Spectral analysis of a Dirac operator with a meromorphic potential

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 $\dim \ker Q(V)$