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

Abstract

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 σ _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 σ _d (H _P ) near the lower bound inf σ_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.