arXiv subjects
Publications and source records attributed to Rodolfo Figari.
We briefly summarize the most relevant steps in the search of rigorous results about the properties of quantum systems made of three bosons interacting with zero-range forces. We also describe recent attempts to solve the unboundedness problem of point-interaction Hamiltonians for a three-boson system, keeping unaltered the spectrum structure at low energies.
We outline an unified introduction to the evolution equations of classical and quantum systems intended for a high school students audience. The attempt consists in circumventing the lack of mathematical knowledge with the use of simplified forms of numerical analysis. The aim is to allow students to approach theoretical features as well as computational aspects of the evolution equations. In particular, the possibility to compute and analyze approximate solutions of the dynamical laws of classical, stochastic and quantum mechanics enables to highlight the distinctive role played by probability in different contexts. As computer support for numerical analysis we selected the spreadsheet. It is a work environment usually presented in high school and it is an ideal tool for an intuitive approach to numerical computation through recursive algorithms. The proposal was presented to an audience composed by students of the course of Didactics of Physics and of high school Science teachers.
We examine the suppression of quantum beating in a one dimensional non- linear double well potential, made up of two focusing nonlinear point interactions. The investigation of the Schr\"odinger dynamics is reduced to the study of a system of coupled nonlinear Volterra integral equations. For various values of the geometric and dynamical parameters of the model we give analytical and numerical results on the way states, which are initially confined in one well, evolve. We show that already for a nonlinearity exponent well below the critical value there is complete suppression of the typical beating behavior characterizing the linear quantum case.
We analyze the paper "The wave mechanics of $\alpha$-ray tracks" (Mott, 1929), published in 1929 by N.F. Mott. In particular, we discuss the theoretical context in which the paper appeared and give a detailed account of the approach used by the author and the main result attained. Moreover, we comment on the relevance of the work not only as far as foundations of Quantum Mechanics are concerned but also as the earliest pioneering contribution in decoherence theory.
We consider a one dimensional evolution problem modeling the dynamics of an acoustic field coupled with a set of mechanical oscillators. We analyze solutions of the system of ordinary and partial differential equations with time-dependent boundary conditions describing the evolution in the limit of a continuous distribution of oscillators.
We examine perturbations of eigenvalues and resonances for a class of multi-channel quantum mechanical model-Hamiltonians describing a particle interacting with a localized spin in dimension $d=1,2,3$. We consider unperturbed Hamiltonians showing eigenvalues and resonances at the threshold of the continuous spectrum and we analyze the effect of various type of perturbations on the spectral singularities. We provide algorithms to obtain convergent series expansions for the coordinates of the singularities.
In dimension $d=1,2,3$ we define a family of two-channel Hamiltonians obtained as point perturbations of the generator of the free decoupled dynamics. Within the family we choose two Hamiltonians, $\hat H_0$ and $\hat H_\ve$, giving rise respectively to the unperturbed and to the perturbed evolution. The Hamiltonian $\hat H_0$ does not couple the channels and has an eigenvalue embedded in the continuous spectrum. The Hamiltonian $\hat H_\ve$ is a small perturbation, in resolvent sense, of $\hat H_0$ and exhibits a small coupling between the channels. We take advantage of the complete solvability of our model to prove with simple arguments that the embedded eigenvalue of $\hat H_0$ shifts into a resonance for $\hat H_\ve$. In dimension three we analyze details of the time behavior of the projection onto the region of the spectrum close to the resonance.
We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions $a_1$, $a_2$, with $a_1 >0$, $|a_2|>a_1$, in the two possible situations: $a_2>0$ and $a_2 <0$. 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. %$\pm M v_0$. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability $\mathcal{P}^{-}_{n_1 n_2} (t)$ (resp. $\mathcal{P}^{+}_{n_1 n_2} (t)$) that both oscillators are in the excited states labelled by $n_1$, $n_2 >0$ at time $t > |a_2| v_0^{-1}$ when $a_2 <0$ (resp. $a_2 >0$). We prove that $\mathcal{P}_{n_1 n_2}^- (t)$ is negligible with respect to $\mathcal{P}_{n_1 n_2}^+ (t)$, 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 in \cite{m}, 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.
We consider a system realized with one spinless quantum particle and an array of $N$ spins 1/2 in dimension one and three. We characterize all the Hamiltonians obtained as point perturbations of an assigned free dynamics in terms of some ``generalized boundary conditions''. For every boundary condition we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin dependent point potentials that could be of interest as multi-component solvable models.
We consider a non relativistic quantum system consisting of $K$ heavy and $N$ light particles in dimension three, where each heavy particle interacts with the light ones via a two-body potential $αV$. No interaction is assumed among particles of the same kind. Choosing an initial state in a product form and assuming $α$ sufficiently small we characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. In the case K=1 the result is extended to arbitrary $α$. The proof relies on a perturbative analysis and exploits a generalized version of the standard dispersive estimates for the Schrödinger group. Exploiting the asymptotic formula, it is also outlined an application to the problem of the decoherence effect produced on a heavy particle by the interaction with the light ones.