Strict positivity of a relativistic Hamiltonian due to Brown and Ravenhall. (English) Zbl 0939.35134

The aim of this paper is the operator \(B:=\Lambda_+(D_0-{e^2 Z\over|x|}) \Lambda_+\) in the Hilbert space \({\mathfrak H}=\Lambda_+ (L^2 (\mathbb{R}) \otimes \mathbb{C}^4)\) for a relativistic electron in the field of a nucleus. Here \(D_0\) is the Dirac operator (see, Hans A. Bethe and Edwin E. Salpeter, [Quantum mechanics of one- and two-electron atoms, Springer, Berlin (1957; Zbl 0089.21006)]). W. D. Evans, P. Perry and H. Siedentop [Commun. Math. Phys. 178, 733-746 (1996; Zbl 0857.47041)] proved that the energy \({\mathcal E}:=(\psi, {\mathcal B}\psi)\) associated with \(B\) is bounded from below by \(\alpha Z(1/\pi-\pi/4)mc^2\) on the set of normalized rapidly decaying smooth spinors, if the nuclear charge does not exceed the critical charge \(Z_c\). This bound is always negative. The author proves that \({\mathcal E}\) is strictly positive for \(Z\leq Z_c\).
Reviewer: B.Dittmar (Halle)


35P15 Estimates of eigenvalues in context of PDEs
35Q40 PDEs in connection with quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Full Text: DOI