Quantitative estimates on the binding energy for hydrogen in non-relativistic QED (Q630034)

It is well-known that the binding energy of the non-relativistic hydrogen-like atom composed of an electron and the nucleus of charge \(Z\) and described by the Schrödinger equation, that is the energy required to remove the electron from the atom to infinity, amounts to \((Z \alpha)^2/4\) where \(\alpha\) is the fine structure constant. It is shown that within the non-relativistic QED, the binding energy of the electron of the hydrogen-like atom, which interacts with the quantized electromagnetic field and is described by the Pauli-Fierz Hamiltonian, is expressed as the expansion series \(\alpha^2/4 + \lambda_1 \alpha^3 + \lambda_2 \alpha^4 + \lambda_3 \alpha^5 \log (\alpha^{-1}) + o(\alpha^5 \log (\alpha^{-1}))\) where \(\lambda_i, i = 1,2,3\) are constants and \(\lambda_1\) is positive. The latter formula particularly implies that the binding energy of the electron is not analytic in \(\alpha\).