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

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Keller and Lieb-Thirring estimates of the eigenvalues

in the gap of Dirac operators

Jean Dolbeault, David Gontier, Fabio Pizzichillo and Hanne Van Den

Bosch

Abstract. 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 Schrödinger oper-

ator, 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.

1 Introduction and main results 2

2 Properties of Dirac operators 8

2.1 A self-adjoint realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 The Birman-Schwinger operator . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Proofs of Proposition 2.1, Lemma 2.3 and Proposition 2.4 ............. 11

3 The variational problem 14

3.1 An auxiliary maximization problem . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Proof of Theorem 3.1 ................................ 17

3.3 Regularity of the solutions of the non-linear Dirac equation . . . . . . . . . . . . 23

4 Lieb-Thirring inequality 24

5 Explicit computations 27

5.1 The case d= 1: proof of Theorem 1.3 ....................... 27

5.2 The radial case in dimension d= 2 ........................ 29

5.3 The radial case in dimension d= 3 ........................ 30

5.4 An explicit bound in the radial case in dimensions d= 2 or d= 3 ........ 32

A Open questions 33

B Is the optimal potential radial? A numerical answer 34

C A nonlinear interpolation inequality for the Dirac operator 34

C.1 Non-relativistic limit and Keller-Lieb-Thirring inequalities . . . . . . . . . . . . 34

C.2 An interpolation inequality for the Dirac operator . . . . . . . . . . . . . . . . . 36

D The case p=d= 1 38

References 41

Mathematics Subject Classiﬁcation (2020): Primary 81Q10; Secondary 49R05,49J35,47A75,47B25.

Keywords: Dirac operators, potential, spectral gap, eigenvalues, ground state, min-max principle,

Birman-Schwinger operator, domain, self-adjoint operators, Keller estimate, Lieb-Thirring inequality,

interpolation, Gagliardo-Nirenberg-Sobolev inequality, Kerr nonlinearity.

arXiv:2210.03091v2 [math.AP] 24 Jul 2023

2 J. Dolbeault, D. Gontier, F. Pizzichillo and H. Van Den Bosch

1. 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,λS(V)of the Schrödinger operator −∆−Vin

dimension d= 1, under a constraint on the Lebesgue norm

∥V∥p=ˆRd|V|pdx1/p

of exponent pof 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 λS(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 /

Dmdenotes the free Dirac operator and Vis a non-negative valued function, /

Dm−V

is not bounded from below. One is actually interested in the lowest eigenvalue λD(V)in

the essential gap (−m c2, m c2), where mdenotes the mass and cthe speed of light. We

shall speak of λD(V)as the ground state energy of /

Dm−V. In the standard setting,

it is expected that λD(V)−m c2converges to λS(V)in the non-relativistic limit, i.e., as

c→+∞. It is therefore a natural question to estimate λD(V)in terms of ∥V∥pand 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∥pfor which λD(V)

reaches, for some potential V≥0, the lower end of the essential gap −m c2. In a linear

setting, a similar question has been raised in [34,35], where the authors ﬁnd a critical value

ν1so that λDµ∗|·|−1>−m c2for all positive measures µwith µ(R3)< ν1, with

2/π/2 + 2/π< ν1≤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 /

Dm−Vin the gap by a min-max

principle according to [28–30] 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 Vunder a constraint on ∥V∥p, except that the Keller type bound on

λD(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 [5–8] by W. Borrelli. In the one-dimensional case, the solu-

tion 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 /

Dm−Vallows us to characterize λD(V)and, except

in Appendix C, we will adopt this point of view.

Keller-Lieb-Thirring estimates and Dirac operators 3

The Keller-Lieb-Thirring inequality for a Schrödinger operator goes as follows. Let us

assume that q > 2, with q < 2∗:= 2 d/(d−2) if d≥3, and let ϑ=d(q−2)/(2 q). For

any function u∈H1(Rd), the Gagliardo-Nirenberg-Sobolev inequality

∥∇u∥ϑ

2∥u∥1−ϑ

2≥Cq∥u∥q

can be rewritten in the non-scale invariant form as

(1.1) ∀(λ, u)∈(0,+∞)×H1(Rd),∥∇u∥2

2+λ∥u∥2

2≥Cqλ1−ϑ∥u∥2

with an optimal constant Cqsuch that C2

q=ϑϑ(1 −ϑ)1−ϑCq. The equivalence of the

two forms can be recovered by optimizing on λin (1.1). There is also an inequality which

is dual of (1.1) and goes as follows. Consider a potential V∈Lp(Rd). Using Hölder’s

inequality with exponents pand qsuch that 1/p + 2/q = 1 and p > d/2, and taking λso

that Cqλ1−ϑ=∥V∥p, we deduce from (1.1) that

ˆRd|∇u|2dx−ˆRd

V|u|2dx≥ ∥∇u∥2

2− ∥V∥p∥u∥2

q≥ −C−1

q∥V∥p1

1−ϑ∥u∥2

This is the Keller-Lieb-Thirring estimate for −∆−V,i.e.,

(1.2) ∀V∈Lp(Rd),0≤λ−

S(V)≤Kp∥V∥η

where η:= 1/(1 −ϑ) = 2 p/(2 p−d)and λ−:= max(0,−λ)denotes the negative part

of λ. See [22–24] for details. An optimization on Vshows that (1.1) and (1.2) are equivalent.

The optimal constant in (1.2) is Kp=C−η

q. In addition, for all λ > 0, if uis a radial positive

solution of

(1.3) −∆u−up+1

p−1=−λ u ,

then (u, λ)is an optimal pair for (1.1), and V:= uq−2=u2/(p−1) is an optimal potential

for (1.2), which moreover satisﬁes λS(V) = −λ. It turns out that the solution of (1.3) is

unique up to translations according to [16,49,55] and can be explicitly computed if d= 1:

see [45], or [23] and references therein for additional related results.

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 Rd. We

refer to [66] for a comprehensive list of results and properties. For simplicity, we choose

units in which c= 1, except in Appendix Cin which we consider the non-relativistic limit

as c→+∞. Let d≥1and set N:= 2⌊(d+1)/2⌋where ⌊x⌋= max{n∈Z:n≤x}denotes

the integer part of x. Let α1,··· , αdand βbe N×NHermitian matrices satisfying the

following anti-commutation rules

(1.4) ∀j, k = 1,...,d, 









αjαk+αkαj= 2 δjk IN

αjβ+β αj= 0

β2=IN

where δjk denotes the Kronecker symbol and INis the N×Nidentity matrix. See,

e.g., [41] for an existence result for such matrices. The free Dirac operator in dimension d

4 J. Dolbeault, D. Gontier, F. Pizzichillo and H. Van Den Bosch

is deﬁned by

Dm:=

j=1

αj(−i∂j) + m β =α·(−i∇) + m β

where we consider Cartesian coordinates (x1, . . . , xd),∂j:= ∂/∂xjand α= (αk)k=1,...,d.

With the Pauli matrices

σ1:= 0 1

1 0, σ2:= 0−i

i 0 and σ3:= 1 0

0−1,

explicit expressions of /

Dmare given

(i) in dimension d= 1, by α=σ2and β=σ3so that

Dm:= σ2(−i∂1) + m σ3,

(ii) in dimension d= 2, by α= (σj)j=1,2and β=σ3so that

Dm:=

j=1

σj(−i∂j) + m σ3,

(iii) in dimension d= 3, by α= (αk)k=1,2,3and βsuch that

αk:= 0σk

σk0and β:= I20

0−I2.

The free Dirac operator satisﬁes /

m=−∆ + m2. It is self-adjoint on L2(Rd,CN), with

domain

Dom( /

Dm) = H1(Rd,CN)

and spectrum

σ(/

Dm) = σess(/

Dm)=(−∞,−m]∪[m, +∞).

Next we consider Dirac operators /

Dm−Vwith potentials V∈Lp(Rd,R+)where the

notation /

Dm−Vdenotes /

Dm−VIN. When switching on a potential V, we expect that

some eigenvalues of /

Dm−Vemerge from the upper essential spectrum [m,+∞). We shall

prove in Section 2that /

Dm−Vcan be deﬁned as a self-adjoint operator with essential

spectrum σess(/

Dm−V) = σess(/

Dm). This allows us to deﬁne the ground state λD(V)as

the lowest eigenvalue in the gap (−m, m).

Our ﬁrst result states that the ground state is bounded by a function of ∥V∥p. Let

(1.5) ΛD(α, p) := inf nλD(V) : V∈Lp(Rd,R+)and ∥V∥p=αo.

Theorem 1.1. Assume that p≥d≥1. There exists α⋆(p)>0such that the map α7→

ΛD(α, p)deﬁned on 0, α⋆(p)is continuous, strictly decreasing, takes values in (−m,m],

and such that

lim

α→0+

ΛD(α, p) = mand lim

α→α⋆(p)ΛD(α, p) = −m .

Keller-Lieb-Thirring estimates and Dirac operators 5

Moreover, if (p, d)̸= (1,1), the inﬁmum (1.5)is attained on 0, α⋆(p)and

∀α∈0, α⋆(p),ΛD(α, p) = λD(Vα,p)

where Vα,p =|Ψ|2/(p−1), and Ψ∈L2(Rd,CN)solves the nonlinear Dirac equation

(1.6) /

DmΨ− |Ψ|2

p−1Ψ=ΛD(α, p) Ψ

and satisﬁes the constraint ´Rd|Ψ|2p/(p−1) dx=∥Vα,p∥p

p=αp.

The proof of Theorem 1.1 is given in Section 3and relies on the properties of the inverse

map of α7→ ΛD(α, p)deﬁned by

(1.7) αD(λ, p) := inf n∥V∥p:V∈Lp(Rd,R+)and λD(V) = λo.

The critical value is α⋆(p) = limλ→(−m)+αD(λ, p). It is such that

lim

α→α⋆(p)−

λD(Vα,p) = −m

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) =

αD(−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 /

Dm−V.

Corollary 1.2. Assume p≥d≥1. For all V∈Lp(Rd,R+)with ∥V∥p< α⋆(p), we have

the optimal bound

(1.8) −m≤ΛD(∥V∥p, p)≤λD(V)≤m .

If (p, d)̸= (1,1), then Vα,p as in Theorem 1.1 realizes the equality case, i.e.,λD(Vα,p) =

ΛD(α, p).

Some plots of α7→ ΛD(α, p)are displayed in Fig. 1(Right).

The nonlinear Dirac equation (1.6) plays for the Dirac operator /

Dm−Vthe same role

as (1.3) for the Schrödinger operator −∆ + V. However, ΛD(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 eﬀect-

ive 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

DmΨ−G(Ψ) = λΨ

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KellerandLieb-ThirringestimatesoftheeigenvaluesinthegapofDiracoperatorsJeanDolbeault,DavidGontier,FabioPizzichilloandHanneVanDenBoschAbstract.WeestimatethelowesteigenvalueinthegapoftheessentialspectrumofaDiracoperatorwithmassintermsofaLebesguenormofthepotential.SuchaboundisthecounterpartforDiracoper...

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

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