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...
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Arai, A. 2019-02-09T17:06:06Z 2019-02-09T17:06:06Z 2006 Mathematical Analysis of a Generalized Chiral Quark Soliton Model / A. Arai // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 7 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81Q10; 81Q05; 81Q60; 47N50 https://nasplib.isofts.kiev.ua/handle/123456789/146440 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. The author would like to thank N. Sawado for kindly informing on typical examples of profile functions and comments. This work was supported by the Grant-In-Aid 17340032 for Scientific Research from the JSPS. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Mathematical Analysis of a Generalized Chiral Quark Soliton Model Article published earlier |
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Mathematical Analysis of a Generalized Chiral Quark Soliton Model |
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Mathematical Analysis of a Generalized Chiral Quark Soliton Model Arai, A. |
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Mathematical Analysis of a Generalized Chiral Quark Soliton Model |
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Mathematical Analysis of a Generalized Chiral Quark Soliton Model |
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Mathematical Analysis of a Generalized Chiral Quark Soliton Model |
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Mathematical Analysis of a Generalized Chiral Quark Soliton Model |
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mathematical analysis of a generalized chiral quark soliton model |
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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.
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Mathematical Analysis of a Generalized Chiral Quark Soliton Model / A. Arai // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 7 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 018, 12 pages
Mathematical Analysis of a Generalized Chiral Quark
Soliton Model
Asao ARAI
Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan
E-mail: arai@math.sci.hokudai.ac.jp
Received October 18, 2005, in final form January 25, 2006; Published online February 03, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper018/
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.
Key words: chiral quark soliton model; Dirac operator; supersymmetry; ground state; sym-
metry reduction
2000 Mathematics Subject Classification: 81Q10; 81Q05; 81Q60; 47N50
1 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 mathemat-
ically 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.
2 A Generalized CQSM
The Hilbert space of a Dirac particle with mass M > 0 and iso-spin 1/2 is taken to be
L2(R3; C4) ⊗ C2. For a generalization, we replace the iso-spin space C2 by an abitrary com-
plex Hilbert space K. Thus the Hilbert space H in which we work in the present paper is given
by
H := L2(R3; C4)⊗K.
We denote by B(K) the Banach space of all bounded linear operators on K with domain K.
Let T : R3 → B(K); R3 3 x = (x1, x2, x3) 7→ T (x) ∈ B(K) be a Borel measurable mapping
mailto:arai@math.sci.hokudai.ac.jp
http://www.emis.de/journals/SIGMA/2006/Paper018/
2 A. Arai
such that, for all x ∈ R3, T (x) is a non-zero bounded self-adjoint operator on K such that
‖T‖∞ := sup
x∈R3
‖T (x)‖ <∞, where ‖T (x)‖ denotes the operator norm of T (x).
Example 1. In the original CQSM, K = C2 and T (x) = τ · n(x), where n : R3 → R3 is a
measurable vector field with |n(x)| = 1, a.e. (almost everywhere) x ∈ R3 and τ = (τ1, τ2, τ3) is
the set of the Pauli matrices.
We denote by {α1, α2, α3, β} the Dirac matrices, i.e., 4× 4-Hermitian matrices satisfying
{αj , αk} = 2δjk, {αj , β} = 0, β2 = 1, j, k = 1, 2, 3,
where {A,B} := AB +BA.
Let F : R3 → R be measurable, a.e., finite and
UF := (cosF )⊗ I + i(sinF )γ5 ⊗ T,
where I denotes identity and γ5 := −iα1α2α3. We set α := (α1, α2, α3) and ∇ := (D1, D2, D3)
with Dj being the generalized partial differential operator in the variable xj . Then the one
particle Hamiltonian of a generalized CQSM is defined by
H := −iα · ∇ ⊗ I +M(β ⊗ I)UF
acting in the Hilbert space H. For a linear operator L, we denote its domain by D(L). It
is well-known that −iα · ∇ is self-adjoint with D(−iα · ∇) = ∩3
j=1D(Dj). Since the operator
M(β ⊗ I)UF is bounded and self-adjoint, it follows that H is self-adjoint with domain D(H) =
∩3
j=1D(Dj ⊗ I) = H1(R3; C4 ⊗ K), the Sobolev space of order 1 consisting of C4 ⊗ K-valued
measurable functions on R3. 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
F (0) = −π, lim
|x|→∞
F (x) = 0.
The following are concrete examples [6]:
(I) F (x) = −π exp(−|x|/R), R = 0.55× 10−15 m;
(II) F (x) = −π{a1 exp(−|x|/R1) + a2 exp(−|x|2/R2
2)},
a1 = 0.65, R1 = 0.58× 10−15 m, a2 = 0.35, R2 =
√
0.3× 10−15 m;
(III) F (x) = −π
(
1− |x|√
λ2 + |x|2
)
, λ =
√
0.4× 10−15 m.
We say that a self-adjoint operator A on H has chiral symmetry if γ5A ⊂ Aγ5.
Proposition 1. The Hamiltonian H has no chiral symmetry.
Proof. It is easy to check that, for all ψ ∈ D(H), γ5ψ ∈ D(H) and [γ5,H]ψ = 2Mγ5βUFψ.
Note that UF 6= 0. Hence, [γ5,H] 6= 0 on D(H). �
We note that, if F and T are differentiable on R3 with sup
x∈R3
|∂jF (x)|<∞ and sup
x∈R3
‖∂jT (x)‖<∞
(j = 1, 2, 3), then the square of H takes the form
H2 = (−∆ +M2)⊗ I − iMβα · (∇UF ) +M2 sin2 F ⊗ (T 2 − I).
This is a Schrödinger operator with an operator-valued potential.
Mathematical Analysis of a Generalized Chiral Quark Soliton Model 3
3 Operator matrix representation
For more detailed analyses of the model, it is convenient to work with a suitable representa-
tion of the Dirac matrices. Here we take the following representation of αj and β (the Weyl
representation):
αj =
(
σj 0
0 −σj
)
, β =
(
0 1
1 0
)
,
where σ1, σ2 and σ3 are the Pauli matrices. Let σ := (σ1, σ2, σ3) and
ΦF := (cosF )⊗ I + i(sinF )⊗ T.
Then we have the following operator matrix representation for H:
H =
(
−iσ · ∇ MΦ∗F
MΦF iσ · ∇
)
.
4 Supersymmetric aspects
Let ξ : R3 → B(K) be measurable such that, for all x ∈ R3, ξ(x) is a bounded self-adjoint
operator on K and ξ(x)2 = I, ∀ x ∈ R3. Let
Γ(x) := iγ5β ⊗ ξ(x), x ∈ R3.
We define an operator Γ̂ on H by
(Γ̂ψ)(x) := Γ(x)ψ(x), ψ ∈ H, a.e. x ∈ R3.
The following fact is easily proven:
Lemma 1. The operator Γ̂ is self-adjoint and unitary, i.e., it is a grading operator on H:
Γ̂∗ = Γ̂, Γ̂2 = I.
Theorem 1. Suppose that ξ is strongly differentiable with sup
x∈R3
‖∂jξ(x)‖ <∞ (j = 1, 2, 3) and
3∑
j=1
αj ⊗Djξ(x) = Mγ5β{ξ(x), T (x)} sinF (x). (1)
Then Γ̂D(H) ⊂ D(H) and {Γ̂,H}ψ = 0, ∀ ψ ∈ D(H).
Proof. For all ψ ∈ D0 := C∞0 (R3) ⊗alg (C4 ⊗ K) (⊗alg denotes algebraic tensor product), we
have
DjΓ̂ψ = iγ5β ⊗ (Djξ)ψ + iγ5β ⊗ ξ(Djψ). (2)
By a limiting argument using the fact that D0 is a core of Dj ⊗ I, we can show that, for
all ψ ∈ D(Dj), Γ̂ψ is in D(Dj) and (2) holds. Hence, for all ψ ∈ D(H), Γ̂ψ ∈ D(H) and (2)
holds. Thus we have for all ψ ∈ D(H) {Γ̂,H}ψ = C1ψ+C2ψ with C1 :=
3∑
j=1
{γ5β⊗ξ, αjDj} and
C2 := iM{γ5β⊗ξ, βUF }. Using the fact that {γ5, β} = 0 and [γ5, αj ] = 0 (j = 1, 2, 3), we obtain
C1ψ = −γ5β(
3∑
j=1
αjDjξ)ψ. Similarly direct computations yield (C2ψ)(x) = −M sinF (x) ⊗
{ξ(x), T (x)}ψ(x). Thus (1) implies {Γ̂,H}ψ = 0. �
4 A. Arai
Theorem 1 means that, under its assumption, H may be interpreted as a generator of a su-
persymmetry with respect to Γ̂.
Example 3. Consider the case K = C2. Let f, g : R3 → R be a continuously differentiable
function such that(
1 + C2
)
f(x)2 + g(x)2 = 1.
with a real constant C 6= 0 and n(x) := (f(x), Cf(x), g(x)). Then |n(x)| = 1, ∀ x ∈ R3. Let
ξ :=
C√
1 + C2
τ1 −
1√
1 + C2
τ2, T (x) := τ · n(x).
Then ξ2 = 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 σp(S). An isolated eigenvalue of S with finite multiplicity is called a discrete
eigenvalue of S. We denote by σd(S) the set of all the discrete eigenvalues of S. The set
σess(S) := σ(S) \ σd(S) is called the essential spectrum of S.
Theorem 2. Under the same assumption as in Theorem 1, the following holds:
(i) σ(H) is symmetric with respect to the origin of R, i.e., if λ ∈ σ(H), then −λ ∈ σ(H).
(ii) σ#(H) (# = p,d) is symmetric with respect to the origin of R with
dim ker(H − λ) = dim ker(H − (−λ))
for all λ ∈ σ#(H).
(iii) σess(H) is symmetric with respect to the origin of R.
Proof. Theorem 1 implies a unitary equivalence of H and −H (Γ̂HΓ̂−1 = −H). Thus the
desired results follow. �
Remark 1. Suppose that the assumption of Theorem 1 holds. In view of supersymmetry
breaking, it is interesting to compute dim kerH. This is related to the index problem: Let
H+ := ker(Γ̂− 1), H− := ker(Γ̂ + 1)
and
H± := H|H±.
Then H+ (resp. H−) is a densely defined closed linear operator from H+ (resp. H−) to H−
(resp. H+) with D(H+) = D(H) ∩H+ (resp. D(H−) = D(H) ∩D(H−)). Obviously
kerH = kerH+ ⊕ kerH−.
The analytical index of H+ is defined by
index(H+) := dim kerH+ − dim kerH∗
+,
provided that at least one of dim kerH+ and dim kerH∗
+ is finite. We conjecture that, for a
class of F and T , index(H+) = 0.
Mathematical Analysis of a Generalized Chiral Quark Soliton Model 5
5 The essential spectrum and finiteness
of the discrete spectrum of H
5.1 Structure of the spectrum of H
Theorem 3. Suppose that dimK <∞ and
lim
|x|→∞
F (x) = 0. (3)
Then
σess(H) = (−∞,−M ] ∪ [M,∞), (4)
σd(H) ⊂ (−M,M). (5)
Proof. We can rewrite H as H = H0 ⊗ I + V with H0 := −iα · ∇+Mβ and V := M(β ⊗ I)
(UF − I). We denote by χR (R > 0) the characteristic function of the set {x ∈ R3| |x| < R}. It
is well-known that, for all z ∈ C \R, (H0− z)−1χR is compact [7, Lemma 4.6]. Since K is finite
dimensional, it follows that (H0 ⊗ I − z)−1χR ⊗ I is compact. We have
‖V (x)‖ ≤M(| cosF (x)− 1|+ | sinF (x)|‖T‖∞) ≤M
(
|F (x)|2
2
+ |F (x)|‖T‖∞
)
.
Hence, by (3), we have lim
R→∞
sup
|x|>R
‖V (x)‖ = 0. Then, in the same way as in the method
described on [7, pp. 115–117], we can show that, for all z ∈ C \ R, (H − z)−1 − (H0 ⊗ I − z)−1
is compact. Hence, by a general theorem (e.g., [7, Theorem 4.5]), σess(H) = σess(H0⊗ I). Since
σess(H0) = (−∞,−M ]∪ [M,∞) ([7, Theorem 1.1]), we obtain (4). Relation (5) follows from (4)
and σd(H) = σ(H) \ σess(H). �
5.2 Bound for the number of discrete eigenvalues of H
Suppose that dimK < ∞ and (3) holds. Then, by Theorem 3, we can define the number of
discrete eigenvalues of H counting multiplicities:
NH := dim RanEH((−M,M)), (6)
where EH is the spectral measure of H.
To estimate an upper bound for NH , we introduce a hypothesis for F and T :
Hypothesis (A).
(i) T (x)2 = I, ∀ x ∈ R3 and T is strongly differentiable with
3∑
j=1
(DjT (x))2 being a multipli-
cation operator by a scalar function on R3.
(ii) F ∈ C1(R3).
(iii) sup
x∈R3
|DjF (x)| <∞, sup
x∈R3
‖DjT (x)‖ <∞ (j = 1, 2, 3).
Under this assumption, we can define
VF (x) :=
√√√√|∇F (x)|2 +
3∑
j=1
(DjT (x))2 sin2 F (x).
6 A. Arai
Theorem 4. Let dimK <∞. Assume (3) and Hypothesis (A). Suppose that
CF :=
∫
R6
VF (x)VF (y)
|x− y|2
dxdy <∞.
Then NH is finite with
NH ≤ (dimK)M2CF
4π2
.
A basic idea for the proof of Theorem 4 is as follows. Let
L(F ) := H2 −M2.
Then we have
L(F ) = −∆ +M
(
0 W ∗
F
WF 0
)
with WF := iσ · ∇ΦF . Note that
W ∗
FWF = WFW
∗
F = V 2
F .
Let
L0(F ) := −∆−MVF .
For a self-adjoint operator S, we introduce a set
N−(S) := the number of negative eigenvalues of S counting multiplicities.
The following is a key lemma:
Lemma 2.
NH ≤ N−(L(F )) ≤ N−(L0(F )). (7)
Proof. For each λ ∈ σd(H) ∩ (−M,M), we have ker(H − λ) ⊂ ker(L(F ) − Eλ) with Eλ =
λ2 − M2 < 0. 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
N−(L0(F )) ≤ (dimK)M2CF
4π2
(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:
Corollary 1. Assume (3) and Hypothesis (A). Let (dimK)M2CF < 4π2. Then σd(H) = ∅,
i.e., H has no discrete eigenvalues.
Mathematical Analysis of a Generalized Chiral Quark Soliton Model 7
6 Existence of discrete ground states
Let A be a self-adjoint operator on a Hilbert space and bounded from below. Then
E0(A) := inf σ(A)
is finite. We say that A has a ground state if E0(A) ∈ σp(A). In this case, a non-zero vector in
ker(A− E0(A)) is called a ground state of A. Also we say that A has a discrete ground state if
E0(A) ∈ σd(A).
Definition 1. Let
E+
0 (H) := inf [σ(H) ∩ [0,∞)] , E−0 (H) := sup [σ(H) ∩ (−∞, 0]] .
(i) If E+
0 (H) is an eigenvalue of H, then we say that H has a positive energy ground state
and we call a non-zero vector in ker(H − E+
0 (H)) a positive energy ground state of H.
(ii) If E−0 (H) is an eigenvalue of H, then we say that H has a negative energy ground state
and we call a non-zero vector in ker(H − E−0 (H)) a negative energy ground state of H.
(iii) If E+
0 (H) (resp. E−0 (H)) is a discrete eigenvalue of H, then we say that H has a discrete
positive (resp. negative) energy ground state.
Remark 2. If the spectrum of H is symmetric with respect to the origin of R as in Theorem 2,
then E+
0 (H) = −E−0 (H), and H has a positive energy ground state if and only if it has a negative
energy ground state.
Assume Hypothesis (A). Then the operators
S±(F ) := −∆±M(D3 cosF )
are self-adjoint with D(S±(F )) = D(∆) and bounded from below.
As for existence of discrete ground states of the Dirac operator H, we have the following
theorem:
Theorem 5. Let dimK < ∞. Assume Hypothesis (A) and (3). Suppose that E0(S+(F )) < 0
or E0(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
σess(L(F )) = [0,∞), σd(L(F )) ⊂ [−M2, 0).
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 f0 ∈ D(∆) such that 〈f0, S+(F )f0〉 < 0 or
〈f0, S−(F )f0〉 < 0. Thus the desired results follow. �
To find a class of F such that E0(S+(F )) < 0 or E0(S−(F )) < 0, we proceed as follows. For
a constant ε > 0 and a function f on Rd, we define a function fε on Rd by
fε(x) := f(εx), x ∈ Rd.
The following are key Lemmas.
8 A. Arai
Lemma 3. Let V : Rd → R be in L2
loc(Rd) and
Sε := −∆ + Vε.
Suppose that:
(i) For all ε > 0, Sε is self-adjoint, bounded below and σess(Sε) ⊂ [0,∞).
(ii) There exists a nonempty open set Ω ⊂ {x ∈ Rd|V (x) < 0}.
Then then there exists a constant ε0 > 0 such that, for all ε ∈ (0, ε0), Sε has a discrete ground
state.
Proof. A basic idea for the proof of this lemma is to use the min-max principle (see [1, Lem-
ma 4.3]). �
Lemma 4. V : Rd → R be continuous with V (x) → 0(|x| → ∞). Suppose that {x ∈ Rd|V (x) <
0} 6= ∅. Then:
(i) −∆ + V is self-adjoint and bounded below.
(ii) σess(−∆ + V ) = [0,∞).
(iii) Sε has a discrete ground state for all ε ∈ (0, ε0) with some ε0 > 0.
Proof. The facts (i) and (ii) follow from the standard theory of Schrödinger operators. Part (iii) fol-
low 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:
Hε := (−i)α · ∇+
1
ε
M(β ⊗ I)UFε .
Theorem 6. Let dimK < ∞. Assume Hypothesis (A) and (3). Suppose that D3 cosF is not
identically zero. Then there exists a constant ε0 > 0 such that, for all ε ∈ (0, ε0), Hε 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, Theo-
rem 4.5]). �
7 Symmetry reduction of H
Let T1, T2 and T3 be bounded self-adjoint operators on K satisfying
T 2
j = I, j = 1, 2, 3,
T1T2 = iT3, T2T3 = iT1, T3T1 = iT2.
Then it is easy to see that the anticommutation relations
{Tj , Tk} = 2δjkI, j, k = 1, 2, 3
hold. Since each Tj is a unitary self-adjoint operator with Tj 6= ±I, it follows that
σ(Tj) = σp(Tj) = {±1}.
We set T = (T1, T2, T3).
Mathematical Analysis of a Generalized Chiral Quark Soliton Model 9
In this section we consider the case where T (x) is of the following form:
T (x) = n(x) · T ,
where n(x) is the vector field in Example 1. We use the cylindrical coordinates for points
x = (x1, x2, x3) ∈ R3:
x1 = r cos θ, x2 = r sin θ, x3 = z,
where θ ∈ [0, 2π), r > 0. We assume the following:
Hypothesis (B). There exists a continuously differentiable function G : (0,∞)× R → R such
that
(i) F (x) = G(r, z), x ∈ R3 \ {0};
(ii) lim
r+|z|→∞
G(r, z) = 0;
(iii) sup
r>0,z∈R
(|∂G(r, z)/∂r|+ |∂G(r, z)/∂z|) <∞.
We take the vector field n : R3 → R3 to be of the form
n(x) :=
(
sinΘ(r, z) cos(mθ), sinΘ(r, z) sin(mθ), cos Θ(r, z)
)
,
where Θ : (0,∞)× R → R is continuous and m is a natural number.
Let L3 be the third component of the angular momentum acting in L2(R3) and
K3 := L3 ⊗ I +
1
2
Σ3 ⊗ I +
m
2
I ⊗ T3 (8)
with Σ3 := σ3 ⊕ σ3. It is easy to see that K3 is a self-adjoint operator acting in H.
Lemma 5. Assume that
Θ(εr, εz) = Θ(r, z), (r, z) ∈ (0,∞)× R, ε > 0. (9)
Then, for all t ∈ R and ε > 0, the operator equality
eitK3Hεe
−itK3 = Hε (10)
holds.
Proof. Similar to the proof of [1, Lemma 5.2]. We remark that, in the calculation of
eitK3T (x)e−itK3 =
3∑
j=1
eitL3nj(x)e−itL3eitmT3Tje
−itmT3 ,
the following formulas are used:
(T1 cosmt− T2 sinmt)eitmT3 = T1, (T1 sinmt+ T2 cosmt)eitmT3 = T2. �
Definition 2. We say that two self-adjoint operators on a Hilbert space strongly commute if
their spectral measures commute.
Lemma 6. Assume (9). Then, for all ε > 0, Hε and K3 strongly commute.
Proof. By (10) and the functional calculus, we have for all s, t ∈ R eitK3eisHεe−itK3 = eisHε ,
which is equivalent to eitK3eisHε = eisHεeitK3 , s, t ∈ R. By a general theorem (e.g., [3, Theo-
rem VIII.13]), this implies the strong commutativity of K3 and Hε. �
10 A. Arai
Lemma 6 implies that Hε is reduced by eigenspaces of K3. Note that
σ(K3) = σp(K3) =
{
`+
s
2
+
mt
2
∣∣∣∣ ` ∈ Z, s = ±1, t = ±1
}
.
The eigenspace of K3 with eigenvalue `+ (s/2) + (mt/2) is given by
M`,s,t := M` ⊗ Cs ⊗ Tt
with Cs := ker(Σ3 − s) and Tt := ker(T3 − t). Then H has the orthogonal decomposition
H = ⊕`∈Z,s,t∈{±1}M`,s,t.
Thus we have:
Lemma 7. Assume (9). Then, for all ε > 0, Hε is reduced by each M`,s,t.
We denote by Hε(`, s, t) by the reduced part of Hε to M`,s,t and set
H(`, s, t) := H1(`, s, t).
For s = ±1 and ` ∈ Z, we define
Ls(G, `) := − ∂2
∂r2
− 1
r
∂
∂r
+
`2
r2
+
∂2
∂z2
+ sMDz cosG
acting in L2((0,∞)× R, rdrdz) with domain
D(Ls(G, `)) := C∞0 ((0,∞)× R)
and set
E0(Ls(G, `)) := inf
f∈C∞0 ((0,∞)×R),‖f‖L2((0,∞)×R,rdrdz)=1
〈f, Ls(G, `)f〉.
The following theorem is concerned with the existence of discrete ground states of H(`, s, t).
Theorem 7. Assume Hypothesis (B) and (9). Fix an ` ∈ Z arbitrarily, s = ±1 and t = ±1.
Suppose that dim Tt <∞ and
E0(Ls(G, `)) < 0.
Then H(`, 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 Tt < ∞ and that Dz cosG is
not identically zero. Then, for each ` ∈ Z, there exists a constant ε` > 0 such that, for all
ε ∈ (0, ε`), each Hε(`, 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 Tt < ∞ and that Dz cosG is
not identically zero. Let ε` be as in Theorem 8 and, for each n ∈ N and k > n (k, n ∈ Z),
νk,n := min
n+1≤`≤k
ε`. Then, for each ε ∈ (0, νk,n), Hε has at least (k − n) discrete eigenvalues
counting multiplicities.
Proof. Note that σp(Hε) = ∪`∈Z,s,t=±1σp(Hε(`, s, t)). �
Mathematical Analysis of a Generalized Chiral Quark Soliton Model 11
8 A unitary transformation
We go back again to the generalized CQSM defined in Section 2. It is easy to see that the
operator
XF :=
1 + γ5
2
exp
(
iF ⊗ T
2
)
+
1− γ5
2
exp
(
−iF ⊗ T
2
)
is unitary. Under Hypothesis (A), we can define the following operator-valued functions:
Bj(x) :=
1
2
Dj [F (x)T (x)], x ∈ R3, j = 1, 2, 3.
We set
B := (B1, B2, B3)
and introduce
H(B) := (−i)α · ∇+Mβ − σ ·B
acting inH. Since σ·B is a bounded self-adjoint operator, H(B) is self-adjoint withD(H(B)) =
∩3
j=1D(Dj ⊗ I).
Proposition 2. Assume Hypothesis (A) and that T (x) is independent of x. Then
XFHX
−1
F = H(B).
Proof. Similar to the proof of [1, Proposition 6.1]. �
Using this proposition, we can prove the following theorem:
Theorem 9. Let dimK < ∞. Assume Hypothesis (A) and that T (x) is independent of x.
Suppose that
lim
|x|→∞
|∇F (x)| = 0.
Then
σess(H) = (−∞,−M ] ∪ [M,∞). (11)
Proof. By Proposition 2, we have σess(H) = σess(H(B)). By the present assumption, Bj(x) =
DjF (x)T (0)/2. Hence
sup
|x|>R
‖σ ·B(x)‖ ≤
3∑
j=1
(‖T (0)‖/2) sup
|x|>R
|DjF (x)| → 0 (R→∞).
Therefore, as in the proof of Theorem 3, we conclude that σess(H(B)) = (−∞,−M ] ∪ [M,∞].
Thus (11) follows. �
12 A. Arai
Acknowledgements
The author would like to thank N. Sawado for kindly informing on typical examples of profile
functions and comments. This work was supported by the Grant-In-Aid 17340032 for Scientific
Research from the JSPS.
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J. Math. Phys., 2005, V.46, N 5, 052360, 12 pages.
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J. Math. Phys., 2001, V.42, 2667–2676.
[3] Reed M., Simon B., Methods of modern mathematical physics I: Functional analysis, New York, Academic
Press, 1972.
[4] Reed M., Simon B., Methods of modern mathematical physics IV: Analysis of operators, New York, Aca-
demic Press, 1978.
[5] Sawado N., The SU(3) dibaryons in the chiral quark soliton model, Phys. Lett. B, 2002, V.524, 289–296.
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1 Introduction
2 A Generalized CQSM
3 Operator matrix representation
4 Supersymmetric aspects
5 The essential spectrum and finiteness of the discrete spectrum of H
5.1 Structure of the spectrum of H
5.2 Bound for the number of discrete eigenvalues of H
6 Existence of discrete ground states
7 Symmetry reduction of H
8 A unitary transformation
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