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Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. (English) Zbl 0927.65102
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 Eigenvalue problems for ODE (numerical methods) 65N25 Numerical methods for eigenvalue problems (BVP of PDE) 35P15 Estimation of eigenvalues and upper and lower bounds for PD operators 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE) 65T50 Discrete and fast Fourier transforms (numerical methods)