28 research outputs found
Exponentially-fitted Gauss-Laguerre quadrature rule for integrals over an unbounded interval
New quadrature formulae are introduced for the computation of integrals over the whole positive semiaxis when the integrand has an oscillatory behavior with decaying envelope. The new formulae are derived by exponential fitting, and they represent a generalization of the usual Gauss-Laguerre formulae. Their weights and nodes depend on the frequency of oscillation in the integrand, and thus the accuracy is massively increased. Rules with one up to six nodes are treated with details. Numerical illustrations are also presented
Antibound poles in cutoff Woods-Saxon and in Salamon-Vertse potentials
The motion of l=0 antibound poles of the S-matrix with varying potential
strength is calculated in a cutoff Woods-Saxon (WS) potential and in the
Salamon-Vertse (SV) potential, which goes to zero smoothly at a finite
distance. The pole position of the antibound states as well as of the
resonances depend on the cutoff radius, especially for higher node numbers. The
starting points (at potential zero) of the pole trajectories correlate well
with the range of the potential. The normalized antibound radial wave functions
on the imaginary k-axis below and above the coalescence point have been found
to be real and imaginary, respectively
Shell model in the complex energy plane and two-particle resonances
An implementation of the shell-model to the complex energy plane is
presented. The representation used in the method consists of bound
single-particle states, Gamow resonances and scattering waves on the complex
energy plane. Two-particle resonances are evaluated and their structure in
terms of the single-particle degreees of freedom are analysed. It is found that
two-particle resonances are mainly built upon bound states and Gamow
resonances, but the contribution of the scattering states is also important.Comment: 20 pages, 9 figures, submitted to Phys.Rev.
Nonlinear Bogolyubov-Valatin transformations and quaternions
In introducing second quantization for fermions, Jordan and Wigner
(1927/1928) observed that the algebra of a single pair of fermion creation and
annihilation operators in quantum mechanics is closely related to the algebra
of quaternions H. For the first time, here we exploit this fact to study
nonlinear Bogolyubov-Valatin transformations (canonical transformations for
fermions) for a single fermionic mode. By means of these transformations, a
class of fermionic Hamiltonians in an external field is related to the standard
Fermi oscillator.Comment: 6 pages REVTEX (v3: two paragraphs appended, minor stylistic changes,
eq. (39) corrected, references [10]-[14], [36], [37], [41], [67]-[69] added;
v4: few extensions, references [62], [63] added, final version to be
published in J. Phys. A: Math. Gen.
Mathematical Properties of a New Levin-Type Sequence Transformation Introduced by \v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la. I. Algebraic Theory
\v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la [J. Math. Phys. \textbf{44}, 962
- 968 (2003)] introduced in connection with the summation of the divergent
perturbation expansion of the hydrogen atom in an external magnetic field a new
sequence transformation which uses as input data not only the elements of a
sequence of partial sums, but also explicit estimates
for the truncation errors. The explicit
incorporation of the information contained in the truncation error estimates
makes this and related transformations potentially much more powerful than for
instance Pad\'{e} approximants. Special cases of the new transformation are
sequence transformations introduced by Levin [Int. J. Comput. Math. B
\textbf{3}, 371 - 388 (1973)] and Weniger [Comput. Phys. Rep. \textbf{10}, 189
- 371 (1989), Sections 7 -9; Numer. Algor. \textbf{3}, 477 - 486 (1992)] and
also a variant of Richardson extrapolation [Phil. Trans. Roy. Soc. London A
\textbf{226}, 299 - 349 (1927)]. The algebraic theory of these transformations
- explicit expressions, recurrence formulas, explicit expressions in the case
of special remainder estimates, and asymptotic order estimates satisfied by
rational approximants to power series - is formulated in terms of hitherto
unknown mathematical properties of the new transformation introduced by
\v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la. This leads to a considerable
formal simplification and unification.Comment: 41 + ii pages, LaTeX2e, 0 figures. Submitted to Journal of
Mathematical Physic
Shell Model in the Complex Energy Plane
This work reviews foundations and applications of the complex-energy
continuum shell model that provides a consistent many-body description of bound
states, resonances, and scattering states. The model can be considered a
quasi-stationary open quantum system extension of the standard configuration
interaction approach for well-bound (closed) systems.Comment: Topical Review, J. Phys. G, Nucl. Part. Phys, in press (2008
Ef-Gaussian direct quadrature methods for Volterra integral equations with periodic solution
A direct quadrature method for the solution of Volterra integral equations with periodic solution is proposed. This method is based on an exponentially fitted quadrature rule of Gaussian type, whose parameters depend on the problem, in order to reproduce the behavior of the analytical solution. The error of the quadrature rule is examined and a convergence analysis of the direct quadrature method is given. Some numerical experiments are presented for comparison with other existing methods