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We prove the existence of a ground state of the Maxwell--Schr\"odingerequations in one spatial dimension, describing a specified amount of freecharge under the influence of a fixed charge. For one case (equal free andfixed charge, i.e., a neutral atom), we introduce a new type of quartic Banachspace, in which the Hamiltonian is naturally coercive. We also show that for apoint charge, for any ratio of charge such that the ground state exists, it issymmetric and decreasing.

Bound States of the One-Dimensional Maxwell--Schr\"odinger Equations