 05286 Joye A., Marx M.
 Semiclassical determination of exponentially small intermode
transitions for $1+1$ spacetime scattering systems
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Aug 24, 05

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Abstract. We consider the semiclassical limit of systems of autonomous PDE's in
1+1 spacetime 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 realvalued modes only with onedimensional polarization subspaces. Hence a BKWtype analysis of the solutions is possible. We typically consider timedependent 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 nondegeneracy of the modes, such transitions between modes are exponentially small in the semiclassical parameter; this is an expression of the LandauZener mechanism.
We completely elucidate the spacetime 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.
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