2007 Fiscal Year Final Research Report Summary
Study of WKB method and saddle point method based on dressed classical dynamics incorporating quantum fluctuations non-perturbatively
| Project/Area Number |
17540350
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Mathematical physics/Fundamental condensed matter physics
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
ADACHI Satoshi Tokyo Institute of Technology, Guraduate School of Science and Engineering, Assistant Professor (90211698)
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| Project Period (FY) |
2005 – 2007
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| Keywords | WKB method / saddle point method / asymptotic analysis / semiclassical mechanics / divergence-free WKB method / divergence-free saddle point method |
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| Research Abstract |
1. The aim of this research is at developing a divergence-free asymptotic analysis and applying it to actual problems in theoretical physics. Any theory of asymptotic analysis appears as a WKB method when it is applied to a differential equation. Similarly, it appears as a saddle point method when applied to an integral. The divergence in WKB method means that the WKB solution diverges at each turning point of the differential equation. The divergence in saddle point method means that the asymptotic evaluation diverges when multiple saddle points collide. Although the asymptotic analysis has been used extensively in a wide range of mathematical sciences, these divergences have restricted usefulness severely.
2. Our preceding research had succeeded in constructing a divergence-free WKB method for differential equations. The key for success was at utilizing a dressed classical dynamics incorporating quantum fluctuations non-perturbatively. The current research started from translating thi
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s divergence-free WKB method for difference equations to the corresponding saddle point method for integrals.
3. For the translation, we constructed a general framework that describes precisely the global aspect of saddle point analysis by utilizing graph theory. The result has already been published as our first paper.
4. Based on it, we translated the most primitive divergence-free WKB method, which is called the cubic WKB method, to the corresponding saddle point method, which is called the cubic saddle point method. This new saddle point method did not become divergence-free. However, the experience of the translation let us notice that the key to obtain a divergence-free saddle point method is at the selection of the integration variable. The result has already been published as our second paper.
5. We have succeeded in constructing a further improved saddle point method, which is called the steepest descent method with the minimal sensitivity. This new steepest descent method achieves a divergence-free saddle point method. Thus, a most important aim of the current research is attained. The result will be published as our third paper. Less
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Research Products
(10 results)
