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Quantum monodromy and semi-classical trace formulæ. (English) Zbl 1038.58033

In this interesting paper a trace formula for semi-classical pseudodifferential operators whose symbol has a close Hamiltonian trajectory is proved. The operators are defined on a compact \(C^\infty\) manifold \(X\). The Poincaré map of trajectories can be quantized by a Fourier integral operator. More generally, the trace formula is proved for a smooth family of operators, that are self-adjoint for real values of the parameter, elliptic off the real axis and of principal type.
The formula is proved with the help of the quantum monodromy operator. The result leads to a new proof of the trace formula of Duistermaat-Guillemin and Gutwiller.

MSC:

58J40 Pseudodifferential and Fourier integral operators on manifolds
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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