Multi-point quasi-rational approximants for the energy eigenvalues of potentials of the form V(x)= Ax^a + Bx^b

Abstract

Analytic approximants for the eigenvalues of the one-dimensional Schrödinger equation with potentials of the form $V(x)=A{x}^a+B{x}^b$ 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 $\hat{I}»=A^{-\frac{b+2}{a+2}}B$, 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 value of $\hat{I}»$. As examples, the technique is applied to the quartic and sextic anharmonic oscillators.