Abstract

Spectral properties of a special class of infinite dimensional Dirac operators on the abstract boson-fermion Fock space associated with the pair of complex Hilbert spaces are investigated, where is a perturbation parameter (a coupling constant in the context of physics) and the unperturbed operator is taken to be a free infinite dimensional Dirac operator. A variety of the kernel of is shown. It is proved that there are cases where, for all sufficiently large with , has infinitely many nonzero eigenvalues even if has no nonzero eigenvalues. Also Fredholm property of restricted to a subspace of is discussed.

1. Introduction

In a previous paper [1] (cf. [2]), the author introduced a general class of infinite dimensional Dirac type operators on the abstract boson-fermion Fock space associated with the pair of complex Hilbert spaces (for the definition of and , see Section 2), where is a densely defined closed linear operator from to . The operator gives an infinite dimensional and abstract version of finite dimensional Dirac type operators. In applications to physics, unifies self-adjoint supercharges (generators of supersymmetry) of some supersymmetric quantum field models (e.g., [3–6]).

In the paper [1], basic properties of are discussed. As for the spectral properties of , only the zero-eigenvalue of is considered. Moreover, a class of perturbations for is introduced and, under a suitable condition, a path (functional) integral representation for the index of the perturbed operator restricted to a subspace of is established. The (essential) self-adjointness of is partially discussed in [7]. Analysis of other properties of including spectral ones except for the zero-eigenvalue has been left open.

In this paper, we undertake a comprehensive operator-theoretical analysis for . But, as a preliminary, we consider, in the present paper, only the case where is a simple form and see what kind of phenomena occurs under the perturbation .

The outline of the present paper is as follows. In Section 2 we review briefly some contents in [1]. In Section 3 we identify the spectra of . This is done via spectral analysis of a second quantization operator on the boson-fermion Fock space . In Section 4, we introduce a perturbation with and , where is a perturbation parameter (a coupling constant in the context of physics). The perturbed operator is the main object of our analysis in the present paper. We prove the self-adjointness of and the essential self-adjointness of it on a suitable dense subspace of with some other properties (Theorem 14). Also the spectra of are identified (Theorem 16). Moreover, it is shown that the domain of is equal to that of for all sufficiently small (Theorem 17). Section 5 is devoted to analysis of the kernel of . We see that the kernel property of may be sensitive to conditions for . In Section 6 we consider Fredholm property of , the restriction of to a subspace of , in comparison with that of . We obtain some classification on Fredholm property of (Theorem 26). An interesting phenomenon occurs in the following sense: for a constant (resp., ), is not semi-Fredholm even if is Fredholm (resp., semi-Fredholm). In the last section, we consider nonzero eigenvalues of . We show that, under a suitable condition, for all sufficiently large with , has infinitely many nonzero eigenvalues even if has no nonzero eigenvalues (Theorem 27). This structure also is interesting, which may be regarded as a strong coupling effect.

2. Brief Review on the Theory of Dirac Operators on the Abstract Boson-Fermion Fock Space

2.1. Definitions

For a linear operator on a Hilbert space, we use the following notation:(i): the domain of ,(ii): the range of ,(iii): the kernel of ,(iv): the resolvent set of ( is injective with being dense and the inverse operator is bounded),(v): the spectrum of ,(vi): the point spectrum of (the set of eigenvalues of ).

For a complex Hilbert space and a nonnegative integer , we denote by (resp., ) the -fold symmetric (resp., antisymmetric) tensor product Hilbert space of with convention (the complex number field) and . In terms of these Hilbert spaces, one can construct two types of larger Hilbert spaces. The one is the boson (symmetric) Fock space over , denoted by , which is defined to be the complete infinite direct sum of , : where, for a complex Hilbert space , we denote its inner product (resp., norm) by (linear in the second variable) (resp., ). The other is the fermion (antisymmetric) Fock space over , denoted by , which is defined to be the complete infinite direct sum of , : For general theories of boson and fermion Fock spaces, see, for example, [8–10].

Let be a complex Hilbert space. Then the abstract boson-fermion Fock space associated with the pair is defined as the tensor product Hilbert space of and .

For each , we denote by the bosonic annihilation operator on with test vector ; that is, is the densely defined closed linear operator acting in such that its adjoint is given by the following form: where is the symmetrization operator acting on the -fold tensor product Hilbert space .

Let called the bosonic Fock vacuum. Then, for all ,

For each , there exists a unique linear operator on , called the fermionic annihilation operator with test vector , which has the following properties: (i) and is bounded with operator norm ; (ii) its adjoint is of the following form: where is the antisymmetrization operator acting on the -fold tensor product Hilbert space . The following canonical anticommutation relations (CAR) hold: where . In particular, one has

The vector is called the fermionic Fock vacuum and obeys

2.2. Infinite Dimensional Exterior Differential Operators

For a subset of a Hilbert space, we denote by the subspace algebraically spanned by all the vectors of .

Let be a densely defined closed linear operator from to and define where for a linear operator on a Hilbert space (the symbol with should read ). It follows that is dense in .

The following proposition is proved in [1].

Proposition 1. There exists a unique densely defined closed linear operator on such that the following (i) and (ii) hold: (i) and is a core of .(ii)For each vector of the form
acts as  where indicates the omission of . In particular, leaves invariant.
Moreover, the following (iii)–(v) hold: (iii) and for all vectors of the form (13) with . In the case , we have . In particular, leaves invariant.(iv) and, for all , .(v)Let be a bounded linear operator from to with . Then, for all and ,

The operator is an abstract version of finite dimensional exterior differential operators, including an infinite dimensional version of them if and are infinite dimensional.

2.3. Infinite Dimensional Dirac Operators

Using the operator , we define a free Dirac operator on by with . We set a Laplace-Beltrami operator on .

For a self-adjoint operator on , we define an operator on by where denotes identity and, for a closable linear operator , denotes the closure of . For each , is self-adjoint. The second quantization of on is defined by which is self-adjoint (see, e.g., [9, Section 5.2], [11, p.302, Example 2]). It follows that

Similarly, for a self-adjoint operator on , we can define the second quantization of on by where acting in . Similarly to the case of , one has

The number operator on is defined by Then the operator on is self-adjoint with . The spectrum of is with eigenspaces giving the orthogonal decomposition

Let and be the orthogonal projections onto and , respectively. Then

For notational simplicity we set We introduce an operator acting in , which is nonnegative and self-adjoint. It follows from (21) and (24) that

Theorem 2 (see [1]). (i) The operator is self-adjoint and essentially self-adjoint on every core of . In particular, is essentially self-adjoint on .
(ii) The operator leaves invariant and
on .
(iii) The following operator equalities hold:

Remark 3. The quadruple is an abstract form of models of free supersymmetric quantum field theory, where , , and are a self-adjoint supercharge, a supersymmetric Hamiltonian, and a sign operator of states (a “fermion number operator”), respectively [1].

For a linear operator , we set the nullity of .

Theorem 4 (see [1, Lemma 3.4]).
Hence the following hold. (i)Let and . Then .(ii)Let and . Then .(iii)Let . Then .

2.4. Fredholm Property and Index

Let be a self-adjoint operator on such that leaves invariant satisfying on . This class of operators may be an abstract category of Dirac operators on the abstract boson-fermion Fock space .

Then it is easy to show that the operator defined by is a densely defined closed linear operator from to . We define an index of by provided that at least one of and is finite. Note that the index of (e.g., [12, Chapter IV, Section 5]).

As for the Fredholm property of , one has the following theorem.

Theorem 5 (see [1, Proposition 3.2]). (i) Let be injective and Fredholm. Then is Fredholm with
(ii) Let be semi-Fredholm with and let be injective. Then is semi-Fredholm with
(iii) Let be semi-Fredholm with and . Then is not semi-Fredholm.

3. Spectra of and

Now we go into analyzing new aspects in the operator theory of and its perturbations. In the previous paper [1], the identification of the spectra of and was not discussed except for zero-eigenvalues. This section is devoted to identifying completely the spectra of and .

In what follows, we assume that and are separable.

We say that a subset of is symmetric with respect to the origin if implies that .

The main results in this section are as follows.

Theorem 6.

Theorem 7. The spectrum and the point spectrum of are symmetric with respect to the origin andwith

To prove these theorems, we need a series of lemmas.

Lemma 8. Let be a self-adjoint operator on a Hilbert space. (i)Suppose that is symmetric with respect to the origin. Then (ii)Suppose that is symmetric with respect to the origin. Then Moreover, for each with , .

Proof. (i) Let be the set on the right hand side of (46). Then, by the spectral mapping theorem, we have . Conversely, let , so that . Hence there exists a sequence such that and . Suppose that . Then is bijective with bounded. Hence , which means that . Hence . Then, by the symmetry of , is in . But this is a contradiction. Hence . Thus .
(ii) Let be the set on the right hand side of (47). Then it is obvious that . Conversely, let , so that . Hence there exists a vector such that and . The second equation is written as . If , then . Hence, by the symmetry of , is in . If , then . Hence . Thus . These arguments also imply the last statement.

A sufficient condition for the spectra of a self-adjoint operator to be symmetric with respect to the origin is given in the following lemma.

Lemma 9. Let be a self-adjoint operator on a Hilbert space . Suppose that there exists a bounded self-adjoint operator on such that and leaves invariant satisfying Then and are symmetric with respect to the origin and, for all ,

Proof. Note that is unitary with . Equation (48) implies the operator equality . Hence is unitarily equivalent to . Thus the desired results follow.

Lemma 10. Let    be a complex Hilbert space and be a self-adjoint operator on . Let be the direct sum of ’s acting in the direct sum Hilbert space . Then, is self-adjoint and

Proof. See, for example, [8, Proposition 3.5-(ii), Theorem 3.7-(i)].

Lemma 11. Let be a self-adjoint operator on . Then

Proof. The following are well known (e.g., [8, Proposition 4.12]): and, for all , By these facts and Lemma 10, we obtain (51) and (52).

For a self-adjoint operator on a Hilbert space and , we define where, for a finite set , denotes the number of the elements of and denotes the discrete spectrum of (the set of isolated eigenvalues of with finite multiplicity).

Lemma 12 (see [13, Corollary 3.2]).

Proof of Theorem 6. By a theorem on the spectrum of the sum of tensor products of self-adjoint operators (e.g., [14, Corollary 7.25]), where we have used the fact that the set on the right hand side is closed (note that , ). By (51) and (55), with By Deift’s theorem [15], we have Hence and . Therefore it follows that . The converse inclusion relation is obvious. Thus . The set on the right hand side is equal to the set on the right hand side of (42). Thus (42) holds.
The proof of (43) is similar to that of (42), where we use another version of Deift's theorem [15]: with for each positive eigenvalue of ().

Proof of Theorem 7. By Theorem 2(ii) and Lemma 9, and are symmetric with respect to the origin and (45) holds. By Lemma 8 and Theorem 2(iii), we have By these facts and Theorem 6, we obtain (44).

4. A Class of Perturbed Infinite Dimensional Dirac Operators

In this section we consider a class of perturbations of . It would be natural to perturb through a perturbation of . Let , and define where is a perturbation parameter (a coupling constant in the context of physics). We have by direct computation The adjoint is densely defined with and Hecne is closable. We denote the closure of by . Then, for all , is in and

The operator yields a perturbed Dirac operator: By an application of [2, Lemma 4.1], one sees that is a closed symmetric operator. We have with

In what follows, we investigate properties of .

4.1. Self-Adjointness of

We define a linear operator from to by It is obvious that is a bounded operator (a one-rank operator). Hence is a densely defined closed linear operator with .

The following lemma is a key fact.

Lemma 13. For all , the following operator equality holds:

Proof. Let be a vector of the form (13). Then, using the canonical commutation relations (CCR) on and (6), we have Hence, by Proposition 1-(v) By a limiting argument, we obtain the operator equality (72).

Theorem 14. (i) For all , is self-adjoint and
(ii) For all , is essentially self-adjoint on .
(iii) For all ,
(iv) The operator leaves invariant and
(v) For all , the vector-valued function: is strongly continuous on . Moreover, for all , is strongly continuous in .

Proof. (i) Operator equality (77) follows from (72) and (67). Then, by Theorem 2-(i) with replaced by ), we conclude that is self-adjoint.
(ii) We have the following operator equalities: where and are bounded linear operators on and , respectively, given as follows: Hence, by the Kato-Rellich theorem ([12, Chapter V, Theorems 4.3, 4.4], [10, Theorem ]), (resp., ) is essentially self-adjoint on each core of (resp., ). In particular, (resp., ) is a core of (resp., ). Hence, by a core property of second quantization operators, is a core of . By this fact and Theorem 2-(i), is a core of .
(iii) This follows from (ii) and (68).
(iv) This follows from (77) and Theorem 2-(ii) with replaced by .
(v) Let . Then, for all , we have Since is a common core for as proved in part (ii), a general fact [11, Theorem VIII.25] implies the strong continuity of () in .

Remark 15. Since the operators and are finite rank, they are compact. Hence it follows from (80) and a general theorem (e.g., [16, p.113, Corollary 2]) that, for all , where, for a self-adjoint operator , denotes the essential spectrum of . In particular, if the spectrum is purely discrete (in this case, also is purely discrete by (60)), then so are and .

4.2. Spectra of

We have (77). Hence, applying Theorem 7 with replaced by , we obtain the following result on the spectra of .

Theorem 16. For all , and are symmetric with respect to the origin andwith

This theorem shows that the spectrum and the point spectrum of are completely determined from those of .

4.3. Identification of the Domain of

The following theorem gives a sufficient condition for the domain of to be equal to that of the unperturbed operator .

Theorem 17. Suppose that is injective and . Then, for all , is self-adjoint with and Moreover, is essentially self-adjoint on any core for .

To prove this theorem, we recall well known facts (e.g., [17, Lemma 2.1], [8]).

Lemma 18. Let be a nonnegative self-adjoint operator on with . Then, for all , with

Lemma 19. The operator is a closed symmetric operator with .

Proof. We need only to prove the closedness of . Using (9) and a limiting argument, we have, for all , Hence, by a general fact [2, Lemma 4.1], is closed.

Lemma 20. Let be injective and . Then .

Proof. Under the present assumption, is injective and . Let . Then, using (87) and the well known fact that we can show that By this estimate and the fact that we obtain Since is a core of , this inequality extends to all showing that . Similarly, using (88), we can show that with Thus .

Proof of Theorem 17. By Lemma 20 and Theorem 14, the operator is essentially self-adjoint with and . By (89), the CCR (73), and the CAR (8), one can show that, for all , Hence, using (87) (cf. the proof of Lemma 20), we obtain Therefore, by the Kato-Rellich theorem, for all satisfying , is self-adjoint on , implying that , and essentially self-adjoint on any core for .

5. Kernel of

In this section we investigate the kernel of . For this purpose, we need some conditions on .  is injective, , and . In this case we introduce a constant   is injective, , and . In this case we introduce a constant   (a) is injective and or (b) is injective and with .  (a) is injective and or (b) is injective and with .

Remark 21. (i) If and hold, then it follows that .
(ii) If is Fredholm, then is Fredholm [12, Chapter IV, Corollary 5.14]. Hence, if and hold and is Fredholm, then , . Hence and are bijective with and . Thus, in this case, conditions and are trivially satisfied.

We first consider the kernel of and .

Lemma 22. (i) Suppose that holds. Then
(ii) Suppose that holds. Then
(iii) Suppose that holds. Then, for all ,
(iv) Suppose that holds. Then, for all ,

Proof. (i) Let . Then and Hence with . This implies
Let . Then, taking the inner product of (106) with and using the fact that , we have . Hence, by (106) again, . Thus (100) follows.
Let . Then Hence . Thus, by (107), we obtain (101).
(ii) We have Hence, by replacing by in the proof of (i), we can easily obtain (102).
(iii) For all , we have (105). Let . Then (105) means that . Hence, if , then , which implies via (105) again that ; that is, . Thus (103) follows.
If with , then and hence . Hence . Thus (103) follows.
(iv) It is similar to the proof of (iii).

Theorem 23. (i) Assume . Then and hence .
Moreover, for all ,
(ii) Assume . Then
(iii) Assume . Then, for all ,
(iv) Assume . Then, for all ,

Proof. Under condition , we have (100) and (101). Hence . Therefore, applying (77) and Theorem 4, we obtain (110).
In the case , (111) follows from (100), (77), and Theorem 4.
Using Lemma 22, one can prove (ii)–(iv) in the same manner as in the proof of (i).

Some compatible compositions of conditions yield the following results.

Corollary 24. (i) Assume and . Then
(ii) Assume and . Then
(iii) Assume and . Then
(iv) Assume and . Then, for all ,

6. Fredholm Property and the Index

Theorem 23 has also implications for Fredholm property of . Since we want to see how Fredholm property of may change under the perturbation , we take our consideration under the same conditions as in Theorem 5.

We first note the following fact.

Lemma 25. Let be semi-Fredholm. Then, for all , is semi-Fredholm and

Proof. We have (71) and is a one-rank operator, which is compact. Thus, by a general theorem [12, Chapter IV, Theorem 5.26], the lemma holds.

Theorem 26. (i) Suppose that holds and is Fredholm. Then is not semi-Fredholm and, for all , is Fredholm with
(ii) Suppose that holds and is Fredholm. Then, for all , is Fredholm and (120) holds.
(iii) Suppose that holds and is semi-Fredholm. Then is not semi-Fredholm and, for all , the following hold.  (a) If is injective, then is Fredholm with . (b) If is not injective, then is semi-Fredholm with .
(iv) Suppose that holds and that is semi-Fredholm.  (a) If is injective, then, for all , is Fredholm and . (b) If is not injective, then, for all , is semi-Fredholm and .
(v) Let be semi-Fredholm with and . Suppose that and . Then is not semi-Fredholm.

Proof. In the proofs below, (77) is used without mentioning explicitly.
(i) By Lemma 22-(i), . By this fact and Lemma 25, we have Hence, applying Theorem 5-(iii) with replaced by , one concludes that is not semi-Fredholm.
Let . Then, by Lemma 25, we have . Hence, by Theorem 5-(i), is Fredholm and . Thus (120) holds.
(ii) It is similar to (i).
(iii) In the present case, we have and . Hence, by Lemma 25, . Hence, by applying Theorem 5-(iii) with replaced by , we conclude that is not semi-Fredholm.
Let . Then, by Lemma 22-(ii), is injective. By Lemma 25, . (a) Let be injective. Then is injective. Hence, we can apply Theorem 5-(i) with replaced by to conclude that is Fredholm with . (b) Let be not injective. Then . Hence, we can apply Theorem 5-(ii) with replaced by to conclude that is semi-Fredholm with .
(iv) In the present case, we have for all . Hence, by Lemma 25, for all . (a) In this case, . Hence, by Theorem 5-(i) with replaced by , is Fredholm and . (b) In this case, we can apply Theorem 5-(ii) with replaced by to conclude that is semi-Fredholm and .
(v) In the present case, we have and . Hence and . Therefore, we can apply Theorem 5-(iii) with replaced by to conclude that is not semi-Fredholm.

7. Nonzero Eigenvalues of

Finally we consider existence of nonzero eigenvalues of . In this section we need the following condition.

Hypothesis (I). (i) ;

(ii) is an injective and nonnegative self-adjoint operator;

(iii) .

Under Hypothesis (I), the constant defined by (98) takes the form

Theorem 27. Let Hypothesis (I) be satisfied and . Then, there exists a unique constant such that and, for all , with eigenvectors where
Moreover, , as a function of , is strictly monotone increasing on with and .

Note that Theorem 27 holds even if has no nonzero eigenvalues. This is an interesting phenomenon. Since the condition implies that , the phenomenon may be regarded as a strong coupling effect.

To prove Theorem 27, we prepare two lemmas.

We consider a condition slightly more general than Hypothesis (I).

Hypothesis (II). In addition to (i) and (ii) in Hypothesis (I), the following holds: for some and

Under Hypothesis (II), the constant defined by (98) takes the form

Lemma 28. Let Hypothesis (II) be satisfied. (i)Let . Then there exist no solutions to the equation (ii)Let . Then there exists a unique constant such that The constant is, as a function of , strictly monotone increasing and differentiable on with and .

Proof. (i) Since is injective, nonnegative, and self-adjoint, we can define a function on by This function is differentiable on with Hence is strictly monotone increasing. It is easy to see that and (note that ). Hence, if , then has no solutions in .
(ii) Let . Then, by the proof of (i), there exists a unique constant such that . The inverse function of exists on and is differentiable on . Hence , . This implies the assertion on .

Lemma 29. Let Hypothesis (II) be satisfied. (i)Let . Then (ii)Let and be defined by (125). Then

Proof. Let . Then there is a nonzero vector such that . It follows that If , then , which is a contradiction. Hence . Taking the inner product of (135) with , we obtain
(i) If , then, by Lemma 28-(i), (136) has no solutions in . Hence (132) holds.
(ii) Let . Then, by Lemma 28-(ii), we have . Thus . It is easy to check that . Thus (133) follows. By (135), every vector satisfies . Hence . Thus (134) follows.

Proof of Theorem 27. Under the present assumption, is self-adjoint. Hence, by Lemma 29, . Hence Theorem 16 implies (123). We have . Hence are eigenvectgors of with eigenvalue . Hence, applying Lemma 8 with , we see that .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Grant-In-Aid no. 24540154 from JSPS.