The Hamiltonians of Pseudorelativistic Atoms with Finite-Mass Nuclei: The Structure of the Discrete Spectrum

We study the structure of the discrete spectrum of pseudorelativistic Hamiltonians H for atoms and positive ions with finite-mass nuclei and with n electrons, where n 1 is arbitrary. The center-of-mass motion cannot be separated, and hence we study the spectrum of the restriction H_P of H to the subspace of states with given value P of the total momentum of the system. For the operators H_P we discover a) two-sided estimates for the counting function of the discrete spectrum _d(H_P) of H_P in terms of the counting functions of some effective two-particle operators; b) the leading term of the spectral asymptotics of _d(H_P) near the lower bound inf _ess(H_P) of the essential spectrum of H_P. The structure of the discrete spectrum of such systems was known earlier only for n=1.