The discovery of a virtual turning point truly is a breakthrough in WKB analysis of higher order differential equations. This monograph expounds the core part of its theory together with its application to the analysis of higher order Painlevé equations of the NoumiYamada type and to the analysis of non-adiabatic transition probability problems in three levels.

As M.V. Fedoryuk once lamented, global asymptotic analysis of higher order differential equations had been thought to be impossible to construct. In 1982, however, H.L. Berk, W.M. Nevins, and K.V. Roberts published a remarkable paper in the Journal of Mathematical Physics indicating that the traditional Stokes geometry cannot globally describe the Stokes phenomena of solutions of higher order equations; a new Stokes curve is necessary.

Is the first book that expounds a virtual turning point, a new and important notion in WKB analysis Contains essential know-how in its concrete treatment of the theory of virtual turning points, written by the founders of that theory Establishes an important bridge between pure mathematics and applied mathematics Includes supplementary material: sn.pub/extras

Autorentext

Sampei Hirose is an associate professor of Shibaura Institute of Technology. Naofumi Honda is a professor of Hokkaido University. Takahiro Kawai is a professor emeritus of Kyoto University. Shinji Sasaki is working at the Osaka Central Advanced Mathematical Institute. Yoshitsugu Takei is a professor of Doshisya University.

Inhalt

1. Definition and basic properties of virtual turning Points.- 2. Application to the Noumi-Yamada system with a large Parameter.- 3. Exact WKB analysis of non-adiabatic transition problems for 3-levels.- A. Integral representation of solutions and the Borel resummed WKBsolutions.