We describe the exact WKB method from the point of view of abelianization, both for Schrödinger operators and for their higher-order analogues (opers). The main new example which we consider is the “T3 equation,” an order 3 equation on the thrice-punctured 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 T3 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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics