Jean-Marie Barbaroux

AND Horia Cornean
email: cornean@math.aau.dk

Edgardo Stockmeyer

SPECTRAL GAPS IN GRAPHENE ANTIDOT LATTICES

Keywords: Mathematics Subject Classification. Primary 81Q10; Secondary 46N50, 81Q37, 34L10, 47A10 Dirac operator, graphene

We consider the gap creation problem in an antidot graphene lattice, i.e. a sheet of graphene with periodically distributed obstacles. We prove several spectral results concerning the size of the gap and its dependence on different natural parameters related to the antidot lattice.

Introduction

Graphene, a two-dimensional material made of carbon atoms arranged in a honeycomb structure, has risen a lot of attention due to its many unique properties. Remarkably, charge carriers close to the Fermi energy behave as massless Dirac fermions. This is due to its energy band structure which exhibits two bands crossing at the Fermi level making graphene a gapless semimetal [START_REF] Castro Neto | The electronic properties of graphene[END_REF]. Many efforts have been carried out for the possibility of tuning an energy gap in graphene [START_REF] Dvorak | Bandgap opening by patterning graphene[END_REF].

The main physical motivation of our work is related to the so-called antidot graphene lattice [START_REF] Pedersen | Graphene antidot lattices: designed defects and spin qubits[END_REF], which consists of a regular sheet of graphene having a periodic array of obstacles well separated from each other. These obstacles can be thought, for instance, as actual holes in the graphene layer [START_REF] Pedersen | Graphene antidot lattices: designed defects and spin qubits[END_REF]. More generally, substrate induced obstacles [START_REF] Hunt | Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure[END_REF] or those created by doping or by mechanical defects have also been considered in the literature (see e.g., [START_REF] Dvorak | Bandgap opening by patterning graphene[END_REF] and references therein). It has been observed both experimentally and numerically that such an array causes a band gap to open up around the Fermi level, turning graphene from a semimetal into a gapped semiconductor (see e.g., [START_REF] Brun | Boron and nitrogen doping in graphene antidot lattices[END_REF], [START_REF] Hunt | Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure[END_REF], and [START_REF] Dvorak | Bandgap opening by patterning graphene[END_REF]).

In [START_REF] Fürst | Electronic properties of graphene antidot lattices[END_REF] and [START_REF] Brun | Electronic and optical properties of graphene antidot lattices: comparison of Dirac and tight-binding models[END_REF] there are given several proposals concerning the modelling of this phenomenon. In one of these proposals the authors replace the usual tight-binding lattice model by a two-dimensional massless Dirac operator, while a hole is modeled with the help of a periodic mass term. For a large mass term held fixed the authors numerically analyse how the gap appearing near the zero energy depends on the natural parameters of the antidot lattice, namely, the area occupied by one mass-insertion versus the area of the super unit-cell which contains only one such hole. For holes with armchair type of boundaries, this model is in very good agreement with tight-binding and density functional ab-initio calculations [START_REF] Brun | Electronic and optical properties of graphene antidot lattices: comparison of Dirac and tight-binding models[END_REF] (see also [START_REF] Brun | Boron and nitrogen doping in graphene antidot lattices[END_REF][START_REF] Pedersen | Graphene antidot lattice waveguides[END_REF][START_REF] Pedersen | Band gaps in graphene via periodic electrostatic gating[END_REF]). Moreover, the Dirac operator with a mass term varying in a superlattice has also been used to explain the gap appearing when a layer of graphene is placed on substrate of hexagonal boron nitride [START_REF] Hunt | Massive Dirac fermions and Hofstadter butterfly in a Van der Waals heterostructure[END_REF] (see also [START_REF] Song | Electron interactions and gap opening in graphene superlattices[END_REF] for the inclusion of electronic interaction).

In this article we consider the Dirac model with a periodic mass term and we estimate the size of the energy gap in terms of the strength and shape of the mass-insertion together with the natural parameters of the antidot lattice. Let us now formulate more precisely the problem we want to investigate. (1.1)

We use the standard notation for the Pauli matrices

σ 1 = 0 1 1 0 , σ 2 = 0 -i i 0 , σ 3 = 1 0 0 -1 .
We define the massless free Dirac operator in L 2 (R 2 , C 2 ) as

H 0 := 1 i σ • ∇ = 0 -i∂ 1 -∂ 2 -i∂ 1 + ∂ 2 0 . (1.2)
Let α ∈ (0, 1] be a dimensionless parameter and let L > 0 have dimensions of length.

In physical units, the operator describing a mass-periodically perturbed graphene sheet is given by

H(α, µ, L) = v F H 0 + σ 3 µ γ∈Z 2 χ x -γL αL , (1.3)
where is Plank's constant divided by 2π and v F is the Fermi velocity in graphene. Here µ 0 has dimensions of energy and represents the strength of the mass-insertion which is LZ 2 -periodic. We note that in order for the continuum Dirac model to hold one needs L to be much larger than the distance between the carbon atoms constituting graphene.

As it is well known, the spectrum of H 0 covers the whole real line. Our main interest in this paper is finding sufficient conditions that the function χ must satisfy in order to create a gap around zero in the spectrum of H and to estimate its size in terms of α, µ, and L. By making the scaling transformation x → Lx one gets that

H(α, µ, L) = v F L H α, µL v F . (1.4)
Note that µL v F is a dimensionless parameter. Here, for β > 0, we define the operator in

L 2 (R 2 , C 2 ) H(α, β) := H 0 + σ 3 β γ∈Z 2 χ x -γ α . (1.5)
This new operator is clearly periodic with respect to Z 2 and (just like H) it is self-adjoint on the first Sobolev space H 1 (R 2 , C 2 ) (see [START_REF] Thaller | The Dirac equation[END_REF]). Given a self-adjoint operator T , we denote by ρ(T ) its resolvent set. Here is the first main result of our paper.

Theorem 1.1. Assume Φ = 0. Then there exist two constants C > 0 and δ ∈ (0, 1) such that for all α ∈ (0, 1/2] and β > 0 obeying αβ < δ we have

-α 2 β(Φ -Cαβ), α 2 β(Φ -Cαβ) ⊂ ρ(H(α, β)).
Remark. Let us comment on the consequences of this result regarding the energy gap, E g , for the family H defined in (1.3). Let us define the area of the supercell A t := L 2 and A r := α 2 L 2 representing the area supporting one mass perturbation. In view of (1.4), Theorem 1.1 states that for α µL v F small enough and for Φ in (1.1) positive

E g µΦ α 2 = µΦ A r A t . (1.6)
Remarkably this estimate does not depend on the side L of the supercell. This is to be contrasted with the regime of µ → ∞ considered in [START_REF] Pedersen | Graphene antidot lattices: designed defects and spin qubits[END_REF] and [START_REF] Brun | Electronic and optical properties of graphene antidot lattices: comparison of Dirac and tight-binding models[END_REF] where it was found that, for α small enough (see e.g., Equation A.8 of [START_REF] Brun | Electronic and optical properties of graphene antidot lattices: comparison of Dirac and tight-binding models[END_REF]),

E g ∼ v F L A r A t . (1.7)
This latter regime can be mathematically investigated using the Dirac operator with infinite mass boundary conditions proposed in [START_REF] Berry | Neutrino billiards: time-reversal symmetry-breaking without magnetic fields[END_REF] (see [START_REF] Benguria | Self-adjointness of twodimensional Dirac operators on domains[END_REF] for its rigorous definition).

In the case Φ = 0 it is still possible to prove the existence of a gap opening at zero. The next result needs some assumptions on χ in terms of its Fourier coefficients

χ(m) = Ω e -2iπx•m χ(x)dx, m ∈ Z 2 .
(1.8)

Note that Φ = 0 means that χ(0, 0) = 0. In terms of the operator H in (1.5), this time we keep α = 1 and we make β smaller than certain constant times the L ∞ -norm of χ. Then the gap can still survive but it scales with β 3 instead of β. As a consequence, the gap for H has the following behaviour, for µL v F small enough,

E g µ µL v F 2 .
(1.9)

Theorem 1.2. Assume Φ = χ(0, 0) = 0, and at the same time:

m =0 m =0 m • m |m| 2 |m | 2 χ(m) χ(m ) χ(m -m ) = 0.
(1.10)

Then there exist two positive numerical constants β 0 and C such that for every 0

< β < β 0 / χ ∞ we have -Cβ 3 , Cβ 3 ⊂ ρ(H(1, β)).
Let us describe a particular class of potentials where assumption (1.10) holds true. Assume that χ(x) is of the form

χ(x) = N -10 |m| N +10 2 cos(2πm • x), N 1.
By construction, all the Fourier coefficients χ(m) = χ(-m) equal either 1 or 0. The non-zero coefficients are those for which m lies in an annulus with outer radius N + 10 and inner radius N -10. When N becomes large enough, the triples of vectors m, m and mm for which the Fourier coefficients in (1.10) are simultaneously non-zero form a triangle which "almost" coincides with an equilateral triangle with side-length equal to N . Here "almost" means that the angle between m and m is close to π/3 when N is large enough. Thus the scalar product m • m is positive whenever the Fourier coefficients are non-zero (provided N is large enough) and the double sum in (1.10) is also positive.

In the rest of the paper we give the proof of the two theorems listed above.

Proof of Theorem 1.1

Throughout this work we use the notation

f p = Ω |f (x)| p dx 1/p , p ∈ [1, ∞).
Note that our conditions on χ imply: In this subsection we start by presenting the main strategy of the proof of Theorem 1.1. It consists of a suitable application of the Feshbach inversion formula to the Bloch-Floquet fiber of H(α, β). The main technical ingredients are Lemmas 2.2 and 2.3 whose proof can be found in the next subsection. At the end of this subsection we present the proof of Theorem 1.1.

0 < Φ χ 1 χ 2 = 1. ( 2
Let S (R 2 , C) denote the Schwartz space of test functions. Consider the map

Ũ : S (R 2 , C) ⊂ L 2 (R 2 , C) -→ L 2 (Ω, L 2 (Ω, C)), ( ŨΨ)(x; k) := γ∈Z 2 e 2πik•(x+γ) Ψ(x + γ), x, k ∈ Ω.
It is well known (see [START_REF] Reed | Methods of Modern Mathematical Physics, volume 4: Analysis of Operators[END_REF]) that Ũ is an isometry in L 2 (R 2 , C) that can be extended to a unitary operator. We denote its unitary extension by the same symbol. We define the Bloch-Floquet transform as U := Ũ ⊗ 1 C 2 . Then we have that

UH(α, β)U * = ⊕ Ω h k (α, β)dk, h k (α, β) := (-i∇ PBC x -2πk) • σ + βχ α (x)σ 3 ,
where each fiber Hamiltonian h k (α, β) is defined in L 2 (Ω, C 2 ). Here ∇ PBC

x is the gradient operator with periodic boundary conditions and

χ α (x) := χ(x/α).
The spectra of H(α, β) and h k (α, β) are related through

σ(H(α, β)) = k∈Ω σ(h k (α, β)).
(2.12)

We will use the standard eigenbasis of ∇ PBC

x given by

ψ m (x) := e 2πim•x , m ∈ Z 2 , x ∈ Ω,
which is periodic and satisfies

-i∇ PBC x ψ m = 2πmψ m .
For m ∈ Z 2 define the projections

P m := |ψ m ψ m | ⊗ 1 C 2 and Q 0 := Id -P 0 . (2.13)
Lemma 2.1. Let α ∈ (0, 1) and β > 0. Then, for every k ∈ Ω and ψ ∈ P 0 L 2 (Ω, C 2 ), we have that

P 0 h k (α, β)P 0 ψ βα 2 Φ ψ .
Proof. For every k ∈ Ω we have:

P 0 h k (α, β)P 0 = (-2πσ • k + α 2 βΦσ 3 )P 0 .
Using the anticommutation relations of the Pauli matrices we get for any ψ ∈ P 0 L 2 (Ω, C 2 ) that

P 0 h k (α, β)P 0 ψ 2 = ((2πk) 2 + (α 2 βΦ) 2 ) ψ 2 (α 2 βΦ) 2 ψ 2 .
The previous lemma shows that h k (α, β) has a spectral gap of order βα 2 on the range of P 0 . In order to investigate whether that is still the case on the full Hilbert space we use the Feshbach inversion formula (see Equations (6.1) and (6.2) in [START_REF] Nenciu | Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians[END_REF]). The latter states in this case that z ∈ ρ(h k (α, β)) if the Feshbach operator

F P 0 (z) := P 0 (h k (α, β) -z)P 0 -β 2 P 0 χ α σ 3 Q 0 (Q 0 (h k (α, β) -z)Q 0 ) -1 Q 0 χ α σ 3 P 0 , is invertible on P 0 L 2 (Ω, C 2 ). Here we used that P 0 h k (α, β)Q 0 = βP 0 χ α σ 3 Q 0 .
The next lemma shows that the inverse of Q 0 (h k (α, β) -z)Q 0 is well defined on the range of Q 0 . Lemma 2.2. There exists a constant δ ∈ (0, 1) such that for all α ∈ (0, 1/2) and β > 0 with αβ < δ, we have that Q 0 (h k (α, β) -z)Q 0 is invertible on the range of Q 0 , for any z ∈ [-π/2, π/2] and k ∈ Ω.

The following lemma controls the second term of the Feshbach operator F P 0 (z)

B P 0 (z) := β 2 P 0 χ α σ 3 Q 0 (Q 0 (h k (α, β) -z)Q 0 ) -1 Q 0 χ α σ 3 P 0 .
Lemma 2.3. There exist two constants δ ∈ (0, 1) and C > 0 such that for all α ∈ (0, 1/2) and β > 0 with αβ < δ we have

B P 0 (z)ψ Cβ 2 α 3 ψ , for any z ∈ [-π/2, π/2], k ∈ Ω, and ψ ∈ P 0 L 2 (Ω, C 2 ).
Having stated all the above ingredients we can proceed to the proof of our first main result.

Proof of Theorem 1.1. In view of (2.12) it is enough to show the invertibility of the Feshbach operator uniformly in k ∈ Ω. Using Lemmas 2.1 and 2.3 we get that for any ψ ∈ P 0 L 2 (Ω, C 2 )

F P 0 (z)ψ (P 0 (h k (α, β) -z)P 0 )ψ -B P 0 (z)ψ (βα 2 Φ -|z| -cα 3 β 2 ) ψ .
This concludes the proof by picking αβ so small that Φ > Cαβ.

Analysis of the Feshbach operator.

In this section we provide the proofs of Lemmas 2.2 and 2.3 from the previous section. For that sake, let us first state some preliminary estimates. Let

h (0) k := (-i∇ PBC x -2πk) • σ (2.14)
Lemma 2.4. There exists a constant C > 0, independent of α ∈ (0, 1), such that for all k ∈ Ω

|χ α |P 0 α (2.15) |χ α | 1/2 (h (0) k ± i) -1 |χ α | 1/2 C(α + α 2 1 -α ) (2.16) |χ α | 1/2 (h (0) k ± i) -1 C( √ α + α 1 -α
).

(2.17)

Proof. In order to show (2.15) we compute for f, g ∈ L 2 (Ω, C 2 ) (see also (2.11)):

| f, |χ α |P 0 g | | f, |χ α |ψ 0 | | ψ 0 , g | χ α 1/2 1 f 2 g 2 α f 2 g 2 .
We now turn to the proof of equations (2.16) and (2.17). Denote the integral kernels of (H 0 -i) -1 and (h

(0) k ± i) -1 by (H 0 -i) -1 (x, x ) and (h (0) k ± i) -1 (x, x ).
In the proof we will use the identity for x = x (h

(0) k -i) -1 (x, x ) = γ∈Z 2 e 2πik•(x+γ-x ) (H 0 -i) -1 (x + γ, x ), x, x ∈ Ω.
(2.18)

Let us first estimate the quadratic form of (h

(0) k -i) -1 for any φ, ψ ∈ L 2 (Ω, C 2 ). According to Lemma A.1 we have | φ, (h (0) k -i) -1 ψ | C γ∈Z 2 Ω×Ω |φ(x)| e -|x-x -γ| |x -x -γ| |ψ(x )|dxdx Ce √ 2 γ∈Z 2 e -|γ| Ω×Ω |φ(x)||ψ(x )| |x -x -γ| dxdx ,
where in the last step we bound the exponential using that |x -x -γ| |γ| -|x -x | |γ| -√ 2. Now assume that the support of φ lies in Ω α := (-α/2, α/2] 2 , i.e., x ∈ Ω α above. Then it is easy to check that if γ = 0 then |x -x + γ| (1 -α). Therefore, we find for such a case that

| φ, (h (0) k -i) -1 ψ | Ce √ 2 Ω×Ω |φ(x)||ψ(x )| |x -x | dxdx + 1 1 -α ( γ =0 e -|γ| ) φ 1 ψ 1 .
Using the Hardy-Littlewood-Sobolev inequality for p = r = 4/3 (see Lieb and Loss Theorem 4.3) we get

Ω×Ω |φ(x)||ψ(x )| |x -x | dxdx C φ 4/3 ψ 4/3 . (2.19)
Thus, denoting the universal constants by C we obtain, for any φ, ψ ∈ L 2 (Ω, C 2 ) with supp(φ) ⊂ Ω α , | φ, (h

(0) k -i) -1 ψ | C φ 4/3 ψ 4/3 + (1 -α) -1 φ 1 ψ 1 .
(2.20)

For some f ∈ L 2 (Ω, C 2 ) we observe that (see Remark 2.11)

|χ α | 1/2 f 1 |χ α | 1/2 2 f 2 α f 2 . (2.21)
Moreover, Hölder's inequality yields

|χ α | 1/2 f 4/3 |χ α | 2/3 3 |f | 4/3 3/2 3/4 = χ α 1/2 2 f 2 √ α f 2 .
(2.22)

In order to get the desired bounds we recall that the norm of an operator T is given by

T = sup f,g =0 | f, T g |/( f
f ∈ Q 0 D(h (0) k ) we have h (0) k Q 0 f π f . (2.23)
Proof. For all m ∈ Z 2 and k ∈ Ω, we have the identity

P m h (0) k P m = 2πσ • (m -k)P m . Thus, we obtain for all f ∈ Q 0 D(h (0) k ), h (0) k Q 0 f 2 = (2π) 2 m =0 σ • (m -k)P m f 2 π 2 m =0 P m f 2 .
Let us introduce some notation: For a self-adjoint operator T and an orthogonal projection Q, we define

ρ Q (T ) := {z ∈ C such that Q(T -z)Q : RanQ → RanQ is invertible} . We set R 0 (z) := Q 0 (h (0) k -z)Q 0 RanQ 0 -1 , z ∈ ρ Q 0 (h (0) k ), R(z) := (Q 0 (h k (α, β) -z)Q 0 RanQ 0 ) -1 , z ∈ ρ Q 0 (h k (α, β)). Let U : L 2 (Ω, C 2 ) → RanQ 0 and W : RanQ 0 → L 2 (Ω, C 2 ) be defined as W := β |χ α |σ 3 Q 0 , U := βQ 0 sgn(χ α ) |χ α | .
Lemma 2.6. There exists C > 0, independent of α ∈ (0, 1/2] and β > 0, such that for all |z| π/2, W R 0 (z)U < Cαβ.

Proof. Note that due to Lemma 2.5 we have for |z| π/2 R 0 (z) 2 π .

(2.24)

Using the first resolvent identity

W R 0 (z)U = W R 0 (i)U + (z -i)W R 0 (i) 2 U + (z -i) 2 W R 0 (i)R 0 (z)R 0 (i)U. ( 2

.25)

We shall estimate separately each term on the right hand side of (2.25). Since h

(0)
k commutes with the projections P m we have R 0 (i) = (h

(0) k -i) -1 -(P 0 (h (0) k -i)P 0 RanP 0 ) -1 .
(2.26) Thus, using (2.15), we obtain the estimate

W (P 0 (h (0) k -i)P 0 RanP 0 ) -1 U β |χ α |P 0 (h (0) k -i) -1 P 0 |χ α | βα 2 .
(2.27)

The identity (2.26) together with the inequalities (2.16) and (2.27), imply that there are universal constants c, C > 0, such that for |α| < 1/2

W R 0 (i)U W (h (0) k -i) -1 U + W (P 0 (h (0) k -i)P 0 RanP 0 ) -1 U cβα + βα 2 Cβα.
(2.28)

This bounds the first term on the right hand side of (2.25). To estimate the second one we first notice that (h

(0) k -i) -2 = (Q 0 (h (0) k -i)Q 0 RanQ 0 ) -2 + (P 0 (h (0) k -i)P 0 RanP 0 ) -2
Therefore, using the same strategy as in (2.27), there exists C > 0 independent of α and β such that

(z -i)W R 0 (i) 2 U = (z -i)W (h (0) k -i) -2 -(P 0 (h (0) k -i)P 0 RanP 0 ) -2 U (z -i)W (h (0) k -i) -2 U + |z -i|βα 2 πβ |χ α |(h (0) k -i) -1 (h (0) k -i) -1 |χ α | + πβα 2 Cβα, (2.29)
where we used (2.17) in the last inequality.

Finally, we bound the last term on the right hand side of (2.25). Observe that from Lemma 2.4 and inequality (2.15), we obtain that there exists c > 0 such that for all α 1/2

|χ α |R 0 (i) |χ α |(h (0) k -i) -1 + |χ α |P 0 (P 0 (h k -i)P 0 ) -1 c √ α.
Therefore, using (2.24) and (2.17)

(z -i) 2 W R 0 (i)R 0 (z)R 0 (i)U β|z -i| 2 |χ α |R 0 (i) R 0 (z) R 0 (i) |χ α | Cβ|z -i| 2 α.
In view of (2.25), the latter bound together with (2.28) and (2.29) concludes the proof.

Before stating the next lemma we define the set

S := {z ∈ ρ Q 0 (h (0) k ) : W R 0 (z)U < 1}.
Lemma 2.7. For any z ∈ S we have that z ∈ ρ Q 0 (h k (α, β)) and

R(z) = R 0 (z) -R 0 (z)U (1 + W R 0 (z)U ) -1 W R 0 (z).
Proof. For z ∈ S we define for short

B(z) := R 0 (z) -R 0 (z)U (1 + W R 0 (z)U ) -1 W R 0 (z).
Since W R 0 (z)U < 1 we may use Neumann series to get

B(z) = R 0 (z) -R 0 (z)U n 0 (-1) n (W R 0 (z)U ) n W R 0 (z) = R 0 (z) 1 - n 0 (-1) n U (W R 0 (z)U ) n W R 0 (z) = R 0 (z) n 0 (-1) n (U W R 0 (z)) n , (2.30)
where the absolute convergence of the last expression is a consequence of the identity:

(U W R 0 (z)) n = U (W R 0 (z)U ) n-1 W R 0 (z), n 1,
and the boundedness of U and W . For z ∈ S ∩ R we define z ε := z + iε. Since S is an open set we may choose ε > 0 so small that z ε ∈ S. We first prove that the claim holds for z ε . A simple iteration of the second resolvent identity gives

R(z ε ) = R 0 (z ε ) N n 0 (-1) n U W R 0 (z ε ) n + T N +1 , (2.31)
where

T N +1 = (-1) N +1 (U W R 0 (z ε )) N U W R(z ε ) = (-1) N +1 U [W R 0 (z ε )U ] N W R(z ε ).
Hence T N +1 converges to zero and R(z ε ) = B(z ε ). Taking the limit ε → 0 on the right hand side of this identity finishes the proof.

Proof of Lemmas 2.2 and 2.3. Notice that the proof of Lemma 2.2 follows from Lemma 2.7 since z ∈ S provided αβ is small enough (see Lemma 2.6).

In order to show Lemma 2.3 observe that

β 2 P 0 χ α R 0 (z)χ α P 0 β P 0 |χ α | 1/2 W R 0 (z)U |χ α | 1/2 P 0 cβ 2 α 3 , (2.32)
where we used Lemma 2.6 and (2.15). Moreover, assuming αβ is so small that W R 0 (z)U < 1/2 we have

β 2 P 0 χ α R 0 (z)U (1 + W R 0 (z)U ) -1 W R 0 (z)χ α P 0 β P 0 |χ α | 1/2 W R 0 (z)U (1 + W R 0 (z)U ) -1 W R 0 (z)U |χ α | 1/2 P 0 cβ 3 α 4 .
The latter inequality together with (2.32) finishes the proof of Lemma 2.3 in view of the resolvent identity of Lemma 2.7.

Proof of Theorem 1.2

In this section we remind that we fix α = 1 in the definition of h k (α, β). Let us redenote the Feshbach operator F P 0 (z) by F k (z) to emphasize its dependence on the vector k ∈ Ω:

F k (z) := P 0 (h k (1, β) -z)P 0 -β 2 P 0 χσ 3 Q 0 (Q 0 (h k (1, β) -z)Q 0 ) -1 Q 0 χσ 3 P 0 .
We shall later on prove that F k (0) is invertible for all k ∈ Ω, with an inverse uniformly bounded in k. We now show that this information is enough for the existence of a gap near zero for the original operator H(1, β). Lemma 3.1. Assume that there exists a constant C > 0 such that

sup k∈Ω F k (0) -1 C/β 3 . (3.33)
Then there exists a constant C > 0 such that for all |z| Cβ 3 , we have z ∈ ρ (H(1, β)).

Proof. There exists C such that

sup k∈Ω F k (z) -F k (0) C |z|. Using (3.33) yields sup k∈Ω (F k (z) -F k (0))F k (0) -1 C |z| C β 3 . Hence, F k (z) = [1 + (F k (z) -F k (0))F k (0) -1 ] F k (0) is invertible for |z| β 3
2CC uniformly in k ∈ Ω. This implies that z ∈ ρ(H(1, β). Now we focus on proving the estimate (3.33). For that sake we consider two regimes in k.

Lemma 3.2. Let β ∈ (0, π/(2 χ ∞ )) and let k ∈ Ω such that |k| > 2β 2 /π 2 . Then F k (0) is invertible and sup k∈Ω, |k|>2β 2 /π 2 F k (0) -1 π 2β 2 . (3.34) Proof. From Lemma 2.5 we have h k (1, 0)f π f for all f ∈ Q 0 D(h k (1, 0)). Thus for β ∈ (0, π/(2 χ ∞ )): Q 0 h k (1, β)f = (h k (1, 0))f + Q 0 βχσ 3 f π f -β χ ∞ f π 2 f .
Thus using χ 2 = 1:

F k (0) + 2πσ • kP 0 = β 2 σ 3 P 0 χQ 0 [Q 0 h k (1, β)Q 0 ] -1 Q 0 χP 0 σ 3 β 2 [Q 0 h k (1, β)Q 0 ] -1 2 π β 2 . (3.35)
We have the identity

(P 0 2πσ • kP 0 ) -1 = P 0 1 2πk 2 σ • k, (3.36)
hence under our assumption on k we get that the operator 1 -(F k (0) + 2πσ • kP 0 )(P 0 2πσ • kP 0 ) -1 is invertible, and

F k (0) -1 = P 0 1 2πk 2 σ • k 1 -F k (0) + 2πσ • kP 0 P 0 1 2πk 2 σ • k -1 π 2β 2
which proves (3.34). Now we focus on the case |k| 2β 2 /π 2 . Applying the resolvent formula to the operator F k (0) yields where K 0 is a modified Bessel function (see Formula 8.432(6) in [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF]). Using this in (A.44) and that K 0 (z) = -K 1 (z) (see Formula 8.486(18) in [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF]) we get that

F k (0) = -2πσ • kP 0 -β 2 σ 3 P 0 χQ 0 (Q 0 h 0 (1, β)Q 0 ) -1 Q 0 χP 0 σ 3 + β 2 σ 3 P 0 χQ 0 (Q 0 h k (1, β)Q 0 ) -1 2πσ • k (Q 0 h 0 (1, β)Q 0 ) -1 Q 0 χP 0 σ 3 =: M k (β) + O(β 2 |k|),
W k (β) := Tr(σ 1 m k (β)), Tr(σ 2 m k (β)), Tr(σ 3 m k (β)) ∈ R 3 , we have m k (β) = W k (β) • σ. ( 3
(H 0 ± i) -1 (x, x ) = i 2π K 1 (|x -x |)σ • ∇ x |x -x | ∓ K 0 (|x -x |) .
Thus |(H 0 ± i) -1 (x, x )| 6) in [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF]).

1. 1 .

 1 Setting and main results. Let χ : R 2 → R be a bounded function supported on a compact subset S included in Ω := (-1/2, 1/2] 2 satisfying Ω χ(x)dx := Φ 0 and Ω |χ(x)| 2 dx = 1.

.11) 2 . 1 .

 21 Bloch-Floquet representation and proof of Theorem 1.1.

( 3 .Lemma 3 . 3 .

 333 37)where we used(3.35) to prove that the third term on the right hand side of the first equality is O(β 2 |k|). The operator M k (β) has the structure P 0 ⊗m k (β) where its matrix part m k (β) is acting on C 2 . The matrix m k (β) is traceless, i.e. Tr(m k (β)) = 0. In particular, defining the three dimensional vector

3 = -β 2 1 e

 321 .38)Proof. Consider the anti-unitary charge conjugation operator U c defined by U c ψ = σ 1 ψ. Then a straightforward computation yields U 2 c = 1 andU c P 0 = P 0 U c , U c σ 3 = -σ 3 U c , U c h 0 (1, β) = -h 0 (1, β)U c .Also:U c σ 3 P 0 χQ 0 (Q 0 (h 0 (1, β)Q 0 ) -1 Q 0 χP 0 σ 3 U c = -σ 3 P 0 χQ 0 (Q 0 (h 0 (1, β)Q 0 ) -1 Q 0 χP 0 σ 3 , Since Tr(U c AU c ) = Tr(A) if A is self-adjoint, we must have Tr(m k (β) + 2πσ • k) = 0. Since Tr(σ • k) = 0,we conclude that Tr(m k (β)) = 0 and the lemma is proved. Lemma 3.4. Assume that hypothesis (1.10) of Theorem 1.2 holds. Then there exists β 0 > 0 and C > 0 such that for all β ∈ (0, β 0 ), and all k ∈ Ω such that |k|2β 2 /π 2 , M k (β) is invertible and M k (β) -1 C/β 3 . (3.39)Proof. Due to (3.38), we haveM k (β) -1 = 1 |W k (β)| 2 M k (β), hence M k (β) -1 = 1 |W k (β)| 1 |Tr(σ 3 M k (β))| . (3.40)Let us now compute (remember that the "third" component of k is by definition equal to zero):Tr(σ 3 M k (β)) = Tr σ 3 -2πσ • kP 0 -β 2 σ 3 P 0 χQ 0 (Q 0 h 0 (1, β)Q 0 ) -1 Q 0 χP 0 σ Tr P 0 χQ 0 (Q 0 h 0 (1, β)Q 0 ) -1 Q 0 χP 0 σ 3 =: -β 2 w(β),(3.41)In order to obtain the kernel for the Laplacian we recall the well-known formula for its heat kernel in dimension n t∆ (x, x ) = 1 (4πt) n/2 e -|x-x | 2 4t . Thus, by the usual integral representation of the resolvent in terms of the heat kernel we get (-∆ + 1) -1 (x, x ) |x -x |),

1 2π

 1 |K 0 (|x -x |)| + |K 1 (|x -x |)| the claim now follows by the asymptotic behaviour of the Bessel functions K 0 and K 1 at zero and infinity (see Formulas 8.447(3), 8.446 and 8.451(

  g ). Hence we find (2.16) by using (2.20) with φ = |χ α | 1/2 f and ψ = |χ α | 1/2 g together with the bounds (2.21) and (2.22). Analogously, we obtain (2.17) using again (2.20) with φ = |χ α | 1/2 f and ψ = g.

	Lemma 2.5. For any

Acknowledgments. It is a pleasure to thank the REB program of CIRM for giving us the opportunity to start this research. Furthermore, we thank the Pontificia Universidad Católica de Chile, Aalborg Universitet and Université de Toulon for their hospitality. E.S has been partially funded by Fondecyt (Chile) project # 114-1008 and Iniciativa Científica Milenio (Chile) through the Nucleus RC120002.

which is independent of k. Using (2.13) we get

Hence we obtain w(0) = 0 because

With equations (3.41)-(3.43) we get

which together with (3.40) concludes the proof of the lemma since we assumed hypothesis (1.10).

Proof of Theorem 1.2. Equation (3.37) together with (3.39) implies that for β ∈ (0, β 0 ) and |k| 2β 2 /π 2 , the operator F k (0) is invertible and

for some constant C > 0 independent of β. Using in addition the estimate (3.34) of Lemma 3.2 it implies that there exists a constant C > 0 independent of β such that

Together with Lemma 3.1, this concludes the proof of Theorem 1.2.

Appendix A. Estimate for the resolvent kernel Lemma A.1. There exists a constant C > 0 such that the following kernel estimate holds for all

Proof. The relation (H 0 + i)(H 0 -i) = (-∆ + 1) implies that (H 0 -i) -1 = (H 0 + i)(-∆ + 1) -1 . (A.44)


Marina Mezhova
email: mezhova75@mail.ru

Angela Repanovici
email: arepanovici@gmail.com

Jurgita Rudžioniene
email: jurgita.rudzioniene@kf.vu.lt

Rene Schneider
email: rene.schneider@hesge.ch

Ana Lucia Terra
email: analuciaterra@yahoo.com

Tania Y Todorova
email: dr.tanya.todorova@gmail.com

Information Behaviors and Information Literacy Skills of LIS Students S81

Librarians are expected to be expert searchers, and developing information literacy skills to navigate the vast world of information is a focus of most library and information science (LIS) programs. It is important to understand the information literacy and behaviors of LIS students to see if they are employing the skills they will need to assist and educate their future patrons, yet there is relatively little research into the information literacy and behaviors of librarians and library students. Using a survey on information behaviors, this study addresses this gap in the literature by examining how LIS students in 18 countries search for, evaluate, and use information in various contexts, and on whom they rely for help. The results of this study demonstrate library science students' information literacy skills within an international context. Faculty in LIS programs will be interested to see how their students approach information problems, and might use this understanding to develop or strengthen courses and inform curricular decisions.

derstand the information literacy and behaviors of LIS students as a group to see if they are employing the skills they will need to assist and educate their future patrons.

On the other hand, it is not advisable to study LIS students as though they are a homogenous group, thereby ignoring possible differences of library and information science curricula as well as culture on information behaviors. In a systematic review of the literature on informationseeking behaviors of graduate students, [START_REF] Catalano | Patterns of graduate students' information seeking behavior: A meta-synthesis of the literature[END_REF] noted some cultural differences in the information-seeking behaviors of international students studying in the United States. As the economy becomes increasingly globalized and as more L ibrarians are expected to be expert searchers of all kinds of information in all formats. Developing the knowledge and skills to navigate the vast world of online and print information is typically a focus of most library and information science (LIS) programs. Understanding information literacy levels and how students approach information seeking tasks can inform curricular decisions. As Chung and Neuman note, "because learning is the primary goal of students' information seeking and use, how students' information seeking and use contributes to learning should be studied in various contexts . . . [and] with different learning tasks" (2007, p. 1516). Thus it is important to un-librarians and library students choose to go abroad to study or work, it will be useful for librarians and LIS faculty to recognize these differences.

While there is a rich literature on information literacy and behaviors in general, there is relatively little research into the information literacy and behaviors of librarians and library students. Further, while some studies have focused on specific geographic areas, including Uganda (Okello-Obura & Ikoja-Odongo, 2010), Greece (Korobili, Malliari, &[START_REF] Korobili | Factors that influence information-seeking be-havior: The case of Greek graduate students[END_REF]Ireland (O'Farrell &[START_REF] O'farrell | Student information behaviours during group projects: A study of LIS students in University College Dublin, Ireland[END_REF], there are no studies that compare information literacy and behaviors of LIS students across different countries. Through a survey of the information literacy skills and information problem-solving habits of LIS students in 18 countries, the authors of this study begin to address this gap in the literature. The study investigates how LIS students search for, evaluate, and use information in various contexts, and on whom they rely for help.

The results of this study demonstrate the current state of library science students' information literacy skills and information problem-solving habits within an international context. Faculty in LIS programs will be interested to see how their students approach information problems especially as they teach greater numbers of international students or consider collaborating with international colleagues. Librarians might be interested to see how current library students in various countries approach information tasks, and how well they are being prepared to serve as intermediaries and instructors in a complex information environment. In addition, an international understanding is important for students and professionals to compete in a global job market.

Literature Review

The body of literature on information literacy and behaviors is extensive and will not be reviewed at length here. The-ories of Information Behavior, edited by Fisher, Erdelez, and McKechnie (2005), offers over 70 chapters, each outlining a theory, model, or conceptual framework related to information-seeking and information behaviors, such as Kuhlthau's Information Search Process (ISP) model of affective behaviors of information seeking, Dervin's theory of Sense-Making, and Bates' berrypicking model. The Pew Research Center's Internet and American Life Project has issued a number of reports that probe the online searching behaviors of the American adult population, showing that the vast majority of American adults use the Internet to search for information, and that they are generally confident in their search skills and happy with the results they get (see, e.g., [START_REF] Estabrook | Information Searches that Solve Problems[END_REF][START_REF] Fox | Pew Research Center's Internet & American Life Project[END_REF][START_REF] Purcell | Search engine use[END_REF]. The OCLC Perceptions of Libraries, 2010: Context andCommunities (2010) similarly explored library usage of adults and college students. The European Council on Information Literacy (ECIL, http://www.ilconf.org/) runs an annual conference with contributed papers that examine information literacy and information behaviors in a wide variety of contexts. The Project Information Literacy (PIL) has been conducting longitudinal studies of college students' information literacy and behaviors since 2008, and has compiled a clearinghouse of reports, videos, and other publications on their findings (see e.g., http://projectinfolit.org; [START_REF] Head | Beyond Google: How do students conduct academic research?[END_REF][START_REF] Head | Lessons learned: How college students seek information in the digital Age[END_REF]. The Project on Information Literacy's work has established college students' frustration with context-gathering and their preference for resources that are familiar and have given them good results in the past. [START_REF] Catalano | Patterns of graduate students' information seeking behavior: A meta-synthesis of the literature[END_REF] offers a comprehensive review of the literature on graduate students' information-seeking behaviors. Other studies have established the importance of convenience in terms of ease of access and use for online searchers [START_REF] Connaway | If it is too inconvenient I'm not going after it:" Convenience as a critical factor in information-seeking behaviors[END_REF][START_REF] Connaway | If it is too inconvenient I'm not going after it:" Convenience as a critical factor in information-seeking behaviors[END_REF].

Librarian and Library Student Information Behavior

A handful of studies have focused on information-seeking or information literacy competencies among librarians in relation to their professional activities. [START_REF] Brown | Informationseeking behavior of physical science librarians: Does research inform practice?[END_REF] found that physical science librarians rely most heavily on personal communications and listservs in order to stay current in their field, followed by scholarly journals, web sites, conferences, and trade and professional magazines. Liaison librarians need detailed knowledge of subject resources, as well as excellent skills in information discovery and literature searching in order to support faculty research [START_REF] Brewerton | Re-skilling for research: Investigating the needs of researchers and how library staff can best support them[END_REF]. [START_REF] Tan | Information literacy competencies among school librarians in Malaysia[END_REF] found that, although most school librarians in Malaysia were familiar with information literacy concepts, they were not comfortable with their own information literacy abilities. Most self-reported their information literacy skills as being "average" to "poor." It is important to note, however, that not all of the respondents in their sample had formal training as librarians.

Studies of library science students reveal a range of behaviors, some of which are similar to those of the general population. LIS students tend to favor online resources [START_REF] Campello | Information literacy and the education of school librarians[END_REF][START_REF] De Jong | How are teacher-librarians finding resources for coursework? Distance learners and the role of university library services[END_REF][START_REF] O'farrell | Student information behaviours during group projects: A study of LIS students in University College Dublin, Ireland[END_REF][START_REF] Tracy | LIS graduate students as library users: A survey study[END_REF], and they consult a wide range of resources (Campello & Abrue, 2005;[START_REF] Tracy | LIS graduate students as library users: A survey study[END_REF]. While LIS students report high self-efficacy and competence with computers, they are generally more comfortable using the computer to communicate or to conduct research for personal rather than academic purposes [START_REF] Malliari | IT self-efficacy and computer competence of LIS students[END_REF], and some report a need for greater exposure to and training in the use of online resources [START_REF] Okello-Obura | Electronic information seeking among LIS postgraduate students at Makerere University, Uganda[END_REF].

De [START_REF] De Jong | How are teacher-librarians finding resources for coursework? Distance learners and the role of university library services[END_REF] found that students in an online library teacher program tended to rely on sources with which they were already familiar, and favored those with online full-text access. Further, those students seemed to be unaware of many of the library services offered by their program, and turned instead to local public libraries for resources. LIS students tend to consult with reference librarians only occasionally [START_REF] Campello | Information literacy and the education of school librarians[END_REF][START_REF] Tracy | LIS graduate students as library users: A survey study[END_REF], with those in an undergraduate LIS program in Brazil relying most heavily on professors and classmates for advice [START_REF] Campello | Information literacy and the education of school librarians[END_REF].

Results of a research study on LIS students' perceived self-efficacy for information literacy in Turkey indicates that students feel confident about performing information literacy related tasks and that their levels of self-efficacy increases slightly over time [START_REF] Kurbanoglu | Self-efficacy: A concept closely linked to information literacy and lifelong learning[END_REF]. However, [START_REF] Conway | How prepared are students for post-graduate study? A comparison of the information literacy skills of commencing undergraduate and post-graduate information studies students at Curtin University[END_REF] found that both undergraduate and post-graduate LIS students in Australia have problems with basic information literacy skills, with students averaging a score of 73% on a survey test. Undergraduates in particular scored poorly on questions related to defining and articulating an information need and understanding the purpose, scope and appropriateness of a variety of information sources. Undergraduate LIS students in Brazil similarly found the initial stages of research, such as choosing and limiting a topic, to be difficult and expressed negative feelings at that stage [START_REF] Campello | Information literacy and the education of school librarians[END_REF]. The lack of information literacy skills displayed by LIS students prompted Campello and Abreu to conclude that "future librarians are not sufficiently prepared to perform the kind of search tasks for which they will be expected to act as mediators for others in the process of learning from information" (p. 49).

Methodology

The purpose of this study is to investigate the information literacy skills and behaviors of LIS students across different countries, both to explore those skills and behaviors generally, and to examine the extent of differences in those skills and behaviors. In particular, this study examines the following questions:

What strategies and sources do LIS students employ when gathering information for course-related information needs? Are there differences in the information seeking behaviors of LIS students in different countries?

The project was initiated and organized by two of the study authors, who recruited LIS professionals and faculty from around the world. Ultimately, 21 researchers from 18 countries participated. Because of the scope of the project, it was determined to use a web survey for data collection. Due to its quantitative nature, and the ability to administer it remotely and online, a survey would allow for a larger sample and greater ease of analysis. A single survey could also be easily translated to different languages and hosted on one site to facilitate the data collection.

One of the organizing authors gained permission from Alison Head to use the PIL survey (available at http://projectinfolit.org/images/pdfs/pil_fall2010_sur-vey_fullreport1.pdf) as the data collection instrument. This survey was chosen for several reasons. As the core instrument for PIL, this survey has the benefit of having been widely used and tested through a series of PIL studies, and therefore is a valid instrument. With its focus on questions of student information behaviors, including where they start their research, what sources they consult, and how they evaluate their sources, the survey was well suited to the research questions for this study. Finally, the data from the previous PIL studies conducted using this survey also offer a baseline of college undergraduate information literacy and behaviors with which the results of this study could be compared. The original PIL survey consists of two parts, one focusing on information behaviors related to everyday information needs, and one centered on information behaviors for course-related assignments. Because this study is concerned with the preparation of LIS students for professional roles, the researchers omitted the section on everyday life information and chose to study only students' behaviors related to course assignments.

Each researcher translated the original study into their own language, in order to allow survey participants to respond in their native language. Researchers made every effort to keep the meaning and intent of the original survey. Changes to the original questions were kept to a minimum and made only when absolutely necessary. Data collection took place in the fall of 2012 and spring of 2013. The researchers distributed the survey to the population of LIS students in their respective schools, with most using some type of internal electronic communication.

Limitations

Like the PIL research on which it is based, this study relies on the perceptions and self-reported behaviors of students. While the survey questions themselves have been widely tested and used, any self-reported data raises questions of reliability and validity. Either intentionally or unintentionally, students might not accurately reflect their information behaviors in their answers. Students might not recall their behaviors clearly, or they might choose answers that they believe are the "correct" answers regardless of whether those answers really describe their own actions. For instance, students could under-report use of resources like Google or Wikipedia because they believe they are not supposed to use them. Further research could supplement this study by using methods such as thinkaloud protocol or diaries to better capture students' actual behavior.

In addition, while this study is international in scope, the majority of the participating countries are concentrated in Europe. Coordinators of the study relied on personal knowledge and referrals to recruit colleagues to participate in the research, which partly accounts for the somewhat limited geographic scope. Further, participants had to commit to a predetermined timeline for completing data collection. Some researchers were interested in participating but were unable to meet the timeframe and therefore were not included in the findings reported here. Thus, it must be acknowledged that the results of the study might reflect particular similarities or differences of the participating countries.

Findings

The purpose of this study is to explore how LIS students from 18 countries approach course-related information tasks. The survey asked students about the kinds of assignments they receive, the sources that they consult, and their approaches to evaluating information. The survey also included questions about research habits, attitudes, and use of productivity tools. The following sections detail the aggregate findings of this study and a cross-country analysis, beginning with an overview of the data collection and response rates.

Demographics

As noted above, data was collected in 18 different countries, and ultimately a total of 1249 responses were received. In some countries, the bachelor's degree is the minimum needed to practice as a professional librarian, while in others the master's is the terminal degree. As such, the survey respondents included a mix of students at both the bachelor's and master's levels, depending on the country. Researchers distributed the survey to the total population of students at their respective institutions, using a variety of distribution methods. Turkey had the largest number of respondents (n = 210), and the United Kingdom had the lowest number (n = 6). Response rates varied from a low of about 10% in Australia and the Netherlands to a high of 92% in Lithuania. Table 1 shows the survey distribution method, degree level of the students surveyed, total number of respondents, and response rate by country.

The United States used a different age scale from the rest of the participants, so the following age breakdown is for the eighteen countries, minus the United States. The majority of respondents were between the ages of 18 and 23 (63%). Only 10% of respondents were over the age of 35, and just seven respondents (0.6%) were under the age of 18. Ages of participants in the United States skewed slightly older, with 65% of respondents between the ages of 22 and 30. Further, 13% of United States respondents were over the age of 40, while only 3% were under the age of 21. Women dominated the survey across countries, with 80% of respondents indicating they were female.

Starting Course-related Assignments and Searching for Resources

Students were asked about the perceived level of difficulty for 13 tasks related to starting a course assignment and searching for resources for that assignment. According to the responses, LIS students generally find the pre-research work to be more difficult than the tasks related to finding and evaluating resources. For instance, 65% of respondents either disagreed or strongly disagreed that finding sources in the library is difficult. Likewise, 76% disagree or strongly disagree that finding resources on the Web is difficult, and 52% disagreed or strongly disagreed that finding articles in library databases is difficult. Further 54% disagreed or strongly disagreed that coming up with search terms is difficult. Just under half (46%) disagreed or strongly disagreed that determining whether Web sites are credible is difficult. On the other hand, students were less confident about the early phases of an assignment. For instance, 56% agreed or strongly agreed that getting started on an assignment is difficult. Similarly 46% and 45% respectively agreed or strongly agreed that defining a topic and narrowing down a topic for an assignment is difficult. Figure 1 shows the aggregate responses across all countries for the students' perceived level of difficulty with each task.

While these statistics describe the behaviors of LIS students generally, a chisquare test revealed correlations across countries for each of the 13 tasks (p = 0.000 for each of the 13 responses) 1 . In other words, LIS students rated the difficulty of each task differently depending on their home country. For instance, defining a topic appears to be most difficult for students in Portugal, Turkey and Singapore (more than 50% in each country either "agree" or "strongly agree"). Students in Turkey also appear to find determining the credibility of a web site more difficult than students from other countries did (58%). Having to sort through irrelevant results to find what is needed is rated as difficult more often in Turkey (71%) and Bulgaria (65%). Narrowing down a topic is rated most difficult by students in France (61%), Turkey (56%), and Singapore (52%), while students in France (52%) and Lithuania (49%) report that deciding which database to use is difficult at higher rates than students from other countries. It is interesting to note that students in Swit- 1 Due to the variable response and participation rates, Finland, Hungary, Malta, the Netherlands, Romania, Russia, and the UK had to be eliminated from the cross-country analysis. Thus, the results reported for each chi-square test describe differences for students in Australia, Bulgaria, Croatia, France, Lithuania, Poland, Portugal, Singapore, Switzerland, Turkey, and the United States. zerland and the United States rated almost all of the 13 tasks as less difficult than students from other countries. Appendix A shows the percentage of respondents either agreeing or strongly agreeing with each statement by country.

Preparing Course-related Assignments

Another question asked respondents to rate the difficulty of tasks related to the mechanics of compiling research and completing assignments. Responses to this question were more mixed. There were not any tasks in this list that stood out as being particularly difficult for all or most students. Indeed, most students indicated that reading through the material, taking notes, integrating different sources, and knowing when to cite a source is not difficult. Overall, the most difficult aspects of the assignments appear to be rephrasing something that is already well expressed, and knowing whether the student has done a good job, with 47% and 45% respectively agreeing or strongly agreeing that these tasks are difficult. Appendix B shows the response rates for students across all countries for this question.

As with initiating an assignment and searching for resources, chi-square tests showed a correlation (p = 0.000) across countries for all responses to this question, except for the response "evaluating the sources I've found is difficult." Students in France expressed relatively more difficulty with deciding whether "the work is done" or not (73% strongly agreed or agreed) and knowing whether use of a source constitutes plagiarism (over 57%). Similarly, students in Turkey (46%) and Portugal (42%) were most likely to agree that knowing when to cite a source is difficult. Students in the United States expressed the least difficulty with knowing whether use of a source constitutes plagiarism (13%), but were somewhat more likely than students in Switzerland to agree that knowing when to cite a source is difficult (12% and 9% respectively). Students in Portugal found rephrasing what is already well expressed in a source more difficult than students from other countries (65%). Students in the United States expressed the least difficulty with evaluating sources (9%), while students in Turkey (45%), Portugal (44%), and Lithuania (41%) expressed the most difficulty. Figure 4 shows the rate at which respondents either strongly agreed or agreed with each statement by country.

Consulting People and Resources

Two questions asked students to rate the frequency with which they consult different resources or people for assistance when completing assignments, ranging from search engines and social networking sites to library databases, gray literature, instructors, classmates, and librarians. Search engines are the most widely consulted resources, with 86% of students indicating that they "almost always" or "often" use them. Search engines are followed by course readings (70%), library catalogs (56%), Wikipedia (54%), and library shelves (51%). LIS students are less likely to use social networking sites, slide sharing sites, video sharing sites, blogs, online forums and gray literature when doing assignments. Research databases fell to the middle, with 43% of students "almost always" or "often" using them for assignments. Figure 5 depicts the frequency with which students report using resources for assignments. Students rely mostly on classmates and instructors for support with assignments, with 70% turning to classmates "almost always" or "often," and 54% consulting with professors. Librarians are the last resource for most students with only 27% "almost always" or "often" meeting with librarians, while 16% report never meeting with a librarian. Figure 6 shows the rate at which students consult with particular people about assignments.

As with previous questions, the frequency with which LIS students consult various resources for course-related assignments showed some variance by country. A chi-square test showed a correlation for each of the fourteen resources by country (p < 0.05). Consulting library shelves is most common in Croatia (75% almost always or often consult library shelves) and Poland (69%), while students in the United States (34%) and France (35%) are least likely to consult library shelves. Students in Bulgaria (61%) and Poland (59%) are more likely to use a personal collection. Wikipedia is used most often by LIS students in France (86% almost always or often use Wikipedia), followed by students in Poland (64%) and Bulgaria (63%), and used least often by students in Australia (14%) and the United States (21%). Students in Australia are most likely to use library catalogs (94%) and online forums (35%), while students in the United States (93%), Australia (90%) and Singapore (85%) report the highest usage of research databases. Course readings are heavily consulted by all students, but are most consulted by students in the United States (90%) and least by students in Poland (44%) and Turkey (51%). Likewise, the frequency with which LIS students consult people-instructors, classmates, librarians, and friends or family-is correlated by country (p < 0.05). All LIS students report consulting classmates first, except in the United States and Australia where instructors are consulted at higher rates. Students in Bulgaria (55%) and Croatia (45%) are the most likely to consult with a librarian, while students in Singapore (12%) and Poland (14%) are the least likely to do so. French students consult friends/family at the highest rates (54%). Figure 5 shows the percentage at which students often or almost always consult certain people by country.

Evaluating Resources

Two questions focused on the criteria students use to evaluate resources they've found either through the library or on the Web. With regard to sources discovered through the library, students are not concerned with who published the information, or whether a librarian ever recommended the source to them. The most important criterion seems to be whether the resource is written in the student's native language, with 71% reporting that they "almost always" or "often" consider that criterion. Students are also concerned with how current the source is (67%), and whether an instructor mentioned the source (67%). The results concerning how students evaluate Web resources were more mixed, with students considering a wider range of criteria (such as domain name, site design, presence of a bibliography, and number of external links) more often. However, the criteria ranked most important were currency (69%), whether the site was written in the student's native language (66%), and whether an instructor had mentioned the site (65%). Students were least interested in whether a librarian had recommended a Web site, with only 33% claiming they "almost always" or "often" consider this factor.

Discussion

The results of this study show that there are some patterns of information literacy levels and behavior common to LIS students in general, but that there are also some significant differences in the ways in which LIS students from different countries find and use information for courserelated assignments. Both the general patterns of behavior and the cross-country differences have implications for LIS faculty as they prepare their students to take on professional roles.

LIS students seem to be comfortable with developing search strategies, choosing search terms, and finding resources in libraries and on the Web. This finding is reassuring since search and location are still core areas for many librarians as they help patrons find resources and facilitate the development of their own search skills. Of course, this study relied solely on students' perceptions and self-reporting of their abilities. While we might expect that LIS students, especially those who might be further along in their program of study, would be effective searchers, this study cannot determine their actual abilities. In fact, other studies have demonstrated that the majority of online searchers enter only one or two search terms per query, and most either do not use Boolean logic at all, or use it incorrectly [START_REF] Bernard | How are we searching the World Wide Web? A comparison of nine search engine transaction logs[END_REF]Brusilovsky, Ahn, & Rasumussen, 2010;[START_REF] Jansen | A review of web searching studies and a framework for future research[END_REF][START_REF] Lau | In search of query patterns: A case study of a university OPAC[END_REF][START_REF] Yu | The impact of web search engines on subject searching in OPAC[END_REF]. Graduate students often are uncomfortable using or are unfamiliar with advanced search strategies [START_REF] Catalano | Patterns of graduate students' information seeking behavior: A meta-synthesis of the literature[END_REF], and most students find choosing keywords and refining searches to be one of the most difficult aspects of information-seeking [START_REF] Hoffman | Library research skills: A needs assessment for graduate student workshops[END_REF]. According to [START_REF] Vezzosi | Doctoral students' information behavior: An exploratory study at the University of Parma (Italy)[END_REF] even doctoral students are not necessarily adept at searching online databases, and rarely take the time to learn advanced search strategies, although when they do learn them they often acknowledge them to be time-savers. A future study might examine LIS student search behaviors directly to determine whether they use more sophisticated strategies and/ or are more efficient and effective in their searches than other populations. In the meantime, it is important for LIS educators to consider the extent to which they are teaching and assessing good search behaviors in their courses.

While literature reviews and other assignments require searching, instructors typically only see the end result in the form of a bibliography. They can assess from the bibliography whether students are finding relevant and authoritative sources, but they cannot necessarily judge whether the searches used to locate the materials were efficient or effective. [START_REF] Bawden | Information seeking and information retrieval: The core of the information curriculum?[END_REF] underscores the importance of search for LIS students, contending that they must be experts at retrieving information, knowing how information is classified and indexed, understanding how retrieval systems are structured, and using sophisticated search strategies to make effective use of information sources and retrieval systems. He further argues that information retrieval and information seeking should form the core curriculum of LIS programs. To that end, faculty members might consider using exercises to practice search techniques, or asking students to outline their search process along with papers in order to determine whether students' confidence in searching is justified, and that they actually are performing high impact searches.

In fact, an analysis of the sources students use suggests that they are not all using resources that would require or even allow for sophisticated searching. For instance, course readings are among the most highly consulted resources. While resources selected and vetted by instructors might be a good starting point for research, [START_REF] Campello | Information literacy and the education of school librarians[END_REF] note that if instructors provide the bulk of the material that students use in assignments, those students get less practice in the essential process of search and discovery. When students do go beyond course readings, there is significant variance by country with regard to resources used. All students report relying heavily on search engines, but students in France, Bulgaria, Turkey, Portugal, and Poland report much higher uses of Wikipedia, while students in the United States, Australia, and Singapore are much more likely to report using research databases and database providers like LISA and EBSCO. Nearly a third or more of students in France, Lithuania, Turkey, and Bulgaria indicated that deciding which database to use and finding articles in a database is difficult.

It is possible that students in some programs have not been exposed to, or perhaps have not had access to, some of the research databases which LIS students in other countries use regularly. On the other hand, it is possible that like most other populations, LIS students are relying on Google and Wikipedia because they are familiar and convenient resources. Understanding the structure of databases and being able to search them effectively is a crucial expectation of professional librarians. If students do not go beyond search engines or Wikipedia-the same resources the general public uses regularly-they will not be able to add value to their patrons' searches. As such, LIS faculty in all programs must incorporate use of databases into their courses and programs and find ways to ensure that students are using them. At the same time, faculty members will need to be aware that some students might have less experience with research databases and will need to offer additional support to those students.

There were several important areas where students from certain countries expressed greater levels of difficulty. Specifically, more than one-third of students in Turkey, Bulgaria, Croatia, Lithuania, Singapore, France, and Portugal express difficulty in evaluating Web sources. As professional librarians, however, these students will be expected to guide their patrons in the selection and evaluation of Web (and other) resources for research and decision-making. School and academic librarians will also be expected to provide instruction on evaluation to students. Indeed, in the United States, in-struction in evaluating Web sites is often begun in elementary school, which might reflect the relative confidence in their abilities that American students in this study express. Nevertheless, this confidence is self-reported and may not be completely reliable. LIS faculty should ensure that all library students build up the confidence and skills to quickly and accurately evaluate Web resources. LIS faculty working with students from countries that have expressed greater difficulty with such evaluation must be careful not to make assumptions about students' previous exposure to source evaluation. Faculty might do pretests, use clickers, or poll their classes to get a sense of student abilities and understanding of source evaluation, and provide extra guidance as needed. Faculty members might incorporate Web evaluation exercises into their courses. Reference, Information Sources and Services, and Information Literacy courses, are a natural fit for such topics as most of these courses devote substantial time to the evaluation of sources generally. Traditionally, however, these courses have often focused on examining sources by type: almanacs, encyclopedias, indexes, and so on. Given the lack of confidence some students feel, it might behoove faculty in these courses to spend more time on Web sources as a whole, with attention to concepts like authority which must be understood differently when evaluating online sources like Wikipedia or other social media.

Plagiarism and source citation was another area of concern for many survey respondents. Understanding what constitutes plagiarism and when and how to properly cite sources is crucial for librarians and library students. These findings confirm other studies that have demonstrated cross-cultural differences in the understanding and application of plagiarism practices [START_REF] Amsberry | Deconstructing plagiarism: International students and textual borrowing practices[END_REF]Sutton, Tayler, & Johnston, 2014). Library students have an ethical obligation to abide by copyright and intellectual property laws. Further, the need to understand plagiarism extends beyond the student to a librarian's role as a professional. Many librarians will be called upon to assist patrons with questions in these areas. Many professional librarians become default copyright consultants, offering guidance to patrons on proper use of sources, delivering instruction on copyright, managing copyright requests, and sometimes even setting local policy. As such, it is important that library students are firmly grounded in how to avoid plagiarism, as well as in related areas of copyright law and proper citation of sources. But to what extent do LIS faculty explicitly teach these concepts, and to what extent do they assume students already know them? This study suggests that faculty might need to put more emphasis on plagiarism and citation.

It is interesting to note that in some cases LIS students engage in behaviors and attitudes very similar to other populations, including the undergraduates that served as the base for the original PIL study. For instance, the LIS students in this study generally find getting started on course assignments, and particularly defining and narrowing a topic, to be difficult. These are the same areas that [START_REF] Head | Beyond Google: How do students conduct academic research?[END_REF] and [START_REF] Head | Lessons learned: How college students seek information in the digital Age[END_REF] found to be most difficult for undergraduates. This finding is especially noteworthy because academic and school librarians often assist students in the processes of gathering context and narrowing and defining a topic, so one might assume that LIS students would have learned (or be learning) techniques that could apply to their own coursework. Perhaps LIS faculty assume that students already understand how to engage in these processes. These findings suggest that, given students' levels of frustration, more time and explicit instruction need to be spent on context-gathering, both to support their own research and to help patrons in future professional practice.

Similarly, it is interesting that although LIS students would, presumably, have a better understanding of the ways in which librarians could help them with tasks from defining a topic to finding and evaluating resources, the students in this study still only turn to librarians for help as a last resort. Like students in other studies [START_REF] Catalano | Patterns of graduate students' information seeking behavior: A meta-synthesis of the literature[END_REF][START_REF] Head | Beyond Google: How do students conduct academic research?[END_REF][START_REF] Head | Lessons learned: How college students seek information in the digital Age[END_REF] they tend instead to rely on their professors. However, students in this study do seem more likely overall to consult library catalogs and library shelves than other searchers. There may not be a real role for LIS faculty to play here, besides perhaps encouraging students to consult with librarians or even directly pointing students to librarians when they come to their instructors with questions related to defining a topic or locating and evaluating sources. The finding does raise some questions, though, as to why even LIS students do not ask librarians for help. Do they feel like it is "cheating" to ask for help on library school assignments? Are instructors discouraging students from consulting with librarians, so that the instructor can better assess the students' skills? Are students embarrassed to ask for assistance, and feel they should be able to accomplish these tasks on their own? Perhaps the reason has more to do with the preference for convenience highlighted by [START_REF] Connaway | If it is too inconvenient I'm not going after it:" Convenience as a critical factor in information-seeking behaviors[END_REF] and Connaway, Lanclos, and Bell (2013). Perhaps even students who presumably understand the value that librarians can bring to an information search find it too time-consuming or inconvenient to have meet with a librarian, and thus prefer to rely on what they can find on their own. However, the types of resources students report difficulty using; the difficulties some students express at finding context; the troubles with defining and narrowing a topic, and evaluating Web sources; suggests that they would benefit from the support of a librarian. In fact, this finding might have more implications for academic librarians than for LIS faculty. Perhaps they need to do more to reach out to students or find ways to make their services more accessible, whether that means more remote services, self-help resources, or a greater presence in student spaces. Perhaps LIS faculty and academic librarians need to work together more closely to ensure that referrals are being made when appropriate.

Conclusion

This study shows that LIS students display at least some of the same informationseeking behaviors and attitudes as other populations. They are confident in their overall searching abilities, but they have a hard time getting started on research assignments, including defining and narrowing down topics. They rely heavily on search engines, and express some concern about their ability to evaluate Web sources. They also report concern with determining what constitutes plagiarism and knowing when to cite sources. While these behaviors and attitudes are understandable, they raise some concerns as to whether LIS students are moving beyond the general population in their location, search, evaluation, and use of resources and building up the information literacy skills necessary to add value to their patrons' seeking and searching as professional librarians. These findings have implications for LIS faculty, who must decide whether their students are achieving the necessary level of mastery in these areas. Faculty will need to set clear goals for learning and assess student progress toward those goals, whether through assignments, tests, discussions, or other measures. Further, faculty will need to consider which areas must be addressed explicitly within the curriculum. For instance, faculty might not directly teach the specifics of plagiarism and citation, although they will have assignments that incorporate these skills. The results of this study suggest that, at least in some countries, students might need more explicit instruction in how to avoid plagiarism.

This study also found some significant differences in behaviors of students from different countries. Those specific findings will speak to LIS faculty in those countries, but they should also serve as a reminder to any LIS faculty who work with students from abroad that those students come to the program with different backgrounds and experiences. Faculty members need to be aware of and attuned to those differences, and be prepared to offer support to those students whose information literacy skills and behaviors or background knowledge may not conform to local expectations.

Figure 1 .

 1 Figure 1. Perceived Difficulty with Course-Related Assignments, Aggregate Responses.

Figure 2 .

 2 Figure 2. Perceived Level of Difficulty in Preparing Course-Related Assignments.

Figure 3 .

 3 Figure 3. Frequency with which Students Report Consulting Resources, Aggregate Responses.

Figure 4 .

 4 Figure 4. Frequency with which Students Report Consulting People, Aggregate Responses.

Figure 5 .

 5 Figure 5. Frequency with which Students Report Consulting People by Country.

Table 1 .

 1 Survey Distribution and Response Rate by Country.

		Students'		Total Number of	Estimated Response
	Country	Level	Distributed via	Respondents	Rate
	Australia	BA	e-learning platform department mailing list and	51	10%
	Bulgaria	BA	e-mail	94	60%
	Croatia	BA & MA	e-learning platform	110	40%
	Finland	BA & MA	student listserv	17	18%
	France	BA	in the classroom	113	85%
	Hungary	BA	internal communication channel	35	78%
	Lithuania	BA	printed version	86	92%
	Malta	BA	email	9	82%
	Netherlands	BA	-	10	10%
	Poland	BA & MA	e-learning platform	176	85%
	Portugal	BA	e-learning platform	66	74%
	Romania	BA & MA	student listserv	13	50%
	Russia	BA	e-mail	29	50%
	Singapore	MSc	e-mail	75	45%
	Switzerland	BA	mailing list	53	17%
	Turkey	BA	internal communication channel	210	80%
	UK	BA	internal communication channel	6	20%
	USA	MA	student listserv	96	17%

Getting started on the assignment is difficult Defining a topic for the assignment is difficult Narrowing down a topic is difficult Coming up with search terms is difficult Building up the search strategy is difficult Deciding which database to use is difficult Finding articles in the library's databases is difficult Finding sources to use "out on the web" is difficult Determining whether a Web site is credible or not is difficult Figuring out where to find sources in the library is difficult Finding up-to-date materials is difficult Finding "gray literature" (e.g. thesis, reports, unpublished papers, etc.) is difficult Having to sort through all the irrelevant results that I get to find what I need is difficult

Appendix B: Perceived Level of Difficulty in Preparing Assignments by Country Information Behaviors and Information Literacy Skills of LIS Students

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Category labels from top:

Evaluating the sources I've found is difficult Reading through the material is difficult Taking notes is difficult Integrating different sources from my research into my assignment is difficult The writing part is difficult Re-phrasing what is already well-expressed in a source is difficult Knowing when I should cite a source is difficult Knowing how to cite a source in the right format is difficult Knowing whether my use of a source, in certain circumstances, constitutes plagiarism or not is difficult Deciding whether "I'm done" or not is difficult Knowing whether I've done a good job on the assignment or not is difficult


Joumana Boustany
email: jboustany@gmail.com

Extremely familiar Moderately familiar Somewhat familiar

Keywords: Copyright literacy, France, public libraries, academic libraries, LIS studies, curricula, education, training

est destinée au dépôt

Introduction

Recently, many events occurred in France regarding copyright issues in libraries that would not have been possible in the past. For example, the French government voted on 1st March 2012 the Act No. 2012-287 which created legal tools for the digital use of 20th century out-of-print books (RELIRE project). This law gave the French National Library (BnF) the right to digitize books without negotiating each publishing contract case by case [1]. Several librarians, within the context of "cities on common"1 , are organizing "copy parties" where patrons can come to the library and scan books for "private use". Noting the complexity of this topic in everyday life and the new challenges related to the digital world, this study questions the readiness of French librarians to raise copyright issues. Is their background, including training and formal education, helping them face changes affecting this topic? Are they aware of international initiatives on copyright issues, or are they focused only on the national level? In this paper, we aim to produce comprehensive knowledge of the copyright literacy of French librarians, considering that this kind of study has never been done.

The French Context Regarding Copyright

In this paper, the word copyright will be used in the French meaning of author rights. In France, copyright and author rights do not have the same connotation even they tend to converge with the economic constraints. In the French law, only an individual can be considered as a work's author. Author rights grant exclusive property and moral rights to the creators of original literary, scientific and artistic works. If the author can transfer later his/her economic and moral rights to a legal entity, this entity could not be considered, in any case, as the author but as the beneficial owner. Copyright in the common-law sense does not make such distinction. An individual as well as a moral entity like publisher or producer could be considered as author.

French literature on copyright topic is abundant. Most publications are guides and handbooks mostly oriented toward learning and practice. We can find also added reports commissioned by the Senate, the Ministry of Higher Education and Research, or the Ministry of Culture and Communication on different issues [START_REF] Racine | Schéma Numérique des Bibliothèques[END_REF][START_REF] Borzeix | La Question du Droit de Prêt Dans les Bibliothèques[END_REF][START_REF] Martin | Commission sur les OEuvres Orphelines[END_REF]. Since 2005, several blogs, specializing in this topic, have arisen. These electronic publications are supposed to contribute actively to build professional copyright knowledge.

In France2 , the protection of author's rights and works can be traced back to the period of the French Revolution (1789) but the copyright issue in libraries is recent. As "fair use" does not exist in French law, libraries, as public bodies under the ministries' or local authorities' supervision, have created, over the years, de facto practices which allow them freedom to give access to all of the works. The need for legal competence in librarianship started to be a sensitive subject with the Council Directive 92/100/EEC concerning rental and lending rights 3 . It was the first attempt to restrict the access to works in libraries access [START_REF] Alix | Les Bibliothèques Saisies par le Droit[END_REF]. Libraries and librarians are now forced, at the risk of becoming marginalized, to integrate the question of copyright in their theoretical reflection, in their practice, and even in the forward planning of their missions and professions [START_REF] Alix | Préface[END_REF]. Nevertheless relations between librarians, or their representatives, and players of copyright are not like a long calm river and are interspersed over the time with exchanges that might show a perpetual conflict [START_REF] Masseron | Droit de Copie et Bibliothèques: Le Grand Malentendu[END_REF]. The latest controversial development was the law on author's rights and related rights in the information society (DADVSI) in 2006, and High Authority of Diffusion of the Art Works and Protection of the (Copy) Rights on the Internet (HADOPI) in 2009 which had a significant impact on libraries [START_REF] Lahary | Clones, Avatars, Lobbies et Incompréhensions Autour de la Loi Dadvsi[END_REF]. And recently, the adoption of RELIRE project which allows the French National Library to digitize out-of-print books is also very controversial.

Methodology

To be able to evaluate copyright literacy, we conducted a web-based survey adapted from an international research project called "Copyright Policies of Libraries and Other Cultural Institutions" (2012-2014) 4 during January-March 2014.

A quantitative survey was created on LimeSurvey a service that allows respondents to complete a questionnaire online. The aim is:

• to comprehend the knowledge and the awareness of the librarians regarding copyright related issues, • to gather the views of respondents regarding institutional level copyright policy, • and to ask librarians for their opinions concerning LIS education.

A 5-point Likert Scale and few open questions has been used to get such assets. The link to the survey was emailed by the author to more than 500 head academic librarians, around 300 Facebook accounts libraries, and three professional mailing lists.

Receivers were asked to complete and spread the questionnaire to their colleagues. Quantitative data were entered, coded, and analyzed using the SPSS statistical package. Descriptive statistics were used to analyze the findings and Chi-square tests of independence to compare different factors.

Respondents Profile

This approach enabled us to collect 329 completed answers, which is a small sample related to the population of librarians in France. 6 157 librarians work for national civil services (2012 estimations) and around 34 000 in local authorities (2010 estimations) [START_REF]Enseignement Supérieur et Recherche : Se Préparer aux Concours de Personnels des Bibliothèques[END_REF]. A higher percentage of survey respondents were female (74.5%) compared to male (25.5%). The breakdown of respondents by age range show predominance of people aged 30 to 39 (34.3%), followed by the 40-49 (28.3%), 50-60 (17%) and younger than 30 (15.8%). Few people are in the older age range, over 60 (4.6%). In France, most of the librarians are civil servants except those who work as contractual employees, volunteers and for private companies. Civil servants could be employed by the State (academic libraries, public libraries in Paris, classified libraries and French National Library) or by the local authorities (public libraries everywhere in France except Paris). In both structures, librarians are classified in three categories: A for manager, B for intermediate profession and C for employees and technicians. One hundred and eighty one (55%) of respondents were managers, 86 (26.1%) at the intermediate level and 33 (10%) employees and technicians. Just 29 (8.8%) work in a private company or as contractual librarians. Over 60% of the respondents have a master level diploma, 63.5 percent studied Library science, 11.9 percent History science. Archive science, Cultural Heritage science and Museology are poorly represented, respectively 4.9 percent, 4 percent and 0.6 percent. 15.1 percent of the respondents are distributed in other disciplines: Literature, Law, Geography, etc. 51.1 percent work in a university library, 24.3 percent in a public library and 6.1 percent in a company library. 18.5 percent work in different organizations. Only one person works in a museum. There is no archivist in our panel.

Findings

Knowledge and Awareness

On a five-point Likert scale ranging from 1 ("not at all familiar") to 5 ("extremely familiar"), the ideal scenario would be a significant number of respondents which are "extremely familiar". However, an examination of figure 1 shows that the level of familiarity of French librarians with issues regarding copyright is weak compared to the importance of the subject addressed. Sometimes the peaks of weakness reached or exceeded 50%. Values for extremely familiar are around 10% except for licensing for information sources (14.3%), licensing conditions in respondent's institution (16.7%), Creative Commons Licenses (17.9%), Open access (15.2%) and Copyright issues regarding public domain (16.7%). The awareness of French librarians is insufficient. The obvious lack in awareness concerns clearing rights (55.9%), licensing for online information via curation (49.8%), fair use (35.6%), copyright related issue at the international level (40.4%), copyright issues regarding virtual services within elearning practices (32.80%) and licensing for cultural action (32.2%). As fair use does not exist in French law, it is expected to have this rate of weakness, but not for the other subject. Likewise, there is a lack of knowledge in copyleft (42.6%) even it is closely linked to creative commons licenses, subject for which respondents consider that they are extremely (17.9% ) and moderately (29.8%) aware. This weakness is noticed as well in questions dealing with the development of institutional repositories (30.1%).

Copy-parties, a movement started in 2012, is still a limited phenomenon. Few librarians are extremely aware (8.3%) or moderately (17%) aware of this subject.

Topic where French librarians consider that they are moderately aware are: copyright and related law at the national level (40.1%), questions dealing with the public domain (37.7%), licensing for information source (31%), open access and open data (30.1%), as well as Licensing conditions in their institution (29.8%) and creative commons (29.8%) which gathered a significant percentage (17.9%) of extremely familiar, which is the highest value in the extremely familiar range. The trend is not very pronounced with the remaining questions.

Fig. 1. Level of knowledge and awareness of French librarians

Figure 2 shows that, for librarians, web sites are the primary source of information (77.5%) followed by books and articles (72.3%), colleagues (57.8%), blogs and wikis (43.8%), professional discussion lists (42.6%), French national library (36.5%) and experts (19.5%). Only 13.4% of librarians refer to lawyers when they need information. Some respondents dissociated lawyers from legal services, in the analysis when we combined them. International resources published by IFLA, World intellectual property organization (12.5%) and eIFL (1.5%) are less used. Respondents are not even interested (83.9%) by the initiatives of those organizations. This disinterest may explain the lack of competencies in copyright law at the international level.

Fig. 2. Information source

A little more than half of the librarians (52.9%) were sure of the presence of a national strategy for copyright in France and more than the quarter (30.4%) was uncertain. However, this strategy does not exist. Only 16.7% of the respondents knew this fact. Is it another proof of lack of competence or is it a lack of concern for copyright issues?

Concerning the national copyright legislation, most librarians (95.7%) knew well the duration of copyright protection, exceptions for private use (78.1%), and exceptions for libraries and educational institutions (61.7%). They knew that orphan works are not included yet in the copyright legislation (72.3%). For the question related to the rights for librarians to provide modified copies of works serving the needs of visual impaired patrons, 53.8% were aware that this exception exists in the French legislation. This exception has been introduced in the French law in August 2006 and does not concern all the libraries, only the accredited ones.

Copyright Policy and Institutional Level

Most respondents (85.4%) answered that their institution has resources protected by copyright, a small percentage (12.5%) was unsure. Few libraries (2.1%) do not have this kind of collection. Over three quarters of respondents (80.9%) think that institutional copyright policy is necessary for libraries and a minority (3%) does not believe that copyright policy is important. Others (16.1%) are uncertain. Despite the importance of issues dealing with law, less than one-twentieth (18.6%) of institutions have a person in charge of questions dealing with copyright issues and more than a quarter (34.3%) does not have a copyright policy in their institution. Only 34.7% of the institutions put this policy in place. What was surprising in these results was the fact that 31% of respondents are uncertain that their institution has its copyright policy. Is this revealing a clear lack of concern of librarians for copyright issues? The same question could be asked for personal digitization purposes. Indeed, slightly fewer than half of the respondents (41.6%) are uncertain if their institution authorizes or prohibits it. In fact, the widespread use of mobile devices makes copying documents easiest. A lot of institutions have a collection protected by copyright. Even private copy is admitted in copyright, the original must be owned by the person who is making the copy and not by the library. Almost half of the libraries (41.6%) authorizes personal digitization against 17% who prohibits it.

Education and Copyright

Almost everyone (97.9%) is in favor of including copyright issues in the curriculum of Library and Information science (LIS) education as well as in continuing education.

For the question concerning the appropriate level of education to introduce intellectual property, respondents were given the possibility to choose more than one answer. Over 70.2% of the respondents indicated undergraduate level, 37.4% master's degree, 17.6% PhD. Some of them (around 75 respondents out of 329) advised to introduce it at the beginning of LIS education.

Regarding the type of appropriate course for continuing education on intellectual property, a multiple choice question, respondents preferred thematic workshops (65%), followed by Internet sources: websites, blogs, wikis, etc. (58.7%) and then training course (54.7%). Less than half selected other options including: distance learning (40.1%), panel conference (40.1), consultation on request (37.7%) and round table discussion (24%).

5

Towards Better Curricula?

Chi-Square tests (cross-tabulation analysis) were performed on different criteria, and statistically significant relations between institution librarians work for, and their copyright awareness were found. The librarians' awareness level was clearly higher at the university library except for competence dealing with licensing for cultural action and out-of-print works. There is no difference regarding copyright at the national level, collective rights management organizations in France and material from public domain. The awareness level of librarians working in other type of institution was not important enough to be taken into consideration. Two hypotheses could explain these results. Most of the librarians working in a public library did not receive education in LIS and the post-recruitment training is insufficient.

To confirm the hypothesis about LIS education the same test was conducted to check if there is a relation between the type of study and copyright awareness.

Librarians who had an LIS degree were more aware than those who didn't except for these three issues: copyright at the international level, collective rights management organizations in France, and copy parties. These results confirm our hypothesis and lead to suggest the creation of the same access pattern for the state or the local authorities' librarians. A LIS education should be a compulsory requirement for recruiting in the local authorities libraries. The chi-test also shows a relation between the level of diploma and the level of awareness. Those who have the highest degree have a better level of competence, which is a good argument as mentioned by our respondents, for the introduction of copyright education upon the entrance of students to the university. This course should be enhanced and enriched every year to cover all the aspects of copyright.

The results show that, Librarians in France are not extremely aware about copyright and intellectual property issues. They had moderate familiarity with this subject, which is very significant progress compared to the situation in year 2000 where "librarians, in the vast majority, do not know a great deal about copyright" [START_REF] Martin | Commission sur les OEuvres Orphelines[END_REF].

To understand the reasons of this weakness, we tried to find an answer from a reading of curricula in some LIS schools and training in relevant bodies. Information on curricula has been gathered from official websites of those organizations.

LIS Education

In France, several diplomas lead students to become librarians. We limited ourselves to the most representative courses, since our study is not curricula oriented. University Diploma of Technology (DUT) is an intermediate two years diploma in Library Science and Heritage. Only 20 hours of a total of 1620 are dedicated to law.

Bachelor professional degree (Licence professionnelle) shows a mean hourly volume of 400. Courses dedicated to copyright do not exceed 20 hours. The same applies for master's degrees.

For a Libraries Curator Diploma (DCB) provided by the French National Library and Information Science School (Enssib), the law courses are optional with an hourly volume of 30.

This quick examination of some degrees at all levels of training demonstrates that copyright in LIS education is inadequate compared to the complexity and vast scope of the subject. There is a need to re-examine curricula to consider the complexity of subject and all the aspect of this issue especially regarding digital libraries and digital resources.

Lifelong Learning and Training

An examination of curricula in lifelong learning and conditions governing access to continue vocational training lead us to the same conclusion that we did for the LIS education. Different organizations are qualified to deliver training to librarians. However, courses in copyright are not that many.

The National Center of Territorial Public service (CNFPT) -a joint decentralized public institution which accompany local authorities and their agents in their public service mission -offers two courses dealing with copyright. Content does not cover all the aspects of copyright and the number of sessions is not enough when we know that around 34 000 librarians work on local authorities.

Regional training centers of library careers have three missions: giving information on libraries and documentation careers, preparing for competitive exams to be librarians, and training people who are already employed [START_REF]Enseignement Supérieur et Recherche : Se Préparer aux Concours de Personnels des Bibliothèques[END_REF]. There are 12 centers all over France, but not all of them offer courses on copyright. Sometimes they are organized jointly with CNFPT or URFIST (regional units for training in scientific and technical information). In general, one course on law in libraries is offered per year with a duration which varies from a minimum of three hours and a maximum of three days.

Access to vocational training is a right covered by law in France. Each civil servant has an annual credit hour of professional training equal to 20 hours for a fulltime agent. Those hours can be held concurrently in the limit of six years or 120 hours. As demonstrated by the examination of curricula of training body, the offering is relatively poor in comparison with the number of librarians and limited by the duration, rarely a training exceeds 30 hours.

Conclusion

Even if the size of the sample is limited, the results are nevertheless very significant.

They converge with what is already known by the relevant administrations. In a report commissioned by Higher Education of the Ministry for Education and Research, and Ministry for culture and communication, the general inspectorate of libraries wrote: "It will also be judicious for curator students to acquire or consolidate elements of basic legal and economic culture, which will be useful in their administrative as well as scientific responsibilities (public and administrative law, labor law, economic law, intellectual property, information economy)" [START_REF] Carbone | Quels Emplois Dans les Bibliothèques? Etat des Lieux et Perspectives[END_REF]. It will be interesting to study the possibility of creating a specialized diploma in legal issues for librarianship; this diploma could be addressed to LIS background students. Aside from the fact that this paper sheds light on the copyright literacy of French librarians, it also highlights the inadequacy of LIS education, and the lack of training. The complexity of law in general and copyright in particular demonstrates that this is not a subject that can be covered in a single course. Furthermore, this state of awareness shows the need for strong emphasis on continuing education, training and copyright law within the library profession. May these results help enhance lifelong learning in LIS education and the curricula in order to give proper answers to the needs in the field.

Villes en biens communs [Cities on commons] is a collective of associations who supports and promote the use of commons like free culture, http://villes.bienscommuns.org/

In what follows, quotes are freely translated by the author.

This directive has been repealed and replaced by Directive 2006/115/EC http://goo.gl/9Gwurj

This project was financed by NSF of the Bulgarian Ministry of Education, Youth and Science, Contract № DFNI-К01/0002-21.11.2012 and headed by Tania Todorova from State University of Library Studies and Information Technologies, Sofia.