I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions?

$$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$

$C$ is a free parameter. So if you know a function that would fulfill this equation only for particular $C$, this would be perfectly fine. I am interested in its solutions on $[-1,1]$.

I should include the motivation/reference here: The equation is motivated by Physics (Quantum Mechanics) and you might want to see this great answer that gives us (meanwhile complete hints) about the structure of the solution (due to O.L. (thank you!))

The problem is that the approach taken by O.L. does not offer an analytical representation of the solution. He was able to show that some types of polynomials will give you the solution, but still you have to throw this ' guess' into the equation. I suspect that the solutions form a nice orthogonal basis of $L^2[-1,1]$, but was incapable of constructing them by recursion or explicit representation.

Now, I also got the hint to consider symmetries in my ODE and separately consider even and odd solutions so that I could reduce my ODE to a confluent Heun's equation. If I follow this hint and substitute $s=t^2$ and $u(t)= v(s)$ I get

$$v''(s)+\frac{1}{2}\left(\frac{1}{s}+ \frac{1}{s-1}\right) v'(s) + \left(n \beta \frac{(2s-1)}{s(1-s)} + \frac{\beta^2(2s-1)^2}{s(1-s)} + C\right) v(s)=0.$$

This is very close to the confluent Heun's equation but not exactly the form that we are looking for (since $s^2$ is appearing in the nominator of the right term in the parenthesis. So probably we need to substitute even more(anything like $v(s) = exp(\alpha s)w(s)$ might help), but actually I don't see how to go further.

So is there anybody who knows how to finish this and who is able to construct the solutions from here?(and how $C$ has to be chosen in order to find a solution)

If anything is unclear, please let me know and just to point this out: If you are able to reduce this ODE to the confluent Heun equation, then this answers my question totally.

  • $\begingroup$ What are the boundary conditions for your problem? Have you tried a Taylor/Fröbenius expansion? $\endgroup$ – Dmoreno May 7 '14 at 17:10

The equation is indeed equivalent to confluent Heun. This follows already from the analysis of its singularity structure: 2 regular singular points and an irregular point of Poincaré rank $1$.

Specifically, setting $v(s)=e^{2\gamma s}u(s)$ with $$\gamma=\sqrt{\beta^2-\frac{C}{4}},$$ transforms your second equation into $$u''(s)+\frac12\left(\frac1s+\frac{1}{s-1}+8\gamma\right)u'(s)+\frac{2(\gamma-n\beta)s+(n\beta-\beta^2-\gamma)}{s(s-1)}u(s)=0.$$ (Compare with 31.12.1 here).

However in my opinion this does not add much new knowledge about the solutions except that they have a name.

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