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

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

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

• $D_0$ is the free Dirac operator
  $$D_0= c {{\boldsymbol{\alpha}}} \cdotp \frac{\hslash}{i} \; \nabla... ...}^3 c \frac{\hslash}{i} \alpha_j \frac{\partial}{\partial x_j} + m c^2 \beta ,$$
  where ${\boldsymbol{\alpha}}=(\alpha_1,\alpha_2,\alpha_3)$ and $\beta$ are the Dirac matrices given by
  $$\alpha_j= \begin{pmatrix} 0 & \sigma_j\\ \sigma_j & 0 \end{pmatrix} , \beta= \begin{pmatrix} 1_2 & 0_2\\ 0_2 & -1_2 \end{pmatrix}$$
  with $0_2 , 1_2$ the zero and unit $2 \times 2$ matrices respectively and $\sigma_j$ the Pauli matrices
  $$\sigma_1= \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}, \sigma_2=... ... i & 0 \end{pmatrix}, \sigma_3= \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}.$$

• $\Lambda_+$ denotes the projection of $\operatorname{L}^2 ({\mathbb{R}}^3)\otimes {\mathbb{C}}^4$ onto the positive spectral subspace of $D_0$, that is $\chi_{(0,\infty)} (D_0)$, where $\chi_{(0,\infty)}$ is the characteristic function of $(0,\infty)$. If we set
  $$\widehat{f}({\bf p}) \equiv {\mathcal{F}}(f) ({\bf p}) = \biggl( ... ...} e^{-i \; {\bf x}\cdotp {\bf p}/ \hslash} f({\bf x}) \operatorname{d} {\bf x}$$
  for the Fourier transform of $f$, then it follows that
  $$(\Lambda_+ f)^{\wedge} ({\bf p}) = \Lambda_+ ({\bf p}) \widehat{f} ({\bf p}),$$
  where

  $$\Lambda_+ ({\bf p})= \frac{1}{2} + \frac{c {\boldsymbol{\alpha}} ... ...\bf p}+ m c^2 \beta}{2 {\bf e}(p)}, \; \; {\bf e}(p) = \sqrt{c^2 p^2 + m^2 c^4}$$ (2)

  
  
  with $p=\lmod {\bf p}\rmod$.
• $2 \pi \hslash$ is Planck's constant, $c$ the velocity of light, $m$ the electron mass, $-e$ the electron charge, and $Z$ the nuclear charge.

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.