Spectral properties of a Dirac operator in the chiral quark soliton model

Abstract

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.

A Class of -Dimensional Dirac Operators with a Variable Mass

“…From this point of view, it would be interesting to investigate if the Hamiltonian of the present model has supersymmetry. Indeed, it was shown that the Hamiltonian of the CQS model as well as that of the GCQS model has supersymmetry [3,4]. In this section we see that a supersymmetric structure similar to that of the CQS (GCQS) model exists in our model.…”

“…In the papers [3,4], it was shown that, under a suitable condition, the Hamiltonian of the CQS (GCQS) model is unitarily transformed to a Dirac operator which is simpler in a sense. In this section, we show that those structures are unified into a simple general structure.…”

“…Hence, the CQS model may be regarded as a model of a Dirac particle with a variable mass. We also note that (x) is not a scalar multiple of a constant matrix in general but may be a nontrivial matrix-valued function on R 3 . This is one of the interesting features of the Dirac operator CQS .…”

“…In a paper [3], Arai et al investigated spectral properties of the Dirac operator CQS . These results have been extended to the case of a generalized CQS (GCQS) model in [4].…”

“…As is pointed out in [3], under a condition for (x) ( = 1, 2, 3), the CQS model has supersymmetry; that is, the Dirac operator CQS may be a supercharge of a supersymmetric 2 ISRN Mathematical Analysis quantum mechanics (e.g., [6,Chapter 5]). This structure is carried over to the GCQS model [4].…”