2004 Fiscal Year Final Research Report Summary
Studies of Correspondence between Classical and Quantum Dynamical Systems
| Project/Area Number |
14540210
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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 |
Global analysis
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| Research Institution | The University of Tokushima |
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Principal Investigator |
KUWABARA Ruishi The University of Tokushima, Dept.of Integrated Arts and Sciences, Prof., 総合科学部, 教授 (90127077)
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| Project Period (FY) |
2002 – 2004
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| Keywords | spectral geometry / Schroedinger operator / quantization condition / semi-classical analysis / Fourier integral operator / magnetic field / gauge field |
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| Research Abstract |
The main result obtained in the research concerns with the classical-quantum correspondence for the mechanical system in a magnetic field. The result was presented at the international workshop ‘Spectral theory of differential operators and the inverse problems' held in the Research Institute for Mathematical Sciences in Kyoto University (October 28-November 1) in 2002. The title of the talk was ‘Quantum energies and classical orbits in a magnetic field'. Later the paper described the details of the results was published in the Proceedings of the Workshop from American Mathematical Society in 2004.
The result of our research is the following. We remark first that a magnetic field on a manifold is regarded as the curvature of a connection given on a principal bundle, and the dynamical flow in the magnetic field can be analyzed as the geodesic flow on the bundle relative to so-called the Kaluza-Klein metric. On the basis of this formulation we considered the relationship between the class
… More
ical flow and the quantum system (the Schroedinger operator), as a result, we clarified that some classical orbit satisfying ‘quantization condition' corresponds to an approximate energy level in a semi-classical sense.
This result is a generalization of the former result by Ralston and Guillemin for the geodesic flow to the case of magnetic flow. It also gives an interesting view to the trace formula. The key tool for the research was the theory of Fourier integral operators of Hermite type which is developed by Boutet de Monvel and Guillemin.
Next we aimed to generalize the results for the magnetic system (the U(1)-gauge system) to the non-abelian gauge systems. We first reconstructed a geometric formulation (originally due to Guillemin, Uribe, Zelditch and so on) of the system in the frame-work of principal G-bundle, and obtained some results for the system in the non-abelian gauge field, which is a generalization of the results for the system in the magnetic field (the U(1)-gauge field). More precisely we clarified the relationship between so-called the Maslov quantization condition (and classical periodic orbits) and the asymptotic properties of quantum energy levels. The paper containing these results is being prepared. Less
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Research Products
(2 results)
