Hmm. It's a tough one. I personally don't see how the alpha is coming in either. What's the name of the paper? Is it accessible through web of science?
I would say don't get too caught up in where they are coming from. It is more important to understand why they are useful. My guess is the author is deriving a set of bases functions to be used in an expansion of [tex]u(x)[/tex]. The Sturm-Liouville equations can deliver a set of functions which can serve this purpose. The nice properties that these bases functions display make them a good choice when you're expanding the unknown u(x).
In particular take a look at (http://en.wikipedia.org/wiki/Asymptotic_expansion
) and (http://en.wikipedia.org/wiki/Perturbation_methods