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

Title A Class of d-Dimensional Dirac Operators with a Variable Mass Author(s) Arai, Asao; Dagva, Dayantsolmon Citation ISRN Mathematical Analysis, 2013: 1-13 Issue Date 2013-06-19 Doc URL http://hdl.handle.net/2115/52894 Rights Copyright © 2013 Asao Arai and Dayantsolmon Dagva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Rights(URL) http://creativecommons.org/licenses/by-nc-sa/2.1/jp/ Type article File Information 913413.pdf

Comparing  CQS with the usual free Dirac operator with mass, one notes that the term  (x) :=   ∑ 3 =1 (x) 5 ⊗    (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 (x) is not a scalar multiple of a constant matrix in general but may be a nontrivial matrix-valued function on R 3 .This is one of the interesting features of the Dirac operator  CQS .From a general point of view, (x) is a special case of the mass deformation of the form   (x) := (x) with  being a mapping from R 3 to the set of linear operators on C 4 ⊗ C 2 .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  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   (x) ( = 1, 2, 3), the CQS model has supersymmetry; that is, the Dirac operator  CQS may be a supercharge of a supersymmetric quantum mechanics (e.g., [6,Chapter 5]).This structure is carried over to the GCQS model [4].
In this paper, for each natural number  ≥ 2, we propose a -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  which may be the Hamiltonian of a -dimensional version of the GCQS model, as mentioned previously.A simple condition for  to be self-adjoint is given.In Section 4 we discuss supersymmetric aspects of .We give a condition for  to be a supercharge of a supersymmetric quantum mechanical model.In that case, ker , the kernel of , describes the supersymmetric states.Hence, it is interesting and important to analyze ker .In Section 5, we prove that, under a condition, ker  is trivial: ker  = {0}.In the case where  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  to a gauge theoretic Dirac operator.This may be physically interesting.Using this structure, we find another condition for the kernel of  to be trivial.In Section 7, we identify the essential spectrum of  under a suitable condition.In the last section, we discuss the number of eigenvalues of  in the interval (−, ) with  > 0 being a constant.

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  on X, we denote its domain by ().If  is densely defined, its adjoint is denoted by  * .For linear operators  and  on X,  ⊂  means that  is an extension of , that is, () ⊂ () and  = , for all  ∈ ().
We denote by B(X) the set of everywhere defined bounded linear operators on X.For  ∈ B(X), we denote the operator norm of  by ‖‖.Definition 1.Let  and  be self-adjoint operators on X.
(i)  and  are said to strongly commute if their spectral measures commute.
(ii)  and  are said to strongly anticommute [7,8] if, for all  ∈ R,    ⊂  − (it is shown that this definition is in fact symmetric in  and ).
The next lemma summarizes some basic facts on strongly commuting (resp., anticommuting) self-adjoint operators.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).

Description of the Model
Let  ≥ 2 be a natural number, and for  even, 2 (+1)/2 for  odd. ( Let K be a separable complex Hilbert space, and where each ≅ means the natural Hilbert space isomorphism and ∫ ⊕ R  C   ⊗ K  denotes the constant fibre direct integral with fiber C   ⊗ K (e.g., [10,section XIII.16]).
Every densely defined closable linear operator  on  2 (R  , C   ) (resp., K) has a tensor product extension  ⊗  (resp.,  ⊗ ) to H ( denotes identity).But we write it  simply if there is no danger of confusion.
We denote by F s.a. the set of mappings Φ(⋅) from R  to the set of self-adjoint operators on C   ⊗ K such that the mapping: R  ∋   → (Φ() + ) −1 is measurable.By a general theorem (e.g., [10, Theorem XIII.85(i)]), for each Φ(⋅) ∈ F s.a., the direct integral is self-adjoint.Let {  } +1 =1 be   ×   Hermitian matrices satisfying Then the free massless Dirac operator on  2 (R  ; C   ) is defined by The operator  0 is self-adjoint with ( 0 ) := ∩  =1 (  ) and To introduce a mass operator, let (⋅) ∈ F s.a.such that, for a.e. ∈ R  , () is a bounded operator on C   ⊗ K, and set We use this self-adjoint operator as an extended mass (variable in the space R  ) of the quantum particle of our model (a Dirac particle).Note that  is not necessarily bounded.
The Hamiltonian  of our model, a -dimensional version of the GCQS model, is defined as follows: with As remarked previously, the mass operator  in  can be variable spatially.This is a point different from the GCQS model.
Remark 3. In the abstract CQS model [5], the strong commutativity of  and  0 as well as the boundedness and the strict positivity of  is assumed.But, in our model, they are not assumed.
(iv) Simillar to the proof of part (ii).
We define If  is a constant operator  > 0, then   represents the free Dirac operator with a constant mass .It is well known (e.g., [6, Theorem Proof.It is well known or easy to see that, for all  ∈ (  ) = ( 0 ), Hence, by (A.4), we have This implies the following: (ii)  is self-adjoint with () = ( 0 ) and the subspace (⊗ means algebraic tensor product) is a core of .
Remark 7. One of the other sufficient conditions for  to be essentially self-adjoint is as follows: assume (A.1)-(A.3)and ess sup ||< ‖()‖ < ∞ for all  > 0. Then  is essentially self-adjoint on D 0 .The proof is similar to that of [6,Theorem 4.3].

Supersymmetric Aspects
As is well known, the standard free Dirac operator − ∑ 3 =1     +  on  2 (R 3 ; C 4 ) with constant mass  ≥ 0 and its suitably perturbed ones have supersymmetry; that is, they are, respectively, a supercharge with the grading operator  5 [6,Section 5.5].From this point of view, it would be interesting to investigate if the Hamiltonian  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  is odd.The matrix Since  is odd, we have Let  : R  → B(K) be Borel measurable such that, for a.e. ∈ R  , () is self-adjoint with () 2 = . (28) We define Γ(⋅) : Then is self-adjoint with Hence Γ is a grading operator on H.  ( In that case, the spectrum () and the point spectrum  p () of  are, respectively, symmetric with respect to the origin 0 ∈ R.
The following lemma gives a sufficient condition for a solution to (35) to be a constant operator.
We have from Proposition 8 and Lemma 10 the following result.

Vanishing Theorems of the Kernel of 𝐻
In supersymmetric quantum mechanics with a supercharge , a nonzero vector in ker  is called a supersymmetric states.If the kernel of  vanishes, that is, ker  = {0}, then the supersymmetry is said to be spontaneously broken.It turns out that, in supersymmetric quantum mechanics, it is importanat to investigate ker .Thus we are led to consider ker  in view of Proposition 8.This would be interesting even if  does not ISRN Mathematical Analysis have supersymmetry (note that  does not necessarily have supersymmetry).
To investigate ker , we also need an additional condition.
Remark 17.Under the same assumption as in Theorem 16,  is Fredholm (the proof is easy).

Unitary Equivalence to a Gauge Theoretic Dirac Operator and a Vanishing Theorem for ker 𝐻
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(⋅): is strongly differentiable and sup where denotes the strong partial derivative of  −Φ()/2 in   .For Φ(⋅) ∈ F, one can define a bounded linear operator Remark 20.If Φ(⋅) ∈ F sucht that Φ() and Φ(  ) commute for a.e.,   ∈ R  , then   () = − −Φ()/2   Φ()/2 and hence Lemma 21.For each  = 1, . . ., ,   is a bounded self-adjoint operator on H.
We note that, if one regards A := ( 1 , . . .,   ) as a (noncommutative) gauge potential, then   is a gauge theoretic Dirac operator with gauge potential A. Let which is unitary.The following theorem shows that, under a suitable condition,  is unitarily equivalent to a gauge theoretic Dirac operator   .
The following theorem gives another sufficient condition for ker  to be trivial.

Essential Spectrum of 𝐻
In this section, we consider the essential spectrum of .For a self-adjoint operator  on a Hilbert space, we denote by  ess () the essential spectrum of .
Lemma 25.Let dim K < ∞ and  > 0 be a constant.Let (⋅) : R  → B(C   ⊗ K) be Borel measurable satisfying the following conditions.

Bounds on the Number of Discrete Eigenvalues
In this section, in view of Theorem 26, we consider the number of eigenvalues of  in the interval (−, ) 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   +  with a scalar potential  : R 3 → R in (−, ) 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.
where we have use the fact that   () =   ( * ) for all compact operators on a Hilbert space [15, (1.3)].
As for  3 , we write is compact and where   1 > 0 is a constant independent of  and  > 0. We have In general, for all compact operators  and bounded operators  on a Hilbert space, where we have used the fact that, for all compact operators  and  on a Hilbert space, Hence where   3 > 0 is a constant independent of ,  and .Thus the desired results follow.
As in Corollaries 1.2 and 1.3 in [13], we have from Theorem 30 the following results.