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We consider a non-relativistic electron bound by an external potential and
coupled to the quantized electromagnetic field in the standard model of
non-relativistic QED. We compute the energy functional of product states of the
form $u\otimes \Psi_f$, where $u$ is a normalized state for the electron and
$\Psi_f$ is a coherent state in Fock space for the photon field. The
minimization of this functional yields a Maxwell–Schr{\”o}dinger system up to
a trivial renormalization. We prove the existence of a ground state under
general conditions on the external potential and the coupling. In particular,
neither an ultraviolet cutoff nor an infrared cutoff needs to be imposed. Our
results provide the convergence in the ultraviolet limit and the second-order
asymptotic expansion in the coupling constant of the ground state energy of
Maxwell–Schr\”odinger systems.

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