Solving simple quaternionic differential equations

The renewed interest in investigating quaternionic quantum mechanics, in particular tunneling effects, and the recent results on quaternionic differential operators motivate the study of resolution methods for quaternionic differential equations. In this paper, by using the real matrix representation of left/right acting quaternionic operators, we prove existence and uniqueness for quaternionic initial value problems, discuss the reduction of order for quaternionic homogeneous differential equations and extend to the non-commutative case the method of variation of parameters. We also show that the standard Wronskian cannot uniquely be extended to the quaternionic case. Nevertheless, the absolute value of the complex Wronskian admits a non-commutative extension for quaternionic functions of one real variable. Linear dependence and independence of solutions of homogeneous (right) H-linear differential equations is then related to this new functional. Our discussion is, for simplicity, presented for quaternionic second order differential equations. This involves no loss of generality. Definitions and results can be readily extended to the n-order case.Comment: 9 pages, AMS-Te

Energy and centrality dependence of particle multiplicity in heavy ion collisions from sNN\sqrt{s_{_{NN}}} = 20 to 2760 GeV

The centrality dependence of midrapidity charged-particle multiplicities at a nucleon-nucleon center-of-mass energy of 2.76 TeV from CMS are compared to PHOBOS data at 200 and 19.6 GeV. The results are first fitted with a two-component model which parameterizes the separate contributions of nucleon participants and nucleon-nucleon collisions. A more direct comparison involves ratios of multiplicity densities per participant pair between the different collision energies. The results support and extend earlier indications that the influences of centrality and collision energy on midrapidity charged-particle multiplicities are to a large degree independent.Comment: 5 pages, 2 figures, 1 table, Replaced with published version, v3 has fixed typ

Right eigenvalue equation in quaternionic quantum mechanics

We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations. Our discussion is also extended to complex linear operators, whose spectrum is characterized by 2n complex eigenvalues. We show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products. Finally, we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution.Comment: 24 pages, AMS-Te

Relativistic tunneling through opaque barriers

We propose an analytical study of relativistic tunneling through opaque barriers. We obtain a closed formula for the phase time. This formula is in excellent agreement with the numerical simulations and corrects the standard formula obtained by the stationary phase method. An important result is found when the upper limit of the incoming energy distribution coincides with the upper limit of the tunneling zone. In this case, the phase time is proportional to the barrier width.Comment: 11 pages, 3 figure

Potential Scattering in Dirac Field Theory

We develop the potential scattering of a spinor within the context of perturbation field theory. As an application, we reproduce, up to second order in the potential, the diffusion results for a potential barrier of quantum mechanics. An immediate consequence is a simple generalization to arbitrary potential forms, a feature not possible in quantum mechanics.Comment: 7 page

Quaternionic eigenvalue problem

We discuss the (right) eigenvalue equation for $\mathbb{H}$, $\mathbb{C}$ and $\mathbb{R}$ linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows to translate the quaternionic problem into an {\em equivalent} real or complex counterpart. Interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics.Comment: 13 pages, AMS-Te

Quaternionic potentials in non-relativistic quantum mechanics

We discuss the Schrodinger equation in presence of quaternionic potentials. The study is performed analytically as long as it proves possible, when not, we resort to numerical calculations. The results obtained could be useful to investigate an underlying quaternionic quantum dynamics in particle physics. Experimental tests and proposals to observe quaternionic quantum effects by neutron interferometry are briefly reviewed.Comment: 21 pages, 16 figures (ps), AMS-Te

Thermostatistics in the neighborhood of the π\pi-mode solution for the Fermi-Pasta-Ulam β\beta system: from weak to strong chaos

We consider a $\pi$-mode solution of the Fermi-Pasta-Ulam $\beta$ system. By perturbing it, we study the system as a function of the energy density from a regime where the solution is stable to a regime, where is unstable, first weakly and then strongly chaotic. We introduce, as indicator of stochasticity, the ratio $\rho$ (when is defined) between the second and the first moment of a given probability distribution. We will show numerically that the transition between weak and strong chaos can be interpreted as the symmetry breaking of a set of suitable dynamical variables. Moreover, we show that in the region of weak chaos there is numerical evidence that the thermostatistic is governed by the Tsallis distribution.Comment: 15 pages, 5 figure

A Sagittarius-Induced Origin for the Monoceros Ring

The Monoceros ring is a collection of stars in nearly-circular orbits at roughly 18 kpc from the Galactic center. It may have originated (i) as the response of the disc to perturbations excited by satellite companions or (ii) from the tidal debris of a disrupted dwarf galaxy. The metallicity of Monoceros stars differs from that of disc stars at comparable Galactocentric distances, an observation that disfavours the first scenario. On the other hand, circular orbits are difficult to accommodate in the tidal-disruption scenario, since it requires a satellite which at the time of disruption was itself in a nearly circular orbit. Such satellite could not have formed at the location of the ring and, given its low mass, dynamical friction is unlikely to have played a major role in its orbital evolution. We search cosmological simulations for low-mass satellites in nearly-circular orbits and find that they result, almost invariably, from orbital changes induced by collisions with more massive satellites: the radius of the circular orbit thus traces the galactocentric distance of the collision. Interestingly, the Sagittarius dwarf, one of the most luminous satellites of the Milky Way, is in a polar orbit that crosses the Galactic plane at roughly the same Galactocentric distance as Monoceros. We use idealized simulations to demonstrate that an encounter with Sagittarius might well have led to the circularization and subsequent tidal demise of the progenitor of the Monoceros ring.Comment: 6 pages, 4 figures, to match version published in MNRAS Letters (http://onlinelibrary.wiley.com/doi/10.1111/j.1745-3933.2011.01035.x/abstract