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

We study the Fröhlich polaron model in $\mathbb{R}^3$, and prove a lower bound on its ground state energy as a function of the total momentum. The bound is asymptotically sharp at large coupling. In combination with a corresponding upper bound proved earlier, it shows that the energy is approximately parabolic below the continuum threshold, and that the polaron’s effective mass (defined as the semi-latus rectum of the parabola) is given by the celebrated Landau–Pekar formula. In particular, it diverges as $α^4$ for large coupling constant $α$.

💡 Research Summary

The paper provides a rigorous analysis of the Fröhlich polaron in three dimensions in the strong‑coupling regime, where the electron–phonon interaction strength α tends to infinity. The authors focus on the ground‑state energy Eα(P) of the Fröhlich Hamiltonian H as a function of the total momentum P, and they establish a sharp lower bound that matches a previously proved upper bound. This result confirms that the energy–momentum relation is asymptotically parabolic below the continuum threshold and that the effective mass of the polaron follows the celebrated Landau–Pekar formula.

The model is defined by the Hamiltonian
H = –Δx – a(w_x) – a†(w_x) + N,
where w_x(k)=π^{-3/2}|k|^{-2}e^{ik·x} and N is the phonon number operator. The total momentum operator is
P = –i∇_x + α²∫k a†_k a_k dk,
which commutes with H, allowing the joint spectrum σ(P,H) to be considered. The ground‑state energy at fixed momentum is Eα(P)=inf{E : (P,E)∈σ(P,H)}.

A central role is played by the Pekar functional
F_Pek(ϕ)=‖ϕ‖² + inf σ(–Δ + V_ϕ), V_ϕ = –2(–Δ)^{-1/2}Re ϕ,
which admits a unique radial minimizer ϕ_Pek, a minimal energy e_Pek, and a Hessian operator H_Pek describing quadratic fluctuations around the minimizer. The constant
m = (2/3)‖∇ϕ_Pek‖²
emerges as the Landau–Pekar effective‑mass coefficient.

Main results.
Theorem 1.1 states that for all momenta P∈ℝ³ and sufficiently large α, \