Paul Abbott

The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations

The sn- and cn-function methods for finding nonsingular periodic-wave solutions to nonlinear evolution equations are described in a form suitable for automation, where sn and cn are the elliptic Jacobi snoidal and cnoidal functions, respectively. Some new solutions are presented.

Theory and computation of spheroidal wavefunctions

In this paper we report on a package, written in the Mathematica computer algebra system, which has been developed to compute the spheroidal wavefunctions of Meixner and Schafke (1954 Mathieusche Funktionen und Spharoidfunktionen) and is available online (physics.uwa.edu.au/~falloon/spheroidal/spheroidal.html). This package represents a substantial contribution to the existing software, since it computes the spheroidal wavefunctions to arbitrary precision for general complex parameters μ, ν, γ and argument z; existing software can only handle integer μ, ν and does not give arbitrary precision. The package also incorporates various special cases and computes analytic power series and asymptotic expansions in the parameter γ. The spheroidal wavefunctions of Flammer (1957 Spheroidal Wave functions) are included as a special case of Meixners more general functions. This paper presents a concise review of the general theory of spheroidal wavefunctions and a description of the formulae and algorithms used in their computation, and gives high precision numerical examples.

Automation of the lifting factorisation of wavelet transforms

Wavelets are sets of basis functions used in the analysis of signals and images. In contrast to Fourier analysis, wavelets have both spatial and frequency localization, making them useful for the analysis of sharply-varying or non-periodic signals. The lifting scheme for finding the discrete wavelet transform was demonstrated by Daubechies and Sweldens (1996). In particular, they showed that this method depends on the factorization of polyphase matrices, whose entries are Laurent polynomials, using the Euclidean algorithm extended to Laurent polynomials. Such factorization is not unique and hence there are multiple factorizations of the polyphase matrix. In this paper we outline a Mathematica program that finds all factorizations of such matrices by automating the Euclidean algorithm for Laurent polynomials. Polynomial reduction using Grobner bases was also incorporated into the program so as to reduce the number of wavelet filter coefficients appearing in a given expression through use of the relations they satisfy, thus permitting exact symbolic factorizations for any polyphase matrix.

Saturation spectroscopy of iodine in hollow-core optical fiber

We present high-resolution spectroscopy of I(2) vapor that is loaded and trapped within the core of a hollow-core photonic crystal fiber (HC-PCF). We compare the observed spectroscopic features to those observed in a conventional iodine cell and show that the saturation characteristics differ significantly. Despite the confined geometry it was still possible to obtain sub-Doppler features with a spectral width of ~6 MHz with very high contrast. We provide a simple theory which closely reproduces all the key observations of the experiment.

Generalized Laguerre polynomials and quantum mechanics

We comment on serious errors in a recently published paper (El-Sayed A M A 1999 J. Phys. A: Math. Gen. 32 8647-54).

Daubechies wavelets and Mathematica

Analysis of time series data using wavelets provides both scale (frequency) and position information. In contrast, the Fourier transform provides frequency information only. We discuss the Daubechies formulation of wavelets, with reference to the WaveletTransform package that calculates the filter coefficients for any Daubechies basis to arbitrary precision. Examples of the wavelet transform applied to selected time series are presented to highlight the advantages of wavelets. We indicate an application of wavelets to sampled Barkhausen noise, a nonlinear phenomenon encountered in magnetic systems. The elements of the WaveletTransform package are discussed, with the emphasis being on the calculation of filter coefficients and their application to the discrete wavelet transform (DWT) and its inverse. With the construction of quadrature mirror filters, an efficient implementation of the DWT is possible and is similar in structure to the fast Fourier transform algorithm.

Efficient computation of Wigner-Eisenbud functions

Abstract The R -matrix method, introduced by Wigner and Eisenbud (1947) [1] , has been applied to a broad range of electron transport problems in nanoscale quantum devices. With the rapid increase in the development and modeling of nanodevices, efficient, accurate, and general computation of Wigner–Eisenbud functions is required. This paper presents the Mathematica package WignerEisenbud , which uses the Fourier discrete cosine transform to compute the Wigner–Eisenbud functions in dimensionless units for an arbitrary potential in one dimension, and two dimensions in cylindrical coordinates. Program summary Program title: WignerEisenbud Catalogue identifier: AEOU_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEOU_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html Distribution format: tar.gz Programming language: Mathematica Operating system: Any platform supporting Mathematica 7.0 and above Keywords: Wigner-Eisenbud functions, discrete cosine transform (DCT), cylindrical nanowires Classification: 7.3, 7.9, 4.6, 5 Nature of problem: Computing the 1D and 2D Wigner–Eisenbud functions for arbitrary potentials using the DCT. Solution method: The R-matrix method is applied to the physical problem. Separation of variables is used for eigenfunction expansion of the 2D Wigner–Eisenbud functions. Eigenfunction computation is performed using the DCT to convert the Schrodinger equation with Neumann boundary conditions to a generalized matrix eigenproblem. Limitations: Restricted to uniform (rectangular grid) sampling of the potential. In 1D the number of sample points, n , results in matrix computations involving n × n matrices. Unusual features: Eigenfunction expansion using the DCT is fast and accurate. Users can specify scattering potentials using functions, or interactively using mouse input. Use of dimensionless units permits application to a wide range of physical systems, not restricted to nanoscale quantum devices. Running time: Case dependent.

Electronic structure calculation for N -electron quantum dots ✩

Abstract The Hartree–Fock method for calculating the electronic structure of N -electron quantum dot systems was implemented in Mathematica using an easily understood modular code. Calculations were performed for quantum dot systems containing up to N =18 electrons. The energy spectra obtained are in good agreement with those previously calculated using density functional theory. Qualitative agreement with an experimental spectrum is also obtained.

Asymptotic expansion of the Keesom integral

The asymptotic evaluation and expansion of the Keesom integral, K(a), is discussed at some length in Battezzati and Magnasco (2004 J. Phys. A: Math. Gen. 37 9677; 2005 J. Phys. A: Math. Gen. 38 6715). Here, using standard identities, it is shown that this triple integral can be reduced to a single integral from which the asymptotic behaviour is readily obtained using Laplaces method.

Expansion of two-body potentials in hyperspherical harmonics

Two-body potentials dependent only on inter-particle separation are shown to have a simple expansion in S-state hyperspherical harmonics. This expansion is helpful when solving the N-particle Schrodinger equation in hyperspherical coordinates.