Multiparticle Brown-Ravenhall operators in external fields

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.