## Abstract

We consider an operator $Q(V)$ of Dirac type with
a meromorphic potential given in terms of a function $V$ of the form
$V(z)=\lambda V_1(z)+\mu V_2(z), \ z\in \BbbC\setminus\{0\}$,
where
$V_1$ is a complex polynomial of $1/z$,
$V_2$ is a polynomial of $z$, and $\lambda$ and $\mu$
are non-zero complex parameters. The operator
$Q(V)$ acts in the Hilbert space
$L^2(\BbbR^2;\BbbC^4)=\oplus^4L^2(\BbbR^2)$.
The main results we prove
include: (i) the (essential) self-adjointness of
$Q(V)$; (ii)
the pure discreteness of the spectrum of $Q(V)$ ;
(iii) if $V_1(z)=z^{-p}$ and
$4 \leq \deg V_2 \leq p+2$, then
$\ker Q(V)\not=\{0\}$ and
$\dim \ker Q(V)$ is independent of $(\lambda,\mu)$ and lower order terms
of $\partial V_2/\partial z$;
(iv) a trace formula for $\dim \ker Q(V)$.