In this paper we present explicit and simple analytical formulae for the energy eigenvaluesE_n (λ) of one-dimensional anharmonic oscillators characterized by the potentials 1/2mω²x²+λx^2α withα=2, 3 and 4. A simple intuitive criterion supplemented by the requirement of correct asymptotic behaviour, has been employed in arriving at the formulae. Our energy values over a wide range ofn andλ are in good agreement with the numerical values computed by earlier workers through very elaborate techniques. To our knowledge this is the first time that formulae of such wide validity have been given. The results for pure power oscillators are trivially obtained by going over to theω→0 limit. Approximate analytic expressions for the low order even moments ofx are also given.

A method is presented for an accurate numerical determination of eigenvalues of real symmetric para-p diagonal matrices. The method takes advantage of the band structure to break up the matrix intop ×p blocks and performing algebraic operations including inversions on these blocks only, no matter what the size of the matrix is. The eigenvalues are determined independently one at a time. Thus any error in the determination of one eigenvalue does not affect the other eigenvalues. The method is ideally suited for the Schrödinger eigen alue problem of the anharmonic potentials, which is taken up in the following paper.

Energy eigenvalues and matrix elements of various anharmonic oscillators are determined to a high accuracy by applying a method for determining the eigenvalues and eigenvectors of real symmetric para-p diagonal matrices (described in the preceding paper). Our results for the 2- and 3-dimensional oscillators are new and complement similar accurate results for the one dimensional oscillators available in the literature.