Robert Seiringer
email: robert.seiringer@mcgill.ca

Jakob Yngvason
email: jakob.yngvason@univie.ac.at

Valentin A Zagrebnov
email: valentin.zagrebnov@univ-amu.fr

DISORDERED BOSE EINSTEIN CONDENSATES WITH INTERACTION IN ONE DIMENSION

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Introduction

While the effects of random potentials on single particle Schrödinger operators [START_REF] Pastur | Spectra of Random and Almost-Periodic Operators[END_REF] and ideal Bose gases [START_REF] Lenoble | Bose-Einstein condensation in random potentials[END_REF]- [START_REF] Th | On the Nature of Bose-Einstein Condensation Enhanced by Localization[END_REF] are rather well explored, the present understanding of such effects on many-body systems of interacting particles is much less complete. In recent years, however, many papers concerning the interplay of Bose-Einstein Condensation (BEC) and disorder have appeared of which references [START_REF] Gimperlein | Ultracold Atoms in Optical lattices with Random One-Site Interactions[END_REF]- [START_REF] Cardoso | Anderson localization of matter waves in chaotic potentials[END_REF] are but a sample.

In this paper we present results on a model that is in a sense the simplest one imaginable where this interplay can be studied by rigorous mathematical means. This is the onedimensional Lieb-Liniger model [START_REF] Lieb | Exact Analysis of an Interacting Bose Gas. I[END_REF] of bosons with contact interaction in a 'flat' trap, augmented by an external random potential that is generated by Poisson distributed point scatterers of equal strength. We study the ground state and prove that, no matter the strength of the random potential, BEC is not destroyed by the random potential in the Gross-Pitaevskii (GP) limit where the particle number tends to infinity while the coupling parameter in a mean-field scaling stays fixed. The character of the wave function of the condensate, however, depends in an essential way on the relative size of the three parameters involved. These are the scaled coupling parameter γ for the interaction among the particles, the density of the scatterers ν, and the strength σ of the scattering potential. All these parameters are assumed to be large in suitable units.

The investigation consists of three parts. In the following Section 2 we prove Bose-Einstein condensation in the ground state of our model in a GP limit. This result holds, in fact, for quite general external potentials, provided they are bounded below. Specific properties of the random potential enter only in the estimate of the energy gap of an Date: July 30, 2012.

effective mean-field Hamiltonian that sets a bound to the depletion of the condensate for finite particle number.

Section 3 is concerned with the properties of the condensate in dependence of the parameters. In particular, for σ ≫ 1 we identify three different "phases" of the condensate. For large γ ≫ ν 2 the condensate is extended over the whole trap (the unit interval in our model). A transition from a delocalized to a localized state takes place when γ is of the order ν 2 , in the sense that for γ ≪ ν 2 the density is essentially distributed among a fraction λ ≪ 1 of the ν ≫ 1 intervals between the obstacles. For γ ≫ ν/(ln ν) 2 we still have λν ≫ 1, but for γ ∼ ν/(ln ν) 2 the fraction of intervals that are significant occupied shrinks to O(ν -1 ). We stress, however, that in all cases there is complete BEC into a single state in the limit when the particle number tends to infinity.

In an appendix, we prove an estimate on the energy gap of one-dimensional Schrödinger Hamiltonians. This estimate, applied to a mean field Hamiltonian, is needed to establish a lower bound for the condensate fraction that depends only on the parameters of the problem and not on the individual realizations of the random potential. The estimate on the energy gap is a generalization of a result from [START_REF] Kirsch | Universal lower bounds on eigenvalue splittings for one dimensional Schrödinger operators[END_REF] and is of independent interest. Finally, for convenience of the reader, we collect in a second appendix some facts about the Poisson distribution of obstacles.

Our study of the Gross-Pitaevskii energy functional in Section 3 can be regarded as complementary to the investigations in [START_REF] Gimperlein | Ultracold Atoms in Optical lattices with Random One-Site Interactions[END_REF][START_REF] Lugan | Ultracold Bose Gases in 1D Disorder: From Lifshits Glass to Bose-Einstein Condensate[END_REF][START_REF] Sanches-Palencia | Disorder-induced trapping versus Anderson localization in Bose-Einstein condensates expanding in disordered potentials[END_REF][START_REF] Pikovsky | Destruction of Anderson Localization by a Weak Nonlinearity[END_REF][START_REF] Lugan | One-dimensional Anderson localization in certain correlated random potentials[END_REF][START_REF] Aleiner | A finite-temperature phase transition for disordered weakly interacting bosons in one dimension[END_REF][START_REF] Piraud | Localization of a matter wave packet in a disordered potential[END_REF] where transitions between delocalization and localization for solutions of the GP equation (also the time-dependent one) are considered from different points of view. Among these papers [START_REF] Lugan | Ultracold Bose Gases in 1D Disorder: From Lifshits Glass to Bose-Einstein Condensate[END_REF] is the one closest to ours as far as the questions asked are concerned, and it is appropriate to make a brief comparison.

One difference is that the random potentials considered in [START_REF] Lugan | Ultracold Bose Gases in 1D Disorder: From Lifshits Glass to Bose-Einstein Condensate[END_REF] are intended to model experimentally generated laser speckles and somewhat different from the present model. In both cases, however, there is a length scale associated with the random potential (ν -1 in our case, σ R in [START_REF] Lugan | Ultracold Bose Gases in 1D Disorder: From Lifshits Glass to Bose-Einstein Condensate[END_REF]), and the strength of the random potential that is denoted by V R in [START_REF] Lugan | One-dimensional Anderson localization in certain correlated random potentials[END_REF] is the analogue of our νσ. Moreover, our coupling parameter γ may be compared with the chemical potential µ in [START_REF] Lugan | Ultracold Bose Gases in 1D Disorder: From Lifshits Glass to Bose-Einstein Condensate[END_REF].

The main difference, however, is that in [START_REF] Lugan | Ultracold Bose Gases in 1D Disorder: From Lifshits Glass to Bose-Einstein Condensate[END_REF] BEC is stated to hold only for large µ (analogous to large γ in our model), while for weaker coupling a breakdown of BEC through a transition to a fragmented condensate and finally to a Lifschitz glass phase is claimed. We, however, prove rigorously that complete BEC holds in the whole parameter range considered when the particle number tends to infinity. Some further comments on this issue will be made at the end of Section 2.

The Many-Body Model and BEC

The model we consider is the Lieb-Liniger model of bosons with contact interaction on the unit interval, with an additional external potential V . In the next section we shall take V to be a sum of delta functions at random points in the unit interval, but for the proof of the energy bounds and BEC in the present section the only assumption used is that V is a nonnegative sum of delta functions and a locally integrable function. The Hamiltonian on the Hilbert space L 2 ([0, 1], dz) ⊗ N symm is

H := N i=1 -∂ 2 z i + V (z i ) + γ N i<j δ(z i -z j ) (1)
with γ ≥ 0, and we assume Dirichlet boundary conditions at the end points of the unit interval. The Hamiltonian has a unique ground state and we denote the ground state energy by E QM 0 . The Gross-Pitaevskii energy functional is defined by

E GP [ψ] := 1 0 |ψ ′ (z)| 2 + V (z)|ψ(z)| 2 + (γ/2)|ψ(z)| 4 dz.
(

) 2
Again we assume Dirichlet boundary conditions. By standard methods one proves that there is a unique nonnegative minimizer ψ 0 ∈ H 1 0 ([0, 1]), normalized so that ψ 2 0 (z)dz = 1, with a corresponding energy e GP . This is equal to the ground state energy e 0 of the meanfield Hamiltonian

h := -∂ 2 z + V (z) + γψ 0 (z) 2 - γ 2 ψ 4 0 (3)
with ψ 0 as the corresponding eigenfunction.

THEOREM 2.1 (Energy bounds).

e 0 ≥ E QM 0 N ≥ e 0 1 -(const.)N -1/3 min{γ 1/2 , γ} . (4)
Note that the error term depends only on γ besides N and is independent of V (as long as V ≥ 0).

We denote the second eigenvalue of h by e 1 and by N 0 the average occupancy of ψ 0 in the many-body ground state Ψ 0 of H, i.e., N 0 := tr ρ (1)

|ψ 0 ψ 0 | (5)
where ρ (1) is the one-particle reduced density matrix of Ψ 0 . The depletion of the condensate is 1

-N 0 N . THEOREM 2.2 (BEC). 1 - N 0 N ≤ (const.) e 0 e 1 -e 0 N -1/3 min{γ 1/2 , γ}. (6)
Theorem 2.2 implies in particular complete BEC in the limit N → ∞ with γ and V fixed. In fact, Eq. [START_REF] Yukalov | Bose-Einstein condensed systems in random potentials[END_REF] implies that the reduced one-particle density matrix divided by N converges in trace norm to the projector on ψ 0 . The energy gap e 1 -e 0 is always strictly positive since h has a unique ground state. Its dependence on γ and V is discussed in the Appendix. For the random external potentials introduced in (20) below, Eq. (103) implies that complete BEC holds not only for almost every random sample of scatterers, but the depletion of the condensate goes to zero even in L p norm on the probability space for every p < ∞.

Our proof of BEC in the GP limit is simpler than the corresponding proof in three and two dimensions [START_REF] Lieb | Proof of Bose-Einstein Condensation for Dilute Trapped Gases[END_REF] because the one-dimensional case considered here corresponds to a high-density, mean-field limit. In contrast, the work in [START_REF] Lieb | Proof of Bose-Einstein Condensation for Dilute Trapped Gases[END_REF] deals with a dilute limit that requires quite different tools.

2.1. Proof of Theorems 2.1 and 2.2. The upper bound in ( 4) is immediately obtained by using ψ ⊗N 0 as a trial function for the expectation value of the Hamiltonian [START_REF] Pastur | Spectra of Random and Almost-Periodic Operators[END_REF]. For the proof of the lower bound in (4), as well as the proof of (6), we begin by splitting off part of the kinetic energy and write, with ε > 0 and p i := -i∂ z i ,

H = N i=1 1 - N -1 2N ε p 2 i + V (z i ) + 1 N i<j ε 2 (p 2 i + p 2 j ) + γδ(z i -z j ) . (7)
Next we estimate the term in square brackets from below by a regular interaction potential using the bound

-ε∂ 2 z + γδ(z) ≥ w(z) (8)
with

w(z) := γ 1 + (bγ/2ε) 1 2b exp(-|z|/b) (9)
for any b > 0, cf. [START_REF] Baumgartner | Atoms in Strong Magnetic Fields: the High Field Limit at Fixed Nuclear Charge[END_REF]Lemma 6.3]. Note that

δ b (z) := 1 2b exp(-|z|/b) → δ(z) (10)
as b → 0, and δ b (z)dz = 1 for all b. From (8) we obtain the following inequality (in the sense of quadratic forms on wave functions that satisfy Dirichlet boundary conditions at the boundary of [0, 1] N and are identically zero outside of [0, 1] N )

H ≥ N i=1 1 - ε 2 p 2 i + V (z i ) + γ N i<j δ b (z i -z j ) (11)
where we have denoted γ/[1 + (bγ/2ε)] by γ for short, and bounded -(N -1)ε/2N from below by -ε/2. Since δ b is of positive type we can, for any density ρ(z), bound the last term by a sum of one-body terms:

γ N i<j δ b (z i -z j ) ≥ N i=1 γ δ b * ρ(z i ) - γ 2 (δ b * ρ)ρ - γ 2N δ b (0) ( 12
)
where we have used that γ ≥ γ. We take ρ to be the GP density ψ 2 0 (and zero outside of [0, 1]). For the comparison of the mean field Hamiltonian (3) with the right side of [START_REF] Radic | Anderson localization of a Tonks-Girardeau gas in potentials with controlled disorder[END_REF] we use the following simple estimates (recall that all integrals are effectively over the unit interval):

δ b * ρ ∞ ≤ ψ 0 2 ∞ ≤ ψ 0 2 ψ ′ 0 2 ≤ e 1/2 0 (13)
and

δ b * ρ -ρ ∞ = sup x [ρ(x -y) -ρ(x)]δ b (y)dy ≤ sup x x x-y |ρ ′ (u)|du δ b (y)dy ≤ 2 ψ 0 ∞ e 1/2 0 |y| 1/2 δ b (y)dy ≤ 2 1/2 π 1/2 e 3/4 0 b 1/2 . ( 14
)
Using γ ≥ γ -bγ 2 /(2ε), Eqs. ( 12)-( 14) and (3) lead to

H ≥ N i=1 h (i) - ε 2 p 2 i - bγ 2 2ε e 1/2 0 - 3π 1/2 2 1/2 γ e 3/4 0 b 1/2 - γ 4Nb . (15)
In the second term we estimate p 2 i by e 0 leading to an optimal ε ∼ b 1/2 γe -1/4 0

. Optimizing the resulting expression

(const.)γ b 1/2 e 3/4 0 + (Nb) -1 (16) over b gives b ∼ N -2/3 e -1/2 0 and E QM 0 N ≥ e 0 1 -(const.)N -1/3 γe -1/2 0 . (17)
For large γ we may use the bound e 0 ≥ (γ/2) ρ 2 ≥ (γ/2) while for γ 1 we use that e 0 ≥ π 2 since π 2 is the lowest eigenvalue of -∂ 2 z on the unit interval with Dirichlet boundary conditions. Altogether we obtain (4).

To prove Theorem 2.2 we trace the one-particle operator on the right side of ( 15) with the one-particle reduced density matrix of the ground state of H and obtain the more precise lower bound

E QM 0 N ≥ N 0 N e 0 + 1 - N 0 N e 1 -(const.) e 0 N -1/3 min{γ 1/2 , γ} (18)
where e 1 is the energy of the first excited state of the mean field Hamiltonian h. Combined with the upper bound E QM 0 ≤ Ne 0 this leads to the claimed estimate [START_REF] Yukalov | Bose-Einstein condensed systems in random potentials[END_REF] for the depletion of the condensate.

2.2.

Remarks. 1. If N ≤k denotes the occupation of the k lowest eigenvalues of h with energies up to e k , then the following generalization of (6) holds:

1 - N ≤k N ≤ C e 0 e k -e 0 N -1/3 min{γ 1/2 , γ}. ( 19
)
The proof is the same way as for (6), using again the bound [START_REF] Piraud | Localization of a matter wave packet in a disordered potential[END_REF]. For finite N the bounds ( 6) and ( 19) are, of course, only useful if their respective right-hand sides are less than 1. This may hold for [START_REF] Baumgartner | Atoms in Strong Magnetic Fields: the High Field Limit at Fixed Nuclear Charge[END_REF] even if it does not for (6).

2. As remarked in the Introduction, the authors of [START_REF] Lugan | Ultracold Bose Gases in 1D Disorder: From Lifshits Glass to Bose-Einstein Condensate[END_REF] claim a transition from a BEC to a fragmented condensate and finally to a Lifschitz glass phase in their model. Since no rigorous limit of large particle numbers is considered in [START_REF] Lugan | Ultracold Bose Gases in 1D Disorder: From Lifshits Glass to Bose-Einstein Condensate[END_REF] there is no contradiction with the complete BEC that is proved in the present paper in the whole parameter range. In fact, for a given finite particle number N a fragmented condensate, i.e., (1-N ≤k /N) small for some 1 ≪ k ≪ N, may be a reasonable substitute for the fully condensed state that emerges according to (6) in the large N limit. We also point out that as far as the density in the position variable is concerned, a fragmented condensate with non-overlapping singleparticle wave functions is indistinguishable from a fully condensed state where the wave function of the condensate is a coherent superposition of the non-overlapping functions. The difference shows up, however, in the momentum distribution.

The Gross-Pitaevskii Theory

Having established the GP minimizer as the wave function of the condensate in the Gross-Pitaevskii limit of the many-body problem we now turn to the study of the dependence of this minimizer on a random external potential. Specifically, we shall take V in (2) to be

V (z) = σV ω (z) (20)
with σ ≥ 0 and

V ω (z) := i δ(z -z i /ν) (21)
where ν > 0 and ω is a variable in a probability space generating the random points z i in R which are assumed to be Poisson distributed with density λ = 1 (see Appendix 2). The distances ℓ i between neighboring points z i /ν and z i+1 /ν are then independent random variables distributed according to the exponential law (cf. Eq.( 107))

dp ν (ℓ) = νe -ℓν dℓ . (22)
Thus, with probability one only finitely many of these points are in [0, 1], and on average there are ν such points. We start by considering the energy in an interval of length ℓ between two of the points, which after a translation and scaling

z → x := z/ℓ, σ → α := ℓσ, γ → κ := ℓγ (23)
can be conveniently taken to be the unit interval.

3.1. An Auxiliary Problem. For κ ≥ 0 and α ≥ 0, let e(κ, α) denote the auxiliary GP energy e(κ, α) := inf

φ 2 =1 E κ,α [φ] , (24)
where

E κ,α [φ] := 1 0 dx |φ ′ (x)| 2 + κ 2 |φ(x)| 4 + α 2 |φ(0)| 2 + |φ(1)| 2 (25) for φ ∈ H 1 ([0, 1]
). Standard methods show that there exists a minimizer for (24), which we denote by φ κ,α , i.e., e(κ, α

) = E κ,α [φ κ,α ] . ( 26
)
The minimizer is unique up to a constant phase factor, which can be chosen such that φ κ,α is non-negative. Note that, as an infimum over linear functions, e(κ, α) is jointly concave in κ and α. For us it will also be important that κ → κ e(κ, α) is strictly convex. This follows from the fact that κ e(κ, α) = inf E 1,α (|φ|) :

1 0 |φ| 2 = κ (27)
together with the strict convexity of |φ| 2 → E 1,α (|φ|).

We have the simple bounds

e(κ, α) = E 0,α (φ κ,α ) + κ 2 1 0 dx |φ κ,α (x)| 4 ≥ e(0, α) + κ 2 (28)
and

e(κ, α) ≤ E κ,α (φ 0,α ) = e(0, α) + κ 2 1 0 dx |φ 0,α (x)| 4 , (29)
where we used the Schwarz inequality to obtain the first inequality. Obviously e(0, 0) = 0 and e(0, ∞) = π 2 . The minimizer φ 0,α for κ = 0 is of the form φ 0,α (x) = a α cos[b α (x -1/2)], where b α ≥ 0 increases from 0 to π as α increases from 0 to ∞, and a α > 0 is determined by the normalization condition φ 0,α 2 = 1. A close inspection shows that φ 0,α 4 is monotone increasing in α, and hence

1 0 dx |φ 0,α (x)| 4 ≤ 1 0 dx |φ 0,∞ (x)| 4 = 3 2 . ( 30
)
In particular, from (28)-(30) we obtain

1 2 ≤ e(κ, α) -e(0, α) κ ≤ 3 4 ( 31
)
for all κ > 0 and α ≥ 0.

Lemma 3.1. For some constant C independent of κ and α we have

e(κ, ∞) ≥ e(κ, α) ≥ e(κ, ∞) 1 -Cα -1/2 . ( 32
)
Proof. The first inequality follows clearly from monotonicity of e(κ, α) in α. To prove the second, it is enough to consider α > C 2 (for otherwise the right side of (32) is negative).

Using a piecewise linear function which is constant for x ∈ (ε, 1 -ε) as a trial function, we obtain (after optimizing over ε) that

e(κ, ∞) ≤ κ 2 1 + Cκ -1/2 (33)
for some C > 0 and κ large enough. In particular, since obviously e(κ, α) ≥ κ/2, the second inequality in (32) follows right away if κ > α.

We are left with the case α ≥ κ. Since

e(κ, ∞) ≥ e(κ, α) ≥ κ 2 + αφ κ,α (0) 2 (34)
(using the symmetry of φ κ,α with respect to reflections at 1/2), we conclude from (33) that

φ κ,α (0) 2 ≤ C κ 1/2 α ≤ Cα -1/2 (35)
in this case. To obtain the desired bound, we use use λ(φ κ,α (x)-φ κ,α (0)) as a trial function for e(κ, ∞), where λ is an appropriate normalization factor. A simple calculation then yields (32).

For later use we shall also introduce the Legendre transform of the map κ → κ e(κ, α),

g(µ, α) := inf n≥0 (n e(n, α) -µn) . (36)
From the concavity of κ → e(κ, α) and the uniqueness of the minimizer of (25) one can easily conclude differentiability of the map κ → e(κ, α). Since κ → κ e(κ, α) is strictly convex, the infimum in (36) is uniquely attained. If it is attained at some n > 0, this n satisfies the equation e(n, α) + n e ′ (n, α) = µ , (37) where we denote e ′ (κ, α) := ∂ κ e(κ, α). We subtract the energy e(0, α) on both sides of (37) for convenience. Consequently, we observe that the infimum in (36) is attained at n satisfying

n = [µ -e(0, α)] + 1 e ′ (n, α) + 1
n (e(n, α) -e(0, α))

.

Here [t] + stands for the positive part of a real number t. The unique solution of (38) will be denoted by n(µ, α). From (31) and concavity of e(κ, α) is follows that 1/2 ≤ e ′ (κ, α) ≤ 3/4. Moreover, the second term in the denominator in (38) lies in [1/2, 3/4], again by (31). We thus conclude that the optimizing n satisfies

2 3 [µ -e(0, α)] + ≤ n(µ, α) ≤ [µ -e(0, α)] + . (39)
In particular, it is non-zero if and only if µ > e(0, α), and we can write

n(µ, α) ∼ [µ -e(0, α)] + , (40)
where a ∼ b means that a/b is bounded from above and below by positive constants.

Finally, we note that since e(0, α) is increasing and concave in α, there exists a constant C > 0 such that e(0, α)

≥ Cα 1 + α . ( 41
)
This bound will be useful later.

3.1.1. The Average Energy. We shall consider the average energy for a particle distributed over intervals of side length ℓ, whose distribution is given by the exponential law (22) with ν > 0. More precisely, we define

e 0 (γ, ν) = inf ν ∞ 0 dp ν (ℓ) n(ℓ) ℓ 2 e(n(ℓ)ℓγ, ∞) : ν ∞ 0 dp ν (ℓ) n(ℓ) = 1 , (42)
where the infimum as over all n : R + → R + satisfying the normalization constraint. With the aid of (36), we can alternatively write

e 0 (γ, ν) = sup µ>0 µ + ν ∞ 0 dp ν (ℓ) 1 ℓ 3 γ g(µℓ 2 , ∞) . ( 43
)
From simple scaling it follows that

e 0 (γ, ν) = γ e 0 (1, ν/ √ γ) . (44)
The infimum in (42) is attained for n(ℓ) = (ℓγ) -1 n(µℓ 2 , ∞) for suitable µ > 0, being equal to the optimal µ in (43). Using (40) as well as the fact that

1 ≤ e x ∞ x e -t t - x 2 t dt ≤ 2 (45)
we see that µ satisfies the relation

1 ∼ µ γ e -πν/ √ µ . ( 46
)
In other words, µ ∼ γ f (ν 2 /γ), where f : R + → R + denotes the function

f (x) = 1 for x ≤ 1 x (1+ln x) 2 for x ≥ 1. (47)
Also e 0 (γ, ν) ∼ γ f (ν 2 /γ).

3.2.

The Gross-Pitaevskii Energy. We now turn to the actual GP energy functional we want to consider,

E GP ω [ψ] := 1 0 dz |ψ ′ (z)| 2 + σV ω (z)|ψ(z)| 2 + γ 2 |ψ(z)| 4 (48)
with V ω as in [START_REF] Hartmann | Ordinary Differential Equations[END_REF], and define the GP energy as

e ω (γ, σ, ν) := inf φ 2 =1 E GP ω [φ] . (49)
THEOREM 3.1 (Convergence of the energy). Assume that ν → ∞, σ → ∞ and γ → ∞ in such a way that

γ ≫ ν (ln ν) 2 and σ ≫ ν 1 + ln (1 + ν 2 /γ) . ( 50
)
Then, for almost every ω,

lim e ω (γ, σ, ν) e 0 (γ, ν) = 1 ( 51
)
where e 0 (γ, ν) is the deterministic function defined in (43), which satisfies e 0 (γ, ν)

= γ e 0 (1, ν/ √ γ) ∼ γ f (ν 2 /γ).
3.2.1. Remark. Without the interaction, i.e., for γ = 0, the energy is not deterministic but is simply the ground state energy in the random potential σV ω . For σ → ∞ it becomes equal to the kinetic energy of a particle localized in the largest subinterval free of obstacles and hence, with probability 1, of the order ν 2 /(ln ν) 2 , cf. Appendix 2.

We shall now investigate the optimal particle number distribution in (42). Since n(µℓ 2 , ∞) > 0 if and only if µℓ 2 > π 2 , the average number of intervals with non-zero occupation numbers is given by

ν ∞ π/ √ µ dp ν (ℓ) = e -πν/ √ µ ν =: λ ν . ( 52
)
Since ν is the total number of available intervals, λ ≤ 1 in (52) defines the fraction of them which are occupied. With the help of the definition of λ in (52) the relation (46) can also be written as

π 2 ν 2 (ln λ -1 ) 2 = µ ∼ γ λ . (53)
In other words, λ is determined by γ and ν via relation

γ ∼ λ ν 2 (ln λ -1 ) 2 . ( 54
)
We can distinguish the following limiting cases:

• If γ ≫ ν 2 then by (54) we get λ → 1, i.e., all the intervals are occupied. The chemical potential satisfies µ ∼ γ in this regime.

• If γ ∼ ν 2 than λ ∼ 1, but λ is strictly less than 1. Again we have µ ∼ γ.

• If γ ≪ ν 2 then λ ≪ 1, i.e., only a small fraction of the intervals are occupied. The relation (53) implies µ ∼ (ν/ ln(ν 2 /γ)) 2 for the chemical potential.

• If γ ∼ ν/(ln ν) 2 then by (54) the fraction λ becomes O(1/ν), i.e., only finitely many intervals are occupied. In this latter case, µ ∼ γν ∼ ν 2 /(ln ν) 2 , which corresponds exactly to the inverse of the square of the size of the largest interval, cf. Appendix 2.

In particular, λ ≫ 1/ν only if γ ≫ ν/(ln ν) 2 , and hence this condition guarantees that many intervals are occupied. In this case the law of large numbers applies and hence the energy becomes deterministic in the limit. If λν = O(1), on the other hand, the value of e ω (γ, σ, ν) is random. This shows, in particular, that our condition (50) on γ is optimal.

Also the second condition in (50) on σ can be expected to be optimal. It can be rephrased as lσ ≫ 1, where l is the (weighted) average interval length

l = ν ∞ 0 dp ν (ℓ) ℓ n(ℓ) (55)
with n(ℓ) = (ℓγ) -1 n(µℓ 2 , ∞) the optimizer in (42). A simple calculation shows that l ∼ ν -1 (1 + ln(1 + ν 2 /γ)).

3.2.2.

Remark. The conclusions above apply to the minimizing particle distribution in (42). In the parameter regime defined in (50), they can be shown to apply also to the actual minimizer of the GP functional (48). This follows in a standard way by perturbing the energy functional, and then taking a derivative with respect to the perturbation. Specifically, one can add a term β R f (|ψ(x)|)dx for suitable functions f to the GP functional, and change the definition of e 0 in (42) accordingly. One then concludes that the energy asymptotics (51) continues to hold in this perturbed case. Taking a derivative with respect to β at β = 0 yields the desired information on the GP minimizer.

E GP [ψ] = m j=0 n j ℓ 2 j E n j ℓ j γ,ℓ j σ [ψ j ] (56)
with

n j = z j+1 z j dx |ψ(x)| 2 (57)
and

ψ j (x) = ℓ j n j ψ (z j + ℓ j x) for x ∈ [0, 1]. ( 58
)
We can choose ψ ∈ H 1 0 ([0, 1]) such that ψ j ∝ φ ℓ j γ,∞ for all j and {n j } is an arbitrary collection of non-negative numbers adding up to 1. In particular, we obtain the upper bound

e ω (γ, σ, ν) ≤ inf {n j } m j=0 n j ℓ 2 j e(n j ℓ j γ, ∞) , (59)
where the infimum is over all collections of non-negative numbers {n 0 , n 1 , . . . , n m } with m j=0 n j = 1.

From (56) we also have the lower bound

e ω (γ, σ, ν) ≥ inf {n j } m j=0 n j ℓ 2 j e(n j ℓ j γ, ℓ j σ) . ( 60
)
From Lemma 3.1 we conclude that the two bounds ( 60) and (59) agree, to leading order, as long as ℓ j σ is large for all the intervals of length ℓ j . In particular, the study of the GP energy reduces to a study of independent intervals in this case. More precisely, it suffices that ℓ j σ be large for those intervals containing most of the particles.

Poisson Distribution.

In this subsection we shall collect a few useful facts about the Poisson distribution, we will be used later on. A more detailed discussion of some of the points can be found in Appendix 2. Let χ m (ω) denote the characteristic function of the set corresponding to having exactly m scatterers inside the interval (0, ν). Its expectation value equals

χ m = e -ν ν m m! . ( 61
)
We shall frequently use the bounds

e -ν m≥λν ν m m! ≤ e -ν m≥0 ν m m! λ m-λν = e -ν(1-λ+λ ln λ) for λ ≥ 1 (62)
and

e -ν m≤λν ν m m! ≤ e -ν m≥0 ν m m! λ m-λν = e -ν(1-λ+λ ln λ) for λ ≤ 1. ( 63
)
Note that 1 -λ + λ ln λ ≥ 0 for λ ≥ 0, with equality only for λ = 1.

For any function depending only on ℓ j for some fixed 1 ≤ j ≤ m -1, we have

f (ℓ j ) = ∞ 0 dp ν (ℓ) f (ℓ) ( 64
)
with dp ν defined in (22). This fact (cf. Eq. ( 107) in Appendix 2) will be used repeatedly below. Moreover, the variables ℓ j and ℓ k for k = j are independent, and hence

f 1 (ℓ j )f 2 (ℓ k ) = f 1 (ℓ j ) f 2 (ℓ k ) in this case.
3.3.3. Upper Bound. Let µ = µ(γ, ν) be the optimizer in (43). With n(µ, α) defined above, we then have

ν ∞ 0 dp ν (ℓ) 1 ℓγ n(µℓ 2 , ∞) = 1 . ( 65
)
In every interval of length ℓ j , 1 ≤ j ≤ m -1, we shall place n j = (ℓ j γ) -1 n(µℓ 2 j , ∞) particles. For simplicity, we shall not place any particles in the first and last interval. The total number of particles in the system is then

N = m-1 j=1 n j . Lemma 3.2. If ν → ∞ and γ → ∞ with γ ≫ ν/(ln ν) 2 then lim N = 1 (66)
for almost every ω.

Proof. We shall show that N → 1 and N 2 → 1 in the limit under consideration, which implies the result. With χ m (ω) denoting the characteristic function of the set corresponding to having exactly m scatterers inside the interval (0, ν), we have, by symmetry,

N = 1 γ m≥2 (m -1) χ m ℓ -1 1 n(µℓ 2 1 , ∞) . ( 67
)
To obtain a lower bound, we can restrict the sum to m ≥ 1 + (1 -ε)ν, for some 0 < ε < 1.

We thus obtain

N ≥ (1 -ε)ν γ m≥1+(1-ε)ν χ m ℓ -1 1 n(µℓ 2 1 , ∞) = (1 -ε)   1 - ν γ m<1+(1-ε)ν χ m ℓ -1 1 n(µℓ 2 1 , ∞)   , (68)
where we used (65) the obtain the last equality. From (39) it follows that n(µℓ 2 1 , ∞) ≤ µℓ 2 1 , and hence

N ≥ (1 -ε) 1 - νµ γ P ℓ 1 , P (ω) = m<1+(1-ε)ν χ m (ω) . (69)
A Schwarz inequality yields

P ℓ 1 ≤ P 1/2 ℓ 2 1 1/2 = P 1/2 √ 2 ν , (70)
where we have evaluated the expectation value of ℓ 2 1 using (64). Moreover, it follows from (63) that P is bounded by a factor that decays exponentially in ν. In particular, using (46), it follows that lim inf N ≥ 1 -ε .

(71) Since ε was arbitrary, this proves the desired lower bound.

For an upper bound, we can proceed similarly. From (67), (65) and n(µℓ

2 1 , ∞) ≤ µℓ 2 1 we have N ≤ 1 + ε + µ γ m>1+(1+ε)ν (m -1) χ m ℓ 1 (72)
for ε > 0. An similar analysis as above, using the Schwarz inequality as in (70), and (62) instead of (63), then yields lim sup N ≤ 1 + ε .

(73) This shows that N → 1, as claimed.

Next we consider N 2 . Similarly to (67), we can write it as

N 2 = 1 γ 2 m≥3 (m -1)(m -2) χ m ℓ -1 1 n(µℓ 2 1 , ∞)ℓ -1 2 n(µℓ 2 2 , ∞) + 1 γ 2 m≥2 (m -1) χ m ℓ -2 1 n(µℓ 2 1 , ∞) 2 . ( 74
)
An analysis as above shows that the first sum can be restricted to an interval m ∈ [(1 -ε)ν, (1 + ε)ν], and therefore yields 1 in the limit considered. Hence it remains to show that last term in (74) goes to zero. If we restrict the sum to m -1 ≤ 2ν, say, then we have

1+2ν m=2 (m -1) χ m l-2 1 n(µℓ 2 1 , ∞) 2 ≤ 2ν l-2 1 n(µℓ 2 1 , ∞) 2 ≤ 2 4 µ 2 ν e -πν/ √ µ (75)
where we used the upper bound in (39) as well as the bound

e x ∞ x e -t t - x 2 t k dt ≤ k! 2 k (76)
for k = 2. It follows from ( 46) that (75), when multiplied by γ -2 , goes to zero as

ν → ∞ if λ ≫ ν/(ln ν) 2 .
For the remaining terms corresponding to m > 1 + 2ν, we use again the Schwarz inequality, which implies that

m>1+2ν (m -1) χ m l-2 1 n(µℓ 2 1 , ∞) 2 ≤ ν √ 2 e (1-ln 4)ν ℓ -4 1 n(µℓ 2 1 , ∞) 4 1/2 . ( 77
)
Because of the upper bound in (39) and (76) for k = 4 the latter expectation value is above bounded by

ℓ -4 1 n(µℓ 2 1 , ∞) 4 ≤ 4! 2 4 µ ν 4 e -πν/ √ µ . (78)
With the aid of (46) one now readily checks that the right side of (77), when divided by γ 2 , goes to zero as ν → ∞. This completes the proof.

We can take n j = (ℓ j γ) -1 n(µℓ 2 j , ∞) for 1 ≤ j ≤ m -1 as above, and divide each n j by N = m-1 j=1 n j in order to have the right particle number. Then (59) implies the upper bound

e ω (γ, σ, ν) ≤ E ≡ m-1 j=1 n j Nℓ 2 j e(n j ℓ j γ/N, ∞) . (79)
The following lemma concludes the proof of the upper bound in Theorem 3.1.

Lemma 3.3. If ν → ∞ and γ → ∞ with γ ≫ ν/(ln ν) 2 then lim E e 0 (γ, ν) = 1 (80)
for almost every ω.

Proof. We proceed as in the proof of Lemma 3.2, and show that both E /e 0 (γ, ν) and E 2 /e 0 (γ, ν) 2 converge to 1 in the limit considered. Because of the result in Lemma 3.2, we can replace N by 1 without loss of generality. Using (40) (for α = ∞) and (31) we see that

n j ℓ 2 j e(n j ℓ j γ, ∞) ∼ µn j . (81)
Since also µ ∼ e 0 (γ, ν) we can proceed exactly as in the proof of Lemma 3.2 to conclude the result.

We thus conclude that, for almost every ω,

lim sup e ω (γ, σ, ν) e 0 (γ, ν) ≤ 1 (82) in the limit ν → ∞, γ → ∞ with γ ≫ ν/(ln ν) 2 .
3.3.4. Lower Bound. We start from (60). Recall the definition of g(µ, α) in (36). We have

e ω (γ, σ, ν) ≥ inf {n j } m j=0 n j ℓ 2 j e(n j ℓ j γ, ℓ j σ) ≥ µ + m j=0 1 γℓ 3 j g(µℓ 2 j , ℓ j σ) (83)
for any µ ∈ R.

Using the lower bound in (31), as well as (41), have

g(µℓ 2 , ℓσ) ≥ - 1 2 µℓ 2 -e(0, ℓσ) 2 + ≥ - µ 2 ℓ 4 2 θ(ℓ -ℓ) , (84)
where

ℓ = - 1 2σ + 1 4σ 2 + C µ (85)
and θ denotes the Heaviside step function. For the first and last interval, corresponding to j = 0 and j = m, respectively, we shall simply use that

1 γℓ 3 j g(µℓ 2 j , ℓ j σ) ≥ - µ 2 ℓ j 2γ . ( 86
)
When divided by µ ∼ e 0 (γ, ν), this goes to zero almost surely in the limit considered, since ℓ j ν -1 and µ ≪ νγ. Pick some small ε > 0, and consider the contribution to the sum in (83) coming from intervals with length ℓ j < (εσ) -1 , for 1 ≤ j ≤ m -1. If εσ ≥ 1/ ℓ, this contribution is zero, hence we can assume εσ < 1/ ℓ from now on, which is equivalent to

σ 2 < µ(1 + ε) Cε 2 . ( 87
)
Because of our assumption (50) on σ, this means that

ν 1 + ln (1 + ν 2 /γ) ≪ √ µ ∼ γ f (ν 2 /γ) , (88)
i.e., ν 2 ≪ γ. We have

1 γℓ 3 j g(µℓ 2 j , ℓ j σ) ≥ - µ 2 2γεσ ∼ - µ ν ν σ(1 + ln(1 + ν 2 /γ)) (89)
in this case, and the last fraction goes to zero in the limit considered.

Finally consider the intervals 1 ≤ j ≤ m -1 with length ℓ j ≥ (εσ) -1 . In this case, we can use Lemma 3.1 to bound

g(µℓ 2 j , ℓ j σ) ≥ 1 -Cε 1/2 g(µℓ 2 j (1 -Cε 1/2 ) -1 , ∞) . ( 90
)
An either case, we thus have the lower bound

1 γℓ 3 j g(µℓ 2 j , ℓ j σ) ≥ 1 -Cε 1/2 γℓ 3 j g(µℓ 2 j (1 -Cε 1/2 ) -1 , ∞) - Cµ σ(1 + ln(1 + ν 2 /γ)) (91) for 1 ≤ j ≤ m -1, for some C > 0. Let μ = µ(1 -Cε 1/2 ) -1 .
With n(µ, ∞) defined in Subsection 3.1 as the optimizer in (36), we can write

g(μℓ 2 , ∞) = n(μℓ 2 , ∞)e(n(μℓ 2 , ∞), ∞) -μℓ 2 n(μℓ 2 , ∞) . (92)
We shall choose μ as in Subsection 3.3.3, i.e., such that

ν ∞ 0 dp ν (ℓ) 1 ℓγ n(μℓ 2 , ∞) = 1 . (93)
Then μ ∼ e 0 (γ, ν). Let n j = (ℓ j γ) -1 n(μℓ 2 j , ∞) and N = m-1 j=1 n j . From (83), (86), ( 91) and (92) we have

e ω (γ, σ, ν) ≥ 1 -Cε 1/2 μ (1 -N) + m-1 j=1 n j ℓ 2 j e(n j ℓ j γ, ∞) - Cµm σ(1 + ln(1 + ν 2 /γ)) - µ 2 (ℓ 0 + ℓ m ) 2γ . ( 94
)
We have already shown in Lemma 3.2 that lim N = 1 almost surely, and in Lemma 3.3 that the ratio of the sum in ( 94) and e 0 (γ, ν) converges to 1 almost surely. Moreover, after division by e 0 (γ, ν) ∼ µ the terms on the last line of (94) vanish almost surely. This proves that lim inf e ω (γ, σ, ν) e 0 (γ, ν)

≥ 1 -Cε 1/2 (95)
for almost every ω. This holds for all ε > 0, hence the proof of the theorem is complete.

Conclusions

The main conclusions that can be drawn from the preceding discussion are as follows: 1. BEC in the ground state of the interacting gas in the Gross-Pitaevskii regime can survive even in a strong random potential with a high density of scatterers. As far as BEC is concerned the interacting gas in this regime thus behaves in a similar way as an ideal gas at zero temperature. The character of the wave function of the condensate, however, is strongly affected by the interaction. 2. A random potential may lead to localization of the wave function of the condensate, even though the density of obstacles is much less than the particle density. The interparticle interaction counteracts this effect and can lead to complete delocalization if the interaction is strong enough. 3. In terms of the interaction strength, γ, and density of scatterers, ν, the transition between localization and delocalization occurs in the model considered when γ ∼ ν 2 . In the localized regime γ ≪ ν 2 the chemical potential is µ ∼ (ν/ ln(ν 2 /γ)) 2 . For γ ν/(ln ν) the condensate is localized in a small number of random intervals.

Appendix 1: Energy Gaps

We consider the Schrödinger operator H = -∂ 2 z + W (z) on L 2 ([0, 1]), with Dirichlet boundary conditions. Assume, for the moment, that W is a bounded and continuous function. Without loss of generality, we may assume W ≥ 0. Let 0 < e 0 < e 1 denote the lowest two eigenvalues of H.

For the proof, we follow closely [START_REF] Kirsch | Universal lower bounds on eigenvalue splittings for one dimensional Schrödinger operators[END_REF].

Proof. For general E > 0, let u(z, E) denote the solution of

-u ′′ + W u = Eu (98)
with u(0, E) = 0 and u ′ (0, E) = 1 (the prime denoting ∂ z ). Pick an η > 0, and introduce the Prüfer variables r(z, E) and θ(z, E) via u = r cos θ , u ′ = -ηr sin θ

The variable θ is only determined modulo 2π. We fix it uniquely be requiring it to be continuous and θ(0, E) = -π/2. We note that θ is increasing in E for fixed z, i.e., ϕ(z, E) = ∂ E θ(z, E) ≥ 0. This follows from

ϕ = θ = η uu ′ -u u′ η 2 u 2 + (u ′ ) 2
where the dot stands for ∂ E . The function χ = uu ′ -u u′ satisfies χ ′ = u 2 and is thus positive since it vanishes at z = 0.

From (98) it follows that

θ ′ = η -1 (E -W ) cos 2 θ + η sin 2 θ (99)
In particular, θ ′ > 0 near points z ∈ [0, 1] where u vanishes. Since u(z, e 0 ) has no zeroes for z ∈ (0, 1), we conclude that θ(1, e 0 ) = π/2. Moreover, u(z, e 1 ) has exactly one zero in (0, 1), hence θ(1, e 1 ) = 3π/2. In particular,

θ(1, e 1 ) -θ(1, e 0 ) = π (100)
We have ϕ(0, E) = 0 for all E. From (99) it follows that

ϕ ′ = η + η -1 (W -E) sin(2θ)ϕ + η -1 cos 2 (θ) ≤ η + η -1 (W -E) ϕ + η -1
From Gronwall's inequality [21, Thm. III.1.1], we thus conclude that

ϕ(z, E) ≤ z η exp z 0 η + η -1 (W (y) -E) dy
In particular, W (y)dy -π 2 η We choose η in order to make the exponent as large as possible, which leads to the choice (96), and concludes the proof. By a simple approximation argument, the bound (97) extends to our case of interest, where

ϕ(1, E) ≤ 1 η exp η + η -1 E + η -1 1 0 W (
W (z) = σ m i=1 δ(z -z i ) + γ|ψ 0 (z)| 2 (102)
for σ ≥ 0 and γ ≥ 0. The result is then e 1 -e 0 ≥ η ln 1 + e -2η with η = π 2 + 3mσ + 3γ .

5.0.5. Remark. While the bound (103) on the gap is uniform in the location of the scatterers, we do not expect it to be optimal for large η. In fact, the gap is presumably smallest when all the z i are equal to 1/2. In this case, a simple calculation (for γ = 0) shows that the gap is asymptotically equal to 32π 2 /(σm) for large σm.

6. Appendix 2: Remarks on the Poisson Random Point Field (RPF) on the Real Axis R.

6.1. Remark 1. (Poisson and exponential distributions, see, e.g., [START_REF] Lenoble | Bose-Einstein Condensation in the Luttinger-Sy Model[END_REF].)

The following theorem is a consequence of the hypothesis of an independent uniform distribution of m point "impurities" on the interval Λ l = [-l/2, l/2] ⊂ R (homogeneous m-point binomial RPF) and the limit:

Λ l → R, m → ∞,
Therefore, the random variables {L ω j } m j=1 are dependent. (c) In the thermodynamic limit: λ = lim m, l→∞ m/l, the intervals L ω j j≥1 form an infinite set of independent random variables and the distribution corresponding to (106) converges (weakly) to the product-measure distribution σ λ defined by the set of consistent marginals: Therefore, by the Borel-Cantelli lemma P((A δ ∞ ) c ) = 0, which is equivalent to (110). This also implies that there exists a subset Ω ⊂ Ω of full measure, P( Ω) = 1, such that for each ω ∈ Ω one can find l 0 (ω) < ∞ with P {ω : ( max (ii) Consider the event (A δ l ) c and note that one can estimate its binomial probability from below taking only one term in the union (113). Then

P bin m-1,l ((A δ l ) c ) ≥ 1 - δ 2λ ln l/| Λ l | m-1 ≥ 1 - δ 2λ ln l/[λ l] [λ l] , (117)
where we put m -1 = [λ l], see (104), and we use (112). Then (117) yields

lim l→∞ P bin [λ l],l ((A δ l ) c )/(l -δ/2 ) ≥ 1 , (118)
By the same arguments as in (i) one concludes from (118) that for δ ≤ 2 Since the events {(A δ l ) c } l>2 are independent, by the Borel-Cantelli lemma we obtain: P((A δ ∞ ) c ) = 1, or equivalently (111). This again can be translated as P {ω : ( max

1 j m L ω j ) l ≥ δ λ ln l} = 1 , for δ ≤ 2 ,
for all l l 0 (ω) and m > [λ l].

Since the Poisson RPF on R conditional to the interval [-l/2, l/2] and to m -1 impurities coincides with the binomial RPF, instead of increasing boxes with finite binomial RPF configurations one can consider restrictions of infinite Poisson RPF configurations to the increasing set of windows [-l/2, l/2], cf [START_REF] Th | On the Nature of Bose-Einstein Condensation Enhanced by Localization[END_REF]. Then Theorem 6.2 claims that almost surely (with respect to the Poisson RPF distribution) the maximal length of impurityfree sub-intervals, (max 1 j m L ω j ) l , in this windows is bounded from below and above by (const) ln l/λ.

3. 3 . 1 . 3 . 3 . 1 .

 31331 Proof of Theorem 3.Preliminaries. Let z 1 ≤ z 2 ≤ • • • ≤ z m denote those points in (21) which lie in the open interval (0, ν). Let also z 0 = 0 and z m+1 = ν, and denote ℓ i = |z i+1 -z i |/ν for 0 ≤ i ≤ m. Then m j=0 ℓ j = 1. For any ψ ∈ H 1 0 ([0, 1]), we have

Lemma 5 . 1 . 2 = π 2 e 1 -

 51221 Define η > 0 by η e 0 ≥ η ln 1 + πe -2η

≤ exp η + η -1 1 0W 1 0

 11 y)dy for E > 0. This implies that π = θ(1, e 1 ) -θ(1, e 0 ) = + η -1 e 0 exp η -1 (e 1 -e 0 ) -1 (101) Using the trial function √ 2 sin(πz), we see that e 0 ≤ π 2 + 2 W . From (101) we thus have e 1 -e 0 ≥ η ln 1 + π exp -η -3 η 1 0

Theorem 6 . 1 .

 61 (a) In the limit (104) the finite-volume binomial RPF of impurities {X ω l } converges (in distribution) to the Poisson RPF {X ω } with the Poisson distributionP{ω : |X ω ∩ Λ a | = n} = (aλ) n n! e -λa ,(105)of the number of impurities in any interval Λ a of the length a. (b) The uniform distribution of m -1 independent points of impurities split up the box Λ l into a set of sub-intervals I ω j m j=1 with ∪ m j=1 I ω j = Λ l and their random lengths {|I ω j | = L ω j } m j=1 , ω ∈ Ω have the joint probability distribution dP l,m (L 1 , ..., L m ) = (m -1)! l m-1 δ(L 1 + ... + L m -l) dL 1 dL 2 . . . dL m .

) 6 . 2 . 1 P

 621 dσ λ,j 1 ,...,j k (L j 1 , . . . , L j k ) = λ k k s=1 e -λL js dL js , k ∈ N .(107Remark 2. (Log intervals[START_REF] Lenoble | Bose-Einstein Condensation in the Luttinger-Sy Model[END_REF],[START_REF] Th | On the Nature of Bose-Einstein Condensation Enhanced by Localization[END_REF].) We can estimate the maximal length, (max 1 j m L ω j ) l , of an interval in the family I ω j m j=1 of random sub-intervals of Λ l = [-l/2, l/2] when l → ∞. To this end we define the eventsA δ l := {ω : (max 1 j m L ω j ) l ln l δ λ } , l ∈ N and l > 2 ,(108)By Theorem 6.1 binomial distribution P bin [λ l]+1,l (•) converges to the Poisson distribution P(•) on R, when l → ∞. Then estimate (115) and (116) imply l ((A δ l ) c ) < ∞.

ln l} = 1 ,

 1 for δ > 4 , for all l l 0 (ω) and m ≥ [λ l].

l 1 P

 1 ((A δ l ) c ) = ∞.

as well as the following "tail" event:

By definition, (109) is the event that for increasing l the length of the largest subinterval in Λ l is bounded from above by the nonrandom number (δ ln l)/λ for all but finitely many l ∈ N. The following theorem holds: Theorem 6.2. Let λ > 0 be a mean concentration of the point Poisson "impurities" on R. Then (i) for any δ > 4, one has:

(ii) for any δ ≤ 2, one has:

Proof: (i) The complementary event to ( 108) is

Let N l := λl/(δ ln l) + 1, where [x] denotes the integer part of x ≥ 0, and define a new interval:

Split this bigger interval into 2N l identical disjoint intervals {J l i } 2N l i=1 of size δ(2λ) -1 ln l. If the event (A δ l ) c occurs, then there exists at least one empty interval J l i (interval without any impurities), i.e.,

Note that for a given number of m -1 impurities on the finite interval Λ l the RPF {X ω l } is binomial for any two sets: J l i and Λ l \ J l i . Since the probability for the interval J l i to be empty depends only on its size δ(2λ) -1 ln l and the size of Λ l , one obtains by ( 112) and (113) the estimate

we obtain that for δ > 4


Aleksandrs ; Versilovskis
email: aleksandrs.versilovskis@yahoo.com

A Veršilovskis

B Huybrecht

E K Tangni

L Pussemier

S De Saeger

A Callebaut

Cross-reactivity of some commercially available deoxynivalenol (DON) and 1 zearalenone (ZEN) immunoaffinity columns to DON and ZEN conjugated 2 forms and metabolites 3 4

Keywords: Mycotoxins -trichothecenes, Mycotoxins -zearalenone, Environmental contaminants, Natural toxicants Food deoxynivalenol, zearalenone, immunoaffinity columns, cross-reactivity, multi analyte 27 UPLC-MS/MS, Fusarium toxins, conjugated mycotoxins 28 29

Seven commercially available deoxynivalenol (DON) and zearalenone (ZEN) immunoaffinity columns (IACs) were tested for cross-reactivity to conjugated forms (3-acetyl-deoxynivalenol, 15-acetyl-deoxynivalenol, DON-3-glucoside, DON-3-glucuronide, ZEN-glucosides, ZEN-glucuronide) and metabolites (de-epoxydeoxynivalenol, α-zearalenol, β-zearalenol) and nivalenol (NIV), using a semi-quantitative multi-mycotoxin ultra performance liquid chromatography -tandem mass spectrometry method. The DON IACs showed cross-reactivity for nearly all DON derivatives tested. The ZEN IACs showed limited cross-reactivity to some of the ZEN derivatives.

Mycotoxins are fungal secondary metabolites toxic for humans and animals and mainly found in 32 food and feed. Mycotoxins also occur in chemically modified forms through acetylation, 33 glucosylation, sulfatation etc. These conjugated mycotoxins emerge after metabolization of the 34 native mycotoxins by plants, animals, during food processing or by the fungi themselves. 35

Awareness of such altered forms of mycotoxins is increasing, but reliable analytical methods, 36 reference standards, occurrence and toxicity data are still lacking [START_REF] Berthiller | Formation, determination and 231 significance of masked and other conjugated mycotoxins[END_REF]. 37

Traditional evaluation of human and animal exposure to mycotoxins is based on analysis of feed 38 and food or biological fluids like blood, serum or urine [START_REF] Turner | Analytical methods for determination of 268 mycotoxins: A review[END_REF][START_REF] Yunus | Blood plasma 285 levels of deoxynivalenol and its de-epoxy metabolite in broilers after a single oral dose of the 286 toxin[END_REF], 39 Köppen et al. 2010, Thieu and Petterson 2009). Due to the complexity of samples, sample clean-40 up is often performed prior to quantification of target mycotoxins. Sample preparation procedures 41 generally include several steps, such as solvent extraction, defatting with hexane, and solid phase 42 extraction (SPE) [START_REF] Köppen | Determination of mycotoxins in 246 foods: current state of analytical methods and limitations[END_REF]. The production of specific antibodies for mycotoxins led 43 to the development of immunoaffinity (IA) based columns, which are now widely used. IA 44 column (IAC) chromatography is based upon molecular recognition and can provide an 45 alternative method to isolate, purify and concentrate target analytes from complex sample 46 matrices (Chuang et al. 2007, Senyuva and[START_REF] Şenyuva | Immunoaffinity column clean-up techniques in food analysis: A 252 review[END_REF]. IACs are easy to use and provide good 47 recoveries, repeatability and reproducibility. The main advantage of IACs is the high selectivity, 48 which allows developing methods with lower limits of detection for the target analytes compared 49 to methods using SPE clean-up. It facilitates the analysis of the most complex matrices starting 50 from cereals-based foods and feeds to biological fluids and animal tissues (Visconti andPascale 51 1998a, Visconti andPascale 1998b). Moreover, IA allows the use of detection methods lacking in 52 separation power such as thin layer chromatography while retaining good method performances 53 (Stroka et al. 2000, Gilbert and[START_REF] Gilbert | Validation of analytical methods for determining mycotoxins in 241 foodstuffs[END_REF]. 54 55 Currently, no antibodies specifically developed against conjugated mycotoxins are available. To 56 cope with this gap, numerous multi-analyte liquid chromatography tandem mass spectrometry 57 (LC-MS/MS) methods are used without any clean-up at all. With highly sensitive LC-MS/MS 58 equipment, it has become feasible to simply dilute the sample extracts to minimize possible 59 matrix effects (dilute and shoot approaches) [START_REF] Vendl | Simultaneous determination of deoxynivalenol, 274 zearalenone, and their major masked metabolites in cereal-based food by LC-MS-MS[END_REF]). However, mycotoxin 60 concentrations in biological fluids are much lower than in food and feed, so the dilute and shoot 61 principle is not possible. This is why for biological fluids IACs would be an ideal approach. 62

Existing IACs for mycotoxins can be evaluated with respect to their ability to bind also 63 mycotoxin conjugates and metabolites. 64 65 Commercially available IACs for zearalenone (ZEN) (Figure 1a) present only cross-reactivity 66 with α-zearalenol (α-ZEL), β-zearalenol (β-ZEL) and zearalanone (ZAN), also enabling their use 67 for the determination of some growth promoters (Visconti andPascale 1998b, Sulyok et al. 68 2006). 69 70 Commercially available IACs for deoxynivalenol (DON) (Figure 1b) present cross-reactivity 71 with de-epoxydeoxynivalenol (DOM-1) [START_REF] Dänicke | Effect of 237 addition of a detoxifying agent to laying hen diets containing uncontaminated or Fusarium 238 toxin-contaminated maize on performance of hens and on carryover of zearalenone[END_REF][START_REF] Valenta | Study on the transmission of deoxynivalenol and deepoxy-270 deoxynivalenol into eggs of laying hens using a high-performance liquid chromatography 271 ultraviolet method with clean-up by immunoaffinity columns[END_REF], Tangni 72 et al. 2009, He et al. 2009), 3-acetylDON (3AcDON), 15-acetylDON (15AcDON) (Sulyok et al. 73 2006[START_REF] Kostelanska | The study 249 of deoxynivalenol and its masked metabolites fate during the brewing process realised by 250 UPLC-TOFMS method[END_REF]) and only in one case also with DON-3-glucoside (DON3G) 74 [START_REF] Kostelanska | The study 249 of deoxynivalenol and its masked metabolites fate during the brewing process realised by 250 UPLC-TOFMS method[END_REF] 4 DON (100 µg.mL -1 ), DON3G (50 µg.mL -1 ), 3AcDON (100 µg.mL -1 ), 15AcDON (100 µg.mL -1 ), 93 DOM-1(100 µg.mL -1 ), NIV (50 µg.mL -1 ), ZEN (100 µg.mL -1 ), α-ZEL (100 µg.mL -1 ) and β-ZEL 94 (100 µg.mL -1 ) were from Biopure (IFA, Tulln, Austria). Zearalenone-diglucoside 95 (ZEN-diG), zearalenone-14-glucoside (ZEN-14-G), zearalenone-16-glucoside , 96 zearalenone-14-glucuronide (ZEN-14-GU) and DON-3-glucuronide (DON3GU) were 97 synthesized in our laboratory on the basis of modified procedures of [START_REF] Zill | Chemically and biologically synthesized 288 zearalenone-4-β-D-glucopyranoside: comparison and convenient determination by gradient 289 HPLC[END_REF] The standard mixture was diluted with 10 mL of PBS buffer in order to obtain a concentration of 127 less than 5% MeOH and allowed to pass the cartridge. Each column was washed twice with 10 128 mL water and finally eluted with 1.5 mL of 100% MeOH. The solvent was evaporated to dryness 129 The Acquity UPLC HSS T3 C 18 100 mm x 2.1mm i.d., 1.8 µm particle size (Waters Corp.,158 Milford, MA. USA) was used. 159 160

Chromatographic conditions 161

The column temperature was kept constant at +40°C. Flow rate was 0.4 mL min -1 , and 10 µL 162 partial loop injection was used (loop of 50 µl). Mobile phase A was Milli-Q water with 0.1% of 163 FA and as mobile phase B MeOH containing 0.1% of FA was used. The gradient used was as 164 follows: 0-8 min A -90%, 8-8.5 min A -65% , 8.5-17 min A -15% , 17-18 min A -0% , 165 18-18.5 min A -90% , 18.5-20 min A -90%. 166 167

Mass spectrometer parameters 168

The main mass spectrometer parameters were as follows: source temperature 140°C, 169 desolvation temperature 450°C, capillary voltage 3 kV, extractor voltage 3 V, RF lens voltage 170 0.3 V, cone gas (nitrogen) flow 50 L.h -1 , desolvation gas (nitrogen) flow 1000 L.h -1 , collision gas 171 (argon) flow 0.15 mL.min -1 . The mass spectrometer was operated in electrospray positive 172 ionization (ESI + ) selected reaction monitoring (SRM) mode. SRM parameters are shown in 173 Table 1. Figure 2 cross-reactivity of some IACs to β-ZEL and α-ZEL (Visconti andPascale 1998b, Sulyok et al. 193 2006), and supports also the study by [START_REF] Vendl | Simultaneous determination of deoxynivalenol, 274 zearalenone, and their major masked metabolites in cereal-based food by LC-MS-MS[END_REF] which shows, that eeZEN IAC does not 194 bind any ZEN and ZEL glucosides. 195 196 DZT MS-prep column which was designed to bind several mycotoxins at once (DON,ZEN,197 T-2 toxin and HT-2 toxin) showed cross-reactivity to all DON derivatives tested, except for NIV,198 and at also binded ZEN, β-ZEL and α-ZEL, but showed no cross-reactivity with the other ZEN 199 α-ZEL, β-ZEL) in serum. No IACs were found suitable for ZEN conjugates .

F o r P e e r R e v i e w O n l y

± 20 < 1 < 1 < 1 8 ZEN-diG < 1 < 1 < 1 < 1 < 1 < 1 < 1 9 ZEN-16-G < 1 < 1 < 1 < 1 < 1 < 1 < 1 10 ZEN-14-GU < 1 < 1 < 1 < 1 < 1 < 1 < 1 11 ZEN-14-G < 1 < 1 < 1 < 1 < 1 < 1 < 1 12 β-ZEL 3

  (used for 98 synthesis of ZEN glucosides) and[START_REF] Wu | Synthesis and characterization of deoxynivalenol 282 glucuronide: Its comparative immunotoxicity with deoxynivalenol[END_REF] (used for synthesis ofDON and ZEN 99 glucuronides). The chemical structures of ZEN-14-G and DON3GU were confirmed by 100 LC-MS/MS and 1 H-NMR (data not shown), while the structure of ZEN-14-GU was identified by 101 fragmentation experiments (MS/MS) in ESI + and ESI -. Furthermore, after incubation with 102 glucuronidase (β-D-Glucuronide glucuronosohydrolase, type B-1, from bovine liver), DON and 103 ZEN were detected as reaction products. Fragmentation of ZEN-16-G in MS/MS gave the same 104 fragments as ZEN-14-G fragmentation. Also, β-glucosidase (β-D-Glucoside glucohydrolase from 105 almonds) treatment released ZEN (data not shown). We tentatively identified this molecule as 106 ZEN-16-G. The fragmentation pattern of ZEN-diG clearly showed the presence of two sugars 107 and β-glucosidase treatment resulted in formation of ZEN (data not shown). We tentatively 108 identified this molecule as ZEN-diG, although it is possible that both sugars are on the same 109 position. Stock solutions of DON and its derivatives were prepared in acetonitrile. Stock 110 solutions of ZEN and its derivatives were prepared in methanol. Working standard mixtures 111 containing 0.1, 0.2 and 0.4 µg.mL -1 of each analyte were prepared in 35% methanol. 112 113 Immunoaffinity columns 114 Seven different immunoaffinity cartridges for DON or ZEN were tested: DON-prep 115 (R-Biopharm, Rhone, Glasgow, UK) (2000 ng capacity for DON), NeoColumn for DON 116 (DONneo) (Neogen Co., Ayr, UK) (1000 ng capacity for DON), AokinImmunoClean C for DON 117 (DONaok) (Aokin AG, Berlin, Germany) (1500 ng capacity for DON), Easi-Extract Zearalenone 118 (eeZEN) (R-Biopharm Rhone, Glasgow, UK) (1500 ng capacity for ZEN), NeoColumn for ZEN 119 (ZENneo) (Neogen Co., Ayr, UK) (2000 ng capacity for ZEN), AokinImmunoCleanC for ZEN 120 (ZENaok)(Aokin AG, Berlin, Germany) (1500 ng capacity for ZEN), DZT MS-prep 121 (R-Biopharm, Rhone, Glasgow, UK) (capacity not indicated). tested by loading a standard mixture containing 40 ng of each analyte 125 (corresponds to 560 ng of total mycotoxins load). 126

  under a nitrogen stream at 40°C, the residue was re-dissolved in 200 µL of 35% MeOH and 130 10 µL was injected in the UPLC-MS/MS system. 131 132 Spiked serum sample 133 One ml of calf serum was spiked with a standard mixture containing 20 ng of each mycotoxin. 134 Two other spiked serum samples were prepared, with 40 and 80 ng of each mycotoxin. The 135 mixtures were vortexed for one minute. One mL of MeOH and two mL of MeCN were added to 136 the spiked serum to precipitate proteins. The solution was vortexed for one minute, centrifuged at 137 14000 rpm for 10 minutes. Three mL of the upper layer were evaporated till dryness at 40°C 138 under a nitrogen stream. Dry residue was dissolved in 150 µL of 35% MeOH and injected in the 139 UPLC-MS/MS system (injection volume -10 µL). Recovery was calculated by comparing the 140 signal of the standard in solvent with the signal obtained in a spiked matrix. 141 142 Serum sample preparation for testing on selected IACs 143 DON-prep, DONneo and DZT MS-prep columns were tested with spiked serum. Serum sample 144preparation was as in previous section, but after centrifugation, 3 mL of upper layer was 145 evaporated till dryeness at 40°C under a nitrogen stream and residue was dissolved in one mL of 146 100% MeOH and then diluted with 20 mL of PBS buffer. The mixture was passed through the 147 IAC column and the column was washed twice with 10 mL of water and finally, the mycotoxins 148 were eluted with 1.5 mL of MeOH. The solvent was evaporated till dryness under a nitrogen 149 stream at 40°C; the dry residue was dissolved in 150 µL of 35% MeOH and injected in the 150 UPLC-MS/MS system (injection volume -10 µL). 151 152 Equipment 153 UPLC-MS/MS was performed on a Waters Quattro Premier XE(Waters Corp., Milford, MA, 154 USA) connected to an Acquity UPLC system(Waters Corp., Milford, MA, USA)

  shows the UPLC-MS/MS chromatogram of the multi-mycotoxin standard 174 mixture. All measurements were carried out in duplicate and given as a mean value. that DONneo IAC cross-reacts with all DON derivatives tested, DON-prep IAC 180 cross-reacts with all DON conjugated forms and metabolites tested, but not with NIV. DONaok 181 IAC does not bind NIV, DON3GU and DON3G. Surprisingly, DONaok IAC also binds ZEN, 182 β-ZEL and α-ZEL, which can be explained by possible interactions of these analytes with IAC 183 resin or other unknown factors. 184 185 Results obtained for DON-prep IAC are in accordance with previous reports on DON-prep IAC 186ability to bind DOM-1 and acetyl-DONs[START_REF] Dänicke | Effect of 237 addition of a detoxifying agent to laying hen diets containing uncontaminated or Fusarium 238 toxin-contaminated maize on performance of hens and on carryover of zearalenone[END_REF][START_REF] Valenta | Study on the transmission of deoxynivalenol and deepoxy-270 deoxynivalenol into eggs of laying hens using a high-performance liquid chromatography 271 ultraviolet method with clean-up by immunoaffinity columns[END_REF], Tangni 187 2009[START_REF] He | Sample clean-up methods, immunoaffinity chromatography and solid 243 phase extraction, for determination of deoxynivalenol and deepoxy deoxynivalenol in swine 244 serum[END_REF][START_REF] Kostelanska | The study 249 of deoxynivalenol and its masked metabolites fate during the brewing process realised by 250 UPLC-TOFMS method[END_REF][START_REF] Tangni | Cross-Reactivity of Antibodies in Some 260 Commercial Deoxynivalenol Test Kits against Some Fusariotoxins[END_REF]), but do not comply with 188[START_REF] Vendl | Simultaneous determination of deoxynivalenol, 274 zearalenone, and their major masked metabolites in cereal-based food by LC-MS-MS[END_REF] who reported no cross-reactivity with DON3G. 189 190 All ZEN IACs also bind β-ZEL and α-ZEL, but showed no cross-reactivity with the other ZEN 191 derivatives tested. Observed results are in accordance with several studies indicating 192

  obtained after application of mycotoxin standards on the IACs, we selected 202 DON-prep, DONneo and DZT MS-prep as clean up step for the DON related mycotoxins in 203 spiked calf's serum sample. Table 3 shows that these three columns are useful as clean-up step 204 for the analysis of DON derivatives. For NIV, only DONneo was useful as clean-up step. For 205 ZEN conjugated mycotoxins only DZT MS-prep was useful as clean-up step, but only for a 206 limited number of ZEN metabolites. 207 208 The recovery of DON3GU, DOM-1, 3AcDON, 15AcDON, ZEN, β-ZEL and α-ZEL in serum 209 was improved compared to analysis without sample clean-up. This resulted in lower limits of 210 detection for most of the DON conjugates (results not shown). No ZEN IACs were suitable for 211the ZEN-diG, ZEN-14-G, ZEN-16-G and ZEN-14-GU clean-up. However the developed 212 UPLC-MS/MS method showed acceptable recovery for those ZEN conjugates without any is possible to use some commercially available IACs for sample clean-up of conjugated 217 mycotoxins and metabolites in biological fluids such as serum. The IAC clean-up significantly 218 of the developed UPLC-MS/MS method for some conjugated 219 mycotoxins (DON3GU, DON3G, 3AcDON and 15AcDON) and some metabolites (DOM-1, 220

Figure 1 .Figure 2 .

 12 Figure 1. Chemical structures and applied numbering of ZEN (a) and DON (b)

to calf serum spiked with native and conjugated mycotoxins. 81 82 Materials and Methods 83 84

		F o
		r
		P
	75	.	e e r
	76 77 78 79 80 85	The aim of this work was to screen seven commercially available IACs for DON and ZEN for their cross-reactivity, sample clean up ability and as a concentration step for analysis of mycotoxins, conjugated forms and metabolites in animal serum. A calf's serum sample was tested for recovery of selected mycotoxins with and without IAC clean-up. The developed R e v i e w method was applied Solvents and reagents O n
	86 87	Methanol (MeOH) and acetonitrile (MeCN) were ultra-performance liquid chromatography -mass spectrometry (UPLC-MS) grade, purchased from Biosolve (Valkenswaard, the l y
	88	Netherlands). Formic acid (FA) was high performance liquid chromatography (HPLC) grade,
	89	purchased from Sigma Chemical Co. (St.-Louis, MO, USA). Water was purified with a Milli-Q
	90	system (Millipore SA, Brussels, Belgium).
	91
	92	Standards

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	ST_MIX_01SST_MIX_01S
	AV_18022011_015									14: MRM of 2 Channels ES+
	0.00 % 100 0	2.00 Table 1 SRM transitions and analyte retention times of the multi-analyte method 4.00 6.00 8.00 10.00 12.00 14.00 Quantifier	16.00 Qualifier 18.00	TIC (ZEN) 2.15e5 20.00
	0.00 % 0 100	Nr.	2.00	4.00 Analyte	6.00	8.00 Precursor Ion (m/z)	10.00 Declustering 12.00 potential (V)	14.00 (eV)) ion (collision energy	16.00 (eV)) ion (collision 18.00 energy	20.00 Retention time (min)
	0.00 % 100 0	1 2 3	2.00 NIV DON DON3GU 4.00	6.00 313 [M+H] + 8.00 297 [M+H] + 473 [M+H] + 490 [M+NH 4 ] +	10.00	13 20 20 15	12.00	14.00 175 (20) 249 (10) -297 (15)	16.00 125 (20) 18.00 203 (14) 297 (18) 249 (15)*	20.00 2.6 5.5 6.0
	% 0 0.00 % 0 100 0.00 % 0 100 AV_18022011_015 2.00 2.00 AV_18022011_015 AV_18022011_015 0.00 2.00 % 0 100 0.00 2.00 % 0 100 0.00 2.00 % 0 100 0.00 2.00 % 100 4 DON3G 4.00 4.00 4.00 4.00 4.00 4.00 0 5 DOM-1 F 6 3AcDON 7 15AcDON o 6.00 6.00 6.00 6.00 6.00 6.00 459 [M+H] + 8.00 8.00 8.00 8.00 8.00 8.00 476 [M+NH 4 ] + 281 [M+H] -339 [M+H] + 339 [M+H] + 643 [M+H] + r 8 ZEN-diG 660 [M+NH 4 ] + 665 [M+Na] + 481 [M+H] + P 9 ZEN-16-G 498 [M+NH 4 ] + 503 [M+Na] + e 10 ZEN-14-GU 495 [M+H] + 512 [M+NH 4 ] + e 481 [M+H] + r 10.00 10.00 10.63 10.62 9.65 10.00 10.00 10.00 10.00 15 15 17 22 15 15 15 60 15 15 50 20 15 20 11 ZEN-14-G 15 498 [M+NH 4 ] + 503 [M+Na] + 50 12 β-ZEL 321 [M+H] + 14 R 13 α-ZEL 321 [M+H] + 11 14 ZEN 319 [M+H] + 20 e 12.00 12.00 w 14.00 14.00 12.00 14.00 12.00 14.00 12.00 14.00 12.00 14.00 -297 (10) 233 (9) -231 (13) 137(10) --323 (40) --341 (30) 319 (20) --319 (10) -175 (29) 285 (11) 187 (22) e i v * additional qualifier ion	16.00 16.00 7: MRM of 2 Channels ES+ 18.00 20.00 18.00 20.00 TIC (15AcDON) 2.58e5 6: MRM of 2 Channels ES+ TIC (3AcDON) 2.24e5 TIC (DOM-1) 4.10e5 5: MRM of 3 Channels ES+ Time 16.00 18.00 20.00 16.00 18.00 20.00 16.00 18.00 20.00 16.00 18.00 20.00 297 (10) 7.5 459 (5)* 137 (13) 109 (19)* 9.6 189 (18) 10.6 261 (10) 10.6 319 (10)* 319 (15) 10.5 -319 (10)* 319 (5) -11.6 283(20) 319 (15)* 12.9 319 (10) -12.9 341 (30)* 285 (9) 14.1 175 (27) 14.9 203 (12) 15.1
	% 100 0.00 % 0 100 0.00 AV_18022011_015 2.00 2.00 AV_18022011_015	4.00 4.00	6.00 6.00 7.26 7.34 5.99 5.93 6.12 6.20	8.00 8.00 7.47 7.54	10.00 10.00		12.00 12.00	14.00 14.00 O 16.00 16.00 4: MRM of 2 Channels ES+ 18.00 20.00 18.00 20.00 TIC (D3G) 2.45e4 3: MRM of 3 Channels ES+ TIC (DGU) 1.75e4 n l y
	0
	0.00		2.00	4.00	6.00		8.00	10.00		12.00	14.00	16.00	18.00	20.00
	AV_18022011_015									2: MRM of 3 Channels ES+
					5.55							TIC (DON)
	%				5.61						1.04e5
	0
	0.00		2.00	4.00	6.00		8.00	10.00		12.00	14.00	16.00	18.00	20.00
	AV_18022011_015									1: MRM of 3 Channels ES+
	% 0 100	1.60	2.64 2.64 2.71 2.85								Time TIC (Nivalenol) 1.51e4
	0.00		2.00	4.00	6.00		8.00	10.00		12.00	14.00	16.00	18.00	20.00

Table 2

 2 Recovery (n = 3) of the analytes after application of the standard solutions on the selected IACs

	Nr.	Compound name			Recovery ± CV, %
			DON-prep	DONneo	DONaok	DZT MS-prep	eeZEN	ZENneo	ZENaok
	1	NIV	< 1	45 ± 10	< 1	< 1	< 1	< 1	< 1
	2	DON	97 ± 12	84 ± 14	102 ± 16	89 ± 15	< 1	< 1	< 1
	3	DON3GU	64 ± 19	88 ± 16	<1	31 ± 13	< 1	< 1	< 1
	4	DON3G	58 ± 18	48 ± 20	<1	41 ± 16	< 1	< 1	< 1
	5	DOM-1	67 ± 15	59 ± 22	51 ± 23	47 ± 12	< 1	< 1	< 1
	6	3AcDON	54 ± 8	41 ± 24	5 ± 28	48 ± 18	< 1	< 1	< 1
	7	15AcDON	13 ± 12	9 ± 23	24 ± 15	11

Table 3

 3 Recovery (n = 3) of the analytes from the spiked serum sample without clean-up and using selected IACs

	Recovery ± CV , %

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Acknowledgements 225

The authors wish to thank Neogen Corporation (Neogen Co., Ayr, UK) for providing 226 immunoaffinity columns. 227 228