Mathematics

Generalizing the exact WKB analysis for one-dimensional Schr\"odinger equations established by Voros, Pham, Delabere and others, Aoki, Kawai and I developed the exact WKB analysis for continuous Painlev\'e equations and clarified, in particular, their Stokes geometry and connection formula. Later Iwaki discussed the wall-crossing formula for Painlev\'e equations as well. In this talk I would like, on one hand, to review these previous works and, on the other hand, to talk about my recent research jointly done in part with N. Joshi (Sydney) on the exact WKB analysis for discrete Painle\'e equations. Main targets of the latter research are discrete Painle\'e equations obtained from continuous Painlev\'e equations through the B\"acklund transformation. In the analysis of such discrete Painlev\'e equations both connection formula and wall-crossing formula for continuous Painlev\'e equations appear as different kinds of connection formula.