2021 Fiscal Year Annual Research Report
Resurgent theory in quantum mechanics and its application
| Project/Area Number |
21F21020
|
|---|
| Allocation Type | Single-year Grants |
|---|
| Research Institution | Tokyo Metropolitan University |
|---|
| Host Researcher |
首藤 啓 東京都立大学, 理学研究科, 教授 (60206258)
|
|---|
| Foreign Research Fellow |
NEMES GERGO 東京都立大学, 理学(系)研究科(研究院), 外国人特別研究員
|
|---|
| Project Period (FY) |
2021-11-18 – 2024-03-31
|
| Keywords | asymptotic analysis / special functions / Stokes phenomenon / exact WKB analysis / Borel summability |
|---|
| Outline of Annual Research Achievements |
We considered the problem of the smooth interpretation of the higher-order Stokes phenomenon that appears in asymptotic problems. This phenomenon can be analysed through certain multidimensional integrals called hyperterminants. We first settled a conjecture of C. J. Howls about these integrals and used it to obtain uniform asymptotic formulae for them. These formuale provide a smooth interpretation of the higher-order Stokes phenomenon. Recently we also gained some results on the Borel summability of certain linear third-order differential equations with a large parameter. Such equations have been studied recently by experts in connection with the wall-crossing of TBA equations. We would like to extend our result to equations as general as possible before submitting it for publication.
|
| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We have made good progress in the asymptotics of multidimensional integrals that describe the higher-order Stokes phenomenon. We have established results on the Borel summability of the WKB solutions of third-order differential equations with a large parameter. We hope to extend our result to more general and of higher-order equations. These results could then be applied to multidimensional integrals that satisfy such equations. There is still a lot to be done but the results thus far are rather promising.
|
| Strategy for Future Research Activity |
We demonstrated the smooth transition of the higher-order Stokes phenomenon in the case that the singularities on the Borel plane are equally spaced and are on a line. In future research we would like to consider a more general case: the singularities are not necessarily equally spaced and lined up. This requires a sophisticated asymptotic analysis of a two-dimensional integral. We shall also continue our exact WKB analysis of third- (and higher-) order equations with a large parameter. We would like to establish explicit, applicable results for the Borel summability of the formal WKB solutions under fairly general circumstances. We shall also consider connection problems through Stokes curves by introducing exact, uniform asymptotic expansions involving new special functions.
|
