Semi classical determination of exponentially small intermode transitions for space time

Semi-classical determination of exponentially small intermode transitions for 1 + 1 space-time scattering systems Alain Joye and Magali Marx Prepublication de l'Institut Fourier no 679 (2005) www-fourier.ujf-grenoble.fr/prepublications.html 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 cor- responding 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 asymptot- ically in the past and as x ? ?∞ along one mode only and determine the piece of the solution that is carried for x ? +∞ along some other mode in the future. Be- cause 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. Keywords: Semi-classical analysis, exponential asymptotics, scattering theory, Landau- Zener mechanism Resume.

• space-time scattering


• limite semi-classique de systemes d'edp autonomes


• semi-classical determination


• dispersion relation


• solutions allow


• space


• small intermode


• solutions dependantes du temps de l'edp


• consider reads


• full time-dependent