  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.
