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

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) |

acting in the Hilbert space The notation in (1) is as follows

- is the free Dirac operator
- denotes the projection of
onto the positive spectral
subspace of , that is
, where
is the characteristic function of
. If we set

with . - is Planck's constant, the velocity of light, the electron mass, the electron charge, and the nuclear charge.

The lecture will discuss spectral properties of operators
appearing in the partial wave decomposition of
**B**: the indices , denote the angular momentum channel
and spin respectively. The following topics will be covered: the
value of the critical charge which yields the
positivity of , 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