A Class of 𝑑-Dimensional Dirac Operators with a Variable Mass

Abstract

A class of d-dimensional Dirac operators with a variable mass is introduced (𝑑≥2), 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 light𝑐and ħ, the Planck constant divided by 2𝜋, are equal to 1): 𝐻_CQS∶=−𝑖³_𝑗=1𝛼_𝑗𝐷_𝑗⊗1₂+𝑚𝛽⊗1₂𝑒^{𝑖∑3𝑗=1𝐹𝛾5⊗𝜏𝑗𝑛𝑗}, (1) acting in the tensor product Hilbert space 𝐿²(ℝ³;ℂ⁴)⊗ℂ² of 𝐿²(ℝ³;ℂ⁴) (the Hilbert space of ℂ⁴-valued square integrable functions on ℝ³) and ℂ². Here𝑖is the imaginary unit, 𝛼₁, 𝛼₂, and 𝛼₃ are 4×4 Hermitian matrices obeying the anticommutation relations 𝛼_𝑗,𝛼_𝑘=2𝛿_𝑗𝑘1₄𝑗,𝑘=1,2,3 (2) ({𝐴,𝐵}∶=𝐴𝐵+𝐵𝐴, 𝛿_𝑗𝑘 is the Kronecker delta and 1_𝑛 denotes the𝑛×𝑛unit matrix), 𝐷_𝑗 (𝑗=1,2,3) is the generalized partial differential operator in the space variable 𝑥_𝑗 (𝐱=(𝑥₁,𝑥₂,𝑥₃)∈ℝ³), 𝑚>0 denotes the mass of a quark, 𝛽 is a 4×4 Hermitian matrix satisfying 𝛼_𝑗,𝛽=0𝑗=1,2,3,𝛽²=1₄,𝛾₅∶=−𝑖𝛼₁𝛼₂𝛼₃, (3)𝐹∶ℝ³→ℝ is a function called a profile function, 𝜏₁,𝜏₂, and 𝜏₃ are the Pauli matrices, and 𝑛_𝑗∶ℝ³→ℝ (𝑗=1,2,3) is a Borel measurable function on ℝ³ such that ∑³_𝑗=1𝑛_𝑗(𝐱)²=1 for a.e. (almost everywhere) 𝐱∈ℝ³.

Comparing𝐻_CQSwith the usual free Dirac operator with mass, one notes that the term 𝑚(𝐱)∶=𝑚𝑒^{𝑖∑3𝑗=1𝐹(𝐱)𝛾5⊗𝜏𝑗𝑛𝑗(𝐱)} (4) corresponds to a mass, although it may depend on the space variable 𝐱 in general. Hence, the CQS model may be regarded as a model of a Dirac particle with a variable mass. We also note that 𝑚(𝐱) is not a scalar multiple of a constant matrix in general but may be a nontrivial matrix-valued function on ℝ³. This is one of the interesting features of the Dirac operator 𝐻_CQS. From a general point of view, 𝑚(𝐱) is a special case of the mass deformation of the form 𝑚_𝐿(𝐱)∶=𝑚𝐿(𝐱) with 𝐿 being a mapping from ℝ³ to the set of linear operators on ℂ⁴⊗ℂ². 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 𝐻_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 𝑛_𝑗(𝐱) (𝑗=1,2,3), the CQS model has supersymmetry; that is, the Dirac operator 𝐻_CQS may be a supercharge of a supersymmetric quantum mechanics (e.g., [6, Chapter5]). This structure is carried over to the GCQS model [4].

In this paper, for each natural number 𝑑≥2, we propose a𝑑-dimensional version of the GCQS model and analyze some mathematical aspects of it including supersymmetric ones.

The present paper is organized as follows. We first recall some basic facts in operator theory in Section 2. In Section 3 we introduce a Dirac operator 𝐻 which may be the Hamiltonian of a𝑑-dimensional version of the GCQS model, as mentioned previously. A simple condition for 𝐻 to be self-adjoint is given. In Section 4 we discuss supersymmetric aspects of 𝐻. We give a condition for 𝐻 to be a supercharge of a supersymmetric quantum mechanical model. In that case, ker 𝐻, the kernel of 𝐻, describes the supersymmetric states. Hence, it is interesting and important to analyze ker 𝐻. In Section 5, we prove that, under a condition, ker 𝐻is trivial: ker 𝐻={0}. In the case where 𝐻 is a supercharge, this means that there is no supersymmetric states; namely, the supersymmetry is spontaneously broken. Section 6 is concerned with a unitary equivalence of 𝐻 to a gauge theoretic Dirac operator. This may be physically interesting. Using this structure, we find another condition for the kernel of 𝐻 to be trivial. In Section 7, we identify the essential spectrum of𝐻under a suitable condition. In the last section, we discuss the number of eigenvalues of𝐻in the interval (−𝑚,𝑚) with 𝑚>0 being a constant.

2. Preliminaries

Let 𝒳 be a complex Hilbert space with inner product ⟨⋅,⋅⟩_𝒳 (linear in the second variable) and norm ‖⋅‖_𝒳 (we sometimes omit the subscript 𝒳 if there is no danger of confusion). For a linear operator 𝐴 on 𝒳, we denote its domain by 𝐷(𝐴). If 𝐴 is densely defined, its adjoint is denoted by 𝐴^∗. For linear operators 𝐴 and 𝐵 on 𝒳, 𝐴⊂𝐵 means that 𝐵 is an extension of 𝐴, that is, 𝐷(𝐴)⊂𝐷(𝐵) and 𝐴𝜓=𝐵𝜓, for all 𝜓∈𝐷(𝐴).

We denote by 𝔅(𝒳) the set of everywhere defined bounded linear operators on 𝒳. For 𝑇∈𝔅(𝒳), we denote the operator norm of 𝑇 by ‖𝑇‖.

Definition 1. Let 𝐴 and 𝐵 be self-adjoint operators on 𝒳. (i)𝐴 and 𝐵 are said to strongly commute if their spectral measures commute. (ii)𝐴 and 𝐵 are said to strongly anticommute [7, 8] if, for all 𝑡∈ℝ, 𝑒^𝑖𝑡𝐵𝐴⊂𝐴𝑒^{−𝑖𝑡𝐵} (it is shown that this definition is in fact symmetric in 𝐴 and 𝐵).

The next lemma summarizes some basic facts on strongly commuting (resp., anticommuting) self-adjoint operators.

Lemma 2. Let 𝐴 and 𝐵 be self-adjoint operators on 𝒳. (i)𝐴 and 𝐵 strongly commute if and only if, for all 𝑡,𝑠∈ℝ, 𝑒^𝑖𝑡𝐵𝑒^𝑖𝑠𝐴=𝑒^𝑖𝑠𝐴𝑒^𝑖𝑡𝐵. (ii)𝐴 and 𝐵 strongly commute if and only if, for all 𝑡∈ℝ, 𝑒^𝑖𝑡𝐵𝐴=𝐴𝑒^𝑖𝑡𝐵. (iii) Let 𝐴 be bounded. Then 𝐴 and 𝐵 strongly commute if and only if 𝐴𝐵⊂𝐵𝐴. (iv) Let 𝐴 be bounded. Then 𝐴 and 𝐵 strongly anticommute if and only if 𝐴𝐵⊂−𝐵𝐴.

Proof. Part (i) is well known (e.g., [9, Theorem VIII.13]). Using (i), functional calculus, and strong differential calculus, one can easily prove (ii) and (iii). A proof of (iv) is similar to the proof of (iii).

3. Description of the Model

Let 𝑑≥2 be a natural number, and 𝑁_𝑑2∶=^𝑑/2for𝑑even,2^(𝑑+1)/2for𝑑odd. (5)

Let 𝒦 be a separable complex Hilbert space, and ℋ∶=𝐿²ℝ^𝑑;ℂ^𝑁𝑑⊗𝒦≅𝐿²ℝ^𝑑;ℂ^𝑁𝑑≅⊗𝒦_⊕ℝ𝑑ℂ^𝑁𝑑⊗𝒦𝑑𝑥, (6) where each ≅ means the natural Hilbert space isomorphism and ∫^⊕_ℝ𝑑ℂ^𝑁𝑑⊗𝒦𝑑𝑥 denotes the constant fibre direct integral with fiber ℂ^𝑁𝑑⊗𝒦 (e.g., [10, sectionXIII.16]).

We denote by𝐷_𝑗the generalized partial differential operator in the variable 𝑥_𝑗 (𝑥=(𝑥₁,…,𝑥_𝑑)∈ℝ^𝑑), acting in 𝐿²(ℝ^𝑑). The𝑑-dimensional generalized Laplacian Δ∶=^𝑑_𝑗=1𝐷²_𝑗 (7) on 𝐿²(ℝ^𝑑) is a nonpositive self-adjoint operator. Each linear operator 𝐴 on 𝐿²(ℝ^𝑑) is extended as the direct sum ⊕^𝑁𝑑𝐴 on 𝐿²(ℝ^𝑑;ℂ^𝑁𝑑)=⊕^𝑁𝑑𝐿²(ℝ^𝑑). For notational simplicity, we denote it by 𝐴 again.

Every densely defined closable linear operator 𝑇 on 𝐿²(ℝ^𝑑,ℂ^𝑁𝑑) (resp., 𝒦) has a tensor product extension 𝑇⊗𝐼 (resp., 𝐼⊗𝑇) to ℋ (𝐼 denotes identity). But we write it𝑇simply if there is no danger of confusion.

We denote by ℱ_s.a. the set of mappings Φ(⋅) from ℝ^𝑑 to the set of self-adjoint operators on ℂ^𝑁𝑑⊗𝒦 such that the mapping: ℝ^𝑑∋𝑥↦(Φ(𝑥)+𝑖)⁻¹ is measurable. By a general theorem (e.g., [10, Theorem XIII.85(i)]), for each Φ(⋅)∈ℱ_s.a., the direct integral Φ∶=^⊕_ℝ𝑑Φ(𝑥)𝑑𝑥 (8) is self-adjoint.

Let {𝛼_𝑗}^𝑑+1_𝑗=1 be 𝑁_𝑑×𝑁_𝑑 Hermitian matrices satisfying 𝛼_𝑗,𝛼_𝑘=2𝛿_𝑗𝑘1_𝑁𝑑,𝑗,𝑘=1,…,𝑑+1. (9) Then the free massless Dirac operator on 𝐿²(ℝ^𝑑;ℂ^𝑁𝑑) is defined by 𝐻₀∶=−𝑖^𝑑_𝑗=1𝛼_𝑗𝐷_𝑗. (10) The operator 𝐻₀ is self-adjoint with 𝐷(𝐻₀)∶=∩^𝑑_𝑗=1𝐷(𝐷_𝑗) and 𝐻²₀=−Δ. (11)

To introduce a mass operator, let 𝑀(⋅)∈ℱ_s.a. such that, for a.e. 𝑥∈ℝ^𝑑, 𝑀(𝑥) is a bounded operator on ℂ^𝑁𝑑⊗𝒦, and set 𝑀∶=^⊕_ℝ𝑑𝑀(𝑥)𝑑𝑥. (12) We use this self-adjoint operator as an extended mass (variable in the space ℝ^𝑑) of the quantum particle of our model (a Dirac particle). Note that 𝑀 is not necessarily bounded.

The Hamiltonian 𝐻 of our model, a 𝑑-dimensional version of the GCQS model, is defined as follows: 𝐻∶=𝐻₀+𝑈, (13) with 𝑈∶=𝛼_𝑑+1𝑒^𝑖Φ𝑀. (14) As remarked previously, the mass operator 𝑀 in 𝑈 can be variable spatially. This is a point different from the GCQS model.

In this work, we do not intend to discuss essential self-adjointness of𝐻in full generality. In the present paper, we assume the following. (A.1)𝛼_𝑑+1Φ⊂−Φ𝛼_𝑑+1.(A.2)𝛼_𝑑+1𝑀⊂𝑀𝛼_𝑑+1. (A.3)Φ and 𝑀 strongly commute. (A.4) The operator 𝑀 is (−Δ)^1/2-bounded: 𝐷((−Δ)^1/2)⊂𝐷(𝑀) and 𝑀𝜓²≤𝑎²(−Δ)^1/2𝜓²+𝑏²𝜓²,∀𝜓∈𝐷(−Δ)^1/2 (15) with constants 0≤𝑎<1 and 𝑏≥0.

Remark 3. In the abstract CQS model [5], the strong commutativity of 𝑀 and 𝐻₀ as well as the boundedness and the strict positivity of 𝑀 is assumed. But, in our model, they are not assumed.

Lemma 4. (i) Condition (𝐴.1) holds if and only if 𝛼_𝑑+1 and Φ strongly anticommute. (ii) Condition (𝐴.1) is equivalent to the operator equality 𝛼_𝑑+1Φ=−Φ𝛼_𝑑+1. (iii) Condition (𝐴.2) holds if and only if 𝛼_𝑑+1 and 𝑀 strongly commute. (iv) Condition (𝐴.2) is equivalent to the operator equality 𝛼_𝑑+1𝑀=𝑀𝛼_𝑑+1.

Proof. (i) This follows from Lemma 2(iv).(ii) Assume (A.1). Let 𝜓∈𝐷(Φ𝛼_𝑑+1). Then 𝜂∶=𝛼_𝑑+1𝜓∈𝐷(Φ). Hence, by (A.1), 𝛼_𝑑+1𝜂∈𝐷(Φ). But, since 𝛼²_𝑑+1=𝐼, we have 𝛼_𝑑+1𝜂=𝜓. Hence 𝜓∈𝐷(Φ). Thereore 𝐷(Φ𝛼_𝑑+1)⊂𝐷(𝛼_𝑑+1Φ). Thus 𝐷(Φ𝛼_𝑑+1)=𝐷(𝛼_𝑑+1Φ). Hence the desired operator equality holds.(iii) This follows from Lemma 2(iii).(iv) Simillar to the proof of part (ii).

We define 𝐻_𝑀∶=𝐻₀+𝛼_𝑑+1𝑀. (16) If 𝑀 is a constant operator 𝑚>0, then 𝐻_𝑚 represents the free Dirac operator with a constant mass𝑚. It is well known (e.g., [6, Theorem1.1]) that 𝐻_𝑚 is self-adjoint with 𝐷(𝐻_𝑚)=𝐷(𝐻₀)=∩^𝑑_𝑗=1𝐷(𝐷_𝑗) and bijective with ‖𝐻_𝑚⁻¹‖=1/𝑚.

Lemma 5. Assume (A.4). Let 𝑚>0 be a constant. Then 𝑀𝐻_𝑚⁻¹ is bounded with 𝑀𝐻_𝑚⁻¹𝑏≤max𝑎,𝑚. (17)

Proof. It is well known or easy to see that, for all 𝜓∈𝐷(𝐻_𝑚)=𝐷(𝐻₀), 𝐻_𝑚𝜓²=(−Δ)^1/2𝜓²+𝑚²𝜓². (18) Hence, by(A.4), we have 𝑀𝜓²≤𝑎²𝐻_𝑚𝜓+𝑏²−𝑎²𝑚²𝜓. (19) This implies the following: (i) if 𝑎𝑚≤𝑏, then ‖𝑀𝐻_𝑚⁻¹‖²≤𝑎²+(𝑏²−𝑎²𝑚²)/𝑚²=𝑏²/𝑚²; (ii) if𝑎𝑚≥𝑏, then ‖𝑀𝐻_𝑚⁻¹‖≤𝑎. Thus (17) follows.

Lemma 6. Assume (𝐴.1)--(𝐴.4). Then (i)𝑈 is self-adjoint with 𝐷(𝑈)=𝐷(𝑀);(ii)𝐻 is self-adjoint with 𝐷(𝐻)=𝐷(𝐻₀) and the subspace 𝒟₀∶=𝐶^∞₀ℝ^𝑑;ℂ^𝑁𝑑⊗𝒦, (20)(⊗means algebraic tensor product) is a core of 𝐻.

Proof. (i) Since 𝐷(𝑈)=𝐷(𝑀) and 𝐷(𝑀) is dense, 𝑈 is densely defined. Since 𝛼_𝑑+1𝑒^𝑖Φ is bounded, it follows that 𝑈^∗=𝑀𝑒^−𝑖Φ𝛼_𝑑+1. By (A.1) and Lemma 2(iv), we have 𝑒^−𝑖Φ𝛼_𝑑+1=𝛼_𝑑+1𝑒^𝑖Φ. (21) By (A.2), (A.3), Lemma 2(ii), and Lemma 4(iv), we have 𝑈^∗=𝑀𝛼_𝑑+1𝑒^𝑖Φ=𝛼_𝑑+1𝑀𝑒^𝑖Φ=𝛼_𝑑+1𝑒^𝑖Φ𝑀=𝑈. (22) Hence𝑈is self-adjoint.
(ii) By (A.4), we have for all 𝜓∈𝐷((−Δ)^1/2)𝑈𝜓=𝑀𝜓≤𝑎(−Δ)^1/2𝜓+𝑏𝜓. (23) Note that (−Δ)^1/2𝜓=𝐻₀𝜓. (24) Hence ‖𝑈𝜓‖≤𝑎‖𝐻₀𝜓‖+𝑏‖𝜓‖. Here0≤𝑎<1. Thus, by the Kato-Rellich theorem (e.g., [11, TheoremX.12]),𝐻is self-adjoint with𝐷(𝐻)=𝐷(𝐻₀)and every core of 𝐻₀ is a core of 𝐻. It is well known that the subspace 𝐶^∞₀(ℝ^𝑑;ℂ^𝑁𝑑) is a core of 𝐻₀ as a linear operator on 𝐿²(ℝ^𝑑;ℂ^𝑁𝑑). Hence the subspace 𝒟₀ defined by (20) is a core of 𝐻₀ as a linear operator on ℋ. Thus it is a core of 𝐻 too.

Remark 7. One of the other sufficient conditions for𝐻to be essentially self-adjoint is as follows: assume (A.1)--(A.3)   and esssup_|𝑥|<𝑅‖𝑀(𝑥)‖<∞  for all 𝑅>0. Then 𝐻is essentially self-adjoint on 𝒟₀. The proof is similar to that of    [6, Theorem4.3].

4. Supersymmetric Aspects

As is well known, the standard free Dirac operator −𝑖∑³_𝑗=1𝛼_𝑗𝐷_𝑗+𝑚𝛽 on 𝐿²(ℝ³;ℂ⁴) with constant mass 𝑚≥0 and its suitably perturbed ones have supersymmetry; that is, they are, respectively, a supercharge with the grading operator 𝑖𝛽𝛾₅ [6, Section 5.5]. From this point of view, it would be interesting to investigate if the Hamiltonian 𝐻 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 this section, we consider only the case where 𝑑 is odd. The matrix 𝛾₅^(𝑑)∶=𝑖^{𝑑(𝑑−1)/2}𝛼₁⋯𝛼_𝑑 (25) is self-adjoint with 𝛾₅^(𝑑)2=1_𝑁𝑑. (26) Since𝑑is odd, we have 𝛼_𝑗𝛾₅^(𝑑)=𝛾₅^(𝑑)𝛼_𝑗𝑗=1,…,𝑑,𝛼_𝑑+1,𝛾₅^(𝑑)=0. (27)

Let 𝜉∶ℝ^𝑑→𝔅(𝒦) be Borel measurable such that, for a.e. 𝑥∈ℝ^𝑑, 𝜉(𝑥) is self-adjoint with 𝜉(𝑥)²=𝐼. (28) Then 𝜉(𝑥)=1,a.e.𝑥∈ℝ^𝑑. (29)

We defineΓ(⋅)∶ℝ^𝑑→𝔅(ℂ^𝑁𝑑⊗𝒦)by Γ(𝑥)∶=𝑖𝛾₅^(𝑑)𝛼_𝑑+1⊗𝜉(𝑥),a.e.𝑥∈ℝ^𝑑. (30) Then Γ∶=^⊕_ℝ𝑑Γ(𝑥)𝑑𝑥 (31) is self-adjoint with Γ²=𝐼. (32) Hence Γ is a grading operator on ℋ. The following proposition shows that, under some additional condition for 𝜉(𝑥), 𝐻 has supersymmetry with respect to Γ.

Proposition 8. Let 𝑑 be odd. Assume (𝐴.1)--(𝐴.4). Suppose that 𝜉 is strongly differentiable on ℝ^𝑑 with 𝐶_𝑗∶=sup_{𝑥∈ℝ𝑑}𝐷_𝑗𝜉(𝑥)<∞,𝑗=1,…,𝑑. (33) Then Γ𝐻⊂−𝐻Γ (34) if and only if ^𝑑_𝑗=1𝛾₅^(𝑑)𝛼_𝑑+1𝛼_𝑗𝐷_𝑗𝜉(𝑥)=𝑖𝛾₅^(𝑑)⊗𝜉(𝑥)𝑒^𝑖Φ(𝑥)𝑀(𝑥)−𝑀(𝑥)𝑒^{−𝑖Φ(𝑥)}𝛾₅^(𝑑)⊗𝜉(𝑥),𝑎.𝑒.𝑥∈ℝ^𝑑. (35) In that case, the spectrum 𝜎(𝐻) and the point spectrum 𝜎_𝑝(𝐻) of 𝐻 are, respectively, symmetric with respect to the origin 0∈ℝ.

Proof. Since the subspace 𝒟₀ given by (20) is a core of 𝐻 by Lemma 6(ii), (34) is equivalent to that, for all 𝜓∈𝒟₀, Γ𝜓∈𝐷(𝐻) and Γ𝐻𝜓=−𝐻Γ𝜓. (36)
Let 𝜓∈𝒟₀. Then Γ𝜓(𝑥)=𝑖𝛾₅^(𝑑)𝛼_𝑑+1⊗𝜉(𝑥)𝜓(𝑥),a.e.𝑥∈ℝ^𝑑. (37) It follows that the ℂ^𝑁𝑑⊗𝒦-valued function: 𝑥↦(Γ𝜓)(𝑥) is strongly differentiable on ℝ^𝑑 with 𝐷_𝑗Γ𝜓(𝑥)=𝑖𝛾₅^(𝑑)𝛼_𝑑+1⊗𝐷_𝑗𝜉(𝑥)𝜓(𝑥)+𝑖𝛾₅^(𝑑)𝛼_𝑑+1⊗𝜉(𝑥)𝐷_𝑗𝜓(𝑥),𝑗=1,…,𝑑. (38) Hence 𝐷_𝑗Γ𝜓(𝑥)²≤2𝐶²_𝑗𝜓(𝑥)²+𝐷_𝑗𝜓(𝑥)², (39) which implies that 𝐷_𝑗Γ𝜓∈ℋ and hence Γ𝜓∈𝐷(𝐻₀)=𝐷(𝐻). Moreover, we have 𝐻₀Γ𝜓(𝑥)=−𝛾₅^(𝑑)𝛼_𝑑+1^𝑑_𝑗=1𝛼_𝑗𝐷_𝑗𝜉(𝑥)𝜓(𝑥)−Γ𝐻₀𝜓(𝑥),(𝑥)(𝑥)𝛾𝑈Γ𝜓=−Γ𝑈𝜓+𝑖₅^(𝑑)⊗𝜉(𝑥)𝑒^𝑖Φ(𝑥)𝑀(𝑥)−𝑀(𝑥)𝑒^{−𝑖Φ(𝑥)}𝛾₅^(𝑑)⊗𝜉(𝑥)𝜓(𝑥). (40) Hence 𝐻Γ𝜓(𝑥)=−Γ𝐻𝜓(𝑥)−^𝑑_𝑗=1𝛾₅^(𝑑)𝛼_𝑑+1𝛼_𝑗⊗𝐷_𝑗𝜉(𝑥)𝜓(𝑥)+𝑖𝛾₅^(𝑑)⊗𝜉(𝑥)𝑒^𝑖Φ(𝑥)𝑀(𝑥)−𝑀(𝑥)𝑒^{−𝑖Φ(𝑥)}𝛾₅^(𝑑)⊗𝜉(𝑥)𝜓(𝑥). (41) Therefore, 𝐻Γ𝜓=−Γ𝐻𝜓 for all 𝜓∈𝒟₀ if and only if ^𝑑_𝑗=1𝛾₅^(𝑑)𝛼_𝑑+1𝛼_𝑗⊗𝐷_𝑗𝜉(𝑥)𝜓(𝑥)=𝑖𝛾₅^(𝑑)⊗𝜉(𝑥)𝑒^𝑖Φ(𝑥)𝑀(𝑥)−𝑀(𝑥)𝑒^{−𝑖Φ(𝑥)}𝛾₅^(𝑑)⊗𝜉(𝑥)𝜓(𝑥),∀𝜓∈𝒟₀. (42) By the original assumption for 𝑀(⋅), 𝑀(𝑥)∈𝔅(ℂ^𝑁𝑑⊗𝒦) for a.e. 𝑥∈ℝ^𝑑. Therefore (42) is equivalent to (35).
By (32) and Γ^∗=Γ, one easily sees that (34) is in fact equivalent to operator equality Γ^∗𝐻Γ=−𝐻. Hence 𝐻 is unitarily equivalent to −𝐻. This implies the symmetry of 𝜎(𝐻) and 𝜎_𝑝(𝐻) with respect to the origin.

Remark 9. Proposition 8 gives a generalization of [4, Theorem1] and clarifies a condition for 𝐻 to have supersymmetry.
It may be difficult in general to show the existence of self-adjoint, unitary solutions 𝜉(𝑥) to operator equation (35). Here we only note the following fact.

Lemma 10. Let𝑑be odd. Assume (𝐴.1)--(𝐴.3). Suppose that 𝛾₅^(𝑑)Φ(𝑥)⊂Φ(𝑥)𝛾₅^(𝑑),𝑎.𝑒.𝑥∈ℝ^𝑑,𝑗=1,…,𝑑, (43)𝜉=𝜉(𝑥) is independent of 𝑥∈ℝ^𝑑 and (𝐼⊗𝜉)Φ(𝑥)⊂−Φ(𝑥)(𝐼⊗𝜉),𝛾₅^(𝑑)⊗𝜉𝑀(𝑥)⊂𝑀(𝑥)𝛾₅^(𝑑)⊗𝜉,𝑎.𝑒.𝑥∈ℝ^𝑑. (44)(45) Then 𝜉 is a solution to (35).

Proof. Since 𝜉 is a constant operator, 𝐷_𝑗𝜉=0. By Lemma 2(iii), (43) implies the strong commutativity of 𝛾₅^(𝑑) and Φ. Hence 𝛾₅^(𝑑)𝑒^𝑖Φ(𝑥)=𝑒^𝑖Φ(𝑥)𝛾₅^(𝑑) for a.e. 𝑥∈ℝ^𝑑. By (44) and Lemma 2(iv), (𝐼⊗𝜉)𝑒^𝑖Φ(𝑥)=𝑒^{−𝑖Φ(𝑥)}(𝐼⊗𝜉). We also have (45) and the strong commutativity of Φ and 𝑀. Hence 𝛾₅^(𝑑)⊗𝜉𝑒^𝑖Φ(𝑥)𝑀(𝑥)=𝑀(𝑥)𝛾₅^(𝑑)⊗𝜉𝑒^𝑖Φ(𝑥)=𝑀(𝑥)𝑒^−𝑖Φ𝛾₅^(𝑑)⊗𝜉. (46) Thus (35) holds with both sides being zero.

Additionally we make a remark on the converse of Lemma 10. For this purpose, we need a lemma.

Lemma 11. Let 𝑇_𝑗 (𝑗=1,…,𝑑) be a densely defined closed linear operator on 𝒦. Suppose that ^𝑑_𝑗=1𝛼_𝑗⊗𝑇_𝑗=0𝑜𝑛∩^𝑑_𝑗=1𝐷𝛼_𝑗⊗𝑇_𝑗. (47) Then, for all 𝑗=1,…,𝑑, 𝑇_𝑗=0 on ∩^𝑑_𝑗=1𝐷(𝑇_𝑗).

Proof. Equation (47) implies that, for all 𝑢∈∩^𝑑_𝑗=1𝐷(𝑇_𝑗) and 𝑣∈𝒦, ∑^𝑑_𝑗=1⟨𝑣,𝑇_𝑗𝑢⟩𝛼_𝑗=0. Since {𝛼_𝑗}^𝑑_𝑗=1 is linearly independent, it follows that ⟨𝑣,𝑇_𝑗𝑢⟩=0,𝑗=1,…,𝑑. Hence 𝑇_𝑗𝑢=0,𝑗=1,…,𝑑.

The following lemma gives a sufficient condition for a solution to (35) to be a constant operator.

Lemma 12. Let 𝑑 be odd. Assume (𝐴.1)--(𝐴.3). Let 𝜉(𝑥) be strongly differentiable on ℝ^𝑑 with (33) and be a solution to (35). Suppose that (43)–(45) hold. Then 𝜉 is independent of 𝑥∈ℝ^𝑑.

Proof. As in the proof of Lemma 10, (43)–(45) imply (46). Hence the right-hand side of (35) vanishes, so that ∑^𝑑_𝑗=1𝛾₅^(𝑑)𝛼_𝑑+1𝛼_𝑗⊗𝐷_𝑗𝜉(𝑥)=0, which implies that ∑^𝑑_𝑗=1𝛼_𝑗⊗𝐷_𝑗𝜉(𝑥)=0. By Lemma 11, 𝐷_𝑗𝜉(𝑥)=0,𝑗=1,…,𝑑, which implies that 𝜉 is independent of 𝑥.

We have from Proposition 8 and Lemma 10 the following result.

Corollary 13. Let 𝑑 be odd. Assume (A.1)--(A.4). Suppose that 𝜉=𝜉(𝑥) is independent of 𝑥∈ℝ^𝑑 and that (43)–(45) hold. Then 𝐻 has supersymmetry with respect to Γ.

5. Vanishing Theorems of the Kernel of 𝐻

In supersymmetric quantum mechanics with a supercharge 𝑄, a nonzero vector in ker𝑄 is called a supersymmetric states. If the kernel of 𝑄 vanishes, that is, ker𝑄={0}, then the supersymmetry is said to be spontaneously broken. It turns out that, in supersymmetric quantum mechanics, it is importanat to investigate ker𝑄. Thus we are led to consider ker𝐻 in view of Proposition 8. This would be interesting even if 𝐻 does not have supersymmetry (note that 𝐻 does not necessarily have supersymmetry).

To investigate ker 𝐻, we also need an additional condition.(A.5)(i) For each 𝑓∈ℂ^𝑁𝑑⊗𝒦, the function: 𝑥↦𝑀(𝑥)𝑓 is strongly differentiable onℝ^𝑑and, for all 𝑥∈ℝ^𝑑, 𝑀(𝑥) commutes with 𝛼_𝑗 (𝑗=1,…,𝑑).  (ii) There exists a constant 𝜇₀>0 such that 𝑀(𝑥)²−𝑖^𝑑_𝑗=1𝛼_𝑗𝛼_𝑑+1𝐷_𝑗𝑀(𝑥)≥𝜇²₀,∀𝑥∈ℝ^𝑑 (48) as an operator inequality on ℂ^𝑁𝑑⊗𝒦 (note that, by the principle of uniform boundedness, the strong partial derivative 𝐷_𝑗𝑀(𝑥) is a bounded operator on ℂ^𝑁𝑑⊗𝒦 for each 𝑥∈ℝ^𝑑 and hence, under (A.2) and condition (A.5)(i), the operator 𝑀(𝑥)²−𝑖∑^𝑑_𝑗=1𝛼_𝑗𝛼_𝑑+1𝐷_𝑗𝑀(𝑥) on ℂ^𝑁𝑑⊗𝒦 is a bounded self-adjoint operator).

For a linear operator 𝐿 on a Hilbert space, we denote the resolvent set of 𝐿 by 𝜌(𝐿).

Lemma 14. Assume (𝐴.2), (𝐴.4), and (𝐴.5). Then 𝐻_𝑀 defined by (16) is self-adjoint with 𝐷(𝐻_𝑀)=𝐷(𝐻₀) and 𝐻₀𝜓²+𝜇²₀𝜓²≤𝐻_𝑀𝜓²,𝜓∈𝐷𝐻₀. (49) In particular, 0∈𝜌(𝐻_𝑀) with operator-norm bound 𝐻_𝑀⁻¹≤1𝜇₀ (50) and 𝐻₀𝐻_𝑀⁻¹ is bounded with 𝐻₀𝐻_𝑀⁻¹≤1. (51) Moreover, 𝑀𝐻_𝑀⁻¹ is bounded with 𝑀𝐻_𝑀⁻¹𝑏≤𝑎+𝜇₀. (52)

Proof. The self-adjointness of 𝐻_𝑀 follows from that of 𝐻 with Φ=0. For all 𝜓∈𝒟₀, using the anticommutativity of 𝛼_𝑗 with 𝛼_𝑑+1 and the commutativity of 𝑀(𝑥) with 𝛼_𝑑+1 and 𝛼_𝑗 (𝑗=1,…,𝑑), we have 𝐻_𝑀𝜓²=𝐻₀𝜓²+𝑀𝜓²+^𝑑_𝑗=1𝜓,−𝑖𝐷_𝑗𝑀𝛼_𝑗𝛼_𝑑+1𝜓≥𝐻₀𝜓²+_ℝ𝑑𝜓(𝑥),𝑀(𝑥)²−𝑖^𝑑_𝑗=1𝛼_𝑗𝛼_𝑑+1𝐷_𝑗𝑀(𝑥)𝜓(𝑥≥𝐻)𝑑𝑥₀𝜓²+𝜇²₀𝜓². (53) Hence (49) holds for all 𝜓∈𝒟₀. Since 𝒟₀ is a core of 𝐻_𝑀, this inequality extends to all 𝜓∈𝐷(𝐻₀). In particular, we have 𝜇₀𝜓≤𝐻_𝑀𝜓,𝜓∈𝐷𝐻₀. (54) This implies that the self-adjoint operator 𝐻_𝑀 is bijective with (50).
Inequality (49) implies also that, for all 𝜓∈𝐷(𝐻₀), ‖𝐻₀𝜓‖≤‖𝐻_𝑀𝜓‖. Hence 𝐻₀𝐻_𝑀⁻¹ is bounded with (51).
By(A.4)and ‖(−Δ)^1/2𝜓‖=‖𝐻₀𝜓‖ for all 𝜓∈𝐷((−Δ)^1/2)=𝐷(𝐻₀), we have ‖𝑀𝜓‖≤𝑎‖𝐻₀𝜓‖+𝑏‖𝜓‖. Hence, for all𝜙∈ℋ, 𝑀𝐻_𝑀⁻¹𝜙≤𝑎𝐻₀𝐻_𝑀⁻¹𝜙+𝑏𝐻_𝑀⁻¹𝜙≤𝑎𝐻₀𝐻_𝑀⁻¹+𝑏𝐻_𝑀⁻¹𝜙≤𝑏𝑎+𝜇₀𝜙. (55) Thus (52) holds.

Lemma 15. Let 𝐴 be a self-adjoint operator on a complex Hilbert space 𝒳. Then 𝑒^𝑖𝐴𝐴−1=2sin2. (56)

Proof. By the functional calculus, one has 𝑒^𝑖𝐴−1=2𝑖𝑒^𝑖𝐴/2sin(𝐴/2). Hence ‖𝑒^𝑖𝐴−1‖=2‖𝑒^𝑖𝐴/2sin(𝐴/2)‖. Since 𝑒^𝑖𝐴/2 is unitary, one has ‖𝑒^𝑖𝐴/2sin(𝐴/2)‖=‖sin(𝐴/2)‖. Thus (56) holds.

Theorem 16. Assume (𝐴.1)--(𝐴.5) and esssup_{𝑥∈ℝ𝑑}Φ(𝑥)sin2<12𝑎+𝑏𝜇₀⁻¹. (57) Then ker 𝐻={0} and 0∈𝜌(𝐻).
Moreover, the constant 𝛾(𝐻)∶=inf_{(𝐻),𝜓𝜓∈𝐷=1}𝐻𝜓 (58) is strictly positive, 𝛾(𝐻)∈𝜎(𝐻) or −𝛾(𝐻)∈𝜎(𝐻), and 𝜎(𝐻)⊂−∞,−𝛾(𝐻)∪𝛾(𝐻),∞. (59)

Proof. The operator 𝐻 is written as 𝐻=𝐻_𝑀+𝛼_𝑑+1𝑒^𝑖Φ−1𝑀=𝐾𝐻_𝑀 (60) with 𝐾∶=𝐼+𝛼_𝑑+1𝑒^𝑖Φ−1𝑀𝐻_𝑀⁻¹. (61) By applying Lemma 15 with 𝐴=Φ(𝑥), we have 𝑒^𝑖Φ(𝑥)Φ(𝑥)−1=2sin2. (62) Therefore, for all 𝜓∈ℋ, 𝛼_𝑑+1𝑒^𝑖Φ−1𝑀𝐻_𝑀⁻¹𝜓≤2esssup_{𝑥∈ℝ𝑑}Φ(𝑥)sin2𝑀𝐻_𝑀⁻¹𝜓. (63) By this estimate and (52), we obtain 𝛼_𝑑+1𝑒^𝑖Φ−1𝑀𝐻_𝑀⁻¹≤2esssup_{𝑥∈ℝ𝑑}Φ(𝑥)sin2𝑎+𝑏𝜇₀⁻¹. (64) Hence, by (57), we obtain ‖𝛼_𝑑+1(𝑒^𝑖Φ−1)𝑀𝐻_𝑀⁻¹‖<1. This implies that 𝐾 is bijective with bounded inverse 𝐾⁻¹. Thus 𝐻 is bijective with 𝐻⁻¹=𝐻_𝑀⁻¹𝐾⁻¹ being bounded. Hence ker𝐻={0} and 0∈𝜌(𝐻).
We set 𝑏_𝐻∶=‖𝐻⁻¹‖. If |𝜆|<1/𝑏_𝐻, then 𝜆 is in 𝜌(𝐻). Therefore 𝜎(𝐻)⊂−∞,−𝑏_𝐻⁻¹∪𝑏_𝐻⁻¹,∞. (65) It is obvious that, for all 𝜓∈𝐷(𝐻) with ‖𝜓‖=1, 1≤𝑏_𝐻‖𝐻𝜓‖. This implies that 𝑏_𝐻𝛾(𝐻)≥1. On the other hand, we have from (58) ‖𝜓‖≥𝛾(𝐻)‖𝐻⁻¹𝜓‖, for all 𝜓∈ℋ. Hence 𝑏_𝐻𝛾(𝐻)≤1. Therefore 𝑏_𝐻⁻¹=𝛾(𝐻). Thus (59) holds and 𝛾(𝐻)>0. Since 𝑏_𝐻∈𝜎(𝐻⁻¹) or −𝑏_𝐻∈𝜎(𝐻⁻¹), it follows that 𝛾(𝐻)=𝑏_𝐻⁻¹∈𝜎(𝐻)or−𝛾(𝐻)∈𝜎(𝐻).

Remark 17. Under the same assumption as in Theorem 16,𝐻is Fredholm (the proof is easy).

We next consider a perturbation of Φ(⋅). Let 𝜂(⋅)∈ℱ_s.a.such that, for a.e. 𝑥∈ℝ^𝑑, 𝜂(𝑥) is bounded and strongly commutes with Φ(𝑥) and 𝑀(𝑥). Then, for a.e. 𝑥∈ℝ^𝑑, Φ_𝜂(𝑥)∶=Φ(𝑥)+𝜂(𝑥) (66) is self-adjoint on ℂ^𝑁𝑑⊗𝒦 and Φ_𝜂∶=^⊕_ℝ𝑑Φ_𝜂(𝑥)𝑑𝑥 (67) is a self-adjoint operator on ℋ.

The quantity 𝜅(𝐻)∶=sup_{(𝐻),𝜓𝜓∈𝐷=1}𝑀𝜓,𝐻𝜓 (68) may be infinite. But we have the following.

Lemma 18. Under the assumption of Theorem 16, 0<𝜅(𝐻)<∞.

Proof. Since𝐻is closed with 𝐷(𝐻)=𝐷(𝐻₀)∩𝐷(𝑈)(=𝐷(𝐻₀)) and ‖𝑈𝜓‖=‖𝑀𝜓‖,𝜓∈𝐷(𝐻), it follows from the closed graph theorem that there exists a constant 𝑐>0 such that 𝑀𝜓≤𝑐𝐻𝜓+𝜓,𝜓∈𝐷(𝐻). (69) Let 𝜓∈𝐷(𝐻) with ‖𝜓‖=1. Then, by Theorem 16, we have ‖𝐻𝜓‖≥𝛾(𝐻)>0. Hence 𝑀𝜓𝑐𝐻𝜓≤𝑐+𝛾(𝐻). (70) Therefore 𝜅(𝐻)≤𝑐+𝑐/𝛾(𝐻)<∞. If 𝜅(𝐻)=0, then ‖𝑀𝜓‖=0 for all 𝜓∈𝐷(𝑀)=𝐷(𝐻₀). Hence 𝑀=0. But this contradicts condition (A.5).

Theorem 19. Assume (A.1)--(A.5) and (57). Suppose that esssup_{𝑥∈ℝ𝑑}𝜂(𝑥)sin2<1(𝐻).2𝜅 (71) Let 𝐻_𝜂∶=𝐻₀+𝛼_𝑑+1𝑒^𝑖Φ𝜂𝑀. (72) Then ker  𝐻_𝜂={0} and 0∈𝜌(𝐻_𝜂). Moreover, the last statement on 𝛾(𝐻) and 𝜎(𝐻) in Theorem 16 holds with 𝐻 being replaced by 𝐻_𝜂.

Proof. We write 𝐻_𝜂=𝐻+𝑊,𝑊∶=𝛼_𝑑+1𝑒^𝑖Φ𝜂−𝑒^𝑖Φ𝑀. (73) By the strong commutativity of Φ(𝑥) and 𝜂(𝑥), we have for a.e. 𝑥∈ℝ^𝑑𝑒^{𝑖Φ𝜂(𝑥)}−𝑒^𝑖Φ(𝑥)=𝑒^𝑖Φ(𝑥)𝑒^{𝑖𝜂(𝑥)}−1=2𝑖𝑒^𝑖Φ(𝑥)𝑒^{𝑖𝜂(𝑥)/2}𝜂(𝑥)sin2. (74) Hence, for all 𝜓∈𝐷(𝐻₀)𝑊𝜓≤2esssup_{𝑥∈ℝ𝑑}𝜂(𝑥)sin2𝑀𝜓. (75) We have ‖𝑀𝜓‖≤𝜅(𝐻)‖𝐻𝜓‖. Hence 𝑊𝜓≤𝐶_𝜂𝐻𝜓 (76) with 𝐶_𝜂∶=2𝜅(𝐻)esssup_{𝑥∈ℝ𝑑}𝜂(𝑥)sin2. (77) Hence 𝑊 is 𝐻-bounded. By Remark 17, 𝐻 is Fredholm. Condition (71) is equivalent to 𝐶_𝜂<1. Hence, by a stability theorem (e.g., [12, ChapterIV, Theorem5.22]), 𝐻_𝜂 is Fredholm and dim ker 𝐻_𝜂≤ dim ker 𝐻=0. Therefore ker 𝐻_𝜂={0}. It follows from this fact and the self-adjointness of 𝐻_𝜂 that Ran (𝐻_𝜂)=ℋ. Hence 0∈𝜌(𝐻_𝜂). Then the last statement of the present theorem can be proved in the same way as in the proof of the corresponding part in Theorem 16.

6. Unitary Equivalence to a Gauge Theoretic Dirac Operator and a Vanishing Theorem for ker𝐻

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.

We introduce a class of Φ(⋅): ℱ∶=Φ(⋅)∈ℱ_s.a.∣𝑒^{±𝑖Φ(⋅)/2}isstronglydifferentiableandsup_{𝑥∈ℝ𝑑}𝐸_𝑗(𝑥)<∞,𝑗=1,…,𝑑, (78) where 𝐸_𝑗(𝑥)∶=𝐷_𝑗𝑒^{−𝑖Φ(𝑥)/2} (79) denotes the strong partial derivative of 𝑒^{−𝑖Φ(𝑥)/2} in 𝑥_𝑗. For Φ(⋅)∈ℱ, one can define a bounded linear operator 𝐴_𝑗∶=𝑖^⊕_ℝ𝑑𝑒^{𝑖Φ(𝑥)/2}𝐸_𝑗(𝑥)𝑑𝑥 (80) on ℋ.

Remark 20. If Φ(⋅)∈ℱ sucht that Φ(𝑥) and Φ(𝑥^′) commute for a.e. 𝑥,𝑥^′∈ℝ^𝑑, then 𝐸_𝑗(𝑥)=−𝑖𝑒^{−𝑖Φ(𝑥)/2}𝐷_𝑗Φ(𝑥)/2 and hence 𝐴_𝑗=12^⊕_ℝ^𝑑𝐷_𝑗Φ(𝑥)𝑑𝑥. (81)

Lemma 21. For each 𝑗=1,…,𝑑, 𝐴_𝑗 is a bounded self-adjoint operator on ℋ.

Proof. Since esssup_{𝑥∈ℝ𝑑}‖𝑒^{𝑖Φ(𝑥)/2}𝐸_𝑗(𝑥)‖=esssup_{𝑥∈ℝ𝑑}‖𝐸_𝑗(𝑥)‖<∞,𝐴_𝑗 is bounded. We have 𝐴^∗_𝑗=(−𝑖)2^⊕_ℝ^𝑑𝐷_𝑗𝑒^{𝑖Φ(𝑥)/2}𝑒^{−𝑖Φ(𝑥)/2}𝑑𝑥. (82) Differentiating the identity 𝑒^{𝑖Φ(𝑥)/2}𝑒^{−𝑖Φ(𝑥)/2}=𝐼 in 𝑥_𝑗, we have 𝑒^{𝑖Φ(𝑥)/2}𝐸_𝑗(𝑥)=−𝐷_𝑗𝑒^Φ(𝑥)/2𝑒^{−𝑖Φ(𝑥)/2}. (83) Hence 𝐴^∗_𝑗=𝐴_𝑗.

For Φ(⋅)∈ℱ, we define an operator: 𝐻^′∶=^𝑑_𝑗=1𝛼_𝑗−𝑖𝐷_𝑗−𝐴_𝑗+𝛼_𝑑+1𝑀=𝐻_𝑀−^𝑑_𝑗=1𝛼_𝑗𝐴_𝑗. (84)

Lemma 22. Assume (𝐴.4). Let Φ(⋅)∈ℱ. Suppose that 𝛼_𝑗Φ⊂Φ𝛼_𝑗,𝑗=1,…,𝑑. (85) Then 𝐻^′ is self-adjoint and every core of 𝐻₀ is a core of 𝐻^′.

Proof. Under condition(A.4), 𝐻_𝑀 is self-adjoint. By (85), we have 𝛼_𝑗𝐴_𝑗=𝐴_𝑗𝛼_𝑗 (𝑗=1,…,𝑑). Hence, by Lemma 21, −∑^𝑑_𝑗=1𝛼_𝑗𝐴_𝑗 is a bounded self-adjoint operator. Hence the Kato-Rellich theorem yields the desired result.

We note that, if one regards 𝐀∶=(𝐴₁,…,𝐴_𝑑) as a (noncommutative) gauge potential, then 𝐻^′ is a gauge theoretic Dirac operator with gauge potential 𝐀.

Let 𝑈∶=𝑒^𝑖Φ/2, (86) which is unitary. The following theorem shows that, under a suitable condition, 𝐻 is unitarily equivalent to a gauge theoretic Dirac operator 𝐻^′.

Theorem 23. Assume (𝐴.1)--(𝐴.4) and (85). Let Φ(⋅)∈ℱ. Then 𝑈𝐻𝑈⁻¹=𝐻^′. (87)

Proof. We have 𝑈𝐻𝑈⁻¹=−𝑖^𝑑_𝑗=1𝑈𝛼_𝑗𝑈⁻¹𝑈𝐷_𝑗𝑈⁻¹+𝑈𝛼_𝑑+1𝑒^𝑖Φ𝑈⁻¹𝑈𝑀𝑈⁻¹. (88) By (85) and Lemma 2, 𝑈𝛼_𝑗𝑈⁻¹=𝛼_𝑗. By(A.3)and Lemma 2, 𝑈𝑀𝑈⁻¹=𝑀. By(A.1)and Lemma 2(iv), 𝑈𝛼_𝑑+1𝑒^𝑖Φ𝑈⁻¹=𝛼_𝑑+1𝑒^−𝑖Φ/2𝑒^𝑖Φ𝑒^𝑖Φ/2=𝛼_𝑑+1. Moreover, 𝑈𝐷_𝑗𝑈⁻¹=𝐷_𝑗−𝑖𝐴_𝑗. (89) Hence (87) holds.

The following theorem gives another sufficient condition for ker𝐻 to be trivial.

Theorem 24. Assume (𝐴.1)--(𝐴.4) and (85). Let Φ(⋅)∈ℱ and ^𝑑_𝑗=1esssup_{𝑥∈ℝ𝑑}𝐸_𝑗(𝑥)<𝜇₀. (90) Then ker𝐻={0}and 0∈𝜌(𝐻).

Proof. We write 𝐻^′=𝐻_𝑀+𝑋 with 𝑋∶=−∑^𝑑_𝑗=1𝛼_𝑗𝐴_𝑗. Then ‖𝑋‖≤^𝑑_𝑗=1𝛼_𝑗𝐴_𝑗≤^𝑑_𝑗=1esssup_{𝑥∈ℝ𝑑}𝐸_𝑗(𝑥). (91) By this estimate and (90), ‖𝑋𝐻_𝑀⁻¹‖<1. Hence 𝐻^′ is bijective and 0∈𝜌(𝐻^′). In particular, ker 𝐻^′={0}. On the other hand, (87) implies that𝜌(𝐻^′)=𝜌(𝐻)and ker 𝐻=𝑈⁻¹ ker 𝐻^′. Thus 0∈𝜌(𝐻) and ker 𝐻={0}.

7. Essential Spectrum of 𝐻

In this section, we consider the essential spectrum of 𝐻. For a self-adjoint operator 𝑆 on a Hilbert space, we denote by 𝜎_ess(𝑆) the essential spectrum of 𝑆.

Lemma 25. Let dim𝒦<∞ and 𝑚>0 be a constant. Let 𝑉(⋅)∶ℝ^𝑑→𝔅(ℂ^𝑁𝑑⊗𝒦) be Borel measurable satisfying the following conditions.(i) The operator 𝑉∶=∫^⊕_ℝ𝑑𝑉(𝑥)𝑑𝑥 is relatively bounded with respect to𝐻₀.(ii)lim_|𝑥|→∞‖𝑉(𝑥)‖=0.(iii) The operator 𝐻_𝑚+𝑉 on ℋ is self-adjoint. Then 𝜎_ess𝐻_𝑚+𝑉=(−∞,−𝑚]∪[𝑚,∞). (92)

Proof. For each 𝑅>0, we denote by 𝜒_𝑅 the characteristic function of the set {𝑥∈ℝ^𝑑∣|𝑥|<𝑅}. As in the case of the 3-dimensional free Dirac operator (cf. [6, Lemma4.6]), one can show that |𝐻_𝑚|^−𝑘𝜒_𝑅 is compact for all 𝑘>0 as an operator on 𝐿²(ℝ^𝑑;ℂ^𝑁𝑑). Sincedim𝒦<∞, it follows that|𝐻_𝑚|^−𝑘𝜒_𝑅is compact as an operator onℋ. Hence, for all 𝑧∈ℂ⧵ℝ, (𝐻_𝑚−𝑧)⁻¹𝜒_𝑅=((𝐻_𝑚−𝑧)⁻¹|𝐻_𝑚|)|𝐻_𝑚|⁻¹𝜒_𝑅 is compact. Since 𝐻_𝑚+𝑉 is self-adjoint with 𝐷(𝐻_𝑚+𝑉)=𝐷(𝐻_𝑚)=𝐷(𝐻₀) and 𝑉 is closed, it follows from the closed graph theorem that 𝑉(𝐻_𝑚+𝑉−𝑧)⁻¹ is bounded (this can be shown by direct estimates too). Therefore we have 𝐻_𝑚+𝑉−𝑧⁻¹−𝐻_𝑚−𝑧⁻¹=−𝐻_𝑚−𝑧⁻¹𝑉𝐻_𝑚+𝑉−𝑧⁻¹=−𝑊_𝑅−𝑋_𝑅, (93) where𝑊_𝑅∶=(𝐻_𝑚−𝑧)⁻¹𝜒_𝑅[𝑉(𝐻_𝑚+𝑉−𝑧)⁻¹], 𝑋_𝑅=(𝐻_𝑚−𝑧)⁻¹(1−𝜒_𝑅)𝑉(𝐻_𝑚+𝑉−𝑧)⁻¹. By the fact mentioned previously, 𝑊_𝑅 is compact. We have 𝑋_𝑅≤1|Im𝑧|²1−𝜒_𝑅𝑉. (94) By condition (ii), for every 𝜀>0, there exists a constant 𝑅>0 such that, for all a.e. 𝑥∈ℝ^𝑑 with |𝑥|≥𝑅, ‖𝑉(𝑥)‖<𝜀, that is, esssup_{|𝑥|≥𝑅}‖𝑉(𝑥)‖≤𝜀, which implies that ‖(1−𝜒_𝑅)𝑉‖≤𝜀. Hence lim_𝑅→∞‖𝑋_𝑅‖=0. Therefore (𝐻_𝑚+𝑉−𝑧)⁻¹−(𝐻_𝑚−𝑧)⁻¹ is compact. Hence, by Weyl’s essential spectrum theorem (e.g., [10, TheoremXIII.14]), 𝜎_ess(𝐻_𝑚+𝑉)=𝜎_ess(𝐻_𝑚). On the other hand, as in the case of the 3-dimensional free Dirac operator [6, Theorem1.1], one can show that 𝜎(𝐻_𝑚)=𝜎_ess(𝐻_𝑚)=(−∞,−𝑚]∪[𝑚,∞). Thus (92) holds.

Theorem 26. Let dim𝒦<∞. Assume (𝐴.1)--(𝐴.4). Suppose that there exists a constant 𝑚∈ℝ satisfying lim_|𝑥|→∞‖𝑀(𝑥)−𝑚‖=0,lim_|𝑥|→∞Φ(𝑥)sin2=0. (95)(96) Then 𝜎_ess(𝐻)=(−∞,−𝑚]∪[𝑚,∞). (97)

Proof. We write 𝐻=𝐻_𝑚+𝑉₁+𝑉₂ (98) with 𝑉₁∶=𝛼_𝑑+1(𝑀−𝑚),𝑉₂∶=𝛼_𝑑+1𝑒^𝑖Φ−1𝑀. (99) It is obvious that 𝑉₁ and 𝑉₂ are relatively bounded with respect to 𝐻₀ and lim_|𝑥|→∞𝑉₁(𝑥)=lim_|𝑥|→∞‖𝑀(𝑥)−𝑚‖=0. (100) As for 𝑉₂, we have 𝑉₂(𝑥)≤‖𝑀(𝑥)‖𝑒^𝑖Φ(𝑥)‖𝑀(𝑥Φ(𝑥)−1≤2)‖sin2. (101) Hence, by (95) and (96), we have lim_|𝑥|→∞‖𝑉₂(𝑋)‖=0. Therefore lim_|𝑥|→∞‖𝑉₁(𝑥)+𝑉₂(𝑥)‖=0. Thus we can apply Lemma 25 to obtain (97).

If Φ(⋅) is in the class ℱ introduced in Section 6, then we can obtain a sufficient condition for (97) to hold.

Theorem 27. Let dim𝒦<∞. Assume (𝐴.1)--(𝐴.4) (85) and (95). Let Φ(⋅)∈ℱ. Suppose that lim_|𝑥|→∞𝐸_𝑗(𝑥)=0. (102) Then (97) holds.

Proof. By (87), we have 𝜎_ess(𝐻)=𝜎_ess(𝐻^′). Hence we need only to prove 𝜎_ess𝐻^′=(−∞,−𝑚]∪[𝑚,∞). (103) We write 𝐻^′=𝐻_𝑚+𝛼_𝑑+1(𝑀−𝑚)−^𝑑_𝑗=1𝛼_𝑗𝐴_𝑗. (104) We have lim_|𝑥|→∞‖𝛼_𝑑+1(𝑀(𝑥)−𝑚)‖=0. Moreover, ‖𝛼_𝑗𝐴_𝑗(𝑥)‖≤‖𝐸_𝑗(𝑥)‖. Hence lim_|𝑥|→∞‖−∑^𝑑_𝑗=1𝛼_𝑗𝐴_𝑗(𝑥)‖=0. Thus we can apply Lemma 25 to obtain (103).

8. Bounds on the Number of Discrete Eigenvalues

In this section, in view of Theorem 26, we consider the number of eigenvalues of𝐻in the interval (−𝑚,𝑚) 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 𝐻_𝑚+𝑉 with a scalar potential 𝑉∶ℝ³→ℝ in (−𝑚,𝑚) is considered. This extension is not difficult. But, for the sake of completeness, we present some details of it. One easily notes that the problem under consideration can be studied in a more general frame work as in Lemma 25. Hence we first discuss the general case.

8.1. A General Case

Let 𝑉 be as in Lemma 25 and 𝐻(𝑉)∶=𝐻_𝑚+𝑉. (105) Then, by (92), an eigenvalue of 𝐻(𝑉) in (−𝑚,𝑚) (if it exists) is an isolated eigenvalue of 𝐻(𝑉) with finite multiplicity. For each 𝜆∈(0,𝑚²), we denote by 𝑁(𝜆,𝑉) the number of eigenvalues in the interval (−√𝑚²−𝜆,√𝑚²−𝜆).

We first note an elementary fact:

Theorem 28. Suppose that the assumption of Lemma 25 holds and that ‖𝑉(𝑥)‖≤𝜆/4𝑚 for a.e. 𝑥∈ℝ^𝑑. Then 𝑁(𝜆,𝑉)=0.

Proof. Suppose that 𝑁(𝜆,𝑉)≥1. Then, it follows from the definition of 𝑁∶=𝑁(𝜆,𝑉) that there exists an𝑁-dimensional subspace 𝐸 of ℋ such that 𝐻(𝑉)𝜓≤√𝑚²−𝜆𝜓,∀𝜓∈𝐸. (106) Hence 𝐻_𝑚𝜓≤𝐻(𝑉)𝜓+𝑉𝜓≤√𝑚²𝜆−𝜆+𝜓≤4𝑚𝑚²−𝜆2𝜓. (107) Hence ‖𝐻_𝑚𝜓‖²≤(𝑚²−(𝜆/2))‖𝜓‖², which is equivalent to ‖(−Δ+(𝜆/2))^1/2𝜓‖²≤0. This implies that 𝜓=0. But this is a contradiction.

In view of Theorem 28, we define, for each 𝜆>0, 𝑉_𝜆∶ℝ^𝑑→𝔅(ℂ^𝑁𝑑⊗𝒦) by 𝑉_𝜆(𝑥)𝑉(𝑥)∶=if‖𝑉(𝑥>𝜆)‖,04𝑚otherwise. (108)

For each 𝜆>0, the operator 𝑅_𝜆𝜆∶=−Δ+2^−1/2 (109) is a bounded self-adjoint operator. Since 𝑉 is 𝐻₀-bounded, where 𝐻₀ is defined by (10) and (𝐻₀+𝑖)𝑅_𝜆 is bounded, it follows that 𝑉𝑅_𝜆 and 𝑉_𝜆𝑅_𝜆 are bounded operators on ℋ. Also 𝐻₀𝑅_𝜆 is bounded with ‖𝐻₀𝑅_𝜆‖≤1. Hence the following operators 𝑇_𝜆𝑗 (𝑗=1,2,3,5) are in 𝔅(ℋ): 𝑇_𝜆1∶=𝐻₀𝑅_𝜆^∗𝑉_𝜆𝑅_𝜆,𝑇_𝜆2∶=𝑉_𝜆𝑅_𝜆^∗𝐻₀𝑅_𝜆,𝑇_𝜆3∶=𝑚𝛼_𝑑+1𝑅_𝜆𝑉_𝜆𝑅_𝜆,𝑇_𝜆4∶=𝑚𝑅_𝜆𝑉_𝜆𝑅_𝜆𝛼_𝑑+1,𝑇_𝜆5∶=𝑉_𝜆𝑅_𝜆^∗𝑉_𝜆𝑅_𝜆. (110) We set 𝑣(𝑥)∶=‖𝑉(𝑥)‖,𝑣_𝜆(𝑥)∶=𝑉_𝜆(𝑥),a.e.𝑥∈ℝ^𝑑. (111)

For a compact operator 𝐴 on a Hilbert space, we denote the nonincreasing sequence of the singular values of 𝐴 (repeated with multiplicity) by 𝜇_𝑛(𝐴) (𝑛∈ℕ). For 𝑓∈𝐿^𝑝(ℝ^𝑑), we set ‖𝑓‖_𝐿𝑝∶=(∫_ℝ𝑑|𝑓(𝑥)|^𝑝𝑑𝑥)^1/𝑝.

Lemma 29. Let 𝑑≥3 and suppose that the assumption of Lemma 25 holds and 𝑣∈𝐿^𝑑(ℝ^𝑑)∩𝐿^𝑑/2(ℝ^𝑑). Then, for all 𝑗=1,2,3,4,5, 𝑇_𝜆𝑗is compact. Moreover, there exists a constant 𝐶>0 independent of 𝑉 and 𝜆>0 such that, for all 𝑛∈ℕ, 𝜇_𝑛𝑇_𝜆𝑗≤𝐶𝑣_𝜆𝐿𝑑𝑛^−1/𝑑,𝜇𝑗=1,2_𝑛𝑇_𝜆𝑗≤𝐶𝑣_𝜆^1/22_𝐿𝑑𝑛^−2/𝑑,𝜇𝑗=3,4_𝑛𝑇_𝜆5≤𝐶𝑣_𝜆²_𝐿𝑑𝑛^−2/𝑑. (112)

Proof. By the weak Hausdorff-Young inequality (e.g., [11, page 32]) and the condition 𝑑≥3, one can easily see that the Fourier transform 𝑔_𝜆 of the function: ℝ^𝑑∋𝑘↦(𝑘²+𝜆/2)^−1/2 is in 𝐿^𝑝′_w(ℝ^𝑑) (the weak 𝐿^𝑝′ space on ℝ^𝑑) with 1/𝑝^′=1−1/𝑑 and ‖𝑔_𝜆‖_𝑝′,w≤𝑐_𝑑, where ‖⋅‖_𝑝′,w denotes the “pseudo” norm of 𝐿^𝑝′_w(ℝ^𝑑) and 𝑐_𝑑 is a constant independent of 𝜆>0. By Cwikel’s theorem [14, Section 3] and the condition 𝑣∈𝐿^𝑑(ℝ^𝑑), which implies that𝑣_𝜆∈𝐿^𝑑(ℝ^𝑑),𝑣_𝜆𝑅_𝜆is compact as an operator on 𝐿²(ℝ^𝑑) and 𝜇_𝑛𝑣_𝜆𝑅_𝜆≤𝐾₁𝑣_𝜆𝐿𝑑𝑛^−1/𝑑,𝑛∈ℕ, (113) where 𝐾₁>0 is a constant independent of 𝑉, 𝜆>0 and 𝑛∈ℕ. Since dim𝒦<∞, it follows that 𝑣_𝜆𝑅_𝜆 is compact also as an operator on ℋ. Let 𝐵_𝜆(𝑥)𝑉(𝑥)∶=𝑣(𝑥)if𝑣(𝑥)>𝜆,04𝑚otherwise. (114) Then 𝐵_𝜆 is bounded with ‖𝐵_𝜆(𝑥)‖≤1. We have 𝑉_𝜆𝑅_𝜆=𝐵_𝜆𝑣_𝜆𝑅_𝜆. Hence 𝑉_𝜆𝑅_𝜆 is compact. This shows that all 𝑇_𝜆𝑗 (𝑗=1,2,3,4,5) are compact.
In general, for all compact operators 𝐴 and bounded operators 𝐵 on a Hilbert space 𝜇_𝑛(𝐵𝐴)≤‖𝐵‖𝜇_𝑛(𝐴). (115) (e.g., see [15, Theorem1.6].). Hence 𝜇_𝑛𝑉_𝜆𝑅_𝜆≤𝐵_𝜆𝜇_𝑛𝑣_𝜆𝑅_𝜆≤𝐾₁𝑣_𝜆𝐿𝑑𝑛^−1/𝑑. (116) Therefore 𝜇_𝑛𝑇_𝜆1≤𝐻₀𝑅_𝜆𝐾₁‖𝑣‖_𝐿^𝑑𝑛^−1/𝑑≤𝐾₁𝑣_𝜆𝐿𝑑𝑛^−1/𝑑. (117) Similarly one can show that 𝑇_𝜆2, is compact and 𝜇_𝑛𝑇_𝜆2≤𝐾₁𝑣_𝜆𝐿𝑑𝑛^−1/𝑑, (118) where we have use the fact that 𝜇_𝑛(𝐴)=𝜇_𝑛(𝐴^∗) for all compact operators on a Hilbert space [15, (1.3)].
As for 𝑇_𝜆3, we write 𝑇_𝜆3=𝑚𝛼_𝑑+1𝑅_𝜆𝑣_𝜆^1/2𝐵_𝜆𝑣_𝜆^1/2𝑅_𝜆. (119) By the condition 𝑣∈𝐿^𝑑/2(ℝ^𝑑), 𝑣_𝜆^1/2∈𝐿^𝑑(ℝ^𝑑). Hence, Cwikel’s theorem again, 𝑣_𝜆^1/2𝑅_𝜆 is compact and 𝜇_𝑛𝑣_𝜆^1/2𝑅_𝜆≤𝐾^′₁𝑣_𝜆^1/2_𝐿𝑑𝑛^−1/𝑑, (120) where 𝐾^′₁>0 is a constant independent of 𝑉 and 𝜆>0. We have 𝜇_𝑛𝑇_𝜆3≤𝑚𝜇_𝑛𝑅_𝜆𝑣_𝜆^1/2𝐵_𝜆𝑣_𝜆^1/2𝑅_𝜆. (121) In general, for all compact operators 𝐴 and bounded operators 𝐷 on a Hilbert space, 𝜇_2𝑛+1𝐴^∗𝐷𝐴≤‖𝐷‖𝜇_𝑛+1(𝐴)²,𝜇_2𝑛𝐴^∗𝐷𝐴≤‖𝐷‖𝜇_𝑛(𝐴)², (122) where we have used the fact that, for all compact operators 𝐴 and 𝐵 on a Hilbert space, 𝜇_𝑛+𝑘+1(𝐴𝐵)≤𝜇_𝑛+1(𝐴)𝜇_𝑘+1(𝐵),𝑛,𝑘≥0. (123) Hence 𝜇_2𝑛+1𝑇_𝜆3≤𝑚𝜇_𝑛+1𝑣_𝜆^1/2𝑅_𝜆²≤𝑚𝐾^′₁²𝑣_𝜆^1/22_𝐿𝑑(𝑛+1)^−2/𝑑,𝜇_2𝑛𝑇_𝜆3≤𝑚𝜇_𝑛𝑣_𝜆^1/2𝑅_𝜆²≤𝑚𝐾^′₁²𝑣_𝜆^1/22_𝐿𝑑𝑛^−2/𝑑, (124) which imply that 𝜇_𝑛𝑇_𝜆3≤𝐾^′₂𝑣_𝜆^1/22_𝐿𝑑𝑛^−2/𝑑,𝑛∈ℕ, (125) where 𝐾^′₂>0 is a constant independent of 𝑉,𝜆, and 𝑛. Similarly we have 𝜇_𝑛𝑇_𝜆4≤𝐾^′₂𝑣_𝜆^1/22_𝐿𝑑𝑛^−2/𝑑,𝜇_𝑛𝑇_𝜆5≤𝐾^′₃𝑣_𝜆²_𝐿𝑑𝑛^−2/𝑑,𝑛∈ℕ, (126) where 𝐾^′₃>0 is a constant independent of 𝑉,𝜆 and 𝑛. Thus the desired results follow.

Theorem 30. Let 𝑑≥3, and suppose that the assumption of Lemma 25 holds and ‖𝑉(⋅)‖∈𝐿^𝑑(ℝ^𝑑)∩𝐿^𝑑/2(ℝ^𝑑). Let 𝜆∈(0,𝑚²). Then, there exists a constant 𝐶₀>0 independent of 𝑉 and 𝜆 such that 𝑁(𝜆,𝑉)≤𝐶_{0‖𝑉(𝑥)‖>𝜆/4𝑚}‖𝑉(𝑥)‖^𝑑/2+‖𝑉(𝑥)‖^𝑑𝑑𝑥. (127)

Proof. We need only to consider the case where 𝑁∶=𝑁(𝜆,𝑉)≥1. Then there exists an𝑁-dimensional subspace𝐸ofℋsuch that (106) holds for all𝜓∈𝐸. It is easy to see that ‖(𝑉_𝜆−𝑉)𝜙‖≤(𝜆/4𝑚)‖𝜙‖, for all 𝜙∈ℋ. Let 𝜓∈𝐸. Then, as in the proof of Theorem 28, we have ‖(𝐻_𝑚+𝑉_𝜆)𝜓‖²≤(𝑚²−(𝜆/2))‖𝜓‖², which is equivalent to the following inequality: 𝜆−Δ+2^1/2𝜓²+𝐻₀𝜓,𝑉_𝜆𝜓+𝑉_𝜆𝜓,𝐻₀𝜓+𝑚𝛼_𝑑+1𝜓,𝑉_𝜆𝜓+𝑚𝑉_𝜆𝜓,𝛼_𝑑+1𝜓+𝑉_𝜆𝜓²≤0. (128) The subspace 𝐹∶=(−Δ+𝜆/2)^1/2  𝐸 is also 𝑁-dimensional. Inequality (128) implies that, for all 𝜙∈𝐹, 𝜙²≤𝜙,𝑇_𝜆𝜙, (129) where 𝑇_𝜆∶=−⁵_𝑗=1𝑇_𝜆𝑗. (130) By Lemma 29, 𝑇_𝜆 is a compact self-adjoint operator on ℋ. Hence, by the Hilbert-Schmidt theorem, there exists a complete orthonormal system {𝜙_𝑛}^∞_𝑛=1 of ℋ and a real sequence {𝑡_𝑛}^∞_𝑛=1 such that 𝑇_𝜆𝜙_𝑛=𝑡_𝑛𝜙_𝑛 and lim_𝑛→∞   𝑡_𝑛=0. Using this fact, one sees that the number of eigenvalues 𝑡_𝑛 of 𝑇_𝜆 with 𝑡_𝑛≥1 is more than or equal to dim𝐹=𝑁. Hence 𝜇_𝑁(𝑇_𝜆)≥1. Let𝑘be the largest natural number not exceeding (𝑁+4)/5. Then 5𝑘−4≤𝑁. Hence 1≤𝜇_𝑁(𝑇_𝜆)≤𝜇_5𝑘−4(𝑇_𝜆). On the other hand, by a general fact on singular values of the sum of two compact operators (e.g., [15, Theorem1.7]), we have 𝜇_5𝑘−4𝑇_𝜆≤⁵_𝑗=1𝜇_𝑘𝑇_𝜆𝑗. (131) Using this fact and Lemma 29, we obtain 1≤2𝐶𝑣_𝜆𝐿𝑑𝑘^−1/𝑑+2𝐶𝑣_𝜆^1/22_𝐿𝑑𝑘^−2/𝑑+𝐶𝑣_𝜆²_𝐿𝑑𝑘^−2/𝑑. (132) We have 𝑘≥𝑁/5. Hence 1≤𝐶^′𝑣_𝜆𝐿𝑑𝑁^−1/𝑑+𝑣_𝜆^1/22_𝐿𝑑𝑁^−2/𝑑+𝑣_𝜆²_𝐿𝑑𝑁^−2/𝑑, (133) where 𝐶^′>0 is a constant independent of 𝑉,𝜆, and 𝑁. This implies that 𝑁≤𝐶₀(‖𝑣_𝜆^1/2‖^𝑑_𝐿𝑑+‖𝑣_𝜆‖^𝑑_𝐿𝑑) with a constant 𝐶₀ independent of 𝑉 and 𝜆. Thus (127) holds.

As in Corollaries 1.2 and  1.3 in [13], we have from Theorem 30 the following results.

Corollary 31. Under the same assumption as in Theorem 30, the number 𝑁(𝑉) of eigenvalues of 𝐻(𝑉) in (−𝑚,𝑚) is finite and 𝑁(𝑉)≤𝐶_0ℝ𝑑‖𝑉(𝑥)‖^𝑑/2+‖𝑉(𝑥)‖^𝑑𝑑𝑥. (134)

Corollary 32. Suppose that the assumption of Theorem 30 holds. Let 𝜆_𝑗(𝑗=1,…,𝑁(𝑉)) be the eigenvalues of 𝐻(𝑉) in (−𝑚,𝑚), counted with multiplicity, and let 𝛾>0 be such that 𝑓_𝛾(𝑉)∶=_ℝ𝑑‖𝑉(𝑥)‖^𝛾‖𝑉(𝑥)‖^𝑑/2+‖𝑉(𝑥)‖^𝑑𝑑𝑥<∞. (135) Then, there exists a constant 𝐶_𝛾>0 such that ^𝑁(𝑉)_𝑗=1𝑚²−𝜆²_𝑗^𝛾≤𝐶_𝛾𝑓_𝛾(𝑉). (136)

8.2. Applications

Now we apply the results in the preceeding section to the Dirac operator 𝐻. For 𝜆∈(0,𝑚²), we denote by 𝑁(𝜆) the number of eigenvalues of 𝐻 in (−√𝑚²−𝜆,√𝑚²−𝜆).

Theorem 33. Let 𝑑≥3 and 𝜆∈(0,𝑚²). Suppose that the assumption of Theorem 26 holds. Let 𝐹_𝑀,Φ(𝑥)‖𝑀(𝑥)‖Φ(𝑥)∶=−𝑚+2𝑚sin2,𝑎.𝑒.𝑥∈ℝ^𝑑. (137)(i) If 𝐹_𝑀,Φ(𝑥)≤𝜆/4𝑚,𝑎.𝑒.𝑥∈ℝ^𝑑, then 𝑁(𝜆)=0. (ii) Suppose that 𝐹_𝑀,Φ∈𝐿^𝑑/2(ℝ^𝑑)∩𝐿^𝑑(ℝ^𝑑). Then there exists a positive constant 𝐶>0 independent of 𝑀,Φ, and 𝜆 such that 𝑁(𝜆)≤𝐶_{𝐹𝑀,Φ(𝑥)>𝜆/4𝑚}𝐹_𝑀,Φ(𝑥)^𝑑/2+𝐹_𝑀,Φ(𝑥)^𝑑𝑑𝑥<∞. (138)
Moreover, the number 𝑁₀ of eigenvalues of 𝐻 in (−𝑚,𝑚) obeys 𝑁₀≤𝐶_ℝ𝑑𝐹_𝑀,Φ(𝑥)^𝑑/2+𝐹_𝑀,Φ(𝑥)^𝑑𝑑𝑥<∞. (139)

Proof. (i) We can write 𝐻=𝐻(𝑉) with 𝑉=𝛼_𝑑+1(𝑀𝑒^𝑖Φ−𝑚). Hence ‖𝑉(𝑥)‖=𝑀(𝑥)𝑒^𝑖Φ(𝑥)≤‖𝑀(𝑥)‖𝑒−𝑚−𝑚+𝑚^𝑖Φ(𝑥)−1=𝐹_𝑀,Φ(𝑥). (140) Hence, the present assumption implies that ‖𝑉(𝑥)‖≤𝜆/4𝑚 a.e. 𝑥∈ℝ^𝑑. Hence, by Theorem 28, 𝑁(𝜆)=0.
(ii) By (140) and the present assumption, ‖𝑉(⋅)‖∈𝐿^𝑑(ℝ^𝑑)∩𝐿^𝑑/2(ℝ^𝑑). Thus we can apply Theorem 30 to obtain (138). Inequality (139) follows from (138) or Corollary 31.

We have from Corollary 32 the following fact.

Corollary 34. Let 𝑑≥3. Suppose that the assumption of Theorem 26 and 𝐹_𝑀,Φ∈𝐿^𝑑/2(ℝ^𝑑)∩𝐿^𝑑(ℝ^𝑑). Let 𝜆_𝑗(𝑗=1,…,𝑁₀) be the eigenvalues of𝐻in (−𝑚,𝑚), counted with multiplicity, and let 𝛾>0 be such that 𝑓_𝛾(𝑀,Φ)∶=_ℝ𝑑𝐹_𝑀,Φ(𝑥)^𝛾𝐹_𝑀,Φ(𝑥)^𝑑/2+𝐹_𝑀,Φ(𝑥)^𝑑𝑑𝑥<∞. (141) Then, there exists a constant 𝐶_𝛾>0 such that ^𝑁0_𝑗=1𝑚²−𝜆²_𝑗^𝛾≤𝐶_𝛾𝑓_𝛾(𝑀,Φ). (142)

We can also use Theorems 23 and 27 to obtain another upper bound for 𝑁(𝜆). Let 𝐺_𝑀,Φ(𝑥)∶=𝑀(𝑥)−𝑚−^𝑑_𝑗=1𝛼_𝑑+1𝛼_𝑗𝑒^{𝑖Φ(𝑥)/2}𝐸_𝑗(𝑥),a.e.𝑥∈ℝ^𝑑. (143)

Theorem 35. Let 𝑑≥3, and let 𝜆∈(0,𝑚²). Suppose that the assumption of Theorem 27 holds. Then one has the following. (i) If 𝐺_𝑀,𝜙(𝑥)≤𝜆/4𝑚 for a.e. 𝑥∈ℝ^𝑑, then 𝑁(𝜆)=0. (ii) Suppose that 𝐺_𝑀,Φ∈𝐿^𝑑/2(ℝ^𝑑)∩𝐿^𝑑(ℝ^𝑑). Then (138) and (139) with 𝐹_𝑀,Φ replaced by 𝐺_𝑀,Φ hold.

Proof. By Theorem 23, 𝑁(𝜆) is equal to the number of eigenvalues of 𝐻^′ in (−√𝑚²−𝜆, √𝑚²−𝜆). One can write 𝐻^′=𝐻_𝑚+𝑉 with 𝑉∶=𝛼_𝑑+1(𝑀−𝑚−∑^𝑑_𝑗=1𝛼_𝑑+1𝛼_𝑗𝐴_𝑗). We have ‖𝑉(𝑥)‖=𝐺_𝑀,Φ(𝑥). Thus, in the same way as in the proof of Theorem 30, we obtain the desired results.

Theorem 35 implies the following result as in Corollary 34.

Corollary 36. Let 𝑑≥3. Suppose that the assumption of Theorem 27 holds and 𝐺_𝑀,Φ∈𝐿^𝑑/2(ℝ^𝑑)∩𝐿^𝑑(ℝ^𝑑). Then (142) with 𝐹_𝑀,Φ replaced by 𝐺_𝑀,Φ holds for all 𝜆>0 such that ∫_ℝ𝑑𝐺_𝑀,Φ(𝑥)^𝛾(𝐺_𝑀,Φ(𝑥)^𝑑/2+𝐺_𝑀,Φ(𝑥)^𝑑)𝑑𝑥<∞.

Acknowledgment

Asao Arai is supported by the Grant-in-Aid 24540154 for Scientific Research from JSPS.