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

Abstract

We study the Fröhlich polaron model in 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 [10], 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.

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

“…The result for |𝑃|/𝛼 ≥ √ 2𝑀 LP can be obtained from equations (1.22) For a long time, the only rigorous lower bound available for nonzero P was the one derived by Lieb and Yamazaki [40] in 1958, which states that 𝐸 𝛼 (𝑃) ≥ 𝑐 1 𝑒 Pek + 𝑐 2 𝑃 2 /(2𝛼 4 𝑀 LP ) with 𝑐 1 ≈ 3.07 and 𝑐 2 ≈ 0.11 (where 𝑒 Pek is negative). After the completion of our paper, a lower bound that matches our upper bound was obtained in [9]. It should be noted that the approach in [9] is different from ours, as it does not utilize the fiber decomposition of 𝐻 𝛼 .…”

“…After the completion of our paper, a lower bound that matches our upper bound was obtained in [9]. It should be noted that the approach in [9] is different from ours, as it does not utilize the fiber decomposition of 𝐻 𝛼 .…”

“…During the publication process of this work, a corresponding lower bound on 𝐸 𝛼 (𝛼𝑃) was presented in [9]. When combined with the upper bound presented in this work, the two bounds establish the validity of equation (1.26).…”

“…Furthermore, we would like to mention the recent progress made in the analysis of the large coupling limit of the effective mass, as reported in [9,55]. These advancements represent a significant step forward towards solving the Landau-Pekar conjecture (1.24).…”