Dirac equation: variational approach and applications

(1)

Strong magnetic fields More on numerical schemes

Dirac equation: variational approach and applications

Jean Dolbeault

http://www.ceremade.dauphine.fr/ ∼ dolbeaul Ceremade, Universit´e Paris-Dauphine

October 14, 2013

Math´ematiques pour le graph`ene, Grenoble, October 14-15, 2013

J. Dolbeault Dirac equation: variational approach and applications

(2)

Outline

1

Eigenvalues of the Dirac operator and minmax characterizations

2

Hardy inequalities, completion of the square

3

Strong magnetic fields

4

Numerical schemes

(3)

Minmax characterizations of the eigenvalues of the Dirac operator

Joint work with:

Maria J. Esteban and Eric S´er´e

(4)

Some notations

Free Dirac operator:

H 0 = − i α · ∇ + β , with α 1 , α 2 , α 3 , β ∈ M ⁴ × 4 ( C ) β =

I 0 0 − I

, α i =

0 σ i

σ i 0

Pauli matrices:

σ 1 =

0 1 1 0

, σ 2 =

0 − i i 0

, σ 3 =

1 0 0 − 1

Decomposition into upper / lower component: Ψ = ϕ

χ

H 0 Ψ =

Pχ + ϕ Pϕ − χ

, with P = − i σ · ∇

(5)

Two of the main properties of H 0 are:

H ₀ ² = − ∆ + 1 and

σ(H 0 ) = ( −∞ , − 1] ∪ [1, + ∞ )

Denote by Y ^± the spaces Λ ^± (H ^1/2 ( R ³ , C ⁴ )), where Λ ^± are the positive and negative spectral projectors on L ² ( R ³ , C ⁴ ) corresponding to the free Dirac operator: Λ ⁺ and Λ ⁻ = I

L2

− Λ ⁺ have both infinite rank and satisfy H 0 Λ ⁺ = Λ ⁺ H 0 = √

1 − ∆ Λ ⁺ = Λ ⁺ √ 1 − ∆ H 0 Λ ⁻ = Λ ⁻ H 0 = − √

1 − ∆ Λ ⁻ = − Λ ⁻ √ 1 − ∆ R = − i σ · ∇ , R ² = − ∆

(6)

Min-max characterization of the discrete spectrum

Kato’s inequality and related inequalities have no evident relation with the spectrum of H 0 : ν = _π ² or ν = _2/π+π/2 ² are not critical for the (point) spectrum of H 0 − _| ^ν x | . The operator

H ν := H 0 − ν

| x | , 0 < ν < 1

has a self-adjoint extension with domain included in H ^1/2 ( R ³ , C ⁴ ) and its spectrum is given by

σ(H ν ) = ( −∞ , − 1] ∪ { λ ^ν ₁ , λ ^ν ₂ , . . . } ∪ [1, ∞ ) , lim

ν → 1 λ ^ν ₁ = 0 H ν is self-adjoint only for ν < 1. The notion of “first eigenvalue” in ( − 1, 1) or “ground state” does not make sense for ν ≥ 1.

(7)

Assume that V ∈ M ³ ( R ³ ) + L ^∞ ( R ³ ) and ∃ δ > 0 such that (H) ± Λ ^± (H 0 + V ) Λ ^± ≥ δ Λ ^± √

1 − ∆ Λ ^± in H ^1/2 ( R ³ , C ⁴ ) With Y ^± = Λ ^± H ^1/2 ( R ³ , C ⁴ ), define

c k (V ) = inf

F⊂Y+ Fvector space

dimF=k

sup

ψ∈F⊕Y−

ψ 6 =0

((H 0 + V )ψ, ψ) (ψ, ψ)

Theorem

[J.D., Esteban, S´er´e] Under assumption (H), if V ∈ L ^∞ ( R ³ \ B R

0

) for some R 0 > 0 is such that

lim R → + ∞ k V k

L∞(|x|>R)

= 0, lim R → + ∞ supess _|x| _>R V (x) | x | ² = −∞ , then { c k (V ) } k ≥ 1 is the non-decreasing sequence of eigenvalues of H 0 + V in the interval [0, 1), counted with multiplicity, and

0 < δ ≤ c 1 (V ) = λ 1 (V ) ≤ ... c k (V ) = λ k (V ) ≤ ... ≤ 1 , lim

k → + ∞ c k (V ) = 1

(8)

Griesemer and Siedentop proved an abstract result which implies the above min-max characterization for the eigenvalues of H 0 + V for a certain class of potentials V (which does not include singularities close to the Coulombic ones).

Remark. Any potential V such that | V | ≤ a | x | ⁻ ^β + C belongs to M ³ ( R ³ ) + L ^∞ ( R ³ ) for all a, C > 0, β ∈ (0, 1). If | V | ≤ a | x | ⁻ ¹ , then (H) is satisfied if a < 2/(π/2 + 2/π) ≈ 0.9. Moreover, any V ∈ L ^∞ ( R ³ ) satisfies (H) if k V k ∞ < 1.

Remark. Assumption (H) implies that for all constants κ > 1, close to 1, there is a positive constant δ(κ) > 0 such that :

± Λ ^± (H 0 + κV )Λ ^± ≥ δ(κ)Λ ^± in H ^1/2 ( R ³ , C ⁴ )

(9)

Further min-max results: Talman’s decomposition

H + ^T = L ² ( R ³ , C ² ) ⊗ 0

0 , H ₋ ^T = 0

0 ⊗ L ² ( R ³ , C ² ) , so that, for any ψ =

ϕ χ

∈ L ² ( R ³ , C ⁴ ),

Λ ^T ₊ ψ = ϕ 0

, Λ ^T ₋ ψ = 0

χ

Assume also that the potential V satisfies

|x|→ lim + ∞ V (x ) = 0 , − ν

| x | − c 1 ≤ V ≤ c 2 = sup(V ) , with ν ∈ (0, 1) and c 1 , c 2 ∈ R . Finally, define the 2-spinor space W := C ₀ ^∞ ( R ³ , C ² ), and the 4-spinor subspaces of L ² ( R ³ , C ⁴ )

W ₊ ^T := W ⊗ 0

0 , W ₋ ^T :=

0 0

⊗ W

(10)

Theorem

[J.D., Esteban, S´er´e] Under the previous assumptions, all eigenvalues of H 0 + V in the interval ( − 1, 1) are given by the following (eventually finite) sequence of real numbers

inf

F⊂W T+ F vector space

dimF=k

sup

ψ∈F⊕W T−

ψ 6 =0

((H 0 + V ) ψ, ψ) (ψ, ψ) ,

assuming that the lowest of these min-max values is larger than − 1 (Talman)

λ 1 (V ) = inf

ϕ 6 =0 sup

χ

(ψ, (H 0 + V )ψ) (ψ, ψ) where both ϕ and χ are in W and ψ =

ϕ χ

, as soon a the above inf-sup takes its values in ( − 1, 1) and W ₊ ^T ⊕ W ₋ ^T is a core

(11)

Abstract min-max approach

Let H be a Hilbert space and A : D(A) ⊂ H → H a self-adjoint operator. F (A) is the form-domain of A. Let H ⁺ , H − be two orthogonal Hilbert subspaces of H such that H = H ⁺ ⊕H − . Λ _± are the projectors on H ± . We assume the existence of a core F such that : (i) F + = Λ + F and F ₋ = Λ ₋ F are two subspaces of F (A).

(ii) a = sup _x

₋

_∈ _F

₋

_\{ ₀ _} ^(x _kx

⁻₋

^,Ax _k

2⁻

⁾

H

< + ∞ . Let c k = inf

V subspace of F

+

dim V =k

sup

x ∈ (V ⊕ F

−

) \{ 0 }

(x, Ax) k x k ²

H

, k ≥ 1.

(iii) c 1 > a , b = inf (σ ess (A) ∩ (a, + ∞ )) ∈ [a, + ∞ ].

Definition: for k ≥ 1, λ k is the k ^th eigenvalue of A in (a, b), counted with multiplicity, if this eigenvalue exists. If not, λ k = b.

(12)

Theorem

[J.D., Esteban, S´er´e] Assume (i)-(ii)-(iii).

c k = λ k , ∀ k ≥ 1 As a consequence, b = lim

k →∞ c k = sup

k

c k > a

References on the min-max approach: Talman and Datta & Deviah for the computation of the first positive eigenvalue of Dirac operators with a potential. Other min-max approaches were proposed by Drake &

Goldman and Kutzelnigg

Griesemer & Siedentop: first abstract theorem on the variational principle, under conditions (i), (ii ), and two additional hypotheses instead of (iii ): (Ax, x) > a k x k ² ∀ x ∈ F + \ { 0 } , the operator

( | A | + 1) ^1/2 P ₋ Λ + is bounded. Here Λ + is the orthogonal projection of H on H + and P ₋ is the spectral projection of A for the interval ( −∞ , a], i .e. P ₋ = χ ( −∞ ,a] (A)

(13)

[Griesemer, Lewis & Siedentop]: an alternative approach which extends the results of Griesemer & Siedentop and applies to Dirac operators with potentials having Coulomb singularities.

Additional comment: a difficult part to apply the previous theorem is condition (iii): the first level of min-max has to be above the lower bound of the gap. A possible method consists in deriving an abstract continuation method when the family of operators depends continuously on a parameter, for instance ν in case of H ν := H 0 − _| ^ν _x _|

Thuis motivates the search for an adapted Hardy inequality

(14)

The case of Talman’s min-max

As a subproduct of our method, we obtain Hardy type inequalities. In the case of the decomposition based on the projectors of the free Dirac operator, one gets a Hardy type inequality. A simpler form is obtained in case of Talman’s decomposition.

For every ϕ ∈ C ₀ ^∞ ( R ³ , C ² ) consider λ(ϕ) = sup

χ

(ψ, (H 0 + V ) ψ)

(ψ, ψ) where ψ = ϕ

χ

This number is achieved by the function

χ(ϕ) := − i (σ · ∇ )ϕ 1 − V + λ(ϕ)

(15)

Moreover, λ = λ(ϕ) is the unique solution to the equation λ

Z

R

³

| ϕ | ² dx = Z

R

³

| (σ · ∇ )ϕ | ²

1 − V + λ + (1 + V ) | ϕ | ²

dx

(uniqueness is an easy consequence of the monotonicity of both sides of the equation in terms of λ). Thus λ 1 (V ) is the solution of the following minimization problem

λ 1 (V ):= inf { λ(ϕ) : ϕ ∈ C ₀ ^∞ ( R ³ , C ² ) } This is by far simpler than working with Rayleigh quotients.

λ 1 (V ) is the best constant in the inequality Z

R

³

| (σ · ∇ )ϕ | ²

1 + λ 1 (V ) − V dx + Z

R

³

(1 − λ 1 (V ) + V ) | ϕ | ² dx ≥ 0

(16)

For any ν ∈ (0, 1), the first eigenvalue of H ν := H 0 − _| ^ν _x _| is explicit:

λ 1

− ν

| x |

= p 1 − ν ² Z

R

³

| (σ · ∇ )ϕ | ²

1+ √

1 − ν

²

+

_|x|^ν

dx + ( ¹ − √ 1 − ν

²

)

Z

R

³

| ϕ | ² dx ≥ ν Z

R

³

| ϕ | ²

| x | dx Moreover this inequality is achieved. In the limit ν → 1, we get the optimal (but not achieved) inequality:

Z

R

³

| (σ · ∇ )ϕ | ² 1 + _| ¹ _x _| dx +

Z

R

³

| ϕ | ² dx ≥ Z

R

³

| ϕ | ²

| x | dx This inequality is not invariant under scaling

(17)

A numerical algorithm for computing the eigenvalues of the Dirac operator

In principle one has to look for the minima of the Rayleigh quotient ((H 0 + V ) ψ, ψ)

(ψ, ψ)

on “well chosen” subspaces of 4-spinors on which the above quotient is bounded from below. Direct approaches may face serious numerical difficulties. Our method is based on finding the best constant λ in the generalized Hardy inequality

Z

R

³

| R φ | ²

λ + 1 − V dx + Z

R

³

V | φ | ² dx + (1 − λ) Z

R

³

| φ | ² dx ≥ 0 ∀ φ

(18)

To do this, we minimize λ = λ(ϕ), given by Z

R

³

| (σ · ∇ )ϕ | ²

1 − V + λ + (1 + V ) | ϕ | ²

dx − λ Z

R

³

| ϕ | ² dx = 0

w.r.t. ϕ. The discretized version of this equation on a finite dimensional space E n of dimension n of 2-spinor functions is

A ⁿ (λ) x n · x n = 0 ,

where x n ∈ E n and A ⁿ (λ) is a λ-dependent n × n matrix. If E n is generated by a basis set { ϕ i , . . . ϕ n } , the entries of the matrix A ⁿ (λ) are the numbers

Z

R

³

(σ · ∇ ) ϕ i , (σ · ∇ ) ϕ j

1 − V + λ + (1 − λ + V ) (ϕ i , ϕ j )

! dx

The matrix is monotone decreasing in λ. The ground state energy will then be approached from above by the unique λ for which the first eigenvalue of A ⁿ (λ) is zero.

(19)

The matrix A ⁿ (λ) is selfadjoint and has therefore n real eigenvalues:

λ 1,n (λ) < λ 2,n (λ) < ... λ 1,n (λ) which are all monotone decreasing functions of λ.

The equation

A ⁿ (λ) x n · x n = 0 ,

means that x n is an eigenvector associated to the eigenvalue λ k ,n (λ) = 0, for some k .

Minimizing λ is therefore equivalent to compute λ 1,n as the solution of the equation

λ 1,n (λ) = 0

The uniqueness of such a λ comes from the monotonicity.

(20)

Moreover, if the approximating finite spaces (E n ) n ∈ N is an increasing family which generates H ¹ ( R ³ ; C ² ), since for a fixed λ

λ 1,n (λ) ց λ 1 (λ) as n → + ∞ we also have

λ 1,n ց λ 1 as n → + ∞

This method has been tested on diatomic configurations (corresponding to a cylindrical symmetry) with B-splines basis sets. Approximations from above of the other eigenvalues of the Dirac operator, or excited levels, can also be computed by requiring successively that the second, third,...

eigenvalues of A ⁿ (λ) are equal to zero

(21)

1 2 3 4 5

c1

c2

c3

c4

c5

¯ µ( ) λ

λ λ λ λ λ

λ

Figure: Each eigenvalue µ

i

(λ) of A (λ), considered as a function of λ, is monotone decreasing. By looking for the zeros of the non-continuous function λ 7→ µ(λ) = inf ¯

i

|µ

i

(λ)|, we obtain an efficient algorithm to compute all eigenvalues of the Dirac operator in the gap (−1, 1) and the corresponding eigenfunctions. The ground state of course corresponds to the smallest zero of

¯

µ(λ) in (−1, 1)

(22)

0

z s= 0.4

0

z

s

Figure: Ground state of Th

⁸⁹⁺

corresponding to Z = 90, one atom, computed with s

_max

= 10, z

_max

= 10, h = 0.4

(23)

0 s= 0.20

z 0

s

z

Figure: Ground state of H

⁺

2

corresponding to Z = 1, two atoms, computed with s

_max

= 700, z

_max

= 820, h = 20

(24)

0 s= 0.4

z 0

z

s

Figure: Ground state of Th

¹⁷⁹⁺

2

corresponding to Z = 90, two atoms, computed with s

_max

= 10, z

_max

= 12, h = 0.4

(25)

Two important issues

1

The issue of spurious eigenvalues: [Lewin-S´er´e (2009)] Spectral pollution and how to avoid it

2

The issue of the self-adjointness of the Dirac-Coulomb operator up to ν = 1: [Esteban-Loss (2007)] Self-adjointness for Dirac operators via Hardy-Dirac inequalities

(26)

General min-max results, sign-changing potentials

Let H be a Hilbert space with scalar product ( · , · ), and

A : D(A) ⊂ H → H be a self-adjoint operator. We denote by F (A) the form-domain of A. Let H ⁺ , H − be two orthogonal Hilbert subspaces of H such that H = H ⁺ ⊕H −

We denote by Λ ⁺ , Λ ⁻ the projectors on H + , H − . We assume the existence of a core F (i.e. a subspace of D(A) which is dense for the norm k·k D(A) ), such that :

(i) F + = Λ ⁺ F and F ₋ = Λ ⁻ F are two subspaces of F (A).

(ii ⁻ ) a ⁻ := sup _x

₋

_∈ _F

₋

_\{ ₀ _} ^(x _kx

⁻₋

^,Ax _k

2⁻

⁾

H

< + ∞ . (ii ⁺ ) a ⁺ := inf _x

₊

_∈ _F

₊

_\{ ₀ _} ^(x _k

⁺

_x ^,Ax

⁺

⁾

+

k

²_H

> −∞ .

(27)

We consider the two sequences of min-max and max-min levels (λ ⁺ _k ) k ≥ 1

and (λ ⁻ _k ) k ≥ 1 defined by λ ⁺ _k := inf

V subspace of F

+

dim V =k

sup

x ∈ (V ⊕ F

−

) \{ 0 }

(x , Ax) k x k ²

H

λ ⁻ _k := sup

V subspace of F

−

dim V =k

x ∈ (V ⊕ inf F

+

) \{ 0 }

(x, Ax) k x k ²

H

Let b ⁻ := inf { σ ess (A) ∩ (a ⁻ , ∞ ) } , b ⁺ := sup { σ ess (A) ∩ ( −∞ , a ⁺ ) } (iii ⁻ ) k ₀ ⁺ := min { k ≥ 1 , λ ⁺ _k > a ⁻ }

(iii ⁺ ) k ₀ ⁻ := min { k ≥ 1 , λ ⁻ _k < a ⁺ } Theorem

If (i)-(ii ⁻ )-(iii ⁻ ) hold, for any k ≥ k ₀ ⁺ , either λ ⁺ _k is the (k − k 0 + 1)-th eigenvalue of A in the interval (a ⁻ , b ⁻ ) or it is equal to b ⁻ . If

(i)-(ii ⁺ )-(iii ⁺ ) hold, for any k ≥ k ₀ ⁻ , either λ ⁻ _k is the (k − k 0 + 1)-th eigenvalue of A (in reverse order) in the interval (b ⁺ , a ⁺ ) or it is equal to b ⁺

(28)

Consider a 1-parameter family of self-adjoint operators A τ := A 0 + τ V , τ ∈ [0, τ ¯ ] = I , V is a bounded scalar potential, A 0 : D(A 0 ) ⊂ H → H a self-adjoint operator. Let H + , H − , Λ ⁺ and Λ ⁻ be above Assume further that there is a space F ⊂ H such that, for all τ ∈ I , F is a core for A τ

and the following hypotheses hold:

(j) F + = Λ ⁺ F and F ₋ = Λ ⁻ F are two subspaces of F (A τ ) (jj ⁻ ) There is a ⁻ ∈ R such that sup _τ _∈I _,x

₋

_∈F

₋

_\{ ₀ _} ^(x

⁻

_k _x ^,A

^τ

^x

⁻

⁾

−

k

²_H

≤ a ⁻ (jj ⁺ ) There is a ⁺ ∈ R such that inf τ ∈I ,x

+

∈ F

+

\{ 0 } (x

+

,A

τ

x

+

)

kx

+

k

²_H

≥ a ⁺ λ ^τ,+ _k := inf

V subspace of F

+

dim V =k

sup

x ∈ (V ⊕ F

−

) \{ 0 }

(x , A τ ) k x k ²

H

λ ^τ, _k ⁻ := sup

V subspace of F

−

dim V =k

x ∈ (V ⊕ inf F

+

) \{ 0 }

(x, A τ ) k x k ²

H

a ⁻ ₁ := inf τ ∈I

h inf

σ(A τ ) ∩ (a ⁻ , + ∞ ) i a ⁺ ₁ := sup _τ _∈I h

sup

σ(A τ ) ∩ ( −∞ , a ⁺ ) i

(29)

A continuation principle...

Theorem

Under the above assumptions, if for some k ₀ ⁺ ≥ 1, λ ^0,+ _k

+

0

> a ⁻ and if a ⁻ ₁ > a ⁻ , for all k ≥ k ₀ ⁺ , the numbers λ ^τ,+ _k are either eigenvalues of A 0 + τ V in the interval (a ⁻ , b ⁻ ) or λ ^τ,+ _k = b ⁻

If for some k ₀ ⁻ ≥ 1, λ ^0, ⁻

k

₀⁻

< a ⁺ and a ⁺ ₁ < a ⁺ , for all k ≥ k ₀ ⁻ , the numbers λ ^τ, _k ⁻ are either eigenvalues of A 0 + τ V in the interval (b ⁺ , a ⁺ ) or λ ^τ, _k ⁻ = b ⁺

(30)

1 − 1 0

a ⁻ _ν = E ⁻ _ν

,1

a _ν = E _ν

+ + ,1

E _ν ⁺

,2

E _ν ⁺

,3

...

ν ...

E _ν ⁻

,3

E _ν ⁻

, 2

ν ₃ ν ₂ ν ₁

= E _ν

, +

ν , +

...

E ⁺

ν , = k ( n )

=

n n

(31)

Hardy inequalities

Joint work with:

Maria J. Esteban and Eric S´er´e Maria J. Esteban and Roberta Bosi Maria J. Esteban, J. Duoandikoetxea and L. Vega

(32)

Standard Hardy inequality

Let u ∈ H ¹ ( R ^N ) and consider Z

R

^N

∇ u + N − 2 2

x

| x | ² u

2 dx ≥ 0

Develop this square and integrate by parts using the identity

∇ ·

x

| x |

²

= ^N _| _x ⁻ _|

2

²

Z

R

^N

|∇ u | ² dx ≥ 1

4 (N − 2) ² Z

R

^N

| u | ²

| x | ² dx

It is optimal (take appropriate truncations of x 7→ | x | ⁻ ^(N ⁻ ^2)/2 ): the operator − ∆ − _|x| ^A

²

is nonnegative if and only if A ≤ ¹ 4 (N − 2) ² . Optimality has to be taken with care: improvements with l.o.t. in L ² by Brezis-Vazquez, in W ^1,q with q < 2 by Vazquez-Zuazua, and logarithmic terms by Adimurthi & al. and many other

N = 3 from now on

(33)

Hardy inequality for the operator P

Proposition

With the above notations, for any ϕ ∈ H ^1/2 ( R ³ , C ² ), Z

R

³

| (σ · ∇ )ϕ | ² 1 + _|x| ¹ dx +

Z

R

³

| ϕ | ² dx ≥ Z

R

³

| ϕ | ²

| x | dx

This is a consequence of the following inequality, which is slightly more general: for all ϕ ∈ H ^1/2 ( R ³ , C ² ) and all ν ∈ (0, 1],

ν Z

R

³

| ϕ | ²

| x | + p 1 − ν ²

Z

R

³

| ϕ | ² ≤ Z

R

³

| (σ · ∇ )ϕ | ²

ν

| x | + 1 + √

1 − ν ² + Z

R

³

| ϕ | ²

(34)

Replace ϕ(x) by ϕ

x µ

and let µ → 0.

Z

R

³

| ϕ | ²

| x | dx ≤ Z

R

³

| x | | (σ · ∇ )ϕ | ² dx for all ϕ ∈ H ^1/2 ( R ³ , C ² )

Actually, taking φ = | x | ^1/2 ϕ, this inequality has to be directly related to the standard Hardy inequality

Z

R

³

| ϕ | ²

| x | = Z

R

³

| φ | ²

| x | ² ≤ 4 Z

R

³

|∇ φ | ² = 4 Z

R

³

| x | ^1/2 ∇ φ + 1 2

x

| x | ^3/2 φ

2 Z

R

³

| ϕ | ²

| x | dx ≤ Z

R

³

| x | |∇ ϕ | ² dx

(35)

Connection with the spectrum of the Dirac operator

Let λ 1 (V ) be the lowest eigenvalue of H 0 + V in the gap ( − 1, 1) of the continuous spectrum of H 0 + V (under appropriate assumptions on V ) Let Ψ =

ϕ χ

be the corresponding eigenfunction

(H 0 + V )Ψ = λ 1 (V )Ψ means, for P = − i c (σ · ∇ )







P χ = (λ 1 (V ) − c ² − V )ϕ P ϕ = (λ 1 (V ) + c ² − V ) χ which can be solved by



 

 

χ = (λ 1 (V ) + c ² − V ) ⁻ ¹ P ϕ P

R ϕ λ

1

(V )+c

²

− V

= (λ 1 (V ) − c ² − V ) ϕ

(36)

Multiplying by ϕ and integrating with respect to x ∈ R ³ , we get : Z

R

³

| P ϕ | ²

λ 1 (V ) + c ² − V dx + Z

R

³

V | ϕ | ² dx + (c ² − λ 1 (V )) Z

R

³

| ϕ | ² dx = 0 Note that for any fixed φ

λ 7→

Z

R

³

| P φ | ²

λ + c ² − V dx + Z

R

³

V | φ | ² dx + (c ² − λ) Z

R

³

| φ | ² dx is monotone decreasing. We shall see that λ 1 (V ) and φ can be characterized as follows

λ 1 (V ) is the smallest λ for which Z

R

³

| P φ | ²

λ + c ² − V dx + Z

R

³

V | φ | ² dx + (c ² − λ) Z

R

³

| φ | ² dx ≥ 0 ∀ φ and ϕ is the corresponding optimal function.

The generalized Hardy inequality is recovered with λ = √ 1 − ν ² .

(37)

Other standard Hardy type inequalities

Define the spectral projectors:

Λ ⁺ = χ (0,+ ∞ ) (H 0 ) and Λ ⁻ = χ ( −∞ ,0) (H 0 ) Using the Fourier transform u(x) 7→ u(ξ), we get ˆ

H ˆ 0 = i α · ξ + β , H ˆ ₀ ² = | ξ | ² + 1 , H ₀ ² = − ∆ + 1

1) Using − ∆ ≥ ¹ ₄ _| _x ¹ _|

²

(Hardy inequality), | H 0 | = Λ ⁺ H 0 Λ ⁺ − Λ ⁻ H 0 Λ ⁻ satisfies

| H 0 | ≥ _| ^κ x | κ = ¹ ₂

Z α ≤ κ with α ⁻ ¹ = 137.037... means Z ≤ 68.

(38)

2) Kato’s inequality.

| H 0 | ≥ _| ^κ _x _| κ = _π ² = 0.63662...

Z α ≤ κ means Z ≤ 87. Optimal [Herbst]

3) An inequality for the Brown & Ravenhall operator [Burenkov, Evans, Perry, Siedentop, Tix]:

Λ ⁺

H 0 − _|x| ^κ

Λ ⁺ ≥ 0 κ =

2

²

π

+

^π₂

= 0.906037...

The above operator (l.h.s.) was introduced by Bethe and Salpeter.

κ is sharp. Z α ≤ κ means Z ≤ 124

(39)

Dirac operator: corresponding Hardy type inequality

Consider the splitting H = H ^f + ⊕ H ^f ₋ , with H ^f _± = Λ ^± H , where Λ ⁺ = χ (0,+ ∞ ) (H 0 ), Λ ⁻ = χ ( −∞ ,0) (H 0 ), i.e.

Λ ^± = 1 2

I ± H 0

√ 1 − ∆

Theorem (J.D., Esteban, S´er´e) If lim

| x |→ + ∞ V (x ) = 0 and − _|x| ^ν − c 1 ≤ V ≤ c 2 with ν ∈ (0, 1), c 1 , c 2 ≥ 0, c 1 + c 2 − 1 < √

1 − ν ² , then

c _k ^f (V ) = λ k (V ) ∀ k ≥ 1

(40)

Q _E,ν ^f (ψ + ):= k ψ + k ² _H

^1/2

− (ψ + , (E − V )ψ + ) +

Λ ⁻ | V | ψ + , Λ ⁻ ( √

1 − ∆ + E + | V | )Λ ⁻ − 1

Λ ⁻ | V | ψ +

≥ 0

∀ ψ + ∈ Λ ⁺ C ₀ ^∞ ( R ³ , C ⁴ )

if E ≥ c ₁ ^f (V )

Proposition

For all ν ∈ [0, 1], ψ + ∈ Λ ⁺ C ₀ ^∞ ( R ³ , C ⁴ ) ,

ν Z

R

³

| ψ + | ²

| x | dx + p 1 − ν ²

Z

R

³

| ψ + | ² dx

≤ R

R

³

(ψ + , √

1 − ∆ ψ + ) dx + ν ² Z

R

³

Λ ⁻

ψ +

| x |

, B ⁻ ¹ Λ ⁻ ψ +

| x |

dx

with B := Λ ⁻ √

1 − ∆ + _| ^ν _x _| + √ 1 − ν ²

Λ ⁻

(41)

Taking functions with support near the origin, we find, after rescaling and passing to the limit, a new homogeneous Hardy-type inequality. If

Λ ⁰ _± := 1 2

I ± α · ˆ p

| ˆ p |

, ˆ p := − i ∇

are the projectors associated with the zero-mass free Dirac operator, then for any ψ + ∈ Λ ⁰ ₊ C ₀ ^∞ ( R ³ , C ⁴ )

Z

R

³

| ψ + | ²

| x | dx ≤ Z

R

³

(ψ + , | ˆ p | ψ + ) dx + Z

R

³

Λ ⁰ ₋

ψ +

| x |

, (B ⁰ ) ⁻ ¹ Λ ⁰ ₋ ψ +

| x |

dx

with B ⁰ := Λ ⁰ ₋

| ˆ p | + _| ¹ _x _| Λ ⁰ ₋

(42)

Generalized Hardy inequality: an analytical proof

[J.D.-Esteban-Loss-Vega] Related works based on commutators of M.J.

Esteban, L. Vega, Adimurthi containing improvements like logarithmic terms.

Proposition

Let g be a bounded radial C ¹ function such that lim x → 0 | x | g (x) is finite.

Then for any ϕ ∈ H ^1/2 ( R ³ , C ² ),

Z

R

³

1 g | (σ · ∇ )ϕ | ² dx + Z

R

³

g | ϕ | ² dx ≥ 2 Z

R

³

| ϕ | ²

| x | dx

(43)

Let φ ∈ H ^1/2 ( R ³ , C ² ) and ε = ± 1. In the case of the generalized Hardy inequality, take g (x) = 1 + _|x| ¹ . From

Z

R

³

√ 1 g (σ · ∇ φ) + ε √ g

σ · x

| x | φ

2 dx ≥ 0, we get Z

R

³

1 g | σ · ∇ φ | ² dx + Z

R

³

g | φ | ² dx ≥ ε

φ, 1

√ g (σ · ∇ ), √ g

σ · x

| x |

φ

L

²

Let L = ix ∧ ∇ . A straightforward computation shows that

(σ · ∇ ),

σ · x

| x |

= 2

| x | (1 + σ · L) h 1

√ g (σ · ∇ ), √ g

σ · _|x| ^x i

= _2g ¹

∇ g · _|x| ^x

+ _|x| ² (1 + σ · L) , h √ 1 g σ ·

∇ − _| ^x x |

∂

∂r

, √ g

σ · _| ^x x |

i

= _| ² _x _| (1 + σ · L)

(44)

Lemma

The spectrum of (1 + σ · L) is Z \ { 0 } and L ² ( R ³ , C ² ) = H − ⊕ H + with H ± = P _± L ² ( R ³ , C ² ), P _± = ¹ ₂

1 ± _| ^1+σ _1+σ ^· _· ^L _L _|

. As a consequence, for any nonnegative radial function h,

if φ ∈ H ± , ± (φ, h (1 + σ · L) φ) _L

2

≥ Z

R

³

h | φ | ² dx

Let ϕ ∈ H ^1/2 ( R ³ , C ² ), ϕ _± = P _± ϕ. Apply Lemma ?? to ϕ + with ε = +1 (resp. to ϕ ₋ with ε = − 1), h = _| ¹ _x _| , ∇ ⊥ =

∇ − _| ^x _x _| _∂r ^∂ : Z

R

³

1 g | σ · ∇ ⊥ ϕ _± | ² dx + Z

R

³

g | φ _± | ² dx ≥ Z

R

³

2 | x | | ϕ _± | ² dx For any radial function h,

R

³

h | ϕ | ² dx = R

R

³

h | ϕ ₋ | ² dx + R

R

³

h | ϕ + | ² dx R

R

³

|∇ ϕ | ² dx ≥ R

R

³

|∇ ⊥ ϕ ₋ | ² dx + R

R

³

|∇ ⊥ ϕ + | ² dx

(45)

Hardy inequality and Dirac: logarithmic improvements

[J.D.-Esteban-Loss-Vega] For some continuous functions W > 1 a.e.

and constants R > 0 and C(R) ≤ 0, the improved inequality Z

R

³



 | σ · ∇ ϕ | ²

1 + ^W _| ⁽ _x ^|x| _| ⁾ + | ϕ | ²



 dx ≥ Z

R

³

W ( | x | )

| x | | ϕ | ² dx + C(R) Z

S

R

| ϕ | ² dµ R

holds for all ϕ ∈ H ¹ ( R ³ , C ² ). Here µ R is the surface measure induced by Lebesgue’s measure on the sphere S R := { x ∈ R ³ : | x | = R }

W _∞ (x) = 1 + 1 8

X ∞

k=1

X 1 ( | x | ) ² · · · X k ( | x | ) ² ,

where X 1 (s) := (a − log(s)) ⁻ ¹ for some a > 1, X k (s) := X 1 ◦ X _k− 1 ).

The functions X k and W ^∞ are well defined for | x | = s < e ^a ⁻ ¹

(46)

Theorem

Assume that for some R > 0 the improved inequality holds for every spinor ϕ ∈ C ₀ ^∞ ( R ³ , C ² ), where W is a radially symmetric continuous function from R ⁺ to R ⁺ . Assume moreover that W (0) > 0 and W is nondecreasing in a neighbourhood of 0 ⁺ . Then W (0) ≤ 1,

lim sup

s → + ∞ W (s)/s ≤ 1 and for all k ≥ 1,

lim inf

s → 0

⁺

W (s) − 1 − 1 8

X k

j=1

X ₁ ² (s) · · · X _j ² (s)

X ₁ ⁻ ² (s) · · · X _k+1 ⁻ ² (s) ≤ 1 8 As soon as W 6≡ 1, C(R) must be negative

(47)

Hardy inequalities for the Schr¨odinger operator and the IMS method

Our goal is to obtain estimates for the lowest eigenvalue of

− ∆ − µ V M (x), where V M (x ) = P M

k=1 1

|x−y

k

|

²

, M ≥ 2 and y 1 , . . . y M are M points of R ^N . Notice that the Hamiltonian is essentially self-adjoint if µ ≤ (N − 2) ² /4 − 1, otherwise one has to use the corresponding Friedrichs extension. Define d by

d := min

1 ≤ j 6 =k ≤ M | y k − y j | /2 Theorem

Consider µ ∈ (0, (N − 2) ² /4]. For any M ≥ 2, there is a nonnegative constant K M < π ² such that, for any u ∈ H ¹ ( R ^N ),

Z

|∇ u | ² dx + K M + (M + 1) µ d ²

Z

| u | ² dx ≥ µ Z

V M (x) | u | ² dx

(48)

Multipolar Hardy inequality for the Dirac operator

We consider an arbitrary configuration of poles y k ∈ R ³ , and define the following quantities :

a := ν M

d , b :=

1 − S(ν) 2

ν ² , c := 2 1 − S (ν ) ν ² , d ^∗ (ν) = 1

2 M ν c + π √ c , λ ^∗ (d, ν, M) := 1

c h

1 + p

c (a ² ν ⁻ ² − π ² d ⁻ ² − a) + 1 − a ² ν ⁻ ² i

− 1 − a 2 , with S(ν) = √

1 − ν ² and d := min _i6 =k | y i − y j | /2 Theorem

With the above notations, for all ν ∈ (0, 1), M ≥ 2, if d ≥ d ^∗ (ν), then for all φ ∈ H ¹ ( R ³ , C ² ) we have

Z

R

³

| ~σ · ∇ φ | ² 1 + λ ^∗ + ν W M

dx + (1 − λ ^∗ ) Z

R

³

| φ | ² dx ≥ ν Z

R

³

W M | φ | ² dx

We observe that λ ^∗ > − 1 for every d

J. Dolbeault

≥

Dirac equation: variational approach and applications

d ^∗ (ν) and that λ ^∗ = 0 for

(49)

Strong magnetic fields

Joint work with:

Maria J. Esteban and Michael Loss

(50)

Outline

Relativistic hydrogenic atoms in strong magnetic fields: min-max and critical magnetic fields

Characterization of the critical magnetic field Proof of the main result

A Landau level ansatz Numerical results

(51)

Relativistic hydrogenic atoms in strong magnetic fields

The Dirac operator for a hydrogenic atom in the presence of a constant magnetic field B in the x 3 -direction is given by

H B − ν

| x | with H B := α · 1

i ∇ + 1

2 B( − x 2 , x 1 , 0)

+ β ν = Z α < 1, Z is the nuclear charge number

The Sommerfeld fine-structure constant is α ≈ 1/137.037 The magnetic field strength unit is ^m _| _q

²

_| ^c _~

²

≈ 4.4 × 10 ⁹ Tesla 1 Gauss = 10 ⁻ ⁴ Tesla

(52)

[R.C. Duncan, Magnetars, soft gamma repeaters and very strong magnetic fields]

(53)

The ground state energy λ 1 (ν, B) is the smallest eigenvalue in the gap As B ր , λ 1 (ν, B) → − 1: we define the critical magnetic field as the field strength B(ν) such that “λ 1 (ν, B(ν)) = − 1”

[J.D., Esteban, Loss, Annales Henri Poincar´e 2007]

Non perturbative estimates based on min-max formulations Theorem

For all ν ∈ (0, 1), there exists a constant C > 0 such that

4 5 ν ² ≤ B(ν ) ≤ min

18 π ν ²

[3 ν ² − 2] ² ₊ , e ^C/ν

²

Relativistic lowest Landau level

ν lim → 0 ν log(B(ν)) = π

(54)

Magnetic Dirac Hamiltonian

H B ψ − _| ^ν _x _| ψ = λ ψ is an equation for complex spinors ψ = ^φ _χ where φ, χ ∈ L ² ( R ³ ; C ² ) are the upper and lower components

P B χ + φ − ν

| x | φ = λ φ P B φ − χ − ν

| x | χ = λ χ with P B := − i σ · ( ∇ − i A B (x))

A B (x) := B 2





− x 2

x 1

0 

 , B(x) :=



 0 0 B





(55)

Min–max characterization of the ground state energy

If ψ = ^φ _χ

is an eigenfunction with eigenvalue λ, eliminate the lower component χ and observe

0 = J[φ, λ, ν, B] :=

Z

R

³

| P B φ | ²

1 + λ + _| ^ν _x _| + (1 − λ) | φ | ² − ν

| x | | φ | ²

! d ³ x

The function λ 7→ J [φ, λ, ν, B] is decreasing: define λ = λ[φ, ν, B] to be the unique solution to

either J [φ, λ, ν, B] = 0 or − 1 Theorem

Let B ∈ R ⁺ and ν ∈ (0, 1). If − 1 < inf φ ∈ C

₀^∞

( R

³

, C

²

) λ[φ, ν, B] < 1, λ 1 (ν, B) := inf

φ λ[φ, ν, B ]

is the lowest eigenvalue of H B − _|x| ^ν in the gap of its continuous spectrum, ( − 1, 1)

(56)

Characterization of the critical magnetic field

Using the scaling properties, we find an eigenvalue problem which characterizes the critical magnetic field

(57)

Notations

Magnetic Dirac operator with Coulomb potential ν/ | x | H B :=

I − ν/ | x | − i σ · ( ∇ − i A)

− i σ · ( ∇ − i A) − I − ν/ | x |

(1) where A is a magnetic potential corresponding to B, and I and σ k are respectively the identity and the Pauli matrices

I =

1 0 0 1

, σ 1 =

0 1 1 0

, σ 2 =

0 − i i 0

, σ 3 =

1 0 0 − 1

(2) Let B = (0, 0, B), A = A B . For any x = (x 1 , x 2 , x 3 ) ∈ R ³ , define

P B := − i σ · ( ∇ − i A B (x)) , A B (x ) := B 2





− x 2

x 1

0 

 (3)

(58)

Consider the functional J[φ, λ, ν, B] :=

Z

R

³

| P B φ | ²

1 + λ + _| ^ν _x _| + (1 − λ) | φ | ² − ν

| x | | φ | ²

!

d ³ x (4) on the set of admissible functions

A (ν, B) := { φ ∈ C ₀ ^∞ : k φ k L

²

= 1, λ 7→ J[φ, λ, ν, B] changes sign in ( − 1, + ∞ ) } λ = λ[φ, ν, B] is either the unique solution to J[φ, λ, ν, B] = 0 if

φ ∈ A (ν, B)

λ 1 (ν, B) := inf

φ ∈A (ν,B) λ[φ, ν, B] (5)

The critical magnetic field is defined by B (ν ) := inf

B > 0 : lim inf

b ր B λ 1 (ν, b) = − 1

(6)

(59)

Auxiliary functional E ^B,ν [φ] :=

Z

R

³

| x |

ν | P B φ | ² d ³ x − Z

R

³

ν

| x | | φ | ² d ³ x (7)

⇐⇒ E B ,ν [φ] + 2 k φ k ² _L

²

₍ R

³

) = J[φ, − 1, ν, B]

Scaling invariance

E ^B,ν [φ B ] = √

B E ^1,ν [φ] φ B := B ^3/4 φ B ^1/2 x

(8) We define

µ(ν) := inf

0 6≡ φ ∈ C

₀^∞

( R

³

)

E 1,ν [φ]

k φ k ² _L

²

(R

³

)

(9) Formally : − 1 = λ 1 (ν, B(ν )), inf

0 6≡ φ ∈ C

₀^∞

( R

³

) J [φ, − 1, ν, B] = 0

= ⇒ p

B(ν) µ(ν ) + 2 = 0

(60)

Main result

Theorem

For all ν ∈ (0, 1),

µ(ν) := inf

0 6≡ φ ∈ C

₀^∞

(R

³

)

E 1,ν [φ]

k φ k ² L

²

(R

³

)

is negative, finite,

B(ν) = 4

µ(ν) ² (10)

and B(ν) is a continuous, monotone decreasing function of ν on (0, 1)

(61)

Proof of the main result

Preliminary results Proof

(62)

Preliminary results

(ν, φ) 7→ ν E ^1,ν [φ] = Z

R

³

| x | | P 1 φ | ² d ³ x − Z

R

³

ν ²

| x | | φ | ² d ³ x (11) is a concave, bounded function of ν ∈ (0, 1), for any φ ∈ C ₀ ^∞ ( R ³ )

φ(x) :=

r B

2 π e ⁻

^B⁴

⁽ ^| ^x

¹

^|

²

⁺ ^| ^x

²

^|

²

⁾ f (x 3 )

0 ∀ x = (x 1 , x 2 , x 3 ) ∈ R ³ (12) f ∈ C ₀ ^∞ ( R , R ) such that f ≡ 1 for | x | ≤ δ, δ > 0, and k f k

_{L2(R+ )}

= 1

E B,ν [φ] ≤ C 1

ν + C 2 ν − C 3 ν log B (13) φ 1/B (x) = B ⁻ ^3/4 φ B ⁻ ^1/2 x

, E 1,ν [φ 1/B ] = B ⁻ ^1/2 E B,ν [φ] < 0 Lemma

On the interval (0, 1), the function ν 7→ µ(ν) is continuous, monotone decreasing and takes only negative real values

(63)

Proofs

B ˜ (ν ) = sup

B > 0 : inf

φ

E ^B,ν [φ] + 2 k φ k ² L

²

(R

³

)

≥ 0

= 4

µ(ν) ² (14) E B ,ν [φ] + 2 k φ k ² L

²

( R

³

) ≥ J [φ, λ 1 (ν, B), ν, B ] ≥ J[φ, λ[φ, ν, B], ν, B] = 0

(15)

= ⇒ B(ν ) ≤ B(ν) ˜

(64)

Proofs

Let B = B (ν ) and consider (ν n ) n ∈ N such that ν n ∈ (0, ν), lim n →∞ ν n = ν,

λ ⁿ := λ 1 (ν n , B ) > − 1 and lim n →∞ λ ⁿ = − 1

Let φ n be the optimal function associated to λ ⁿ : J[φ n , λ ⁿ , ν n , B] = 0 Let χ ≥ 0 on R ⁺ such that χ ≡ 1 on [0, 1], 0 ≤ χ ≤ 1 and χ ≡ 0 on [2, ∞ ), and χ n (x) := χ( | x | /R n ), lim n →∞ R n = ∞ , φ ˜ n := φ n χ n

P B φ n = (P B φ ˜ n ) χ n

| {z }

=a

+

− (P 0 χ n ) φ n

| {z }

=b

(16)

Using | a | ² ≥ ^|a+b| _1+ε

²

− ^|b| _ε

²

, we get

| P B φ n | ² ≥ | (P B φ ˜ n ) χ n | ²

1 + ε n − | (P 0 χ n ) φ n | ² ε n

(17)

(65)

1) The function ˜ φ n is supported in the ball B(0, 2 R n ): with µ n := (1 + ε n )

2(1 + λ ⁿ )R n + ν n , 1

1 + ε n

Z

R

³

| P B φ ˜ n | ²

1 + λ ⁿ + _| ^ν _x

ⁿ

_| d ³ x ≥ 1 µ n

Z

R

³

| x | | P B φ ˜ n | ² d ³ x (18) 2) Supp(P 0 χ n ) ⊂ B(0, 2 R n ) \ B (0, R n ), | P 0 χ n | ² ≤ κ R _n ⁻ ²

1 ε n

Z

R

³

| (P 0 χ n ) φ n | ²

1 + λ ⁿ + _|x| ^ν

ⁿ

d ³ x ≤ κ ε n R n

(1 + λ ⁿ ) R n + ν n Z

R

³

| φ n | ² d ³ x (19)

(66)

With η n = κ/(ε n R n ((1 + λ ⁿ ) R n + ν n )) + ν n /R n , we can write 0 = J [φ n , λ ⁿ , ν n , B] ≥ 1

µ n

Z

R

³

| x | | P B φ ˜ n | ² d ³ x − ν n

Z

R

³

| φ ˜ n | ²

| x | d ³ x +(1 − λ ⁿ − η n )

Z

R

³

| φ ˜ n | ² d ³ x Let ˜ ν n = √ µ n ν n

1 ˜ ν n

Z

R

³

| x | | P B φ ˜ n | ² d ³ x − ν ˜ n

Z

R

³

| φ ˜ n | ²

| x | d ³ x + 2 Z

R

³

| φ ˜ n | ² d ³ x

≤

2 − r µ n

ν n

(1 − λ ⁿ − η n ) Z

R

³

| φ ˜ n | ² d ³ x → 0

B(ν ˜ ^′ ) ≤ B = B(ν) ∀ ν ^′ > ν (20)

By continuity, B(ν) ˜ ≤ B(ν) ⊔ ⊓

(67)

Dirac equation: variational approach and applications