Dept. Math, Hokkaido Univ. EPrints Server

Spectral Properties of a Dirac Operator in the Chiral Quark Soliton Model

Preprint Series # 678
Arai, Asao and Hayashi, Kunimitsu and Sasaki, Itaru Spectral Properties of a Dirac Operator in the Chiral Quark Soliton Model. (2004);

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Abstract

We consider a Dirac operator $H$ acting in the Hilbert space $L^2(\BbbR^3;\BbbC^4)\otimes \BbbC^2$, 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 $F:\BbbR^3 \to \BbbR$, called a profile function, and a vector field ${\bf n}$ on $\BbbR^3$, which fixes pointwise a direction in the iso-spin 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 spetra of $H$ are symmetric with respect to the origin of $\BbbR$. 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.

Item Type:Preprint
Uncontrolled Keywords:Dirac operator, chiral quark soliton model, supersymmetry, spectrum, ground state
Subjects:81-xx QUANTUM THEORY
ID Code:645