The scaled Hermite-Weber basis in the spectral and pseudospectral pictures

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Taseli H., Alici H.

JOURNAL OF MATHEMATICAL CHEMISTRY, vol.38, no.3, pp.367-378, 2005 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38 Issue: 3
  • Publication Date: 2005
  • Doi Number: 10.1007/s10910-005-5826-5
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.367-378
  • Keywords: Schrodinger operator, quantum mechanical oscillators, singular Sturm-Liouville problems on the real line, spectral and pseudospectral methods, Hermite-Weber functions, Hermite collocation points, DIFFERENTIATION MATRICES
  • Middle East Technical University Affiliated: Yes


Computational efficiencies of the discrete (pseudospectral, collocation) and continuous (spectral, Rayleigh-Ritz, Galerkin) variable representations of the scaled Hermite-Weber basis in finding the energy eigenvalues of Schrodinger operators with several potential functions have been compared. It is well known that the so-called differentiation matrices are neither skew-symmetric nor symmetric in a pseudospectral formulation of a differential equation, unlike their Rayleigh-Ritz counterparts. In spite of this fact, it is shown here that the spectra of matrix Hamiltonians generated by Hermite collocation method may be determined by way of diagonalizing symmetric matrices. Furthermore, the symmetric matrix elements do not require the evaluation of Hermite polynomials at the grid points. Surprisingly, the present numerical results suggest that the convergence rates of collocation and Rayleigh-Ritz methods are entirely the same.