| Research Abstract |
The purpose of the research project is to investigate the relationships between the properties of classical mechanics and the spectrum of the associated Schrodinger operator on the Riemannian manifolds.
Particularly, we have payed attention to the mechanics in a magnetic field on the Riemannian manifold. A magnetic field is regarded as a closed two-form on the manifold, and the motion of a charged particle in the magnetic field is formulated as the flow of the Hamiltonian system with the symplectic structure twisted by the two-form. On the other hand, the associated quantum system or the Schrodinger operator is the Laplacian on the complex line bundle naturally defined by the integral closed two-form (the magnetic field) on the manifold. In this context, we have obtained the following results :
1. We have considered the quantization condition for the invariant torus of the Hamiltonian system of magnetic flow, and have clarified by virtue of the theory of Fourier integral operators that the (semi-classical) energy levels determined by the quantization condition give a "good" approximation of the true energies of the quantum system.
2. We have, moreover, considered the "quantization condition" for the stable periodic orbits of the magnetic flow, and have similarly clarified that the classical "energy" of the a suitable periodic orbit gives an approximation of the energy of the associated quantum system. The tool used in this research if the theory of Fourier integral operators of Hermite type which are the operator corresponding to the isotropic submanifold.
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