Spectrum of the semi-relativistic Pauli-Fierz model II

We consider the ground state of the semi-relativistic Pauli-Fierz Hamiltonian (equation presented) Here A.x/ denotes the quantized radiation field with an ultraviolet cutoff function and Hf the free field Hamiltonian with dispersion relation jkj. The Hamiltonian H describes the dynamics of a massless and semi-relativistic charged particle interacting with the quantized radiation field with an ultraviolet cutoff function. In 2016, the first two authors proved the existence of the ground state m of the massive Hamiltonian Hm is proven. Here, the massive Hamiltonian Hm is defined by H with dispersion relation pk2 C m2 (m > 0). In this paper, the existence of the ground state of H is proven. To this aim, we estimate a singular and non-local pull-through formula and show the equicontinuity of {a(k)φm}0<m<m0 with somem0, where a(k) denotes the formal kernel of the annihilation operator. Showing the compactness of the set m0<m<m0 , the existence of the ground state of H is shown.