- 05-286 Joye A., Marx M.
- Semi-classical determination of exponentially small intermode
transitions for $1+1$ space-time scattering systems
(746K, post script)
Aug 24, 05
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Abstract. We consider the semiclassical limit of systems of autonomous PDE's in
1+1 space-time dimensions in a scattering regime. We assume the matrix
valued coefficients are analytic in the space variable and we further suppose that the corresponding dispersion relation admits real-valued modes only with one-dimensional polarization subspaces. Hence a BKW-type analysis of the solutions is possible. We typically consider time-dependent solutions to the PDE which are carried asymptotically in the past and as $x\rightarrow -\infty$ along one mode only and determine
the piece of the solution that is carried for $x\rightarrow +\infty$ along some other mode in the future.
Because of the assumed non-degeneracy of the modes, such transitions between modes are exponentially small in the semiclassical parameter; this is an expression of the Landau-Zener mechanism.
We completely elucidate the space-time properties of the leading term
of this exponentially small wave, when the semiclassical parameter
is small, for large values of $x$ and $t$, when some avoided
crossing of finite width takes place between the involved modes.