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Semiclassical determination of exponentially small intermode transitions for \(1 + 1\) spacetime scattering systems. (English) Zbl 1387.35439

Summary: We consider the semiclassical limit of systems of autonomous PDEs 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 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 that 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 Landau-Zener 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.

MSC:

35P25 Scattering theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
35Q53 KdV equations (Korteweg-de Vries equations)
35Q60 PDEs in connection with optics and electromagnetic theory
47A55 Perturbation theory of linear operators
47N50 Applications of operator theory in the physical sciences
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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