Persistence of the spectral gap for the Landau–Pekar equations

Abstract

The Landau–Pekar equations describe the dynamics of a strongly coupled polaron. Here, we provide a class of initial data for which the associated effective Hamiltonian has a uniform spectral gap for all times. For such initial data, this allows us to extend the results on the adiabatic theorem for the Landau–Pekar equations and their derivation from the Fröhlich model obtained in previous works to larger times.

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

“…Further improvements have been obtained for confined polarons or polaron models with more regular interaction [13,16,43]. For completeness, let us also mention recent progress in the understanding of the polaron path measure [3,42] as well as the increased interest in the analysis of the Schrödinger time evolution of strongly coupled polarons [11,14,15,23,30,31,38].…”

“…for all v ∈ L 2 (R 3 ) with v|∇ϕ L 2 = 0 (note that this also implies (iii)). For that purpose we quote [11,Lemma 2.7] stating that there exists a constant τ > 0 such that…”

Derivation of the Maxwell-Schrödinger Equations: A note on the infrared sector of the radiation field

“…In [23] the Nelson model with ultraviolet cutoff has been studied in a limit of many weakly interacting fermions. The classical behavior of quantum fields has also been proven in different scaling regimes [3,5,6,7,8,9,12,13,14,16,17,19,21,26,28,34]. We also would like to mention [32] which derives the Maxwell-Schrödinger equations in a nonrigorous manner by neglecting certain terms in the Pauli-Fierz Hamiltonian.…”

Bogoliubov Dynamics and Higher-order Corrections for the Regularized Nelson Model

“…In this appendix we give further details about the unitary, defined by (19), and the derivation of the excitation Hamiltonian from (24). To this end, we closely follow [42,Chapter 4.1] and [43,Chapter 2.3].…”

“…In [15,32,60] it has, moreover, been shown that the effect of the quantum field on the particles can in certain situations be approximated by a direct pair interaction. Let us also mention further works that focus on related models [37,38,40], a partial limit [9,12,13,14] and the strong coupling limit of the polaron [19,20,21,26,36,41,46].…”