We consider a charged particle, spin 1 2 , with relativistic kinetic energy and minimally coupled to the quantized Maxwell field. Since the total momentum is conserved, the Hamiltonian admits a fiber decomposition as H (P ), P ∈ R 3 . We study the spectrum of H (P ). In particular we prove that, for non-zero photon mass, the ground state is exactly two-fold degenerate and separated by a gap, uniformly in P , from the rest of the spectrum.

Within the framework of nonrelativisitic quantum electrodynamics we consider a single nucleus and N electrons coupled to the radiation field. Since the total momentum P is conserved, the Hamiltonian H admits a fiber decomposition with respect to P with fiber Hamiltonian H (P ). A stable atom, respectively ion, means that the fiber Hamiltonian H (P ) has an eigenvalue at the bottom of its spectrum. We establish the existence of a ground state for H (P ) under (i) an explicit bound on P , (ii) a binding condition, and (iii) an energy inequality. The binding condition is proven to hold for a heavy nucleus and the energy inequality for spinless electrons.

The Holstein model has been widely accepted as a model comprising electrons interacting with phonons; analysis of this model's ground states was accomplished two decades ago. However, the results were obtained without completely taking repulsive Coulomb interactions into account. Recent progress has made it possible to treat such interactions rigorously; in this paper, we study the Holstein-Hubbard model with repulsive Coulomb interactions. The ground state properties of the model are investigated; in particular, the ground state of the Hamiltonian is proven to be unique for an even number of electrons on a bipartite connected lattice. In addition, we provide a rigorous upper bound on charge susceptibility.

The bipolaron are two electrons coupled to the elastic deformations of an ionic crystal. We study this system in the Fröhlich approximation. If the Coulomb repulsion dominates, the lowest energy states are two well separated polarons. Otherwise the electrons form a bound pair. We prove the validity of the Pekar-Tomasevich energy functional in the strong coupling limit, yielding estimates on the coupling parameters for which the binding energy is strictly positive. Under the condition of a strictly positive binding energy we prove the existence of a ground state at fixed total momentum P , provided P is not too large.

In condensed matter physics, the polaron has been fascinating subject. It is described by the Hamiltonian of H. Fröhlich. In this paper, the Fröhlich Hamiltonian is investigated from a viewpoint of operator inequalities proposed in [36]. This point of view clarifies the monotonicity of polaron energy, i.e., denoting the lowest energy of the Fröhlich Hamiltonian with the ultraviolet cutoff Λ by E Λ , we prove E Λ > E Λ ′ for Λ < Λ ′ .