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

TL;DR

This paper develops a multi-point quasi-rational approximation method to accurately estimate eigenvalues of 1D Schrödinger equations with potentials combining different powers, validated on anharmonic oscillators.

Contribution

It introduces a novel multi-point quasi-rational approximation technique based on power series and asymptotic expansions for eigenvalues of specific potentials.

Findings

Approximants are valid for all values of the parameter λ.

Method applied successfully to quartic and sextic anharmonic oscillators.

High accuracy of eigenvalue estimates across potential parameters.

Abstract

Analytic approximants for the eigenvalues of the one-dimensional Schr\"odinger equation with potentials of the form $V(x)= Ax^a + Bx^b$ 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 $\lambda=A^{-\frac{b+2}{a+2}}B$, 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 $\lambda$. As examples, the technique is applied to the quartic and sextic anharmonic oscillators.