The Polaron Problem. A Journey through Stochastic and Analytical methods.

A charged particle, such as an electron, travelling through a periodic lattice produces a polarization field that induces oscillations in the crystal structure. A polaron is defined as the dressing of the charged particle by the quantized lattice oscillations, modelled by bosonic fields. The polaron was originally introduced in 1933 by L.  Landau, and later considered in 1946 by S. Pekar, to study the selftrapping phenomenon that an electron incurs into by deforming a lattice structure with its polarization field. The energy spectrum and the dynamics of a polaron are described by an Hamiltonian operator in second quantization, the coupling interaction being linear in creation and annihilation operators. In recent times, significant advances have been made toward the understanding of the mathematical properties of polaron Hamiltonians and, in particular, for the Fröhlich polaron where the dressing is described by a dipolar interaction. The mathematical description of polarons admits both an analytic and a probabilistic approach. In fact, the spectral properties of a polaron Hamiltonian can be derived through the analysis of the partition function of the model by means of the theory of Brownian motion or by means of methods purely from functional analysis and the spectral calculus. I will discuss the modern mathematical techniques to study polaron problems and discuss the state of the art.