Abstract
We describe the exact WKB method from the point of view of abelianization, both for Schrödinger operators and for their higherorder analogues (opers). The main new example which we consider is the “T_{3} equation,” an order 3 equation on the thricepunctured sphere, with regular singularities at the punctures. In this case the exact WKB analysis leads to consideration of a new sort of Darboux coordinate system on a moduli space of flat SL(3)connections. We give the simplest example of such a coordinate system, and verify numerically that in these coordinates the monodromy of the T_{3} equation has the expected asymptotic properties. We also briefly revisit the Schrödinger equation with cubic potential and the Mathieu equation from the point of view of abelianization.
Original language  English 

Pages (fromto)  131–186 
Number of pages  56 
Journal  Communications in Mathematical Physics 
Volume  380 
Issue number  1 
Early online date  13 Oct 2020 
DOIs  
Publication status  Published  Nov 2020 
ASJC Scopus subject areas
 Statistical and Nonlinear Physics
 Mathematical Physics
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Lotte Hollands
 School of Mathematical & Computer Sciences  Associate Professor
 School of Mathematical & Computer Sciences, Mathematics  Associate Professor
Person: Academic (Research & Teaching)