Spectral analysis of a Dirac operator with a meromorphic potential

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 ) = λ V 1 ( z ) + μ V 2 ( z ) , z ∈ C ∖ { 0 } , where V 1 is a complex polynomial of 1 / z , V 2 is a polynomial of z, and λ and μ are nonzero complex parameters. The operator Q ( V ) acts in the Hilbert space L 2 ( R 2 ; C 4 ) = ⊕ 4 L 2 ( R 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 ⩽ deg V 2 ⩽ p + 2 , then ker Q ( V ) ≠ { 0 } and dim ker Q ( V ) is independent of ( λ , μ ) and lower order terms of ∂ V 2 / ∂ z ; (iv) a trace formula for dim ker Q ( V ) .