# An alternative derivation of the Fern\'andez-Castro analytic approximate expression for the eigenvalues of the bounded quartic oscillator

## TL;DR

This paper demonstrates that the Rayleigh-Schrödinger perturbation method and the hypervirial perturbative method yield the same approximate eigenvalues for the bounded quartic oscillator, revealing an unnoticed connection between the two approaches.

## Contribution

It shows the equivalence of RS and HPM methods for this problem and clarifies how certain sums can be expressed in closed form.

## Key findings

- RS and HPM produce identical eigenvalue approximations
- Polygamma sums in RS can be expressed in closed form
- Uncovered a long-overlooked connection between perturbation methods

## Abstract

In this note we show that the standard \mbox{Rayleigh-Schr\"odinger} (RS) perturbation method gives the same result as the hypervirial pertubative method (HPM), for an approximate analytic expression for the energy eigenvalues of the bounded quartic oscillator. This connection between the HPM and the RS method went unnoticed for a long time, apparently because it was not obvious that the resulting polygamma sums to be evaluated in the RS method could, in fact, be expressed in closed form.