10 research outputs found
Density of States Extracted from Modified Recursion Relations
We evaluate the density of states (DOS) associated with tridiagonal symmetric
Hamiltonian matrices and study the effect of perturbation on one of its
entries. Analysis is carried out by studying the resulting three-term recursion
relation and the corresponding orthogonal polynomials of the first and second
kind. We found closed form expressions for the new DOS in terms of the original
one when perturbation affects a single diagonal or off-diagonal site or a
combination of both. The projected DOS is also calculated numerically and its
relation to the average DOS is explored both analytically and numerically.Comment: 15 pages including 8 figures (one in color
J-matrix method of scattering in one dimension: The nonrelativistic theory
We formulate a theory of nonrelativistic scattering in one dimension based on
the J-matrix method. The scattering potential is assumed to have a finite range
such that it is well represented by its matrix elements in a finite subset of a
basis that supports a tridiagonal matrix representation for the reference wave
operator. Contrary to our expectation, the 1D formulation reveals a rich and
highly non-trivial structure compared to the 3D formulation. Examples are given
to demonstrate the utility and accuracy of the method. It is hoped that this
formulation constitutes a viable alternative to the classical treatment of 1D
scattering problem and that it will help unveil new and interesting
applications.Comment: 24 pages, 9 figures (3 in color
Analytical Treatment of the Oscillating Yukawa Potential
Using a suitable Laguerre basis set that ensures a tridiagonal matrix
representation of the reference Hamiltonian, we were able to evaluate in closed
form the matrix elements of the generalized Yukawa potential with complex
screening parameter. This enabled us to treat analytically both the cosine and
sine-like Yukawa potentials on equal footing and compute their bound states
spectrum as the eigenvalues of the associated analytical matrix representing
their Hamiltonians. Finally we used a carefully designed complex scaling method
to evaluate the resonance energies and compared our results satisfactorily with
those obtained in the literature.Comment: 8 pages 2 table
Scattering theory with a natural regularization: Rediscovering the J-matrix method
In three dimensional scattering, the energy continuum wavefunction is
obtained by utilizing two independent solutions of the reference wave equation.
One of them is typically singular (usually, near the origin of configuration
space). Both are asymptotically regular and sinusoidal with a phase difference
(shift) that contains information about the scattering potential. Therefore,
both solutions are essential for scattering calculations. Various
regularization techniques were developed to handle the singular solution
leading to different well-established scattering methods. To simplify the
calculation the regularized solutions are usually constructed in a space that
diagonalizes the reference Hamiltonian. In this work, we start by proposing
solutions that are already regular. We write them as infinite series of square
integrable basis functions that are compatible with the domain of the reference
Hamiltonian. However, we relax the diagonal constraint on the representation by
requiring that the basis supports an infinite tridiagonal matrix representation
of the wave operator. The hope is that by relaxing this constraint on the
solution space a larger freedom is achieved in regularization such that a
natural choice emerges as a result. We find that one of the resulting two
independent wavefunctions is, in fact, the regular solution of the reference
problem. The other is uniquely regularized in the sense that it solves the
reference wave equation only outside a dense region covering the singularity in
configuration space. However, asymptotically it is identical to the irregular
solution. We show that this natural and special regularization is equivalent to
that already used in the J-matrix method of scattering.Comment: 10 page
He Structure and Mechanisms of He Backward Elastic Scattering
The mechanism of He backward elastic scattering is studied.
It is found that the triangle diagrams with the subprocesses He,
He and He, where and
denote the singlet deuteron and diproton pair in the state,
respectively, dominate in the cross section at 0.3-0.8 GeV, and their
contribution is comparable with that for a sequential transfer of a pair
at 1-1.5 GeV.
The contribution of the , estimated on the basis of the spectator
mechanism of the He reaction, increases the HeHe cross section by one order of magnitude as compared to the
contribution of the deuteron alone.
Effects of the initial and final states interaction are taken into account.Comment: 17 pages, Latex, 4 postscript figures, expanded version, accepted by
Physical Review
Singular Short Range Potentials in the J-Matrix Approach
We use the tools of the J-matrix method to evaluate the S-matrix and then
deduce the bound and resonance states energies for singular screened Coulomb
potentials, both analytic and piecewise differentiable. The J-matrix approach
allows us to absorb the 1/r singularity of the potential in the reference
Hamiltonian, which is then handled analytically. The calculation is performed
using an infinite square integrable basis that supports a tridiagonal matrix
representation for the reference Hamiltonian. The remaining part of the
potential, which is bound and regular everywhere, is treated by an efficient
numerical scheme in a suitable basis using Gauss quadrature approximation. To
exhibit the power of our approach we have considered the most delicate region
close to the bound-unbound transition and compared our results favorably with
available numerical data.Comment: 14 pages, 5 tables, 2 figure
Information entropies for the Morse potential using the J-matrix method
Information entropies, such as Shannon, Fisher and Rényi entropies, with their associated quantities, are calculated and discussed for the Morse potential in the position (r) and momentum (p) spaces using the J-matrix method. The entropies, for the s-wave, are numerically obtained for the diatomic molecules H2 and LiH. It is the first information entropy calculation for real diatomic molecules using the Morse potential. Interesting characteristic features of the entropy densities are shown, where the position and momentum space information entropies satisfy the uncertainty relations. Keywords: Shannon entropy, Rényi entropy, Fisher information, J-matrix method, Diatomic molecule