Abstract
We state a theorem that relates the theory of dimensional reduction in Hamiltonian mechanics to the spectral properties of elliptic operators with symmetries on compact manifolds. As an application, we show that the spectrum of the Schrödinger operator, -[unk]hΔ + V, as [unk]h → 0, contains geometric information about the closed trajectories of a classical particle with Hamiltonian ǁpǁ2 + V(q). More generally, we show that this is true for particles with internal degrees of freedom and subject to an external Yang-Mills field, the classical limit being the Wong-Sternberg-Weinstein system for such particles.
Keywords: elliptic operators, spectral theory, Fourier integral operators
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Selected References
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- Sternberg S. Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field. Proc Natl Acad Sci U S A. 1977 Dec;74(12):5253–5254. doi: 10.1073/pnas.74.12.5253. [DOI] [PMC free article] [PubMed] [Google Scholar]