1,366 research outputs found
Similarity, attraction and initial conditions in an example of nonlinear diffusion
Similarity solutions play an important role in many fields of science. The
recent book of Barenblatt (1996) discusses many examples. Often, outstanding
unresolved issues are whether a similarity solution is dynamically attractive,
and if it is, to what particular solution does the system evolve. By recasting
the dynamic problem in a form to which centre manifold theory may be applied,
based upon a transformation by Wayne (1997), we may resolve these issues in
many cases. For definiteness we illustrate the principles by discussing the
application of centre manifold theory to a particular nonlinear diffusion
problem arising in filtration. Theory constructs the similarity solution,
confirms its relevance, and determines the correct solution for any compact
initial condition. The techniques and results we discuss are applicable to a
wide range of similarity problems
Gell-Mann - Low Function in QED for the arbitrary coupling constant
The Gell-Mann -- Low function \beta(g) in QED (g is the fine structure
constant) is reconstructed. At large g, it behaves as \beta_\infty g^\alpha
with \alpha\approx 1, \beta_\infty\approx 1.Comment: 5 pages, PD
The Degenerate Parametric Oscillator and Ince's Equation
We construct Green's function for the quantum degenerate parametric
oscillator in terms of standard solutions of Ince's equation in a framework of
a general approach to harmonic oscillators. Exact time-dependent wave functions
and their connections with dynamical invariants and SU(1,1) group are also
discussed.Comment: 10 pages, no figure
The Minimum-Uncertainty Squeezed States for for Atoms and Photons in a Cavity
We describe a six-parameter family of the minimum-uncertainty squeezed states
for the harmonic oscillator in nonrelativistic quantum mechanics. They are
derived by the action of corresponding maximal kinematical invariance group on
the standard ground state solution. We show that the product of the variances
attains the required minimum value 1/4 only at the instances that one variance
is a minimum and the other is a maximum, when the squeezing of one of the
variances occurs. The generalized coherent states are explicitly constructed
and their Wigner function is studied. The overlap coefficients between the
squeezed, or generalized harmonic, and the Fock states are explicitly evaluated
in terms of hypergeometric functions. The corresponding photons statistics are
discussed and some applications to quantum optics, cavity quantum
electrodynamics, and superfocusing in channeling scattering are mentioned.
Explicit solutions of the Heisenberg equations for radiation field operators
with squeezing are found.Comment: 27 pages, no figures, 174 references J. Phys. B: At. Mol. Opt. Phys.,
Special Issue celebrating the 20th anniversary of quantum state engineering
(R. Blatt, A. Lvovsky, and G. Milburn, Guest Editors), May 201
Spin Polarization Phenomena and Pseudospin Quantum Hall Ferromagnetism in the HgTe Quantum Well
The parallel field of a full spin polarization of the electron gas in a
\Gamma8 conduction band of the HgTe quantum well was obtained from the
magnetoresistance by three different ways in a zero and quasi-classical range
of perpendicular field component Bper. In the quantum Hall range of Bper the
spin polarization manifests in anticrossings of magnetic levels, which were
found to strongly nonmonotonously depend on Bper.Comment: to be published in AIP Conf. Proc.: 15-th International Conference on
Narrow Gap Systems (NGS-15
Renormalization Group Functions for Two-Dimensional Phase Transitions: To the Problem of Singular Contributions
According to the available publications, the field theoretical
renormalization group (RG) approach in the two-dimensional case gives the
critical exponents that differ from the known exact values. This fact was
attempted to explain by the existence of nonanalytic contributions in the RG
functions. The situation is analysed in this work using a new algorithm for
summing divergent series that makes it possible to analyse dependence of the
results for the critical exponents on the expansion coefficients for RG
functions. It has been shown that the exact values of all the exponents can be
obtained with a reasonable form of the coefficient functions. These functions
have small nonmonotonities or inflections, which are poorly reproduced in
natural interpolations. It is not necessary to assume the existence of singular
contributions in RG functions.Comment: PDF, 11 page
Quantum Electrodynamics at Extremely Small Distances
The asymptotics of the Gell-Mann - Low function in QED can be determined
exactly, \beta(g)= g at g\to\infty, where g=e^2 is the running fine structure
constant. It solves the problem of pure QED at small distances L and gives the
behavior g\sim L^{-2}.Comment: Latex, 6 pages, 1 figure include
Unusual phase transition in 1D localization and its observability in optics
Localization of electrons in 1D disordered systems is usually described in
the random phase approximation, when distributions of phases \varphi and
\theta, entering the transfer matrix, are considered as uniform. In the general
case, the random phase approximation is violated, and the evolution equations
are written in terms of the Landauer resistance \rho and the combined phases
\psi=\theta-\varphi and \chi=\theta+\varphi. The distribution of the phase \psi
is found to exhibit an unusual phase transition at the point E_0 when changing
the electron energy E, which manifests itself in the appearance of the
imaginary part of \psi. The distribution of resistance P(\rho) has no
singularity at the point E_0, and the transition seems unobservable in the
framework of condensed matter physics. However, the theory of 1D localization
is immediately applicable to the scattering of waves propagating in a
single-mode optical waveguide. Modern optical methods open a way to measure
phases \psi and \chi. As a result, the indicated phase transition becomes
observable.Comment: Latex, 9 pages, 6 figures include
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