39 research outputs found
Automatic computation of quantum-mechanical bound states and wavefunctions
We discuss the automatic solution of the multichannel Schr\"odinger equation.
The proposed approach is based on the use of a CP method for which the step
size is not restricted by the oscillations in the solution. Moreover, this CP
method turns out to form a natural scheme for the integration of the Riccati
differential equation which arises when introducing the (inverse) logarithmic
derivative. A new Pr\"ufer type mechanism which derives all the required
information from the propagation of the inverse of the log-derivative, is
introduced. It improves and refines the eigenvalue shooting process and implies
that the user may specify the required eigenvalue by its index
On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation
The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schrödinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897–907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials, since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods
Solution of Sturm-Liouville problems using modified Neumann schemes
The main purpose of this paper is to describe the extension of the successful modified integral series methods for Schrodinger problems to more general Sturm-Liouville eigenvalue problems. We present a robust and reliable modified Neumann method which can handle a wide variety of problems. This modified Neumann method is closely related to the second-order Pruess method but provides for higher-order approximations. We show that the method can be successfully implemented in a competitive automatic general-purpose software package
Study of special algorithms for solving Sturm-Liouville and Schrodinger equations
In dit proefschrift beschrijven we een specifieke klasse van numerieke methoden voor het oplossen van Sturm-Liouville en Schrodinger vergelijkingen. Ook de Matlab-implementatie van de methoden wordt besproken
Computing stability of multi-dimensional travelling waves
We present a numerical method for computing the pure-point spectrum
associated with the linear stability of multi-dimensional travelling fronts to
parabolic nonlinear systems. Our method is based on the Evans function shooting
approach. Transverse to the direction of propagation we project the spectral
equations onto a finite Fourier basis. This generates a large, linear,
one-dimensional system of equations for the longitudinal Fourier coefficients.
We construct the stable and unstable solution subspaces associated with the
longitudinal far-field zero boundary conditions, retaining only the information
required for matching, by integrating the Riccati equations associated with the
underlying Grassmannian manifolds. The Evans function is then the matching
condition measuring the linear dependence of the stable and unstable subspaces
and thus determines eigenvalues. As a model application, we study the stability
of two-dimensional wrinkled front solutions to a cubic autocatalysis model
system. We compare our shooting approach with the continuous orthogonalization
method of Humpherys and Zumbrun. We then also compare these with standard
projection methods that directly project the spectral problem onto a finite
multi-dimensional basis satisfying the boundary conditions.Comment: 23 pages, 9 figures (some in colour). v2: added details and other
changes to presentation after referees' comments, now 26 page
On the use of CP methods for the numerical solution of the Schrödinger equation
One of the main scientific achievements of L. Gr. Ixaru concerns the formulation of CP methods for the numerical solution of the Schrodinger equation. The first ideas were described in the seventies but the CP methods still prove to be very efficient for not only the one-dimensional time-independent Schrodinger equation but also for Sturm-Liouville problems, coupled channel Schrodinger equations and two-dimensional Schrodinger problems. We summarize here the main ideas and discuss the use of a CP method for the single channel (one-dimensional) equation as well as for the multichannel equation