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Global study of differential equations via the exact WKB - from Schrödinger and Panlevé

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ARA - Applicable resurgent asymptotics: towards a universal theory

The exact WKB analysis provides a powerful tool for the

global study of differential equations. In this talk we would like to give a

brief review of this analysis and discuss an important problem related to it.

 First we review

the exact WKB analysis for one-dimensional stationary Schrödinger

equations. In this case the exact WKB analysis gives a quite satisfactory

answer, that is, global behavior of solutions such as the monodromy group, the

exact quantization condition, etc are described by contour integrals of

logarithmic derivative of WKB solutions. Next we consider its generalization to

Panlevé equations. Even for such nonlinear equations the exact WKB analysis

is successful and we can obtain an explicit connection formula for Stokes

phenomena of Panlevé equations in terms of their formal power series

solutions and transseries solutions. However, to complete the global study of

Panlevé equations, we need to deal with the instanton-type solutions (or

two-parameter transseries solutions), which are purely formal and whose

behavior are much wilder than transseries solutions. It is really a big and

important problem to give an analytic interpretation to instanton-type

solutions in the exact WKB analysis. In the latter half of the talk we discuss

our recent trial to attack this challenging problem.

This talk is part of the Isaac Newton Institute Seminar Series series.

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