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Morozov, Sergey (2008): Multiparticle Brown-Ravenhall operators in external fields. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
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Abstract

The Brown–Ravenhall model is used in quantum physics and chemistry to describe relativistic multiparticle systems, particularly atoms and molecules. In this dissertation we analyse some general properties of this model on the mathematically rigorous level. We show that under very general assumptions on the interaction potentials the essential spectrum of multiparticle Brown–Ravenhall operators is the right semiaxis starting from the minimal energy possible for the decompositions of the system into two clusters. This result, usually called HVZ theorem, is the fundamental starting point in the spectral analysis of multiparticle Hamiltonians with decaying potentials. Suppose now that the particles constituting the system repel each other but are confined by an external field decaying at infinity. In this situation we prove that the eigenfunctions corresponding to the eigenvalues below the essential spectrum decay exponentially. If some particles of the system are identical, the laws of quantum mechanics often require to reduce the operator to the subspace of functions which transform according to some irreducible representation of the group of permutation of identical particles. On the other hand, the interactions are often invariant under some rotations and reflections. We prove that both the HVZ theorem and the exponential decay of eigenfunctions hold true for operators reduced to the irreducible representations of the above groups. Our results are potentially important in further studies of the spectrum and in the scattering theory of Brown–Ravenhall operators.