Polaron models with regular interactions at strong coupling

TL;DR

This paper rigorously analyzes polaron models with regular interactions at strong coupling, confirming semiclassical approximations for ground state energy and effective mass divergence across all spatial dimensions.

Contribution

It provides the first rigorous confirmation of the semiclassical Landau--Pekar formula for the effective mass in regular polaron models at strong coupling.

Findings

Ground state energy bounds match semiclassical predictions

Effective mass diverges in the strong coupling limit

Asymptotic formula for effective mass when phonon dispersion is linear or faster

Abstract

We study a class of polaron-type Hamiltonians with sufficiently regular form factor in the interaction term. We investigate the strong-coupling limit of the model, and prove suitable bounds on the ground state energy as a function of the total momentum of the system. These bounds agree with the semiclassical approximation to leading order. The latter corresponds here to the situation when the particle undergoes harmonic motion in a potential well whose frequency is determined by the corresponding Pekar functional. We show that for all such models the effective mass diverges in the strong coupling limit, in all spatial dimensions. Moreover, for the case when the phonon dispersion relation grows at least linearly with momentum, the bounds result in an asymptotic formula for the effective mass quotient, a quantity generalizing the usual notion of the effective mass. This asymptotic form agrees with the semiclassical Landau--Pekar formula and can be regarded as the first rigorous confirmation, in a slightly weaker sense than usually considered, of the validity of the semiclassical formula for the effective mass.