# Quantum pure quartic oscillator

It was recently brought to my attention, that there exist analytic solutions for the quantum pure-quartic oscillator with the hamiltonian

$$\hat{H} = \frac{1}{2m} \hat{p}^2 + \frac{\lambda}{24} \hat{x}^4.$$

The person who ascertained that, however, couldn't find proof of his claims on the internet. But he claims that was what he was told in his quantum mechanics class.

I would appreciate any hints on: how to derive analytical expressions for energy spectrum $E_n$ and the corresponding eigenfunctions $\Psi_n(x)$, satisfying

$$- \frac{\hbar^2}{2m} \Psi_n''(x) + \frac{\lambda}{24} x^4 \, \Psi_n(x) = E_n \Psi_n(x).$$

• Wolfram Mathematica's DSolve gives solutions in the form of its DifferentialRoot functions, so I'd suppose there're no closed-form solutions in terms of well-known special functions. – Ruslan Jan 26 '18 at 11:17