The effective mass problem for the Landau–Pekar equations

Abstract

We provide a definition of the effective mass for the classical polaron described by the Landau-Pekar equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by Landau and Pekar.

Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron

“…Mathematically rigorous results for the leading order asymptotics of E α (0), for α large, were obtained by Lieb and Yamazaki [36] (with non-matching upper and lower bounds) and by Donsker and Varadhan [9] as well as Lieb and Thomas [35]. The effective mass has been studied in [4,10,12,33,34,48]. Further improvements have been obtained for confined polarons or polaron models with more regular interaction [13,16,43].…”

“…As explained in the previous section, the phonon field behaves classically for large coupling, and thus it is expected that M eff (α) should asymptotically tend to the expression that follows from the corresponding semiclassical counterpart of the problem. This semiclassical theory of the effective mass was introduced by Landau and Pekar in 1948 [27], and, based on this work (see also [12,48]), it is conjectured that…”

The Fröhlich Polaron at Strong Coupling -- Part II: Energy-Momentum Relation and Effective Mass

“…In order to formulate our main Theorem 1.1, let us further introduce the minimal Pekar energy e Pek := inf ϕ F Pek (ϕ) as well as the Hessian H Pek of F Pek at the minimizer ϕ Pek restricted to real-valued functions ϕ ∈ L 2 R R 3 , i.e. we define H Pek as the unique self-adjoint operator on L 2 R 3 satisfying ϕ|H Pek |ϕ = lim ǫ→0 1 ǫ 2 F Pek ϕ Pek + ǫϕ − e Pek for all ϕ ∈ L 2 R R 3 . With this notation at hand, we can state our main new result in Theorem 1.1.…”

“…of the number operator N , is essential for this argument to work. In contrast, in the classical case the effective mass is infinite since there nothing prevents a priori the wavenumber from escaping to infinity without an energy penalty, and one has to introduce a suitable regularization in order to observe the expected asymptotics M eff = α 4 m + o α→∞ α 4 , see [3].…”

“…In order to find a good lower bound on Ψ α |H Λ + p m (p − Υ Λ ) |Ψ α , and therefore on E α,Λ (α 2 p), it is natural to conjugate H Λ + p m (p−Υ Λ ) with the Weyl transformation W ϕ Pek −iξ = W ϕ Pek W −iξ , since ϕ Pek −iξ is close to the minimizer ϕ Pek −i p m ∇ x 1 ϕ Pek of the corresponding classical problem, see [3]. Since iξ is purely imaginary, the interaction term in H Λ is invariant under the transformation W −iξ , i.e.…”