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 $n\ge1$ 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 $\sigma_d(H_P)$ of $H_P$ in terms of the counting functions of some effective two-particle operators; b) the leading term of the spectral asymptotics of $\sigma_d(H_P)$ near the lower bound $\inf\sigma_{\operatorname{ess}}(H_P)$ of the essential spectrum of $H_P$. The structure of the discrete spectrum of such systems was known earlier only for $n=1$.