Quasi-classical ground states. I. Linearly coupled Pauli–Fierz Hamiltonians

We consider a spinless, non-relativistic particle bound by an external potential and linearly coupled to a quantized radiation field. The energy $\mathcal{E}(u,f)$ of product states of the form $u\otimes \Psi_f$, where $u$ is a normalized state for the particle and $\Psi_f$ is a coherent state in Fock space for the field, gives the energy of a Klein-Gordon--Schr\''odinger system. We minimize the functional $\mathcal{E}(u,f)$ on its natural energy space. We prove the existence and uniqueness of a ground state under general conditions on the coupling function. In particular, neither an ultraviolet cutoff nor an infrared cutoff is imposed. Our results establish the convergence in the ultraviolet limit of both the ground state and ground state energy of the Klein-Gordon--Schr\''odinger energy functional, and provide the second-order asymptotic expansion of the ground state energy at small coupling.