Lemoine, Didier Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. (English) Zbl 0927.65102 Comput. Phys. Commun. 99, No. 2-3, 297-306 (1997). Summary: Optimal discrete transforms based upon the radial Laplacian eigenfunctions in cylindrical and spherical coordinates are presented, featuring the following properties: (1) bound state boundary conditions are enforced; (2) in the case of cylindrical or spherical symmetry, the relevant discrete Bessel transform (DBT) is analogous to the discrete Fourier transform in Cartesian coordinates; (3) the underlying quadrature algorithms achieve a Gaussian-like accuracy; (4) orthogonality of the transform can be ensured even in the absence of symmetry. Efficient multidimensional pseudospectral schemes are thus enabled in either direct or nondirect product representations. The illustrative program computes the various DBTs and applies them to the eigenvalue calculation for the two- and three-dimensional harmonic oscillator. MSC: 65L15 Numerical solution of eigenvalue problems involving ordinary differential equations 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs 35P15 Estimates of eigenvalues in context of PDEs 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 65T50 Numerical methods for discrete and fast Fourier transforms Keywords:discrete Bessel transform; cylindrical and spherical coordinates; radial symmetry; generalized finite basis and discrete variable representations; nondirect product representation; pseudospectral scheme; radial Laplacian eigenfunctions; discrete Fourier transform; eigenvalue; harmonic oscillator PDFBibTeX XMLCite \textit{D. Lemoine}, Comput. Phys. Commun. 99, No. 2--3, 297--306 (1997; Zbl 0927.65102) Full Text: DOI Digital Library of Mathematical Functions: 4th item ‣ §10.77(ix) Integrals of Bessel Functions ‣ §10.77 Software ‣ Computation ‣ Chapter 10 Bessel Functions References: [1] Lill, J. V.; Parker, G. A.; Light, J. C., Chem. Phys. Lett., 89, 483 (1982) [2] Light, J. C.; Hamilton, I. P.; Lill, J. V., J. Chem. Phys., 82, 1400 (1985) [3] Lemoine, D., J. Chem. Phys., 101, 3936 (1994) [4] Corey, G. C.; Lemoine, D., J. Chem. Phys., 97, 4115 (1992) [5] Corey, G. C.; Tromp, J. W.; Lemoine, D., (Cerjan, C., Numerical Grid Methods and Their Application to Schrödinger’s Equation (1993), Kluwer Academic: Kluwer Academic Dordrecht), 1 [6] Lemoine, D., Comput. Phys. Commun., 97, 331 (1996) [7] Corey, G. C.; Tromp, J. W., J. Chem. Phys., 103, 1812 (1995) [8] J. Comp. Phys., 8, 417 (1971) [9] Johnson, H. F., Comput. Phys. Commun., 43, 181 (1987) [10] Layton, E. G.; Stade, E., J. Phys. B, 26, L489 (1993) [11] Stade, E.; Layton, E. G., Comput. Phys. Commun., 85, 336 (1995) [12] Lemoine, D., Chem. Phys. Lett., 224, 483 (1994) [13] G.C. Corey and R.J. Le Roy, J. Chem. Phys., submitted.; G.C. Corey and R.J. Le Roy, J. Chem. Phys., submitted. [14] Sun, Y.; Mowrey, R. C.; Kouri, D. J., J. Chem. Phys., 87, 339 (1987) [15] Persson, M.; Jackson, B., J. Chem. Phys., 102, 1078 (1995) [16] Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T., Numerical Recipes (1994), Cambridge University: Cambridge University Cambridge, 3rd printing [17] Besprozvannaya, A.; Tannor, D. J., Comput. Phys. Commun., 63, 569 (1991) [18] Colbert, D. T.; Miller, W. H., J. Chem. Phys., 96, 1982 (1992) [19] Abramowitz, M.; Stegun, I. A., (Handbook of Mathematical Functions (1970), Dover: Dover New York), 9th printing This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.