Renewal approach for the energy-momentum relation of the FrÃ¶hlich polaron

The FrÃ¶hlich polaron is a model for the interaction of an electron with a polar crystal. We study the energy-momentum relation E(P) which is the bottom of the spectrum of the fixed total momentum Hamiltonian H(P). An application of the Feynman-Kac formula leads to Brownian motion perturbed by a pair potential. The point process representation introduced by Mukherjee and Varadhan represents this path measure as a mixture of Gaussian measures, the respective mixing measure can be interpreted in terms of a perturbed birth and death process. We apply the renewal structure of this point process representation in order to obtain a representation of a diagonal element of the resolvent of H(P). This then yields several properties of the energy-momentum relation, such as monotonicity in |P| and that the correction to the quasi-particle energy is negative. _{Contact: Benjamin Hinrichs}