# Exact solutions of the sextic oscillator from the bi-confluent Heun equation

## TL;DR

This paper derives exact solutions for the sextic oscillator by transforming the bi-confluent Heun equation, expressing solutions as Hermite function series, and identifying conditions for quasi-exact solvability and energy eigenvalues.

## Contribution

It provides a new analytical approach to solve the sextic oscillator using bi-confluent Heun functions and Hermite polynomial expansions, connecting to quasi-exact solvability.

## Key findings

- Exact solutions expressed via Hermite functions and polynomials.
- Conditions for quasi-exact solvability identified.
- Energy eigenvalues obtained from polynomial roots.

## Abstract

The sextic oscillator is discussed as a potential obtained from the bi-confluent Heun equation after a suitable variable transformation. Following earlier results, the solutions of this differential equation are expressed as a series expansion of Hermite functions with shifted and scaled arguments. The expansion coefficients are obtained from a three-term recurrence relation. It is shown that this construction leads to the known quasi-exactly solvable form of the sextic oscillator when some parameters are chosen in a specific way. By forcing the termination of the recurrence relation, the Hermite functions turn into Hermite polynomials with shifted arguments, and, at the same time, a polynomial expression is obtained for one of the parameters, the roots of which supply the energy eigenvalues. With the $\delta=0$ choice the quartic potential term is cancelled, leading to the {\it reduced} sextic oscillator. It was found that the expressions for the energy eigenvalues and the corresponding wave functions of this potential agree with those obtained from the quasi-exactly solvable formalism. Possible generalizations of the method are also presented.