Philippe Briet
email: briet@univ-tln.fr

J Dittrich
email: dittrich@ujf.cas.cz

E Soccorsi
email: eric.soccorsi@univ-amu.fr

Scattering through a straight quantum waveguide with combined boundary conditions

Keywords: numbers: 03.65.Db, 03.65.Nk, 73.63.Fg quantum waveguide, scattering, Dirichlet and Neumann boundary conditions

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Scattering through a straight quantum waveguide with combined boundary conditions

I. INTRODUCTION

Free motion of the non-relativistic quantum mechanical particle is described by Laplace operator as the Hamiltonian up to unessential constants, i.e. if units where = 1 and particle mass m = 1 2 are used. Impenetrable walls guiding the particle motion are described by boundary conditions on the wave function annulating the current through the walls and at the same time making the Laplacian to be a self-adjoint operator with the suitable domain.

Restricting ourselves to the local boundary conditions, they are of the Robin form with the real coefficient. The important special cases are Dirichlet and Neumann boundary conditions known also from other parts of physics. They can also effectively appear for the wave functions of special symmetry, e.g. 1,2 . If walls with different types of boundary conditions can be realized e.g. in some semiconductor materials we would have at our disposal a new type of electron motion control giving perspectives of new microelectronic elements. The study of quantum mechanical problems with combined Dirichlet and Neumann boundary conditions is also a mathematical challenge which can lead to further solvable or nearly solvable models. Two-dimensional straight quantum waveguides with the combined Dirichlet and Neumann boundary conditions are studied for years. They were examined as auxiliary problems in 1,2 and the existence of bound states was proved for some configurations. The existence of bound states in a Dirichlet planar waveguide with Neumann window was shown in 3 . In 4 , the existence or non-existence of bound states for the cases of Dirichlet and Neumann boundary conditions on half-lines of each boundary were shown in dependence of the overlap of Neumann part projections to the waveguide axis. The Hamiltonian domain was also studied thoroughly. The existence of discrete spectrum for the same system was reconsidered also in 5 where a lower bound on the Hamiltonian of the Hardy inequality type was proved.

Further details on the discrete spectrum were obtained in 6 . Paper 7 show the existence of bound states for the 3-dimensional layer with Dirichlet boundary conditions outside one or two Neumann disc-shaped windows. The time decay of heat equation solution in twodimensional waveguides with combined boundary conditions was studied in 8 . The cases with infinitely many changes of boundary condition type were studied in 9 . The limit of infinitely thin waveguide was investigated in 10 and the Dirichlet-like decoupling of parts with different boundary conditions proved in the limit. In the above mentioned papers, mostly spectral properties were studied.

In the present paper we investigate a simple situation of scattering occuring in the planar straight strip-like domain that is displayed on Figure ??. The Hamiltonian H is the (minus) Laplace operator in the waveguide Ω = R × (0, d) with the Dirichlet boundary condition on (R * -× {0}) ∪ (R * + × {d}) and the Neumann boundary condition on (R * -× {d}) ∪ (R * + × {0}). Its domain reads (see 4 )

D(H) = { ψ ∈ H 1 (Ω)| -∆ψ ∈ L 2 (Ω), ψ(-x, 0) = ψ(x, d) = ∂ψ ∂y (x, 0) = ∂ψ ∂y (-x, d) = 0 for x > 0 } .
The domain of H is not H 2 (Ω) but it is contained in H 2 loc (Ω). In fact, functions from D(H) are in H 2 (Ω 1 ) for any open Ω 1 ⊂ Ω such that Ω 1 ∩ {(0, 0) , (0, d)} = ∅. This is an analogy of the well known situation for the Dirichlet Laplacian on a planar domains with angles larger than π on the boundary 11 .

χ - n (y) = √ 2 d sin ( (2n -1) πy 2d ) , χ + n (y) = √ 2 d cos ( (2n -1) πy 2d ) (1)
where n ∈ N * , with eigenvalues µ n = (2n -1) 2 π 2 4d 2 . We obtain the following results.

f (k, x, y) ≈ e ikx χ - n 0 (y) + n 1 ∑ l=1 r l (k)e -ik l x χ - l (y) as x → -∞ , f (k, x, y) ≈ n 1 ∑ l=1 t l (k)e ik l x χ + l (y) as x → +∞ .
Here E > 0 is the energy of scattered particle, k > 0 the longitudinal momentum of incoming particle, k l > 0 outgoing momenta, χ - n 0 incoming transversal mode, χ ± l outgoing transversal modes, r l and t l reflection and transmission coefficients. The result is formulated below as Theorem 1, the proof consists of formulating the matching conditions at x = 0 and showing the existence of their solution.

II. STATIONARY SCATTERING WAVE FUNCTION

Given an initial transversal mode n 0 ∈ N * , we fix the energy E ∈ (µ N , µ N +1 ) for some natural number N ≥ n 0 and pick a longitudinal momentum k ∈ R * + such that

E = µ n 0 + k 2 . ( 2
)
We skip the case of E equal to the energy of some transversal mode µ n for some n ∈ N * , whose set has zero Lebesgue measure on the real line generated by k. Let us denote the maximal excitable transversal mode as

n 1 (k) =   1 2 + √ ( n 0 - 1 2
) 2 + d 2 π 2 k 2   , ( 3
)
where the symbol [•] stands for the integer part, and set

k n = √ µ n 0 -µ n + k 2 > 0, for n = 1, 2, . . . , n 1 (k), (4)
and

κ n = √ µ n -µ n 0 -k 2 > 0 for n ≥ n 1 (k) + 1. (5)
In light of ( 5) we may find constants c 0 (k), c 1 (k) > 0, depending only on k, such that the estimates

c 0 (k) ≤ κ n (k) n ≤ c 1 (k) (6)
hold for all n sufficiently large.

We note

F E,n 0 ,k the set of functions f (k, •, •) of the form f (k, x, y) = e ikx χ - n 0 (y) + n 1 ∑ l=1 r l (k)e -ik l x χ - l (y) + ∞ ∑ n=n 1 +1
r n (k)e κnx χ - n (y) (7) for x < 0, and

f (k, x, y) = n 1 ∑ l=1 t l (k)e ik l x χ + l (y) + ∞ ∑ n=n 1 +1 t n (k)e -κnx χ + n (y) (8)
for x > 0. Here r n (k) and t n (k), for n ∈ N * , are complex coefficients.

We look for a function f (k, •, •) ∈ F E,n 0 ,k , which is solution to the eigenvalue equation

-∆f (k, •, •) = Ef (k, •, •)
in Ω, and satisfies the following matching conditions

f (k, 0 -, •) = f (k, 0 + , •), ∂f ∂x (k, 0 -, •) = ∂f ∂x (k, 0 + , •). ( 9
)
We shall justify and precise (9) in subsection II B. We require in addition that f (k, •, •) be locally in D(H), entailing that the restrictions of f (k, •, •) to (-L, 0)×(0, d) and (0, L)×(0, d)

are respectively in H 1 ((-L, 0) × (0, d)) and H 1 ((0, L) × (0, d)), for all L > 0. Namely, this yields that

∞ ∑ n=1 n|r n | 2 < ∞ and ∞ ∑ n=1 n|t n | 2 < ∞. ( 10
)

A. Suitable functional spaces

For any α ∈ R, the space

H α ± = { g = ∞ ∑ n=1 c n χ ± n ∥g∥ 2 α,± = ∞ ∑ n=1 n 2α |c n | 2 < ∞ }
is Hilbertian for the scalar product (•, •) α,± induced by the above defined norm ∥ • ∥ α,± . For α ≥ 0, H α ± ⊂ L 2 (0, d) while for α < 0 this is a space of functionals defined below. For all f (k, •, •) ∈ F E,n 0 ,k , it can be checked that the trace from left at x = 0,

f (k, 0 -, y) = χ - n 0 (y) + n 1 ∑ l=1 r l (k)χ - l (y) + ∞ ∑ n=n 1 +1 r n (k)χ - n (y), (11)
belongs to

H 1 2
-, while the trace from right

f (k, 0 + , y) = ∞ ∑ n=1 t n (k)χ + n (y), (12)
is in H

1 2
+ . Identity (11) (resp., (12)) may be verified upon taking finite partial sums in the right hand side of (7) (resp., (8)) as approximating C ∞ -functions in H 1 ((-L, 0)×(0, d)) norm (resp., H 1 ((0, L) × (0, d)) norm) and combining the exponential decay of the corresponding terms for x ̸ = 0 with (10).

Imposing on f (k, •, •) that the two expressions in the right hand side of (11) and (12) coincide in L 2 (0, d), the common value f (k, 0 -, •) = f (k, 0 + , •) then belongs to the space

H 1 2 = H 1 2 -∩ H 1 2
+ . More generally, for any α > 0, the space H α = H α -∩ H α + , endowed with the scalar product, (f, g) α = (f, g) α,-+ (f, g) α,+ , is Hilbertian and thus a reflexive Banach space. Moreover, the dual space (H α ± ) ′ of H α ± , and H -α ± are isometric. These two spaces may be identified through the duality bracket

(f, g) H -α ± ,H α ± = +∞ ∑ n=1 f n g n , f = +∞ ∑ n=1 f n χ ± n , g = +∞ ∑ n=1 g n χ ± n .
Analogously, we put for every α > 0,

H -α = (H α ) ′ ⊃ H -α -∪ H -α + ,
and we write (f, g) -α,α instead of (f, g) H -α ,H α .

B. Matching conditions

As we seek for generalized eigenfunctions of the operator H, we look for the solution f to the equation (∆ + E)f = 0 in Ω, where E ∈ R.

For k ∈ R fixed, we know from (6) that κ n scales like n so the series appearing in (7) (resp., in (8)) converges locally uniformly in (-∞, 0) × (0, d) (resp., on (0, +∞) × (0, d)).

Differentiating term by term in ( 7)-( 8) we obtain through direct computation that any

function f (k, •, •) ∈ F E,n 0 ,k verifies (∆ + E)f (k,
x, y) = 0 for every (x, y) ∈ R * ± × (0, d) pointwise as well as distributionally. For every φ ∈ C ∞ 0 (Ω), it follows from this upon applying Stokes theorem that the scalar product (f (k,

•, •), (∆ + E)φ) L 2 (Ω) expresses as: lim ε↓0 (∫ d 0 f (k, -ε, y) ∂φ ∂x (-ε, y)dy - ∫ d 0 f (k, ε, y) ∂φ ∂x (ε, y)dy - ∫ d 0 ∂ f ∂x (k, -ε, y)φ(-ε, y)dy + ∫ d 0 ∂ f ∂x (k, ε, y)φ(ε, y)dy ) . ( 13
)
Taking the limit in the above identity and bearing in mind that

f (k, 0 ± , •) ∈ H 1 2 ± ⊂ L 2 (0, d), we find out that (f (k, •, •), (∆ + E)φ) L 2 (Ω) = ( f (k, 0 -, •) -f (k, 0 + , •), ∂φ ∂x (0, •) ) L 2 (0,d) +(f x (k, 0 + , •) -f x (k, 0 -, •), φ(0, •)) -1 2 , 1 2 , ( 14
)
for all φ ∈ C ∞ 0 (Ω). Here φ(0, •) is regarded as an element of H 1 2 , and

f x (k, 0 -, •) = ikχ - n 0 -i n 1 ∑ l=1 k l r l (k)χ - l + ∞ ∑ n=n 1 +1 κ n r n (k)χ - n ∈ H -1 2 -, (15)
f x (k, 0 + , •) = i n 1 ∑ l=1 k l t l (k)χ + l - ∞ ∑ n=n 1 +1 κ n t n (k)χ + n ∈ H -1 2 + . ( 16
)
The two above expressions are obtained upon formally differentiating at x = 0 each term in the series ( 7)-( 8) with respect to x. Actually, (15)-( 16) may be rigorously justified with the help of ( 5) and (10). We refer to Appendix B for more details. In light of ( 14) we have obtained the following result.

Proposition 1 For n 0 ∈ N * fixed, pick E ∈ (µ n 0 , ∞) and k ∈ R * + in accordance with (2) and let f (k, •, •) ∈ F E,n 0 ,k .
Assume moreover that the two following conditions holds simultaneously:

f (k, 0 -, •) = f (k, 0 + , •) in H 1 2 (17) f x (k, 0 -, •) = f x (k, 0 + , •) in H -1 2 (18)
where f x (k, 0 ± , •) is defined in ( 15)-( 16). Then we have

(∆ + E)f (k, •, •) = 0 in the distri- butional sense in Ω.

Remark 1 The equality (17) is a necessary condition for the conclusion of Proposition

1 to hold, but we do not know whether this is the case for ( 18). This can be seen upon taking φ(x, y) = φ 1 (x)φ 2 (y) in ( 14) for two arbitrary functions

φ 1 ∈ C ∞ 0 (R) and φ 2 ∈ C ∞ 0 (0, d),
getting that both terms in the right hand side of ( 14) vanish. This entails that 17), and that the restrictions of f x (k, 0 -, •)

f (k, 0 -, •) = f (k, 0 + , •) in L 2 (0, d), which yields (
and f x (k, 0 + , •) to C ∞ 0 (0, d) coincide. As we did not succeed to prove that C ∞ 0 (0, d) is dense in H 1 2
, it is not clear whether (18) can be derived from this. Notice that it is not clear whether our H

1 2 = H 1 2 -∩H 1 2
+ given as intersection with the above defined norm is the standard Sobolev space H

= f (k, 0 -, •) = f (k, 0 + , •) the common value in H 1 2
of traces from left and right at x = 0 of f (k, •, •). In view of ( 7)-( 8), we have

φ = χ - n 0 + ∞ ∑ n=1 r n (k)χ - n = ∞ ∑ n=1 t n (k)χ + n ∈ H 1 2 . ( 19
)
Relation (19) holds necessarily in L 2 (0, d), expressing the function φ in two different Hilbertian basis of this space, so there is one-to-one correspondence between the coefficients r n (k) and t n (k). We are thus left with the task of finding a function φ of the form prescribed by (19) which fulfills (18). To this purpose we put for all n ∈ N * ,

p ± n = (χ ± n , •) L 2 (0,d) χ ± n , ( 20
)
and we recall (15)-( 16) and ( 19) in order to re-express f x (k, 0 ± , •) in terms of φ, getting:

f x (k, 0 -, •) = ( -i n 1 ∑ l=1 k l p - l + ∞ ∑ n=n 1 +1 κ n p - n ) φ + 2ikχ - n 0 , f x (k, 0 + , •) = ( i n 1 ∑ l=1 k l p + l - ∞ ∑ n=n 1 +1 κ n p + n ) φ.
In light of the two above identities, the matching condition (18) may be equivalently reformulated as

Dφ = -2ikχ - n 0 , ( 21
)
where D :

H 1 2 → H -1 2
is the operator

D = -iD 1 + D 2 , D 1 = n 1 ∑ l=1 k l (p - l + p + l ), D 2 = ∞ ∑ n=n 1 +1 κ n (p - n + p + n ). ( 22
)
Since k n /n and κ n /n are both majorized by 2 √ E, uniformly in n ∈ N * , then the estimate

∥Dψ∥ -1 2 ≤ 2 √ E∥ψ∥1 2 holds true for any ψ = ∑ n≥1 a n χ - n = ∑ n≥1 b n χ + n ∈ H 1 2 . As a consequence, the linear operator D is bounded from H 1 2 into H -1 2 , with ∥D∥ B(H 1 2 ,H -1 2 ) ≤ 2 √ E.
The proof of the existence of φ obeying (21) boils down to the following technical but essential result, whose proof is postponed to Appendix A 2.

Lemma 1

The operator D 2 is a strictly positive and boundedly invertible from

H 1 2 onto H -1 2 . Moreover, the restriction (D -1 2 ) |L 2 (0,d) of D -1 2 to L 2 (0, d), is symmetric.
In light of (22), the matching condition (21) reads -iD 1 φ + D 2 φ = -2ikχ - n 0 so it may be equivalently rephrased as

φ -iD -1 2 D 1 φ = -2ikD -1 2 χ - n 0 , ( 23
)
according to Lemma 1.

For every n ∈ N * , we denote by P n the orthogonal projection onto the finite dimensional subspace P n = span{χ ± j , j = 1, . . . , n}. Due to (4), D 1 is a positive self-adjoint bounded operator in P n 1 and the same is true for D . Further, putting

φ 1 = D 1/2 1 P n 1 φ, we may rewrite (23) as φ -iD -1 2 D 1/2 1 φ 1 = -2ikD -1 2 χ - n 0 . Applying D 1/2
1 P n 1 and I -P n 1 successively to both sides of the above equation, we end up getting that (23) is equivalent to the system formed by the two following equations:

(I -iD 1/2 1 P n 1 D -1 2 P n 1 D 1/2 1 )φ 1 = -2ikD 1/2 1 P n 1 D -1 2 χ - n 0 (24) φ 2 -i(I -P n 1 )D -1 2 P n 1 D 1/2 1 φ 1 = -2ik(I -P n 1 )D -1 2 χ - n 0 , ( 25
)
where

φ 2 = (I -P n 1 )φ.
Notice from the second part of Lemma 1 that the operator M = D

1/2 1 P n 1 D -1 2 P n 1 D 1/2 1 is symmetric in P n 1 .
As a consequence, all the eigenvalues λ j , j = 1, . . . , n 1 , of M , are real and det(I

-iM ) = ∏ n 1 j=1 (1 -iλ j ) ̸ = 0,
showing that (24) admits a unique solution φ 1 ∈ P n 1 ⊂ L 2 (0, d). From this and (25) then follows that φ 2 ∈ L 2 (0, d), which in turn

yields φ = D -1 2 1 φ 1 + φ 2 = iD -1 2 (P n 1 D 1/2 1 φ 1 -2kχ - n 0 ) ∈ H 1 2 , since L 2 (0, d) ⊂ H -1 2
. This proves the existence of φ and consequently of f . Summing up, we have obtained the:

Theorem 1 Let n 0 , E and k be the same as in Proposition 1. Then there exists f (k,

•, •) ∈ F E,n 0 ,k satisfying the equation (∆ + E)f (k, •, •) = 0 in the distributional sense in Ω.
Remark 2 Let ψ ∈ H -1 2 be arbitrary. It is clear that the above reasonning remains valid upon substituting ψ for -2ikχ - n 0 in the right hand side of (21). Therefore the equation Dφ = ψ admits a unique solution solution φ ∈ H 1 2 . This entails that the linear bounded

operator D is invertible from H 1 2 onto H -1 2
. By the inverse mapping theorem, its inverse

D -1 is thus bounded from H -1 2 onto H 1 2 .

D. Smooth k-dependence

For all (x, y) ∈ R * × (0, d), it is apparent from ( 7)-( 8) that k → f (k, x, y) is continuously differentiable in R \ M , where the set

M = {k ∈ R * + | ∃n ∈ N * , µ n = k 2 + µ n 0 }
is discrete. Moreover, for x = 0 we have the:

Theorem 2 The solution to (21), regarded as a function of the parameter k

∈ R * + , belongs to C 1 (R * + \ M ; H 1 2 ).
Proof. We prove that k → f (k, 0, y) is continuously differentiable for any y ∈ (0, d). To this purpose we first estimate the variation of the operators D = D(k) with respect to k.

To do that we fix k ∈ R and refer to (4)-( 5) in order to choose ε 1 = ε 1 (k) > 0 so small that

n 1 (k + ε) = n 1 (k) for ε ∈ (-ε 1 , ε 1
). Then we apply (22), getting

D(k + ε) -D(k) = -i n 1 ∑ l=1 (k l (k + ε) -k l (k))(p - l + p + l ) + ∞ ∑ n=n 1 +1 (κ n (k + ε) -κ n (k))(p - n + p + n ), (26)
where n 1 stands for n 1 (k) = n 1 (k + ε). Next we recall from ( 4)-( 5) that there exist a continuous functions c n (k, •) obeying

k n (k + ε) -k n (k) = k k n (k) ε + c n (k, ε)ε 2 , n = 1, .., n 1 , κ n (k + ε) -κ n (k) = - k κ n (k) ε + c n (k, ε)ε 2 , n ≥ n 1 + 1,
for (-ε 1 , ε 1 ). Moreover, we know from ( 6) that the estimate

|c n (k, ε)| ≤ c(k)/n holds uniformly in ε ∈ (-ε 1 , ε 1 )
, where the constant c(k) > 0 depends only on k and ε 1 . Hence

C(k, ε) = ∞ ∑ n=1 c n (k, ε)(p - n + p + n ) ∈ B(H 1 2 , H -1 2 ),
with

∥C(k, ε)∥ B(H 1 2 ,H -1 2 ) ≤ c(k), ε ∈ (-ε 1 , ε 1 ). ( 27
)
for some positive constant c(k) independent of ε. Arguing in the same way as above we obtain that the linear term with respect to ε, appearing in (26), satisfies -ik

n 1 ∑ l=1 1 k l (p - l + p + l ) -k ∞ ∑ n=n 1 +1 1 κ n (p - n + p + n ) B(H 1 2 ,H -1 2 ) ≤ c(k), ( 28
)
where c(k) is another constant depending only on k. Putting ( 26) and ( 27)-(28) together we obtain for all |ε| < ε 1 that

∥D(k + ε) -D(k)∥ B(H 1 2 ,H -1 2 ) ≤ c(k)ε(1 + ε), hence ∥D(k) -1 (D(k + ε) -D(k))∥ B(H 1 2 ) < 1, upon eventually shortening ε 1 . Therefore I + D(k) -1 (D(k + ε) -D(k)) is invertible in B(H 1 2 ) for every ε ∈ (-ε 1 , ε 1 ) and the same is true for D(k + ε) = D(k)(I + D(k) -1 (D(k + ε) -D(k))) in B(H 1 2 , H -1 2 ). Moreover ∥D(k + ε) -1 ∥ B(H -1 2 ,H 1 2
) is uniformly bounded by, say, c ′ = c ′ (k) > 0, in ε ∈ (-ε 1 , ε 1 ). Thus, bearing in mind that

∥D(k + ε) -1 -D(k) -1 ∥ B(H -1 2 ,H 1 2 ) = ∥D(k) -1 (D(k) -D(k + ε))D(k + ε) -1 ∥ B(H -1 2 ,H 1 2 ) ≤ c ′2 ∥(D(k) -D(k + ε))∥ B(H 1 2 ,H -1 2 ) ,
we end up getting that

lim ε→0 ∥D(k + ε) -1 -D(k) -1 ∥ B(H -1 2 ,H 1 2 ) = 0. ( 29
) For ε ∈ (-ε 1 , ε 1 ), let φ(k + ε) = D(k + ε) -1 (-2i(k + ε)χ - n 0 ) denotes the solution to (21). Then we have D(k + ε)(φ(k + ε) -φ(k)) = -(D(k + ε) -D(k))φ(k) -2iεχ - n 0 by direct calculation, hence φ(k + ε) -φ(k) ε = -D(k + ε) -1 ∂ k D(k)φ(k) -2iD(k + ε) -1 χ - n 0 -εD(k + ε) -1 C(k, ε)φ(k),
for any 0 < |ε| < ε 1 , where

∂ k D(k) = -ik n 1 ∑ l=1 1 k l (p - l + p + l ) -k ∞ ∑ n=n 1 +1 1 κ n (p - n + p + n ).
From this, (27) and (29) then follows that φ is differentiable in

H 1 2 at k, with ∂φ ∂k (k) = -D(k) -1 ∂ k D(k)φ(k) -2iD(k) -1 χ - n 0 ∈ H 1 2 .
Notice that this expression coincides with the one obtained by formal differentiation of (21).

Its continuity with respect to k was actually established in the proof.

Remark 3 By induction, the continuity for higher derivatives of k → φ(k) follows from the reasoning developped in the proof of Theorem 2.

III. TIME-DEPENDENT SOLUTIONS

We now characterize the scattering properties of the physical model under study with the aid of the generalized eigenfunctions of H, described in section II (see Theorem 1).

For the sake of simplicity we consider sufficiently small quasi-momenta intervals [α, β], with 0 < α < β, so that the index n 1 (k) of the highest excited transversal mode remains constant with respect to k ∈ [α, β]. For any fixed k 0 ∈ R, it is clear from (3) that this can be achieved

upon imposing √ µ n 1 (k 0 ) -µ n 0 < α < β < √ µ n 1 (k 0 )+1 -µ n 0 . ( 30
) Let a ∈ C ∞ 0 (R) be such that supp a ⊂ [α, β] ⊂ (0, ∞) \ M, ( 31
)
where M is as in Theorem 2, and let f ∈ F E,n 0 ,k be defined by Theorem 1. In view of characterizing the asymptotic behaviour of the time-evolution generated by H of

ψ(t, x, y) = ∫ R a(k)e -i(µn 0 +k 2 )t f (k, x, y)dk, (32)
we introduce the two following states,

ψ -(t, x, y) = ∫ R a(k)e -i(µn 0 +k 2 )t e ikx χ - n 0 (y) dk (33)
and

ψ + (t, x, y) =    ψ DN (t, x, y) if x < 0, ψ N D (t, x, y) if x > 0, (34)
where we have set

ψ DN (t, x, y) = ∫ R a(k)e -i(µn 0 +k 2 )t n 1 ∑ l=1 r l (k)e -ik l x χ - l (y)dk, ( 35
)
ψ N D (t, x, y) = ∫ R a(k)e -i(µn 0 +k 2 )t n 1 ∑ l=1 t l (k)e ik l x χ + l (y)dk. ( 36
)
As can be seen from the following statement, ψ DN (resp.

Proposition 2 For every t ∈ R, the function ψ(t, •, •), defined by (32), belongs to D(H).

Moreover ψ is a solution to the equation

( i ∂ ∂t + H ) ψ = 0. ( 37
)
Similarly ψ -(t, •, •) ∈ D(∆ DN ) for each t ∈ R and we have

( i ∂ ∂t + ∆ DN ) ψ -= 0. ( 38
)
Finally ψ DN (t, •, •) and ψ N D (t, •, •) are respectively in D(∆ DN ) and in D(∆ N D ) for all t ∈ R, with ( i ∂ ∂t + ∆ DN ) ψ DN = ( i ∂ ∂t + ∆ N D ) ψ N D = 0. ( 39
)
Proof. We only show the first part of the statement, the remaining part being obtained by arguing in the same way.

From the expression (19 Therefore there exists a constant c 1 > 0 such that we have

) of φ = f (k, 0 -, •) = f (k, 0 + , •) we get that r n (k) = (χ - n (•), φ(k, •)) L 2 (0,d) (up to some additive constant for n = n 0 ) and t n (k) = (χ + n (•), φ(k, •)) L 2 (0,
|r n (k)| ≤ c 1 n -1 2 and |t n (k)| ≤ c 1 n -1 2 , ( 40
)
for all n ≥ 1 and k ∈ [α, β], by (10). On the other hand (5) yields κ n ≥ c 2 n uniformly in

n ≥ n 1 + 1 and k ∈ [α, β],
for another positive constant c 2 . As a consequence the function defined by

ψND (t, x, y) = ∫ R a(k)e -i(µn 0 +k 2 )t ∞ ∑ n=n 1 +1 t n (k)e -κnx χ + n (y)dk, (41)
for every (t, x, y

) ∈ R × R * + × (0, d), satisfies ∫ d 0 | ψND (t, x, y)| 2 dy = ∞ ∑ n=n 1 +1 ∫ R a(k)e -i(µn 0 +k 2 )t t n (k)e -κnx dk 2 ≤ c 2 1 ∥a∥ 2 L 1 (R) ∞ ∑ n=n 1 +1 e -2c 2 nx n , (t, x) ∈ R × R * + , whence ∫ ∞ 0 ∫ d 0 | ψND (t, x, y)| 2 dydx ≤ c 2 1 ∥a∥ 2 L 1 (R) 2c 2 ( ∞ ∑ n=n 1 +1 1 n 2 ) < ∞. Therefore ψND (t, •, •) ∈ L 2 (R * + × (0, d))
for any t ∈ R, and we get in the same way that ψDN (t, •, •) ∈ L 2 (R * -× (0, d)), where we have set

ψDN (t, x, y) = ∫ R a(k)e -i(µn 0 +k 2 )t ∞ ∑ n=n 1 +1 r n (k)e κnx χ - n (y)dk, (42)
for all (t, x, y) ∈ R × R * -× (0, d). By performing the change of integration variable k → k l in each term of the sum appearing in ( 35)-(36), we obtain the Fourier transform of an L 2 -function (with respect to the variable k l ), which is consequently square integrable with respect to x. Therefore the functions defined in ( 35)-( 36) are lying in L 2 (Ω), and we have ψ ∈ L 2 (Ω) from ( 7)-( 8) and (32). The next step of the proof is to check out that the first order partial derivatives of ψ(t, •, •) are square integrable in Ω for any t ∈ R. We shall do it for ∂ψ/∂y, the case of ∂ψ/∂x being handled in the same manner. We start by computing ∂ψ/∂y in the distributional sense. We have

⟨ ∂ψ ∂y (t, •, •), ζ⟩ (C ∞ 0 (Ω)) ′ ,C ∞ 0 (Ω) = - ∫ Ω (∫ R a(k)e -i(µn 0 +k 2 )t f (k, x, y)dk ) ∂ζ ∂y (x, y)dxdy, ( 43
)
for any arbitrary test function ζ ∈ C ∞ 0 (Ω). For y ∈ (0, d) and k ∈ R fixed, we deduce from (8) that |f (k, x, y)| is majorized by

( 2 d ) 1/2 ( n 1 ∑ l=1 |t l (k)| + ∞ ∑ n=n 1 +1 |t n (k)|e -c 2 nx ) ≤ ( 2 d ) 1/2   n 1 ∑ l=1 |t l (k)| + (1 -e -2c 2 x ) -1 2 ( ∞ ∑ n=n 1 +1 |t n (k)| 2 ) 1/2   ,
for all x > 0, and a similar estimate holds from (7) for x < 0. Hence the integral over (supp ζ) × (supp a) in the right hand side of (43) converges so we may apply Fubini's theorem. By integrating by parts over y, we find out by direct calculation that

∂ψ ∂y (t, x, y) = ∫ R a(k)e -i(µn 0 +k 2 )t f y (k, x, y)dk,
in the distributional sense, where f y (k, x, y) expresses as

µ 1/2 n 0 e ikx χ + n 0 (y) + n 1 ∑ l=1 µ 1/2 l r l (k)e -ik l x χ + l (y) + ∞ ∑ n=n 1 +1 µ 1/2 n r n (k)e κnx χ + n (y),
for x < 0, and as

- ( n 1 ∑ l=1 µ 1/2 l t l (k)e ik l x χ - l (y) + ∞ ∑ n=n 1 +1 µ 1/2 n t n (k)e -κnx χ - n (y)
) , for x > 0. We are thus left with the task of checking out that the two above expressions lead to square integrable functions in Ω. Since this is obviously the case for all terms expressed by a finite sum, we shall only examine the one of

ψND (t, x, y) = ∞ ∑ n=n 1 +1 µ 1/2 n χ - n (y) ∫ R a(k)e -i(µn 0 +k 2 )t t n (k)e -κnx dk.
Actually, it turns out that

∫ d 0 ψND (t, x, y) 2 dy = ∞ ∑ n=n 1 +1 µ n ∫ R a(k)e -i(µn 0 +k 2 )t t n (k)e -κnx dk 2 so we have ∫ ∞ 0 ∫ d 0 ψND (t, x, y) 2 dxdy ≤ ∥a∥ 2 L 2 (R) ∫ supp a ∞ ∑ n=n 1 +1 µ n 2κ n (k) |t n (k)| 2 dk < ∞,
the last series being majorized (up to some multiplicative constant) by ∥φ(k)∥ 2 , which is bounded on supp a. Now, bearing in mind that f (k, •, •) is locally in the domain of the Laplacian (this is guaranteed by the matching conditions (17)-( 18)), so that x → f (k, •, y) ∈ C 1 (R) for almost every y ∈ (0, d) and any arbitrary fixed k, we may apply the same method to ∂ψ/∂x, proving that ψ(t,

•, •) ∈ H 1 (Ω).
Further, as ψ evidently satisfies the specified boundary conditions, it remains to prove (37), which involves calculating ∆ψ in the distributional sense. Arguing as above, we find out that,

-∆ψ(t, x, y) = ∫ R (µ n 0 + k 2 )a(k)e -i(µn 0 +k 2 )t f (k, x, y)dk, ( 44
)
with the integrand lying in L 2 (Ω). Since the r.h.s. of (44) coincides with -i∂ψ(t, x, y)/∂t (the derivative is computed for the topology of the norm in L 2 (Ω)), we end up getting (37).

Having established the main properties of the functions ψ defined by (32) and ψ ± defined by ( 33)-(36) in Proposition 2, we are in position to prove the main result of this section.

Theorem 3 Let a ∈ C 1 0 (R) obey (30)-(31), let ψ be the same as in (32) and let ψ ± be defined by (33)-(36). Then we have

lim t→±∞ ∥ψ(t, •, •) -ψ ± (t, •, •)∥ L 2 (Ω) = 0. (45)
Proof. We first examine the case of t > 0. In light of ( 7)-( 8), ( 32)-( 36) and ( 41)-( 42), the function ψ decomposes for any x ∈ R as

ψ(•, x, •) = θ(-x) ( ψ -(•, x, •) + ψ DN (•, x, •) + ψDN (•, x, •) ) + θ(x) ( ψ N D (•, x, •) + ψND (•, x, •) ) , ( 46
)
where θ stands for the usual Heaviside function, and ψ -, ψ DN , ψ N D , ψDN and ψND are respectively defined by (33), ( 35), ( 36), ( 42) and (41).

Let us now establish that the transversal modes associated with n ≥ n 1 + 1, appearing in the decomposition of ψ(t, •, •), vanish as t → ±∞. In view of (32), ( 35)-( 36) and ( 46), it is enough to examine the functions ψDN and ψND . Applying Fubini Theorem, we obtain that

∥ ψND (t, •, •)∥ 2 L 2 (Ω) = ∫ R + ∞ ∑ n=n 1 +1 ∫ R a(k)e -i(µn 0 +k 2 )t t n (k)e -κn(k)x dk 2 dx, (47)
since {χ

(+) n , n ≥ 1} is an orthornomal basis of L 2 (0, d). Further we get ∫ R a(k)e -i(µn 0 +k 2 )t t n (k)e -κn(k)x dk = 1 2it ∫ R e -i(µn 0 +k 2 )t ∂ ∂k ( k -1 a(k)t n (k)e -κn(k)x ) dk, x ∈ R + , t > 0,
by integrating by parts. Taking into account that a ∈ C 1 (R) is supported in [α, β], with α > 0, that t n (k) satisfies the estimate (40), that

dt n dk (k) = d dk (χ + n , φ(k)) L 2 (0,d) = ( χ + n , ∂φ ∂k (k) ) L 2 (0,d)
,

where k → ∂φ ∂k (k) ∈ C 0 (R * + , H 1
2 ) from Theorem 2, and that

κ n (k) ≥ κ n (β) for all k ∈ [α, β],
we find out two positive constants d 1 and d 2 , both of them being independent of n, such

that the estimate ∫ R a(k)e -i(µn 0 +k 2 )t t n (k)e -κn(k)x dk ≤ d 1 + d 2 x 2|t| √ n e -κn(β)x ,
holds uniformly in t ∈ R * and x ∈ R * + . From this and (47) then follows that

∥ ψND (t, •, •)∥ 2 L 2 (Ω) ≤ c |t| , t ∈ R * , (48)
for some constant c > 0, which does not depend on t. Arguing in the same way we get in addition that

∥ ψDN (t, •, •)∥ 2 L 2 (Ω) ≤ c |t| , t ∈ R * . ( 49
)
The next step of the proof involves estimating ∥ψ

-(t, •, •) -ψ N D (t, •, •)∥ L 2 (R * + ×(0,d))
for t < 0. Recalling (33) and (36), we get

ψ -(t, x, y) -ψ N D (t, x, y) = ∫ R a(k)e -i(µn 0 +k 2 )t n 1 ∑ l=1 e ik l x ( δ ln 0 χ - l (y) -t l (k)χ + l (y) ) dk = - n 1 ∑ l=1 ∫ R e i(k l x-(µn 0 +k 2 )t) ∂q l ∂k (k, t, x, y)dk,
upon integrating by parts, where

q l (k, t, x, y) = a(k) i(-2kt + kk -1 l x) (δ ln 0 χ - l (y) -t l (k)χ + l (y)).
From this and (4) then follows that

|ψ -(t, x, y) -ψ N D (t, x, y)| ≤ e 1 x -2k n 1 (α)t + e 2 x -e 3 t (x -2k n 1 (α)t) 2 ,
for t < 0, x > 0 and y ∈ (0, d), where e 1 , e 2 and e 3 are three positive constants which are independent of t, x and y. As a consequence we may find c > 0, such that for all t < 0, we have

∥ψ -(t, •, •) -ψ N D (t, •, •)∥ 2 L 2 (R * + ×(0,d)) ≤ c |t| . ( 50
)
Analogously, recalling from Theorem 2 that k → r l (k) and k → t l (k) are smooth for all l = 0, . . . , n 1 , we get for every t < 0 that

∥ψ DN (t, •, •)∥ 2 L 2 (R * -×(0,d)) ≤ c |t| . ( 51
)
Now, since

∥ψ(t, •, •) -ψ -(t, •, •)∥ 2 L 2 (Ω) ≤ ∥ψ -(t, •, •) -ψ N D (t, •, •)∥ 2 L 2 (R * + ×(0,d)) + ∥ψ DN (t, •, •)∥ 2 L 2 (R * -×(0,d)) + ∥ ψDN (t, •, •)∥ 2 L 2 (Ω) + ∥ ψND (t, •, •)∥ 2 L 2 (Ω) ,
from (46), we obtain the desired result for t → -∞ by putting (48)-(51) together.

Similarly, arguing as above for t > 0, by means of the decomposition

ψ(•, x, •) = ψ + (•, x, •) + θ(x) ψND (•, x, •) + θ(-x) ( ψ -(•, x, •) + ψDN (•, x, •) ) ,
we obtain the desired result for t → +∞.

As seen in the proof of (45), ∥ψ(t, •, •) -

ψ ± (t, •, •)∥ L 2 (Ω) scales like O(t -1/2 ). Actually,
this is due to the fact that a is C 1 , as a smoother a would allow for several integrations by parts, and consequently for a faster time decay.

Further, assumption (30) guarantees that the quasi-momenta involved in the framework of in Theorem 3 remain separated from the discrete threshold momentum values of M . This is not a very strong restriction when studying the scattering of particles with momentum close to given value of k > 0, which is sufficient for the definition of reflection and transmission coefficients, as the skipped threshold points form a zero measure set where the reflection and transmission coefficients need not necessarily be defined. However, their behavior near the thresholds, which we do not attempt to study in this paper, is certainly a point of interest.

IV. NUMERICAL RESULTS

For the illustration, we present here also numerically computed transition probabilities.

Recall that reflection and transmission probability densities from the initial state of transversal mode n 0 and longitudinal momentum k to the final transversal mode m are

P Rm = k m (k) k |r m (k)| 2 , P T m = k m (k) k |t m (k)| 2 .
The matching conditions were cut to the finite number of transversal modes, projected on a suitable finite-dimensional basis, and the resulting system of linear equations for the reflection and transmission coefficients solved numerically. No attempt to prove the convergence of this procedure was done, the precision was tested only numerically.

The computed probability densities for n 0 = 1, 2 are shown in Figures 2 and3. They seem to be continuous but not smooth at the thresholds where excitations of the further transversal modes in the final state are opened. The total reflection at k → 0 is seen in accordance with the Dirichlet decoupling at d → 0 proved by Borisov and Cardone 10 (notice, that the dimesionless value of kd is a relevant parameter).

Thus we have P +

N φ 0 = φ 0 = P - N φ 0 . Since {χ ± n , n = 1, . . . , N } is linearly independent, then φ 0 = 0, contradicting the fact that ∥φ 0 ∥ L 2 (0,d) = 1. As a consequence the statement is true for every φ ∈ P N such that ∥φ∥ L 2 (0,d) = 1. Therefore it holds for any ϕ ∈ P N since it is valid for ϕ = 0 and for ϕ/∥ϕ∥ L 2 (0,d) when ϕ ̸ = 0. Lemma 3 Let N ∈ N * . Then there exists η N ∈ (0, 1) such that the estimate

max ( ∥(I -P + N )ϕ∥ L 2 (0,d) , ∥(I -P - N )ϕ∥ L 2 (0,d) ) ≥ η N ∥ϕ∥ L 2 (0,d) ,
holds for all ϕ ∈ L 2 (0, d).

Proof. The proof boils down to the fact that

∥(I -P ± N )ϕ∥ 2 L 2 (0,d) = ∥ϕ∥ 2 L 2 (0,d) -∥P ± N ϕ∥ 2 L 2 (0,d) = ∥ϕ∥ 2 L 2 (0,d) -∥P ± N P N ϕ∥ 2 L 2 (0,d) ,
and

∥ϕ∥ 2 L 2 (0,d) = ∥(I -P N )ϕ∥ 2 L 2 (0,d) + ∥P N ϕ∥ 2 L 2 (0,d) . Indeed, this entails max ( ∥(I -P + N )ϕ∥ 2 L 2 (0,d) , ∥(I -P - N )ϕ∥ 2 L 2 (0,d) ) = ∥(I -P N )ϕ∥ 2 L 2 (0,d) + ∥P N ϕ∥ 2 L 2 (0,d) -min ( ∥P + N P N ϕ∥ 2 L 2 (0,d) , ∥P - N P N ϕ∥ 2 L 2 (0,d)
) ,

which, together with Lemma 2, yields max

( ∥(I -P + N )ϕ∥ 2 L 2 (0,d) , ∥(I -P - N )ϕ∥ 2 L 2 (0,d) ) ≥ ∥(I -P N )ϕ∥ 2 L 2 (0,d) + (1 -ϵ 2 N )∥P N ϕ∥ 2 L 2 (0,d) .
Since the right hand side of the above estimate is lower bounded by (1 -ϵ 2 N )∥ϕ∥ 2 L 2 (0,d) , we obtain the desired result.

Completion of the proof

For all ϕ ∈ H 1 2 , we have

(D 2 ϕ, ϕ) -1 2 , 1 2 = ∞ ∑ n=n 1 +1 κ n ( (p - n ϕ, ϕ) L 2 (0,d) + (p + n ϕ, ϕ) L 2 (0,d)
) , from ( 22), so we get

(D 2 ϕ, ϕ) -1 2 , 1 2 ≥ κ n 1 +1 ( ∥(I -P - n 1 )ϕ∥ 2 L 2 (0,d) + ∥(I -P + n 1 )ϕ∥ 2 L 2 (0,d) ) ≥ κ n 1 +1 η 2 n 1 ∥ϕ∥ 2 L 2 (0,d) , (A1)
by Lemma 3, showing that D 2 is injective.

To prove that D 2 is a surjection, we first establish the two following lemmas.

Lemma 4

For each ψ ∈ H -1 2 , the functional

F ψ (ϕ) = (D 2 ϕ, ϕ) -1 2 , 1 2 -2ℜ(ψ, ϕ) -1 2 , 1 2
is real-valued, strictly convex, continuous and coercive on H Proof. First, it is apparent from the definition of

F ψ that F ψ (ϕ) ∈ R for all ϕ ∈ H 1 2 .
Next, we get through direct calculation that

tF ψ (ϕ 1 ) + (1 -t)F ψ (ϕ 2 ) -F ψ (tϕ 1 + (1 -t)ϕ 2 ) = t(1 -t)(D 2 (ϕ 1 -ϕ 2 ), ϕ 1 -ϕ 2 ) -1 2 , 1 2 , for each t ∈ [0, 1] and all ϕ 1 , ϕ 2 ∈ H 1 2
. This and (A1) yield that F ψ is strictly convex.

Further, ∥D 2 δ∥ -1 2 being majorized, up to some multiplicative positive constant, by ∥δ∥1

2 , uniformly in δ = ∑ +∞ n=1 δ ± n χ ± n ∈ H 1 2
, as can be seen from the following computation arising from (5),

∥D 2 δ∥ 2 -1 2 = +∞ ∑ n=n 1 +1 κ 2 n n ( |δ + n | 2 + |δ - n | 2 ) ≤ π 2 d 2 +∞ ∑ n=n 1 +1 n 2 -n 2 0 n ( |δ + n | 2 + |δ - n | 2 ) ≤ π 2 d 2 +∞ ∑ n=n 1 +1 n ( |δ + n | 2 + |δ - n | 2 ) ≤ π 2 d 2 ∥δ∥ 2 1 2
, the continuity of F ψ follows readily from the basic estimate

|F ψ (ϕ + δ) -F ψ (ϕ)| ≤ 2 ℜ(D 2 ϕ, δ) -1 2 , 1 2 -ℜ(ψ, δ) -1 2 , 1 2
+ (D 2 δ, δ) -1 2 , 1 2 ≤ 2 ( ∥D 2 ϕ∥ -1 2 + ∥ψ∥ -1 2 + ∥D 2 δ∥ -1 2 ) ∥δ∥1 2 ,
which holds true for any ϕ ∈ H for every ε ∈ (0, ε 0 ), we get that

c -1 ∥f (k, -ε, •) -f (k, 0 -, •)∥1 2 ≤ ∥f (k, •, •) -f n ∥ H 1 ((-L-ε,-ε)×(0,d)) + ∥f n (• -ε, •) -f n (•, •)∥ H 1 ((-L,0)×(0,d)) + ∥f n -f (k, •, •)∥ H 1 ((-L,0)×(0,d)) . (B2)
In light of (B1), the first and third terms in the right hand side of (B2) can de made arbitrarily small upon chosing n sufficiently large. For such an n, using the uniform continuity of the function f n or applying [START_REF] Zuily | Éléments de distributions et d'équations aux dérivées partielles[END_REF] [Chapter 1, Lemma 3.5], it is possible to make the second term arbitrarily small by taking ε > 0 small enough. As a consequence we have

lim ε↓0 f (k, -ε, •) = f (k, 0 -, •) in H 1 2
, and thus in L 2 (0, d), which is the statement of (ii).

Putting (i) and (ii) together, we obtain that .

(B3)

The rest of the proof is to compute ∫ d 0 ∂ f ∂x (k, ±ε, •)φ(±ε, y)dy as ε ↓ 0. We shall actually restrict ourselves to calculating the limit from the left, the other case being fully analogical.

To do that we start by decomposing φ(-ε, •) ∈ H

FIG. 1 .

 1 FIG. 1. The shape of waveguide.

  (i) There exists scattering stationary wave function f (k, x, y) satisfying the stationary Schrödinger equation (∆ + E)f (k, x, y) = 0 in the distributional sense inside the strip Ω, Dirichlet, respectively Neumann, boundary conditions at the appropriate parts of the boundary of Ω, and having the scattering asymtotics

(

  ii) From the non-integrable functions f (k, •, •), time-dependent wave packets ψ(t, •, •) are formed belonging to L 2 (Ω) and the Hamiltonian domain and satisfying the time-dependent Schrödinger equation (Proposition 2). Their scattering behavior is shown as asymptotics (Theorem 3) lim t→±∞ ∥ψ(t, •, •) -ψ ± (t, •, •)∥ L 2 (Ω) = 0 where ψ ± are solutions of the free Schrödinger equations, i.e., ψ -satisfies Dirichlet boundary condition at y = 0 and Neumann boundary condition at y = d for arbitrary x, and vice versa for ψ + .

  This section is devoted to proving the existence of an actual f (k, •, •) ∈ F E,n 0 ,k satisfying the stationary Schrödinger equation (∆ + E)f = 0 in the distributional sense in Ω. In light of Proposition 1, it is enough to seek for a function f (k, •, •) in F E,n 0 ,k fulfilling the matching conditions (17)-(18). With reference to (17), we denote by φ

  ψ N D ) is a weak solution to the Schrödinger equation associated with the Laplace operator ∆ DN (resp. ∆ N D ) acting in L 2 (Ω) with the Dirichlet (resp. Neumann) boundary conditions at y = 0 and Neumann (resp. Dirichlet) boundary conditions at y = d.

  d) for any n ∈ N * . Each r n and t n as well as ∥φ∥ 1 2 is thus a continuous function of k ∈ [α, β] by Theorem 2.

FIG. 2 .

 2 FIG.2. Reflection (dashed) and transmission probabilities P Rm and P T n from the initial transversal state n 0 = 1 and longitudinal momentum k (in units of π 2d ) to the transversal states m and n respectively.

FIG. 3 .

 3 FIG.3. Reflection (dashed) and transmission probabilities P Rm and P T n from the initial transversal state n 0 = 2 and longitudinal momentum k (in units of π 2d ) to the transversal states m and n respectively.

1 2 . 1 2 2 ≤ 2 + 2 +

 121222 For the second statement (ii), we shall only prove that lim ε↓0 f (k, -ε, •) = f (k, 0 -, •) in L 2 (0, d), the case of the limit from the right being fully analogical. From the definition of the trace f (k, 0 -, •), we may find L > 0, ε 0 > 0 and a sequence of functions(f n ) n≥1 in C ∞ (Ω) satisfying lim n→+∞ ∥f n -f (k, •, •)∥ H 1 ((-L-ε 0 ,0)×(0,d)) = 0, (B1)such that f n (0, •) tends to f (k, 0 -, •) in H as n goes to infinity. Moreover we have∥f n (0, •) -f (k, 0 -, •)∥1 c∥f n -f (k, •, •)∥ H 1 ((-L,0)×(0,d)) ,for some positive constant c which is independent of f and f n . Further, since ∥f (k, -ε, •) -f (k, 0 -, •)∥1 2 is majorized by the sum ∥f (k, -ε, •) -f n (-ε, •)∥1 ∥f n (-ε, •) -f n (0, •)∥1 ∥f n (0, •) -f (k, 0 -, •)∥1 2 ,

  0 -, •) -f (k, 0 + , •),

1 2 - 1 2 , 1 2 = 1 ∑∑ n=n 1 κ n |r n | µ 2 n. 1 2 , 1 2 ,

 121211212 , asφ(-ε, •) = ∞ ∑ n=1 b n (-ε)χ - n , b n (-ε) = ∫ d 0 χ - n (y)φ(-ε, y)dy, n ∈ N * ,and notice from (1) upon integrating several times by parts, that|b n (-ε)| = -ε, y)φ(-ε, y)dy -(f x (k, 0 -, •), φ(0, •)) --ik(b n 0 (0) -b n 0 (-ε)e -ikε ) + i n l=1 k l r l (b l (0) -b l (-ε))e ik l ε ) -∞ +1 κ n r n (b n (0) -b n (-ε)e -knε ), (B4)from where we get∫ d 0 ∂ f ∂x (k, -ε, y)φ(-ε, y)dy -(f x (k, 0 -, •), φ(0, •))Bearing in mind that µ n scales like n 2 , and recalling (5) and (10), we see that the series in the above expression is convergent. Thus we obtain lim -ε, y)φ(-ε, y)dy = (f x (k, 0 -, •), φ(0, •)) -by taking the limit ε → 0 + term by term in (B4) and using the continuity of ε → b n (-ε) at 0. Now the desired result follows from this and (B3).

ACKNOWLEDGMENTS

The work was partly supported by the Czech Science Foundation project 14-06818S and by the NPI ASCR institutional support RVO 61389005. J.D. is indebted to CPT CNRS Marseille for the hospitality extended to him during several visits.

Appendix A: Proof of lemma 1

The proof is by means of the two following technical lemmas. With the notation (20), we shall also write

Preliminary results

Lemma 2 Let N ∈ N * . Then there exists ϵ N ∈ (0, 1) such that we have

Proof. Let us first establish that {χ ± n , n = 1, . . . , N } is linearly independent. We consider 2N complex numbers α ± n for n = 1, . . . , N such that

Evaluating this identity at x = 0, we obtain ∑ N n=1 µ r n α + n = 0. The determinant of the system formed by the above equations with r = 0, . . . , N -1, is Vandermonde and equals ∏ 1≤r<s≤N (µ s -µ r ) > 0. Therefore α + n = 0 for all n = 1, . . . , N . Arguing as before with x = d we obtain that α - n = 0 for n = 1, . . . , N , showing that the vectors χ ± n , for n = 1, . . . , N , are linearly independent.

Assuming that the statement of the Lemma does not hold, there would be a sequence

for every n ∈ N * . As the unit sphere in the finite dimensional linear space P N is compact, we may find a subsequence that converges to a limit φ 0 ∈ P N obeying

Let us now prove that F ψ is coercive. We use (5), getting κ n ≥

we derive from Lemma 3 that

Bearing in mind that η n 1 ∈ 1), this entails that

, for all ϕ ∈ H 1 2 . From this and (A2) then follows that

showing that F ψ is coercive.

Lemma 5 Let ψ and F ψ be the same as in Lemma 4. Assume that F ψ has its minimum at

. Then we have

has a minimum at t = 0, by assumption, so its derivative is zero there. As

= 0. Further, substituting iδ for δ in this equation, we find out that

We are now in position to prove the first assertion in Lemma 1. For ψ in H -1 2 fixed, we know from Lemma 4 that F ψ admits a unique minimum ϕ ∈ H 1 2 (see e.g. [START_REF] Jaffe | Vortices and Monpoles[END_REF] ), which is solution to (A3), according to Lemma 5. This establishes that D 2 is surjective, and thus one-to-one from H

, since it is already known to be injective. Moreover, the boundedness of D -1 2 follows from the inverse mapping theorem (e.g. 12 [Theorem III.11]). Last, we prove that the operator (D -1

2

2 g in this equation we get that

In the peculiar case where f, g

A4), which is the required symmetry relation.

Appendix B: Limits in matching conditions

This appendix is devoted to proving that the limit (13) coincides with the right hand side of (14). To do that we start by establishing that (i) ∂φ ∂x (±ε, •) goes to ∂φ ∂x (0, •) in L 2 (0, d) as ε ↓ 0;

(ii) f (k, ±ε, •) tends to f (k, 0 ± , •) in L 2 (0, d) as ε ↓ 0.

First, we notice for every y ∈ (0, d) and ε > 0 that

ε, so we get ∂φ ∂x (±ε, •) -∂φ ∂x (0, •)

which entails (i).


Zeinah Hmedeh

Cedric Du Mouza

Nicolas Travers

Cedric Du Mouza

Nicolas Travers An

Cédric Du Mouza

An Intelligent PubSub Filtering System

Motivation

Web 2.0 technologies have transformed the Web from a publishing-only environment into a living information place where passive readers have become active information collectors and content generators themselves. In this context, Web syndication formats such as RSS or Atom emerge as a popular mean for timely delivery of frequently updated Web content. According to these formats, information publishers provide brief summaries, called information items, of the content they deliver on websites [HVT + 11], while information consumers subscribe to a number of RSS/Atom feeds and get informed about newly published items. Today, almost every personal weblog, news portal, discussion forum, user group or social media (e.g., Facebook, Twitter, Flickr) on the Web employs RSS/Atom feeds.

Given that the amount of the information generated in Web 2.0 is growing dramatically nowadays. An active information consumer becomes rapidly overwhelmed by the amount of information he receives. There is an urgent need for efficient real-time filtering methods across feeds. We focus in this work on a content-based Publish/Subscribe paradigm for Web syndication in which information consumers are decoupled (in both space and time) from information providers and they can express their interest on specific information from items by using content-based subscriptions. Rather than overwhelming users with all items of a feed, keyword-based subscriptions can be matched on the fly against the content of incoming items originating from different feeds. Keyword-based subscriptions essentially allow specifying feeds on a custom-basis (according to the Web 2.0 philosophy) from information available "out there" and essentially act as continuous queries running into the future rather than trying to go "back in time" through searching information stored in a local store.

We have previously proposed three filtering structures [HKC + 12] for this paradigm. Each of these structures reveals interesting characteristics in broad-match semantics. Now, we improve our system by integrating partial matching and diversity/novelty filtering services. In fact, over millions of subscriptions, we show that two large specific sets of non conventional subscriptions are noticed during the filtering process; (a) long subscriptions are never notified since very few items contain all subscription's terms. For this we proposed to relax the broadmatch semantic using partial matching, (b) most of short subscriptions are still overwhelmed by too many matching items, especially by duplicate or similar items. To deal with this issue we integrated a diversity and novelty filtering step to our system in order to improve user satisfaction. By doing so, we reduce the amount of duplicate notifications corresponding to similar items. This last step needs to take into account the history of items already received by the user.

This demonstration details the architecture and items life cycle in a Publish/Subscribe system which integrates keyword-based subscriptions. We describe our new index for partial matching with a diversity/novelty filtering step.

Architecture

Our system is a Publish/Subscribe engine which gathers information (items) from feeds, filters them by keywordbased subscriptions and notifies the corresponding users with the resulting items. It is based on a push-oriented architecture which allows users to be notified on real-time. The global architecture of our system is depicted in Figure 1 showing its main modules. It decouples sources from users in order to push directly notifications to them. The Acquisition module converts feed items to useful items in our system. In fact, the filtering module is based on set of terms, so a simple transformation is applied on items. An extraction from title and description is necessary to produce this set of terms. According to this, term identifiers are provided in correlation to the Vocabulary Manager. Then, every new item is pushed to the Keyword Matching index.

The Vocabulary manager gives for each term its identifier and weight. It helps the system to make faster comparison and search during the process. Term weights are useful during the partial matching and novelty/diversity filtering steps.

The Subscription Manager links each subscription to the corresponding user information, and stores personal parameters for partial matching and novelty/diversity filtering processes. It also deals with updates of subscriptions on different structures of our system.

The Publish/Subscribe Service is in charge the communications between users and the system. It provides subscription management (insert, update, delete, parameters), and notification services. In fact, some specific notification services are necessary to provide new items to the user. It handles notifications by push (on real-time) and/or pull connections (on demand/refresh). Matching items from the PubSub Filtering Engine are notified to users, according to linked subscriptions.

The Keyword Match filtering system is an index that filters items coming from the Acquisition process. It is based on our previous work on the Broad Matching semantic [HKC + 12], in which we have integrated a partial matching capability (see Section 3.1). Then, the item is pushed to the Diversity & Novelty index in order to filter it for every subscription that it satisfies. It can be also directly notified to the PubSub service if no further filters are required by a given user.

Diversity and Novelty filtering is a way to remove two kinds of duplicates in a set of documents. In our case, this filter is applied to each subscription according to its own history of notifications. In order to filter out duplicates for each subscription, a history of already delivered items is necessary to provide new information. Further details are provided in Section 3.2. Every new filtered item is notified to the PubSub Service according to linked subscriptions to be delivered to the corresponding users.

Filtering in PubSub

Our system is based on a two filtering steps, the first one on partial matching of keywords subscription, and the second on window-based history for filtering items by novelty and diversity according to their subscription. We will detail each filtering step and show how items can be notified.

Partial Matching

The broad-match semantic can be to strict for few notifications, especially for long subscriptions. We propose to extend our previous work [HKC + 12] with the partial matching semantic. It relaxes this constraint by taking into account the weight of each term in a subscription, allowing a subscription to be notified if the "most characteristic" terms are present in the item. Thus, we study in this section the modifications to bring. We proposed subscriptions decomposition approach that consists of indexing all sub-subscriptions of interest. While the decomposition approach is the simplest way to handle partial matching for all structures, Countbased Inverted List (CIL) can also be extended to efficiently support partial matching. In this demonstration, we present only CIL for partial matching (CIL p ).

Term weight

Our partial matching relies on a matching score between the subscriptions and items, thus we need to define how to weight terms of the subscriptions. We assign a weight to each term in the vocabulary of items which represents their importance. Several term weighting models are proposed in literature like the Term Frequency (TF) combined with Inverse Documents Frequency (IDF) [START_REF] Baeza-Yates | Modern Information Retrieval[END_REF], the Term Discrimination Value (TDV) [START_REF] Gerard Salton | A Vector Space Model for Automatic Indexing[END_REF] or the Term Precision [START_REF] Bookstein | Probabilistic Models for Automatic Indexing[END_REF].

In our context, the TDV weighting function is more adapted for items and subscriptions. In fact, an item is a short set of terms (52 terms in average [HVT + 11]) where term frequencies cannot be used, so the TF-IDF standard function is inappropriate. Moreover, the TDV weighting function measures how a term helps to distinguish a set of documents (i.e. the term influence on the global entropy). So basically for our subscriptions set, neither a very frequent term (not a selective filter), nor a very uncommon term (never lead to a notification) have an important TDV value.

Each term weight is the TDV value normalized by the sum of TDV values of the subscription terms. So, for a subscription s={t 1 , t 2 , . . . , t }, the subscription term weight ̺(t k , s) is equal to:

tdv(t k ) t i ∈s tdv(ti)

Extending Count-based Inverted List: CIL p

In order to take into account partial matching, we need to define a satisfaction threshold κ p which determines if an item is close enough to a given subscription. An item is notified if the sum of ̺(t k , s) of all matched terms ( t∈s∩I ̺(t, s) > κ p ) exceeds κ p .

The main difference between CIL and CIL p is the Counter which is replaced by a decrementing counter with entries equal to κ p for each subscription, and term's weight ̺(t, s) is stored in postings lists for each subscription's id. Then during the matching attempt, for each term t j of the item the corresponding value in the Counter copy of each subscription s in its postings list is decremented with ̺(t j , s). Whenever a counter reaches zero, a matching is reported.

Novelty and Diversity

The filtering items by Novelty or Diversity is based on the history of received items per subscription. We are not looking of finding the k newest and most dissimilar items among a set of items to notify it to the user. User has already received a set of items (history) and we aim to check if the new published item is new (contains new information) and upgrade the diversity (makes the items less similar to each other). These definitions are based on the weight of item terms, we will use the TDV value for same reasons as explained previously.

Novelty

The objective of filtering items by novelty is to not send to the user the same information several times (identical or contained). Thus, an item I is said to have new information compared to another item I ′ if it contains more important terms which are not contained in I ′ . It is measured by the proportion of new terms weight that should exceed a novelty threshold κ n : t∈(I\I∩I ′ ) tdvt t ′ ∈I tdv t ′ . Notice that the novelty measure should not be symmetric. Regarding the example of a document d and a paragraph p in d, p is not new compared with d, but d is new compared with p. That is why using an asymmetric one, as Jaccard measure, is not suitable where the distance between two items is the ratio of their intersection on the set of their union.

So an item I is called new, according to a subscription with a corresponding history of items H, if there is no item I ′ ∈ H that does not prove the novelty of I:

∀I ′ ∈ H, t∈(I\I∩I ′ ) tdvt t ′ ∈I tdv t ′ ≥ κ n .

Diversity

At the opposite of novelty, diversity deals with the global information contained into subscriptions' history. An item is diverse if the information that it contains is not present in the set of items already notified to the user. It measures how much the incoming item can increase the average pairwise distance D(H) between the history's items. Given a distance metric dist between two items:

D(H) = 2 |H| * (|H|-1) I∈H (I ′ ∈H∧I ′ =I) dist(I, I ′ ).

So, an item I improves the diversity of H if and only if: D(H ∪ {I}) > D(H) (increases the average pairwise distance).

Several approaches were used in the literature to compute the distance between two items: Cosine [ZCM02], Euclidean [START_REF] Drosou | Disc diversity: result diversification based on dissimilarity and coverage[END_REF][START_REF] Pripužić | Top-k/w publish/subscribe: finding k most relevant publications in sliding time window w[END_REF], and Jaccard [START_REF] Drosou | Dynamic diversification of continuous data[END_REF]. . . Since the importance of diversity is bounded by measuring the amount of non-common terms (dissimilarity), the Euclidean distance is the more appropriate metric (Cosine and Jaccard are based on common terms). So, dist is measured by: dist(I,

I ′ ) = √ t∈(I∪I ′ \I∩I ′ ) tdv 2 (t) I 2 × I ′ 2
(To take into account items size and the importance of other terms in the items, a normalisation by the product of their Euclidean norm is applied).

Sliding window-based filtering

Our PubSub system is faced to a large amount of subscriptions (more than 10M), and must deal with a heavy bandwidth of incoming items (tens thousands of items per hour). The CIL p index was design to efficiently match items and subscriptions. For diversity and novelty computation, we take into account history of each subscription. Over time, this requirement is highly memory consuming. Fortunately, old information is no more relevant in term of filtering, that is why we adapted a sliding window-based on time to manage the history of items. All old items are removed from the system wherever they are older than a given period of time.

Rather than storing a history for each subscription, we will store each item once. This will reduce the memory space used by histories. Also, we will benefit to optimize the filtering process by computing once the novelty and distance between stored items and the new published item. For each subscription, we keep links between the subscription and the corresponding items of its history. An example of a common history is illustrated in Figure 2 where common items between subscriptions S 1 and S 2 are stored only once. Links between items and subscriptions means that those items have been notified by the corresponding subscription.

Demonstration

In our demonstration, we will show real-time impact of different parameters on our system. We will use a real data set of items composed of a total number of 10,7M items collected originating from 8K productive feeds (spanning over 2K different hosting sites) [HVT + 11]. The subscriptions have been automatically generated based on the Alias sampling method [START_REF] Walker | An efficient method for generating discrete random variables with general distributions[END_REF]. This method aims to generate subscriptions whose distinct terms follow occurrences in a real distribution, and their size respect the distribution of web queries size reported in [BJC + 04]. It is characterized among others by a maximal size equal to 12 and an average equal to 2.2. For all sketches, the same set of subscriptions and items will be played in order to show the impact of each modification for different parameters.

Subscription Scenario

The demonstration setting is corresponding to the user interaction on the system. As we said above, we will play the same set of subscriptions, so we do not allow user to add or remove subscriptions. But, user can choose the subscription to be considered and, add or remove its terms to see the impact of their weight on matching rate. All interaction will be shown on the user interface (Figure 3), and see on real-time notifications of items after the keyword-match index and the Novelty/Diversity filter. This scenario will show each step of tuning an intelligent filtering system based on keyword subscriptions.

In the top left part of the interface, the subscription size will be displayed to see the impact of short and long subscriptions on notifications rate. The partial matching threshold, the novelty threshold and take into account diversity in the filtering can be modified by the user in the part just below. Note that, these values will be the same for all subscriptions indexed.

For each play of items (partial matching or filtering by novelty and diversity) in our system, for the specified subscription, the results of filtering will be shown on the right hand-side with a list of items on which we can see its information (title, description, link). Each play of items will generate a new tab containing the new corresponding set of items. This will help users to compare the different sets of items to notify for different parameters.

Workload Scenario

This scenario of our demonstration focuses on system administration. We would like to show real-time workload and memory space on the whole system by varying all parameters. In the left part of the management interface (Figure 4), all system's parameters (number of subscriptions to index, number of items to match, satisfaction threshold, novelty threshold, take into account diversity or not, and the size of the window to manage histories) can be specified. Also in this part, one of the three functions of our system (indexing, partial matching and filtering by novelty and diversity) can be executed. The right part shows information about the filtering system: memory, workload, subscriptions statistics, thresholds, histories statistics, number of items in the sliding window and emulation speed.

A first sketch will show the indexing phase in our system. We will set the number of subscriptions to index. This setting will show the evolution of memory and workload, according to different number of subscriptions. Generation of subscriptions and corresponding statistics can be seen on the top-left hand side of the interface (memory space used, average indexing time by subscription, number of postings lists and their average size).

Then, a partial matching for a set of items on the index is done. The impact satisfaction threshold will be shown by varying it. For each play, a tab in the right part of the interface is created to visualize the matching time and the number of satisfied subscriptions (top part). This will help users to compare the impact of satisfaction threshold on matching time and number of subscriptions to notify.

The filtering step can be executed after each matching step, by default it is computed for the last matched set of items. The statistics of the filtering is shown in the bottom-right part of the interface. We are interested in shown the processing time of items and the number of subscriptions to notify, in order to compare it to the number of satisfied by the partial matching. Also, we will show the size of the global history, and repartition of histories for subscription (less than 5 items, between 6 and 10, between 11 and 20, and more than 20). The impact of diversity, novelty threshold and size of the window will be shown by several executions of filtering by varying their values.

Figure 1 :

 1 Figure 1: Architecture of the PubSub Filtering System

Figure 2 :

 2 Figure 2: Common History

Figure 3 :

 3 Figure 3: User interface Figure 4: Admin interface


Heitor Evangelista
email: evangelista.uerj@gmail.com

Marcio Gurgel

Abdelfettah Sifeddine

Rodolfo Rigozo

Mohammed Boussafir

South Tropical Atlantic anti-phase response to Holocene Bond Events

Keywords:

Records of the climatic impacts of the North Atlantic Bond cycles over the subtropical Southern Hemisphere remain scarce, and their mechanism is a topic of active discussion. We present here an alkenone-based reconstructed sea surface temperature (SST) of a sediment core retrieved from the Brazilian Southwestern Tropical Atlantic (SWTA), Rio de Janeiro, together with a sediment SST record from the Cariaco Basin. The sediment cores span the period 2,100 B.P. -11,100 B.P. Morlet-wavelet analysis detected marked periodic signals of ~0.8, ~1.7 and ~2.2 kyr, very similar and with comparable phases to the hematite-stained-grain time series from the Northern North Atlantic in which the cyclic pattern was recognized as Bond cycles. Our result corroborates the modeled surface ocean anti-phase thermal relation between the North and the South Atlantic. We attribute this behavior to the slowing of the Atlantic Meridional Overturning Circulation. The relative SST warming at Rio de Janeiro and the relative cooling at Cariaco were comparatively more pronounced during the early Holocene (from 11 to 5 kyr B.P.) than in more recent time.

Introduction

The Atlantic Meridional Overturning Circulation (AMOC) is considered to be the primary ocean process in which heat is delivered meridionally between the two hemispheres through the Atlantic Ocean. It carries warm surface waters towards the North Atlantic, where they cool and sink. It is estimated that the AMOC redistributes as much as ~25% of the global combined atmosphere-ocean heat flux [START_REF] Cunningham | The present and future system for measuring the Atlantic meridional overturning circulation and heat transport[END_REF] and that it explains the mild climate conditions observed at the Eastern North Atlantic Ocean sector. It has been speculated that a weakening of the AMOC is likely to occur in the near future due to the effects of anthropogenic greenhouse gas emissions [START_REF] Cubasch | Projections of future climate change[END_REF][START_REF] Bryden | Slowing of the Atlantic Meridional Overturning Circulation at 26.5[END_REF][START_REF] Meehl | Global climate projections[END_REF]. This weakening would result in excess melt-water influx from the Arctic and Greenland, which in turn would trigger further climate changes. Nevertheless, the additional meltwater influx is probably not the only cause for AMOC weakening. Results from modeling experiments of [START_REF] Gregory | ThorpeA model intercomparison of changes in the Atlantic thermohaline circulation in response to increasing atmospheric CO 2 concentration[END_REF] have indicated that weakening would be caused more by changes in ocean surface heat flux than by changes in surface water flux. In all of the 11 climate models investigated by their work, the AMOC does not collapse strongly despite the final concentrations of CO 2 reaching 4 times modern values.

During the last glacial, rapid air temperature increases followed by intense cooling are clearly evident in NGRIP Greenland ice core [START_REF] Andersen | A 60000 year Greenland stratigraphic ice core chronology[END_REF], North Atlantic sediment cores [START_REF] Clement | Mechanisms of abrupt climate change of the last glacial period[END_REF] and Northern Hemisphere palynology [START_REF] Correa-Metrio | Rapid climate change and no-analog vegetation in lowland Central America during the last 86,000 years[END_REF]. These events were named Dansgaard-Oeschger (D-O) and Heinrich events. They were attributed to influx of meltwater that promoted significant reduction in the strength of AMOC in which effects were globally synchronous. It has been proposed that D-O events were pretty regular with cyclicity around 1470 years [START_REF] Rahmstorf | Timing of abrupt climate change: a precise clock[END_REF]. Although much less intense, an apparently analogous climate fluctuation during the Holocene was also recorded from petrologic tracers present in sediment cores from the North Atlantic with a cyclicity of ~1470 ± 500 years and named Bond cycles [START_REF] Bond | A pervasive millennial-scale cycle in North Atlantic Holocene and glacial climates[END_REF]. The same cyclicity was found by [START_REF] Bianchi | Holocene periodicity in North Atlantic climate and deep ocean flow south of Iceland[END_REF] using sediment grain-size data that they employed as a proxy of the deep-water flow, an important component of the thermohaline circulation in the Iceland basin. Their grain-size data indicated increased water speeds when the climate of Northern Europe was warmer.

During the last 12 kyr, eight occurrences of hematite-stained-grain (HSG) peaks were recognized as ice-rafted debris. Significant correlations obtained between the stacked marine records and cosmogenic radionuclides, whose activity concentrations are thought to be modulated by the solar-terrestrial geomagnetic activity, suggested originally that changes in solar activity could be the major forcing agent of these events [START_REF] Bond | Persistent solar influence on north Atlantic climate during the Holocene[END_REF]. A later study conducted on the cyclicity pattern of the time series using spectral wavelet analysis was able to distinguish between the solar forcing and the oceanic forcing to show that the 1500-year cycle was basically an ocean-related climate phenomenon [START_REF] Debret | The origin of the 1500-year climate cycles in Holocene North-Atlantic records[END_REF].

The consensus is that the influx of excess meltwater into the Northern North Atlantic, whether forced by anthropogenic causes or due to natural forcings like the Bond events, would have important implications to the regional ocean circulation dynamics. It is proposed that if the AMOC collapses, it would reduce the inter-hemispheric northward heat transport and consequently induce a positive temperature anomaly in the tropical Southern Ocean and induce a bipolar see-saw pattern characterized by a cooling of the northern Atlantic Ocean and warming of the South Atlantic [START_REF] Crowley | North Atlantic Deep Water cools the Southern Hemisphere[END_REF]. Nevertheless, no robust experimental evidence from marine records from the Southern Atlantic Ocean exists to show this north-south synchronicity in ocean heat transport. In this work we present reconstructed SST record from Rio de Janeiro that may help to clarify the South Hemisphere response to Holocene Bond cyclicity and that extends our knowledge of the global impact of the Pleistocene ~1500 yr cyclicity.

Methods

Site description

The oceanographic system at the coring site has two main components: (1) the prevailing action of the Brazil and the North Brazil Currents (BC and NBC) that constitute low density tropical waters flowing south/southwest along the Brazilian coast partially over and mostly off the continental shelf; and (2) the South Atlantic Central Water (SACW) that rises at the coast coming from the west, which is most pronounced during austral spring and summer seasons [START_REF] Coelho-Souza | Biophysical interactions in the Cabo Frio upwelling system, Southeastern Brazil[END_REF] (Fig. 1). The SACW component is responsible for an important quasi-seasonal upwelling phenomenon that occurs in Cabo Frio at around latitude 23°S. The upwelling itself occurs when a prevailing northeasterly trade wind dominates in the region [START_REF] Calado | Feature-oriented regional modeling and simulations (FORMS) for the western South Atlantic: Southeastern Brazil region[END_REF] as a result of a north-south to east-west change in the orientation of the South American shoreline. The intense and constant flow of the BC and NBC waters, accompanied by a relatively low volume of regional fluvial terrigenous sediment delivery, makes the Southwestern Atlantic coast a region of low sedimentation rates [START_REF] Mahiques | Modern sedimentation in the Cabo Frio upwelling system, Southeastern Brazilian shelf[END_REF]. Along the Southwestern Brazilian coast the BC occupies the top 200 m of the water column. The SACW is located just below it to the domain of the Antarctic Intermediary Waters. The sediment core was retrieved on the continental shelf, which is the domain of the Plataform Waters (mostly confined near shore) that is fed by meanders of the BC and is seasonally influenced by the oligotrophic waters of the SACW.

Sediment core recovery

The sediment core (ID: CF02-02A) used in the study was collected by the PSV Astro Garoupa Vessel (Astromarítima/PETROBRAS), on 15th August 2002, at 124 m water depth (23°16′00″S, 41°48′02″W). A Kullenberg piston corer was used to recover 228 cm of sediment from a shelf mud patch made up of grayish hemipelagic mud. The core was sub-sampled continuously at 1 cm intervals; the sub-samples were oven dried at b50 °C and ground in an agate mortar. Core X ray stratigraphy was performed by the SCOPIX method [START_REF] Migeon | SCOPIX: a new X-ray imaging system for core analysis[END_REF].

Sea surface temperature (SST) reconstruction method

Past SSTs were obtained from the unsaturations ratio of long-chain C 37 alkenones: the alkenone unsaturation index, U K′ 37 . These compounds are biosynthesised exclusively by a group of prymnesiophyte algae, principally the coccolithophorid Emiliania huxleyi [START_REF] Prahl | Further evaluation of long-change alkenones as indicators of paleoceanographic conditions[END_REF][START_REF] Marlowe | Long-chain alkenones and alkyl alkenoates and the fossil coccolith record of marine sediments[END_REF][START_REF] Volkman | Alkenones in Gephyrocapsa oceanica: implications for studies of paleoclimate[END_REF]. Because of the presence of a natural compound in the samples with the same retention time as the 5-α-cholestane employed as a quantitative standard and in order to avoid quantification problems, U K′ 37 was calculated using the peak areas of C 37:2 and C 37:3 instead of their absolute concentrationstherefore, U K′ 37 = (C 37:2 peak area) ÷ (C 37:2 peak area + C 37:3 peak area). Alkenone compositions were obtained using an improved method [START_REF] Peltzer | Methodology for sampling and analysis of lipids in aerosols from the remote marine atmosphere[END_REF][START_REF] Ternois | Hydrocarbons, sterols and alkenones in sinking particles in the Indian Ocean sector of the Southern Ocean[END_REF]. Approximately 2 to 4 g of dried sample were solvent extracted three times, first with 20 ml of methylene chloride, a second time with 20 ml of methylene chloride and methanol (1:1), and finally a third time with 20 ml of methanol.

For each extraction, the samples were ultrasonicated for 15 min and centrifuged at 2500-3000 rpm for 10 min. All extracts were mixed and concentrated just to dryness using a rotary evaporator and transferred to 4 ml vials. Lipids were partitioned into compound classes [START_REF] Peltzer | Methodology for sampling and analysis of lipids in aerosols from the remote marine atmosphere[END_REF]. Compound classes were separated into 7 fractions using Pasteur pipettes filled with silica gel (Merck, 0.063-0.100 mm, 5% deactivated) and eluting with increasing polarity solvents and then stored in 4 ml vials. All fractions were evaporated under a nitrogen stream. Alkenones (fraction 4) were first identified on a ThermoQuest Trace 2000 Series gas chromatograph equipped with a Varian coated fused silica CPSil-5-CB capillary column (60 m × 0.32 mm i.d., 0.25 μm film thickness) coupled to a Finnigan Polaris mass spectrometer.

Alkenones were then quantified on a Perkin Elmer Autosytem XL gas chromatograph equipped with the same column and a flame ionization detector. The oven temperature was programmed from 50 to 150 °C at 30 °C min -1 , followed by 7 °C min -1 heating until 250 °C and a final ramp of 1.5 °C min -1 until 310 °C and an isothermal hold for 60 min.

Helium was used as a carrier gas for both types of analysis. Compounds were identified and quantified using Scalibur and Turbochrom PC-based chromatographic software. Reconstructed Rio de Janeiro SSTs (SSTRJ) are based on alkenones and calculated using the equation T (°C) = (U K′ 37 -0.044) / (0.033) [START_REF] Müller | Calibration of the alkenone paleotemperature index UK37′ based on core-tops from the eastern South Atlantic and the global ocean (60°N-60°S)[END_REF]. Analytical error was 0.9 °C, calculated as three times the standard deviation of random replicate analysis.

Sediment core dating and stratigraphy

Chronology was established using bulk organic matter radiocarbon AMS dating [START_REF] Angulo | Reservoir effect of the southern and southeastern Brazilian coast[END_REF]. The age control was made using CALIB software. Twelve samples from the core CF02-02A were dated at the L.M.C.14 Laboratory (UMS2572-Laboratoire de Mesure du Carbone 14, CNRS/IRD/CEA/IRSN/MC, Saclay-France). Radiocarbon dating was based on bulk organic matter, and radiocarbon ages were calibrated using CALIB 5.0.1 software (http://calib.qub.ac.uk/calib/) and applying the regional reservoir effect correction (ΔR = 8 ± 17), according to [START_REF] Angulo | Reservoir effect of the southern and southeastern Brazilian coast[END_REF]. This correction is very similar to that estimated by the Marine Reservoir Correction Database (http://calib.qub. ac.uk/calib/), which was ΔR = -10 ± 48.

Digital core stratigraphy was used to detail the internal structure of sedimentary or biological unity by a high resolution X-ray method, the SCOPIX analysis [START_REF] Migeon | SCOPIX: a new X-ray imaging system for core analysis[END_REF][START_REF] Lofi | SCOPIX-digital processing of X-ray images for the enhancement of sedimentary structures in core slabs[END_REF]. The core is macroscopically composed of a mix of carbonaceous concretions, shells and other carbonate debris (reworked material) from the base to 195 cm. This composition gradually changes to mud-sands, and from 130 cm to the core top it becomes a homogeneous dark-gray to olive-gray mud (5GY-4/1 to 5G-4/2). The core log, based on macroscopical observations, SCOPIX gray level values and Rock-Eval results, shows six sedimentary sequences (Fig. 2)

Cyclicity patterns from SST reconstruction

We have conducted two methods of spectral analysis to explore statistically significant cyclicities from reconstructed SSTs (Rio de Janeiro/ Brazil), Cariaco Basin on the Northern Venezuelan Shelf inferred from Mg/Ca ratios in surface-dwelling planktonic foraminifera [START_REF] Lea | Synchroneity of tropical and highlatitude Atlantic temperatures over the last glacial termination[END_REF] and HGS time series [START_REF] Bond | Persistent solar influence on north Atlantic climate during the Holocene[END_REF] downloaded from an official data repository/NOAA World Data Center for Paleoclimatology. The methods were the iterative regression analysis (named here ARIST "Análise por Regressão Iterativa de Séries Temporais") [START_REF] Rigozo | Prediction of solar minimum and maximum epochs on the basis of spectral characteristics for the next millennium[END_REF], and the Morlet wavelet analysis [START_REF] Torrence | A practical guide to wavelet analysis[END_REF][START_REF] Rigozo | Prediction of solar minimum and maximum epochs on the basis of spectral characteristics for the next millennium[END_REF]. The ARIST method has as a basic concept the fitting of sine functions over time series. The simple sine function used in ARIST has 3 parameters to be determined: amplitude, angular frequency and phase, where the starting point of the method is the definition of the so-called conditional function defined as F = Ya 0 sin(a 1 t + a 2 ) in which Y is the signal (time series) and a 0 , a 1 , a 2 are the three unknown parameters. For the determination of each parameter we have used 200 iterations for each time series. From the total range of cyclicities detected in each time series by this method, we have taken into consideration, for the interpretation of data, only those cyclicities in which the amplitude/deviation ratio, that is, the cycle amplitude is 2 times higher than the standard deviation. This amplitude corresponds to cyclicities within 95% significance level. The complex Morlet wavelet analysis was used here to detect important changes in time series and cyclicities. The Morlet Wavelet is a plane wave modulated by a Gaussian function. DWT (discrete wavelet transform) as used here deals with time series defined essentially over a range of integers (t = 0, 1, … N -1), where N is the number of values in the time series. Therefore if a time series x(t) is formed by N data points, the DWT{x} will also have N points or N coefficients. The 2 j -1 scales are the wavelet coefficients. The wavelet coefficients are proportional to the difference of averages of the original data and reflect long term variations.

Results

Reconstructed SSTRJ varied from 24.2 °C to 28.1 °C with a mean value of 25.9 °C and a standard deviation of 0.87 °C, for the period 2100 to 11,100 years B.P. For the same chronology, a mean value of 0.8 °C higher was estimated for the Cariaco Basin on Fig. 3. During the periods of Bond peaks (1, 2,…, 8), SSTs have, on average, decreased from -0.6 °C to -0.3 °C with respect to the mean from the early Holocene (~11 ka to 5 ka) to late Holocene (5 ka to ~2 ka) in the Cariaco Basin and from +1.5 °C to +0.8 °C at Rio de Janeiro. For the SST variance, 0.74 (n = 59) for Rio de Janeiro and 0.22 (n = 45) for Cariaco Basin, the Test-F has yielded evidence of statistical difference at 95% confidence level (F critic = 1.61 and F = 3.28) between the sites.

Results of Morlet wavelet analysis reinforce the apparent Northern-Central-Southern Atlantic synchronicity (Fig. 4). The spectral analysis revealed two statistically significant oscillatory bands (~0.8 kyr and one extending from ~1.7 to 2.5 kyr) dominating the 3 investigated sites, that is consistent to previously reported spectral analysis [START_REF] Debret | The origin of the 1500-year climate cycles in Holocene North-Atlantic records[END_REF].

From the ARIST spectral method we were able to retrieve main oscillatory signals from the 3 time series, identified phases and correlations. The technique allowed a deconvolution of the periodic bands detected by the wavelet spectral analysis so that the 1.7 kyr and 2.2 kyr signals were recognized apart. Fig. 5 depicts the total cyclicities detected by the ARIST method with intervals of statistical significance with respect to 1 and 2 standard deviation.

Discussion

A comparison between the time series of the reconstructed SSTs and the Holocene Bond Stages (HBS) (1, 2, …, 8) verifies that almost all HBS were well recognized. The ARIST method indicated a bi-polar synchronicity between the increasing HSG delivery into the North Atlantic Ocean floor and the increasing SST in the tropical Southwestern Atlantic. Extending the comparison to the Cariaco Basin, SST peaks are predominantly in opposite phase with respect to Bond events, indicating that cooler periods in the Cariaco Basin coincided with cooling in the Northern North Atlantic.

Considering the different reconstruction methods employed in each sediment core, the analytical error for alkenone determination, as described here, and the amplitude of the variance, the mean SSTs derived from Cariaco and Rio de Janeiro cannot be assumed as statistically different, a result that can be also achieved with the use of a two-tailed Student's t-test at 95% confidence level. Nevertheless, during the Bond events, Rio de Janeiro SSTs may reach levels comparable to the Cariaco average despite the difference of the mean direct solar radiation between the two sites.

The mechanism responsible for the anti-phase relation between the Northern and Southern Hemispheres of the Atlantic is well documented. The AMOC transports heat and salt towards the Northern Hemisphere, but when increased melt-water is delivered to Northern subpolar regions and vicinities, a slowing of the AMOC may result in the accumulation of heat and salt in the Southern Hemisphere, a process called the bipolar seesaw [START_REF] Crowley | North Atlantic Deep Water cools the Southern Hemisphere[END_REF][START_REF] Manabe | Simulation of abrupt climate change induced by freshwater input to the North Atlantic Ocean[END_REF][START_REF] Broecker | Paleocean circulation during the last deglaciation: a bipolar seesaw?[END_REF]. The reconstructed SSTRJ data seem to support this mechanism. The described mechanism is also consistent with coupled ocean-atmosphere general circulation models (e.g.: [START_REF] Dahl | Assessing the role of North Atlantic freshwater forcing in millennial scale climate variability: a tropical Atlantic perspective[END_REF]. In their model they added 0.1 Sv (1Sv: 1Sverdrup = 10 6 m 3 s -1 ) of freshwater into the North Atlantic for 100 years. Under such condition the AMOC is estimated to be reduced by roughly 40%, cooling the North Atlantic region up to 8°, and both air temperature and SST are expected to increase from the North towards the Equator. One consequence of this scenario is the southwards shift of the ITCZ, along with changes in the trade winds and an asymmetric response of the Hadley circulation [START_REF] Broccoli | Response of the ITCZ to Northern Hemisphere cooling[END_REF]. In the Cariaco Basin, the observed response to these Holocene oscillations can be interpreted as a small scale response to the climate oscillations that took part during last glacial termination, as revealed by the Greenland Ice Sheet Project 2, when a quasi synchronicity response was recorded [START_REF] Lea | Synchroneity of tropical and highlatitude Atlantic temperatures over the last glacial termination[END_REF].

With regard to the alkenone-derived Holocene SST trends for the Atlantic, data from Rio de Janeiro presented a near neutral pattern. In particular, the last 5 kyr showed a decrease of ~-1 °C to the present. In this context the [START_REF] Leduc | Holocene and Eemian sea surface temperature trends as revealed by alkenone and Mg/Ca paleothermometry[END_REF] study found robust cooling trends towards the present for the Northern Atlantic as far south as ~30°N. Between 15°N and 15°S a predominant SST reversal (warming trend) was observed. However, SST reconstructions south of 15°S are lacking in the Atlantic. SST reconstructions for the Pacific and Indian Ocean present high prevalence of cooling southwards of this latitudinal band [START_REF] Leduc | Holocene and Eemian sea surface temperature trends as revealed by alkenone and Mg/Ca paleothermometry[END_REF]. It is clear from Fig. 1 that a change in variability pattern at the 3 time series occurs before and after ~5 yr-B.P. This mid-Holocene transition accompanies a worldwide climate variability well documented for the Polar Regions in Antarctica and Greenland. In the tropicas and subtropics the climate signals of this transition are more ambiguous, varying from cooling to aridity regimes according to palynology records, speleothems, or lake sediment data [START_REF] Steig | Changes in climate, ocean and ice-sheet conditions in the Ross embayment, Antarctica, at 6 ka[END_REF].

The cross-wavelet analysis performed between the tropical sites and HSG time series indicated that the ~1.7-2.5 kyr band was persistent during a significant time interval of the Holocene (~3.5 kyr and 9 kyr), but it became weaker towards the modern epoch, while the synchronicity was more prominent between ~4 and 9 kyr. The response to the ~0.8 kyr oscillatory response was dominant through the late Holocene. Combining the Holocene and the last glacial period, HSG data from North Atlantic sediment cores exhibit a cyclicity centered at 1470 ± 532 years. Therefore, the Bond event cyclicity essentially spanned ~938 years to 2002 years. Based on the wavelet Morlet analysis the main cyclic pattern obtained for Rio de Janeiro and Cariaco corresponds to a broad interval from ~1700 to 2500 years and therefore lies mostly within the cyclicity interval detected by Bond et al. From our observations, we conclude that the transition from a neutral response to Bond events to warming of surface water masses that is caused by the retention of heat in the Tropical-to-Polar Atlantic waters as a consequence of the AMOC slowing may take place well below the Equator and likely in tropical-subtropical latitudes. There is a clear need for better definition of this geographical boundary, which may be under the influence of North Atlantic Bond events, using new sediment core analyses. Time lags between oscillatory signals were found to vary between 50 and 200 yr. This time interval is comparable to the estimates of inter-hemispheric transport of water masses. Nevertheless, this interpretation should be considered with caution considering the resolution of the data of this study. continental shelf to investigate the Southern Tropical Atlantic response to Bond events.

Conclusions

Fig. 1 .

 1 Fig.1. Main oceanographic features and marine currents at Rio de Janeiro core site. The 500 m isoline limits the continental shelf.

Fig. 2 .

 2 Fig.2. Core general lithology, core depth vs. calibrated age curve and X-ray imaging scanning ("SCOPI data").

Fig. 6

 6 shows superimposed 0.8 kyr, 1.7 kyr and 2.2 kyr signals of Bond and SSTRJ and Bond and Cariaco Basin, and r-Pearson correlation coefficients among oscillatory signals. An interesting finding here is the high correlation values varying from 0.89 to 0.99 between the retrieved cycles. This correlation means a great coherency between the signals when the time lag is adequately adjusted.

Fig. 3 .

 3 Fig. 3. Sedimentary records at 3 sites along the Atlantic: (A) North Atlantic stacked marine record. Data refers to percent hematite-stained grains (%HSG); (B) reconstructed Central Atlantic SST at Cariaco Basin based on Mg/Ca ratios[START_REF] Lea | Synchroneity of tropical and highlatitude Atlantic temperatures over the last glacial termination[END_REF]; and (C) Southwestern Atlantic SST/Rio de Janeiro derived from alkenones (this paper). Numbers denote the "Holocene Bond Stages" (the millennial-scale cycles recognized in the North Atlantic of nearly 1500-year cyclicity). Vertical boxes highlight, approximately, each Bond event interval.

Fig. 4 .

 4 Fig. 4. Spectral analysis (Morlet wavelet analysis) for North Atlantic stacked marine record (A), reconstructed SST at the Southwestern tropical Atlantic (SSTRJ) (B), and reconstructed SST at Cariaco Basin (C). Cross wavelet spectrum between Bond and SSTRJ (D) and between Bond and Cariaco Basin (E). Main cyclicities bands are indicated by arrows. White contour line defines the cyclicities within 95% confidence level and black continuous line defines the cone of significance of the analysis.

Fig. 5 .

 5 Fig.5. Cyclicities (yr) and intervals of statistical significance from ARIST method for Cariaco and Rio de Janeiro SST reconstructions and hematite-stained grains (%HSG).

  This work presents a new alkenone-based SST record covering the part of the Holocene from 2.1 kyr B.P. to 11.1 kyr B.P. from a shelf site in the Southwestern South Atlantic and compares it to the SST record from the Carioca Basin. Spectral analyses performed with Morletwavelet and ARIST methods reproduced marked oscillatory signals of ~0.8 kyr, ~1.7 kyr and ~2.2 kyr for reconstructed SSTs off Rio de Janeiro and from Cariaco and the hematite-stained-grain time series in the Northern North Atlantic that are recognized as Bond cycles within statistic significance. Time lags between events (Bond peaks) in the North Atlantic and the South Atlantic at the tropics were estimated to occur at a maximum interval of 200 yr. We attribute the anti-phase thermal relation to the slowing of the Atlantic Meridional Overturning Circulation as described previously by several authors. The relative SST warming at Rio de Janeiro and the relative cooling at Cariaco seemed to be more pronounced during the early Holocene (from 11 kyr to 5 kyr B.P.). Although our analyses for the Southwestern Atlantic reproduce the Northern Hemisphere main cyclicity patterns they are one of the initial alkenone-based SST reconstructions inside the vast Southwestern Atlantic sector and have the additional caveat that they have low-to-moderate resolution. Our results nonetheless provide strong encouragement towards improving SST reconstructions on the Brazilian

Fig. 6 .

 6 Fig. 6. Superimposed SST oscillatory signals corresponding to the cyclicities of 0.8 kyr (A and B), 1.7 kyr (C and D) and 2.5 kyr (E and F) for Bond, Rio e Janeiro and Cariaco Basin; (Right) regressions between the oscillatory signals. The temporal lag between two oscillatory signals was defined as the time interval in which the higher r-Pearson value is obtained while performing a lag correlation.

Acknowledgments

We acknowledge support from LMI PALEOTRACES (IRD/UFF/Uantof/ UPCH/UPMC), Id. 300448/00; CLIMPAST Project (CNPq/IRD 490735/ 2006) and PRIMO Project (CNPq/IRD 590172/2011-5). We also thank two reviewers of Palaeogeography, Palaeoclimatology, Palaeoecology for their valuable suggestions that considerably improved this contribution.