ON THE SPECTRAL PROPERTIES OF THE BROWN-RAVENHALL OPERATOR


A A BALINSKY

BalinskyA@cardiff.ac.uk



W D EVANS

EvansWD@cardiff.ac.uk

School of Mathematics
Cardiff University
23 Senghennydd Road
P O Box 926
Cardiff CF24 4YH, UK






The fact that the Dirac is unbounded below creates problems if it is used to describe multi-particle relativistic systems since the resulting operator has a spectrum which covers the whole of the real line. To overcome this difficulty Brown and Ravenhall proposed the following one-particle model. To describe an electron in the field of its nucleus and subject to relativistic effects, the operator of Brown and Ravenhall is

(1) $\displaystyle {{\bf B}} := \Lambda_+ \biggl(D_0 - \frac{e^2 Z}{\lmod \cdotp \rmod}\biggr) \Lambda_+ .$

acting in the Hilbert space $ {\mathcal{H}} :=
\Lambda_+(\operatorname{L}^2 ({\mathbb{R}}^3)\otimes
{\mathbb{C}}^4).$ The notation in (1) is as follows

The lecture will discuss spectral properties of operators $ b_{l,s}$ appearing in the partial wave decomposition of B: the indices $ l,s$, denote the angular momentum channel and spin respectively. The following topics will be covered: the value of the critical charge $ Z_c(l,s)$ which yields the positivity of $ b_{l,s}$, the charge range for essential self-adjointness, and the charge range for the absence of embedded eigenvalues.



Eastham Meeting at Gregynog, 26-27 July 2002

Gregynog Abstracts