516 research outputs found
Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures
Asymptotic approximations to the zeros of Hermite and Laguerre polynomials
are given, together with methods for obtaining the coefficients in the
expansions. These approximations can be used as a standalone method of
computation of Gaussian quadratures for high enough degrees, with Gaussian
weights computed from asymptotic approximations for the orthogonal polynomials.
We provide numerical evidence showing that for degrees greater than the
asymptotic methods are enough for a double precision accuracy computation
(- digits) of the nodes and weights of the Gauss--Hermite and
Gauss--Laguerre quadratures.Comment: Submitted to Studies in Applied Mathematic
Computation of the Marcum Q-function
Methods and an algorithm for computing the generalized Marcum function
() and the complementary function () are described.
These functions appear in problems of different technical and scientific areas
such as, for example, radar detection and communications, statistics and
probability theory, where they are called the non-central chi-square or the non
central gamma cumulative distribution functions.
The algorithm for computing the Marcum functions combines different methods
of evaluation in different regions: series expansions, integral
representations, asymptotic expansions, and use of three-term homogeneous
recurrence relations. A relative accuracy close to can be obtained
in the parameter region ,
, while for larger parameters the accuracy decreases (close to
for and close to for ).Comment: Accepted for publication in ACM Trans. Math. Soft
Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters
The determination of the physical entropies (R\'enyi, Shannon, Tsallis) of
high-dimensional quantum systems subject to a central potential requires the
knowledge of the asymptotics of some power and logarithmic integral functionals
of the hypergeometric orthogonal polynomials which control the wavefunctions of
the stationary states. For the -dimensional hydrogenic and oscillator-like
systems, the wavefunctions of the corresponding bound states are controlled by
the Laguerre () and Gegenbauer
() polynomials in both position and momentum
spaces, where the parameter linearly depends on . In this work we
study the asymptotic behavior as of the associated
entropy-like integral functionals of these two families of hypergeometric
polynomials
Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions
Conical functions appear in a large number of applications in physics and
engineering. In this paper we describe an extension of our module CONICAL for
the computation of conical functions. Specifically, the module includes now a
routine for computing the function , a
real-valued numerically satisfactory companion of the function for . In this way, a natural basis for solving
Dirichlet problems bounded by conical domains is provided.Comment: To appear in Computer Physics Communication
Fermions on one or fewer Kinks
We find the full spectrum of fermion bound states on a Z_2 kink. In addition
to the zero mode, there are int[2 m_f/m_s] bound states, where m_f is the
fermion and m_s the scalar mass. We also study fermion modes on the background
of a well-separated kink-antikink pair. Using a variational argument, we prove
that there is at least one bound state in this background, and that the energy
of this bound state goes to zero with increasing kink-antikink separation, 2L,
and faster than e^{-a2L} where a = min(m_s, 2 m_f). By numerical evaluation, we
find some of the low lying bound states explicitly.Comment: 7 pages, 4 figure
Non-equilibrium dynamics of a Bose-Einstein condensate in an optical lattice
The dynamical evolution of a Bose-Einstein condensate trapped in a
one-dimensional lattice potential is investigated theoretically in the
framework of the Bose-Hubbard model. The emphasis is set on the
far-from-equilibrium evolution in a case where the gas is strongly interacting.
This is realized by an appropriate choice of the parameters in the Hamiltonian,
and by starting with an initial state, where one lattice well contains a
Bose-Einstein condensate while all other wells are empty. Oscillations of the
condensate as well as non-condensate fractions of the gas between the different
sites of the lattice are found to be damped as a consequence of the collisional
interactions between the atoms. Functional integral techniques involving
self-consistently determined mean fields as well as two-point correlation
functions are used to derive the two-particle-irreducible (2PI) effective
action. The action is expanded in inverse powers of the number of field
components N, and the dynamic equations are derived from it to next-to-leading
order in this expansion. This approach reaches considerably beyond the
Hartree-Fock-Bogoliubov mean-field theory, and its results are compared to the
exact quantum dynamics obtained by A.M. Rey et al., Phys. Rev. A 69, 033610
(2004) for small atom numbers.Comment: 9 pages RevTeX, 3 figure
Resonantly enhanced pair production in a simple diatomic model
A new mechanism for the production of electron-positron pairs from the
interaction of a laser field and a fully stripped diatomic molecule in the
tunneling regime is presented. When the laser field is turned off, the Dirac
operator has resonances in both the positive and the negative energy continua
while bound states are in the mass gap. When this system is immersed in a
strong laser field, the resonances move in the complex energy plane: the
negative energy resonances are pushed to higher energies while the bound states
are Stark shifted. It is argued here that there is a pair production
enhancement at the crossing of resonances by looking at a simple 1-D model: the
nuclei are modeled simply by Dirac delta potential wells while the laser field
is assumed to be static and of finite spatial extent. The average rate for the
number of electron-positron pairs produced is evaluated and the results are
compared to the single nucleus and to the free cases. It is shown that
positrons are produced by the Resonantly Enhanced Pair Production (REPP)
mechanism, which is analogous to the resonantly enhanced ionization of
molecular physics. This phenomenon could be used to increase the number of
pairs produced at low field strength, allowing the study of the Dirac vacuum.Comment: 11 pages, 4 figure
Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments
- …