Eigenvalues of the Schrödinger equation with rational potentials

A matrix-continued-fraction algorithm which basically relies on an expansion of the eigenfunctions into complete sets is presented for calculating the eigenvalues of the one-dimensional Schr\"odinger equation with rational potentials. The method also applies to the corresponding three-dimensional central-force potentials. In particular, we investigate the one-dimensional nonpolynomial-oscillator potential VNPO(x)=x²+{λ}x²/(1+gx²), the eigenvalues of which are obtained for a large range of parameters {λ} and g. In the three-dimensional case we check the shifted 1/N expansion which has been applied to the nonpolynomial potential VNPO(r) very recently. In addition to the numerical results, analytically exact solutions are given for a restricted class of such potentials. Moreover, we point out that the continued-fraction algorithm applies to a wider class of polynomial and nonpolynomial (rational) potentials. In summary, the matrix-continued-fraction algorithm is shown to produce the eigenvalues of essentially one-dimensional Schr\"odinger equations for a wide class of rational potentials with very high accuracy, and the proposed technique seems to offer several computational advantages compared to other numerical approaches applied in the field. Finally, an application of the continued-fraction algorithm to the two-variable Schr\"odinger equation is suggested.