Spectrum of the semi-relativistic Pauli-Fierz model II

We consider the ground state of the semi-relativistic Pauli-Fierz Hamiltonian H = vertical bar p A(x)vertical bar + H-f + V(x). Here A(x) denotes the quantized radiation field with an ultraviolet cutoff function and H-f the free field Hamiltonian with dispersion relation vertical bar k vertical bar. 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 Phi(m) of the massive Hamiltonian H-m is proven. Here, the massive Hamiltonian Hm is defined by H with dispersion relation root k(2) + m(2) (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)Phi(m)}(0<m< m0) with some m(0), where a(k) denotes the formal kernel of the annihilation operator. Showing the compactness of the set {Phi(m)}(0<m< m0), the existence of the ground state of H is shown.