How much is known about ~x^4 potential

[jostpuur]

How much is known about eigenstates and values of the Hamiltonian

$$H=-\frac{1}{2m}\partial_x^2 + c x^4$$

I understand the fact that we cannot find precise solutions to everything, but what I find disturbing here is that I haven't even encountered approximation methods that would be useful here. Do approximate solutions to this exist in mainstream knowledge?

 

[Avodyne]

Sure. For low-lying states, variational methods would work. For high energy states, the WKB approximation would work well. I'm not an expert on numerical methods, but I'm sure you could figure out anything you wanted to know to high precision.

 

[Count Iblis]

http://arxiv.org/abs/math-ph/9903045

http://arxiv.org/abs/hep-th/0008039