Keller and Lieb-Thirring estimates of the eigenvalues in the gap of Dirac operators

We estimate the lowest eigenvalue in the gap of the essential spectrum of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schr\"odinger operator, which are equivalent to Gagliardo-Nirenberg-Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. The Keller estimate is then extended to a Lieb-Thirring inequality for the eigenvalues in the gap. Most of our result are established in the Birman-Schwinger reformulation.


Introduction and main results
In 1961, J.B. Keller established in [45] the expression of the potential which minimizes the lowest eigenvalue, or ground state, λ S (V ) of the Schrödinger operator −∆ − V in dimension d = 1, under a constraint on the Lebesgue norm of exponent p of V .This estimate was later extended in [53] by E.H. Lieb and W. Thirring to higher dimensions and to a sum of the lowest eigenvalues.During the last forty years, various refinements were published.As an example, we quote stability results for λ S (V ) proved in [12] by E.A. Carlen, R.L. Frank, and E.H. Lieb.Although Dirac operators inherit many qualitative properties of Schrödinger operators, dealing with Dirac operators turns out to be a delicate issue.If / D m denotes the free Dirac operator and V is a non-negative valued function, / D m − V is not bounded from below.One is actually interested in the lowest eigenvalue λ D (V ) in the essential gap (−m c 2 , m c 2 ), where m denotes the mass and c the speed of light.We shall speak of λ D (V ) as the ground state energy of / D m − V .In the standard setting, it is expected that λ D (V ) − m c 2 converges to λ S (V ) in the non-relativistic limit, i.e., as c → +∞.It is therefore a natural question to estimate λ D (V ) in terms of V p and identify the corresponding optimal potential.This question is the main purpose of our paper.A new critical value appears, which corresponds to the smallest value of V p for which λ D (V ) reaches, for some potential V ≥ 0, the lower end of the essential gap − m c 2 .In a linear setting, a similar question has been raised in [34,35], where the authors find a critical value ν 1 so that λ D µ * | • | −1 > − m c 2 for all positive measures µ with µ(R 3 ) < ν 1 , with 2/ π/2 + 2/π < ν 1 ≤ 1. Going back to [21,27,28], it is known that Hardy inequalities play an essential role in the analysis of the spectrum of Dirac-Coulomb operators.In the present article, except for the case p = d = 1, we rather find a nonlinear functional inequality of Gagliardo-Nirenberg-Sobolev nature, instead of a Hardy inequality (see comments in Appendix C.2).
It is possible to characterize the eigenvalues of / D m − V in the gap by a min-max principle according to [28][29][30] but this raises delicate issues involving the domain of the operator and its self-adjoint extensions addressed respectively in [30,33,36,37,63].Applied with a Coulombian potential V , the method gives rise, after the maximising step in the min-max method, to a lower bounded quadratic form which amounts to a kind of Hardy inequality for the upper component: see [10,21,27] for details.The same strategy applies to a general potential V under a constraint on V p , except that the Keller type bound on λ D (V ) is given by an implicit condition: see Appendix C. The optimal potential solves a nonlinear Dirac equation with Kerr-type nonlinearity.For the two-dimensional case, this equation has been studied in [5][6][7][8] by W. Borrelli.In the one-dimensional case, the solution is explicit, which allows us to identify it as in the case of the Schrödinger operator studied in [45].Alternatively to the min-max principle, the properties of the Birman-Schwinger operator corresponding to / D m − V allows us to characterize λ D (V ) and, except in Appendix C, we will adopt this point of view.
In order to state a Keller-Lieb-Thirring inequality for the Dirac operator, we need some definitions and preliminary properties.Let us start with the free Dirac operator on R d .We refer to [66] for a comprehensive list of results and properties.For simplicity, we choose units in which c = 1, except in Appendix C in which we consider the non-relativistic limit as c → +∞.Let d ≥ 1 and set N := 2 (d+1)/2 where x = max{n ∈ Z : n ≤ x} denotes the integer part of x.Let α 1 , • • • , α d and β be N × N Hermitian matrices satisfying the following anti-commutation rules (1.4) ∀ j, k = 1, . . ., d , where δ jk denotes the Kronecker symbol and I N is the N × N identity matrix.See, e.g., [41] for an existence result for such matrices.The free Dirac operator in dimension d When switching on a potential V , we expect that some eigenvalues of / D m − V emerge from the upper essential spectrum [m, +∞).We shall prove in Section 2 that / D m − V can be defined as a self-adjoint operator with essential spectrum σ ess ( / ).This allows us to define the ground state λ D (V ) as the lowest eigenvalue in the gap (− m, m).
Our first result states that the ground state is bounded by a function of The proof of Theorem 1.1 is given in Section 3 and relies on the properties of the inverse map of α → Λ D (α, p) defined by and this limit is the upper bound of the lower essential spectrum (−∞, −m] or, equivalently, the lower end of the gap.For sake of simplicity, we adopt the convention that α (p) = α D (− m, p).In the subcritical range of potentials, a simple consequence of Theorem 1.1 is the following Keller-Lieb-Thirring estimate for the Dirac operator If (p, d) = (1, 1), then V α,p as in Theorem 1.1 realizes the equality case, i.e., λ D (V α,p ) = Λ D (α, p).
The nonlinear Dirac equation (1.6) plays for the Dirac operator / D m − V the same role as (1.3) for the Schrödinger operator −∆ + V .However, Λ D (α, V ) is not obtained as the infimum but as a critical point of a Rayleigh quotient with infinitely many negative directions corresponding to a min-max principle (see [28]) and for this reason there is no simple interpolation inequality such as (1.1) in the case the Dirac operator.A more involved functional inequality holds: see Appendix C.
Nonlinear Dirac equations have been introduced to model extended fermions, as effective operators for nonlinear effects in graphene-like materials or Bose-Einstein condensates: see [32,Section 1.6] and [5,Introduction] for an introduction to the literature.Since the spinors in the Dirac equation have at least two components, many types of nonlinearities can be considered (see, e.g., [59] and references therein) and give rise to various phenomena.For instance, localized solutions to a nonlinear equation of the form for some function G : C N → C N correspond to solitary wave solutions to the time-dependent nonlinear Dirac equation and have attracted considerable attention: see, e.g., [4,15,38,56].
It is a common assumption to consider a nonlinearity that preserves Lorentz, or particlehole, symmetry.Such a non-linearity takes the form and is called the Soler-type nonlinearity.The Soler nonlinearity formally appears when minimizing the first positive eigenvalue of / D m − β V but will not be studied in this paper.In contrast, the nonlinearity that appears in (1.6) is of the form F ( Ψ, Ψ C N ) Ψ, which is sometimes called a Kerr-type nonlinearity as in [5], apparently by extension of the cubic nonlinearity used in optics.Existence of localized solutions for (1.6) is studied in [5] in the critical exponent case p = d = 2, and in [8] in the critical exponent case p = d for all dimensions d ∈ N with m = 0. Our results give an independent proof of the existence of a localized solution.
In [4,38], the authors proved that equations of the form (1.9) have many solutions if d ≥ 2 by looking for solutions of (1.9) in subspaces of fixed angular momentum.It seems that similar techniques could also be applied to (1.6).While it is reasonable to expect that the optimal potential is radially symmetric and the corresponding ground state Ψ is the solution with lowest positive angular momentum and smallest number of oscillations, this is so far an open question: see Appendix A. In Appendix B, we also give numerical results that point in this direction.
We now focus on the dimension d = 1.It turns out that one can completely solve (1.6) using special functions.Explicit formulae are given below, where B and 2 F 1 respectively denote the Euler Beta function and the hypergeometric function.
, up to a phase factor and a translation.Up to a translation, V = |Ψ| 2/(p−1) is even, decreasing on R + and such that α  See Fig. 1.With the notations of Theorem 1.1 and α = α D (λ, p), up to translations, we know that V = V α,p in (1.10) and (1.11).For the proof of Theorem 1.3 and some additional details, see Section 5.1.Formally as p → 1 + , the potential given by (1.10) converges to a delta Dirac distribution at x = 0 of mass arccos(λ/m) (see [67] for the study of selfadjoint extensions of / D m − α δ 0 ).A remarkable consequence of the estimate in the case p = d = 1 is the Keller-Lieb-Thirring inequality See Appendix D for a result on optimality cases in the case p = d = 1, with a proof.The case of Theorem 1.3 presents some similarities with the results of [35]: in the case p = 1, it is expected that optimality is achieved only by singular measures.Our goals differ from those of [35] as we adopt the point of view of functional interpolation inequalities with Keller-type estimates as a subproduct, while [35] is concerned with the issue of the optimal charge distribution for a Dirac-Coulomb equation.In terms of methods, there are many similarities since we use Birman-Schwinger reformulations as well as classical tools of the concentration-compactness method.However, there are also significant differences because requesting that the potential is in L p (R d ) means that the optimal V is obtained through a non-linear Dirac equation which is not measure-valued as soon as p > 1.
Our results are not limited to estimates for the ground state and we also have a Lieb-Thirring inequality for the sum of eigenvalues in the gap (− m, m) of Dirac operators of the form / D m − V with V ∈ L p (R d , R + ).We denote by − m < λ 1 ≤ λ 2 ≤ • • • < m the possibly infinite sequence of eigenvalues in the gap (− m, m), and write The quantity e k is the distance between the eigenvalue λ k and the bottom of the upper essential spectrum + m.Theorem 1.4.For all γ > d/2 and p ∈ (d, γ + d/2], there is a constant L γ,d,p > 0 so that, for all V ∈ L p (R d , R + ), and all m > 0, we have The inequality (1.13) is, in some sense, an interpolation between these two critical cases.
In the proof, we use rough estimates: the method is constructive but there is a lot of space for improving on the constant L γ,d,p .
Structure of the paper.This paper is organized as follows.In Section 2, we establish some properties of the operator / D m − V with V ∈ L p (R d ): domain, associated Birman-Schwinger operator and self-adjointness.Section 3 is devoted to the variational problem associated with (1.5), after reformulation in the Birman-Schwinger framework.Theorem 3.1 is devoted to the existence of an optimal potential V by concentrationcompactness methods (Section 3.2).The regularity of the optimizers is studied in Section 3.3.Section 4 is devoted to the proof of Theorem 1.4.Explicit and numerical computations are performed in Section 5 in dimensions d = 1, 2 and 3. Open questions, numerical observations, remarks on the non-relativistic limit and Gagliardo-Nirenberg-Sobolev inequalities, and a result in the case p = d = 1 are collected in Appendices A, B, C and D respectively.

Properties of Dirac operators 2.1. A self-adjoint realization
We assume that V ∈ L p (R d , R + ) is positive valued and deal with the self-adjoint extensions of / D m − V .
This is the unique self-adjoint realisation verifying: Moreover, we have the following properties.
We call the extension of Proposition 2.1 the distinguished extension, as it is the unique one whose domain is included in the formal form domain ).We will consider only this extension in what follows, so that the operator / D m − V is selfadjoint under the condition p ≥ d ≥ 1.The proof of the first part of Proposition 2.1 follows from [57].For completeness, we provide a short proof using the associated Birman-Schwinger operator.Under Condition (2.1), the point (i) comes from the usual Kato-Rellich theorem [43,62]

The Birman-Schwinger operator
The Birman-Schwinger operator is a powerful tool for analysing the spectral properties of / D m − V when V belongs to a large class of perturbations.In the relativistic case, Klaus in [46] used it extensively to characterize and study the first eigenvalue of Dirac operators when proving the existence of a distinguished self-adjoint extension.For non-Hermitian potentials V , it can be employed to locate the eigenvalues of / D m − V , as shown for example by Cuenin, Laptev and Tretter in [18], and by Fanelli and Krejčiřík in [39].Furthermore, it can be applied to discuss properties of the ground state of / D m − V when V is a generalised Coulomb-type potential, see, e.g., [14,34,35,46].Throughout this paper, following the approach by Kato [44] and by Konno and Kuroda [48], the Birman-Schwinger operator is used to define the self-adjoint extension of the operator / D m − V .Then, with this rigorous definition at hand, we prove the existence of the optimization problem which defines the ground state by applying variational methods directly on the Birman-Schwinger reformulation of the problem. For Let us point out some differences with Birman-Schwinger operators associated with Schrödinger operators (see Figure 2).In the Schrödinger case, the Birman-Schwinger operator is of the form For any λ < 0, the operator K V (λ) is a positive compact operator and the map λ , ranked in decreasing order and counted with multiplicities, all functions λ → µ j (λ) are increasing on R − .In addition, the first eigenvalue µ 1 is simple because the kernel K V (x, y) is pointwise positive, together with Krein-Rutman theorem: see [11,Theorem 6.13] for a statement and also [61,Section XIII.12].
In the Dirac case, the operator K V (λ) with λ ∈ R is defined only in the gap (− m, m) of the essential spectrum.It is compact by Lemma 2.3 and symmetric because λ is real, but it is not a positive operator.Its eigenvalues are real valued, and can be ranked as ) λ, all maps λ → µ j (λ) and λ → ν j (λ) are increasing.This explains in particular why we expect eigenvalues to emerge from the upper essential spectrum in this setting.We do not know whether µ 1 (λ) is always a simple eigenvalue or not (see Appendix A for more details on open questions).

Proofs of Proposition 2.1, Lemma 2.3 and Proposition 2.4
We start by establishing that K V defined by (2.3) is a compact operator (Lemma 2.3) before proving Propositions 2.1 and 2.4.
Proof of Lemma 2.3.Assume first that p > d.We claim that, for z / ∈ σ( / D m ), the operator with obvious notation.Let us focus on the g 1 z (k) term.We write

All components of the functions g
, we can use the Kato-Seiler-Simon inequality [65, Chapter 4, Theorem 4.1] and conclude that the operator In addition, we have g 1 z=is p → 0 as s → ±∞.Similar computation for the other terms shows that K V is in the Schatten class S p , and that lim s→±∞ K V (i s) op = 0.
In the case p = d with d ≥ 2, we use that all components of the functions g A z and g B z are in the weak-Sobolev space are in the weak Schatten class S 2d,w .In particular, they are compact operators.This already proves that K V is compact as well.Note that g 1 z=i s d,w does not converge to 0 as s → ±∞.
For any R > 0, We have For R large enough, the second term is small in the Schatten space S 2d,w , and for z = i s with |s| large, the first term is small in S q with q > p. Hence lim s→±∞ K V (i s) op = 0.
Let us finally assume p = d = 1.In this case, with explicit computations, the kernel of the Birman-Schwinger operator K V (z) is given by where k = √ m 2 − z 2 is chosen with a positive real part.Thus, K V is a Hilbert-Schmidt operator (hence it is compact) and by the dominated convergence theorem we can conclude that lim s→±∞ K V (i s) op = 0.
Proof of Proposition 2.1.We divide the proof in several steps.

Distinguished self-adjoint extension.
We define the domain of self-adjointness for the operator / D m − V as a perturbation of / D m by applying the method of G. Nenciu in [57].Using similar techniques as in the proof of Lemma 2.3, one can show that the operators R 0 (z) √ V and √ V R 0 (z) can be extended into bounded linear operators on L 2 (R d , C N ).These operators are compact operators, in the Schatten class S 2p .We are now in the setting of [57].
According to [57], the operator / D m − V has a unique self-adjoint extension whose resolvent is the operator

Domain of the distinguished extension. Define the maximal domain as
Then, the set This proves that Dom( / From the relation we obtain √ V φ = 0, and from the relation Self-adjointness on H 1 (R d , C N ).Let us prove (i).Assume that p satisfies (2.1).Thanks to (1.4) we have This shows that the graph norm of / D m is equivalent to the usual We write where, in the last inequality, we used Sobolev's embedding We choose R large enough so that C S V 1 p < 1 and conclude with the Kato-Rellich theorem (see [60,Theorem X.12 ).Since any self-adjoint operator only admits trivial self-adjoint extensions, we can conclude that Regularity for d = 1.Let us focus on (ii) and assume that d = 1 and We recall the following negative Sobolev embeddings: for all ).We now bootstrap the argument.For p > 1, we have 1 well, and we obtain ψ ∈ follows from [66, Theorem 4.5] (also see [61, Theorem XIII.14 and Corollary 1]).Such a result is known in the literature as Weyl's theorem.
Moreover, by construction, the Birman-Schwinger principle holds for the distinguished self-adjoint extension defined as in Proposition 2.1: for a similar application of the Birman-Schwinger principle in a non-relativistic setting.
Remark 2.5.The self-adjointness of Dirac operators involving potentials with one Coulomb singularity or several Coulomb singularities has been intensively studied in respectively [3,44,57,64,[69][70][71] (with additional references therein) and [46,58].In the alternative strategy of [36,37] based on [28], a distinguished self-adjoint extension is built using the underlying Hardy inequality, which was related with the other constructions for Dirac-Coulomb operators in [33,34].Also see [31,63] for further considerations on min-max principles, Hardy inequalities and self-adjointness issues.Optimal Hardy inequalities have been repeatedly use to establish optimal conditions for the existence of a ground state.For instance, in presence of a magnetic field as in [20,25,26], a critical magnetic field is obtained as the ground state energy approaches − m c 2 , which determines the optimal constant of the corresponding Hardy inequality.In the approach of [34,35] as well as in our paper, the Birman-Schwinger formula is essential as it was in [46,47,57].Notice that we do not rely on Nenciu's method [57, Corollary 2.1], but instead use the method of Konno and Kuroda [48] and Kato's approach [44].

The variational problem
In this section, we consider the minimization problem (2.4) and prove Theorem 1.1 in a reformulation which relies on the Birman-Schwinger operator associated to / D m − V , as introduced in Section 2.2.The proof of Theorem 1.1 is given below, right after the statement of Corollary 3.2, as a simple consequence of previous results in the Birman-Schwinger framework.

An auxiliary maximization problem
First, we notice that, for all t > 0, we have we deduce that In what follows, we study the maximization problem (3.1).We perform several changes of variables to study this problem.First, the min-max principle shows that N (λ, p) equals We make the change of variable w := √ W φ so that, by Hölder's inequality, w ∈ L , and, with the convention that w r = |w| C N r , In addition, there is equality if and only if W p is proportional to |φ| 2 , both proportional to |w| this shows that N (λ, p) is also solution to the optimization problem In addition, if w ∈ L q (R d , R + ) is an optimizer of (3.3), then the corresponding optimal W and φ are given by Thus, by showing the existence of an optimizer for (3.3), we solve problem (3.1), and by definition of the Birman-Schwinger operator, find an optimal potential and eigenfunction for our original problem (1.7).Since α → Λ D (α, p) is the inverse map of λ → α D (λ, p) according to (3.2), and since α D (λ, p) = 1/N (λ, p), it is enough to focus on the properties of N (•, p).Theorem 3.1.Let us consider N defined by (3.3).For all λ ∈ (− m, m) and all p > d, we have N (λ, p) > 0. All maximizing sequences for (3.3) are precompact up to translations, hence (3.3) has maximizers.If w is such an optimizer, then w satisfies the Euler-Lagrange equation Finally, the map λ → N (λ, p) is continuous, strictly increasing, and satisfies The proof of the first part relies on the profile decomposition method (concentrationcompactness) used by Lions [54], and is given in the next section.Theorem 3.1 implies the existence of an optimal potential and an optimal spinor.Proof of Corollary 3.2.First, we translate the Euler-Lagrange equation for w into an equation for the potential V and an eigenfunction (not normalized) Ψ.We set Applying / D m − λ to (3.4) shows that Ψ satisfies the nonlinear Dirac equation (3.5).The optimal potential W for the and finally, the optimal potential V for the α D (λ, p) problem is, as wanted, We recover the value of N (λ, p) and α D (λ, p) from the solution Ψ because ˆRd |Ψ| Among all solutions of (3.5), Ψ is the one with the smallest L ) norm so that λ = Λ D (α, p) and Ψ actually solves (1.6).

Proof of Theorem 3.1
We now prove Theorem 3.1.We consider a more general case, and study a general optimization problem.In what follows, we use the notation and define for any s > 0 the maximization problem Here, K is a convolution operator, or equivalently a multiplication operator in Fourier space.In our case, K(x − y) = R 0 (λ)(x − y) is the kernel of the Dirac resolvent, but we state a more general result.
Before proving this result, we make several remarks.
Remark 3.4.Lemma 3.3 fails at the endpoint q = 2. Indeed, by applying the Fourier transform we have This means that all optimizing sequences must concentrate on Dirac masses in Fourier space at locations where k → sup spec K(k) has maxima.Since the Fourier transform is an isometry on L 2 (R d ), we deduce that the maximization problem has no maximum in general.The same argument shows that the existence of optimizers is closely related to the fact that the Fourier transform is not a bĳection between L q (R d ) and Remark 3.5.In the case of the Dirac operator, one has an explicit expression for The function k → g λ (k) is analytic on R d because there is no singularity in the denominator since |λ| < m, so its Fourier transform is exponentially decaying in x.Actually, we have and K ν is the modified Bessel function of the second kind.In particular, there is C ≥ 0 so that In particular, the case (ii) is not covered by (i) only in the case where r = d d−1 , which corresponds to the critical exponent case p = d ≥ 2, that is, q = 2 d d+2 in (3.3).Remark 3.6.Let us consider the case s = 1 in (3.6).In order to see that J(1) > 0 in the Dirac case with λ ∈ (− m, m), let f ∈ L q (R d , C) be a normalized function and let φ + ∈ C N be a normalized vector such that β φ + = φ + .We find that Moreover, by (1.4), we have that φ + , α j φ + C N = 0. Thus: Proof of Lemma 3.3.First, we note that the condition In the first part of the proof, we cover both cases (i) and (ii) by assuming From the Hardy-Littlewood-Sobolev inequality, and since K ∈ L r w (R d ), we have In particular, w → w, K * w is well-defined and real valued on L q (R d ).
Let (w n ) n∈N be a maximizing sequence for J(1).Our argument relies on the concentration-compactness method for the sequence (w n ) n∈N , following the approach of Lions [54] and using Levy's functional.It differs from the concentration-compactness method used in [34], as we work directly with the Birman-Schwinger operator instead of the min-max quadratic form.We set It is clear from the definition that ρ → Q(ρ) is non-decreasing, and that Q(ρ) ≤ 1 for all ρ > 0. We set We divide the proof in the classical steps of the concentration-compactness method and start by discarding the cases µ = 0 (vanishing) and µ < 1 (dichotomy).
• Vanishing.Fix ε := J(1)/4 > 0. Since K ∈ L r (B c 1 ), there is R > 1 large enough so that By Young's inequality, since 2 q + 1 r = 2, we get that for all w ∈ L q (R d ) with w q = 1, We now estimate the contribution of d} using again the Hardy-Littlewood-Sobolev inequality.The double sum can be seen as a discrete convolution, and we apply Young's inequality with z → w L q (Cz) ∈ 2 (Z d ) and where C R is a positive constant which is independent of w: for all w ∈ L q (R d ) with w q = 1, we have Applying this estimate to a maximizing sequence (w n ) n∈N for J(1) = 4 ε, we obtain that, up to a subsequence, This implies and finally µ > 0, which discards the vanishing case of the concentration-compactness method.
• Dichotomy.By definition of Q n , there are sequences of centers x n ∈ R d and radii ρ n > 0 going to infinity so that lim n→+∞ ˆB(xn,ρn) Without loss of generality, by translating the functions w n , we may assume x n = 0.In addition, up to a non-displayed subsequence, we have that for all ε > 0, there is n 0 large enough so that, for all n ≥ n 0 , we have := w n 1 {|x|>2 ρn} .
Introducing E(w 1 , w 2 ) := w 1 , K * w 2 and E(w) := E(w, w), we have n n n n From (3.9), and the fact that w (2) n q ≤ ε 1/q , we get that Finally, we have for n large enough, where in the last line we used Young's inequality, and the fact that ρ n → +∞.Thanks to these facts, we can conclude that In the limit as ε → 0, we obtain J(1) ≤ J(µ) + J(1 − µ), which contradicts (3.11) if µ = 1.So µ = 1, which discards the dichotomy case of the concentration-compactness method.
• Convergence for tight sequences.At this point, we proved that for all ε > 0 there is ρ > 0 and n 0 large enough so that, for all n > n 0 , and after appropriate translations and subsequences, (3.12) In other words, the sequence (w n ) n∈N is tight in L q (R d , C N ).The sequence (w n ) n∈N is bounded in the reflexive Banach space L q (R d , C N ).Hence, up to a non-displayed subsequence, (w n ) n∈N converges weakly to some w ∈ L q (R d , C N ), and we have w q ≤ 1.
Let us prove that E(w) = J(1).Let ε > 0, and let ρ > 0 be large enough so that (3.12) holds.In particular, by Hardy-Littlewood-Sobolev, we have and we have a similar inequality with w instead of w n .On the other hand, we have where T is the operator from L q (R d ) to L q (R d ) with kernel T (x, y) = 1 Bρ (x) K(x − y).The operator We claim that T is a compact operator.In the Dirac case (ii), this comes from the fact that K * w n ∈ W 1,q with K * w n W 1,q (R d , C N ) ≤ C w n q together with the Rellich-Kondrachov compact embedding theorem.In the case (i), where K ∈ L r (R d ), setting τ h f (x) := f (x − h), we have Since K ∈ L r (R d ), we have τ h K − K r → 0 as h → 0, and we conclude with the Kolmogorov-Riesz-Fréchet theorem (see for instance [11,Theorem 4.26]).As a consequence, (T w n ) n∈N converges strongly to T w in L q (R d , C N ).In particular, we obtain that Gathering the two inequalities gives Sending first n to +∞, and then ε to 0 shows that w, K * w = J(1).Finally, since w q ≤ 1, by (3.10) we deduce that w q = 1.This proves that (w n ) n∈N converges strongly to w in L q (R d , C N ) and that w is an optimizer.
It is an open question to decide whether T is compact or not under the condition (3.8).
The fact that λ → N (λ, p) is strictly increasing comes from the fact that λ → R 0 (λ) is operator strictly increasing: for instance, we have ∂ λ R 0 (λ) = (R 0 (λ)) 2 > 0. Let us prove the continuity.Let − m < λ < λ < m, and let w λ be the optimizer for N (λ, p).Using that N (•) is strictly increasing and the resolvent identity Using that R 0 is a bounded operator from L q (R d ) to L q (R d ), and from L q (R d ) into itself, with uniform bounds in a neighborhood of λ, we deduce that there is C > 0 so that This proves that N (•, p) is locally Lipschitz, hence continuous.
We now prove the bounds on lim λ→±m N (λ, p).To prove that lim λ→m N (λ, p) = +∞, we go back to (3.7) and take a function f = L −d/q g(•/L), where g is an arbitrary test function that is normalized in L q (R d ).This gives We bound the resolvent as and change variables to obtain Since 1 − 2 q = p, we may take L = (m − λ) −α for any α ∈ (1/2, p/d) and conclude that lim λ→m N (λ, p) = +∞.
Finally, to prove that lim λ→−m N (λ, p) > 0, we claim that (3.13) there exists a function w ∈ L 2 ∩ L q (R d , C N ) such that w q = 1 and P w = w, where P := 1 m< / D m <2 m is the spectral projection of the free Dirac operator onto (m, 2 m).This would give The second term is null since P ⊥ w = 0.For the first term, we have m < / D m < 2 m on the range of P , and in particular P ( / D m − λ) It remains to prove (3.13).Recall that / D m = FM (k) F * , where F denotes the Fourier transform and To construct such a local family of spinors, one can consider v 0 a normalized eigenfunction of M (k = 0), and set, Since k → P (k) is smooth locally around 0 (P (k) can be written as a Cauchy integral Let also χ(k) : R d → R + be a non null smooth compactly supported function, with χ(k) = 0 for |k| > ε.We consider the function By construction, we have w = 0, and since w has a Fourier transform which is smooth and compactly supported, it belongs to the Schwartz class S(R d , C N ).Finally, since on the support of χ, we have , we deduce that which concludes the proof of (3.13).

Regularity of the solutions of the non-linear Dirac equation
Under Condition (2.1), solutions of (3.5) with Let us consider the other cases of Proposition 2.1.If d = 1 and 1 < p ≤ 2, any optimal function for (3.3) obtained in Theorem 3.1 gives rise to a solution Ψ ∈ W 1,q (R, C 2 ) of (3.5) with q = 2 p/(p + 1).We conclude that Ψ is continuous.If p = d = 2 and q = 4/3, the corresponding solution In dimension d = 1, an explicit expression of the solutions of (3.5) such that by general arguments, as pointed out in [6].However, any solution to (3.5) (and not only the ones found in Theorem 3.1) have additional regularity properties under Condition (2.1).
otherwise.As a first step of an iteration scheme, we proved that if For the initialization, we note that . We easily deduce that there is n ∈ N so that

Lieb-Thirring inequality
This section contains the proof of Theorem 1.4.We closely follow the original proof by Lieb and Thirring [51,53] (see also [52]).This is possible since we are assuming V ≥ 0. In the general case where V has no sign, some results can be found in the works of Cuenin [17], and Frank-Simon [40], where the authors control the Riesz-mean that is, the distance to the whole spectrum.Actually, without assuming a sign on V , one cannot expect to control the sums in (1.13), since, for V ≤ 0 small, the eigenvalues of / D m − V emerge from the bottom essential spectrum (hence have a distance of order 2 m > 0 to the upper essential spectrum).Here, since V is nonnegative, the eigenvalues emerge from the upper essential spectrum as the strength of the potential increases.
Proof of Theorem 1.4.It is sufficient to prove the result for V bounded and compactly supported.By the Birman-Schwinger principle introduced in Section 2.2, we know that λ is an eigenvalue for / D m − V acting on C N valued spinors if and only if 1 is an eigenvalue of K V (λ) defined by (2.3): see Proposition 2.4.We also proved that λ → K V (λ) is operator increasing.In particular, if we set and B e (V ) := number of eigenvalues of K V (m − e) which are greater or equal than 1 , then we have N e (V ) ≤ B e (V ).We have equality if the highest eigenvalues of K V (λ) gets strictly smaller than 1 as λ → − m.This happens for instance if V p ≤ α * (p).With R 0 defined by (2.2), using the operator inequality we can estimate B e (V ) by N B pr e (V ), where B pr e (V ) is the number of eigenvalues above 1 of the pseudo-relativistic Birman-Schwinger operator In addition, with the definition In other words, the Riesz-mean of the eigenvalues increases when one replaces the Dirac operator by the pseudo-relatisvistic one (up to the N factor).Lieb-Thirring inequalities for the last sum have been derived by Daubechies in [19] (and used, e.g., in [50]).In what follows, we derive another inequality specifically for the Dirac operator.We use in particular the fact that the integral in (4.2) only runs for e in the bounded interval (0, 2 m) instead of R + .
• Bound for B pr e (V ).Assume V ∈ L p (R d ) with d < p.The number of eigenvalues above 1 of K pr V (m − e) is bounded from above by K pr V (m − e) p S p .We estimate this norm using the Kato-Simon-Seiler inequality (see [65,Theorem 4.2]).Using a decomposition similar to the one in the proof of Lemma 2.3, we obtain where we introduced the function .
Note that g m,e ∈ L p (R d ) since p > d, and To estimate this norm, we make the change of variable The last integral is an increasing function of e (and has a finite value as e → 0 by the monotone convergence theorem).Since e ∈ (0, 2 m), we can bound this integral by its value at e = 2 m.We deduce that there is a constant C p,d such that • Proof of the Lieb-Thirring estimate.We now follow [51][52][53].The min-max principle for the pseudo-relativistic operator shows that its eigenvalues are decreasing when V increases.Since V ≤ [V − e/2] + + e/2, we may bound For any p > d, we can apply the bound in (4.3) to estimate B pr e/2 [V − e 2 ] + .Inserting this estimate into (4.2),we get where s * (x) := min{m/V (x), 1}, with the convention that s * (x) = 1 if V (x) = 0.The second integral converges whenever p < γ + d/2.We can simply use the bound (1 − s) p ≤ 1 in the last integral, and finally obtain (4.4) This inequality is valid for all d < p < γ + d/2.Note that C γ,d,p stays bounded in the limit as p → γ + d/2, so a similar inequality also holds if p = γ + d/2.Remark 4.1.The result of Theorem 1.4 can be extended to the case of a potential ) by noticing that the right-hand side of (4.4) is continuous for V in this space.In this section, we prove the uniqueness and the symmetry up to translations of the solution of the nonlinear Dirac equation (3.5).We also compute the map α D (λ, p).

Explicit computations
Proof of Theorem 1.3.In the one-dimensional case, Equation (3.5) can be rewritten for the components of Ψ =: (ϕ, χ) as (5.1) Since we are looking for solutions vanishing at ±∞, they satisfy H ϕ(x), χ(x) = 0, G(ϕ(x), χ(x)) = 0 for all x ∈ R.This second condition shows that solutions can be chosen real valued.For real valued variables in the (ϕ, χ)-plane, the level set H(ϕ, χ) = 0 has the shape of an infinity sign.Among real valued functions, uniqueness up to translations follows from the phase plane analysis.We can choose the unique solution with χ(0) = 0, ϕ(0) > 0, given.For this solution, ϕ is even and χ is odd and positive on R + .Hence symmetry and uniqueness, up to translations and multiplication by a phase, are granted by elementary considerations.Next, we have which proves that V is increasing in the quadrant {χ < 0, ϕ > 0} and decreasing in the quadrant {χ > 0, ϕ > 0}.Hence V is even and decreasing on R + , while on R + both χ and ϕ are positive valued.Now let us compute V p .It is enough to do the computation on R + .First, the equation H(ϕ, χ) = 0 can be rewritten as and so ϕ = Next, from the equation Finally, we have Collecting the three last equalities shows that V solves the autonomous differential equation At x = 0, we have V (0) = 0, which implies • Subcritical regime λ > − m.The function One can directly check that the solution of (5.2) is This gives (1.10).The L p (R) norm of V is computed as Using that Z is even, monotone decreasing on R + , with the change of variable z = Z(x) and t = z/z 0 , we obtain, using (5.2), See [1, 15.3.1 p. 558] for the last equality.This completes the computation of α D (λ, p).By taking the limit as p → 1 + , we obtain α D (λ, 1) = arccos(λ/m).

The radial case in dimension d = 2
We now provide some numerical simulations to obtain upper bounds for the maps Λ D (α, p).
First, we restrict the minimization problem (1.5) to radial potentials, that is, we compute Below in Appendix B, we provide some numerical evidences that the optimal potentials are radial.We abusively write V (x) = V (r) with r = |x|, x ∈ R 2 , use polar coordinates (x, y) = (r cos θ, r sin θ), and write In these coordinates, the Dirac operator becomes This suggests to decompose a spinor Ψ in Fourier modes with the convention Ψ(r, θ) = n∈Z ϕ n (r) e i n θ i χ n (r) e i (n+1) θ .

The radial case in dimension d = 3
As in the two dimensional case, we restrict the minimization problem (1.5) to radially symmetric decreasing potentials.The corresponding Dirac operator decomposes as a direct sum in eigenspaces of the spin-orbit operator We have It is an open question to decide whether the above inequalities are in fact equalities or not.If κ = 1, we look for an eigenstate of / D m − V in the Wakano form of [68], that is,   Let us assume that m = 1 and consider at λ = −1 (lower end of the gap) the system According to the previous section, the radial case d = 2 corresponds to δ = 1 (that is n = 0 in (5.4)), and the radial case d = 3 to δ = 2 (that is κ = 1 in (5.5)).Writing χ(r) = f (r) ϕ(r), the equation becomes We now notice that this system admits a solution with f (r) = r/µ (so that all functions in the middle equality are constant functions).Explicitly, assuming δ < p − 1, with µ := In particular, The upper bound given by this expression for d = 1 (with δ = 0) coincides with the expression found in Theorem 1. an open question to decide whether ϕ p , χ p and W p is the unique solution of (5.6) and if it is optimal among radial optimal functions, and also among non-radial optimal functions (see Appendix B).

A. Open questions
In this article, we study the ground state defined the lowest eigenvalue in the gap λ D (V ) of a general Dirac operator / D m − V with V ∈ L p (R d , R + ) using Birman-Schwinger techniques, and prove that this quantity always makes sense if the L p (R d ) norm of V is small enough.To our knowledge, there are several open questions concerning this lowest eigenvalue, which we recall here.
• Is the map V → λ D (V ) concave?
• Is λ D (V ) always a simple eigenvalue, or equivalently, is µ 1 (K V ) always simple?
Concerning the variational problem associated with (1.5), we recall two questions that were already raised earlier: • Is the optimal potential V radial (decreasing) if d ≥ 2?
• If so, is the corresponding ground state Ψ the solution with lowest angular momentum and smallest number of oscillations, as it is suggested in Sections 5.2 and 5.3?

B. Is the optimal potential radial? A numerical answer
In dimension d = 2, we investigate numerically whether the optimal potential V for (1.5) is radial, or equivalently whether the optimal potential W for (3.1) is radial.In order to do so, we run the following self-consistent algorithm .Recall that K W := √ W R 0 (λ) √ W where R 0 denotes the resolvent of the free Dirac operator.For p > d = 2 and λ ∈ [− m, m), we choose an initial potential W 0 at random, and set In practice, the potential W k+1 is also translated so that its maximum is at the origin.We can check that the quantity µ 1 (K W k ) is increasing, and that the sequence (W k ) k∈N converges to some limit potential W * in L p (R 2 ).A typical run of the algorithm is displayed in Fig 5 .In order to check whether W * is radial or not, we compute the L p (R 2 ) norm of its angular derivative.For λ ∈ [−0.9, 0.9], m = 1 and p ∈ (2, 8), this norm is always much smaller than 1 and usually of the order of 10 −2 or 10 −3 , after less than 100 iterations, depending on the parameters we chose.These numerical results suggest that the optimal potentials might be radial, up to translations.

C. A nonlinear interpolation inequality for the
The code is available upon request to the authors.
by expanding the expression of α D (λ, p) given in Theorem 1.3 as λ → 1 − .This is consistent with K p = C −η q and the expression of the explicit, optimal value of the constant C q in (1.1) if the dimension is d = 1: we refer to [23] and references therein for details. 2) in the non-relativistic limit.In dimension d = 1, a tedious but elementary computation directly shows that the constant obtained by taking the non-relativistic limit in the Keller-Lieb-Thirring inequality for the Dirac operator written with optimal constant is the optimal constant in the Keller-Lieb-Thirring inequality for the Schrödinger operator, as it can be deduced for instance from [23,45].

C.2. An interpolation inequality for the Dirac operator
Using a min-max principle as in [28], it is possible to write an optimal interpolation inequality of Gagliardo-Nirenberg-Sobolev type which plays for the free Dirac operator the same role as (1.1).The inequality is somewhat involved, but Inequality (1.1) is recovered in the non-relativistic limit as c → +∞.For sake of simplicity, we consider only the case d = 1.
Let us start by a short and formal summary of the min-max principle applied to the determination of the ground state of the Dirac operator.If (ϕ, χ) is an eigenspinor of the operator / D where ν ≥ 0 is now the Lagrange multiplier for the constraint W p = α.Note that for all fixed a, b, c ≥ 0, the equation has a unique solution in X ν ≥ 0, as the left-hand side is an increasing function of X, while the right-hand side is decreasing, and that ν → X ν is increasing.So for fixed ν ≥ 0, there is a unique W = V ν [φ] satisfying (C.4) and the map ν → V ν is pointwise decreasing, hence so is the map ν → V ν p .With α ,c D (λ, p) given by (C.3), we define ν * (λ, p, φ) := inf ν > 0 : V ν [φ] p ≤ α ,c D (λ, p) .
Summarizing, we proved that for all φ ∈ C ∞ 0 (R) and all ν ≥ ν * (λ, p, φ), which can be interpreted as a Gagliardo-Nirenberg type inequality for φ alone.Such an inequality is known for a fixed, given potential V from [21,27,28] and it is then of Hardytype, as for instance the new Hardy inequality in [35], but the novelty in this paper is that we take V = V ν [φ] thus making it a non-linear interpolation inequality.While the form (C.5) is non-explicit, it allows to recover the usual Gagliardo-Nirenberg inequality in the non-relativistic limit as c → ∞.By writing λ = m c 2 + E for some E < 0, (C.5) becomes Let us choose ν = φ This inequality is the Gagliardo-Nirenberg inequality (1.1) written in non-scale invariant form, for an appropriate choice of the parameter λ in (1.1).

D. The case p = d = 1
This appendix deals with the limit case p = 1 of Theorem 1.3 devoted to the one-dimensional Keller estimates.We give a computation of α D (λ, 1) which is not based on the limit as p → 1 + of the nonlinear estimates and prove that any sequence of optimizing potentials concentrates into a Dirac δ distribution.In addition, any sequence of nonnegative potentials (V n ) n∈N with V n 1 = α and eigenvalues λ n approaching m cos α, converges as n → +∞ to a Dirac δ distribution.
According to [67], "the method of directly solving the Dirac equation with a δ-function potential and the method of obtaining the solution by first solving the Dirac equation with a short-range potential and afterward taking the δ-function limit, lead to different results" [concerning the spectrum].This issue is known as Klein's paradox.Although the Keller-Lieb-Thirring (1.12) makes sense for any nonnegative potential V ∈ L 1 (R d ), it is a natural question to investigate by direct methods whether the bound is achieved in the larger set of bounded nonnegative measures and consider sequences of optimizing potentials.
In order to approximate unbounded potentials, we need an estimate on the negative term in (D.2).Take any number c > 1.Since t is continuous, there is an interval I V (c) ⊂ (−∞, x 1 ] such that t(x) ∈ (1/c, c) for all x ∈ I V (c).We have the bound (note that the integrand is symmetric under t → 1/t) ˆIV V (s) ds .Now, assume that (V n ) n∈N is a sequence of potentials with V n 1 = α and eigenvalues λ n converging to λ := m cos α.Without loss of generality, we may assume that each V n is bounded and compactly supported.By (D.3), in order to approach the equality case, we need that |I Vn (c)| tends to zero for each c > 1.We now use (D.4) to show that this implies the convergence (after suitable translations) to a Dirac δ distribution.
Fix > 0 and r > 0. Fix c > 1 and n 0 such that for all n ≥ n 0 , we have Since r and are arbirary, we have shown that Q Vn converges pointwise to α.In the language of concentration-compactness, this excludes vanishing and dichotomy and implies that, after a sequence of translations, V n converges to a measure of total mass α supported at the origin, hence, to a Dirac δ distribution.

1
by the Hölder inequality and the Sobolev embedding theorem we get that H 1 (R d , C N ) ⊆ Dom( / D m − V ) and this concludes the first part of the proof.

4
V ) := number of eigenvalues of −∆ + m 2 − m − V less or equal than − e , the usual Birman-Schwinger principle shows that B pr e (V ) = N pr e (V ).To sum up, we have (4.1)N e (V ) ≤ B e (V ) ≤ N B pr e (V ) = N N pr e (V ) .The operator √ −∆ + m 2 − m is sometimes called the Chandrasekhar (or pseudo-relativistic) kinetic energy operator.It is a positive operator, √ −∆ + m 2 − m − V is bounded from below, and the min-max formula applies.We can now repeat the usual arguments of Lieb and Thirring for the pseudo-relativistic operator.First, for γ > 0, the cake-layer representation gives (1 N e (V ) de ≤ γ N ˆ2 m 0 e γ−1 B pr e (V ) de .Note that for the pseudo-relativistic model, if − e pr 1 ≤ − e pr 2 ≤ • • • < 0 are the negative eigenvalues of √ −∆ + m 2 − m − V , we have k≥1 (e pr k ) γ = γ ˆ∞ 0 e γ−1 N pr e (V ) de = γ ˆ∞ 0 e γ−1 B pr e (V ) de , and the integral runs over e ∈ R + instead of e ∈ (0, 2 m).Actually, the previous two inequalities together with (4.1) show that
It is an open question to decide whether Λ rad D (α, p) is attained by Λ rad,(n) D (α, p) with n = 0 or not, and if equality holds in (5.3) so that Λ D (α, p) = Λ rad,(0) D

5. 4 .
An explicit bound in the radial case in dimensions d = 2 or d = 3

3 ,
and we conjecture that we actually have equality in d = 2 and d = 3 as well.Numerically, the curve p → W p p coincides with the numerical solution p → α rad,(n=0) (p) if d = 2 and p → α rad,(κ=1) (p) if d = 3 of Figs. 3 and 4. It is however

Dirac operator C. 1 .
Non-relativistic limit and Keller-Lieb-Thirring inequalities In order to consider the non-relativistic limit c → +∞, it is interesting to reintroduce the parameters , m and c.The eigenvalue problem / D ,c m − W ψ = µ ψ where / D ,c m := − i c α • ∇ + m c 2 β is reduced to the eigenvalue problem corresponding to = c = m = 1 by the change of variables ψ

Figure 5 .Proposition C. 1 .
Figure 5. Contour lines of the potential W k during the iterations, for p = 3 and λ = 1/2, for some W0 chosen at random.The quantities W k and φ k are computed on a square [−a, a] 2 with a = 6, L = 100 discretization points per direction and periodic boundary conditions.The Dirac operator and its inverse are computed in Fourier space and the L p (R 2 ) integrals in direct space.

Proof. 2 2
Let us consider the general case d ≥ 1.The non-relativistic limit of the ground state λ ,c D (W ) of the Dirac operator / D ,c m − W is, up to the mass energy m c 2 , given by the ground state of the Schrödinger operator − m ∆ − W by standard results: see for instance [32, Section 2.4].Hence lim c→+∞ m c 2 − λ ,c D (W ) = λ − S (W µ ) where W µ (x) := W (µ x) and µ = √ 2 m .Here − λ − S (W µ ) denotes, if it exists, the negative ground state of the Schrödinger operator − ∆ − W µ .The factor µ = / √ 2 m arises from a scaling argument.By definition (1W p , p ≤ K p W µ η p = K p µ − d η p W η p but there is in fact equality if we use as test function an optimal function W for (1.2).Taking α = − d p m d p −1 c d p −2 W p in the limit as c → +∞ concludes the proof.physical constants.In other words, we recover a standard Keller-Lieb-Thirring inequality for the Schrödinger operator (1. p) stated in Theorem 1.1 and (C.1).If α guarantees that λ ,c D (W ) ≥ λ.Notice that p ≥ d implies that lim c→∞ W p ≤ α ,c D (λ, p) = ∞ .
. The result in (ii) is derived by bootstrapping the Sobolev embedding theorem.See Section 2.3 for the proof of Proposition 2.1.