Alain Joye
email: joye@cpt.univ-mrs.fr

Exponential Asymptotics in a Singular Limit for n-Level Scattering Systems

Keywords: Singular perturbations, semiclassical and adiabatic approximations, n-level Smatrix, turning point theory. AMS classification: 34E20, 34L25, 81Q20

The singular limit ε → 0 of the S-matrix associated with the equation iεdψ(t)/dt = H(t)ψ(t) is considered, where the analytic generator H(t) ∈ M n (C) is such that its spectrum is real and non-degenerate for all t ∈ R. Sufficient conditions allowing to compute asymptotic formulas for the exponentially small off-diagonal elements of the S-matrix as ε → 0 are explicited and a wide class of generators for which these conditions are verified is defined. These generators are obtained by means of generators whose spectrum exhibits eigenvalue crossings which are perturbed in such a way that these crossings turn to avoided crossings. The exponentially small asymptotic formulas which are derived are shown to be valid up to exponentially small relative error, by means of a joint application of the complex WKB method together with superasymptotic renormalization. The application of these results to the study of quantum adiabatic transitions in the time dependent Schrödinger equation and of the semiclassical scattering properties of the multichannel stationary Schrödinger equation closes this paper. The results presented here are a generalization to n-level systems, n ≥ 2, of results previously known for 2-level systems only.

Introduction

Several problems of mathematical physics lead to the study of the scattering properties of linear ordinary differential equations in a singular limit iεψ ′ (t) = H(t)ψ(t), t ∈ R, ε → 0,

(1.1)

where the prime denotes the derivative with respect to t, ψ(t) ∈ C n , H(t) ∈ M n (C), for all t. A system described by such an equation will be called a n-level system. Let us mention for example the study of the adiabatic limit of the time dependent Schrödinger equation or the semiclassical limit of the one-dimensional multichannel stationary Schrödinger equation at energies above the potential barriers, to which we will come back below. When the generator H(t) is well behaved at +∞ and -∞, the scattering properties of the problem can be described by means of a matrix naturally associated with equation (1.1), the so-called S-matrix. This matrix relates the behavior of the solution ψ(t) as t → -∞ to that of ψ(t) as t → +∞. Assuming that the spectrum σ(t) of H(t) is real and non-degenerate,

σ(t) = {e 1 (t) < e 2 (t) < • • • < e n (t)} ∈ R, (1.2)
the S-matrix is essentially given by the identity matrix

S = diag (s 11 (ε), s 22 (ε), • • • s nn (ε)) + O(ε ∞ ), where s jj (ε) = 1 + O(ε) as ε → 0, (1.3)
provided H(t) is C ∞ , see e.g. [F1], [F2], [W]. Moreover, if H(t) is assumed to be analytic, it was proven in various situations that the off-diagonal elements s jk of S are exponentially decreasing [FF], [W], [F1], [F2], [JKP], [START_REF] Joye | Semi-Classical Asymptotics beyond All Orders for Simple Scattering Systems[END_REF] s jk = O e -κ/ε , ∀j = k, (1.4) as ε → 0. See also [START_REF] Joye | Exponentially Small Adiabatic Invariant for the Shhrödinger Equation[END_REF], [N], [M], [Sj] for corresponding results in infinite dimensional spaces. Since the physical information is often contained in these off-diagonal elements, it is of interest to be able to give an asymptotic formula for s jk , rather than a mere estimate.

For 2-level systems (or systems reducible to this case, see [START_REF] Joye | Superadiabatic Evolution and Adiabatic Transition Probability between Two Non-degenerate Levels Isolated in the Spectrum[END_REF], [J], [MN]), the situation is now reasonably well understood, at least under generic circumstances. Indeed, a rigorous study of the S-matrix associated with (1.1) when n = 2, under the hypotheses loosely stated above, is provided in the recent paper [START_REF] Joye | Semi-Classical Asymptotics beyond All Orders for Simple Scattering Systems[END_REF]. The treatment presented unifies in particular earlier results obtained either for the time dependent adiabatic schrödinger equation, see e.g [START_REF] Joye | Quantum Adiabatic Evolution[END_REF] and references therein, or for the study of the above barrier reflexion in the semiclassical limit, see e.g. [FF], [O]. Further references are provided in [START_REF] Joye | Semi-Classical Asymptotics beyond All Orders for Simple Scattering Systems[END_REF]. As an intermediate result, the asymptotic formula

s jk = g jk e -Γ jk /ε (1 + O(ε)) , ε → 0,
(1.5) for j = k ∈ {1, 2}, with g jk ∈ C and Re Γ jk > 0 is proven in [START_REF] Joye | Semi-Classical Asymptotics beyond All Orders for Simple Scattering Systems[END_REF]. As is well known, to get an asymptotic formula for s jk , one has to consider (1.1) in the complex plane, in particular in the vicinity of the degeneracy points of the analytic continuations of eigenvalues e 1 (z) and e 2 (z). Provided the level lines of the multivalued function Im z 0 e 1 (z ′ )e 2 (z ′ )dz ′ = cst, (1.6) called Stokes lines, naturally associated with (1.1) behave properly in the complex plane, the so-called complex WKB method allows to prove (1.5). But more important, it is also shown in [START_REF] Joye | Semi-Classical Asymptotics beyond All Orders for Simple Scattering Systems[END_REF] how to improve (1.5) to an asymptotic formula accurate up to exponentially small relative error s jk = g * jk (ε)e -Γ * jk (ε)/ε 1 + O e -κ/ε , ε → 0, (1.7) with g * jk (ε) = g jk +O(ε) and Γ * jk (ε) = Γ jk +O(ε 2 ). This is achieved by using a complex WKB analysis jointly with the recently developed superasymptotic theory [Be], [N], [START_REF] Joye | Superadiabatic Evolution and Adiabatic Transition Probability between Two Non-degenerate Levels Isolated in the Spectrum[END_REF]. Note that when given a generator, the principal difficulty in justifying formulas (1.5) and (1.7) is the verification that the corresponding Stokes lines (1.6) display the proper behavior globally in the complex plane, which may or may not be the case [JKP]. However, this condition is always satisfied when the complex eigenvalue degeneracy is close to the real axis, as shown in [J]. See also [MN] and [R] for recent related results.

For n-level systems, with n ≥ 3, the situation is by no means as well understood. There are some results obtained with particular generators. In [D], [START_REF] Carroll | Generalization of the Landau-Zener Calculation to Three-Level Systems[END_REF], [START_REF] Carroll | Transition Probabilities for the Three-Level Landau-Zener Model[END_REF] and [BE], certain elements of the S-matrix are computed if H(t) = H * (t) depends linearly on t, H(t) = A+tB, for some particular matrices A and B. The choices of A and B are such that all components of the solution ψ(t) can be deduced from the first one and an exact integral representation of this first component can be obtained. The integral representation is analyzed by standard asymptotic techniques and this leads to results which are valid for any ε > 0, as for the classical Landau-Zener generator. The study of the three-level problem when H(t) = H * (t) ∈ M 3 (R) is tackled in the closing section of the very interesting paper [HP]. A non rigorous and essentially local discussion of the behavior of the level lines of Im z 0 e j (z ′ )e k (z ′ )dz ′ , j = k = 1, 2, 3, is provided and justifies in very favorable cases an asymptotic formula for some elements of the S-matrix. See also the review [So], where a non rigorous study of (1.1) is made close to a complex degeneracy point of a group of eigenvalues by means of an exact solution to a model equation. However, no asymptotic formula for s jk , j = k, can be found in the literature for general n-level systems, n ≥ 3. This is due to the fact that the direct generalization of the method used successfully for 2-level systems may lead to seemingly inextricable difficulties for n = 3 already. Indeed, with three eigenvalues, one has to consider three sets of level lines Im z 0 e j (z ′ )e k (z ′ )dz ′ to deal with (1.1) in the complex plane, and the conditions they have to fulfill in order that the limit ε → 0 can be controlled may be incompatible for a given generator, see [F1], [F2] and [HP]. It should be mentioned however, that there are specific examples in which this difficult problem can be mastered. Such a result was recently obtained in the semiclassical study [Ba] of a particular problem of resonances for which similar considerations in the complex plane are required.

The goal of this paper is to provide some general insight on the asymptotic computation of the S-matrix associated with n-level systems, n ≥ 3, based on a generalization of the techniques which proved to be successful for 2-level systems. The content of this paper is twofold. On the one hand we set up a general framework in which asymptotic formulas for the exponentially small off-diagonal coefficients can be proven. On the other hand we actually prove such formulas for a wide class of n-level systems. In the first part of the paper, we give our definition of the S-matrix associated with equation (1.1) and explicit the symmetries it inherits from the symmetries of H(t), for t ∈ R (proposition 2.1). Then we turn to the determination of the analyticity properties of the eigenvalues and eigenvectors of H(z), z ∈ C, which are at the root of the asymptotic formulas we derive later (lemma 3.1). The next step is the formulation of sufficient conditions adapted to the scattering situation we consider, under which a complex WKB analysis allows to prove a formula like (1.5) (proposition 4.1). The conditions stated are similar but not identical to those given in [JKP] or [HP]. As a final step, we show how to improve the asymptotic formula (1.5) to (1.7) by means of the superasymptotic machinery (proposition 5.2 and lemma 5.2). Then we turn to the second part of the paper, where we show that a wide class of generators fits into our framework and satisfy our conditions. These generators are obtained by perturbation of generators whose eigenvalues display degeneracies on the real axis, in the spirit of [J]. We prove that for these generators, in absence of any symmetry of the generator H(t), one element per column at least in the S-matrix can be asymptotically computed (theorem 6.1). This is the main technical section of the paper. The major advantage of this construction is that it is sufficient to look at the behavior of the eigenvalues on the real axis to check if the conditions are satisfied. The closing section contains an application of our general results to the study of quantum adiabatic transitions in the time dependent Schrödinger equation and of the semiclassical scattering properties of the multichannel stationary Schrödinger equation. In particular, explicit use of the symmetries of the S-matrix is made to increase the number of its elements for which an asymptotic formula holds. In the latter application, further specific symmetry properties of the S-matrix are derived (lemma 7.1).

Acknowledgments: It is a great pleasure to thank Charles-Edouard Pfister for many enlightening and fruitful discussions which took place in Marseille and Lausanne. The hospitality of the Institut de Physique Théorique de l'EPFL where part of this work was accomplished is acknowledged.

2 Definition and properties of the S-matrix.

We consider the evolution equation

iεψ ′ (t) = H(t)ψ(t), t ∈ R, ε → 0, (2.1)
where the prime denotes the derivative with respect to t, ψ(t) ∈ C n , H(t) ∈ M n (C), for all t. We make some assumptions on the generator H(t). The first one is the usual analyticity condition in this context.

H1There exists a strip

S α = {z ∈ C| |Im z| ≤ α} , α > 0, (2.2)
such that H(z) is analytic for all z ∈ S α .

Since we are studying scattering properties, we need sufficient decay at infinity. H2 There exist two nondegenerate matrices H(+), H(-) ∈ M n (C) and a > 0 such that

lim t→±∞ |t| 1+a sup |s|≤α H(t + is) -H(±) < ∞. (2.3)
We finally give a condition which has to do with the physics behind the problem. H3 For t ∈ R, the spectrum of H(t), denoted by σ(t), is real and non-degenerate

σ(t) = {e 1 (t) < e 2 (t) < • • • < e n (t)} ⊂ R (2.4)
and there exists g > 0 such that

inf j =k t∈R |e j (t) -e k (t)| ≥ g. (2.5)
As a consequence of H3, for each t ∈ R, there exists a complete set of projectors

P j (t) = P 2 j (t) ∈ M n (C), j = 1, 2, • • • , n such that n j=1 P j (t) ≡ I (2.6) H(t) = n j=1 e j (t)P j (t) (2.7)
and there exists a basis of C n of eigenvectors of H(t). We determine these eigenvectors ϕ j (t), j = 1, 2, • • • , n uniquely (up to a constant) by requiring them to satisfy

H(t)ϕ j (t) = e j (t)ϕ j (t) (2.8) P j (t)ϕ ′ j (t) ≡ 0, j = 1, 2, • • • , n.
(2.9)

Explicitly, if ψ j (t), j = 1, 2, • • • , n form a complete set of differentiable eigenvectors of H(t), the eigenvectors

ϕ j (t) = e -t 0 ξ j (t ′ )dt ′ ψ j (t), s.t. ϕ j (0) = ψ j (0) (2.10) with ξ j (t) = ψ j (t)|P j (t)ψ ′ j (t) ψ j (t) 2 , j = 1, • • • , n (2.11)
verify (2.9). That this choice leads to an analytic set of eigenvectors close to the real axis will be proven below. We expand the solution ψ(t) along the basis just constructed, thus defining unknown coefficients c j (t), j = 1, 2, • • • , n to be determined,

ψ(t) = n j=1 c j (t)e -i t 0 e j (t ′ )dt ′ /ε ϕ j (t).
(2.12)

The phases e -i t 0 e j (t ′ )dt ′ /ε (see H3) are introduced for convenience. By inserting (2.12) in (2.1) we get the following differential equation for the c j (t)'s

c ′ j (t) = n k=1 a jk (t)e i∆ jk (t)/ε c k (t) (2.13)
where

∆ jk (t) = t 0 (e j (t ′ ) -e k (t ′ ))dt ′ (2.14)
and

a jk (t) = - ϕ j (t)|P j (t)ϕ ′ k (t) ϕ j (t) 2 .
(2.15)

Here •|• denotes the usual scalar product in C n . Our choice (2.9) implies a jj (t) ≡ 0. It is also shown below that the a jk (t)'s are analytic functions in a neighborhood of the real axis and that hypothesis H2 imply that they satisfy the estimate

lim t→±∞ sup j =k |t| 1+a |a jk (t)| < ∞. (2.16)
As a consequence of this last property and of the fact that the eigenvalues are real by assumption, the following limits exist lim t±∞ c j (t) = c j (±∞).

(2.17)

We are now able to define the associated S-matrix, S ∈ M n (C), by the identity

S      c 1 (-∞) c 2 (-∞) . . . c n (-∞)      =      c 1 (+∞) c 2 (+∞) . . . c n (+∞)     
.

(2.18) Such a relation makes sense because of the linearity of the equation (2.13). It is a well known result that under our general hypotheses, the S-matrix satisfies

S = I + O(ε). (2.19)
Note that the j th column of the S-matrix is given by the solution of (2.13) at t = ∞ subjected to the initial conditions

c k (-∞) = δ jk , k = 1, 2, • • • , n.
In general, the S-matrix defined above has no particular properties besides that of being invertible. However, when the generator H(t) satisfies some symmetry properties, the same is true for S. As such properties are important in applications, we show below that if H(t) is self-adjoint with respect to some indefinite scalar product, then S is unitary with respect to another indefinite scalar product. Let J ∈ M n (C) be an invertible self-adjoint matrix. We define an indefinite metric on C n by means of the indefinite scalar product

(•, •) J = •|J• .
(2.20)

It is easy to check that the adjoint A # of a matrix A with respect to the (•, •) J scalar product is given by

A # = J -1 A * J.
(2.21)

Proposition 2.1 Let H(t) satisfy H1, H2 and possess n distinct eigenvalues ∀t ∈ R. Further assume H(t) is self-adjoint with respect to the scalar product (•, •) J ,

H(t) = H # (t) = J -1 H * (t)J, ∀t ∈ R, (2.22)
and the eigenvectors ϕ j (0) of H(0) satisfy

(ϕ j (0), ϕ j (0)) J = ρ j , ρ j ∈ {-1, 1}, ∀j = 1, • • • , n.
(2.23)

Then the eigenvalues of H(t) are real ∀t ∈ R and the S-matrix is unitary with respect to the scalar product (•, •) R , where

R = R * = R -1 is the real diagonal matrix R = diag (ρ 1 , ρ 2 , • • • , ρ n ), S # = RS * R = S -1 .
(2.24)

Remark:

The condition (ϕ j (0), ϕ j (0)) J = ±1 can always be satisfied by suitable renormalization provided (ϕ j (0), ϕ j (0)) J = 0.

The main interest of this proposition is that when the S-matrix possesses symmetries, some of its elements can be deduced from resulting identities, without resorting to their actual computations.

A simple proof of the proposition making use of notions discussed in the next section can be found in appendix. The above proposition can actually be used for the two main applications we deal with in section 7. Note that in specific cases, further symmetry property can be derived for the S-matrix, see section 7.

Analyticity properties

The generator H(z) is analytic in S α , hence the solution of the linear equation (2.1) ψ(z) is analytic in S α as well. However, the eigenvalues and eigenprojectors of H(z) may have singularities in S α . Let us recall some basic properties, the proofs of which can be found in [K]. The eigenvalues and eigenprojectors of a matrix analytic in a region of the complex plane have analytic continuations in that region with possible singularities located at points z 0 called exceptional points. In a neighborhood free of exceptional points, the eigenvalues are given by branches of analytic functions and their multiplicities are constant. One eigenvalue can therefore be analytically continued until it coincides at z 0 with one or several other eigenvalues. The set of such points defines the set of exceptional points. The eigenvalues may possess branching points at an exceptional z 0 , where they are continuous, whereas the eigenprojectors are also multivalued but diverge as z → z 0 . Hence, by hypothesis H3, the n distinct eigenvalues e j (t) defined on the real axis are analytic on the real axis and possess multivalued analytic continuations in S α , with possible branching points at the set of degeneracies Ω, given by Ω = {z 0 | e j (z 0 ) = e k (z 0 ), for some k, j and some analytic continuation} .

(3.1)

By assumption H2, Ω is finite, by H3, Ω ∩ R = ∅ and Ω = Ω, due to Schwarz's principle.

Similarly, the eigenprojectors P j (t) defined on the real axis are analytic on the real axis and possess multivalued analytic continuations in S α , with possible singularities at Ω. To see more precisely what happens to these multivalued functions when we turn around a point z 0 ∈ Ω, we consider the construction described in figure 1. Let f be a multivalued analytic function in S α \Ω. We denote by f (z) the analytic continuation of the restriction of f around 0 along some path β ∈ S α \Ω from 0 to z. Then we perform the analytic continuation of f (z) along a negatively oriented loop δ based at z around a unique point z 0 ∈ Ω, and denote by f (z) the function we get when we come back at the starting point (if δ is positively oriented, the construction is similar). For later purposes, we define η 0 as the negatively oriented loop homotopic to the loop based at the origin encircling z 0 obtained by following β from 0 to z, δ from z back to z and β in the reverse sense from z back to the origin. We'll keep this notation in the rest of this section. It follows from the foregoing that if we perform the analytic continuation of the set of eigenvalues {e j (z)} n j=1 , along a negatively oriented loop around z 0 ∈ Ω, we get the set { e j (z)} n j=1 with

e j (z) = e σ 0 (j) (z), j = 1, • • • , n, (3.2)
where

σ 0 : {1, 2, • • • , n} → {1, 2, • • • , n} (3.3)
is a permutation which depends on η 0 . Similarly, and with the same notations, we get for the analytic continuations of the projectors around z 0

P j (z) = P σ 0 (j) (z), j = 1, • • • , n. (3.4)
Let us consider now the eigenvectors ϕ j (t). We define W (t) as the solution of

W ′ (t) = n j=1 P ′ j (t)P j (t)W (t) (3.5) ≡ K(t)W (t), W (0) = I,
where t ∈ R. It is well known [K], [Kr], that W (t) satisfies the intertwining identity

W (t)P j (0) = P j (t)W (t), j = 1, 2, • • • , n , ∀t ∈ R, (3.6)
so that, if {ϕ j (0)} n j=1 denotes a set of eigenvectors of H(0), the vectors defined by

ϕ j (t) = W (t)ϕ j (0) (3.7)
are eigenvectors of H(t). Moreover, using the identity Q(t)Q ′ (t)Q(t) ≡ 0 which is true for any differentiable projector, it is easily checked that condition (2.9) is satisfied by these vectors. The generator K(t) is analytic on the real axis and can be analytically continued in S α \Ω.

Actually, K(z) is single valued in S α \Ω. Indeed, let us consider the analytic continuation of K(z) around z 0 ∈ Ω. We get from (3.4) that

P ′ j (z) = P ′ σ 0 (j) (z), (3.8) so that K(z) = n j=1 P ′ j (z) P j (z) = n j=1 P ′ σ 0 (j) (z)P σ 0 (j) (z) = n k=1 P ′ k (z)P k (z) = K(z). (3.9)
Consequently, W (t) can be analytically continued in S α \Ω, where it is multivalued and satisfies both (3.5) and (3.6) with z ∈ S α \Ω in place of t ∈ R. Moreover, the relation between the analytic continuation W (z) from 0 to some point z ∈ S α \Ω and the analytic continuation W (z) is given by a monodromy matrix W (η 0 ) such that

W (z) = W (z)W (η 0 ), (3.10)
where η 0 is the negatively oriented loop based at the origin which encircles z 0 ∈ Ω only, (see figure 1). Note also that the analytic continuation W (z) is invertible in S α \Ω and W -1 (z) satisfies

W -1 ′ (z) = -W -1 (z)K(z), W -1 (0) = I. (3.11)
As a consequence, the eigenvectors (3.7) possess multivalued analytic extensions in S α \Ω. Indeed, it is easily seen that the analytic continuation of ϕ j (z) along a negatively oriented loop around z 0 ∈ Ω, ϕ j (z), is proportional to ϕ σ 0 (j) (z). Hence we introduce the quantity θ j (η 0 ) ∈ C by the definition

ϕ j (z) = e -iθ j (η 0 ) ϕ σ 0 (j) (z), j = 1, 2, • • • , n . (3.12)
Note that this is equivalent to (see (3.10)) W (η 0 )ϕ j (0) = e -iθ j (η 0 ) ϕ σ 0 (j) (0). Let us consider the couplings (2.15). Using the definition (3.7), the invertibility of W (t) and the identity (3.6), it's not difficult to see that we can rewrite

a jk (t) = - ϕ j (0)|P j (0)W (t) -1 K(t)W (t)ϕ k (0) ϕ j (0) 2 , t ∈ R, (3.13)
which is analytic on the real axis and can be analytically continued in S α \Ω, where it is multivalued. Thus, the same is true for the coefficients c j (t) which satisfy the linear differential equation (2.13) and their analytic continuations satisfy the same equation with z ∈ S α \Ω in place of t ∈ R. We now come to the main identity of this section, regarding the coefficients c j (z). Let us denote by c j (z) the analytic continuation of c j (0) from 0 to some z ∈ S α \Ω. We perform the analytic continuation of c j (z) along a negatively oriented loop around z 0 ∈ Ω and denote by c j (z) the function we get when we come back at the starting point z.

Lemma 3.1 For any j = 1, • • • , n, we have

c j (z)e -i η 0 e j (u)du/ε e -iθ j (η 0 ) = c σ 0 (j) (z) (3.14)
where η 0 , θ j (η 0 ) and σ 0 (j) are defined as above.

Proof:

It follows from hypothesis H1 that ψ(z) is analytic in S α so that n j=1 c j (z)e -i z 0 e j (u)du/ε ϕ j (z) = n j=1 c j (z) e -i z 0 e j (u)du/ε ϕ j (z) = n j=1 c j (z)e -i η 0 e j (u)du/ε e -i z 0 e σ 0 (j) (u)du/ε e -iθ j (η 0 ) ϕ σ 0 (j) (z). (3.15)
We conclude by the fact that {ϕ j (z)} n j=1 is a basis. ✷ Remark: It is straightforward to generalize the study of the analytic continuations around one singular point of the functions given above to the case where the analytic continuations are performed around several singular points, since Ω is finite. The loop η 0 can be rewritten as a finite succession of individual loops encircling one point of Ω only, so that the permutation σ 0 is given by the composition of a finite number of individual permutations. Thus the factors e -iθ j (η 0 ) in (3.12) should be replaced by a product of such factors, each associated with one individual loop and the same is true for the factors exp(-i η 0 e j (z)dz/ε) in lemma 3.1. This process is performed in the proof of theorem 6.1.

Complex WKB analysis

This section is devoted to basic estimates on the coefficients c j (z) in certain domains extending to infinity in both the positive and negative directions inside the strip S α . We first consider what happens in neighborhoods of ±∞. It follows from assumption H2 by a direct application of the Cauchy formula that (possibly by reducing α by an arbitrarily small amount)

lim t→±∞ sup |s|≤α |t| 1+a H ′ (t + is) < ∞. (4.1)
Hence the same is true for the single valued matrix K(z) All quantities encountered so far are analytic in S α ∩ {z||Re z| > T }, and we denote by a " " any analytic continuation in that set. As noticed earlier

lim t→±∞ sup |s|≤α |t| 1+a K(t + is) < ∞. (4.2) Let 0 < T ∈ R
W ′ (z) = K(z) W (z), z ∈ S α ∩ {z||Re z| > T } (4.4)
so that it follows from (4.2) that the limits

lim t→±∞ W (t + is) = W (±∞) (4.5) exist uniformly in s ∈] -α, α[. Consequently, see (3.13), lim t→±∞ |t| 1+a sup |s|≤α | a jk (t + is)| < ∞, ∀j, k ∈ {1, • • • , n}. (4.6)
Finally, for |t| > T , we can write

Im ∆ jk (t + is) = Im η e j (z)dz - η e k (z)dz + s 0 Re (e σ(j) (t + is ′ ) -e σ(k) (t + is ′ ))ds ′ , (4.7)
where this equation is obtained by deforming the path of integration from 0 to z = t + is into a loop η based at the origin, which may encircle points of Ω, followed by the real axis from 0 to Re z and a vertical path from Re z to z, see figure 2, and σ is the corresponding permutation. Hence we have sup

z∈Sα∩{z||Re z|>T } Im ∆ jk (z) < ∞, (4.8)
which, together with (4.6) yields the existence of the limits

lim t→±∞ c j (t + is) = c j (±∞) (4.9)
uniformly in s ∈]α, α[. We now define the domains in which useful estimates can be obtained.

Definition: Let j ∈ {1, • • • , n} be fixed. A dissipative domain for the index j, D j ⊂ S α \Ω, is such that sup z∈D j Re z = ∞, inf z∈D j Re z = -∞, (4.10)
and is defined by the property that for any z ∈ D j and any k ∈ {1, • • • , n}, there exists a path with sup

γ k ⊂ D j parameterized by u ∈] -∞, t] which links -∞ to z lim u→-∞ Re γ k (u) = -∞, γ k (t) = z (4.11)
z∈D j sup u∈]-∞,t] d du γ k (u) < ∞ (4.12)
and satisfies the monotonicity condition

Im ∆ jk (γ k (u)) is a non decreasing function of u ∈] -∞, t]. (4.13)
Such a path is a dissipative path for {jk}. Here ∆ jk (z) is the analytic continuation of

∆ jk (t) = t 0 (e j (t ′ ) -e k (t ′ ))dt ′ , t ∈ R, (4.14)
in D j along a path β described in figure 3 going from 0 to -T ∈ R along the real axis and then vertically up or down until it reaches D j , where T > 0 is chosen as in (4.3).

Let c k (z), k = 1, 2, • • • , n, z ∈ D j
, be the analytic continuations of c k (t) along the same path β which are solutions of the analytic continuation of (2.13) in

D j along β c ′ k (z) = n l=1 a kl (z)e i ∆ kl (z)/ε c l (z). (4.15)
We take as initial conditions in

D j lim Re z→-∞ c k (z) = lim t→-∞ c k (t) = δ jk , k = 1, • • • , n. (4.16)
and we define

x k (z) = c k (z)e i ∆ jk (z)/ε , z ∈ D j , k = 1, • • • , n. (4.17)
Lemma 4.1 In a dissipative domain for the index j we get the estimates

sup z∈D j |x j (z) -1| = O(ε) (4.18) sup z∈D j |x k (z)| = O(ε), ∀k = j. (4.19)

Remark:

The real axis is a dissipative domain for all indices. In this case we have c j (t) ≡ c j (t). Hence we get from the application of the lemma for all indices successively that S = I + O(ε).

The estimates we are looking for are then just a direct corollary.

Proposition 4.1 Assume there exists a dissipative domain D j for the index j. Let η j be a loop based at the origin which encircles all the degeneracies between the real axis and D j and let σ j be the permutation of labels associated with η j , in the spirit of the remark ending the previous section. The loop η j is negatively, respectively positively, oriented if D j is above, respectively below, the real axis. Then the solution of (2.13) subjected to the initial conditions

c k (-∞) = δ jk satisfies c σ j (j) (+∞) = e -iθ j (η j ) e -i η j e j (z)dz/ε (1 + O(ε)) (4.20) c σ j (k) (+∞) = O εe Im η j e j (z)dz/ε+h j (e σ j (j) (+∞)-e σ j (k) (+∞))/ε , (4.21) with h j ∈ [H - j , H + j ]
, where H ± j is the maximum, respectively minimum, imaginary part of the points at +∞ in D j

H + = lim sup t→+∞ sup s|t+is∈D j s, H -= lim inf t→+∞ inf s|t+is∈D j s. (4.22)
Thus we see that it is possible to get the (exponentially small) asymptotic behavior of the element s σ j (j),j of the S-matrix, provided there exists a dissipative domain for the index j.

The difficult part of the problem is of course to prove the existence of such domains D j , which do not necessarily exist, and to have enough of them to compute the asymptotic of the whole S-matrix. This task is the equivalent for n-level systems to the study of the global behavior of the Stokes lines for 2-level systems. We postpone this aspect of the problem to the next section. Note that we also get from this result an exponential bound on the elements s σ j (k),j of the S-matrix, k = j, which may or may not be useful. If η j encircles no point of Ω, we cannot get the asymptotic behavior of s σ j (j),j but we only get the exponential bounds. Since our main concern is asymptotic behaviors, we call the corresponding dissipative domain trivial.

Remark:

In contrast with the 2-level case, see [START_REF] Joye | Semi-Classical Asymptotics beyond All Orders for Simple Scattering Systems[END_REF], we have to work with dissipative domains instead of working with one dissipative path for all indices. Indeed, it is not difficult to convince oneself with specific 3-level cases that such a dissipative path may not exist, even when the eigenvalue degeneracies are close to the real axis. In return, we prove below the existence of dissipative domains in this situation.

Proof:

The asymptotic relation is a direct consequence of lemma 3.1, (4.9), (4.17) and the first part of the lemma. The estimate is a consequence of the same equations, the second estimate of the lemma and the identity, for t > T ,

Im ∆ jk (t + is) = Im η j e j (z)dz - η j e k (z)dz + s 0 Re (e σ j (j) (t + is ′ ) -e σ j (k) (t + is ′ ))ds ′ . (4.23)
The path of integration from 0 to z for ∆ jk (z) is deformed into the loop η j followed by the real axis from 0 to Re z and a vertical path from Re z to z. It remains to take the limit t → +∞. ✷ Proof of lemma 4.1: We rewrite equations (4.15) and (4.16) as an integral equation and perform an integration by parts on the exponentials

c k (z) = δ jk -iε n l=1 a kl (z) e k (z) -e l (z) e i ∆ kl (z)/ε c l (z) + iε n l=1 z -∞ a kl (z ′ ) e k (z ′ ) -e l (z ′ ) ′ e i ∆ kl (z ′ )/ε c l (z ′ )dz ′ + iε n l,m=1 z -∞ a kl (z ′ ) a lm (z ′ ) e k (z ′ ) -e l (z ′ ) e i ∆ km (z ′ )/ε c m (z ′ )dz ′ . (4.24)
Since all eigenvalues are distinct in S α \Ω, the denominators are always different from 0. In terms of the functions x k we get

x k (z) = δ jk -iε n l=1 a kl (z) e k (z) -e l (z)
x l (z)

+ iε n l=1 z -∞ a kl (z ′ ) e k (z ′ ) -e l (z ′ ) ′ e i( ∆ jk (z)-∆ jk (z ′ ))/ε x l (z ′ )dz ′ + iε n l,m=1 z -∞ a kl (z ′ ) a lm (z ′ ) e k (z ′ ) -e l (z ′ ) e i( ∆ jk (z)-∆ jk (z ′ ))/ε x m (z ′ )dz ′ . (4.25)
We introduce the quantity

|||x||| j = sup z∈D j l=1,•••,n |x l (z)| (4.26)
and consider for each k the equation (4.25) along the dissipative path γ k (u) described in the definition of D j , such that

e i( ∆ jk (γ k (t))-∆ jk (γ k (u)))/ε ≤ 1 (4.27)
when u ≤ t along that path. Due to the integrability of the a kl (z) at infinity and the uniform boundedness of dγ k (u)/du, we get the estimate |x k (z)δ kj | ≤ ε|||x||| j A for some constant A uniform in z ∈ D j , hence |||x||| j ≤ 1 + ε|||x||| j A. Consequently, for ε small enough |||x||| j ≤ 2 and the result follows. ✷

Superasymptotic improvement

All results above can be substantially improved by using the so-called superasymptotic renormalization method [Be], [N], [START_REF] Joye | Superadiabatic Evolution and Adiabatic Transition Probability between Two Non-degenerate Levels Isolated in the Spectrum[END_REF]. The joint use of complex WKB analysis and superasymptotic renormalization is very powerful, as demonstrated recently in [START_REF] Joye | Semi-Classical Asymptotics beyond All Orders for Simple Scattering Systems[END_REF] for 2-level systems, and allows, roughly speaking, to replace all remainders O(ε) by O(e -κ/ε ), where κ > 0. We briefly show how to achieve this improvement in the case of n-level systems.

Let H(z) satisfy H1, H2 and H3 in S α and let

S α = S α \ ∪ r=1,•••,p (J r ∪ J r ), (5.1)
where each J r is an open domain containing one point of Ω only in the open upper half plane. Hence, any analytic continuation e j (z) of e j (t), t ∈ R, in S α is isolated in the spectrum of H(z) so that e j (z) is analytic and multivalued in S α , and the same is true for the corresponding analytic continuation P j (z) of P j (t), t ∈ R. Let σ r be the permutation associated with the loop ζ r based at the origin which encircles J r once, such that e j (z) = e σr(j) (z), (5.2)

with the convention of section 3. The matrix K(z) is analytic and single valued in S α .

Consider the single valued analytic matrix

H 1 (z, ε) = H(z) -iεK(z), z ∈ S α .
(5.3)

For ε small enough, the spectrum of H 1 (z, ε) is non degenerate ∀z ∈ S α so that its eigenvalues e 1 j (z, ε) and eigenprojectors P 1 j (z, ε) are multivalued analytic functions in S α . Moreover, for ε small enough, the analytic continuations of e 1 j (z, ε) and P 1 j (z, ε) around J r satisfy e 1 j (z) = e 1 σr(j) (z) and P 1 j (z) = P 1 σr (j) (z), as can be easily deduced from (5.2) by perturbation theory. Consequently the matrix

K 1 (z, ε) = m j=1 P 1 j ′ (z, ε)P 1 j (z, ε) (5.4)
is analytic and single valued in S α . Defining the single valued matrix

H 2 (z, ε) = H(z) -iεK 1 (z, ε), z ∈ S α , (5.5)
we can repeat the argument, for ε small enough. By induction we set for any q ∈ N,

H q (z, ε) = H(z) -iεK q-1 (z, ε) (5.6) K q-1 (z, ε) = m j=1 P q-1 j ′ (z, ε)P q-1 j (z, ε), z ∈ S α (5.7)
for ε is small enough. We have

H q (z, ε) = m j=1
e q j (z, ε)P q j (z, ε), (5.8)

where the eigenvalues and eigenprojections are multivalued in S α and satisfy e q j (z, ε) = e q σr(j) (z, ε) (5.9)

P q j (z, ε) = P q σr(j) (z, ε), j = 1, • • • , n, (5.10)
with the notations of (5.2). We quote from [START_REF] Joye | Semi-Classical Asymptotics beyond All Orders for Simple Scattering Systems[END_REF], [START_REF] Joye | Superadiabatic Evolution and Adiabatic Transition Probability between Two Non-degenerate Levels Isolated in the Spectrum[END_REF] the main proposition regarding this construction.

Proposition 5.1 Let H(z) satisfy H1, H2 and H3 in S α and let S α be defined as above. Then there exist constants c > 0, ε * > 0 and a real function b(t) with lim t→±∞ |t| 1+a b(t) < ∞, such that

K q (z, ε) -K q-1 (z, ε) ≤ b(Re z)ε q c q q! (5.11) K q (z, ε) ≤ b(Re z),
(5.12) for all z ∈ S α , all ε < ε * and all q ≤ q * (ε) ≡ [1/ecε], where [y] denotes the integer part of y and e is the basis of the neperian logarithm.

We can deduce from this that in S α e q j (z, ε) = e j (z) + O(ε 2 b(Re z)) (5.13) P q j (z, ε) = e j (z) + O(ε(Re z)), ∀q ≤ q * (ε).

(5.14)

We introduce the notation f q * (ε) ≡ f * for any quantity f q depending on the index q and we drop from now on the ε in the arguments of the functions we encounter. We define the multivalued analytic matrix W * (z) for z ∈ S α by

W * ′ (z) = K * (z)W * (z), W * (0) = I.
(5.15) Due to the above observations and proposition 5.1, W * (z) enjoys all properties W (z) does, such as

W * (z)P * j (0) = P * j (z)W * (z) (5.16) W * (z) = W * (z)W * (ζ r ) (5.17)
and, uniformly in s,

lim t±∞ W * (t + is) = W * (∞).
(5.18)

Thus we define for any z ∈ S α a set of eigenvectors of

H * (z) by ϕ * j (z) = W * (z)ϕ * j (0), where H * (0)ϕ * j (0) = e * j (0)ϕ * j (0), j = 1, • • • , n, and which satisfy ϕ * j (0) = exp{-iθ * j (ζ r )}ϕ * σr (j) (0), with θ * j (ζ r ) = θ(ζ r ) + O(ε) ∈ C.
Let us expand the solution of (2.1) on this multivalued set of eigenvectors as

ψ(z) = n j=1 c * j (z)e -i z 0 e * j (z ′ )dz ′ /ε ϕ * j (z). (5.19)
Since the analyticity properties of the eigenvectors and eigenvalues of H * (z) are the same as those enjoyed by the eigenvectors and eigenvalues of H(z), we get as in lemma 3.1

c * j (z)e -i ζr e * j (u)du/ε e -iθ * j (ζr ) = c * σr (j) (z), ∀z ∈ S α .
(5.20) Substituting (5.19) in (2.1), we see that the multivalued coefficients c * j (z) satisfy in S α the differential equation

c * j ′ (z) = n k=1 a * jk (z)e i∆ * jk (z)/ε c * k (z) (5.21)
where

∆ * jk (z) = z 0 e * j (z ′ ) -e * k (z ′ )dz ′ (5.22) and a * jk (z) = ϕ * j (z)(0)|P * j (z)(0)W * (z) -1 (K q * -1 (z) -K q * (z))W * (z)ϕ * k (0) ϕ * j (0) 2 , (5.23)
to be compared with (3.13). The key point of this construction is that it follows from proposition 5.1 with q = q * (ε) that

|a * jk (z)| ≤ 2b(Re z)e -κ/ε , ∀z ∈ S α (5.24)
where κ = 1/ec > 0, and from (5.13) that

Im ∆ * jk (z) = Im ∆ jk (z) + O(ε 2 ), (5.25)
uniformly in z ∈ S α . Thus, we deduce from (5.24) that the limits

lim t→±∞ c * j (t + is) = c * j (±∞), j = 1, • • • , n, (5.26)
exist for any analytic continuation in S α . Moreover, along any dissipative path γ k (u) for {jk}, as defined above, we get from (5.25)

e i( ∆ * jk (γ k (t))-∆ * jk (γ k (u)))/ε = O(1), ∀u ≤ t (5.27)
so that, reproducing the proof of lemma 4.1 we have

Lemma 5.1 In a dissipative domain D j , if c * k (-∞) = c * k (-∞) = δ kj , then c * j (z) = 1 + O(e -κ/ε ) (5.28) e i ∆ jk (z)ε c * k (z) = O e -κ/ε , ∀k = j, (5.29) uniformly in z ∈ S α .
This lemma yields the improved version of our main result.

Proposition 5.2 Under the conditions of proposition 4.1 and with the same notations. If

c * k (-∞) = δ jk , then c * σ j (j) (+∞) = e -iθ * j (η j ) e -i η j e * j (z)dz/ε 1 + O e -κ/ε (5.30) c * σ j (k) (+∞) = O e -κ/ε e
Im η j e j (z)dz/ε+h j (e σ j (j) (+∞)-e σ j (k) (+∞))/ε .

(5.31)

Note that we may or may not replace e j (z) by e * j (z) in the estimate without altering the result. It remains to make the link between the S-matrix and the c * k (+∞)'s of the proposition explicit. We define β * ± j by the relations (H * (z) and H(z) coincide at ±∞), ϕ * j (±∞) = e -iβ * ± j ϕ j (±∞).

(5.32) By comparison of (5.19) and (2.12) we deduce the lemma

Lemma 5.2 If c k (t) and c * k (t) satisfy c k (-∞) = c * k (-∞) = δ jk ,
then, the element kj of the S-matrix is given by

s kj = c k (+∞) = e -i(β * + k -β * - j ) e -i +∞ 0 e * k (t ′ )-e k (t ′ )dt ′ /ε e -i 0 -∞ e * j (t ′ )-e j (t ′ )dt ′ /ε c * k (+∞) ≡ e -iα * kj c * k (+∞), (5.33) with β * ± j = O(ε) and 0 ±∞ e * j (t ′ ) -e j (t ′ )dt ′ /ε = O(ε), i.e. e -iα * kj = 1 + O(ε).
Remarks: i) Proposition 5.2 together with lemma 5.2 are the main results of the first part of this paper. ii) As a direct consequence of these estimates on the real axis we have

s jk = O e -κ/ε , ∀k = j, (5.34)
and

s jj = e -iα * jj 1 + O e -κ/ε . (5.35)
iii) It should be clear from the analysis just performed that all results obtained hold if the generator H(z) in (2.1) is replaced by

H(z, ε) = H 0 (z) + O(εb(Re z)),
(5.36) with b(t) = O(1/t 1+a ), provided H 0 (z) satisfies the hypotheses we assumed.

Avoided crossings

We now come to the second part of the paper in which we prove asymptotic formulas for the off-diagonal elements of the S-matrix, by means of the general set up presented above.

To start with, we define a class of n-level systems for which we can prove the existence of one non trivial dissipative domain for all indices. They are obtained by means of systems exhibiting degeneracies of eigenvalues on the real axis, hereafter called real crossings, which we perturb in such a way that these degeneracies are lifted and turn to avoided crossings on the real axis. When the perturbation is small enough, this process moves the eigenvalue degeneracies off the real axis but they remain close to the place where the real crossings occurred. This method was used successfully in [J] When δ = 0, the functions e j (t, 0) are analytic on the real axis and there exists a finite set of crossing points

{t 1 ≤ t 2 ≤ • • • ≤ t p } ∈ R, p ≥ 0, such that i) ∀t < t 1 , e 1 (t, 0) < e 2 (t, 0) < • • • < e n (t, 0). (6.3) ii) ∀j < k ∈ {1, 2, • • • , n}
, there exists at most one t r with e j (t r , 0)e k (t r , 0) = 0, (6.4)

and if such a t r exists we have ∂ ∂t (e j (t r , 0)e k (t r , 0)) > 0. (6.5)

Figure 4: A pattern of eigenvalue crossings (bold curves) with the corresponding pattern of avoided crossings (fine curves) satisfying H5.

iii) ∀j ∈ {1, 2, • • • , n}, the eigenvalue e j (t, 0) crosses eigenvalues whose indices are all superior to j or all inferior to j.

Remarks:

i) The parameter δ can be understood as a coupling constant controlling the strength of the perturbation.

ii)The eigenvalues e j (t, 0) are assumed to be analytic on the real axis, because of the degeneracies on the real axis. However, if H(t, δ) is self adjoint for any δ ∈ [0, d], this is true for an indexation, as follows from a theorem of Rellich, see [K].

iii) We give in figure 4 an example of pattern of crossings with the corresponding pattern of avoided crossings for which the above conditions are fulfilled. iv) The crossings are assumed to be generic in the sense that the derivative of e je k are non zero at the crossing t r . v) The crossing points {t 1 , t 2 , • • • , t p } need not be distinct, which is important when the eigenvalues possess symmetries. However, for each j = 1, • • • , n, the eigenvalue e j (t, δ) experiences avoided crossings with e j+1 (t, δ) and/or e j-1 (t, δ) at a subset of distinct points

{t r 1 , • • • , t r j } ⊆ {t 1 , t 2 , • • • , t p }.
We now state the main lemma of this section regarding the analyticity properties of the perturbed levels and the existence of dissipative domains for all indices in this perturbative context. Lemma 6.1 Let H(t, δ) satisfy H4 and H5. We can choose α > 0 small enough so that the following assertions are true for sufficiently small δ > 0: i) Let {t r 1 , • • • , t r j } be the set of avoided crossing points experienced by e j (t, δ), j = 1, • • • , n. For each j, there exists a set of distinct domains J r ∈ S α , where r ∈ {r 1 , • • • , r j },

J r = {z = t + is| 0 ≤ |t -t r | < L, 0 < g < s < α ′ }, (6.6)
with L small enough, α ′ < α and g > 0 such that e j (-∞, δ) can be analytically continued in

S j α = S α \ ∪ r=r 1 ,•••,r j (J r ∪ J r ). (6.7)
ii) Let t r be an avoided crossing point of e j (t, δ) with e k (t, δ), k = j ± 1. Then the analytic continuation of the restriction of e j (t, δ) around t r along a loop based at t r ∈ R which encircles J r once yields e j (t r , δ) back at t r with e j (t r , δ) = e k (t r , δ). (6.8)

iii) For each j = 1, • • • , n, there exists a dissipative domain D j above or below the real axis in S α ∩ {z = t + is| |s| ≥ α ′ }. The permutation σ j associated with these dissipative domains (see proposition 4.1) are all given by σ j = σ where σ is the permutation which maps the index of the k th eigenvalue e j (∞, 0) numbered from the lowest one on k, for all k ∈ {1, 2, • • • , n}.

Remarks:

i) In part ii) the same result is true along a loop encircling J r .

ii) The dissipative domains D j of part iii) are located above (respectively below) all the sets J r (resp. J r ), r = 1, • • • , p.

iii) The main interest of this lemma is that the sufficient conditions required for the existence of dissipative domains in the complex plane can be deduced from the behavior of the eigenvalues on the real axis. iv) We emphasize that more general types of avoided crossings than those described in H5 may lead to the existence of dissipative domains for certain indices but we want to get dissipative domains for all indices. For example, if part iii) of H5 is satisfied for certain indices only, then part iii) of lemma 6.1 is satisfied for those indices only. v) Note also that there are patterns of eigenvalue crossings for which there exist no dissipative domain for some indices. For example, if e j (t, 0) and e k (t, 0) display two crossings, it is not difficult to see from the proof of the lemma that no dissipative domains can exist for j or k.

We postpone the proof of this lemma to the end of the section and go on with its consequences. By applying the results of the previous section we get Theorem 6.1 Let H(t, δ) satisfy H4 and H5. If δ > 0 is small enough, the elements σ(j)j of the S-matrix, with σ(j) defined in lemma 6.1, are given in the limit ε → 0 for all j = 1, • • • , n, by

s σ(j)j = σ(j)∓1 k=j e -iθ k (ζ k ) e -i ζ k e k (z,δ)dz/ε (1 + O(ε)) , σ(j) > j < j (6.9)
where, for σ(j) > j (respectively σ(j

) < j), ζ k , k = j, • • • , σ(j) -1 (resp. k = j, • • • , σ(j) + 1
), denotes a negatively (resp. positively) oriented loop based at the origin which encircles the set J r (resp. J r ) corresponding to the avoided crossing between e k (t, δ) and e k+1 (t, δ) (resp. e k-1 (z, δ)) at t r , ζ k e k (z, δ)dz denotes the integral along ζ k of the analytic continuation of e k (0, δ) and θ k (ζ k ) is the corresponding factor defined by (3.12), see figure 5. More accurately, with the notations of section 5, we have the improved formula

s σ(j)j = e -iα * σ(j)j σ(j)∓1 k=j e -iθ * k (ζ k ) e -i ζ k e * k (z,δ)dz/ε 1 + O e -κ/ε , σ(j) > j < j (6.10)
The element σ(l)j, l = j, are estimated by

s σ(l)j = O   εe h(e σ(j) (∞,δ)-e σ(l) (∞,δ))/ε σ(j)∓1 k=j e Im ζ k e k (z,δ)dz/ε   , σ(j) > j < j (6.11)
where h is strictly positive (resp. negative) for σ(j) > j (respectively σ(j) < j). Remarks: i) As the eigenvalues are continuous at the degeneracy points, we have that

lim δ→0 Im ζ k e k (z, δ)dz = 0, ∀k = 1, • • • , p. (6.12)
ii) The remainders O(ε) depend on δ but it should be possible to get estimates which are valid as both ε and δ tend to zero, in the spirit of [J], [MN] and [R].

iii) This result shows that one off-diagonal element per column of the S-matrix at least can be computed asymptotically. However, it is often possible to get more elements by making use of symmetries of the S-matrix. Moreover, if there exist dissipative domains going above or below other eigenvalue degeneracies further away in the complex plane, other elements of the S-matrix can be computed. iv) Finally, note that all starred quantities in (6.10) depend on ε.

Proof: The first thing to determine is whether the loops ζ k are above or below the real axis.

Since the formulas we deduce from the complex WKB analysis are asymptotic formulas, it suffices to choose the case which yields exponential decay of s σ(j)j . It is readily checked in the proof of lemma 6.1 below that if σ(j) > j, D j is above the real axis and if σ(j) < j, D j is below the real axis. Then it remains to explain how to pass from the loop η j given in proposition 4.1 to the set of loops ζ k , k = j, • • • , σ(j) -1. We briefly deal with the case σ(j) > j, the other case being similar. It follows from lemma 6.1 that we can deform η j into the set of loops ζ k , each associated with one avoided crossing, as described in figure 5. Thus we have

η j = σ(j)-1 k=j ζ k (6.13)
for the decay rate and, see 3.10,

W (η j ) = W (ζ σ(j)-1 ) • • • W (ζ j+1 )W (ζ j ) (6.14)
for the prefactors. Let ν j be a negatively oriented loop based at t r which encircles J r as described in lemma 6.1. Consider now the loop ζ j associated with this avoided crossing and deform it to the path obtained by going from 0 to t r along the real axis, from t r to t r along ν j and back from t r to the origin along the real axis. By the point ii) of the lemma we get e j (0, δ) = e j+1 (0, δ) (6.15) along ζ j , and, accordingly (see (3.12)), (6.16) This justifies the first factor in the formula. By repeating the argument at the next avoided crossings, keeping in mind that we get e j+1 (0, δ) at the end of ζ j and so on, we get the final result. The estimate on s σ(l)j is obtained by direct application of lemma 6.1. ✷ Proof of lemma 6.1: In the sequel we shall denote " ∂ ∂t " by a "′ ". We have to consider the analyticity properties of e j (z, δ) and define domains in which every point z can be reached from -∞ by means of a path γ(u), u ∈] -∞, t], γ(t) = z such that Im ∆ jk (γ(u), δ) is non decreasing in u for certain indices j = k when δ > 0 is fixed. Note that by Schwarz's principle

ϕ j (0, δ) = e -iθ j (ζ j ) ϕ j+1 (0, δ).
if γ(u) is dissipative for {jk}, then γ(u) is dissipative for {kj}. When γ(u) = γ 1 (u) + iγ 2 (u) is differentiable, saying that γ(u) is dissipative for {jk} is equivalent to Re ( e j (γ(u), δ) -e k (γ(u), δ)) γ2 (u) + Im ( e j (γ(u), δ) -e k (γ(u), δ)) γ1 (u) ≥ 0 ∀u ∈] -∞, t]
(6.17)

where " ˙" denotes the derivative with respect to u. Moreover, if the eigenvalues are analytic in a neighborhood of the real axis, we have the relation in that neighborhood (6.18) which is a consequence of the Cauchy-Riemann identity. We proceed as follows. We construct dissipative domains above and below the real axis when δ = 0 and we show that they remain dissipative for the perturbed quantities ∆ jk (z, δ) provided δ is small enough. We introduce some quantities to be used in the construction. Let C r ∈ {1, • • • , n} 2 denote the set of distinct couples of indices such that the corresponding eigenvalues experience one crossing at t = t r . Similarly, N ∈ {1, • • • , n} 2 denotes the set of couples of indices such that the corresponding eigenvalues never cross. We further introduce

Im ( e j (t + is, δ) -e k (t + is, δ)) = s 0 Re ( e ′ j (t + is ′ , δ) -e ′ k (t + is ′ , δ))ds ′ ,
Let I r = [t r -L, t r + L] ∈ R, r = 1, • • • , p, with L so small that min r∈{1,•••,p} min {jk}∈Cr, j<k inf t∈Ir (e ′ j (t, 0) -e ′ k (t, 0)) ≡ 4c > 0. ( 6
I α r = {z = t + is|t ∈ I r , |s| ≤ α}, r = 1, • • • , p. (6.22)
Then we choose α small enough so that the only points of degeneracy of eigenvalues in S α are on the real axis and That this choice is always possible is a consequence of the analyticity of e j (z, 0) close to the real axis and of the fact that we can work essentially in a compact, because of hypothesis H4. Let a(t) be integrable on R and such that

a(t) 2 > max j<k∈{1,•••,n} sup |s|≤α Re (e ′ j (t + is, 0) -e ′ k (t + is, 0)) . (6.26)
It follows from H4 that such functions exist. Let r ∈ {1, • • • , p} and γ 2 (u) be a solution of (6.28) and since a(u) is integrable, the limits

     γ2 (u) = -γ 2 (u)a(u) b u ∈] -∞, t r -L] γ2 (u) = 0 u ∈]t r -L, t r + L[ γ2 (u) = + γ 2 (u)a(u) b u ∈ [t r + L, ∞[ (6.27) with γ 2 (t r ) > 0. Then γ 2 (u) > 0 for any u, since      γ 2 (u) = γ 2 (t r )e - u tr -L a(u ′ )du ′ /b u ∈] -∞, t r -L] γ 2 (u) = γ 2 (t r ) u ∈]t r -L, t r + L[ γ 2 (u) = γ 2 (t r )e u tr +L a(u ′ )du ′ /b u ∈ [t r + L, ∞[
lim u→±∞ γ 2 (u) = γ 2 (±∞) (6.29)
exist. Moreover, we can always choose γ 2 (t r ) > 0 sufficiently small so that γ r (u) ≡ u + iγ 2 (u) ∈ S α , for any real u. Let us verify that this path is dissipative for all {jk} ∈ C r , j < k.

For u ∈] -∞, t r -L],
we have, using

Re (e j (z, 0)e k (z, 0)) < -2b < 0, ∀z ∈ S α ∩ {z|Re z ≤ t r -L} (6.30)

and

|Im (e j (t + is, 0) -e k (t + is, 0))| < |s| sup s ′ ∈[0,s]
Re (e ′ j (t + is ′ , 0)e ′ k (t + is ′ , 0)) , (6.31) (see (6.18)), and the definition (6.26)

Re (e j (γ r (u), 0)e k (γ r (u), 0)) γ2 (u) + Im (e j (γ r (u), 0)e k (γ r (u), 0)) γ1 (u) = -Re (e j (γ r (u), 0)e k (γ r (u), 0))

γ 2 (u)a(u) b + Im (e j (γ r (u), 0) -e k (γ r (u), 0)) > 2γ 2 (u)a(u) -γ 2 (u)a(u)/2 > γ 2 (u)a(u) > 0. (6.32)
Similarly, when u ≥ t r + L we get, using

Re (e j (z, 0)

-e k (z, 0)) > 2b > 0, ∀z ∈ S α ∩ {z|Re z ≥ t r + L}, (6.33)
Re (e j (γ r (u), 0)e k (γ r (u), 0)) γ2 (u) + Im (e 1 (γ r (u), 0)e k (γ r (u), 0)) γ1 (u) =

Re (e j (γ r (u), 0)e k (γ r (u), 0))

γ 2 (u)a(u) b + Im (e j (γ r (u), 0) -e k (γ r (u), 0)) > 2γ 2 (u)a(u) -γ 2 (u)a(u)/2 > γ 2 (u)a(u) > 0. (6.34) Finally, for s ∈ [t r -L, t r + L],
we have with (6.23)

Im (e j (γ r (u), 0) -e k (γ r (u), 0)) = γ 2 (u) 0 Re (e ′ j (t ′ + is, 0) -e ′ k (t ′ + is, 0)) ≥ γ 2 (u)2c > γ 2 (u)c > 0.
(6.35) Thus, γ r (u) is dissipative for all {jk} ∈ C r , j < k. Note that the last estimate shows that it is not possible to find a dissipative path for {jk} ∈ C r , j < k below the real axis. Consider now {jk} ∈ N, j < k and let γ + 2 (u) be a solution of (6.36) i.e.

γ+ 2 (u) = - γ + 2 (u)a(u) b , γ + 2 (0) > 0, u ∈] -∞, +∞[,
γ + 2 (u) = γ + 2 (0)e -u 0 a(u ′ )du ′ /b . (6.37)
As above, we have γ + 2 (u) > 0 for any u and we can choose γ + 2 (0) > 0 small enough so that

γ + (u) ≡ u + iγ + 2 (u) ∈ S α for any u ∈ R. Since Re (e j (z, 0) -e k (z, 0)) > -2b ∀z ∈ S α (6.38)
we check by a computation analogous to (6.32

) that γ + (u) is dissipative for {jk} ∈ N, j < k. Similarly, one verifies that if γ - 2 (u) is the solution of γ- 2 (u) = γ - 2 (u)a(u) b , γ - 2 (0) < 0, u ∈] -∞, +∞[ (6.39) with |γ - 2 (0)| small enough, the path γ -(u) ≡ u + iγ - 2 (u)
below the real axis is in S α for any u ∈ R and is dissipative for {jk} ∈ N, j < k as well.

Finally, the complex conjugates of these paths yield dissipative paths above and below the real axis for {jk} ∈ N, j > k.

We now define the dissipative domains by means of their borders. Let γ + (u) and γ -(u), u ∈ R, two dissipative paths in S α defined as above with |γ - 2 (0)| sufficiently small so that γ -is below γ + . We set 1) If k ∈ X j , then there exists a dissipative path γ r ∈ D for {jk} ∈ C r , j < k constructed as above which links -∞ to z. It is enough to select the initial condition γ 2 (t r ) suitably, see figure 6.

D = {z = t + is|0 < -γ - 2 (t) ≤ s ≤ γ + 2 (t), t ∈ R}. ( 6
2) Similarly, if j < k ∈ X j , there exists a dissipative path γ + ∈ D for {jk} constructed as above which links -∞ to z obtained by a suitable choice of γ + 2 (0). 3) Finally, if k > j, we can take as a dissipative path for {jk}, the path γ -∈ D constructed as above which links -∞ to z with suitable choice of γ - 2 (0). Hence D is dissipative for the index j, when δ = 0. If j is such that the set X j consists in points k with k > j, a similar argument with the complex conjugates of the above paths shows that the domain D below the real axis is dissipative for j when δ = 0.

Let us show that these domains remain dissipative when δ > 0 is not too large. We start by considering the analyticity properties of the perturbed eigenvalues e j (z, δ), δ > 0. Let 0 < α ′ < α be such that

I α ′ r ∩ (D ∪ D) = ∅ , ∀r = 1, • • • , p. (6.41)
The analytic eigenvalues e j (z, 0), j ∈ {1, • • • , n}, are isolated in the spectrum of H(z, 0) for any z ∈ S α , where

S α = S α \ ∪ r=1,•••,p I α ′ r .
(6.42)

Figure 6: The dissipative domain D and some dissipative paths.

For any j = 1, • • • , n we get from perturbation theory [K], that the analytic continuations e j (z, δ) of e j (t 1 -L, δ) in S α are all distinct in S α , provided δ is small enough. This is due to the fact that assumption H4 implies the continuity of H(z, δ) in δ uniformly in z ∈ S α , as is easily verified. More precisely, for any fixed index j, the eigenvalue e j (t, δ) experiences avoided crossings at the points {t r 1 , • • • , t r j }. We can assume without loss of generality that

I α ′ k ∩ I α ′ l = ∅ , ∀k = l ∈ {r 1 , • • • , r j }. (6.43)
Hence, for δ > 0 small enough, the analytic continuation e j (z, δ) is isolated in the spectrum of H(z, δ), uniformly in z ∈ S α \ ∪ r=r 1 ,•••,r j I α ′ r . Since by assumption H5 there is no crossing of eigenvalues on the real axis when δ > 0, there exists a 0 < g < α ′ , which depends on δ, such that e j (z, δ) is isolated in the spectrum of H(z, δ), uniformly in z ∈ S j α , where Hence the singularities of e j (z, δ) are located in ∪ r=r 1 ,•••,r j (J r ∪ J r ), which yields the first assertion of the lemma. Consider a path ν r from t r -L to t r + L which goes above J r , where t r is an avoided crossing between e j (t, δ) and e k (t, δ), k = j ± 1. By perturbation theory again, e j (t r -L, δ) and e k (t r -L, δ) tend to e j ′ (t r -L, 0) and e k ′ (t r -L, 0) as δ → 0, for some j ′ , k ′ ∈ 1, • • • , n, whereas e j (t r + L, δ) and e k (t r + L, δ) tend to e k ′ (t r + L, 0) and e j ′ (t r + L, 0) as δ → 0, see figure 4. Now, the analytic continuations of the restrictions of e j (t, δ) and e k (t, δ) around t r -L along ν r , e j (z, δ) and e k (z, δ) tend to the analytic functions e j ′ (z, 0) = e j ′ (z, 0) and e k ′ (z, 0) = e k ′ (z, 0) as δ → 0, for all z ∈ ν r . Thus, we deduce that for δ small enough e j (t r + L, δ) ≡ e k (t r + L, δ), (6.46) since we know that e j (t r + L, δ) = e σ(j) (t r + L, δ), for some permutation σ. Hence the point iii) of the lemma follows.

S j α = S α \ ∪ r=r 1 ,•••,r j (J r ∪ J r ) (6.
Note that the analytic continuations e j (z, δ) are single valued in S α . Indeed, the analytic continuation of e j (t r -L, δ) along ν r , denoted by e j (z, δ), ∀z ∈ ν r , is such that e j (t r + L, δ) = e j (t r + L, δ) = e j (t r + L, δ) = e k (t r + L, δ), (6.47) due to Schwarz's principle. We further require δ to be sufficiently small so that the following estimates are satisfied (6.52)

The simultaneous requirements (6.26) and (6.50) is made possible by the continuity properties of H ′ (z, δ) and the uniformity in δ of the decay at ±∞ of H(z, δ) assumed in H4 together with the fact that a(t) can be replaced by a multiple of a(t) if necessary, to satisfy both estimates. The condition on δ is given by the first inequalities in (6.51) and (6.52), whereas the second ones are just recalls.

Then it remains to check that the paths γ r , γ + and γ -defined above satisfy the dissipativity condition (6.17) for the corresponding indices. This is not difficult, since the above estimates are precisely designed to preserve inequalities such as (6.32), (6.34) and (6.35). However, it should not be forgotten that in the sets I α ′ r the eigenvalues may be singular so that (6.18) cannot be used there. So when checking that a path parameterized as above by u ∈ R is dissipative, it is necessary to consider separately the case u ∈ R\(∪ r=1,•••,p I r ), where we proceed as above with (6.48), (6.49), (6.50) and (6.18) and the case u ∈ ∪ r=1,•••,p I r , where we use use (6.51) and (6.52) instead of (6.18) as follows. If u ∈ I r for r such that t r is a crossing point for e j (t, 0) and e k (t, 0), one takes (6.51) to estimate Im ( e j ′ (z, δ)-e k ′ (z, δ)) for the corresponding indices j ′ and k ′ , and if t r is not a crossing point for e j (t, 0) and e k (t, 0), one uses (6.52) to estimate Im ( e j ′ (z, δ)e k ′ (z, δ)). Consequently, the domains D and D defined above keep the same dissipativity properties when δ > 0 is small enough.

Let us finally turn to the determination of the associated permutation σ. As noticed earlier, the eigenvalues e j (z, δ) are continuous in δ, uniformly in z ∈ S α . Hence, since the eigenvalues e j (z, 0) are analytic in S α , we have

Applications

Let us consider the time-dependent Schrödinger equation in the adiabatic limit. The relevant equation is then (2.1) where H(t) = H * (t) is the time-dependent self-adjoint hamiltonian.

Thus we can take J = I in proposition 2.1 to get

H(t) = H * (t) = H # (t). (7.1)
The norm of an eigenvector being positive, it remains to impose the gap hypothesis in H3 to fit in the framework and we deduce that the S-matrix is unitarity, since R = I. In this context, the elements of the S-matrix describe the transitions between the different levels between t = -∞ and t = +∞ in the adiabatic limit. We now specify a little more our concern and consider a three-level system, i.e. H(t) = H * (t) ∈ M 3 (C). We assume that H(t) satisfies the hypotheses of corollary 6.1 with an extra parameter δ which we omit in the notation and displays two avoided crossings at t 1 < t 2 , as shown in figure 7. The corresponding permutation σ is given by

σ(1) = 3, σ(2) = 1, σ(3) = 2.
(7.2) By corollary 6.1, we can compute asymptotically the elements s 31 , s 12 , s 23 and s jj , j = 1, 2, 3. Using the unitarity of the S-matrix, we can get some more information. Introducing

Γ j = Im ζ j e j (z)dz , j = 1, 2, (7.3)
where ζ j is in the upper half plane, with the notation of section 6, it follows that

s 31 = O e -(Γ 1 +Γ 2 )/ε , s 12 = O e -Γ 1 /ε , s 23 = O e -Γ 2 /ε (7.4)
and

s jj = 1 + O(ε), j = 1, 2, 3. (7.5)
Expressing the fact that the first and second columns as well as the second and third rows are orthogonal, we deduce

s 21 = -s 12 s 11 s 22 1 + O e -2Γ 2 /ε (7.6) s 32 = -s 23 s 33 s 22 1 + O e -2Γ 1 /ε . (7.7)
Finally, the estimate in corollary 6.1 yields

s 13 = O εe -|h|(e 2 (∞,δ)-e 1 (∞,δ))/ε e -Γ 2 /ε = O e -(Γ 2 +Γ 2 +K)/ε (7.8)
where K > 0, since we have that Γ j → 0 as δ → 0. Hence we get

S =     s 11 s 12 O e -(Γ 2 +Γ 2 +K)/ε -s 12 s 11 s 22 1 + O e -2Γ 2 /ε s 22 s 23 s 31 -s 23 s 33 s 22 1 + O e -2Γ 1 /ε s 33     (7.9)
where all s jk above can be computed asymptotically up to exponentially small relative error, using (6.10).

The smallest asymptotically computable element s 31 describes the transition from e 1 (-∞, δ) to e 3 (+∞, δ). The result we get for this element is in agreement with the rule of the thumb claiming that the transitions take place locally at the avoided crossings and can be considered as independent. Accordingly, we can only estimate the smallest element of all, s 13 , which describes the transition from e 3 (-∞, δ) to e 1 (+∞, δ), for which the avoided crossings are not encountered in "right order", as discussed in [HP]. It is possible however to get an asymptotic expression for this element in some cases. When the unperturbed levels e 2 (z, 0) and e 3 (z, 0) possess a degeneracy point in S α and when there exists a dissipative domain for the index 3 of the unperturbed eigenvalues going above this point, one can convince oneself that s 13 can be computed asymptotically for δ small enough, using the techniques presented above.

Our second application is the study of the semi-classical scattering properties of the multichannel stationnary Schrödinger equation with energy above the potential barriers. The relevant equation is then

-ε 2 d 2 dt 2 Φ(t) + V (t)Φ(t) = EΦ(t), (7.10)
where t ∈ R has the meaning of a space variable, Φ(t) ∈ C m is the wave function, ε → 0 denotes Planck's constant, V (t) = V * (t) ∈ M m (C) is the matrix of potentials and the spectral parameter E is kept fixed and large enough so that (7.16) then the spectrum of the generator H(t) given by (7.13) consists in 2m real distinct eigenvalues

U (t) ≡ E -V (t) > 0. (7.11) Introducing ψ(t) = Φ(t) iεΦ(t) ∈ C 2m , ( 7
(t) < k 2 2 (t) < • • • < k 2 m (t),
-k m (t) < -k m-1 (t) < • • • < -k 1 (t) < k 1 (t) < k 2 (t) < • • • < k m (t) (7.17)
associated with the 2m eigenvectors

χ ± j (t) = u j (t) ±k j (t)u j (t) ∈ C 2m , H(t)χ ± j (t) = ±k j (t)χ ± j (t). (7.18)
We check that 7.19) where u j (0) is computed in C m , so that proposition 2.1 applies. Before dealing with its consequences, we further explicit the structure of S. Adopting the notation suggested by (7.17) and (7.18) we write

(χ ± j (0), χ ± j (0)) J = ±2k j (0) u j (0) = 0, j = 1, . . . , m (
H(t) = n j=1 k j (t)P + j (t) - n j=1 k j (t)P - j (t) (7.20) ψ(t) = n j=1 c + j (t)ϕ + j (t)e -i t 0 k j (t ′ )dt ′ /ε + n j=1 c - j (t)ϕ - j (t)e i t 0 k j (t ′ )dt ′ /ε (7.21) and introduce c ± (t) =      c ± 1 (t) c ± 2 (t) . . . c ± m (t)      ∈ C m . (7.22)
Hence we have the block structure

S c + (-∞) c -(-∞) ≡ S ++ S +- S -+ S -- c + (-∞) c -(-∞) = c + (+∞) c -(+∞) (7.23) where S στ ∈ M m (C), σ, τ ∈ {+, -}.
Let us turn to the symmetry properties of S. We get from (7.19) and proposition 2.1 that related to the transmission and reflexion coefficients associated with a wave incoming from the left. It should be noted that in case of equation (7.10) another convention is often used to define an S-matrix, see [F1], for instance. This gives rise to a different S-matrix with similar interpretation. However it is not difficult to establish a one-to-one correspondence between the two definitions. If the matrix of potentials V (t) is real symmetric, we have further symmetry in the S-matrix.

S ++ S +- S -+ S -- -1 = I O O -I S ++ S +- S -+ S -
Lemma 7.1 Let S given by (7.23) be the S-matrix associated with (7.10) under condition (7.11). If we further assume V (t) = V (t), then, taking ϕ ± j (0) ∈ R 2m , j = 1, • • • , m, in (7.21), we get S ++ = S --, S +-= S -+ .

(7.28)

The corresponding results for the S-matrix defined in [F1] are derived in [MN]. The proof of this lemma can be found in appendix. We consider now (7.10) the case U (t) = U * (t) = U (t) ∈ M 2 (R), which describes a two-channel Schrödinger equation. We assume that the four-level generator H(t) displays three avoided crossings at t 1 < t 2 , two of which take place at the same point t 1 , because of the symmetry of the eigenvalues, as in figure 8. By lemma 7.1, it is enough to consider the blocks S ++ and S +-. The transitions corresponding to elements of these blocks which we can compute asymptotically are from level 1 + to level 2 + and from level 2 -to level 1 + . They correspond to elements s ++ 21 and s +- 12 respectively. With the notations Γ j = Im (7.38) where all elements s στ jk can be asymptotically computed up to exponentially small relative corrections using (6.10). We get no information on the first column of S +-but the estimate (5.34) where necessarily, (7.36) holds. However, if there exists one or several other dissipative domains for certain indices, it is then possible to get asymptotic formulas for the estimated terms.

A Proof of proposition 2.1

A direct consequence of the property

H(t) = H # (t) = J -1 H * (t)J (A.1)
is the relation σ(H(t)) = σ(H(t)). Thus, if σ(H(0)) ⊂ R, then σ(H(t)) ⊂ R, for all t ∈ R, since the analytic eigenvalues are assumed to be distinct and nondegenerate for all t ∈ R. Let e j (0) be the eigenvalue associated with ϕ j (0). Then, due to the property H(0) = H # (0) (ϕ j (0), H(0)ϕ k (0)) J = e k (0)(ϕ j (0), ϕ k (0)) J = e j (0)(ϕ j (0), ϕ k (0)) J , (A.2) for any j, k = 1, • • • , n. For j = k we get from the assumption (ϕ j (0), ϕ j (0)) J = 0 that e j (0) ∈ R and from the fact that the eigenvalues of H(0) are distinct (ϕ j (0), ϕ k (0)) J = 0, j = k. As the matrix of potentials U (0) is real symmetric, its eigenvectors u j (0) may be chosen real, so that we can assume that ϕ ± j (0) = u j (0) ±k j (0)u j (0) ∈ R 2m . (B.7)

Thus it follows from the foregoing that ϕ ± j (t) = W (t)ϕ ± j (0) ∈ R 2m ∀t ∈ R (B.8) and satisfies

Gϕ ± j (t) = GW (t)GGϕ ± j (0) = W (t)Gϕ ± j (0) = ϕ ∓ j (t). (B.9)

Finally, the main consequence of (

Figure 1 :

 1 Figure 1: The paths β, δ and η 0 in S α \Ω.

Figure 2 :

 2 Figure2: The path of integration for ∆ jk (z) (the x's denote points of Ω).

Figure 3 :

 3 Figure 3: The path β along which the analytic continuation of ∆ jk (t) in D j is taken.

Figure 5 :

 5 Figure 5: The loops η j and ζ k , k = j, • • • , σ(j) -1.

  .19) This relation defines the constant c and we also define b by min r∈{1,•••,p} min {jk}∈Cr, j<k inf t∈R\Ir |e j (t, 0)e k (t, 0)| ≥ 4b > 0 (6.20) min {jk}∈N, j<k inf t∈R |e j (t, 0)e k (t, 0)| ≥ 4b > 0. (6.21)

  j (z, 0)e k (z, 0))| > 2b > 0 (6.24) min {jk}∈N, j<k inf z∈Sα |Re (e j (z, 0)e k (z, 0))| > 2b > 0.(6.25)

  .40) Let z ∈ D, and j ∈ {1, • • • , n} be fixed. By assumption H5, the set X j of indices k such that {jk} ∈ C r for some r ∈ {1, • • • , p} consists in values k satisfying j < k or it consists in values k satisfying j > k. Let us assume the first alternative takes place. Now for any k ∈ {1, • • • , n}, there are three cases.

  44)andJ r = I α ′ r ∩ {z| Im z > g} , r = 1, • • • , p. (6.45)

  j (z, δ)e k (z, δ))| > b > 0 ( e j (z, δ)e k (z, δ))| > b > 0 j (z, δ)e k (z, δ)j (z, 0)e k (z, 0))| > |Im z|c (6.51) max r∈{1,•••,p} max j<k∈{1,•••,n} sup z∈ I α r |Im ( e j (z, δ)e k (z, δ))| < 2 max r∈{1,•••,p} max j<k∈{1,•••,n} sup z∈ I α r|Im ( e j (z, 0)e k (z, 0))| < |Im z|a(Re z).

Figure 7 :

 7 Figure 7: The pattern of avoided crossings in the adiabatic context.

S

  Figure 8: The pattern of avoided crossings in the semiclassical context.

  is in the upper half plane, we have the estimatess ++ 21 = O e -Γ 1 /ε , s +- 12 = O e -(Γ 1 +Γ 2

  Let G = G * = G -1 be given in block structure by t) be given by (7.13) withU (t) = U (t) = U * (t). Since GH(t)G = -H(t) , H(t) = H(t) , (B.2)and the eigenvalues of H(t) are real, it is readily verified thatGP ± j (t)G = P ∓ j (t) , P ± j (t) = P ± j (t) , j = 1, • • • , m. that the solution W (t) of W ′ (t) = K(t)W (t) , W(0) = I (B.5) satisfies W (t) = W (t) = GW (t)G. (B.6)

  Fig. 1. The paths β, δ and η 0 in S α \Ω.

Fig. 2 .

 2 Fig.2. The path of integration for ∆ jk (z) (the x's denote points of Ω).

Fig. 3 .

 3 Fig. 3. The path β along which the analytic continuation of ∆ jk (t) in D j is taken.

Fig. 4 .

 4 Fig. 4. A pattern of eigenvalue crossings (bold curves) with the corresponding pattern of avoided crossings (fine curves) satisfying H5.

Fig. 5 .

 5 Fig. 5. The loops η j and ζ k , k = j, • • • , σ(j) -1.

Fig. 6 .

 6 Fig. 6. The dissipative domain D and some dissipative paths.

Fig. 7 .

 7 Fig. 7. The pattern of avoided crossings in the adiabatic context.

Fig. 8 .

 8 Fig. 8. The pattern of avoided crossings in the semiclassical context.

  Concerning the spectrum of H(t), it should be remarked that if the real and positive eigenvalues of U (t), k 2 j (t), j = 1, • • • , m associated with the eigenvectors u j (t) ∈ C m are assumed to be distinct, i.e.

	0 < k 2 1
					.12)
	we cast equation (7.10) into the equivalent form (2.1) for ψ(t) with generator
	H(t) =	O U (t) O I	∈ M 2m (C).	(7.13)
	It is readily verified that
	H(t) = J -1 H * (t)J	(7.14)
	with
	J =	O I I O	.	(7.15)

  )/ε , s ++ jj = 1 + O(ε), j = 1, 2.(7.30)It follows from (7.26) and lemma 7.1 that the matrix S ++ S T +-is symmetric. Hence

	whereas we get from (7.25)
	s ++ 11 s ++ 21 + s ++ 12 s ++ 22 = s +-11 s +-21 + s +-12 s +-22 .	(7.32)
	The only useful estimate we get with corollary 6.1 is
	s +-22 = O e -(Γ 1 +Γ 2 +K)/ε , K > 0,	(7.33)
	which yields together with (7.30) in (7.31)
	s +-21 = s ++ 21 s +-11 /s ++ 11 + O e -(Γ 1 +Γ 2 )/ε .	(7.34)
	Thus, from (7.32) and (5.34) for s +-11 ,
	s ++ 12 = -s ++ 21	s ++ 11 22 s ++	1 + O e -κ/ε ,	(7.35)
	with
	0 < κ < min(Γ 1 , Γ 2 ).	(7.36)
	Summarizing, we have
	s ++ 11 s +-21 + s ++ 12 s +-22 = s ++ 21 s +-11 + s ++ 22 s +-12 ,	(7.31)

The resulting reality of e j (t) for all t ∈ R and j = 1, • • • , n yields together with (A.1) P j (t) = J -1 P * j (t)J.

(A.4)

Hence, using the fact that the P * j are projectors,

Let Φ, Ψ ∈ C n and W (t) be defined by (see (3.5))

Then we have

Thus, in the indefinite metric, the scalar products of the eigenvectors of

We can then normalize the ϕ j (0) in such a way that

with ρ j ∈ {+1, -1}. Let ψ(t) and χ(t) be two solutions of (2.1). By a argument similar to the one above using (A.1), we deduce (χ(t), ψ(t)) J ≡ (χ(0), ψ(0)) J .

(A.10)

Inserting the decompositions .12) in this last identity yields n j,k=1 which is equivalent to the assertion S -1 = RS * R. ✷


Claudio Gutiérrez-Soto
email: claudio-orlando.gutierrez-soto@irit.fr

Gilles Hubert
email: gilles.hubert@irit.fr

Evaluating the Interest of Revamping Past Search Results

In this paper we present two contributions: a method to construct simulated document collections suitable for information retrieval evaluation as well as an approach of information retrieval using past queries and based on result combination. Exponential and Zipf distribution as well as Bradford's law are applied to construct simulated document collections suitable for information retrieval evaluation. Experiments comparing a traditional retrieval approach with our approach based on past queries using past queries show encouraging improvements using our approach.

Introduction

Information retrieval (IR) is finding information (usually text) that satisfies an information need from within large collections [START_REF] Manning | Introduction to Information Retrieval[END_REF]. An information retrieval system (IRS) represents and stores large amounts of information in order to facilitate and accelerate determination of information estimated relevant to a user's query. A common IRS relies on two main processes: indexing and matching. Indexing intends to construct comparable representations of documents and queries. Matching intends to estimate the extend to which a given document is relevant to a query, usually represented via a score. Documents are then returned to the user in the form of list of results ranked by decreasing score.

IR systems emerged in the late 1940s but improvements really appeared from the late 1950s. Improvements first concerned indexing, ranking functions, weighting schemes, relevance feedback and then models [START_REF] Sanderson | The history of information retrieval research[END_REF]. IR improvements are highly related to evaluation of IRS dating back from the late 1950s [START_REF] Cleverdon | The evaluation of systems used in information retrieval[END_REF]. The IR community notably benefited from TREC evaluation campaigns and workshops [START_REF] Voorhees | TREC: Experiment and Evaluation in Information Retrieval[END_REF]. These have been offering researchers means to measure system effectiveness and compare approaches.

The literature of IR is crammed with different contributions such as indexing approaches, matching functions, formal models and relevance feedback approaches. However, few approaches gain advantage from searches performed in the past by previous users. Past searches constitute though a useful source of information for new users for example. For example, a user searching about a new subject could benefit from past searches led by previous users about the same subject.

The weak interest of IR in past queries may be understandable because of the lack of suitable IR collections. Indeed, most of the existing IR collections are composed of independent queries. These collections are not usable to evaluate approaches based on past queries since they do not gather similar queries for which ground thruth relevance judgments are provided. In addition, elaborating such collections is difficult due to the cost and time needed. An alternative is to simulate such collections. Simulation in IR dates back to at least the 1980s [START_REF] Huurnink | Validating query simulators: An experiment using commercial searches and purchases[END_REF].

In this paper, on the one hand we propose an approach of creating simulated IR collections dedicated to evaluating IR approaches based on past searches. We simulated several scenarios under different probability distributions to build the collection of documents and determine the relevant documents given a query. On the other hand, we introduce a first approach of information retrieval using past queries based on result combination. We experimented this approach on different simulated IR collections using the aforementioned approach. We evaluated the effectiveness of our approach and compared it with a traditional retrieval approach.

This paper is organized as follows. In Sect. 2, related work on simulation in IR evaluation and the use of Past Searches in IR is presented. In Sect. 3 we present our approach to create a simulated IR collection described and the approach of retrieval based on past queries is presented in Sect. 4. Experiments are detailed in Sect. 5. Finally, conclusion and future work are presented in Sect. 6.

Related Work

Simulation in IR Evaluation

The use of simulation in IR is not recent, several approaches have been presented since 1980s. Simulation in IR can be seen as a method where a large collection of queries together with their judgments can be obtained without user interaction [START_REF] Huurnink | Validating query simulators: An experiment using commercial searches and purchases[END_REF]. Simulation have been used in different contexts in IR. In [START_REF] Tague | Simulation of user judgments in bibliographic retrieval systems[END_REF], an algorithm was developed with the purpose to simulate relevance judgments of users. Here, simulated precision is compared with real precision. In [START_REF] Azzopardi | Building simulated queries for known-item topics: an analysis using six european languages[END_REF], queries to find known-item are simulated. The authors proposed a model to build simulated topics that are comparable to real topics. Works dedicated to simulate the interaction among queries, click log, and preferences of users have been built [START_REF] Dang | Query reformulation using anchor text[END_REF][START_REF] Huurnink | Validating query simulators: An experiment using commercial searches and purchases[END_REF]. In addition, today with the exponential growth of the Web, simulation provides interesting approximations of performance on the Web [START_REF] Baeza-Yates | Crawling a country: better strategies than breadth-first for web page ordering[END_REF][START_REF] Marin | Sync/async parallel search for the efficient design and construction of web search engines[END_REF].

Past Searches in IR

Many approaches intending to improve results of queries using past searches can be found in the literature. A framework dedicated to improve effectiveness measures such as Average Precision (AP), where a set of documents is assigned to the best system cluster (i.e. best answer given a query) can be found in [START_REF] Bigot | Fusing different information retrieval systems according to query-topics: a study based on correlation in information retrieval systems and trec topics[END_REF]. Several approaches in IR use past queries for query expansion. For instance, similarity measures are defined in [START_REF] Raghavan | On the reuse of past optimal queries[END_REF] to retrieve past optimal queries that are used to reformulate new queries or to propose the results of past optimal queries. [START_REF] Klink | Improving document transformation techniques with collaborative learned term-based concepts[END_REF] proposed to learn from old queries and their retrieved documents to expand a new submitted query. In [START_REF] Shen | Exploiting query history for document ranking in interactive information retrieval[END_REF], historical queries, in particular terms defining queries, are combined to improve average precision for new queries. In [START_REF] Shen | Context-sensitive information retrieval using implicit feedback[END_REF], feedback information, including clickthrough data and previous queries, are used to improve retrieval effectiveness. Another approach aiming to improve retrieval effectiveness is presented in [START_REF] Cetintas | Using past queries for resource selection in distributed information retrieval[END_REF]. A new selection technique using past queries is proposed to estimate utility of available information sources for a given user query. Here, relevance judgments of past queries were not used.

Simulating an Information Retrieval Collection

A usual IR collection is composed of three parts: a set of documents,a set of queries and a set of relevance judgments per query (i.e. indications on documents considered relevant or not relevant) [START_REF] Voorhees | TREC: Experiment and Evaluation in Information Retrieval[END_REF]. Consequently, our approach aims at defining by simulation. Our method is split in two steps. The first step concerns the creation of terms, documents and queries. The second step involves the simulation of relevance judgments using Bradford's law.

Creation of Documents and Queries

In this first step, we use an alphabet in to build a set of terms. Each term is composed of letters of this alphabet. This set of terms can be split in subsets called topics in order to represent different subjects. Each letter is chosen using uniform distribution with the purpose to build a term. Thus, each term is unique.

In addition, each document is defined according to all the topics. In order to build a document, topics are selected using either the exponential or Zipf distribution and then terms constituting the document are chosen using uniform distribution. Thus, a document is constructed with terms from one topic mainly but not exclusively.

Past queries are built from documents. To built a past query, a document is chosen under uniform distribution. The terms that constitute the query are chosen from the document under uniform distribution. It is important to emphasize that the intersection among past queries is empty, that is, they have no terms in common. New queries are then built from past queries. For each past query a new query is built, either by changing or adding another term. Thus, the most similar query for the new query is its corresponding past query.

Simulating Relevant Judgments

In order to simulate the decision given by a user about if a document is relevant or not relevant for a given query, we relied on the zeta distribution. Zeta distribution gives a discrete approximation of Bradford's law [START_REF] Garfield | Bradford's Law and Related Statistical Patterns[END_REF]. Bradford's law says that among the production of journal papers, there is an heterogeneous number of papers where most relevant papers are in few journals, while a few number of relevant papers are spread on a high quantity of journals. In our case, for a given query, it means that the most relevant documents should be at the top of the list (most relevant papers are in few journal), while a few relevant documents should be spread at down of the list document given a query.

In addition, we assume that for two very similar queries q and q ′ , when a document is relevant for a query, it could be also relevant for the other query. In an intuitive way, there is a subset of common relevant documents for both queries. This does not implies that all relevant documents for query q, are relevant documents for query q ′ . With the objective to simulate this scenario, we use zeta distribution as follows. We retrieve documents for queries q ′ and q. Zeta distribution is applied to the set of common documents to the two queries to determine a subset of common relevant documents to q ′ and q. Eventually, zeta distribution is applied again to retrieved documents of each query q ′ and q (other relevant documents may be added), preserving the relevant common documents to q and q ′ . Therefore, the set of relevant documents for q ′ differs from the set of relevant documents for q.

Retrieval Using Past Queries

The basic idea behind of our approach is to incorporate to the system every query with its set of associated documents (query plus its list of documents, which are part of the answer of this query). Thus, the system has not only the set of documents but also the queries executed by user (past queries) with the set of associated documents them. At the beginning just there are documents without the queries, but every time a query is processed by the system, it is aggregated with its documents to the system. When a new query is submitted, first, it is checked and compared with the past queries in the system. When no similar query is found a traditional retrieval can be performed. When similar past queries are found, relevant documents of past queries can be used to build the result for the new query as well as combined with documents retrieved using traditional retrieval. Different strategies combining results from past queries and a traditional result. As first tested approach, we built the result for a new query by adding first the relevant documents from the most similar past query and then documents from a traditional retrieval.

Experiments

Experimental Environment

For this series of experiments, we used the English alphabet in order to build a set of terms. In a general way, the length of a term |t| was between 3 and 7. This length was selected under uniform distribution. The number of terms |T | was 700 for each experiment. We generated documents comprising between 300 to 900 words from a vocabulary composed of between 15 to 30 words for each document [START_REF] Heaps | Information Retrieval: Computational and Theoretical Aspects[END_REF][START_REF] Navarro | Adding compression to block addressing inverted indexes[END_REF][START_REF] Silva De Moura | Fast and flexible word searching on compressed text[END_REF]. We defined 7 topics, each topic comprising 100 terms. When a document is built, terms of the other topics are chosen using either exponential distribution or Zipf distribution. Therefore, the most words to compose a document are chosen specific topic. For each experiment we generated five sets of documents comprising 700, 1400, 2100, 2800, and 3500 documents.

On the other hand, we defined 30 queries comprising between 3 to 8 terms for each experiment. The terms of a query were chosen from a particular document. Both terms and documents were chosen using uniform distribution to build the 15 past queries. It is important to mention that intersection among pairs of queries is empty.

In order to simulate judgments of users on documents retrieved given a query q, we implemented the zeta distribution with the purpose to represent the Bradford's law. We applied zeta distribution on the top 30 retrieved documents for each query. Each experiment gathers three different scenarios of zeta distribution for determining relevant documents by varying the parameter s with the values 2, 3, and 4.

Experimental Results

In this section, three experiments are detailed. We used exponential distribution to build the collection of documents D in the two first experiments and we used Zipf distribution [START_REF] Zipf | Human Behavior and the Principle of Least Effort[END_REF] in the third experiment. We computed P@10 (precision at ten retrieved documents) on the results returned by our approach using past results and the traditional cosine retrieval, for each of the thirty queries. Then, we applied the Student's paired sample t-test to test if the difference between the two compared approaches with regards to P@10 was statistically significant.

Experiment 1. Exponential distribution with parameter equal to 1.5 was used to build the dataset D. When using zeta distribution with parameter s = 2 for relevance, our approach improved P@10 for 26 over 30 queries on average over the five sets of documents generated (i.e. comprising 700, 1400, 2100, 2800, and 3500 documents). The average P@10 improvement was +35.00 % over all queries. Statistical significance was reached in all cases, the highest p-value for the Student's t-test being 0.00128 for the set of 700 documents. When using zeta distribution with parameter s = 3, our approach improved P@10 for 24.4 over 30 queries on average over the five sets of documents. The average P@10 improvement was +33.99 %. Statistical significance was reached in all cases, the highest p-value for the Student's t-test being 0.00056 for 3500 documents. When using zeta distribution with parameter s = 4, our approach improved P@10 for 21.4 over 30 queries on average over the five sets of documents. The average P@10 improvement was +38.48 %. Statistical significance was reached in all cases, the highest p-value for the Student's t-test being 0.00732 for 1400 documents.

Experiment 2. Exponential distribution with parameter equal to 1.0 was used to build the dataset D. When using zeta distribution with parameter s = 2 for relevance, our approach improved P@10 for 26.2 over 30 queries on average over the five sets of documents. The average P@10 improvement was +31.12 %. Statistical significance was reached in all cases, the highest p-value for the Student's t-test being 0.00002 for 1400 documents. When using zeta distribution with s = 3, our approach improved P@10 for 21.6 over 30 queries on average. The average P@10 improvement was +26.10 %. Statistical significance was reached in all cases, the highest p-value for the Student's t-test being 0.01306 for 3500 documents. When using zeta distribution with parameter s = 4, our approach improved P@10 for 21.6 over 30 queries on average. The average P@10 improvement was +33.76 %. Statistical significance was reached in all cases, the highest p-value for the Student's t-test being 0.00122 for 2800 documents.

Experiment 3. Zipf distribution with parameter equal to 1.6 was used to build the dataset D. When using zeta distribution with parameter s = 2 for relevance, our approach improved P@10 for 24.8 over 30 queries on average over the five sets of documents. The average P@10 improvement was +25.50 %. Statistical significance was reached in all cases, the highest p-value for the Student's t-test being 0.00005 for 700 documents. When using zeta distribution with parameter s = 3, our approach improved P@10 for 22.6 over 30 queries on average. The average P@10 improvement was +22.77 %. Statistical significance was reached in all cases, the highest p-value for the Student's t-test being 0.00034 for 1400 documents. When using zeta distribution with parameter s = 4, our approach improved P@10 for 24.8 over 30 queries on average. The average P@10 improvement was +27.92 %. Statistical significance was reached in all cases, the highest p-value for the Student's t-test being 0.00031 for 2100 documents.

Discussion

Due to space limitations, some experiment details were not presented, however additional observations are reported in this section. Using different parameters in zeta distribution (s = 2, 3 and 4) for relevance judgments allowed us to analyze how function influences on the average P@10. According to the evaluations we observed that average P@10 decreases for both compared approaches when the parameter s is increases in zeta distribution. In addition, we observed that there is not a radical tendency in the differences of average P@10 between the approach using past queries and the traditional retrieval approach when the number of documents increases.

Summarizing the results reported in this paper, each experiment and each scenario showed that the approach based on past queries always overcomes the traditional approach (> +22.50 %). Statistical significance was always reached since the highest p-value for the Student's t-test was 0.01306.

In addition one should notice that we applied the Zipf distribution with value 1.6 only to simulate distribution frequencies of terms from topics for document creation. It would be interesting to test other Zipf distributions to simulate the distribution of the frequencies of terms [START_REF] Zipf | Human Behavior and the Principle of Least Effort[END_REF]. It is for this reason that we used not only the Zipf distribution but also exponential distribution to build collections of documents.

Eventually, our experiments used simulated datasets which allow us to have promising preliminary results. However, we should test other datasets using other distributions such as power law for term selection or using queries generated from different document collections.

Conclusion and Future Work

In this paper, we have presented on the one hand an approach of information retrieval using past queries. This approach is based on the reuse, for a new submitted query, of relevant documents retrieved for the most similar past query. On the other hand, due to the lack of available existing IR collections suitable for evaluating this kind of approach, we proposed an approach to creating simulated IR collections. We simulated several scenarios under different probability distributions to build the collection of documents and determine the relevant documents given a query. We experimented our approach of retrieval based on past queries on different simulated IR collections using the aforementioned approach. We evaluated the effectiveness of our approach and compared it with a traditional retrieval approach. Experiments showed encouraging results when evaluating the results for the top ten retrieved documents (P@10).

Future work will be devoted first to define more real evaluation datasets suitable for IR approaches based on past queries, adapting TREC collections for instance. We will also develop other approaches to construct the retrieved documents for a new query from various results of past queries, based on clustering methods [START_REF] Xu | Survey of clustering algorithms[END_REF] and based on diversification methods [START_REF] Drosou | Search result diversification[END_REF].


Pavel Exner

Alain Joye

H Kovařík

Edge currents in the absence of edges

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