Peter D Hislop
email: peter.hislop@uky.edu

Nicolas Popoff
email: nicolas.popoff@cpt.univ-mrs.fr

AND Nicolas Raymond
email: nicolas.raymond@univ-rennes1.fr

Mikael P Sundqvist

Band functions in presence of magnetic steps

Keywords: AMS 2000 Mathematics Subject Classification: 35J10, 81Q10, 35P20 magnetic Schrödinger operators, edge currents, band functions

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Introduction and motivation

In this paper, we investigate the spectral properties of magnetic Hamiltonians

L b = (-i∇ -A) 2 = D 2 x + (D y -a b (x)) 2 , (1.1)
in L 2 (R 2 ) with the following discontinuous magnetic field

B(x, y) = b 1 1 R-(x) + b 2 1 R+ (x),
where b = (b 1 , b 2 ) ∈ R 2 , and 1 X (x) is the characteristic function on X ⊂ R. An associated vector potential is given by :

A(x, y) = (0, a b (x)), a b (x) = b 1 x1 R-(x) + b 2 x1 R+ (x).
These operators naturally appear in several models involved in nanophysics such as the Ginzburg-Landau model of superconductivity (see [START_REF] Saint-James | Type II Superconductivity[END_REF]) and in quantum transport in a bidimensional electron gas. In particular in an unbounded system, inhomogenous magnetic fields may play the role of a quantum waveguide ( [START_REF] Peeters | Quantum structures created by nonhomogeneous magnetic fields[END_REF][START_REF] Reijniers | Snake orbits and related magnetic edge states[END_REF]).

More recently it appears that such magnetic barriers also induce transport and act as waveguides in graphene by creating snake states along magnetic discontinuities (see [START_REF] Oroszlány | Theory of snake states in graphene[END_REF] and [START_REF] Ghosh | Conductance quantization and snake states in graphene magnetic waveguides[END_REF] where piecewise constant magnetic fields are considered). These physical properties can be described with the corresponding magnetic Schrödinger operators that exhibit a variety of interesting spectral and transport properties. In particular, there may exist edge currents that propagate along the edge at x = 0.

The conductivity corresponding to these edge currents is, in fact, quantized. Most of these properties can be deduced from the study of the band functions associated with the Hamiltonian (see for instance [START_REF] Mantoiu | Some propagation properties of the Iwatsuka model[END_REF][START_REF] Dombrowski | Quantization of edge currents along magnetic barriers and magnetic guides[END_REF]). We will describe later some consequences of our analysis of band functions.

The symmetries of the Hamiltonian allow us to classify various configurations of the magnetic field (b 1 , b 2 ). We notice that L b = L -b so that we may assume that b 2 ≥ 0. If S denotes the symmetry (x, y) → (-x, -y), we have SL b1,b2 S = L -b2,-b1 = L b2,b1 . For B > 0, we introduce the L 2 -unitary transform U B ψ(x, y) = B -1/2 ψ(B -1/2 x, B -1/2 y) and we have U -1 B L b U B = B -1 L Bb . These considerations allow the following reductions:

(1) If b 1 or b 2 is zero we may assume that b 1 = 0 and, if b 2 = 0, we may assume that b 2 = 1. We call the case (b 1 , b 2 ) = (0, 1) the "magnetic wall". ). This last one may be called the "symmetric trapping magnetic step" or "magnetic barrier". (4) If b 1 and b 2 have the same sign, we may assume that 0 < b 1 < b 2 = 1. We call this case the "non-trapping magnetic step". A physical description of the distinct classical orbits associated with these magnetic field configurations is provided in [START_REF] Peeters | Quantum transport of a two-dimensional electron gas in a spatially modulated magnetic field[END_REF][START_REF] Reijniers | Snake orbits and related magnetic edge states[END_REF]. The last two cases, (3) and (4), have already been studied in the mathematical literature (see the overview below). Our goal is to provide a mathematical treatment of the unstudied cases (1) and [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]. We will also give the consequences of our results on the behavior of quantum particles submitted to the magnetic field b.

1.1. The band functions and their link with the magnetic Laplacian.

• Direct fiber decomposition. In order to perform the spectral analysis, we can use the translation invariance in the y-direction and thus the direct integral decomposition (see [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]XIII.16]) associated with the Fourier transform with respect to y, denoted by F y ,

F y L b F -1 y = ⊕ k∈R h b (k) dk,
where

h b (k) = D 2 x + V b (x, k), with V b (x, k) = (k -a b (x)) 2 .
Note that the effective potential V b (x, k) has the form

V b (x, k) = (k -b 1 x) 2 , x < 0, (k -b 2 x) 2 , x > 0. (1.2) k < 0 k k > 0 Figure 1. The potential V b (x, k) corresponding to case (1). k < 0 k > 0 k k b 1 Figure 2. The potential V b (x, k) corresponding to case (2). k < 0 k > 0 k k b 1 Figure 3. The potential V b (x, k) corresponding to case (3) with b 1 = -1 and b 2 = 1. k < 0 k b 1 k Figure 4. The potential V b (x, k) corresponding to case (4).
The domain of h b (k) is given by

Dom(h b (k)) = {ψ ∈ Dom(q b (k)) : (D 2 x + V b (x, k))ψ ∈ L 2 (R)}
, where the quadratic form q b (k) is defined by

q b (k)(ψ) = x∈R |ψ (x)| 2 + V b (x, k)|ψ| 2 dx. The spectrum of L b , sp(L b ), satisfies sp(L b ) = k∈R sp(h b (k)).
We now define the n-th band function below the essential spectrum: Notation 1.1. We denote by λ b,n (k) the n-th Rayleigh quotient of h b (k). We recall that if λ b,n (k) is strictly less than the infimum of the essential spectrum, it coincides with the n-th eigenvalue of h b (k).

• State of the art. The cases (3) and ( 4) have already been extensively studied. Case [START_REF] Bonnaillie-Noël | Ground state energy of the magnetic laplacian on general three-dimensional corner domains[END_REF] for which (b 1 , b 2 ) = (1, 1) is just the Landau Hamiltonian. Its spectrum is pure point and consists of the so-called Landau levels; infinitely degenerated eigenvalues (2n -1), for n ≥ 1. The band functions are constants λ b,n (k) = 2n -1.

The case (3), for which (b 1 , b 2 ) = (-1, 1), can be linked to operators acting on a half-line. We first need to introduce the standard de Gennes operators, that will also be used in some results of this article. We recall below well-known properties of the spectrum of these operators. Notation 1.2. We let E 0 = 0 and for n ≥ 1, 2 and its eigenvalues (µ • (k)) ≥1 . For all ≥ 1, the function µ N admits a unique and non-degenerate minimum at

E n = 2n -1. Notation 1.3. For • = N, (resp. • = D), let us introduce h • (k) the Neumann (resp. Dirichlet) realization on R + of D 2 t + (t -k)
k = ξ -1 , denoted by Θ -1 , with Θ -1 ∈ (E -1 , E ). Moreover, ξ -1 is also the unique solution of the equation µ N (k) = k 2 . The function µ D (k) is a decreasing function of k. Both µ N (k) and µ D (k) go to +∞ as k → -∞ and go to E as k → +∞.
We refer mainly to [START_REF] De Bievre | Propagating edge states for a magnetic hamiltonian[END_REF] for the Dirichlet operator and [START_REF] Dauge | Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators[END_REF][START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF] for the Neumann one. All these properties can also be found in [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF][START_REF] Ivrii | Microlocal Analysis, Sharp Spectral Asymptotics and Applications[END_REF].

Using symmetry arguments, it is proved in [START_REF] Dombrowski | Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators[END_REF][START_REF] Popoff | Sur l'opérateur de Schrödinger magnétique dans un domaine diédral[END_REF] that for b = (-1, 1) and ≥ 1,

λ b,2 -1 (k) = µ N (k) and λ b,2 (k) = µ D (k) k ∈ R,
linking the de Gennes operator with the model with symmetric trapping magnetic steps. This analysis of the band functions provides a description of a quantum particle localized in energy far from the thresholds {E n } and submitted to the system: they can exhibit edge current that may be viewed as quantum manifestations of the classical snake orbits [START_REF] Reijniers | Snake orbits and related magnetic edge states[END_REF] along the edge x = 0. These edge currents are exponentially decreasing far from the edge. The corresponding edge currents and the asymptotics of the band functions are described in [START_REF] Dombrowski | Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators[END_REF]. The effective potential (see (1.2)) has the form of two symmetric double wells for k > 0. The fact that each band function is exponentially close to a Landau level as k → +∞ is called a splitting of the band function. The analysis of quantum states localized in energy near the thresholds is made in [START_REF] Hislop | Characterization of bulk states in one-edge quantum hall systems[END_REF] for one edge quantum hall system and relies on a precise asymptotic analysis of the band function µ D (k) as k → +∞. The band functions associated to this case are displayed in Figure 5.

The case (4), a non-trapping magnetic step, was treated in the initial paper by Iwatsuka [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF] (along with the more general situation for which the magnetic field depends only on one direction and simply has constant values at ±∞). In this case, all the band functions are monotone nondecreasing. The analysis of the associated edge current is done in [START_REF] Hislop | Edge states induced by Iwatsuka hamiltonians with positive magnetic fields[END_REF]. Notice that all cases (1)-( 4) are described from a physical point of view in [START_REF] Reijniers | Snake orbits and related magnetic edge states[END_REF].

1.2. Main results. We first consider case [START_REF]Digital library of mathematical functions[END_REF] for which (b 1 , b 2 ) = (0, 1). This was not treated in [START_REF] Hislop | Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields[END_REF] or in [START_REF] Dombrowski | Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators[END_REF]. It is physically interesting because the classical orbits of an electron originating in the half-plane x < 0 are unbounded straight half-lines or lines (depending upon whether the orbit intersects x = 0 or not), whereas those in the half-plane x > 0 are bounded circles provided the Lamor orbit does not cross the edge at x = 0.

Theorem 1.4. Let b = (0, 1). If k < 0, then the fiber operator h b (k) has only purely absolutely continuous spectrum

[k 2 , ∞). Consequently, the spectrum of L b is [0, ∞). If k > 0, the essential spectrum of h b (k) is [k 2 , ∞).
The band functions λ b,n are nondecreasing functions with E n-1 < λ b,n (k) < E n and lim k→∞ λ b,n (k) = E n = 2n -1. For n ≥ 1 and ξ n-1 < k < ξ n (recall Notation 1.3), the operator h b (k) admits exactly n simple eigenvalues below the threshold of its essential spectrum.

The case (2) of a trapping magnetic step is technically more challenging. It relies on the asymptotics of the eigenfunctions of a Schrödinger operator with an asymmetric double-well potential. The results concerning the splitting of two energy levels depends upon the ratio of the two constant magnetic fields b 1 /b 2 = b 1 since b 2 = 1 by normalization. Definition 1.5. We define L = {E n , n ≥ 1} {|b 1 |E n , n ≥ 1} and we define the splitting set S = {E n , n ≥ 1} {|b 1 |E n , n ≥ 1}. We will say that there is a splitting of levels if S = ∅ i.e. if there exist positive integers m < n such that

b = 2n -1 2m -1 , with b = |b 1 |. (1.3)
As we shall see, the problem of the behavior for large k of the band functions can be reformulated into a semiclassical problem for a one dimensional Schrödinger operator with multiple wells (see for instance [START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF][START_REF] Helffer | Multiple wells in the semi-classical limit III-interaction through non-resonant wells[END_REF]). In particular the limits of the band functions are precisely the elements of L, counted with multiplicty. Roughly speaking, either one or two band functions converge toward each element of L.

However our investigation will not rely on the semiclassical tools: We will use asymptotic expansions of special functions in order to derive the asymptotic behavior of the band functions. Let us now state our main theorem.

Theorem 1.6. Let b = (b 1 , 1), with -1 < b 1 < 0. For k ∈ R, the fiber operator h b (k) has only purely discrete spectrum λ b,n (k) with simple eigenvalues, and lim k→-∞ λ b,n (k) = +∞. The set of limit points of the band functions λ b,n (k) as k → +∞ is precisely L. Moreover, we may describe the asymptotics of the band functions in the following way.

(i) Non-splitting case. Let λ ∈ L \ S. We can write in a unique way λ = E n or λ = bE n with n ≥ 1. Then there exists a unique

p n ∈ N * such that: (a) λ b,pn (k) -E n = ε n (k) → k→+∞ 0 (in the case λ = E n ), (b) λ b,pn (k) -bE n = ε n (k) → k→+∞ 0 (in the case λ = bE n ).
In the first case,

ε n (k) = - 1 √ π 2 n-2 (1 + b) 1 (n -1)! k 2n-3 e -k 2 (1 + o(1)),
and in the second case,

ε n (k) = - 1 √ π 2 n-2 b 3/2-n (1 + b) 1 (n -1)! k 2n-3 e -k 2 /b (1 + o(1)).
(ii) Splitting case. Assume that the splitting set is not empty and consider λ ∈ S.

Let us consider n ≥ 1 and m ≥ 1 such that λ = E n = bE m . There exists

p n ∈ N * such that λ b,pn (k) -E n = ε - n (k) λ b,pn+1 (k) -E n = ε + n (k) with, as k tends to +∞, ε - n (k) = - 2 n-3/2 (1 + b) (n -1)! √ π k 2n-3 e -k 2 (1 + o(1)), ε + n (k) = 2 m+ 3 2 b -m+ 3 2 (m -1)!(1 + b) √ π k 2m+1 e -k 2 /b (1 + o(1)).
The energy levels for which the exponentially small splitting occurs when k → ∞ behave like the band functions in the symmetric magnetic field case case b 1 = -1 and b 2 = 1. In that case (see [START_REF] Dombrowski | Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators[END_REF]), two band functions approach the Landau level 2n -1 as k → ∞, one is monotone decreasing and the other has a unique, nondegenerate minimum (see Figure 5). In the present article, with the help of our asymptotic expansions, we may deduce the existence of a minimum for some band functions. Notice that the first band function always admits a global minimum. Moreover if |b 1 | is not a ratio of odd integers, all the band functions admit a global minimum. We conjecture that the only case when monotonicity occurs is Case (ii) of Theorem 1.6 and corresponds to λ pn+1 . In that case, the non-monotonicity of each band function depends on a non trivial way on the index of the eigenvalue. This situation seems striking to us, especially compared to other known case (see [START_REF] Dombrowski | Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators[END_REF][START_REF] Hislop | Edge states induced by Iwatsuka hamiltonians with positive magnetic fields[END_REF][START_REF] Yafaev | On spectral properties of translationally invariant magnetic schrödinger operators[END_REF]). 1 3 , 1). In dash line: the asymptotics from Therorem 1.6, case (ib). Here n = 3. 1.4. Consequence of the analysis of the band functions. In this section we qualitatively present possible consequences of our results.

• Standard perturbation of the Hamiltonian. If we perturb the Hamiltonian L b by a suitable electric potential, discrete spectrum might appear under the essential spectrum, which is the infimum of the first band function. In particular, the existence of a minimum is useful if one wants to study the counting function of created eigenvalues. Here we know that the infimum is reached for some finite frequency k * . Then, following the standard procedure described for example in [START_REF] Raikov | Eigenvalue asymptotics for the Schrödinger operator with perturbed periodic potential[END_REF], the number of discrete eigenvalues can be described, depending on the degeneracy of the minimum and on the decay of the electric perturbation.

In the same way, it is possible to derive a limiting absorption principle at energies close to the bottom of the spectrum of L b (and also near each band minimum), as in [START_REF] Soccorsi | Analyticity and asymptotic properties of the Maxwell operator's dispersion curves[END_REF]. The case of thresholds corresponding to the limits of band functions can also be studied in the spirit of [START_REF] Popoff | Limiting absorption principle for the magnetic dirichlet laplacian in a half-plane[END_REF]. In particular the asymptotics of the band functions for large k allow to describe the suitable functional spaces when writing a limiting absorption principle near the Landau levels.

• Semiclassical magnetic Laplacian with piecewise constant magnetic field. Let us consider the semiclassical magnetic Laplacian (-i ∇ + A) 2 in the plane with a magnetic field such that

∇ × A = B = b 1 1 Ω + b 2 1 Ω ,
where Ω is a smooth and bounded domain. It is well-known that the semiclassical analysis of the low lying spectrum (motivated by the analysis of the third critical field in the Ginzburg-Landau theory) is governed by the spectral behavior of some local models (see [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF][START_REF] Bonnaillie-Noël | Ground state energy of the magnetic laplacian on general three-dimensional corner domains[END_REF]). When the magnetic field is not continuous, models operators in the form of L b have to be considered. It is known that the minima (which are often non degenerate) of the corresponding band functions may combine with the curvature (of the boundary for instance) to generate the semiclassical asymptotics. In particular, the existence of a minimum of the band functions implies a microlocalization of the magnetic eigenfunctions (see [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF][START_REF] Raymond | Little Magnetic Book[END_REF][START_REF] Bonnaillie-Noël | Magnetic WKB constructions[END_REF]).

• Bulk states localized in energy near the thresholds. In general Iwatsuka Hamiltonians (i.e. Hamiltonians associated with magnetic fields depending only on one variable), the analysis of quantum states submitted to the magnetic field and localized in energy in an interval I depends on the behavior of the band functions when crossing the energy interval I, see [START_REF] Mantoiu | Some propagation properties of the Iwatsuka model[END_REF]. For example if I does not contain any threshold, then the quantum states are localized in space near the variations of the magnetic field (here, the jump line {x = 0}) and their current (defined below) is not small. This corresponds to the fact that a classical charged particle moves along the edge {x = 0}, in a "snake orbit".

One may also show, as in [START_REF] De Bievre | Propagating edge states for a magnetic hamiltonian[END_REF][START_REF] Hislop | Characterization of bulk states in one-edge quantum hall systems[END_REF], that there exist states, called bulk states, localized in energy near the thresholds, bearing small current. If the threshold corresponds to a limit of a band function, these bulk states are exponentially small near the jump line. The precise (de)localization of these states and the current they bear can be derived from the asymptotics of the band function by the following method (see [START_REF] Hislop | Characterization of bulk states in one-edge quantum hall systems[END_REF]). Assume that I = (λ -δ, λ + δ) where λ ∈ S and δ ↓ 0. The current of a state ψ ∈ L 2 (R) along the jump line is given by J y ψ, ψ where

J y := i 2 [L b , y] = -i∂ y -a b (x)
is the current operator. This quantity may be linked to the derivative of the band functions by using the Feynman-Hellmann formula. Using the asymptotics from Theorem 1.6, one may show that the derivative of the band functions is small on the large k interval corresponding to the values where λ n (k) ∈ I. This would provide an upper bound on the current carried by bulk states, depending on how far from the threshold is their energy. Then follows an analysis in phase space of these states in order to show that delocalization in the frequency k involves delocalization in the x-variable.

Note finally that the sign of the derivative of the band function is related to the direction of propagation of the particle along the jump line of the magnetic field. In particular quantum states related to non-monotonic band functions (see Corollary 1.7) may flow along the line in two possible directions. This corresponds to the possible negative velocity described in [START_REF] Reijniers | Snake orbits and related magnetic edge states[END_REF]. We refer to [START_REF] Yafaev | On spectral properties of translationally invariant magnetic schrödinger operators[END_REF] for an analysis of the direction of propagation of quantum states in a translation invariant magnetic field.

Magnetic wall

This section is devoted to the case b = (0, 1). Away from the edge at x = 0, a classical electron moves in a straight line for x < 0 and in a closed circular orbit for x > 0. If the orbit intersects the edge, then the classical electron escapes to x = -∞. The absence of magnetic field for x < 0 results in a half-line of essential spectrum for the quantum electron. As we will see, the spectrum varies with the sign of k.

The reduced potential V b satisfies V b (x, k) = k 2 for x ≤ 0, and V b (x, k) → +∞ as x → +∞. As a consequence, we easily deduce the following proposition.

Proposition 2.1. Assume that b = (0, 1) and k ∈ R. The essential spectrum of h b (k) is given by sp ess (h b (k)) = [k 2 , +∞). Moreover, sp(L b ) = [0, +∞).
In fact, we can prove slightly more.

-ψ + (k -a b (x)) 2 ψ = λψ.
(2.4)

For x < 0, we have -ψ = (λ -k 2 )ψ whose only solution in L 2 (R -) is zero. But since the solutions of (2.4) belongs to H 2 loc and are in C 1 (R), this implies that ψ(0) = ψ (0) = 0 and thus ψ ≡ 0 by the Cauchy-Lipschitz theorem.

Let us now describe the discrete spectrum, that is the eigenvalues λ < k 2 . Since, for k ≤ 0, we have q b (k) ≥ k 2 , we deduce the following proposition by the min-max principle.

Proposition 2.3. Assume that b = (0, 1) and k ≤ 0. Then

sp(h b (k)) = sp ess (h b (k)) = [k 2 , +∞).
Therefore we must only analyze the case when k > 0. The following lemma is a reformulation of the eigenvalue problem.

L 2 (R + ) satisfying -ψ (x) + (x -k) 2 ψ = λψ(x), (2.5a
) Proof. We use the translation

ψ (0) -k 2 -λ ψ(0) = 0 . ( 2
x = y + k to see that h b (k) is unitarily equivalent to D 2 y + Ṽ (y, k) with Ṽ (y, k) = 1 (-∞,-k) (y)k 2 + 1 (-k,+∞) (y)y 2 . For 0 < k 1 < k 2 , we have: Ṽ (y, k 2 ) -Ṽ (y, k 1 ) = 1 (-k2,-k1) (y)y 2 + 1 (-∞,-k2) (y)k 2 2 -1 (-∞,-k1) (y)k 2 1 ≥ 1 (-k2,-k1) (y)k 2 1 + 1 (-∞,-k2) (y)k 2 2 -1 (-∞,-k1) (y)k 2 1 = 1 (-∞,-k2) (y)k 2 2 -1 (-∞,-k2) (y)k 2 1 ≥ 0.
By the min-max principle, we infer the desired monotonicity.

The next lemma is a consequence of the Sturm-Liouville theory (see for instance [START_REF] Simon | Sturm oscillation and comparison theorems[END_REF]). Lemma 2.6. Assume that b = (0, 1), and let n ∈ N be such that λ b,n (k) < k 2 . Then, the corresponding eigenspace is one dimensional and is generated by a normalized function ψ b,n (k), depending analytically on k. Moreover, ψ b,n (k) has exactly n -1 zeros, all located on the positive half line.

Proof. We have only to explain the part of the statement concerned with the zeros. Thanks to Lemma 2.4, one knows that the zeros are necessarily positive. Let us briefly recall the main lines of the Sturm's oscillation theorem. Let us denote by m -1 the number of zeros of ψ = ψ b,n (k) and let us check that m ≤ n. For j = 1, . . . , m -1, z n,j denotes the j-th zero of ψ b,n (k) (the zeros are simple and isolated by the Cauchy-Lipschitz theorem) and we let ϕ j = ψ |(zn,j ,zn,j+1) extended by zero. By definition,

q b (k)(ϕ j , ϕ ) = R ϕ j ϕ + V b (x, k)ϕ j ϕ dx, so, q b (k)(ϕ j , ϕ ) = δ j, R ϕ 2 j + V b (x, k)ϕ 2 j dx = λ b,n δ j, ϕ j 2 .
We apply the min-max principle to the space F m = span 1≤j≤m ϕ j to get λ b,m ≤ λ b,n and thus m ≤ n. In particular we recover that ψ b,1 does not vanish. Let us explain why ψ b,2 vanishes exactly once. Introducing the Wronskian W

1,2 = ψ 1 ψ 2 -ψ 2 ψ 1 , we notice that W 1,2 = (λ b,1 -λ b,2 )ψ b,1 ψ b,2 . If ψ b,2 > 0, then W 1,2 < 0.
Since the eigenfunctions belong to the Schwartz class at infinity, W 1,2 (x) → 0 as |x| → +∞. This is a contradiction. With the same kind of monotonicity arguments, we can prove that the zeros of the eigenfunctions are interlaced and that ψ b,n has exactly n -1 zeros.

By using the harmonic approximation in the semiclassical limit (see [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] and the Section 3.1 below), we can prove the following lemma. Lemma 2.7. Assume that b = (0, 1). For all n ≥ 1, we have

lim k→+∞ λ b,n (k) = E n .
In particular, for k large enough, we have λ b,n (k) ≤ E n < k 2 .

Let us now prove that the n-th band function lies between the two consecutive Landau levels E n-1 and E n .

Proposition 2.8. Assume that b = (0, 1). For all n ≥ 1 and for all k > 0 such that λ b,n (k

) < k 2 we have λ b,n (k) ∈ (E n-1 , E n ).
Proof. By Lemmas 2.5 and 2.7, λ b,n (k) < E n (the strict inequality comes from the analyticity). It remains to prove that λ b,n (k) > E n-1 . Clearly λ b,1 (k) > E 0 . We rely on Notation 1.3. For n ≥ 2, we consider the function ϕ n (x) = ψ b,n (x + z n,1 (k)) which satisfies h D (k -z n,1 (k))ϕ n = λ b,n (k)ϕ n and has exactly n-2 zeros on R + , cf. Lemma 2.6. By the Sturm's oscillation theorem, ϕ n is the (n -1)-th eigenfunction of h

D (k -z n,1 (k)). Therefore we have λ b,n (k) = µ D n-1 (k -z n,1 (k))
. Moreover for all ≥ 1 and k ∈ R, µ D (k) > E . This follows by domain-monotonicity, once we notice that h D (k), after a translation, is given by the harmonic oscillator D 2

x + x 2 on (-k, +∞) with Dirichlet boundary condition at x = -k, together with the fact that the harmonic oscillator D 2

x + x 2 on R has simple eigenvalues E . This provides the desired conclusion.

We recall that the sequence (ξ n ) n≥1 is defined in Notation 1.3. For the next proposition, we will need to use the so-called parabolic cylinder functions. We denote by U (a, x) the first Weber parabolic cylinder function (whose properties are summarized in the Appendix, section A; for more details see, for example, [1, section 12] or [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]) which is a solution of the linear ordinary differential equation:

-y (x) + 1 4
x 2 y(x) = -ay(x).

(2.6)

The Weber function U (a, x) decays exponentially when x → +∞.

Proposition 2.9. Assume that b = (0, 1). For all n ≥ 1, the equation

λ b,n (k) = k 2 has a unique non negative solution, k = ξ n-1 , such that, locally, for k > k n , λ b,n (k) < k 2 . Moreover we have λ b,n (ξ n-1 ) = Θ n-1 .
Proof. Thanks to Lemma 2.7, we can define . Let us now consider the eigenvalue equation

k n = max{k ≥ 0 : λ b,n (k) = k 2 }.
D 2 x ϕ n,p + ( kn,p -a b (x)) 2 ϕ n,p = λ n,p ϕ n,p , (2.7)
where ϕ n,p = ψ b,n ( kn,p ), kn,p = kn + 1 p and λ n,p = λ b,n ( kn,p ). Let us investigate the limit p → +∞. As seen in the proof of Lemma 2.4, we know that there exists α ∈ R * such that, for x ≤ 0,

ϕ n,p (x) = αe x k2 n,p -λn,p .
By solving (2.7) on x ≥ 0 and using the parabolic cylinder function U , we find that there exits β such that, for x ≥ 0,

ϕ n,p (x) = βU - λ n,p 2 ; √ 2(x -kn,p ) .
Since ϕ n,p = 0, we have (α, β) = (0, 0) and ϕ n,p is C 1 at x = 0, we get the transmission condition:

k2 n,p -λ n,p Û - λ n,p 2 ; -kn,p √ 2 - √ 2U - λ n,p 2 ; -kn,p √ 2 = 0.
By continuity and taking the limit p → +∞, we get

U - k2 n 2 ; -kn √ 2 = 0.
Notice that the function x → U -k2 n 2 ;

√ 2(x -kn ) solves the differential equation -y (x) + (x -kn ) 2 y(x) = k2 n y(x) y (0) = 0 and y(0) = 0 Moreover it belongs to S(R + ). Therefore, there exists ≥ 1 such that µ N ( kn ) = k2 n , and therefore kn = ξ -1 and k2 n = Θ -1 . By Proposition 2.8, we know that

k2 n = λ b,n ( kn ) ∈ [E n-1 , E n ]. Moreover, we recall that Θ -1 ∈ (E -1 , E
). This implies that = n.

Trapping magnetic step

In this section, we focus on the case b = (b 1 , 1) with -1 < b 1 < 0. The case of b 1 = -1 is treated in [START_REF] Dombrowski | Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators[END_REF]. In that case, due to the symmetry, all levels are split as k → +∞. The asymptotic expansion comes from [START_REF] Hislop | Characterization of bulk states in one-edge quantum hall systems[END_REF][START_REF] Popoff | Sur l'opérateur de Schrödinger magnétique dans un domaine diédral[END_REF]. The band functions for this case are shown in Figure 5.

When b 1 < 0, the effective potential V b (x; k), described in (1.2), diverges to infinity as x → ±∞, so that for all k ∈ R, the operator h b (k) has compact resolvent. The spectrum of h b (k) is discrete and the eigenvalues (λ b,n (k)) n≥1 are all simple.

For k < 0, the effective potential V b (x; k) has the form of an asymmetric potential well, see Figure 2

. It follows that V b (•; k) ≥ k 2 so that, for k ≤ 0, λ b,n (k) ≥ k 2 . In particular, we have, for all n ≥ 1, lim k→-∞ λ b,n (k) = +∞, k < 0.
For k > 0, the effective potential has the form of an asymmetric double-well:

V b (x; k) = (k -b 1 x) 2 x < 0 (k -x) 2 x > 0 (3.8)
This asymmetric double-well admits two nondegenerate minima at x = k b1 and x = k, for k > 0 and b 1 < 0.

Let us now investigate the behavior of the band functions in the limit k → +∞ and prove Theorem 1.6. Due to the asymmetry of the double-well potential (3.8), we will prove, among other results, that not all the Landau levels exhibit an exponentially small splitting as in the case of (b 1 , b 2 ) = (-1, 1).

3.1. Limits of the band functions. Let us first describe the set of the limits of the band functions λ b,n (k) as k → +∞. As noted above, the limit as k → -∞ of any band function is plus infinity. Lemma 3.1. Assume that b = (b 1 , 1), with -1 < b 1 < 0. Recall from Definition 2.7 that L denotes the union of the Landau levels for magnetic fields b 1 ∈ (-1, 0) and b 2 = 1, and that S denotes the splitting set {E N , n ≥ 1}∩{|b 1 |E n , n ≥ 1}.Then the set of limits as k → +∞ of the band functions is precisely the set of Landau levels L:

lim k→+∞ λ b,n (k), n ≥ 1 = L.
The band functions exhibit the following behavior as k → +∞:

(1) Non-splitting case. If a band function limit λ ∈ L \ S, then for any

0 > 0, there exists a K > 0 such that if k ≥ K then it holds that rank 1 [λ-ε0,λ+ε0] (h b (k)) = 1
and there exists a unique n ≥ 1 such that λ b,n converges towards λ. (2) Splitting case. If a band function limit λ ∈ S, then for any 0 > 0, there exists a K > 0 such that if k ≥ K then it holds that rank 1 [λ-ε0,λ+ε0] (h b (k)) = 2 and there exists a unique n ≥ 1 such that λ b,n and λ b,n+1 converge towards λ.

Proof. Let us first provide a semiclassical reformulation and summarize the standard procedure of the harmonic approximation for multiple wells (see [START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF]Chapter 4] for example). For k > 0, we let x = ky and the operator h b (k) is unitarily equivalent to k 2 hb,k -2 , where hb, is the self-adjoint realization on L 2 (R) of the differential y + V (y; 1) on L 2 ((-∞, b 1 + η)) and on L 2 ((1 -η, +∞)) respectively. Let us now consider an arbitrary constant C > 0. It is well-known (see Helffer's lecture notes [START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF]) that any eigenfunction of hb, ,η or of hb, associated with an eigenvalue λ ≤ C satisfies decay estimates of Agmon type away from the minima. In particular, this allows to prove the existence of c > 0 such that sp( hb, ) ∩ {λ ∈ R : λ ≤ C } and sp( hb, ,η ) ∩ {λ ∈ R : λ ≤ C } coincides modulo O e -c -1 . Then, by the harmonic approximation, the spectrum of hb, ,η is well-known in the semiclassical limit: the n-th eigenvalue of hle b, ,η is equivalent to |b 1 |E n and the one of hri b, ,η to E n . The desired conclusion follows. The last part of the statement follows from the fact that the spectra (with multiplicity) of hb, and hb, ,η coincide modulo an exponentially small error.

As described in [START_REF] Hislop | Characterization of bulk states in one-edge quantum hall systems[END_REF], the convergence of the band functions toward these limits corresponds to the existence of bulk states with energies close to the Landau levels. We next provide a more accurate description of the convergence of the band functions towards their limits. This is important for two reasons. First, this allows a more quantitative characterization of the associated bulk states following the analysis of [START_REF] Hislop | Characterization of bulk states in one-edge quantum hall systems[END_REF]. Second, the manner in which the band function approaches its limit as k → +∞, that is, whether it approaches from above or below the Landau level, provides a hint as to whether the band function is globally monotonic or not. In order to describe the asymptotic behavior of the band function, we will leave the semiclassical techniques and use the characterization of the eigenvalues λ b,n (k) as zeros of certain special functions.

3.2. Splitting and non-splitting of levels. We consider k > 0 and calculate the behavior of the energy levels in the splitting and the non-splitting cases. This is a refinement of Lemma 3.1, where we have shown that each band function λ b,n tends to a threshold that is an element of the set of Landau levels L. There is no explicit formula for the index of the corresponding Landau level as a function of n, the band index.

In this section, for the sake of simplicity, we will consider a converse strategy: Given a threshold E n or bE n , where n ∈ N * is fixed, we describe the asymptotic behavior of the band functions converging toward this threshold. As described in Lemma 3.1, there are either one or two band functions λ b,pn (k) converging toward a threshold E n or bE n in L. Although the indices p n of these band functions are not explicit in terms of the index n, the asymptotics of the related band functions λ b,pn (k), as k → +∞, are explicit functions of n.

Let us notice that this procedure will provide the asymptotics of all the band functions since there is a bijection between their limits and the set of the thresholds L (counted with multiplicity).

D b (λ, k) := U (-λ 2b , - √ 2 k √ b )U (-λ 2 , - √ 2k) + √ bU (-λ 2b , - √ 2 k √ b )U (-λ 2 , - √ 2k
)

and b = |b 1 |.
Proof. From the eigenvalue equation and the form of the potential (3.8), we are led to solve for x > 0:

-y (x) + (x -k) 2 y(x) = λy(x)
with the condition that the solution is in L 2 (R + ). The only such solution is

x → U (-λ 2 , √ 2(x -k)).
In particular, when λ = 2n -1 is a Landau level, the solution is the n-th Hermite function centered at x = k. On the half line x < 0, we solve

-y (x) + (b 1 x -k) 2 y(x) = λy(x)
with the condition that the solution is in L 2 (R -). This solution may be expressed as

x → U (-λ 2b , √ 2(- √ bx -k √ b )).
Therefore, if ψ b,n (•, k) is an eigenfunction associated with λ = λ b,n (k), it may be written as (up to an overall multiplicative constant):

ψ b,n (x, k) = U (-λ 2b , √ 2(- √ bx -k √ b )) if x < 0 αU (-λ 2 , √ 2(x -k)) if x > 0
where α ∈ R is a constant. Since the eigenfunction is C 1 at 0, we have

   U (-λ 2b , - √ 2 k √ b ) = αU (-λ 2 , - √ 2k) - √ 2bU (-λ 2b , - √ 2 k √ b ) = α √ 2U (-λ 2 , - √ 2k)
It is possible to solve the above conditions if and only if the 2 × 2 matrix

U (-λ 2b , - √ 2 k √ b ) U (-λ 2 , - √ 2k) - √ bU (-λ 2b , - √ 2 k √ b ) U (-λ 2 , - √ 2k)
is non-invertible.

We will use the following result on the asymptotic expansion of the determinant D b (λ, k) as k → +∞.

D b (λ, k) = e (1+ 1 b ) k 2 2 T 1 (λ, k) + e (-1+ 1 b ) k 2 2 T 2 (λ, k) + e (1-1 b ) k 2 2 T 3 (λ, k) + e (-1-1 b ) k 2 2 T 4 (λ, k), (3.9)
where the factors T j (λ, k) admit the following asymptotic expansions as k → +∞:

                           T 1 (λ, k) = 1 Γ( b-λ 2b )Γ( 1-λ 2 ) k -b+1 2b λ 2 3 2 - (1+b)λ 4b b b+λ 4b π 1 + O(k -2 ) , T 2 (λ, k) = 1 Γ( b-λ 2b ) k b-1 2b λ-2 √ π 2 2 λ b-1 4b b b+λ 4b (1 + b)(-sin λπ 2 ) 1 + O(k -2 ) , T 3 (λ, k) = 1 Γ( 1-λ 2 ) k -b+1 2b λ-2 √ π 2 2 λ -b+1 4b b b-λ 4b (1 + b)(-sin λπ 2b ) 1 + O(k -2 ) , T 4 (λ, k) = √ 2k b+1 2b λ 2 λ b+1 4b b b-λ 4b cos (1 + b)λπ 2b 1 + O(k -2 ) .
Proof. These expansions follow by the asymptotic expansion of the Weber function U . We refer the appendix, section A, and to [2, Chapter 19].

Remark 3.4. We have controlled these rather tedious calculations with the computer software Mathematica.

:= e -(1+ 1 b ) k 2 2 D b (λ, k), where Db (λ, k) = T 1 (λ, k) + e -k 2 T 2 (λ, k) + e -k 2 b T 3 (λ, k) + e -(1+ 1 b )k 2 T 4 (λ, k). (3.10)
We next note that the functions T 1 and T 2 are bounded as k → +∞. The coefficient T 3 grows like k ( 1-b 2b )λ-2 for λ large enough. Similarly, the term T 4 is increasing in k. However, the prefactors of these terms exhibit Gaussian decay.

The strategy of the proof of Theorem 1.6 consists of writing the band function λ b,pn (k) as

λ b,pn (k) = Ẽn + ε n (k),
where lim k→+∞ λ b,pn (k) = Ẽn , with either Ẽn = E n or Ẽn = bE n , and

lim k→+∞ ε n (k) = 0. For all k ∈ R, we have Db (λ b,pn (k), k) = Db ( Ẽn + ε n (k), k) = 0.
The distinction between the non-splitting and splitting cases is encoded in the gamma function coefficients of the T j (λ, k) terms in Lemma 3.3.

In the non-splitting case, we have for one of the gamma functions

Γ b -λ b,pn (k) 2b → Γ b -Ẽn 2b .
In case (ia), the argument of the gamma function is ((b + 1)/2 -n)/b that is never a negative integer so Γ -1 does not have a zero and does not contribute to the asymptotics. In case (ib), the argument of this gamma functions is (1 -n) so there are poles of the gamma function. The corresponding zeros of Γ -1 contribute to the vanishing rate of T 1 and T 2 . As for the other gamma function, we have

Γ 1 -λ b,pn (k) 2 → Γ (1 -n) (ia) Γ b+1 2 -nb (ib)
, so there is a pole in case (ia). These asymptotics are developed in Section 3.2.2.

In the splitting case, we have that b = E n /E m , and it follows that both gamma functions have poles. Consequently, both factors Γ -1 have a zero as k → +∞ and these contribute to the asymptotics as developed in Section 3.2.3. This analysis also shows that T 1 ( Ẽn , k) = 0 since for (ia) the inverse of the gamma function Γ( 1-λ 2 ) vanishes for λ = E n and in the case (ib) the inverse of the other gamma function Γ( b-λ 2b ) = 0 for λ = bE n . 3.2.2. No splitting of levels. Let us prove Item (i) in Theorem 1.6. The existence of p n comes from Lemma 3.1. Let us assume case (ia), the proof of case (ib) being symmetric. Then (1 + b) sin E n π 2 e -k 2 , obtained by using Lemma 3.3. We deduce the asymptotic behavior of ε n (k) as k → +∞. Since ε n (k) is negative the corresponding band functions λ b,n (k) approach the Landau level E n or bE n as k → +∞ from below. Taking into account the fact that λ b,pn (k) → +∞ as k → -∞, this means that these band functions are not monotonic and have at least one minimum.

3.2.3. Splitting of levels. Let us finally give the proof of Item (ii) in Theorem 1.6. When the condition for splitting is achieved, two band functions have the same limit as k → +∞. We prove that one level approaches the common Landau level from below, and hence has a minimum, whereas the other band function tends to its limit by above.

The existence of p n is a consequence of Lemma 3.1. We also know from this lemma that both functions ε ± n tends to 0 as k → +∞. We use the generic notation λ(k) = E n + ε(k) satified by these two functions. Recalling that b = En Em , we note that as k tends to +∞:

( 2 )

 2 If b 1 and b 2 have opposite signs and |b 1| = |b 2 |, it is sufficient to consider the case when -1 < b 1 < 0 < b 2 = 1.We call this case the "trapping magnetic step" (the term trapping will appear clearer after stating our results). (3) If |b 1 | = |b 2 |, we may restrict ourselves to the cases (b 1 , b 2 ) = (1, 1) and (b 1 , b 2 ) = (-1, 1

Figure 5 .

 5 Figure 5. Band functions for the symmetric double-well case b = (-1, 1).

Corollary 1 . 7 .

 17 With the same assumptions and notations as in Theorem 1.6, the following statements hold:(i) Let λ ∈ L \ S. Then each band function converging to λ admits a global minimum. (ii) Let λ ∈ S. Then k → λ pn (k) admits a global minimum.

1. 3 .Figure 6 .

 36 Figure 6. Band functions k → λ b,n (k) for b = (-1 3 , 1).

Figure 7 .

 7 Figure 7. Zoom near λ = 5 3 on the band function k → λ b,4 (k) for b = (-13 , 1). In dash line: the asymptotics from Therorem 1.6, case (ib). Here n = 3.

Figure 8 .

 8 Figure 8. Zoom near λ = 1 on the band functions k → λ b,p (k) for b = (-13 , 1) and p = 2, 3 (splitting case). In dash line: the asymptotics from Therorem 1.6, case (ii). Here λ = 1, n = 1 and m = 2.

Proposition 2 . 2 .

 22 Assume that b = (0, 1) and k ∈ R. The operator h b (k) has no embedded eigenvalues in its essential spectrum. Proof. Let us consider λ ≥ k 2 and ψ ∈ Dom(h b (k)) such that:

Lemma 2 . 4 .

 24 Assume that b = (0, 1) and k > 0. The number λ < k 2 is an eigenvalue of h b (k) if and only if there exists a non zero function ψ ∈

Lemma 2 . 5 .

 25 .5b) Moreover, the eigenfunctions of h b (k) can only vanish on R + . The eigenvalues of h b (k) are simple. Proof. We consider h b (k)ψ = λψ. Since ψ ∈ L 2 (R -), there exists A ∈ R such that, for x ≤ 0, ψ(x) = Ae x √ k 2 -λ . Then we have to solve -ψ + V b (x, k)ψ = λψ for x ≥ 0 with the transmission conditions ψ(0) = A and ψ (0) = A k 2 -λ or equivalently ψ (0) -k 2 -λ ψ(0) = 0. In particular, A cannot be zero. The simplicity is a consequence of the Cauchy-Lipschitz theorem. Assume that b = (0, 1). The functions R + k → λ b,n (k) are non decreasing.

operator 2 D 2 y

 2 + V b (y; 1). The minima of V b (•; 1) are reached for x = b -1 1 < 0 and x = 1. We have V b (1; 1) = 2 and V b (b -1 1 ; 1) = 2|b 1 |. Let us just recall the spirit of the analysis. Let us consider η > 0 such that b -1 1 + η < 0 and 1 -η > 0. Then, let us consider the operator hb, ,η = hle b, ,η ⊕ hri b, ,η where hle b, and hri b, are the Dirichlet realizations of 2 D 2

3. 2 . 1 .Proposition 3 . 2 .

 2132 Algebraic characterization of the eigenvalues of h b (k). We begin with an algebraic characterization of the eigenvalues of h b (k). Recall that U (a, x) is the first Weber parabolic cylindrical function introduced in Section 2 as a solution of (2.6). Assume that b = (b 1 , 1), with -1 < b 1 < 0. A number λ is an eigenvalue of h b (k) for k ∈ R if and only if it is a solution of D b (λ, k) = 0, where

Lemma 3 . 3 .

 33 Assume that b = (b 1 , 1), with -1 < b 1 < 0, and let b = |b 1 |. The determinant D b (λ, k) of Proposition 3.2 can be written as

Remark 3 . 5 . 1 b ) k 2 2

 3512 To obtain the large k asymptotics of D b (λ, k), we factor out the leading exponential factor e (1+ from the right side of (3.9). The zeros of D b (λ, k) are the same as the zeros of Db (λ, k)

2 / 1 Γ ( 1 -

 211 coming from the facts that the only roots of 1/Γ are the negative integers and that b-En ∈ -N, since this is not a splitting case. Recall that the roots of 1/Γ are simple. In particular, as k gets large, n)(1 + O(ε n (k)) = (-εn(k)2 )(-1) n-1 (n -1)!(1 + o(1)).

(3. 12 )

 12 As mentioned above, we replace λ b,pn (k) by E n +ε n (k) in the equation Db (λ b,pn (k), k) = 0 and obtain an equation in the formα(k)ε n (k) + β(k) = 0,where α and β are analytic functions such that α

1 Γ2b

 1 n) + O(ε(k) 2 ).Substituting these Taylor expansions into (3.9), the condition D b (λ(k), k) = 0 implies that ε ± n are the roots of the equationÂε 2 + ε( B + Ĉ) + D = 0,(3.13)where ( Â, B, Ĉ, D) are analytic functions of k respectively equivalent to the following functions as k → +∞:) = (-1) n+m (n -1)!(m -1)!π2 -En e -k 2 (1+1/b) . (3.14) The discriminant ∆ = ( B + Ĉ) 2 -4 Â D

  By continuity, we have, for all k > k n , λ b,n (k) < k 2 . Let us now prove the uniqueness. Let us consider a solution kn ≥ 0. For all integer p ≥ p 0 with p 0 large enough, we

	have λ b,n kn + 1 p < kn + 1 p	2

PDH was partially supported by NSF through grant DMS-1103104. Version of January 13, 2015.

is positive for k large enough since there hold B(k) Ĉ(k) and B(k) 2 D(k) as k goes to +∞ (notice also that Â(k) = o(1)). In particular, the two roots of (3.13) admit the following expansions:

Therefore, using the equivalents given in (3.14), we get as k gets large:

By definition, ε - n < ε + n . Due to the obvious sign of ε + and ε -for k large enough, we get ε ± = ε ± n and the result follows from the above asymptotic expansions.

Appendix A. Parabolic cylinder functions

We list some properties of parabolic cylinder functions, and refer to [2, Chapter 19]. Given a ∈ R, denote by U (a, x) and V (a, x) the so-called standard solutions of the differential equation

If follows by the series expansions of U and V that they are real-valued if a ∈ R and x ∈ R. Moreover, as x → +∞,

and

Here (x) j = x(x -1)(x -2) • • • (x -j + 1) denotes the Pochhammer symbol. It also holds that πV (a, x) = Γ(a + 1 2 ){sin(πa)U (a, x) + U (a, -x)}. (A.17)

This enables us to give an asymptotic expansion of U (a, x) as x → -∞. Indeed, as x → +∞,

If we keep only two terms in the sum,

Next, it holds that

and thus

Combining this with (A.18), we have, as x → +∞,


Franc ¸ois Laviolette
email: francois.laviolette@ift.ulaval.ca

Emilie Morvant
email: emilie.morvant@univ-st-etienne.fr

Liva Ralaivola
email: liva.ralaivola@lif.univ-mrs.fr

Jean-Francis Roy
email: jean-francis.roy@ift.ulaval.ca

On Generalizing the C-Bound to the Multiclass and Multi-label Settings

, gives a tight upper bound on the risk of a binary majority vote classifier. In this work, we present a first step towards extending this work to more complex outputs, by providing generalizations of the C-bound to the multiclass and multi-label settings.

Introduction

In binary classification, many state-of-the-art algorithms output prediction functions that can be seen as a majority vote of "simple" classifiers. Ensemble methods such as Bagging [START_REF] Breiman | Bagging predictors[END_REF], Boosting [START_REF] Schapire | Improved boosting algorithms using confidence-rated predictions[END_REF] and Random Forests [START_REF] Breiman | Random Forests[END_REF] are well-known examples of learning algorithms that output majority votes. Majority votes are also central in the Bayesian approach (see Gelman et al. [START_REF] Gelman | Bayesian Data Analysis[END_REF] for an introductory text); in this setting, the majority vote is generally called the Bayes Classifier. It is also -interesting to point out that classifiers produced by kernel methods, such as the Support Vector Machine [START_REF] Cortes | Support-vector networks[END_REF], can also be viewed as majority votes. Indeed, to classify an example x, the SVM classifier computes sign |S| i=1 α i y i k(x i , x) . Hence, as for standard binary majority votes, if the total weight of each α i y i k(x i , x) that votes positive is larger than the total weight for the negative choice, the classifier will output a +1 label (and a -1 label in the opposite case).

Most bounds on majority votes take into account the margin of the majority vote on an example (x, y), that is the difference between the total vote weight that has been given to the winning class minus the weight given to the alternative class. As an example, PAC-Bayesian bounds give bounds on majority votes classifiers by relating it to a stochastic classifier, called the Gibbs classifier which is, up to a linear transformation equivalent to the first statistical moment of the margin when (x, y) is drawn i.i.d. from a distribution [START_REF] Laviolette | From PAC-Bayes Bounds to Quadratic Programs for Majority Votes[END_REF]. Unfortunately, in most ensemble methods, the voters are weak and no majority vote can obtain high margins. Lacasse et al. [START_REF] Lacasse | PAC-bayes bounds for the risk of the majority vote and the variance of the Gibbs classifier[END_REF] proposed a tighter relation between the risk of the majority vote that take into account both the first and the second moments of the margin: the C-bound. This sheds a new light on the behavior of majority votes: it is not only how good are the voters but also how they are correlated in their voting. Namely, this has inspired a new learning algorithm named MinCq [START_REF] Laviolette | From PAC-Bayes Bounds to Quadratic Programs for Majority Votes[END_REF], whose performance is state-of-the-art. In this work, we generalize the C-bound for multiclass and multi-label weighted majority votes as a first step towards the goal of designing learning algorithms for more complex outputs. This paper is organized as follows. Section 2 recalls the C-bound in binary classification. We generalize it to the multiclass and multi-label settings in Sections 3 and 4. We conclude in Section 5.

The C-bound for Binary Classification

In this section, we recall the C-bound [START_REF] Lacasse | PAC-bayes bounds for the risk of the majority vote and the variance of the Gibbs classifier[END_REF][START_REF] Laviolette | From PAC-Bayes Bounds to Quadratic Programs for Majority Votes[END_REF] in the binary classification setting.

Let X ⊆ R d be the input space of dimension d, and let Y = {-1, +1} be the output space. The learning sample S = {(x i , y i )} m i=1 is constituted by m examples drawn i.i.d. from a fixed but unknown distribution D over X ×Y . Let H be a set of real-valued voters from X to Y . Given a prior distribution π on H and given S, the goal of the PAC-Bayesian approach is to find the posterior distribution ρ on H which minimizes the true risk of the ρ-weighted majority vote B ρ (•) given by

R D (B ρ ) = E (x,y)∼D I (B ρ (x) = y) , where B ρ (x) = sign E h∼ρ h(x) ,
and where I(a) = 1 if predicate a is true and 0 otherwise.

It is well-know that minimizing R D (B ρ ) is NP-hard. To get around this problem, one solution is to make use of the C-bound which is a tight bound over R D (B ρ ). This bound is based on the notion of margin of B ρ (•) defined as follows.

Definition 1 (the margin). Let M D ρ be the random variable that, given an example (x, y) drawn according to D, outputs the margin of B ρ (•) on that example, defined by M ρ (x, y) = y E h∼ρ h(x).

We then consider the first and second statistical moments of the random variable M D ρ , respectively given by µ

1 (M D ρ ) = E (x,y)∼D M ρ (x, y) and µ 2 (M D ρ ) = E (x,y)∼D (M ρ (x, y)) 2 .
According to the definition of the margin, B ρ (•) correctly classifies an example (x, y) when its margin is strictly positive, i.e. R D (B ρ ) = Pr (x,y)∼D (M ρ (x, y) ≤ 0). This equality makes it possible to prove the following theorem.

Theorem 1 (The C-bound of Laviolette et al. [START_REF] Laviolette | From PAC-Bayes Bounds to Quadratic Programs for Majority Votes[END_REF]). For every distribution ρ on a set of real-valued functions H, and for every distribution

D on X × Y , if µ 1 (M D ρ ) > 0, then we have R D (B ρ ) ≤ 1 - µ 1 (M D ρ ) 2 µ 2 (M D ρ )
.

Proof. The Cantelli-Chebyshev inequality states that any random variable Z and any a > 0, we have that Pr (Z ≤ E [Z]a) ≤ Var Z Var Z+a 2 . We obtain the result by applying this inequality with Z = M ρ (x, y), and with a = µ 1 (M D ρ ), and by using the definition of the variance.

Note that the minimization of the empirical counterpart of the C-bound is a natural solution for learning a distribution ρ that leads to a ρ-weighted majority vote B ρ (•) with low error. This strategy is justified thanks to an elegant PAC-Bayesian generalization bound over the C-bound, and have led to a simple learning algorithm called MinCq [START_REF] Laviolette | From PAC-Bayes Bounds to Quadratic Programs for Majority Votes[END_REF].

In the following, we generalize this important theoretical result in the PAC-Bayesian literature to the multiclass setting.

Generalizations of the C-bound for Multiclass Classification

In this section, we stand in the multiclass classification setting where the input space is still X ⊆ R d , but the output space is Y = {1, . . . , Q}, with a finite number of classes Q ≥ 2. Let H be a set of multiclass voters from X to Y . We recall that given a prior distribution π over H and given a sample S, i.i.d. from D, the PAC-Bayesian approach looks for the ρ distribution which minimizes the true risk of the majority vote B ρ (•). In the multiclass classification setting, B ρ (•) is defined by

B ρ (x) = argmax c∈Y E h∼ρ I(h(x) = c) . (1)
As in binary classification, the risk R D (B ρ ) of a ρ-weighted majority vote can be related to the notion of margin realized on an example (x, y). However, in multiclass classification, such a notion can be expressed in a variety of manners. In the next section, we present three versions of multiclass margins that are equivalent in binary classification.

Margins in Multiclass Classification

We first make use of the multiclass margin proposed by Breiman [START_REF] Breiman | Random Forests[END_REF] for the random forests, which can be seen as the usual notion of margin. Note that when Y = {-1, +1}, we recover the usual notion of binary margin of Definition 1. Definition 2 (the multiclass margin). Let D be a distribution over X ×Y , let H be a set of multiclass voters. Given a distribution ρ on H, the margin of the majority vote B ρ (•) realized on (x, y) ∼ P is

M ρ (x, y) = E h∼ρ I(h(x) = y) -max c∈Y,c =y E h∼ρ I(h(x) = c) .
Like in the binary classification framework presented in Section 2, the majority vote B ρ (•) correctly classifies an example if its ρ-margin is strictly positive, i.e., R D (B ρ ) = Pr (x,y)∼D (M ρ (x, y) ≤ 0).

We then consider the relaxation proposed by Breiman [START_REF] Breiman | Random Forests[END_REF] that is based on a notion of strength of the majority vote in regard to a class c. Definition 3 (the strength). Let H be a set of multiclass voters from X to Y and let ρ be a distribution on H. Let S D ρ,c be the random variable that, given an example (x, y) drawn according to a distribution D over X × Y , outputs the strength of the majority vote B ρ (•) on that example according to a class c ∈ Y , defined by S ρ,c (x, y)

= E h∼ρ I(h(x) = y) -E h∼ρ I(h(x) = c).
From this definition, one can show that

R D (B ρ ) = Pr (x,y)∼D (M ρ (x, y) ≤ 0) ≤ Q c=1 Pr (x,y)∼D (S ρ,c (x, y) ≤ 0) -1 , (2)
where we have the equality in the binary classification setting. Lastly, we consider a relaxation of the notion of margin, that we call the ω-margin. Definition 4 (the ω-margin). Let H be a set of multiclass voters from X to Y , let ρ be a distribution on H and let ω ≤ 1. Let M D ρ,ω be the random variable that, given an example (x, y) ∼ D over X × Y , outputs the ω-margin of the majority vote B ρ (•) on that example, defined by

M ρ,ω (x, y) = E h∼ρ I(h(x) = y) -1/ω . (3)
This notion of margin can be seen as the difference between the weight given by the majority vote to the correct class y and a certain threshold 1/ω. In the case of the binary classification, we have that the sign of the ω-margin with ω = 2 is the same than the sign of the binary margin. This observation comes from the fact that E h∼ρ I(h(x) = y) is the proportion of voters that vote y. In the binary case, this proportion is ≤ 1 2 when the majority vote makes a mistake, and > 1 2 otherwise. The following theorem relates the risk of B ρ (•) and the ω-margin associated to ρ. Theorem 2. Let Q ≥ 2 be the number of classes. For every distribution D over X × Y and for every distribution ρ over a set of multiclass voters H, we have Pr

(x,y)∼D (M ρ,Q (x, y) ≤ 0) ≤ R D (B ρ ) ≤ Pr (x,y)∼D (M ρ,2 (x, y) ≤ 0) . (4)
Proof. First, let us prove the left-hand side inequality. We have

RD(Bρ) = Pr (x,y)∼D (Mρ(x, y) ≤ 0) = Pr (x,y)∼D E h∼ρ I(h(x) = y) ≤ max c∈Y,c =y E h∼ρ I(h(x) = c) ≥ Pr (x,y)∼D E h∼ρ I(h(x) = y) ≤ E c∈Y,c =y E h∼ρ I(h(x) = c) = Pr (x,y)∼D E h∼ρ I(h(x) = y) ≤ 1 Q -1 1 -E h∼ρ I(h(x) = y) = Pr (x,y)∼D (Mρ,Q(x, y) ≤ 0) .
The right-hand side inequality is easily verified by observing that the majority vote necessarily makes a correct prediction if the weight given to the correct class y is higher than 1 2 .

All the above-mentioned notions of margin are equivalent if we stand in the binary classification setting. However, they differ in the multiclass setting. The multiclass margin of Definition 2 is associated to the true decision function in multiclass classification, and is calculated considering all other classes. The strength of Definition 3 depends on the true class y of x and corresponds to a combination of binary margins (one class versus another class) for c = y. The ω-margin of Definition 4 also depends on the true class y of x, but does not consider the other classes. This measure is easier to manipulate, but implies a higher indecision region (see Theorem 2).

P y y -1 2 1 E h∼ρ h(x) -1 2 1
Figure 1: Graphical representation of label y, hyperplane P y and vector E h∼ρ h(x).

Definition 5 (the multi-label margin). Let D be a distribution over X × Y , let H be a set of multilabel voters. Given a distribution ρ on H, the margin of the majority vote B ρ (•) on (x, y) is

M ρ (x, y) = E h∼ρ h(x) - 1 2 1 • y - 1 2 1 -max c∈Y ,c =y E h∼ρ h(x) - 1 2 1 • c - 1 2 1 .
As we did in the multiclass setting with the margin of Definition 2, we can upper bound the risk of the multi-label majority vote classifier by developing a C-bound using the margin of Definition 5. However, as this margin also depends on a max term, the derivation of a learning algorithm minimizing the resulting C-bound remains hard. To overcome this, we generalize M ρ,2 , the ω-margin with ω = 2 of Definition 4, to the multi-label setting, as follows. Definition 6. Let D be a distribution over X × Y , let H be a set of multi-label voters. Given a distribution ρ on H, the 2-margin of the majority vote

B ρ (•) on (x, y) is M ρ,2 (x, y) = E h∼ρ h(x) - 1 2 1 -y i→1/2 - 1 2 1 • y - 1 2 1 = y • E h∼ρ h(x) - 1 2 1 -E h∼ρ h(x) • 1 2 1 - 1 4
,

where i ∈ {1, .., Q} and y i→1/2 is obtained from y by replacing its i th coordinate by 1/2.

The second equality of the definition is obtained by straightforward calculation. Now, let P y , be the only hyperplane on which lies all the points of the form y i→1/2 for i = 1, . . . , Q. Since this hyperplane has normal y -1 2 1 , it follows from basic linear algebra that if M ρ,2 > 0, then vectors E h∼ρ h(x) and y will be on the same side of P y . It is also easy to see that in this case, we have B ρ (x) = y. Figure 1 shows an example in the case where Q = 2. Thus, we have that R D (B ρ ) ≤ Pr (x,y)∼D M ρ,2 (x, y) ≤ 0 , and following the same arguments as in Theorem 5, one can derive the following multi-label C-bound. Theorem 6. For every distribution ρ on a set of multi-label voters H and for every distribution D on X × Y , if µ 1 (M ρ,2 (x, y)) > 0, we have

R D (B ρ ) ≤ E (x,y)∼D I M ρ,2 (x, y) ≤ 0 ≤ 1 - µ 1 (M D ρ,2 ) 2 µ 2 (M D ρ,2 )
.

Conclusion and Outlooks

In this paper, we extend an important theoretical result in the PAC-Bayesian literature to the multiclass and multi-label settings. Concretely, we prove three multiclass versions and one multi-label version of the C-bound, a bound over the risk of the majority vote, based on generalizations of the notion of margin for multiclass and multi-label classification. These results open the way to extending the theory to more complex outputs and developing new algorithms for multiclass and multi-label classification with PAC-Bayesian generalization guarantees.

Generalizations of the C-bound in the Multiclass Setting

The following bound is based on the definition of the multiclass margin in multiclass (Definition 2). Theorem 3 (the multiclass C-bound). For every distribution ρ on a set of multiclass voters H, and for every distribution D on X × Y , such that µ 1 (M D ρ ) > 0, we have

.

Proof. The proof is the same than the one of the binary C-bound (see Theorem 1), by considering the multiclass majority-vote of Equation ( 1) and the multiclass margin of Definition 2.

This bound offers an accurate relation between the risk of the majority vote and the margin. However, the max term in the definition of the multiclass margin makes the derivation of an algorithm to minimize this bound much harder than in binary classification.

Thanks to the definition of the strength of Definition 3 and according to the proof process of the C-bound, we obtain the following relation. Theorem 4. For every distribution ρ on a set of multiclass voters H, and for every distribution

The result is obtained by using Inequality (2) in the proof of the C-bound.

This result can be seen as a sum of C-bounds for every class. A practical drawbacks of this bound in order to construct a minimization algorithm is that we have to minimize a sum of ratios. Finally, the C-bound obtained by using the ω-margin is given by the following theorem. Theorem 5. For every distribution ρ on a set of multiclass voters H, for every ω ≥ 1, and for every distribution D on X × Y , if µ 1 (M D ρ,ω ) > 0, we have

Proof. The result is obtained with the same proof process than the C-bound, by replacing the use of the random variable M D ρ by M D ρ,ω .

The ω-margin being linear, we are now able to build a bound minimization algorithm as in Laviolette et al. [START_REF] Laviolette | From PAC-Bayes Bounds to Quadratic Programs for Majority Votes[END_REF] for the multiclass classification setting.

Extending the ω-margin to the Multi-label Setting

In this section, we will extend the ω-margin with ω = 2 to the more general multi-label classification setting. Doing so, we will be able to upper bound the risk of the multi-label majority vote classifier. We stand in the multi-label classification setting where the input space is still X ⊆ R d , the space of possible labels is Y = {1, . . . , Q} with a finite number of classes Q ≥ 2, but we consider the output space Y = {0, 1} Q that contains vectors y of length Q where the i th element is 1 if example i is among the labels associated to the example x, and 0 otherwise. We consider a set H of multilabel voters h : X → Y . As usual in structured output prediction, given a distribution ρ over H, the multi-label majority vote classifier B ρ chooses the label c ∈ Y that has the lowest squared Euclidean distance with the ρ-weighted cumulative confidence,

where 1 is a vector of length Q containing ones. The multi-label margin is given by Definition 5.


Marie Pancher

Nathalie Désiré

Yen Ngo

Sepideh Akhavan

Coralie Pallier

Thierry Poynard

Dr Vincent Thibault
email: vincent.thibault@psl.aphp.fr

Akhavan Sepideh

Thibault Vincent

Coexistence of circulating HBsAg and anti-HBs antibodies in chronic hepatitis B carriers is not a simple analytical artifact and does not influence HBsAg quantification

Keywords:

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Background

HBsAg is a key marker for the diagnosis of acute or chronic HBV infection. Available qualitative or quantitative assays detect all circulating HBsAg containing particles, mainly infectious virions (Dane particles) but in a large excess non infectious/defective particles [START_REF] Seeger | Hepatitis B virus biology[END_REF]. While chronic infection is defined by the persistence of HBsAg over 6 months, HBsAg loss and appearance of anti-HBs Ab (anti-HBs) usually sign viral clearance. Yet, occurrence of anti-HBs is sometimes observed despite persistence of HBsAg and viral replication. The prevalence of this entity has been reported in less than 5 % to more than 25 % of CHB cases, depending on the studies [START_REF] Liu | Coexistence of hepatitis B surface antigen and anti-HBs in Chinese chronic hepatitis B virus patients relating to genotype C and mutations in the S and P gene reverse transcriptase region[END_REF][START_REF] Zhang | Coexistence of hepatitis B surface antigen (HBsAg) and heterologous subtype-specific antibodies to HBsAg among patients with chronic hepatitis B virus infection[END_REF][START_REF] Lada | Coexistence of hepatitis B surface antigen (HBs Ag) and anti-HBs antibodies in chronic hepatitis B virus carriers: influence of "a" determinant variants[END_REF][START_REF] Colson | Clinical and virological significance of the co-existence of HBsAg and anti-HBs antibodies in hepatitis B chronic carriers[END_REF].

The molecular and immunologic mechanisms leading to this serologic profile remain unclear but could be due to the selection of HBV escape mutants by immune pressure [START_REF] Pondé R A | The underlying mechanisms for the "simultaneous HBsAg and anti-HBs serological profile[END_REF][START_REF] Gerlich | The enigma of concurrent hepatitis B surface antigen (HBsAg) and antibodies to HBsAg[END_REF][START_REF] Dienstag | Concurrent hepatitis B surface antigen and antibody and the clonal selection theory of antibody diversity[END_REF]. In this scenario, the detected circulating antibodies are directed against an HBV strain not present anymore. Indeed, several reports have shown an association of this pattern with an increased proportion of HBsAg mutants characterized by specific substitution or deletion in the preS/S gene, especially in the "a" determinant region. Amino acid substitutions within this epitope could modify the antigenic conformation of HBsAg and alter the binding of both monoclonal and polyclonal anti-HBs antibodies; the main consequences being the emergence of an immune escape variant and an alteration of HBsAg detection assay [START_REF] Zhang | Coexistence of hepatitis B surface antigen (HBsAg) and heterologous subtype-specific antibodies to HBsAg among patients with chronic hepatitis B virus infection[END_REF][START_REF] Lada | Coexistence of hepatitis B surface antigen (HBs Ag) and anti-HBs antibodies in chronic hepatitis B virus carriers: influence of "a" determinant variants[END_REF][START_REF] Kohno | Mutations in the envelope gene of hepatitis B virus variants co-occurring with antibody to surface antigen in sera from patients with chronic hepatitis B[END_REF][START_REF] Huang | PreS deletion mutations of hepatitis B virus in chronically infected patients with simultaneous seropositivity for hepatitis-B surface antigen and anti-HBS antibodies[END_REF][START_REF] Chen | Mutations in hepatitis B virus DNA from patients with coexisting HBsAg and anti-HBs[END_REF][START_REF] Weber | Genetic variability of the S gene of hepatitis B virus: clinical and diagnostic impact[END_REF][START_REF] Servant-Delmas | Variable capacity of 13 hepatitis B virus surface antigen assays for the detection of HBsAg mutants in blood samples[END_REF].

Most recently, the amount of circulating HBsAg has been recognized as a potential biomarker linked to the liver disease stage, the risk of HCC or the chance to respond to interferon therapy [START_REF] Nguyen | Hepatitis B surface antigen levels during the natural history of chronic hepatitis B: a perspective on Asia[END_REF][START_REF] Jaroszewicz | Hepatitis B surface antigen (HBsAg) levels in the natural history of hepatitis B virus (HBV)-infection: a European perspective[END_REF][START_REF] Martinot-Peignoux | Prediction of disease reactivation in asymptomatic hepatitis B e antigen-negative chronic hepatitis B patients using baseline serum measurements of HBsAg and HBV-DNA[END_REF][START_REF] Tseng | Serum hepatitis B surface antigen levels help predict disease progression in patients with low hepatitis B virus loads[END_REF]. Beside the most widely used Architect assay (Abbott), two other CE marked assays are also available for the quantification of HBsAg, the Elecsys platform (Roche) and the Liaison-XL from DiaSorin. These assays use both mononoclonal and/or polyclonal antibodies to enhance recognition of HBsAg and HBsAg mutants. Several studies have convincingly demonstrated the comparable performances of Roche and Abbott assays [START_REF] Wursthorn | Correlation between the Elecsys HBsAg II assay and the Architect assay for the quantification of hepatitis B surface antigen (HBsAg) in the serum[END_REF][START_REF] Sonneveld | A comparison of two assays for quantification of Hepatitis B surface Antigen in patients with chronic hepatitis B[END_REF][START_REF] Maylin | Quantification of hepatitis B e antigen between Elecsys HBeAg and Architect HBeAg assays among patients infected with hepatitis B virus[END_REF]. More recently, the DiaSorin assay was shown to perform comparably to the Architect assay in sequential serum samples from 14 HBe-negative patients [START_REF] Burdino | Quantification of hepatitis B surface antigen with the novel DiaSorin LIAISON XL Murex HBsAg Quant: Correlation with the ARCHITECT quantitative assays[END_REF].

Objectives

The aims of our study were first to document the performances of the DiaSorin assay compared to our routine Abbott's Architect HBsAg quantification. Secondly, the robustness of HBsAg quantification using the three CE marked available assays on a panel of CHB seras containing both HBsAg and anti-HBs was challenged. One main issue was to determine whether concomitant detection of both HBsAg/anti-HBs could solely be an analytical artifact linked to a specific assay. Likewise, a further goal was to assess the ability of these assays to quantify potential variants more prone to be selected in these peculiar serological profiles.

Study design

Patients and sera

In the first part of the study comparing Abbott and DiaSorin on HBsAg positive samples, 227 sera successively tested for HBsAg quantification between September and November 2012 on Architect (Abbott, Rungis, France) were tested on DiaSorin Liaison-XL (Antony, France).

One hundred and one sera corresponding to 62 CHB patients identified HBsAg/anti-HBs positive on Architect between January 2008 and June 2012 and stored at -20°C were selected. Exclusion criteria were any HIV, hepatitis C or hepatitis D coinfection, HBIg passive immunization or reactivation of past HBV infection. All sera were subsequently tested for HBsAg and anti-HBs on the Roche Elecsys system (Meylan, France) and DiaSorin Liaison-XL, according to each manufacturer's recommendation. The measurements were performed within 5 days on all systems. Among the 62 selected patients, 31 had also an HBV viral load above 100 IU/mL on the same day as HBsAg quantification and sequencing of the HBsAg gene could be performed on -80°C stored plasma.

Unless specified, HBsAg quantification was reported in log 10 IU/mL and anti-HBs in IU/L. The study general outline is presented on figure 1.

HBsAg gene sequencing method HBV-DNA was purified from 500µL of plasma using Nuclisens EasyMAG (Biomérieux, Marcilly, France) and resuspended in 70µL of sterile water. Amplification of part of the HBsAg coding region (nt. 186-1196, aa 100-227) was performed as previously described (4). Sequences were aligned and compared to selected Genbank reference sequences of known genotype using SeqScape (Applied Biosytems) and Mega (v5) softwares [START_REF] Tamura | MEGA5: molecular evolutionary genetics analysis using maximum likelihood, evolutionary distance, and maximum parsimony methods[END_REF]. For analysis purpose, HBsAg was divided into 3 regions: residues 100-123, 124-147 ("a" determinant) and 148-221. Substitutions were determined manually and with the help of two online tools (http://hbv.bioinf.mpi-inf.mpg.de/ and http://www.hiv-grade.de).

Statistical analysis

Statistical analysis, were performed using Analyze-it software. Chisquare and Fischer tests were used for categorical variables. Mean and medians were analyzed with Mann Whitney and Kruskal tests.

Bland-Altman analysis was used to compare methods. Significance was defined by a P value below 0.05.

Results

Overall performances of Liaison-XL HBsAg quantification on 227 HBsAg positive samples

On these 227 samples, there was a strong correlation (p<0.001) between both Abbott and DiaSorin systems with an r 2 of 0.96 (fig. 2A) despite a mean quantification difference of -0.16 log 10 IU/mL (95% CI: -0.19; -0.14) throughout the quantification range (fig. 2B). The results provided by the Liaison-XL were constantly and significantly lower than those by the Architect assay (p=0.04).

Characteristics of patients carrying both HBsAg and anti-HBs

Among 2578 chronic HBsAg carriers followed between 2008 and 2012, 129 (5%) patients were concomitantly HBsAg/anti-HBs positive and 62 (2.4%) were included in the study (fig. 1). Except from above mentioned inclusion criteria, available stored sample volume also restricted inclusion into the study. The main characteristics of the patients selected for the principal and molecular studies are summarized in table 1. Noteworthy, most of the samples had an anti-HBs concentration lower than 50 IU/L (median: 24 IU/L) but 31 (30.7%) and 14 (13.9%) had values above 50 and 100 IU/L, respectively. No correlation was found between the concentration of HBsAg and anti-HBs (data not shown). The median viral load of the patients selected for S-gene sequencing was 3.6 log IU/mL; the best performance for sequencing being usually obtained above 1000 IU/mL.

Qualitative analysis of the HBsAg results obtained with the three assays

Concurrent HBsAg and anti-HBs reactivity was confirmed unequivocally with the three techniques for 65 % of the 101 samples (table 2). The concordance between all techniques was better for HBsAg than for anti-HBs. Noteworthy, the median anti-HBs values obtained with Abbott for concordant (32.5 IU/L; IQR 69.4) versus discordant (19 IU/L; IQR 21.8) results were statistically different (p=0.01).

The HBsAg discordant samples (1 for DiaSorin and 2 for Roche) belong to the same patient who was followed at several visits and had persistently very low HBsAg levels: 0.07 and 0.05 IU/mL on Architect (table 3). Noteworthy, a third sample with a value of 0.1 IU/mL with Abbott was correctly detected with Roche and DiaSorin, respectively at 0.041 UI/mL and 0.057 IU/mL.

Quantitative analysis of HBsAg results obtained with the three assays

All HBsAg positive data for HBsAg/anti-HBs samples were considered in the quantitative assessment of these assays.

The overall correlation of all assays was satisfactory with r 2 ranging from 0.93 for DiaSorin (n=100)

and Roche (n=99) to 0.97 for Abbott and Roche assays (fig. 3). The best concordance was obtained between Roche and Abbott with a mean bias of 0.022 log IU/mL (95% CI: -0.024 to 0.067) in favor of Roche while the most divergent quantification was observed between DiaSorin and Roche (bias of 0.130 log IU/mL in favor of Roche with 95% CI from 0.048 to 0.212). The mean bias between Architect and Liaison-XL was -0.113 log IU/mL; the lower values being reported by DiaSorin. A comparison of the biases was performed between samples concordant for the detection of both Ag and Ab on all techniques (n=66) to the discordant ones (n=33); no statistically significant difference was observed according to the discordant status.

Quantitative analysis of the results obtained with the three assays for anti-HBs antibodies

To limit any bias, only samples reactive (anti-HBs > 10 IU/L) with the three techniques (n=68) were taken into consideration for this analysis. Each technique value was compared to the obtained mean value on the three techniques. The best correlation was obtained with DiaSorin assay (R2 = 0.81). The mean differences (95% CI) to the mean for Abbott, DiaSorin and Roche were, -18.9 (-31 to -6.8); 21.2 (8.4 to 33.9) and -2.30 (-12.1 to 7.5) IU/L, respectively (data not shown).

Variability of HBsAg quantification according to genotypic features

Thirty one samples could be sequenced for genotype determination and HBsAg substitution analysis (table 1). Ten genotype C, 8 D, 7 E, 4 B, and 1 each genotype A and F were detected. To compare HBsAg quantification between all techniques, the mean value obtained by the 3 techniques for each sample was calculated. Then, the percentage of HBsAg quantification over the mean was calculated for each technique and analyzed according to the genotype or the presence of substitution.

Sequence analyses showed that 77% (24/31 cases) of the HBsAg/anti-HBs samples harbored amino acid substitutions on residues 100-221. Substitutions previously described to modify HBsAg antigenicity were closely examined [START_REF] Zaaijer | Substitution rate of the hepatitis B virus surface gene[END_REF]. Forty five percent (n=14) of the strains had such substitutions, with the following amino-acid changes: sT126I (n=3), sQ129G (n=1), sG130N (n=1), sN131H (n=1), sF134S (n=1), sC137S (n=1), sS143L (n=3), sG145R (n=2) (table 4). The most described sG145R substitution was detected in 2 cases (6%). All but one genotype D strains carried S substitution, while only 3 (30%) variants were found among the 10 genotype C strains.

Quantification differences between the three assays according to genotype and "a" determinant substitutions were specifically analyzed on the three systems (figure 4 and5). Although the size of our population was limited, we found a slight influence of HBV genotype on HBV quantification whatever the considered technique. Regarding the "a" determinant substitutions, Roche and DiaSorin techniques seemed to be the most affected. Noteworthy, all variants were correctly detected but DiaSorin tended to underestimate the variant concentration while Roche gave higher than expected values.

DISCUSSION

HBsAg concentration measurement has become a clinically relevant marker in addition to the well standardized panel of serological and molecular markers. As for HBV viral load, standardization of both the measurement unit and the techniques is mandatory to provide clinicians with comparable quantitative HBsAg values whatever the utilized method. Both Roche and Abbott HBsAg quantification techniques have been extensively studied with the conclusion that both techniques are overall very comparable and can be used indifferently to monitor patients [START_REF] Wursthorn | Correlation between the Elecsys HBsAg II assay and the Architect assay for the quantification of hepatitis B surface antigen (HBsAg) in the serum[END_REF][START_REF] Sonneveld | A comparison of two assays for quantification of Hepatitis B surface Antigen in patients with chronic hepatitis B[END_REF][START_REF] Maylin | Comparison between Elecsys HBsAg II and Architect HBsAg QT Assays for Quantification of Hepatitis B Surface Antigen among Patients Coinfected with HIV and Hepatitis B Virus[END_REF]. By contrast, the newest DiaSorin assay has only been studied in two small studies comparing its performance to the Abbott system [START_REF] Burdino | Quantification of hepatitis B surface antigen with the novel DiaSorin LIAISON XL Murex HBsAg Quant: Correlation with the ARCHITECT quantitative assays[END_REF][START_REF] Krawczyk | Clinical performance of the novel DiaSorin LIAISON(®) XL murex: HBsAg Quant, HCV-Ab, HIV-Ab/Ag assays[END_REF]. Our study first confirms these previous results and indicates that the Architect values are always 0.16 log IU/mL higher than those from the Liaison-XL. From all published studies, one may conclude that a patient could be followed indifferently with Abbott or Roche assay but an adjustment may be required if the Liaison-XL is used during follow up.

Simultaneous detection of both HBs Ag and Ab is often puzzling in clinical situations. Indeed, although this peculiar profile has been described by several authors in different settings, one may wonder whether it corresponds to a true analytical observation [START_REF] Zhang | Coexistence of hepatitis B surface antigen (HBsAg) and heterologous subtype-specific antibodies to HBsAg among patients with chronic hepatitis B virus infection[END_REF][START_REF] Lada | Coexistence of hepatitis B surface antigen (HBs Ag) and anti-HBs antibodies in chronic hepatitis B virus carriers: influence of "a" determinant variants[END_REF][START_REF] Colson | Clinical and virological significance of the co-existence of HBsAg and anti-HBs antibodies in hepatitis B chronic carriers[END_REF][START_REF] Tabor | Coincident hepatitis B surface antigen and antibodies of different subtypes in human serum[END_REF][START_REF] Shiels | Frequency and significance of concurrent hepatitis B surface antigen and antibody in acute and chronic hepatitis B[END_REF][START_REF] Yamamoto | Naturally occurring escape mutants of hepatitis B virus with various mutations in the S gene in carriers seropositive for antibody to hepatitis B surface antigen[END_REF][START_REF] Zhang | Elimination of hepatitis B virus surface antigen and appearance of neutralizing antibodies in chronically infected patients without viral clearance[END_REF]. Overall, we show that the qualitative agreement is 65% for the coexistence of both Ag and Ab, providing a strong argument that these detections are not a simple artifact. These data are comparable to the 78.6% concordance rate reported in a small study using only two first generation qualitative assays [START_REF] Cheng | Discrepancies between two automated immunoassay systems in determining hepatitis B virus markers in serum samples with concomitant presence of antigens and antibodies[END_REF]. Each of these techniques uses different combination of Ag and antibodies in different format and confirmation by three independent techniques is reassuring. According to our experience, Ag/Ab samples may sometimes pose problem when performing an HBsAg neutralization assay as proposed by manufacturers. One should stress that a thorough selection was performed in our population to not include clinical situations where such profiles may simply be a consequence of therapeutic interventions. Indeed, coexistence of both Ag and Ab markers is common during HBV reactivation (occurrence of HBsAg in a patient with resolved HBV infection) or administration of immunoglobulins; in these settings coexistence of both markers are usually only transient. In our set of samples, the agreement for anti-HBs detection screened by Abbott, is not as convincing as for HBsAg but reach 86% and 71% with DiaSorin and Roche assays, respectively. In terms of correlation between anti-HBs measurement, the results are concordant with what observed on classical samples, anti-HBs positive but HBsAg negative [START_REF] Kinn | Performance of the DiaSorin LIAISON(®) anti-HBs II for the detection of hepatitis B surface antibodies: comparison with the Abbott Architect anti-HBs assay[END_REF]. Taking into account the mean of the three techniques for anti-HBs determination, it should be pointed out that 15.8% (n=16) samples were above 100 IU/L, a threshold often considered as highly protective in exposed persons. Yet, it emphasizes the risk of considering someone protected against HBV infection if anti-HBs are solely looked for in a screening program;

obviously, anti-HBs should not be used as a single screening test as it may falsely misdiagnose a vaccine status in a patient with an active HBV infection! [START_REF] Prevention | Immunization of health-care personnel: recommendations of the Advisory Committee on Immunization Practices (ACIP)[END_REF] Correlations between HBsAg measurement assays were very satisfactory even in these peculiar sera where anti-HBs could potentially interfere with measurement (fig. 3). Astonishingly, all assays showed performances very similar to what observed in classical HBsAg positive samples by us and others [START_REF] Wursthorn | Correlation between the Elecsys HBsAg II assay and the Architect assay for the quantification of hepatitis B surface antigen (HBsAg) in the serum[END_REF][START_REF] Burdino | Quantification of hepatitis B surface antigen with the novel DiaSorin LIAISON XL Murex HBsAg Quant: Correlation with the ARCHITECT quantitative assays[END_REF][START_REF] Maylin | Comparison between Elecsys HBsAg II and Architect HBsAg QT Assays for Quantification of Hepatitis B Surface Antigen among Patients Coinfected with HIV and Hepatitis B Virus[END_REF][START_REF] Krawczyk | Clinical performance of the novel DiaSorin LIAISON(®) XL murex: HBsAg Quant, HCV-Ab, HIV-Ab/Ag assays[END_REF][START_REF] Verheyen | The detection of HBsAg mutants expressed in vitro using two different quantitative HBsAg assays[END_REF]. The concordance for HBsAg detection was also very strong and only 2 samples at the limit of detection by all techniques were not detected on the Elecsys. The above conclusions, regarding the lower values obtained on the Liaison-XL, were still valid on these Ag/Ab positive samples. One rationale for testing such Ag/Ab samples was the likelihood to artificially enrich our set with HBsAg variants [START_REF] Gerlich | The enigma of concurrent hepatitis B surface antigen (HBsAg) and antibodies to HBsAg[END_REF][START_REF] Brunetto | Chance and necessity of simultaneous HBsAg and anti-HBs detection in the serum of chronic HBsAg carriers[END_REF]. Actually, variant strains were detected in 77% (24/31) of the samples and 45% (14/31) of the substitutions could potentially modify HBsAg antigenicity [START_REF] Zaaijer | Substitution rate of the hepatitis B virus surface gene[END_REF]. Although our study was not primarily designed to demonstrate an influence of HBV genotype on HBsAg quantification and the tested numbers are rather limited, some unexpected differences between techniques for the most represented genotypes, B, C, D and E were observed. This observation is currently being further documented in a collaborative study on more samples. Many studies have shown that "a" determinant variants were incorrectly detected or quantified by commercially available assays in the past [START_REF] Servant-Delmas | Variable capacity of 13 hepatitis B virus surface antigen assays for the detection of HBsAg mutants in blood samples[END_REF]. The attractive strategy of protein unfolding proposed by DiaSorin did not seem to significantly enhance the detection of variants and the Architect system was the less affected by the substitutions present on our samples (fig. 5) [START_REF] Krawczyk | Clinical performance of the novel DiaSorin LIAISON(®) XL murex: HBsAg Quant, HCV-Ab, HIV-Ab/Ag assays[END_REF]. Noteworthy, the 2 samples carrying the sG145R variant were quantified to the same level by all three techniques. Overall, it was not possible to identify a specific protein pattern that would particularly affect one technique and the most divergent results were not specifically observed for HBsAg variants.

In summary, this study demonstrates the overall good performance of quantitative assays for HBsAg even in patients with coexisting HBs and anti-HBs. Although, the precise reasons for this entity remain elusive, confirmation on three techniques tend to prove that a technical artifact is not to be blamed in the majority of the cases. The difference observed between techniques should remind us that sequential monitoring should ideally be performed with a unique assay even though all techniques provide well correlated quantification. HBsAg values expressed as IU/mL were expressed as a percentage to the mean values obtained with the 3 assays and grouped according to the genotype B (n=4), C (n=10), D (n=8) and E (n=7). Statistically significant differences according to genotype are indicated for each assay.

Figure legends

Figure 5: Box plot representation of HBsAg quantification difference according to the presence of "a" determinant substitution. HBsAg values expressed as IU/mL were expressed as a percentage to the mean values obtained with the 3 assays and grouped according to the presence (Y, n=11) or absence (N, n=20), of "a" substitution. Statistically significant differences according to "a" substitution are indicated for each assay.

Figure 1 :Figure 2 :

 12 Figure legendsFigure 1: General outline of the study Figure 2: Correlation between HBsAg quantification using Abbott and DiaSorin assays (n=227) (A) and Bland Altman representation (B). The mean bias between both techniques was -0.162 logIU/mL.

Figure 3 :Figure 4 :

 34 Figure 3: Correlation between the mean HBsAg values obtained by the three techniques (x axis) and values obtained by each technique (y axis) on 99 samples detected by all techniques. Abbott: closed circles, plain line (y = 0.9724x + 0.0361, R² = 0.9877); DiaSorin: open squares, interrupted line (y = 1.0457x -0.2432, R² = 0.9576); Roche: closed triangles, dotted line: (y = 1.0042x -0.0031, R² = 0.9801) Figure 4: Box plot representation of HBsAg quantification difference according to HBV genotype.HBsAg values expressed as IU/mL were expressed as a percentage to the mean values obtained with the 3 assays and grouped according to the genotype B (n=4), C (n=10), D (n=8) and E (n=7). Statistically significant differences according to genotype are indicated for each assay.

Figure 2 - 1 20 Figure 3 :

 21203 Figure 2

Table 1 :

 1 Studied population main characteristics

	Parameters: median (range)	Main cohort	Molecular study selected patients
	Sex ratio M/F	37/25	18/13
	Age (y)	44.9	45.2
		(22.1-79.6)	(22.1-72.2)
	HBsAg Abbott (IU/mL)	556	1545
		(0.1-233 185)	(4.7-233 185)
	Anti-HBs Ab Abbott IU/mL	24	22
		(11-257)	(11-115)

Table 2 :

 2 qualitative agreement between all techniques (n=101)

	Abbott Reactivity	DiaSorin	Roche	DiaSorin and Roche agreement
	HBsAg	100 (99%)	99 (98%)	99 (98%)
	Anti-HBs	87 (86%)	72 (71%)	68 (67%)
	Ag/Ab	86 (85%)	70 (69%)	66 (65%)

Table 3 :

 3 Main results for the two HBsAg discrepant samples (samples below detection limit are underlined)

	Sample	Marker	Abbott DiaSorin Roche
	82	HBsAg (IU/mL)	0.07	0.032	<0.05
		Anti-HBs (IU/L)	21	41	32
	91	HBsAg (IU/mL)	0.05	<0.030	<0.05
		Anti-HBs (IU/L)	32	57	61

Table 4 :

 4 Main substitutions known to influence HBsAg antigenicity detected on each strain

	HBsAg residues

Acknowledgements: Authors are grateful to Odile Pertrizeard, Lydie Postic, Hélène Tsyrina-Kouyoumdjian and Raphael Crosnier for their excellent technical expertise.

Funding: Reagents and technical support were kindly provided by DiaSorin for the Liaison-XL and Roche for the Elecsys.

Competing interests: All authors but VT declare to have no conflict of interest in the presented work.

VT declares to have received speaker fees and travel support from Abbott, DiaSorin and Roche.

Ethical approval: this non interventional study has been performed according to the national current health regulations and in accordance with the ethics principles of the Declaration of Helsinki.