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The subject of this thesis is the investigation of the stability of a system composed of an arbitrary number of non-relativistic electrons and an arbitrary number of static nuclei interacting through Coulomb forces, where in addition the eleetrons are coupled to the photon field, or more precisely, to the quantized ultraviolet-cutoff electromagnetic field. Stability means that the energy per particle in the system is bounded below uniformly in the number of particles (and in the nuclear configuratibns). For this system of matter and radiation we prove two results. The first one asserts that stability holds if the values of the fine structure constant, a, and of the product of its square with the largest nuclear charge, Z, are not too large. These constraints are of the same kind as the ones occurring in a system of nuclei and electrons coupled to a classical external field, which are caused first of all by the fact that the Pauli operator has zero-modes. The second result establishes stability for every value of a and Za , showing in this way that actually stability can no longer manifest, once an ultraviolet cutoff is imposed on the quantized electromagnetic field, Both results are presented in Section 1.7. The proofs of these two results rest on two magnetic Lieb-Thirring type inequalities. By this we mean a lower bound on the sum of the negative eigenvalues of a Pauli operator, that describes the motion of one eleetron in an external electromagnetic field, including also the interaction of its magnetic moment (i.e., spin) with the external magnetic field. The original inequality stems from Elliot H. Lieb and Walter Thirring, who 1975 derived such an estimate for the Schrödinger operator describing the motion of a non-relativistic spinless eleetron in an external potential. The proofs of our two LiebThirring inequalities require the use of effective magnetic fields (i.e., smeared magnetic fields) in order to account for the semilocal cooperation between the physical magnetic field and the potential in creating new bound states, and to them are devoted the second and third chapter.

Stability of quantum electrodynamics with non-relativistic matter and magnetic Lieb-Thirring estimates