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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

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

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    $\begingroup$ 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. $\endgroup$ – Ruslan Jan 26 '18 at 11:17
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The pure quartic oscillator is an unsolved problem with a rich overmature bibliography.

So it is inevitable that, beyond numerical studies, there are partial analytic results, which are inspired iterative guesses, but not expansions in the divergent perturbation series... See Liverts, Mandelzweig & Tabakin, 2006 and references therein, where they do guess some analytic wave functions; preprint.

Eremenko & Gabrielov, 2008 -- preprint-- study the analytic structure of the problem in the complex plane, and all hell breaks loose as the quadratic coupling vanishes leaving the quartic behind... but still, something can be learned...

There is no point in outlining results... and could not be covered in a serious class, either. But lots is understood.

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