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

Abstract

Computational efficiencies of the discrete (pseudospectral, collocation) and continuous (spectral, Rayleigh-Ritz, Galerkin) variable representations of the scaled HermiteWeber 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. KEY WORDS:Schrödinger operator, quantum mechanical oscillators, singular SturmLiouville problems on the real line, spectral and pseudospectral methods, HermiteWeber functions, Hermite collocation points

Anharmonic oscillator via Legendre and Chebyshev pseudo-spectral methods

“…As a result, the relevant boundary or eigenvalue problem for the differential equation is transformed into an unsymmetric matrix eigenvalue problem. However, using a suitable similarity transformation, it can be converted into a symmetric eigenvalue problem [6,7,8,9,10].…”

“…Eigenvalues of v(x) = x 2 + 0.1x8 with Chebyshev E C i and Legendre E L i pseudospectral methods for α = 1; 0.3 and 0.4.…”