00-112 Bily, J.M., Robert, D.

The Semi-classical Van-Vleck Formula. Application to the Aharonov-Bohm Effect (51K, Latex 2e) Mar 14, 00

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Abstract. At the very beginning of the quantum theory, Van-Vleck (1928) proposed a nice approximation formula for the integral kernel of the time dependent propagator for the Schr\"odinger equation. This formula can be deduced from the Feynman path integral by a formal stationary phase argument. After the fondamental works by H\"ormander and Maslov on Fourier-integral operators, it became possible to give a rigorous mathematical proof of the Van Vleck formula. We present here a more direct and elementary proof, using propagation of coherent states. We apply this result to give a mathematical proof of the Aharonov-Bohm effect observed on the time dependent propagator. This effect concerns a phase factor depending on the flux of a magnetic field, which can be non trivial, even if the particle never meets the magnetic field.

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