The eigenenergies of the wave function through the non-variational Galerkin-B-spline approach B. Nine, O. Haif-Khaif, A. Zerarka * Labo de Mathe ´matiques Applique ´es, Universite ´ Med Khider, BP 145, 07000 Biskra, Algeria Abstract The non-variational Galerkin-B-spline method has been applied to the solution of the Schro ¨ dinger equation for the vibration and rotation bound states of diatomic molecules. This method allows the construction of the full Hamiltonian matrix of relatively modest size. The eigenspectra obtained are compared with the analytic and modified shifted 1/N expan- sion cases. Ó 2005 Elsevier Inc. All rights reserved. Keywords: B-spline; Variational; Galerkin; Wave function; Eigenenergies 1. Introduction Many powerful methods have been developed for quantum energy calculation and the estimation of its associated wave function have shown that qualitative and significant results can be explored, both for bound and scattering problems [1–12]. If such problems are linear, Landtman et al. [13] have shown that the intro- duction of orthogonality conditions in the treatment of a basis leads normally to a non-Hermitian eigenvalue problem. B-Spline functions have been evolved successfully in the major areas of quantum and solid mechanics and nuclear, atomic and molecular physics during the past several years [2,3,14,15]. Furthermore, it is shown that, the approximation of the state wave function constructed by the spline basis must be always equal to the wave function at points which are commonly chosen in each segment in consideration. In other words, the B-spline functions provide a reasonable and consistent basis, and can offer superior accuracy for the construction of a large variety of numerical problems. The B-spline functions constitute a complete set of basis functions that will approximate a given function in the referred range of interest. For a more in depth review of B-splines, refer to [16,17]. Also the famous book by de Boor [18], which con- tains a certain number of useful definitions and routines to perform spline functions of a given order. In addi- tion, for an interested reader, other packages which underlie this method are also available in most texts on the 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.11.067 * Corresponding author. E-mail address: abzerarka@yahoo.fr (A. Zerarka). Applied Mathematics and Computation 178 (2006) 486–492 www.elsevier.com/locate/amc