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)=λV1(z)+μV2(z), z∈C∖{0}, where V1 is a complex polynomial of 1/z, V2 is a polynomial of z, and λ and μ are nonzero complex parameters. The operator Q(V) acts in the Hilbert space L2(R2;C4)=⊕4L2(R2). 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 V1(z)=z−p and 4⩽degV2⩽p+2, then kerQ(V)≠{0} and dimkerQ(V) is independent of (λ,μ) and lower order terms of ∂V2/∂z; (iv) a trace formula for dimkerQ(V).