Polaron Models with Regular Interactions at Strong Coupling

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.

Automated summary

In this study, we investigate the characteristics of a class of polaron-type Hamiltonians under the strong-coupling limit. The results show that the ground state energy of the system is bounded by the total momentum, in agreement with the semiclassical approximation. Additionally, we demonstrate that the effective mass diverges in the strong coupling limit for all models in all spatial dimensions. Moreover, for certain models with a phonon dispersion relation that grows at least linearly with momentum, we obtain an asymptotic formula for the effective mass quotient, which agrees with the semiclassical Landau-Pekar formula and provides a rigorous confirmation of its validity.