Studies of Correspondence between Classical and Quantum Dynamical Systems
Project/Area Number 
14540210

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Global analysis

Research Institution  The University of Tokushima 
Principal Investigator 
KUWABARA Ruishi The University of Tokushima, Dept.of Integrated Arts and Sciences, Prof., 総合科学部, 教授 (90127077)

Project Period (FY) 
2002 – 2004

Project Status 
Completed (Fiscal Year 2004)

Budget Amount *help 
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2004: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2003: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2002: ¥500,000 (Direct Cost: ¥500,000)

Keywords  spectral geometry / Schroedinger operator / quantization condition / semiclassical analysis / Fourier integral operator / magnetic field / gauge field / Schrodinger operator / spectrum / magnetic flow / semiclasssical analysis / periodic orbit / Fourier integral operator 
Research Abstract 
The main result obtained in the research concerns with the classicalquantum 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 28November 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 socalled the KaluzaKlein 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 semiclassical 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 nonabelian gauge systems. We first reconstructed a geometric formulation (originally due to Guillemin, Uribe, Zelditch and so on) of the system in the framework of principal Gbundle, and obtained some results for the system in the nonabelian 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 socalled 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

Report
(4 results)
Research Products
(3 results)