応用特異点ラボ・シンポジウム：Mini Workshop on WKB Analysis and Singularity Theory

開催日時

2018年12月10日 10時 00分 ～ 2018年12月10日 15時 30分

場所

理学部３号館３ー２０５

主催者

世話人：泉 屋周一、大本 亨、寺本 央

Mini Workshop on WKB Analysis and Singularity Theory

Program

10:00 – 10:30
Hiroshi Teramoto (Hokkaido University)
Welcome Remarks

Abstract
Non-adiabatic transitions play important roles in chemical reaction dynamics. In this mini-workshop, we invite Professor George Hagedorn who are a pioneer of mathematical WKB analysis of non-adiabatic transitions along with two experts of WKB analysis, Professor Akira Shudo and Professor Takuya Watanabe.
WKB analysis is deeply related to singularity theory and the aim of this workshop is to deepen the relation even further for mutual developments. In this short remark, I will present some motivating examples and our ongoing projects.

10:30 – 11:30
George Hagedorn (Virginia Polytechnic Institute and State University)
Theory and Numerics for Semiclassical Quantum Mechanics

Abstract
We present numerical algorithms for solving the Schrödinger equation based on semiclassical wave packets.

13:00 – 14:00
Akira Shudo (Tokyo Metropolitan University)
Role of virtual turning points and new Stokes curves in multi-state nonadiabatic models

Abstract
Although enormous efforts were made for finding solutions of generalized Landau-Zener model, it was difficult to find the WKB solution for multi-state nonadiabatic models in general since the WKB analysis for higher-order differential equations had been almost unexplored until recently. On the basis of the exact WKB analysis, Aoki-Kawai-Takei gave a closed expression for the S-matrix of a time-dependent multi-state nonadiabatic system, in which diabatic potential terms are given as polynomial functions of time. Their result is completely explicit, and the model covers wider classes of systems compared to multi-state nonadiabatic models studied so far.

Here we briefly introduce an idea of the exaxt WKB analysis, and show how their exact WKB formula works well as long as turning points are well separated, including a model with diabatic potential functions being nonlinear. We also discuss the situations where virtual turning points and new Stokes curves, new components in the Stokes geometry appearing only for higher-order differential equations, come into play. In particular we consider the case where turning points are connected by a Stokes curve, which is beyond the exact WKB treatment, and also examine the observability of new Stokes curves in a couple of concrete systems.

14:30 – 15:30
Takuya Watanabe (Ritsumeikan University)
Two-level transition problem for avoided crossings in a non-adiabatic regime
Abstract We study a two-level adiabatic transition probability for a finite number of avoided crossings with a small interaction through a first order 2 x 2 system.
The asymptotic behavior of the transition probability when two parameters (an adiabatic parameter \(h\) and an energy-gap parameter \(\varepsilon\)) tend to 0 depends on the ratio \(h/\varepsilon^2\).
The exponential decaying property in an adiabatic regime (\(h/\varepsilon^2 \to 0\)) has been studied by many researchers and also in a critical regime (\(h/\varepsilon^2\) is \({\mathcal O}(1)\)) had been obtained by G.-A. Hagedorn in 1991.
In this talk, we give the asymptotic behavior of the transition probability in the other regime (\(\varepsilon^2/h \to 0\)) called “non-adiabatic” regime. In this regime, an exact WKB method can not be adapted because of the confluence of turning points. We would like to report to overcome the difficulty by means of a microlocal analysis in stead of an exact WKB method.
This is the joint work with M. Zerzeri (Paris Nord).

応用特異点ラボHP