Derivation of an Effective Evolution Equation for a Strongly Coupled Polaron

1.1. Setting of the problem. In this paper we are interested in the dynamics of a strongly coupled polaron. A polaron is a model of an electron in an ionic lattice interacting with its surrounding polarization field. In 1937 Fröhlich [8] proposed a quantum-mechanical Hamiltonian, given in (1.1) below, in order to describe the dynamics of a polaron. In this model the phonon field is treated as a quantum field. The Fröhlich Hamiltonian depends on a single parameter α > 0 which describes the strength of the coupling between the electron and the phonon field. In 1948 Landau and Pekar [11] proposed a system of non-linear PDEs, see (1.8), (1.9) below, to describe the dynamics of a polaron and used this in their famous computation of the effective polaron mass (see [16] for an alternative approach). They treat the phonons as a classical field. The derivation of their equations is phenomenological and they do not comment on the relation between their equations and the dynamics generated by Fröhlich’s Hamiltonian. Our purpose in this paper is to establish a connection between the two dynamics and to rigorously derive the Landau–Pekar equations from the Fröhlich dynamics in the strong coupling limit α → ∞ for a natural class of initial conditions and on certain time scales. In order to describe this result in detail, we recall that the Fröhlich Hamiltonian acts in L2(R3)⊗F , where L2(R3) corresponds to the electron and F = F(L2(R3)), the bosonic Fock space over L2(R3), corresponds to the phonon field. The Hamiltonian is given by