Reduction, the trace formula, and semiclassical asymptotics.
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]hDelta + V, as [unk]h --> 0, contains geometric information about the closed trajectories of a classical particle with Hamiltonian p(2) + 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.
Mathematics Department, Massachusetts Institute of Technology, Cambridge, MA 02139.
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Guillemin
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Uribe