No abstract

Bethe and Salpeter introduced a relativistic equation -different from the Bethe-Salpeter equation -which describes relativistic multi-particle systems. Here we will begin some basic work concerning its mathematical structure. In particular we show self-adjointness of the one-particle operator which will be a consequence of a sharp Sobolev type inequality yielding semi-boundedness of the corresponding sesquilinear form. Moreover we locate the essential spectrum of the operator and show the absence of singular continuous spectrum.

The increasing interest in the Müller density-matrix-functional theory has led us to a systematic mathematical investigation of its properties. This functional is similar to the Hartree-Fock ͑HF͒ functional, but with a modified exchange term in which the square of the density matrix ␥͑x , xЈ͒ is replaced by the square of ␥ 1/2 ͑x , xЈ͒. After an extensive introductory discussion of density-matrix-functional theory we show, among other things, that this functional is convex ͑unlike the HF functional͒ and that energy minimizing ␥'s have unique densities ͑r͒, which is a physically desirable property often absent in HF theory. We show that minimizers exist if N Յ Z, and derive various properties of the minimal energy and the corresponding minimizers. We also give a precise statement about the equation for the orbitals of ␥, which is more complex than for HF theory. We state some open mathematical questions about the theory together with conjectured solutions.

A minimax principle is derived for the eigenvalues in the spectral gap of a possibly non-semibounded selfadjoint operator. It allows the nth eigenvalue of the Dirac operator with Coulomb potential from below to be bound by the nth eigenvalue of a semibounded Hamiltonian which is of interest in the context of stability of matter. As a second application it is shown that the Dirac operator with suitable non-positive potential has at least as many discrete eigenvalues as the Schro$ dinger operator with the same potential.

Introducing the Hellmann-Weizsacker functional for large angular momenta and the orbitals of the Bohr atom for small angular momenta we obtain an upper bound on the quantum mechanical ground state energy of atoms that proves Scott's conjecture.