MPEJ Volume 5, No.6, 11pp
Received: Sep 21, 1999, Accepted: Dec 7, 1999
Georg Hoever, Heinz Siedentop
Stability of the Brown-Ravenhall Operator
ABSTRACT: The Brown-Ravenhall Hamiltonian is a model for the behavior of
$N$ electrons in a field of $K$ fixed nuclei having the atomic numbers
${\bf Z}=(Z_1,\ldots,Z_K)$, which is written, in appropriate units, as
$$B=\Lambda_{+,N}\left(\sum_{n=1}^N D_0^{(n)} +\alpha V_c\right)\Lambda_{+,N}$$
acting on the $N$-fold antisymmetric tensor product $\mathfrak H_N$ of
$\Lambda_+(L^2({\mathbb R}^3)\otimes {\mathbb C}^4)$, where $D_0^{(n)}$
denotes the free Dirac operator $D_0$ acting on the $n$-th particle,
$\Lambda_+$ denotes the projection onto the positive spectral subspace of
$D_0$, $\Lambda_{+,N}$ the projection onto $\mathfrak H_N$ and the potential
$V_c$ is the usual Coulomb interaction of the particles, coupled by
the constant $\alpha$. It is proved in the massless case that for any
$\gamma <2/(2/\pi+\pi/2)$ there exists an $\alpha_0$ such that for all
$\alpha<\alpha_0$ and $\alpha Z_k\leq\gamma$ $(k=1,\ldots K)$ we have stability,
i.e., $B\geq 0$. Using numerical calculations we get stability for the
physical value $\alpha\approx 1/137$ up to $Z_k\leq 88$ $(k=1,\ldots K)$.