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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.

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closed as too broad by ACuriousMind, Neuneck, Ruslan, Kyle Kanos, JamalS Feb 20 '15 at 16:10

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ This seems too broad of a question (and partially opinion-based). Could you refine the question to not be asking for a list, but asking for a specific concept instead? $\endgroup$ – Kyle Kanos Feb 20 '15 at 13:42
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    $\begingroup$ Have a look at the lectures by Carl Bender at pirsa.org; he discusses asymptotics, the Schrodinger equation, Pade approximations and the Shanks transform. He also addresses the issue of when $Q(x)$ is zero (which is a problem since the regular WKB approximation becomes singular). $\endgroup$ – JamalS Feb 20 '15 at 16:09
  • $\begingroup$ @JamalS Something tells me this question was actually motivated by those [wonderful] lectures. $\endgroup$ – Arturo don Juan Feb 26 '16 at 0:10