Seminarium KMMF "Teoria Dwoistości"

The Fröhlich polaron is a model for an electron in a polarizable crystal described by a continuous quantum field. Despite being introduced by Landau almost 90 years ago, there are still some basic aspects of the polaron that are not fully understood mathematically. In particular, the connection between Pekar's semi-classical analysis, in which the field is treated as a classical variable, and the quantum model at strong coupling has posed interesting mathematical problems. In this talk, we will review the definition of the polaron model and some classic results, and thenfocus on recent results concerning the spectrum of the Fröhlich Hamiltonian. These include an asymptotic formula for the ground state energy as a function of the conserved total momentum and the abundance of eigenvalues below the essential spectrum at fixed total momentum. If time permits, we will also discuss the dynamical properties of the polaron. For suitable initial conditions, the quantum dynamics can be approximated by the time-dependent Landau-Pekar equations, a set of coupled partial differential equations that describe the evolution of an electron in a slowly varying classical polarization field. The talk is based on results obtained together with J. Lampart, N. Leopold, K. Mysliwy, S. Rademacher, B. Schlein and R. Seiringer.