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**The scaled Hermite-Weber basis in the spectral and pseudospectral pictures.**
*(English)*
Zbl 1217.65153

Summary: 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 Schrödinger 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.

### MSC:

65L15 | Numerical solution of eigenvalue problems involving ordinary differential equations |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |

42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |

### Keywords:

Schrödinger operator; quantum mechanical oscillators; singular Sturm-Liouville problems on the real line; spectral and pseudospectral methods; Hermite-Weber functions; Hermite collocation points
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\textit{H. Taşeli} and \textit{H. Alıcı}, J. Math. Chem. 38, No. 3, 367--378 (2005; Zbl 1217.65153)

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