AN OVERVIEW on LINEAR and NONLINEAR DIRAC EQUATIONS 1. Introduction and Basic Properties of the Dirac Operator. in Relativistic

DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS Volume 8, Number 2, April 2002 pp. 381–397

AN OVERVIEW ON LINEAR AND NONLINEAR DIRAC EQUATIONS

MARIA J. ESTEBAN AND ERIC SER´ E´

Abstract. In this paper we review the variational treatment of linear and nonlinear eigenvalue problems involving the Dirac operator. These problems arise when searching for bound states and bound energies of electrons sub- mitted to external or self-consistent interactions in a relativistic framework. The corresponding energy functionals are totally indefinite and that creates difficulties to define variational principles related to those equations. Here we describe recent works dealing with this kind of problems. In particular we describe the solutions found to linear Dirac equations with external potential, the Maxwell-Dirac equations, some nonlinear Dirac equations with local nonlinear terms and the Dirac-Fock equations arising in the description of atoms and molecules.

1. Introduction and basic properties of the dirac operator. In relativistic quantum mechanics [11], the state of a free electron is represented by a wave function Ψ(t, x) with Ψ(t, .) ∈ L2(IR3, CI 4) for any t. This wave satisﬁes the free Dirac equation:

3 X 2 i ∂t Ψ = H0Ψ, with H0 = −ic~ αk∂k + mc β, (1.1) k=1 where c denotes the speed of light, m > 0, the mass of the electron, and ~ is Planck’s constant. In the Dirac equation, α1, α2, α3 and β are 4×4 complex matrices, whose standard form (in 2 × 2 blocks) is I 0 0 σk β = , αk = (k = 1, 2, 3), 0 −I σk 0 with 0 1 0 −i 1 0 σ = , σ = , σ = . 1 1 0 2 i 0 3 0 −1 One can easily check the following relations α = α∗ , β = β∗, k k (1.2) αkα` + α`αk = 2δk` , αkβ + βαk = 0.

1991 Mathematics Subject Classiﬁcation. 49S05, 35J60, 35P30, 35P05, 35Q75, 49R05, 49R10, 81Q05, 81V70, 81V45. Key words and phrases. Relativistic quantum mechanics, Dirac operator, eigenvalues, variational methods, indeﬁnite integrals, nonlinear eigenvalue problems, nonlinear elliptic equations, min-max.

381 382 MARIA J. ESTEBAN AND ERIC SER´ E´

These algebraic conditions are here to ensure that H0 is a symmetric operator, such that 2 2 2 4 H0 = −c ∆ + m c . (1.3) Unless otherwise stated, we shall work with a system of physical units such that m = c = ~ = 1. Before going further, let us ﬁx some notations. All throughout these notes, the   z1 ∗  .  4 conjugate of z ∈ CIwill be denoted by z . For X =  .  a column vector in CI , z4 ∗ ∗ ∗ we denote by X the row covector (z1 , ..., z4 ). Similarly, if A = (aij) is a 4 × 4 ∗ ∗ ∗ complex matrix, we denote by A its adjoint, (A )ij = aji. We denote by (X,X0) the Hermitian product of two vectors X, X0 in CI 4, and by 4 4 2 X ∗ |X| , the norm of X in CI , i. e. |X| = XiXi . The usual Hermitian product i=1 in L2(IR3, CI 4) is denoted Z (ϕ, ψ) = ϕ(x), ψ(x) d3x. (1.4) L2 IR3 This course will be devoted to the presentation of various results concerning the existence of stationary solutions of linear and nonlinear Dirac equations. This is not an exhaustive description of the existing literature. Our main purpose is to describe some problems, methods, results and references which will make it possible for graduate students and researchers to be initiated to the study of some linear and nonlinear partial diﬀerential equations arising in models of the Relativistic Quantum Mechanics. Linear Dirac equations like

(H0 + V )ψ = λψ (1.5) appear when looking for bound states of an electron subjected to the action of an external electrostatic potential V . Chapter 2 will be devoted to the study of (1.5). Nonlinear Dirac equations appear in different contexts. For instance: • When looking for solutions of (1.5) in special classes of functions ψ. We will see this in detail when describing the Dirac-Fock equations (Chapter 5). • When building relativistic models of extended particles by means of nonlinear Dirac fields. This will be described via the Soler model in Chapter 3. • When analyzing the iteraction of an electron with its own electrostatic potential. This problem of self-interaction can be approximately described by the so-called Maxwell-Dirac equations, which are nonlinear Dirac equations, where the nonlinearities are nonlocal (of convolution type). This will be treated in Chapter 4. Let us now list some basic important properties of the free Dirac operator. 2 3 4 1 3 4 (P1) H0 is a self-adjoint operator on L (IR , CI ), with domain D(H0) = H (IR , CI ). Its spectrum is (−∞, −1] ∪ [1, +∞). There are two orthogonal projectors on 2 3 4 + − + L (IR , CI ) , Λ and Λ = 1 2 − Λ , both with infinite rank, and such that L √ √ + + + + H0Λ = Λ H0 = √1 − ∆ Λ = Λ 1 −√∆ − − − − (1.6) H0Λ = Λ H0 = − 1 − ∆ Λ = −Λ 1 − ∆ . AN OVERVIEW ON LINEAR AND NONLINEAR DIRAC EQUATIONS 383

(P2) The coulombic potential V (x) = 1|x| satisﬁes the following Hardy-type in- equalities: 1 π 2 ϕ, (µ ∗ V ) ϕ ≤ + ϕ, |H0|ϕ , (1.7) L2 2 2 π L2 for all ϕ ∈ Λ+(H1/2) ∪ Λ−(H1/2) and for all probability measures µ on IR3. Moreover,

π 1/2 ϕ, (µ ∗ V ) ϕ ≤ ϕ, |H0|ϕ , ∀ϕ ∈ H (1.8) L2 2 L2

k (µ ∗ V ) ϕk ≤ 2k∇ϕk , ∀ϕ ∈ H1 (1.9) L2 L2 In the particular case where µ is equal to the Dirac mass at the origin δ0, an inequality more precise than (1.8) was proved by Tix ([76]) and by Burenkov-Evans ([14]). This inequality reads as follows αZ H − ϕ, ϕ ≥ ((1 − αZ)ϕ, ϕ) , 0 |x| 2 + 1/2 3 4 for all Z ≤ Zc := π 2 , for all ϕ ∈ Λ (H (IR , CI )), where α ∼ 1/137 is ( 2 + π )α the so-called fine structure constant. The technique used to prove this inequality is based on ideas introduced by Evans-Perry and Siedentop in [34]. We refer to [48], [52] for inequality (1.8) in the case µ = δ0. Thaller’s book [75] gathers many results on the Dirac operator, including (P1) and the standard Hardy inequality (1.9) for µ = δ0, with references. The extension of (1.7), (1.8) and (1.9) from µ = δ0 to a general probability measure µ is immediate, since the projectors Λ±, the gradient ∇ and the free Dirac operator H0 commute with translations. For ϕ ∈ L2(IR3, CI 4), let us denote ϕ+ = Λ+ϕ, ϕ− = Λ−ϕ. Let E = H1/2(IR3, CI 4),E+ = Λ+E,E− = Λ−E. E is a Hilbert space with Hermitian product √ ϕ, ψ = ϕ, 1 − ∆ ψ = ϕ+, ψ+ + ϕ−, ψ− . (1.10) E L2 E E 2. Eigenvalue problems for dirac operators. In this chapter we are interested in perturbed Dirac operators of the form H0 + V , V being a scalar potential. Our goal is to find variational characterizations for the eigenvalues of such operators. One expects the eigenvalues of H0 +V to be critical points (in an appropriate sense) of the Rayleigh quotient ((H + V )ϕ, ϕ) R (ϕ) := 0 , V (ϕ, ϕ) 2 3 where (·, ·) denotes the inner product in L (IR ). If H0 + V were bounded below, like Schrödinger operators −∆ − Z/|x| for instance, minimizing the above quotient would give the least eigenvalue (ground state energy). This standard approach fails for Dirac operators: for physically relevant potentials V , the Rayleigh quotient is neither bounded from above nor from below (for instance, [50]). This is the main difficulty we have to face. Many efforts have been devoted to the characterization and computation of Dirac eigenvalues. Some works deal with approximate potentials acting on 2-spinors. The idea is to write Dirac’s equation for the upper spinor in ψ by performing 384 MARIA J. ESTEBAN AND ERIC SER´ E´ some adequate transformation. Then, different procedures have been produced to find approximate Hamiltonians, which “become exact” in the nonrelativistic limit (c → +∞). For references in this direction see, for instance, Durand [27], Durand- Malrieu [28], van Lenthe-Van Leeuwen-Baerends-Snijders [56]. Other approximate Hamiltonians are constructed by using projectors. One of the first attempts in this direction was made by Brown and Ravenhall. In [12], they proposed the following (usually called the Brown-Ravenhall “no-pair”) Hamiltonian + + B = Λ (H0 + V )Λ (2.1) + where Λ = χ(0,∞)(H0) denotes the projection on the positive spectral subspace of the free Dirac operator. If 0 is not in the spectrum of H0 + V , then instead of defining B as above, one + can use the exact positive projector corresponding to H0 + V , denoted by ΛV . The new operator + + BeV = ΛV (H0 + V )ΛV (2.2) coincides with H0 + V in its positive spectral subspace. For attractive potentials V that are not “too strong”, minimizing the Rayleigh quotient for BeV yields the smallest positive eigenvalue of H0 + V . This eigenvalue may be interpreted as + the ground state energy. Unfortunately, the exact projector ΛV is not known a priori. This makes its use difficult in numerical computations. On the contrary, the approximate Brown-Ravenhall Hamiltonian B is known explicitly. Several recent papers have been devoted to the mathematical study of the Brown- Ravenhall Hamiltonian for one-electron atoms with nuclear charge Z > 0. Its precise form is Zα BZ := Λ H − Λ (2.3) + 0 |x| + where α is the fine structure constant. Hardekopf and Sucher [46], on the basis of numerical computations, conjectured that the operator BZ is strictly positive for −1 Z < Zc = 2α /(π/2 + 2/π), that its first eigenvalue vanishes for Z = Zc, and that BZ is unbounded from below for Z > Zc. Evans, Perry and Siedentop [34] proved Z that the operator B is bounded from below if and only if Z ≤ Zc, as predicted in Z [47]. Tix [76], [77] proved that the operator B is strictly positive for Z ≤ Zc (see also Burenkov-Evans [14]). Note that the integer part of Zc is 124, strictly below the critical value Z∗ = 137, for which the first eigenvalue of the Dirac-Coulomb Hamiltonian H0 − Zα/|x| vanishes (see for instance [64], [75]). As we already pointed out above, one the most important properties of the free Dirac operator H0 is that 2 H0 = −∆ + 1 (2.4) Another Hamiltonian having the same property is: √ −∆ + 1 (2.5) This Hamiltonian is positive definite, and so, it does not pose the problems related to the unboundedness of the Dirac operator. But it is nonlocal, which contradicts the Causality Principle in special relativity and does not preserve the symmetry in the t and x variables for the evolution problem (covariance of the equation under the action of Lorentz boosts). Nevertheless, the study of the spectrum of √ −∆ + 1 + V (2.6) AN OVERVIEW ON LINEAR AND NONLINEAR DIRAC EQUATIONS 385 and its use as an approximation model for multi-particle systems is interesting. For references in this direction, let us mention [22], [48], [55], [57] and references therein. In 1994, Hill and Krauthauser ([50]) proposed a variational characterization of eigenvalues of H0 + V based on maximization of quotients like

(ψ, (H0 + V )ψ) 2 . (ψ, (H0 + V ) ψ)

This method works when the spectrum of H0 + V has a gap around 0. The mathematical study of this approach has still to be done. The characterization of the stable eigenvalues of H0 +V by a minimax method is a quite natural idea and it has been attempted both analytically and numerically: in 1986, Talman [74] proposed a minimax procedure to find the first eigenvalue of H0 + V for negative potentials V (see also [21]). His proposal consisted in first maximizing in the lower component of the spinor ψ, and then minimizing over the upper spinor. Various results concerning the spectrum of H0 + V can be found in [75] and papers quoted therein. Minimax approaches to characterize the lowest eigenvalue of H0 + V have been proposed in the Physics literature (see for instance [26], [54], [74] and [21]). Mathematical rigorous results on this topic can be found in [31], [45] and [24]. In [31] was proved a minimax characterization of all eigenvalues of H0 + V for potentials V which are less singular at the origin than Coulomb potentials −1 1 V = −C|x| , for 0 < C < 2 :

((H0 + V )ϕ, ϕ) λk = inf sup . (2.7) + (ϕ, ϕ) F ⊂ Y ϕ ∈ F ⊕ Y − F vector space ϕ6=0 dim F = k Here, Y ± are the spaces Λ±(H1/2(IR3)). In [45], a minimax approach close to the above one is shown to provide all the eigenvalues of operators with spectral gaps which satisfy a number of conditions. This allows the authors to bound the n-th eigenvalue of the Dirac operator A = H0 + V with Coulomb potential V by the n-th eigenvalue of the Brown-Ravenhall + + + operator Λ (H0 + V )Λ , where Λ = χ(0,∞)(H0). Moreover, for certain bounded non-positive potentials, the number of eigenvalues of H0 + V is shown to be larger than or equal to those of the operator −∆ + V . Finally, also in [45], deﬁning ϕ ψ = , ϕ, χ ∈ IR3 → CI2, the equality χ

((H0 + V )ψ, ψ) λ1(H0 + V ) = min max , (2.8) ϕ6=0 χ (ψ, ψ) and its corresponding counterpart for the n-th eigenvalue, are proved to hold for a particular class of bounded potentials V . The formula (2.8) had been proposed (without proof) by Talman [74] and Datta-Deviah [21] for numerical and computational purposes, without asking ϕ to be different from 0. In [24], two approaches are proposed. The first one characterizes all eigenvalues of H0 + V by a minimax formula for all potentials V which are nonpositive, small at infinity and which have a singularity at 0 less strong than −C |x|−1, C < 0.9. 386 MARIA J. ESTEBAN AND ERIC SER´ E´

To prove this, the authors strongly use an inequality proved in [14] and [76], [77]. Finally, for potentials which behave at 0 less singularly than −|x|−1, a minimization characterization of λ1(H0 + V ) is shown to hold. Moreover, one can prove that Talman’s formula (2.8) (and the corresponding one for the n-th eigenvalue) holds true for all potentials which are nonpositive, small at inﬁnity and which have a singularity at 0 less strong than −|x|−1 (condition which is known to be optimal). One can also show that there is a strong link between Talman’s formula and the minimization procedure proposed in section 4 of [24] to characterize λ1(H0 + V ). All this is done in [25].

3. The nonlinear local dirac equations: the soler model. generalized extended particles models. Nonlinear Dirac equations have been widely used to build relativistic models of extended particles by means of nonlinear Dirac fields. A general form of this equation in the case of an elementary fermion is µ 0 iγ ∂µψ − mψ + γ ∇F (ψ) = 0. (3.1) 4 4 ∂ Here, ψ : IR → CI , ∂µψ = ∂xµ ψ, 0 ≤ µ ≤ 3, we have used Einstein’s convention for summation over µ. Note that x0 plays here the role of time. Throughout this chapter we assume that F satisfies F ∈ C2(CI 4, IR) and F (eiθψ) = F (ψ) for all θ. (3.2) Different functions F have been used to model various types of self-couplings. For a review on this and historical background see for instance [69]. Stationary states of the nonlinear Dirac equation are considered as particle-like solutions. These solutions are in some sense solitons which propagate without changing their shape. Stationary solutions are functions of the type

0 ψ(x0, x) = e−iωx ϕ(x) (3.3) where by x we denote (x1, x2, x3) ∈ IR3, and such that ϕ is a non-zero localized solution of the following stationary nonlinear Dirac equation k 0 0 3 iγ ∂kϕ − mϕ + ωγ ϕ + γ ∇F (ϕ) = 0 in IR . (3.4) Here, we have used the repeated index convention for summation over the three values of k. By “localized”, we mean that ϕ ∈ W 1,q(IR3, CI 4) for any 2 ≤ q < ∞ and F (ϕ) ∈ L1(IR3, IR). The above equation has a variational structure. Indeed it is the Euler-Lagrange equation corresponding to the following functional Z ω 1 0 k m ω 2 I (ϕ) = − (iγ γ ∂kϕ, ϕ) + ϕϕ − |ϕ| − F (ϕ) dx (3.5) IR3 2 2 2 where ( , ) is the usual scalar product in CI 4 and ϕϕ denotes (γ0ϕ, ϕ). Existence of solutions of (3.4), i.e. of critical points of Iω has been proved in [7, 8, 16, 65] in the particular case when 1 F (ϕ) = G(ϕϕ) ,G ∈ C2(IR, IR) ,G(0) = 0 (3.6) 2 AN OVERVIEW ON LINEAR AND NONLINEAR DIRAC EQUATIONS 387 under suitable conditions on G, with ω ∈ (0, m). When (3.6) holds, one can use a particular ansatz for the solutions:  1  v(r) 0 ϕ(x) =   (3.7)  cos θ   iu(r)  sin θ eiφ and then equation (3.4) reduces to the O.D.E. system u0 + 2u = vg(v2 − u2) − (m−ω) r (3.8) v0 = ug(v2 − u2) − (m+ω) where g(s) = G0(s). In the above papers the system (3.8) is solved by a shooting method which yields an infinity of localized solutions for (3.8). Below we will state the precise results of [7, 16, 65]. The particular nonlinearity (3.6) corresponds to the so-called Soler model, and has been widely studied (see [69, 70]). There are other models of selfcoupling for which the ansatz (3.7) is no more valid, for instance the following nonlinearity (see [36, 69]) 1 F (ϕ) = |ϕϕ|2 + b |ϕγ5ϕ|2 (3.9) 2 with b 6= 0, ϕ = γ0ϕ and γ5 = γ0γ1γ2γ3. We can prove, by a variational method, the existence of non-zero critical points of Iω, and therefore of localized solutions of (3.4), both in the case when F satisfies (3.6) and in the general case. When F satisfies (3.6), we actually solve the O.D.E. system (3.8) by a variational method to obtain the existence of an infinity of solutions of (3.8) under assumptions slightly less general than those in [7, 16, 65]. The main assumption we have to add is that G satisfies G0(x) · x ≥ θ G(x) , θ > 1 , ∀ x ∈ IR (3.10) This is a superlinearity condition on G. This kind of technical condition often appears as necessary to use variational techniques when solving nonlinear PDE’s and was first introduced by Ambrosetti and Rabinowitz in [3]. If F is a general function which does not necessarily satisfy (3.6), we also obtain an existence result for (3.4) but only for functions F which grow more slowly than |ϕ|3 at infinity. This limitation comes from the fact that we work in the space H1/2(IR3, CI 4). Hence we are not able yet to treat nonlinearities like those satisfying (3.9). Nevertheless we believe that the method used in our paper ([29]) should be useful to extend our results and treat more general nonlinearities. Our results are concerned, for instance, with nonlinearities as 1 3 |ϕϕ|α + b |ϕγ5ϕ|β , 1 < α, β < , b ≥ 0. (3.11) 2 2 Let us now state our main results about equations (3.4) and (3.8). Theorem 3.1 ([29]). Let F : CI 4 → IR satisfy (3.6) with G ∈ C2(IR, IR) . Denoting by g the first derivative of G, we make the following assumptions: (H1) g(x) · x ≥ θ G(x) , θ > 1 , ∀x ∈ IR (H2) G(0) = g(0) = 0. 388 MARIA J. ESTEBAN AND ERIC SER´ E´

(H3) G(x) ≥ 0 (∀x ∈ IR) and G(A0) > 0 for some A0 > 0. (H4) 0 < ω < m. T 1,q 3 4 Then there is an inﬁnity of solutions of equation (3.4) in 2≤q<+∞ W (IR , CI ). Each of them is found by a min-max on the functional Iω. They are of the form (3.7), so they correspond to classical solutions of (3.8) on IR+, decreasing exponentially at inﬁnity. For more general nonlinearities F , not compatible with the ansatz (3.7), we have

α1 5 β 3 Theorem 3.2 ([29]). Let F (ϕ) = λ |ϕϕ| + b|ϕγ ϕ| , with 1 < α, β < 2 ; λ, b > 0. T 1,q 3 4 Then there exists a non-zero solution of (3.4) in 2≤q<+∞ W (IR , CI ) for every ω ∈ (0, m). In fact, in [29] we prove a more general result: Theorem 3.3 ([29]). Assume that F : CI2 → IR satisﬁes:

α1 α2 4 (H5) 0 ≤ F (ϕ) ≤ a1 (|ϕ| + |ϕ| ) , ∀ϕ ∈ CI .

Here, a1 > 0 and 2 < α1 ≤ α2 < 3. F ∈ C2(CI 4, IR) ,F 0(0) = F 00(0) = 0 , (H6) 00 α2−2 |F (ϕ)| ≤ a2|ϕ| , a2 > 0 , for |ϕ| large. (H7) ∇F (ϕ) · ϕ ≥ αF (ϕ) , α > 2 , ∀ ϕ ∈ CI 4. 4 ∃ β > 3 , ∀ δ > 0 , ∃ Cδ > 0 , ∀ ϕ ∈ CI , (H8) 1 |∇F (ϕ)| ≤ δ + CδF (ϕ) β |ϕ|.

ν 4 (H9) F (ϕ) ≥ a3|ϕϕ| − a4 , ν > 1 , a3, a4 > 0 , ∀ ϕ ∈ CI . Then, for every ω ∈ (0, m), we can ﬁnd a non-zero solution of (3.4) in T 1,q 3 4 2≤q<+∞ W (IR , CI ). Theorem 3.2 is clearly a consequence of Theorem 3.3. To prove Theorems 3.1 to 3.3 one can use a linking argument. For a description of the general linking method see [10]. In the case of Theorems 3.2, 3.3, we need some ideas introduced in [51] in the context of Hamiltonian systems, and we use the concentration-compactness theory [62].

4. The maxwell-dirac equations: a model for self interaction of an electron and its own electromagnetic field. The Maxwell-Dirac system of classical field equations describes the interaction of an electron with its own electromagnetic field. In the Lorentz gauge it can be written as follows µ µ 3 (iγ ∂µ − γ Aµ)ψ − mψ = 0 in IR × IR (M-D) µ µ ν ν 3 ∂µA = 0 , 4π ∂µ∂ A = J in IR × IR I 0 0 σk where m > 0, ψ(x , x) ∈ CI 4, γ0 = , γk = , ψ = γ0ψ, 0 0 −I −σk 0 J µ = (ψ, γµψ) and σk are the Pauli matrices 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = . 1 0 i 0 0 −1 AN OVERVIEW ON LINEAR AND NONLINEAR DIRAC EQUATIONS 389

Stationary solutions of classical nonlinear wave equations which are localized in space are sometimes used to represent particles. The most precise results in QED have been obtained after ﬁeld quantization. But one can obtain at least qualitative results by using classical ﬁelds (see [42], chapter 7). In this respect, we think that our result could be useful for further understanding of the interaction of matter and radiation. The Klein-Gordon-Dirac system arises in the so-called Yukawa model (see [11]). It can be written as

µ µ 3 (iγ ∂µ −γ Aµ)ψ − mψ = 0 in IR × IR (KG-D) µ 2 1 3 ∂ ∂µχ + M χ = 4π (ψ, ψ) in IR × IR .

The above systems have been extensively studied and many results are available concerning the Cauchy problem ([30] and [42], chapter 7, for detailed references). But the existence of soliton-like solutions of the Maxwell-Dirac system has been an open problem for a long time (see [42], p. 235). Indeed, the interaction between the spinor and the electromagnetic field makes equations (M-D) nonlinear. Thus, no solution of (M-D) can be found by a superposition argument. Concerning previous works on solitary waves of (M-D), let us mention the pi- onneering works of Finkelstein et al [37] and Wakano [79]. The latter considered the approximation in which A0 6≡ 0,A1 = A2 = A3 ≡ 0 (Dirac-Poisson). The approximate problem (D-P) can be reduced to a system of three coupled differential equations by using the spherical spinors. Wakano obtained numerical evidence for the existence of stationary solutions of (D-P). Garrett Lisi (see [39]) used a numerical method to obtain an axially symmetric solution of (M-D). Moreover, in [39] solutions of (D-P) are also numerically com- puted. The computation of the magnetic part of the field A for these solutions shows that the hypothesis of Wakano is quite reasonable, since these computations show that the fields Ai, i = 1, 2, 3 are small compared to A0. By using variational methods, we can give a the rigorous proof of existence of such a soliton-like solution of (M-D). For 0 < ω < m it is proved that there are exact solutions (ψ, A) : IR × IR3 → CI 4 × IR4 of (M-D) of the form

( iωx0 3 4 ψ(x0, x) = e ϕ(x), ϕ : IR → CI (S) µ µ 1 R dy µ A (x0, x) = J ∗ |x| = IR3 |x−y| J (y).

These solutions can be chosen to be axially symmetric, but this symmetry assumption is not essential in our proof. A more recent result by S. Abenda ([1]) extends the above result to the case −m < ω < m. This new result was obtained by studying the corresponding functional in a particular class of functions. Many questions are open by the existence of this type of solitary-wave solutions for (M-D). On one side, it is easy to see that they have a negative “mass”. Wakano already observed this phenomenon for the soliton-like solutions of (D-P). However, it was shown in [79] that a positive mass can be reached by taking into account the vacuum polarization eﬀect. 390 MARIA J. ESTEBAN AND ERIC SER´ E´

Let us note that if (ψ, A) is a solution of (M-D) of the form (S), then formally, ϕ is a critical point of the functional Z 1 0 k m ω 2 Iω(ϕ) = iγ γ ∂kϕ, ϕ − (ϕ, ϕ) − |ϕ| IR3 2 2 2 1 ZZ J µ(x) J (y) − µ dx dy. 4 IR3×IR3 |x−y| We use the above remark to obtain a solution of (M-D) of the form (S). Indeed, we look for a critical point of the functional Iω in the appropriate space of functions, 0 i.e. a function ψ such that Iω(ψ) = 0. As it is well known, the presence of the negative spectrum for the Dirac operator implies that no critical point of Iω can be found by minimization. Actually, one has to use a sort of min-max argument with a complicated topology to find this solution. This kind of argument used to treat problems which have infinite negative and positive spectrum have been already used under the name of linking. What makes the use of variational arguments nonstandard in this case is: (1) the equations are translation invariant, which µ creates losses of compactness. (2) The interaction term J Aµ is not positive definite. We deal with all this by defining a particular min-max argument which enables us to prove the following facts, whose proofs can be found in [30]. – For any ω strictly between 0 and m, there exists a non-zero critical point ϕω of Iω. This function ϕω is smooth in x and exponentially decreasing. Finally, the iωx0 µ µ 1 fields ψ(x0, x) = e ϕω, A (x0, x) = Jω ∗ |x| are solutions of the Maxwell-Dirac system (M-D). Similarly, the stationary solutions of the Klein-Gordon-Dirac equations are given by the critical points ϕ of the functional Z 1 0 k m ω 2 Jω(ϕ) = (iγ γ ∂kϕ, ϕ) − (ϕ, ϕ) − |ϕ| − IR3 2 2 2 1 ZZ (ϕ, ϕ)(x)(ϕ, ϕ)(y) e−M|x−y| dx dy. 4 IR3×IR3 |x−y| In this case for every ω strictly between 0 and m, there is an infinity of solutions of (KG-D). These functions have the form  1  v(r) 0 ϕ(x) =   ,  cos θ   iu(r)  eiφ sin θ where (r, θ, φ) are the spherical coordinates of x ∈ IR3. Note that the above ansatz, which simplifies the analysis, cannot be used for (M-D). This is why many solutions are found for (KG-D), but for the moment, only one for (M-D). The multiplicity problem for (M-D) seems very difficult, because of a lack of compactness of the functional Iω which is created by the translational invariance of the Maxwell-Dirac equations. Maybe the axial symmetry ansatz introduced in [39] could help to obtain multiple solutions of (M-D).

5. The dirac-fock equations in models of relativistic atoms and molecules. The Dirac-Fock (DF) functional was ﬁrst introduced by Swirles [73] as an approximation for the energy of a system of N electrons in an atom of large nuclear charge AN OVERVIEW ON LINEAR AND NONLINEAR DIRAC EQUATIONS 391

Z. In such atoms, the inner-shell electrons have relativistic energies, and the standard Hartree-Fock (HF) approximation, based on the nonrelativistic Schr¨odinger equation, is no longer valid. The Euler-Lagrange equations of the DF energy functional can be solved numerically. The solutions represent stationary states of the electrons in the atom. The numerical results are in very good agreement with experimental data (see e.g. [53, 43, 23, 61, 67, 41]). In [71], [72], [66], [44], [17], the relationship between Dirac-Fock and quantum electrodynamics is studied. In the Dirac-Fock model, the N electrons are represented by a Slater determinant of N functions ϕk ∈ E, subjected to the normalization constraints ϕ`, ϕk = δk` . (5.1) L2

We shall denote Φ = (ϕ1, ··· , ϕN ), and the constraints above will be written in the shorter form Gram Φ = 1I , with h i Gram Φ := ϕ`, ϕk . (5.2) k` L2 We consider a molecule, with: • nuclear charge density Zµ, where Z > 0 is the total nuclear charge and µ is a probability measure defined on IR3 . In the particular case of m point-like nu- m X clei, each one having atomic number Zi at a fixed location xi , Zµ = Zi δxi i=1 m X and Z = Zi. i=1 • N relativistic electrons. We assume that the interaction between these particles is purely electrostatic. The DF energy of the N electrons in the molecule, is N N X X 1 E(Φ) = ϕ` ,H0ϕ` − αZ ϕ` , µ ∗ ϕ` L2 |x| L2 `=1 ZZ `=1 (5.3) α h i 3 3 + 2 V (x − y) ρ(x)ρ(y) − tr R(x, y)R(y, x) d xd y IR3×IR3 Here, ρ is a scalar and R is a 4 × 4 complex matrix, given by N N X X ρ(x) = ϕ (x), ϕ (x) ,R(x, y) = ϕ (x) ⊗ ϕ∗(y) , (5.4) ` ` ` ` `=1 `=1 ρ is the electronic density, R is the exchange matrix which comes from the an- tisymmetry of the Slater determinant. Note that R(y, x) = R(x, y)∗, so that X 2 tr R(x, y)R(y, x) = |R(x, y)ij | . i,j The main difference with the more standard HF functional, is that the kinetic energy term (ϕ , −∆ϕ ) in HF is replaced by (ϕ ,H ϕ ) in DF. This changes k k L2 k 0 k L2 completely the nature of the functional, which becomes indefinite: it is not bounded below, and any of its critical points has an infinite Morse index. The DF functional is invariant under the action of the group U(N):

X X N u · Φ = u1` ϕ` , ..., uN` ϕ` , u ∈ U(N) , Φ ∈ E . (5.5) ` ` 392 MARIA J. ESTEBAN AND ERIC SER´ E´

We denote n o Σ = Φ ∈ EN / Gram Φ = 1I . (5.6) Using inequality (1.8), one can easily prove that the DF functional E is smooth N on E . A critical point of E|Σ is a weak solution of the following Euler-Lagrange equations: N X H Φ ϕk = λk` ϕ` , k = 1, ..., N. (5.7) `=1 Here, 1 H ψ = H ψ −αZ µ ∗ ψ Φ 0 |x| Z (5.8) +α(ρ ∗ V )ψ − α R(x, y)ψ(y)V (x − y). IR3 1/2 −1/2 Since H Φ is self-adjoint from H to its dual H , Λ = (λk` ) is a self-adjoint (N × N) complex matrix. It is the matrix of Lagrange multipliers associated to the constraints (ϕ , ϕ ) = δ . ` k L2 k` For Φ ∈ Σ a critical point whose matrix of multipliers is Λ, and u ∈ U(N), the matrix of multipliers of the critical point Φe = u · Φ is Λe = uΛu∗. So any U(N)- orbit of critical points of E|Σ contains a weak solution of the following system of nonlinear eigenvalue problems, called the Dirac-Fock equations:

H Φ ϕk = k ϕk , k = 1, ..., N. (5.9)

Physically, H Φ represents the Hamiltonian of an electron in the mean field due to the nuclei and the electrons. The eigenvalues 1, ..., N are the energies of each electron in this mean field. In the HF model, the Euler-Lagrange equations have a form similar to (5.9), with −∆ instead of H0 in the expression of HΦ. The physically interesting states correspond to 1 ≤ ... ≤ N < 0, and the ground state minimizes EHF on Σ, which implies that 1, ..., N are the N first eigenvalues of HΦ (see [60]). In the DF model, the physically interesting states correspond to 0 < k < 1: a positive energy inferior to the rest mass of the electron. The definition of a ground state is less clear: the DF functional has no minimum on Σ. This fact is at the origin of serious difficulties in the numerical implementation, as well as the interpretation, of the DF equations (see [17] and the references therein). One way to deal with this problem, is to restrict the energy functional to the space (Λ+E)N ,Λ+ being, as defined above, the projector on the space of positive states of the free Dirac operator [71, 72]: this corresponds to a Hartree-Fock reduction of the already mentioned Brown-Ravenhall model. The associated Euler-Lagrange equations are the “projected” Dirac-Fock equations + + Λ HΦ Λ ϕk = kϕk. (5.10)

Note that, in the case k > 0, (5.9) can be written formally as + + ΛΦ HΦ ΛΦ ϕk = kϕk. (5.11) + Here, ΛΦ is the projector on the positive space associated to HΦ. Numerical computations using (5.9) rather than (5.10), give results that are in very good agreement with experimental data (see e.g. [43, 61]). This is not very surprising: in the + presence of strong electric fields, the projector ΛΦ seems physically more adequate than the free-energy projector Λ+ (see [49]). In [66] Mittleman derived the DF AN OVERVIEW ON LINEAR AND NONLINEAR DIRAC EQUATIONS 393 equations with “self-consistent projector” (5.11), from a variational procedure ap- plied to a QED Hamiltonian in Fock space, followed by the standard Hartree-Fock approximation. Important existence results are known on the HF equations. Lieb and Simon [60] proved the existence of a ground state of EHF on Σ, provided N < Z + 1, where Z is the total nuclear charge, P.-L. Lions [63] proved the existence of infinitely many excited states if N ≤ Z. Using inequality (1.7), one can easily extend the results of [60, 63] to the projected equations (5.10), assuming that 2 α max(Z,N) < , N < Z + 1. π/2 + 2/π 1 The only difference is that |x| is not a compact perturbation of H0, but this does not create any important difficulty. In [32], we give the first existence result for solutions of the DF equations “without projector” (5.9). Our assumptions are 2 α max(Z, 3N − 1) < , N < Z + 1. π/2 + 2/π

Since we find positive eigenvalues k, the equations we solve are formally equiv- alent to the DF equations with “self-consistent projector” (5.11). 2 The condition α(3N − 1) < π/2+2/π is rather restrictive, and we do not have a clear definition of the “ground state”. But we hope that this first study will stimulate further mathematical research on Dirac-Fock. Our main theorem about the Dirac-Fock equations is the following: 2 Theorem 5.1 ([32]). Assume that α max(Z, 3N − 1) < π/2+2/π , N < Z + 1, with Z > 0 the total nuclear charge. The nuclear charge density is Zµ, where µ is a fixed 3 j probability measure on IR . Then, there is an infinite sequence (Φ )j≥0 of critical points of the DF functional E on n o Σ = Φ ∈ EN / Gram Φ = 1I .

j j The functions ϕ1, ··· , ϕN satisfy the normalization constraints (5.1) and they are 1/2 3 4 T 1,q 3 4 strong solutions, in H (IR , CI ) ∩ 1≤q<3/2 W (IR , CI ), of the Dirac-Fock equations H ϕj = j ϕj , 1 ≤ k ≤ N, (5.12) Φj k k k 0 < j ≤ ... ≤ j < 1. (5.13) 1 N Moreover, 0 < E(Φj) < N, (5.14) lim E(Φj) = N. (5.15) j→∞ 1 Remark 5.2. With the physical value α ≈ 137 and Z an integer, our conditions become Z ≤ 124 ,N ≤ 41 ,N ≤ Z. Remark 5.3. Since µ is arbitrary, our assumptions contain the case of point-like X 1 nuclei as well as more realistic nuclear potentials, of the form −α ρ (x) ∗ , i |x| i Z ∞ 1 X where ρi ∈ L ∩ L , ρi ≥ 0 , ρi = Z. 3 i IR 394 MARIA J. ESTEBAN AND ERIC SER´ E´

Remark 5.4. The first solution Φ0 is a good candidate for a ground state. Indeed, in the nonrelativistic limit (α → 0), it converges, after rescaling, to a ground state of Hartree-Fock. This will be proved in a forthcoming paper ([33]). Remark 5.5. In the case N = 1, the Dirac-Fock equations are linear and correspond to an eigenvalue problem for Dirac operators with scalar potentials. For variational results in this case, see [31, 45, 24]. j j Remark 5.6. The functions ϕ1, ··· , ϕN of Theorem 5.1 are smooth outside supp µ. Moreover, if supp µ is compact, they decay exponentially fast, as well as their derivatives, when |x| goes to infinity. Our main theorem on the Dirac-Fock equations is the analogue, for the DF model, of P.-L. Lions’ result for HF [63]. In order to control the Lagrange multipliers k (they should be negative in his case), Lions uses an estimate on the Morse index of the critical points. Such an estimate can be obtained for a critical point associated to a “finite-dimensional” min-max, if the functional satisfies the Palais-Smale compactness condition (see e.g. [4, 5, 6, 18, 78]). However, the HF functional does not satisfy the Palais-Smale compactness condition: since the essential spectrum of −∆ is [0, ∞), only Palais-Smale sequences with negative Lagrange multipliers k are precompact. Lions works on approximate functionals that satisfy Palais-Smale, finds critical points of these functionals with Morse index estimates, and passes to the limit. In [35, 40], Fang and Ghoussoub give general existence results on Palais- Smale sequences with Morse-type information, for functionals that do not satisfy Palais-Smale. As an application, they rewrite Lions’ proof, working directly with the HF functional. For DF, we also need a control on k: 0 < k < 1. Moreover, the essential spectrum of H0 is IR \ (−1, 1), so that the only precompact Palais-Smale sequences for DF, are such that |k| < 1. So a natural approach is to adapt the above ideas to DF. To realize this program, we faced several difficulties. 1 The first (and smallest) difficulty is that µ ∗ |x| is not a compact perturbation of H0. This creates some technical problems. They are easily solved, replacing 1 V = |x| by a regularized potential Vν . At the end of the proof, we can pass to the limit ν → 0, thanks to inequality (1.7). The second difficulty is that the Morse index estimates can only give upper bounds on the multipliers in [63]. But in the DF case, we want to ensure that k > 0. To overcome this problem, we replace the constraint Gram (Φ) = 1I by a penalization term πp(Φ), subtracted from the energy functional. The new Euler equations are HΦϕk = ∂ϕk πp (no more Lagrange multipliers). The eigenvalue k is now an explicit function of ϕk, which appears in the expression of the derivative

∂ϕk πp. This function has only positive values, so we automatically get k > 0. The third difficulty with DF, is that all critical points have an infinite Morse index. This kind of problem is often encountered in the theory of Hamiltonian systems and in certain elliptic PDEs. One way of dealing with it, is to use a concavity property of the functional, to get rid of the “negative directions”: see e.g. [2, 13, 15]. We shall use this method. We get a reduced functional Iν,p. A min- max argument gives us Palais-Smale sequences Φn,ν,p for Iν,p with finite “Morse index”, thanks to [35]. Adapting the arguments of [63], we prove that the k’s of such sequences are smaller than 1. Then we pass to the limit (ν, n, p) → (0, ∞, ∞), and get the desired solutions of DF, with 0 < k < 1. AN OVERVIEW ON LINEAR AND NONLINEAR DIRAC EQUATIONS 395

2 Our concavity argument works only if α(3N −1) < π/2+2/π . In the last 20 years, very powerful methods have been developped to deal with indeﬁnite functionals, that do not present any concavity property [68, 20, 38, 51]. This suggests that it might be possible to weaken the assumptions on N in Theorem 5.1.

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