Mathematical Analysis of a Generalized Chiral Quark Soliton Model

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.


Introduction
The chiral quark soliton model (CQSM) [5] is a model describing a low-energy effective theory of the quantum chromodynamics, which was developed in 1980's (for physical aspects of the model, see, e.g., [5] and references therein). The Hamiltonian of the CQSM is given by a Dirac type operator with iso-spin, which differs from the usual Dirac type operator in that the mass term is a matrix-valued function with an effect of an interaction between quarks and the pion field. It is an interesting object from the purely operator-theoretical point of view too. But there are few mathematically rigorous analyses for such Dirac type operators (e.g., [2], where the problem on essential self-adjointness of a Dirac operator with a variable mass term given by a scalar function is discussed).
In the previous paper [1] we studied some fundamental aspects of the CQSM in a mathematically rigorous way. In this paper we present a slightly general form of the CQSM, which we call a generalized CQSM, and report that results similar to those in [1] hold on this model too, at least, as far as some general aspects are concerned.

A Generalized CQSM
The Hilbert space of a Dirac particle with mass M > 0 and iso-spin 1/2 is taken to be L 2 (R 3 ; C 4 ) ⊗ C 2 . For a generalization, we replace the iso-spin space C 2 by an abitrary complex Hilbert space K. Thus the Hilbert space H in which we work in the present paper is given by H := L 2 (R 3 ; C 4 ) ⊗ K.
We denote by B(K) the Banach space of all bounded linear operators on K with domain K. Let T : R 3 → B(K); R 3 ∋ x = (x 1 , x 2 , x 3 ) → T (x) ∈ B(K) be a Borel measurable mapping A. Arai such that, for all x ∈ R 3 , T (x) is a non-zero bounded self-adjoint operator on K such that Example 1. In the original CQSM, K = C 2 and T (x) = τ · n(x), where n : R 3 → R 3 is a measurable vector field with |n(x)| = 1, a.e. (almost everywhere) x ∈ R 3 and τ = (τ 1 , τ 2 , τ 3 ) is the set of the Pauli matrices.
, the Sobolev space of order 1 consisting of C 4 ⊗ K-valued measurable functions on R 3 . In the context of the CQSM, the function F is called a profile function. In what follows we sometimes omit the symbol of tensor product ⊗ in writing equations down.
Example 2. Usually profile functions are assumed to be rotation invariant with boundary conditions The following are concrete examples [6]: We say that a self-adjoint operator A on H has chiral symmetry if γ 5 A ⊂ Aγ 5 . We note that, if F and T are differentiable on R 3 with sup , then the square of H takes the form This is a Schrödinger operator with an operator-valued potential.

Operator matrix representation
For more detailed analyses of the model, it is convenient to work with a suitable representation of the Dirac matrices. Here we take the following representation of α j and β (the Weyl representation): where σ 1 , σ 2 and σ 3 are the Pauli matrices. Let σ := (σ 1 , σ 2 , σ 3 ) and Then we have the following operator matrix representation for H:

Supersymmetric aspects
Let ξ : We define an operatorΓ on H by The following fact is easily proven: Lemma 1. The operatorΓ is self-adjoint and unitary, i.e., it is a grading operator on H: Γ * =Γ,Γ 2 = I.
Example 3. Consider the case K = C 2 . Let f, g : R 3 → R be a continuously differentiable function such that Then ξ 2 = I and (ξ, T ) satisfies (1).
To state spectral properties of H, we recall some definitions. For a self-adjoint operator S, we denote by σ(S) the spectrum of S. The point spectrum of S, i.e., the set of all the eigenvalues of S is denoted σ p (S). An isolated eigenvalue of S with finite multiplicity is called a discrete eigenvalue of S. We denote by σ d (S) the set of all the discrete eigenvalues of S. The set σ ess (S) := σ(S) \ σ d (S) is called the essential spectrum of S. Theorem 2. Under the same assumption as in Theorem 1, the following holds: for all λ ∈ σ # (H). provided that at least one of dim ker H + and dim ker H * + is finite. We conjecture that, for a class of F and T , index(H + ) = 0. 5 The essential spectrum and f initeness of the discrete spectrum of H 5.1 Structure of the spectrum of H Then Proof . We can rewrite H as Hence, by (3), we have lim

Bound for the number of discrete eigenvalues of H
Suppose that dim K < ∞ and (3) holds. Then, by Theorem 3, we can define the number of discrete eigenvalues of H counting multiplicities: where E H is the spectral measure of H.
To estimate an upper bound for N H , we introduce a hypothesis for F and T : (D j T (x)) 2 being a multiplication operator by a scalar function on R 3 .
Under this assumption, we can define

A. Arai
Theorem 4. Let dim K < ∞. Assume (3) and Hypothesis (A). Suppose that Then N H is finite with A basic idea for the proof of Theorem 4 is as follows. Let Then we have The following is a key lemma: Proof .
Hence the first inequality of (7) follows. The second inequality of (7) can be proven in the same manner as in the proof of [1, Lemma 3.3], which uses the min-max principle.
On the other hand, one has (the Birman-Schwinger bound [4, Theorem XIII.10]). In this way we can prove Theorem 4. As a direct consequence of Theorem 4, we have the following fact on the absence of discrete eigenvalues of H:  is finite. We say that A has a ground state if E 0 (A) ∈ σ p (A). In this case, a non-zero vector in ker(A − E 0 (A)) is called a ground state of A. Also we say that A has a discrete ground state if E 0 (A) ∈ σ d (A).  As for existence of discrete ground states of the Dirac operator H, we have the following theorem: Theorem 5. Let dim K < ∞. Assume Hypothesis (A) and (3). Suppose that E 0 (S + (F )) < 0 or E 0 (S − (F )) < 0. Then H has a discrete positive energy ground state or a discrete negative ground state.
Proof . We describe only an outline of proof. We have Hence, if L(F ) has a discrete eigenvalue, then H has a discrete eigenvalue in (−M, M ). By the min-max principle, we need to find a unit vector Ψ such that Ψ, L(F )Ψ < 0. Indeed, for each f ∈ D(∆), we can find vectors Ψ ± f ∈ D(L(F )), such that Ψ ± f , L(F )Ψ ± f = f, S ± f . By the present assumption, there exists a non-zero vector f 0 ∈ D(∆) such that f 0 , S + (F )f 0 < 0 or f 0 , S − (F )f 0 < 0. Thus the desired results follow.
To find a class of F such that E 0 (S + (F )) < 0 or E 0 (S − (F )) < 0, we proceed as follows. For a constant ε > 0 and a function f on R d , we define a function f ε on R d by The following are key Lemmas.
Proof . A basic idea for the proof of this lemma is to use the min-max principle (see [1, Lemma 4.3]). Then: (i) −∆ + V is self-adjoint and bounded below.
Proof . The facts (i) and (ii) follow from the standard theory of Schrödinger operators. Part (iii) follow from a simple application of Lemma 3 (for more details, see the proof of [1, Lemma 4.4]).
We now consider a one-parameter family of Dirac operators: Theorem 6. Let dim K < ∞. Assume Hypothesis (A) and (3). Suppose that D 3 cos F is not identically zero. Then there exists a constant ε 0 > 0 such that, for all ε ∈ (0, ε 0 ), H ε has a discrete positive energy ground state or a discrete negative ground state.
Proof . This follows from Theorem 5 and Lemma 4 (for more details, see the proof of [1, Theorem 4.5]).

Symmetry reduction of H
Let T 1 , T 2 and T 3 be bounded self-adjoint operators on K satisfying T 2 j = I, j = 1, 2, 3, Then it is easy to see that the anticommutation relations {T j , T k } = 2δ jk I, j, k = 1, 2, 3 hold. Since each T j is a unitary self-adjoint operator with T j = ±I, it follows that We set T = (T 1 , T 2 , T 3 ).
We take the vector field n : R 3 → R 3 to be of the form n(x) := sin Θ(r, z) cos(mθ), sin Θ(r, z) sin(mθ), cos Θ(r, z) , where Θ : (0, ∞) × R → R is continuous and m is a natural number. Let L 3 be the third component of the angular momentum acting in L 2 (R 3 ) and with Σ 3 := σ 3 ⊕ σ 3 . It is easy to see that K 3 is a self-adjoint operator acting in H.
Then, for all t ∈ R and ε > 0, the operator equality holds.
Proof . Similar to the proof of [1, Lemma 5.2]. We remark that, in the calculation of the following formulas are used: (T 1 cos mt − T 2 sin mt)e itmT 3 = T 1 , (T 1 sin mt + T 2 cos mt)e itmT 3 = T 2 .
Definition 2. We say that two self-adjoint operators on a Hilbert space strongly commute if their spectral measures commute.
Proof . By (10) and the functional calculus, we have for all s, t ∈ R e itK 3 e isHε e −itK 3 = e isHε , which is equivalent to e itK 3 e isHε = e isHε e itK 3 , s, t ∈ R. By a general theorem (e.g., [3, Theorem VIII.13]), this implies the strong commutativity of K 3 and H ε .
We denote by H ε (ℓ, s, t) by the reduced part of H ε to M ℓ,s,t and set H(ℓ, s, t) := H 1 (ℓ, s, t).
For s = ±1 and ℓ ∈ Z, we define and set The following theorem is concerned with the existence of discrete ground states of H(ℓ, s, t).
Then H(ℓ, s, t) has a discrete positive energy ground state or a discrete negative ground state.
Proof . Similar to the proof of Theorem 5 (for more details, see the proof of [1, Theorem 5.5]).
Theorem 8. Assume Hypothesis (B) and (9). Suppose that dim T t < ∞ and that D z cos G is not identically zero. Then, for each ℓ ∈ Z, there exists a constant ε ℓ > 0 such that, for all ε ∈ (0, ε ℓ ), each H ε (ℓ, s, t) has a discrete positive energy ground state or a discrete negative ground state.
Proof . Similar to the proof of Theorem 6 (for more details, see the proof of [1, Theorem 5.6]).
Theorem 8 immediately yields the following result: Corollary 2. Assume Hypothesis (B) and (9). Suppose that dim T t < ∞ and that D z cos G is not identically zero. Let ε ℓ be as in Theorem 8 and, for each n ∈ N and k > n (k, n ∈ Z), ν k,n := min n+1≤ℓ≤k ε ℓ . Then, for each ε ∈ (0, ν k,n ), H ε has at least (k − n) discrete eigenvalues counting multiplicities.

A unitary transformation
We go back again to the generalized CQSM defined in Section 2. It is easy to see that the operator Proof . Similar to the proof of [1, Proposition 6.1].
Using this proposition, we can prove the following theorem: Theorem 9. Let dim K < ∞. Assume Hypothesis (A) and that T (x) is independent of x.