# A quantum-mechanical anharmonic oscillator with a most interesting spectrum

## TL;DR

This paper analyzes a specific anharmonic quantum oscillator with a polynomial potential, revealing exact ground state solutions, summability of perturbation series for positive parameters, and complex spectral behavior including avoided crossings for negative parameters.

## Contribution

It provides an exact ground state solution for a sixth-degree polynomial anharmonic oscillator and studies the spectral properties and perturbation series behavior depending on the parameter .

## Key findings

- Exact ground state energy independent of  for  > 0
- Perturbation series are Pade9 and Borel-Pade9 summable for  > 0
- Spectrum exhibits avoided crossings and dramatic eigenfunction changes for  < 0

## Abstract

We revisit the problem posed by an anharmonic oscillator with a potential given by a polynomial function of the coordinate of degree six that depends on a parameter $\lambda $. The ground state can be obtained exactly and its energy $E_{0}=1$ is independent of $\lambda $. This solution is valid only for $\lambda >0$ because the eigenfunction is not square integrable otherwise. Here we show that the perturbation series for the expectation values are Pad\'{e} and Borel-Pad\'{e} summable for $\lambda >0$. When $% \lambda <0$ the spectrum exhibits an infinite number of avoided crossings at each of which the eigenfunctions undergo dramatic changes in their spatial distribution that we analyze by means of the expectation values $\langle x^{2}\rangle $.