On the Single Layer Boundary Integral Operator for the Dirac Equation

This paper is devoted to the analysis of the single layer boundary integral operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_z$$\end{document}Cz for the Dirac equation in the two- and three-dimensional situation. The map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_z$$\end{document}Cz is the strongly singular integral operator having the integral kernel of the resolvent of the free Dirac operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_0$$\end{document}A0 and z belongs to the resolvent set of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_0$$\end{document}A0. In the case of smooth boundaries fine mapping properties and a decomposition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_z$$\end{document}Cz in a ‘positive’ and ‘negative’ part are analyzed. The obtained results can be applied in the treatment of Dirac operators with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions that are combined in a critical way.


Introduction
In the analysis of boundary value and transmission problems for partial differential equations associated potential and boundary integral operators often play an important role. These objects are well understood for uniformly elliptic second order differential expressions, cf. the monograph [20] and the references therein. We remark that the transmission problems are closely related to differential operators with singular interactions like, e.g., δ-potentials. In the recent years it turned out that similar objects are also of importance in the study of boundary value and transmission problems for the Dirac equation, which is the relativistic counterpart of the Laplace equation and for which the associated differential expression is of first order and lacks ellipticity. While the potential operator is sufficiently well investigated, the subtle examination of the associated boundary integral operator is less complete, see the review paper [11] and the references therein. It is the main goal in this article to make some further contributions to this study that are, in particular, necessary to examine Dirac operators with singular δ-potentials, where the involved parameters are combined in a critical way.
To introduce the problem setting in a more detailed way, consider first a formally symmetric and strongly elliptic second order partial differential operator P in R q , q ≥ 2, let P 0 be the self-adjoint realization of P defined on H 2 (R q ), and let E z (x, y) be the integral kernel of (P 0 −z) −1 , z ∈ ρ(P 0 ). Moreover, let Ω ⊂ R q be a sufficiently smooth domain with unit normal vector field ν that is pointing outwards of Ω and let B ν be the conormal derivative at ∂Ω associated with P. Then, the single layer potential SL(z) and the double layer potential DL(z) applied to a sufficiently smooth function ϕ : ∂Ω → C evaluated at x ∈ R q \ ∂Ω are SL(z)ϕ(x) = ∂Ω E z (x, y)ϕ(y)dσ(y), DL(z)ϕ(x) = ∂Ω ∂ Bν ,y E z (x, y)ϕ(y)dσ(y).
It is known that all solutions of the partial differential equation (1.1) (P − z)f = g in R q \ ∂Ω 1 can be described with the help of SL(z) and DL(z). To analyze boundary value or transmission problems associated with the above equation, it is common to employ the Dirichlet trace operator γ D and B ν to SL(z) and DL(z). Of particular interest is the single layer boundary integral operator S(z) := γ D SL(z). Its properties are closely related to the solvability of the Dirichlet boundary value problem for (1.1), and relevant properties, as, e.g., its ellipticity, are inherited from those of P, cf. [20,Chapter 7]. Moreover, a good understanding of S can be useful for a thorough understanding of different problems related to P in an operator theoretic language, for instance to show the self-adjointness and compute spectral properties of operators associated with boundary value or transmission problems for P including perturbations of P 0 by singular interactions, cf. [12,17,18,20]. It is the main goal in this paper to provide some properties of the counterpart of the single layer boundary integral operator for the Dirac operator in the two-and three-dimensional situation. For m ≥ 0 the free Dirac operator in dimension two is given by where σ 1 , σ 2 , σ 3 ∈ C 2×2 are the Pauli spin matrices defined in (2.1), and in dimension three it is where α 1 , α 2 , α 3 , β ∈ C 4×4 are the Dirac matrices in (2.2). Here, we used the notation H k (R q ; C l ) = H k (R q ) ⊗ C l for the L 2 -based Sobolev space of k times weakly differentiable vector valued functions. The free Dirac operator is used to describe the propagation of a spin 1 2 particle in vacuum taking effects of the special theory of relativity into account [24]. Moreover, the two-dimensional Dirac operator appears in the mathematical description of graphene [1]. It is known that A 0 is self-adjoint and that its spectrum is cf. [24]. This shows, in particular, that A 0 is not semi-bounded from above or below. For z ∈ ρ(A 0 ) = C \ ((−∞, −m] ∪ [m, ∞)) the resolvent of A 0 can be expressed via the convolution with a function G z,q given in (2.3) below, i.e. as an integral operator. With the help of this function we can formally introduce for a smooth and closed curve Σ ⊂ R 2 or a smooth and closed surface Σ ⊂ R 3 the boundary integral operator acting on sufficiently smooth functions ϕ : Σ → C N as where B(x, ε) is the ball centered at x with radius ε and N = 2 for q = 2 and N = 4 for q = 3. It is known that C z gives rise to a bounded operator in L 2 (Σ; C N ), but in a similar way as A 0 also C z lacks ellipticity [2,5,10]. The operator C z plays an important role in the analysis of boundary value and transmission problems for the Dirac equation and motivated by this C z was studied intensively in the recent years, cf. [2,4,6,9,10,11,13,14,15,22]. In the present paper this study is continued and, in particular, detailed mapping properties and a more detailed analysis of the contribution of C z associated with the 'positive' and the 'negative' part of its spectrum are provided. More precisely, as a consequence of the main results, it is shown in dimension q = 2 in Corollary 3.8 that and similarly in dimension q = 3 in Corollary 4.12 that where ν = (ν 1 , . . . , ν q ) is the unit normal vector on Σ pointing outwards of the bounded domain with boundary Σ, V is a unitary matrix-valued function defined in (3.14) and (4.15) below, K is an operator with good mapping properties between different Sobolev spaces (and hence, K is compact in H s (Σ; C N ) for s ∈ [−1, 1]), and P ± are self-adjoint operators in L 2 (Σ; C N/2 ) that satisfy (1.5) P + + P − = I, dim ran P ± = ∞, P 2 ± = P ± , and P ± P ∓ = 0, i.e. P ± are orthogonal projections in L 2 (Σ; C N/2 ). While for dimension q = 2 the operators P ± are also orthogonal projections in H s (Σ; C), s ∈ R, that satisfy (1.5), in dimension q = 3 we prove that P ± give rise to bounded maps in H s (Σ; C 2 ), s ∈ [−1, 1], that fulfil (1.5) in H s (Σ; C 2 ). We remark that in dimension q = 3 similar, but not exactly the same decompositions of C z were considered in [4,22] with skew projections onto Hardy spaces on Σ; cf. Remark 4.13 for details. However, we believe that the above representations are of interest, as the projections in [4,22] are not self-adjoint in L 2 (Σ; C N/2 ) unless Σ is a sphere, but which is useful in some applications. One possible application of this fact is shown in the very recent paper [11], where Dirac operators perturbed by singular potentials involving electrostatic, Lorentz scalar, and anomalous magnetic interactions combined in a critical way are studied.
It should be remarked that the above mentioned results are proved in different ways in dimension two and three. In dimension two C z is closely related to the Cauchy transform on Σ, for its analysis we follow ideas from [10] and apply the theory of periodic pseudodifferential operators. In dimension three C z is closely related to the Riesz transform on Σ and results on pseudo-homogeneous kernels from [21] are employed to show similar results as in dimension two. In particular, the latter approach would allow to weaken the geometric assumptions on Σ and to consider also higher space dimensions. We remark that in the recent paper [16], which was written independently of this article, the operator C z is studied in the three-dimensional situation with the help of pseudodifferential techniques and results that are related to the ones in this paper are recovered there as well.
The paper is organized as follows. In Section 2 we rigorously introduce the operator C z formally given by (1.4) and discuss its basic properties. Then, in Section 3 we recall some results about periodic pseudodifferential operators and do a more detailed analysis of C z in the two-dimensional case. Finally, in Section 4 we revise basic notions and results on pseudo-homogeneous kernels and use them to investigate C z in dimension three.
Acknowledgement. The author is grateful to the anonymous referee for helpful suggestions to improve the paper. He also thanks Jussi Behrndt, Dale Frymark, Christian Stelzer, and Georg Stenzel for fruitful discussions. Moreover, he gratefully acknowledges financial support by the Austrian Science Fund (FWF): P33568-N. This publication is based upon work from COST Action CA 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology), www.cost.eu.

2.
Notations and basic properties of C z Throughout this paper, let q ∈ {2, 3} be the space dimension and define the number N = N (q) by N (2) = 2 and N (3) = 4. Moreover, we assume that Ω ⊂ R q is a bounded and simply connected domain with C ∞ -smooth boundary Σ := ∂Ω and unit normal vector field ν which is pointing outwards of Ω. Let be the Pauli spin matrices and define the 4 × 4 Dirac matrices by where I n is the n × n-identity matrix. We will often use for x = (x 1 , x 2 ) ∈ C 2 the notation Next, we introduce the function G z,q evaluated at x ∈ R q \ {0} by where we write K j for the modified Bessel functions of the second kind and choose √ w for w ∈ C \ [0, ∞) such that Im √ w > 0. It is well-known that G z,q is the integral kernel of the resolvent of the free Dirac operator A 0 in (1.2) & (1.3); cf. [10,11,24]. Now, we are prepared to rigorously introduce and discuss the properties of the operator C z , z ∈ ρ(A 0 ) = (−∞, −m] ∪ [m, ∞), formally given by (1.4), i.e. we consider now the strongly singular integral operator here B(x, ε) is the ball of radius ε centered at x. The basic properties of C z are summarized in the following proposition; in the proof we mostly refer to [11], but parts of the results were shown before in [10] in dimension q = 2 and in [2,3,8,9,15,22] in dimension q = 3. Below, H s (Σ), s ∈ R, are the Sobolev spaces on Σ defined as in [20] and we denote by (·, ·) H s (Σ;C N )×H −s (Σ;C N ) the sesquilinear duality product in H s (Σ; C N ) × H −s (Σ; C N ). (ii) For ϕ ∈ H s (Σ; C N ) and ψ ∈ H −s (Σ; C N ) one has In particular, for the realization of C z in L 2 (Σ; C N ) the relation C * z = C z holds.
Proof. The mapping properties of C z in item (i) follow from [ [11, equations (4.6) and (4.24)], the statement for general s ∈ R follows then by restriction for s > 1 2 and duality for s < − 1 2 .

3.
A more detailed analysis in dimension q = 2 In this section we investigate C z in space dimension q = 2 in a more detailed way. For this, we make use of the theory of periodic pseudodifferential operators. Their definition and some basic properties are recalled in Subsection 3.1. With the help of these results we analyze C z in Subsection 3.2.
3.1. Periodic pseudodifferential operators. In this subsection we follow closely the short exposition on periodic pseudodifferential operators in [10, Section 2.1], for a more comprehensive presentation see [23]. In order to define periodic pseudodifferential operators, some notations are necessary. We set T := R/Z and denote by D(T) the set of all 1-periodic test functions and by D ′ (T) the set of all 1-periodic distributions. We will often make use of the functions e n ∈ D(T) defined by (3.1) e n (t) := e 2πint , n ∈ Z.
For f ∈ D ′ (T) we introduce the Fourier coefficients by where ·, · D ′ (T)×D(T) denotes the bilinear duality product in D ′ (T) × D(T). Set With the help of the Fourier coefficients one can define for s ∈ R the Sobolev space H s (T) of order s on T by The Fourier coefficients in (3.2) allow us also to establish periodic pseudodifferential operators. Definition 3.1. A linear map A defined on D(T) is called periodic pseudodifferential operator of order s ∈ R, if there exists a function h : T × Z → C such that the following holds: (i) For any fixed n ∈ Z one has h(·, n) ∈ D(T).
(ii) The action of A is given by The set of all periodic pseudodifferential operators of order s is denoted by Ψ s . Moreover, we set Ψ −∞ := s∈R Ψ s .
In the following let Σ ⊂ R 2 be the boundary of a simply connected C ∞ -domain, let ℓ be the length of Σ, and let γ : [0, ℓ] → R 2 be an arc-length parametrization of Σ that is orientated in the counter clockwise way. Similarly as above, we denote by D(Σ) the set of all C ∞ -smooth functions defined on Σ and by D ′ (Σ) the set of all distributions on Σ. In order to define periodic pseudodifferential operators on Σ, we introduce the mapping U : If f is a regular distribution, then U f is also a regular distribution generated by U f = f (γ(ℓ·)). With the map U one can translate the Sobolev spaces H s (T) on T to Sobolev spaces on Σ by and endow them with the inner product By this construction the map U defined in (3.4) is for any s ∈ R a unitary operator from H s (Σ) to H s (T) and the corresponding norm is equivalent to the norm in H s (Σ) defined as in [20].
Σ . An important feature of periodic pseudodifferential operators is the fact that they are automatically bounded in Sobolev spaces [23, Theorems 7.3.1 and 7.8.1].
Then, the following is true: (i) For any r ∈ R the map A can be extended to a well-defined and bounded operator A : There are three types of periodic pseudodifferential operators that we are going to use frequently. First, if f ∈ D(Σ), then one has for the associated multiplication operator Then it is not difficult to see that L t1 L t2 = L t1+t2 holds for all t 1 , t 2 ∈ R and L t ∈ Ψ t/2 . Hence, we can define Σ . Taking the definition of H r (Σ) in (3.3) and (3.5) into account it is not difficult to show that for any r ∈ R the map Λ t : H r (Σ) → H r−t/2 (Σ) is bijective and that the realization of Λ t as a possibly unbounded operator in L 2 (Σ) defined on . The third periodic pseudodifferential operator that will play a crucial role in our analysis is a multiple of the Cauchy transform on Σ. We define formally the integral operator R acting on ϕ ∈ D(Σ) by We also consider the formal adjoint R * of R in L 2 (Σ), which acts on ϕ ∈ D(Σ) as Let C Σ be the Cauchy transform on Σ, i.e. the strongly singular boundary integral operator that acts on ϕ ∈ D(Σ) evaluated at where the first integral is a complex line integral, cf. [23, Section 5.9]. As we have By [10, Proposition 2.8] this implies that (3.11) R, R * ∈ Ψ 0 Σ . In particular, R, R * give rise to bounded operators in H s (Σ) for all s ∈ R. Since R and R * are adjoint to each other, a continuity argument shows for all ϕ ∈ H s (Σ) and ψ ∈ H −s (Σ), s ∈ R, that where (·, ·) H s (Σ)×H −s (Σ) denotes the sesquilinear duality product.

3.2.
Analysis of C z . In the following proposition, which is the starting point for our further considerations and which is a direct consequence of [10, Proposition 3.3], we provide a link of R and R * defined in (3.8) and (3.9), respectively, and the operator C z given in (2.4). Recall that the map Λ is introduced in (3.7).
In the following lemma we state a variant of (3.13) that is particularly useful in the application in the boundary triple framework, as it is done, e.g., in [11]. Define the matrix V ∈ C 2×2 by (3.14) .
Then we have the following result: Lemma 3.5. Let R and R * be defined by (3.8) and (3.9), respectively, let C z , z ∈ (−m, m), be given by (2.4), let Λ be as in (3.7), and let s ∈ R. Then there exists an operator K that is compact in H s (Σ; C 2 ) such that the bounded and everywhere defined operator can be written as In particular, the realization of Proof. The claim follows immediately from Proposition 3.4 and a direct calculation taking the identity Proposition 3.6. Let R and R * be defined by (3.8) and (3.9), respectively. Then, the following holds: Σ . Eventually, we show that R + R * can be written as the difference of two projections, which is useful in the analysis of boundary value and transmission problems for the Dirac equation with critical combinations of the coefficients, see, e.g., [11] for an application. In order to formulate the result, we define the operators P ± ∈ Ψ 0 Σ by (3.16) where U is the map in (3.4). Remark that P ± are orthogonal projections in H s (Σ) defined by (3.5) for all s ∈ R and with H ± := ran P ± one has H s (Σ) = H + ⊕ H − , dim H ± = ∞, and H ± ⊂ H r (Σ) for any r > s. Moreover, P ± commute with the operator Λ t defined in (3.7). The map P + can be interpreted as the projection onto the Hardy space on the unit circle.
Theorem 3.7. Let R and R * be defined by (3.8) and (3.9), respectively, and let P ± be given as above. Then, there exists K ∈ Ψ −∞ Σ such that In particular, for all s, t ∈ R the operator K : Proof. Let C ⊂ R 2 be the circle of radius 1 centered at the origin and let U C be the map in (3.4) defined for C instead of Σ. Then, by [23, Section 5.8] one has for the Cauchy transform C C on C Moreover, by [23, Section 5.9 and Theorem 7.6.1] there exists K 1 ∈ Ψ −∞ such that With (3.10) and the last two displayed formulas one gets By adding the last two displayed formulas one gets the claim.
Finally, combining the results from Lemma 3.5 and Theorem 3.7, we get the following decomposition of C z into a positive and a negative part. In particular, this result implies that ± 1 2 belongs to the essential spectrum of C z , when this operator is viewed as a mapping in H s (Σ; C 2 ). Corollary 3.8. Let C z , z ∈ (−m, m), be defined by (2.4), let P ± be as in (3.16), and let V be given by (3.14). Then, there exists an operator K ∈ Ψ −1 Σ such that In particular, the realization of K in L 2 (Σ; C 2 ) is self-adjoint.

4.
A more detailed analysis in dimension q = 3 In this section we are going to prove similar results as in Section 3, but instead of pseudodifferential operators we use the theory of pseudo-homogeneous kernels from [21]. Throughout this section we assume that Σ ⊂ R 3 is the boundary of a bounded and simply connected C ∞ -smooth domain. First, in Subsection 4.1 we follow closely [21,Section 4.3] and recall the main definitions and results related to pseudo-homogeneous kernels, which are employed in Subsection 4.2 to analyze C z .

4.1.
Pseudo-homogeneous kernels. With the notion of pseudo-homogeneous kernels one can easily describe mapping properties of the associated integral operators. In the following, we restrict our attention to the three-dimensional case, but a similar construction can be done in any space dimension, cf. [21,Section 4.3]. To define pseudo-homogeneous kernels, one first introduces homogeneous kernels [21,Section 4.3.2].
is called a homogeneous kernel of class −m, if the following conditions hold: (i) For all α, β ∈ N 3 0 there exists a constant C α,β such that (ii) For any β ∈ N 3 0 with |β| = m the function ∂ |β| ∂ζ β K(y, ζ) is homogeneous of degree −2 with respect to ζ. In the following lemma we consider a type of homogeneous kernels that will play an important role in the analysis of C z in the next section.
Proof. First, one shows as in [21,Example 4.2] that K(y, ζ) = ζ α |ζ| 2k−1 is a homogeneous kernel of class −(|α| + 2k + 1). Since the multiplication by a smooth function depending only on y does not affect the conditions in Definition 4.1, the claim on K follows.
With the help of homogeneous kernels we can introduce now pseudo-homogeneous kernels [21,Section 4.3.3].
is called a pseudo-homogeneous kernel of class −m, if for any s ∈ N there exist l ∈ N, homogeneous kernels K m+j of class −(m + j), j ∈ {0, . . . , l − 1}, and a function K m+l that is s times differentiable such that The important property of pseudo-homogeneous kernels is that one can provide the mapping properties of the associated integral operators [21,Theorem 4.3.2].
Proposition 4.4. Let m ∈ N 0 and let K be a pseudo-homogeneous kernel of class −m. Then, for any r ∈ R the operator that is formally defined by gives rise to a bounded operator K : H r (Σ) → H r+m (Σ).

4.2.
Analysis of C z . Let −∆ be the free Laplacian defined on H 2 (R 3 ). We use for µ ∈ ρ(−∆) = C \ [0, ∞) the notation S(µ) for the single layer boundary integral operator associated with −∆ − µ, which acts on a sufficiently smooth function ϕ : Σ → C as where √ µ is the complex square root satisfying Im √ µ > 0 for µ / ∈ [0, ∞), cf. [20]. If µ < 0, then it is well-known that S(µ) : H s (Σ) → H s+1 (Σ) is bounded and bijective for all s ∈ R and that the realization of S(µ) in L 2 (Σ) is non-negative and self-adjoint; cf. [7, Lemma 2.6] for a similar argument for µ = −1, space dimension 2, and s ≥ − 1 2 ; the property for negative s follows then by duality due to the formal symmetry of S(µ) for µ < 0. For a constant c Λ > 0 we define The following mapping properties Λ should be well-known, but for the sake of completeness we present a proof: Proof.
Step 2: Next, we show the claim for s = l + 1 2 with l ∈ N. Since S(−1) : H r (Σ) → H r+1 (Σ) is bijective for all r ∈ R and c Λ > 0, it is not difficult to see that S(−1) −1 + c Λ gives for any r ∈ R rise to a bounded and bijective operator In particular, the mapping is bounded and bijective for any r ∈ R. Hence, we conclude with the result from Step 1 and the spectral theorem that is a bounded and bijective map that is a restriction of (S(−1) −1 + c Λ ) 1/2 .
Step 3: To conclude, we mention first that the statement for s = −l with l ∈ N 0 follows from the results in Steps 1 & 2 by duality and the formal symmetry of Λ. An interpolation argument yields the claim for arbitrary s ∈ R.
The main object in this section is the operator R formally acting on a sufficiently smooth function ϕ : Σ → C 2 as where the integral kernel r is the C 2×2 -valued function The map R is closely related to the Riesz transform on Σ. We define also the formal adjoint of R with respect to the inner product in L 2 (Σ; C 2 ) by r(y, x) * ϕ(y)dσ(y).
We remark that the definitions of R and R * imply the simple, but useful relations In the following proposition we provide a link of R and R * and the operator C z given in (2.4). In particular, this implies the mapping properties of R and R * .
Proposition 4.6. Let R and R * be defined by (4.3) and (4.4), respectively, let Λ be as in (4.2), let C z , z ∈ (−m, m), be given by (2.4), and let s ∈ R. Then R and R * give rise to bounded operators in H s (Σ; C 2 ) and there exists an operator K that is compact from H s (Σ; C 4 ) to H s+1 (Σ; C 4 ) such that In particular, the realization of K in L 2 (Σ; C 4 ) is self-adjoint.
Proof. Denote by T the integral operator that is formally acting on a sufficiently smooth function ϕ : Σ → C 2 as where the integral kernel t : R 3 \ {0} → C 2×2 is given by Recall that S(z 2 − m 2 ) denotes the single layer boundary integral operator for −∆ + m 2 − z 2 . With these notations we conclude from (2.3) that Note that (4.8) implies that T gives rise to a bounded operator in H s (Σ; C 2 ), as C z in bounded in H s (Σ; C 4 ) by Proposition 2.1. First, we analyze the diagonal terms and show that With (4.1) we get that K 2 is an integral operator with kernel k 2 (x − y), where By making a power series expansion we see that the first term in the sums cancels, with some suitable coefficients c k , d k ∈ R, and that both sums converge absolutely, as the power series expansion of the exponential function has this property as well. Hence, the first sum is an analytic function, while the second sum is a pseudohomogeneous kernel of class −3 in the sense of Definition 4.3, cf. Lemma 4.2. Thus, we conclude with Proposition 4.4 that K 2 : H s (Σ) → H s+3 (Σ) is bounded for any s ∈ R and in particular, K 2 is compact as an operator from H s (Σ) to H s+1 (Σ). Next, using the resolvent identity we see that (4.10) which is due to the mapping properties of S(−1) and Λ bounded from H s (Σ) to H s+2 (Σ) and hence compact from H s (Σ) to H s+1 (Σ). Thus, we have shown (4.9). It remains to consider the anti-diagonal blocks. The integral kernel t of the operator T in (4.7) has the series expansion (4.11) t with some suitable coefficientsc k ,d k ∈ C, as the terms with |x| −2 cancel. Since the power series for the exponential function is absolutely converging, both sums in (4.11) are absolutely converging as well for any x = 0. In particular, the first sum defines an analytic function. Moreover, by Lemma 4.2 the second sum is a pseudo-homogeneous kernel of class −2 in the sense of Definition 4.3. Hence, we conclude with Proposition 4.4 that where K 4 is bounded from H s (Σ; C 2 ) to H s+2 (Σ; C 2 ) and, in particular, compact from H s (Σ; C 2 ) to H s+1 (Σ; C 2 ). Combining (4.12) with (4.5), we can also write In particular, as T is bounded in H s (Σ; C 2 ) and σ · ν is a bijection in H s (Σ; C 2 ), we conclude that R and R * give rise to bounded operators in H s (Σ; C 2 ). Finally, equations (4.8), (4.9), (4.12), and (4.13) imply (4.6). Furthermore, as the realization of C z in L 2 (Σ; C 4 ) is self-adjoint for z ∈ (−m, m) by Proposition 2.1 (ii), also the operator K in (4.6) must be self-adjoint in L 2 (Σ; C 4 ).
Let us mention that, since R and R * are formally adjoint to each other with respect to the inner product in L 2 (Σ; C 2 ), a continuity argument shows for all ϕ ∈ H s (Σ; C 2 ) and ψ ∈ H −s (Σ; C 2 ), s ∈ R, that (4.14) ( where (·, ·) H s (Σ;C 2 )×H −s (Σ;C 2 ) denotes the sesquilinear duality product in H s (Σ; C 2 )× H −s (Σ; C 2 ). In the following lemma we state a variant of (4.6) that is particularly useful in the application in the boundary triple framework, as it is done, e.g., in [11]. Define the matrix V ∈ C 4×4 by Then we have the following result: Lemma 4.7. Let R and R * be defined by (4.3) and (4.4), respectively, let Λ be as in (4.2), and let C z , z ∈ (−m, m), be given by (2.4). Then, for any s ∈ R there exists an operator K that is compact in H s (Σ; C 4 ) such that the bounded and everywhere defined operator can be written as In particular, the realization of K in L 2 (Σ; C 4 ) is self-adjoint.
Proof. First, equation (4.6) and a simple calculation imply where K 1 is a compact operator in H s (Σ; C 4 ) due to the mapping properties of Λ. By (4.2) we can write where T is the integral operator acting as and K 2 is given by A similar argument as in (4.10) shows that K 2 is compact as a mapping from H s−1/2 (Σ; C 2 ) to H s+1/2 (Σ; C 2 ). Next, we show that T in (4.18) is compact as an operator from H s−1/2 (Σ; C 2 ) to H s+1/2 (Σ; C 2 ). Denote byν = (ν 1 ,ν 2 ,ν 3 ) a smooth and compactly supported extension of ν that is defined on R 3 . Then, this function has a Taylor series expansion of the form where Dν and Hν j are the Jacobi matrix and the Hessian ofν andν j , respectively, and e j are the canonical basis vectors in R 3 . Using this and the power series expansion of the exponential function we find that the function t defined in (4.19) can be written as Since the second term is a smooth function, we find with Lemma 4.2 that t(x, y) is a pseudo-homogeneous kernel of class −2. Therefore, we get with Proposition 4.4 that T is bounded from H s−1/2 (Σ; C 2 ) to H s+3/2 (Σ; C 2 ) and, in particular, compact from H s−1/2 (Σ; C 2 ) to H s+1/2 (Σ; C 2 ). Hence, we conclude that . Together with (4.17) this implies (4.16). Finally, equation (4.16) and the fact that the realization of C z in L 2 (Σ; C 4 ) is selfadjoint for z ∈ (−m, m) by Proposition 2.1 yield that also K must be self-adjoint in L 2 (Σ; C 4 ).
In the following proposition some further properties of R and R * that follow from known properties of C z are stated:  3) and (4.4), respectively, and let s ∈ R. Then, the following holds: (i) The operator R − R * : H s−1 (Σ; C 2 ) → H s (Σ; C 2 ) gives rise to a bounded operator. (ii) One has R 2 = 4I 2 and (R * ) 2 = 4I 2 . In particular, this implies that gives rise to a bounded operator.
Proof. (i) Taking the definitions of R and R * into account, we see that their difference is a singular integral operator with kernel (4.21) Following ideas from the proof of [22, Proposition 2.8], we remark first that the kernel s 2 (x, y) := −2 ν(x)·(x−y) π|x−y| 3 is pseudo-homogeneous of class −1, as ν(x)·(x−y) ∼ |x − y| 2 for x → y (for details see the considerations on the kernel K 1 in the proof of [22,Proposition 2.8]). Concerning the first term in the sum in (4.21), one gets in a similar way as in the study of (4.19) with Lemma 4.2 that s 1 (x, y) is a pseudohomogeneous kernel of class −1 in the sense of Definition 4.3. Summing up, we find that r(x, y) − r(y, x) * is a kernel of class −1 and hence, by Proposition 4.4 the operator R − R * : H s−1 (Σ; C 2 ) → H s (Σ; C 2 ) is bounded, which is the claim of item (i).
(ii) First, we show that R 2 = 4I 2 holds. It is known from [6, Proposition 3.5 (i)] that C m := lim z→m C z exists with respect to the norm in the space of bounded operators in L 2 (Σ; C 4 ) and the limit is given by Combining this with Proposition 2.1 (iii) yields which shows R 2 = 4I 2 . By taking the adjoint this implies (R * ) 2 = 4I 2 . Using this we also find that RR * − 4I 2 = R R * − R holds. This and the result from item (i) yield the mapping properties in (4.20).
(iii) First we note that Proposition 4.6 implies that is compact. Using this representation in −4(C z (α · ν)) 2 = I 4 , see Proposition 2.1 (iii), we find that the upper right block of this equation has the form to H s+1 (Σ; C 2 ) by item (i) and the mapping properties of Λ, the last equation and Rellich's embedding theorem yield the claim.
Next, we show that R + R * can be written as the difference of two projections, which is useful in the analysis of boundary value and transmission problems for the Dirac equation with critical combinations of the coefficients, see, e.g., [11] for an application.
where P ± is the projection onto H ± .
Proof. The proof of this result is separated in four steps. In Step 1 we consider the realization of R+R * in the space L 2 (Σ; C 2 ) and prove a spectral representation that is suitable to define the map K. In Step 2 we show an auxiliary embedding result, which is then used in Step 3 to conclude the mapping properties of K. Finally, in Step 4 we show the claims about H ± and P ± . Throughout the proof, we will denote by (·, ·) the inner product in L 2 (Σ; C 2 ).
Thus, if we denote by µ k = sign µ k · 16 + λ 2 k all eigenvalues of R + R * with some λ k ∈ σ p (A) and by e ± k the corresponding eigenfunctions (where the superscript corresponds to the sign of the associated eigenvalue and which are by (4.23) also eigenfunctions of A), and that are by the spectral theorem an orthonormal basis in L 2 (Σ; C 2 ), we can write (4.27) It remains to show that P ± give rise to bounded operators P ± in H s (Σ; C 2 ), that H ± := ran P ± fulfil H s (Σ; C 2 ) = H ++ H − , and that K can be extended to a bounded operator from H s (Σ; C 2 ) to H 1 (Σ; C 2 ).
Step 2: Define the space G := ran A and endow it with the norm Remark that (A ↾ (L 2 (Σ; C 2 ) ⊖ ker A)) −1 is self-adjoint in the Hilbert space ran A and hence, it is closed and G is a Hilbert space. Clearly, those eigenfunctions e ± k of A that correspond to eigenvalues λ k = 0 are an orthogonal basis in G. Note that Proposition 4.8 (i) implies that G = ran (R − R * ) ⊂ H 1 (Σ; C 2 ). In this step, we show that the embedding ι : G → H 1 (Σ; C 2 ) is continuous.
For that, we show that ι is closed. So let (ϕ n ) ⊂ G such that ϕ n → ϕ in G and ιϕ n = ϕ n → ψ in H 1 (Σ; C 2 ). Then, as G is complete ϕ ∈ G and by the same argument ψ ∈ H 1 (Σ; C 2 ). Since both G and H 1 (Σ; C 2 ) are boundedly embedded in L 2 (Σ; C 2 ), we have that (ϕ n ) converges to ϕ and ψ in L 2 (Σ; C 2 ). This can only be true if ϕ = ψ. This finishes the proof that ι is closed and thus, continuous.
Step 3: In this step we prove that the map K introduced in (4.27) can be extended to a bounded operator from H −1 (Σ; C 2 ) to H 1 (Σ; C 2 ), which gives then rise to a bounded mapping K : H s (Σ; To show the stated boundedness property, let ϕ ∈ L 2 (Σ; C 2 ) be fixed. We claim that Kϕ ∈ G. To see this, we use that the definition of K in (4.27) yields |( Kϕ, e ± k )| = ( 16 + λ 2 k − 4)|(ϕ, e ± k )| and thus, (4.28) Therefore, we have shown Kϕ ∈ G. Combining now (4.28) with Proposition 4.8 (i) (applied with s = 0) and the result in Step 2 we conclude . Since L 2 (Σ; C 2 ) is a dense subspace of H −1 (Σ; C 2 ), the latter estimate shows that K can indeed be extended to a bounded map from H −1 (Σ; C 2 ) to H 1 (Σ; C 2 ).
Step 4: It remains to show that P ± give rise to bounded operators P ± in H s (Σ; C 2 ) and that H ± := ran P ± fulfil H s (Σ; C 2 ) = H ++ H − . To see the first claim, remark that the eigenfunctions e ± k of R + R * form an orthonormal basis of L 2 (Σ; C 2 ) and that the definition of P ± in (4.27) implies P + + P − = I 2 . Hence, formula (4.26) yields (4.29) Since the operator on the right hand side gives rise to a continuous map in H s (Σ; C 2 ), the same is true for the left hand side and we denote this realization by P ± . It remains to show that the direct sum decomposition H s (Σ; C 2 ) = H ++ H − holds. First, we note that (4.29) implies P + + P − = I 2 and hence, H s (Σ; C 2 ) = H + + H − . To see that the sum is direct, assume that ϕ ∈ H + ∩ H − . Then, there exist ϕ − , ϕ + ∈ H s (Σ; C 2 ) such that Note that the definition of P ± in (4.27) implies that P ± P ∓ = 0. Thus, also P ± P ∓ = 0. An Application of P ± to (4.30) shows then P ± ϕ = 0, i.e. ϕ = P + ϕ + P − ϕ = 0. Therefore, H s (Σ; C 2 ) = H ++ H − as a direct sum and all claims are shown.
Eventually, we state a result about a special operator in L 2 (Σ; C 2 ) involving the realizations R, R * of R, R * in H −1/2 (Σ; C 2 ). In the following proposition we show that zero belongs to the essential spectrum of Λ( R R * − 4I 2 )Λ under some suitable assumptions on the surface Σ. Remark that in the recent paper [16], that was developed independently from this note, a related result was shown with pseudodifferential techniques. Proposition 4.11. Assume that there exists an open subset Σ 0 ⊂ Σ such that Σ 0 is contained in a plane. Then, there exists a sequence (ϕ n ) ⊂ L 2 (Σ; C 2 ) such that ϕ n L 2 (Σ;C 2 ) = 1, ϕ n converges weakly to zero, and Λ( R R * − 4I 2 )Λϕ n → 0 in L 2 (Σ; C 2 ). It is possible to choose this sequence such that (Λϕ n ) ⊂ H ± .
Proof. The proof of this proposition is split into two steps. In Step 1 we follow closely arguments from [8, Theorem 5.9 (i)] to show that there exists a singular sequence (ϕ n ) for Λ( R R * − 4I 2 )Λ, while in Step 2 we prove that this sequence (ϕ n ) can be chosen such that Λϕ n ∈ H ± .
Finally, combining the results from Lemma 4.7 and Theorem 4.9, we get the following decomposition of C z into a positive and a negative part. In particular, this result implies that ± 1 2 belongs to the essential spectrum of C z , when this operator is viewed as a mapping in H s (Σ; C 4 ), s ∈ [−1, 1]. Corollary 4.12. Let C z , z ∈ (−m, m), be defined by (2.4), let P ± be as in Theorem 4.9, and let V be given by (4.15). Then, for any s ∈ [−1, 1] there exists a bounded operator K : H s (Σ; C 4 ) → H τ (Σ; C 4 ) with τ = min {1, s + 1} such that In particular, the realization of K in L 2 (Σ; C 4 ) is self-adjoint. were considered and they were identified as skew projections onto Hardy spaces. Note that Theorem 4.9 and P + + P − = I 2 imply and similarly Q * ± = 1 2 I 2 ± 1 4 (R + R * ) ∓ 1 8 (R − R * ) = P ± ∓ 1 8 (R − R * − K).
Hence, the projections P ± constructed in Theorem 4.9 coincide, up to the map ± 1 8 (R − R * + K), which is by Proposition 4.8 bounded from H s−1 (Σ; C 2 ) to H s (Σ; C 2 ) and hence compact in H s (Σ; C 2 ) for any s ∈ R, with the skew projections onto Hardy spaces considered in [4,Section 4.2]. However, in contrast to P ± the projections Q ± are not self-adjoint in L 2 (Σ; C 2 ), unless Σ is a sphere. This is a drawback, as Q ± Q * ∓ = 0, unless Σ is a sphere, which makes the application of Q ± in problems involving the Schur complement of C z and related operators more involved.
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