Multi-point quasi-rational approximants for the energy eigenvalues of two-power potentials

Abstract

Analytic approximants for the eigenvalues of the one-dimensional Schrödinger equation with potentials of the form $V(x) = x^a + \lambda x^b$ are found using a multi-point quasi-rational approximation technique. This technique is based on the use of the power series and asymptotic expansion of the eigenvalues in $\lambda$, as well as the expansion at intermediate points. These expansions are found through a system of differential equations. The approximants found are valid and accurate for any values of $\lambda>0$ (with $b>a$). As an example, the technique is applied to the quartic anharmonic oscillator.