Spectral theoretic characterization of the massless Dirac action

We consider an elliptic self-adjoint first order differential operator L acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of the operator L is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Given a positive scalar weight function, we study the weighted eigenvalue problem for the operator L. The corresponding counting function (number of eigenvalues between zero and a positive lambda) is known to admit, under appropriate assumptions on periodic trajectories, a two-term asymptotic expansion as lambda tends to plus infinity and we have recently derived an explicit formula for the second asymptotic coefficient. The purpose of this paper is to establish the geometric meaning of the second asymptotic coefficient. To this end, we identify the geometric objects encoded within our eigenvalue problem - metric, nonvanishing spinor field and topological charge - and express our asymptotic coefficients in terms of these geometric objects. We prove that the second asymptotic coefficient of the counting function has the geometric meaning of the massless Dirac action.

infinitely smooth. We also assume that the operator L is formally self-adjoint (symmetric): M u * Lv dx = M (Lu) * v dx for all infinitely smooth u, v : M → C 2 .
Here and further on the superscript * in matrices, rows and columns indicates Hermitian conjugation in C 2 and dx := dx 1 dx 2 dx 3 , where x = (x 1 , x 2 , x 3 ) are local coordinates on M .
Let L prin (x, p) be the principal symbol of the operator L, i.e. matrix obtained by leaving in L only the leading (first order) derivatives and replacing each ∂/∂x α by ip α , α = 1, 2, 3. Here p = (p 1 , p 2 , p 3 ) is the variable dual to the position variable x; in physics literature the p would be referred to as momentum. Our principal symbol L prin (x, p) is a 2 × 2 Hermitian matrix-function on the cotangent bundle T * M , linear in every fibre T * x M (i.e. linear in p). We assume that det L prin (x, p) = 0 for all (x, p) ∈ T ′ M := T * M \ {p = 0} (cotangent bundle with the zero section removed), which is a version of the ellipticity condition.

Remark 1.1
The tradition in microlocal analysis is to denote momentum by ξ. We choose to denote it by p instead because we will need the letter ξ for the spinor.
We now make two additional assumptions: • we assume the principal symbol to be trace-free and • we assume the subprincipal symbol of the operator L to be zero (see formula (2.1.9) in [8] for the definition of the subprincipal symbol).
The latter condition implies that our differential operator L is completely determined by its principal symbol. Namely, in local coordinates our operator reads where the subscripts indicate partial derivatives and the repeated index α indicates summation over α = 1, 2, 3. We study the eigenvalue problem where w(x) is a given infinitely smooth positive scalar weight function. Obviously, the problem (1.1) has the same spectrum as the problem so it may appear that the weight function w(x) is redundant. We will, however, work with the eigenvalue problem (1.1) rather than with (1.2) because we want our problem to possess a gauge degree of freedom (5.1). This gauge degree of freedom will eventually manifest itself as the conformal invariance of the massless Dirac action, see Section 5 for details. The problem (1.1) has a discrete spectrum accumulating to ±∞. We define the counting function N (λ) := 0<λ k <λ 1 as the number of eigenvalues λ k of the problem (1.1), with account of multiplicities, between zero and a positive λ. Theorem 8.4 from [3] states that under appropriate assumptions on periodic trajectories our counting function admits a two-term asymptotic expansion as λ → +∞. If one wishes to reformulate the asymptotic formula (1.3) in such a way that it remains valid without assumptions on periodic trajectories, this can easily be achieved, say, by taking a convolution with a function from Schwartz space S(R); see Theorem 7.2 in [3] for details. Alternatively, one can look at the eta function η(s) := |λ| −s sign λ , where summation is carried out over all nonzero eigenvalues λ and s ∈ C is the independent variable. The results of Section 10 of [3] imply that the series converges absolutely for Re s > 2 and defines a holomorphic function in this half-plane. Moreover, it is known [1] that the eta function extends meromorphically to the whole s-plane with simple poles. It is easy to see that the residue of the eta function at the point s = 2 is 4b, where b is the coefficient from (1.3).
There is an extensive bibliography devoted to the subject of two-term spectral asymptotics for first order systems. This bibliography spans a period of over three decades. Unfortunately, all publications prior to [3] gave formulae for the second asymptotic coefficient that were either incorrect or incomplete (i.e. an algorithm for the calculation of the second asymptotic coefficient rather than an actual formula). The appropriate bibliographic review is presented in Section 11 of [3]. The correct explicit formula for the coefficient b is given in Section 1 of [3].
The objective of this paper is to establish the geometric meaning of the coefficient b. The logic behind restricting our analysis to the case when the manifold is 3-dimensional and L is a 2 × 2 matrix differential operator with trace-free principal symbol and zero subprincipal symbol is that this is the simplest eigenvalue problem for a system of partial differential equations. Hence, it is ideal for the purpose of establishing the geometric meaning of the coefficient b.
In order to establish the geometric meaning of the coefficient b we first need to identify the geometric objects encoded within our eigenvalue problem (1.1).
Geometric object 1: the metric. Observe that the determinant of the principal symbol is a negative definite quadratic form in the dual variable (momentum) p, 4) and the coefficients g αβ (x) = g βα (x), α, β = 1, 2, 3, appearing in (1.4) can be interpreted as the components of a (contravariant) Riemannian metric.
Geometric object 2: the nonvanishing spinor field. The determinant of the principal symbol does not determine the principal symbol uniquely. In order to identify a further geometric object encoded within the principal symbol L prin (x, p) we will now start varying this principal symbol, assuming the metric g, defined by formula (1.4), to be fixed (prescribed).
Let us fix a reference principal symbolL prin (x, p) corresponding to the prescribed metric g and look at all principal symbols L prin (x, p) which correspond to the same prescribed metric g and are sufficiently close to the reference principal symbol. Restricting our analysis to principal symbols which are close to the reference principal symbol allows us to avoid dealing with certain topological issues; this restriction will be dropped in Section 4. It turns out, see Section 2, that the principal symbols L prin (x, p) andL prin (x, p) are related as is a unique infinitely smooth special unitary matrix-function which is close to the identity matrix. Thus, special unitary matrix-functions R(x) provide a convenient parametrization of principal symbols with prescribed metric g. LetL be the differential operator with principal symbolL prin (x, p) and zero subprincipal symbol. It is important to emphasize that for the operators L and L themselves, as opposed to their principal symbols, we have, in general, the inequality L = RLR * (1.7) because according to formula (9.3) from [3] the operator RLR * has nontrivial subprincipal symbol i Hence, the transformation of operatorsL → L specified by formula (1.5) and the conditions that the subprincipal symbols of L andL are zero does, in general, change the spectrum.
The choice of reference principal symbolL prin (x, p) in our construction is arbitrary, as long as this principal symbol corresponds to the prescribed metric g, i.e. as long as we have detL prin (x, p) = −g αβ (x) p α p β for all (x, p) ∈ T * M . It is natural to ask the question: what happens if we choose a different reference principal symbolL prin (x, p)? The freedom in choosing the reference principal symbol L prin (x, p) is a gauge degree of freedom in our construction and our results are invariant under changes of the reference principal symbol. This issue will be addressed in Section 6.
In order to work effectively with special unitary matrices we need to choose coordinates on the 3-dimensional Lie group SU (2). It is convenient to describe a 2 × 2 special unitary matrix by means of a spinor ξ, i.e. a pair of complex numbers ξ a , a = 1, 2. The relationship between a matrix R ∈ SU(2) and a nonzero spinor ξ is given by the formula where the overline stands for complex conjugation and ξ := |ξ 1 | 2 + |ξ 2 | 2 . Formula (1.8) establishes a one-to-one correspondence between SU(2) matrices and nonzero spinors, modulo a rescaling of the spinor by an arbitrary positive real factor. We choose to specify the scaling of our spinor field ξ(x) in accordance with (1.9) Remark 1.2 In [4] we chose to work with a teleparallel connection (metric compatible affine connection with zero curvature) rather than with a spinor field. These are closely related objects: locally a teleparallel connection is equivalent to a normalized ( ξ(x) = 1) spinor field modulo rigid rotations (7.3) of the latter.
Geometric object 3: the topological charge. It is known, see Section 3 in [4], that the existence of a principal symbol implies that our manifold M is parallelizable. Parallelizability, in turn, implies orientability. Having chosen a particular orientation, we allow only changes of local coordinates x α , α = 1, 2, 3, which preserve orientation.
We define the topological charge as with the subscripts p α indicating partial derivatives. As explained in Section 3 of [4], the number c defined by formula (1.10) can take only two values, +1 or −1, and describes the orientation of the principal symbol relative to the chosen orientation of local coordinates.
We have identified three geometric objects encoded within the eigenvalue problem (1.1) -metric, nonvanishing spinor field and topological charge -defined in accordance with formulae (1.4)-(1.10). Consequently, one would expect the coefficients a and b from formula (1.3) to be expressed via these three geometric objects. This assertion is confirmed by the following theorem which is the main result of our paper.
where S(ξ) is the massless Dirac action (A.3) with Pauli matrices (1.13) Theorem 1.1 warrants the following remarks. Firstly, recall that theL appearing in Theorem 1.1 is our reference operator which we need to describe all possible operators L with given metric g. What happens if we take L =L? In this case formula (1.12) holds with spinor field ξ 1 (x) = w(x), ξ 2 (x) = 0. This, on its own, is a nontrivial result.
Secondly, the topological charge c does not appear explicitly in Theorem 1.1. Nevertheless, it is implicitly present in our Pauli matrices (1.13). Indeed, formula (1.5) implies that the integer quantity − i 2 det g αβ tr (L prin ) p1 (L prin ) p2 (L prin ) p3 has the same value as (1.10).
Thirdly, it is tempting to apply Theorem 1.1 in the case when the operator L is itself a massless Dirac operator. This cannot be done because a massless Dirac operator acts on spinors rather than on pairs of half-densities. This impediment can be overcome by switching to a massless Dirac operator on half-densities, see formula (A.19) in [4]. However, we cannot take L to be a massless Dirac operator on half-densities either because, according to Lemma 6.1 from [4], the latter has a nontrivial subprincipal symbol. Furthermore, it is known [2,4] that for the massless Dirac operator the coefficient b is zero.
Finally, Theorem 1.1 provides a fresh perspective on the history of the subject of two-term spectral asymptotics for first order systems, see Section 11 of [3] for details. Namely, Theorem 1.1 shows that even in the simplest case the second asymptotic coefficient for a first order system has a highly nontrivial geometric meaning. At a formal level, the application of microlocal techniques does not require the use of advanced differential geometric concepts. However, the calculations involved are so complicated that it is hard to avoid mistakes without an understanding of the differential geometric content of the spectral problem.
The paper is organized as follows. In Section 2 we explain the origins of formula (1.5) and in Section 3 we give the proof of Theorem 1.1. In Section 4 we introduce the concept of spin structure which allows us to drop the restriction that our principal symbol L prin (x, p) is sufficiently close to the reference principal symbol L prin (x, p). And in Sections 5-7 we show that our formula (1.12) is invariant under the action of certain gauge transformations.

Spinor representation of the principal symbol
Let L prin (x, p) andL prin (x, p) be a pair of trace-free Hermitian 2 × 2 principal symbols and let g be a prescribed Riemannian metric. Both L prin (x, p) and L prin (x, prin ) are assumed to be linear in p: prin (x), α = 1, 2, 3, are some trace-free Hermitian 2×2 matrixfunctions. The assumption that our principal symbols L prin (x, p) andL prin (x, p) are linear in p means, of course, that we are dealing with differential operators as opposed to pseudodifferential operators. The principal symbols L prin (x, p) and L prin (x, p) are assumed to satisfy for all (x, p) ∈ T * M , and are also assumed to be sufficiently close in terms of the C ∞ (M ) topology applied to the matrix-functions L prin (x), α = 1, 2, 3. Our task in this section is to show that there exists a unique infinitely smooth special unitary matrix-function (1.6) which is close to the identity matrix and which relates the principal symbols L prin (x, p) andL prin (x, p) in accordance with formula (1.5).
We follow the convention of [5,4] in denoting the elements of the matrices L As explained in Section 3 of [4], formula (2.4) defines a frame -a triple of infinitely smooth real orthonormal vector fields e j (x), j = 1, 2, 3, on the manifold M -and, moreover, the principal symbol L prin (x, p) is equivalent to the frame e j in the sense that the principal symbol uniquely determines the frame and the frame uniquely determines the principal symbol. Similarly, formula (2.5) defines a framee j which is equivalent to the principal symbolL prin (x, p). Condition (2.3) implies that the frames e j ande j are orthonormal with respect to the same metric. Hence, this pair of orthonormal frames is related as As we assumed the principal symbols L prin (x, p) andL prin (x, p) to be close, the frames e j ande j are also close. Consequently, the matrix-function O(x) is close to the identity matrix and, hence, special orthogonal. It is well-known that the Lie group SO (3) is locally (in a neighbourhood of the identity) isomorphic to the Lie group SU(2). According to formulae (A.15) and (A.2) from [4], a 3 × 3 special orthogonal matrix O is expressed via a 2 × 2 special unitary matrix R as where prin are the matrix-functions from formula (2.2). We could stick with the notationL (α) prin but we choose to switch to σ α because this is how Pauli matrices are traditionally denoted in the subject.
It is easy to see that formula (2.3) implies where I is the 2 × 2 identity matrix. Formula (2.10) means that our σ α satisfy the defining relation for Pauli matrices. Formulae (2.6)-(2.8), (1.8), (2.5) and (2.9) allow us to express the frame e j via the spinor field ξ and Pauli matrices σ α . We took great care to choose coordinates on the Lie group SU(2) (i.e. structure of the matrix in the RHS of formula (1.8)) so that the resulting expressions agree with formulae (B.3), (B.4) and (B.1) from [5]. The only difference is in notation: the ϑ j in Appendix B of [5] stands for ϑ j α = δ jk g αβ e k β (compare with formula (3.4)). The fact that our construction agrees with that in [5] will become important in the next section when we will make use of a particular formula from [5].
Remark 2.1 As explained in Section 3 of [4], the topological charge, initially defined in accordance with formula (1.10), can be equivalently rewritten in terms of frames as c = sgn det e j α = sgn dete j α . (2.11) The paper [5] was written under the assumption that c = +1 , (2.12) see formula (A.1) in [5]. This means that care is required when using the results of [5]. Namely, in the next section we will first prove Theorem 1.1 for the case (2.12) and then provide a separate argument explaining why formula (1.12) remains true in the case c = −1 . (2.13) 3 Proof of Theorem 1.1 We prove Theorem 1.1 by examining the equivalent spectral problem (1.2). Note that transition from (1.1) to (1.2) is a special case of the gauge transformation (5.1) with ϕ = ln w. As explained in the beginning of Section 5, the transformation (5.1) preserves the structure of our eigenvalue problem: the principal symbol of the operator w −1/2 Lw −1/2 is trace-free and its subprincipal symbol is zero. We now apply Theorem 1.
The real scalar field * T ax (x) has the geometric meaning of the Hodge dual of axial torsion of the teleparallel connection, see [4] for details. Let us now drop the assumption (3.1). The introduction of a weight function is equivalent to a scaling of the principal symbol L prin (x, p) → (w(x)) −1 L prin (x, p), which, in view of formulae (2.4) and (1.4), leads to a scaling of the frame e j → w −1 e j (3.5) and corresponding scaling of the metric g αβ → w −2 g αβ . Here the remarkable fact is that we do not get derivatives of the weight function because these cancel out due to the antisymmetric structure of the RHS of formula (3.3). In other words, axial torsion, defined by formulae (1.20) and (3.12) from [4], has the remarkable property that it scales in a covariant manner under scaling of the frame. Note that the full torsion tensor, defined by formula (3.12) from [4], does not possess such a covariance property. Formula (3.7) tells us that in order to accommodate an arbitrary weight function w(x) we need to multiply the integrand in formula (3.2) by (w(x)) 2 , which gives us b = 3c 8π 2 M w 2 * T ax det g αβ dx . (3.8) Let us emphasize that the metric and torsion appearing in formula (3.8) are the original, unscaled metric and torsion determined by the original, unscaled principal symbol L prin (x, p). The scaling argument has been incorporated into the factor (w(x)) 2 .
We now need to express the integrand in (3.8) in terms of the spinor field ξ. We already have an expression for the weight function in terms of the spinor field, see formula (1.9). So we only need to express the Hodge dual of axial torsion in terms of the spinor field. Formulae (2.4), (2.1), (2.2), (2.9), (1.5) and (1.8) allow us to express the frame e j via the spinor field ξ and Pauli matrices σ α . Hence one needs to combine all these formulae to get explicit expressions for the vector fields e j , j = 1, 2, 3, and substitute these into (3.4) and (3.3). This is a massive calculation. Fortunately, for the case (2.12) this calculation was carried out in Appendix B of [5]: formula (B.5) from [5] where W is the massless Dirac operator (A.1). Formulae In order to prove formula (1.12) for the case (2.13), we invert coordinates (x α → −x α ), which changes the sign of topological charge and allows us to use formula (1.12). We then invert coordinates again and use the facts that

Spin structure
In stating our results in Section 1 we assumed the principal symbols L prin (x, p) and L prin (x, p) to be sufficiently close. This was done in order to ensure that equation (1.5) could be resolved with respect to the special unitary matrix-function R(x).
The restriction of closeness of principal symbols can be overcome by means of the introduction of the concept of spin structure.  The closeness of the principal symbols L prin (x, p) andL prin (x, p) was never used in the proof of Theorem 1.1. All that is needed for Theorem 1.1 to be true is for the principal symbols L prin (x, p) andL prin (x, p) to have the same spin structure, i.e. belong to the same equivalence class.
Hence, it would have been more logical to identify the spin structure as a separate geometric object from the very start, in Section 1, and avoid arguments relying on the closeness of the principal symbols. We chose not to proceed along this route in order to make the exposition in Section 1 as simple and clear as possible.
The only difference between the "local" setting (the principal symbols L prin (x, p) andL prin (x, p) are assumed to be close) and the "global" setting (the principal symbols L prin (x, p) andL prin (x, p) are assumed to have the same spin structure) is that we can no longer claim that the special unitary matrix-function R(x) appearing in formula (1.5) is unique. In the "local" setting uniqueness was achieved by requiring R(x) to be close to the identity matrix, whereas in the "global" setting R(x) is defined modulo sign (not surprising as SU (2) is the double cover of SO (3)). This sign indeterminacy does not affect formula (1.12) because the massless Dirac action is quadratic in the spinor field.
The number of different spin structures (i.e. number of equivalence classes of principal symbols) depends on the topology of the manifold. Say, the torus T 3 admits eight different spin structures, whereas the sphere S 3 admits a unique spin structure. See Appendices A and B in [4] and further bibliographic references therein for more details.
It may seem that our Definition 4.1 is different from the definition of spin structure in differential geometric literature. Indeed, differential geometers do not operate with concepts such as the principal symbol, using frames instead. However, it has been shown in Section 3 of [4] that a principal symbol is equivalent to a frame, so our "microlocal" definition of spin structure can be easily recast in terms of frames, bringing it in agreement with the traditional differential geometric one.
Here it is important to emphasize that we do not claim to have redefined the notion of spin structure for the most general case. We work in the very specific setting of dimension three.

Conformal invariance
Let us transform the differential operator L and weight w(x) as where ϕ : M → R is an arbitrary infinitely smooth real-valued scalar function. The transformation (5.1) does not change the spectrum of our eigenvalue problem (1.1) and, moreover, preserves its structure: the principal symbol remains trace-free and the subprincipal symbol remains zero. The fact that the subprincipal symbol remains zero is established by using formula (9.3) from [3]. Here, of course, it is important that we are conjugating the operator by a real-valued scalar function e −ϕ/2 rather than a complex-valued matrix-function R.
In this section we examine how the gauge transformation (5.1) works its way into scalings of the metric, Pauli matrices and spinor field.
Formulae (1.4) and (5.1) imply that the metric transforms as which means that we are looking at a conformal scaling of the metric. We scale the reference principal symbolL prin (x, p) the same way as the principal symbol L prin (x, p), i.e. asL prin → e −ϕL prin , Of course, the scaling of Pauli matrices (5.4) agrees with the scaling of the metric (5.2) in the sense that the scaled Pauli matrices and metric satisfy the identity (2.10).
Formulae (1.9) and (5.1) imply that the spinor field scales as ξ → e −ϕ ξ. We first look at the expression W ξ. Examination of formulae (A.1) and (A.2) show that the expression W ξ transforms as We see that the expression W ξ scales in a covariant way, with "covariant" meaning that the derivatives of ϕ do not appear in the RHS of (5.6). Of course, the covariance of the massless Dirac operator under conformal scalings of the metric is a know differential geometric fact: see Theorem 4.3 in [6]. Formulae (5.5) and (5.6) imply Re(ξ * W ξ) → e −3ϕ Re(ξ * W ξ). (5.7) Formula (5.2) implies g αβ → e 2ϕ g αβ and, as we are working in dimension 3, this, in turn, implies that the Riemannian density scales as det g αβ → e 3ϕ det g αβ .

SU(2) invariance
Let us transform the reference principal symbolL prin (x, p) as

Invariance under rigid rotations
Let us transform the differential operator L as is a constant special unitary matrix. The transformation (7.1) does not change the spectrum of our eigenvalue problem (1.1) and, moreover, preserves its structure: the principal symbol remains trace-free and the subprincipal symbol remains zero. We refer to the transformation (7.1) as a rigid rotation because it describes a rotation of the frame (2.4), with this rotation being the same at all points of the manifold M . The transformation (7.1) is equivalent to the following transformation of the special unitary matrix-function R(x) appearing in formula (1.5): Formulae (1.8) and (1.9) give us a one-to-one correspondence between special unitary matrix-functions and weight functions on the one hand and nonvanishing spinor fields on the other. In terms of the spinor field the transformation (7.2) reads Note that, unlike (6.3), this transformation is not linear because of the complex conjugation. The transformation (7.3) can be written as a sum of linear and antilinear transformations: is the charge conjugation operator, see formula (A.9) in [4].
Let us show, by performing explicit calculations, that the massless Dirac action (A.3) is invariant under the transformation (7.4). Using the fact that the massless Dirac operator commutes with the charge conjugation operator, see Property 3 in Appendix A of [4], we get But the expression in the curly brackets is purely imaginary, so

A Massless Dirac action
In this appendix we define, in a concise manner, the massless Dirac action. For more details see Appendix A in [4].
The massless Dirac operator is the matrix operator where summation is carried out over α, β, γ = 1, 2, 3, and are the Christoffel symbols. The operator (A.1) acts on a 2-component complexvalued spinor field ξ which we write as a 2-column, ξ = ξ 1 ξ 2 T . We chose the letter "W " for denoting the massless Dirac operator because in theoretical physics literature it is often referred to as the Weyl operator. Note that one should really be referring here to the static Weyl operator because we have excluded time, which is natural in the setting of spectral theory.
We define the massless Dirac action as where the star indicates Hermitian conjugation. This is the variational functional corresponding to the operator (A.1). Here, of course, we use the fact that in view of the self-adjointness of the operator W we have

B Example
In this appendix we consider an explicit example illustrating the use of Theorem 1.1. Consider the unit torus T 3 parameterized by cyclic coordinates x α , α = 1, 2, 3, of period 2π. Let L be the differential operator with principal symbol and zero subprincipal symbol. We examine below the eigenvalue problem (1.1) for this particular operator L and trivial weight function (3.1). Substituting (B.1) into (1.4) we see that the above principal symbol generates the Euclidean metric Hence, as the reference principal symbol it is natural to take Substituting (B.3) into (1.13) we get standard Pauli matrices It is not a priori obvious that the principal symbols L prin (x, p) andL prin (x, p) have the same spin structure. The only way to establish that they do indeed have the same spin structure is to resolve equation (1.5) with respect to the special unitary matrix-function R(x). Straightforward calculations give Of course, the underlying reasons why in this particular case we do not encounter topological obstructions are that both principal symbols have the same (positive) topological charge and that the frame encoded in (B.1) makes an even number of turns (two turns) as x 3 runs from 0 to 2π. See Appendix A in [4] for more details. as λ → +∞. Note that the nonperiodicity condition (see Definitions 8.3 and 8.4 in [3]) is fulfilled in our example, so, according to Theorem 8.4 from [3], the asymptotic formula (B.7) holds as it is, without mollification. Observe now that in our example the spectrum of the operator L can be evaluated explicitly. Indeed, letL be the differential operator with principal symbol (B.3) and zero subprincipal symbol. In other words, letL =L prin (x, −i∂/∂x). Consider now the operator RLR * , where R is the matrix-function (B.5) It is easy to check that the subprincipal symbol of the operator RLR * is −I, where I is the 2 × 2 identity matrix. Hence, L = RLR * + I, (B.8) compare with formula (1.7). But the operator RLR * is unitarily equivalent to the operatorL and the spectrum ofL is known, see Appendix B in [4]. Using (B.8), we conclude that the eigenvalues of our operator L are as follows.
• The number 1 is an eigenvalue of multiplicity two.
• For each m ∈ Z 3 \ {0} we have the eigenvalue 1 + m and unique (up to rescaling) eigenfunction, with eigenfunctions corresponding to different m being linearly independent.
• For each m ∈ Z 3 \ {0} we have the eigenvalue 1 − m and unique (up to rescaling) eigenfunction, with eigenfunctions corresponding to different m being linearly independent.
Thus, N (λ) − 1 is the number of integer lattice points inside a 2-sphere of radius λ − 1 in R 3 centred at the origin. Applying the result from [7] we get as λ → +∞, with ε being an arbitrary positive number. The more advanced number theoretic result (B.9) agrees with our asymptotic formula (B.7). Note that the calculations presented in this section remain unchanged if we replace everywhere p 1 ∓ ip 2 by p 1 ± ip 2 . This is in agreement with the fact that the topological charge c does not appear in our formula (1.12).