A note on the three dimensional Dirac operator with zigzag type boundary conditions

In this note the three dimensional Dirac operator $A_m$ with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that $A_m$ is self-adjoint in $L^2(\Omega;\mathbb{C}^4)$ for any open set $\Omega \subset \mathbb{R}^3$ and its spectrum is described explicitly in terms of the spectrum of the Dirichlet Laplacian in $\Omega$. In particular, whenever the spectrum of the Dirichlet Laplacian is purely discrete, then also the spectrum of $A_m$ consists of discrete eigenvalues that accumulate at $\pm \infty$ and one additional eigenvalue of infinite multiplicity.


Introduction
In the recent years Dirac operators with boundary conditions, which make them self-adjoint, gained a lot of attention. From the physical point of view, they appear in various applications such as in the description of relativistic particles that are confined in a box Ω ⊂ R 3 ; in this context the MIT bag model is a particularly interesting example, cf. [2]. Moreover, in space dimension two the spectral properties of self-adjoint massless Dirac operators play an important role in the mathematical description of graphene, see, e.g., [7] and the references therein. On the other hand, from the mathematical point of view, self-adjoint Dirac operators with boundary conditions are viewed as the relativistic counterpart of Laplacians with Robin type and other boundary conditions.
To set the stage, let Ω ⊂ R 3 be an open set, let be the Pauli spin matrices, and let α j := 0 σ j σ j 0 and β := be the C 4×4 -valued Dirac matrices, where I d denotes the identity matrix in C d×d . For m ∈ R we introduce the differential operator τ m acting on distributions by τ m := −i 3 j=1 α j ∂ j + mβ =: −iα · ∇ + mβ. (1. 3) The main goal in this short note is to study the self-adjointness and the spectral properties of the Dirac operator A m in L 2 (Ω; C 4 ) := L 2 (Ω) ⊗ C 4 which acts as τ m on functions f = (f 1 , f 2 , f 3 , f 4 ) ∈ L 2 (Ω; C 4 ) which satisfy τ m f ∈ L 2 (Ω; C 4 ) and the boundary conditions in a suitable sense specified below. Note that no boundary conditions are imposed for the components f 1 and f 2 . If m > 0, then the solution of the evolution equation with Hamiltonian A m describes the propagation of a quantum particle with mass m and spin 1 2 in Ω taking these boundary conditions and relativistic effects into account.
The motivation to study the operator A m is twofold. Firstly, in the recent paper [5] Dirac operators in L 2 (Ω; C 4 ) acting as τ m on function satisfying the boundary conditions were studied in the case that m > 0 and that Ω is a C 2 -domain with compact boundary and unit normal vector field ν; in (1.5) the convention α · The authors were able to prove the self-adjointness and to derive the basic spectral properties of these operators, whenever the parameter ϑ appearing in (1.5) is a real-valued Hölder continuous function of order a > 1 2 satisfying ϑ(x) = ±1 for all x ∈ ∂Ω, and it is shown that the domain of definition of these self-adjoint operators is contained in the Sobolev space H 1 (Ω; C 4 ). For bounded domains Ω this implies, in particular, that the spectrum is purely discrete. The case when ϑ(x) = ±1 for some x ∈ ∂Ω remained open and it is conjectured that different spectral properties should appear. We note that ϑ ≡ 1 corresponds to the boundary conditions (1.4). Let us mention that the self-adjointness and spectral properties of Dirac operators with boundary conditions of the form (1.5) for special realizations ϑ = 1 were studied in 3D in [1,2,12] and in 2D in [6,7,10,11].
The second main motivation for this study is the paper [14], where the two dimensional counterpart of A m was investigated in the massless case (m = 0). It was shown in [14] that the two dimensional Dirac operator with similar boundary conditions as in (1.4), which are known as zigzag boundary conditions, is selfadjoint on a domain which is in general not contained in H 1 (Ω) and that for any bounded domain Ω zero is an eigenvalue of infinite multiplicity. In particular, the spectrum of the operator is not purely discrete. Let us mention here that the two dimensional zigzag boundary conditions have a physical relevance, as they appear in the description of graphene quantum dots, when a lattice in this quantum dot is terminated and the direction of the boundary is perpendicular to the bonds [9].
The goal in the present note is to prove similar and even more explicit results as those in [14] also in the three dimensional setting, which complement then the results from [5] in the critical case ϑ ≡ 1 at least for constant boundary parameters. In Lemma 3.2 we will see that the operator corresponding to ϑ ≡ −1 in (1.5) is unitarily equivalent to −A m and hence this case is also contained in the analysis in this note. In the formulation of the following main result of the present paper we denote by −∆ D the self-adjoint realization of the Dirichlet Laplacian in L 2 (Ω). Theorem 1.1. The operator A m is self-adjoint in L 2 (Ω; C 4 ) and its spectrum is The value m always belongs to the essential spectrum of A m , while for m = 0 the number −m is not an eigenvalue of A m . Theorem 1.1 is proved in a series of results in Section 3. It gives a full description of the spectrum of A m in terms of the spectrum of the Dirichlet Laplacian in Ω, which is well-studied in many cases. For bounded domains Ω it follows from the Rellich embedding theorem that the spectrum of −∆ D is purely discrete and therefore, the spectrum of A m consists of an infinite sequence of discrete eigenvalues accumulating at ±∞ and the eigenvalue m, which has infinite multiplicity. In particular, the essential spectrum of A m is not empty, which is in contrast to the case of non-critical boundary values in [5]. Moreover, if Ω is a bounded Lipschitz domain, then the non-emptiness of the essential spectrum implies that the domain of A m is not contained in the Sobolev space H s (Ω; C 4 ) for any s > 0.
If Ω is unbounded, then there are different ways how the spectrum of the Dirichlet Laplacian and hence also the spectrum of A m may look like. On the one hand it is known that for some special horn shaped domains Ω, which have infinite measure, the spectrum of −∆ D is purely discrete, cf. [13,15]. Therefore, by Theorem 1.1 also in this case the spectrum of A m consists only of eigenvalues and it follows from the spectral theorem that the multiplicity of m is again infinite. On the other hand, for many unbounded domains it is known that σ(−∆ D ) = [0, ∞) and thus, σ(A m ) = (−∞, −|m|] ∪ [|m|, ∞) for such Ω. The simplest example for this case is when Ω is the complement of a bounded domain.
Let us finally collect some basic notations that are frequently used in this note. If not stated differently Ω is an arbitrary open subset of R 3 . For n ∈ N we write L 2 (Ω; C n ) := L 2 (Ω) ⊗ C n . The inner product and the norm in L 2 (Ω; C n ) are denoted by (·, ·) and · , respectively. We use for k ∈ N the symbol H k (Ω) for the L 2 -based Sobolev spaces of k times weakly differentiable functions and H 1 0 (Ω) for the closure of the test functions C ∞ 0 (Ω) in H 1 (Ω). For a linear operator A its domain is dom A and its Hilbert space adjoint is denoted by A * . If A is a closed operator, then σ(A) is the spectrum of A, and if A is self-adjoint, then its essential spectrum is σ ess (A).

Some auxiliary operators
In this section we introduce and discuss two auxiliary operators T min and T max in L 2 (Ω; C 2 ) which will be useful to study the Dirac operator A m with zigzag type boundary conditions. Let Ω ⊂ R 3 be an arbitrary open set and let σ = (σ 1 , σ 2 , σ 3 ) be the Pauli spin matrices defined by (1.1). In the following we will often use the notation σ · ∇ = σ 1 ∂ 1 + σ 2 ∂ 2 + σ 3 ∂ 3 . We define the set D max ⊂ L 2 (Ω; C 2 ) by where the derivatives are understood in the distributional sense, and the operators T max and T min acting in L 2 (Ω; C 2 ) by and T min := T max ↾ H 1 0 (Ω; C 2 ), which has the more explicit representation In the following lemma we summarize the basic properties of T min and T max . In the next lemma it is shown that T min does not have eigenvalues. If Ω is a C 2 -domain with compact boundary, then this would follow from the simplicity of T min , which can be proved in the same way as for the minimal Dirac operator in Ω in [5,Proposition 3.2]. However, for our purposes also the weaker statement of absence of eigenvalues is sufficient. Proof. Assume that λ is an eigenvalue of T min with corresponding eigenfunction f = 0 and let ϕ ∈ C ∞ 0 (R 3 ; C 2 ) be arbitrary. In the following we will denote by −∆ the free Laplace operator on R 3 , which is defined on H 2 (R 3 ; C), and by f the extension of f by zero onto R 3 . Note that σ j σ k + σ k σ j = 2δ jk I 2 holds by the definition of the Pauli matrices in (1.1) and hence, (σ · ∇) 2 ϕ = ∆ϕ. Using this and f, T min f = λf ∈ dom T min = H 1 0 (Ω; C 2 ) we find that i.e. −∆ f = λ 2 f in L 2 (R 3 ; C 2 ). Therefore, we conclude that f is an eigenfunction of −∆ and hence, f = 0. This is a contradiction to the assumption and completes the proof of this lemma.
Eventually, we show that 0 always belongs to the spectrum of T max . This result will be of importance to prove that m is in the essential spectrum of A m . Proposition 2.3. There exists a sequence (f n ) ⊂ dom T max with f n = 1 converging weakly to zero such that T max f n → 0, as n → ∞. In particular, 0 ∈ σ(T max ).

Definition of A m and its spectral properties
This section is devoted to the study of the operator A m and the proof of the main result of this note, Theorem 1.1. First, we introduce A m rigorously and show its self-adjointness, then we investigate its spectral properties.
Let Ω ⊂ R 3 be an arbitrary open set and let T max and T min be the operators defined in (2.1) and (2.2), respectively. We define for m ∈ R the Dirac operator A m with zigzag type boundary conditions, which acts in L 2 (Ω; C 4 ), by The operator in ( Before we start analyzing A m we remark that this operator is unitarily equivalent with the operator −B m , where B m is defined by Note that B m is the Dirac operator acting on spinors f = (f 1 , f 2 , f 3 , f 4 ) ∈ L 2 (Ω; C 4 ) satisfying the boundary conditions f 1 | ∂Ω = f 2 | ∂Ω = 0. In particular, the following Lemma 3.2 shows that all results which are proved in this paper for A m can be simply translated to corresponding results for B m . In order to formulate the lemma we recall the definition of the Dirac matrix β from (1.2), define the matrix and note that βγ 5 is a unitary matrix. 1 it is not difficult to show that A m is self-adjoint. In the proof, we use in a similar way as in [14] that the operator A 0 (i.e. A m for m = 0) has a supersymmetric structure.

Proof.
We use for f ∈ L 2 (Ω; C 4 ) the splitting f = (f 1 , f 2 ) with f 1 , f 2 ∈ L 2 (Ω; C 2 ), that means f 1 and f 2 are the upper and lower two components of the Dirac spinor, respectively. It suffices to consider m = 0, as mβ is a bounded self-adjoint perturbation. Moreover, we note that T max = T * min and T * max = T min hold by Lemma 2.1; this will be used several times throughout this proof.
First we show that A 0 is symmetric. Indeed for f = (f 1 , f 2 ) ∈ dom A 0 a simple calculation shows Next, one has for f = (f 1 , f 2 ) ∈ dom A * 0 and g = ( Choosing g 1 = 0 we get from (3.2) (A * 0 f ) 2 , g 2 = (f 1 , T min g 2 ) for all g 2 ∈ dom T min and hence f 1 ∈ dom T max and T max f 1 = (A * 0 f ) 2 . Similarly, choosing g 2 = 0 we obtain from (3.2) that f 2 ∈ dom T min and T min f 2 = (A * 0 f ) 1 . Therefore, we conclude f ∈ dom A 0 and A * 0 f = A 0 f , that means A * 0 ⊂ A 0 . This finishes the proof of this theorem.
In the following theorem we state the spectral properties of A m . We will see that they are closely related to the spectral properties of the Dirichlet Laplacian −∆ D , which is the self-adjoint operator in L 2 (Ω; C) that is associated to the closed and non-negative sesquilinear form In particular, σ(A m ) ∩ (−|m|, |m|) = ∅.
We note that Theorem 3.4 applied for m = 0 shows that the spectrum of A 0 is symmetric w.r.t. λ = 0. This observation would also follow from the stronger fact that A 0 = −βA 0 β, i.e. A 0 is unitarily equivalent to −A 0 .

Proof of Theorem 3.4. (i) Consider the nonlinear time reversal operator
and let f λ be a corresponding eigenfunction. Then, one can show in the same way as in [3,Proposition 4.2 (ii)] that also T f λ is a linearly independent eigenfunction of A m for the eigenvalue λ. This shows the claim of statement (i).
(ii) According to Proposition 2.3 there exists a sequence (g n ) ⊂ dom T max , which converges weakly to zero, such that g n = 1 and T max g n → 0, as n → 0. Define f n := (g n , 0) ∈ L 2 (Ω; C 4 ), i.e. the first two components of f n ∈ L 2 (Ω; C 4 ) are g n ∈ L 2 (Ω; C 2 ) and the last two components are zero. Then f n ∈ dom A m , f n = 1, (f n ) converges weakly to zero, and as n → ∞. Hence (f n ) is a singular sequence for A m and λ = m and thus, m ∈ σ ess (A m ).
(iii) Assume that −m ∈ σ p (A m ) and that f is a nontrivial eigenfunction. According to the definition of A m we can write f = (f 1 , f 2 ) with f 1 ∈ D max and f 2 ∈ dom T min = H 1 0 (Ω; C 2 ). Then i.e. T max f 1 = 0 and T min f 2 = −2mf 1 . With Lemma 2.1 this implies i.e. f 1 = 0. Thus, f 2 ∈ ker T min and since T min has no eigenvalues by Lemma 2.2, we conclude that also f 2 = 0. Therefore, we have shown f = (f 1 , f 2 ) = 0, i.e. −m / ∈ σ p (A m ).
(iv) First, we prove the inclusion For this, due to the results from items (ii) and (iii), it suffices to prove A simple direct calculation shows that Hence, . To see this, we note that T max T min is the unique self-adjoint operator corresponding to the quadratic form . Since σ j σ k + σ k σ j = 2δ jk I 2 holds by the definition of the Pauli matrices in (1.1), which extends by density to all f ∈ dom a = H 1 0 (Ω; C 2 ). Therefore, a is the quadratic form associated to −∆ D and hence, by the first representation theorem we conclude T max T min = −∆ D . This implies (3.4).
Let us end this note with a short discussion of the spectral properties of A m for some special domains Ω and some consequences of that. In many situations it is known that the Dirichlet Laplacian has purely discrete spectrum. Then, by Theorem 3.4 (iv) also the spectrum of A m consists only of eigenvalues and, as a consequence of the spectral theorem, m is an eigenvalue of infinite multiplicity. Moreover, in a similar way as in (3.6) and (3.7) one can construct eigenfunctions of A m . The spectrum of the Dirichlet Laplacian is purely discrete, e.g., when Ω is a bounded subset of R 3 , as then the space H 1 0 (Ω; C) is compactly embedded in L 2 (Ω; C) by the Rellich embedding theorem, and hence the Dirichlet Laplacian −∆ D associated to the sesquilinear form a D in (3.3) has a compact resolvent. In this situation let us denote by 0 < µ D 1 ≤ µ D 2 ≤ µ D 3 ≤ . . . the discrete eigenvalues of −∆ D , where multiplicities are taken into account. Then one immediately has the following result. If the Sobolev space H s (Ω; C) is compactly embedded in L 2 (Ω; C) for some s > 0, then the above result implies that dom A m can not be contained in H s (Ω; C 4 ) for any s > 0. This is, e.g., the case, when Ω is a bounded Lipschitz domain.
Corollary 3.6. Assume that Ω is a bounded subset of R 3 with a Lipschitz-smooth boundary. Then dom A m ⊂ H s (Ω; C 4 ) for all s > 0.