Abstract
The Schr\"odinger equation with minimal coupling for a nonrelativistic electron interacting with a single-mode photon field is not satisfied by the nonrelativistic limit of the exact solutions to the corresponding Dirac equation. A Schr\"odinger-like equation valid for arbitrary photon intensity is derived from the Dirac equation without the weak-field assumption. The ``eigenvalue'' in the present equation is an operator in a Cartan subalgebra. An approximation consistent with the nonrelativistic energy level derived from its relativistic value replaces the eigenvalue operator by an ordinary number, recovering the Schr\"odinger eigenvalue equation used in the formal scattering formalism. The Schr\"odinger-like equation for the multimode case is also presented.
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