Spectral: Solving Schroedinger and Wheeler–DeWitt equations in the positive semi-axis by the spectral method

Abstract

The Galerkin spectral method can be used for approximate calculation of eigenvalues and eigenfunctions of unidimensional Schroedinger-like equations such as the Wheeler–DeWitt equation. The criteria most commonly employed for checking the accuracy of results is the conservation of norm of the wave function, but some other criteria might be used, such as the orthogonality of eigenfunctions and the variation of the spectrum with varying computational parameters, e.g. the number of basis functions used in the approximation. The package Spectra, which implements the spectral method in Maple language together with a number of testing tools, is presented. Alternatively, Maple may interact with the Octave numerical system without the need of Octave programming by the user. Program summaryProgram title: SpectralCatalogue identifier: AEQQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEQQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 20417No. of bytes in distributed program, including test data, etc.: 2149904Distribution format: tar.gzProgramming language: Maple, GNU Octave 3.2.4Computer: Any supporting MapleOperating system: Any supporting MapleRAM: About 4 GbytesClassification: 1.9, 4.3, 4.6.Nature of problem:Numerical solution of Schrödinger-like eigenvalue equations (especially the Wheeler–DeWitt equation) in the positive semi-axisSolution method:The unknown wave function is approximated as a linear combination of a suitable set of functions, and the continuous eigenvalue problem is mapped into a discrete (matricial) eigenvalue problemRestrictions:Limitations are due to memory usage onlyUnusual features:The package may not work properly in older versions of Maple, due to a bug in that CAS; for that reason an interface with the GNU Octave system is provided, requiring no user intervention or Octave programming during calculationsRunning time:Seconds to hours, depending on the number of basis functions used and on the complexity of the potential used