- 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.