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) = 4 L2(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 dim kerQ(V ) is independent of (λ,μ) and lower order terms of ∂V2/∂z; (iv) a trace formula for dim kerQ(V ).  2005 Elsevier Inc. All rights reserved