1,721,216 research outputs found
Multi-scale analysis in the occupation numbers of particle states: an application to three-modes Bogoliubov Hamiltonians
No full textIn this paper, we describe a multi-scale technique introduced in Pizzo (Bose particles in a box I. A convergent expansion of the ground state of a three-modes Bogoliubov Hamiltonian, [32]) to study many-body quantum systems. The method is based on the Feshbach–Schur map and the scales are represented by occupation numbers of particle states. The main purpose of this method is to implement singular perturbation theory to deal with large field problems. A simple model to apply our method is the three-modes (including the zero mode) Bogoliubov Hamiltonian that here we consider for a sufficiently small ratio between the kinetic energy and the Fourier component of the (positive type) potential corresponding to the two nonzero modes. In space dimension d≥3, for an arbitrarily large box and at fixed, large particle density ρ (i.e., ρ is independent of the size of the box), this method provides the construction of the ground state vector of the system and its expansion, up to any desired precision, in terms of the bare operators and the ground state energy. In the mean field limiting regime (i.e., at fixed box volume |Λ| and for a number of particles, N, sufficiently large), this method provides the same results in any dimension d≥1. Furthermore, in the mean field limit, we can replace the ground state energy with the Bogoliubov energy in the expansion of the ground state vector
Coulomb scattering in the massless Nelson model III: ground state wave functions and non-commutative recurrence relations
No full textLet HP,σbe the single-electron fiber Hamiltonians of the massless Nelson model at total momentum P and infrared cut-off σ> 0. We establish detailed regularity properties of the corresponding n-particle ground state wave functions fP,σn as functions of P and σ. In particular, we show that (Formula presented.) where c is a numerical constant, λ00 is a positive function of the maximal admissible coupling constant which satisfies limλ0→0δλ0=0 and χ[σ,κ)is the (approximate) characteristic function of the energy region between the infrared cut-off σ and the ultraviolet cut-off κ. While the analysis of the first derivative is relatively straightforward, the second derivative requires a new strategy. By solving a non-commutative recurrence relation, we derive a novel formula for fP,σn with improved infrared properties. In this representation ∂Pj′∂PjfP,σn is amenable to sharp estimates obtained by iterative analytic perturbation theory in part II of this series of papers. The bounds stated above are instrumental for scattering theory of two electrons in the Nelson model, as explained in part I of this series
Coulomb scattering in the massless Nelson model II. Regularity of ground states
No full textFor the massless Nelson model, we provide detailed information about the dependence of the normalized ground states ψP,σ of the fiber single-electron Hamiltonians HP,σ on the total momentum P and the infrared cut-off σ. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form ∥∂PiψP,σ∥,∥∂Pi∂PjψP,σ∥≤ c σδΛ0, for some constant c and a function of the maximal admissible coupling constant Λ0→δΛ0 such that limΛ0→0δΛ0 = 0. These results hold both in the infrared-regular and infrared-singular cases. They are exploited in part I of this series to construct the two-electron scattering states in the infrared-regular massless Nelson model (in the absence of an infrared cut-off) along the lines of the Haag-Ruelle scattering theory. They should also be relevant to the problem of scattering of two infraparticles in the infrared-singular Nelson model, whose solution is the goal of this series of papers. Although a part of a larger investigation, the present work is written in a self-contained fashion
Lie-Schwinger block-diagonalization and gapped quantum chains
No full textWe study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain
The time-evolution of states in quantum mechanics according to the ETH-approach
Full text linkIt is argued that the Schrodinger equation does not yield a correct description of the quantum-mechanical time evolution of states of isolated physical systems featuring events. A general statistical law replacing unitary Schrodinger evolution of states is then formulated within the so-called ETH-Approach to Quantum Mechanics. This law eliminates the infamous "measurement problem." Our general concepts and results are illustrated by an analysis of simple models describing a very heavy atom coupled to the quantized radiation field. In the limit where the speed of light tends to infinity these models can be treated quite explicitly
Coulomb scattering in the massless Nelson model IV. Atom-electron scattering
No full textWe consider the massless Nelson model with two types of massive particles which we call atoms and electrons. The atoms interact with photons via an infrared regular form-factor and thus they are Wigner-type particles with sharp mass-shells. The electrons have an infrared singular form-factor and thus they are infraparticles accompanied by soft-photon clouds correlated with their velocities. In the weak coupling regime, we construct scattering states of one atom and one electron, and demonstrate their asymptotic clustering into individual particles. The proof relies on the Cook's argument, clustering estimatesblack, and the non-stationary phase method. The latter technique requires sharp estimates on derivatives of the ground state wave functions of the fiber Hamiltonians of the model, which were proven in the earlier papers of this series. Although we rely on earlier black studies of the atom-atom and electron-photon scattering in the Nelson model, the paper is written in a self-contained manner. A perspective on the open problem of the electron-electron scattering in this model is also given
Diffusione COVID-19: il trade off tra contact tracing e trattamento dei dati personali degli individui
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