All publications from Gianfausto Dell’Antonio
Contact interactions and gamma convergence
Dell'Antonio G.
We introduce contact interactions defined by boundary conditions at the contact manifold Γ=Ui, j(xi = xj). There are two types of contact interactions, weak and strong. Both provide self-adjoint extensions of H0 the free hamiltonian restricted away from G. We analyze both of them by "lifting" the system to a space of more singular functions: the map is fractioning and mixing. In the new space we use tools of Functional Analysis. After returning to physical space we use Gamma convergence, a well-known variational tool. We prove that contact interactions are strong resolvent limits of potentials with finite range. Weak contact of one boson with two other bosons leads to the low-density Bose-Einstrin condensate. Simultaneous weak contact of three bosons produces the high-density condensate which has an Efimov sequence of bound states. In Low Energy Physics strong contact of one particle with another two produces an Efimov sequence of bound states (we will comment briefly on the relation with the effect with the same name in Quantum Mechanics). For N bosons strong contact gives a lower bound -CN for the energy. A system of fermions in strong contact (unitary gas) has a positive hamiltonian. We give several examples in dimension 3,2,1. In the Appendix we describe the ground state of the Polaron.
Schrödinger operators on half-line with shrinking potentials at the origin
Dell'Antonio G., Michelangeli A.
We discuss the general model of a Schrödinger quantum particle constrained on a straight half-line with given selfadjoint 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.
Existence of an isolated band in a system of three particles in an optical lattice
Lakaev S., Dell'Antonio G., Khalkhuzhaev A.
We prove the existence of two- and three-particle bound states of the Schrödinger operators hμ (k), k ∈ Td and Hμ (K), K ∈ Td associated to Hamiltonians hμ and Hμ of a system of two and three identical bosons on the lattice ℤd, d = 1, 2 interacting via pairwise zero-range attractive μ < 0 or repulsive μ > 0 potentials. As a consequence, we show the existence of an isolated band in the two- and three-bosonic systems in an optical lattice.
Quantum mechanics: Light and shadows (ontological problems and epistemic solutions)
Dell'Antonio G.
We discuss several problems that arise in the Copenhagen interpretation of quantum mechanics, in an attempt to come to grips with what E. T. Jaynes has called the quantum omelette.
A Class of Hamiltonians for a Three-Particle Fermionic System at Unitarity
Correggi M., Dell’Antonio G., Finco D., Michelangeli A., Teta A.
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∗ ≃ (13.607)−1 a self-adjoint and lower bounded Hamiltonian H0 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 ∈ (m∗,m∗∗), where m∗∗ ≃ (8.62)−1, there is a further family of self-adjoint and lower bounded Hamiltonians H0,β, β ∈ 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.
On Tracks in a Cloud Chamber
Dell’Antonio G.F.
It is an experimental fact that (Formula presented.) -decays produce in a cloud chamber at most one track (sequence of liquid droplets) and that this track points in a random direction. This seems to contradict the description of decay in Quantum Mechanics: according to Gamow a spherical wave is produced and moves radially according to Schrödinger’s equation. It is as if the interaction with the supersaturated vapor turned the wave into a particle. The aim of this note is to place this effect in the context of Schrödinger’s Quantum Mechanics. We shall see that the properties of the initial wave function suggest the introduction of a semiclassical formalism in which the (Formula presented.) -wave can be described as a collection of semiclassical (probability) wavelets; each of them interacts with an atom and forms an entangled state. The interaction can be regarded as a semiclassical inelastic scattering event. The measurement (of the position of the first droplet of the track) selects the wave function of one of the ions, with a probability given by Born’s rule. This ion interacts with the atoms nearby leading to the formation of a droplet. One can reasonably assume that also the wavelet entangled with the selected ion has probability one to remain as part of the description of the system. The measurement process is therefore represented by a (non local) unitary operator. The (semiclassical) wavelet remains sharply localized on a classical path (Formula presented.) It is still a probability wave: it determines the probability that another atom be ionized. This probability is essentially zero unless the atom lies on (Formula presented.). This gives the visible track.
Lectures on the mathematics of quantum mechanics I
Dell’Antonio G.
The first volume (General Theory) differs from most textbooks as it emphasizes the mathematical structure and mathematical rigor, while being adapted to the teaching the first semester of an advanced course in Quantum Mechanics (the content of the book are the lectures of courses actually delivered). It differs also from the very few texts in Quantum Mechanics that give emphasis to the mathematical aspects because this book, being written as Lecture Notes, has the structure of lectures delivered in a course, namely introduction of the problem, outline of the relevant points, mathematical tools needed, theorems, proofs. This makes this book particularly useful for self-study and for instructors in the preparation of a second course in Quantum Mechanics (after a first basic course). With some minor additions it can be used also as a basis of a first course in Quantum Mechanics for students in mathematics curricula. The second part (Selected Topics) are lecture notes of a more advanced course aimed at giving the basic notions necessary to do research in several areas of mathematical physics connected with quantum mechanics, from solid state to singular interactions, many body theory, semi-classical analysis, quantum statistical mechanics. The structure of this book is suitable for a second-semester course, in which the lectures are meant to provide, in addition to theorems and proofs, an overview of a more specific subject and hints to the direction of research. In this respect and for the width of subjects this second volume differs from other monographs on Quantum Mechanics. The second volume can be useful for students who want to have a basic preparation for doing research and for instructors who may want to use it as a basis for the presentation of selected topics.
Stability for a system of N fermions plus a different particle with zero-range interactions
Correggi M., Dell'Antonio G., Finco D., Michelangeli A., Teta A.
We study the stability problem for a non-relativistic quantum system in dimension three composed by N < 2 identical fermions, with unit mass, interacting with a different particle, with mass m, via a zero-range interaction of strength α ∈ . We construct the corresponding renormalized quadratic (or energy) form $\mathcal{F}-$ and the so-called SkornyakovTerMartirosyan symmetric extension H α, which is the natural candidate as Hamiltonian of the system. We find a value of the mass m*(N) such that for m > m*(N) the form $\mathcal{F}- $ is closed and bounded from below. As a consequence, $\mathcal{F}-$ defines a unique self-adjoint and bounded from below extension of H α and therefore the system is stable. On the other hand, we also show that the form $\mathcal{F}-$ 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. © 2012 World Scientific Publishing Company.
Some remarks on Quantum mechanics
Dell'Antonio G.
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. © 2012 World Scientific Publishing Company.
On the number of eigenvalues of a model operator related to a system of three particles on lattices
Dell'Antonio G., Muminov Z., Shermatova Y.
We consider a quantum mechanical system on a lattice ℤ3 in which three particles, two of them being identical, interact through a zero-range potential. We admit a very general form for the 'kinetic' part H 0γ of the Hamiltonian, which contains a parameter γ to distinguish the two identical particles from the third one (in the continuum case this parameter would be the inverse of the mass). We prove that there is a value γ*. of the parameter such that only forγ < γ*. the Efimov effect (infinite number of bound states if the two-body interactions have a resonance) is absent for the sector of the Hilbert space which contains functions which are antisymmetric with respect to the two identical particles, while it is present for all values of γ on the symmetric sector. We comment briefly on the relation of this result with previous investigations on the Thomas effect. We also establish the following asymptotics for the number N(z) of eigenvalues z below Emin, the lower limit of the essential spectrum of H0. In the symmetric subspace lim z→Emin- Ns(z)/|log|Emin-z∥ = u0s(γ), ∀ γ, whereas in the antisymmetric subspace lim z→Emin- N as(z)/|log|Emin-z∥ = u0as(γ), ∀ γ > γ*, where Uas0 (γ), Us0(γ) are written explicitly as a function of the integral kernel of operators acting on L2((0, r) × (L2(2) ⊗ L 2(2)) (2 is the unit sphere in ℝ3). © 2011 IOP Publishing Ltd.
Effective Schrödinger dynamics on ε-thin dirichlet waveguides via quantum graphs: I. Star-shaped graphs
Dell'Antonio G., Costa E.
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. © 2010 IOP Publishing Ltd.
A time-dependent perturbative analysis for a quantum particle in a cloud chamber
Dell'Antonio G., Figari R., Teta A.
We consider a simple model of a cloud chamber consisting of a test particle (the α-particle) interacting with two quantum systems (the atoms of the vapor) initially confined around a1, a2 ∈ R3. At time zero, the α-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 a2lies on the line joining the origin with a1. The work is a fully time-dependent version of the original analysis proposed by Mott in 1929. © 2010 Springer Basel AG.
Limit motion on metric graphs
Dell'Antonio G.
We study the solutions to the Schrödinger equation in a neighborhood of a metric graph, in the limit when the radius of the neighborhood tends to zero. We give conditions under which these solutions converge (in a suitable sense) to solutions of the Schrödinger equation on the metric graph, and prove that the limit unitary group depends on the spectral properties of an intermediate operator, obtained by restricting the Schrödinger operator to a "mesoscopic" neighborhood of the vertices.
Spectral analysis of a two body problem with zero-range perturbation
Correggi M., Dell'Antonio G., Finco D.
We consider a class of singular, zero-range perturbations of the Hamiltonian of a quantum system composed by a test particle and a harmonic oscillator in dimension one, two and three and we study its spectrum. In fact we give a detailed characterization of point spectrum and its asymptotic behavior with respect to the parameters entering the Hamiltonian. We also partially describe the positive spectrum and scattering properties of the Hamiltonian. © 2008 Elsevier Inc. All rights reserved.
Joint excitation probability for two harmonic oscillators in one dimension and the Mott problem
Dell'Antonio G., Figari R., Teta A.
We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions a1 and a2, with a1 >0 and ∫ a2 ∫ > a1, in the two possible situations: a2 >0 and a2 <0. At time zero, the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability P n1 n2 - (t) [P n1 n2 + (t)] that both oscillators are in the excited states labeled by n1 and n2 >0 at time t> ∫ a2 ∫ v0 -1 when a2 <0 (a2 >0). We prove that P n1 n2 - (t) is negligible with respect to P n1 n2 + (t) up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott ["The wave mechanics of α -ray tracks," Proc. R. Soc. London, Ser. A 126, 79 (1929)], where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schrödinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schrödinger equation for the three-particle system, with no reference to the wave packet collapse postulate. © 2008 American Institute of Physics.
A brief review on point interactions
Dell'Antonio G., Figari R., Teta A.
We review properties and applications of point interaction Hamiltonians. This class of operators is first defined following a classical presentation and then generalized to cases in which some dynamical and/or geometrical parameters are varying with time. We recall their relations with smooth short range potentials. © 2008 Springer-Verlag Berlin Heidelberg.
Dynamics on quantum graphs as constrained systems
Dell'Antonio G.F.
We study the possibility of regarding the dynamics on a quantum graph as limit, as a small parameter ∈ → O, of a dynamics with a strong confining potential. We define a projection operator along the first eigenfunction of a transversal operator and, under suitable assumptions, we prove that the projection of the solution strongly converges along subsequences to a function that satisfies the Schrödinger equation on each open edge of the graph. Moreover the limit dynamics is unitary. If the limit is independent of the subsequence, one has a limit one-parameter group, generated by one of the self-adjoint extensions of a symmetric operator defined on the open graph (with the vertices deleted). The crucial role of the shape of the confining potential at the vertices is pointed out. © 2007 Polish Scientific Publishers PWN, Warszawa.
Quantum graphs as holonomic constraints
Dell'Antonio G., Tenuta L.
We consider the dynamics on a quantum graph as the limit of the dynamics generated by a one-particle Hamiltonian in ℝ2 with a potential having a deep strict minimum on the graph, when the width of the well shrinks to zero. For a generic graph we prove convergence outside the vertices to the free dynamics on the edges. For a simple model of a graph with two edges and one vertex, we prove convergence of the dynamics to the one generated by the Laplacian with Dirichlet boundary conditions in the vertex. © 2006 American Institute of Physics.
Introductory Article: Quantum Mechanics
dell'Antonio G.F.
Decay of a bound state under a time-periodic perturbation: A toy case
Correggi M., Dell'Antonio G.
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 'strength' α(t). Under very weak generic conditions on the Fourier coefficients of α(t), we prove complete ionization as t → ∞. We prove also that, under the same conditions, all the states of the system are scattering states. © 2005 IOP Publishing Ltd.

End of content

No more pages to load