Accurate eigenvalues of the Schr\"odinger equation with the potential $V(r)=V_{0}r^{\alpha}$

TL;DR

This paper presents a highly accurate method for calculating eigenvalues of the Schrödinger equation with power-law potentials using the Riccati-Padé approach, applicable to rational exponents.

Contribution

It introduces the Riccati-Padé method for precise eigenvalue computation of Schrödinger equations with power-law potentials, extending applicability to rational exponents.

Findings

Achieved high-precision eigenvalues for various power-law potentials.

Demonstrated the method's effectiveness for rational exponents.

Provided a new computational approach for quantum systems with power-law potentials.

Abstract

We calculate accurate eigenvalues of the Schr\"odinger equation with the potential $V(r)=V_{0}r^{\alpha}$, $\alpha \geq -1$, $V_{0}\alpha >0$. We resort to the Riccati-Pad\'e method that is based on a rational approximation to the logarithmic derivative of the wavefunction. This approach applies when $\alpha $ is a rational number.