Yves Colin

Yves Colin De Verdière

A Semi-classical calculus of correlations

Keywords: Imagerie passive, semi-classique, ondes de surfaces. English passive imaging, semi-classics, surface waves

Français.

La méthode d'imagerie passive en sismologie a été développée récemment en vue d'imager la croûte terrestre à partir d'enregistrements du bruit sismique. Elle repose sur le calcul des fonctions de corrélation de ce bruit. Nous donnons dans cet article des formules explicites pour cette corrélation dans le régime "semi-classique". Pour cela, nous définissons le spectre de puissance d'un champ aléatoire comme l'espérance de sa mesure de Wigner, ce qui permet d'utiliser un calcul dans l'espace des phases : le calcul pseudo-différentiel et la théorie des "rays". Nous obtenons ainsi une formule pour la corrélation du bruit sismique dans le régime "semiclassique" avec une source de bruit qui peut être localisée et non homogène.

Nous montrons ensuite comment l'utilisation des ondes guidées de surface permet d'imager la croûte terrestre.

Mots clés :

Introduction

Correlations of the noisy wave fields is used as a new tool in seismic imaging and monitoring, starting from the pioneering work of Campillo and Paul [START_REF] Campillo | Long range correlations of seismic codas[END_REF] (similar tools have been used in helio-seismology [START_REF] Duvall | Time-distance in helioseismology[END_REF]) and followed by many works [START_REF] Derode | Recovering the Green's function from field-field correlations in an open scattering medium[END_REF][START_REF] Derode | How to estimate the Green's function of a heterogeneous medium between to passive sensors? Application to acoustic waves[END_REF][START_REF] Ph | Ambient noise cross correlation in free space: Theoretical approach[END_REF][START_REF] Sabra | Using Ocean Ambient Noise for Array Self-Localization and Self-Synchronization[END_REF][START_REF] Sanchez-Sesma | The elastodynamic 2D Green's function retrieval from cross-correlation: the canonical inclusion problem[END_REF][START_REF] Shapiro | Emergence of broadband Rayleigh waves from correlations of the ambient seismic noise[END_REF][START_REF] Shapiro | High Resolution Surface Wave Tomography From Ambient Seismic Noise[END_REF][START_REF] Weaver | Diffuse fields in open systems and the emergence of the Green's functions (L)[END_REF][START_REF] Weaver | Information from seismic noise[END_REF]. See also the review paper [START_REF] Gouédard | Cross-correlation of random fields: mathematical approach and applications[END_REF]. It has also been used in the monitoring of the deformations of volcanoes [START_REF] Brenguier | Towards Forecasting Volcanic Eruptions using Seismic Noise[END_REF]. Because it is a very powerful method and, hopefully, in order to make it more efficient, it is quite challenging to give mathematical supports to this method, now called "passive imaging". This has been done in a rather great generality in [START_REF] De Verdière | Mathematical models for passive imaging I: general background[END_REF][START_REF] De Verdière | Semi-classical analysis and passive imaging[END_REF] using semi-classical analysis (see also [START_REF] Lobkis | On the emergence of Green's function in the correlations of a diffuse field[END_REF][START_REF] Ph | Ambient noise cross correlation in free space: Theoretical approach[END_REF][START_REF] Bardos | Identification of Green's Functions Singularities[END_REF][START_REF] Gouédard | Cross-correlation of random fields: mathematical approach and applications[END_REF]).

Exact formulas for the correlations of the fields are known if the source noise is homogeneous (a white noise). This assumption is not satisfied in applications. It is therefore desirable to get formulate valid for more general source noises, in particular if the source noise is localized in some part of the domain. This turns out to be possible in the so-called semi-classical regime where the wavelengths are negligible with respect to the size of the propagation domain. The field correlation admits a general expression in terms of the Green's function and the source correlation (Equation ( 3)). The idea is to find the asymptotics of this expression in the semi-classical regime.

I will present in this paper approximate formulas which are valid in the range of high frequency wave propagation and for which the source noise is localized in some part of the domain of propagation. The correlation is explicitly given in term of the decomposition of the Green's function as a sum over rays and the (phase-space) power spectrum of the source noise. I can use ray theory if I assume that the source noise has a short correlation distance of the same order of magnitude than the wavelengths. This calculus can be presented in a very geometric way using rays propagation as well as a re-interpretation of the source correlation in terms of the phase space power spectra. I use the calculus of pseudo-differential operators in a very essential way. I will not reproduce the mathematical arguments which are presented in my paper [START_REF] De Verdière | Semi-classical analysis and passive imaging[END_REF], but I will try not only to give explicit formulas, but also to present the main ideas and tools.

Here is a more precise description of the content: the goal is to get the formula given in Theorem 4.1 which gives the modification of the correlation of the seismic noise induced by the non-homogeneity of the source noise. The modification is given in terms of the power spectrum of the source noise, the attenuation and the ray dynamics associated to the deterministic wave equation.

I first give a review of the pseudo-differential calculus (section 1): this allows to put the basic terminology of rays dynamics and to define power spectra of arbitrary random fields (section 2).

I then introduce the simplest mathematical model where the source noise is simply the right-handside of the wave equation (section 3) and I present our main formula in section 4. The interest of the result depends of the relation between 2 time scales discussed in section 5: the Ehrenfest time given in terms of the Lyapounov exponent and the attenuation time.

How to use all of this in imaging problems? I do that (section 6) in the case of seismology using the effective wave equation for the guided surface waves. The final problem turns out to be an inverse spectral problem whose mathematical solution is known.

Finally, I discuss in section 7 a related issue, namely the calculus of the correlations of plane waves scattered by an obstacle or an inhomogeneity viewed as random waves: the direction of the waves is supposed to be random and uniform. This way, I show that the result of [START_REF] Sanchez-Sesma | The elastodynamic 2D Green's function retrieval from cross-correlation: the canonical inclusion problem[END_REF] is completely general.

A short review of the pseudo-differential calculus and Wigner measures

For the mathematics of pseudo-differential operators, see [START_REF] Dimassi | Spectral Asymptotics in The Semi-Classical Limit[END_REF][START_REF] Evans | Lectures on semi-classical analysis (Version 0.2[END_REF][START_REF] Folland | Harmonic analysis in phase space[END_REF][START_REF] Trèves | Introduction to pseudo differential and Fourier integral operators[END_REF]. The pseudo-differential operators (ΨDO's) were introduced in the sixties by Calderon, Zygmund, Nirenberg, Hörmander and others as a tool in the study of linear partial differential equations with non constant coefficients. They provide also the geometrical extension of Hamiltonian formalism of classical mechanics to wave mechanics (see [START_REF] Duistermaat | Fourier Integral Operators[END_REF]). In applications to physics, it is often called the ray theory (see [START_REF] Popov | Ray method and Gaussian beams for geophysicists[END_REF]). The same tools apply to the study of the semi-classical limit of quantum mechanics and to the high frequency limit of wave equations (acoustic, electromagnetic or seismic waves).

There is a small parameter ε > 0 in the theory which is the Planck "constant" in quantum mechanics and the wave length or more precisely the dimensionless ratio between the wave length and the size of the propagation domain for wave equations. Most results are only valid in the limit ε → 0, but, for simplicity, the reader can think of ε as a fixed, small enough, number.

ΨDO's

A pseudo-differential operator (ΨDO) on R d is a linear operator on functions f : R d → C, A ε := Op ε (a), defined using a suitable function defined on the phase space, a(x, ξ) :

R d x ⊕ R d ξ → C, called the symbol of A ε , by the formula (Weyl quantization) A ε (f )(x) = 1 (2π) d R d ×R d e i(x-y).ξ a x + y 2 , εξ f (y)dydξ .
The function a is assumed to be smooth and homogeneous near infinity in ξ. The Schwartz kernel

1 [A ε ](x, y) of A ε is located near the diagonal x = y and is of the form [A ε ](x, y) ∼ = ε→0 k(x, (x -y)/ε)
where k(x, z) is a smooth function outside z = 0 going to 0 as z → ∞.

Simple examples are

• Op ε (1) = Id by the Fourier inversion formula

• Op ε (ξ j ) = ε i ∂ ∂x j • Op ε (x j ) is the multiplication by x j
• If χ is a positive function with bounded support, the operator Op ε (χ(ξ)) is a frequency filter

• Op (|ξ| 2 + V (x)) = -2 ∆ + V (x): the Schrödinger operator • Op ε (n(x)|ξ| 2 ) = -ε 2 div (n(x) grad):
the acoustic wave operator.

The main properties are the following ones which hold as ε → 0:

• Composition: Op ε (a) • Op ε (b) ≈ Op ε (ab) • Brackets: [Op ε (a), Op ε (b)] ≈ ε i Op ε {a, b} where
{a, b} = d j=1 ∂a ∂ξ j ∂b ∂x j - ∂a ∂x j ∂b ∂ξ j
is the Poisson bracket. This last property is very important because it relates the algebra of ΨDO's to the geometry of the phase space given by the Poisson bracket.

Wigner functions

Wigner functions define the localization of energy in the phase space

R d x × R d ξ
for a wave function u = u(x). They involve the scale ε. The Wigner function W ε u (x, ξ) of u is the function on the phase space defined by the identities

∀a ∈ C ∞ 0 R 2d , R 2d a(x, ξ)W ε u (x, ξ)dxdξ = Op ε (a)u|u , where u|v = u(x)v(x)dx, or W ε u (x, ξ) = 1 (2π) d R d e -iv.ξ u x + εv 2 ū x - εv 2 dv . I have R d W ε u (x, ξ)dξ = |u(x)| 2 , R d W ε u (x, ξ)dx = |F ε u(ξ)| 2 ,
where F ε u(ξ) = is the ε-Fourier transform of u given by

F ε u(ξ) = 1 (2πε) d/2 e -ix.ξ/ε u(x)dx .
This means that the marginals of the Wigner measure

W ε u (x, ξ)dxdξ are |u(x)| 2 dx and |F ε u(ξ)| 2 dξ.

Hamiltonian dynamics and ray method

Let us consider the wave equation u tt -Lu = 0 where L is an elliptic ΨDO like the acoustic operator L = div(n grad). The symbol, usually called the dispersion relation, of this equation is ω 2n(x) ξ 2 = 0. To this relation is associated a dynamics called the ray dynamics given by the Hamilton equations:

dx j dt = ∂H ∂ξ j , dξ j dt = - ∂H ∂x j (1)
with H = √ n ξ . The main result (Theorem 4.1 below) uses the "Hamiltonian flow" Φ t : Φ t (x, ξ) is the value at time t of the previous differential system (1) with data (x, ξ) at the time t = 0. In the case of an homogeneous medium, n = n 0 =constant, I have

Φ t (x, ξ) = (x + t √ n 0 ξ/ ξ , ξ) .
In the ray theory, this correspond to the group velocity of waves √ n 0 .

The mathematical theory of rays is called the theory of Fourier Integral Operators and has been developed in the seventies by Hörmander and Duistermaat (see [START_REF] Duistermaat | Fourier Integral Operators[END_REF]) following some pioneering work of Lax and Maslov. A presentation more adapted to physicists is given in [START_REF] Popov | Ray method and Gaussian beams for geophysicists[END_REF]. Unfortunately, the geometric background is rather sophisticated and cannot be presented in a few pages. However, explicit formulas in terms of oscillatory functions and oscillatory integrals are available.

In what follows, I will use the fact that the Green's function G(t, x, y) of wave equation admits, in the semi-classical regime (short wave-length), a decomposition as a sum of contributions of rays γ going from y to x in time t: G = γ G γ .

2 Random fields: power spectra and correlations The power spectrum of the random field f is the function on the phase space given by the expectation of the Wigner functions

Let f = f (x), x ∈ R d ,
p ε := E(W ε f ) .
The power spectrum and the correlation contain the same information:

• The correlation C(x, y) is ((2πε) d times) the operator kernel of Op ε (p) or C(x, y) = e i x-y|ξ /ε p ε x + y 2 , ξ dξ .
• p ε is ((2πε) -d times) the symbol of the operator whose integral kernel is C.

Example 2.1 : the white noise

C = δ(x -y), p ε = 1/(2πε) d . Example 2.2 : a stationary noise on R with ε = 1, C(s, t) = F (s -t) and p 1 (s, ω) is the Fourier transform F (F )(ω).

A mathematical model

I will now discuss the basic mathematical model: it consists of 2 parts :

• A deterministic wave equation which could be the elastic wave equation or more simply here the acoustic wave equation. Because the source of noise will be permanent, some attenuation in the equation is needed.

• A source noise assumed to be stationary and ergodic in time. In seismology, this source is usually created by the interaction of the fluids surrounding the earth crust (atmosphere or ocean) with the crust itself. This source is modeled by a random field which I put on the right-handside of the equation.

For simplicity, I will discuss only the case of a scalar acoustic wave equation on some domain in R d

x with a random source field f = f (x, t) (t is the time):

u tt + a(x)u t -Lu = f (2)
where

• The field u = u(x, t) is scalar
• a, the attenuation, is a smooth > 0 function. I will assume for simplicity that a is time independent, but it is not really necessary

• L is a self-adjoint pseudo-differential operator of symbol -ε -2 l 2 0 (x, ξ). Usually, l 0 is homogeneous of degree 1 which makes L independent of ε. This will not be the case for dispersive waves like surface waves. Typical examples are the Laplace-Beltrami operator of a Riemannian metric on X with l 0 (x, ξ) = g ij (x)ξ i ξ j and the acoustic wave operator div (n(x) grad)) with

l 0 (x, ξ) = n(x) ξ . I introduce L 0 := Op ε (l 0 ) = ε √ -L • f = f (x, t
) is a stationary and ergodic (in time) random field with correlation E(f (s, x)f (s ′ , y)) = δ(ss ′ )Γ(x, y) and power spectrum p(x, ξ); I assume that p(x, ξ) has bounded support and that f is real valued and hence that p(x, ξ) is even w.r. to ξ. I assume that p is independent of ε, this implies that the correlation is ε-dependent: in particular, Γ(x, y) << |x -y|/ε. The source noise decorrelates rapidly as |x -y| >> ε.

The Green's function is the integral kernel G giving the causal solution of Equation ( 2) in terms of f : Hence I can (and will!) restrict ourselves to τ > 0.

u(x, t) = ∞ 0 ds X G(s, x, y)f (t -s, y)dy .

Our goal is to compute the correlation

Using the fact that the source noise is ergodic and stationary, I get the following result

Theorem 3.1 The field correlation is given by Equation (3) only in terms of the Green's function G and the correlation Γ of the source noise

C A,B (τ ) = ∞ 0 ds X×X dxdyG(s + τ, A, x)G(s, B, y)Γ(x, y). ( 3
)
All the work is now concentrated to get a more explicit and more geometric expression: this will be done using an expression of the Green's function as a sum over rays going from B to A in time τ and using the power spectrum p of f which is a semi-classical expression of the correlation of the source noise.

The main formula

Let us denote by Ω ± (t) the "one-parameter groups" of linear operators generated by ±iL 0εa/2: Ω + (t)u 0 is the solution of the differential equation u = ( i ε L 0a/2)u with u(0) = u 0 , and similarly for Ω -(t). The use of Ω ± (t) is a way to split the Green function of the wave equation usually given by some "sinus" function into 2 exponentials: this way, I reduce the wave equation from an equation with of second order in time to a diagonal system of first order in time.

Ω ± (t) = e t(± i ε L 0 -a/2) = e t(±i √ -L-a/2) .
I will express the result in terms of operators instead of expressing them in terms of their kernels (matrices). This gives a much more compact expression! The symbol • means the composition of operators while Ĉ(τ ) is the operator whose integral kernel (matrix) is C A,B (τ ):

( Ĉ(τ )u)(A) = X C A,B (τ )u(B)dB .
The main result is Theorem 4.1 The correlation is given, for τ > 0, as ε goes to 0, by

Ĉ(τ ) ∼ = [Ω + (τ ) + Ω -(τ )] • Π , (4)
with Π = Op ε (π) and

π(x, ξ) = ε 2 4l 2 0 0 -∞ e -0 t a(Φs(x,ξ))ds p(Φ t (x, ξ))dt , (5)
and if a = a 0 is constant

π(x, ξ) = ε 2 4l 2 0 0 -∞
e -a 0 |t| p(Φ t (x, ξ))dt .

I will compare our result (Equations ( 4) and ( 5)) to the Green's function.

In the semi-classical regime, i.e. as ε → 0, I have

G(t, A, B) ∼ = ε 2i (Ω + (t) -Ω -(t)) • L -1 0 (A, B) ,
Let us now compute the τ -derivative of C A,B (τ ):

d dτ C(τ ) ∼ = - ε i (Ω + (τ ) -Ω -(τ )) • L 0 • Π .
In the case of white noise and constant attenuation a 0 , I know (see for example [START_REF] De Verdière | Semi-classical analysis and passive imaging[END_REF] Section 5.1 for a derivation)) that

d dτ C(τ ) = - 1 2a 0 G(τ ) (6)
which is consistent with the previous semi-classical formula. I can now give a more concrete formula:

Corollary 4.1 Writing G(τ, A, B) as a sum γ G γ of contributions of rays γ(s) with γ(0) = (B, ξ B ) and γ(τ ) = Φ τ (B, ξ B ) = (A, ξ A ), I get d dτ C(τ, A, B) ∼ = γ M γ G γ , with M γ = - 1 2 0 -∞
e -0 t a(γ(s))ds p(γ(t))dt .

In the case of the white noise p = 1 and a = a 0 , I recover the formula

M γ = -1/2a 0 . (7)
Let us also remark that, if there is an unique trajectory from B to A in time τ , the prefactor M γ applies to the Green's function itself. It is the case, if I work with wave equations with constant coefficients in R n .

The previous formula is consistent with the observations of the paper [START_REF] Stehly | A Study of the seismic noise from its long range correlation properties[END_REF]: the correlation C A,B (τ ) is not always an even function of τ as it is if the source is a white noise. The evenness is valid only up to scaling of C A,B (τ ):

C B,A (τ ) = C A,B (-τ ) ∼ kC A,B (τ ) .
The factor k is the ratio of the integrals giving M γ for the ray γ(t) going from B to A and γ(-t) going from A to B.

Time scales

As I see from the general expression of the correlation given in Equation (3), the proof of the main theorem 4.1 involves the knowledge of the Green's function at large times. This is a well known difficulty and the semi-classical expansions of the Green's functions are valid up to to the so-called Lyapounov time which involves the Lyapounov exponent measuring the rate of instability of the ray dynamics. Roughly speaking, the Lyapounov exponent is the smallest number λ so that the distance between any to rays γ 1 (t) and γ 2 (t) satisfies the estimates

d(γ 1 (t), γ 2 (t)) ≤ Ce λt d(γ 1 (0), γ 2 (0)
with C independent of γ 1 (0) and γ 2 (0). There is an associated time scale T Lyap = 1/λ. On the other hand there is an attenuation time scale for the wave dynamics expressed in terms of the decay of the Green's function

|G(t, x, y)| ≤ Ce -T /Tatt .
T att satisfies the estimate T att ≥ 2/ inf a. The approximation given in Theorem 4.1 is better when T att >> T Lyap . In particular, this condition is necessary in order to get point-wise convergence (i.e. convergence for A and B fixed).

The use of surface waves for passive imaging

A remarkable application of the previous tool is to the imaging of the earth crust [START_REF] Lobkis | On the emergence of Green's function in the correlations of a diffuse field[END_REF][START_REF] Weaver | On the emergence of Green's function in the correlations of a diffuse field: pulse echo using thermal phonon's[END_REF][START_REF] Weaver | Ultrasonics without a source: thermal fluctuation correlations at MHz frequencies[END_REF][START_REF] Campillo | Long range correlations of seismic codas[END_REF][START_REF] Shapiro | Emergence of broadband Rayleigh waves from correlations of the ambient seismic noise[END_REF][START_REF] Shapiro | High Resolution Surface Wave Tomography From Ambient Seismic Noise[END_REF]. This is done using the part of the Green's function associated to the surface waves: the earth crust acts as a wave guide on elastic waves and these waves follow an effective wave equation. The effective Hamiltonian is described now: let us start with the acoustic wave equation u tt -div(n gradu) = 0 with the function n coming from a stratified medium n = n(x, z) (here z = 0 is the surface) where n is weakly dependent of x (this can be formalized as n(x, z) = N(εx, z) with N smooth and ε small). Using the adiabatic separation of variables u ∼ U(εx, z)e i x|ξ with U weakly dependent of x, I can operate as if n was independent of x and I get the reduced equation

U tt + Op 1 (λ(x, ξ))U = 0
where λ(x, ξ) is an eigenvalue of the Sturm-Liouville operator

L x,ξ = - d dz n(x, z) d dz + n(x, z) ξ 2
with appropriate boundary conditions at z = 0. From the correlation, I get the ray dynamics of the surface waves and hence the effective Hamiltonians λ(x, ξ). The inverse problem to be solved is the following inverse spectral problem: from the fundamental mode (or any other available mode) of L x 0 ,ξ in some range of wave numbers |ξ|, recover n(x 0 , z). This is the kind of well posed inverse problem for which analytical/numerical method can be used (see [START_REF] De Verdière | A semi-classical inverse problem II: reconstruction of the potential[END_REF]).

A formula for the scattering of random plane waves

I have seen an exact formula for the correlation of the wave field when the attenuation a is constant and the source noise is a white noise. I will see another exact formula in the context of wave scattering by a perturbation sitting in a bounded domain of R d (see [START_REF] De Verdière | Mathematical models for passive imaging I: general background[END_REF]). This formula is very general and applies in all situations of wave scattering (scalar or elastic waves), i.e. for any medium which is homogeneous near infinity: non-homogeneity's lies at finite distances or there is a scattering by a bounded obstacle. This calculus was motivated by the result of [START_REF] Sanchez-Sesma | The elastodynamic 2D Green's function retrieval from cross-correlation: the canonical inclusion problem[END_REF], showing that this result is completely general. Let us consider for example an acoustic wave equation ( 2) with n = n 0 outside a bounded set of R d . I will consider scattering solutions of the stationary wave equation div(n gradu)ω 2 u = 0 [START_REF] Derode | How to estimate the Green's function of a heterogeneous medium between to passive sensors? Application to acoustic waves[END_REF] which are of the following form: let us define, for k ∈ R d , the plane wave e 0 (x, k) = e ik.x .

I am looking for solutions e(x, k) = e 0 (x, k) + e s (x, k)

of equation (8) in R d , with n 0 k 2 = ω 2 ( 2 )
, where e s , the scattered wave, satisfies the so-called Sommerfeld radiation condition:

e s (x, k) = e ik|x| |x| (d-1)/2 e ∞ x |x| , k + O 1 |x| , x → ∞ .
The complex function e ∞ (x, k) is usually called the scattering amplitude and is a signature of the inhomogeneities. The functions e(x, k) are deformed exponentials and allow to write an explicit spectral decomposition of our wave operator, which is a "deformation" of the Fourier transform.

Let us look at e(x, k) as a random wave with k = ω/ √ n 0 fixed. The pointpoint correlation of such a random wave C scatt ω (x, y) is given by:

C scatt ω (x, y) = k √ n 0 =ω
e(x, k)e(y, k)dσ( k).

It is proved in [START_REF] De Verdière | Mathematical models for passive imaging I: general background[END_REF], section 8, that

C scatt ω (x, y) = - 2 d+1 π d-1 n d/2 0 ω d-2 ℑ(G(ω + i0, x, y)) ,
where G(ω, x, y) is the stationary Green's function, i.e. the Schwartz kernel of (ω2 + div(n grad)) -1 .

Conclusions

I hope to have convinced the reader, even if he is not very much involved in mathematics, that it is possible to derive rather explicit asymptotic formulas for the correlation C A,B (τ ) of seismic noise. The main conclusion is that, in the semiclassical regime, even if the source noise is not homogeneous, the field correlation is very close to the Green's function; in many cases, there is only a prefactor which I computed and which introduces no phase shift. This prefactor vanishes if the support of the source noise does not meet the rays from B to A.

Many other ideas and applications remains to be exploited: Is it possible to use the previous tools in order to get informations on the source noise? Can I extend the previous calculus to the case where the source noise is located on a surface? Can I do something similarly in other regimes of propagation, in particular in non-smooth media? Can I get applications of the general formula to monitoring?

Definition 2 . 1

 21 be a random complex-valued field with zero mean value. Let us denote by E the expectation or ensemble average. The correlation of the random field f is the 2-points function given by C(x, y) := E(f (x) f (y))

  t)u(B, tτ )dt . Lemma 3.1 The following relation holds: C A,B (-τ ) = C B,A (τ ).

The "Schwartz kernel" of a linear operator A is the "continuous matrix" of A, we will denote it by [A](x, y) and it is characterized by Af (x) = X [A](x, y)f (y)dy.

As often, I denote k := |k| and k := k/k


Yves Colin

Yves Colin De Verdière
email: yves.colin-de-verdiere@ujf-grenoble.fr

† ‡ March

Bohr-Sommerfeld phases for avoided crossings

Keywords: adiabatic limit, semi-classical analysis, Landau-Zener formula, avoided crossings, pseudo-differential operators 34E05, 35S30, 81Q20, 81Q70

published or not. The documents may come

Bohr-Sommerfeld phases for avoided crossings

Introduction 0.1 Adiabatic limit in quantum mechanics

The problem of adiabatic limit in quantum mechanics with avoided eigenvalues crossings will be the basic example in our paper.

Let us describe it in a precise way. We start with a smooth family of N × N Hermitian matrices A(t) with a ≤ t ≤ b and consider the following linear system of differential equations:

h i dX dt = A(t)X (1)
Solving this equation gives an unitary map S h defined by S h (X(a)) = X(b). We call it the scattering matrix of the problem. Adiabatic Theorems describe the asymptotic behaviour of S h when h → 0.

1. The clasical adiabatic theorem [START_REF] Born | Beweis des Adiabatensatzes[END_REF][START_REF] Kato | On the adiabatic theorem of quantum mechanics[END_REF] concerns the case where the eigenvalues of A(t) satisfy, for all t:

λ 1 (t) < λ 2 (t) < • • • < λ N (t) . (2)
In this case, if we start with X j (a) an eigenvector of A(a) with eigenvalue λ j (a), we have X j (b) = exp(i b a λ j (s)ds)Y j (b) + O(h) where Y j (b) is obtained by parallel transporting X j (a) along [a, b] in the eigenbundle L j (t) = ker(A(t) -λ j (t)) w.r. to the geometric (or Berry) connection defined by

∇ ∂/∂t Y (t) = proj L j (t)
dY dt 1 2. Another case which is well known [START_REF] Avron | An Adiabatic Theorem without a gap condition[END_REF][START_REF] Born | Beweis des Adiabatensatzes[END_REF][START_REF] Hagedorn | Adiabatic Expansions near Eigenvalue Crossings[END_REF][START_REF] Rousse | Landau-Zener transitions for eigenvalue avoided crossings in the adiabatic and Born-Oppenheimer approximations[END_REF], is the case where the eigenvalues of A(t) cross transversally. Then the previous results remains true with a less good remainder term O( √ h) but we need to label the eigenbundle by following smoothly the eigenvalues at the crossings points. In other words, there is some exchange between L j and L j±1 at each crossing point.

So the scattering matrix remains diagonal but with some relabeling of the indices j = 1, • • • , N.

We want to consider the bifurcation between the 2 situations. For that purpose, we allow A(t) to depend smoothly of a d-dimensional parameter µ close to 0 in R d . We will assume that A 0 (t) admits eigenvalues crossings transversally. For µ small, A µ will have avoided eigenvalue crossings. We will derive in this paper a way to get the asymptotic behaviour of S µ,h w.r. to both small parameter µ and h.

Basically the result uses 2 parts :

1. The local situation where a suitable Landau-Zener formula can be used describing the local 2 × 2 scattering matrix (see [START_REF] Joye | Proof of the Landau-Zener formula[END_REF][START_REF] De Verdière | The microlocal Landau-Zener formula[END_REF]).

2. The global problem where we have to take into account interferences given by what we called Bohr-Sommerfeld phases. In the case of several crossings (analytic case), using Stokes lines, they are many works where the interferences effects are exponentially small [START_REF] Joye | Exponential asymptotics in a singular limit for n-level scattering systems[END_REF][START_REF] Martin | Semi-classical inelastic S-matrix for one dimensional N-states systems[END_REF].

Born-Oppenheimer approximation

Another important example comes from the Born-Oppenheimer approximation. We consider, on the real line a Schrdinger equation with matrix valued potential as follows:

Ĥ = -h 2 d 2 dx 2 + V (x) (3)
where V is an Hermitian matrix depending smoothly on x ∈ R. We are interested in the eigenvalue equation ( Ĥ -E) u(x) = 0 with u : R → C n . If there is no degenerate eigenvalues of V (x) smaller than E, then eigenfunctions can be foud using a WKB-Maslov Ansatz in a very similar way to the scalar case. It is the generic situation for a single V [START_REF] Hagedorn | A time-Independent Born-Oppenheimer Approximation with Exponentially Accurate Error Estimates[END_REF]20]). The most precise results are shown in [START_REF] Emmrich | Geometry of the Transport Equation in Multicomponent WKB Approximation[END_REF]. However, if V depends on some external parameters µ close to 0 in R d , it is possible that, in a stable way, V µ (x) admits degenerate eigenvalues. The problem we adress in this case is the description of Bohr-Sommerfeld rules uniformly w.r. to µ.

The general setting

Both examples can be put together as follows: we consider a self-adjoint semiclassical system of N (pseudo-)differential equations of order 0 with N complex valued unknown functions

U = (U 1 , • • • , U N ), H U = 0(h ∞ ) (4)
on the real line. Here Ĥ = ( Ĥi,j ) 1≤i,j≤N is a matrix of semi-classical (pseudo-)differential operators of order 0 in 1 variable x with ( Ĥi,j ) ⋆ = Ĥi,j . Viewing the revious ewmples, it will be important to consider the case where Ĥ = Ĥµ depends smoothly on a germ of parameters µ ∈ (R d , 0). In our previous papers [START_REF] De Verdière | The level crossing problem in semi-classical analysis I. The symmetric case[END_REF][START_REF] De Verdière | The level crossing problem in semi-classical analysis II. The Hermitian case[END_REF], we derived normal forms near the eigenvalues crossings which allow to compute a local scattering matrix including the Landau-Zener amplitude. The goal of this paper is to compute global objects including interferences effects. The general picture is already provided by the study of the scalar case [START_REF] De Verdière | Singular Bohr-Sommerfeld rules[END_REF] from which we know that we need to define ad hoc Bohr-Sommerfeld phases.

The general terminology is the same as in [START_REF] De Verdière | The level crossing problem in semi-classical analysis I. The symmetric case[END_REF] and [START_REF] De Verdière | The level crossing problem in semi-classical analysis II. The Hermitian case[END_REF], but in the present paper, our phase space will always be 2 dimensional:

H µ class : T ⋆ R → Herm(C N ) ,
the (matrix valued) principal symbol, is the dispersion matrix, and C µ = p -1 µ (0) with p µ = det(H µ class ) the dispersion relation. We first recall the local normal form as derived in our previous papers [START_REF] De Verdière | The level crossing problem in semi-classical analysis I. The symmetric case[END_REF][START_REF] De Verdière | The level crossing problem in semi-classical analysis II. The Hermitian case[END_REF] and we solve it. After that, we come to the new part which consists in deriving global objects in the spirit of [START_REF] De Verdière | Singular Bohr-Sommerfeld rules[END_REF].

We will need another piece of information which we call Bohr-Sommerfeld phase; let us take any simple cycle c (with singular vertices z 1 , • • • , z j , • • • , z p ) of the dispersion relation C 0 . We will associate to c a real valued symbol S h (µ) ∼ ∞ j=0 S j (µ)h j were the S j 's are formal power series in µ. S 0 is a purely classical object which involves regularized action integrals. S 1 (0) is computed using the transport equation which is smooth along the edges [z j , z j+1 ] (Berry phases) and singular Maslov indices. From the Bohr-Sommerfeld phases we recover the global objects mod O(h ∞ ).

The local normal form

Let us recall the following result from [START_REF] De Verdière | Le lemme de Morse isochore[END_REF][START_REF] De Verdière | The level crossing problem in semi-classical analysis I. The symmetric case[END_REF][START_REF] De Verdière | The level crossing problem in semi-classical analysis II. The Hermitian case[END_REF] (see also [START_REF] De Verdière | The microlocal Landau-Zener formula[END_REF]):

Theorem 1 Let us assume that the function p 0 (x, ξ) = det(H class,µ=0 ) admits at the point z 0 ∈ T ⋆ R a non degenerated critical point of Morse index 1 (also called hyperbolic critical point, because the Hamiltonian vector field of p 0 is hyperbolic at the singular point z 0 ) and with critical value p 0 (z 0 ) = 0.

Then, we can find the following objects which depends smoothly on µ close enough to 0:

• A smooth family of germs of canonical transformations χ µ : (T ⋆ R, 0) → (T ⋆ R, z 0 ) such that p µ (χ µ (x, ξ)) = e µ (x, ξ)(xξ -γ 0 (µ))
with e µ an invertible germ function and γ 0 a germ of ≥ 0 function of µ satisfying γ 0 (0) = 0. Moreover, the Taylor expansion of γ 0 is unique.

• A smooth family of unitary FIO's U µ associated to χ µ and N × N matrix of ΨDO 's A µ so that, we have the following normal form (called the Landau-Zener normal form) near z 0 :

A ⋆ µ U ⋆ µ Ĥµ U µ A µ =   D α ᾱ x 0 0 Q   with D = h i ∂ ∂x , α(µ, h) ∼ Σ ∞ j=0 a j (µ)h j a symbol and Q is elliptic. Moreover, a 0 is a complex valued function of µ which satisfies |a 0 | 2 (µ) = γ 0 (µ) ,
and we have γ 0 (µ) = -p µ (z 0 )

|detp ′′ 0 (z 0 )| + O(µ 3 ) .
Remark 1 In [START_REF] De Verdière | The level crossing problem in semi-classical analysis II. The Hermitian case[END_REF], Theorem 1 is proved under the following transversality hypothesis:

(⋆) if W ⊂ Herm(C N )
is the submanifold defined by dim ker H = 2, we assumed there that (µ, z) → H class,µ (z) is transversal to W at the point (0, z 0 ).

This hypothesis can be restored using more parameters, so that Theorem 1 is also correct. For simplicity, we will assume that hypothesis (⋆) holds true in what follows.

It implies that µ → a 0 (µ) is a submersion from (R d , 0) onto (C, 0). We will denote by Z = a -1 0 (0). Z is a smooth germ of codimension 2 manifold of (R d , 0).

Formal expansions w.r. to the parameter µ mean formal expansions along Z.

Remark 2 Contrary to the scalar case, there is no arbitrary choice concerning the images of the half axes {ξ = 0, x > 0}, ... by χ. The smooth arcs of the dispersion relation are oriented in the following way: there is a change of the Morse index of the quadratic form associated to H class from m to m ± 1 while crossing these arcs. The sign of this change is preserved by the gauge transform which acts directly on the previous quadratic form by an invertible linear change of variable. We choose to orient the arcs so that the Morse index is bigger on the right than on the left of the path.

Remark 3

The symbol α is not uniquely defined because a diagonal unitary gauge transform preserves the normal form while changing α by some phase shift exp(iϕ(h)). Its modulus γ(µ, h) = |α(µ, h)| 2 is uniquely defined from the Landau-Zener coefficient given in Equation [START_REF] De Verdière | Singular Bohr-Sommerfeld rules[END_REF].

The matrix A is defined up to matrices which will change the phase of α.

More precisely, if A 0 is the principal symbol of A at the crossing point, the only prescription is that

A 0 is a map from C N to C N which sends C 2 ⊕ 0 into ker H 0 (z 0 ) and satisfies A 0 ((C ⊕ 0) ⊕ 0) = E 1 and A 0 ((0 ⊕ C) ⊕ 0) = E 2
where E j = lim z→z 0 ,z∈Λ j \z 0 ker H 0 with Λ 1 = χ 0 ({ξ = 0}) and Λ 2 = χ 0 ({x = 0}). The choice of (A 0 ) |C 2 ⊕0 will be important in the computation of S 1 (0) in section 5.

The previous result is a microlocal result and the subject of the present paper is to get a global result.

In the adiabatic case (see Section 6.1), we get

γ 0 (µ) = gap(µ) 2 4(|λ ′ + -λ ′ -|) + O(µ 3 )
where gap(µ) is the minimal gap of the avoided crossing and λ ′ ± are the slopes of the unperturbed eigenvalues at the crossing point.

The local scattering matrix for the Landau-Zener normal form

The goal of this section is to compute in a very explicit way the local 2 × 2 scattering matrix T for the Landau-Zener normal form :

(LZ) Du + αv = 0 ᾱu + xv = 0 (5) with D = h i ∂ ∂x .
Let us put γ = |α| 2 and let us choose some small a > 0 and assume γ < a 2 . Let C LZ α = {xξ = γ} be the characteristic manifold. The set C LZ α ∩{max(|x|, |ξ|) ≥ a} is the union of 4 connected arcs. These arcs are labelled Λ in,out ± as follows:

• Λ in + = {(x, ξ) ∈ C LZ α | x ≤ -a} • Λ out + = {(x, ξ) ∈ C LZ α | ξ ≤ -a} • Λ in -= {(x, ξ) ∈ C LZ α | ξ ≥ a} • Λ out -= {(x, ξ) ∈ C LZ α | x ≥ a}.
The meaning of the labels is as follows:

• "in" (resp. "out") means that the arc oriented according to remark 2 is incoming (resp. outgoing).

• "+" (resp. "-") means that the vanishing eigenvalue of the dispersion matrix is the largest (resp. smallest) one.

We start defining 4 WKB (exact) solutions of the previous system associated to the 4 Lagrangian arcs Λ in,out

± : Λ out - x ξ Λ in - Λ out + Λ in + m = 0 m = 1 m = 1 m = 2
           W out - : u out -(x) = x i γ h + , v out -(x) = -ᾱx i γ h -1 + W in + : u in + (x) = x i γ h -, v in + (x) = ᾱx i γ h -1 - W in -: u in -(ξ) = -αξ -i γ h -1 + , v in -(ξ) = ξ -i γ h + W out + : u out + (ξ) = αξ -i γ h -1 - , v out + (ξ) = ξ -i γ h - (6)
where f (ξ) is the h-Fourier transform of f (x) defined by

f (ξ) = 1 √ 2πh R e -i xξ h f (x)|dx| ,
and x ± = Y (±x)|x| with Y the Heaviside function.

Computing the Fourier transforms of u ± and v ± , we get the following compatibility conditions in order to get microlocal solutions of (5) near the origin: 5) outside the origin, we get, for any microlocal solution near the origin,

W out -(x) ↔ h 1 2 +i γ h Γ(1+i γ h ) √ 2πα ie π γ 2h W in -(x) + ie -π γ 2h W out + (x) W in + (x) ↔ h 1 2 +i γ h Γ(1+i γ h ) √ 2πα -ie -π γ 2h W in -(x) -ie π γ 2h W out + (x) If W in := y + W in + + y -W in -and W out := z + W out + + z -W out -are WKB-solutions of Equation (
z - z + = T y + y -
where T is the unitary matrix defined by:

T = 1 A -B 1 B 2 -A 2 -B (7)
with

A = ih 1 2 +i γ h Γ(1 + i γ h ) √ 2πα e π γ 2h , B = ih 1 2 +i γ h Γ(1 + i γ h ) √ 2πα e -π γ 2h . (8)
The matrix T will be called the local scattering matrix associated to the singular point (and the choice of a normal form). Unitarity of T is checked using the well known formula

Γ(1 + ix)Γ(1 -ix) = πx sinh πx which implies |A| 2 = |B| 2 + 1, A B = ĀB.
The transmission coefficient

τ = B A = exp(-π γ h ) (9)
gives the Landau-Zener formula. The previous explicit expression for the scattering matrix will allow to define in the next section the Bohr-Sommerfeld phases and to take into account interferences patterns due to several (avoided) crossings.

3 Singular Bohr-Sommerfeld phases

Outline

To each cycle c of the dispersion relation C 0 , we can associate, using the recipe of [START_REF] De Verdière | Singular Bohr-Sommerfeld rules[END_REF], a singular phase of the form

S h (µ) = S 0 (µ) + hS 1 (µ) + • • •
where the S j 's are smooth w.r. to µ.

In this section, we will define precisely these phases. We show that they are uniquely defined as formal power series w.r. to (µ, h). More precisely, each S j (µ) is well defined modulo flat functions on Z (see Remark 1). We will give more precise properties of S 0 in section 4 and S 1 in section 5: S 0 (µ) is, as a formal power series, a purely classical object derived from the dispersion relation, while S 1 (0) is a semi-classical object associated to phases given by the transport equation which in the adiabatic case are Berry phases.

Bohr-Sommerfeld phases: a definition

Let us take a simple oriented cycle c of the dispersion relation C 0 (boundary of a bounded component of

T ⋆ R \ C 0 ). Let z 1 , z 2 , • • • , z n be
the singular points of c ordered cyclically around c. For each singular point z j , let us build a FIO U j and a ΨDO gauge transform A j (all depending smoothly on µ) which give the normal form of Theorem 1 with α j = α j (µ, h) a full symbol.

We will define H µ (c) = exp(iS h (µ)/h) as follows: we will denote by W in, out ±,j the images of W in, out ± by the operators U µ j A µ j . These functions are WKB solutions of equation ( 4) associated to arcs of C µ near z j . We introduce also WKB solutions u j of (4) along arcs of C µ close to ]z j , z j+1 [. From those objects we get a global holonomy H µ (c) of the cycle c defined as follows: we have (by uniqueness, modulo multiplication by a full symbol, of WKB solutions) for example near z j :

u j = x j W in -,j , u j-1 = y j W in +,j .
We define H µ (c) = Π n j=1 y j x -1 j . In other words, H µ (c) is the holonomy of a sheaf on c given by the WKB solutions on the smooth part of the cycle and whose jumps of section are given from the normal forms. In our previous example

W in +,j → W in -,j . Lemma 1 We have |H µ (c)| = 1 + O(h ∞ ). Proof.- W in - W in + u j-1 u j z j+1 z j-1 z j c Figure 3: defining H µ (c)
Following [START_REF] Monvel | Singular Lagrangian manifolds and semi-classical analysis[END_REF] section 11.2.1. and Figure 4, we associate to the cycle c an unitary scattering matrix S c which is computable from the local unitary scattering matrices associated to the singular points and the holonomy H µ (c). If this holonomy does not satisfy |H µ (c)| = 1 + O(h ∞ ), the global scattering matrix would not be unitary: the previous matrix is the product of (unitary) local scattering matrices and a diagonal matrix whose unique nonzero entry is H µ (c). Taking the Logarithms, we get the phase S h (µ)/h = ∞ j=0 S j (µ)h j-1 which is well defined modulo a multiple of 2π.

Lemma 2 Given the gauge transforms used in the normal form of Theorem 1, the Taylor expansions of the S j 's at µ = 0 are uniquely defined.

Proof.-We will use the fact that the local scattering matrix computed in Section 2 is irreducible in the domain h N ≤ |α(µ)| ≤ √ h, meaning that none of the entries are O(h ∞ ) in this domain. It implies that, for each j, the W in, out ±,j 's can be, up to a global multiplication by a symbol, defined as sets of WKB solutions for which the scattering matrix is given by Equations ( 7) and ( 8) with the value of α given by the normal form at the point z j . Proof.-From the definition, S 0 depends only on the terms in 1/h in the phases of the images by our normal form transformations of the explicit solutions of the normal form. Those terms depends only on the canonical transformations used in the normal form and the associated generating functions via stationnary phases (the Lagrangian manifolds).

S 0 as a regularized action integral

Let us denote by |µ| = d(µ, Z). As in [START_REF] De Verdière | Singular Bohr-Sommerfeld rules[END_REF], it would be nice to get S 0 (µ) as a regularisation of an usual action integral. A basic fact in [START_REF] De Verdière | Singular Bohr-Sommerfeld rules[END_REF] was that any simple cycle c is a limit of a cycle of C µ as µ → 0 ± . This is no longer the case here because γ 0 ≥ 0; one can see an example in section 4.4. The idea is now to forget the initial problem and to work only with the dispersion relation C µ which can be embedded into a larger family C t,µ for which we can define action integral in some suitable sectors of the (t, µ) space. We can then restrict to t = 0 and get our actions S 0 .

We will calculate S 0 by first computing the same object Σ 0 (t, µ) for C t,µ = {p(x, ξ, µ) -t = 0} and taking S 0 (µ) = Σ 0 (0, µ).

The cycle c is a limit of a cycle c(t, µ) of C t,µ as (t, µ) → 0 in some sector

Ω ± := {(t, µ)| ± t > 0, |µ| << |t|} . We have, for (t, µ) ∈ Ω, Σ 0 (t, µ) = ct,µ ξdx + p j=1 ±γ 0,j (t, µ)(ln |γ 0,j (t, µ)| -1) ,
where the ± signs depends on orientation and can be determined from the Logarithmic singularities of the action integrals. The contributions ±γ 0,j (t, µ)(ln |γ 0,j (t, µ)|-1) come from the phase shift between x iγ/h + and ξ -iγ/h + expressed as a WKB function of the single variable x.

Knowing that Σ 0 is smooth, the previous formula defines the Taylor expansion of Σ 0 w.r. to (t, µ) and hence the Taylor expansion of S 0 w.r. to µ.

The analytic case

In the analytic case, we could also consider the Riemann surfaces X µ = {p µ = 0} and look at some complex cycles c µ on X µ whose limit is c. Those cycles are not unique, but the real part of their action integrals are well defined and we can then take directly the previous regularisation.

An example

Let us consider the adiabatic equation:

h i dX dt = A µ (t)X with A µ (t) = t 2 µ µ 2 -t 2
and the only cycle c 0 of C 0 passing by the singular points (±1, ±1). It is clear that c 0 is not a limit of real cycle c µ of C µ , because the matrix A µ (t) has real eigenvalues for each t and so C µ is the union of 2 disjoint graphs and has no real cycle.

The subprincipal action

We know that the Landau-Zener coefficient given by Equation ( 9

) is 0(h ∞ ) if |µ| >> √ h.
It implies that in order to solve our problem up to O( √ h) terms it is enough to know S 0 mod O(|µ| 3 ) and S 1 for µ ∈ Z. Let us assume that we have local coordinates so that 0 ∈ Z. We will describe below the calculus of S 1 (0). Lemma 3 Assuming |µ| = 0, the principal part a(x)exp(iS(x)/h) of the WKB solutions of Equation ( 4) associated to arcs ]z j , z j+1 [ of C 0 can be smoothly extended beyond the singular vertices as WKB functions.

Proof.-

The property is invariant by FIO and it is enough to prove it for the solutions of the normal form given in Equation ( 6). The point is that a 0 (0) = 0, hence γ = O(h 2 ).

The previous result is related to the fact that the adiabatic theorem is still valid in case of eigenvalue crossings (see [START_REF] Avron | An Adiabatic Theorem without a gap condition[END_REF][START_REF] Born | Beweis des Adiabatensatzes[END_REF]).

We will define on c a piecewise smooth Hermitian line bundle L with a connection as follows:

• On each arc [z j , z j+1 ], L z = ker H class (z) with the connection given by the transport equation as in [START_REF] Emmrich | Geometry of the Transport Equation in Multicomponent WKB Approximation[END_REF] (in the case of the adiabatic limit, it is the so called geometric connection or Berry phase [START_REF] Berry | Quantal phase factors accompanying adiabatic changes[END_REF]).

• At each singular point, there are 2 limit fibers L ±,j and from A 0 (defined in Remark 3) we have an isomorphism between both limits given by transporting the isomorphism (1, 0)

→ (0, 1) of C ⊕ 0 on 0 ⊕ C by (A 0 ) |C 2 ⊕0 .
Definition 1 The phase exp(iS ∇ 1 (0)) is the holonomy of the discontinuous line bundle L. The Maslov index of any cycle is defined by linearity from the previous Maslov indices, so it gives a cocycle. For example, the Maslov index of a smooth cycle (even if not simple) is the usual one, namely ±2.

+ 1 2 + 1 2 + 1 2 + 1 2 c m(c) = 2 -3 × 1 2
Theorem 3 Using the previous definitions, we have:

S 1 (0) = S ∇ 1 (0) + m(c) π 2 .
Proof.-

The proof follows essentially the lines of [START_REF] De Verdière | Singular Bohr-Sommerfeld rules[END_REF] p. 474-476.

Let us give some details. A priori, there are several cases to check depending on the position of the cycle c at the singular points w.r. to the verticals. We will assume that the matrix

χ ′ (O) = a b c d
of the canonical transformation χ = χ 0 satisfies a = 0 and b = 0, this is the generic case. We define

ε = +1 if ab > 0 -1 if ab < 0 .
The generating function ϕ(x, y) = ϕ 2 (x, y)+O(|x| 3 +|y| 3 ) of χ, defined by χ(y, -∂ y ϕ) = (x, ∂ x ϕ), satisfies ϕ 2 (x, y) = 1 2b (dx 2 -2xy + ay 2 ). We need to compute mod o h (1) the values for x close to 0 of the images by the normal form transform of

W out -(y) = Y (y) 0 and W in -(η) = 0 Y (η)
.

Let us assume that the principal symbol of the ΨDO gauge transform is the N × N matrix

σ(A)(y, η) = α 1 (y, η) α 2 (y, η) • • • .
We get for the components of both WKB solutions for x small but nonzero: If we evaluate the integrals by stationnary phase, the dominant contributions come from the critical points and not from the boundary. The determinant of both Hessians are the same, while the signature differs by 1. The final result follows then by

W out -,j (
• Looking at the value of the stationnary phase calculations as x is close to 0: the limits are respectively C(0, 0)e iεπ/4 α 1 (0, 0) and C(0, 0) α 2 (0, 0)

• if ε > 0, one should add a contribution of the smoothed c, while if ε < 0 there is no such contribution.

• Remembering that

A 0|C 2 ⊕0 = α 1 (0, 0) α 2 (0, 0) .
6 Application 1: adiabatic limit with avoided crossings

Adiabatic limit

We consider the following equation:

1 i dX dτ = A µ (hτ )X (10)
where A µ (t), 0 ≤ t ≤ a, is a self-adjoint matrix which is smooth w.r. to (t, µ) and we consider 0 ≤ τ ≤ a/h. The limit h → 0 of this equation is called the adiabatic limit.

We can rewrite Equation ( 10) in a standard semi-classical form by puting t = hτ :

h i dX dt = A µ (t)X (11)
where 0 ≤ t ≤ a.

We will assume that the eigenvalues of A µ (0) and A µ (a) are all non degenerate. The scattering matrix S(µ, h) : C N → C N is defined by X(0) → X(a) where X is a solution of Equation (11 ).

Outside eigenvalues crossings

Let λ(t) be an eigenvalue of multiplicity 1 of A 0 (t) for t in some open intervall I. Then Equation ( 11) admits a unique (up to multiplication by some function of h) formal WKB solution given by

X(t) = e iΛ(t)/h ∞ j=0 a j (t)h j
where Λ ′ (t) = λ(t) and a 0 (t) satifies:

• a 0 (t) ∈ ker (A 0 (t) -λ(t))

• ∇a 0 (t) = 0 where ∇ is the geometric or Berry connection.

Let us recall that ∇ ∂/∂t a(t) = Π t a ′ (t) where Π t is the orthogonal projection of C N onto the eigenspace ker(A 0 (t) -λ(t)).

The previous statement is the content of the so called quantum adiabatic theorem and goes back to [START_REF] Born | Beweis des Adiabatensatzes[END_REF].

Avoided crossings

What happens when eigenvalues become degenerate at some values of t?

Let us try to understand the generic situation. It is well known that eigenvalue crossings for a real symmetric (resp. complex Hermitian) matrix is a codimension 2 (resp. 3) property. It is the content of the well known Wigner-Von Neumann theorem [START_REF] Neuman | Uber das Verhalten von Eigenwerten bei adiabatischen Prozessen[END_REF].

Physically, eigenvalues crossings can still occur for symmetry reasons. But, if we break the symmetry by a small perturbation of size µ, we will get the socalled avoided crossings. We have now two small parameters: the semi-classical (adiabatic) parameter h and the perturbation parameter µ. The previous results allow to discuss the uniform expansion of the scattering matrix w.r. to both small parameters.

Precise assumptions

We will assume that the eigenvalues of A 0 (t) cross transversally only by pairs on ]0, a[. The dispersion relation C µ ⊂ T ⋆ [O, a] is defined by p µ (t, τ ) = det(τ Id -A µ (t)). So that C µ is exactly the union of the graphs of the eigenvalues of A µ (t). We try to describe a global solution of our system which is given from WKB solutions associated to each arc of a maximal tree of C 0 . We have 10 equations with 13 unknowns which allow to compute y from x.

Calculation of the scattering matrix

                               w -= H µ (c 1 )w + v -= H µ (c 2 )v + z w - = S 1 x 3 x 2 v - y 3 = S 2 w + p y 1 y 2 = S 3 v + u u p = S 4 z x 1
It turns out that the global scattering matrix is the product of 5 unitary matrices as follows : We consider a Born-Oppenheimer Hamiltonian of the following form:

x →   x 1 z w -   →   u p w +   →   u v - y 3   →   u v + y 3   → y .
K ν = -h 2 d 2 dx 2 ⊗ Id + V ν (x)
where V ν : R → Herm(C N ) is smooth w.r. to (x, ν). We assume:

• The eigenvalues of V 0 (x) are of multiplicities at most 2 and cross transversally.

• The following properness condition:

V ν (x) ≥ p(x)Id
where lim |x|→∞ p(x) = +∞.

• We choose E so that, for any x ∈ R, E is not a degenerate eigenvalue of V 0 (x).

• If the eigenvalue λ j (x) of V 0 (x) satisfies λ j (x 0 ) = E, then λ ′ j (x 0 ) = 0.

We can apply the previous method in order to compute EBK quantization rules for the equation ( K ν -E) U = O(h ∞ ) .

EBK quantization can be solved following the same path; but is this case we have the same number of equations than of unknowns and EBK rule is given by the vanishing of a suitable determinant as in [START_REF] De Verdière | Singular Bohr-Sommerfeld rules[END_REF].

Figure 1 :

 1 Figure 1: the jumps of the Morse index of the dispersion matrix

Figure 2 :

 2 Figure 2: the 4 arcs of the characteristic manifold and Morse indices

3 cFigure 4 :

 34 Figure 4: the scattering matrix associated to a cycle

4

  The classical part S 0 4.1 S 0 is classical We have the following: Theorem 2 For any simple cycle c of C 0 , the function S 0 (µ) depends only on the dispersion relation C µ .

Figure 5 :Definition 2

 52 Figure 5: the singular Maslov indices

  x) = (2πh) -3/2 y ′ ≥O e i h (ϕ(x,y)+(y-y ′ )η) C(x, y) α 1 (y, η)dydy ′ dη , with C(0, 0) = |b| -1 2 , and W in -,j (x) = (2πh) -1 η≥O e i h (ϕ(x,y)+yη) C(x, y) α 2 (y, η)dydη .

  Let us describe how to compute the global scattering matrix in the case of Figure 7. Let us start with the 4 local scattering matrices S j , j = 1, • • • , 4 and the 2 holonomies H µ (c k ), k = 1, 2.

Figure 6 : 3 pFigure 7 :

 637 Figure 6: the dispersion relation for the adiabatic limit

7 Application 2 :Figure 8 :

 28 Figure 8: the dispersion relation for the Born-Oppenheimer Hamiltonian