Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics

Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behaviour of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss some techniques that yield uniform approximation over the whole eigenvalue spectrum and can take large steps even for high eigenvalues. In particular, we will focus on methods based on coefficient approximation which replace the coefficient functions of the Sturm-Liouville problem by simpler approximations and then solve the approximating problem. The use of (modified) Magnus or Neumann integrators allows to extend the coefficient approximation idea to higher order methods

Phase-fitted Discrete Lagrangian Integrators

Phase fitting has been extensively used during the last years to improve the behaviour of numerical integrators on oscillatory problems. In this work, the benefits of the phase fitting technique are embedded in discrete Lagrangian integrators. The results show improved accuracy and total energy behaviour in Hamiltonian systems. Numerical tests on the long term integration (100000 periods) of the 2-body problem with eccentricity even up to 0.95 show the efficiency of the proposed approach. Finally, based on a geometrical evaluation of the frequency of the problem, a new technique for adaptive error control is presented

Some new uses of the η_m(Z) functions

We present a procedure and a MATHEMATICA code for the conversion of formulae expressed in terms of the trigonometric functions sin(omega x), cos(omega x) or hyperbolic functions sinh(lambda x), cosh(lambda x) to forms expressed in terms of eta(m)(Z) functions. The possibility of such a conversion is important in the evaluation of the coefficients of the approximation rules derived in the frame of the exponential fitting. The converted expressions allow, among others, a full elimination of the 0/0 undeterminacy, uniform accuracy in the computation of the coefficients, and an extended area of validity for the corresponding approximation formulae

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

A New Family of Multistep Methods with Improved Phase Lag Characteristics for the Integration of Orbital Problems

In this work we introduce a new family of ten-step linear multistep methods for the integration of orbital problems. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. The efficiency of the new family of methods is proved via error analysis and numerical applications.Comment: 21 pages, 3 figures, 1 tabl

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

Using Continuum Level Density in the Pairing Hamiltonian: BCS and Exact Solutions

Pairing plays a central role in nuclear systems. The simplest model for the pairing is the constant-pairing Hamiltonian. The aim of the present paper is to include the continuum single particle level density in the constant pairing Hamiltonian and to make a comparison between the approximate BCS and the exact Richardson solutions. The continuum is introduced by using the continuum single particle level density. It is shown that the continuum makes an important contribution to the pairing parameter even in those case when the continuum is weakly populated. It is shown that while the approximate BCS solution depends on the model space the exact Richardson solution does not.Comment: 15 pages, 5 figures, accepted in Nucl. Phys.

Using a (Higher-Order) Magnus Method to Solve the Sturm-Liouville Problem

The main purpose of this paper is to describe techniques for the numerical solution of a Sturm-Liouville equation (in its Schrodinger form) by employing a Magnus expansion. With a suitable method to approximate the highly oscillatory integrals which appear in the Magnus series, high order schemes can be constructed. A method of order ten is presented. Even when the solution is highly-oscillatory, the scheme can accurately integrate the problem using stepsizes typically much larger than the solution "wavelength". This makes the method well suited to be applied in a shooting process to locate the eigenvalues of a boundary value problem

Systematics of proton emission

A very simple formula is presented that relates the logarithm of the half-life, corrected by the centrifugal barrier, with the Coulomb parameter in proton decay processes. The corresponding experimental data lie on two straight lines which appear as a result of a sudden change in the nuclear shape marking two regions of deformation independently of the angular momentum of the outgoing proton. This feature provides a powerful tool to assign experimentally quantum numbers in proton emitters.Comment: 4 pages, 3 figure