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I would like to know how one solves Sturm-Liouville problems on $(0,\infty)$ NUMERICALLY for the eigenvalues that are of the form

$$-f''(x)+\frac{1}{\sinh(x)^2}f(x)=\lambda f(x).$$

So even if there is a closed form solution to this problem, I would like to know how to treat singularities like this one with a computer. I also do not want to use programs, I am just interested in how to do this "by hand with a computer".

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  • $\begingroup$ Presumably on $(0,\infty)$ rather than all of $\mathbb R$, since there's a singularity at $0$? $\endgroup$
    – Robert Israel
    Commented Aug 2, 2017 at 1:35
  • $\begingroup$ You may want to look at the existing SLEIGN2 library. $\endgroup$
    – Igor Khavkine
    Commented Aug 2, 2017 at 1:53
  • $\begingroup$ I don't know much about Sturm-Liouville theory, but wouldn't you expect this equation to admit a nonzero eigenfunction for every $\lambda > 0$? Physically, this is the Schrodinger equation for a particle on $(0, \infty)$ experiencing a $1/\sinh^2$ potential spike at the origin. It is entirely unconfined to the right, and hence should happily be able to move to the right with any desired momentum $p$. This corresponds to solutions $f(x)$ asymptotic to $e^{ipx}$ as $x \to \infty$. $\endgroup$
    – David Zhang
    Commented Aug 2, 2017 at 18:00

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