On the strong coupling limit of many-polaron systems in electromagnetic fields

Abstract

In this paper estimates on the ground state energy of Fröhlich N -polarons in electromagnetic fields in the strong coupling limit, α → ∞, are derived. It is shown that the ground state energy is given by α 2 multiplied by the minimal energy of the corresponding Pekar-Tomasevich functional for N particles, up to an error term of order α 42/23 N 3 . The potentials A, V are suitably rescaled in α. As a corollary, binding of N -polarons for strong magnetic fields for large coupling constants is established.

On the ground state energy of the multipolaron in the strong coupling limit

“…Remark. In contrast to [Wel15], the growth of the error in the lower bound now is of order N 2 and not N .…”

“…They combined the methods of Lieb and Thomas and also applied Feynman-Kac formulas, which were applied in [FLST11] to show that bipolarons do not form a bound state above a threshold of ν. A few months later Wellig extended in [Wel15] this result by including external fields. Wellig followed [LT97] as well, and he generalized a localization method of the phonons, originally devepoled in [FLST11] for bipolarons.…”

“…Proof. As in [Wel15] we will subdivide the proof into two steps: In the first step, we will construct an appropriate localized normalized function Φ 0 which satisfies Estimate (3.2) and having (electronic) support within Ωp Ś N i"1 B R py i qq for some y 1 , y 2 , . .…”

“…As in [Wel15], we introduce annihilation operators restricted to S i Ě B i byâ i :" pâ`g i q1 S i with g i py, x 1 , . .…”

“…We introduce some further notation to apply the result given in [Wel15]. First we define operators contributing to the inter-ball exchange energy…”