Small amplitude solitary waves in the Dirac-Maxwell system

We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system proving the existence of solutions in which the Dirac wave function is of the form $\phi(x,\omega)e^{-i\omega t}$, $\omega\in(-m,\omega_*)$, with some $\omega_*>-m$, such that $\phi_\omega\in H^1(\mathbb{R}^3,\mathbb{C}^4)$, $\Vert\phi_\omega\Vert^2_{L^2}=O(m-|\omega|)$, and $\Vert\phi_\omega\Vert_{L^\infty}=O(m-|\omega|)$. The method of proof is an implicit function theorem argument based on an identification of the nonrelativistic limit as the ground state of the Choquard equation.


Introduction and results
The Dirac equation, which appeared in [Dir28] just two years after the Schrödinger equation, is the correct Lorentz invariant equation to describe particles with nonzero spin when relativistic effects cannot be ignored. The Dirac equation predicts accurately the energy levels of an electron in the Hydrogen atom, yielding relativistic corrections to the spectrum of the Schrödinger equation. Further higher order corrections arise on account of interactions with the electromagnetic field, described mathematically by the Dirac-Maxwell Lagrangian, which aims to provide a selfconsistent description of the dynamics of an electron interacting with its own electromagnetic field. The perturbative treatment of the Dirac-Maxwell system in the framework of second quantization allows computation of quantities such as the energy levels and scattering cross-sections, which have been compared successfully with experiment; of course this quantum formalism does not provide the type of tangible description of particles and dynamical processes familiar from classical physics. Mathematically, the quantum theory (QED) has not been constructed, and indeed may not exist in the generally understood analytical sense. In particular it is a curious fact that although the electron is the most stable elementary particle known to physicists today, there is no mathematically precise formulation and proof of its existence and stability. This has resulted in an enduring interest in the classical Dirac-Maxwell system, both in the physics and mathematics literature. Regarding the former, the relevance of the classical equations of motion for QED has been widely debated. The prevalent view seems to be that the Dirac fermionic field does not have a direct meaning or limit in classical physics, and hence that the classical system is not really directly relevant to the world of observation. Nevertheless, there have been numerous attempts, both by Dirac himself and by many others -see [Dir62,Wak66,Lis95] and references therein -to construct localized solutions of the classical system or some modification thereof, with the aim of obtaining a more cogent mathematical description of the electron (or other fundamental particles). We consider the system of Dirac-Maxwell equations, where the electron, described by the standard "linear" Dirac equation, interacts with its own electromagnetic field which is in turn required to obey the Maxwell equations: (1.1) with the charge-current density J µ = (ρ, J) generated by the spinor field: (1.2) Above, ρ and J are the charge and current respectively. We denoteψ = (γ 0 ψ) * = ψ * γ 0 , with ψ * the hermitian conjugate of ψ. The charge is denoted by e (so that for the electron e < 0); the fine structure constant is the dimensionless coupling constant α ≡ e 2 c ≈ 1/137. We choose the units so that = c = 1. We have written the Maxwell equations using the Lorentz gauge condition ∂ µ A µ = 0. The Dirac γ-matrices satisfy the anticommutation relations {γ µ , γ ν } = 2g µν , with g µν = diag[1, −1, −1, −1]. The four-vector potential A µ has components (A 0 , A), with A = (A 1 , A 2 , A 3 ), so that the lower index version A µ = g µν A ν has components (A 0 , −A) so A 0 = A 0 . Following [BD64] and [BS77], we define the Dirac γ-matrices by where I 2 is the 2 × 2 unit matrix and σ j are the Pauli matrices: After introduction of a space time splitting, the system (1.1) takes the form Here α = (α 1 , α 2 , α 3 ), and α j and β are the 4 × 4 Dirac matrices: with {σ j } 3 j=1 the Pauli matrices. We will not distinguish lower and upper indices j of α and σ, so that α j = α j , σ j = σ j . The α-matrices and γ-matrices are related by Numerical justification for the existence of solitary wave solutions to the Dirac-Maxwell system (1.1) was obtained in [Lis95], where it was suggested that such solutions are formed by the Coulomb repulsion from the negative part of the essential spectrum (the Klein paradox). The numerical results of [Lis95] showed that the Dirac-Maxwell system has infinitely many families of solitary wave solutions φ N (x, ω)e −iωt , ω −m. Here the nonnegative integer N denotes the number of nodes of the positronic component of the solution (number of zeros of the corresponding spherically symmetric solution to the Choquard equation; see §3). A variational proof of existence of solitary waves for ω ∈ (−m, 0) and with N = 0 first appeared in [EGS96], and the generalization to handle ω ∈ (−m, m) is in [Abe98].
In the present paper, we give a proof of existence of solitary wave solutions to the Dirac-Maxwell system based on the perturbation from the nonrelativistic limit and also obtain the precise asymptotics for the solution in this limit. The physical significance of these types of solitary wave solutions requires not only their existence but also stability, and it is to be hoped that the type of detailed information about the solutions which is a consequence of the existence proof in this article, but does not seem to be so easily accessible from the original variational constructions, will be helpful in future stability analysis (see Remark 3.6 below). The second motivation for presenting this proof is to realize mathematically the physical intuition explained in [Lis95] which explains the existence of these bound state solutions in terms of the Klein paradox ([BD64, §3.3]). Moreover, once one knows that the excited eigenstates of the Choquard equation are nondegenerate (currently this nondegeneracy is established only for the ground state, N = 0 [Len09]), our argument will yield the existence of excited solitary wave solutions in Dirac-Maxwell system, extending the results of [EGS96] to N ≥ 1. We will construct solitary wave solutions by deforming the solutions to the nonrelativistic limit (represented by the Choquard equation) via the implicit function theorem. Such a method was employed in [Oun00,Gua08] for the nonlinear Dirac equation and in [RN10a,Stu10,RN10b] for Einstein-Dirac and Einstein-Dirac-Maxwell systems.
The solitary wave (φe −iωt , A µ (x)) satisfies the stationary system (1.6) Theorem 1.1. There exists ω * > −m such that for ω ∈ (−m, ω * ) there is a solution to (1.6) of the form are of Schwartz class. The solutions could be chosen so that in the nonrelativistic limit ǫ = 0 one haŝ where n ∈ C 2 , |n| = 1, and ϕ 0 is a strictly positive spherically symmetric solution of Schwartz class to the Choquard equation (1.8) Remark 1.2. The existence of a positive spherically-symmetric solution ϕ 0 ∈ S (R 3 ) to (1.8) was proved in [Lie77].
Here is the plan of the paper. We give the heuristics in §2. The Choquard equation, which is the nonrelativistic limit of the Dirac-Maxwell system, is considered in §3. In §4, we complete the proof of existence of solitary waves via the implicit function theorem.

Heuristics on the nonrelativistic limit
The small amplitudes waves constructed in Theorem 1.1 are best understood physically in terms of the non-relativistic limit. Since we have set the speed of light and other physical constants equal to one, the relevant small parameter is the excitation energy (or frequency) as compared to the mass m. To develop some preliminary intuition regarding the non-relativistic limit, following [Lis95], we neglect the magnetic field described by the vector-potential A j , getting Let us consider a solitary wave solution ψ( , where φ 1 , φ 2 ∈ C 2 and A 0 = A 0 (x) only. Then φ 1 , φ 2 , and A 0 satisfy where σ = (σ 1 , σ 2 , σ 3 ), the vector formed from the Pauli matrices. Consider small amplitude solitary waves with ω ≈ −m. Then A 0 is small and −2mφ 1 ≈ −iσ·∇φ 2 , Denoting ǫ 2 = m 2 − ω 2 , 0 < ǫ ≪ m, the above suggests the following scaling: Note that since φ j and A 0 depend on ω and x, the scaled functions A 0 and Φ j are functions of y and of ǫ. In the limit which can be rewritten as the following equation for Φ 2 only: with the understanding that Φ 1 is then obtained from the first equation of (2.3).
Remark 2.1. Regarding self-consistency of this approximation: one can check that, when using the scaling (2.2), the magnetic field vanishes to higher order in the limit ǫ → 0, in agreement with [Lis95].
. The second equation from (2.1) would then take the form where A·σφ 1 = O(ǫ 6 ) while other terms are O(ǫ 4 ). Thus the approximation is al least self-consistent, and the analysis in §4 justifies this rigorously.
Remark 2.2. Regarding symmetry: while it is clear that radial symmetry of both φ 1 and φ 2 is inconsistent with (2.3), are permitted in principle, suggesting that in the non-relativistic limitΦ 2 could be radial, or to be more precise of the The starting point for our perturbative construction of solitary wave solutions to (1.4) is indeed a radial solution of (2.6), although the exact form of these solitary waves has to be modified from (2.5) when the effect of the magnetic The method of proof we employ does not require any particular symmetry class of the solitary wave.
The above discussion suggests that the system (2.6) will determine the non-relativistic limit to highest order. The system (2.6) describes a Schrödinger wave function with an attractive self-interaction determined by the Poisson equation. Because the sign of the interaction is attractive it is often referred to as the stationary Newton-Schrödinger system. It is equivalent to a nonlocal equation for Φ known as the Choquard equation, which is the subject of the next section.

The nonrelativistic limit: the Choquard equation
The system (2.6) can also be obtained by looking for solitary wave solutions in the system This is the time-dependent Newton-Schrödinger system. If φe −iωt , V (x) is a solitary wave solution, then φ and V satisfy the stationary system We rewrite the system (3.1) in the non-local form, called the Choquard equation: where ∆ −1 is the operator of convolution with − 1 4π|x| . The solitary waves are solutions are of the form ψ(x, t) = φ ω (x)e −iωt , with φ ω satisfying the non-local scalar equation This suggests the following variational formulation for the problem: find critical points of subject to the constraint |φ(x)| 2 dx = const. This formulation is the basis of the existence and uniqueness proofs in the references which are summarized in the following theorem. Remark 3.2. Together with the heuristics in the previous section, this result suggests that for ω sufficiently close to −m there might exist infinitely many families of solitary waves to the Dirac-Maxwell system, which differ by the number of nodes. As mentioned in [EGS96], the variational methods used in that paper are hard to generalize to prove the existence of multiple solitary waves for each ω (such a multiplicity result is obtained in [EGS96] for the Dirac -Klein-Gordon system).
Remark 3.3. The φ(x) and V (x) for different values of ω < 0 can be scaled to produce a standard form as follows.
Let ζ > 0 satisfy ζ 2 = −ω and write y = ζx , φ(x) = ζ 2 u(ζx) , and V (x) = ζ 2 v(ζx). Then (3.2) is equivalent to the following system for u(y), v(y): In the remainder of this section we summarize the properties of the linearized Choquard equation which follow from [Len09] and are needed in §4. Consider a solution to the Choquard equation of the form with R, S real-valued. The linearized equation for R, S is: Both L 0 and L 1 are unbounded operators L 2 → L 2 which are self-adjoint with domain H 2 ⊂ L 2 . Clearly L 0 ϕ 0 = 0, with 0 ∈ σ d (L 0 ) an eigenvalue corresponding to a positive eigenfunction ϕ 0 ; it follows that 0 is a simple eigenvalue of L 0 , with the rest of the spectrum separated from zero. The range of L 0 is {ϕ 0 } ⊥ , the L 2 orthogonal complement of the linear span of ϕ 0 . Notice that . Lemma 3.4. The self-adjoint operator L 1 : H 2 → L 2 has exactly one negative eigenvalue, which we denote −Λ 0 , and has a three dimensional kernel Ker L 1 spanned by {∂ j ϕ 0 } 3 j=1 . The range of L 1 is (Ker L 1 ) ⊥ , the L 2 orthogonal complement of the linear span of the {∂ j ϕ 0 } 3 j=1 . Proof. We proceed similarly to [Kik08, Lemma 5.4.3]. The n = 0 ground state solution ϕ 0 to (3.4) is characterized in [Lie77] as the solution, unique up to translation and phase rotation, to the following minimization problem: for certain µ > 0. We claim that this implies that establishing the claim. We took into account that ϕ 0 satisfies the stationary equation E ′ (ϕ 0 ) = ω 0 Q ′ (ϕ 0 ) and also that Q ′ (ϕ 0 ), ϕ 0 = 2 ϕ 0 2 So L 1 is non-negative on a codimension one subspace. On the other hand, since the integral kernel of ∆ −1 is strictly negative, while ϕ 0 is strictly positive and L 0 ϕ 0 = 0, it follows that ϕ 0 L 1 ϕ 0 < 0 so that there certainly exists one negative eigenvalue characterized as Let η 0 be the corresponding eigenfunction, L 1 η 0 = −Λ 0 η 0 . To prove that (−Λ 0 , 0) ⊂ ρ(L 1 ), the resolvent set, consider the minimization problem (3.9) Now the relation L 0 ϕ 0 = 0, together with translation invariance, implies that L 1 ∂ j ϕ 0 = 0. Moreover, it is proved in [Len09] that ϕ 0 is nondegenerate, in the sense that the kernel of L 1 is spanned by the ∂ j ϕ 0 , 1 ≤ j ≤ 3. Hence, by consideration of linear combinations of the eigenfunctions η 0 and ∂ j ϕ 0 , that the number defined by (3.9) is ≤ 0. In fact it must equal zero since if it were negative a simple compactness argument (based on the negativity of ω 0 ) would imply the existence of a negative eigenvalue in the interval (−Λ 0 , 0) with corresponding eigenfunction η 1 orthogonal to η 0 . But since η 0 , η 1 would then be an orthogonal pair of eigenfunctions of L 1 with negative eigenvalues, and both having non-zero inner product with ϕ 0 , this would immediately contradict the fact that L 1 is non-negative on {ϕ 0 } ⊥ .
We conclude with a few remarks on the stability of solitary waves to the Choquard equation. By Remark 3.3 we know the ω-dependence of a localized solution φ ω (x)e −iωt to (3.3): one has φ ω (x) = ζ 2 u(ζ|x|), where ζ = √ −ω. From this we can obtain the frequency dependence of the charge: It follows that for all negative frequencies dQ dω < 0 . By the Vakhitov-Kolokolov stability criterion ( [VK73]), this leads us to expect the linear stability of no-node solitary waves (the ground states) in the Choquard equation. To determine the point spectrum of is an eigenfunction corresponding to the eigenvalue λ ∈ C, then −λ 2 R = L 0 L 1 R. If λ = 0, then one concludes that R is orthogonal to Ker L 0 = {ϕ 0 }, hence we can apply L −1 0 ; taking then the inner product with R, we deduce that: which implies that λ 2 ∈ R. Moreover, by (3.9), λ 2 ≤ 0, leading to the conclusion that the point spectrum σ d (JL) ⊂ iR, and hence the absence of growing modes at the linearized level. The (nonlinear) orbital stability of the ground state solitary wave was proved in [CL82].
Remark 3.6. In view of [CGG12,BC12], one expects that the linear stability or instability of small amplitude solitary waves is directly related to the linear stability or instability of the corresponding nonrelativistic limit, which for Dirac-Maxwell is given by the Choquard equation. We hope that this may provide a route to understanding stability of small solitary waves solutions for the Dirac-Maxwell system.

Proof of existence of solitary waves in Dirac-Maxwell system
In this section, we complete the proof of Theorem 1.1. It is obtained as a consequence of Proposition 4.6 after the application of a rescaling motivated by the discussion in §3.
, where for j = 1, 2 the φ j ∈ C 2 are essentially the components of φ in the range of the projection operators Π 1 = 1 2 (1 + β), and Π 2 = 1 2 (1 − β) (under obvious isomorphisms of these subspaces with C 2 ). Applying Π 1 and Π 2 to (1.6), we have: We write (4.3) as and regard the potentials A 0 and A = (A j ) as non-local functionals of φ = φ 1 φ 2 . Above, N(x) = (4π|x|) −1 is the Newtonian potential. In abstract terms, the equations are of the form ωQ ′ = E ′ where the charge functional is and, regarding A 0 , A as fixed non-local functionals (4.4) of φ, the Hamiltonian E(φ) is given by For future reference we recall the following trick from [Stu99]: Lemma 4.1. Let ξ α be a finite collection of vector fields on the phase space which are infinitesimal symmetries, in the sense that Q ′ , ξ α = 0 = E ′ , ξ α . Then any solution of the equation ωQ ′ − E ′ − a α ξ α = 0 , for some set a α ∈ R, is also a solution of ωQ ′ − E ′ = 0, as long as the matrix ξ α , ξ β is well defined and nondegenerate.
Proof. For sufficiently regular ξ β it is possible to take the inner product, yielding a α ξ α , ξ β = 0 which gives the result. (The precise meaning of sufficiently regular is just that this computation is valid; it would be sufficient for ξ α to lie in a subspace F of L 2 with the property that the equation ωQ ′ − E ′ − a α ξ α = 0 holds in the dual of F .) Example 4.2. For ψ : R → C and Q = 1 2 |ψ| 2 and E = 1 2 |∇ψ| 2 − 1 p+1 |ψ| p+1 the symmetry of phase rotation corresponds to the infinitesimal symmetry ξ(ψ) = iψ, and it is easy to check that given an H 1 distributional solution of ωQ ′ − E ′ − aξ = 0, i.e. a weak solution of −∆ψ − |ψ| p ψ = ωψ − iaψ, for any a ∈ R, one necessarily has a = 0. The same holds in higher dimensions as long as p is such that the equation holds as an equality in H −1 .
Remark 4.3. The advantage of solving the more general equation with the unknown "multipliers" a α is that in an implicit function theorem setting the multipliers can be varied to fill out the part of the cokernel corresponding to the symmetries. It is then shown after the fact that the multipliers are in fact zero. The choice of ξ α is determined by the symmetry group; in the case of Dirac-Maxwell the relevant group is the seven dimensional group generated by translations, rotations and phase rotation. The infinitesimal versions of these actions give the following vector fields ( [BD64]): In accordance with the heuristics in §2 we introduce functions Φ 1 (y, ǫ), Φ 2 (y, ǫ) ∈ C 2 and A µ (y, ǫ) by the following scaling relations: where ǫ and ω are related by ω = − √ m 2 − ǫ 2 . Then, writing ∇ y for the gradient with respect to y j = ǫx j , 1 ≤ j ≤ 3, we have: Let ϕ 0 ∈ S (R 3 ) be the ground state solution to the Choquard equation with ω 0 = − 1 2m : That is, ϕ 0 is a strictly positive, spherically symmetric, smooth, and exponential decaying function. As discussed in the previous section, such a solution exists by [Lie77]; the value ω 0 = −(2m) −1 is chosen for our convenience. Using ϕ 0 , we can produce a solution to (4.9)-(4.11) in the nonrelativistic limit ǫ = 0: (4.14) The symmetry of this configuration is axial, with the magnetic field along the z axis of symmetry. C 4 ). Then A µ defined by (4.11) satisfy Proof. The functions A µ defined by (4.11) are of the form N * h with h := f g, where f, g ∈ H 1 (R 3 ). Due to the Sobolev embedding H 1 (R 3 ) ⊂ L 6 (R 3 ), we have h ∈ L p (R 3 ) , 1 ≤ p ≤ 3. By the Hölder inequality, one has where B 1 is the unit ball in R 3 and χ B1 is its characteristic function, hence |x| −1 * h ∈ L ∞ (R 3 ). Furthermore the structure of (4.11) makes it clear that the mappings (4.16) Introducing P / = −iσ·∇ y and substituting ω = − √ m 2 − ǫ 2 , we rewrite (4.9), (4.10) as the equation (4.17) As above, we regard the A µ = (A 0 , A), A = (A j ), as non-local functionals A µ = A µ (Φ, ǫ) determined by (4.11).
With this understood, the entire system is encapsulated in the equation F (Φ, ǫ) = 0 for Φ = Φ 1 Φ 2 only. In terms of the original variables: where the functionals Q, E are defined by (4.5), (4.6). The nonrelativistic limit satisfies F (Φ, 0) = 0 (cf. (4.13), (4.14)), so that to obtain solutions for small ǫ it is necessary to compute the derivative of F at the point (Φ, 0). This is determined by the set of directional derivatives. Let e 1 = 1 0 and e 2 = 0 1 , and let g ∈ H 1 (R 3 , C 2 ). To compute the directional derivatives first note that A j drops out on putting ǫ = 0, and then note further that by (4.11) only the derivative of A 0 at (Φ 1 , 0) with respect to Φ 2 is nonzero, with derivative given by We deduce that for C 2 -valued functions U and V , Thus, the derivative of F at the nonrelativistic limit point (Φ, 0) is the linear map DF (Φ, 0) given by the matrix M. This is a differential operator, which we consider as an unbounded operator on L 2 (R 3 ; C 2 ) ⊕ L 2 (R 3 ; C 2 ).
1. The map M : is a Hermitian operator with domain X.

The kernel of M is given by
4. The range of M : is closed in the topology of Y and is given by where ⊥ means the orthogonal complement with respect to the inner product in L 2 ⊕ L 2 .

The inverse of M :
where the definitions and properties of the operators L 0 , L 1 are given in §3.
Proof. The proof depends on some properties of the linearized Choquard equation from [Len09] which are stated in §3. The fact in (1) that M is Hermitian follows from the fact that P / is Hermitian. From Lemma 4.4 the assertion (2) is immediate from the properties of N and the fact that ϕ 0 and its partial derivatives are smooth and exponentially decreasing. To prove (3),(4) and (5) we consider how to solve M U V = F G , i.e. the system We first express U in terms of V by U = 1 2m (F − P / V ) , and, writing V = V 1 e 1 + V 2 e 2 , Referring to the definitions in §3 of L 0 and L 1 , with ω 0 set equal to −(2m) −1 , we arrive at the following equations: (4.20) It is useful here that the components with respect to e 1 and e 2 are decoupled. Noting also from the form of L 0 , L 1 that these operators take real/imaginary valued functions to real/imaginary valued functions, and further that L 1 = L 0 on pure imaginary functions, we obtain the given formula for V , and hence for U , immediately from §3. The identification of the kernel in (3) is then a specialization of this, given the information on Ker L 0 and Ker L 1 in §3, and also (4) is a consequence of the identification of the ranges of L 0 and L 1 given in §3.
The statement of Theorem 1.1 will follow from the following result.
Proof. Solutions of (4.1)-(4.3) for small ǫ can be produced by solving F = 0. The proof of existence of solutions to this equation is by the implicit function theorem and Lemma 4.1, perturbing from the nonrelativistic limit point F (Φ, 0) = 0. To start we claim that F , as defined in (4.17), is a C ∞ function X × (−m, +m) → Y . To prove this notice that the expression for F is manifestly smooth in ǫ for ǫ 2 < m 2 , and its dependence on Φ j is built up from compositions of certain multilinear maps and linear operators; the structure of the expressions obtained after successive differentiation is the same. Referring to the specific formulae, the fact that these expressions are all C ∞ is an immediate consequence of the fact that multiplication gives continuous bilinear ( =⇒ smooth) maps H 1 × H 2 → H 1 and H 2 × H 2 → H 2 (Moser inequalities) and Lemma 4.4. We are looking for Φ(ǫ) in the form We use the same component notation as above We apply the implicit function theorem to the function G : (4.23) Remark 4.7. Referring to (4.3) we have introduced a linear combination of the six infinitesimal symmetries corresponding to translation and rotation. The action of phase rotation is not independent of rotation in the nonrelativistic limit, which is why the seventh parameter does not appear. In terms of the original variables (cf. (4.7)): Computing the derivatives of (4.23) at ǫ = 0, we see that the linear span {∂ aj G, ∂ bj G; 1 ≤ j ≤ 3} is equal to Ker M. Referring to Lemma 4.5, this establishes that the derivative of G at ǫ = 0, This latter condition serves to divide out by the action of the symmetry group, giving a local slice. Referring to Lemma 4.1, to deduce that these in fact generate solutions of F = 0, for sufficiently small ǫ > 0, it is sufficient to verify that a(ǫ) = 0, b(ǫ) = 0, which is in turn a consequence of the nondegeneracy of the matrix of inner products of the infinitesimal vector fields, scaled as above. This amounts to the need to verify nondegeneracy of the 6 × 6 matrix for small ǫ. (In the matrix (4.25) the indices j, j ′ , k, k ′ run between 1 and 3.) Lemma 4.8. The matrix given by (4.25), evaluated at φ(x) = Proof. Clearly the dominant terms arise from the second ("large") component giving rise to diagonal matrix elements which, referring to the block form in (4.25), are O(ǫ 4 ). Since Ψ j = O(ǫ), the result will follow from nondegeneracy of the matrix with Ψ j set equal to zero. Using ǫ −2 φ = − ǫ 2m P /Φ 2 Φ 2 andΦ 2 = ϕ 0 0 , we calculate the first diagonal term: where we took into account the spherical symmetry of ϕ 0 , which leads to ∂ y 1 ϕ 0 , ∂ y 1 ϕ 0 = 1 3 ϕ 0 , (−∆ y )ϕ 0 . Next for the off-diagonal terms we compute, again using the same expression for ǫ −2 φ: The first two terms are identically zero since ϕ 0 is spherically symmetric (so that by parity considerations it is L 2 orthogonal to all of its first partial derivatives, which are in turn orthogonal to all of the second partial derivatives). Finally, for the second diagonal term: The non-degeneracy of the matrix (4.25) for small ǫ follows.
Remark 4.10. We briefly consider the symmetry properties of the solitary wave solutions: in [Lis95, Section 5] Lisi gives an ansatz for the solitary waves, using cylindrical coordinates (ρ, z, θ), from which symmetry properties can be deduced. For our situation the relevant ansatz for the Dirac wave function is φ =     ψ 1 (ρ, z) ψ 2 (ρ, z)e iθ ψ 3 (ρ, z) ψ 4 (ρ, z)e iθ     . (4.28) It seems likely that the solutions constructed via Proposition 4.6 have this symmetry and that this fact could be proved via an application of the implicit function theorem within the symmetry class of (4.28).