%0 Journal Article
%T A Class of -Dimensional Dirac Operators with a Variable Mass
%A Asao Arai
%A Dayantsolmon Dagva
%J ISRN Mathematical Analysis
%D 2013
%R 10.1155/2013/913413
%X A class of d-dimensional Dirac operators with a variable mass is introduced ( ), which includes, as a special case, the 3-dimensional Dirac operator describing the chiral quark soliton model in nuclear physics, and some aspects of it are investigated. 1. Introduction In the chiral quark soliton (CQS) model in nuclear physics (see, e.g., [1] and references therein), a Dirac operator of the following form appears (we use the physical unit system where the speed of light and , the Planck constant divided by , are equal to ): acting in the tensor product Hilbert space of (the Hilbert space of -valued square integrable functions on ) and . Here is the imaginary unit, , , and are Hermitian matrices obeying the anticommutation relations ( , is the Kronecker delta and denotes the unit matrix), ( ) is the generalized partial differential operator in the space variable ( ), denotes the mass of a quark, is a Hermitian matrix satisfying is a function called a profile function, , and are the Pauli matrices, and ( ) is a Borel measurable function on such that for a.e. (almost everywhere) . Comparing with the usual free Dirac operator with mass, one notes that the term corresponds to a mass, although it may depend on the space variable in general. Hence, the CQS model may be regarded as a model of a Dirac particle with a variable mass. We also note that is not a scalar multiple of a constant matrix in general but may be a nontrivial matrix-valued function on . This is one of the interesting features of the Dirac operator . From a general point of view, is a special case of the mass deformation of the form with being a mapping from to the set of linear operators on . To our best knowledge, mathematically rigorous analysis on Dirac operators with such a mass deformation seems to be few, although a Dirac operator with a mass given by a scalar function has been studied (e.g., [2]). In a paper [3], Arai et al. investigated spectral properties of the Dirac operator . 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 Schr£żdinger operator with a binding potential was derived. As is pointed out in [3], under a condition for ( ), the CQS model has supersymmetry; that is, the Dirac operator may be a supercharge of a supersymmetric quantum mechanics (e.g., [6, Chapter ]). This structure is carried over to the GCQS model [4]. In this paper, for each natural number , we propose a
%U http://www.hindawi.com/journals/isrn.mathematical.analysis/2013/913413/