Semi-classical determination of exponentially small intermode transitions for $1+1$ space-time scattering systems

Alain Joye (1) , Magali Marx (1)

1 IF - Institut Fourier

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.

Mots clés: Semi-classical analysis; exponential asymptotics; scattering theory; Landau-Zener mechanism

Domaines: Physique mathématique [math-ph]

Alain Joye, Magali Marx. Semi-classical determination of exponentially small intermode transitions for $1+1$ space-time scattering systems. Communications on Pure and Applied Mathematics, 2007, 60, pp.1189-1237. ⟨hal-00008156⟩