We consider a Dirac operator H acting in the Hilbert space L 2 ͑R 3 ; C 4 ͒ C 2 , which describes a Hamiltonian of the chiral quark soliton model in nuclear physics. The mass term of H is a matrix-valued function formed out of a function F : R 3 → R, called a profile function, and a vector field n on R 3 , which fixes pointwise a direction in the isospin space of the pion. We first show that, under suitable conditions, H may be regarded as a generator of a supersymmetry. In this case, the spectra of H are symmetric with respect to the origin of R. We then identify the essential spectrum of H under some condition for F. For a class of profile functions F, we derive an upper bound for the number of discrete eigenvalues of H. Under suitable conditions, we show the existence of a positive energy ground state or a negative energy ground state for a family of scaled deformations of H. A symmetry reduction of H is also discussed. Finally a unitary transformation of H is given, which may have a physical interpretation.

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) = λV 1 (z) + µV 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 ).