Semiclassical determination of exponentially small intermode transitions for 1 + 1 spacetime scattering systems

Abstract

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 → −∞ along one mode only and determine the piece of the solution that is carried for x → +∞ 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.

Exponentially accurate semiclassical tunneling wavefunctions in one dimension

“…Analogous results for exponentially small reflected waves when the energy is strictly above a potential bump are presented in [2]. Similar results for non-adiabatic transitions in the Born-Oppenheimer approximation are presented in [8] and [11].…”

Emergence of exponentially small reflected waves

“…In particular these formulas show where, when and how the non-classical reflected wave emerges. Let us note that similar questions can be asked for more general dispersive evolution equations, in the same spirit as the systems considered in [16]. However, the missing piece of information that forbids us to deal with such systems is an equivalent of the analysis performed in [5,6], and adapted here to the scattering setup, that yields the exponentially small leading order of the solutions to (2), for all xs.…”

“…When integrated against a suitable energy density Q(E, ε) like the one we will specify below, this reflected wave gives rise to a freely propagating Gaussian, for large values of x and t, as can be shown by the methods of [16].…”

“…Since this has essentially been done before [16], we will be much briefer here. We use Lemma 2 in order to write…”

“…This will allow us to expand the reflected wave around its maximum in the variable x, and get the result. Since we already know that χ far is a Gaussian centred around the asymptotically freely moving classical reflected wave, [16], it will be negligible for the times t considered. We start by proving concentration in x for finite times.…”