Jean-Marie Barbaroux

Josef Mehringer

Edgardo Stockmeyer

Amal Taarabt

DYNAMICAL LOCALIZATION OF DIRAC PARTICLES IN ELECTROMAGNETIC FIELDS WITH DOMINATING MAGNETIC POTENTIALS

We consider two-dimensional massless Dirac operators in a radially symmetric electromagnetic field. In this case the fields may be described by one-dimensional electric and magnetic potentials V and A. We show dynamical localization in the regime when lim r→∞ |V |/|A| < 1, where dense point spectrum occurs.

Introduction

Graphene is a two dimensional material consisting of carbon atoms arranged in a honeycomb lattice which was isolated in 2004 [START_REF] Novoselov | Electric field effect in atomically thin carbon films[END_REF]. Behind its remarkable properties such as Klein tunneling and finite minimal conductivity [START_REF] Katsnelson | Chiral tunnelling and the Klein paradox in graphene[END_REF] stays the fact that at low excitations energies the dynamics of charge carriers is described by the massless two-dimensional Dirac operator [START_REF] Castro Neto | The electronic properties of graphene[END_REF]. For technological devices based on graphene one needs the ability to confine and control the mobility of charge carriers. However, confining Dirac particles is not an easy task due the so-called Klein effect, where particles are able to penetrate electric potential walls [START_REF] Katsnelson | Chiral tunnelling and the Klein paradox in graphene[END_REF] with very little reflexion index. In [START_REF] Giavaras | Magnetic field induced confinement-deconfinement transition in graphene quantum dots[END_REF] it was argued that in presence of rotational symmetric electric and magnetic fields one could confine or deconfine Dirac particles by manipulating the strength of the fields at infinity, i.e., far away from the sample. Our main result is a dynamical statement on this effect and a continuation of a recent work [START_REF] Mehringer | Ballistic dynamics of Dirac particles in electro-magnetic fields[END_REF] by two of the present authors. Before presenting the result let us explain this phenomenon with more mathematical details.

Denote by H the two-dimensional massless Dirac operator coupled to a radially symmetric field E = E r on the plane and a radially symmetric transversal magnetic field B. If the fields are sufficiently regular H is a self-adjoint operator densely defined in L 2 (R 2 , C 2 ) acting as

(1) H = σ • (-i∇ -A) + V,
where V : R 2 → R is the electric potential satisfying

V (x) = - |x| 0 E(s)ds ≡ V (r)
Date: April 14 th 2015. 2010 Mathematics Subject Classification. Primary 81Q10; Secondary 46N50, 81Q37, 34L10, 47A10.

Key words and phrases. Dirac operator, graphene, dynamical localization, dense point spectrum.

(abusing notation, we write V (r) to denote V (x), where r = |x| ∈ [0, ∞) is the standard radial variable). Here σ = (σ 1 , σ 2 ) is a matrix-valued vector whose components are the first two Pauli matrices

σ 1 = 0 1 1 0 , σ 2 = 0 -i i 0 .
The vector potential A = (A 1 , A 2 ) : R 2 → R 2 generates the magnetic field B, with

B = ∂ 1 A 2 -∂ 2 A 1 .
We choose the rotational gauge, i.e., we set

A(x) := 1 r A(r) -x 2 x 1 , with A(r) = 1 r r 0 B(s)sds.
We note that, besides some local regularity requirements for (V, A), H is essentially self-adjoint on C ∞ 0 (R 2 , C 2 ) independently of the growth rate of A and V (see [START_REF] Chernoff | Schrödinger and Dirac operators with singular potentials and hyperbolic equations[END_REF]). In this setting there exists a unitary transform [START_REF] Thaller | The Dirac equation[END_REF] (see also [START_REF] Könenberg | Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot[END_REF]Section 6]

) (2) U : L 2 (R 2 , C 2 ) -→ L 2 (R + , C 2 ) ⊗ ℓ 2 (Z) ,
such that the operator H can be written as a direct sum of operators on the half-line

(3) UHU * = j∈Z h j , where (4)
h j = -iσ 2 ∂ r + σ 1 (A - mj r ) + V on L 2 (R + , C 2 ),
with m j = j + 1 2 for j ∈ Z. Clearly, the spectra of H and h j are related through

σ(H) = j∈Z σ(h j ), σ c (H) = j∈Z σ c (h j ), and σ pp (H) = j∈Z σ pp (h j ). ( 5
)
For the operators h j we have the following properties, assuming sufficiently regular fields:

If A(r) → ∞ as r → ∞ and lim r→∞ V (r) A(r) < 1, (6)
then the spectrum of h j is discrete for each j ∈ Z (see [START_REF] Mehringer | Confinement-deconfinement transitions for two-dimensional Dirac particles[END_REF]Proposition 1] for the precise regularity conditions). In contrast, if

V (r) → ∞ as r → ∞ and lim r→∞ A(r) V (r) < 1, (7)
then the spectrum of h j equals the whole real line and it is purely absolutely continuous [START_REF] Schmidt | Spherically symmetric Dirac operators with variable mass and potentials infinite at infinity[END_REF]Propsition 2]. This suggests delocalized particles in the regime given by [START_REF] Giavaras | Magnetic field induced confinement-deconfinement transition in graphene quantum dots[END_REF] and confined particles in the one given by [START_REF] Del Rio | Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization[END_REF]. However, the latter is not obvious since, for fields satisfying [START_REF] Del Rio | Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization[END_REF], H may have dense point spectrum (see [START_REF] Miller | Quantum magnetic hamiltonians with remarkable spectral properties[END_REF] and [START_REF] Thaller | The Dirac equation[END_REF]Theorem 7.10] for the case when B decays at infinity and [START_REF] Mehringer | Confinement-deconfinement transitions for two-dimensional Dirac particles[END_REF] for the case when B is not decaying at infinity). We recall that dense point spectrum may lead to non-trivial dynamics. In fact, in this case, it is only known that the wave-packet spreading is sub-ballistic [START_REF] Simon | Absence of ballistic motion[END_REF] (the result stated in [START_REF] Simon | Absence of ballistic motion[END_REF] is for Laplace-type operators but can easily be adapted for the Dirac case). Moreover, there are examples of systems with pure point spectrum where the spreading rate is arbitrarily close to the ballistic one [START_REF] Del Rio | What is localization?[END_REF][START_REF] Del Rio | Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization[END_REF] (see also [START_REF] Barbaroux | Universal lower bounds for quantum diffusion[END_REF]).

Concerning dynamical results we know that particles in electromagnetic fields satisfying [START_REF] Giavaras | Magnetic field induced confinement-deconfinement transition in graphene quantum dots[END_REF] behave ballistically, i.e., for any finite energy state ψ ∈ L 2 (R 2 , C 2 ) and κ > 0 one has (see [START_REF] Mehringer | Ballistic dynamics of Dirac particles in electro-magnetic fields[END_REF])

1 T ∞ 0 |x| κ/2 e -iHt ψ 2 dt ∼ T κ , for large T > 0.
The main result of this work is that under condition [START_REF] Del Rio | Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization[END_REF] the operator H exhibits dynamical localization, i.e, for any κ > 0, for any finite energy interval I, and for any state ψ ∈ L 2 (R 2 , C 2 ), with sufficient regularity in the angular variable (depending on κ; see [START_REF] Mehringer | Confinement-deconfinement transitions for two-dimensional Dirac particles[END_REF]), holds sup

t≥0 |x| κ/2 e -itH P I (H)ψ 2 < ∞,
where P I (H) is the spectral projection of H onto I.

Let us now state our assumptions and result more precisely:

Hypothesis 1. A, V ∈ C 1 (R + , R
) and they satisfy

|A(r)| → ∞ as r → ∞, (8)
lim r→∞ V (r) A(r) < 1, (9)
lim r→∞ A ′ (r) A 2 (r) = 0. ( 10
)
Recall that U (see [START_REF] Bargmann | On the number of bound states in a central field of force[END_REF]) is the unitary map that decomposes H in the direct sum of the operators h j . For a given ψ ∈ L 2 (R 2 , C 2 ) we write

Uψ = ⊕ j∈Z ϕ j , with ϕ j ∈ L 2 (R + , C 2 ). ( 11
)
Our main result is the following theorem.

Theorem 1.1. Let κ > 0, I ⊂ R be a bounded energy interval and let P I (H) be the spectral projection of H onto I. Assume that A and V fulfill Hypothesis 1 and

let ψ ∈ L 2 (R 2 , C 2 ) be a normalized state such that Uψ = ⊕ j∈Z ϕ j satisfies j∈Z |j| κ ϕ j 2 < ∞. ( 12
)
Then we have [START_REF] Miller | Quantum magnetic hamiltonians with remarkable spectral properties[END_REF] sup

t≥0 |x| κ/2 e -itH P I (H)ψ 2 < ∞.
We note that the condition ( 12) is related to regularity of the initial state ψ in the angular variable. Indeed, let r -1/2 ψ with ψ

∈ L 2 (R + × [0, 2π), C 2 ) be equal to ψ ∈ L 2 (R 2 , C 2 ) expressed in polar coordinates. Then, (12) follows if ψ ∈ H κ/2 ([0, 2π), L 2 (R + )) ⊕ H κ/2 ([0, 2π), L 2 (R + )). Here, for κ > 0 H κ/2 [0, 2π), L 2 (R + ) := {u ∈ L 2 ([0, 2π) × R + ), ℓ∈Z (1 + |ℓ|) κ ûℓ 2 L 2 (R + ) < ∞},
is the fractional Sobolev space on the torus [START_REF] Grubb | Distributions and operators[END_REF] and ûℓ is the ℓ-th Fourier coefficient of u with respect to the variable θ. To be more explicit, note that F ψ = ⊕ j ϕ j where,

for g ∈ L 2 ([0, 2π), C 2 ), (F g)(j) := 1 √ 2π 2π 0
M θ e -imj θ g(θ)dθ, and

M θ = e iθ/2 0 0 ie -iθ/2 .
We notice that

(F g)(j) = ĝ1 (j) iĝ 2 (j + 1)
.

The angular momentum operator J := -i∂ θ +σ 3 /2 satisfies F JF * = m j . Assuming for simplicity that κ = 2n with n ∈ N, we have

j∈Z |m j | 2n ϕ j 2 1/2 = J n ψ = (-i∂ θ + σ 3 2 ) n ψ ≤ n k=0 n k (-i∂ θ ) k ψ ≤ 2 n ψ H n ([0,2π),L 2 (R + )) 2
.

We can also derive a sufficient condition on ψ for ( 12) to hold. To avoid unnecessary complications, we stick to the case of even values of κ.

Since ∂ θ = -x 2 ∂x 1 + x 1 ∂x 2 , we have (∂ θ ) k ψ 2 = (-x 2 ∂ x1 + x 1 ∂ x2 ) k ψ 2 ≤ C k |α|≤k (1 + |x| 2 ) k 2 ∂ α x ψ 2 ,
where for the multi-index α = (α 1 , α 2 ) we used the notation

∂ α x = ∂ α1 x1 ∂ α2 x2 .
Hence, in Cartesian coordinates, condition [START_REF] Mehringer | Confinement-deconfinement transitions for two-dimensional Dirac particles[END_REF] holds if ψ belongs to the weighted Sobolev space

H κ/2 κ/2 (R 2 , C 2 ) := {g ∈ L 2 (R 2 , C 2 ) | (1 + |x| 2 ) κ 4 g(x) ∈ H κ/2 (R 2 , C 2 )}.

Proof of the main result

The strategy of the proof is as follows. We first rewrite the moment in (13) using the decomposition of H as a direct sum of h j 's and the representation ψ ≃ ⊕ ϕ j (see [START_REF] Novoselov | Electric field effect in atomically thin carbon films[END_REF]). The main idea consists in taking advantage of the discrete spectrum of the operator h j and the exponential decay of its eigenfunctions to compensate the growth of the moment. However, the decay is not uniform in j, it turns out (see (29) in Lemma 4.1) that the exponential decay estimate can only be derived outside an interval which grows with j. This is due to the fact that the term A -m j /r in h j can be controlled by a fraction of A only outside a ball of radius r j with r j |A(r j )| ∼ |m j |. Hence, we split the space R + × Z into two regions through the function f , given in Definition 2.2. In the first region, where the values of r|A(r)| are sufficiently large compared to those of j, we control the growth of the moment in r with the exponential decay of the eigenfunctions obtained in Section 4 by an Agmon-type argument. We also need to control the number of these eigenfunctions in the spectral region we consider. This latter bound is derived in Section 3 by using arguments due to Bargmann. In the second region where the values of j are large compared to r|A(r)|, the regularity assumption [START_REF] Mehringer | Confinement-deconfinement transitions for two-dimensional Dirac particles[END_REF] in the angular variable for our initial state yields a decay in the variable j which is used to control the growth of r κ j κ .

We now turn to the detailed proof of Theorem 1.1. Since Uψ = ⊕ j ϕ j we have ( 14)

|x| κ/2 e -itH P I (H)ψ 2 = j∈Z r κ/2 e -ithj P I (h j )ϕ j 2 .
We note that it is enough to consider only the sum for |j| > J 0 , for sufficiently large J 0 > 1. Indeed, let N j be the number of eigenvalues of h j in the interval I. Let ξ (k) j , k = 1, 2, . . . , N j denote the corresponding eigenfunctions of h j . Then, expanding P I (h j )ϕ j in terms of the ξ

(k) j , we have |j|≤J0 r κ/2 e -ithj P I (h j )ϕ j 2 ≤ |j|≤J0 Nj k=1 r κ/2 ξ (k) j 2 < ∞.
where the last inequality holds in view of Remark 4.2 below. To estimate the right hand side of ( 14) we split the integration j ∞ 0 into two regions characterized by [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF] r j ≡ r j (δ 0 ) :

= sup{r ∈ R + | |m j | ≥ δ 0 r|A(r)|},
for some δ 0 ∈ (0, 1) which is chosen in the proof of Lemma 4.1 below.

Remark 2.1. Note that since A is continuous and A(r) → ∞ as r → ∞ we have that r j < ∞ and that r j → ∞ as |j| → ∞. Moreover, the supremum in (15) is attained and hence

|m j | = δ 0 r j |A(r j )|. ( 16
)
Moreover, we note that

|m j | ≤ δ 0 r|A(r)|, r ≥ r j . ( 17
) Definition 2.2. Let θ ∈ C ∞ (R + , [0, 1]
) with θ(r) = 0 for r < 1 and θ(r) = 1 for r > 2. We set f (r, j) = f j (r) := θ(r/3r j ) and f c j := 1 -f j . Proof of Theorem 1.1. We have j∈Z r κ/2 e -ithj P I (h j )ϕ j

2 ≤ 2 j∈Z f j r κ/2 e -ithj P I (h j )ϕ j 2 + f c j r κ/2 e -ithj P I (h j )ϕ j 2 . ( 18
)
We first estimate the second term of the right hand side of (18) using the regularity in the angular variable for the initial state as given by [START_REF] Mehringer | Confinement-deconfinement transitions for two-dimensional Dirac particles[END_REF]. In what follows we pick J 0 > 1 so large that |A(r)| > 1 for all r > r j and |j| > J 0 . In particular, we have that

r j < |m j |/δ 0 . ( 19
)
Using this and the support properties of f c j , we have

|j|>J0 f c j r κ/2 e -ithj P I (h j )ϕ j 2 ≤ |j|>J0 f c j (6r j ) κ/2 e -ithj P I (h j )ϕ j 2 ≤ |j|>J0 (6|m j |/δ 0 ) κ e -ithj P I (h j ) 2 ϕ j 2 < ∞ , (20)
where we used [START_REF] Mehringer | Confinement-deconfinement transitions for two-dimensional Dirac particles[END_REF] in the last bound.

We now estimate the first term in the right hand side of [START_REF] Thaller | The Dirac equation[END_REF]. For |j| > J 0 we compute

f j r κ/2 P I (h j )e ithj ϕ j 2 ≤ sup φ =1   Nj k=1 f j r κ/2 ξ (k) j , φ ξ (k) j   2 ≤ Nj k=1 f j r κ/2 ξ (k) j 2 sup φ =1 Nj l=1 | ξ (l) j , φ | 2 = Nj k=1 f j r κ/2 ξ (k) j 2 .
Consider the function fj defined at the beginning of Section 4 below. Since f j = f j fj we have, choosing also J 0 > J 2 (see Lemma 4.1)

f j r κ/2 ξ (k) j ≤ f j r κ/2 e -γ̺ e γ̺ fj ξ (k) j ≤ f j r κ/2 e -γ̺ C r j e γ̺(2rj ) ,
where ρ(r) = r 0 |A(s)|ds is the exponential weight defined in Lemma 4.1. Since r κ/2 e -γ̺ decays monotonically at infinity, we may choose J 0 > 1 to be so large that the supremum of f j r κ/2 e -γ̺ is bounded above by (3r j ) κ/2 e -γ̺(3rj ) . Hence

f j r κ/2 ξ (k) j ≤ C r j (3r j ) κ/2 e -γ(̺(3rj)-̺(2rj )) .
Note that due to ( 17)

̺(3r j ) -̺(2r j ) = 3rj 2rj r|A(r)| dr r ≥ |m j | δ 0 ln( 3 2 ) > |m j | 3δ 0 ,
for |j| large enough. Thus, using that r j ≤ |m j | and Lemma 3.1, we get for > J 2

f j r κ/2 P I (h j )e ithj ϕ j 2 ≤ C 2 Nj k=1 e -γ |m j | 3δ 0 (3|m j |) κ/2 2 ≤ 3 κ C 2 C I |m j | κ+1 ln|m j | e -γ 2|m j | 3δ 0 .
Since the last bound is summable for |j| = |m j -1/2| > J 0 we get the expected result.

Estimate on the number of eigenvalues of h j

Let T be a self-adjoint operator on a Hilbert space H with purely discrete spectrum. We set for an interval I ⊂ R N I (T ) := dim P I (T )H, i.e. N I (T ) denotes the number of eigenvalues of T in I counted with multiplicity. Lemma 3.1 (Bound on the number of eigenvalues for h j ). There is a J 1 > 1 such that for any E > 0 there is a constant C E > 0 so that

(21) N [-E, E] (h j ) ≤ C E |m j | ln|m j |, for |j| ≥ J 1 .
Proof. We first note that

h j = -iσ 2 ∂ r + σ 1 A(r) - mj r + V (r) is essentially self-adjoint on C ∞ 0 (R + , C 2 ).
In addition, we obtain by the spectral theorem

N [-E, E] (h j ) = N [0, E 2 ] (h 2 j
). In the sense of quadratic forms on C ∞ 0 (R + , C 2 ) we obtain for any ǫ ∈ (0, 1) the estimate 9) and [START_REF] Könenberg | Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot[END_REF] we have that δA 2 -σ 3 A ′ and ǫA 2 -V 2 /ǫ are positive outside a large ball B for ǫ ∈ (0, 1) sufficiently close to 1. Let

h 2 j ≥(1 -ǫ) -iσ 2 ∂ r -σ 1 mj r + σ 1 A(r) 2 + 1 -1 ǫ V 2 (r) =(1 -ǫ) -iσ 2 ∂ r -σ 1 mj r 2 + A 2 (r) -1 ǫ V 2 (r) -σ 3 A ′ (r) - 2mj r A(r) . Let δ := (1 -ǫ)/2. Due to (
C := V 2 /ǫ + σ 3 A ′ L ∞ (B)
. Then we find

h 2 j ≥ (1 -ǫ) -iσ 2 ∂ r -σ 1 mj r 2 + δA 2 (r) -C - 2|mj| r |A(r)| . We write h 2 j -E 2 ≥ (1 -ǫ) -iσ 2 ∂ r -σ 1 mj r 2 + W j ,
where

W j (r) = δA 2 (r) -C -(1 -ǫ) -1 E 2 - 2|mj| r |A(r)|. Let R j := r j (δ/4
), where r j (•) is defined in [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]. This yields, |m j | ≤ δ 4 r|A(r)| for all r > R j . Moreover, we may pick J 1 so large that (recall that R j → ∞ as |j| → ∞)

δ 2 A 2 (r) -C -(1 -ǫ) -1 E 2 > 0, for all r > R j , |j| > J 1 . Thus , W j 1 (Rj ,∞) ≥ 0 and h 2 j -E 2 ≥ (1 -ǫ) -iσ 2 ∂ r -σ 1 mj r 2 + W j 1 (0,Rj ] . (22) Define D j := {r ∈ (0, R j ) | |m j | ≥ δ 4 r|A(r)|}. (23) Note that if r ∈ (0, R j ) ∩ (R + \ D j ) then δ 2 A(r) 2 - 2|mj| r |A(r)| ≥ 0. Hence we have h 2 j -E 2 ≥ (1 -ǫ) -iσ 2 ∂ r -σ 1 mj r 2 + W < j , (24)
where

W < j (r) := (δA 2 (r) - 2|mj| r |A(r)|)1 Dj -(C + (1 -ǫ) -1 E 2 )1 (0,Rj ] . (25)
An application of the min-max principle leads to

N [0,E 2 ] h 2 j = N (-∞,0] h 2 j -E 2 ≤ N (-∞,0] -iσ 2 ∂ r -σ 1 mj r 2 + W < j . A direct computation shows that -iσ 2 ∂ r -σ 1 mj r 2 = -∂ 2 r + 1 r 2 m j (m j -σ 3 ) = -∂ 2 r + 1 r 2 m j (m j -1) 0 0 -∂ 2 r + 1 r 2 m j (m j + 1)
.

Note that m j (m j ± 1) > 0 for |j| > J 1 . Using the generalized Bargmann estimate [START_REF] Bargmann | On the number of bound states in a central field of force[END_REF] (see also [15, Theorem XIII.9]) we obtain for

|m j | > 1/2 N (-∞,0] -∂ 2 r + 1 r 2 m j (m j -1) + W < j ≤ 1 2mj -1 ∞ 0 r|W < j (r)|dr if m j > 1 2 1 2|mj |+1 ∞ 0 r|W < j (r)|dr if m j < -1 2 and N (-∞,0] -∂ 2 r + 1 r 2 m j (m j + 1) + W < j ≤ 1 2mj +1 ∞ 0 r|W < j (r)|dr if m j > 1 2 1 2|mj |-1 ∞ 0 r|W < j (r)|dr if m j < -1 2 ,
and therefore

N (-∞,0] -iσ 2 ∂ r -σ 1 mj r 2 + W < j ≤ 1 |m j | -1/2 ∞ 0 r|W < j (r)|dr. ( 26
)
Now we estimate using the definition of

D j ∞ 0 r|W < j (r)|dr ≤ (C + (1 -ǫ) -1 E 2 )R 2 j 2 + Dj r δA 2 (r) - 2|mj | r |A(r)| dr ≤ (C + (1 -ǫ) -1 E 2 )R 2 j 2 + Dj 2|m j ||A(r)|dr. (27) Furthermore
2|m j ||A(r)|dr ≤ 2|m j | A L ∞ [0,1] + 8m 2 j δ ln(R j ). ( 28
)
Note that in view of Remark 2.1, and the fact that |A(r)| grows at infinity, we have for sufficiently large J 1

R j = 4|m j | δ|A(R j )| ≤ 4|m j | δ , |j| > J 1 .
This together with (26), (27), and (28) yields the result.

Exponential decay of eigenfunctions of h j

Let r j ≡ r j (δ 0 ) be given as in [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]. We note that δ 0 ∈ (0, 1) will be fixed throughout the proof of the next lemma. For the function θ as defined in Definition 2.2, we set f (r, j) = fj (r) := θ(r/r j ). Lemma 4.1. There exist γ > 0 and J 2 > 1 such that for all |j| > J 2 the following holds: Let ξ j ∈ L 2 (R + , C 2 ) be a normalized eigenfunction of h j with energy E ∈ I. Then, for some C > 0 (independent of j), Remark 4.2. It is clear from the proof that the exponential decay of the eigenfunctions of h j remains true for |j| ≤ J 2 , however, in this case we get a different constant in front of the exponential. Remark 4.3. Throughout the proof of Lemma 4.1 we use that h j and k j := h j -V are essentially self-adjoint operators on C ∞ 0 (R + , C 2 ) (see for instance [START_REF] Mehringer | Ballistic dynamics of Dirac particles in electro-magnetic fields[END_REF] and references therein). Moreover, we also use that V is a perturbation with respect to the magnetic Dirac operator k j in the sense that D(V ) ⊃ C ∞ 0 (R + , C 2 ) and there exists C such that

V ϕ ≤ C( k j ϕ + ϕ ) f or all ϕ ∈ C ∞ 0 (R + , C 2 ). Indeed, let ϕ ∈ C ∞ 0 (R + , C
) and χ R be a smooth characteristic function of a ball of radius R > 0. We set

χ c R = 1 -χ R and V = V < + V > where V < := V χ R and V > := V χ c R . We thus have V ϕ ≤ V < ϕ + V > ϕ ≤ C R ϕ + Aχ c R ϕ , since V < is
bounded and V is dominated by A at infinity according to assumption [START_REF] Katsnelson | Chiral tunnelling and the Klein paradox in graphene[END_REF]. Moreover, for R large enough, we use [START_REF] Könenberg | Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot[END_REF] and the identity k

2 j = -∂ 2 r + A 2 j - σ 3 A ′ j to write 1 2 Aχ c R ϕ ≤ k j χ c R ϕ ≤ (∇χ c R )ϕ + χ c R k j ϕ ≤ ∇χ c R ϕ + k j ϕ .
Proof of Lemma 4.1. In order to derive the Agmon-type estimates, we follow [START_REF] Könenberg | Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot[END_REF]. We set (30)

A j := (A - m j r ),
and we notice that |A j | ≥ (1 -δ 0 )|A| on the support of fj . Let ξ j be a normalized eigenfunction of h j associated to an energy E. We define (31) g j := e γ̺ǫ fj ξ j , where γ ∈ (0, 1) and

̺ ǫ = ̺ 1 + ǫ̺ such that ̺(r) = r 0 |A (
k j = -iσ 2 ∂ r + σ 1 A j = h j -V,
and we define (33) Q j := Re k j e γ̺ǫ g j , k j e -γ̺ǫ g j .

The task is to obtain bounds for Q j that will allow us to bound g uniformly in ǫ.

Lower bound. Notice that (34) [k j , e γ̺ǫ ] = -iσ 2 γ̺ ′ ǫ e γ̺ǫ , so that we rewrite Q j = Re e -γ̺ǫ k j e γ̺ǫ g j , e γ̺ǫ k j e -γ̺ǫ g j

= Re (k j -iγ̺ ′ ǫ σ 2 )g j , (k j + iγ̺ ′ ǫ σ 2 )g j = k j g j 2 -γ 2 ̺ ′ ǫ g j 2 .
Moreover, we have

(35) k 2 j = -∂ 2 r + A 2 j -σ 3 A ′ j .
In view of Remark 2.1 and (10) for any ǫ > 0 there exists J ǫ > 0 such that for all |j| > J ǫ one has g j , A ′ j g j ≤ ǫ g j , A 2 g j and therefore

(36) g j , (A 2 j -σ 3 A ′ j )g j ≥ ((1 -δ 0 ) 2 -ǫ) g j , A 2 g j . Now we drop the term -∂ 2 r of (35). This together with (36) yields

Q j ≥ ((1 -δ 0 ) 2 -ǫ) Ag j 2 -γ 2 ̺ ′ ǫ g j 2 ≥ ((1 -δ 0 ) 2 -ǫ) Ag j 2 -γ 2 ̺ ′ g j 2 = ((1 -δ 0 ) 2 -ǫ -γ 2 ) Ag j 2 . ( 37
)
Upper bound. We rewrite Q j = Re k j e γ̺ǫ g j , fj (E -V )ξ j + Re k j e γ̺ǫ g j , -iσ 2 f ′ j ξ j = Re e γ̺ǫ g j , fj (E -V ) 2 ξ j + Re e γ̺ǫ g j , k j , fj (E -V ) ξ j + Re e γ̺ǫ g j , -iσ 2 f ′ j (E -V )ξ j + Re e γ̺ǫ g j , k j , -iσ 2 f ′ j ξ j

= Re e γ̺ǫ g j , fj (E -V ) 2 ξ j + Re e γ̺ǫ g j , k j , -iσ 2 f ′ j ξ j , since Re e γ̺ǫ g j , -iσ 2 f ′ j (E -V )ξ j = Re e γ̺ǫ g j , [V, k j ] fj ξ j = 0. Furthermore, we use Re e γ̺ǫ g j , fj (E -V ) 2 ξ j = (E -V ) g j 2 .

In addition, we find some C > 0 such that | e γ̺ǫ g j , k j , -iσ 2 f ′ j ξ j | = | e γ̺ǫ g j , (-f ′′ j + 2σ 3 A j f ′ j )ξ j | ≤ C e γ̺(2rj ) r j ( 1 rj g j + A j g j )

≤ C e γ̺(2rj ) r j ( g j + (1 + δ 0 ) Ag j ),

where in the last inequality we use that r j > 1 (for sufficiently large |j|) and (30) together with the support properties of fj . We thus get Q j ≤ (E -V ) g j 2 + C e γ̺(2rj ) r j ( g j + (1 + δ 0 ) Ag j ). (38) Then, combining (37) and (38) we get for |j| > J ǫ (39) g j , (((1

-δ 0 ) 2 -ǫ -γ 2 )A 2 -(E -V ) 2 )g j ≤ C
e γ̺(2rj) r j ( g j + (1 + δ 0 ) Ag j ).

According to (9) and Remark 2.1 we may pick δ 0 , ǫ and γ so small that there are constants J δ0,ǫ,γ , c δ0,ǫ,γ > 0 such that |A| > 1 on the support of fj and, for all |j| > J δ0,ǫ,γ ,

g j , [((1 -δ 0 ) 2 -ǫ -γ 2 )A 2 -(E -V ) 2 ]g j ≥ c δ0,ǫ,γ Ag j 2 . ( 40
)
This together with (39) yields Ag j ≤ C c δ0,ǫ,γ e γ̺(2rj ) r j ( g j / Ag j + (1 + δ 0 )) ≤ C c δ0,ǫ,γ e γ̺(2rj) r j (2 + δ 0 ). (41)

The claim follows using the theorem of monotonic convergence for the limit ǫ → 0 of (41).

,

  Dj 2|m j ||A(r)|dr = Dj ∩(0,1) 2|m j ||A(r)|dr + Dj ∩(1,∞)

  Ae γ̺ fj ξ j ≤ C r j e γ̺(2rj ) (29)where for r ≥ 0, ̺(r) := r 0 |A(s)|ds.

Acknowledgments. The authors want to thank Rafael Benguria for useful discussions and remarks. J.-M.B has been supported by the project SQFT ANR-12-JS01-0008-01. J.M. has been supported by SFB-TR12 "Symmetries and Universality in Mesoscopic Systems" of the DFG. E.S. has been supported by Fondecyt (Chile) project 1141008 and Iniciativa Científica Milenio (Chile) through the Millenium Nucleus RC120002 "Física Matemática". A.T. has been supported by the Millenium Nucleus RC120002 "Física Matemática".

Jean-Marie Barbaroux, Aix-Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France, and Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France.

E-mail address: barbarou@univ-tln.fr Josef Mehringer, Mathematisches Institut, Ludwig-Maximilians-Universität, Theresienstraße 39, D-80333 München, Germany.

E-mail address: josef.mehringer@math.lmu.de

Edgardo Stockmeyer, Instituto de Física, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago 7820436, Chile.

E-mail address: stock@fis.puc.cl Amal Taarabt, Instituto de Física, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago 7820436, Chile.

E-mail address: ataarabt@fis.puc.cl