ISRN Mathematical Analysis

Volume 2013 (2013), Article ID 913413, 13 pages

http://dx.doi.org/10.1155/2013/913413

Research Article

A Class of -Dimensional Dirac Operators with a Variable Mass

Asao Arai and Dayantsolmon Dagva

Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Received 3 May 2013; Accepted 9 June 2013

Academic Editors: S. Ravindran and C. Zhu

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.

Abstract.

A class of d-dimensional Dirac operators with a variable mass is introduced (), 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 lightand , the Planck constant divided by , are equal to ):

acting in the tensor product Hilbert space of (the Hilbert space of -valued square integrable functions on ) and . Hereis the imaginary unit, , , and are Hermitian matrices obeying the anticommutation relations

(, is the Kronecker delta and denotes theunit matrix), () is the generalized partial differential operator in the space variable (), denotes the mass of a quark, is a Hermitian matrix satisfying

is a function called a profile function, , and are the Pauli matrices, and () is a Borel measurable function on such that for a.e. (almost everywhere) .

Comparingwith the usual free Dirac operator with mass, one notes that the term

corresponds to a mass, although it may depend on the space variable in general. Hence, the CQS model may be regarded as a model of a Dirac particle with a variable mass. We also note that is not a scalar multiple of a constant matrix in general but may be a nontrivial matrix-valued function on . This is one of the interesting features of the Dirac operator . From a general point of view, is a special case of the mass deformation of the form with being a mapping from to the set of linear operators on . 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 . 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 (), the CQS model has supersymmetry; that is, the Dirac operator may be a supercharge of a supersymmetric quantum mechanics (e.g., [6, Chapter]). This structure is carried over to the GCQS model [4].

In this paper, for each natural number , 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 . 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 ofunder a suitable condition. In the last section, we discuss the number of eigenvalues ofin the interval with being a constant.

2. Preliminaries

Let be a complex Hilbert space with inner product (linear in the second variable) and norm (we sometimes omit the subscript if there is no danger of confusion). For a linear operator on , we denote its domain by . If is densely defined, its adjoint is denoted by . For linear operators and on , means that is an extension of , that is, and , for all .

We denote by the set of everywhere defined bounded linear operators on . For , we denote the operator norm of by .

Definition 1. Let and be self-adjoint operators on .

and are said to strongly commute if their spectral measures commute. and are said to strongly anticommute [7, 8] if, for all , (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.

Lemma 2. Let and be self-adjoint operators on .

and strongly commute if and only if, for all , . and strongly commute if and only if, for all , . Let be bounded. Then and strongly commute if and only if . Let be bounded. Then and strongly anticommute if and only if .

Proof. Part (i) is well known (e.g., [9, Theorem ]). 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 be a natural number, and

Let be a separable complex Hilbert space, and

where each means the natural Hilbert space isomorphism and denotes the constant fibre direct integral with fiber (e.g., [10, ]).

We denote bythe generalized partial differential operator in the variable (), acting in . The-dimensional generalized Laplacian

on is a nonpositive self-adjoint operator. Each linear operator on is extended as the direct sum on . For notational simplicity, we denote it by again.

Every densely defined closable linear operator on (resp., ) has a tensor product extension (resp., ) to ( denotes identity). But we write itsimply if there is no danger of confusion.

We denote by the set of mappings from to the set of self-adjoint operators on such that the mapping: is measurable. By a general theorem (e.g., [10, Theorem .85(i)]), for each , the direct integral

is self-adjoint.

Let be Hermitian matrices satisfying

Then the free massless Dirac operator on is defined by

The operator is self-adjoint with and

To introduce a mass operator, let such that, for a.e. , is a bounded operator on , and set

We use this self-adjoint operator as an extended mass (variable in the space ) 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.

In this work, we do not intend to discuss essential self-adjointness ofin full generality. In the present paper, we assume the following.

.. and strongly commute. The operator is -bounded: and

with constants and .

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

Lemma 4.

Condition holds if and only if and strongly anticommute. Condition is equivalent to the operator equality . Condition holds if and only if and strongly commute. Condition is equivalent to the operator equality .

Proof.

This follows from Lemma 2(iv). Assume . Let . Then . Hence, by , . But, since , we have . Hence . Thereore . Thus . Hence the desired operator equality holds. This follows from Lemma 2(iii). Simillar to the proof of part (ii).

We define

If is a constant operator , then represents the free Dirac operator with a constant mass. It is well known (e.g., [6, Theorem]) that is self-adjoint with and bijective with .

Lemma 5. Assume . Let be a constant. Then is bounded with

Proof. It is well known or easy to see that, for all ,

Hence, by, we have

This implies the following: (i) if , then ; (ii) if, then . Thus (17) follows.

Lemma 6. Assume . Then

is self-adjoint with ; is self-adjoint with and the subspace

means algebraic tensor product is a core of .

Proof. (i) Since and is dense, is densely defined. Since is bounded, it follows that . By and Lemma 2(iv), we have

By , , Lemma 2(ii), and Lemma 4(iv), we have

Henceis self-adjoint.

(ii) By , we have for all

Note that

Hence . Here. Thus, by the Kato-Rellich theorem (e.g., [11, Theorem]),is self-adjoint withand every core of is a core of . It is well known that the subspace is a core of as a linear operator on . Hence the subspace defined by (20) is a core of as a linear operator on . Thus it is a core of too.

Remark 7. One of the other sufficient conditions forto be essentially self-adjoint is as follows: assume    and   for all . Then is essentially self-adjoint on . The proof is similar to that of    [6, Theorem].

4. Supersymmetric Aspects

As is well known, the standard free Dirac operator on with constant mass and its suitably perturbed ones have supersymmetry; that is, they are, respectively, a supercharge with the grading operator [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

is self-adjoint with

Sinceis odd, we have

Let be Borel measurable such that, for a.e. , is self-adjoint with

Then

We defineby

Then

is self-adjoint with

Hence is a grading operator on . The following proposition shows that, under some additional condition for , has supersymmetry with respect to .

Proposition 8. Let be odd. Assume . Suppose that is strongly differentiable on with

Then

if and only if

In that case, the spectrum and the point spectrum of are, respectively, symmetric with respect to the origin .

Proof. Since the subspace given by (20) is a core of by Lemma 6(ii), (34) is equivalent to that, for all , and

Let . Then

It follows that the -valued function: is strongly differentiable on with

Hence

which implies that and hence . Moreover, we have

Hence

Therefore, for all if and only if

By the original assumption for , for a.e. . Therefore (42) is equivalent to (35).

By (32) and , one easily sees that (34) is in fact equivalent to operator equality . Hence is unitarily equivalent to . This implies the symmetry of and with respect to the origin.

Remark 9. Proposition 8 gives a generalization of [4, Theorem] and clarifies a condition for to have supersymmetry.

It may be difficult in general to show the existence of self-adjoint, unitary solutions to operator equation (35). Here we only note the following fact.

Lemma 10. Letbe odd. Assume . Suppose that

is independent of and

Then is a solution to (35).

Proof. Since is a constant operator, . By Lemma 2(iii), (43) implies the strong commutativity of and . Hence for a.e. . By (44) and Lemma 2(iv), . We also have (45) and the strong commutativity of and . Hence

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 () be a densely defined closed linear operator on . Suppose that

Then, for all , on .

Proof. Equation (47) implies that, for all and , . Since is linearly independent, it follows that . Hence .

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

Lemma 12. Let be odd. Assume . Let be strongly differentiable on with (33) and be a solution to (35). Suppose that (43)–(45) hold. Then is independent of .

Proof. As in the proof of Lemma 10, (43)–(45) imply (46). Hence the right-hand side of (35) vanishes, so that , which implies that . By Lemma 11, , which implies that is independent of .

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

Corollary 13. Let be odd. Assume . Suppose that is independent of and that (43)–(45) hold. Then has supersymmetry with respect to .

5. Vanishing Theorems of the Kernel of

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

To investigate ker , we also need an additional condition.

(i) For each , the function: is strongly differentiable onand, for all , commutes with ().   There exists a constant such that

as an operator inequality on (note that, by the principle of uniform boundedness, the strong partial derivative is a bounded operator on for each and hence, under and condition (i), the operator on is a bounded self-adjoint operator).

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

Lemma 14. Assume , , and . Then defined by (16) is self-adjoint with and

In particular, with operator-norm bound

and is bounded with

Moreover, is bounded with

Proof. The self-adjointness of follows from that of with . For all , using the anticommutativity of with and the commutativity of with and (), we have

Hence (49) holds for all . Since is a core of , this inequality extends to all . In particular, we have

This implies that the self-adjoint operator is bijective with (50).

Inequality (49) implies also that, for all , . Hence is bounded with (51).

Byand for all , we have . Hence, for all,

Thus (52) holds.

Lemma 15. Let be a self-adjoint operator on a complex Hilbert space . Then

Proof. By the functional calculus, one has . Hence . Since is unitary, one has . Thus (56) holds.

Theorem 16. Assume and

Then ker  and .

Moreover, the constant

is strictly positive, or , and

Proof. The operator is written as

with

By applying Lemma 15 with , we have

Therefore, for all ,

By this estimate and (52), we obtain

Hence, by (57), we obtain . This implies that is bijective with bounded inverse . Thus is bijective with being bounded. Hence and .

We set . If , then is in . Therefore

It is obvious that, for all with , . This implies that . On the other hand, we have from (58) , for all . Hence . Therefore . Thus (59) holds and . Since or , it follows that or.

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

We next consider a perturbation of . Let such that, for a.e. , is bounded and strongly commutes with and . Then, for a.e. ,

is self-adjoint on and

is a self-adjoint operator on .

The quantity

may be infinite. But we have the following.

Lemma 18. Under the assumption of Theorem 16, .

Proof. Sinceis closed with and , it follows from the closed graph theorem that there exists a constant such that

Let with . Then, by Theorem 16, we have . Hence

Therefore . If , then for all . Hence . But this contradicts condition .

Theorem 19. Assume and (57). Suppose that

Let

Then ker   and . Moreover, the last statement on and in Theorem 16 holds with being replaced by .

Proof. We write

By the strong commutativity of and , we have for a.e.

Hence, for all

We have . Hence

with

Hence is -bounded. By Remark 17, is Fredholm. Condition (71) is equivalent to . Hence, by a stability theorem (e.g., [12, Chapter, Theorem]), is Fredholm and dim ker dim ker . Therefore ker . It follows from this fact and the self-adjointness of that Ran . Hence . 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

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 :

where

denotes the strong partial derivative of in . For , one can define a bounded linear operator

on .

Remark 20. If sucht that and commute for a.e. , then and hence

Lemma 21. For each , is a bounded self-adjoint operator on .

Proof. Since is bounded. We have

Differentiating the identity in , we have

Hence .

For , we define an operator:

Lemma 22. Assume . Let . Suppose that

Then is self-adjoint and every core of is a core of .

Proof. Under condition, is self-adjoint. By (85), we have (). Hence, by Lemma 21, is a bounded self-adjoint operator. Hence the Kato-Rellich theorem yields the desired result.

We note that, if one regards as a (noncommutative) gauge potential, then is a gauge theoretic Dirac operator with gauge potential .

Let

which is unitary. The following theorem shows that, under a suitable condition, is unitarily equivalent to a gauge theoretic Dirac operator .

Theorem 23. Assume and (85). Let . Then

Proof. We have

By (85) and Lemma 2, . Byand Lemma 2, . Byand Lemma 2(iv), . Moreover,

Hence (87) holds.

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

Theorem 24. Assume and (85). Let and

Then and .

Proof. We write with . Then

By this estimate and (90), . Hence is bijective and . In particular, ker . On the other hand, (87) implies thatand ker ker . Thus and ker .

7. Essential Spectrum of

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

Lemma 25. Let and be a constant. Let be Borel measurable satisfying the following conditions.

The operator is relatively bounded with respect to.. The operator on is self-adjoint.

Then

Proof. For each , we denote by the characteristic function of the set . As in the case of the 3-dimensional free Dirac operator (cf. [6, Lemma]), one can show that is compact for all as an operator on . Since, it follows thatis compact as an operator on. Hence, for all , is compact. Since is self-adjoint with and is closed, it follows from the closed graph theorem that is bounded (this can be shown by direct estimates too). Therefore we have

where, . By the fact mentioned previously, is compact. We have

By condition (ii), for every , there exists a constant such that, for all a.e. with , , that is, , which implies that . Hence . Therefore is compact. Hence, by Weyl’s essential spectrum theorem (e.g., [10, Theorem]), . On the other hand, as in the case of the 3-dimensional free Dirac operator [6, Theorem], one can show that . Thus (92) holds.

Theorem 26. Let . Assume . Suppose that there exists a constant satisfying

Then

Proof. We write

with

It is obvious that and are relatively bounded with respect to and

As for , we have

Hence, by (95) and (96), we have . Therefore . Thus we can apply Lemma 25 to obtain (97).

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

Theorem 27. Let . Assume (85) and (95). Let . Suppose that

Then (97) holds.

Proof. By (87), we have . Hence we need only to prove

We write

We have . Moreover, . Hence . 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 ofin 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 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.

8.1. A General Case

Let be as in Lemma 25 and

Then, by (92), an eigenvalue of in (if it exists) is an isolated eigenvalue of with finite multiplicity. For each , we denote by the number of eigenvalues in the interval .

We first note an elementary fact:

Theorem 28. Suppose that the assumption of Lemma 25 holds and that for a.e. . Then .

Proof. Suppose that . Then, it follows from the definition of that there exists an-dimensional subspace of such that

Hence

Hence , which is equivalent to . This implies that . But this is a contradiction.

In view of Theorem 28, we define, for each , by

For each , the operator

is a bounded self-adjoint operator. Since is -bounded, where is defined by (10) and is bounded, it follows that and are bounded operators on . Also is bounded with . Hence the following operators () are in :

We set

For a compact operator on a Hilbert space, we denote the nonincreasing sequence of the singular values of (repeated with multiplicity) by (). For , we set .

Lemma 29. Let and suppose that the assumption of Lemma 25 holds and . Then, for all , is compact. Moreover, there exists a constant independent of and such that, for all ,

Proof. By the weak Hausdorff-Young inequality (e.g., [11, page 32]) and the condition , one can easily see that the Fourier transform of the function: is in (the weak space on ) with and , where denotes the “pseudo” norm of and is a constant independent of . By Cwikel’s theorem [14, Section 3] and the condition , which implies that,is compact as an operator on and

where is a constant independent of , and . Since , it follows that is compact also as an operator on . Let

Then is bounded with . We have . Hence is compact. This shows that all () are compact.

In general, for all compact operators and bounded operators on a Hilbert space

(e.g., see [15, Theorem].). Hence

Therefore

Similarly one can show that , is compact and

where we have use the fact that for all compact operators on a Hilbert space [15, ].

As for , we write

By the condition , . Hence, Cwikel’s theorem again, is compact and

where is a constant independent of and . 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

which imply that

where is a constant independent of , and . Similarly we have

where is a constant independent of , and . Thus the desired results follow.

Theorem 30. Let , and suppose that the assumption of Lemma 25 holds and . Let . Then, there exists a constant independent of and such that

Proof. We need only to consider the case where . Then there exists an-dimensional subspaceofsuch that (106) holds for all. It is easy to see that , for all . Let . Then, as in the proof of Theorem 28, we have , which is equivalent to the following inequality:

The subspace    is also -dimensional. Inequality (128) implies that, for all ,

where

By Lemma 29, is a compact self-adjoint operator on . Hence, by the Hilbert-Schmidt theorem, there exists a complete orthonormal system of and a real sequence such that and    . Using this fact, one sees that the number of eigenvalues of with is more than or equal to . Hence . Letbe the largest natural number not exceeding . Then . Hence . On the other hand, by a general fact on singular values of the sum of two compact operators (e.g., [15, Theorem]), we have

Using this fact and Lemma 29, we obtain

We have . Hence

where is a constant independent of , and . This implies that with a constant independent of and . 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 of eigenvalues of in is finite and

Corollary 32. Suppose that the assumption of Theorem 30 holds. Let be the eigenvalues of in , counted with multiplicity, and let be such that

Then, there exists a constant such that

8.2. Applications

Now we apply the results in the preceeding section to the Dirac operator . For , we denote by the number of eigenvalues of in .

Theorem 33. Let and . Suppose that the assumption of Theorem 26 holds. Let

If , then . Suppose that . Then there exists a positive constant independent of , and such that

Moreover, the number of eigenvalues of in obeys

Proof. (i) We can write with . Hence

Hence, the present assumption implies that a.e. . Hence, by Theorem 28, .

(ii) By (140) and the present assumption, . 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 . Suppose that the assumption of Theorem 26 and . Let be the eigenvalues ofin , counted with multiplicity, and let be such that

Then, there exists a constant such that

We can also use Theorems 23 and 27 to obtain another upper bound for . Let

Theorem 35. Let , and let . Suppose that the assumption of Theorem 27 holds. Then one has the following.

If for a.e. , then . Suppose that . Then (138) and (139) with replaced by hold.

Proof. By Theorem 23, is equal to the number of eigenvalues of in , . One can write with . We have . 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 . Suppose that the assumption of Theorem 27 holds and . Then (142) with replaced by holds for all such that .

Acknowledgment

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

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