6,085 research outputs found
High-energy asymptotic behavior of the Bourrely-Soffer-Wu model for elastic scattering
Some time ago, an accurate phenomenological approach, the BSW model, was
developed for proton-proton and antiproton-proton elastic scattering cross
sections at center-of-mass energies above 10 GeV. This model has been used to
give successful theoretical predictions for these processes, at successive
collider energies. The BSW model involves a combination of integrals that,
while computable numerically at fairly high energies, require some mathematical
analysis to reveal the high-energy asymptotic behavior. In this paper we
present a high-energy asymptotic representation of the scattering amplitude at
moderate momentum transfer, for the leading order in an expansion parameter
closely related to the logarithm of the center-of-mass energy. The fact that
the expansion parameter goes as the logarithm of the energy means that the
asymptotic behavior is accurate only for energies greatly beyond any
foreseeable experiment. However, we compare the asymptotic representation
against the numerically calculated model for energies in a less extreme region
of energy. The asymptotic representation is given by a simple formula which, in
particular, exhibits the oscillations of the differential cross section with
momentum transfer. We also compare the BSW asymptotic behavior with the
Singh-Roy unitarity upper bound for the diffraction peak.Comment: 27 pages, 11 figures,revised version to appear in Physical Review
On scattering of solitons for the Klein-Gordon equation coupled to a particle
We establish the long time soliton asymptotics for the translation invariant
nonlinear system consisting of the Klein-Gordon equation coupled to a charged
relativistic particle. The coupled system has a six dimensional invariant
manifold of the soliton solutions. We show that in the large time approximation
any finite energy solution, with the initial state close to the solitary
manifold, is a sum of a soliton and a dispersive wave which is a solution of
the free Klein-Gordon equation. It is assumed that the charge density satisfies
the Wiener condition which is a version of the ``Fermi Golden Rule''. The proof
is based on an extension of the general strategy introduced by Soffer and
Weinstein, Buslaev and Perelman, and others: symplectic projection in Hilbert
space onto the solitary manifold, modulation equations for the parameters of
the projection, and decay of the transversal component.Comment: 47 pages, 2 figure
Virtual photon structure functions and positivity constraints
We study the three positivity constraints among the eight virtual photon
structure functions, derived from the Cauchy-Schwarz inequality and which are
hence model-independent. The photon structure functions obtained from the
simple parton model show quite different behaviors in a massive quark or a
massless quark case, but they satisfy, in both cases, the three positivity
constraints. We then discuss an inequality which holds among the unpolarized
and polarized photon structure functions , and
, in the kinematic region , where is the mass squared of the probe (target) photon, and we examine
whether this inequality is satisfied by the perturbative QCD results.Comment: 24 pages, 13 eps figure
A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation
We give a short proof of asymptotic completeness and global existence for the
cubic Nonlinear Klein-Gordon equation in one dimension. Our approach to dealing
with the long range behavior of the asymptotic solution is by reducing it, in
hyperbolic coordinates to the study of an ODE. Similar arguments extend to
higher dimensions and other long range type nonlinear problems.Comment: To appear in Lett. Math. Phy
Theory of Nonlinear Dispersive Waves and Selection of the Ground State
A theory of time dependent nonlinear dispersive equations of the Schroedinger
/ Gross-Pitaevskii and Hartree type is developed. The short, intermediate and
large time behavior is found, by deriving nonlinear Master equations (NLME),
governing the evolution of the mode powers, and by a novel multi-time scale
analysis of these equations. The scattering theory is developed and coherent
resonance phenomena and associated lifetimes are derived. Applications include
BEC large time dynamics and nonlinear optical systems. The theory reveals a
nonlinear transition phenomenon, ``selection of the ground state'', and NLME
predicts the decay of excited state, with half its energy transferred to the
ground state and half to radiation modes. Our results predict the recent
experimental observations of Mandelik et. al. in nonlinear optical waveguides
Carleman estimates and absence of embedded eigenvalues
Let L be a Schroedinger operator with potential W in L^{(n+1)/2}. We prove
that there is no embedded eigenvalue. The main tool is an Lp Carleman type
estimate, which builds on delicate dispersive estimates established in a
previous paper. The arguments extend to variable coefficient operators with
long range potentials and with gradient potentials.Comment: 26 page
Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field
The long-time asymptotics is analyzed for all finite energy solutions to a
model U(1)-invariant nonlinear Klein-Gordon equation in one dimension, with the
nonlinearity concentrated at a single point: each finite energy solution
converges as time goes to plus or minus infinity to the set of all ``nonlinear
eigenfunctions'' of the form \psi(x)e\sp{-i\omega t}. The global attraction
is caused by the nonlinear energy transfer from lower harmonics to the
continuous spectrum and subsequent dispersive radiation.
We justify this mechanism by the following novel strategy based on inflation
of spectrum by the nonlinearity. We show that any omega-limit trajectory has
the time-spectrum in the spectral gap [-m,m] and satisfies the original
equation. This equation implies the key spectral inclusion for spectrum of the
nonlinear term. Then the application of the Titchmarsh Convolution Theorem
reduces the spectrum of each omega-limit trajectory to a single harmonic in
[-m,m].
The research is inspired by Bohr's postulate on quantum transitions and
Schroedinger's identification of the quantum stationary states to the nonlinear
eigenfunctions of the coupled U(1)-invariant Maxwell-Schroedinger and
Maxwell-Dirac equations.Comment: 29 pages, 1 figur
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