On the dynamics of the mean-field polaron in the high-frequency limit

Abstract

We consider the dynamics of the mean-field polaron in the weak-coupling limit of vanishing electron-phonon interaction, ε → 0. This is a singular limit formally leading to a Schrödinger-Poisson system that is equivalent to the nonlinear Choquard equation. By establishing estimates between the approximation obtained via the Choquard equation and true solutions of the original system we show that the Choquard equation makes correct predictions about the dynamics of the polaron mean-field model for small values of ε > 0.

Effective Slow Dynamics Models for a Class of Dispersive Systems

“…in the V -equation there are terms of order O(ε −1 ) on the right hand side which can lead to growth rates of order O(e ε −1 t ) and which would make it impossible to prove the bounds for the error on an O(1) time scale. At a second view one finds such approximation results scattered in the literature, for the KGZ system in [DSS16], for the mean field polaron model in [GSS17], and for the Zakharov system for instance in [SW86,AA88]. In fact the underlying idea to prove such approximation results is rather simple.…”

On the dynamics of polarons in the strong-coupling limit

“…For such initial data the system (42), (43) has a unique global solution ϕ t , η t , t ∈ R, with ϕ t ∈ H 2 (R 3 ) depending continuously on time [6,9]. This well-posedness result is only available in d = 3 space dimensions so far.…”

“…which apparently was first written down, in an equivalent form, by Landau and Pekar [10,4]. The mathematical analysis of this system was initiated in [2] and continued in [6,9]. Note that (4) reduces to (3) in the stationary case where V is independent of time and ϕ t = e −iλt ϕ 0 .…”

The Dirac-Klein-Gordon system in the strong coupling limit

“…So, while in general (1.6) does not hold even approximately, Ψ is still well described by (1.7) in the limit. Similar singular limits have been studied in the literature [22,1,8,13,2], and we will comment in Section 1.3 below on the particularity of our situation compared to some of these works.…”

“…with respect to the H s+1 -norm would require an H s+1 -bound on S. Hence, the chain of estimates does not close due to a loss of derivatives. In the works [8,13,2], it turns out that this loss of derivatives does not occur due to an additional regularizing effect, and in these cases one can use such an argument. This loss of derivatives can be dealt with by considering the differences, or reduced variables,…”