I'm looking to do a small personal project regarding the time independent Schrödinger equation in 1-D:

$$y'' +V(x)y=Ey$$

$$y''=Q(x)y$$

where $ Q(x):=E-V(x) $.

There is obviously nothing stopping me now from choosing arbitrary $ Q(x) $ and finding solutions. However, I'm interested in coefficient functions that are actually relevant. I, being of weak physics background don't particularly know which $Q(x)$ would actually be both mathematically and physically intriguing.
I'm most interested in evaluating the asymptotic behavior of solutions about some irregular singular point as well as one and two turning point problems. I then want to use Padé approximations and then potentially a Shanks transform on the asymptotic series associated with the asymptotic behavior.
So my question stands, which $Q(x)$ should I choose? Many different recommendations would be much appreciated.

This post imported from StackExchange Physics at 2015-02-20 16:50 (UTC), posted by SE-user Seenathin