Mathematical Analysis of a Generalized Chiral Quark Soliton Model

Abstract

Abstract. A generalized version of the so-called chiral quark soliton model (CQSM) in nuclear physics is introduced. The Hamiltonian of the generalized CQSM is given by a Dirac type operator with a mass term being an operator-valued function. Some mathematically rigorous results on the model are reported. The subjects included are: (i) supersymmetric structure; (ii) spectral properties; (iii) symmetry reduction; (iv) a unitarily equivalent model.

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.…”

“…In this section, in view of Theorem 26, we consider the number of eigenvalues of in the interval (− , ) and establish upper bounds on it. This aspect has been considered in the CQS model [3] as well as the GCQS model [4]. In this paper, we take another method, which is an extension of the method used in [13] where the number of eigenvalues of the three-dimensional Dirac operator + with a scalar potential :…”

“…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]. Miyao [5] proposed an abstract version of the CQS model and investigated a nonrelativistic limit of it; as an application of the abstract result to the CQS model, a Schrödinger operator with a binding potential was derived.…”

“…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].…”