@mikestone I am interested in two reasons, 1 for a heuristic example to better learn how to apply the wkb method for QNMs, and 2 to see how accurate the WKB method is against an analytically solvable example. In the literature I have sometimes read that WKB is sommetimes "more accurate than one would expect". For better context I am an undergraduate research assistant in the field of black hole spectroscopy. The team I am on specializes in the continued fraction method, and is not really familiar with the WKB method. Also can you explain the significance of PT being "repulsive"?

Can you provide a link to some of the relevant literature? I'm very curious to see the methods and results. In Black hole Spectroscopy there are analytical solutions so I'm curious how the implementation is different in string theory

@FilipMilovanović yes this was precisely my confusion. I didn't realize that you have to expand it into separate expressions, then sum over beta. Thank you for confirming this as well, the responses have been very helpful in correcting this misunderstanding.

Aha, okay so this makes more sense to me. I appreciate your candor and feedback. I will set the question as answered!