M. Rizea

A numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies

Abstract Expressions for the coefficients of the Numerov scheme are found such as to ensure the optimal approximation to the Schrodinger equation in the deep continuum spectrum of energies.

Comparison of some four-step methods for the numerical solution of the Schrödinger equation

Abstract Three four-step methods for the numerical solution of the radial Schrodinger equation are here compared theoretically and experimentally.

Piecewise perturbation methods for calculating eigensolutions of a complex optical potential

Abstract It is shown that the piecewise perturbation methods provide a powerful numerical tool for solving the complex eigenvalue problem of the Schrodinger equation in a complex optical potential. One of them, the CPM(2), is compared with the existing NAG subroutines and found substantially faster. We also present appropriate methods to compute the solution in the vicinity of the origin and in the asymptotic region.

Numerov method maximally adapted to the Schro¨dinger equation

Abstract We construct and investigate the two-step scheme which is maximally specialized for the Schrodinger equation. The new scheme is found to preserve all attractive features of the standard Numerov method (ease in programming, flexibility in application, speed in run, etc.) with the important advantage of being more efficient for problems involving high values of energy. Its efficiency is actually found to be close to that of the piecewise perturbation methods, a fact which strongly recommends it for further applications.

Solution of the Schrödinger equation by a high order perturbation method based on a linear reference potential

The paper is devoted to the enhancement of the accuracy of the line-based perturbation method via the introduction of the perturbation corrections. We effectively construct the first and the second order corrections. We also perform the error analysis to predict that the introduction of successive corrections substantially enhances the order of the method from four, for the zeroth order version, to six and ten when the first and the second-order corrections are included. In order to remove the effect of the accuracy loss due to near-cancellation of like-terms when evaluating the perturbation corrections we construct alternative asymptotic formulae using a Maple code. We also propose a procedure for choosing the step size in terms of the preset accuracy and give a number of numerical illustrations.

Weights of the exponential fitting multistep algorithms for first-order ODEs

Abstract We describe a numerical method for the calculation of the weights of the linear multistep algorithms for solving first-order differential equations. The main novelties are that (i) we admit nonequidistant mesh points in the partition and (ii) the weights are determined on the basis of the exponential functions exp (λ i x), i=1,2,3,… rather than on the power function set, as it is done for the classical weights. In this way the method allows computing not only the weights of the well-established algorithms but also those of new ones. Another novelty consists in the construction of a general scheme for the error analysis of this kind of algorithms. Some relevant numerical illustrations are given.

Solution of the schrodinger equation over an infinite integration interval by perturbation methods, revisited

We consider the solution of the one-dimensional Schrodinger problem over an infinite integration interval. The infinite problem is regularized by truncating the integration interval and imposing the appropriate boundary conditions at the truncation points. The Schrodinger problem is then solved on the truncated integration interval using one of the piecewise perturbation methods developed for the regular Schrodinger problem.We select the truncation points using a procedure based on the WKB approximation. However for problems which behave as a Coulomb problem both around the origin and in the asymptotic range, a more accurate treatment of the numerical boundaries is possible. Taking into account the asymptotic form of the Coulomb equation, adapted boundary conditions can be constructed and as a consequence smaller truncation points can be chosen. To deal with the singularity of the Coulomb-like problem around the origin, a special perturbative algorithm is applied in a small interval around the origin.

Finite difference approach for the two-dimensional Schrodinger equation with application to scission-neutron emission

We present a grid-based procedure to solve the eigenvalue problem for the two-dimensional Schrodinger equation in cylindrical coordinates. The Hamiltonian is discretized by using adapted finite difference approximations of the derivatives and this leads to an algebraic eigenvalue problem with a large (sparse) matrix, which is solved by the method of Arnoldi. By this procedure the single particle eigenstates of nuclear systems with arbitrary deformations can be obtained. As an application we have considered the emission of scission neutrons from fissioning nuclei.

Four step methods for y ′′ =f x,y

Abstract A systematic investigation is undertaken here on the possible practical consequences of the fact that the linear four step methods for y ″= f ( x , y ) form a family with a certain structure. The mentioned property is found to be surprisingly rich in consequences, and three of them are of particular importance: (i) drastic reduction in the number of iterations at each step, (ii) increased flexibility with respect to the start of integration and to the modification of the step size during the integration process, (iii) possibility of direct and simple appraisal of the local truncation error.

Proton emission from Gamow resonance

We developed two computer codes: CCGAMOW and NONADI for calculating the complex energy eigenvalues and eigenfunctions of deformed Gamow resonances with high accuracy by using the piecewise perturbation method. The code CCGAMOW calculates resonant Nilsson orbitals using the adiabatic approximation in which the energies of the ground and excited rotational states of the daughter nucleus are degenerate. In the code NONADI this approximation and the rotational degree of freedom of the core and the Coriolis coupling in the parent nucleus are taken into account. The difference between adiabatic and non-adiabatic approaches is found to be non-negligible for the proton emission from the ground state of 141Ho.