Feynman-Kac formulas for semigroups generated by multi-polaron Hamiltonians in magnetic fields and on general domains

Benjamin Hinrichs, Oliver Matte

Published 2024-03-18Version 1

We prove Feynman-Kac formulas for the semigroups generated by selfadjoint operators in a class containing Fr\"ohlich Hamiltonians known from solid state physics. The latter model multi-polarons, i.e., a fixed number of quantum mechanical electrons moving in a polarizable crystal and interacting with the quantized phonon field generated by the crystal's vibrational modes. Both the electrons and phonons can be confined to suitable open subsets of Euclidean space. We also include possibly very singular magnetic vector potentials and electrostatic potentials. Our Feynman-Kac formulas comprise Fock space operator-valued multiplicative functionals and can be applied to every vector in the underlying Hilbert space. In comparison to the renormalized Nelson model, for which analogous Feynman-Kac formulas are known, the analysis of the creation and annihilation terms in the multiplicative functionals requires novel ideas to overcome difficulties caused by the phonon dispersion relation being constant. Getting these terms under control and generalizing other construction steps so as to cover confined systems are the main achievements of this article.