1963/2298}, author = {Michele Correggi and Gianfausto Dell{\textquoteright}Antonio} } @article {2005, title = {Ionization for Three Dimensional Time-dependent Point Interactions}, journal = {Comm. Math. Phys. 257 (2005) 169-192}, number = {SISSA;11/2004/FM}, year = {2005}, abstract = {We 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 {\textquoteleft}{\textquoteleft}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.}, doi = {10.1007/s00220-005-1293-x}, url = {http://hdl.handle.net/1963/2297}, author = {Michele Correggi and Gianfausto Dell{\textquoteright}Antonio and Rodolfo Figari and Andrea Mantile} } @article {2004, title = {Blow-up solutions for the Schr{\"o}dinger equation in dimension three with a concentrated nonlinearity}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 21 (2004) 121-137}, year = {2004}, publisher = {Elsevier}, abstract = {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.}, doi = {10.1016/j.anihpc.2003.01.002}, url = {http://hdl.handle.net/1963/2998}, author = {Riccardo Adami and Gianfausto Dell{\textquoteright}Antonio and Rodolfo Figari and Alessandro Teta} } @article {2004, title = {Rotating Singular Perturbations of the Laplacian}, journal = {Ann. Henri Poincare 5 (2004) 773-808}, number = {SISSA;66/2003/FM}, year = {2004}, publisher = {Springer}, abstract = {We 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).}, doi = {10.1007/s00023-004-0182-8}, url = {http://hdl.handle.net/1963/2945}, author = {Michele Correggi and Gianfausto Dell{\textquoteright}Antonio} } @article {2004, title = {Semiclassical analysis of constrained quantum systems}, journal = {J. Phys. A 37 (2004) 5605-5624}, number = {arXiv.org;math-ph/0312034v2}, year = {2004}, publisher = {IOP Publishing}, abstract = {We 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.}, doi = {10.1088/0305-4470/37/21/007}, url = {http://hdl.handle.net/1963/2997}, author = {Gianfausto Dell{\textquoteright}Antonio and Lucattilio Tenuta} } @article {1998, title = {Diffusion of a particle in presence of N moving point sources}, journal = {Annales Poincare Phys.Theor.69:413-424,1998}, number = {SISSA;9/96/ILAS/FM}, year = {1998}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/134}, author = {Gianfausto Dell{\textquoteright}Antonio and Rodolfo Figari and Alessandro Teta} } @article {1997, title = {Statistics in space dimension two}, journal = {Lett. Math. Phys. 40 (1997), no. 3, 235-256}, number = {SISSA;5/96/ILAS/FM}, year = {1997}, publisher = {SISSA Library}, abstract = {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).}, doi = {10.1023/A:1007361832622}, url = {http://hdl.handle.net/1963/130}, author = {Gianfausto Dell{\textquoteright}Antonio and Rodolfo Figari and Alessandro Teta} } @article {1995, title = {Classical solutions for a perturbed N-body system}, journal = {Topological nonlinear analysis, II (Frascati, 1995), 1--86, Progr. Nonlinear Differential Equations Appl., 27, Birkhauser Boston, Boston, MA, 1997}, number = {SISSA;1/96/ILAS/FM}, year = {1995}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/126}, author = {Gianfausto Dell{\textquoteright}Antonio} } @article {1993, title = {Workshop on point interactions, Trieste, 21-23 December 1992}, number = {SISSA;8/93/ILAS/MS}, year = {1993}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/71}, author = {Gianfausto Dell{\textquoteright}Antonio} } @article {1989, title = {On the number of families of periodic solutions of a Hamiltonian system near equilibrium. II. (English. Italian summary)}, journal = {Boll. Un. Mat. Ital. B (7) 3 (1989), no. 3, 579-590}, number = {SISSA;13/88/FM}, year = {1989}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/609}, author = {Gianfausto Dell{\textquoteright}Antonio and Biancamaria D{\textquoteright}Onofrio} } @article {1988, title = {Methods of stochastic stability and properties of the Gribov horizon in the stochastic quantization of gauge theories}, journal = {Stochastic processes, physics and geompetry (Ascona and Locarno, 1988), 302, World Sci.Publishing,NJ(1990)}, number = {SISSA;57/89/FM}, year = {1988}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/817}, author = {Gianfausto Dell{\textquoteright}Antonio} }