Brief communications

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

G. M. Zhislin

Scientific Research Institute of Radio Physics

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 $n\ge1$ 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 $n=1$.

Keywords: pseudorelativisic Hamiltonian, discrete spectrum, spectral asymptotics.

UDC: 517.9

Received: 24.01.2003

DOI: 10.4213/faa111

 English version: Functional Analysis and Its Applications, 2004, 38:2, 151–156

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