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

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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 non- linear 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 func- tion 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, varia- tional 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 po- tential. 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 Schr¨odinger 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.
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